3, choose index j such that 1 < j < m− 1. Let a = tj and b = n− tj , and define compositions µ = (a, b), α = (a, λj+1, . . . , λm), and β = (λ1, . . . , λj , b), all of which have weight n. Consider the hyperplane H ′ = {x ∈ Rn : i=1 xi = 1}. Then H ′ ∩ Q(Mλ) = Q(Mµ), giving us a hyperplane split Q(Mλ) = Q(Mα) ∪ Q(Mβ). It follows from the above that F (Mλ) = F (Mα) + F (Mβ) − F (Mµ), and F (Mλ) = F (Mα) + F (Mβ). We can summarize this in the following proposition. The relations given in the proposition remain true even if λ has only two or three parts, but in that case the resulting relations are trivial. Proposition 6.1. Let λ = (λ1, . . . , λt) be a composition with at least two parts. Let 1 ≤ s < t, a = i=1 λi, and b = i=s+1 λi. Consider compositions α = (a, λs+1, . . . , λt), β = (λ1, . . . , λs, b), and µ = (a, b). We then have F (Mλ) = F (Mα) + F (Mβ)− F (Mµ), and modulo m2, F (Mλ) = F (Mα) + F (Mβ). Moreover there is a split of matroid base polytopes Q(Mλ) = Q(Mα) ∪Q(Mβ). The splitting process can be repeated on the constituent matroid base polytopes until we have decomposed Q(Mλ) into the union of matroid base polytopes of type Q(Mα) where ℓ(α) = 3. Consequently, modulo m 2, F (Mλ) can be written as a positive sum F (Mλ) = F (Mi), where each Mi is a loopless rank 2 matroid indexed by a partition of length 3. In this setting, Billera, Jia, and Reiner , pose the following question: [3, Question 7.10] Fix n and consider the semigroup generated by F (M) within QSymn/m 2 as one ranges over all matroids M of rank 2 on n elements. Is the Hilbert basis for this semigroup indexed by those M for which λ(M) has exactly 3 parts? By repeated application of Proposition 6.1, the set {F (Mλ) : ℓ(λ) = 3} generates the semigroup in question, so the point of the question is whether this generating set is minimal, and whether distinct indices yield distinct functions. We prove that this is the case as a corollary of Theorem 6.2. 6.2 Results for rank two matroids In this section, we prove that the morphism F : Mat → QSym distinguishes isomorphism classes of rank two matroids and that decomposability of F (M) for a rank two matroid M implies decomposability of Q(M), as stated in the following theorem. Theorem 6.2. Let λ ⊢ n with ℓ(λ) ≥ 3, and let J be a multiset of partitions of n, all of length three or more, such that F ([Mλ]) = F ([Mµ]), (16) where [Mτ ] denotes the isomorphism class of (loopless) rank two matroids on n elements indexed by the partition τ . Then, taking the set of standard basis vec- tors of Rn as the common ground set, there exists a collection of representative matroids on this ground set, Mλ ∈ [Mλ] and Mµ ∈ [Mµ] for all µ ∈ J which form a decomposition of matroid base polytopes Q(Mλ) = Q(Mµ). (17) Before the main proof of this theorem, we establish some preliminary results. We begin by developing a formula for F (Mλ) in terms of the new basis {Nα}. We define the following quasisymmetric functions in V n2 = span{Nα : |α| = n, r(α) = 2}. For all 1 ≤ k ≤ n− 1 let T nk := N(2,n−2) + k − 1 N(1,j,1,n−2−j), (18) where we understand N(1,j,1,n−2−j) to be N(1,n−2,1) when j = n − 2. We also define quasisymmetric functions Unk := k(n− k)T Note that each of the sets {T nk } and {U k } forms a basis for the subspace V where we consider QSym to have rational coefficients. Lemma 6.3. Let Mλ be the rank two matroid on n elements indexed by the partition λ = (λ1, . . . , λm). Then F (Mλ) = Unλi . (19) Proof. We write c(λi) to denote the parallelism class of elements in Mλ cor- responding to the part λi. A typical base B ∈ B(Mλ) is B = {ei, ej}, where ei ∈ c(λi) and ej ∈ c(λj) are in distinct parallelism classes. The Hasse diagram of PB has two minimal elements, ei and ej. There are edges from ei to all elements of the cobase Bc = E(Mλ) − B except for the λj − 1 elements which are in the same parallelism class c(λj) as ej. Similarly, there are edges from ej to all elements of the cobase except for the λi − 1 elements which are in the same parallelism class c(λi) as ei. We can analyze F (PB) as in the proof of Lemma 5.1 by applying a strict labeling γ : E(Mλ) → [n] such that γ(ei) = n, γ(ej) = n − 1, and the cobase elements are arbitrarily labeled with {1, 2, . . . , n − 2}. We take T = {B,Bc} to be our antichain-inducing partition of (PB , γ). There is one induced ordered partition (of [n]) of type (2, n − 2), namely K = (B,Bc), classifying one set of permutations in L(PB , γ), and thus contributing one N(2,n−2) term to the expansion of F (PB). For each 1 ≤ k < λj , and for each k-set A ⊂ B c(λj), there is an induced ordered partition K = ({ej}, A, {ei}, B c −A) of type (1, k, 1, n − 2 − k) contributing a term N(1,k,1,n−2−k) to the expansion. Thus there are such terms N(1,k,1,n−2−k) corresponding to ordered partitions K of type (1, k, 1, n−2−k) withK1 = {ej}. Likewise there are such terms N(1,k,1,n−2−k) corresponding to ordered partitions K of type (1, k, 1, n− 2− k) with K1 = {ei}. All the Nα ∈ N 2 are of one of these types, and we know that the terms of F (PB) must lie in V 2 , so these are the only types appearing in the expansion for F (PB). There can be no other terms than these due to the order relations in PB. Thus F (PB) = N(2,n−2) + λi − 1 λj − 1 N(1,k,1,n−2−k). (20) Using Equation (18), we can rewrite this as F (PB) = T + T nλj . Finally, there are λiλj such bases B ∈ c(λi)× c(λj). Summing over all pairs of parallelism classes of the matroid yields the formula F (Mλ) = λi(n− λi)T Unλi . Next we develop a similar formula for F (Mλ) in QSymn/m 2. Our starting point is the following corollary. Corollary 6.4. Let a and b be positive integers such that a+ b = n. Then ab ·N(1,a−1) ·N(1,b−1) = U a + U Proof. Let λ = (a, b). ThenMλ = U1,a⊕U1,b, where U1,m is the uniform matroid of rank one onm elements. As discussed in Example 5.3, F (U1,m) = mN(1,m−1). Therefore by the Hopf algebra morphism, we have F (Mλ) = F (U1,a) · F (U1,b) = aN(1,a−1) · bN(1,b−1). On the other hand, by Lemma 6.3 we have F (Mλ) = U a + U b . Equating right hand sides yields the desired formula. Since QSym with respect to its product structure is graded by composition rank as well as degree, the vector subspace V n2 ∩m 2 is spanned by the vectors {N(1,a−1) ·N(1,b−1) : a+ b = n}. Thus a basis for V n2 ∩ m 2 is {Unk + U n−k : 1 ≤ k ≤ }. For expressing our formula for F (Mλ), we find it convenient to define vectors U k as follows: Unk = Unk if k < 0 if k = n −Unn−k if k > Thus the set {Unk : 1 ≤ k < } forms a basis (over rational coefficients) for V n2 /m 2. We have the immediate corollary of Lemma 6.3: Corollary 6.5. Let Mλ be the rank two matroid on n elements indexed by the partition λ = (λ1, . . . , λm). Then F (Mλ) = Unλi . (22) The next proposition provides a necessary step for the main result, but may be of interest in its own right. Proposition 6.6. Let M2 be the set of matroid isomorphism classes (including those with loops) of rank two matroids. Let Matc be the vector subspace of Mat spanned by the isomorphism classes of connected matroids, and let M2c be the set of matroid isomorphism classes of connected rank two matroids. Then the algebra morphism F : Mat → QSym is injective when restricted to M2, and the induced quotient map of vector spaces F : Matc → QSym/m 2 is injective when restricted to M2c. Proof. We show that we can recover the isomorphism class of the matroid from its respective function. Suppose we are given F (M) for a rank two matroid M . We know that F (M) is a non-zero homogeneous function of degree n = |E(M)|, and so we recover the size of the ground set. Clearly, n ≥ 2. It is possible that M may have loops or coloops. By Lemma 5.4 we can recover the total number s of loops and coloops of M from F (M) by Equation (13). If s = n, then M consists of two coloops and n − 2 loops. Otherwise s ≤ n− 2, and we may factor F (M) as F (M) = N(s) · F (M where (s) is the one-part composition of s, and M ′ is the matroid obtained from M by removing all loops and coloops. If now F (M ′) ∈ V n−s1 , we have M ′ ∼= U1,n−s and M has one coloop and s− 1 loops. Otherwise M has s loops, no coloops, F (M ′) ∈ V n−s2 , and M ′ is a loopless rank two matroid on n − s elements. So now without loss of generality, we assume that M has no loops or coloops and thus is isomorphic to Mλ for some λ ⊢ n. We expand F (M) as F (M) = k . (23) This expansion can be determined since the set of {Unk } form a basis of V 2 . Per Lemma 6.3, for each k, the coefficient tk is the number of parts of λ that are equal to k, and so we recover λ from F (Mλ). The argument for recoveringM from F (M) for a connected rank two matroid M is similar. Since M is connected, it has no loops or coloops, and so again M is isomorphic to Mλ for some λ ⊢ n with ℓ(λ) ≥ 3, where n is the degree of F (M). We expand F (M) = ⌊(n−1)/2⌋∑ k . (24) This expansion can be determined since the set {Unk : 1 ≤ k < } forms a basis for the subspace V n2 /m 2. Note that λ cannot have a pair of parts with values k and n − k. Using this fact together with Corollary 6.5, we see that if the coefficient tk is nonnegative, then λ has exactly tk parts with value k. From this we can determine all the parts of λ which are < n . Since λ cannot have more than one part ≥ n , this allows us to determine the remaining part of λ, if Proof of Theorem 6.2. We write A⊔B to denote the disjoint union of multisets A and B. Note that a partition may be considered to be a multiset of integers. We fix n > 2 and λ ⊢ n with ℓ(λ) ≥ 3, and proceed by induction on |J |. The base case |J | = 1 follows from Proposition 6.6. So we assume that the statement holds for |J | < m for some fixed m > 1. Suppose now that F ([Mλ]) = F ([Mµ]), (25) where |J | = m. Say that a pair of elements µ, ν ∈ J are matching if for some value 1 < k < n − 1 we have k ∈ µ and n − k ∈ ν. If µ, ν are a matching pair, then we can apply Proposition 6.1 to form a new relation of type (25) by replacing J with J ′ = (J − {µ, ν}) ⊔ {τ}, where τ = (µ ⊔ ν) − {k, n − k}. At the same time, Proposition 6.1 tells us that we also have a decomposition of base polytopes Q(Mτ ) = Q(Mµ) ∪ Q(Mν). Since |J ′| < m, we can apply our induction hypothesis, and we are done. It remains to show that there exists a matching pair in J . For a partition τ ⊢ n, define the multiset g(τ) = {τi : τi > 1, τi 6= Define multisets L = g(λ) and R = µ∈J g(µ). Per Corollary 6.5 we expand F (Mλ) = ℓ(λ)∑ Unλi , and we similarly expand each F (Mµ) on the right hand side of (25). Since the set {Unk : 1 ≤ k < } forms a basis for V n2 /m 2, with Unk = −U n−k, we conclude that L ⊆ R and that the parts in R − L can be matched into complementary pairs of the form (k, n− k). Since no partition in J can contain both parts of a complementary pair, there exists a matching pair in J if R− L 6= ∅. We are assuming that |J | ≥ 2, and that R and L contain all the parts not equal to n or 1 on the respective sides of (25). The parts equal to 1 on both sides must match since all of the partitions have at least three parts and hence no part equal to (n− 1). The only way to have R−L = ∅ is if there exist µ, ν ∈ J each of which contains a part equal to n , in which case they are matching. Thus in all cases, there exists a matching pair µ, ν ∈ J , and the result follows by induction. Now we can give an affirmative answer to [3, Question 7.10]. Corollary 6.7. For a fixed n, the Hilbert basis for the semigroup in QSym/m2 generated by the set S = {F (Mλ) : λ ⊢ n, ℓ(λ) ≥ 3} is indexed by those Mλ for which ℓ(λ) = 3. Proof. Let T = {F (Mλ) : λ ⊢ n, ℓ(λ) = 3}. It follows from Proposition 6.1 that for ℓ(λ) > 3, F (Mλ) is decomposable into a sum µ F (Mµ), where for all µ, ℓ(µ) < ℓ(λ). Hence T generates the same semigroup as S. As noted in [3, Section 7], if ℓ(λ) = 3, then Q(Mλ) is indecomposable. Theorem 6.2 then implies that F (Mλ) must also be indecomposable, so T is the minimal generating set, i.e. the Hilbert basis of the semigroup. By Proposition 6.6, distinct indexing partitions yield distinct images, establishing the claim. 7 Additional observations In this section we discuss additional aspects of our new basis, especially regard- ing the expansion of F (M) for a matroid M . 7.1 Matroid duality, loops, and coloops Although we describe the basis {Nα} as ‘matroid-friendly’, things are slightly less friendly when considering matroid duality in the presence of coloops. This is due to the fact, mentioned in Section 5.1, that the mapping F : Mat → QSym factors through the quotient Mat → Mat/∼ → QSym, where ∼ denotes loop- coloop equivalence. For example, a fact proved in [3] is that, for any matroid M , in terms of the monomial basis for QSym the following relationship holds: F (M) = α =⇒ F (M∗) = α∗ . (26) where α∗ is the reversal of α, obtained by writing the parts of α in reverse order. If M be is a matroid of rank r on n elements having no loops or coloops, then we have the analogous relationship F (M) = mαNα =⇒ F (M mαNα∗ . (27) However this relationship breaks down if M has loops or coloops. We showed in Theorem 5.2 that if M is a loopless matroid of rank r on n elements, then F (M) ∈ V nr . More generally, if M is of rank r on n elements and has exactly ℓ loops, then F (M) ∈ V nr+ℓ. Thus if M has exactly c coloops, then we have the duality relationship F (M) ∈ V nr =⇒ F (M ∗) ∈ V nn−r+c. 7.2 Comultiplication The matroid Hopf algebra is graded by matroid rank as well as ground set size. Let Wnr be the subspace of Mat spanned by the classes of matroids of rank r on n elements. Then Wnr ·W s ⊂ W r+s . For any matroid M and A ⊆ E(M), r(M) = r(M |A) + r(M/A). So comultiplication in Mat also respects these gradings. (For general background on Hopf algebras, see [8].) That is, ∆Wnr ⊆ a+b=n, s+t=r W as ⊗W . (28) One might wonder whether the standard comultiplication of the Hopf algebra QSym respects the grading by the rank function for our new basis, that is, whether ∆V nr ⊆ a+b=n, s+t=r V as ⊗ V . (29) This is not the case. For the simplest example, consider n = 2 and r = 1. We have N 00 = {N0} = {1} and N 1 = {N11} = {x 11}. Note that there is no Nm0 (or rather, Nm0 = ∅) for m > 0. The basis vectors corresponding to the right hand side of (29) are N11 ⊗N0 = x 11 ⊗ 1, and N0 ⊗N11 = 1⊗ x However, ∆N11 = ∆x 11 = x11 ⊗ 1 + x1 ⊗ x1 + 1⊗ x11, which clearly does not lie in the span of the above vectors. The failure of the comultiplication to respect the rank grading can be viewed as another artifact of loop-coloop equivalence under the morphism F , as evi- denced by the fact that the rank grading is respected by comultiplication in the quotient space corresponding to matroids with neither loops nor coloops. Let J ⊂ QSym be the ideal generated by degree one elements, i.e. by {N1} = {x Similarly, let I ⊂ Mat be the ideal generated by degree one elements, i.e. by {[U0,1], [U1,1]}. Both I and J are Hopf ideals in their respective Hopf algebras, hence Mat/I and QSym/J (with their naturally induced comultiplications) are Hopf algebras. Moreover, I = F−1(J), so F : Mat → QSym induces a sur- jective Hopf algebra morphism Mat/I → QSym/J . Note that a natural basis for Mat/I is the set of all matroid isomorphism classes that have neither loops nor coloops, while a natural basis for QSym/J is {Nα : ℓ(α) is even}. Taking appropriate images under the quotient map, the relation (29) holds in QSym/J . The duality formula (27) also holds in QSym/J . 7.3 Comparison with other QSym bases In the course of their proof in Section 10 of [3], the authors introduce two new Z-bases for QSym. They also compare their bases to another Z-basis due to Stanley [19]. Our new basis is different from these three, as evidenced by the report by those authors that all three of these bases have some negative structure con- stants, whereas our new basis does not. However, of the three, ours most closely resembles that of Stanley. Stanley’s basis element indexed by a composition α = (α1, . . . , αm) is F (P ) where, as with our basis, P = A1 ⊕ · · · ⊕ Am, is the ordered sum of antichains A1, · · · , Am on α1, . . . , αm elements respectively. However, Stanley applies a natural labeling to P , whereas we apply an alter- nating labeling to the ranks in the poset for our basis. 7.4 Surjectivity of the Hopf algebra morphism Billera, Jia, and Reiner devote [3, Section 10] to showing that the morphism F : Mat → QSym is surjective over rational coefficients. In this subsection we sketch one way to shorten their proof somewhat using our new basis. The reader will need to consult [3] to have the full context. Define an ordering on compositions as follows. To each composition α we assign the binary word b(α) that begins with α1 zeros followed by α2 ones, then α3 zeros, then α4 ones, etc. We then linearly order compositions according to their binary words: α < β if b(α) <|startoftext|> Bremsstrahlung Radiation At a Vacuum Bubble Wall Jae-Weon Lee∗ School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea Kyungsub Kim and Chul H. Lee Department of Physics, Hanyang University, Seoul 133-791, Korea Ji-ho Jang Korea Atomic Energy Research Institute Yuseong, Daejeon 305-353, Korea When charged particles collide with a vacuum bubble, they can radiate strong electromagnetic waves due to rapid deceleration. Owing to the energy loss of the particles by this bremsstrahlung radiation, there is a non-negligible damping pressure acting on the bubble wall even when thermal equilibrium is maintained. In the non-relativistic region, this pressure is proportional to the velocity of the wall and could have influenced the bubble dynamics in the early universe. PACS numbers: 12.15.Ji, 98.80.Cq There have been many studies on cosmological roles of first-order phase transitions, which proceed by nucleations and collisions of vacuum bubbles[1]. For example, in electroweak baryo- genesis models[2] rapid bubble expansion can provide a non-equilibrium environment, which may result in asymmetry between matter and antimatter. Furthermore, in some inflation- ary models[3, 4, 5], the speed of expanding vacuum bubbles determines how long the infla- tion period lasts. To understand the bubble kinematics in a hot plasma, it is important to study particle scatterings at a moving bubble wall. To calculate the velocity of electro-weak bubbles[6, 7, 8, 9, 10, 11] and the CP violating charge transport rate by the wall available for baryogenesis[2], one should know the reaction force acting on the wall due to the scattered par- ticles, such as quarks and gauge bosons[12, 13]. (For a supersymmetric model see, for example, Ref. 14) At the first order cosmological phase transition, the false vacuum decays to the true vacuum, which has lower energy, by making a vacuum bubble. When it is created, the wall of the bubble is at rest. As the free energy difference between the inner and the outer parts of the bubble fuels the wall, the velocity of the wall increases to the light velocity unless there is a damping force. In the literature, it is generally believed that the non-trivial damping force is caused by a deviation of the particle population from a thermal equilibrium one. In this paper, we study the effect of bremsstrahlung radiations emitted by particles on the pressure acting on a bubble wall (not necessary electroweak bubbles) during cosmological first order phase transitions. The aim of this work is to show that, contrary to the usual arguments, the radiation damping could give a non-negligible pressure even when the particles maintain thermal equilibrium. Bremsstrahlung (braking radiation) is a radiation due to the acceleration or deceleration of a charged particle[15]. Entering a true vacuum through a bubble wall, particles interact with the wall and could acquire mass and be decelerated. For example, a fermion field ψ can get mass through the well-known Yukawa term gψ̄φψ = mψ̄ψ, where φ is a Higgs field. At this time, if the particle is charged electromagnetically, it can radiate strong electromagnetic waves due to the deceleration. Let us calculate the pressure from the scattering. For simplicity, we assume a linear profile for the bubble wall, i.e., gφ(x) ≡ m(x) = m0x/d when 0 < x < d. (See Fig.1.) and choose the coordinates of the rest frame of the bubble wall. ∗Electronic address: scikid@kias.re.kr http://arxiv.org/abs/0704.0837v1 mailto:scikid@kias.re.kr This approximation is good for the usual tanh profile of the wall. The radiation power of an accelerated particle is given by a relativistic version of the Larmor’s formula[16]: dErad , (1) where A = 2e2/3c3 ≃ 0.0611 in the natural units (~ = c = k = 1) and ~k = (kx, ky, kz) is the 3-momentum of a particle. We assume a situation where this classical description of bremsstrahlung is good enough. Also, assuming that the wall is planar and parallel to the y-z plane, we can treat the bubble as a 1-dimensional one along the x-axis. The energy, momentum, and mass of the particle satisfy the usual relation E2 ≡ m2(x) + ~k2(x). (2) Let us denote the x-component of the momentum (kx) as k from now on. Differentiating the above equation with time t and using dx/dt ≡ v and k = Ev, we get the force acting on the wall due to the particles , (3) which is the starting point of the pressure calculation[6]. However, if we also consider the energy carried away by the radiation Erad, then the total energy conserved is Etot ≡ E+Erad and the force and, hence, the pressure should be changed. From dEtot/dt = 0, we obtain = 0, (4) which has a solution for the force . (5) Up to O(A), one can expand the square root term and obtain . (6) The second term represents the radiation damping. Then, the total pressure due to the collisions of the particles in the plasma is given by[6] (2π)3 f(E(k)) , (7) where f(E) = (exp(βE) ± 1)−1 is a distribution function of fermions and bosons, respectively. First, let us briefly review the well-known results without radiation damping. When the mean velocity of the plasma fluid V relative to the wall (or the negative of the bubble wall velocity relative to the fluid ) is zero, the first term of Eq. (6) contributes dm2(x) (2π)3 eβE ± 1 = F (m0, T )− F (0, T ), (8) where F (φ, T ) is a free energy of φ at a temperature T = β−1. When V 6= 0, the distribution function is changed to f [γ(E − V k)] = eβγ(E−V k) ± 1 . (9) Here, γ = (1−V 2)− 2 . However, using the fact that the phase factor d3~k/E is a Lorentz invariant and changing the integration variable to k′ = γ(k − V E) and defining E′ ≡ γ(E − V k), one can find that the V dependency of P1 disappears [6]. From this, it is generally believed that to get non-trivial pressure on the wall, one needs to consider a non-equilibrium deviation of f [7]. Our work indicates this is not necessarily true for some phase transitions. To see this, consider the effect of the radiation (the second term of Eq. (6)). When V = 0, the term contributes to the pressure P2 = 2A dm(x) (2π)3 Ek(eβE ± 1) which also vanishes because the second integrand is an odd function of k. However, when V 6= 0, one can easily check that, due to the 1/k term, the V dependency survives even under the change of the integration variable. Thus, in this case, P2 = 2A dm(x) (2π)3 Ek(eβγ(E−V k) ± 1) dm(x) dx I2(x). (10) To be more concrete, let us calculate an approximate value of the integration when V ≪ 1 for fermions. In this case, we can expand f [γ(E − V k)] ≃ f(E) − V βkf(E)[f(E) − 1] = f(E) + V βkf2(E)exp(−βE). The integration of the first term gives zero, and the second term contributes I2 = V β (2π)3 f2(E)exp(−βE) ≃ (ln2)TV because (2π)3 f2(E)exp(−βE) ≃ (ln2)T 2 , (12) to lowest oder in (m/T )2 (See Ref. 7). Therefore, for the wall described in Fig. 1 the pressure by the radiation is (ln2)Am20T V, (13) which is comparable to the result of numerical integration of Eq. (10) for V ≪ 1, as shown in Fig. 2. (During the numerical study it is useful to change the measure from dkydkz to 2πEdE.) This pressure is proportional to the wall velocity up to the moderately relativistic case and exists even when the system is in a thermal equilibrium. During the electroweak phase transition, a particle’s electromagnetic charge is not definite, so the A value in Lamor’s formula can not be a constant. In this paper , however, to perform a rough calculation, we have assumed that A is a constant during the phase transition. For illustration of high temperature effects on electric charges, now we consider a Debye screening of electric charge by plasma during the phase transition, which is given by effective coupling αeff = α/(1 − 2α ln(k/Λ)/3π) ≃ 0.97α, where we used averaged momentum 〈k〉 ≃ 3T and Λ of order electron mass at the last approximation (see Eq. (42) of [17]). Thus, we obtain A = 0.0599 which is slightly smaller than the zero temperature value. We also plot the pressure with this A value. It is noteworthy that the pressure caused by the radiation damping (Eq. (13)) is of order O(α), which is bigger than the pressure due to a departure from thermal equilibriums[7, 18, 19] (O(α2))[18], and hence non-negligible. Here, α is the fine structure constant. Note also that the power of bremsstrahlung due to bubble walls is much stronger (O(α)) than that of ordinary bremsstrahlung of electrons colliding with ions in a plasma (O(α3))[20]. Since the electroweak phase transition is a complicated phenomenon, by no means is our work a full calculation of the pressure acting on the electroweak bubbles. The purpose of this paper is to present a general idea that radiation damping (although usually ignored in the many related works for bubble wall velocity calculations) could give rise to significant frictional forces even in thermal equilibrium states at some cosmological phase transitions. To include the effects of other particles (e.g., gluon and W/Z particles) in our work, we need to modify Larmor’s formula by using some sort of group factor. Even in this case, it is hardly probable that the pressure from the radiation damping from different gauge sectors exactly cancel each other. Hence, one can expect that a O(α) viscosity to survive. Since bubbles are slow initially, they are supposed to be in a thermal equilibrium state initially. An ordinary calculation shows no friction at this time, but the radiation damping force exists in this stage, and hence, this pressure can significantly change the early evolution of the vacuum bubbles, and the nature of electroweak baryogenesis or inflationary cosmology. ACKNOWLEDGEMENTS The authors are thankful to Myongtak Choi for helpful discussions. This work was supported in part by the Korean Science and Engineering Foundation and Korea Research Foundation (BSRI-98-2441). [1] C. H. Lee, J. Korean Phys. Soc. 33, 588 (1998). [2] M. Trodden, Rev. Mod. Phys. 71, 1463 (1999). [3] D. La and P. J. Steinhardt, Phys. Rev. Lett. 62, 376 (1989). [4] D. S. Goldwirth and H. W. Zaglauer, Phys. Rev. Lett. 67, 3639 (1991). [5] S. Koh, J. Korean Phys. Soc. 49, 787 (2005). [6] N. Turok, Phys. Rev. Lett. 68, 1803 (1992). [7] G. Moore and T. Prokopec, Phys. Rev. Lett. 75, 777 (1995). [8] P. J. Steinhardt, Phys. Rev. D 25, 2074 (1982). [9] K. Enqvist, J. Ignatius, K. Kajantie, and K. Rummukainen, Phys. Rev. D 45, 3415 (1992). [10] M. Dine, R. G. Leigh, P. Huet, A. Linde, and D. Linde, Phys. Rev. D 46, 550 (1992). [11] C. H. Lee, J. Korean Phys. Soc. 32, 861 (1998). [12] D. B. K. Andrew G. Cohen and A. E. Nelson, Nuc. Phys. B 349, 727 (1991). [13] G. R. Farrar and M. E. Shaposhnikov, Phys. Rev. D 50, 774 (1994). [14] P. John and M. G. Schmidt, Nucl. Phys. B 598, 291 (2001). [15] K. T. Byun, K. Y. Kim, and H. Y. Kwak, J. Korean Phys. Soc. 47, 1010 (2005). [16] J. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975). [17] R. A. Schneider, Phys. Rev. D66, 036003 (2002). [18] G. D. Moore, JHEP 0003, 006 (2000). [19] G. D. Moore and T. Prokopec, Phys. Rev. D 52, 7182 (1995). [20] S. Ichimaru, Basic Principles of Plasma Physics (W. A. Benjamin, Reading, MA., 1973). m ( x ) FIG. 1: The effective mass of the particle m(x) in the wall rest frame. 0.2 0.4 0.6 0.8 1 0.001 0.002 0.003 0.004 FIG. 2: The pressure by the radiation damping of fermions colliding with the linear bubble wall as a function of the wall velocity. The thick line shows numerical integration of Eq. (10) and the dotted line shows the approximate formula in Eq. (13). Here we set 1/d = 1 = m0 = T for simplicity. The dashed line represents the result with Debye screening of charge. ACKNOWLEDGEMENTS References ABSTRACT When charged particles collide with a vacuum bubble, they can radiate strong electromagnetic waves due to rapid deceleration. Owing to the energy loss of the particles by this bremsstrahlung radiation, there is a non-negligible damping pressure acting on the bubble wall even when thermal equilibrium is maintained. In the non-relativistic region, this pressure is proportional to the velocity of the wall and could have influenced the bubble dynamics in the early universe. <|endoftext|><|startoftext|> Introduction The classical setting of the universal lossless compression problem [5], [8], [9] assumes that a se- quence xn of length n that was generated by a source θ is to be compressed without knowledge of the particular θ that generated xn but with knowledge of the class Λ of all possible sources θ. The average performance of any given code, that assigns a length function L(·), is judged on the basis of the redundancy function Rn (L,θ), which is defined as the difference between the expected code length of L (·) with respect to (w.r.t.) the given source probability mass function Pθ and the nth-order entropy of Pθ normalized by the length n of the uncoded sequence. A class of sources is said to be universally compressible in some worst sense if the redundancy function diminishes for this worst setting. Another approach to universal coding [29] considers the individual sequence redundancy R̂n (L, x n), defined as the normalized difference between the code length obtained by L(·) for xn and the negative logarithm of the maximum likelihood (ML) probability of the sequence xn, where the ML probability is within the class Λ. We thereafter refer to this negative logarithm as the ML description length of xn. The individual sequence redundancy is defined for each sequence that can be generated by a source θ in the given class Λ. Classical literature on universal compression [5], [8], [9], [23], [29] considered compression of sequences generated by sources over finite alphabets. In fact, it was shown by Kieffer [15] (see also [13]) that there are no universal codes (in the sense of diminishing redundancy) for sources over infinite alphabets. Later work (see, e.g., [21], [25]), however, bounded the achievable redundancies for identically and independently distributed (i.i.d.) sequences generated by sources over large and infinite alphabets. Specifically, while it was shown that the redundancy does not decay if the alphabet size is of the same order of magnitude as the sequence length n or greater, it was also shown that the redundancy does decay for alphabets of size o(n). 1 While there is no universal code for infinite alphabets, recent work [20] demonstrated that if one considers the pattern of a sequence instead of the sequence itself, universal codes do exist in the sense of diminishing redundancy. A pattern of a sequence, first considered, to the best of our knowledge, in [1], is a sequence of indices, where the index ψi at time i represents the order of first occurrence of letter xi in the sequence x n. Further study of universal compression of patterns [20], [21], [26], [28] provided various lower and upper bounds to various forms of redundancy in universal 1For two functions f(n) and g(n), f(n) = o(g(n)) if ∀c,∃n0, such that, ∀n > n0, f(n) < cg(n); f(n) = O(g(n)) if ∃c, n0, such that, ∀n > n0, 0 ≤ f(n) ≤ cg(n); f(n) = Θ(g(n)) if ∃c1, c2, n0, such that, ∀n > n0, c1g(n) ≤ f(n) ≤ c2g(n). compression of patterns. Another related study is that of compression of data, where the order of the occurring data symbols is not important, but their types and empirical counts are [30]-[31]. This paper considers universal compression of data sequences generated by distributions that are known a-priori to be monotonic. Hence, the order of probabilities of the source symbols is known in advance to both encoder and decoder and can be utilized as side information to improve universal compression performance. Monotonic distributions are common for distributions over the integers, including the geometric distribution and others. Such distributions do occur in image compression problems (see, e.g., [18], [19]), and in other applications that compress residual signals. A specific application one can consider for the results in this paper is compression of the list of last or first names in a given city of a given population. One can usually find some monotonicity for such a distribution in the given population, which both encoder and decoder may be aware of a-priori . For example, the last name “Smith” can be expected to be much more common than the last name “Shannon”. Another example is the compression of a sequence of observations of different species, where one has prior knowledge which species are more common, and which are rare. Finally, one can consider compressing data for which side information given to the decoder through a different channel gives the monotonicity order. Unlike compression of patterns, Foster, Stine, and Wyner, showed in [10] that there are no universal block codes in the standard sense for the complete class of monotonic distributions. The main reason is that there exist such distributions, for which much of the statistical weight lies in symbols that have very low probability, and most of which will not occur in a given sequence. Thus, in practice, even though one has the prior knowledge of the monotonicity of the distribution, this monotonicity is not necessarily retained in an observed sequence. Therefore, actual coding can be very similar to compressing with infinite alphabets, and the additional prior knowledge of the monotonicity is not very helpful in reducing redundancy. Despite that, Foster, Stine, and Wyner demonstrated codes that obtained universal per-symbol redundancy of o(1) as long as the source entropy is fixed (i.e., neither increasing with n nor infinite). However, instead of considering redundancy in the standard sense, the study of monotonic distributions resorted to studying relative redundancy , which bounds the ratio between average assigned code length and the source entropy. This approach dates back to work by Elias [7], Rissanen [22], and Ryabko [24]. The work in [10] studied coding sequences (or blocks) generated by i.i.d. monotonic distributions, and designed codes for which the relative block redundancy could be (upper) bounded. Unlike that work, the focus in [7], [22], and [24] was on designing codes that minimize the redundancy or relative redundancy for a single symbol generated by a monotonic distribution. Specifically, in [22], minimax codes, which minimize the relative redundancy for the worst possible monotonic distribution over a given alphabet size, were derived. In [24], it was shown that redundancy of O(log log k), where k is the alphabet size, can be obtained with minimax per-symbol codes. Very recent work [16] considered per-symbol codes that minimize an average redundancy over the class of monotonic distributions for a given alphabet size. Unlike [10], all these papers study per-symbol codes. Therefore, the codes designed always pay non-diminishing per-symbol redundancy. A different line of work on monotonic distributions considered optimizing codes for a known monotonic distribution but with unknown parameters (see [18], [19] for design of codes for two-sided geometric distributions). In this line of work, the class of sources is very limited and consists of only the unknown parameters of a known distribution. In this paper, we consider a general class of monotonic distributions that is not restricted to a specific type. We study standard block redundancy for coding sequences generated by i.i.d. monotonic distributions, i.e., a setting similar to the work in [10]. We do, however, restrict ourselves to smaller subsets of the complete class of monotonic distributions. First, we consider monotonic distributions over alphabets of size k, where k is either small w.r.t. n, or of O(n). Then, we extend the analysis to show that under minimal restrictions of the monotonic distribution class, there exist universal codes in the standard sense, i.e., with diminishing per-symbol redundancy. In fact, not only do universal codes exist, but under mild restrictions, they achieve the same redundancy as obtained for alphabets of size O(n). The restrictions on this subclass imply that some types of fast decaying monotonic distributions are included in it, and therefore, sequences generated by these distributions (without prior knowledge of either the distribution or of its parameters) can still be compressed universally in the class of monotonic distributions. The main contributions of this paper are the development of codes and derivation of their upper bounds on the redundancies for coding i.i.d. sequences generated by monotonic distributions. Specifically, the paper gives complete characterization of the redundancy in coding with monotonic distributions over “small” alphabets (k = o(n1/3)) and “large” alphabets (k = O(n)). Then, it shows that these redundancy bounds carry over (in first order) to fast decaying distributions. Next, a code that achieves good redundancy rates for even slower decaying monotonic distributions is derived, and is used to study achievable redundancy rates for such distributions. Lower bounds are also presented to complete the characterization, and are shown to meet the upper bounds in the first three cases (small alphabets, large alphabets, and fast decaying distributions). The lower bounds turn out to result from lower bounds obtained for coding patterns. The relationship to patterns is demonstrated in the proofs of those lower bounds. Finally, individual sequences are considered. It is shown that under mild conditions, there exist universal codes w.r.t. the monotonic ML description length for sequences that contain the O(n) more likely symbols, even if their empirical distributions are not monotonic. The outline of this paper is as follows. Section 2 describes the notation and basic definitions. Then, in section 3, lower bounds on the redundancy for monotonic distributions are derived. Next, in Section 4, we propose codes and upper bound their redundancy for coding monotonic distribu- tions over small and large alphabets. These bounds are then extended to fast decaying monotonic distributions in Section 5. Finally, in Section 6, we consider individual sequence redundancy. 2 Notation and Definitions Let xn = (x1, x2, . . . , xn) denote a sequence of n symbols over the alphabet Σ of size k, where k can go to infinity. Without loss of generality, we assume that Σ = {1, 2, . . . , k}, i.e., it is the set of positive integers from 1 to k. The sequence xn is generated by an i.i.d. distribution of some source, determined by the parameter vector θ = (θ1, θ2, . . . , θk), where θi is the probability of X taking value i. The components of θ are non-negative and sum to 1. The distributions we consider in this paper are monotonic. Therefore, θ1 ≥ θ2 ≥ . . . ≥ θk. The class of all monotonic distributions will be denoted by M. The class of monotonic distributions over an alphabet of size k is denoted by Mk. It is assumed that prior to coding xn both encoder and decoder know that θ ∈ M or θ ∈ Mk, and also know the order of the probabilities in θ. In the more restrictive setting, k is known in advance and it is known that θ ∈ Mk. We do not restrict ourselves to this setting. In general, boldface letters will denote vectors, whose components will be denoted by their indices in the vector. Capital letters will denote random variables. We will denote an estimator by the hat sign. In particular, θ̂ will denote the ML estimator of θ which is obtained from xn. The probability of xn generated by θ is given by Pθ (x = Pr (xn | Θ = θ). The average per-symbol2 nth-order redundancy obtained by a code that assigns length function L(·) for θ is Rn (L,θ) EθL [X n]−Hθ [X] , (1) where Eθ denotes expectation w.r.t. θ, and Hθ [X] is the (per-symbol) entropy (rate) of the source 2In this paper, redundancy is defined per-symbol (normalized by the sequence length n). However, when we refer to redundancy in overall bits, we address the block redundancy cost for a sequence. (Hθ [X n] is the nth-order sequence entropy of θ, and for i.i.d. sources, Hθ [X n] = nHθ [X]). With entropy coding techniques, assigning a universal probability Q (xn) is identical to designing a uni- versal code for coding xn where, up to negligible integer length constraints that will be ignored, the negative logarithm to the base of 2 of the assigned probability is considered as the code length. The individual sequence redundancy (see, e.g., [29]) of a code with length function L (·) per sequence xn is R̂n (L, x {L (xn) + log PML (xn)} , (2) where the logarithm function is taken to the base of 2, here and elsewhere, and PML (x n) is the probability of xn given by the ML estimator θ̂Λ ∈ Λ of the governing parameter vector Θ. The negative logarithm of this probability is, up to integer length constraints, the shortest possible code length assigned to xn in Λ. It will be referred to as the ML description length of xn in Λ. In the general case, one considers the i.i.d. ML. However, since we only consider θ ∈ M, i.e., restrict the sequence to one governed by a monotonic distribution, we define θ̂M ∈ M as the monotonic ML estimator. Its associated shortest code length will be referred to as the monotonic ML description length. The estimator θ̂M may differ from the i.i.d. ML θ̂, in particular, if the empirical distribution of xn is not monotonic. The individual sequence redundancy in M is thus defined w.r.t. the monotonic ML description length, which is the negative logarithm of PML (x xn | Θ = θ̂M ∈ M The average minimax redundancy of some class Λ is defined as R+n (Λ) = min Rn (L,θ) . (3) Similarly, the individual minimax redundancy is that of the best code L (·) for the worst sequence R̂+n (Λ) = min {L (xn) + logPθ (xn)} . (4) The maximin redundancy of Λ is R−n (Λ) = sup w (dθ)Rn (L,θ) , (5) where w(·) is a prior on Λ. In [5], it was shown that R+n (Λ) ≥ R−n (Λ). Later, however, [6], [11], [24] the two were shown to be essentially equal. 3 Lower Bounds Lower bounds on various forms of the redundancy for the class of monotonic distributions can be obtained with slight modifications of the proofs for the lower bounds on the redundancy of coding patterns in [14], [20], [21], and [26]. The bounds are presented in the following three theorems. For the sake of completeness, the main steps of the proofs of the first two theorems are presented in appendices, and the proof of the third theorem is presented below. The reader is referred to [14], [20], [21], [25] and [26] for more details. Theorem 1 Fix an arbitrarily small ε > 0, and let n → ∞. Then, the nth-order average max- imin and minimax universal coding redundancies for i.i.d. sequences generated by a monotonic distribution with alphabet size k are lower bounded by R−n (Mk) ≥ log n + k−1 log πe log k , for k ≤ πn1−ε )1/3 · (1.5 log e) · n(1−ε)/3 , for k > πn1−ε )1/3 . (6) Theorem 2 Fix an arbitrarily small ε > 0, and let n→ ∞. Then, the nth-order average universal coding redundancy for coding i.i.d. sequences generated by monotonic distributions with alphabet size k is lower bounded by Rn (L,θ) ≥ log n − k−1 log 8π log k , for k ≤ 1 1.5 log e 2π1/3 · n(1−ε)/3 , for k > 1 )1/3 (7) for every code L(·) and almost every i.i.d. source θ ∈ Mk, except for a set of sources Aε (n) whose relative volume in Mk goes to 0 as n→ ∞. Theorems 1 and 2 give lower bounds on redundancies of coding over monotonic distributions for the class Mk. However, the bounds are more general, and the second region applies to the whole class of monotonic distributions M. As in the case of patterns [20], [26], the bounds in (6)-(7) show that each parameter costs at least 0.5 log(n/k3) bits for small alphabets, and the total universality cost is at least Θ(n1/3−ε) bits overall for larger alphabets. Unlike the currently known results on patterns, however, we show in Section 4 that for k = O(n) these bounds are achievable for monotonic distributions. The proofs of Theorems 1 and 2 are presented in Appendix A and in Appendix B, respectively. Theorem 3 Let n → ∞. Then, the nth-order individual minimax redundancy for i.i.d. sequences with maximal letter k w.r.t. the monotonic ML description length with alphabet size k is lower bounded by R̂+n (Mk) ≥ log n log e 23/12 log k , for k ≤ e5/18 (2π)1/3 · n1/3 e5/18 (2π)1/3 (log e) · n1/3 , for n > k > e (2π)1/3 · n1/3 (log e) · n1/3 , for k ≥ n. Theorem 3 lower bounds the individual minimax redundancy for coding a sequence believed to have an empirical monotonic distribution. The alphabet size is determined by the maximal letter that occurs in the sequence, i.e., k = max {x1, x2, . . . , xn}. (If k is unknown, one can use Elias’ code for the integers [7] using O(log k) bits to describe k. However this is not reflected in the lower bound.) The ML probability estimate is taken over the class of monotonic distributions, i.e., the empirical probability (standard ML) estimate θ̂ is not θ̂M in case θ̂ does not satisfy the monotonicity that defines the class M. While the average case maximin and minimax bounds of Theorem 1 also apply to R̂+n (Mk), the bounds of Theorem 3 are tighter for the individual redundancy and are obtained using individual sequence redundancy techniques. Proof of Theorem 3: Using Shtarkov’s normalized maximum likelihood (NML) approach [29], one can assign probability Q (xn) yn Pθ̂M maxθ′∈M Pθ′ (x yn maxθ′∈M Pθ′ (y to sequence xn. This approach minimizes the individual minimax redundancy, giving individual redundancy of R̂n (Q,x maxθ′∈M Pθ′ (x Q (xn) Pθ′ (y to every xn, specifically achieving the individual minimax redundancy. It is now left to bound the logarithm of the sum in (10). For the first two regions, we follow the approach used in Theorem 2 in [21] for bounding the redundancy for standard compression of i.i.d. sequences over large alphabets, but adjust it to monotonic distributions. Alternatively, one can derive the same bounds following the approach used for bounding the individual minimax redundancy of patterns in proving Theorem 12 in [20]. Let nℓx = (nx(1), nx(2), . . . , nx(ℓ)) denote the occurrence counts of the first ℓ letters of the alphabet Σ in xn. For ℓ = k, i=1 nx(i) = n. Now, following (10), nR̂+n (Mk) ≥ log yn:θ̂(yn)∈M ≥ log ny(1), . . . , ny(ℓ) ny(i) )ny(i) ≥ log ny(1), . . . , ny(k) ny(i) )ny(i) ≥ log ek/12 · (2π)k/2 nx(i) ≥ log k − 1 ek/12 ≥ k − 1 + k log e23/12√ −O (log k) (11) where (a) follows from including only sequences yn that have a monotonic empirical (i.i.d. ML) distribution in Shtarkov’s sum. Inequality (b) follows from partitioning the sequences yn into types as done in [21], first by the number of occurring symbols ℓ, and then by the empirical distribution. Unlike standard i.i.d. distributions though, monotonicity implies that only the first ℓ symbols in Σ occur, and thus the choice of ℓ out of k in the proof in [21] is replaced by 1. Like in coding patterns, we also divide by ℓ! because each type with ℓ occurring symbols can be ordered in at most ℓ! ways, where only some retain the monotonicity. (Note that this step is the reason that step (b) produces an inequality, because more than one of the orderings may be monotonic if equal occurrence counts occur.) Except the division by ℓ!, the remaining steps follow those in [21]. Retaining only the term ℓ = k yields inequality (c). Inequality (d) follows from Stirling’s bound 2πm · ≤ m! ≤ 2πm · · exp . (12) Then, (e) follows from the relation between arithmetic and geometric means, and from expressing the number of types as the number of ordered partitions of n into k parts k − 1 . Finally, (f) follows from applying (12) again and by lower bounding k − 1 The first region in (8) results directly from (11). The behavior is similar to patterns as shown in [1] for this region. As mentioned in [20], to obtain the second region, the bound is maximized by retaining ℓ̂ = n1/3e5/18 /(2π)1/3 instead of k in step (c) of (11), for every k ≥ ℓ̂. The bounds obtained are equal to those obtained for patterns because the first step (a) in (11) discards all the sequences whose contributions to Shtarkov’s sum are different between patterns and monotonic distributions. A similar step is effectively done deriving the bounds for patterns. The difference is that in the case of patterns, components of Shtarkov’s sum are reduced, but all are retained in the sum, while here, we omit components from the sum, corresponding to sequences with non- monotonic i.i.d. ML estimates. The analysis in [20] that also attains the second region of the bound in (8) is still valid here. It differs from the steps taken above by lower bounding a pattern probability by a larger probability than the ML i.i.d. probability corresponding to the pattern. The bound used in the derivation of Theorem 12 in [20] adds a multiplicative factor to each pattern probability which equals the number of sequences with the same pattern and an equal i.i.d. ML probability. However, this similar effect is included in Shtarkov’s sum for monotonic distributions since all these sequences do have a corresponding i.i.d. ML estimate which is monotonic, and are thus not omitted by step (a) of the derivation. The analysis in [14] yields the third region of the bound in (8), since, for k ≥ n, R̂+n (Mk) = Ψ(yn) 1.5n1/3 log e log n , (13) where Ψ(yn) is the pattern of the sequence yn. Inequality (a) holds because each pattern cor- responds to at least one sequence whose ML probability parameter estimates are ordered, i.e., θ̂i ≥ θ̂i+1,∀i, where the most probable index represents i = 1, the second most probable index i = 2, and so on. Note that the sum element on the right hand side is for a probability of a sequence, not a pattern, but the sum is over all patterns. The left hand side also includes sequences for which the probabilities are unordered. Furthermore, exchanging the letters that correspond to two indices with the same occurrence count will not violate monotonicity. Thus the inequality follows. Step (b) in (13) is taken from [14], where the sum on the left hand side was shown to equal the right hand side. This was true when summing over all patterns with up to n indices, thus requiring k ≥ n. Note that this requirement does not mean that n distinct symbols must occur in xn, only that the maximal symbol in xn is n or greater. This concludes the proof of Theorem 3. � 4 Upper Bounds for Small and Large Alphabets In this section, we demonstrate codes that asymptotically achieve the lower bounds for θ ∈ Mk and k = O(n). We begin with a theorem that shows the achievable redundancies, and devote the remainder of the section to describing the codes and deriving upper bounds on their redundancies. The theorem is stated assuming no initial knowledge of k. The proof first considers the setting where k is known, and then shows how the same bounds are achieved even when k is unknown in advance, but as long as it satisfies the conditions. Theorem 4 Fix an arbitrarily small ε > 0, and let n → ∞. Then, there exist a code with length function L∗ (·) that achieves redundancy Rn (L ∗,θ) ≤ (1 + ε) k−1 n(logn)2 , for k ≤ n1/3, (1 + ε) (log n) log k n1/3−ε , for n1/3 < k = o(n), (1 + ε) 2 (log n) 2 n1/3 , for n1/3 < k = O(n), for i.i.d. sequences generated by any source θ ∈ Mk. Slightly tighter bounds are possible in the first and second regions and between them. The bounds presented, however, are inclusive for each of the regions. Note that the third region con- tains the second, but if k = o(n), a tighter bound is possible in the second region. The code designed to code a sequence xn is a two part code [23] that quantizes a distribution that minimizes the cost, and uses it to code xn. The total redundancy cost consists of the cost of describing the quantized distribution and the quantization cost. The second is bounded through the quantized true distribution of the sequence, which cannot result in lower cost than that of the chosen dis- tribution (which minimizes the cost). In order to achieve the low costs of the lower bound, the probability parameters are quantized non-uniformly, where the smaller the probability the finer the quantization. This approach was used in [25] and [26] to obtain upper bounds on the redundancy for coding over large alphabets and for coding patterns, respectively. The method used in [25] and [26], however, is insufficient here, because it still results in too many quantization points due to the polynomial growth in quantization spacing. Here, we use an exponential growth as the parameters increase. This general idea was used in [28] to improve an upper bound on the redundancy of coding patterns. Here, however, we improve on the method presented in [28]. Another key step in the proof here is the fact that since both encoder and decoder know the order of the probabilities a-priori , this order need not be coded. It is sufficient to encode the quantized probabilities of the monotonic distribution, and the decoder can identify which probability is associated with which symbol using the monotonicity of the distribution. Proof of Theorem 4: We start with k ≤ n1/3 assuming k is known. Let β = 1/(log n) be a parameter (note, that we can choose other values). Partition the probability space into J1 = ⌈1/β⌉ intervals, n(j−1)β , 1 ≤ j ≤ J1. (15) Note that I1 = [1/n, 2/n), I2 = [2/n, 4/n), . . . , Ij = [2 j−1/n, 2j/n). Let kj = |θi ∈ Ij| denote the number of probabilities in θ that are in interval Ij. In interval j, take a grid of points with spacing . (16) Note that to complete all points in an interval, the spacing between two points at the boundary of an interval may be smaller. There are ⌈log n⌉ intervals. Ignoring negligible integer length constraints (here and elsewhere), in each interval, the number of points is bounded by |Ij | ≤ , ∀j : j = 1, 2, . . . , J1, (17) where | · | denotes the cardinality of a set. Let the grid τ = (τ1, τ2, . . .) = , . . . , , . . . be a vector that takes all the points from all intervals, with cardinality = |τ | ≤ 1 ⌈log n⌉ . (19) Now, let ϕ = (ϕ1, ϕ2, . . . , ϕk) be a monotonic probability vector, such that ϕi = 1, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕk ≥ 0, and also the smaller k−1 components of ϕ are either 0 or from τ , i.e., ϕi ∈ (τ ∪ {0}), i = 2, 3, . . . , k. One can code xn using a two part code, assuming the distribution governing xn is given by the parameter ϕ. The code length required (up to integer length constraints) is L (xn|ϕ) = log k + LR(ϕ)− log Pϕ (xn) , (20) where log k bits are needed to describe how many letter probabilities are greater than 0 in ϕ, and LR(ϕ) is the number of bits required to describe the quantized points of ϕ. The vector ϕ can be described by a code as follows. Let k̂ϕ be the number of nonzero letter probabilities hypothesized by ϕ. Let bi denote the index of ϕi in τ , i.e., ϕi = τbi . Then, we will use the following differential code. For ϕ we need at most 1 + log b + 2 log(1 + log b bits to code its index in τ using Elias’ coding for the integers [7]. For ϕi−1, we need at most 1 + log(bi−1 − bi + 1) + 2 log[1 + log(bi−1 − bi + 1)] bits to code the index displacement from the index of the previous parameter, where an additional 1 is added to the difference in case the two parameters share the same index. Summing up all components of ϕ, and taking b k̂ϕ+1 LR(ϕ) ≤ k̂ϕ − 1 + log (bi − bi+1 + 1) + 2 log [1 + log (bi − bi+1 + 1)] ≤ (k − 1) + (k − 1) log B1 + k − 1 + 2(k − 1) log log B1 + k − 1 + o(k) = (1 + ε) k − 1 n (log n) . (21) Inequality (a) is obtained by applying Jensen’s inequality once on the first sum, twice on the second sum utilizing the monotonicity of the logarithm function, and by bounding k̂ϕ by k and absorbing low order terms in the resulting o(k) term. Then, low order terms are absorbed in ε, and (19) is used to obtain (b). To code xn, we choose ϕ which minimizes the expression in (20) over all ϕ, i.e., L∗ (xn) = min L (xn|ϕ) △= L (xn|ϕ̂) . (22) The pointwise redundancy for xn is given by nRn (L ∗, xn) = L∗ (xn) + log Pθ (x n) = log k + L∗R (ϕ̂) + log Pθ (x Pϕ̂ (x . (23) Note that the pointwise redundancy differs from the individual one, since it is defined w.r.t. the true probability of xn. To bound the third term of (23), let θ′ be a quantized still monotonic version of θ onto τ , i.e., θ′i ∈ (τ ∪ {0}), i = 2, 3, . . . , k, where if θi > 0 ⇔ θ′i > 0 as well. Define the quantization error, δi = θi − θ′i. (24) The quantization is performed from the smallest parameter θk to the largest, where monotonicity is retained, as well as minimal absolute quantization error. This implies that θi will be quantized to one of the two nearest grid points (one smaller and one greater than it). It also guarantees that |δ1| ≤ ∆ , where j2 is the index of the interval in which θ2 is contained, i.e., θ2 ∈ Ij2 . Now, since θ′ is included in the minimization of (22), we have, for every xn, L∗ (xn) ≤ L xn|θ′ , (25) and also nRn (L ∗, xn) ≤ log k + LR + log Pθ (x Pθ′ (x . (26) Averaging over all possible xn, the average redundancy is bounded by nRn (L ∗,θ) = log k + EθL R (ϕ̂) + Eθ log Pθ (X Pϕ̂ (X ≤ log k + EθLR ′)+ Eθ log Pθ (X Pθ′ (X . (27) The second term of (27) is bounded with the bound of (21), and we proceed with the third term. Eθ log Pθ (X Pθ′ (X θi log θ′i + δi ≤ n(log e) θ′i + δi = n(log e) ≤ k log e+ 2(log e)k kj · njβ ≤ 5(log e)k. (28) Equality (a) is since the argument in the logarithm is fixed, thus expectation is performed only on the number of occurrences of letter i for each letter. Representing θi = θ i + δi yields equation (b). We use ln(1+x) ≤ x to obtain (c). Equality (d) is obtained since all the quantization displacements must sum to 0. The first term of inequality (e) is obtained under a worst case assumption that θi ≪ 1/n for i ≥ 2. Thus it is quantized to θ′i = 1/n, and the bound |δi| ≤ 1/n is used. The second term is obtained by separating the terms into their intervals. In interval j, the bounds θ′i ≥ n(j−1)β/n, and |δi| ≤ knjβ/n1.5 are used, and also nβ = 2. Inequality (f) is obtained since j ≤ 2n. (29) Inequality (29) is obtained since k1 ≤ n, k2 ≤ (n− k1)/2, k3 ≤ (n− k1)/4− k2/2, and so on, until kJ1 ≤ 2J1−1 2J1−ℓ j ≤ 2n. (30) The reason for these relations are the lower limits of the J1 intervals that restrict the number of parameters inside the interval. The restriction is done in order of intervals, so that the used probabilities are subtracted, leading to the series of equations. Plugging the bounds of (21) and (28) into (27), we obtain, nRn (L ∗,θ) ≤ log k + (1 + ε) k − 1 n (log n) + 5(log e)k 1 + ε′ ) k − 1 n (log n) , (31) where we absorb low order terms in ε′. Replacing ε′ by ε normalizing the redundancy per symbol by n, the bound of the first region of (14) is proved. We now consider the larger values of k, i.e., n1/3 < k = O(n). The idea of the proof is the same. However, we need to partition the probability space to different intervals, the spacing within an interval must be optimized, and the parameters’ description cost must be bounded differently, because now there are more parameters quantized than points in the quantization grid. Define the jth interval as n(j−1)β , 1 ≤ j ≤ J2, (32) where J2 = ⌈2/β⌉ = ⌈2 log n⌉. Again, let kj = |θi ∈ Ij| denote the number of probabilities in θ that are in interval Ij. It could be possible to use the intervals as defined in (15), but this would not guarantee bounded redundancy in the rate we require if there are very small probabilities θi ≪ 1/n. Therefore, the interval definition in (15) can be used for larger alphabets only if the probabilities of the symbols are known to be bounded. Define the spacing in interval j as , (33) where α is a parameter to be optimized. Similarly to (17), the interval cardinality here is |Ij| ≤ 0.5 · nα, ∀j : j = 1, 2, . . . , J2, (34) In a similar manner to the definition of τ in (18), we define η = (η1, η2, . . .) = , . . . , , . . . . (35) The cardinality of η is = |η| ≤ 0.5 · nα ⌈2 log n⌉ ≤ nα ⌈log n⌉ . (36) We now perform the encoding similarly to the small k case, where we allow quantization to nonzero values to the components of ϕ up to i = n2. (This is more than needed but is possible since η1 = 1/n 2.) Encoding is performed similarly to the small k case. Thus, similarly to (27), we nRn (L ∗,θ) ≤ 2 log n+ EθLR ′)+ Eθ log Pθ (X Pθ′ (X , (37) where the first term is due to allowing up to k̂ = n2. Since usually in this region k ≥ B2 (except the low end), the description of vectors ϕ and θ′ is done by coding the cardinality of |ϕi = ηj | and |θ′i = ηj |, respectively, i.e., for each grid point the code describes how many letters have probability quantized to this point. This idea resembles coding profiles of patterns, as done in [20]. However, unlike the method in [20], here, many probability parameters of symbols with different occurrences are mapped to the same grid point by quantization. The number of parameters mapped to a grid point of η is coded using Elias’ representation of the integers. Hence, in a similar manner to (21), 1 + log |θ′i = ηj |+ 1 + 2 log 1 + log |θ′i = ηj |+ 1 ≤ B2 +B2 log k +B2 + 2B2 log log k +B2 + o (B2) (1 + ε)(log n) log k nα, for nα < k = o(n), (1 + ε)(1 − α) (log n)2 nα, for nα < k = O(n). The additional 1 term in the logarithm in (a) is for 0 occurrences, (b) is obtained similarly to step (a) of (21), absorbing all low order terms in the last term. To obtain (c), we first assume, for the first region, that knε ≫ B2 (an assumption that must be later validated with the choice of α). Then, low order terms are absorbed in ε. The extra nε factor is unnecessary if k ≫ B2. The second region is obtained by upper bounding k without this factor. It is possible to separate the first region into two regions, eliminate this factor in the lower region, and obtain a more complicated, yet tighter, expression in the upper region, where k ∼ Θ(n1/3). Now, similarly to (28), we obtain Eθ log Pθ (X Pθ′ (X ≤ n(log e) ≤ O(1) + 2 log e n1+2α ≤ 4(log e)n1−2α +O(1). (39) The first term of inequality (a) is obtained under the assumption that k = O(n), θ′i ≥ 1/n2, and |δi| ≤ 1/n2. For the second term |δi| ≤ njβ/n2+α, and θ′i ≥ n(j−1)β/n2. Inequality (b) is obtained in a similar manner to inequality (f) of (28), where the sum is shown similarly to be 2n2. Summing up the contributions of (38) and (39) in (37), it is clear that α = 1/3 minimizes the total cost (to first order). This choice of α also satisfies the assumption of step (c) in (38). Using α = 1/3, absorbing all low order terms in ε and normalizing by n, we obtain the remaining two regions of the bound in (14). It should be noted that the proof here would give a bound of O(n1/3+ε) up to k = O(n4/3). If the intervals in (15) were used for bounded distributions, the coefficients of the last two regions will be reduced by a factor of 2. Additional manipulations on the grid η may reduce the coefficients more (see, e.g., [28]). The proof up to this point assumes that k is known in advance. This is important for the code resulting in the bound for the first region because the quantization grid depends on k. Specifically, if in building the grid, k is underestimated, the description cost of ϕ increases. If k is overestimated, the quantization cost will increase. Also, if the code of the second region is used for a smaller k, a larger bound than necessary results. To solve this, the optimization that chooses L∗ (xn) is done over all possible values of k (greater than or equal to the maximal symbol occurring in xn), i.e., every greater k in the first region, and the construction of the code for the other regions. For every k in the first region, a different construction is done, using the appropriate k to determine the spacing in each interval. The value of k yielding the shortest code word is then used, and O(log n) additional bits are used at the prefix of the code to inform the decoder which k is used. The analysis continues as before. This does not change the redundancy to first order, giving all three regions of the bound in (14), even if k is unknown in advance. This concludes the proof of Theorem 4. � 5 Upper Bounds for Fast Decaying Distributions This section shows that with some mild conditions on the source distribution, the same redundancy upper bounds achieved for finite monotonic distributions can be achieved even if the monotonic distribution is over an infinite alphabet. The key observation that allows this is that a distribution that decays fast enough will result in only a small number of occurrences of unlikely letters in a sequence. These letters may very likely be out of order, but since there are very few of them, they can be handled without increasing the asymptotic behavior of the coding cost. More precisely, fast decaying monotonic distributions can be viewed as if they have some effective bounded alphabet size, where occurrences of symbols outside this limited alphabet are rare. We present two theorems and a corollary that show how one can upper bound the redundancy obtained when coding with some unknown distribution. The first theorem provides a slightly stronger bound (with smaller coefficient) even for k = O(n), where the smaller coefficient is attained by improved bounding, that more uniformly weights the quantization cost for minimal probabilities. In the weaker version of the results presented here, if the distribution decays slower and there are more low probability symbols, the redundancy order does increase due to the penalty of identifying these symbols in a sequence. However, we show, consistently with the results in [10], that as long as the entropy of the source is finite, a universal code, in the sense of diminishing redundancy per symbol, does exist. We begin with stating the two theorems and the corollary, then the proofs are presented. The section is concluded with three examples of typical monotonic distributions over the integers, to which the bounds are applied. 5.1 Upper Bounds We begin with some notation. Fix an arbitrary small ε > 0, and let n→ ∞. Define m △= mρ as the effective alphabet size, where ρ > ε. (Note that ρ = (logm)/(log n).) Let Rn(m) log n , for m = o ρ+ ε− 1 (log n) n1/3, otherwise. Theorem 5 I. Fix an arbitrarily small ε > 0, and let n → ∞. Let xn be generated by an i.i.d. monotonic distribution θ ∈ M. If there exists m∗, such that, nθi log i = o [Rn (m∗)] , (41) then, there exists a code with length function L∗(·), such that Rn (L ∗,θ) ≤ (1 + ε) Rn (m∗) (42) for the monotonic distribution θ. II. If there exists m∗ for which ρ∗ = o n1/3/(log n) , such that, θi log i = o(1), (43) then, there exists a universal code with length function L∗(·), such that Rn (L ∗,θ) = o(1). (44) Theorem 5 implies that if a monotonic distribution decays fast enough, its effective alphabet size does not exceed O(nρ), and, as long as ρ is fixed, bounds of the same order as those obtained for finite alphabets are achievable. Specifically, very fast decaying distributions, although over infinite alphabets, may even behave like monotonic distributions with o symbols. The condition in (41) merely means that the cost that a code would obtain in order to code very rare symbols, that are larger than the effective alphabet size, is negligible w.r.t. the total cost obtained from other, more likely, symbols. Note that for m = n, the bound is tighter than that of the third region of Theorem 4, and a constant of 5/9 replaces 2/3. The second part of the theorem states that if the decay is slow, but the cost of coding rare symbols is still diminishing per symbol, a universal code still exists for such distributions. However, in this case the redundancy will be dominated by coding the rare (out of order) symbols. This result leads to the following corollary: Corollary 1 As n → ∞, sequences generated by monotonic distributions with Hθ(X) = O(1) are universally compressible in the average sense. Corollary 1 shows that sequences generated by finite entropy monotonic distributions can be com- pressed in the average with diminishing per symbol redundancy. This result is consistent with the results shown in [10]. While Theorem 5 bounds the redundancy decay rate with two extremes, a more general theorem can be used to provide some best redundancy decay rate that a code can be designed to adapt to for some unknown monotonic distribution that governs the data. As the examples at the end of this section show, the next theorem is very useful for slower decaying distributions. Theorem 6 Fix an arbitrarily small ε > 0, and let n → ∞. Let xn be generated by an i.i.d. monotonic distribution θ ∈ M. Then, there exists a code with length function L∗(·), that achieves redundancy nRn (L ∗,θ) ≤ (1 + ε) · α,ρ:ρ≥α+ε · (ρ+ 2α) (ρ− α) (log n)2nα + 5(log e)n1−2α + θi log i for coding sequences generated by the source θ. We continue with proving the two theorems and the corollary. Proof : The idea of the proof of both theorems is to separate the more likely symbols from the unlikely ones. First, the code determines the point of separation m = nρ. (Note that ρ can be greater than 1.) Then, all symbols i ≤ m are considered likely and are quantized in a similar manner as in the codes for smaller alphabets. Unlike bounded alphabets, though, a more robust grid is used here to allow larger values of m. Coding of occurrences of these symbols uses the quantized probabilities. The unlikely symbols are coded hierarchically. They are first merged into a single symbol, and then are coded within this symbol, where the full cost of conveying to the decoder which rare symbols occur in the sequence is required. Thus, they are presented giving their actual value. As long as the decay is fast enough, the average cost of conveying these symbols becomes negligible w.r.t. the cost of coding the likely symbols. If the decay is slower, but still fast enough, as the case described in condition (43), the coding cost of the rare symbols dominates the redundancy, but still diminishing redundancy can be achieved. In order to determine the best value of m for a given sequence, all values are tried and the one yielding the shortest description is used for coding the specific sequence xn. Let m ≥ 2 determine the number of likely symbols in the alphabet. For a given m, define θi, (46) as the total probability of the remaining symbols. Given θ, m and Sm, a probability P (xn|m,Sm,θ) nx(i) · Snx(x>m)m · nx(i) nx(x > m) )nx(i) , (47) can be computed for xn, where nx(i) is the occurrence count of symbol i in x n, and nx(x > m) is the count of all symbols greater than m in xn. This probability mass function clusters all large symbols (with small probabilities) greater than m into one symbol. Then, it uses the ML estimate of each of the large symbols to distinguish among them in the clustered symbol. For every m, we can define a quantization grid ξm for the first m probability parameters of θ. The idea is similar to that used for all probability parameters in the proof of Theorem 4. If m = o(n1/3), we use ξm = τm, where τm is the grid defined in (18) where m replaces k. Otherwise, we can use the definition of η in (35). However, to obtain tighter bounds for large m, we define a different grid for the larger values of m following similar steps to those in (32)-(36). First, define the jth interval as n(j−1)β nρ+2α nρ+2α , 1 ≤ j ≤ Jρ, (48) where ρ = (logm)/(log n) as defined above, α is a parameter, and β = 1/(log n) as before. Within the jth interval, we define the spacing in the grid by nρ+3α . (49) As in (34), |Ij | ≤ 0.5 · nα, ∀j : j = 1, 2, . . . , Jρ, (50) and the total number of intervals is Jρ = ⌈(ρ+ 2α) log n⌉ . (51) Similarly to (35), ξm is defined as ξm = (ξ1, ξ2, . . .) = nρ+2α nρ+2α nρ+3α , . . . , nρ+2α nρ+2α nρ+3α , . . . . (52) The cardinality of ξm is thus = |ξm| ≤ 0.5 · nα ⌈(ρ+ 2α) log n⌉ . (53) An mth order quantized version θ′m of θ is obtained by quantizing θi, i = 2, 3, . . . ,m onto ξm, such that θ′i ∈ ξm for these values of i. Then, the remaining cluster probability Sm is quantized into S′m ∈ [1/n, 2/n, . . . , 1]. The parameter θ′1 is constrained by the quantization of the other parameters. Quantization is performed in a similar manner as before, to minimize the accumulating cost and retain monotonicity. Now, for any m ≥ 2, let ϕm be any monotonic probability vector of cardinality m whose last m− 1 components are quantized into ξm, and let σm ∈ [1/n, 2/n, . . . , 1] be a quantized estimate of the total probability of the remaining symbols, such that i=1 ϕi,m+σm = 1, where ϕi,m is the ith component of ϕm. If m, σm and ϕm are known, a given x n can be coded using P (xn|m,σm,ϕm) as defined in (47), where σm replaces Sm, and the m components of ϕm replace the first m com- ponents of θ. However, in the universal setting, none of these parameter are known in advance. Furthermore, neither the symbols greater than m nor their conditional ML probabilities are known in advance. Therefore, the total cost of coding xn using these parameters requires universality costs for describing them. The cost of universally coding xn assigning probability P (xn|m,σm,ϕm) to it thus requires the following five components: 1) m should be described using Elias’ representation with at most 1+ ρ log n+2 log(1+ ρ log n) bits. 2) The value of σm in its quantization grid should be coded using log n bits. 3) The m components of ϕm require LR (ϕm) (which is bounded below) bits. 4) The number cx(x > m) of distinct letters in x n greater than m is coded using log n bits. 5) Each letter i > m in xn is coded. Elias’ coding for the integers using 1 + log i+ 2 log(1 + log i) bits can be used, but to simplify the derivation we can also use the code, also presented in [7], that uses no more than 1 + 2 log i bits to describe i. In addition, at most log n bits are required for describing nx(i) in x n. For n→ ∞, m≫ 1, and ε > 0 arbitrarily small, this yields a total cost of L (xn|m,σm,ϕm) ≤ − log P (xn|m,σm,ϕm) + LR (ϕm) + [(1 + ε)ρ+ cx(x > m) + 2] log n +cx(x > m) + 2 i>m,i∈xn log i, (54) where we assume m is large enough to bound the cost of describing m by (1 + ε)ρ log n. The description cost of ϕm for m = o(n 1/3) is bounded by LR (ϕm) ≤ (1 + ε) using (21), where m replaces k. The (log n)2 factor in (21) can be absorbed in ε since we limit m to o(n1/3), unlike the derivation in (21). For larger values of m, we describe symbol probabilities of ϕm in the grid ξm in a similar manner to the description of O(n) symbol probabilities in the grid η. Similarly to (38), we thus have LR(ϕm) ≤ Bρ +Bρ log nρ +Bρ + 2Bρ log log nρ +Bρ + o (Bρ) ≤ (1 + ε) (ρ+ 2α) (ρ+ ε− α) (log n)2nα (56) where to obtain inequality (a), we first multiply nρ by nε in the numerator of the argument of the logarithm. This is only necessary for ρ→ α to guarantee that nρ+ε ≫ Bρ. Substituting the bound on Bρ from (53), absorbing low order terms in the leading ε, yields the bound. A sequence xn can now be coded using the universal parameters that minimize the length of the sequence description, i.e., L∗ (xn) = min σm′∈[ ,...,1] ϕm′ :ϕi∈ξm′ ,i≥2 xn|m′, σm′ ,ϕm′ xn|m,S′m,θ′m , (57) where θ′m and S m are the true source parameters quantized as described above, and the inequality holds for every m. Note that the maximization on m′ should be performed only up to the maximal symbol the occurs in xn. Following (54)-(57), up to negligible integer length constraints, the average redundancy using L∗(·) is bounded, for every m ≥ 2, by nRn (L ∗,θ) = Eθ [L ∗ (Xn) + log Pθ (X Xn | m,S′m,θ′m + logPθ (X ≤ Eθ log Pθ (X Xn | m,S′m,θ′m ) + LR Pθ (i ∈ Xn) log i +(1 + ε) [EθCx (X > m) + ρ+ 2] log n (58) where (a) follows from (57), and (b) follows from averaging on (54) with σm = S m, and ϕm = θ where the average on cx(x > m) is absorbed in the leading ε. Expressing Pθ (x n) as Pθ (x nx(i)  · Snx(x>m)m · )nx(i) , (59) and defining δS = Sm − S′m, the first term of (58) is bounded, for the upper region of m, by Eθ log Pθ (X Xn | m,S′m,θ′m ) ≤ Eθ Nx(i) log θ′i,m +Nx (X > m) log Nx(i) log θi/Sm Nx(i)/Nx(X > m) ≤ n · θi log θ′i,m + nSm log ≤ n(log e) θ′i,m ≤ (log e) · n · nρ nρ+2α + 2(log e)n1−ρ−4α · jβ + log e ≤ 5(log e)n1−2α + log e, (60) where (a) is since for the third term, the conditional ML probability used for coding is greater than the actual conditional probability assigned to all letters greater than m for every xn. Hence, the third term is bounded by 0. For the other terms expectation is performed. Inequality (b) is obtained similarly to (28) where quantization includes the first m components of θ and the parameter Sm. Then, inequality (c) follows the same reasoning as step (a) of (39). The first term bounds the worst case in which all nρ symbols are quantized to 1/nρ+2α with |δi| ≤ 1/nρ+2α. The second term is obtained where θ′i,m ≥ n(j−1)β/nρ+2α and |δi| ≤ njβ/nρ+3α for θi ∈ Ij , and kj = |θi ∈ Ij| as before. The last term is since S′m ≥ 1/n and |δS | ≤ 1/n. Finally, (d) is obtained similarly to step (b) of (39), where as in (29), jβ ≤ 2nρ+2α. For m = o(n1/3), the same initial steps up to step (b) in (60) are applied, and then the remaining steps in (28) are applied to the left sum with m replacing k, yielding a total quantization cost of 5(log e)m+ log e. To bound the third and fourth terms of (58), we realize that Pθ (i ∈ Xn) = 1− (1− θi)n ≤ nθi. (61) Similarly, EθCx(X > m) = Pθ (i ∈ Xn) ≤ nSm. (62) Combining the dominant terms of the third and fourth terms of (58), we have Pθ (i ∈ Xn) log i+ (1 + ε)EθCx(X > m) log n Pθ (i ∈ Xn) [2 log i+ (1 + ε) log n] 1 + ε Pθ (i ∈ Xn) log i 1 + ε θi log i (63) where (a) is because EθCx(X > m) = i>m Pθ (i ∈ Xn), (b) is because for i > m = nρ, log i > ρ log n, and (c) follows from (61). Given ρ > ε for an arbitrary fixed ε > 0, the resulting coefficient above is upper bounded by some constant κ. Summing up the contributions of the terms of (58) from (28), (55), and (63), absorbing low order terms in a leading ε′, we obtain that for m = o(n1/3), nRn (L ∗,θ) ≤ 1 + ε′ ) m− 1 θi log i. (64) For the second region, substituting α = 1/3, and summing up the contributions of (60), (56), and (63) to (58), absorbing low order terms in ε′, we obtain nRn (L ∗,θ) ≤ (1 + ε′) ρ+ ε′ − (log n) n1/3 + κn θi log i. (65) Since (64)-(65) hold for every m > nε, there exists m∗ for which the minimal bound is obtained. To bound the redundancy, we choose this m∗. Now, if the condition in (41) holds, then the second term in (64) and (65) is negligible w.r.t. the first term. Absorbing it in a leading ε, normalizing by n, yields the upper bound of (42), and concludes the proof of the Part I of Theorem 5. For Part II of Theorem 5, we consider the bound of the second region in (65). If there exists ρ∗ = o n1/3/(log n) for which the condition in (43) holds, then both terms of (65) are of o(n), yielding a total redundancy per symbol of o(1). The proof of Theorem 5 is concluded. � To prove Corollary 1, we use Wyner’s inequality [32], which implies that for a finite entropy monotonic distribution, θi log i = Eθ [logX] ≤ Hθ [X] . (66) Since the sum on the left hand side of (66) is finite if Hθ[X] is finite, there must exist some n0 such θi log i = o(1). Let n > n0, then for m ∗ = n and ρ∗ = 1, condition (43) is satisfied. Therefore, (44) holds, and the proof of Corollary 1 is concluded. � We now consider only the upper region in (58) with parameters α and ρ taking any valid value. (The code leading to the bound of the upper region can be applied even if the actual effective alphabet size is in the lower region.) We can sum up the contributions of (60), (56), and (63) to (58), absorbing low order terms in ε. Equation (56) is valid without the middle ε term as long as ρ ≥ α + ε. Since, in the upper region of m, i ≥ m is large enough, Elias’ code for the integers can be used costing (1 + ε) log i to code i, with ε > 0 which can be made arbitrarily small. Hence, the leading coefficient of the bound in (63) can be replaced by (1 + ε)(1 + 1/ρ). This yields the expression bounding the redundancy in (45). This expression applies to every valid choice of α and ρ, including the choice that minimizes the expression. Thus the proof of Theorem 6 is concluded. � 5.2 Examples We demonstrate the use of the bounds of Theorems 5 and 6 with three typical distributions over the integers. We specifically show that the redundancy rate of O n1/3+ε bits overall is achievable when coding many of the typical monotonic distributions, and, in fact, for many distributions faster convergence rates are achievable with the codes provided in proving the theorems above. The assumption that very few unlikely symbols are likely to appear in a sequence generated by a monotonic distribution, which is reflected in the conditions in (41) and (43), is very realistic even in practical examples. Specifically, in the phone book example, there may be many rare names, but only very few of them may occur in a certain city, and the more common names constitute most of any possible phone book sequence. 5.2.1 Fast Decaying Distributions Over the Integers Consider the monotonic distributions over the integers of the form, , i = 1, 2, . . . , (67) where γ > 0, and a is a normalization coefficient that guarantees that the probabilities over all integers sum to 1. It is easy to show by approximating summation by integration that for some m→ ∞, Sm ≤ (1 + ε) θi log i ≤ (1 + ε) a logm . (69) For m = nρ and fixed ρ, the sum in (41) is thus O n1−ργ log n , which is o n1/3(log n)2 for every ρ ≥ 2/(3γ). Specifically, as long as γ ≤ 2 (slow decay), the minimal value of ρ required to guarantee negligibility of the sum in (41) is greater than 1/3. Using Theorem 5, this implies that for γ ≤ 2, the second (upper) region of the upper bound in (42) holds with the minimal choice of ρ∗ = 2/(3γ). Plugging in this value in the second region of (40) (i.e., in (42)) yields the upper bound shown below for this region. For γ > 2, 2/(3γ) < 1/3. Hence, (41) holds for m∗ = o . This means that for the distribution in (67) with γ > 2, the effective alphabet size is o , and thus the achievable redundancy is in the first region of the bound of (42). Thus, even though the distribution is over an infinite alphabet, its compressibility behavior is similar to a distribution over a relatively small alphabet. To find the exact redundancy rate, we balance between the contributions of (55) and (63) in (58). As long as 1 − ργ < ρ, condition (41) holds, and the contribution of small letters in (63) is negligible w.r.t. the other terms of the redundancy. Equality, implying ρ∗ = 1/(1 + γ), achieves the minimal redundancy rate. Thus, for γ > 2, nRn (L ≤ (1 + ε) a(2ρ∗ + 1) ∗γ log n+ (1− 3ρ∗) log n = (1 + ε) 1+γ log n (70) where the first term in (a) follows from the bounds in (63) and (69), with m = nρ , and the second term from that in (55), and (b) follows from ρ∗ = 1/(1 + γ). Note that for a fixed ρ∗, the factor 3 in the first term can be reduced to 2 with Elias’ coding for the integers. The results described are summarized in the following corollary: Corollary 2 Let θ ∈ M be defined in (67). Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ (1 + ε) 1 1 + 1 + ε− 1 n1/3(logn)2 , for γ ≤ 2, (1 + ε) 1+γ logn , for γ > 2. Corollary 2 gives the redundancy rates for all distributions defined in (67). For example, if γ = 1, the redundancy is O n1/3(log n)2 bits overall with coefficient 2/9. For γ = 3, O(n1/4 log n) bits are required. For faster decays (greater γ) even smaller redundancy rates are achievable. 5.2.2 Geometric Distributions Geometric distributions given by θi = p (1− p)i−1 ; i = 1, 2, . . . , (72) where 0 < p < 1, decay even faster than the distribution over the integers in (67). Thus their effective alphabet sizes are even smaller. This implies that a universal code can have even smaller redundancy than that presented in Corollary 2 when coding sequences generated by a geometric distribution (even if this is unknown in advance, and the only prior knowledge is that θ ∈ M). Choosing m = ℓ · log n, the contribution of low probability symbols in (63) to (58) can be upper bounded by θi (log i+ log n) ≤ 2n(1− p)m log n+O (n(1− p)m logm) = 2n1+ℓ log(1−p)(log n) +O n1+ℓ log(1−p) log log n where (a) follows from computing Sm using geometric series, and bounding the second term, and (b) follows from substituting m = ℓ log n and representing (1 − p)ℓ logn as nℓ log(1−p). As long as ℓ ≥ 1/(− log(1− p)), the expression in (73) is O(log n), thus negligible w.r.t. the redundancy upper bound of (42) with m∗ = ℓ∗ log n = (log n)/(− log(1 − p)). Substituting this m∗ in (42), we obtain the following corollary: Corollary 3 Let θ ∈ M be a geometric distribution defined in (72). Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ 1 + ε −2 log(1− p) · (log n) . (74) Corollary 3 shows that if θ parameterizes a geometric distribution, sequences governed by θ can be coded with average universal coding redundancy of O (log n)2 bits. Their effective alphabet size is O(log n), implying that larger symbols are very unlikely to occur. For example, for p = 0.5, the effective alphabet size is log n, and 0.5(log n)2 bits are required for a universal code. For p = 0.75, the effective alphabet size is (log n)/2, and (log n)2/4 bits are required by a universal code. 5.2.3 Slow Decaying Distributions Over the Integers Up to now, we considered fast decaying distributions, which all achieved the O(n1/3+ε/n) redun- dancy rate. We now consider a slowly decaying monotonic distribution over the integers, given i (log i) , i = 2, 3, . . . , (75) where γ > 0 and a is a normalizing factor (see, e.g., [12], [27]). This distribution has finite entropy only if γ > 0 (but is a valid infinite entropy distribution for γ > −1). Unlike the previous distributions, we need to use Theorem 6 to bound the redundancy for coding sequences generated by this distribution. Approximating the sum with an integral, the order of the third term of (45) θi log i = O (logm)γ . (76) In order to minimize the redundancy bound of (45), we define ρ = nℓ. For the minimum rate, all terms of (45) must be balanced. To achieve that, we must have α+ 2ℓ = 1− 2α = 1− γℓ. (77) The solution is α = γ/(4 + 3γ), and ℓ = 2/(4 + 3γ). Substituting these values in the expression of (45), with ρ = nℓ, results in the first term in (45) dominating, and yields the following corollary: Corollary 4 Let θ ∈ M be defined in (75) with γ > 0. Then, there exists a universal code with length function L∗(·) that has only prior knowledge that θ ∈ M, that can achieve universal coding redundancy Rn (L ∗,θ) ≤ (1 + ε) 3γ+4 (log n)2 . (78) Due to the slow decay rate of the distribution in (75), the effective alphabet size is much greater here. For γ = 1, for example, it is nn . This implies that very large symbols are likely to appear in xn. As γ increases though, the effective alphabet size decreases, and as γ → ∞, m → n. The redundancy rate increases due to the slow decay. For γ ≥ 1, it is O n5/7(log n)2/n . As γ → ∞, since the distribution tends to decay faster, the redundancy rate tends to the finite alphabet rate n1/3(log n)2/n . However, as the decay rate is slower γ → 0, a non-diminishing redundancy rate is approached. Note that the proof of Theorem 6 does not limit the distribution to a finite entropy one. Therefore, the bound of (78) applies, in fact, also to −1 < γ ≤ 0. However, for γ ≤ 0, the per-symbol redundancy is no long diminishing. 6 Individual Sequences In this section, we first show that individual sequences whose empirical distributions obey the monotonicity constraints can be universally compressed as well as the average case. We then study compression of sequences whose empirical distributions may diverge from monotonic. We demonstrate that under mild conditions, similar in nature to those of Theorems 5 and 6, redundancy that diminishes (slower than in the average case) w.r.t. the monotonic ML description length can be obtained. However, these results are only useful when the monotonic ML description length diverges only slightly from the (standard) ML description length of a sequence, i.e., the empirical distribution of a sequence only mildly violates monotonicity. Otherwise, the penalty of using an incorrect monotone model overwhelms the redundancy gain. We begin with sequences that obey the monotonicity constraints. Theorem 7 Fix an arbitrarily small ε > 0, and let n→ ∞. Let xn be a sequence for which θ̂ ∈ M, i.e., θ̂1 ≥ θ̂2 ≥ . . .. Let k = k̂ be the number of letters occurring in xn. Then, there exists a code L∗ (·) that achieves individual sequence redundancy w.r.t. θ̂M = θ̂ for xn which is upper bounded R̂n (L ∗, xn) ≤ (1 + ε) k−1 n(logn) , for k ≤ n1/3, (1 + ε) (log n) log k n1/3−ε , for n1/3 < k = o(n), (1 + ε) 1 (log n) 2 n1/3 , for n1/3 < k = O(n). Note that by the monotonicity constraint, the number of symbols k̂ occurring in xn also equals to the maximal symbol in xn. Since, in the individual sequence case, this maximal symbol defines the class considered and also to be consistent with Theorem 3, we use k to characterize the alphabet size of a given sequence. (The maximal symbol in the individual sequence case is equivalent to the alphabet size in the average case.) Finally, since θ̂ is monotonic, θ̂M = θ̂. Proof of Theorem 7: The result in Theorem 7 follows directly from the proof of Theorem 4. Both regions of the proof apply here, where instead of quantizing θ to θ′, we quantize θ̂ to θ̂ similar manner, and do not need to average over all sequences. In fact, instead of using any general ϕ̂ to code xn, we can use θ̂ without any additional optimizations, where log n bits describe k. The description costs of θ̂ are almost the same as those of θ′. The factor 2 reduction in the last region is because it is sufficient here to replace n2 by n in the denominators of (32). This is because for every occurring symbol θ̂′i ≥ 1/n and δi ≤ 1/n, thus the first term of step (a) in (39) holds with the new grid, and B2 in (36) reduces by a factor of 2. The quantization costs bounded in (28) and (39) are thus bounded similarly, where θ̂ replaces θ and θ̂ replaces θ′. This results in the bounds in (79) and concludes the proof of Theorem 7. � If one a-priori knows that xn is likely to have been generated by a monotonic distribution, the case considered in Theorem 7 is with high probability the typical one. However, a typical sequence can also be one for which θ̂ 6∈ M, where θ̂ mildly violates the monotonicity. In the pure individual sequence setting (where no underlying distribution is assumed but some monotonicity assumption is reasonable for the empirical distribution of xn), one can still observe sequences that have empirical distributions that are either monotonic or slightly diverge from monotonic. Coding for this more general case can apply the methods described in Section 5 to the individual sequence case. If the divergence from monotonicity is small, one may still achieve bounds of the same order of those presented in Theorem 7 with additional negligible cost of relaying which symbols are out of order. The next theorem, however, provides a general upper bound in the form of the bounds of Theorems 5 and 6 for the individual sequence redundancy w.r.t. the monotonic ML description length, as defined in (10). We begin, again, with some notation. Recall the definition of an effective alphabet size m = nρ (where ρ = (logm)/(log n).) Now, use this definition for a specific individual sequence xn. Let R̂n(m) log n , m ≤ n1/3, m log n , n1/3 < m = o ( minα<ρ ρ+1+α (ρ− α) (log n)2 nα + 3(log e)n1−α , otherwise. Theorem 8 Fix an arbitrarily small ε > 0, and let n→ ∞. Then, there exists a code with length function L∗(·), that achieves individual sequence redundancy w.r.t. the monotonic ML description length of xn (as defined in (10)) bounded by R̂n (L ∗, xn) ≤ 1 + ε R̂n (nρ) + i>nρ,i∈xn log i for every xn. Theorem 8 shows that if one can find a relatively small effective alphabet of the symbols that occur in xn, and the symbols outside this alphabet are small enough, xn can be described with diminishing per-symbol redundancy w.r.t. its monotonic ML description length. This implies that as long as the occurring symbols are not too large, there exist a universal code w.r.t. a monotonic ML distribution for any such sequence xn. This is unlike standard individual sequence compression w.r.t. the i.i.d. ML description length. Specifically, if the effective alphabet size is O(n), and only a small number of symbols which are only polynomial in n occur, the universality cost is n(log n)2) bits overall, which gives diminishing per-symbol redundancy of O((log n)2/ This redundancy is much better than what can be achieved in standard compression. The penalty, of course, is when the empirical distribution of an individual sequence diverges significantly away from a monotonic one. While the monotonic redundancy can be made diminishing under mild conditions, there is a non-diminishing divergence cost by using the monotonic ML description length instead of the ML description length in that case. This implies that one should compress a sequence as generated by a monotonic distribution only if the total description length required to code xn as such is shorter than the total description length required to code xn with standard methods. As shown in the proof of Theorem 8, one prefix bit can inform the decoder which type of description is used. Theorem 8 shows that as long as the effective alphabet size is polynomial in n, α = 0.5 optimizes the third region of the upper bound, thus yielding the rate shown above, unless very large symbols occur in xn. For small effective alphabets (the first region), there is no redundancy gain in using the monotonic ML description length over the ML description length. The reason, again, is that the bound is obtained for cases where the actual empirical distribution of a sequence may not be monotonic. One can still use an i.i.d. ML estimate w.r.t. only the effective alphabet, if the additional cost of symbols outside this alphabet is negligible, to better code such sequences. Theorem 8 also shows that if a very large symbol, such as i = an; a > 1, occurs in xn, xn cannot be universally compressed even w.r.t. its monotonic ML description length. This is because it is impossible to avoid the cost of (1+ε) log i = (1+ε)n log a bits to describe this symbol to the decoder. The bound above and its proof below give a very powerful method to individually compress sequences that have an almost monotonic empirical distribution but may have some limited disorder, for which the monotonic ML description length diverges only negligibly from the ML description length. Proof of Theorem 8: The proof follows the same steps as the proof of Theorems 5 and 6. Each value of m is tested and the best one is chosen, where the same coding costs described in the mentioned proof are computed for each m. In addition, one can test the cost of coding xn using the description lengths for both θ̂ and θ̂M. Then, one bit can be used to relay which ML estimator is used. If θ̂ is used, the codes for coding individual sequences over large alphabets in either [21] or [25] can be used. In the first region in (81), the bound in [25] is obtained since log P (xn) ≥ log P for every xn. This bound yields smaller redundancy for this region than that obtained using θ̂M if θ̂M 6= θ̂. It implies that for small alphabets, if xn does not have an empirical monotonic distribution, it is better coded, even in terms of universal coding redundancy, using standard universal compression methods without taking advantage of a monotonicity assumption. For the other two regions, we start with a lemma. Lemma 6.1 Let θ̂M = θ̂1,M, θ̂2,M, . . . , θ̂k,M be the monotonic ML estimator of θ from xn, i.e., θ̂1,M ≥ θ̂2,M ≥ · · · ≥ θ̂k,M, where k = max {x1, x2, . . . , xn}. Then, θ̂k,M ≥ . (82) Lemma 6.1 provides a lower bound on the minimal nonzero probability component of the monotonic ML estimator. This bound helps in designing the grid of points used to quantize the monotonic ML distribution of xn, while maintaining bounded quantization costs. The proof of Lemma 6.1 is in Appendix C. For m in the second region, we cannot use the grid in (18). The reason is that, here, the quantization cost is affected by both θ̂ and θ̂M. This is unlike the average case, where the av- erage respective vectors merge. To limit the quantization cost for very small probabilities, using Lemma 6.1, the minimal grid point must be 1/n2 or smaller. To make the quantization cost neg- ligible w.r.t. the cost of describing the quantized ML, the ratio ∆j/ϕi,M between the spacing in interval j, and a quantized version ϕi,M of θ̂i,M in the jth interval, must be O(m/n). Hence, using the same methodology of the proof of Theorems 5 and 6, we define the jth interval for an effective alphabet m = nρ = o ( n) as Îj = n(j−1)β , 1 ≤ j ≤ Ĵρ. (83) The spacing in the jth interval is . (84) This gives a total of B̂ρ ≤ log n (85) quantization points. Using the same methodology as in (21), this yields a representation cost of LR (ϕm) ≤ (1 + ε)m log where ϕm is the quantized version of θ̂M in which only the firstm components of θ̂M are considered. Using the quantization with the grid defined in (83)-(86) in a code similar to the one used in the proof of Theorems 5 and 6, the individual quantization cost is given by P (xn|m,S′m,ϕm) θ̂i log θ̂i,M + log e ≤ n(log e) + log e ≤ (log e) · n ·mn+ (log e) · n · mn + log e = 3m(log e) + log e. (87) where (a) follows the same steps as in (60), (b) follows from ln(1+x) ≤ x, and then x ≤ |x|, where = θ̂i,M − ϕi,m, and (c) follows from Lemma 6.1 and the definition of Îj in (83) (for the worst case first term, |δi| ≤ 1/n2 and ϕi,m ≥ 1/(mn)), from (84) and (83) (the second term), and since θ̂i = 1. The only additional non-negligible cost of coding sequences using a code as defined in the proof of Theorems 5 and 6 for a given m is the cost of coding all symbols i > m that occur in xn. Using a similar derivation to (54), with Elias’ asymptotic code for the integers, this yields an additional cost of (1 + ε) (1 + 1/ρ) i>nρ,i∈xn log i code bits. Combining all costs, absorbing low order terms in ε, and normalizing by n, yields the second region of the bound in (81). Note that this bound also applies to the first region, but in that region, a tighter bound is obtained by using a code that uses the standard i.i.d. ML estimator θ̂. This is because very fine quantization is needed to offset the cost of mismatch between θ̂ and θ̂M. This quantization requires higher description costs than the description of a quantized type of a sequence when using standard compression. (This is not the case when θ̂ obeys the monotonicity, as in Theorem 7. Even if θ̂ does not obey monotonicity in the upper regions of the bound, this is not the case.) For the last region of the bound, we follow the same steps above as was done for the upper region of the bound in Theorem 5 with a parameter α. The intervals are chosen, again, to guarantee bounded quantization costs. Hence, Îj = n(j−1)β nρ+1+α nρ+1+α , 1 ≤ j ≤ Ĵρ. (88) The spacing in the jth interval is nρ+1+2α . (89) This gives a total of B̂ρ ≤ 0.5nα ⌈(ρ+ 1 + α) log n⌉ (90) quantization points. Using the same methodology as in (56), this yields a representation cost of LR (ϕm) ≤ (1 + ε) ρ+ 1 + α (ρ+ ε− α) (log n)2nα. (91) Similarly to (87), P (xn|m,S′m,ϕm) ≤ (log e) nρ+1+α + (log e)2n1−α + log e = 3(log e)n1−α + log e (92) where (a) follows from similar steps to (a)-(c) of (87). Using Lemma 6.1, ϕi,m ≥ 1/nρ+1 and |δi| ≤ 1/nρ+1+α, leading to the first term. Bounding |δi| ≤ njβ/nρ+1+2α and ϕi,m ≥ n(j−1)β/nρ+1+α leads to the second term. Note that as before, m is used here in place of k, because using an ef- fective alphabet m, all greater symbols are packed together as one symbol, and the additional cost to describe them is reflected in an additional term. Adding this additional term with an identical expression to that in the lower regions, absorbing low order terms in ε, and normalizing by n, yields the third region of the bound in (81). Since the bound holds for every α and every ρ > α, it can be optimized to give the values that attain the minimum, concluding the proof of Theorem 8. � 7 Summary and Conclusions Universal compression of sequences generated by monotonic distributions was studied. We showed that for finite alphabets, if one has the prior knowledge of the monotonicity of a distribution, one can reduce the cost of universality. For alphabets of o(n1/3) letters, this cost reduces from 0.5 log(n/k) bits per each unknown probability parameter to 0.5 log(n/k3) bits per each unknown probability parameter. Otherwise, for alphabets of O(n) letters, one can compress such sources with overall redundancy of O(n1/3+ε) bits. This is a significant decrease in redundancy from O(k log n) or O(n) bits overall that can be achieved if no side information is available about the source distribution. Redundancy of O(n1/3+ε) bits overall can also be achieved for much larger alphabets including infinite alphabets for fast decaying monotonic distributions. Sequences generated by slower decaying distributions can also be compressed with diminishing per-symbol redundancy costs under some mild conditions and specifically if they have finite entropy rates. Examples for well-known monotonic distributions demonstrated how the diminishing redundancy decay rates can be computed by applying the bounds that were derived. Finally, the average case results were extended to individual sequences. Similar convergence rates were shown for sequences that have empirical monotonic distributions. Furthermore, universal redundancy bounds w.r.t. the monotonic ML description length of a sequence were also derived for the more general case. Under some mild conditions, these bounds still exhibit diminishing per-symbol redundancies. Appendix A – Proof of Theorem 1 The proof follows the same steps used in [25] and [26] to lower bound the maximin redundancies for large alphabets and patterns, respectively, using the weak version of the redundancy-capacity theorem [5]. This version ties between the maximin universal coding redundancy and the capacity of a channel defined by the conditional probability Pθ (x n). We define a set ΩMk of points θ ∈ Mk. Then, show that these points are distinguishable by observing Xn, i.e., the probability that Xn generated by θ ∈ ΩMk appears to have been generated by another point θ ′ ∈ ΩMk diminishes with n. Then, using Fano’s inequality [3], the number of such distinguishable points is a lower bound on R−n (Mk). Since R+n (Mk) ≥ R−n (Mk), it is also a lower bound on the average minimax redundancy. The two regions in (6) result from a threshold phenomenon, where there exists a value km of k that maximizes the lower bound, and can be applied to all Mk for k ≥ km. We begin with defining ΩMk . Let ω be a vector of grid components, such that the last k − 1 components θi, i = 2, . . . , k, of θ ∈ ΩMk must satisfy θi ∈ ω. Let ωb be the bth point in ω, and define ω0 = 0 and 2(j − 1 , b = 1, 2, . . . . (A.1) Then, for the bth point in ω, . (A.2) To count the number of points in ΩMk , let us first consider the standard i.i.d. case, where there is no monotonicity requirement, and count the number of points in Ω, which is defined similarly, but without the monotonicity requirement (i.e., ΩMk ⊆ Ω). Let bi be the index of θi in ω, i.e., θi = ωbi . Then, from (A.1)-(A.2) and since the components of θ are probabilities, ωbi = θi ≤ 1. (A.3) It follows that for θ ∈ Ω, b2i ≤ n1−ε. (A.4) Hence, since the components bi are nonnegative integers, = |Ω| ≥ n1−ε⌋ n1−ε−b22 · · · n1−ε− i=2 b n1−ε−x22 · · · n1−ε− i=2 x dxk · · · dx3dx2 (A.5) where Vk−1 is the volume of a k − 1 dimensional sphere with radius , (a) follows from monotonic decrease of the function in the integrand for all integration arguments, and (b) follows since its left hand side computes the volume of the positive quadrant of this sphere. Note that this is a different proof from that used in [25]-[26] for this step. Applying the monotonicity constraint, all permutations of θ that are not monotonic must be taken out of the grid. Hence, = |ΩMk | ≥ k! · 2k−1 , (A.6) where dividing by k! is a worst case assumption, yielding a lower bound and not an equality. This leads to a lower bound equal to that obtained for patterns in [26] on the number of points in ΩMk . Specifically, the bound achieves a maximal value for km = πn1−ε/2 and then decreases to eventually become smaller than 1. However, for k > km, one can consider a monotonic distribution for which all components θi; i > km, of θ are zero, and use the bound for km. Distinguishability of θ ∈ ΩMk is a direct result of distinguishability of θ ∈ Ω, which is shown in Lemma 3.1 in [25], i.e., there exits an estimator Θ̂g(X n) ∈ Ω for which the estimate θ̂g satisfies limn→∞ Pθ θ̂g 6= θ = 0 for all θ ∈ Ω. Since this is true for all points in Ω, it is also true for all points in ΩMk ⊆ Ω, where now, θ̂g ∈ ΩMk . Assuming all points in ΩMk are equally probable to generate Xn, we can define an average error probability Pe Θ̂g(X n) 6= Θ θ∈ΩMk θ̂g 6= θ /MMk . Using the redundancy-capacity theorem, nR−n [Mk] ≥ C [Mk → Xn] ≥ I[Θ;Xn] = H [Θ]−H [Θ|Xn] = logMMk −H [Θ|X ≥ (1− Pe) (logMMk)− 1 ≥ (1− o(1)) logMMk , (A.7) where C [Mk → Xn] denotes the capacity of the respective channel and I[Θ;Xn] is the mutual information induced by the joint distribution Pr (Θ = θ) · Pθ (Xn). Inequality (a) follows from the definition of capacity, equality (b) from the uniform distribution of Θ in ΩMk , inequality (c) from Fano’s inequality, and (d) follows since Pe → 0. Lower bounding the expression in (A.6) for the two regions (obtaining the same bounds as in [26]), then using (A.7), normalizing by n, and ab- sorbing low order terms in ε, yields the two regions of the bound in (6). The proof of Theorem 1 is concluded. � Appendix B – Proof of Theorem 2 To prove Theorem 2, we use the random-coding strong version of the redundancy-capacity theorem [17]. The idea is similar to the weak version used in Appendix A. We assume that grids ΩMk of points are uniformly distributed over Mk, and one grid is selected randomly. Then, a point in the selected grid is randomly selected under a uniform prior to generate Xn. Showing distinguishability within a selected grid, for every possible random choice of ΩMk , implies that a lower bound on the cardinality of ΩMk for every possible choice is essentially a lower bound on the overall sequence redundancy for most sources in Mk. The construction of ΩMk is identical to that used in [26] to construct a grid of sources that generate patterns. We pack spheres of radius n−0.5(1−ε) in the parameter space defining Mk. The set ΩMk consists of the center points of the spheres. To cover the space Mk, we randomly select a random shift of the whole lattice under a uniform distribution. The cardinality of ΩMk is lower bounded by the relation between the volume of Mk, which equals (as shown in [26]) 1/[(k− 1)!k!], and the volume of a single sphere, with factoring also of a packing density (see, e.g., [2]). This yields eq. (55) in [26], MMk ≥ (k − 1)! · k! · Vk−1 n−0.5(1−ε) · 2k−1 , (B.1) where Vk−1 n−0.5(1−ε) is the volume of a k−1 dimensional sphere with radius n−0.5(1−ε) (see, e.g., [2] for computation of this volume). For distinguishability, it is sufficient to show that there exists an estimator Θ̂g(X n) ∈ ΩMk such that limn→∞ PΘ Θ̂g(X n) 6= Θ = 0 for every choice of ΩMk and for every choice of Θ ∈ ΩMk . This is already shown in Lemma 4.1 in [25] for a larger grid Ω of i.i.d. sources, which is constructed identically to ΩMk over the complete k−1 dimensional probability simplex. Therefore, by the monotonicity requirement, for every ΩMk , there exists such Ω, such that ΩMk ⊆ Ω. Since Lemma 4.1 in [25] holds for Ω, it then must also hold for the smaller grid ΩMk . Note that distinguishability is easier to prove here than for patterns because Θ̂g(X n) is obtained directly form Xn and not from its pattern as in [26]. Now, since all the conditions of the strong random- coding version of the redundancy-capacity theorem hold, taking the logarithm of bound in (B.1), absorbing low order terms in ε, and normalizing by n, leads to the first region of the bound in (7). More detailed steps follow those found in [26]. The second region of the bound is handled in a manner related to the second region of the bound of Theorem 1. However, here, we cannot simply set the probability of all symbols i > km to zero, because all possible valid sources must be included in one of the grids ΩMk to generate a complete covering of Mk. As was done in [26], we include sources with θi > 0 for i > km in the grids ΩMk , but do not include them in the lower bound on the number of grid points. In- stead, for k > km, we bound the number of points in a km-dimensional cut of Mk for which the remaining k− km components of θ are very small (and insignificant). This analysis is valid also for k > n. Distinguishability for k > km is shown for i.i.d. non-monotonically restricted distributions in the proof of Lemma 6.1 in [26]. As before, it carries over to monotonic distributions, since as before, for each ΩMk , there exists an unrestricted corresponding Ω, such that ΩMk ⊆ Ω. The choice of km = 0.5(n 1−ε/π)1/3 gives the maximal bound w.r.t. k. Since, again, all conditions of the strong version of the redundancy-capacity theorem are satisfied, the second region of the bound is obtained. Again, more detailed steps can be found in [26]. This concludes the proof of Theorem 2. � Appendix C – Proof of Lemma 6.1 For cardinality k, we consider the largest component of θ̂M; θ̂1,M, as the constraint component, i.e., θ̂1,M = 1− i=2 θ̂i,M. For any given probability parameter ϕ of cardinality k with ϕ1 > 0, we Pϕ (x n) = ϕ nx(1) 1 (1− ϕ1) n−nx(1) · 1− ϕ1 )nx(i) △ nx(1) 1 (1− ϕ1) n−nx(1) nx(i) i (C.1) where we recall that nx(i) is the occurrence count of i in x n. Therefore, maximization of Pϕ (x w.r.t. ϕ1 is independent of the maximization over ϑi; i > 1, and is obtained for ϕ1 = θ̂1 = nx(1)/n. Since for all i > 1, θ̂1,M ≥ θ̂i,M, θ̂1,M can thus only increase from θ̂1 by the monotonicity constraint. (Note that the monotonicity constraint implies a water filling [3] optimization to achieve θ̂M.) Hence, θ̂1,M ≥ nx(1)/n. Now, using the result above, we show that the derivative of lnPϕM (x n) w.r.t. ϕk,M is positive for ϕk,M < 1/(kn) and a monotonic ϕM. A component of a parameter vector ϕM, which is monotonic, can be expressed as ϕi,M = ϕ′ℓ, ϕ ℓ ≥ 0. (C.2) Hence, ∂ lnPϕM (x ∂ϕk,M ϕ1,M=θ̂1,M ∂ lnPϕM (x ϕ1,M=θ̂1,M nx(i) − (k − 1)nx(1) θ̂1,M knx(k) − knx(1) = 0 (C.3) where (a) follows from ϕk,M being the smallest nonzero component of ϕM, (b) is since by (C.2), ϕ′k is included in all terms, and ϕ1,M = 1− ϕi,M = 1− (i− 1)ϕ′i − (k − 1)ϕk,M, (C.4) where the last equality follows from (C.2), (c) follows by omitting all terms of the sum except i = k, from the assumption that ϕk,M < 1/(nk) ≤ θ̂k/k, and since θ̂1,M ≥ nx(1)/n = θ̂1, and (d) follows since its left hand side is 0 for the (i.i.d.) ML parameter values. Hence, PϕM (x n) must increase, with ϕ1,M taking its optimal value, for all ϕM for which ϕk,M < 1/(nk), and the maximum is thus achieved for θ̂k,M ≥ 1/(nk). � References [1] J. Åberg, Y. M. Shtarkov, and B. J. M. Smeets, “Multialphabet coding with separate alphabet description,” in Proceedings of Compression and Complexity of Sequences, pp. 56-65, Jun. 1997. [2] J. H. Conway, N. J. A. 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Introduction Notation and Definitions Lower Bounds Upper Bounds for Small and Large Alphabets Upper Bounds for Fast Decaying Distributions Upper Bounds Examples Fast Decaying Distributions Over the Integers Geometric Distributions Slow Decaying Distributions Over the Integers Individual Sequences Summary and Conclusions – Proof of Theorem ?? – Proof of Theorem ?? – Proof of Lemma ?? ABSTRACT We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size $k$, each probability parameter costs essentially $0.5 \log (n/k^3)$ bits, where $n$ is the coded sequence length, as long as $k = o(n^{1/3})$. Otherwise, for $k = O(n)$, the total average sequence redundancy is $O(n^{1/3+\epsilon})$ bits overall. We then show that there exists a sub-class of monotonic distributions over infinite alphabets for which redundancy of $O(n^{1/3+\epsilon})$ bits overall is still achievable. This class contains fast decaying distributions, including many distributions over the integers and geometric distributions. For some slower decays, including other distributions over the integers, redundancy of $o(n)$ bits overall is achievable, where a method to compute specific redundancy rates for such distributions is derived. The results are specifically true for finite entropy monotonic distributions. Finally, we study individual sequence redundancy behavior assuming a sequence is governed by a monotonic distribution. We show that for sequences whose empirical distributions are monotonic, individual redundancy bounds similar to those in the average case can be obtained. However, even if the monotonicity in the empirical distribution is violated, diminishing per symbol individual sequence redundancies with respect to the monotonic maximum likelihood description length may still be achievable. <|endoftext|><|startoftext|> Introduction: smooth tropical varieties In this section we follow the definitions of [5] and [4]. The underlying algebra of tropical geometry is given by the semifield T = R∪{−∞} of tropical numbers. The tropical arithmetic operations are “a + b” = max{a, b} and “ab” = a + b. The quotation marks are used to distinguish between the tropical and classical operations. With respect to addition T is a commutative semigroup with zero “0T” = −∞. With respect to multiplication T× = T r {0T} ≈ R is an honest commutative group with the unit “1T” = 0. Furthermore, the addition and multiplication satisfy to the distribution law “a(b+c)” = “ab+ac”, a, b, c ∈ T. These operations may be viewed as a result of the so-called dequantization of the classical arithmetic operations that underlies the patchworking construction, see [3] and [8]. These tropical operations allow one to define tropical Laurent poly- nomials. Namely, a tropical Laurent polynomial is a function f : Rn → f(x) = “ j” = max (aj + jx), where jx denotes the scalar product, x ∈ (T×)n ≈ Rn, j ∈ Zn and only finitely many coefficients aj ∈ T are non-zero (i.e. not −∞). http://arxiv.org/abs/0704.0839v1 2 GRIGORY MIKHALKIN Affine-linear functions with integer slopes (for brevity we call them simply affine functions) form an important subcollection of all Laurent polynomials. Namely, these are such functions f : Rn → R that both f and “1T ” = −f are tropical Laurent polynomials. We equip Tn ≈ [−∞,∞)n with the Euclidean topology. Let U ⊂ Tn be an open set. Definition 1.1. A continuous function f : U → T is called regular if its restriction to U ∩ Rn coincides with a restriction of some tropical Laurent polynomial to U ∩ Rn. We denote the sheaf of regular functions on Tn with O (or sometimes OTn to avoid confusion). Any subset X ⊂ T n gets an induced regular sheaf OX by restriction. For our purposes we restrict our attention only to the case when X is a polyhedral complex, i.e. when X is the closure of a union of convex polyhedra (possibly unbounded) in Rn such that the intersection of any number of such polyhedra is their common face. We say that X is an k-dimensional polyhedral complex if it is obtained from a union of k-dimensional polyhedra. These polyhedra are called the facets of X . Let V ⊂ X be an open set and f ∈ OX(V ) be a regular function in V . A point x ∈ V is called a “zero point” of f if the restriction of ” = −f toW ⊂ V is not regular for any open neighborhoodW ∋ x. Note that it may happen that x is a “zero point” for φ : U → T, but not for φ|X∩U . It is easy to see that if X is a k-dimensional polyhe- dral complex then the “zero locus” Zf of f is a (k − 1)-dimensional polyhedral subcomplex. To each facet of Zf we may associate a natural number, called its weight (or degree). To do this we choose a “zero point” x inside such a facet. We say that x is a “simple zero” for f if for any local de- composition into a sum (i.e. the tropical product) of regular function f = “gh” = g+h on V near x we have either g or h affine (i.e. without a “zero”). We say that the weight is l if f can be locally decomposed into a tropical product of l functions with a simple zero at x. A regular function f allows us to make the following modification on its domain V ⊂ X ⊂ Tn. Consider the graph Γf ⊂ V × T ⊂ T It is easy to see that the “zero locus” Γ̄f ⊂ V × T of the (regular) function “y+ f(x)” (defined on V ×T), where x is the coordinate on V and y is the coordinate on T, coincides with the union MODULI SPACES OF RATIONAL TROPICAL CURVES 3 of Γf and the undergraph UΓf,Z = {(x, y) ∈ V × T | x ∈ Zf , y ≤ f(x)}. Furthermore, the weight of a facet F ⊂ Γ̄f is 1 if F ∈ Γf (recall that as V is an unweighted polyhedral complex all the weights of its facets are equal to one) and is the weight of the corresponding facet of Zf if F ∈ UΓf,Z . We view Γ̄f as a “tropical closure” of the set-theoretical graph Γf . Note that we have a map Γ̄f → V . We set Ṽ = Γ̄f to be the result of the tropical modification µf : Ṽ → V along the regular function f . The locus Zf is called the center of tropical modification. The weights of the facets of Ṽ supplies us with some inconvenience as they should be incorporated in the definition of the regular sheaf OṼ on Ṽ . Namely, the affine functions defined by OṼ on a facet of weight w should contain the group of functions that come as restrictions to this facet of the affine functions on Tn+1 as a subgroup of index w. Sometimes one can get rid of the weights of Ṽ by a reparameteriza- tion with the help of a map V̄ → Ṽ that is given by locally linear maps in the corresponding charts. Indeed, the restriction of µg : V̄ → Ṽ to a facet is locally given by a linear function between two k-dimensional affine-linear spaces defined over Z. If its determinant equals to w then the push-forward of OV̄ supplies an extension of OṼ required by the weights. Note however that if w > 1 then the converse map is not defined over Z and thus is not given by elements of OṼ . Tropical modifications give the basic equivalence relation in Tropical Geometry. It can be shown that if we start from Tk and do a number of tropical modifications on it then the result is a k-dimensional polyhe- dral complex Y ⊂ Tn that satisfies to the following balancing property (cf. Property 3.3 in [4] where balancing is restated in an equivalent way). Property 1.2. Let E ⊂ Y ∩ RN be a (k − 1)-dimensional face and F1, . . . , Fl be the facets of Y adjacent to F whose weights are w1, . . . , wl. Let L ⊂ RN be a (N−k)-dimensional affine-linear space with an integer slope and such that it intersects E. For a generic (real) vector v ∈ RN the intersection Fj ∩ (L + v) is either empty or a single point. Let ΛFj ⊂ Z N be the integer vectors parallel to Fj and ΛL ⊂ Z N be the integer vectors parallel to L. Set λj to be the product of wj and the index of the subgroup ΛFj +ΛL ⊂ Z N . We say that Y ⊂ Tn is balanced if for any choice of E, L and a small generic v the sum j | Fj∩(L+v)6=∅ 4 GRIGORY MIKHALKIN is independent of v. We say that Y is simply balanced if in addition for every j we can find L and v so that Fj ∩ (L + v) 6= ∅, ιL = 1 and for every small v there exists an affine hyperplane Hv ⊂ L such that the intersection Y ∩ (L + v) sits entirely on one side of Hv + v in L + v while the intersection Y ∩ (Hv + v) is a point. Definition 1.3 (cf. [5],[4]). A topological space X enhanced with a sheaf of tropical functions OX is called a (smooth) tropical variety of dimension k if for every x ∈ X there exist an open set U ∋ x and an open set V in a simply balanced polyhedral complex Y ⊂ TN such that the restrictions OX |U and OY |V are isomorphic. Tropical varieties are considered up to the equivalence generated by tropical modifications. It can be shown that a smooth tropical variety of dimension k can be locally obtained from Tk by a sequence of tropical modifications centered at smooth tropical varieties of dimension (k−1). This follows from the following proposition. Proposition 1.4. Any k-dimensional simply balanced polyhedral com- plex X ⊂ Rn can be obtained from Tk by a sequence of consecutive trop- ical modifications whose centers are simply balanced (k−1)-dimensional polyhedral complexes. Proof. We prove this proposition inductively by n. Without the loss of genericity we may assume that X is a fan, i.e. each convex polyhedron of X is a cone centered at the origin. The base of the induction, when n = k, is trivial. If n > k let us take a (n− k)-dimensional affine-linear subspace L ⊂ Rn given by Property 1.2. Choose a linear projection λ : Rn → Rn−1 defined over Z and such that ker(λ) is a line contained in L. The image λ(X) ⊂ Rn−1 is a k-dimensional polyhedral complex since L is transversal to some facets of X . We claim that λ|X : X → λ(X) is a tropical modification once we identify Rn and Rn−1×R. The center of this modification is the locus Zf = {x ∈ R n−1 | dim(λ−1(x) ∩X) > 0}. Here we use the dimension in the usual topological sense. Note that the (k − 1)-dimensional complex Zf ⊂ R n−1 is simply balanced, existence of the needed (n− k)-dimensional affine-linear spaces follows from the fact that X ⊂ Rn is simply balanced. MODULI SPACES OF RATIONAL TROPICAL CURVES 5 To justify our claim we note that near any point x ∈ Zf the sub- complex Y ⊂ X obtained as the (Euclidean topology) closure of X r λ−1(Zf) is a (set-theoretical) graph of a convex function. This, once again, follows from the fact that X ⊂ Rn is simply balanced, this time applied to the points in the facets on X r Y . Thus it gives a regular tropical function f and it remains only to show that the the weight of any facet of E ⊂ Zf is 1. But this follows, in turn, from the balancing condition at λ−1(E) ∩ Y . � 2. Tropical curves and their moduli spaces The definition of tropical variety is especially easy in dimension 1. Tropical modifications take a graph into a graph (with arbitrary va- lence of its vertices) and the tropical structure carried by the sheaf OX amounts to a complete metric on the complement of the set of 1-valent vertices of the graph X (cf. [5], [6], [1]). Thus, each 1-valent vertex of a tropical curve X is adjacent to an edge of infinite length. A tropical modification allows one to contract such an edge or to attach it at any point of X other than a 1-valent vertex. If we have a finite collection of marked points on X then by passing to an equivalent model if needed we may assume that the set of marked points coincides with the set of 1-valent vertices. (Of course, if X is a tree then we have to have at least two marked points to make such assumption.) The genus of a tropical curve X is dimH1(X). Let Mg,n be the set of all tropical curves X of genus g with n distinct marked points. Fixing a combinatorial type of a graph Γ with n marked leaves defines a subset UΓ ⊂ Mg,n consisting of marked tropical curves with this combinatorics. A length of any non-leaf edge of Γ defines a real-valued function on UΓ. Such functions are called edge-length functions. To avoid difficulties caused by self-automorphisms of X from now on we restrict our attention to the case g = 0. Definition 2.1. The combinatorial type of a tropical curve X is its equivalence class up to homeomorphisms respecting the markings. Combinatorial types partite the set M0,n into disjoint subsets. The edge-length functions define the structure of the polyhedral cone RM≥0 in each of those subsets (as the lengths have to be positive). The number M here is the number of the bounded (non-leaf) edges in X . By the Euler characteristic reasoning it is equal to n − 3 if X is (1- and) 3-valent, it is smaller if X has vertices of higher valence. Furthermore, any face of the polyhedral cone RM≥0 coincides with the cone corresponding to another combinatorial type, the one where we 6 GRIGORY MIKHALKIN contract some of the edges of X to points. This gives the adjacency (fan-like) structure on M0,n, so M0,n is a (non-compact) polyhedral complex. In particular, it is a topological space. Theorem 1. The set M0,n for n ≥ 3 admits the structure of an (n−3)- dimensional tropical variety such that the edge-length functions are reg- ular within each combinatorial type. Furthermore, the space M0,n can be tropically embedded in RN for some N (i.e. M0,n can be presented as a simply balanced complex). Proof. This theorem is trivial for n = 3 as M0,3 is a point. Otherwise, any two disjoint ordered pairs of marked points can be used to define a global regular function on M0,n with values in R = T ×. Namely, each such ordered pair defines the oriented path on the tropical curve X connecting the corresponding marked points. These paths can be embedded. Since the two pairs of marked points are disjoint the intersection of the two corresponding paths has to have finite length. We take this length with the positive sign if the orientations agree and with the negative sign otherwise. This defines a function on M0,n. We call such functions the double ratio functions. Take all possible disjoint pairs of marked points and use them as coordinates for our embedding ι : M0,n → R where N is the number of all possible decompositions of n into two disjoint pairs. The theorem now follows from the following two lemmas. Note that, strictly speaking, each coordinate in RN depends not only on the choice of two disjoint pairs of marked points but also on the order of points in each pair. However, changing the order in one of the pairs only reverses the sign of the double ratio. Taking an extra coordinate for such a change of order would be redundant. Indeed, for any balanced complex Y ⊂ RN and any affine-linear function λ : RN → R with an integer slope the graph of λ is a balanced complex in RN+1 isomorphic to the initial complex Y . Lemma 2.2. The map ι is a topological embedding. Proof. First, let us prove that ι is an embedding. The combinatorial type of X is determined by the set of the coordinates that do not vanish on X . Indeed, any non-leaf edge E of the tree X separates the leaves (i.e. the set of markings) into two classes corresponding to the components of XrE. Let us take a coordinate in Rn that corresponds MODULI SPACES OF RATIONAL TROPICAL CURVES 7 to four marking points (union of the two disjoint pairs) such that two of these points belong to one class and two to the other class. We call such a coordinate an E-compatible coordinate. Note that an E-compatible coordinate vanishes on X if and only if the pairs of markings defined by the coordinate agree with the pairs defined by the classes. This observation suffices to reconstruct the combinatorial type of X . Furthermore, the length of E equals to the minimal non-zero abso- lute value of the E-compatible coordinates. This implies that ι is an embedding. � Lemma 2.3. The image ι(M0,n) is a simply balanced complex in R Proof. This is a condition on codimension 1 faces of M0,n. First we shall check it for the case n = 4. There are three ways to split the four marking points into two disjoint pairs. Accordingly, there are three combinatorial types of 3-valent trees with three marked leaves. Thus our space M0,4 is homeomorphic to the tripod, or the “interior” of the letter Y , see Figure 1. Each ray of this tripod correspond to a combinatorial type of a 3-valent tree with 4 leaves while the vertex correspond to the 4-valent tree. Figure 1. The tropical moduli space M0,4 and its points on the corresponding edges. 8 GRIGORY MIKHALKIN Up to the sign we have the total of three double ratios for n = 4. Let us e.g. take those defined by the following ordered pairs: {(12), (34)}, {(13), (24)} and {(14), (23)} Each is vanishing on the corresponding ray of the tripod. Let us parameterize each ray of the tripod by its only edge-length t ≥ 0 and compute the corresponding map to R3. We have the following embeddings on the three rays t 7→ (0, t, t), t 7→ (t, 0,−t), t 7→ (−t,−t, 0). The sum of the primitive integer vectors parallel to the resulting direc- tions is 0 and thus ι(M0,4) is balanced. In the case n > 4 the codimension 1 faces of M0,n correspond to the combinatorial types of X with a single 4-valent vertex. Near a point inside of such face F the space M0,n looks like the product of M0,4 and R n−4. The factor Rn−4 comes from the edge-lengths on F (its combinatorial type has n−4 bounded edges) while the factor M0,4 comes from perturbations of the 4-valent vertex (which result in a new bounded edge in one of the three possible combinatorial types of the result). We have a well-defined map from the union U of the F -adjacent facets to F by contracting the new edge to a point. Note that the edge-length functions exhibit F as the positive quadrant in Rn−4. Fur- thermore, in the combinatorial type of F we may choose 4 leaves such that contracting all other leaves will take place outside of the 4-valent vertex (see Figure 2). This contraction defines a map U → M0,4. Figure 2. One of the possible contractions of a tree with a 4-valent vertex to the tree corresponding to the origin O ∈ M0,4. The lemma now follows from the observation that the resulting de- composition into M0,4 × R n−4 agrees with the double ratio functions. Indeed, note that the complement of the 4-valent vertex for a curve in the combinatorial type F is composed of four components. If the double ratio is such that its four markings are in one-to-one correspon- dence with these components then at U it coincides with sum of the pull-back of the corresponding double ratio in F with the pull-back of the corresponding double ratio in M0,4. If one of the four components MODULI SPACES OF RATIONAL TROPICAL CURVES 9 is lacking a marking from the double ratio ρ then ρ|U coincides with the corresponding pull-back from F . � Remark 2.4. The functions Zxi,xj from [4] do not define regular func- tions on M0,n, contrary to what is written in [4]. These functions were a result of an erroneous simplification of the double ratio functions. But these functions cannot be regular as they are always positive and Proposition 5.12 of [4] is not correct. Even the projectivization of the embedding is not a balanced complex already for M0,5. One should use the (non-simplified) double ratios instead. Clearly, the space M0,n is non compact. However it is easy to com- pactify it by allowing the lengths of bounded edges to assume infinite values. Let M0,n be the space of connected trees with n (marked) leaves such that each edge of this tree is assigned a length 0 < l ≤ +∞ so that each leaf has length necessarily equal to +∞. Corollary 2.5. The space M0,n is a smooth compact tropical variety. To verify that M0,n is smooth near a point x at the boundary ∂M0,n = M0,n rM0,n we need to examine those double ratios that are equal to ±∞ at x. There we use only those signs that result in −∞ do that the map takes values in TN . Remark 2.6. Note that the compactification M0,n ⊃ M0,n corresponds to the Deligne-Mumford compactification in the complex case as under the 1-parametric family collapse of a Riemannian surface to a tropical curve the tropical length of an edge corresponds to the rate of growth of the complex modulus of the holomorphic annulus collapsing to that edge. Furthermore, similarly to the complex story the infinite edges de- compose a tropical curve into components (where the non-leaf edges are finite). Any tropical map from an infinite edge which is bounded would have to be constant and thus the image would have to split as a union of several tropical curves in the target. Such decompositions were used by Gathmann and Markwig in their deduction of the tropical WDVV equation in R2, see [1]. 3. Tropical ψ-classes Note that we do have the forgetting maps ftj : M0,n+1 → M0,n 10 GRIGORY MIKHALKIN for j = 1, . . . , n + 1 by contracting the leaf with the j-marking. This map is sometimes called the universal curve. Each marking k 6= j defines a section σk of ftj. The conormal bundle to σk defines the ψk- class in complex geometry (to avoid ambiguity we take j = n+1). This notion can be adapted to our tropical setup. Recall that so far our choice of tropical models in their equivalence class was such that the leaves of the tropical curves were in 1-1 cor- respondence with the markings. For this choice we have the images σk(M0,n) contained in the boundary part of M0,n+1. This presenta- tion is compatible with the point of view when we think about line bundles in tropical geometry to be given by H1(X,O×). Here X is the base of the bundle and O× is the sheaf of “non-vanishing” tropi- cal regular functions. Such functions are given in the charts to RN by affine-linear functions with integer slopes, see [6]. (Recall that T× = R is an honest group with respect to tropical multiplication, i.e. the classical addition.) However, the following alternative construction allows one to obtain the ψ-classes more geometrically (as we’ll illustrate in an example in the next section). This approach is based on contracting the leaves marked by number k. The canonical class of a tropical curve is supported at its vertices, namely we take each vertex with the multiplicity equal to its valence minus 2, cf. [6]. Furthermore, the cotangent bundle near a 3-valent vertex point can be viewed as a neighborhood of the origin for the line given by the tropical polynomial “x + y + 1T” in R 2, so the +1 self- intersection of the line gives the required multiplicity for the canonical class at any 3-valent vertex. Thus we can use the intersections with the corresponding codimension 1 faces inM0,n to define the ψ-classes there. In other words, tropical ψ-classes will be supported on the (n − 4)- dimensional faces in M0,n. Namely, for a ψk-class we have to collect those codimension 1 faces in M0,n whose only 4-valent vertex is adjacent to the leaf marked by k. After a contraction of this leaf we get a 3-valent vertex, thus the multiplicity of every face in a ψ-divisor is 1. We arrive to the following definition. Definition 3.1. The tropical ψk-divisor Ψk ⊂ M0,n is the union of those (n−4)-dimensional faces that correspond to tropical curves with a 4-valent vertex adjacent to the leaf marked by k, k = 1, . . . , n. Each such face is taken with the multiplicity 1. Proposition 3.2. The subcomplex Ψk is a divisor, i.e. satisfies the balancing condition. MODULI SPACES OF RATIONAL TROPICAL CURVES 11 Proof. Recall that the balancing condition is a condition at (n − 5)- dimensional faces. In M0,n there are two types of such faces, one corresponding to tropical curves with two 4-valent vertices and one corresponding to a tropical curve with a 5-valent vertex. Near the faces of the first type the moduli space M0,n is locally a product of two copies of M0,4 and R n−5. The Ψ-divisor is a product of Rn−5, one copy of M0,4 and the central (3-valent) point in the other copy of M0,4 (this is the point corresponding to the 4-valent vertex adjacent to the leaf marked by k). Thus the balancing condition holds trivially in this case. Near the faces of the second type the moduli space M0,n is locally a product of M0,5 and R n−5. As in the proof of Theorem 1 each double ratio decomposes to the sum of the corresponding double ration inM0,5 (perhaps trivial if two of the markings for the double ratio correspond to the same edge adjacent to the 5-valent vertex) and an affine-linear function in Rn−5. Thus it suffices to check only the balancing condition for the Ψ-divisors in M0,5. This example is considered in details in the next section. The balancing condition there follows from Proposition 4.1. � Conjecturally, the tropical Ψ-divisors are limits of some natural rep- resentatives of the divisors for the complex ψ-classes under the collapse of the complex moduli space onto the corresponding tropical moduli space M0,n. Note that our choice for the tropical Ψ-divisor is not con- tained in the boundary ∂M0,n ⊂ M0,n (cf. the calculus of the complex boundary classes in [2]), but comes as a closure of a divisor in M0,n. 4. The space M0,5 We have already described the moduli space M0,4 as the tripod of Figure 1. It has only one 0-dimensional face O ∈ M0,4. This point (considered as a divisor) coincides with the divisors Ψ1 = Ψ2 = Ψ3 = Ψ4. The description of M0,5 is somewhat more interesting. There are 15 combinatorial types of 3-valent trees with 5 marked leaves. If we forget about the markings there is only one homeomor- phism class for such a curve (see Figure 3). To get the number of non-isomorphic markings we take the number all possible reordering of vertices (equal to 5! = 120) and divide by 23 = 8 as there is an 8-fold symmetry of reordering. Indeed there is one symmetry interchanging the left two leaves, one interchanging the right two leaves and the cen- tral symmetry around the central leave of the 3-valent tree on top of Figure 3. 12 GRIGORY MIKHALKIN 1 15 5 Figure 3. Adjunction of combinatorial types corre- sponding to the quadrant connecting the rays (45) and (12). (25) (13) (15) (23) Figure 4. The link of the origin in M0,5. MODULI SPACES OF RATIONAL TROPICAL CURVES 13 Thus the space M0,5 is a union of 15 quadrants R ≥0. These quad- rants are attached along the rays which correspond to the combinato- rial types of curves with one 4-valent vertex. Such curves also have one 3-valent vertex which is adjacent to two leaves and the only bounded edge of the curve, see the bottom of Figure 3. Such combinatorial types are determined by the markings of the two leaves emanating from the 3-valent vertex. Thus we have a total of = 10 of such rays. The two boundary edges of the quadrant correspond to contractions of the bounded edges of the combinatorial type as shown on Figure 3. The global picture of adjacency of quadrants and rays is shown on Figure 4 where the reader may recognize the well-known Petersen graph, cf. the related tropical Grassmannian picture in [7]. Vertices of this graph correspond to the rays of M0,5 while the edges correspond to the quadrants. Thus the whole picture may be interpreted as the link of the only vertex O ∈ M0,5 (the point O corresponds to the tree with a 5-valent vertex adjacent to all the leaves). To locate the Ψk-divisor we recall that the kth leaf has to be adjacent to a 4-valent vertex if it appears in Ψk. This means that Ψk consists of 6 rays that are marked by pairs not containing k. Proposition 4.1. The subcomplex Ψk ⊂ M0,5 is a divisor. Proof. Since the whole M0,5 is S5-symmetric it suffices to check the balancing condition only for Ψ1. The embedding M0,5 ⊂ R N is given by the double ratios, so it suffices to check that for each double ratio function the sum of its gradients on the six rays of Ψ1 vanishes. If the double ratio is determined by two pairs disjoint from the mark- ing 1, e.g. by {(23), (45)} then its restriction onto the six rays of Ψ1 is the same as its restriction to the three rays M0,4 taken twice and thus balanced. Namely its gradient is 1 on the rays (24) and (35); −1 on the rays (25) and (34); and 0 on the rays (23) and (45). If the four markings of the double ratio contain the marking 1 then thanks to the symmetry we may assume that the double ratio is given by {(12), (34)}. It vanishes on the rays (34), (35), (45) and (25); it has gradient +1 on the ray (24) and the gradient −1 on the ray (23). Once again, the balancing condition holds. � As our final example of the paper we would like to describe explicitly the universal curve ft5 : M0,5 → M0,4. This is presented on Figure 5. Once again, we interpret the Peterson graph as the link L of the vertex O ∈ M0,5. Similarly, the link of the 14 GRIGORY MIKHALKIN Figure 5. The three fibers and four sections of the universal curve ft5 : M0,5 → M0,4. origin in M0,4 consists of three points. Thus L is the union of the fibers of ft5 (away from a neighborhood of infinity) over these three points and four copies of a neighborhood of the origin in M0,4 corresponding to the four sections σ1, σ2, σ3 and σ4 of the universal curve. Figure 5 depicts the fibers in L with solid lines and the sections with dashed lines. Acknowledgements. I am thankful to Valery Alexeev and Kristin Shaw for discussions related to geometry of tropical moduli spaces. My research is supported in part by NSERC. References [1] Gathmann, A., Markwig, H., Kontsevich’s formula and the WDVV equations in tropical geometry, http://arxiv.org/abs/math.AG/0509628. [2] Keel, S., Intersection theory of moduli space of stable N -pointed curves of genus zero, Transactions of the AMS 330 (1992), 545–574. [3] Litvinov, G. L., The Maslov dequantization, idempotent and tropical mathe- matics: a very brief introduction. In Idempotent mathematics and mathemat- ical physics, Contemp. Math., 377, Amer. Math. Soc., Providence, RI, 2005, 1–17. [4] Mikhalkin, G., Tropical Geometry and its application, to appear in the Pro- ceedings on the ICM-2006, Madrid; http://arxiv.org/abs/math/0601041. [5] Mikhalkin, G., Tropical Geometry, book in preparation. http://arxiv.org/abs/math.AG/0509628 http://arxiv.org/abs/math/0601041 MODULI SPACES OF RATIONAL TROPICAL CURVES 15 [6] Mikhalkin, G., Zharkov, I., Tropical curves, their Jacobians and Theta func- tions, http://arxiv.org/abs/math/0612267. [7] Speyer, D., Sturmfels, B., The tropical Grassmannian. Adv. Geom. 4 (2004), no. 3, 389–411. [8] Viro, O. Ya., Dequantization of real algebraic geometry on logarithmic paper. In European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, 135–146. Department of Mathematics, University of Toronto, 40 St George St, Toronto ON M5S 2E4 Canada http://arxiv.org/abs/math/0612267 1. Introduction: smooth tropical varieties 2. Tropical curves and their moduli spaces 3. Tropical -classes 4. The space M0,5 References ABSTRACT This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical $\psi$-class divisors. <|endoftext|><|startoftext|> Difermion condensates in vacuum in 2-4D four-fermion interaction models Bang-Rong Zhou† College of Physical Sciences, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China In any four fermion (denoted by q) interaction models, the couplings of (qq)2-form can always coexist with the ones of (q̄q)2-form via the Fierz transformations. Hence, even in vacuum, there could be interplay between the condensates 〈q̄q〉 and 〈qq〉. Theoretical anal- ysis of this problem is generally made by relativistic effective potentials in the mean field approximation in 2D, 3D and 4D models with two flavor and Nc color massless fermions. It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio GS/HS for 2D and 4D case or GS/HP for 3D case, where GS , HS and HP are respectively the coupling constants in a scalar (q̄q), a scalar (qq) and a pseudoscalar (qq) channel. In ground states of all the models, only pure 〈q̄q〉 condensates could exist if GS/HS or GS/HP is bigger than the critical value 2/Nc, the ratio of the color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉. Below it, differences of the models will manifest themselves. In the 4D Nambu-Jona-Lasinio (NJL) model, as GS/HS decreases to the region below 2/Nc, one will first have a coexistence phase of the two condensates then a pure 〈qq〉 con- densate phase. Similar results come from a renormalized effective potential in the 2D Gross- Neveu model, except that the pure 〈qq〉 condensates could exist only if GS/HS = 0. In a 3D Gross-Neveu model, when GS/HP < 2/Nc, the phase transition similar to the 4D case can arise only if Nc > 4, and for smaller Nc, only a pure 〈qq〉 condensate phase exists but no coexistence phase of the two condensates happens. The GS −HS (or GS − HP ) phase diagrams in these models are given. The results deepen our understanding of dynamical phase structure of four-fermion inter- action models in vacuum. In addition, in view of absence of difermion condensates in vacuum of QCD, they will also imply a real restriction to any given two-flavor QCD-analogous NJL model, i.e. in the model, the derived smallest ratio GS/HS via the Fierz transformations in the Hartree approximation must be bigger than 2/3. The project supported by the National Natural Science Foundation of China under Grant No.10475113. Electronic mailing address: zhoubr@163bj.com http://arxiv.org/abs/0704.0841v3 I. MAIN RESULTS We have researched interplay between the fermion(q)-antifermion (q̄) condensates 〈q̄q〉 and the difermion condensates 〈qq〉 in vacuum in 2D, 3D and 4D four-fermion interaction models with two flavor and Nc color massless fermions. It is found that the ground states of the systems could be in different phases shown in the following GS − HS and GS − HP phase diagrams [Fig.(a)–Fig.(d)], where GS — coupling constant of scalar (q̄q) 2 channel HS — coupling constant of scalar color Nc(Nc − 1) −plet (qq)2 channel (4D, 2D case) HP — coupling constant of pseudoscalar color Nc(Nc − 1) −plet (qq)2 channel (3D case) Λ — Euclidean Momentum cutoff of loop integrals (4D,3D case) (σ1, 0) — pure 〈q̄q〉 phase (0,∆1) — pure 〈qq〉 phase (σ2,∆2) — mixed phase with both 〈q̄q〉 and 〈qq〉 Fig.(a)-Fig.(d) (pages 3-6) 3 4D NJL Model y=GSΛ 2/π2 y =2x/ Nc (σ1, 0) (σ2,Δ2) 1/Nc (0 ,Δ1) R 0 1/2 x =HSΛ R: y=x/[1+(Nc-2)x] Fig. (a) 4 2D GN Model y=GS /π y=2x /Nc (σ1, 0) (σ2 ,Δ2) x=HS /π (0,Δ1) Fig. (b) 3D GN Model, Nc≤4 y=GSΛ/π y=2x/ Nc (σ1, 0) 1/4Nc (0,Δ1) 0 1/8 x=HPΛ/π Fig. (c) 3D GN Model, Nc≥5 y=GSΛ/π y =2x/ Nc (σ1, 0) 1/4Nc (σ2,Δ2) (0,Δ1) 0 1/8 x =HPΛ/π Fig. (d) Main Conclusions 1. In all the models, pure 〈q̄q〉 phase happens if ) > 2 (also GS must be large enough in 3D and 4D model). 2. The phases with condensates 〈qq〉, including pure 〈qq〉 phase and mixed phase with 〈q̄q〉 and 〈qq〉, arise only if ) < 2 3. In 3D Gross-Neveu model, no mixed phase with 〈q̄q〉 and 〈qq〉 exists for Nc ≤ 4. II. Motive and general approach • In any four-fermion interaction model [1, 2], the couplings of (qq)2-form and (q̄q)2-form can always coexist via the Fierz transformations, hence there must be interplay between the condensates 〈q̄q〉 and 〈qq〉 in ground state of the system. • In the vacuum, despite of absence of net fermions, based on a relativistic quantum field theory, it is possible that the condensates 〈qq〉 and 〈q̄q̄〉 are generated simultaneously. • The mean field approximation has been taken. In this case, we have used the Fierz transformed four-fermion couplings in the Hartree approximation to avoid double counting [3]. • In selecting the couplings of (qq)2-form, we always simulate SU(Nc) gauge interaction, where two fermions are attractive in the antisymmetric Nc(Nc − 1) -plet. • Euclidean momentum cutoffs in 3D and 4D models have been used so as to maintain Lorentz invariance of effective potentials in the vacuum. • In massless fermion limit, all the discussions can be made analytically. • The coupling constants GS and HS (or HP ) are viewed as independent parameters. III. 4D Nambu-Jona-Lasinio model With 2 flavors and Nc color massless fermions, the Lagrangian L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄iγ5~τq) 2] +HS (q̄iγ5τ2λAq C)(q̄C iγ5τ2λAq), (1) where the fermion fields q are in the doublet of SUf (2) and the Nc-plet of SUc(Nc), i.e. i = 1, · · · , Nc, (2) http://arxiv.org/abs/0704.0841v3 qC is the charge conjugate of q and ~τ = (τ1, τ2, τ3) are the Pauli matrices acting in two-flavor space. The matrices λA run over all the antisymmetric generators of SUc(Nc). Assume that the four-fermion interactions can lead to the scalar condensates 〈q̄q〉 = φ (3) with all the Nc color fermion entering them, and the scalar color Nc(Nc − 1) 2 -plet difermion and di-antifermion condensates (after a global SUc(Nc) transformation) 〈q̄Ciγ5τ2λ2q〉 = δ, 〈q̄iγ5τ2λ2q C〉 = δ∗, (4) with only two color fermions enter them. The corresponding symmetry breaking is that SUfL(2) ⊗ SUfR(2) → SUf (2), SUc(Nc) → SUc(2), and a ”rotated” electric charge UQ̃(1) and a ”rotated” quark number U ′q(1) leave unbroken. It should be indicated that in the case of vacuum, the Goldstone bosons induced by spontaneous breaking of SUc(Nc) could be some combinations of difermions and di-antifermions. Define that σ = −2GSφ, ∆ = −2HSδ, ∆ ∗ = −2HSδ ∗. (5) With standard technique and a 4D Euclidean momentum cutoff Λ [4], we obtain the relativistic effective potential V4(σ, |∆|) = 2 + 2|∆|2)Λ2 − (Nc − 2) −(σ2 + |∆|2)2 σ2 + |∆|2 . (6) The ground states of the system, i.e. the minimum points of V4(σ, |∆|), will be at (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) , 0 ≤ 1 + (Nc − 2) 1 + (Nc − 2) , (7) Eq.(7) gives the phase diagram Fig.(a) of the 4D NJL model. IV. 2D Gross-Neveu model The Lagrangian is expressed by L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄iγ5~τq) 2] +HS(q̄iγ5τSλAq C)(q̄Ciγ5τSλAq), (8) All the denotations are the same as ones in 4D NJL model, except that in 2D space-time , γ1 = = −C, γ5 = γ and τS = (τ0 ≡ 1, τ1, τ3) are flavor-triplet symmetric matrices. It is indicated that the product matrix Cγ5τSλA is antisymmetric. Assume that the four-fermion interactions could lead to the scalar quark-antiquark conden- sates 〈q̄q〉 = φ, (9) which will break the discrete symmetries χD : q(t, x) → γ5q(t, x), P1 : q(t, x) → γ1q(t,−x), and that the coupling with HS can lead to the scalar color Nc(Nc − 1) -plet difermion con- densates and the scalar color anti- Nc(Nc − 1) -plet di-antifermion condensates (after a global transformation in flavor and color space) 〈q̄C iγ51fλ2q〉 = δ, 〈q̄iγ51fλ2q C〉 = δ∗ (10) which will break discrete symmetries Zc (center of SUc(3)) and Z (center of SUf (2)), besides χD and P1. Noting that in a 2D model, no breaking of continuous symmetry needs to be considered on the basis of Mermin-Wagner-Coleman theorem [5]. The model is renormalizable. In the space-time dimension regularization approach, we can write down the renormalized L in D = 2− 2ε dimension space-time by the replacements GS → GSM 2−DZG, HS → HSM 2−DZH , with the scale parameter M , the renormalization constants ZG and ZH . In addition, the γ L will become 2D/2 × 2D/2 matrices. Define the order parameters σ = −2GSM 2−DZGφ, ∆ = −2HSM 2−DZHδ, (11) which will be finite if ZG and ZH are selected so as to cancel the UV divergences in φ and δ. In the minimal substraction scheme, ZG = 1− 2NcGS , ZH = 1− . (12) By similar derivation to the one made in Ref.[6], the corresponding renormalized effective po- tential in the mean field approximation up to one-loop order becomes V2(σ, |∆|) = σ2 + |∆|2 + (Nc − 2) ln σ2 + |∆|2 , M̄2 = 2πe−γM2, (13) where γ is the Euler constant. The ground states of the system i.e. the minimal points of V2(σ, |∆|) will be at (σ, |∆|) = (0, ∆1) (σ2, ∆2) (σ1, 0) GS/HS = 0 0 < GS/HS < 2/Nc GS/HS > 2/Nc Eq.(14) gives the phase diagram Fig.(b) of 2D GN model. In 2D case, the GS -HS phase structure has the following feature: 1. The pure 〈qq〉 phase (0,∆1) could appear only if GS/HS = 0; 2. Formations of the condensates do not call for that the coupling constant GS and HS have some lower bounds. V. 3D Gross-Neveu model The Lagrangian is expressed by L = q̄iγµ∂µq +GS [(q̄q) 2 + (q̄~τq)2] +HP (q̄τ2λAq C)(q̄Cτ2λAq), (15) where γµ(µ = 0, 1, 2) are taken to be 2× 2 matrices , γ1 = , γ2 = It is noted that the product matrix Cτ2λA is antisymmetric, and since without the ”γ5” matrix, the only possible color Nc(Nc − 1) -plet difermion interaction channel is pseudoscalar one. The condensates 〈q̄q〉 will break time reversal symmetry T : q(t, ~x) → γ2q(−t, ~x), special parity P1 : q(t, x 1, x2) → γ1q(t,−x1, x2), special parity P2 : q(t, x 1, x2) → γ2q(t, x1,−x2). The difermion condensates 〈q̄Cτ2λ2q〉 (after a global rotation in the color space) will break SUc(Nc) → SUc(2) and leave a ”rotated” electrical charge U (1) and a ”rotated” fermion number U ′q(1) unbroken. It also breaks parity P : q(t, ~x) → γ0q(t,−~x) and this shows pseudoscalar feature of the difermion condensates. Define the order parameters in the 3D GN model σ = −2GS〈q̄q〉, ∆ = −2HP 〈q̄ Cτ2λ2q〉, (16) on bases of the same method used in Ref.[7], we find out the effective potential in the mean field approximation V3(σ, |∆|) = 2 + 2|∆|2)Λ 6σ2|∆|+ 2|∆|3 + (Nc − 2)σ 3 + 2θ(σ − |∆|)(σ − |∆|)3 , (17) where Λ is a 3D Euclidean momentum cutoff. The ground states of the system correspond to the least value points of V3(σ, |∆|) which will respectively be at (σ, |∆|) = (0,∆1), (0,∆1), (σ2,∆2), (σ1, 0), , for Nc ≤ 4 for Nc > 4 , for all Nc Eq.(18) gives the GS −HP phase diagrams Fig.(c) and Fig.(d) of the 3D GN model. VI. Summary • Present research deepens our theoretical understanding of the four-fermion interaction models: 1. Even in vacuum, it is possible that the difermion condensates are generated as long as the coupling constants of the difermion channel are strong enough (bigger than zero or some finite values). 2. Interplay between the condensates 〈q̄q〉 and 〈qq〉 mainly depends on GS/HS (or GS/HP ), the ratio of the coupling constants of scalar fermion-antifermion channel and scalar (or pseudoscalar ) difermion channel. 3. In all the discussed 2-flavor models, if GS/HS (GS/HP ) > 2/Nc, the ratio of the color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉, (and also with sufficiently large GS in 4D and 3D model), then only pure 〈q̄q〉 condensates phase may exist. Below 2/Nc, (and also with sufficiently large HS or HP in 4D or 3D model), one will always first have a mixed phase with condensates 〈q̄q〉 and 〈qq〉, then a pure 〈qq〉 condensate phase, except that in the 3D GN model, no the mixed phase appears when Nc ≤ 4. • In view of absence of 〈qq〉 condensates in vacuum of QCD, the result here also implies a real restriction to any given two-flavor QCD-analogue NJL model: in such model, the derived smallest ratio GS/HS via the Fierz transformation in the Hartree approximation must be bigger than 2/3 [4]. [1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [2] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235. [3] M. Buballa, Phys. Rep. 407 (2005) 205. [4] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95. [5] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; S. Coleman, Commun. Math. Phys. 31 (1973) 259. [6] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 520. [7] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 695. Main results References ABSTRACT Theoretical analysis of interplay between the condensates $<\bar{q}q>$ and $$ in vacuum is generally made by relativistic effective potentials in the mean field approximation in 2D, 3D and 4D models with two flavor and $N_c$ color massless fermions. It is found that in ground states of these models, interplay between the two condensates mainly depend on the ratio $G_S/H_S$ for 2D and 4D case or $G_S/H_P$ for 3D case, where $G_S$, $H_S$ and $H_P$ are respectively the coupling constants in a scalar $(\bar{q}q)$, a scalar $(qq)$ and a pseudoscalar $(qq)$ channel. In ground states of all the models, only pure $<\bar{q}q>$ condensates could exist if $G_S/H_S$ or $G_S/H_P$ is bigger than the critical value $2/N_c$, the ratio of the color numbers of the fermions entering into the condensates $$ and $<\bar{q}q>$. As $G_S/H_S$ or $G_S/H_P$ decreases to the region below $2/N_c$, differences of the models will manifest themselves. Depending on different models, and also on $N_c$ in 3D model, one will have or have no the coexistence phase of the two condensates, besides the pure $$ condensate phase. The $G_S-H_S$ (or $G_S-H_P$) phase diagrams in these models are given. The results also implicate a real constraint on two-flavor QCD-analogous NJL model. <|endoftext|><|startoftext|> Oscillation bands of condensates on a ring: Beyond the mean field theory C. G. Bao Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collisions, Lanzhou 730000, P. R. China The State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan University, Guangzhou, 510275, P.R. China Abstract: The Hamiltonian of a N-boson system confined on a ring with zero spin and repulsive interaction is diagonalized. The excitation of a pair of p-wave-particles rotating reversely appears to be a basic mode. The fluctuation of many of these excited pairs provides a mechanism of oscillation, the states can be thereby classified into oscillation bands. The particle correlation is studied intuitively via the two-body densities. Bose-clustering originating from the symmetrization of wave functions is found, which leads to the appearance of 1-, 2-, and 3-cluster structures. The motion is divided into being collective and relative, this leads to the establishment of a relation between the very high vortex states and the low-lying states. After the experimental realization of the Bose-Einstein condensation1, various condensates confined under dif- ferent circumstances have been extensively studied the- oretically and experimentally. Mostly, the condensates are considered to be confined in a harmonic trap. Con- densates trapped by periodic potential have also been studied due to the appearance of optical lattices.2 It is believed that the appearance of condensates confined in particular geometries is possible. Experimentally, the particle interactions can now be tuned from very weak to very strong,3−8 it implies that the particle correlation may become important. Theoretically, to respond, go- ing beyond the mean field Gross-Pitaevskii (GP) theory is desirable, and the condensates confined in particular geometries are also deserved to be considered. Along this line, in addition to the ground state, the yrast states have been studied both analytically and numerically.9−17 The condensation on a ring has also been studied recently.12 The present paper is also ded- icated to the N−boson systems confined on a ring with weak interaction, its scope is broader and covers the whole low-lying spectra. A similar system has been in- vestigated analytically by Lieb and Liniger16,17. How- ever, the emphasis of their papers is different from the present one, which is placed on analyzing the structures of the excited states to find out their distinctions and sim- ilarities, and to find out the modes of excitation. Based on the analysis, an effort is made to classify the ex- cited states. Traditionally, the particle correlation and its effect on the geometry of N−boson systems is a topic scarcely studied if N is large. In this paper, the corre- lation is studied intuitively so as the geometric features inherent in the excited states can be understood. Tradi- tionally, a separation between the collective and internal motions is seldom to be considered if N is large. In this paper such a separation is made and leads to the estab- lishment of a relation between the vortex states and the low-lying states. It is assumed that the N identical bosons confined on a ring have mass m, spin zero, and square-barrier inter- action. The ring has a radius R, N is given at 100, 20 and 10000. Let G = ~2/(2mR2) be the unit of energy. The Hamiltonian then reads H = − i<|startoftext|> Kadowaki-Woods Ratio of Strongly Coupled Fermi Liquids Takuya Okabe Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu 432-8561,Japan (Dated: November 28, 2018) On the basis of the Fermi liquid theory, the Kadowaki-Woods ratio A/γ2 is evaluated by using a first principle band calculation for typical itinerant d and f electron systems. It is found as observed that the ratio for the d electron systems is significantly smaller than the normal f systems, even without considering their relatively weak correlation. The difference in the ratio value comes from different characters of the Fermi surfaces. By comparing Pd and USn3 as typical cases, we discuss the importance of the Fermi surface dependence of the quasiparticle transport relaxation. PACS numbers: 71.10.Ay, 71.18.+y, 71.20.Be, 71.27.+a, 72.15.-v It is widely known as a universal feature of heavy fermion systems that there holds the Kadowaki-Woods (KW) relation A/γ2 ≃ 1 × 10−5µΩ cm(mol K/mJ)2 be- tween the electronic specific heat coefficient γ of C = γT and the coefficient A of the resistivity ρ = AT 2 in the clean and low temperature limit.[1] According to the Fermi liquid theory, this is interpreted as an indication of the fact that A is squarely proportional to quasiparti- cle mass enhancement due to strong electron correlation. On the other hand, transition metal systems are reported since before to obey a similar relation with a more than an order of magnitude smaller value of A/γ2.[2, 3] In view of the observation that there seems to exist several types of systems in this regard, the recent finding by Tsujii et al.[4] is quite impressive that many Yb-based compounds show the KW ratio A/γ2 as small as the transition met- als. Kontani derived the small ratio as a result of the large orbital degeneracy of the the 4f13 state of trivalent Yb by applying the dynamical mean field approximation to a periodic Anderson model of an orbitally degenerate f electron states coupled with a single conduction band.[5] To discuss the KW ratio A/γ2 and the many-body mass enhancement effect, a simple model is usually adopted at the cost of neglecting material specific individ- ual factors. In the present work, we are interested in such an effect as caused by a system-dependent factor, that is, the Fermi surface dependence of quasiparticle current re- laxation. The system should have a large enough Fermi surface relative to the Brillouin zone boundary in order for the quasiparticle current to dissipate effectively into an underlying lattice through mutual quasiparticle scat- terings. In other words, the effectiveness of the trans- port relaxation may depend on the size and shape of the Fermi surface. To investigate this point definitely, we discuss the quasiparticle transport by taking account of the momentum dependence of quasiparticle scattering on the basis of realistic band structures. This has been ham- pered so far by a task required for not so simple Fermi surfaces of many band systems as could be simply mod- elled analytically. In terms of fairly realistic energy bands obtained from a first principle calculation, we evaluate those quantities which are not affected severely by the electron correlation effect. The theory in use is essen- tially within the phenomenological Fermi liquid theory described by renormalized quantities, and unlike a model calculation no bare microscopic quantities appear explic- itly. Schematic results using simple abstract models have been given before, in which a tight binding square lattice model and a two-band model are investigated.[6, 7, 8] For the ratio A/γ2 we make use of the expression, = 21.3αFa [µΩ (mol K/mJ) ], (1) which corresponds to Eq. (4.11) in Ref. 7 where we set a = 4Å for the lattice constant. In what follows we sub- stitute a calculated value for a. Below we follow how to derive αF , where α is a coupling constant, and F is a factor determined by the Fermi surface. Following a microscopic analysis of the quasiparticle transport with vertex corrections properly taken into account,[9] we may derive a phenomenological linearized Boltzmann equation.[7] Generalizing the theory to take a many-band effect into account, in the low temperature T → 0 we end up with the equation ∗Electronic address: ttokabe@ipc.shizuoka.ac.jp vipµ = (πT ) pp′kρ p′+k(l pµ + l p′µ − l p′+kµ − l p−kµ), (2) where vipµ and ρ p = δ(µ − ε p) are the velocity compo- nent and the local density of state of the renormalized http://arxiv.org/abs/0704.0843v2 mailto:ttokabe@ipc.shizuoka.ac.jp (mass-enhanced) quasiparticle with the crystal momen- tum p in the i-th band. The superscripts i and j are the band indices, while the subscript µ = x, y, z are Carte- sian coordinates. In the right hand side of Eq. (2), the 2nd to 4th terms in the parenthesis represent vertex cor- rections in the microscopic formulation. In terms of the solution lipµ, which physically represents stationary devi- ation of the Fermi surface in an applied electric field Eµ, the conductivity is given by σ ≡ σµ = 2e pµ, (3) The above equations (2) and (3) correspond to Eqs. (3.10) and (3.15) of Ref. 8 respectively. We may suppress the index µ (= x) in Eq. (3) as we discuss the cubic systems in what follows. Instead of solving the simultaneous matrix equations (2) exactly, we use trial functions for lipµ as commonly ap- plied in a variational principle formulation of the trans- port problems.[10] Assuming lipµ ∝ e |vipµ| we obtain αijci,j ρ2|vx| , (4) where ci,j = k1,k2,k3,k4 k1+k2=k3+k4 ρik1ρ ρik4(e −eik4) 2/4ρiρj , ρ|vx| ≡ ρip|v px|. (6) We define coupling constants αij = ρiρj〈W ij〉/π, where p, is the density of states of the i-th band at the Fermi level and 〈W ij〉 denotes the quasiparticle scat- tering probability W pp′k averaged over the momenta p, p and k. As the double sum in (2), dominated by Umk- lapp processes, covers a complicated shaped phase space over the Fermi surface, it is generally a good approxi- mation to take W pp′k out of the momentum sum as an averaged quantity. The total density of states ρ = is substituted for γ = 2π2ρ/3. In heavy fermion systems, the momentum dependence pp′k could be generally neglected, for the quasiparti- cle scattering W pp′k is primarily caused by strong on-site Coulomb repulsion U . Then we can make an order of magnitude estimate of αii in terms of Landau parame- ters F and F . For an anisotropic Fermi liquid, as in an isotropic case, one can derive that the charge and spin susceptibilities are given by χic = 2ρi/(1+F ) and χis = 2ρi/(1 + F ), respectively. Thus, for the systems in which charge fluctuations are suppressed, χic → 0, we obtain F ≫ 1. On the other hand, in terms of A /(1 +F ), one obtains a rough estimate of the cou- pling αii = 1 )2 + 1 . There- fore, under the normal condition that the spin enhance- ment is moderate, (1+F )−1 ∼ 1, αii should universally stay around a constant of an order of unity.[7] This corre- sponds to the condition to make the Wilson ratioRW = 2 in the impurity model.[11, 12] We discuss a normal state that the system is well away from critical instabilities, around which A/γ2 will be strongly enhanced at vari- ance with experimental results under consideration.[13] We evaluate F numerically for α = αij = 1 to obtain A/γ2, and investigate the Fermi surface dependence. It is noted that the factor F is determined by the shape and extent of the Fermi surfaces relative to the Brillouin zone boundary. Microscopically, the mass enhancement due to the many-body effect is represented by the ω- derivative of the electron self-energy Σ(q, ω), or by the renormalization factor zip as ρ p = ρ 0,p/z p, where ρ 0,p is a bare density of states. It is easily checked that the factor z cancels in F when zip is independent of i. Oth- erwise, in case that a dominant contribution to the re- sistivity comes from an electron-correlated main band, then the other bands may be neglected and A/γ2 be- comes independent of z of the main band. As we see below numerically, it is found indeed that F is domi- nated by a few scattering channels within a main band or two. Hence, we elaborate on a numerical estimate of F on the basis of a realistic band calculation reproducing reliable Fermi surfaces of relevant bands, even if it may not take account of local many-body correlation effects fully enough for the renormalized quantities like ρi and vip to be separately compared with experiments. As a matter of course, we must exclude the extreme case in which strong correlation modifies electron states around the Fermi level qualitatively from those of a band calcu- lation. We apply our theory to those itinerant electron systems in which correlation strength is not negligible but not so strong. To calculate F for some typical cubic d and f itiner- ant electron systems in the fcc and Cu3Au structures, we have performed ab initio band calculations within den- sity functional theory using the plane wave pseudopoten- tial code VASP with the Perdew-Wang 1991 generalized gradient approximation to the exchange correlation func- tional Exc.[14, 15, 16, 17] By minimizing the total energy we obtain the lattice constant a, which is accurate enough to be used in Eq. (1). To evaluate F numerically, we have to broaden the delta function ρip = δ(µ − ε p) by ∆ to pick up electron states around the Fermi level. The width ∆ of the order of real temperature should be decreased as the number of the k-points is increased until we confirm to have a con- vergent result. For the number L of subdivisions along re- TABLE I: Calculated results. a (Å) ρ|vx| a F N A/γ2 b USn3 4.60 3.1 4.0 3 0.39 UIn3 4.61 4.9 1.6 3 0.16 UGa3 4.24 3.9 2.5 3 0.23 Pd 3.86 7.4 0.23 3 0.019 Pt 3.91 8.4 0.15 4 0.012 aIn unit of a = 1. bIn unit of [10−5 µΩ cm (mol K/mJ)2]. ciprocal lattice vectors, band calculations are performed with Lband ∼ 50, from which we obtain the band energies εik on the finer k-mesh of L ∼ 200 by interpolation. As the four-fold k-sum in the numerator of Eq. (4), especially for the most important terms coming from the main d or f correlated bands, constitutes the most time consuming part of the calculation, we have to reduce the numerical task by some symmetry considerations not only on the cubic symmetry of the quasiparticle states, but on the relative directions of the four momentum vectors of the scattering quasiparticle states and the x-direction of the current flow. The reduction is particularly effective for the intra-band scatterings i = j. The calculated results are shown in Table I, where F and A/γ2 for α = αij = 1 are shown along with the lattice constant a, the number N of metallic bands con- tributing to the resistivity, and ρ|vx| defined in Eq. (6). We find that our results explain well the experimental tendency of an order of magnitude small values of the ratio A/γ2 for the transition metal systems. As for the absolute values of the ratio, our results are a few times smaller than observed evenly, but the accuracy of this order should not be taken seriously here. Among other things, the results indicate that different characters of the Fermi surfaces play an important role. To show the relative contribution to the resistivity from relevant bands, relative magnitudes of ci,j in the nu- merator of Eq. (4) are shown for Pd and USn3 in Figs. 1 and 2, respectively. For Pd, the contribution to F comes from the 4th to 6th bands, among which dominant is the 5th hole band of the 3d character. Similarly, the 5th band contributes majorly not only to ρ, i.e., ρ5 ≃ 5.4ρ4 ≃ 12ρ6, but to ρ|vx| in Eq. (6). On the other hand, for USn3, while the 14th heavy electron band plays a central role, the 12th and 13th hole bands also make non-negligible contributions through the inter-band scatterings. Hence, as the first point to note, numerical importance of the inter-band contributions makes F large in the f electron system. This is partly because ρi for i = 12, 13, 14 are comparable with each other, namely, ρ14 ≃ 2ρ13 ≃ 3ρ12. Moreover, it is remarked that the large and nearly spher- ical shape of the Fermi surfaces are essential too. As the second point to note, the importance of the Fermi surface geometry can be understood within a single band model by comparing contribution from the main band. We find that c5,5/ρ 5 = 0.097 for Pd is an order of magnitude 6 4 FIG. 1: cij (i, j = 4, 5, 6) for Pd. The contribution from the 5th band is dominant for the resistivity. 14 12 FIG. 2: cij (i, j = 12, 13, 14) for USn3. The interband con- tribution with the 14th band is important too. smaller than c14,14/ρ 14 = 0.93 for USn3. The difference comes from the different characters of the Fermi surfaces. According to an elementary formula σ = e2ρv2τ = e2ρvl, the conductivity σ depends on ρv as well as l. In this context, the mean free path l is not a single particle property determined by a lifetime of the particle state, but it is the transport property which characterizes how efficiently the total electric current decays into a lattice system, e.g., in our case, through mutual Umklapp scat- tering processes between the current carriers. In partic- ular, regardless of interaction, electrons in free space will not have resistivity.[9] Thus, to evaluate the transport property l correctly, it is crucial to take account of the momentum dependence of the scattering states and their conservation modulo the reciprocal lattice vectors. Note that ρ|vx| defined in Eq. (6) is related to the sur- face area S of the Fermi surfaces, as ρdε = Sdk⊥/(2π) Hence, ρ|vx| too is independent of the mass renormaliza- tion z as F is, and for free electrons we obtain ρ|vx| ∝ ∝ n2/3. One can see a correlation between F and ρ|vx| in Table I. In fact, Pd and Pt have twice as large ρ|vx| as the uranium compounds. The difference can- not be simply explained by the difference in the Fermi surface volume n. It is caused by the fact that the f - electron systems have the nearly isotropic Fermi surfaces while the d-electron systems have complicated ones with FIG. 3: The intersection of the Fermi surfaces of Pd d-hole states with the (111̄) plane. FIG. 4: The intersection of the Fermi surfaces of USn3 with the (100) plane relatively large area compared to their total volume, as indicated in Figs. 3 and 4. The different characters of the surfaces affect not only the single particle quantity ρ|vx| but also the transport property of the total cur- rent relaxation. As the order of magnitude difference in F is not explained merely by ρ|vx|, we have to have re- sort to the other factor, that is, the transport property depending on the Fermi surfaces. It originates from the detailed k-dependence of the scattering states, as repre- sented in ci,j , or by the phase space volume available for all possible scattering channels under strict restrictions of energy and momentum conservations. Thus our quan- titative analysis concludes the important effect on the quasiparticle transport due to the shape and complexity of the Fermi surfaces. In summary, we evaluated the Kadowaki-Woods ratio A/γ2 of some itinerant d and f electron systems numeri- cally on the basis of the Fermi liquid theory using quasi- particle Fermi surfaces obtained by band calculations. In a single framework, we find the d electron systems have smaller ratio than the f systems, as observed, and among others we pointed out an important effect to the transport coefficient A originating from a commonly ne- glected specific feature depending on the characters of the Fermi surfaces. The effect is not understood fully as a single-particle property of interacting systems, but we stress the importance of the phase space restriction due to momentum conservation in two-body scattering processes to dissipate a total electric current. In short, to realize effective dissipation, the system should have a large and regular shaped Fermi surface. In future we will examine that the Fermi-surface dependent efficiency of mutual quasiparticle scatterings may depend on a type of transport current to be relaxed. Acknowledgment The author is grateful to N. Fujima, S. Kokado and T. Hoshino for providing assistance in the numerical cal- culations. He also acknowledges computational resources offered from YITP computer system in Kyoto University. [1] K. Kadowaki and S. B. Woods, Solid State Commun. 58, 507 (1986). [2] M. J. Rice, Phys. Rev. Lett. 20, 1439 (1968). [3] K. Miyake, T. Matsuura, and C. M. Varma, Solid State Commun. 71, 1149 (1989). [4] N. Tsujii, H. Kontani, and K. Yoshimura, Phys. Rev. Lett. 94, 057201 (2005). [5] H. Kontani, J. Phys. Soc. Jpn. 73, 515 (2004). [6] T. Okabe, J. Phys. Soc. Jpn. 67, 2792 (1998). [7] T. Okabe, J. Phys. Soc. Jpn. 67, 4178 (1998). [8] T. Okabe, J. Phys. Soc. Jpn. 68, 2721 (1999). [9] K. Yamada and K. Yosida, Prog. Theor. Phys. 76, 621 (1986). [10] J. M. Ziman, Electrons and Phonons (Clarendon Press, Oxford, 1960). [11] P. Nozières, J. Low. Temp. Phys. 17, 31 (1974). [12] K. Yosida and K. Yamada, Prog. Theor. Phys. 53, 1286 (1975). [13] T. Takimoto and T. Moriya, Solid State Commun. 99, 457 (1996). [14] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996). [15] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [16] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [17] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992). ABSTRACT On the basis of the Fermi liquid theory, the Kadowaki-Woods ratio $A/\gamma^2$ is evaluated by using a first principle band calculation for typical itinerant $d$ and $f$ electron systems. It is found as observed that the ratio for the $d$ electron systems is significantly smaller than the normal $f$ systems, even without considering their relatively weak correlation. The difference in the ratio value comes from different characters of the Fermi surfaces. By comparing Pd and USn$_3$ as typical cases, we discuss the importance of the Fermi surface dependence of the quasiparticle transport relaxation. <|endoftext|><|startoftext|> Structure of Strange Dwarfs with Color Superconducting Core Masayuki Matsuzaki∗ and Etsuchika Kobayashi Department of Physics, Fukuoka University of Education, Munakata, Fukuoka 811-4192, Japan Abstract We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the strange quark matter core. We find that unpaired quark matter is a good approximation to the core of strange dwarfs. PACS numbers: 95.30.-k ∗matsuza@fukuoka-edu.ac.jp http://arxiv.org/abs/0704.0844v1 mailto:matsuza@fukuoka-edu.ac.jp Witten made a conjecture that the absolute ground state of quantum chromodynamics (QCD) is not 56Fe but strange quark matter, which is a plasma composed of almost equal number of deconfined u, d, and s quarks [1]. Although this conjecture has been neither confirmed nor rejected, if this is true, since deconfinement is expected in high density cores of compact stars, there could exist stars that contain strange quark matter converted from two-flavor quark matter via weak interaction. Strange quark stars whose radii are about 10 km, with or without thin nuclear crust, have long been investigated. Glendenning et al. proposed a new class of compact stars containing strange quark matter and thick nuclear crust ranging from a few hundred to ten thousand km [2, 3, 4]. They named them the strange dwarfs because their radii correspond to those of white dwarfs. Alcock et al. discussed the mechanism that the strange quark core supports the cruct [5]. Since the mass of s quark is larger than those of u and d, strange quark matter is positively charged. In order to electrically neutralize the core, electrons are bound to the surface of the core. They estimated that the thickness of this electric dipole layer is a few hundred fm. Then this layer can support a nuclear crust. Although Alcock et al. considered only thin crusts, Glendenning et al. considered thick crusts up to about ten thousand km. Very recently, Mathews et al. identified eight candidates of strange dwarfs from observed data [6]. A theoretical facet whose importance in nuclear physics was recognized later is color superconductivity in quark matter. At asymptotically high density, the color-flavor locking (CFL) is believed to be the ground state [7]. At realistic densities, however, the two-flavor color superconductivity (2SC) is thought to be realized even when electric neutrality is imposed if the coupling constant is strong [8]. Thus, in the present paper, we discuss effects of the 2SC phase in the strange quark matter core on the structure of strange dwarfs. In order to determine the structure of compact stars, we solve the general relativistic Tolman-Oppenheimer-Volkoff (TOV) equation, dp(r) Gǫ(r)M(r) 4πr3p(r) M(r)c2 2GM(r) , (1) M(r) = 4π ǫ(r′) r′2dr′, (2) for the pressure p(r), the energy density ǫ(r), and the mass enclosed within the radius r, M(r). Here G is the gravitational constant and c is the speed of light. The equation is closed when an equation of state (EOS), a relation between p and ǫ, is specified. In the present case, strange dwarfs are composed of the strange quark matter core and the nuclear crust. Accordingly two parameters, the pressure at the center and at the core-crust boundary, must be specified to integrate the TOV equation. The latter must be equal or less than that corresponds to the nucleon drip density ǫdrip. Otherwise neutrons drip and gravitate to the core. In the present calculation we take a pcruct calculated from ǫcrust = ǫdrip. We assume zero temperature throughout this paper. As for the EOS of the quark core, we adopt the MIT bag model without any QCD corrections (see Ref. [4], for example). For unpaired free quark matter, p = −B + µfkFf µ2f − m4f ln µf + kFf , (3) ǫ = B + µfkFf µ2f − m4f ln µf + kFf , (4) where mf , kFf , and µf = m2f + k Ff are the mass, the Fermi momentum, and the chemical potential of quarks of each flavor, respectively, and f runs u, d, and s. Hereafter we put c = h̄ = 1. The quantity B is the bag constant. The effect of color superconductivity is incorporated as a chemical potential dependent effective bag constant. In the 2SC case [9], Beff = B − ∆2(µ)µ2, (5) where ∆(µ) is the quark pairing gap as a function of a chemical potential µ, whose relation to µf is specified later. The pairing gap is obtained as a function of the Fermi momentum by solving the gap equation [10] ∆(kF) = − v̄(kF, k) E ′(k) k2dk, (6) E ′(k) = (Ek − EkF) 2 + 3∆2(k), (7) with kF = kFu = kFd, Ek = k2 +m2q , and mq = mu = md. The one gluon exchange pairing interaction is given by v̄(p, k) = − pkEpEk 2EpEk + 2m q + p 2 + k2 +m2E (p+ k)2 +m2E (p− k)2 +m2E 6EpEk − 6m q − p 2 − k2 m2E = 2, (8) where p and k are the magnitudes of 3-momenta. The running coupling constant is given by [11] q2max+q q = p− k, qmax = max{p, k}. (9) As for the EOS of the crust, we adopt the tabulated one for β-equilibrium nuclear matter of Baym, Pethick, and Sutherland [12] (BPS) conforming to Refs. [2, 3, 4]. The positively charged strange quark matter in the core is simply approximated by µ = µu = µd = µs. Quark masses are given by mu = md = 10 MeV, ms = 150 MeV. The bag constant is chosen to be B1/4 = 160 MeV. Parameters entering into the pairing interaction are q2c = 1.5Λ QCD and ΛQCD = 400 MeV. The nucleon drip density is ǫdrip = 4.3×10 11 g/cm3. 26 28 30 32 34 36 38 log� (J/m free quark 2SC quark FIG. 1: Equations of state of free and 2SC quark matter and β-equilibrium nuclear matter. The latter is tabulated in Refs. [12] and [4]. The adopted EOS is displayed in Fig. 1. The logarithm is to base 10 throughout this paper. The quark matter EOS describes the core and the BPS EOS describes the crust. At the boundary, the pressure is common whereas the energy density jumps discontinuously. In order to obtain the EOS for 2SC matter, the pairing gap must be calculated at each kF beforehand. This is shown in Fig. 2 left. The effective bag constant determined by the pairing gap is shown in Fig. 2 right. The resulting 2SC EOS is included in Fig. 1. Figure 3 presents the mass-radius relation obtained by integrating the TOV equation with a fixed pcruct, determined from ǫcrust = ǫdrip, and various central pressures. This result can 200 300 400 500 600 700 800 µ (MeV) 200 300 400 500 600 700 800 µ (MeV) FIG. 2: Left: color superconducting pairing gap and right: effective bag constant, as functions of the quark chemical potential. 0 1 2 3 4 5 logR (km) FIG. 3: Mass-radius relation of strange dwarfs and white dwarfs. be classified into three regions. The first region (larger central pressures), almost vertical curve at around R ∼ 10 km, describes strange stars with thin crusts. In this region, color superconductivity makes the maximum mass and radius larger because the pairing gap reduces the bag constant and consequently the energy density decreases and the pressure increases. This is consistent with another calculation with the CFL phase [13]. The second region, horizontal at around M/Msun ∼ 10 −2, and the third region, vertical at around R ∼ 104 km up to the maximum mass, correspond to strange dwarfs. In the second region, color superconducting quark cores support slightly larger masses than unpaired free quark cores. In the third region, effect of color superconductivity is negligible. In Fig. 3, The mass-radius relation of ordinary white dwarfs without quark matter cores calculated by adopting the BPS EOS is also shown although it is known that the BPS EOS is not very suitable for white dwarfs. As the central pressure decreases, the quark matter core shrinks (Fig. 4 left) and eventually strange dwarfs reduce to ordinary white dwarfs. When their masses are the same, the former is more compact than the latter (see also Fig. 5 right) because of the gravity of the core. Mathews et al. paid attention to this difference in the mass-radius relation and classified the observed data of dwarfs [6]. According to their work, eight of them are classified into strange dwarfs. 28 29 30 31 32 33 34 35 logp0 (J/m 28 29 30 31 32 33 34 35 logp0 (J/m FIG. 4: Left: core radius and right: mass of strange dwarfs, as functions of the central pressure. 14 15 16 log�0 (g/cm -1 0 1 2 3 4 logr (km) SD(free) FIG. 5: Left: mass of strange dwarfs as a function of the central energy density. Right: energy profile of a strange dwarf with M/Msun = 0.465 and that of a white dwarf with M/Msun = 0.466. Figure 4 right indicates that strange dwarfs, in particular those of 103 km < R < 104 km, are realized in a very narrow range of the central pressure. This is reflected in the density of calculated points. During this rapid structure change from the second to the third region, the core radius almost does not change, see Fig. 4 left. Figure 5 left also graphs M/Msun as Fig. 4 right but as a function of the central energy density. The difference between these two figures at the low pressure/energy density side can be understood from the quark matter EOS in Fig. 1 such that the pressure decreases steeply at the lowest energy density. Figure 5 left indicates that strange dwarfs have central energy densities just below the lowest stable compact strange stars and several orders of magnitude larger than those of ordinary white dwarfs. This is clearly demonstrated in Fig. 5 right. To summarize, we have solved the Tolman-Oppenheimer-Volkoff equation for strange dwarfs with ǫcrust = ǫdrip and a wide range of the central pressure. We have examined effects of the two-flavor color superconductivity in the strange quark matter core in a simplified manner. The obtained results indicate that, aside from a slight increase of the minimum mass, effect of color superconductivity is negligible in the mass-radius relation. This is consistent with the conjecture given in Ref. [6]. As a function of the central energy density, however, strange dwarfs are realized at slightly lower energy densities than the unpaired free quark case reflecting the effect on the equation of state. Recently Usov discussed that electric fields are also generated on the surface of the color-flavor locked matter [14]. This suggests that strange dwarfs with color-flavor locked cores might also be possible although this is expected only at relatively high densities. Since the pairing gap enters into the calculation only through the effective bag constant, aside from a possible slight change in chemical potentials, it can surely be expected that the effect of color-flavor locking does not differ much from that of the two-flavor color superconductivity. In conclusion, unpaired quark matter is a good approximation to the core of strange dwarfs. Another aspect that might be affected by color superconductivity is the cooling [15]. This is beyond the scope of the present study. [1] E. 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Althaus, Astrophys. J. 462 (1996), 364. References ABSTRACT We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the strange quark matter core. We find that unpaired quark matter is a good approximation to the core of strange dwarfs. <|endoftext|><|startoftext|> Information entropic superconducting microcooler A. O. Niskanen,1, 2 Y. Nakamura,1, 3, 4 and J. P. Pekola5 1CREST-JST, Kawaguchi, Saitama 332-0012,Japan 2VTT Technical Research Centre of Finland, Sensors, PO BOX 1000, 02044 VTT, Finland 3NEC Fundamental Research Laboratories, Tsukuba, Ibaraki 305-8501, Japan 4The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan 5Low Temperature Laboratory, Helsinki University of Technology, PO BOX 3500, 02015 TKK, Finland (Dated: October 25, 2018) We consider a design for a cyclic microrefrigerator using a superconducting flux qubit. Adiabatic modulation of the flux combined with thermalization can be used to transfer energy from a lower temperature normal metal thin film resistor to another one at higher temperature. The frequency selectivity of photonic heat conduction is achieved by including the hot resistor as part of a high frequency LC resonator and the cold one as part of a low-frequency oscillator while keeping both circuits in the underdamped regime. We discuss the performance of the device in an experimentally realistic setting. This device illustrates the complementarity of information and thermodynamic entropy as the erasure of the quantum bit directly relates to the cooling of the resistor. PACS numbers: 74.50.+r,85.80.Fi,03.67.-a For the purpose of quantum computing, the coher- ence properties of superconducting quantum bits (qubits) should be optimized by decoupling them from all noise sources as well as possible. However, many interesting experiments can be envisioned also when the decoupling is far from perfect. One such experiment closely related to coherence optimization is using a qubit as a spectrom- eter [1, 2, 3] for the environmental noise by monitoring the effect of the environment on the quantum two-level system. Here we focus on the opposite phenomenon, i.e. the effect of a qubit on the environment. Recently a superconducting flux qubit [4, 5] with a quite small tun- neling energy from the point of view of quantum com- puting was cooled using sideband cooling and a third level [6] from about 400 mK down to 3 mK. Motivated by this experiment we consider the possibility of using a single quantum bit as a cyclic refrigerator for environ- mental degrees of freedom. The utilized heat conduction mechanism is photonic which was recently studied also in experiment [7]. Besides the possible practical uses, the device is interesting physically as it directly illus- trates the connection between information entropy and thermodynamical entropy. For related superconducting high-frequency cooler concepts see eg. Refs. [8, 9]. Here we study a flux qubit coupled inductively to two different loops shown in Fig. 1a. In loop j (j = 1, 2) we have a resistor Rj in series with an inductance Lj and a capacitance Cj . These form two damped harmonic os- cillators. The resistors are in general at different tem- peratures T1 and T2. The coupling of the qubit to these two admittances Y1 and Y2 is assumed to be sufficiently large to dominate the relaxation of the qubit. This as- sumption can be easily validated by e.g. increasing the mutual inductance. The flux qubit is an otherwise su- perconducting loop except for three or four Josephson junctions with suitably picked parameters. In particu- FIG. 1: (color online) Principle of the flux-qubit cooler. (a) Layout of the circuit. (b) Energy band diagram. (c) Schematic of the cooling cycle in the qubit temperature- entropy plane. lar one of the junctions is made smaller than others to form a two-level system. When biased close to half of the flux quantum Φ0 = h/2e, the qubit can be described (in persistent current basis) by the Hamiltonian H/~ = −1 (∆σx + εσz) (1) where σx and σz are Pauli matrices, ~ε = 2Ip(Φ−Φ0/2) is the flux-tunable energy bias and Φ is the controllable flux threading the qubit loop. Away from Φ = Φ0/2 the eigenstates have the persistent currents ±Ip circulating in the loop. The tunneling energy ~∆ results in an an- http://arxiv.org/abs/0704.0845v1 ticrossing at Φ = Φ0/2 and there the energy eigenstates do not carry average current. The resonant angular fre- quency of the qubit is ω = ε2 +∆2. Consider the ideal cycle shown in Fig. 1b-c where the bias of the flux qubit is swept slowly (slower than ∆/2π) between two extreme values ε1 and ε2 corresponding to two different energy level separations ~ω1 and ~ω2. Let us further assume that ωj ≈ ωLCj and Qj ≫ 1, where ωLCj = 1/ LjCj and Qj = Lj/Cj/Rj. This choice guarantees that the qubit mainly couples to resistor R1 (R2) at bias point 1 (2). The cooling cycle consists of steps O, P, Q and R. First in step O the qubit has the angular frequency ω2 and is allowed to thermalize. Be- cause of the bandwidth limitations imposed by the reac- tive elements, the qubit tends to thermalize with resistor R2 to temperature T2. In the next step P the flux bias is adiabatically changed to point 1 such that the level popu- lations do not change but the energy eigenstates do. The sweep is assumed to be however faster than relaxation. In point 1 the angular frequency is reduced to ω1. Be- cause the level populations and therefore the Boltzmann factors do not change the qubit must now be at lower temperature T̃2 given by T̃2 = T2ω1/ω2 in order to com- pensate for the change of the qubit splitting. Note that the quantum mechanical adiabaticity implies also ther- modynamical adiabaticity: while the energy eigenbasis changes the level populations and thus also entropy do not change. In step Q the qubit is allowed to thermalize to temperature T1 which results in heating of the qubit and in cooling of resistor 1 if T̃2 < T1. At this point the ideally pure quantum state of the qubit gets erased and information stored is lost. The entropy of the qubit increases, but locally the entropy of resistor 1 decreases such that one can say that some information is “stored” in the resistor as it cools but naturally with some loss. Finally in step R the qubit is adiabatically shifted back to frequency ω2 which results in heating of the qubit to the effective temperature T̃1 = T1ω2/ω1 which is assumed to be higher than T2. The excess energy is dumped to admittance 2 when the cycle starts again from the be- ginning. Note that due to the condition T̃2 < T1 resistor 1 can never be cooled below T2ω1/ω2. Since there is no isothermal stage in the above cycle the present device is not even in principle a Carnot cooler but rather an Otto-type device.[10] The density matrix of the qubit with the resonant an- gular frequency ω at temperature T (β = (kBT ) −1) is given by ρeq(β, ε) = . (2) Using this the cooling power and the efficiency of the ideal cycle in Fig. 1c can be easily calculated. It is given by the area of the shaded region in the entropy-temperature plane below points P and Q. In principle one could solve for the effective temperature of the qubit along the line between points P and Q as a function of entropy given by S = −kBTr(ρ ln ρ). Alternatively, we can simply note that the expectation value of the energy stored in the qubit in point P is EP = Tr(ρeq(β2ω2/ω1, ε1)H1) while after relaxation we have EQ = Tr(ρeq(β1, ε1)H1), where H1 = H(ε1) is the Hamiltonian at point 1. We thus get for the ideal cooling power P/f = EQ − EP = −β1~ω1 e−β1~ω1 + 1 − ~ω1e −β2~ω2 e−β2~ω2 + 1 ≤ ~ω1 where f is the pump frequency. The cooling power achieves the maximum value of ~ω1f/2 when the ther- mal population in step O (and P) is small and when the population in step Q is large, i.e. when β2~ω2 ≫ 1 and β1~ω1 ≪ 1. Naturally a practical device has to be designed to fulfill the first condition always, in which case the smallest achievable temperature is on the or- der of ~ω1/kB below which the cooling power decreases rapidly. The dynamic range could be made wider by a tunable ∆ which can be achieved by splitting the small- est junction into a dc SQUID geometry. Another figure of merit is the ratio η of the heat removed from resis- tor 1 divided by the heat added to resistor 2. It can be obtained as the ratio of the shaded area divided by the sum of the hatched area and the shaded area, i.e., η = (EQ−EP )/(ER−EO) where EO = Tr(ρeq(β2, ε2)H2) and ER = Tr(ρeq(β1ω1/ω2, ε2)H2). This simplifies neatly to η = ω1/ω2 < 1 which is in harmony with the second law of thermodynamics. For more quantitative analysis we have to consider the details of the relaxation rates due to the baths. The Golden Rule transition rates due to resistor j are given ↓,↑ = |〈0|dH/dΦ|1〉|2M2j S I (±ωj) M2j S I(±ωj) (4) where the positive sign corresponds to relaxation. The total thermalization rate is Γ ↑ + Γ ↓. Here the unsymmetrized noise spectrum is given by I(ω) = e−iωt〈δIj(0)δIj(t)〉dt 2~ωReYj(ω) 1− exp(−βj~ω) . (5) where ReYj(ω) = R j /[1+Q − ωLCj )2] is the real part of admittance of circuit j. The total relaxation rate is thus 2(Ip∆Mj) 2 coth 1 +Q2j − ωLCj ) . (6) To model the behavior of the device we utilize the Bloch master equation (see e.g Ref. [11]) given in our case by ~̇M = − ~B× ~M−Γ1th( ~M‖− ~MT1)−Γ2th( ~M‖− ~MT2)−Γ2 ~M⊥, where ~M = Tr(~σρ) is the “magnetization” of the qubit, and ~B = ∆~x + ε~z is the fictitious magnetic field. Note however that the z-component of ~B and ~M do correspond to real magnetic field and magnetization, respectively. In Eq. (7) ~M‖ and ~M⊥ are the components of the magnetiza- tion parallel and perpendicular to ~B, respectively. These are explicitly ~M‖ = (∆Mx + εMz)(∆~x + ε~z) (8) ~M⊥ = ε2Mx−∆εMz ~x+My~y + 2Mz−∆εMx ~z. (9) Here ~MT stands for the ε-dependent equilibrium mag- netization of a qubit at temperature T given explicitly ~MT = and Γ2 = (Γ th + Γ th)/2 + Γϕ is the dephasing rate. The possibility of pure dephasing at the rate Γϕ has been included. In the simulation we neglect pure dephasing due to the intentionally large dominating thermalization rate. Equation (7) describes relaxation towards instan- taneous equilibrium with two competing rates due to two different thermal baths. Equations of this type are usually used in the stationary case, but for driving fre- quencies slower than ∆/~ it should be also valid. As is obvious from Eq. (7), the qubit actually tends to re- lax towards an effective ε-dependent equilibrium mag- netization (Γ1th ~MT1 + Γ ~MT2)/(Γ th + Γ th) at the rate To illustrate the practical potential of the device we show in Fig. 2 the simulated cooling power with si- nusoidal driving of ǫ(t) compared to the ideal case along with the actual loop in the entropy temperature plane. The heat flow Pj from resistor j to the qubit is simply obtained by integrating the product of the thermalization rate and the energy deficit, i.e., Pj = ∫ 1/f [Tr(ρeq(βj , ǫ(t))H)− Tr(ρ(t)H)] . The den- sity matrix ρ(t) = 1 ~M(t) · ~σ is solved numerically using the Bloch equation (system is followed over a few periods until it has converged to the limit cycle). We see that the actual simulated behavior does not significantly deviate at low f from the ideal behavior and that cooling pow- ers on the order of fW can be achieved with reasonable sample parameters. The oscillatory behavior at high f is interpreted as Landau-Zener interference [12, 13]. However, the cooling power has to be compared with realistic heat loads to evaluate the utility of the flux qubit cooler. On one hand, resistor 1 is subject to heat load 0 ln 2 (a) (c) (b) (d) 0 0.5 1 1.5 f (GHz) 0 ln 2 0 0.5 1 1.5 f (GHz) FIG. 2: (color online) Example of the simulated cooling power with ω1/2π = ∆/2π = 5 GHz (ǫ = 0 GHz), ω2/2π = 20.62 GHz (ǫ = 20 GHz), Q1 = Q2 = 10, ωj = ωLCj and 2(Ip∆Mj) 2/(Rj~ωj) = 20 × 10 9s−1. This can be achieved e.g. with Ip = 200 nA, M1 = 29 pH, M2 = 59 pH and R1 = R2 = 1 Ω. The driving is sinusoidal. (a) The solid line illustrates the path in the T − S plane for the ideal cy- cle described in the text while the dashed (dotted) line is a result of simulation for f = 0.05 GHz (f = 1 GHz) with T1 = T2 = 0.3 × ~ω2/kB ≈ 300 mK. (b) Simulated cooling power vs. f for the same temperatures as in (a) is shown with the dashed line while the solid line is the ideal result of Eq. 3 (c-d) Same as (a-b) but with T1 = 0.5× T2 ≈ 150 mK. The cooling threshold at 0.14 GHz in (d) is caused by finite Q-factor. from the phonons of the substrate on which the device rests. On the other hand, resistor 2 should be coupled well enough to phonon bath such that the unavoidable work done on it does not raise T2 excessively. The heat flow between the electron system of resistor j and the phonon system is given by Pel−ph = ΣV (T j −T 5ph) where Vj is the volume of resistor j and Σ is typically on the order of 109 Wm−3K−5. Thus resistor 1 needs to have a sufficiently small volume while resistor 2 should be large enough physically in order to serve as a heat sink. In ad- dition the photonic heat conduction between the resistors due to temperature gradient may in principle contribute also. Following an analysis similar to Ref. [14], the heat flow from admittance Y2(ω) to Y1(ω) can be written as ReY1(ω)ReY2(ω)(n2(ω)− n1(ω)) where nj(ω) = [exp(βj~ω − 1)]−1 are the boson occu- pation factors and M is the mutual inductance between the loops. For detuned high-Q resonators the photonic heat conduction turns out to be quite negligible. For instance for the values of Fig. 3 with M = 5 pH and R1 = R2 = 1 Ω we get only Pγ = 2 × 10−18 W even 0 0.5 1 1.5 f (GHz) FIG. 3: (color online) Equilibrium temperature as a function of pump frequency for three different phonon bath temper- atures. The temperature of resistor 1 (volume 10−21m3) is shown with dashed line while the temperature of resistor 2 (volume 10−18m3) is shown with solid line. The bath tem- peratures Tph ≈ T2 from top to bottom are 0.3 × ~ω2/kB, 0.2 × ~ω2/kB and 0.1 × ~ω2/kB. Otherwise the parameters are like in Fig. 2. if T1 = 0 K and T2 = 300 mK. Figure 2 illustrates the calculated equilibrium temperature versus operation fre- quency obtained numerically by finding the balance be- tween the dominating phononic heat conduction and the integrated cooling power. We see that almost a factor of 2 reduction of T1 is possible with realistic parameters. In practice the drop of T1 can be measured e.g. using an additional SINIS thermometer, in which resistor 1 will serve as the normal metal N. Its reading is sensitive to the electronic temperature of N only, and self-heating can be made very small. The resistors should be made out of thin film normal metal such as copper or gold with typ- ically sub 1 Ω square resistance. Volume can be picked freely. To get the resonant frequencies and quality factor as above we need L1 = 320 pH, C1 = 3.2 pF, L2 = 80 pH and C2 = 0.8 pF which are also realistic. For the inductor one may use either Josephson or the kinetic in- ductance of superconducting wire while the capacitance values are similar to those in typical flux qubits [2]. To satisfy the conditions of the above numerical example we need quite large mutual inductances which however can be easily achieved using e.g. kinetic inductance [15]. The strong driving requires also rather large inductance be- tween the microwave line and the qubit, which should not result in uncontrolled relaxation. For instance, Mmw=5 pH coupling to the control line is acceptable as it would result in at most 3 × 107 s−1 relaxation rate assuming a 50 Ω environment at 0.3 K. This choice will not degrade the performance of the device significantly since driving is much faster. Yet sufficiently strong driving can be achieved with a modest 3 µA ac current. Fabrication process will require most likely three lithography steps. 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Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, Science 310, 1653 (2005). [14] D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland, Phys. Rev. Lett. 93, 045901 (2004). [15] A. O. Niskanen, K. Harrabi, F. Yoshihara, Y. Nakamura, and J. S. Tsai, Phys. Rev. B 74, 220503(R) (2006). http://arxiv.org/abs/cond-mat/0701041 http://arxiv.org/abs/quant-ph/0611275 http://arxiv.org/abs/cond-mat/0309049 ABSTRACT We consider a design for a cyclic microrefrigerator using a superconducting flux qubit. Adiabatic modulation of the flux combined with thermalization can be used to transfer energy from a lower temperature normal metal thin film resistor to another one at higher temperature. The frequency selectivity of photonic heat conduction is achieved by including the hot resistor as part of a high frequency LC resonator and the cold one as part of a low-frequency oscillator while keeping both circuits in the underdamped regime. We discuss the performance of the device in an experimentally realistic setting. This device illustrates the complementarity of information and thermodynamic entropy as the erasure of the quantum bit directly relates to the cooling of the resistor. <|endoftext|><|startoftext|> Introduction Presented here is a new technique for analyzing skew polynomial rings satisfying a poly- nomial identity with an eye toward discovering their PI degrees. It combines and extends the methods of Jøndrup [21] and Cauchon [5], who introduced techniques of “deleting derivations” in skew polynomial rings, by means of which they showed that some proper- ties of certain types of iterated skew polynomial ring A = k[x1][x2; τ2, δ2] · · · [xn; τn, δn] are determined by the corresponding ring A′ = k[x1][x2; τ2] · · · [xn; τn]. Jøndrup’s re- sults imply that A and A′ have the same PI degree under certain hypotheses, including characteristic zero for the base field. Cauchon developed an algorithm that gives an isomorphism between certain localizations of A and A′, but this requires a qi-skew condition on each (τi, δi) with qi not a root of unity, which usually precludes A from satisfying a polynomial identity. We relax the restrictions placed on the base field and its chosen scalars by Jøndrup and Cauchon, respectively, by introducing the notion of a higher q-skew τ -derivation. If we “twist” the multiplication in the (commutative) coordinate ring of affine, symplec- tic, or Euclidean n-space over a field k, we get a (noncommutative) quantized coordinate ring which has the structure of an iterated skew polynomial ring with coefficients in k. This structure is also exhibited in the quantized Weyl algebras and in the quantized coordinate ring of n×n matrices over k. Letting A represent one of these k-algebras, the quantum Gel’fand-Kirillov conjecture asserts that FractA is isomorphic to the quotient division ring of a quantum affine space over a purely transcendental extension of k. For 1991 Mathematics Subject Classification. 16R99; 16S36; 81R50; 16P40. Key words and phrases. noncommutative rings; skew polynomial rings; quantum algebras. This research will form a part of the author’s PhD dissertation at the University of California at Santa Barbara. http://arxiv.org/abs/0704.0846v1 2 HEIDI HAYNAL more information on the quantum Gel’fand-Kirillov conjecture and proofs of conditions under which the result holds, see [1] [7] [23] [28] [32] [33]. We will confirm some of these cases in a new way. The first section sets up the conventions under which we work, including definitions and an established result concerning the PI degree of quantum affine space. We assume that the reader has some familiarity with the subject, so we do not give an exhaustive collection of definitions. A comprehensive discussion of any unfamiliar terms can be found in [16] [4] and [27]. In the second section we define higher τ -derivations and give necessary and sufficient conditions for their existence. Of particular interest are higher τ -derivations which satisfy a q-skew relation. In the third section we present a structure theorem for a localization of q-skew polynomial rings. This extends the work of Cauchon [5], and the calculations are simplified by the presence of higher q- skew τ -derivations. In the fourth section we deal with the structure of iterated skew polynomial rings. Sometimes it is advantageous to rearrange the order in which the indeterminates appear, so we establish a sufficient condition that allows such reordering. The main theorem there asserts that if A is an iterated q-skew polynomial ring with certain higher τ -derivations, then there is a finitely generated Ore set T ⊆ A such that AT−1 is isomorphic to a localization of a much “nicer” iterated skew polynomial ring. In the fifth section, we use the tools developed in the previous sections to confirm certain cases of the quantum Gel’fand-Kirillov conjecture and to find the PI degree of some quantized coordinate rings and quantized Weyl algebras. In the last section, we follow up with a structure theorem for completely prime factors of iterated skew polynomial rings. We also present an open question which, if answered positively, would show that the quantum Gel’fand-Kirillov conjecture holds for certain of the prime factor algebras we study. Throughout, k will denote a field of arbitrary characteristic, q ∈ k a nonzero ele- ment. The following assumptions apply to all skew polynomial rings that we will con- sider: • all coefficient rings are k-algebras • all automorphisms are k-algebra automorphisms • all skew derivations are k-linear • in all skew polynomial rings R[x; τ, δ], τ is an automorphism, not just an endo- morphism. To say that R[x; τ, δ] is a q-skew polynomial ring means that the auomorphism and skew derivation satisfy the relation δτ = qτδ. The reader will note that this is opposite to Cauchon’s conventions, but it matches the presentation in [10] and others. To say that δ is locally nilpotent means that for every r ∈ R there is an integer nr ≥ 0 such that δnr(r) = 0, and δp(r) 6= 0 for p < nr. Such nr is called the δ-nilpotence index of r. The PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 3 symbol N refers to the set of positive integers. For a real number m we use the notation ⌊m⌋ in section five to indicate the integer part of m. Definition 1.1. We say that two rings R and S exhibit PI degree parity when these two conditions are satisfied: (1) R is a PI ring if and only if S is a PI ring, (2) PIdegR = PIdegS. For a field k and multiplicatively antisymmetric λ ∈ Mn(k), the corresponding mul- tiparameter quantum affine space is the k-algebra Oλ(kn) with generators x1, . . . , xn and relations xixj = λijxjxi for all i, j. The corresponding multiparameter quantum torus is the k-algebra Oλ((k×)n) given by generators x±11 , . . . , x±1n and the same rela- tions. The multiplicative set generated by x1, . . . , xn in Oλ(kn) is a denominator set, and Oλ((k×)n) is a localization of Oλ(kn) with respect to this set. In this paper we’ll show that iterated skew polynomial algebras covering a large class of standard examples have PI degree parity with Oλ(kn) for an appropriately chosen λ. To find out what that PI degree may be, we utilize a result of De Concini and Procesi. In [8, Proposition 7.1], they establish the following formula for calculating the PI degree of a quantum affine space Oλ(kn). Their assumption of characteristic zero from [8, Section 4] is not used in this result. Theorem 1.2. [De Concini - Procesi] Let λ = (λij) be a multiplicatively antisymmetric n× n matrix over k. (1) The quantum affine space Oλ(kn) is a PI ring if and only if all the λij are roots of unity. In this case, there exist a primitive root of unity q ∈ k× and integers aij such that λij = q aij for all i, j. (2) Suppose λij = q aij for all i, j, where q ∈ k is a primitive ℓth root of unity and the aij ∈ Z. Let h be the cardinality of the image of the homomorphism n (aij)−−−−−−→ Zn π−−−−→ (Z/ℓZ)n where π denotes the canonical epimorphism. Then PI-deg (Oλ(kn)) = 2. Higher q-Skew τ-Derivations Before the featured definition, a brief discussion of a tool used to study q-skew polyno- mial rings is needed. Having the q-skew relation δτ = qτδ in place allows us to group terms of the same degree when we do skew polynomial arithmetic. The means to do this are provided by the q-Liebnitz rules. 4 HEIDI HAYNAL Definition 2.1. For an indeterminate t, and integers n ≥ m ≥ 0, we define the following polynomial functions: (m)t = t m−1 + tm−2 + · · ·+ t + 1 (1) (m)!t = (m)t(m− 1)t · · · (1)t, and (0)!t = 1 (2)( (n)!t (m)!t(n−m)!t The expressions are called the t-binomial coefficients, or Gaussian polynomials. The t-binomial coefficients have properties similar to those of the regular binomial coefficients. Two that will be useful for this work are:( = 1 for all n ≥ 0 (4) + tn−m for all 0 < m < n Proofs for these identities may be found in combinatorics texts such as [39]. When we evaluate the t-binomial coefficients at t = q, we obtain the q-binomial coefficients that we need for studying q-skew polynomial rings. As shown in [10, Section 6], the following q-Liebnitz rules hold for any q-skew polynomial ring R[x; τ, δ]: δn(rs) = τn−iδi(r)δn−i(s) for all r, s ∈ R and n = 0, 1, 2, ... xnr = τn−iδi(r)xn−i for all r ∈ R and n = 0, 1, 2, ... Now, taking a cue from the study of Schmidt differential operator rings, for instance [25], we define a sequence of k-linear maps that allows us to broaden the class of rings for which we may derive results like those of Jøndrup and Cauchon. Definition 2.2. A higher q-skew τ -derivation (h.q-s.τ -d.) on a k-algebra R is a sequence d0, d1, d2, . . . of k-linear operators on R such that d0 is the identity dn(rs) = τn−idi(r)dn−i(s) for all r, s ∈ R and all n diτ = q iτdi for all i. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 5 If a sequence of k-linear maps satisfies the first two conditions, we refer to it as a higher τ -derivation. We abbreviate the sequence {di}∞i=0 usually as just {di}. A h.q-s.τ -d is locally nilpotent if for all r ∈ R, there exists an integer n ≥ 0 such that di(r) = 0 for all i ≥ n, and dp(r) 6= 0 for p < n. In this case, n is called the d-nilpotence index of r. A h.q-s.τ -d is iterative if didj = di+j for all i, j. This implies that the di commute with each other. A q-skew τ -derivation δ on R extends to a h.q-s.τ -d. if there is a h.q-s.τ -d {di} on R with d1 = δ. For example, consider the k-algebra with two generators x and y, and one relation xy − qyx = 1, where q ∈ k×. We’ll assume that q 6= 1 and recognize this algebra as a q-skew polynomial ring k[y][x; τ, δ] with τ(y) = qy and δ(y) = 1, commonly known as a quantized Weyl algebra and denoted A 1(k). If q is not a root of unity, then the (i)!q comprise an iterative higher q-skew τ -derivation that extends δ on k[y]. The prop- erties of a higher q-skew τ -derivation follow directly from the fact that δ is a q-skew τ -derivation and the first q-Liebnitz rule. This particular h.q-s.τ -d. is also locally nilpotent because yn−i when i ≤ n, 0 when i > n. Proposition 2.3. Let {di} be a sequence of k-linear maps on a k-algebra R with d0 = idR, and let R[[x; τ −1]] be the skew power series ring where τ is a k-linear automor- phism of R, the coefficients are written on the right of the variable x, and rx = xτ(r) for all r ∈ R. (a) Then {di} is a higher τ -derivation on R if and only if the map Ψ : R → R[[x; τ−1]] given by r 7→ i=0 x idi(r) is a ring homomorphism. (b) Extend τ to an automorphism of R[[x; τ−1]] such that τ(x) = xq. Assume that {di} is a higher τ -derivation. Then the sequence {di} is a h.q-s.τ -d. if and only if this diagram is commutative: R[[x; τ−1]] // R[[x; τ−1]] 6 HEIDI HAYNAL Proof. (a) Suppose {di} is a higher τ -derivation on R. Consider any r, s ∈ R. It is clear that Ψ is additive and Ψ(1) = 1. Applying the definition 2.2 gives Ψ(rs) = xidi(rs) = τ i−mdm(r)di−m(s) Power series multiplication, with rx = xτ(r), gives Ψ(r)Ψ(s) = xidi(r) )( ∞∑ xidi(s) τ i−mdm(r)di−m(s) So Ψ preserves products. Therefore, Ψ is a ring homomorphism. To demonstrate the other implication, suppose Ψ is a ring homomorphism. Then Ψ(r)Ψ(s) = Ψ(rs) implies that dn(rs) = i=0 τ n−idi(r)dn−i(s) for all r, s ∈ R. There- fore, {di} is a higher τ -derivation. (b) Suppose that {di} is a h.q-s.τ -d. Then the relations diτ = qiτdi imply that τΨ(r) = i=0 x iqiτdi(r) = i=0 x idi(τ(r)) = Ψτ(r), for all r ∈ R. Now if the diagram is commutative, then comparing the coefficients of τΨ(r) = i=0 x iqiτdi(r) and Ψτ(r) = i=0 x idi(τ(r)) for all r ∈ R yields that diτ = q iτdi. � Remark 2.4. If {di} is locally nilpotent on R, we observe that claims analogous to the proposition can be made for the map Ψ : R → R[x; τ−1]. Proposition 2.5. Let {di} be a h.q-s.τ -d. on a k-algebra R, where τ is an automor- phism, and let S be a right denominator set in R with τ(S) = S. Then {di} can be uniquely extended to a h.q-s.τ -d. on RS−1. Proof. It has been established that τ and d1 extend uniquely to RS −1 by τ(rs−1) = τ(r)τ(s)−1 and d1(rs −1) = d1(r)s −1 − τ(rs−1)d1(s)s−1 in [10, Lemma 1.3]. Suppose that {di} extends to a h.q-s.τ -d. on RS−1. For r ∈ R and s ∈ S, we apply dn to the equation r1−1 = (rs−1)(s1−1) to get dn(r)1 −1 = dn (rs−1)(s1−1) τn−jdj(rs −1)dn−j(s1 = τn(rs−1)dn(s)1 −1 + · · ·+ dn(rs−1)s1−1. This implies that dn(rs −1) = dn(r)− τn−jdj(rs −1)dn−j(s) So we have uniqueness in case of existence. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 7 To show existence, let Ψ : R → R[[x; τ−1]] be the map defined in Proposition 2.3, and let φ : R[[x; τ−1]] → RS−1[[x; τ−1]] be the natural map. Consider the composite map Φ = φΨ : R → RS−1[[x; τ−1]]. For any s ∈ S, the constant term of Φ(s) is a unit. So we may inductively solve for the coefficients of an inverse for Φ(s) in RS−1[[x; τ−1]]. Details, as in [37, 1.2], are left to the reader. Hence, Φ extends to a ring homomorphism Φ′ : RS−1 → RS−1[[x; τ−1]] such that Φ′(rs−1) = Φ(r)Φ(s)−1, and we consider the diagram: RS−1[[x; τ−1]] // RS−1[[x; τ−1]] // RS−1 where τ has been extended to an automorphism of RS−1[[x; τ−1]] as in Proposition 2.3. Since Φ(r) = i=0 x idi(r)1 −1, and {di} is a h.q-s.τ -d. on R, we have τΦ(r) = xiqiτdi(r)1 1−1 = Φτ(r) for all r ∈ R. It follows directly that τΦ′(rs−1) = Φ′τ(rs−1). So, indeed, the diagram is commutative. Define a sequence {di} on RS−1 such that di(t) equals the coefficient of xi in Φ′(t) for all t ∈ RS−1. Then by Proposition 2.3 we conclude that this sequence is a h.q-s.τ -d. on RS−1 extending {di} on R. � Lemma 2.6. Let A be a k-algebra, B ⊆ A a k-subalgebra generated by {b1, b2, . . . }, τ a k-linear automorphism of A, and {di} a higher τ -derivation on A. If di(bj) ∈ B and τ(bj) ∈ B, for all i, j ∈ N, then di(B) ⊆ B for all i. Proof. First, observe that τ(bj) ∈ B for all j implies that τ(B) ⊆ B. Since the di are k-linear maps, it suffices to check monomials in the bj , using induction on their length. Suppose, inductively, that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have di(bj1 · · · bjℓ) ∈ B for all i and all j1, . . . , jℓ. Then using the product rule for h.q-s.τ -d. gives dn(bj1 · · · bjm) = τn−1di(bj1 · · · bjm−1)dn−i(bjm) ∈ B for all n and all j1, . . . , jm, by the induction hypothesis. � 8 HEIDI HAYNAL Lemma 2.7. Let A be a k-algebra with a set {xj} of generators, τ an automorphism of A, and {di} a h.q-s.τ -d. on A. If {di} is locally nilpotent for all xj, then {di} is locally nilpotent on A. Proof. It suffices to check monomials in the xj because the di are k-linear maps. We proceed by using induction on the length of such monomials. For a given xn, let i(n) be its nilpotence index, so di(xn) = 0 for all i ≥ i(n). Suppose inductively that for n ≥ 2, all integers ℓ with 1 ≤ ℓ ≤ n− 1, and all choices of j1, . . . , jℓ, there exists an integer m such that di(xj1 · · ·xjl) = 0 for all i ≥ m. For instance, m = i(j1)+ · · ·+ i(jℓ) will suffice, although the d-nilpotence index of xj1 · · ·xjℓ may be less than this sum. Then, for p ≥ m+ i(jn), we have dp(xj1 · · ·xjn) = τ p−idi(xj1 · · ·xjn−1)dp−i(xjn) = 0, completing the induction. � Consider again the quantized Weyl algebra A 1(k). In case q is an ℓ-th root of unity, the dℓ given in (7) would be undefined due to the occurrence of a zero denominator. However, realizing A 1(k) as a factor of a quantized Weyl algebra over k[t ±1] allows us to define a h.q-s.τ -d. on A 1(k) nonetheless. The k[t ±1]-algebra At1(k[t ±1]) has generators x and y and one relation xy− tyx = 1. This is a t-skew polynomial ring k[t±1][y][x; τ̄ , δ̄] where τ̄ (y) = ty, τ̄(t) = t, δ̄(y) = 1, and δ̄(t) = 0. Note that δ̄i(yn) = (n)!t (n−i)!t yn−i when i ≤ n 0 when i > n implying that δ̄i k[t±1][y] ⊆ (i)!tk[t±1][y]. So the assignment d̄i = (i)!t defines an iterative, locally nilpotent h.t-s.τ̄ -d. {d̄i} on k[t±1][y]. Now, the relation xy − tyx = 1 is equivalent to the relation xy − qyx = 1 modulo 〈t − q〉. Hence we k[t±1] /〈t− q〉 ∼= Aq1(k). When q is an ℓth root of unity, we have δ̄ℓ k[t±1][y] ⊆ 〈t−q〉k[t±1][y]. Nonetheless, the h.t-s.τ̄ -d. {d̄i} on k[t±1][y] induces a h.q-s.τ -d. {di} on k[y], also iterative and locally nilpotent, with d1 = δ. Note that even though δ ℓ = 0 in this algebra, we have di(y i) = 1 for all i. This phenomenon is not unique to the quantized Weyl algebras. The conditions that drive it are codified in the following theorem. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 9 Theorem 2.8. Let R be a k-algebra and R[x; τ, δ] a q-skew polynomial ring where q ∈ k, q 6= 1. Suppose there exists a torsion-free k[t±1]-algebra R and R[x; τ̄ , δ̄] a t-skew polynomial ring such that R/〈t− q〉R ∼= R, with τ̄ and δ̄ reducing to τ and δ. Suppose further that δ̄i(R) ⊆ (i)!tR for all i. Then δ extends to an iterative h.q-s.τ -d. {di} on R. If δ̄ is locally nilpotent, then so is {di}. If q is not a root of unity, then di = δ (i)!q all i. If q is a primitive ℓth root of unity, then di = (i)!q for i < ℓ. Proof. The assumption δ̄i(R) ⊆ (i)!tR for all i implies that the sequence of maps d̄i = (i)!t make up a well-defined iterative h.t-s.τ̄ -d. on R, and also implies that δ̄ℓ(R) ⊆ 〈t − q〉R because (ℓ)t ≡ (ℓ)q = 0 modulo 〈t − q〉. Since τ̄ and δ̄ reduce to τ and δ modulo 〈t− q〉, we have an isomorphism R/〈t− q〉[x; τ̄ , δ̄] ∼= R[x; τ, δ] whereby {d̄i} induces an iterative h.q-s.τ -d. {di} on R. The reduction of the maps from R to R also implies the remaining results. � We will find that all of the conditions assumed above are satisfied by the common quantized coordinate rings and related examples, which will be discussed in a subsequent section. 3. The τ-Derivation Removing Homomorphism Following the pattern in [5], let A = R[x; τ, δ], and suppose that δ is locally nilpotent. Set S = {xn | n ∈ N ∪ {0}} ⊂ A. Lemma 3.1. The set S is a denominator set in A. Proof. Clearly, S is a multiplicative set inA. And, since S contains only regular elements of A, it is left and right reversible. It remains to show that S is an Ore set. Let a = i=0 rix i be an element of A with rn 6= 0. For each ri in the expression of a, and each mi ≥ 0, we have xmiri = τmi−jδj(ri)x = a′ix+ δ mi(ri) for some a i ∈ A. Since δ is locally nilpotent, we may choose mi to be the δ-nilpotence index of ri to conclude that xmiri = a ix for some a i ∈ A. Set ma = max{mi | 0 ≤ i ≤ n}. Then for each ri, we have x mari = ãix, and hence x maa = ãx for some ã ∈ A. Now suppose, inductively, that for a given a ∈ A and xp ∈ S we can find elements xma ∈ S and ā ∈ A such that xmaa = āxp, say ā = i=0 r̄ix i. We know that there 10 HEIDI HAYNAL exists an element xmā such that xmā ā = a′x for some a′ ∈ A. So, xmaa = āxp implies xmā+maa = a′xp+1, completing the induction. Hence, for any a ∈ A and s ∈ S, we have Sa∩As 6= ∅. So S is a left Ore set in A. We see that S is a right Ore set by applying the same argument to Aop = Rop[x; τ−1,−δτ−1]. � Suppose also that the derivation δ extends to an iterative, locally nilpotent higher q- skew τ -derivation {di} on R and that q 6= 1. Denote  = AS−1 = S−1A, the localization of A with respect to S, and define a map f : R −→  by f(r) = n(n+1) 2 (q − 1)−ndnτ−n(r)x−n, noting that {di} is locally nilpotent and that q − 1 is invertible. If q is not a root of unity and {di} is obtained from a q-skew τ -derivation δ as in (7), the formula for f can be rewritten as f(r) = n(n+1) (q − 1)−n (n)!q δnτ−n(r)x−n. The rewritten formula matches the one presented in [5, Section 2] when q is replaced by q−1 to account for the difference between δτ = qτδ (used here) and τδ = qδτ (used in [5]). We will show that f is a homomorphism and that the the multiplication in imf is made simpler than that in A by removing the derivation, as seen in the following. Proposition 3.2. If r ∈ R, then xf(r) = f x in Â. Proof. Using the hypothesis that {di} is iterative, we compute that xf(r) = n(n+1) 2 (q − 1)−nxdnτ−n(r)x−n n(n+1) 2 (q − 1)−n −n(r)x+ d1dnτ −n(r) n(n+1) 2 (q − 1)−nq−ndnτ−n+1(r)x−n+1 n(n+1) 2 (q − 1)−n(n+ 1)qdn+1τ−n(r)x−n n(n+1) 2 (q − 1)−nq−ndnτ−n(τ(r))x−n+1 n(n−1) 2 (q − 1)−n+1(n)qdnτ−n(τ(r))x−n+1 PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 11 = τ(r)x n(n+1) 2 (q − 1)−nq−n + q n(n−1) 2 (q − 1)−n+1(n)q −n(τ(r))x−n+1 = τ(r)x (q − 1)−n 2 + q 2 (qn − 1) −n(τ(r))x−n+1 = τ(r)x+ (q − 1)−nq n(n+1) 2 dnτ −n(τ(r))x−n+1 n(n+1) 2 (q − 1)−ndnτ−n(τ(r))x−n x = f which gives the result. � From Proposition 3.2, it follows by routine induction that xmf(r) = f τm(r) xm ∀m ∈ Z. (8) This is what we need in order to show that our map is indeed a k-algebra homomor- phism. Proposition 3.3. The map f : R −→  is a k-algebra homomorphism. Proof. It is immediate that f is k-linear (τ and {di} are k-linear), and that f(1) = 1. We’ll show that f is multiplicative. If r, s ∈ R, then using Prop. 3.2, f(r)f(s) = i(i+1) 2 (q − 1)−idiτ−i(r)x−if(s) i(i+1) 2 (q − 1)−idiτ−i(r)f(τ−i(s))x−i i≥0, j≥0 i(i+1)+j(j+1) 2 (q − 1)−(i+j)diτ−i(r)djτ−(i+j)(s)x−(i+j). For n ∈ N, the coefficient of x−n in the sum above is i≥0, j≥0, i+j=n i(i+1)+j(j+1) 2 (q − 1)−ndiτ−i(r)djτ−n(s) (n−p)2+p2+n 2 (q − 1)−ndn−pτ p−n(r)dpτ−n(s) 12 HEIDI HAYNAL n(n+1) 2 (q − 1)−n qp(p−n)dn−pτ pτ−n(r)dpτ −n(s) n(n+1) 2 (q − 1)−n τ pdn−p(τ −n(r))dp(τ −n(s)) n(n+1) 2 (q − 1)−ndn(τ−n(r)τ−n(s)) n(n+1) 2 (q − 1)−ndnτ−n(rs), computed by putting p = j and using the second condition in the Definition 2.2. In summary, f(r)f(s) = n=0 q n(n+1) 2 (q − 1)−ndnτ−n(rs)x−n = f(rs). � Proposition 3.4. (1) The map f extends uniquely to an algebra homomorphism, also denoted f , of R[y; τ ] to  satisfying f(y)=x. (2) The extended homomorphism is injective. Proof. (1) This result follows from Proposition 3.2 and the universal property of Ore extensions. (2) Let P = pmy m + · · ·+ p1y + p0 be a nonzero element of R[y; τ ], where each pi ∈ R, m ≥ 0, pm 6= 0. Then f(P ) = f(pm)xm + · · ·+ f(p1)x+ f(p0). Since f(pi) = n(n+1) 2 (q − 1)−ndnτ−n(pi)x−n ∈ AS−1, we know that there exists an integer l ≥ 0 such that each f(pi)xl is a nonzero element of A of positive degree l (in x) whenever pi 6= 0. (Because {di} is locally nilpotent, we may choose an l large enough.) It follows that f(P )xl is a nonzero element of  of degree m+ l, hence f(P ) 6= 0. � Definition 3.5. The algebra homomorphism f : R[y; τ ] −→  = AS−1 is called the derivation removing homomorphism. The image of f , call it A′, is the subalgebra of  = AS−1 generated by x and f(R), and is isomorphic (as an algebra) to R[y; τ ] by the derivation removing homomorphism f . Observe that A′ contains the multiplicative system S = {xn | n ∈ N ∪ {0}}. Since equation (8) holds and f(y) = x, the elements of this set are normal in A′. Hence, S satisfies the (two-sided) Ore condition in A′. The elements of S are regular in A′ because they are regular in Â, and thus: Proposition 3.6. A′S−1 = AS−1 Proof. We have A′S−1 ⊆ AS−1 because A′ = im(f) ⊆ AS−1. To show the other inclusion, it suffices to show that R ⊆ A′S−1. (This suffices because A is built up from PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 13 R by x, x2, . . . . So if R ⊆ A′S−1, then AS−1 ⊆ A′S−1.) Consider any r ∈ R and let ℓ be the d-nilpotence index of r. We show that r ∈ A′S−1 with an induction argument on ℓ. If ℓ ≤ 1, then d1(r) = 0, whence f(r) = r ∈ A′ ⊆ AS−1. If ℓ ≥ 2, we write f(r) = r + −n, with rn = q n(n+1) 2 (q − 1)−ndnτ−n(r) ∈ R. We’ll show that n=1 rnx −n ∈ A′S−1 in order to conclude that r ∈ A′S−1, because f(r) − n=1 rnx −n = r. That is, we need to show that each rn ∈ A′S−1. Suppose, inductively, that for any element r̃ ∈ R with d-nilpotence index m such that m < ℓ, we have r̃ ∈ A′S−1. Note that for n ∈ {1, . . . , ℓ}, we have dℓ−n(rn) = q n(n+1) 2 (q − 1)−n −n(r) = q n(n+1) 2 (q − 1)−n q−nℓτ−ndℓ(r) = 0 because dℓ(r) = 0 by hypothesis. Hence, by the induction hypothesis, each rn ∈ A′S−1 for 1 ≤ n ≤ ℓ− 1. It follows that r = f(r)− n=1 rnx −n also belongs to A′S−1. � This equality of quotient rings reveals that if A is a PI ring, then PIdegA = PIdegA′ = PIdegR[y; τ ], with the second equality arising from the derivation removing homomorphism f . This recovers the result of Jøndrup [21] without the assumption that k has characteristic zero. We summarize the results of this section in the following theorem. Theorem 3.7. Let k be a field, R a k-algebra and A = R[x; τ, δ] a q-skew polynomial ring in which δ extends to a locally nilpotent, iterative h.q-s.τ -d. {di} on R for some q ∈ k×, q 6= 1. Let S be the Ore set in A generated by x, and define a map f : R −→ AS−1 by f(r) = n=0 q n(n+1) 2 (q − 1)−ndnτ−n(r)x−n. Then f is a k-algebra homomorphism, and it extends to an injective homomorphism f : R[y; τ ] −→ AS−1 sending y to x. Furthermore, the extension f : R[y±1; τ ] −→ AS−1 is an isomorphism. So there is PI degree parity between A and R[y; τ ]. Moreover, if R is a noetherian domain, then FractA ∼= FractR[y; τ ]. 14 HEIDI HAYNAL 4. Main Theorem In the case where A is an iterated skew polynomial ring, we would like to apply re- peatedly the method presented above to remove all of the derivations and compare the resulting Ore localizations. We must first establish some facts about the behavior of h.q-s.τ -d. when the variables adjoined to the coefficient ring are rearranged, and about iterated localization. The results of these lemmas will ensure that after the induction step in the proof of the main theorem we are left with a ring to which the method of the preceding section applies. The first parts of the following lemmas hold in a broader class of skew polynomial rings and also when the q-skew condition is imposed. The final parts assert that h.q-s.τ -d. are preserved when rearranging of the variables is permissible. Lemma 4.1. Let S = R[x; τ, δ], A = R[x; τ, δ][y; σ], and  = R[x; τ, δ][y±1; σ], where σ(R) = R and σ(x) = λx for some λ ∈ k×. (1) Then A = R[y; σ′][x; τ ′; δ′], and  = R[y±1; σ′][x; τ ′; δ′], where σ′ = σ , τ ′ = τ , = δ, τ ′(y) = λ−1y, and δ′(y) = 0. (2) If (τ, δ) is q-skew, then so is (τ ′, δ′). (3) Suppose further that δ extends to a h.q-s.τ -d. {di} on R, and that σdi = λidiσ for all i. Then the τ ′-derivation δ′ extends to a h.q-s.τ ′-d. {d′i} on R[y±1; σ′] such that the restrictions of the d′i to R coincide with di, and d i(y) = 0 for all i ≥ 1. Moreover, {d′i} restricts to a h.q-s.τ ′-d. on R[y; σ′]. (a) If {di} is iterative, then {d′i} is iterative. (b) If {di} is locally nilpotent, then {d′i} is locally nilpotent. Proof. (1) Routine details omitted so as not to try the patience of the reader. (2) Suppose that (τ, δ) is q-skew on R. We’ll check that the two τ ′-derivations τ ′−1δ′τ ′ and qδ′ agree on R[y±1; σ′]. It suffices to check their agreement on a set of generators, R ∪ {y, y−1}. It is clear that τ ′−1δ′τ ′(r) = qδ′(r) for all r ∈ R. Since δ′(y) = 0, they agree on {y, y−1} as well. So (τ ′, δ′) is q-skew. (3) Define a sequence of maps d′i : R[y ±1; σ′] → R[y±1; σ′] by di(rj)y Clearly these are k-linear maps, d′i(r) = di(r) for all r ∈ R; also d′i(y) = di(1)y = 0 for i ≥ 1, and d′0 is the identity on R[y±1; σ′]. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 15 Because δ extends to {di} on R, we get d1(rj)y δ(rj)y j = δ′ for all rj ∈ R. So d′1 = δ′ on R[y±1; σ′]. Now, for integers j,m, n, and elements r, s ∈ R, (ryj)(sym) = d′n rσj(s)yj+m rσj(s) τn−idi(r)dn−i(σ j(s))yj+m τn−idi(r)y jσ−jdn−i(σ j(s))ym τn−idi(r)y jλ−j(n−i)dn−i(s)y (τ ′)n−i di(r)y d′n−i(sy (τ ′)n−id′i(ry j)d′n−i(sy So {d′i} satisfies the product rule for a higher τ -derivation on R[y±1; σ′]. Furthermore, τ ′d′i = τ ′ di(rj)y τdi(rj)λ −jyj, and d′iτ = d′i τ(rj)λ diτ(rj)λ τdi(rj)λ −jyj, giving the q-skew relation d′iτ ′ = qiτ ′d′i on R[y ±1; σ′]. It follows directly from the definition of the maps {di} that their restrictions to the k-subalgebra R[y; σ′] also exhibit the properties of definition 2.2. 16 HEIDI HAYNAL If {di} is iterative on R, then d′ℓd′i(rym) = d′ℓ di(r)y = dℓdi(r)y dℓ+i(r)y d′ℓ+i(ry m) for all r ∈ R, m ∈ Z, and non-negative integers ℓ, i. Hence, {d′i} is iterative on R[y±1; σ′]. Suppose that {di} is locally nilpotent on R. By Lemma 2.7 we need only check that {d′i} is locally nilpotent on R ∪ {y, y−1}, a set of generators for R[y±1; σ′]. This is clear because d′i(r) = di(r) for all r ∈ R, and d′i(y) = 0 for all i by construction. � Lemma 4.2. Let A = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y; σ],  = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y±1; σ], where σ(R) = R, and for all i ∈ {1, . . . , n}, σ(xi) = λixi for some nonzero λi ∈ k. Let Aj = R[x1; τ1; δ1][x2; τ2, δ2] · · · [xj ; τj, δj ] for j = 1, 2, . . . , n, and A0 = R. (1) Then A = R[y; σ∗][x1; τ 1][x2; τ 2] · · · [xn; τ ′n, δ′n],  = R[y±1; σ∗][x1; τ 1][x2; τ 2] · · · [xn; τ ′n, δ′n], where σ∗ = σ , τ ′i = τi, δ = δi, τ i(y) = λ i y, and δ i(y) = 0 for all 1 ≤ i ≤ n and j ≤ i− 1. (2) If (τi, δi) is qi-skew for any 1 ≤ i ≤ n, then (τ ′i , δ′i) is also qi-skew. (3) Suppose that each δi extends to an h.qi-s.τi-d. {di,p}∞p=0, and that σdi,p = λ i di,pσ on Ai−1 for all i and p. Then each δ i extends to a h.qi-s.τ i -d. {d′i,p}∞p=0 on the algebra R〈y, y−1, x1, . . . , xi−1〉, where d′i,p coincides with di,p on Aj, for j < i, and d′i,p(y) = 0 for p ≥ 1. Moreover, {d′i,p} restricts to a h.qi-s.τ ′i -d. on R〈y, x1, . . . , xi−1〉. (a) If {di,p} is iterative for any 1 ≤ i ≤ n, then {d′i,p} is iterative. (b) If {di,p} is locally nilpotent for any 1 ≤ i ≤ n, then {d′i,p} is locally nilpotent. Proof. (1) The condition σ(xi) = λixi for all i implies that σ(Ai) = Ai. We will use induction on n to prove the result. Lemma 4.1 proves the case n = 1. Suppose the result holds for all m < n, and consider A = An−1[xn; τn, δn][y; σ]. Application of Lemma 4.1, and then the induction hypothesis, gives A = An−1[xn; τn, δn][y; σ] = An−1[y; σ ′][xn; τ = R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y; σ′][xn; τ ′n, δ′n] = R[y; σ∗][x1; τ 1] · · · [xn; τ ′n, δ′n], PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 17 with the desired conditions met by the automorphisms and derivations, completing the induction. Similarly,  = R[y±1; σ∗][x1; τ 1] · · · [xn; τ ′n, δ′n]. (2) Consider the two τ ′i -derivations τ i and qiδ i on the ring R[y±1; σ∗][x1; τ 1] · · · [xi−1; τ ′i−1, δ′i−1] for 1 ≤ i ≤ n. Since (τi, δi) is q-skew, it is clear that these two τ ′i derivations agree on Ai−1. And since δ i(y) = 0 for all i = 1, . . . , n, these two τ i -derivations agree on a full set of generators of R[y±1; σ∗][x1; τ 1] · · · [xi−1; τ ′i−1, δ′i−1]. Hence, δ′iτ ′i = qiτ ′iδ′i. (3) Suppose the result holds for the algebra R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y±1; σ]. Then Lemma 4.1 may be applied, with An−1 providing the coefficients, to get An−1[xn; τn, δn][y ±1; σ] = An−1[y ±1; σ′][xn; τ where δ′n extends to a h.qn-s.τ n-d. {d′n,p} on An−1[y±1]. The induction hypothesis gives the result. � Definition 4.3. For a k-algebra A and a, b ∈ A, we say that a and b scalar commute if there is an element α ∈ k× such that ab = αba. We may also say that a and b α-commute. In the following two lemmas, we let D denote the division ring of fractions for the noetherian domain A. When comparing localizations of A, we identify them as subrings of D. Lemma 4.4. Let A be a noetherian domain, S ⊆ A \ {0} an Ore set. Let T be an Ore set in AS−1 \ {0} with S ⊆ T . (1) Then there exists an Ore set T̃ ⊆ A\{0} with S ⊆ T̃ such that AT̃−1 = (AS−1)T−1. (2) Suppose A is a k-algebra and S is generated by s1, . . . , sn satisfying sisj = γijsjsi for all i, j and some γij ∈ k×. Further suppose that T is generated by S ∪ t for some t ∈ AS−1 that satisfies sit = λitsi for all i and some λi ∈ k×. Then there exist a cyclic Ore set T̂ ⊆ A \ {0} and an (n + 1)-generator Ore set Ŝ ⊆ A \ {0} such that S ⊆ Ŝ, and (AS−1)T−1 = AT̂−1 = AŜ−1. Proof. (1) Consider T ∩A, the subset in T of elements with a denominator of 1. Clearly, this is a multiplicative set in A which contains S. Set T̃ = T ∩A. Let a ∈ T̃ and α ∈ A. Then a ∈ T , and since α ∈ AS−1, there exist b′ ∈ T and β ′ ∈ AS−1 such that aβ ′ = αb′. By [16, 10.2], there exist y ∈ S, and b, β ∈ A such that β ′ = βy−1 and b′ = by−1; hence, aβy−1 = αby−1 in AS−1. It follows that aβ = αb in A. So T̃ satisfies the right Ore condition in A, and the left Ore condition by symmetry. By the universal property, AT̃−1 ∼= (AS−1)T−1. As subrings of D, we have AT̃−1 = (AS−1)T−1. 18 HEIDI HAYNAL (2) The generating element t has the form t = ā(sm11 s 2 · · · smnn )−1 for some mi ∈ N, and ā ∈ A. For any si ∈ S, we have siā(s 2 · · · smnn )−1 = λiā(sm11 sm22 · · · smnn )−1si = µλiāsi(sm11 sm22 · · · smnn )−1, where µ is a product of powers of the γij. So ā scalar commutes with the genera- tors of S via the relations siā = µλiāsi. Let Ŝ be the multiplicative set generated by ā, s1, . . . , sn in A, and T̂ the multiplicative set generated by ās1s2 · · · sn in A. Recall that (AS−1)T−1 = AT̃−1, where T̃ = T ∩ A from part (1). From the scalar com- muting relations it follows that any element at̃−1 ∈ AT̃−1 may be written in the form b(ās1, · · · sn)−m for some m ∈ N ∪ {0}, b ∈ A, or the form cā−ℓn+1s−ℓ11 · · · s−ℓnn , for ℓj ∈ N ∪ {0}, c ∈ A. So we conclude that Ŝ and T̂ are Ore sets in A and that (AS−1)T−1 = AT̂−1 = AŜ−1. � Lemma 4.5. Let A be a noetherian domain, S1 ⊆ A \ {0} an Ore set, and for integers j = 2, . . . , n let Sj be an Ore set in ((AS 1 ) · · · )S−1j−1 \ {0} with Sj−1 ⊆ Sj. (1) Then there exists an Ore set T ⊆ A \ {0} such that AT−1 = (((AS−11 )S−12 ) · · · )S−1n . (2) Suppose A is a k-algebra, S1 is generated by s1, and for j = 2, . . . , n, Sj is generated by Sj−1 ∪ {sj}, where sisj = γijsjsi for some multiplicatively antisymmetric matrix (γij) ∈ Mn(k×). Then there are a cyclic Ore set T̂ ⊆ A and an n-generator Ore set Ŝ ⊆ A such that S1 ⊆ Ŝ, and ((AS−11 )S−12 ) · · ·S−1n = AT̂−1 = AŜ−1. Proof. (1) The proof proceeds by induction on n. The case n = 1 is covered in the lemma above. Suppose that for all j ≤ n− 1 there exists an Ore set Tj ⊆ A \ {0} such that AT−1j = (((AS 2 ) · · · )S−1j . Then the equality AT−1n−1 = (((AS 2 ) · · · )S−1n−1 identifies an Ore set Tn ⊆ AT−1n−1 \ {0} such that (AT−1n−1)T n = (((AS 2 ) · · ·S−1n−1)S−1n . Furthermore, Lemma 4.4 implies the existence of an Ore set T ⊆ A \ {0} such that AT−1 = (AT−1n−1)T n = (((AS 2 ) · · ·S−1n−1)S−1n . (2) Suppose, inductively, that there exist (i) a cyclic Ore set T̂n−1 ⊆ A \ {0} generated by s1ā2 · · · ān−1 (ii) an (n− 1)-generator Ore set Ŝn−1 ⊆ A \ {0} with S1 ⊆ Ŝn−1 and generators s1, ā2, ā3, . . . , ān−1 (iii) the āi scalar commute with s1 and with each other (iv) ((AS−11 )S 2 ) · · ·S−1n−1 = AT̂−1n−1 = AŜ−1n−1 as subrings of D. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 19 Then sn = ān(s1ā2 · · · ān−1)−r for some ān ∈ A and r ∈ N. Using the relations sisj = γijsjsi, routine calculations show that the āi scalar commute with the sj , and also with each other, for all i, j. Let T̂ be the multiplicative set generated by s1ā2 · · · ān, and let Ŝ be the multiplicative set generated by s1, ā2, ā3, . . . , ān. Then ((AS−11 )S 2 ) · · ·S−1n = (AT̂−1n−1)S−1n = AT−1 from part (1). Using Lemma 4.4, we con- clude that T̂ and Ŝ are Ore sets in A and that AT−1 = AT̂−1 = AŜ−1. � In the proof of the main theorem, we will use without mention the facts gathered here. For greater details on these statements, see [16, 10X, 10Y] and [10, 1.4]. (1) Given a noetherian ring A and a normal element x ∈ A, the multiplicative set generated by x is an Ore set. (2) The multiplicative set generated by a nonempty family of right Ore sets is right (3) Let A = R[x; τ, δ], and S a right denominator set in R such that τ(S) = S. Then S is a right denominator set in A and the identity map on AS−1 extends to an isomorphism of AS−1 onto (RS−1)[x; τ, δ] sending x1−1 to x. Note that if A is a k-algebra, τ , δ are k-linear, and τ(k×S) = k×S, then the result holds because S is a denominator set if and only if k×S is a denominator set. Theorem 4.6. Let R be a k-algebra and noetherian domain, A = R[x1; τ1, δ1] · · · [xn; τn, δn], where each τi is a k-linear automorphism of R〈xi, . . . , xi−1〉 such that τi(xj) = λijxj for all i, j with 1 ≤ j < i ≤ n and some λij ∈ k×, and where each δi is a k-linear τi- derivation. Assume that there exist elements qi ∈ k× with qi 6= 1 such that δiτi = qiτiδi, and that δi extends to a locally nilpotent, iterative h.qi-s.τi-d. on R〈xi, . . . , xi−1〉 for i = 1, . . . , n. (1) Then there exists an Ore set T ⊆ A generated by n elements of A such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n] where τ ′i |R = τi and τ ′i(yj) = λijyj for all i, j with 1 ≤ j < i ≤ n (2) There is PI degree parity between A and R[y1; τ1][y2; τ 2] · · · [yn; τ ′n]. Moreover, these algebras have isomorphic division rings of fractions. Proof. (a) Suppose, inductively, that we have R[x1; τ1, δ1][y 2 ; τ2] · · · [y±1n ; τ ′n] ∼= AS−12 where the restriction of τ ′i to R〈x1〉 coincides with τi, τ ′i(ym) = λimym for 2 ≤ i ≤ n and 1 < m < i, and S2 is an Ore set in A generated by n − 1 elements from A. Then by 20 HEIDI HAYNAL Lemma 4.2 AS−12 ∼= R[y±12 ; τ ′′2 ] · · · [y±1n ; τ ′′n ][x1; τ ′1, δ′1] (9) where the restrictions of τ ′1 and δ 1 to R coincide with τ1 and δ1, τ 1(yj) = λ j1 yj, δ 1(yj) = 0, and τ ′′i coincides with the restriction of τi to R〈y2, . . . , yi−1〉 for 2 ≤ i ≤ n. Observe that by Lemmas 4.2 and 2.7 we also have δ′1τ 1 = q1τ 1, and that δ 1 extends to a locally nilpotent iterative h.q1-s.τ -d. on R〈y±12 , . . . , y±1n 〉. Then applying the derivation removing homomorphism to the right hand side of (9) gives an isomorphism (AS−12 )T ∼= R[y±12 ; τ ′2] · · · [y±1n ; τ ′n][y±11 ; τ ′1] where T1 ⊆ AS−12 is an Ore set generated by one element of AS−12 . Then Lemma 4.5 and a reordering of variables shows the existence of an Ore set T ⊆ A, generated by n elements of A, such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n]. (2) This follows from part (1). � Corollary 4.7. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] with the hypotheses as in Theorem 4.6. Set λ = (λij). Then (1) A and Oλ(kn) have isomorphic division rings of fractions. (2) A is a PI-algebra if and only if all the λij are roots of unity, in which case A and Oλ(kn) have the same PI degree. In general, identification of the generators for the Ore set T in Theorem 4.6 is very cumbersome. To illustrate the computations on a fairly short iterated skew polynomial ring, we consider the multiparameter second quantized Weyl algebra A 2 (k). Here, Q = (q1, q2) ∈ (k×)2, qi 6= 1 for all i, and Γ = (γij) ∈ M2(k×) with γii = 1 and γ21 = γ 12 . The algebra A 2 (k) may be presented as an iterated skew polynomial ring of the form k[y1][x1; τ2, δ2][y2; τ3][x2; τ4, δ4], where the τi are k-linear automorphisms and the δ2i are k-linear τ2i-derivations such that τ2(y1) = q1y1, δ2(y1) = 1 τ3(y1) = γ 12 y1 τ3(x1) = γ12x1 τ4(y1) = q1γ12y1, δ4(y1) = 0 τ4(x1) = q 1 γ21x1, δ4(x1) = 0 τ4(y2) = q2y2, δ4(y2) = (q1 − 1)y1x1 + 1. For greater detail about this algebra, the reader is referred to [1], [23], [12], and [15]. Routine computations show that the pair (τ2, δ2) is a q1-skew derivation and that (τ4, δ4) is a q2-skew derivation. To show that δ2 and δ4 are locally nilpotent, it suffices to check PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 21 for local nilpotence on a set of generators. Given their definitions, this is accomplished by verifying their action on powers of y1 and y2: δi2(y 1 ) = (n)!q1 (n−i)!q1 yn−i1 i ≤ n 0 i > n δi4(y 2 ) = (n)!q2 (n−i)!q2 [δ4(y2)] iyn−i2 i ≤ n 0 i > n Using Theorem 2.8 we have a h.q1-s.τ2-d. {d2,i} extending δ2, and a h.q2-s.τ4-d. {d4,i} extending δ4, both of which are iterative and locally nilpotent. Let S2 ⊆ AQ,Γ2 (k) be the multiplicative set generated by x2. The derivation removing homomorphism induces an isomorphism Φ : k[y1][x1; τ2, δ2][y2; τ3][z 2 ; τ4] −→ A 2 (k)S whose action on generators is given by y1 7→ y1 x1 7→ x1 z2 7→ x2 y2 7→ y2 + (q2 − 1)−1 (q1 − 1)y1x1 + 1 x−12 . For simplicity, label the domain of Φ as BZ−1. Let X1 ⊆ BZ−1 be the Ore set generated by z2 and x1. Applying the derivation removing homomorphism to BZ −1 induces an isomorphism Ψ : k[y1][z 1 ; τ2][y2; τ3][z 2 ; τ 4] −→ (BZ−1)X−11 whose action on generators is given by z1 7→ z1 z2 7→ z2 y2 7→ y2 y1 7→ y1 + (q1 − 1)−1x−11 . The derivation removing homomorphism need not be employed again to achieve the result. Through iterated localization we find that there is an Ore set T ⊆ AQ,Γ2 (k) such 2 (k)T −1 ∼= k[y±11 ][x±11 ; τ2][y±12 ; τ3][x±12 ; τ4] and T is generated by the four elements x2, x1, y2x2(q2 − 1) + y1x1(q1 − 1) + 1, and y1x1(q1 − 1) + 1. Note that we recover the result of [22, Theorem 5]. 22 HEIDI HAYNAL 5. Examples We will demonstrate how each of the following k-algebras satisfies all the conditions of Theorem 2.8. Then Corollary 4.7 is applied to obtain an isomorphism of quotient division rings (thereby confirming the quantum Gel’fand-Kirillov conjecture) and PI degree parity with a multiparameter quantum affine space. When calculating the PI degree of a quantum affine space, we encounter an antisymmetric, or skew-symmetric, integral matrix. As proved in [30, Theorem IV.1], such a matrix is congruent to a matrix in skew normal form. Theorem 5.1. [Newman] Let A be a skew-symmetric matrix of rank r which belongs to Mn(R), where the commutative principal ideal domain R is not of characteristic 2. Then r = 2s and A is congruent to a matrix in block diagonal form  −h1 0 0 −h2 0 . . . 0 −hs 0  where hi | hi+1, 1 ≤ i ≤ s− 1. The same result, in the language of alternating bilinear forms, can be found in [3, Section 5.1]. The matrix S in Theorem 5.1 is clearly equivalent to the more familiar Smith normal form, diag(h1, h1, h2, h2, . . . , hs, hs, 0, 0, . . . , 0), where the diagonal entries are the in- variant factors of the matrix A. In the examples that follow, we outline the operations necessary to obtain the Smith normal form. Definition 5.2. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] and A′ = k[x1; τ1] · · · [xn; τn] be iterated skew polynomial rings. (1) If there exists Q = (q1, . . . , qn) ∈ (k×)n such that δiτi = qiτiδi for i = 1, . . . , n, then A is called an iterated Q-skew polynomial ring. (2) If there exist λji ∈ k× such that τj(xi) = λjixi for all i < j, then set λij = λ−1ji and λii = 1 for all i. We call Λ = (λij) ∈ Mn(k×) the matrix of relations for A′. Lemma 5.3. Let C be a commutative k-algebra, A a C-algebra, B ⊆ A a C-subalgebra generated by {b1, b2, . . . }. Let τ be a C-algebra automorphism of A, and δ a u-skew τ -derivation on A for some unit u ∈ C. If τ(bj) ∈ B and δn(bj) ∈ (n)!uB for all j, n, then δn(B) ⊆ (n)!uB for all n. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 23 Proof. Note that τ(bj) ∈ B for all j implies that τ(B) ⊆ B and hence we have (j)!uB ⊆ (j)!uB for all j. Suppose that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have δi(bj1 · · · bjℓ) ∈ (i)!uB for all i, and all choices of j1, . . . , jℓ. Then δn(bj1 · · · bjm) = τn−iδi(bj1 · · · bj(m−1))δ n−i(bjm) (i)!u(n− i)!uB ⊆ (n)!uB for all n and all j1, . . . , jm by induction. � For a first family of examples, we take odd-dimensional quantum Euclidean spaces. The even-dimensional ones will be covered in Example 5.4. 5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq(ok2n+1). For q ∈ k×, assuming q has a (fixed) square root q1/2 ∈ k, the k-algebra Oq(ok2n+1) may be presented as an iterated skew polynomial ring k[w][y1; σ1][x1; τ1, δ1] · · · [yn; σn][xn; τn, δn] with automorphisms σi, τi and derivations δi defined by σi(w) = q −1w all i τi(w) = qw all i σi(yj) = q −1yj j < i σi(xj) = q −1xj j < i τi(yj) = qyj i 6= j τi(xj) = qxj j < i τi(yi) = yi all i δi(w) = δi(xj) = δi(yj) = 0 j < i δi(yi) = (q 1/2 − q3/2)w2 + (1− q2) yℓxℓ all i. Quantum Euclidean spaces have been studied since 1990 when they were introduced by Reshetikhin et al. in [36]. The three-dimensional case has applications to the structure of space-time at small distances. Musson simplified the original set of relations in [29], and Oh further simplified them, renaming the generators ω, xi, yi in [31]. Here, we have made a change to Oh’s variables, yi 7→ qiyi, to obtain the relations in our presentation of Oq(ok2n+1). Routine computations show that τ−1i δiτi(yi) = q −2δi(yi) for all i, and so we conclude that each (τi, δi) is a q −2-skew derivation. We may present the analogous k[t±1]-algebra 24 HEIDI HAYNAL Ot(ok[t±1]2n+1) as an iterated skew polynomial ring with coefficient ring k[t±1] and generators w, yi, xi for i = 1, . . . , n, k[t±1][w][y1; σ̄1][x1; τ̄1, δ̄1] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of the algebra Oq(ok2n+1) with t ∈ k[t±1] replacing q ∈ k×. So each (τ̄i, δ̄i) is a t−2-skew derivation. It is immediate that Ot(ok[t±1]2n+1)/〈t− q〉 ∼= Oq(ok2n+1) with each τ̄i and δ̄i reducing to τi and δi respectively. Let Aj denote the k[t ±1]-subalgebra generated by w, ym, xm for m < j, and yj. To show that δ̄ij(Aj) ⊆ (i)!t−2Aj, we apply Lemma 5.3 noting that δ̄ij(yj) has been given for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of Oq(ok2n+1) extends to an iterative, locally nilpotent h.q−2-s.τi-d. on an appropriate subalgebra. Then Corollary 4.7 gives FractOq(ok2n+1) ∼= FractOB(k2n+1), where the matrix of relations is  1 q q−1 q q−1 · · · q q−1 q−1 1 1 q q−1 · · · q q−1 q 1 1 q q−1 · · · q q−1 q−1 q−1 q−1 1 1 · · · q q−1 q q q 1 1 · · · q q−1 . . . q−1 q−1 q−1 q−1 q−1 · · · 1 1 q q q q q · · · 1 1  If q ∈ k× is a root of unity, we may assume without loss of generality that it is a primitive rth root of unity. Then the powers of q from the matrix B become the entries of a (2n+ 1)× (2n+ 1) integer matrix  0 1 −1 1 −1 · · · 1 −1 −1 0 0 1 −1 · · · 1 −1 1 0 0 1 −1 · · · 1 −1 −1 −1 −1 0 0 · · · 1 −1 1 1 1 0 0 · · · 1 −1 . . . −1 −1 −1 −1 −1 · · · 0 0 1 1 1 1 1 · · · 0 0  Now, PIdegOq(ok2n+1) can be computed from Theorem 1.2(2) using the matrix B′. The cardinality of the image will not be changed if we first perform some row reductions on B′. Letting N = 2n+ 1, n > 2, we manipulate the rows as follows. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 25 • For i = 2, 4, 6, . . . , N − 1, replace row i with row i + row (i+ 1). • For i = N,N − 2, N − 4, . . . , 5, replace row i with row i − row (i− 2). • Replace row 5 with row 5 − row 1. • For i = 2, 4, 6, . . . , N − 5, replace row i with row i − 2row (i+ 5). • Multiply the even numbered rows, except row 2n− 2, by −1. The resulting matrix has 2n pivots and one zero row. We put the rows in this order 3, 1, 5, 7, 2, 9, 4, 11, 6, 13, . . . , 2i, 2i+ 7, . . . , N,N − 5, N − 3, N − 1 to place the pivots on the main diagonal and the zero row in the last position. Then we have a matrix of this form  1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 −1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 . . . ∗ ∗ ∗ ∗ 0 0 0 0 0 0 0 0 1 1 ∗ ∗ 0 0 0 0 0 0 0 0 0 4 ∗ ∗ 0 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0  The diagonal entries of this echelon matrix do not yet reveal the size of its image because the pivot in row three does not divide all of the (suppressed) entries in its row when n ≥ 3. So more row reduction is needed. First replace row 3 with row 3 + row(4i+ 2). For n even and j = 5, 7, 9, . . . , 2n− 3, replace row j as follows: for j = 4p+ 1, p ≥ 1, use row j + i=p+1 2 · row(4i) + row(2n); for j = 4p+ 3, p ≥ 1, use row j + i=p+1 2 · row(4i+ 2). 26 HEIDI HAYNAL For n odd and j = 5, 7, 9, . . . , 2n− 5, replace row j as follows: for j = 4p+ 1, p ≥ 1, use row j + i=p+1 2 · row(4i) + 2 · row(2n); for j = 4p+ 3, p ≥ 1, use row j + i=p+1 2 · row(4i+ 2) + row(2n). Then add row(2n) to row(2n − 3), and add 2·row(2n) to row(2n − 1). For integers 4 ≤ j ≤ 2n − 1, with j 6≡ 2(mod 4), add (−1)jcol 3 to col j. Subtract col(2n + 1) from col 3; add row 3 to row(2n − 2); and subtract 2·row 3 from row(2n). The result is an upper echelon matrix in which each pivot divides all the nonzero entries in its row. So it is trivial to diagonalize by column operations. The Smith normal form for n odd is diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0) with n+1 ones and n− 1 fours. The Smith normal form for n even is diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0) with n ones, two twos, and n − 2 fours. For the cases n = 1, 2, the row-reduced matrices are, respectively, 1 0 0 0 1 −1 0 0 0  1 0 0 1 −1 0 1 −1 1 −1 0 0 2 −2 2 0 0 0 2 −2 0 0 0 0 0  Hence we have, for all n > 0, PIdegOq(ok2n+1) = rn, r odd rn/2⌊ ⌋, r even, r /∈ 4Z rn/2n−1, r ∈ 4Z 5.2. The multiparameter quantized Weyl algebras; AQ,Γn (k). For a fixed n-tuple Q = (q1, . . . , qn) ∈ (k×)n and Γ = (γij) a multiplicatively antisymmetric n × n matrix over k, the algebra AQ,Γn (k), studied in [23] and [26], may be presented as an iterated skew polynomial ring k[y1][x1; τ1, δ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn] PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 27 where the automorphisms and derivations are defined by σi(yj) = γjiyj j < i σi(xj) = γijxj j < i τi(yj) = qjγjiyj j < i τi(xj) = q j γijxj j < i τi(yi) = qiyi all i δi(xj) = δi(yj) = 0 j < i δi(yi) = 1 + (qℓ − 1)yℓxℓ all i. Routine computations show that τ−1i δiτi(yi) = qiδi(yi) for all i, and so we conclude that each (τi, δi) is a qi-skew derivation. We may present the k[t 1 , . . . , t n ]-algebra AT,Γn (k[t 1 , . . . , t n ]) as an iterated skew polynomial ring k[t±11 , . . . , t n ][y1][x1; τ̄1, δ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of AQ,Γn (k) with ti ∈ k[t±11 , . . . , t±1n ] replacing qi ∈ k. So each (τ̄i, δ̄i) is a ti-skew derivation. It is immediate that AT,Γn (k[t 1 , . . . , t n ])/〈t1 − q1, . . . , tn − qn〉 ∼= AQ,Γn (k) with each τ̄i and δ̄i reducing to τi and δi respectively. Let Aj denote the k[t 1 , . . . , t n ]-subalgebra generated by ym, xm for m < j, and yj. To show that δ̄ij(Aj) ⊆ (i)!tjAj, it suffices to check δ̄ij(yj) by Lemma 5.3. But this is given by definition for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of AQ,Γn (k) extends to an iterative, locally nilpotent h,qi-s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives FractAQ,Γn (k) ∼= FractOΛ(k2n), where the 2n× 2n matrix of relations Λ is comprised of 2× 2 blocks Bii = 1 q−1i , for all i; Bij = γji q i γji γij qiγij , for i < j; Bij = γji γij qjγji q j γij , for i > j. If γij and qi are roots of unity for all i, j, then OΛ(k2n) is a PI algebra. Assuming that γij is an r ij root of unity and that qi is an r i root of unity, we let r = lcm{rij, ri | i, j = 1, . . . , n}. 28 HEIDI HAYNAL Then there exists a primitive rth root of unity q ∈ k and integers bi, bij such that qi = qbi and γij = q bij for i, j = 1, . . . , n. The powers of this q from the matrix Λ give a 2n× 2n integer matrix Λ′ comprised of 2× 2 blocks B′ii = 0 −bi , for all i; B′ij = bji bji − bi bij bij + bi , for i < j; B′ij = bji bij bj + bji bij − bj , for i > j. Then PIdegAQ.Γn (k) can be computed using the matrix Λ ′ in Theorem 1.2 (2). Consider the single parameter case, denoted Aqn(k), where qi = q for all i, and γij = 1 for i < j, relegating the σi to identity maps. Assuming that q is a primitive r th root of unity, then δi(y i ) = 0 and τi(y i ) = y i for all i, implying that y i is central. The definition of the τi, along with the q-Liebnitz rule, implies that x i is central for all i. So the algebra Aqn(k) is a finitely generated module over the central subring k[y i , x 1, . . . , y n]. To find the PI degree in this case, the integer matrix becomes  0 −1 0 −1 . . . 0 −1 1 0 0 1 0 1 0 0 0 −1 0 −1 1 −1 1 0 0 1 . . . 0 0 0 0 . . . 0 −1 1 −1 1 −1 . . . 1 0  which is seen to have a trivial kernel after these row reductions: • Replace row 2n with row 2n− row (2n− 2)− row (2n− 3) • For j = n−1, n−2, . . . , 2, replace row 2j with row 2j−row (2j−2)−row (2j−3) • Rearrange the rows to order 2, 1, 4, 3, 6, 5 . . . , 2n, 2n− 1. PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 29 The resulting matrix has the form  0 −1 ∗ . . . 0 1 0  thus verifying that PIdegAqn(k) = r 5.3. The multiparameter coordinate ring of quantum n×nmatrices; Oλ,p Mn(k) The multiparameter coordinate ring of quantum n×n matrices was introduced by Artin, Schelter, and Tate in [2]. The k-algebra Oλ,p Mn(k) is defined by generators xij for i, j = 1, . . . , n and relations xℓmxij = pℓipjmxijxℓm + (λ− 1)pℓiximxlj (ℓ > i, m > j) λpℓipjmxijxℓm (ℓ > i, m ≤ j) pjmxijxℓm (ℓ = i, m > j), where λ ∈ k× and p = (pij) ∈ Mn2(k×) is multiplicatively antisymmetric. It can also be presented as an iterated skew polynomial ring k[x11][x12; τ12] · · · [xij ; τij, δij ] · · · [xnn; τnn, δnn] where each τℓm and δℓm is k-linear and satisfies τℓm(xij) = pℓipjmxij when ℓ > i and m 6= j λpℓipjmxij when ℓ > i and m = j pjmxij when ℓ = i and m > j, δℓm(xij) = (λ− 1)pℓiximxℓj when ℓ > i and m > j 0 otherwise. Routine computations show τ−1ℓm δℓmτℓm(xij) = λ −1δℓm(xij) as in [9, Section 5], and so we conclude that each (τℓm, δℓm) is a λ −1-skew derivation. We may present the k[t±1]- algebra Ot,p Mn(k[t as an iterated skew polynomial ring with generators xij for i, j = 1, . . . , n k[t±1][x11][x12, τ̄12] · · · [xij ; τ̄ij, δ̄ij ] · · · [xnn; τ̄nn, δ̄nn] 30 HEIDI HAYNAL where the automorphisms and derivations are defined analogously to those of the algebra Mn(k) with t ∈ k[t±1] replacing λ ∈ k. So each (τ̄ℓm, δ̄ℓm) is a t−1-skew derivation. It is immediate that Mn(k[t /〈t− λ〉 ∼= Oλ,p Mn(k) with each τ̄ℓm and δ̄ℓm reducing to τℓm and δℓm respectively. Let A−ℓm denote the k[t ±1]-subalgebra generated by the xij with (i, j) < (ℓ,m) in the lexicographic order. Lemma 5.3 allows us to to verify that δ̄sℓm(A ℓm) ⊆ (s)!t−1(A−ℓm) by checking only that δ̄sℓm(xij) is contained in A ℓm. This is immediate from the formula for δ̄ℓm given above. Thus, by Theorem 2.8, each δℓm in our presentation of Oλ,p Mn(k) extends to an iterative, locally nilpotent h.λ−1-s.τℓm-d. on the appropriate k-subalgebra. Then Corollary 4.7 gives FractOλ,p Mn(k) ) ∼= FractOΛ(kn where the matrix of relations Λ = (bij) ∈ Mn2(k) is comprised of n× n blocks Bii =  1 p21 p31 · · · pn1 p12 1 p32 · · · pn2 p13 p23 1 · · · pn3 . . . p1n p2n p3n · · · 1  for all i, Bij =  λ−1pij pijp21 pijp31 · · · pijpn1 λ−1pijp12 λ −1pij pijp32 · · · pijpn2 λ−1pijp13 λ −1pijp23 λ −1pij · · · pijpn3 . . . λ−1pijp1n λ −1pijp2n λ −1pijp3n · · · λ−1pij  , for i < j, Bij =  λpij λpijp21 λpijp31 · · · λpijpn1 pijp12 λpij λpijp32 · · · λpijpn2 pijp13 pijp23 λpij · · · λpijpn3 . . . pijp1n pijp2n pijp3n · · · λpij  , for i > j. If λ and pij are roots of unity for all i, j, then OΛ(kn ) is a PI algebra. In this case we may assume that λ is an sth root of unity and that pij is an r ij root of unity, and let r = lcm{s, rij | i, j = 1, . . . , n}. Then there exists a primitive rth root of unity q ∈ k and integers b, bij such that λ = q b and pij = q bij . The powers of this q from the matrix PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 31 Λ provide entries for an n2 × n2 integer matrix Λ′ made up of n× n blocks B′ii =  0 b21 b31 · · · bn1 b12 0 b32 · · · bn2 b13 b23 0 · · · bn3 . . . b1n b2n b3n · · · 0  for all i, B′ij =  bij − b bij + b21 bij + b31 · · · bij + bn1 bij + b12 − b bij − b bij + b32 · · · bij + bn2 bij + b13 − b bij + b23 − b bij − b · · · bij + bn3 . . . bij + b1n − b bij + b2n − b bij + b3n − b · · · bij − b  , for i < j, B′ij =  bij + b bij + b21 + b bij + b31 + b · · · bij + bn1 + b bij + b12 bij + b bij + b32 + b · · · bij + bn2 + b bij + b13 bij + b23 bij + b · · · bij + bn3 + b . . . bij + b1n bij + b2n bij + b3n · · · bij + b  , for i > j. Then PIdegOλ,p Mn(k) can be calculated using Λ′ in Theorem 1.2 (2). The single parameter quantized coordinate ring of n × n matrices, Oq(Mn(k)), is de- fined over k analogously to Oλ,p(Mn(k)), but with relations that are recovered by setting λ = q−2 and pij = q for all i > j. When k has characteristic zero and q is a primitive m root of unity for m odd, Jakobsen and Zhang found in [20] that PIdegOq(Mn(k)) = m n(n−1) 2 by using De Concini’s and Procesi’s tool given in Theo- rem 1.2. This result is reproved in [19] using results of De Concini and Procesi and also Jøndrup’s work from [21]. Now we can recover PIdegOq(Mn(k) without the assumption that k has characteristic zero. In the single parameter case of n × n quantum matrices, the matrix that we use to calculate the PI degree is  An In In In · · · In −In An In In · · · In −In −In An In · · · In −In −In −In −In · · · An  32 HEIDI HAYNAL where  0 1 1 1 · · · 1 −1 0 1 1 · · · 1 −1 −1 0 1 · · · 1 −1 −1 −1 · · · −1 0  is n× n and In is the n× n identity matrix. For any n, the characteristic polynomial of An is the sum of the terms of degree ≡ n (mod 2) in the binomial expansion of (x+1)n, so in fact χn(x) = (x+1)n+ 1 (x− 1)n. But there is also a recursion formula for the characteristic polynomial for n ≥ 3 given χn(x) = χn−1(x)(x+ 1)− (x− 1)n−1, which will be useful in the linear algebra that follows. We will perform the following row reductions on the rows of blocks of Λ′. For ease of notation, we’ll denote the jth row of blocks as BRj , the interchange of BRi and BRj as BRi ↔ BRj , and the addition of a multiple of BRi to BRj as MBRi + BRj 7→ BRj , where M ∈ Mn(Z). • BR1 ↔ BRn. • −InBR1 7→ BR1. • For i = 2, . . . , n− 1, BR1 +BRi 7→ BRi. • BRn − AnBR1 7→ BRn. This yields the matrix  In In In In · · · −An 0 An + In 2In 2In · · · In −An 0 0 An + In 2In · · · In −An . . . 0 0 0 · · · An + In In −An 0 In −An In −An In −An · · · In + A2n  which can be reduced further by n − 2 block row operations, each of which produces one zero block in the nth row. We list the first three here along with the resulting (n, n) block. • (An + In)BRn − (In − An)BR2 7→ BRn : A3n + 3An • (An + In)BRn + (In −An)2BR3 7→ BRn : A4n + 6A2n + In • (An + In)BRn − (In − An)3BR4 7→ BRn : A5n + 10A3n + 5An PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 33 In general, the block row operations that we need to perform in order to obtain a block upper triangular matrix are: • For i = 2, . . . , n− 1, (An + In)BRn + (−1)i−1(In − An)i−1BRi 7→ BRn. These row operations are justified when m is odd because An + In is invertible in Mn(Z/mZ) in that case, as will be shown below. After applying this step to the i th row, the (n, n) block is χi+1(An). So the resulting block upper triangular matrix is  In In In In · · · −An 0 An + In 2In 2In · · · In −An 0 0 An + In 2In · · · In −An . . . 0 0 0 · · · An + In In −An 0 0 0 0 · · · χn(An)  where χn(An) is the n× n zero matrix. Each block on the diagonal is An + In =  1 1 1 1 · · · 1 −1 1 1 1 · · · 1 −1 −1 1 1 · · · 1 −1 −1 −1 −1 · · · 1  which can be row reduced just by adding row 1 to rows 2 through n to yield the matrix   1 1 1 1 · · · 1 0 2 2 2 · · · 2 0 0 2 2 · · · 2 . . . 0 0 0 0 · · · 2  In particular this shows that An + In is invertible in Mn(Z/mZ) for m odd. Hence Λ′ can be reduced through row operations to an upper triangular n2 × n2 matrix with 2n− 2 ones, (n− 1)(n− 2) twos, and n zeroes on the diagonal. Assuming that q ∈ k is a primitive mth root of unity, and recalling Theorem 1.2, the cardinality of the image in (Z/mZ)n is mn 2−n if m is odd. Thus we conclude that PIdegOqMn(k) = m n(n−1) recovering the result of Jakobsen and Zhang [20] in characteristic zero. By similar methods, one can show that PIdegOqMn(k) = m n(n−1) (n−1)(n−2) 2 when m is even. For details on this result see [20] or [17]. 5.4. The algebra K n,Γ (k), which generalizes the coordinate rings of even- dimensional quantum Euclidean space and quantum symplectic space. For P = (p1, . . . , pn) and Q = (q1, . . . , qn) in (k ×)n with pi 6= qi for all i = 1, . . . , n, and 34 HEIDI HAYNAL Γ = (γij) ∈ Mn(k×) multiplicatively antisymmetric, the k-algebra KP,Qn,Γ (k) introduced in [18] is defined by generators xi, yi for i = 1, . . . , n and relations yiyj = γijyjyi all i, j xixj = qip j γijxjxi i < j xiyj = pjγjiyjxi i < j xiyj = qjγjiyjxi i > j xiyi = qiyixi + (qℓ − pℓ)yℓxℓ all i. This algebra may be presented in the form of an iterated skew polynomial ring k[y1][x1; τ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn] where the automorphisms τi, σi and derivations δi are defined by σi(yj) = γijyj j < i σi(xj) = p i γjixj j < i τi(yj) = qjγjiyj j < i τi(xj) = q j piγijxj j < i τi(yi) = qiyi all i δi(xj) = δi(yj) = 0 j < i δi(yi) = (qℓ − pℓ)yℓxℓ all i. Routine computations show that τ−1i δiτi(yi) = qip i δi(yi) for all i, and so we conclude that each (τi, δi) is a qip i -skew derivation. For ease of notation we now shall let k = k[t±11 , . . . , t n , u 1 , . . . , u n ] with T = (t1, . . . , tn) ∈ k and U = (u1, . . . , un) ∈ k. We may present the k-algebra K n,Γ (k) as an iterated skew polynomial ring k[y1][x1; τ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n] where the automorphisms and derivations are defined analogously to those of K n.Γ (k) with ti replacing pi and ui replacing qi. Let I ⊆ KT,Un,Γ (k) be the ideal generated by the 2n monomials ti − pi, ui − qi for i = 1, . . . , n. It is immediate that n,Γ (k)/I ∼= KP,Qn,Γ (k), with each τ̄i, δ̄i, σ̄i reducing to τi, δi, σi respectively. Let Aj denote the subalgebra of K n,Γ (k) generated by ym, xm for m < j and yj. To show that δ̄ij(Aj) ⊆ (i)!ujt−1j Aj , it suffices to check that δ̄ j(yj) is an element of (i)!ujt−1j PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 35 by Lemma 5.3. This is given for i = 1 by the formula for δ̄j and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of K n,Γ (k) extends to an iterative, locally nilpotent h.qip i -s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives FractK n,Γ (k) ∼= FractOΛ(k2n) where the 2n× 2n matrix of relations Λ = (Bij) is comprised of 2× 2 blocks Bii = 1 q−1i , for all i; Bij = γij q i γji pjγji qip j γij , for i < j; Bij = γij p i γij qjγji q j piγij , for i > j. If the qi, pi and γi are all roots of unity, then OΛ(k2n) is a PI algebra. Suppose qi is an rthi root of unity, pi is an s i root of unity, and γij is an r ij root of unity for all i, j. Let r = lcm{ri, si, γij | i, j = 1, . . . , n}. Then there extsis a primitive rth root of unity q ∈ k and integers bi, ci, bij such that qi = q bi , pi = q ci, and γij = q bij for all i, j. The powers of q from the matrix Λ provide the entries for an integer matrix Λ′ comprised of 2 × 2 blocks B′ii = 0 −bi , for all i; B′ij = bij bji − bi bji + cj bi + bij − cj , for i < j; B′ij = bij bij − ci bji + bj bij + ci − bj , for i > j. Then PIdegK n,Γ (k) can be calculated using Λ ′ in Theorem 1.2 (2). The coordinate ring of quantum Euclidean 2n-space over k, Oq(ok2n), is formed by setting qi = 1, pi = q −2 for all i, and γij = q −1 for i < j in the parameters Q, P , and Γ 36 HEIDI HAYNAL (see [18], Example 2.6). Then the integer matrix, Λ′, is  0 0 −1 1 −1 1 . . . −1 1 0 0 −1 1 −1 1 . . . −1 1 1 1 0 0 −1 1 −1 1 −1 −1 0 0 −1 1 ... 1 1 1 1 0 0 −1 −1 −1 −1 0 0 . . . 1 1 1 1 1 . . . 0 0 −1 −1 −1 −1 −1 . . . 0 0  We perform the following row reductions that preserve the size of the image of the homomorphism Z2n −→ Z2n given by Λ′: • For j = 2n, 2n− 1, 2n− 2, . . . , 4, replace row j with row j + row (j − 1) • Replace row 2 with row 2− row 1 • Replace the (new) row 5 with row 5 + row 1 • For j = 4, 6, 8, . . . , 2n− 4, replace row j with row j + 2row (j + 3) • For n ≥ 4, rearrange the rows to order 3, 1, 5, 7, 4, 9, 6, 11, . . . , 2i, 2i+ 5, . . . , 2n− 4, 2n− 2, 2, 2n. The resulting matrix has the form  0 −1 1 0 2 ∗ 0 1 1 . . . 0 1 1 0 0 4 0 −2 2  When n is even, the pivot in the third row does not divide all the entries in its row, so more elementary row and column operations are needed before it becomes clear that the matrix can be diagonalized. By a method similar to that used in Example 5.1, suppressed here in the interest of saving space but listed explicitly in [17], we obtain the Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0, 0) with n ones and n− 2 fours when n PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 37 is even; and diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0, 0), with n− 1 ones and n− 3 fours when n is odd. Thus we have PIdegOq(ok2n) = rn−1, r odd rn−1/2⌊ ⌋, r even /∈ 4Z rn−1/2n−2, r ∈ 4Z . (10) The low-dimensional cases do not fit the same pattern, but the matrices for the cases n = 2 and n = 3 are readily transformed to 1 1 0 0 0 0 −1 1 0 0 0 0 0 0 0 0  and  1 1 0 0 −1 1 0 0 −1 1 −1 1 0 0 0 2 0 0 0 0 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0 0  respectively. Therefore, formula (10) holds for all n ≥ 2. As a specific case ofK n,Γ (k), quantum symplectic spaceOq(sp(k2n)) is formed by setting qi = q −2 and pi = 1 for all i, and γij = q for i < j (see [18], Example 2.4). With these parameters, the 2n× 2n integer matrix Λ′ is  0 2 1 1 1 1 . . . 1 1 −2 0 −1 −1 −1 −1 . . . −1 −1 −1 1 0 2 1 1 1 1 −1 1 −2 0 −1 −1 −1 −1 −1 1 −1 1 0 2 ... −1 1 −1 1 −2 0 . . . −1 1 −1 1 −1 1 . . . 0 2 −1 1 −1 1 −1 1 . . . −2 0  We perform the following row reductions that preserve the size of the image of the homomorphism Z2n −→ Z2n given by Λ′: • For j = 2n, 2n− 1, . . . , 4, replace row j with row j − row (j − 1) • Replace row 2 with −(row 2− 2row 3 + row 1) • For j = 4, 6, 8, . . . , 2n− 2, replace row j with row j + 2row (j + 1) • For n ≥ 3, order the rows 3, 1, 5, 2, 7, 4, 9, . . . , 2j, 2j + 5, . . . , 2n− 4, 2n, 2n− 2. 38 HEIDI HAYNAL This yields a matrix whose image is more easily measured:  0 1 1 ∗ . . . 0 0 4 −2 −2  But the pivot in row 2 is problematic because it does not always divide the other entries in its row. With further elementary row and column operations, full details of which can be found in [17], we can bring this matrix into Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4) with n ones and n fours when n is even; or the form diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4) with n − 1 ones, two twos, and n − 1 fours when n is odd. For n = 1, 2, the row reduced matrices are, respectively, −1 1 0 2 0 1 1 1 0 0 −4 −4 0 0 0 −4  . Hence we have, for all n, PIdegOq(sp(k2n)) = rn, r odd rn/2⌊ ⌋, r even, r /∈ 4Z rn/2n, r ∈ 4Z 6. Prime Factor Localizations In this section we present a structure theorem for completely prime factors of iterated skew polynomial rings analogous to the main theorem of section four. Applying this result to the algebras studied in section five, we’d like to strengthen it to the form of the quantum Gel’fand-Kirillov conjecture. Recall that the assumptions about skew polynomial rings from section one are still in effect. Theorem 6.1. Let A = R[x; τ, δ], where R is noetherian and δτ = qτδ for some q ∈ k×. Assume that δ extends to a locally nilpotent, iterative h.q-s.τ -d., {di}, on R. Let P ∈ specA be completely prime. Then PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 39 (1) there exists a cyclic Ore set S in A/P such that (A/P )S−1 ∼= R[y; τ ]/Q Y −1 for some completely prime Q ∈ specR[y; τ ] and cyclic Ore set Y , (2) FractA/P ∼= FractR[y; τ ]/Q. Proof. The completely prime ideal P naturally satisfies one of two cases: x ∈ P or x /∈ P . If x ∈ P , then xA ⊆ P and Ax ⊆ P . So the relation xr = τ(r)x + δ(r) implies that δ(r) ∈ P for all r ∈ R. Hence, there is a completely prime ideal I ∈ R such that A/P ∼= R/I ∼= R[y; τ ]/(I + 〈y〉). In this case, we can take S = Y = {1} and localize. If x /∈ P , then xi /∈ P for all i ∈ N ∪ {0} because A/P is a domain. Letting S = {1, x, x2, . . . }, which is a known denominator set in A, we have P ∩ S = ∅. Since extension and contraction provide inverse bijections between the sets specAS−1 and {I ∈ specA | I ∩ S = ∅}, we know that P e ∈ specAS−1. From Theorem 3.7, we have AS−1 ∼= R[y±1; τ ], a localization of R[y; τ ]. So there is a completely prime ideal Q̄⊳R[y±1; τ ] such that AS−1/P e ∼= R[y±1; τ ]/Q̄. Setting Y = {1, y, y2, . . . , }, contraction to R[y; τ ] gives a completely prime ideal Q, where Q∩ Y = ∅, such that R[y±1; τ ]/Q̄ is isomorphic to (R[y; τ ]/Q)Y −1. The canonical projection π : AS−1 −→ (A/P )S−1 gives AS−1/P e ∼= (A/P )S−1. Thus (A/P )S−1 ∼= R[y; τ ]/Q Y −1. � Theorem 6.2. Let R be a noetherian k-algebra, and let A = R[x1, τ1, δ1] · · · [xn; τn, δn] be an iterated skew polynomial ring where, for j < i and λij ∈ k×, τi(xj) = λijxj, and δi is a qi-skew τi-derivation, qi 6= 1, which extends to a locally nilpotent, iterative h.qi-s.τi-d. {di,p}∞p=0 on R[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1] for all i. Let A′ = R[y1; τ ′1][y2; τ ′2] · · · [yn; τ ′n] where τ ′i(yj) = λijyj for all i with j < i and the same units λij as above. Let P be a completely prime ideal in A. Then (1) there exists a finitely generated Ore set Sn in A/P such that (A/P )S n is isomorphic Y −1n for some completely prime ideal Q ⊆ A′ and finitely generated Ore set (2) FractA/P ∼= FractA′/Q. Proof. The case n = 1 has been established in Theorem 6.1. Suppose the result holds for the case n− 1, and let An−1 = R[x1, τ1, δ1] · · · [xn−1; τn−1, δn−1] ⊆ A. Then we have A = An−1[xn; τn, δn]. If xn ∈ P , then as in Theorem 6.1 there is a completely prime ideal I ⊆ An−1 such that A/P ∼= An−1/I ∼= An−1[yn; τ ′n]/(I + 〈yn〉). The induction hypothesis and Lemma 4.2 imply that An−1[yn; τ n]/(I + 〈yn〉) S−1 ∼= Y −1 for some finitely generated Ore sets S and Y . Hence there is a finitely generated Ore set Sn in A such that (A/P )S ∼= (A′/Q)Y −1. If xn /∈ P , let Sn = {1, xn, x2n, . . . } ⊆ A and Yn = {1, yn, y2n, . . . } ⊆ An−1[yn; τn]. Then from the single-variable result, it follows that ( An−1[yn; τ n]/Q̄ Y −1n , (11) 40 HEIDI HAYNAL for a completely prime ideal Q̄ ⊆ An−1[yn; τ ′n]. From Lemma 4.2, we have An−1[yn; τ n] = R[yn; τ n][x1; τ 1] · · · [xn−1; τ ′n−1, δ′n−1], which is an iterated skew polynomial ring in n − 1 variables over the coefficient ring R[yn; τ n] that satisfies the current assumptions. So, we apply the induction hypothesis and rearrange variables to obtain An−1[yn; τn]/Q̄ Y −1n R[yn; τ n][y1; τ 1] · · · [yn−1; τ ′n−1]/Q R[y1; τ 1][y2; τ 2] · · · [yn; τ ′n]/Q for a completely prime ideal Q ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n] and a denominator set Z ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n]/Q. This, along with isomorphism (11) gives the re- sult. � When R is replaced by k, we have the following result. Corollary 6.3. Let A = k[x1, τ1, δ1] · · · [xn; τn, δn], where τi(xj) = λijxj and δiτi = qiτiδi, qi 6= 1, for λij , qi ∈ k× and all i with j < i. Assume that each δi extends to a locally nilpotent, iterative h.qi-s.τi-d. {di,m}∞m=0 on the subalgebra k[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1]. Let P be a completely prime ideal in A and set λii = 1 and λji = λ ij . Then for λ = (λij) ∈ Mn(k), and an appropriate completely prime ideal Q ⊆ Oλ(kn), we have FractA/P ∼= FractOλ(kn)/Q. We summarize how this applies to the k-algebras of quantized coordinate type. Corollary 6.4. Let A be any of the examples discussed in sections 5.1 - 5.4, and let P be a completely prime ideal of A. Then there exist a positive integer N , a multiplicatively antisymmetric N ×N matrix λ over k, and a completely prime ideal Q ∈ Oλ(kN) such that FractA/P ∼= FractOλ(kN)/Q. To complete the question posed by the corollary, one might ask how far the quantum Gel’fand-Kirillov conjecture extends to prime factor algebras. For instance: Question 6.5. Find conditions under which we can conclude that for any positive in- teger n, multiplicatively antisymmetric matrix λ ∈ Mn(k×), and completely prime ideal Q ∈ specOλ(kn), we have FractOλ(kn)/Q ∼= FractOp(Km) for some field extension K ⊇ k, integer m ≤ n, and m×m matrix p over K. The case n = 1 is trivial. When n = 2 and Q contains x1 or x2, then FractOλ(k2)/Q is isomorphic either to FractOp(k(y)) where p = (1), or to k itself. In fact, for any n, if Q is generated by a subset S of {x1, . . . , xn}, then the result holds, with p the submatrix of λ formed by deleting the ith row and column for xi ∈ S, and K = k. When xi /∈ Q PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 41 for all i, answering the question fully will likely require different methods depending on the presence of roots of unity among the λij. A positive answer in the generic case has been provided in the proof of [13, Theorem 2.1]: Theorem 6.6. [Goodearl - Letzter] Let k be a field, λ = (λij) a multiplicatively anti- symmetric n× n matrix over k×, and Λ the subgroup of k× generated by the λij. If Λ is torsionfree, then all of the prime ideals Q of Oλ(kn) are completely prime. In their proof, they showed that FractOλ(kn)/Q ∼= FractOp(Km), and identified K as the quotient field of a commutative domain embedded in the center of Oλ((k×)n)/Q′, where Q′ is the prime ideal in Oλ((k×)n) induced by localization. Quantum affine space is included in a class called quantum solvable algebras by A. N. Panov. The main theorem of [34, Section 3], states that when the group generated by the λij is torsionfree, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quan- tum torus. The main theorem of [35, Section 3], allows roots of unity and states that when Q satisfies the extra condition of being stable under a certain set of derivations, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quantum torus. Cauchon’s work may also be specialized to apply to quantum affine space when the group generated by the λij is torsionfree. The result of [5, Theorem 6.1.1], indicates that FractOλ(kn)/Q is isomorphic to FractOp(Km) which specializes to this result. But the division ring of real quaternions provides an example showing that Question 6.5 needs to have some conditions imposed. Note that H ∼= Oλ(R3)/Q, where λ = 1 −1 −1 −1 1 −1 −1 −1 1  , andQ = 〈x21 + 1, x22 + 1, x23 + 1〉. Therefore, we cannot obtain the desired isomorphism of quotient division rings in this case, illustrating the necessity of an extra condition such as the one imposed by Panov in [35]. acknowledgments The author thanks her dissertation advisor, Ken Goodearl, for his direction that was so freely given in many inspiring discussions. References [1] J. Alev and F. Dumas, Sur le corps de fractions de certaines algèbres quantiques, J. Algebra 170 (1994), 229-265 [2] M. Artin, W. Schelter, and J. 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Smith, Quantum groups: An introduction and survey for ring theorists, in Noncommutative Rings (S. Montgomery and L. W. Small, eds.), pp131-178, MSRI Publ. 24, Springer-Verlag, Berlin (1992) [39] R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, Department of Mathematics, University of California, Santa Barbara, California 93106 E-mail address : heidi@softerhardware.com 1. Introduction 2. Higher q-Skew -Derivations 3. The -Derivation Removing Homomorphism 4. Main Theorem 5. Examples 5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq (o k2n+1) 5.2. The multiparameter quantized Weyl algebras; AnQ, (k) 5.3. The multiparameter coordinate ring of quantum n n matrices; O, bold0mu mumu pppppp(to.Mn(k))to. 5.4. The algebra Kn, P, Q (k), which generalizes the coordinate rings of even-dimensional quantum Euclidean space and quantum symplectic space 6. Prime Factor Localizations acknowledgments References ABSTRACT For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;\tau_1,\delta_1]...b[x_n;\tau_n,\delta_n] agrees with the PI degree of R[x_1;\tau_1]...b[x_n;\tau_n] when each (\tau_i,\delta_i) satisfies a q_i-skew relation for q_i \in k^{\times} and extends to a higher q_i-skew \tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras. <|endoftext|><|startoftext|> Semi-spheroidal Quantum Harmonic Oscillator D. N. Poenaru,1, 2, ∗ R. A. Gherghescu,1, 2 A. V. Solov’yov,1 and W. Greiner1 1Frankfurt Institute for Advanced Studies, J. W. Goethe Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2 Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), P.O. Box MG-6, RO-077125 Bucharest-Magurele, Romania (Dated: November 15, 2018) A new single-particle shell model is derived by solving the Schrödinger equation for a semi- spheroidal potential well. Only the negative parity states of the Z(z) component of the wave function are allowed, so that new magic numbers are obtained for oblate semi-spheroids, semi-sphere and prolate semi-spheroids. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... The superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator: 2, 8, 20, 40, 70, 112, 168 ... PACS numbers: 03.65.Ge, 21.10.Pc, 31.10.+z, The spheroidal harmonic oscillator have been used in various branches of Physics. Of particular interest was the famous single-particle Nilsson model [1] very success- ful in Nuclear Physics and its variants [2, 3, 4] for atomic clusters. Major spherical-shells N = 2, 8, 20, 40, 58, 92 have been found [2] in the mass spectra of sodium clus- ters of N atoms per cluster, and the Clemenger’s shell model [3] was able to explain this sequence of spherical magic numbers. In the present paper we would like to write explicitly the analytical relationships for the energy levels of the spheroidal harmonic oscillator and to derive the corre- sponding solutions for a semi-spheroidal harmonic oscil- lator which may be useful to study atomic cluster de- posited on planar surfaces. For spheroidal equipotential surfaces, generated by a potential with cylindrical symmetry the states of the va- lence electrons were found [3] by using an effective single- particle Hamiltonian with a potential Mω20R 2 + δ 2 + δ In order to get analytical solutions we shall neglect an additional term proportional to (l2 − 〈l2〉n). We plan to include in the future such a term which needs a numerical solution. K. L. Clemenger introduced the deformation δ by expressing the dimensionless two semiaxes (in units of the radius of a sphere with the same volume, R0 = 1/3, where rs is the Wigner-Seitz radius, 2.117 Å for Na [5, 6]) as 2 + δ ; c = 2 + δ The spheroid surface equation in dimensionless cylindri- cal coordinates ρ and z is given by = 1 (3) where a is the minor (major) semiaxis for prolate (oblate) spheroid and c is the major (minor) semiaxis for prolate (oblate) spheroid. Volume conservation leads to a2c = 1. One can separate the variables in the Schrödinger equation, HΨ = EΨ, written in cylindrical coordinates. As a result the wave function [7, 8] may be written as Ψ(η, ξ, ϕ) = ψmnr (η)Φm(ϕ)Znz (ξ) (4) where each component of the wave function is ortonor- malized leading to Φm(ϕ) = e 2π (5) ψ(η) = Nmnrη |m|/2e−η/2L nr (η) Nmnr = α⊥(nr+|m|)! in which η = R20ρ 2/α2⊥ and the quantum numbers m = (n⊥−2i) with i = 0, 1, ... up to (n⊥−1)/2 for an odd n⊥ or to (n⊥− 2)/2 for an even n⊥. Lmn (x) is the associated Laguerre polynomial and the constant α⊥ = h̄/Mω⊥ has the dimension of a length. Znz(ξ) = Nnze −ξ2Hnz (ξ) Nnz = π2nznz!) 1/2 (7) where ξ = R0z/αz, αz = h̄/Mωz, and the main quan- tum number n = n⊥ + nz = 0, 1, 2, .... The eigenvalues are En = h̄ω⊥(n⊥ + 1) + h̄ωz(nz + 1/2) (8) The parity of the Hermite polynomials Hnz (ξ) is given by (−1)nz meaning that the even order Hermite poly- nomials are even functions H2nz(−ξ) = H2nz (ξ) and the odd order Hermite polynomials are odd functions H2nz+1(−ξ) = −H2nz+1(ξ). There is a recurrence rela- tionship 2zHn = Hn+1+2nHn−1. One hasH0 = 1, H1 = http://arxiv.org/abs/0704.0847v1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 (spheroidal deformation) -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 (spheroidal deformation) FIG. 1: LEFT: Spheroidal harmonic oscillator energy levels in units of h̄ω0 vs. the deformation parameter δ. Only 6 major shells (N = 0, 1, 2, ..., 5) have been considered. Each level is labeled by n, n⊥ quantum numbers and is (2n⊥+2)-fold degenerate. The labels are 0, 0; 1, 0, 1, 1; 2, 0, 2, 1, 2, 2; 3, 0, 3, 1, 3, 2, 3, 3; 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, etc. RIGHT: Semi-spheroidal harmonic oscillator energy levels in units of h̄ω0 vs. the deformation coordinate δ. Only 9 major shells (N = 0, 1, 2, ..., 8) have been considered. Each level is labeled by n, n⊥ quantum numbers (with nz = n− n⊥ = 1, 3, 5, ... and is (2n⊥ + 2)-fold degenerate. The labels are 1, 0; 2, 1; 3, 2, 3, 0; 4, 3, 4, 1; 5, 4, 5, 2, 5, 0; 6, 5, 6, 3, 6, 1, etc. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape (δ = −2/3): 2, 6, 14, 26, 44, 68, 100, 140, ... 2z, H2 = 4z 2−2, H3 = 8z3−12z, H4 = 16z4−48z2+12, H5 = 32z 5 − 160z3 + 120z, etc. In units of h̄ω0 the eigenvalues, ǫ = E/(h̄ω0), are given (2− δ)1/3(2 + δ)2/3 For a prolate spheroid, δ > 0, at n⊥ = 0 the energy level decreases with deformation except for n = 0, but when n⊥ = n it increases. For a given prolate deformation and z0=0.5 z1=1.5 z2=2.5 z3=3.5 z4=4.5 z5=5.5 FIG. 2: LEFT: Harmonic oscillator potential V = V (ξ), the wave functions Znz = Znz (ξ) for nz = 0, 1, 2, 3, 4, 5 and the corresponding contributions to the total energy levels ǫz nz = Enz/h̄ωz = (nz + 1/2) for spherical shapes, δ = 0. ξ = h̄/Mωz. RIGHT: The similar functions for a semi- spherical harmonic oscillator potential. Only negative parity states are retained which are vanishing at ξ = 0 where the potential wall is infinitely high . a maximum energy ǫm, there are nmin closed shells and other levels for high-order shells up to nmax: nmin = (2− δ)1/3(2 + δ)2/3ǫm − 2 + δ nmax = (2− δ)1/3(2 + δ)2/3ǫm − 2− δ (11) and similar formulae for oblate deformations, δ < 0. The low lying energy levels for the six shells (main quantum number n = 0, 1, 2, 3, 4, 5) can be seen in figure 1. Each level, labelled by n⊥, n, may accomodate 2n⊥ + 2 parti- cles. One has 2 (n⊥+1) = (n+1)(n+2) nucleons in a completely filled shell charcterized by n, and the total number of states of the low-lying n + 1 shells is n=0(n+1)(n+2) = (n+1)(n+2)(n+3)/3 leading to the magic numbers 2, 8, 20, 40, 70, 112, 168... for a spheri- cal shape. Besides the important degeneracy at a spher- ical shape (δ = 0), one also have degeneracies at some superdeformed shapes, e.g. for prolate shapes at the ra- tio c/a = (2 + δ)/(2 − δ) = 2 i.e. δ = 2/3. More details may be found in the Table I. The first five shells can reproduce the experimental magic numbers mentioned above; in order to describe the other shells Clemenger introduced the term proportional to (l2 − 〈l2〉n). Let us consider a particular shape (half of an oblate or prolate spheroid) of a semi-spheroidal cluster deposited on a surface with the z axis perpendicular on the sur- face and the ρ axis in the surface plane. Then the semi- 20 40 60 80 100 120 140 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 = - 1 = - 2/3 = - 0.4 = - 0.8/3 = 2/3 = 0.4 = 0.8/3 FIG. 3: Variation of shell corrections with N for Na clusters. TOP: δ = 0. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... For a prolate superdeformed (δ = 2/3) shape the magic numbers are identical with those obtained at the spherical shape: 2, 8, 20, 40, 70, 112, 168, ... Other magic numbers are given in Table I. Oblate and prolate shapes are considered on the left-hand side and right-hand side, respectively. spheroidal surface equation is given by (a/c)2(c2 − z2) z ≥ 0 0 z < 0 The radius of the semi-sphere obtained for the defor- mation δ = 0 is Rs, given by the volume conservation, (1/2)(4πR3s/3) = 4πR 0/3, leading to Rs = 2 1/3R0. We shall give ρ, z, a, c in units of Rs instead of R0. Accord- ing to the volume conservation, a2cR3s/2 = R 0 so that a2c = 1. Other kind of shapes obtained from a spheroid by removing less or more than its half (as in the liquid drop calculations [9]) will be considered in the future; in this case it is not possible to obtain analytical solutions. The new potential well we have to consider in order to solve the quantum mechanical problem is shown in the right-hand side of the figure 2. The potential along the symmetry axis, Vz(z), has a wall of an infinitely large height at z = 0, and concerns only positive values of z ∞ z = 0 MR2sω 2/2 z ≥ 0 (13) In this case the wave functions should vanish in the origin, where the potential wall is infinitely high, so that only negative parity Hermite polynomials (nz odd) should be taken into consideration. From the energy levels given in figure 1 we have to select only those corresponding to this condition. In this way the former lowest level with n = 0, n⊥ = 0 should be excluded. From the two leveles with n = 1 we can retain the level with n⊥ = 0 i.e. nz = 1. This will be the lowest level for the semi-spherical harmonic oscillator and will accomodate 2n⊥+2 = 2 atoms. From the three levels with n = 2 only the one with nz = n⊥ = 1 with 2n⊥ + 2 = 4 degeneracy is retained so that the first two magic numbers at spherical shape (δ = 0) are now 2 followed by 6, etc. Some deformed magic numbers may be found in the Table I and as position of minima in Fig. 3. Each level, labelled by n⊥, n, may accomodate 2n⊥+2 particles. When n is an odd number, one should only have even n⊥ in order to select the odd nz = n − n⊥. The contribution of the shells with odd n to the semi- spherical magic numbers will be neven (2n⊥ + 2) = (n+ 1)2 leading to the sequence 2, 8, 18 for n = 1, 3, 5. The con- tribution of the shells with even n to the semi-spherical TABLE I: TOP: Deformed magic numbers of the spheroidal harmonic oscillator. BOTTOM: Deformed magic numbers of the semi-spheroidal harmonic oscillator. OBLATE PROLATE δ a/c Magic numbers δ a/c Magic numbers −0.8/3 17/13 2, 8, 18, 20, 34, 38, 58, 64, 92, 100, 136, 148, ... 0.8/3 13/17 2, 8, 20, 22, 42, 46, 76, 82, 124, 134 ... −0.4 1.5 2, 6, 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, ... 0.4 2/3 2, 8, 10, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, ... −2/3 2 2, 6, 14, 26, 44, 68, 100, 140, ... 2/3 0.5 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, ... −1 3 2, 6, 12, 22, 36, 54, 78, 108, 144, 1 1/3 4, 12, 18, 24, 36, 48, 60, 80, 100, 120, 150, ... −0.8/3 17/13 2, 6, 12, 22, 26, 36, 42, 56, 64, 82, 92, 114, 126, 154, ... 0.8/3 13/17 2, 6, 8, 14, 18, 28, 34, 48, 58, 76, 90, 114, 132, ... −0.4 1.5 2, 6, 12, 22, 36, 54, 78, 108, 144, 0.4 2/3 2, 8, 18, 20, 34, 38, 50, 58, 64, 80, 92, 100, ... −2/3 2 2, 6, 12, 20, 32, 48, 68, 92, 122, 158, ... 2/3 0.5 2, 8, 20, 40, 70, 112, 168, ... −1 3 2, 6, 20, 30, 42, 58, 78, 102, 130, 1 1/3 2, 8, 10, 14, 22, 26, 46, 54, 66, 84, 96, 114, 138, 156, ... magic numbers will be (2n⊥ + 2) = n(n+ 2) which gives the sequence 4, 12, 24 for n = 2, 4, 6. This should be interlaced with the preceding one so that the magic numbers will be 2, 2+4 = 6, 6+8 = 14, 14+12 = 26, 26+18 = 44, 44+24 = 68, as shown at the right-hand side of the Fig. 1. The equation (9) from the harmonic oscillator, in units of h̄ω0 is still valid, but one should only allow the values of n and n⊥ for which nz = n−n⊥ ≥ 1 are odd numbers. The ortonormalization condition of the Znz component of the wave function became Zn′z(z)Znz(z)dz = δn′znz (16) with nz = 1, 3, 5, ..., n for odd n and nz = 1, 3, 5, ..., n− 1 for even n. Consequently the normalization factor is times the preceding one Znz(ξ) = 2Nnze −ξ2Hnz(ξ) Nnz = π2nznz!) 1/2 (17) For a nucleus with mass number A the shell gap is given by h̄ω00 = 41A 1/3 MeV. For an atomic cluster [10] the single-particle shell gap is given by h̄ω0(N) = 13.72 eV Å rsN1/3 which is 3.0613N−1/3 eV in case of Na clusters. Since we consider solely monovalent elements, N in this eq. is the number of atoms and t denotes the electronic spillout for the neutral cluster according to [10]. The shell correction energy, δU [11], in figure 3 shows minima at the oblate and prolate magic numbers given in the lower part of the table I. The striking result is that the superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscil- lator. We expect that this kind of symmetry will not be present anylonger for the Hamiltonian including the term proportional to (l2−〈l2〉n) and/or the more complex equipotential surface we shall study in the future. ∗ poenaru@fias.uni-frankfurt.de [1] S. G. Nilsson, Det Kongelige Danske Videnskabernes Sel- skab (Dan. Mat. Fys. Medd.) 29 (1955). [2] W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou, and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). [3] K. L. Clemenger, Phys. Rev. B 32, 1359 (1985). [4] S. M. Reimann, M. Brack, and K. Hansen, Z. Phys. D 28, 235 (1993). [5] M. Brack, Phys. Rev. B 39, 3533 (1989). [6] C. Yannouleas and U. Landman, Phys. Rev. B 51, 1902 (1995). [7] A. J. Rassey, Phys. Rev. 109, 949 (1958). [8] D. Vautherin, Phys. Rev. C 7, 296 (1973). [9] V. V. Semenikhina, A. G. Lyalin, A. V. Solov’yov, and W. Greiner, to be published (2007). [10] K. L. Clemenger, Ph. D. Dissertation (1985), University of California, Berkeley. [11] V. M. Strutinsky, Nuclear Physics, A 95, 420 (1967). mailto:poenaru@fias.uni-frankfurt.de ABSTRACT A new single-particle shell model is derived by solving the Schr\"odinger equation for a semi-spheroidal potential well. Only the negative parity states of the $Z(z)$ component of the wave function are allowed, so that new magic numbers are obtained for oblate semi-spheroids, semi-sphere and prolate semi-spheroids. The semi-spherical magic numbers are identical with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... The superdeformed prolate magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape of the spheroidal harmonic oscillator: 2, 8, 20, 40, 70, 112, 168 ... <|endoftext|><|startoftext|> Introduction to mathematics of quasicrys- tals, edited by M. V. Jaric (Academic Press, Boston, 1989), pp. 53–80. [6] S. Dworkin and J.-I. Shieh, Commun. math. Phys. 168, 337 (1995). [7] R. Penrose, Bull. Inst. Math. and Its Appl. 10, 266 (1974). [8] M. V. Jaric and M. Ronchetti, Phys. Rev. Lett. 62, 1209 (1989). [9] G. van Ophuysen, M. Weber, and L. Danzer, J. Phys. A 28, 281 (1995). [10] J. E. S. Socolar, in Quasicrystals, The State of Art (2nd Ed.), edited by D. DiVincenzo and P. J. Steinhardt (World Scientific, Singapore, 1999), pp. 225–250. [11] M. Gardner, Sci. Am. 236, 110 (1977). [12] H.-C. Jeong and E. D. Williams, Surface Science Reports 34, 171 (1999). [13] R. Penrose, Emperor’s New Mind (Oxford University Press, New York, 2002), p. 640ff. [14] T. Dotera, H.-C. Jeong, and P. J. Steinhardt, in Methods of structural analysis of modulated structures and qua- sicrystals, edited by et. al.. J. M. Perez-Mato (World Scientific, Singapore, 1991), pp. 660–663. [15] K. Ingersent, Ph.D. thesis, University of Pennsylvania, 1990. [16] We introduce the concept of sticky sites as a mathemat- ical devise to avoid the mistakes of copying a tile in a flipped worm but it may mimic the real growth process of quasicrystals. Recently, Fournée et al. observed “traps” or sticky sites on which adatoms are easily captured for some quasicrystal surfaces [21]. [17] Covering of all ends of the semi-infinite worms further ensures that all sticky sites in the decapod tiling stay outside of the semi-infinite worms. [18] From the Fig 2(b), one can see that an active decapod tiling is obtained by flipping a semi-infinite worm in a cartwheel tiling while flipping a half of an infinite worm results in an inactive decapod tiling. [19] The requirement of the underneath tiles is added to en- sure that the nucleation happens from the third layer. In real growth, this requirement may not be needed. Nu- cleation is likely to happen only on the top layer (hence from the third layer) which can be large enough to wait the slow necleation process. [20] The sticky sites (Fig. 2(a)) are formed only on a compact cluster of tiles. Therefore, the vertical growth by Rule-V, which produces isolated tiles, cannot be continue more than one layer height without nucleation process. [21] V. Fournée et al., Phys. Rev. B 67, 033406 (2003). http://arxiv.org/abs/cond-mat/9903074 ABSTRACT A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of non-local information for a 2D PPT growth. <|endoftext|><|startoftext|> Introduction Cosmological models representing the early stages of the Universe have been studied by several authors. An LRS (Locally Rotationally Symmetric) Binachi type-V spatially homogeneous space-time creates more interest due to its richer structure both physically and geometrically than the standard perfect fluid FRW models. An LRS Bianchi type-V universe is a simple generalization of the Robertson-Walker metric with negative curvature. Most cosmological models assume that the matter in the universe can be described by ’dust’ (a pressure- less distribution) or at best a perfect fluid. However, bulk viscosity is expected to play an important role at certain stages of expanding universe [1]−[3]. It has been shown that bulk viscosity leads to inflationary like solution [4] and acts like a negative energy field in an expanding universe [5]. Furthermore, there are several processes which are expected to give rise to viscous effects. These are the decoupling of neutrinos during the radiation era and the decoupling of 1Corresponding Author http://arxiv.org/abs/0704.0849v2 radiation and matter during the recombination era. Bulk viscosity is associ- ated with the Grand Unification Theories (GUT) phase transition and string creation. Thus, we should consider the presence of a material distribution other than a perfect fluid to have realistic cosmological models (see Grøn [6] for a review on cosmological models with bulk viscosity). A number of authors have discussed cosmological solutions with bulk viscosity in various context [7]−[9]. Models with a relic cosmological constant Λ have received considerable at- tention recently among researchers for various reasons (see Refs.[10]−[14] and references therein). Some of the recent discussions on the cosmological constant “problem” and consequence on cosmology with a time-varying cosmological con- stant by Ratra and Peebles [15], Dolgov [16]−[18] and Sahni and Starobinsky [19] have pointed out that in the absence of any interaction with matter or radi- ation, the cosmological constant remains a “constant”. However, in the presence of interactions with matter or radiation, a solution of Einstein equations and the assumed equation of covariant conservation of stress-energy with a time- varying Λ can be found. For these solutions, conservation of energy requires decrease in the energy density of the vacuum component to be compensated by a corresponding increase in the energy density of matter or radiation. Earlier researchers on this topic, are contained in Zeldovich [20], Weinberg [11] and Carroll, Press and Turner [21]. Recent observations by Perlmutter et al. [22] and Riess et al. [23] strongly favour a significant and positive value of Λ. Their finding arise from the study of more than 50 type Ia supernovae with redshifts in the range 0.10 ≤ z ≤ 0.83 and these suggest Friedmann models with negative pressure matter such as a cosmological constant (Λ), domain walls or cosmic strings (Vilenkin [24], Garnavich et al. [25]) Recently, Carmeli and Kuzmenko [26] have shown that the cosmological relativistic theory (Behar and Carmeli [27]) predicts the value for cosmological constant Λ = 1.934 × 10−35s−2. This value of “Λ” is in excellent agreement with the measurements recently obtained by the High-Z Supernova Team and Supernova Cosmological Project (Garnavich et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al. [28]). The main conclusion of these observations is that the expansion of the universe is accelerating. Several ansätz have been proposed in which the Λ term decays with time (see Refs. Gasperini [29, 30], Berman [31], Freese et al. [14], Özer and Taha [14], Peebles and Ratra [32], Chen and Hu [33], Abdussattar and Viswakarma [34], Gariel and Le Denmat [35], Pradhan et al. [36]). Of the special interest is the ansätz Λ ∝ S−2 (where S is the scale factor of the Robertson-Walker metric) by Chen and Wu [33], which has been considered/modified by several authors ( Abdel-Rahaman [37], Carvalho et al. [14], Waga [38], Silveira and Waga [39], Vishwakarma [40]). Recently Bali and Yadav [41] obtained an LRS Bianchi type-V viscous fluid cosmological models in general relativity. Motivated by the situations discussed above, in this paper, we focus upon the exact solutions of Einstein’s field equa- tions in presence of a bulk viscous fluid in an expanding universe. We do this by extending the work of Bali and Yadav [41] by including a time dependent cosmological term Λ in the field equations. We have also assumed the coefficient of bulk viscosity to be a power function of mass density. This paper is organized as follows. The metric and the field equations are presented in section 2. In section 3 we deal with the solution of the field equations in presence of viscous fluid. The sections 3.1 and 3.2 contain the two different cases and also con- tain some physical aspects of these models respectively. Section 4 describe two models under suitable transformations. Finally in section 5 concluding remarks have been given. 2 The Metric and Field Euations We consider LRS Bianchi type-V metric in the form ds2 = −dt2 +A2dx2 +B2e2x(dy2 + dz2), (1) where A and B are functions of t alone. The Einstein’s field equations (in gravitational units c = 1, G = 1) read as i + Λg i = −8πT i , (2) where R i is the Ricci tensor; R = g ijRij is the Ricci scalar; and T i is the stress energy-tensor in the presence of bulk stress given by i = (ρ+ p)viv j + pg i − (v i; + v ;i + v jvℓvi;ℓ + viv ξ − 2 vℓ;ℓ(g i + viv j). (3) Here ρ, p, η and ξ are the energy density, isotropic pressure, coefficients of shear viscosity and bulk viscous coefficient respectively and vi the flow vector satisfying the relations ivj = −1. (4) The semicolon (; ) indicates covariant differentiation. We choose the coordinates to be comoving, so that vi = δi4. The Einstein’s field equations (2) for the line element (1) has been set up as = −8π p− 2ηA4 ξ − 2 − Λ, (5) = −8π p− 2η − Λ, (6) 2A4B4 = −8πρ− Λ, (7) = 0. (8) The suffix 4 after the symbols A, B denotes ordinary differentiation with respect to t and θ = vℓ;ℓ 3 Solutions of the Field Eqations In this section, we have revisited the solutions obtained by Bali and Yadav [41]. Equations (5) - (8) are four independent equations in seven unknowns A, B, p, ρ, ξ, η and Λ. For complete determinacy of the system, we need three extra conditions. Eq. (8), after integration, reduce to A = Bk, (9) where k is an integrating constant. Equations (5) and (6) lead to − A44 − A4B4 = −16πη . (10) Using Eq. (9) in (10), we obtain k + 1 f = −16πη, (11) where B4 = f(B). Eq. (11) leads to f = − 16πη (k + 2) , (12) where L is an integrating constant. Eq. (12) again leads to B = (k + 2) k1 − k2e−16πηt k+2 , (13) where , (14) , (15) N being constant of integration. From Eqs. (9) and (13), we obtain A = (k + 2) k1 − k2e−16πηt k+2 . (16) Hence the metric (1) reduces to the form ds2 = −dt2 + (k + 2) k1 − k2e−16πηt k+2 dx2 + e2x(k + 2) k1 − k2e−16πηt k+2 (dy2 + dz2). (17) The pressure and density of the model (17) are obtained as 8πp = (8π)(16πη)k2e −16πηt 3(k + 2)2(k1 − k2e−16πηt)2 k1(k + 2) 2(4η + 3ξ)− {k2(4η + 3ξ) +4k(η+3ξ)+2(5η+6ξ)}k2e−16πηt [(k + 2)(k1 − k2e−16πηt)] −Λ, (18) 8πρ = − (2k + 1) (k + 2)2 (16πη)2k22 e−32πηt (k1 − k2e−16πηt)2 [(k + 2)(k1 − k2e−16πηt)] + Λ. (19) The expansion θ in the model (17) is obtained as (16πη)k2e −16πηt (k1 − k2e−16πηt) . (20) For complete determinacy of the system we have to consider three extra condi- tions. Firstly we assume that the coefficient of shear viscosity is constant, i.e., η = η0 (say). For the specification of Λ(t), we secondly assume that the fluid obeys an equation of state of the form p = γρ, (21) where γ(0 ≤ γ ≤ 1) is a constant. Thirdly bulk viscosity (ξ) is assumed to be a simple power function of the energy density [42]−[45]. ξ(t) = ξ0ρ n, (22) where ξ0 and n are constants. For small density, n may even be equal to unity as used in Murphy’s work [46] for simplicity. If n = 1, Eq. (22) may correspond to a radiative fluid [47]. Near the big bang, 0 ≤ n ≤ 1 is a more appropriate assumption [48] to obtain realistic models. For simplicity and realistic models of physical importance, we consider the following two cases (n = 0, 1): 3.1 Model I: Solution for n = 0 When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs. (18) and (19), with the use of (21), lead to 8π(1 + γ)ρ = k1(k + 2) 2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0) + 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t − (2k + 1)M . (23) Eliminating ρ(t) between Eqs. (19) and (23), we obtain (1 + γ)Λ = k1(k + 2) 2(4η0 + 3ξ0)− {k2(4η0 + 3ξ0) + 4k(η0 + 3ξ0) + 2(5η0 + 6ξ0)}k2e−16πη0t + (2k + 1)γ (1− 3γ) , (24) where M = 16πk2η0e −16πη0t, N = (k + 2)(k1 − k2e−16πη0t), P = 2k2 + 2k + 5, Q = k2 + 4k + 4. (25) 3.2 Model II: Solution for n = 1 When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (18) and (19), with the use of (21), leads to 8πρ = 16πM{2k1(k + 2)2η0 − Pk2η0e−16πη0t} 3 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}] k+2 − (2k + 1)M2 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}] . (26) Eliminating ρ(t) between Eqs. (19) and (26), we get Λ = 16πM [2k1(k + 2) 2η0 − Pk2η0e−16πη0t] + γ(2k + 1) (1 + γ) (1− 3γ) (1 + γ)N M [k1(k + 2) 2ξ0 −Qk2ξ0e−16πη0t]{4N k+2 − (2k + 1)M2} (1 + γ)N2 [(1 + γ)N2 −M{k1(k + 2)2ξ0 −Qk2ξ0e−16πη0t}] From Eqs. (23) and (26), we note that ρ(t) is a decreasing function of time and ρ > 0 for all time in both models. The behaviour of the universe in these models will be determined by the cosmological term Λ; this term has the same effect as a uniform mass density ρeff = −Λ/4πG, which is constant in space and time. A positive value of Λ corresponds to a negative effective mass density (repulsion). Hence, we expect that in the universe with a positive value of Λ, the expansion will tend to accelerate; whereas in the universe with negative value of Λ, the expansion will slow down, stop and reverse. From Eqs. (24) and (27), we observe that the cosmological term Λ in both models is a decreasing function of time and it approaches a small positive value as time increase more and more. This is a good agreement with recent observations of supernovae Ia (Garnavich et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al. [28]). The shear σ in the model (17) is given by (k − 1)M√ . (28) The non-vanishing components of conformal curvature tensor are given by C2323 = −C1414 = (k − 1)M [kM − 16πη0k1(k + 2)], (29) C1313 = −C2424 = (k − 1)M [16πη0k1(k + 2)− kM ], (30) C1212 = −C3434 = (k − 1)M [16πη0k1(k + 2)− kM ]. (31) Equations (20) and (28) lead to (k − 1)√ 3(k + 2) = constant. (32) The model (17) is expanding, non-rotating and shearing. Since σ = conatant, hence the model does not approach isotropy. The space-time (17) is Petrov type D in presence of viscosity. 4 Other Models After using the transformation k1 − k2e−16πηt = sin (16πητ), k + 2 = 1/16πη, (33) the metric (17) reduces to ds2 = − cos (16πητ) k1 − sin (16πητ) dτ2 + sin (16πητ) ]2(1−32πη) + e2x sin (16πητ) ](32πη) (dy2 + dz2). (34) The pressure (p), density (ρ) and the expansion (θ) of the model (34) are ob- tained as 8πp = (16πη)2{k1 − sin (16πητ) 3 sin2 (16πητ) 2k1−2(1−48πη+1152π2η2){k1−sin (16πητ)} (16πη)(8πξ){k1 − sin (16πητ)} sin (16πητ) sin (16πητ) ]2(1−32πη) − Λ, (35) 8πρ = 2(24πη − 1)(16πη)3{k1 − sin (16πητ)}2 sin2 (16πητ) sin (16πητ) ]2(1−32πη) (16πη){k1 − sin (16πητ)} sin (16πητ) . (37) 4.1 Model I: Solution for n = 0 When n = 0, Eq. (22) reduces to ξ = ξ0 = constant. Hence, in this case Eqs. (35) and (36), with the use of (21), lead to 8π(1 + γ)ρ = 2(16πη0) 3 sin2(16πη0τ) [k1 − P1M1] + (16πη0)(8πξ0)M1 sin(16πη0τ) + 4N1 + 2(24πη0 − 1)(16πη0)3M21 sin2(16πη0τ) . (38) Eliminating ρ(t) between Eqs. (36) and (38), we obtain (1 + γ)Λ = 2(16πη0) 3 sin2(16πη0τ) [k1 − P1M1] + (16πη0)(8πξ0)M1 sin(16πη0τ) + (1− 3γ)N1 + 2γ(24πη0 − 1)(16πη0)3M21 sin2(16πη0τ) . (39) 4.2 Model II: Solution for n = 1 When n = 1, Eq. (22) reduces to ξ = ξ0ρ . Hence, in this case Eqs. (35) and (36), with the use of (21), lead to 8πρ = 2(16πη0) 2M1[(k1 − P1M1) + 3(24πη0 − 1)(16πη0)M1] 3 sin(16πη0τ)[(1 + γ) sin(16πη0τ) − 16πη0ξ0M1] 4N1 sin(16πη0τ) [(1 + γ) sin(16πη0τ) − 16πη0ξ0M1] . (40) Eliminating ρ(t) between Eqs. (36) and (40), we obtain 2(16πη0) 2M1(k1 − P1M1) 3 sin(16πη0τ)[(1 + γ) sin(16πη0τ)− 16πη0ξ0M1] [3(16πη0ξ0)M1 + (1− 3γ) sin(16πη0τ)] [(1 + γ) sin(16πη0τ)− 16πη0ξ0M1] 2(24πη0 − 1)(16πη0)3M21 [γ(1 + γ) sin(16πη0τ) − (1− γ)(16πη0ξ0)M1] (1 + γ) sin2(16πη0τ)[(1 + γ) sin(16πη0τ)− (16πη0ξ0)M1] , (41) where M1 = k1 − sin(16πη0τ), 16πη0 sin (16πη0τ) ]2(1−32πη) P1 = 1− 48πη0 + 1152π2η20 . (42) The shear (σ) in the model (34) is obtained as (1− 48πη0)(16πη0)[k1 − sin(16πη0τ)√ 3 sin(16πη0τ) . (43) The models descibed in cases 4.1 and 4.2 preserve the same properties as in the cases of 3.1 and 3.2. 5 Conclusions We have obtained a new class of LRS Bianchi type-V cosmological models of the universe in presence of a viscous fluid distribution with a time dependent cosmological term Λ. We have revisited the solutions obtained by Bali and Ya- dav [41] and obtained new solutions which also generalize their work. The cosmological constant is a parameter describing the energy density of the vacuum (empty space), and a potentially important contribution to the dy- namical history of the universe. The physical interpretation of the cosmological constant as vacuum energy is supported by the existence of the “zero point” energy predicted by quantum mechanics. In quantum mechanics, particle and antiparticle pairs are consistently being created out of the vacuum. Even though these particles exist for only a short amount of time before annihilating each other they do give the vacuum a non-zero potential energy. In general relativity, all forms of energy should gravitate, including the energy of vacuum, hence the cosmological constant. A negative cosmological constant adds to the attractive gravity of matter, therefore universes with a negative cosmological constant are invariably doomed to re-collapse [49]. A positive cosmological constant resists the attractive gravity of matter due to its negative pressure. For most universes, the positive cosmological constant eventually dominates over the attraction of matter and drives the universe to expand exponentially [50]. The cosmological constants in all models given in Sections 3.1 and 3.2 are decreasing functions of time and they all approach a small and positive value at late times which are supported by the results from recent type Ia supernova observations recently obtained by the High-z Supernova Team and Supernova Cosmological Project (Garnavich et al. [25], Perlmutter et al. [22], Riess et al. [23], Schmidt et al. [28]). Thus, with our approach, we obtain a physically rele- vant decay law for the cosmological term unlike other investigators where adhoc laws were used to arrive at a mathematical expressions for the decaying vacuum energy. Our derived models provide a good agreement with the observational results. We have derived value for the cosmological constant Λ and attempted to formulate a physical interpretation for it. Acknowledgements The authors wish to thank the Harish-Chandra Research Institute, Allahabad, India, for providing facility where part this work was done. 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Introduction The Metric and Field Euations Solutions of the Field Eqations Model I: Solution for n = 0 Model II: Solution for n = 1 Other Models Model I: Solution for n = 0 Model II: Solution for n = 1 Conclusions ABSTRACT An LRS Bianchi type-V cosmological models representing a viscous fluid distribution with a time dependent cosmological term $\Lambda$ is investigated. To get a determinate solution, the viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. It turns out that the cosmological term $\Lambda(t)$ is a decreasing function of time, which is consistent with recent observations of type Ia supernovae. Various physical and kinematic features of these models have also been explored. <|endoftext|><|startoftext|> Introduction The spin-1/2 antiferromagnetic Heisenberg XXZ chain is one of the most fundamental models for one-dimensional quantum magnetism, which is given by the Hamiltonian Sxj S j+1 + S j+1 +∆S , (1.1) where Sαj = σ j /2 with σ j being the Pauli matrices acting on the j-th site and ∆ is the anisotropy parameter. For ∆ > 1, it is called the massive XXZ model where the system is gapful. Meanwhile for −1 < ∆ ≤ 1 case, the system is gapless and called the massless XXZ model. Especially we call it XXX model for the isotropic case ∆ = 1. The exact eigenvalues and eigenvectors of this model can be obtained by the Bethe Ansatz method [1, 2]. Many physical quantities in the thermodynamic limit such as specific heat, magnetic susceptibility, elementary excitations, etc..., can be exactly evaluated even at finite temperature by the Bethe ansatz method [2]. The exact calculation of the correlation functions, however, is still a difficult problem. The exceptional case is ∆ = 0, where the system reduces to a lattice free-fermion model by the Jordan-Wigner transformation. In this case, we can calculate arbitrary correlation functions by means of Wick’s theorem [3, 4]. Recently, however, there have been rapid developments in the exact evaluations of correlation functions for ∆ 6= 0 case also, since Kyoto Group (Jimbo, Miki, Miwa, Nakayashiki) derived a multiple integral representation for arbitrary correlation functions. Using the representation theory of the quantum affine algebra Uq(ŝl2), they first derived a multiple integral representation for massive XXZ antiferromagnetic chain in 1992 [5, 6], which is before long extended to the XXX case [7, 8] and the massless XXZ case [9]. Later the same integral representations were reproduced by Kitanine, Maillet, Terras [10] in the framework of Quantum Inverse Scattering Method. They have also succeeded in generalizing the integral representations to the XXZ model with an external magnetic field [10]. More recently the multiple integral formulas were extended to dynamical correlation functions as well as finite temperature correlation functions [11, 12, 13, 14]. In this way it has been established now the correlation functions for XXZ model are represented by multiple integrals in general. However, these multiple integrals are difficult to evaluate both numerically and analytically. For general anisotropy ∆, it has been shown that the multiple inetegrals up to four- dimension can be reduced to one-dimensional integrals [15, 16, 17, 18, 19, 20, 21]. As a result all the density matrix elements within four lattice sites have been obtained for general anisotropy [21]. To reduce the multiple integrals into one-dimension, however, involves hard calculation, which makes difficult to obtain correlation functions on more than four lattice sites. On the other hand, at the isotropic point ∆ = 1, an algebraic method based on qKZ equation has been devised [22] and all the density matrix elements up to six lattice sites have been obtained [23, 24]. Moreover, as for the spin-spin correlation functions, up to seventh-neighbour correlation 〈Sz1Sz8〉 for XXX chain have been obtained from the generating functional approach [25, 26]. It is desirable that this algebraic method will be generalized to the case with ∆ 6= 1. Actually, Boos, Jimbo, Miwa, Smirnov and Takeyama have derived an exponential formula for the density matrix elements of XXZ model, which does not contain multiple integrals [27, 28, 29, 30, 31]. It, however, seems still hard to evaluate the formula for general density matrix elements. Among the general ∆ 6= 0, there is a special point ∆ = 1/2, where some intriguing prop- erties have been observed. Let us define a correlation function called Emptiness Formation Probability (EFP) [8] which signifies the probability to find a ferromagnetic string of length P (n) ≡ + Szj . (1.2) The explicit general formula for P (n) at ∆ = 1/2 was conjectured in [33] P (n) = 2−n (3k + 1)! (n + k)! , (1.3) which is proportional to the number of alternating sign matrix of size n × n. Later this conjecture was proved by the explicit evaluation of the multiple integral representing the EFP [34]. Remarkably, one can also obtain the exact asymptotic behavior as n → ∞ from this formula, which is the unique valuable example except for the free fermion point ∆ = 0. Note also that as for the longitudinal two-point correlation functions at ∆ = 1/2, up to eighth-neighbour correlation function 〈Sz1Sz9〉 have been obtained in [32] by use of the multiple integral representation for the generating function. Most outstanding is that all the results are represented by single rational numbers. These results motivated us to calculate other correlation functions at ∆ = 1/2. Actually we have obtained all the density matrix elements up to six lattice sites by the direct evaluation of the multiple integrals. All the results can be written by single rational numbers as expected. A direct evaluation of the multiple integrals is possible due to the particularity of the case for ∆ = 1/2 as is explained below. 2 Analytical evaluation of multiple integral Here we shall describe how we analytically obtain the density matrix elements at ∆ = 1/2 from the multiple integral formula. Any correlation function can be expressed as a sum of density matrix elements P ,··· ,ǫ′n ǫ1,··· ,ǫn , which are defined by the ground state expectation value of the product of elementary matrices: ,··· ,ǫ′n ǫ1,··· ,ǫn ≡ 〈E 1 · · ·Eǫ n 〉, (2.1) where E j are 2× 2 elementary matrices acting on the j-th site as E++j = + Szj , E − Szj , E+−j = = S+j = S j + iS j , E = S−j = S j − iS The multiple integral formula of the density matrix element for the massless XXZ chain reads [9] ,··· ,ǫ′n ǫ1,··· ,ǫn =(−ν)−n(n−1)/2 · · · sinh(xa − xb) sinh[(xa − xb − ifabπ)ν] sinhyk−1 [(xk + iπ/2)ν] sinh n−yk [(xk − iπ/2)ν] coshn xk , (2.2) where the parameter ν is related to the anisotropy as ∆ = cosπν and fab and yk are determined as fab = (1 + sign[(s ′ − a + 1/2)(s′ − b+ 1/2)])/2, y1 > y2 > · · · > ys′, ǫ′yi = + ys′+1 > · · · > yn, ǫn+1−yi = −. (2.3) In the case of ∆ = 1/2, namely ν = 1/3, the significant simplification occurs in the multiple integrals due to the trigonometric identity sinh(xa−xb) = 4 sinh[(xa−xb)/3] sinh[(xa−xb+iπ)/3] sinh[(xa−xb−iπ)/3]. (2.4) Actually if we note that the parameter fab takes the value 0 or 1, the first factor in the multiple integral at ν = 1/3 can be decomposed as sinh(xa − xb) sinh[(xa − xb − iπ)/3] = 4 sinh xa − xb xa − xb + iπ = −1 + ωe (xa−xb) + ω−1e− (xa−xb), (2.5) sinh(xa − xb) sinh[(xa − xb)/3] = 4 sinh xa − xb + iπ xa − xb − iπ = 1 + e (xa−xb) + e− (xa−xb), (2.6) where ω = eiπ/3. Expanding the trigonometoric functions in the second factor into exponen- tials sinhy−1 [(x+ iπ/2)/3] sinhn−y [(x− iπ/2)/3] = 21−n ω1/2ex/3 − ω−1/2e−x/3 )y−1 ( ω−1/2ex/3 − ω1/2e−x/3 = 21−n (−1)l+m y − 1 ωy−l+m−(n+1)/2e (n−2l−2m−1)x, (2.7) we can explicitly evaluate the multiple integral by use of the formula eαxdx coshn x = 2n−1B , Re(n± α) > 0, (2.8) where B(p, q) is the beta function defined by B(p, q) = tp−1(1− t)q−1dt, Re(p),Re(q) > 0. (2.9) Table 1: Comparison with the asymptotic formula of the transverse correlation function 〈Sx1Sx2 〉 〈Sx1Sx3 〉 〈Sx1Sx4 〉 〈Sx1Sx5 〉 〈Sx1Sx6 〉 Exact −0.156250 0.0800781 −0.0671234 0.0521997 −0.0467664 Asymptotics −0.159522 0.0787307 −0.0667821 0.0519121 −0.0466083 In this way we have succeeded in calculating all the density matrix elements up to six lattice sites. All the results are represented by single rational numbers, which are presented in Appendix A. As for the spin-spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse two-point correlation function 〈Sx1Sx2 〉 = − = −0.15625, 〈Sx1Sx3 〉 = = 0.080078125, 〈Sx1Sx4 〉 = − 65536 = −0.0671234130859375, 〈Sx1Sx5 〉 = 1751531 33554432 = 0.0521996915340423583984375, 〈Sx1Sx6 〉 = − 3213760345 68719476736 = −0.046766368104727007448673248291015625. The asymptotic formula of the transverse two-point correlation function for the massless XXZ chain is established in [35, 36] 〈Sx1Sx1+n〉 ∼ Ax(η) (−1)n − Ãx(η) + · · · , η = 1− ν, Ax(η) = 8(1− η)2 sinh(ηt) sinh(t) cosh[(1− η)t] − ηe−2t Ãx(η) = 2η(1− η) cosh(2ηt)e−2t − 1 2 sinh(ηt) sinh(t) cosh[(1− η)t] sinh(ηt) η2 + 1 , (2.10) which produces a good numerical value even for small n as is shown in Table 1. Note that the longitudinal correlation function was obtained up to eighth-neighbour correlaion 〈Sz1Sz9〉 from the multiple integral representation for the generating function [32]. Note also that up to third-neighbour both longitudinal and transverse correlation functions for general anisotropy ∆ were obtained in [21]. 3 Reduced density matrix and entanglement entropy Below let us discuss the reduced density matrix for a sub-chain and the entanglement entropy. The density matrix for the infinite system at zero temperature has the form ρT ≡ |GS〉〈GS|, (3.1) 0 10 20 30 40 50 60 0 10 20 30 Figure 1: Eigenvalue-distribution of density matrices Table 2: Entanglement entropy S(n) of a finite sub-chain of length n S(1) S(2) S(3) S(4) 1 1.3716407621868583 1.5766810784924767 1.7179079372711414 S(5) S(6) 1.8262818282012363 1.9144714710902746 where |GS〉 denotes the ground state of the total system. We consider a finite sub-chain of length n, the rest of which is regarded as an environment. We define the reduced density matrix for this sub-chain by tracing out the environment from the infinite chain ρn ≡ trEρT = ,··· ,ǫ′n ǫ1,··· ,ǫn ǫj ,ǫ . (3.2) We have numerically evaluate all the eigenvalues ωα (α = 1, 2, · · · , 2n) of the reduced density matrix ρn up to n = 6. We show the distribution of the eigenvalues in Figure 1. The distribution is less degenerate comapared with the isotropic case ∆ = 1 shown in [24]. In the odd n case, all the eigenvalues are two-fold degenerate due to the spin-reverse symmetry. Subsequently we exactly evaluate the von Neumann entropy (Entanglement entropy) defined as S(n) ≡ −trρn log2 ρn = − ωα log2 ωα. (3.3) The exact numerical values of S(n) up to n = 6 are shown in Table 2. By analyzing the behaviour of the entanglement S(n) for large n, we can see how long quantum correlations reach [37]. In the massive region ∆ > 1, the entanglement entropy will be saturated as n grows due to the finite correlation length. This means the ground state is well approximated by a subsystem of a finite length corresponding to the large eigenvalues of reduced density matrix. On the other hand, in the massless case −1 < ∆ ≤ 1, the conformal field theory predict that the entanglement entropy shows a logarithmic divergence [38] S(n) ∼ 1 log2 n + k∆. (3.4) 1 2 3 4 5 6 Exact Asymptotics Figure 2: Entanglement entropy S(n) of a finite sub-chain of length n Our exact results up to n = 6 agree quite well with the asymptotic formula as shown in Figure 2. We estimate the numerical value of the constant term k∆=1/2 as k∆=1/2 ∼ S(6)− 13 log2 6 = 1.0528. This numerical value is slightly smaller than the isotropic case ∆ = 1, where the constant k∆=1 is estimated as k∆=1 ∼ 1.0607 from the exact data for S(n) up to n = 6 [24]. At free fermion point ∆ = 0, the exact asymptotic formula has been obtained in [39] S(n) ∼ 1 log2 n+ k∆=0, k∆=0 = 1/3− t sinh2(t/2) − cosh(t/2) 2 sinh3(t/2) / ln 2. (3.5) In this case the numerical value for the constant term is given by k∆=0 = 1.0474932144 · · · . 4 Summary and discussion We have succeeded in obtaining all the density matrix elements on six lattice sites for XXZ chain at ∆ = 1/2. Especially we have newly obtained the fourth- and fifth-neighbour transverse spin-spin correlation functions. Our exact results for the transverse correlations show good agreement with the asymptotic formula established in [35, 36]. Subsequently we have calculated all the eigenvalues of the reduced density matrix ρn up to n = 6. From these results we have exactly evaluated the entanglement entropy, which shows a good agreement with the asymptotic formula derived via the conformal field theory. Finally, we remark that similar procedures to evaluate the multiple integrals are also possible at ν = 1/n for n = 4, 5, 6, · · · , since there are similar trigonometric identities as (2.4). We will report the calculation of correlation functions for these cases in subsequent papers. Acknowledgement The authors are grateful to K. Sakai for valuable discussions. This work is in part sup- ported by Grant-in-Aid for the Scientific Research (B) No. 18340112. from the Ministry of Education, Culture, Sports, Science and Technology, Japan. Appendix A Density matrix elements up to n = 6 In this appendix we present all the independent density matrix elements defined in eq. (2.1) up to n = 6. Other elements can be computed from the relations ,··· ,ǫ′n ǫ1,··· ,ǫn = 0 if ǫj 6= ǫ′j , (A.1) ,··· ,ǫ′n ǫ1,··· ,ǫn = P ǫ1,··· ,ǫn ,··· ,ǫ′n ,··· ,−ǫ′n −ǫ1,··· ,−ǫn ǫ′n,··· ,ǫ ǫn,··· ,ǫ1 , (A.2) ,··· ,ǫ′n +,ǫ1,··· ,ǫn ,··· ,ǫ′n −,ǫ1,··· ,ǫn ,··· ,ǫ′n,+ ǫ1,··· ,ǫn,+ ,··· ,ǫ′n,− ǫ1,··· ,ǫn,− ,··· ,ǫ′n ǫ1,··· ,ǫn , (A.3) and the formula for the EFP [33, 34] P (n) = P +,··· ,+ +,··· ,+ = 2 (3k + 1)! (n+ k)! . (A.4) Appendix A.1 n ≤ 4 P−++− = − = −0.3125, P−++++− = = 0.0800781, P−++++−++ = − = −0.0269775, P−+++++−+ = 65536 = 0.0240936, P−++++++− = − 32768 = −0.00881958, P+−+++−++ = 16384 = 0.0632935, P+−++++−+ = − 32768 = −0.0611877, P−−+++−+− = − 65536 = −0.0583038, P−−++++−− = 65536 = 0.0212555, P−+−++−+− = 32768 = 0.149017, P−++−+−−+ = 32768 = 0.0943298. Appendix A.2 n = 5 P−+++++−+++ = − 14721 8388608 = −0.00175488, P−++++++−++ = 37335 16777216 = 0.00222534, P−+++++++−+ = − 48987 33554432 = −0.00145993, P−++++++++− = 13911 33554432 = 0.00041458, P+−++++−+++ = 179699 33554432 = 0.00535545, P+−+++++−++ = − 120337 16777216 = −0.00717264, P+−++++++−+ = 165155 33554432 = 0.004922, P++−++++−++ = 168313 16777216 = 0.0100322, P−−++++−−++ = 31069 2097152 = 0.0148149, P−−++++−+−+ = − 411583 16777216 = −0.0245323, P−−++++−++− = 196569 16777216 = 0.0117164, P−−+++++−+− = − 281271 33554432 = −0.00838253, P−−++++++−− = 79673 33554432 = 0.00237444, P−+−+++−−++ = − 1441787 33554432 = −0.0429686, P−+−+++−++− = − 1261655 33554432 = −0.0376002, P−+−++++−+− = 59459 2097152 = 0.0283523, P−++−++−++− = 1575515 33554432 = 0.046954, P−+++−+−−++ = − 696151 33554432 = −0.0207469, P−+++−+−+−+ = 1366619 33554432 = 0.0407284. Appendix A.3 n = 6 P−++++++−++++ = − 1546981 34359738368 = −0.0000450231, P−+++++++−+++ = 5095899 68719476736 = 0.0000741551, P−++++++++−++ = − 2366275 34359738368 = −0.0000688677, P−+++++++++−+ = 2455833 68719476736 = 0.0000357371, P−++++++++++− = − 284577 34359738368 = −8.28228× 10−6, P+−+++++−++++ = 2927709 17179869184 = 0.000170415, P+−++++++−+++ = − 20086627 68719476736 = −0.000292299, P+−+++++++−++ = 19268565 68719476736 = 0.000280395, P+−++++++++−+ = − 10295153 68719476736 = −0.000149814, P++−+++++−+++ = 17781349 34359738368 = 0.000517505, P++−++++++−++ = − 35087523 68719476736 = −0.000510591, P−−+++++−−+++ = 48421023 34359738368 = 0.00140924, P−−+++++−+−++ = − 214080091 68719476736 = −0.00311528, P−−+++++−++−+ = 88171589 34359738368 = 0.00256613, P−−+++++−+++− = − 57522267 68719476736 = −0.000837059, P−−++++++−−++ = 56776545 34359738368 = 0.00165241, P−−++++++−+−+ = − 154538459 68719476736 = −0.00224883, P−−++++++−++− = 60809571 68719476736 = 0.000884896, P−−+++++++−−+ = 6708473 8589934592 = 0.000780969, P−−+++++++−+− = − 33366621 68719476736 = −0.000485548, P−−++++++++−− = 3860673 34359738368 = 0.00011236, P−+−++++−−+++ = − 85706851 17179869184 = −0.0049888, P−+−++++−+−++ = 12211375 1073741824 = 0.0113727, P−+−++++−++−+ = − 332557469 34359738368 = −0.0096787, P−+−++++−+++− = 56183761 17179869184 = 0.00327033, P−+−+++++−−++ = − 430452959 68719476736 = −0.00626391, P−+−+++++−+−+ = 606065059 68719476736 = 0.00881941, P−+−+++++−++− = − 123612511 34359738368 = −0.0035976, P−+−++++++−−+ = − 108202041 34359738368 = −0.00314909, P−+−++++++−+− = 70061315 34359738368 = 0.00203905, P−++−+++−−+++ = 7860495 1073741824 = 0.00732066, P−++−+++−+−++ = − 591759525 34359738368 = −0.0172225, P−++−+++−++−+ = 1044016671 68719476736 = 0.0151924, P−++−+++−+++− = − 367905053 68719476736 = −0.00535372, P−++−++++−−++ = 676957849 68719476736 = 0.00985103, P−++−++++−+−+ = − 988973861 68719476736 = −0.0143915, P−++−++++−++− = 6581795 1073741824 = 0.00612977, P−++−+++++−−+ = 363618785 68719476736 = 0.00529135, P−+++−++−−+++ = − 185522333 34359738368 = −0.00539941, P−+++−++−+−++ = 901633567 68719476736 = 0.0131205, P−+++−++−++−+ = − 103539423 8589934592 = −0.0120536, P−+++−++−+++− = 38524625 8589934592 = 0.00448486, P−+++−+++−−++ = − 267901987 34359738368 = −0.00779697, P−+++−+++−+−+ = 12750645 1073741824 = 0.011875, P−+++−++++−−+ = − 309855965 68719476736 = −0.004509, P−++++−+−−+++ = 29410257 17179869184 = 0.0017119, P−++++−+−+−++ = − 296882461 68719476736 = −0.00432021, P−++++−+−++−+ = 35985105 8589934592 = 0.00418922, P−++++−++−−++ = 92176287 34359738368 = 0.00268268, P+−−++++−−+++ = 202646807 34359738368 = 0.0058978, P+−−++++−+−++ = − 972245985 68719476736 = −0.014148, P+−−++++−++−+ = 217687057 17179869184 = 0.0126711, P+−−+++++−+−+ = − 211696415 17179869184 = −0.0123224, P+−−++++++−−+ = 78922695 17179869184 = 0.00459391, P+−+−+++−+−++ = 1196499417 34359738368 = 0.0348227, P+−+−+++−++−+ = − 2209522727 68719476736 = −0.0321528, P+−+−++++−+−+ = 1108384987 34359738368 = 0.0322582, P+−++−++−++−+ = 530683585 17179869184 = 0.0308899, P+−++−+++−−++ = 347202525 17179869184 = 0.0202098, P−−−++++−−++− = − 268623007 68719476736 = −0.00390898, P−−−++++−+−+− = 46285135 8589934592 = 0.0053883, P−−−++++−++−− = − 136974885 68719476736 = −0.00199325, P−−−+++++−+−− = 19939391 17179869184 = 0.00116063, P−−−++++++−−− = − 18442085 68719476736 = −0.000268368, P−−+−+++−−++− = 1018463205 68719476736 = 0.0148206, P−−+−+++−+−+− = − 1454513249 68719476736 = −0.021166, P−−+−+++−++−− = 277721503 34359738368 = 0.00808276, P−−+−++++−+−− = − 335265249 68719476736 = −0.00487875, P−−++−++−−++− = − 369408975 17179869184 = −0.0215024, P−−++−++−+−+− = 1104236607 34359738368 = 0.0321375, P−−++−++−++−− = − 880560357 68719476736 = −0.0128138, P−−++−+++−−+− = − 876924641 68719476736 = −0.0127609, P−−+++−+−−−++ = 113631201 17179869184 = 0.00661421, P−−+++−+−−+−+ = − 292857807 17179869184 = −0.0170466, P−−+++−+−+−−+ = 548645951 34359738368 = 0.0159677, P−−+++−++−−−+ = − 377925345 68719476736 = −0.00549954, P−+−+−++−−++− = 1719255909 34359738368 = 0.0500369, P−+−+−++−+−+− = − 5350158879 68719476736 = −0.0778551, P−+−++−+−−+−+ = 1565770597 34359738368 = 0.0455699, P−+−++−+−+−−+ = − 3059753503 68719476736 = −0.0445253, P−++−−++−−++− = − 2117554719 68719476736 = −0.0308145. 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Lukyanov, A. Zamolodchikov, Nucl. Phys. B 493 (1997) 571. [36] S. Lukyanov, V. Terras, Nucl. Phys. B 654 (2003) 323. [37] G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90 (2003) 227902. [38] C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424 (1994) 443. [39] B.-Q. Jin, V.E. Korepin, J. Stat. Phys. 116 (2004) 79. http://arxiv.org/abs/hep-th/0507290 Introduction Analytical evaluation of multiple integral Reduced density matrix and entanglement entropy Summary and discussion Density matrix elements up to n=6 ABSTRACT We have analytically obtained all the density matrix elements up to six lattice sites for the spin-1/2 Heisenberg XXZ chain at $\Delta=1/2$. We use the multiple integral formula of the correlation function for the massless XXZ chain derived by Jimbo and Miwa. As for the spin-spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse correlation functions. We have calculated all the eigenvalues of the density matrix and analyze the eigenvalue-distribution. Using these results the exact values of the entanglement entropy for the reduced density matrix up six lattice sites have been obtained. We observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory. <|endoftext|><|startoftext|> Counting on Rectangular Areas Milan Janjić, Faculty of Natural Sciences and mathematics, Banja Luka, Republic of Srpska, Bosnia and Herzegovina. Counting on Rectangular Areas Abstract In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. A special case, concerning (0, 1)-matrices, is also stated. In the next section we apply this theorem to derive several combina- torial identities by counting specified subsets of a finite set. This means that the obtained identities will involve binomial coefficients only. We start with a simple equation which is, in fact, an immediate consequence of Binomial theorem, but it is derived independently of it. The second result concerns sums of binomial coefficients. In a special case we obtain one of the best known binomial identity dealing with alternating sums. Klee’s identity is also obtained as a special case as well as some formu- lae for partial sums of binomial coefficients, that is, for the numbers of Bernoulli’s triangle. 1 A counting theorem The set of natural numbers {1, 2, . . . , n} will be denoted by [n], and by |X | will be denoted the number of elements of the set X. For the proof of the main theorem we need the following simple result: (−1)|I| = 0, (1) where I run over all subsets of [n] (empty set included). This may be easily proved by induction or using Binomial theorem. But the proof by induction makes all further investigations independent even of Binomial theorem. Let A be an m× n rectangular matrix filled with elements which belong to a set Ω. By the i-column of A we shall mean each column of A that is equal to [c1, c2, . . . , cm] T , where c1, c2, . . . , cm of Ω are given. We shall denote the number of i-columns of A by νA(c) or simply by ν(c). For I = {i1, i2, . . . , ik} ⊂ [m], by A(I) will be denoted the maximal number of columns j of A such that aij 6= cj , (i ∈ I). http://arxiv.org/abs/0704.0851v1 We also define A(∅) = n. Theorem 1. The number ν(c) of i-columns of A is equal ν(c) = (−1)|I|A(I), (2) where summation is taken over all subsets I of [m]. Proof. Theorem may be proved by the standard combinatorial method, by counting the contribution of each column of A in the sum on the right side of We give here a proof by induction. First, the formula will be proved in the case ν(c) = 0 and ν(c) = n. In the case ν(c) = n it is obvious that for I 6= ∅ we have A(I) = 0, which implies (−1)|I|A(I) = n+ I 6=∅ (−1)|I|A(I) = n. In the case ν(c) = 0 we use induction on n. If n = 1 then the matrix A has only one column, which is not equal c. It yields that there exists i0 ∈ {1, 2, . . . ,m} such that ai0,1 6= ci0 . Denote by I0 the set of all such numbers. Then A(I) = 1 if and only if I ⊂ I0. From this and (1) we obtain (−1)|I|A(I) = (−1)|I| = 0. Suppose now that the formula is true for matrices with n columns and that A has n+ 1-columns, and νA(c) = 0. Omitting the first column, the matrix B with n columns remains. If I0 is the same as in the case n = 1, then (−1)|I|A(I) = I 6⊂I0 (−1)|I|A(I) + (−1)|I|A(I) = I 6⊂I0 (−1)|I|B(I) + (−1)|I|(B(I) + 1) = (−1)|I|B(I) + (−1)|I| = 0, since the first sum is equal zero by the induction hypothesis, and the second by For the rest of the proof we use induction on n again. For n = 1 the matrix A has only one column which is either equal c or not. In both cases theorem is true, from the preceding. Suppose that theorem holds for n, and that the matrix A has n+1 columns. We may suppose that ν(c) ≥ 1. Omitting one of the i-columns we obtain the matrix B with n columns. By the induction hypothesis theorem is true for B. On the other hand it is clear that A(I) = B(I) for each nonempty subset I. Furthermore A has one i-column more then B, which implies ν(c) = νA(c) = νB(c) + 1 = 1 + (−1)|I|B(I) = = 1 + n+ I 6=∅ (−1)|I|B(I) = 1 + n+ I 6=∅ (−1)|I|A(I). ν(c) = (−1)|I|A(I), and theorem is proved. If the number A(I) does not depend on elements of the set I, but only on its number |I| then the equation(2) may be written in the form ν(c) = (−1)i A(i), (3) where |I| = i. Our object of investigation will be (0, 1) matrices. Let c be the i- column of a such matrix A. Take I0 ⊆ [m], |I0| = k such that 1 i ∈ I0 0 i 6∈ I0 Then the number A(I) is equal to the number of columns of A having 0’s in the rows labelled by the set I ∩ I0, and 1’s in the rows labelled by the set I \ I0. Suppose that the number A(I) depends only on |I ∩ I0|, |I \ I0|. If we denote |I ∩ I0| = i1, |I \ I0| = i2, A(I) = A(i1, i2), then (2) may be written in the form ν(c) = (−1)i1+i2 A(i1, i2). (5) 2 Counting subsets of a finite set Suppose that a finite set X = {x1, x2, . . . , xn} is given. Label by 1, 2, . . . , 2 n all subsets of X arbitrary and define an n× 2n matrix A in the following way aij = 1 if xi lies in the set labelled by j 0 otherwise . (6) Take I0 ⊆ [n], |I0| = k, and form the submatrix B of A consisting of those rows of A which indices belong to I0. Let c be arbitrary i-column of B. Define = {i ∈ I0 : ci = 1}, I ′′0 = {i ∈ I0 : ci = 0} . (7) The number ν(c) is equal to the number of subsets that contain the set {xi, i ∈ I 0}, and do not intersect the set {xi : i ∈ I 0 }. There are obviously ν(c) = 2n−k, such sets. Furthermore, if I ⊆ I0 then the number B(I) is equal to the number of subsets that contain the set {xi : i ∈ I ∩ I }, and do not meet the set {xi : i ∈ I ∩ I ′0}. It is clear that there are B(I) = 2n−|I| such subsets, so that the formula (2) may be applied. It follows 2n−k = (−1)i 2n−i. Thus we have Proposition 2.1. For each nonnegative integer k holds (−1)i 2k−i. Note 2.1. The preceding equation is a trivial consequence of Binomial theorem. But here it is obtained independently of this theorem. The preceding Proposition shows that counting i-columns over all subsets of X always produce the same result. We shall now make some restrictions on the number of subsets of X . Take 0 ≤ m1 ≤ m2 ≤ n fixed, and consider the submatrix C of A consisting of rows whose indices belong to I0, and columns corresponding to those subsets of X that have m, (m1 ≤ m ≤ m2) elements. Let c be an i-column of C. Define I ′0 = {i ∈ I0 : ci = 1}, |I 0| = l. The number ν(c) is equal to the number of sets that contain {xi : i ∈ I and do not intersect the sets {xi : i ∈ I0 \ I }. We thus have m2−|I i=m1−|I n− |I0| On the other hand, for I ⊆ I0 the number C(I) corresponds to the number of sets that contain {xi : i ∈ I \ I }, and do not intersect {xi : i ∈ I ∩ I }. Its number is equal m2−|I\I i3=m1−|I\I n− |I| It follows that the formula (5) may be applied. We thus have Proposition 2.2. For 0 ≤ m1 ≤ m2 ≤ n, and 0 ≤ l ≤ k holds i=m1−l m2−i2 i3=m1−i2 (−1)i1+i2 k − l n− i1 − i2 In the special case when one takes k = l, m1 = m2 = m we obtain Corollary 2.1. For arbitrary nonnegative integers m,n, k holds (−1)i . (9) Note 2.2. The preceding is one of the best known binomial identities. It appears in the book [1] in many different forms. Taking m1 = m2 = m, in (8) one gets Corollary 2.2. For arbitrary nonnegative integer m,n, k, l, (l ≤ k) holds (−1)i1+i2 k − l n− i1 − i2 m− i2 , (10) For l = 0 we obtain (−1)i , (11) which is only another form of (9). Taking n = 2k, l = k in (10)we obtain (−1)i1 2k − i1 Substituting k − i1 by i we obtain Corollary 2.3. Klee’s identity,([2],p.13) (−1)k (−1)i k + i From (8) we may obtain different formulae for partial sums of binomial coefficients, that is, for the numbers of Bernoulli’s triangle. For instance, taking l = 0, m1 = 0, m2 = m we obtain Corollary 2.4. For any 0 ≤ m ≤ n and arbitrary nonnegative integer k holds (−1)i1 n+ k − i1 . (12) Note 2.3. The number k in the preceding equation may be considered as a free variable that takes nonnegative integer values. Specially, for k = 1 the equa- tion represents the standard recursion formula for the numbers of Bernoulli’s triangle. Taking k = l = m1, m2 = m one obtains (−1)i1 n+ k − i1 Note 2.4. The formulae (12) and (13) differs in the range of the index i2. References [1] J. Riordan, Combinatorial Identities. New York: Wiley, 1979. A counting theorem Counting subsets of a finite set ABSTRACT In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting specified subsets of a finite set. <|endoftext|><|startoftext|> Introduction Photons have an extremely long mean free path length and escape from the hot matter without rescattering. By measuring their Bose-Einstein (or Hanbury-Brown Twiss, HBT) correlations one can extract the space-time dimensions of the hottest central part of the collision1,2,3,4,5 in contrast to hadron HBT correlations which measure the size of the system at the moment of its freeze-out. Moreover, photons emitted at different stages of the collision dominate in different ranges of trans- verse momentum6, therefore measuring photon correlation radii at various average transverse momenta (KT ) one can scan the space-time dimensions of the system at various times and thus trace the evolution of the hot matter. Photons emitted directly by the hot matter – direct photons – constitute only a small fraction of the total photon yield while the dominant contribution comes from decays of the final state hadrons, mainly π0 → 2γ and η → 2γ mesons. Fortunately, the lifetime of these hadrons is extremely large and the width of the Bose-Einstein correlations between the decay photons is of the order of a few eV and cannot obscure the direct photon correlations. This feature can be used to extract the direct photon yield3: assuming that direct photons are emitted incoherently, the photon correlation strength parameter can be related to the proportion of direct photons as λ = 1/2(Ndirγ /N 2. This approach is probably the only way to experimentally measure direct photon yield at very small pT . Presently, the only experiment to ∗For the full list of the PHENIX collaboration and acknowledgments, see9. http://arxiv.org/abs/0704.0852v1 November 9, 2018 19:7 WSPC/INSTRUCTION FILE DPeressounko- ggHBT-T 2 D. Peressounko have measured direct photon Bose-Einstein correlations in ultrarelativistic heavy ion collisions is WA987. An invariant correlation radius was extracted and the direct photon yield was measured in Pb+Pb collisions at sNN = 17 GeV. Since the strength of the direct photon Bose-Einstein correlation is typically a few tenths of a percent, it is important to exclude all background contributions which could distort the photon correlation function. These contributions can be classified as following: apparatus effects (close clusters interference – attraction of close clusters in the calorimeter during reconstruction) and correlations caused by real particles. The latter in turn can be divided into contribution due to ”splitting” of particles – processes like antineutron annihilation in the calorimeter and photon conversion on detector material in front of the calorimeter; contamination by corre- lated hadrons (e.g. Bose-Einstein-correlated π±); background correlations of decay photons. In this paper we consider all of these contributions in detail and describe how to control for them in the PHENIX experiment. 2. Analysis This analysis is based on the data taken by PHENIX in Run3 (d+Au) and Run4 (Au+Au). The total collected statistics is ≈ 3 billion d+Au events and ≈ 900 M Au+Au events. Details of the PHENIX configuration in these runs can be found in references 8 and 9, respectively. 2.1. Apparatus effects Since correlation functions are rapidly rising functions at small relative momenta any small distortion of the relative momentum for real pairs, because of errors in reconstruction of close clusters in the calorimeter (”cluster attraction”) for example, can lead to the appearance of a fake bump in the correlation function. To explore the influence of cluster interference in the calorimeter EMCAL, we construct a set of correlation functions by applying different cuts on the minimal distance between photon clusters in EMCAL. To quantify the difference between these correlation functions we fit them with a Gaussian and compare the extracted correlation parameters. We find that for correlation functions that include clusters with small relative distances there is strong dependence on minimal distance cut, but for distance cuts above 24 cm (4-5 modules) the correlation parameters are independent of the relative distance cut. This implies that with this distance cut the apparatus effects are sufficiently small. 2.2. Photon conversion, n̄ annihilation, and similar backgrounds The next class of possible backgrounds are processes in which one real particle produces several clusters in the calorimeter close to each other. These are processes like n̄ annihilation in the calorimeter producing several separated clusters, or photon conversion in front of calorimeter, or residual correlations between photons that November 9, 2018 19:7 WSPC/INSTRUCTION FILE DPeressounko- ggHBT-T Bose-Einstein correlations of direct photons in Au+Au collisions 3 (GeV) 0 0.05 0.1 0.15 0.2 0.25 0.3 C Min.Bias, Au+Au Min.Bias, d+Au, scaled Fig. 1. Two-photon correlation function measured in d+Au collisions at sNN = 200 GeV scaled to reproduce the height of the π0 peak in Au+Au collisions compared to the same correlation function measured in Au+Au collisions at sNN = 200 GeV. Absolute vertical scale is omitted in this technical plot. belong to different π0 in decays like η → 3π0 → 6γ. The common feature of this type of process is that their strength is proportional to the number of particles per event and not to the square of the number of particles, as would be the case for Bose-Einstein correlations. To estimate the upper limit on these contributions, we compare two-photon correlation functions, calculated in d+Au and Au+Au collisions. For the moment we assume, that all correlations at small relative momenta seen in d+Au collisions are due to the background effects under consideration. Then we scale the correlation function obtained in d+Au collisions with the number of π0 (that is we reproduce the height of the π0 peak in Au+Au collisions): Cscaled2 = 1− hAu+Auπ hd+Auπ (C2 − 1). (1) The result of this operation is shown in Fig. 1. We find that the scaled d+Au correlation function lies well below (close to unity) the correlation function calcu- lated for Au+Au collisions at small relative momenta. From this we conclude that the contribution from effects with strength proportional to the first power of the number of particles is negligible in Au+Au collisions. 2.3. Charged and neutral hadron contamination Another possible source of distortion of the photon correlation function is a contam- ination by (correlated) hadrons. Although we use rather strict identification criteria November 9, 2018 19:7 WSPC/INSTRUCTION FILE DPeressounko- ggHBT-T 4 D. Peressounko for photons there still may be some admixture of correlated hadrons contributing to the region of small relative momenta. (GeV) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 C Converted + EMCAL EMCAL + EMCAL Fig. 2. Comparison of two-photon correlation functions measured in Au+Au collisions at sNN = 200 GeV by two different methods: both photons are registered in the EMCAL (closed) and one photon is registered in EMCAL while the other is reconstructed through its external conversion (open). Absolute vertical scale is omitted in this technical plot. To exclude this possibility, we construct the two-photon correlation function using one photon registered in the calorimeter EMCAL and reconstructing the sec- ond photon from its conversion into an e+e− pair on the material of the beam pipe. The photon sample, constructed using external conversions is completely free from hadron contamination, so comparison of the standard correlation function with the pure one allows to estimate the contribution from non-photon contami- nation. This comparison is shown in Fig. 2. We find that the correlation function constructed with the more pure photon sample demonstrates a slightly larger cor- relation strength. This demonstrates that the observed correlation is indeed a pho- ton correlation, while hadron contamination in the photon sample just increases combinatorial background and reduces the correlation strength. In addition, this comparison shows that we have properly excluded the region of cluster interference. Due to deflection by the magnetic field the electrons of the e+e− conversion pair hit the calorimeter far from the location of the pair photon used in the correlation function and thus effects related to the interference of close clusters are absent. 2.4. Photon residual correlations The last possible source of the distortion of the photon correlation function are residual correlations between photons. We have already demonstrated that the con- November 9, 2018 19:7 WSPC/INSTRUCTION FILE DPeressounko- ggHBT-T Bose-Einstein correlations of direct photons in Au+Au collisions 5 tributions of residual correlations between photons in decays like η → 3π0 → 6γ, with strength proportional to Npart and not N part is negligible in Au+Au collisions. Below we consider other effects, which may cause photon correlations. These are collective flow (and jet-like correlations) and correlations between photons, origi- nated from decays of Bose-Einstein correlated mesons. Collective (elliptic) flow as well as jet-like correlations are long-range effects, resulting in correlations at relative angles much larger than under consideration here (for example, the opening angle of a photon pair with 20 MeV mass and KT = 500 MeV is ∼ 5 degrees). Monte-Carlo simulations demonstrate that flow and jet-like contribution are indeed negligible. (GeV) 0 0.05 0.1 0.15 0.2 0.25 0.3 C Min.Bias Au+Au, Data HBT resid.corr., Sim.0π Fig. 3. Comparison of two-photon correlation functions measured in Au+Au collisions at sNN = 200 GeV with Monte-Carlo simulations of the contribution of residual correlations due to decays of Bose-Einstein-correlated neutral pions. Absolute vertical scale is omitted in this technical plot. Potentially, the most serious distortion of the photon correlation function are residual correlations between decay photons of HBT-correlated π0s. Monte-Carlo simulations show that this contribution is not negligible, but has a rather specific shape (see Fig. 3), so that it does not distort the photon correlation function at small Qinv. This result can be explained as follows. Let us consider two π 0s with zero relative momentum. The distribution of decay photons is isotropic in their rest frame, and the probability to find a collinear photon pair (Qinv = 0) is suppressed due to phase space reasons. The photon pair mass distribution has a maximum at 2/3mπ, not at zero. After convoluting with the pion correlation function we find a step-like two-photon correlation function3. On the other hand, if one artificially chooses photons with momentum along the direction of the parent π0 (e.g. by looking at photon pairs at very large KT ), then the shape of the decay photon correlation function will reproduce the shape of the parent π0 correlation. This November 9, 2018 19:7 WSPC/INSTRUCTION FILE DPeressounko- ggHBT-T 6 D. Peressounko probably explains the different shape of the residual correlations due to decays of HBT-correlated π0 found in10. 3. Conclusions We have presented the current status of analysis of direct photon Bose-Einstein correlations in the PHENIX experiment. We are able to measure the two-photon correlation function with a precision sufficient to extract the direct photon corre- lations. Correlation measurements in which one of the photon pair has converted to an e+e− pair have been used to provide an important cross-check. We have demonstrated that all known backgrounds are under control. The extraction of the correlation parameters of direct photon pairs is in progress. References 1. A.N. Makhlin, JETP Lett. 46:55 (1987); A.N. Makhlin, Sov.J.Nucl.Phys. 49:151,(1989). 2. D.K. Srivastava, J. Kapusta, Phys.Rev. C48:1335 (1993); D.K. Srivastava, C. Gale, Phys.Lett. B319:407 (1993); D.K. Srivastava, Phys.Rev. D49:4523 (1994); D.K. Srivas- tava, J. Kapusta, Phys.Rev. C50:505 (1994). D.K. Srivastava, Phys.Rev. C71:034905 (2005); S. Bass, B. Muller, D.K. Srivastava, Phys.Rev. Lett. 93:162301 (2004). 3. D. Peressounko, Phys.Rev. C67:014905 (2003). 4. J. Alam et al., Phys.Rev. C67:054902 (2003); J. Alam et al., Phys.Rev. C70:054901 (2004). 5. T. Renk, Phys.Rev. C71:064905 (2005); hep-ph/0408218. 6. D. d’Enterria and D.Peressounko, Eur.Phys.J.C46:451 (2006). 7. M.M. Aggarwal et al., Phys.Rev.Lett. 93:022301 (2004). 8. S.S.Adler et al., (PHENIX collaboration), Phys.Rev.Lett. 98:012002. 9. S.Bathe et al., (PHENIX collaboration), Nucl.Phys. A774:731 (2006). 10. D.Das et al., nucl-ex/0511055. http://arxiv.org/abs/hep-ph/0408218 http://arxiv.org/abs/nucl-ex/0511055 Introduction Analysis Apparatus effects Photon conversion, annihilation, and similar backgrounds Charged and neutral hadron contamination Photon residual correlations Conclusions ABSTRACT The current status of the analysis of direct photon Bose-Einstein correlations in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV done by the PHENIX collaboration is summarized. All possible sources of distortion of the two-photon correlation function are discussed and methods to control them in the PHENIX experiment are presented. <|endoftext|><|startoftext|> Introduction Let (M, g) be a Riemannian manifold of dimension 2. The normalized ricci flow = (r −R)gij , where R is the scalar curvature and r is some constant. For compact surface, r is the average of scalar curvature. In this case, Hamilton [4] and Chow [2] proved the normalized Ricci flow from any initial metric will exist for all time and converge to a metric of constant curvature. It’s therefore nature to ask if such result holds for non-compact surfaces. Recently, a preprint of Ji and Sesum [14] generalized the above result to complete surfaces with logarithmic ends. Such surfaces have infinities like hyperbolic cusps. In particular, they have finite volume, therefore are parabolic, in the sense that there exists no positive Green’s function. One of their result shows that the normalized Ricci flow from such a metric will exist for all time and converge to hyperbolic metric. In this paper, we study nonparabolic complete surfaces, i.e. surfaces admitting positive Green’s function. In contrast to [14], such surfaces have at least one nonparabolic end and have infinite volume. For a discussion of parabolic and nonparabolic ends and their geometric characterization, see Li’s survey paper [6]. Here we choose r = −1 because if the flow converges, the limit metric will be of constant curvature r. Since we are considering noncompact surfaces, r can’t be positive. If r = 0, the limit will be flat R2 or its quotient. However, it’s well known that these flat surfaces are parabolic. On the other hand, whether a surface is parabolic or nonparabolic is invariant under quasi-isometries. Since if the normalized Ricci flow converges, then the limit metric will be quasi-isometric to the initial one, we know r can’t be zero.(For the definition of quasi-isometry, see also [6].) If r < 0, we can always assume r = −1 by a scaling. The main result of this paper is Theorem 1.1. Let (M, g) be a nonparabolic surface with bounded curvature. If the infinity is close to a hyperbolic metric in the sense that |R+ 1| dV < +∞. http://arxiv.org/abs/0704.0853v2 2 HAO YIN Then, the normalized Ricci flow will converge to a metric of constant scalar curva- ture −1. As in [14], we try to apply the above result to prove results along the line of Uniformization theorem. That amounts to prove the existence of a complete hyperbolic metric within a given conformal class of a noncompact surface. In [14], the authors proved that there is a uniformization theorem for Riemann surfaces obtained from compact Riemann surface by removing finitely many points and remarked that similar result should be true for Riemann surfaces obtained from compact ones by removing finitely many disjoint disks and points. Our theorem can be used to prove the same result in the case there is at least one disk removed. In fact, we will give a unified proof, which includes and simplifies the proof of [14]. Precisely, we will show Corollary 1.2. Let M be a Riemann surface obtained from compact Riemann sur- face by removing finitely many disjoint disks and/or points. If no disk is removed, then we further assume that the Euler number of M is less than zero. Then there exists on M a complete hyperbolic metric compatible with the conformal structure. The proof of Theorem 1.1 is along the same line as [14]. The method was initiated by Hamilton in [4]. There, Hamilton considered only compact case. for the purpose of generalizing this method to complete case, we need to overcome some analytic difficulties. Precisely, one need to solve Poisson equations and obtain estimates for the solutions, for all t. Those growth estimates for the solution are needed to apply the maximum principle. As for the maximum principle, there are many versions of maximum principle on complete manifolds. Since we will be working on complete manifold with a changing metric, the closest version for our need is in [1]. We still need a little modification. Theorem 1.3. Suppose g(t) is a smooth family of complete metrics defined on M , 0 ≤ t ≤ T with Ricci curvature bounded from below and ∣ ≤ C on M × [0, T ]. Suppose f(x, t) is a smooth function defined on M × [0, T ] such that △tf − whenever f(x, t) > 0 and exp(−ar2t (o, x))f2+(x, t)dVt < ∞ for some a > 0. If f(x, 0) ≤ 0 for all x ∈ M , then f ≤ 0 on M × [0, T ]. Although there is no detail in [1], one can prove it using the method of Ecker and Huisken in [3] and Ni and Tam in [12]. To solve the Poisson equation△u = R+1 for t = 0. We use a result of Ni[10], See Theorem 3.1. That’s the reason why we assume |R+ 1| dV < +∞. Moreover, we prove a growth estimate of the solution under the further assumption that Ricci curvature bounded from blow. This result is true for all dimensions. For the growth estimate, an estimate of Green’s function is proved under the assumption that Ricci curvature bounded from below. This estimate may be of independent interests, see the discussion in Section 2. Instead of solving △tu(x, t) = R(x, t) + 1 for later t. We solve an evolution equation for u. Thanks to the recent preprint of Chau, Tam and Yu [1], we can RICCI FLOW ON SURFACES 3 solve this evolution equation with a changing metric. Following a method in [11], we show that u, |∇u| and △u satisfy the growth estimate like in equation (1). With these preperation, we proceed to show that u(x, t) is indeed the potential functions we need. Now the Theorem 1.1 follows from the approach of Hamilton and repeated use of Theorem 1.3. The paper is organized as follows: In Section 2, we prove the crucial estimate of Green’s function needed for the growth estimate. In Section 3, we solve the Poisson equation and prove the relevant growth estimates. In the last section, we prove Theorem 1.1 and discuss results related to Uniformization theorem. 2. An estimate of Green’s function In this section we prove that Theorem 2.1. Let (M, g) be a complete noncompact manifold with Ricci curvature bounded from below by −K. Assume that M admits a positive Green’s function G(x, y). Let x0 be a fixed point in M . Then there exists constant A > 0 and B > 0, which may depend on M and x0, so that {G(x,y)>eAr(y,x0)} G(x, y)dx ≤ BeAr(y,x0), where r(y, x0) is the distance from y to x0. Remark 2.2. It’s impossible to get an estimate of this kind with constant depending only on K. Considering a family of nonparabolic manifolds Mi, which are becoming less and less ’nonparabolic’, i.e. their infinities are closing up. For any A,B > 0, there exists Mi and some xi ∈ Mi such that {Gi(x,xi)>A} Gi(x, xi)dx > B. See [8]. Remark 2.3. To the best of the author’s knowledge, known estimates on Green’s function in terms of volume of balls require Ricci curvature to be non-negative, See [9]. There could be one estimate of such type for Ricci curvature bounded from below, in light of [1]. If so, our relative estimate should be a corollary. The following proof is a direct one. We begin with a lemma, Lemma 2.4. There is a constant C depending only on K and the dimension, such that if Ricci curvature on B(x, 1) is bounded from below by −K and G(x, y) is the Dirichlet Green’s function on B(x, 1), then B(x,1) G(x, y)dy < C. Proof. Let H(x, y, t) be the Dirichlet heat kernel of B(x, 1). It’s easy to see B(x,1) H(x, y, t)dy ≤ 1, for all t > 0. 4 HAO YIN Now we prove that H(x, y, 2) is bounded from above. The proof is Moser itera- tion, which has appeared several times. Here we follow computations in [17]. Since we have Dirichlet boundary condition, we don’t need cut off function of space. Let 0 < τ < 2 and 0 < δ ≤ 1/2 be some positive constants, σk = (1− (1/2)kδ)τ and ηi be smooth function on [0,∞) such that 1) ηi = 0 on [0, σi], 2) ηi = 1 on [σi+1,∞) and 3) η′i ≤ 2i+3(δτ)−1. Let pi = (1 + 2n ) i. Since H is a solution to the heat equation, it’s easy to know Hp is a subsolution to the heat equation for p > 1. −△y)Hp(x, y, t) ≤ 0. Multiply by η2iH pi and integrate B(x,1) −△y)Hpidydt ≤ 0. Routine computation gives B(x,1) |∇yHpi |2 dydt+ B(x,1) H2pi(x, y, T )dy ≤ 2i+3(τδ)−1 B(x,1) H2pidydt. The sobolev inequality in [13] implies B(x,1) (Hpi) n−2 dy ≤ CV −2/n B(x,1) |∇yHpi |2 +H2pidy, where V is the volume of B(x, 1). By Hölder inequality, B(x,1) H2pi+1dy ≤ B(x,1) (Hpi) n−2 dy B(x,1) H2pidy)2/n ≤ (CV −2/n B(x,1) |∇yHpi |2 +H2pidy)( B(x,1) H2pidy)2/n. By (2), integrate over time B(x,1) H2pi+1dydt ≤ CV −2/nci+30 (στ)−(1+2/n)( B(x,1) H2pidydt)1+ where c0 = 2 1+2/n. A standard Moser iteration gives t∈[τ,2] y∈B(x,1) H2(x, y, t) ≤ CV −1(στ)− (1−δ)τ B(x,1) H2(x, y, t)dydt. An iteration process as given in [7] implies the L1 mean value inequality. In par- ticular, y∈B(x,1) H(x, y, 2) ≤ CV −1 B(x,1) H(x, y, t)dydt ≤ CV −1. Hence, B(x,1) H2(x, y, 2)dy ≤ CV −1. RICCI FLOW ON SURFACES 5 Due to a Poincaré inequality in [7], B(x,1) H2(x, y, t)dy = B(x,1) 2H△yHdy B(x,1) |∇yH |2 dy B(x,1) H2(x, y, t)dy. This differential inequality implies B(x,1) H2(x, y, t)dy ≤ B(x,1) H2(x, y, 2)dy × e−C(t−2) ≤ CV −1e−C(t−2). Hölder inequality shows B(x,1) H(x, y, t) ≤ V (B(x, 1)) B(x,1) H2(x, y, t)dy ≤ Ce−C(t−2), for t ≥ 2. The lemma follows from B(x,1) G(x, y)dy = B(x,1) H(x, y, t)dydt. Now let’s turn to the proof of Theorem 2.1. Proof. The key tool in the proof is Gradient estimate for harmonic function. Recall that if u is a positive harmonic function on B(x, 2R), then B(x,R) |∇ log u(x)|2 ≤ C1K + C2R−2 This is to say outside B(x, 0.1), the Green function as a function of y decays or increases at most exponentially with a factor C1K + 100C2. (1) Consider G(x0, y), Set p = max y∈∂B(x0,1) G(x0, y). As pointed out in Li and Tam, in the paper constructing Green function, G(x0, y) ≤ p for y /∈ B(x0, 1). Since the Green function is symmetric, for any point y far out in the infinity, G(y, x0) ≤ p. (2) If the theorem is not true, then for any big A and B, there is a point y (far away) so that {G(x,y)>eAr(y,x0)} G(x, y) > BeAr(y,x0). We will derive a contradiction with (1). Claim: {x|G(x, y) > eAr} ⊂ B(y, 1) is not true. If true, then consider the Dirichlet Green function G1(z, y) on B(y, 1). It’s well known that G(z, y) − G1(z, y) is a harmonic function. Notice that this harmonic function has boundary value less than eAr. Therefore, its integration on B(y, 1) is less than eAr × V ol(B(y, 1)). Since we assume Ricci lower bound, V ol(B(y, 1)) is less than a universal constant depending on K. 6 HAO YIN Therefore, {G(x,y)>eAr} G(x, y)dx ≤ B(y,1) G(x, y)dx ≤ V ol(B(y, 1))× eAr + B(y,1) G1(x, y)dx ≤ C(K,n)× eAr, where we used Lemma 2.4 for the last inequality. If we choose B to be any number larger than C(K,n) in the above equation, then the choice of y gives an contradic- tion and implies that the claim is true. (3) There is a z ∈ {x|G(x, y) > eAr} so that d(z, y) = 1 because the set {G(x, y) > eAr} is connected. This follows from the maximum principle and the construction of Green’s function. (3.1) If |d(y, x0)− d(z, x0)| < 0.3, then Let σ be the minimal geodesic connecting z and x0. Claim: the nearest distance from y to σ is no less than 0.1. If not, let w be the point in σ such that d(y, w) < 0.1. Since d(y, z) > 1, we d(w, z) > 0.9 Now, w is on the minimal geodesic from z to x0, so d(w, x0) ≤ d(z, x0)− 0.9 d(y, x0) < d(w, x0) + d(y, w) < d(z, x0)− 0.8 This is a contradiction , so the claim is true. We can use the gradient estimate along the segment σ. (Notice that d(z, x0) < r(x, x0) + 1) G(y, x0) > G(y, z) C1K + 100C2(r + 1)) This is a contradiction if we choose A >> C1K + 100C2. (3.2) If d(z, x0) ≤ d(y, x0)− 0.3, then The distance from y to the minimal geodesic connecting z and x0 will be larger than 0.1. The above argument gives a contradiction. (3.3) If d(z, x0) ≥ d(y, x0) + 0.3, then Since G(z, y) > eAr, we move the center to z, by symmetry of Green function. G(y, z) > eAr(y,x0)) > eA ′r(z,x0). This is case (3.2). We get a contradiction at z. This finishes the proof of estimate of Green function. � 3. Poisson equations △u = R+ 1 This section is divided into two parts. The first part solves the Poisson equation for t = 0. The second part solves for t > 0 before the maximum time using an indirect way. First, we use Theorem 2.1 to obtain an growth estimate of the solution of the Poisson equation △u = R + 1 for t = 0. The existence part without curvature restriction and boundedness of f of the following theorem is due to Lei Ni in [10]. RICCI FLOW ON SURFACES 7 Theorem 3.1. Let M be a complete nonparabolic manifold with Ricci curvature bounded from below by −K. For non-negative bounded continuous function f the Poisson equation △u = −f has a non-negative solution u ∈ W 2,nloc (M) ∩ C loc (M)(0 < α < 1) if f ∈ L1(M). Moreover, for any fixed x0 ∈ M , there exists A > 0 and C > 0 such that u(x) ≤ CeAr(x,x0). Proof. Let G(x, y) be the positive Green’s function. G(x, y)f(y)dy = {G(x,y)≤eAr(x,x0)} G(x, y)f(y)dy {G(x,y)>eAr(x,x0)} G(x, y)f(y)dy ≤ CeAr(x,x0). For the first term, we use the assumption that f is integrable, for the second term, we use the boundedness of f and the Theorem 2.1. The estimate above shows the Poisson equation is solvable with the required estimate. � Corollary 3.2. Let M be a surface satisfying the assumptions in Theorem 1.1. There exists a solution u0 to the equation △u0 = R(x) + 1 satisfying exp(−ar2(x, x0))u20(x)dV < ∞ exp(−br2(x, x0)) |∇u0|2 (x)dV < ∞ where a and b are some positive constants. Proof. Solve the Poisson equation for the positive part and the negative part of R+ 1 respectively. Then subtract the solutions. The first integral estimate follows from the pointwise growth estimate and volume comparison. Let R > 1. Choose a cut-off function ϕ such that ϕ(x) = 1 x ∈ B(x0, R) 0 x /∈ B(x0, 2R) |∇ϕ|2 ≤ C1ϕ. Multiply the equation by ϕu0 and integrate over M , ϕu0△u0dV = (R + 1)ϕu0dV, which implies ϕ |∇u0|2 dV + u0∇ϕ · ∇u0dV = − (R+ 1)ϕu0dV. 8 HAO YIN Hence (ϕ− |∇ϕ| ) |∇u0|2 dV ≤ C B(x0,2R) u20dV + C B(x0,2R) |u0| dV. B(x0,2R) u20dV + CV ol(B(x0, 2R)). From the integration estimate of u0, B(x0,2R) u20dV ≤ Ce4aR By choice of ϕ, B(x0,R) |∇u0|2 dV ≤ CeãR From here, it’s not difficult to see the estimate we need. � Now let’s look at the case of t > 0. In fact, it’s not difficult to show the above method can be used for t > 0. This amounts to show that M is still nonparabolic for t > 0 and |R+ 1| dV is still finite. The first claim is trivial and the second follows from the evolution equation and maximum principle. Assume the solutions are u(t). We have trouble in deriving the evolution equation for u(t), due to the possible existence of nontrivial harmonic functions. This explains why we use the following indirect way. Lemma 3.3. Assume the normalized Ricci flow exists for t ∈ [0, Tmax). The following equation has a solution u(x, t) (0 ≤ t < Tmax) with initial value u0, = △u− u, where △ is the Laplace operator of metric g(t). Moreover, there exists a > 0 depending on T such that for any T < Tmax exp(−ar2(x, x0))u2(x, t)dVt < ∞. Similar estimates hold for |∇u| and △u with different constants. Remark 3.4. Since g(0) and g(t) are equivalent up to a constant depending on T , it doesn’t matter whether we estimate ∇u or ∇tu and whether we use r to stand for distance at g(0) or g(t) if t ∈ [0, T ]. Proof. In [1], the authors considered a class of evolution equation with changing metric. ∂u = △u− u with the underling metric evolving by normalized Ricci flow is in this class. They proved, among other things, that the fundamental solution Z(x, t; y, s) has a Gaussian upper bound, i.e Z(x, t; y, x) ≤ C t− s) r2(x,y) D(t−s) . These constants depends on the solution of normalized Ricci flow and T . See Corollary 5.2 in [1]. For simplicity, denote Z(x, t; y, 0) by H(x, y, t), then to solve the equation, it suffices to show the following integral converges, u(x, t) = H(x, y, t)u0(y)dy. RICCI FLOW ON SURFACES 9 Bt(x,1) H(x, y, t)u0(y)dy ≤ CeAr(x,x0), because the integral of H on Bt(x, 1) is less than 1 and u0 grows at most exponen- tially by Theorem 2.1. M\Bt(x,1) H(x, y, t)u0(y)dy M\Bt(x,1) r2(x,y) Dt |u0(y)| dy. By volume comparison, Vx(1) ≥ C1e−A1r(x,x0)Vx0(1) t) ≥ C2e−A1r(x,x0) min(1, tn/2). Therefore M\Bt(x,1) H(x, y, t)u0(y)dy M\Bt(x,1) CeA1r(x,x0)e− r2(x,y) 2DT |u0(y)| dy M\Bt(x,1) CeA2r(x,x0)eAr(x,y)e− r2(x,y) 2DT dy ≤ CeA2r(x,x0). In summary, |u(x, t)| ≤ CeAr(x,x0), where A means a different constant. Volume comparison then implies exp(−ar2(x, x0))u2(x, t)dVt < ∞. For estimates on derivatives, note first that etu(x, t) is a solution of heat equation (with evolving metric) with initial value u0. Since we allow constants depend on T , it’s equivalent to prove estimates for etu(x, t). Therefore, from now on, to the end of this proof, we assume u(x, t) is a solution of heat equation. Then (2) (△− ∂ )u2 = 2 |∇u| . Assume that ϕ : R+ → R+ satisfies 1) ϕ(x) = 1 for x ≤ 1; 2) ϕ(x) = 0 for x ≥ 2. Choose the cut-off function ϕ( r(x,x0) )(R > 1). Multiplying this to the equation (2) and integrate, r(x, x0) ) |∇u|2 dVtdt ≤ r(x, x0) )u2dVtdt. △ϕ( r ) = div(ϕ′( = ϕ′′( |∇r|2 + ϕ′( 10 HAO YIN By definition of ϕ, we know ϕ( r ) vanishes unless R ≤ r(x, x0) ≤ 2R. Laplacian comparison implies (curvature is bounded from below −k) △r ≤ (n− 1) kcoth( kr) ≤ C. Therefore, ϕ△u2dVt ≤ C B(x0,2R) u2dVt. Let dVt = e FdV0, (ϕu2)eF dV0dt ≥ (ϕu2eF )dtdV0 − Cϕu2dVtdt ϕu2(x, T )dVT − ϕu2(x, 0)dV0 − C ϕu2dVtdt ϕu20(x)dV0 − C ϕu2dVtdt. Here we have used the fact that ∂e is bounded. Combined with equation (3) and B(x0,R) |∇u|2 dVtdt ≤ C B(x0,2R) u2dVtdt+ B(x0,2R) u20(x)dV0. From here it’s easy to see the type of estimate in Theorem 1.3. For △u, it suffices to consider ∣. The Bochner formula in this case is (remember we have assumed that u is a solution of the heat equation), (△− ∂ ) |∇u|2 = 2 2 − |∇u|2 . The same argument as before works for Lemma 3.5. For t ∈ [0, Tmax), △tu(x, t) = R(x, t) + 1. Proof. We know for t = 0 it’s true. Calculation shows (△tu−R(t)− 1) = (R+ 1)△tu+△t(△tu− u)−△tR−R(R+ 1) = △t(△tu−R(t)− 1) +R(△tu−R− 1) By previous lemma, we have growth estimate for △tu−R(t)−1. If △tu−R−1 ≥ 0, −△t)(△tu−R(t)− 1) ≤ C(△tu−R− 1). If △tu−R− 1 ≤ 0, then −△t)(△tu−R(t)− 1) ≥ C(△tu−R− 1). Apply maximum principle for △tu − R − 1, which is zero at t = 0. We know it’s zero forever. � RICCI FLOW ON SURFACES 11 4. Proof of the main theorem and the corollary Assume we have a surface satisfying the assumptions of Theorem 1.1. Short time existence is known, see [15]. The long time existence and convergence follows exactly by an argument of Hamilton in [4]. For completeness, we outline the steps. Solve Poisson equations△tu(x, t) = R(x, t)+1 as we did. Consider the evolution equation for H = R+ 1 + |∇u|2, H = △H − 2 |M |2 −H, where M = ∇∇u − 1 △f · g. Since we have growth estimate for H , maximum principle says R+ 1 ≤ H ≤ Ce−t. Therefore, after some time R will be negative everywhere. Applying maximum principle again to the evolution equation of scalar curvature R = △R+ R(R+ 1) will prove Theorem 1.1. Next, we discuss the application of the above theorem to Uniformization theorem. Let S be a compact Riemann surface. Let p1, · · · , pk be k different points in S and D1, · · · , Dl be l domains on S such that all of them are disjoint and Di is diffeomorphic to disk. Denote S \ ∪iDi \ {p1, · · · , pk} by M . The aim is to show there exists a complete hyperbolic metric on M compatible with the conformal structure. The approach is to construct an initial metric g0 on M compatible with the conformal structure so that the normalized Ricci flow starting from g0 will converge to a hyperbolic metric. Assume there is metric h in the given conformal class of S. Note that h is incomplete as a metric on M . For pi, there is an isothermal coordinate (x, y) around pi. By a conformal change of h, one can ask g0 to be (x2 + y2) log2(x2 + y2) (dx2 + dy2) in a small neighborhood Ui of pi. Remark 4.1. This is called hyperbolic cusp metric in [14] and it has scalar curva- ture −1. For Dj, let r be the distence to ∂Dj on M with respect to h. Let Vj be a neighborhood of ∂Dj in M . Let (r, θ) be the Fermi coordinates for ∂Dj so that h0 = dr 2 +A(r, θ)dθ2. We will find ρ = ρ(r, θ) such that 1) ρ = 0 on ∂Dj; 2) dρ 6= 0 on ∂Dj; 12 HAO YIN is asymptoticly hyperbolic in high order. Let K and K0 be the Gaussian curvature of h and g0 respectively. We have the formula, K0 = ρ 2(△h log ρ+K). In order that K0 = −1, 1− |∇ρ|2 + ρ△ρ+ ρ2K = 0. In terms of r and θ, |∇ρ|2 = (∂ρ )2 +A−1(r, θ)( △ρ = ∂ Here A, B, C and D are smooth functions of r and θ. The equation now becomes (5) ρ + 1− ( )2 −A−1( )2 + ρ2K = 0. If equation (5) is true at r = 0, then (r, θ) = 1. Here we used that fact that ρ > 0. Set η(r, θ) = ρ . Equation (6) implies η(0, θ) = 1. Equation (5) becomes +Brη +Br2 ∂η + Cr2 ∂η +Dr2 ∂ + 1−η )2 −A−1 r )2 + ηr2K = 0 For the convinience of formal calculation, this equation is rewritten as (7) (D2 −D − 2)η + F [r, η] = 0, where D = r ∂ F [r, η] = Brη +Br2 + Cr2 (1− η)2 )2 −A−1 r )2 + ηr2K. Equation (7) is a very typical form of Fuchsian type PDE. Formal solutions of this kind of equation has been discussed many times. For example, Kichenassamy [5] and Yin [16]. We will only outline the main steps here, for details see [5] and [16]. Consider formal solution with the following expansion, (8) η(r, θ) = 1 + aij(θ)r i(log r)j . We will call the sum j=0 aijr i(log r)j the i-level of the expansion. Note that D maps i-level to i-level. Details on formal calculation could be find in [5] and [16]. A common feature of all terms in F [r, η], which is crutial in obtaining a formal solution, is that the k-level of F [r, η] could be calculated with knowledge of only l-level of η with l < k. For example, consider (1 − η)2/η. It’s the multiplication of three formal series, two 1 − η and 1/η. In order the k-level of η appears in the RICCI FLOW ON SURFACES 13 k-level of (1 − η)2/η, the only possibility is that two of the three series contribute zero level and one k-level. However, the zero level of 1− η vanishes. The only thing we need is that there exists a formal solution and furthermore due to Borel’s Lemma as in [16], there is an approximate solution so that (D2 −D − 2)η + F [r, η] = o(rk) for any k. In terms of ρ, (9) K0 + 1 = 1 + ρ 2(△h log ρ+K) = o(ρk) for any k. This metric g0 near ∂Dj has Gaussian curvature -1 asymptotically. By a scaling, we assume it has scalar curvature -1 asymptotically. We construct g0 by doing the above to every point Pi and disk Dj. If there is at least one disk removed, we know M is nonparabolic. |R+ 1| dV is finite because of equation (9). Therefore, Theorem 1.1 proves the Uniformization in this case. If there is no disk removed, i.e. M = S \ {p1, · · · , pk} and M has negative Euler number, then it’s proved in [14] that there exists a hyperbolic metric in the conformal class. A large part of [14] is devoted to solve (10) △g0u = Rg0 + 1 with |∇u| < ∞. Observe that the above equation is equivalent to (11) △hu = (Rg0 + 1). Since every end of (M, g0) is a hyperbolic cusp, Gauss-Bonnet theorem says Rg0dV0 = 2πχ(M) < 0. There exists a function f of compact support on M such that the volume of (M, efg0) is −2πχ(M), because (M, g0) has finite volume. Denote efg0 by g0, since the infinity is not changed, equation (12) is still true. Now, the volume of (M, g0) is −2πχ(M). This implies (Rg0 + 1)dV0 = 0. Therefore (Rg0 + 1)dVh = 0. By construction of g0, we know (Rg0+1) is zero near Pi. So (Rg0+1) is a smooth function on S. Therefore, equation (11) is solvable. Since u is a smooth function on compact surface S, u has bounded gradient with respect to h. The relation of h and g0 near Pi is explicit. It’s straight forward to check u has bounded gradient as a function of (M, g0). This simplifies the proof in [14]. 14 HAO YIN Remark 4.2. In the case that there is at least one disk removed, by construction of g0, Rg0 +1 vanishes at high order near ∂Dj. Then one can extend the definition (Rg0 + 1) to S so that (Rg0 + 1)dVh = 0. The rest is the same as in the previous case. This method of solving Poisson equation depends on the conformal structure of M , therefore Theorem 3.1 and Theorem 1.1 are not coverd by the above discussion. References [1] Chau, A., Tam, L.-F. and Yu, C., Pseudolocality for the Ricci flow and applications, preprint, DG/0701153. [2] Chow, B., The Ricci flow on the 2-sphere, J. Diff. Geom. 33(1991), 325-334. [3] Ecker, K. and Huisken, G., Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105(1991), 547-569. [4] Hamilton, R., The Ricci flow on surfaces. Mathematics and General Relativity, Contemporary Mathematics, 71(1988), 237-261. [5] Kichenassamy, S., On a conjecture of Fefferman and Graham, Adv. In Math., 184(2004), 268-288. [6] Li, P., Curvature and function theorey on Riemannian manifolds, Surveys in Differential Ge- ometry, Vol VII, International Press(2000), 375-432. [7] Li, P. and Schoen, R., Lp and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. , 153(1984), no.3-4, 279-301. [8] Li, P. and Tam, L.-F., Symmetric Green’s function on complete manifolds, Amer. J. Math., 109(1987), 1129-1154. [9] Li,P. and Yau, S.-T., On the parabolic kernel of the Schrödinger operator, Acta. Math., 156(1986), 139-168. [10] Ni, L. Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kähler manifolds, Indiana Univ. Math. Jour., 51 (2002), 670-703. [11] Ni, L. and Tam, L.-F., Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvture, J. Diff. Geom., 64(2003), 457-524. [12] Ni, L. and Tam, L.-F., Kähler Ricci flow and Poincaré Lelong equation, Comm. Anal. Geom., 12(2004), no 1. 111-141. [13] Saloff-Coste, L. Uniform elliptic operators on Riemannian manifolds, J. Diff. Geom., 36(1992), 417-450. [14] Ji, L.-Z. and Sesum, N. Uniformization of conformally finite Riemann surfaces by the Ricci flow, preprint, DG/0703357. [15] Shi, W.-X., Deforming the metric on complete Riemannian manifolds, J. Diff. Geom., 30(1989), 223-301. [16] Yin, H., Boundary regularity of harmonic maps from Hyperbolic space into nonpositively curved manifolds, to appear in Pacific. J. Math.. [17] Zhang, Q.-S., Some gradient estimates for the heat equation on domains and for an equation by Perelman, preprint, DG/0605518. 1. Introduction 2. An estimate of Green's function 3. Poisson equations u=R+1 4. Proof of the main theorem and the corollary References ABSTRACT This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool. <|endoftext|><|startoftext|> myjournal manuscript No. (will be inserted by the editor) Polarization properties of subwavelength hole arrays consisting of rectangular holes Xi-Feng Ren, Pei Zhang, Guo-Ping Guo⋆, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of China Received: date / Revised version: date Abstract Influence of hole shape on extraordinary op- tical transmission was investigated using hole arrays con- sisting of rectangular holes with different aspect ratio. It was found that the transmission could be tuned con- tinuously by rotating the hole array. Further more, a phase was generated in this process, and linear polar- ization states could be changed to elliptical polarization states. This phase was correlated with the aspect ratio of the holes. An intuitional model was presented to explain these results. PACS numbers:78.66.Bz,73.20.MF, 71.36.+c 1 introduction In metal films perforated with a periodic array of sub- wavelength apertures, it has long been observed that there is an unusually high optical transmission[1]. It is ⋆ E-mail: e-mail: :gpguo@ustc.edu.cn believed that metal surface plays a crucial role and the phenomenon is mediated by surface plasmon polaritons (SPPs) and there is a process of transforming photon to SPP and back to photon[2,3,4]. This phenomenon can be used in various applications, for example, sen- sors, optoelectronic device, etc[5,6,7,8,9,10]. Polariza- tion properties of nanohole arrays have been studied in many works[11,12,13]. Recently, orbital angular momen- tum of photons was explored to investigate the spatial mode properties of surface plasmon assisted transmis- sion [14,15]. It is also showed that entanglement of pho- ton pairs can be preserved when they respectively travel through a hole array [15,16,17]. Therefore, the macro- scopic surface plasmon polarizations, a collective excita- tion wave involving typically 1010 free electrons propa- gating at the surface of conducting matter, have a true quantum nature. However, the increasing use of EOT requires further understanding of the phenomenon. http://arxiv.org/abs/0704.0854v2 2 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo The polarization of the incident light determines the mode of excited SPP which is also related to the periodic structure. For the manipulation of light at a subwave- length scale with periodic arrays of holes, two ingredi- ents exist: shape and periodicity[2,3,4,11,18,19,20]. In- fluence of unsymmetrical periodicity on EOT was dis- cussed in [21]. Influence of the hole shape on EOT was also observed recently[18,20], in which the authors mainly focused on the transmission spectra. In this work, we used rectangle hole arrays to investigate the influence of hole shape on the polarization properties of EOT. It is found that linear polarization states could be changed to elliptical polarization states and a phase could be added between two eigenmode directions. The phase was changed when the aspect ratio of the rectangle holes was varied. The hole array was also rotated in the plane per- pendicular to the illuminate beam. The optical transmis- sion was changed in this process. It strongly depended on the rotation angle, in other words, the angle between polarization of incident light and axis of hole array, as in the case with unsymmetrical hole array structure[21]. 2 experimental results and modeling 2.1 Relation between transmission efficiency and photon polarization Fig. 1(a) is a scanning electron microscope picture of part of our hole arrays. The hole arrays are produced as follows: after subsequently evaporating a 3-nm tita- nium bonding layer and a 135-nm gold layer onto a 0.5- mm-thick silica glass substrate, a focused ion beam etch- ing system is used to produce rectangle holes (100nm× 100nm, 100nm × 150nm, 100nm × 200nm, 100nm × 300nm respectively) arranged as a square lattice (520nm period). The area of the hole array is 10µm× 10µm. Transmission spectra of the hole arrays were recorded by a silicon avalanche photodiode single photon counter couple with a spectrograph through a fiber. White light from a stabilized tungsten-halogen source passed though a single mode fiber and a polarizer (only vertical polar- ized light can pass), then illuminated the sample. The hole arrays were set between two lenses of 35mm focal length, so that the light was normally incident on the hole array with a cross sectional diameter about 10µm and covered hundreds of holes. The light exiting from the hole array was launched into the spectrograph. The hole arrays were rotated anti-clockwise in the plane per- pendicular to the illuminating light, as shown in Fig. (a) (b) Fig. 1 (Color online)The rectangle hole arrays. (a) Scanning electron microscope pictures. (b) Rotation direction. S (L) is the axis of short (long) edge of rectangle hole; H(V) is horizontal (vertical) axis. Polarization properties of subwavelength hole arrays consisting of rectangular holes 3 1(b). Transmission spectra of the hole arrays for rota- tion angle θ = 0o and 90o were given in Fig. 2. There were large difference between the two cases, which was also observed in [18]. Further, the typical hole array(100nm×300nm holes) was rotated anti-clockwise in the plane perpendicular to the illuminating light(see Fig.1 (b)). Transmission effi- ciencies of H and V photons(702nm wavelength) were measured with rotation angle θ = 0o, 30o, 45o, 60o, and 90o respectively, as shown in Fig. 3. They were varied with θ. To explain the results, we gave a simple model. For our sample, photons with 702nm wavelength will excite the SPP eigenmodes (0,±1) and (±1, 0). Since the SPPs were excited in the directions of long (L) and short (S) edges of rectangle holes, we suspected that this 550 600 650 700 750 800 850 550 600 650 700 750 800 850 550 600 650 700 750 800 850 550 600 650 700 750 800 850 Wavelength(nm) Fig. 2 (Color online)Hole array transmittance as a function of wavelength for rotation angle θ = 0o(black square dots) and 90o(red round dots)(holes for a, b, c, and d are 100nm× 100nm, 100nm × 150nm, 100nm × 200nm, and 100nm × 300nm respectively). The dashed vertical lines indicate the wavelength of 702nm used in the experiment. two directions were eigenmode-directions for our sample. The polarization of illuminating light was projected into the two eigenmode-directions to excite SPPs. After that, the two kinds of SPPs transmitted the holes and irritated light with different transmission efficiencies TL and TS respectively. For light whose polarization had an angle θ with the S direction, the transmission efficiency Tθ will Tθ = TS cos 2(θ) + TL sin 2(θ). (1) This equation was also given in the works[20,21]. Due to the unequal values of TL and TS, the whole transmis- sion efficiency was varied with angle θ. So if we know the transmission spectra for enginmode-directions (here L and S), we can calculate out the transmission spectra (including the heights and locations of peaks) for any 0 15 30 45 60 75 90 10000 20000 30000 40000 50000 60000 70000 Tilt angle (degree) Fig. 3 (Color online)Transmittance as a function rotation angle θ for photons in 702nm wavelength(100nm × 300nm holes). Red round dots and black square dots are the counts for V and H photons respectively. The lines come from the- oretical calculation. 4 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo θ. The theoretical calculations were also given in Fig. 3, which agreed well with the experimental data. The similar results were also observed when the hole arrays (100nm× 150nm and 100nm× 200nm) were used. With this model, the transmission efficiency can be continu- ously tuned in a certain range. 2.2 Influence of hole shape on photon polarization To investigate the polarization property of the hole ar- ray, we used the method of polarization state tomog- raphy. Experimental setup was shown in Fig. 4. White light from a stabilized tungsten-halogen source passed though single mode fiber and 4nm filter (center wave- length 702 nm) to generate 702nm wavelength photons. Polarization of input light was controlled by a polarizer, a HWP (half wave plate, 702nm) and a QWP (quar- ter wave plate, 702nm). The hole array was set between two lenses of 35mm focal length. Symmetrically, a QWP, a HWP and a polarizer were combined to analyze the polarization of transmitted photons. For arbitrary in- put states, the output states were measured in the four bases: H , V , 1/ 2(|H〉 + |V 〉), and 1/ 2(|H〉 + i|V 〉). With these experimental data, we could get the density matrix of output states, which gave the full polarization characters of transmitted photons. For example, in the case of θ = 0o, for input state 1/ 2(|H〉 + eI∗0.5π|V 〉), four counts (8943, 31079, 3623 and 21760) were recorded when we used the four detection bases. The density ma- trix was calculated as: 0.223 −0.410− 0.043i −0.410 + 0.043i 0.777 , (2) which had a fidelity of 0.997 with the pure state 0.472|H〉+ 0.882eI∗0.967π|V 〉. Compared this state with the input state, we found that not only the ratio of |H〉 and |V 〉 was changed, but also a phase ϕ = 0.467π was added between them. The similar phenomenon was also ob- served when the input state was 1/ 2(|H〉 + |V 〉) and in this case ϕ = 0.442π. We also considered the cases for θ = 30o, 45o, 60o, and 90o. The experimental density matrices had the fidelities all larger than 0.960 with the theoretical calculations, where ϕ = (0.462 ± 0.053)π. It can be seen that the phase ϕ was hardly influenced by the rotation. To study the dependence of phase ϕ with the hole shape, we performed the same measurements on other hole arrays which were shown in Fig. 1. It was found that ϕ was changed with the aspect ratio of the rectan- Filter Detector HA SMF Source SMF Polarization Controller Polarization Analyzer Fig. 4 (Color online)Experimental setup to investigate the polarization property of our rectangle hole array. Polariza- tion of input light was controlled by a polarizer, a HWP and a QWP. The hole array was set between two lenses of 35mm focal length. Symmetrically, a QWP, a HWP and a polar- izer were combined to analyze the polarization of transmitted photons. Polarization properties of subwavelength hole arrays consisting of rectangular holes 5 gle holes. Fig. 5 gave the relation between ϕ and aspect ratio. The phases are 0, (0.227±0.032)π, (0.357±0.020)π and (0.462±0.053)π for aspect ratio 1, 1.5, 2.0 and 3.0 re- spectively. As mentioned above, period is another impor- tant parameter in the EOT experiments. Since no similar result was observed for hole arrays with symmetrical pe- riods, a special quadrate hole array(see Fig. 1 of [21]) was also investigated to show the influence of the hole period. We found that even the periods were different in two di- rections, there was no birefringent phenomenon(ϕ = 0). This birefringent phenomenon might be explained with the propagating of SPPs on the metal surface. As we know, the interaction of the incident light with sur- face plasmon is made allowed by coupling through the grating momentum and obeys conservation of momen- k sp = k 0 ± i Gx ± j Gy, (3) 1.0 1.5 2.0 2.5 3.0 Aspect ratio Fig. 5 (Color online)Relation between birefringent phase ϕ and hole shape aspect ratio. ϕ becomes lager when the aspect ratio increases. where k sp is the surface plasmon wave vector, k 0 is the component of the incident wave vector that lies in the plane of the sample, Gx and Gy are the reciprocal lattice vectors, and i, j are integers. Usually, Gx = Gy = 2π/d for a square lattice, and relation k sp ∗ d = mπ was sat- isfied, where m was the band index[22]. While for our rectangle hole arrays, the length of holes in L direction was changed form 150nm to 300nm, which was not as same as it in S direction. Though Gx = Gy = 2π/d for our rectangle hole array, the time for surface plasmon polariton propagating in the L direction must be influ- enced by the aspect ratio of hole shape, which could not be same as that in the S direction. A phase difference ϕ was generated between the two directions, leading the birefringent phenomenon. Due to the absorption or scat- tering of the SPPs and scattering at the hole edges, it is hard to give the accurate value of the phase or the exact relation between the phase and aspect ratio of holes. Even so, ϕ could be controlled by changing the hole shape. As a contrast, there was no birefringent phe- nomenon observed when the quadrate hole array(see Fig. 1 of [21]) was used. The reason was that phase Gx ∗ dx always equal to Gy ∗ dy, even Gx 6= Gy for the quadrate hole array. 3 conclusion In conclusion, rectangle hole array was explored to study the influence of hole shape on EOT, especially the prop- erties of photon polarization. Because of the unsymmet- 6 Xi-Feng Ren, Pei Zhang, Guo-Ping Guo, Yun-Feng Huang, Zhi-Wei Wang, Guang-Can Guo rical of the hole shape, a birefringent phenomenon was observed. The phase was determined by the hole shape, which gave us a potential method to control this bire- fringent process. It was also found that the transmission efficiency can be tuned continuously by rotating the hole array. These results might be explained using an intu- itional model based on surface plasmon eigenmodes. This work was funded by the National Fundamental Research Program, National Natural Science Foundation of China (10604052), Program for New Century Excel- lent Talents in University, the Innovation Funds from Chinese Academy of Sciences, the Program of the Educa- tion Department of Anhui Province (Grant No.2006kj074A). Xi-Feng Ren also thanks for the China Postdoctoral Sci- ence Foundation (20060400205) and the K. C. Wong Ed- ucation Foundation, Hong Kong. References 1. T.W. Ebbesen, H. J. Lezec, H. 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X. F. Ren, G. P. Guo, Y. F. Huang, Z. W. Wang, and G. C. Guo, Appl. Phys. Lett. 90, 161112 (2007). 22. F. L. Tejeira, S. G. Rodrigo, L. M. Moreno, F. J. G. Vi- dal, E. Devaux, T. W. Ebbesen, J. R. Krenn, I. P. Radko, S. I.Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, Nature Physics 3, 324 (2007). introduction experimental results and modeling conclusion ABSTRACT Influence of hole shape on extraordinary optical transmission was investigated using hole arrays consisting of rectangular holes with different aspect ratio. It was found that the transmission could be tuned continuously by rotating the hole array. Further more, a phase was generated in this process, and linear polarization states could be changed to elliptical polarization states. This phase was correlated with the aspect ratio of the holes. An intuitional model was presented to explain these results. <|endoftext|><|startoftext|> Introduction Cooling and trapping alkaline-earth atoms offer interest- ing alternatives to alkaline atoms. Indeed, the singlet- triplet forbidden lines can be used for optical frequency measurement and related subjects [1]. Moreover, the spin- less ground state of the most abundant bosonic isotopes can lead to simpler or at least different cold collisions prob- lems than with alkaline atoms [2]. Considering fermionic isotopes, the long-living and isolated nuclear spin can be controlled by optical means [3] and has been proposed to implement quantum logic gates [4]. It has also been shown that the ultimate performance of Doppler cooling can be greatly improved by using narrow transitions whose pho- ton recoil frequency shifts ωr are larger than their natural widths Γ [5]. This is the case for the 1S0 →3 P1 spin- forbidden line of Magnesium (ωr ≈ 1100Γ ) or Calcium (ωr ≈ 36Γ ). Unfortunately, both atomic species can not be hold in a standard magneto-optical trap (MOT) be- cause the radiation pressure force is not strong enough to overcome gravity. This imposes the use of an extra quenching laser as demonstrated for Ca [6]. For Stron- tium, the natural width of the intercombination transition (Γ = 2π×7.5 kHz) is slightly broader than the recoil shift (ωr = 2π×4.7 kHz). The radiation pressure force is higher than the gravity but at the same time the final tempera- ture is still in the µK range [7,8]. In parallel, the narrow transition partially prevents multiple scattering processes and the related atomic repulsive force [10]. Hence impor- tant improvements on the spatial density have been re- ported [7]. However, despite experimental efforts, such as adding an extra confining optical potential, pure optical methods have not allowed yet to reach the quantum de- generacy regime with Strontium atoms [9]. In this paper, we will discuss some performances, es- sentially in terms of temperatures, sizes and loading rates, of a Strontium 88 MOT using the 689 nm 1S0 →3P1 in- tercombination line. Initially the atoms are precooled in a MOT on the spin-allowed 461 nm 1S0 →1P1 transition (natural width Γ = 2π × 32MHz) as discussed in [11]. Then the atoms are transferred into the 689 nm intercombination MOT. To achieve a high loading rate, Katori et al. [7] have used laser spectrum, broadened by frequency modulation. Thus the velocity capture range of the 689 nm MOT matches the typical velocity in the 461 nm MOT. They report a transfer efficiency of 30%. The same value of transfer effi- ciency is also reported in reference [8]. In our set-up, 50% of the atoms initially in the blue MOT are transferred into the red one. In section 3 we present a systematic study of the transfer efficiency as function of the parameters of the frequency modulation. In order to discuss the intrin- sic limitations of the loading efficiency, we compare our experimental results to a simple model. In particular, we demonstrated that our transfer efficiency is limited by the size of the red MOT beams. We show that it could be op- timized up to 90% with realistic laser power (25mW per beams). The minimum temperature achieved in the broadband MOT is about 2.5µK. In order to reduce the tempera- ture down to the photon recoil limit (0.5µK), we apply a second cooling stage, using a single frequency laser and observe similar temperatures, detuning and intensity de- pendencies as reported in the literature (see references [7], http://arxiv.org/abs/0704.0855v2 2 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line [8], [12] and [13]). In those publications, the role of gravity on the cooling and trapping dynamics along the z vertical direction has been discussed. In this paper we compare the steady state behaviour along vertical (z) direction to that along the horizontal plane (x−y) where gravity plays indirectly a crucial role (section 4). Details about the dynamics are given in references [8],[12]. In particular the authors establish three regimes. In regime (I) the laser detuning |δ| is larger than the power-broadened linewidth ΓE = Γ (1 + s). Regime (II) on the contrary corresponds to ΓE > |δ|. In both regimes (I) and (II) ΓE ≫ Γ, ωr and the semiclassical limit is a good approximation. In regime (III) the saturation pa- rameter is small and a full quantum treatment is required. We will focus here on the semiclassical regime (I). In this regime, we confirm that the temperature along the z di- rection is independent of the detuning δ. Following Loftus et al. [12], we have also found (see section 4.1) that this behavior is due to the balance of the gravitational force and the radiation pressure force produced by the upward pointing laser (the gravity defining the downward direc- tion). The center of mass of the atomic cloud is shifted downward from the magnetic field quadrupole center. As a consequence, cooling and trapping in the horizontal plane occur at a strong bias magnetic field mostly perpendicu- lar to the cooling plane. This unusual situation is studied in detail (section 4.2). Despite different friction and dif- fusion coefficients along the horizontal and the vertical directions, the horizontal temperature is found to be the same as the vertical one (see section 4.3). In reference [12], the trapping potential is predicted to have a box shape whose walls are given by the laser detuning. This is in- deed the case without a bias magnetic field along the z axis. It is actually different for the regime (I) described in this paper. Here we have found that the trapping poten- tial remain harmonic. This leads to a cloud width in the horizontal direction which is proportional to |δ| (section 4.2). 2 Experimental set-up Our blue MOT setup (on the broad 1S0 →1 P1 transi- tion at 461 nm) is described in references [14,15]. Briefly, it is composed by six independent laser beams typically 10mW/cm2 each. The magnetic field gradient is about 70G/cm. The blue MOT is loaded from an atomic beam extracted from an oven at 550 ◦C and longitudinally slowed down by a Zeeman slower. The loading rate of our blue MOT is of 109 atoms/s and we trap about 2.106 in a 0.6mm rms radius cloud when no repumping lasers are used [16]. To optimize the transfer into the red MOT, the temperature of the blue MOT should be as small as possi- ble. As previously observed [11], this temperature depends strongly on the optical field intensity. We therefore de- crease the intensity by a factor 5 (see figure 1) 4ms before switching off the blue MOT. The rms velocity right before the transfer stage is thus reduced down to σb = 0.6m/s whereas the rms size remains unchanged. Similar two stage cooling in a blue MOT is also reported in reference [13]. The 689 nm laser source is an anti-reflection coated laser diode in a 10 cm long extended cavity, closed by a diffraction grating. It is locked to an ULE cavity using the Pound-Drever-Hall technique [17]. The unity gain of the servo loop is obtained at a frequency of 1MHz. From the noise spectrum of the error signal, we derive a frequency noise power. It shows, in the range of interest, namely 1Hz − 100 kHz, an upper limit of 160 Hz2/Hz which is low enough for our purpose. The transmitted light from the ULE cavity is injected into a 20mW slave laser diode. Then the noise components at frequencies higher than the ULE cavity cut-off (300 kHz) are filtered. It is important to note that the lateral bands used for the lock-in are also removed. Those lateral bands, at 20MHz from the carrier, are generated modulating directly the current of the master laser diode. A saturated spectroscopy set-up on the 1S0 →3P1 intercombination line is used to compensate the long term drift of 10−50Hz/s mainly due to the daily temperature change of the ULE cavity. The slave beam is sent through an acousto-optical mod- ulator mounted in a double pass configuration. The laser detuning can then be tuned within the range of a few hundreds of linewidth around the resonance. This acousto- optical modulator is also used for frequency modulation (FM) of the laser, as required during the loading phase (see section 3). The red MOT is made of three retroreflected beams with a waist of 0.7 cm. The maximum intensity per beam is about 4mW/cm2 (the saturation intensity being Is = 3µW/cm2). The magnetic gradient used for the red MOT is varied from 1 to 10G/cm. To probe the cloud (number of atoms and tempera- ture) we use a resonant 40µs pulse of blue light (see fig 1). The total emitted fluorescence is collected onto an avalanche detector. From this measurement, we deduce the number of atoms and then evaluate the transfer rate into the red MOT. At the same time, an image of the cloud is taken with an intensified CCD camera. The typical spa- tial resolution of the camera is 30µm. Varying the dark period (time-of-flight) between the red MOT phase and the probe, we get the ballistic expansion of the cloud. We then derive the velocity rms value and the corresponding temperature. 3 Broadband loading of the red MOT The loading efficiency of a MOT depends strongly on the width of the transition. With a broad transition, the max- imum radiation pressure force is typically am = 104 × g, where vr is the recoil velocity [18]. Hence, on l ≈ 1 cm (usual MOT beam waist) an atom with a veloc- ity vc = 2aml ≈ 30m/s can be slowed down to zero and then be captured. During the deceleration, the atom re- mains always close to resonance because the Doppler shift is comparable to the linewidth. Thus MOTs can be di- rectly loaded from a thermal vapor or a slow atomic beam using single frequency lasers. Moreover typical magnetic field gradients of few tens of G/cm usually do not dras- T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 3 tically change the loading because the Zeeman shift over the trapping region is also comparable to the linewidth. An efficient loading is more complex to achieved with a narrow transition. For Strontium, the maximum radia- tion pressure force of a single laser is only am ≈ 15 × g. Assuming the force is maximum during all the capture process, one gets vc = 2aml ≈ 1.7m/s. Hence, precool- ing in the blue MOT is almost mandatory. In that case the initial Doppler shift will be vcλ −1 ≈ 2.5MHz, 300 times larger than the linewidth. In order to keep the laser on resonance during the capture phase, the red MOT lasers must thus be spectrally broadened. Because of the low value of the saturation intensity, the spectral power den- sity can easily be kept large enough to maintain a maxi- mum force with a reasonable total power (few milliwatts). The magnetic field gradient of the MOT may also affect the velocity capture range. To illustrate this point, let us consider an atom initially in the blue MOT at the center of the trap with a velocity vc = 1.7m/s. During the decel- eration, the Doppler shift decreases whereas the Zeeman shift increases. However, the magnetic field gradient does not affect the capture velocity as far as the total shift (Doppler+Zeeman) is still decreasing. This condition is fulfilled if the magnetic field gradient is lower than [19]: λgeµbvc ≈ 0.6G/cm (1) where ge = 1.5 is the Landé factor of the 3P1 level and µb = 1.4MHz/G is the Bohr magneton. In practice we use a magnetic field gradient which is larger than bc. In that case, it is necessary to increase the width of the laser spec- trum so that the optimum transfer rate is not limited by the Zeeman shift (see section 3.2). An alternative solution may consist of ramping the magnetic field gradient during the loading [7]. 3.1 Transfer rate: experimental results In this section we will present the experimental results re- garding the loading efficiency of the red MOT from the blue MOT. To optimize the transfer rate, the laser spec- trum is broadened using frequency modulation (FM). Thus the instantaneous laser detuning is ∆(t) = δ+∆ν. sin νmt. ∆ν and νm are the frequency deviation and modulation frequency respectively, δ is the carrier detuning. Here, the modulation index ∆ν/νm is always larger than 1, thus the so-called wideband limit is well fulfilled. Hence one can assume the FM spectrum to be mainly enclosed in the interval [δ −∆ν; δ +∆ν]. As shown in figure 2, the transfer rate increases with νm up to 15 kHz where we observe a plateau at 45% trans- fer efficiency. On the one hand when νm is larger than the linewidth, the atoms are in the non-adiabatic regime where they interact with all the Fourier components of the laser spectrum. Moreover, the typical intensity per Fourier component remains always higher than the saturation in- tensity Is = 3µW/cm 2. As a consequence, the radiation pressure force should be close to its maximum value for any atomic velocity. On the other hand when νm < Γ/2π, the atoms interact with a chirped intense laser where the mean radiation pressure force (over a period 2π/νm) is clearly smaller than in the case νm > Γ/2π. As a conse- quence, the transfer rate is reduced when νm decreases. In figure 3, the transfer rate is measured as a func- tion of ∆ν. The carrier detuning is δ = −1MHz and the modulation frequency is kept larger than the linewidth (νm = 25 kHz). Starting from no deviation (∆ν = 0), we observe (fig. 3) an increase of the transfer rate with ∆ν (in the range 0 < ∆ν < 500 kHz). After reaching its maximum value, the transfer rate does not depend on ∆ν anymore. Thus the capturing process is not limited by the laser spectrum anymore. If we further increase the frequency deviation ∆ν, the transfer becomes less efficient and fi- nally decreases again down to zero. This reduction occurs as soon as ∆ν > |δ|, i.e. some components of the spectrum are blue detuned. This frequency configuration obviously should affect the MOT steady regime adding extra heat- ing at zero velocity (see section 3.3). We can see that it is also affecting the transfer rate. To confirm that point, figure 4 shows the same experiment but with a larger de- tuning δ = −1.5MHz and δ = −2MHz for the figures 4a and 4b respectively. Again the transfer rate decreases as soon as ∆ν > |δ|. The transfer rate is also very small on the other side for small values of ∆ν. In that case the entire spectrum of the laser is too far red detuned. The radiation pressure forces are significant only for velocities larger than the capture velocity and no steady state is expected. Keeping now the deviation fixed and varying the detuning as shown in figure 5, we observe a maximum transfer rate when the detuning is close to the deviation frequency ∆ν ≃ |δ|. Closer to resonance (∆ν < |δ|), the blue detuned components prevent an efficient loading of the MOT. The magnetic field gradient plays also a crucial role for the loading. We indeed observe (fig. 6) that the transfer rate decreases when the magnetic field gradient increases. At very low magnetic field (b < 1G/cm) the reduction of the transfer rate is most likely due to a lack of stabil- ity within the trapping region. In that case we actually observe a strong displacement of the center of mass of the cloud. This is induced by imperfections of the set-up such as non-balanced laser intensities which are critical at low magnetic gradient. Hence, the optimum magnetic field gradient is found to be the smallest one which ensure the stability of the cloud in the MOT. 3.2 Theoretical model and comparison with the experiments To clearly understand the limiting processes of the transfer rate, we compare the experimental data to a simple 1D theoretical model based on the following assumptions: - An atom undergoes a radiation pressure force and thus a deceleration if the modulus of its velocity is between vmax and vmin with vmax = λ(|δ|+∆ν), vmin = max{λ(|δ|−∆ν);λ(−|δ|+∆ν)} 4 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line am = 0 elsewhere. We simply write that the Doppler shift is contained within the FM spectrum. We add the condi- tion vmin = λ(−|δ|+∆ν) when some components are blue detuned ∆ν > |δ|. In this case, we consider the simple ideal situation where the two counter-propagating lasers are assumed perfectly balanced and then compensate each other in the spectral overlapping region. - Even in the semiclassical model, it is difficult to calculate the acceleration as a function of the velocity for a FM spec- trum. However for all the data presented here, the satura- tion parameter is larger than one. Hence the deceleration is set to a constant value − 1 am when vmin < |v| < vmax. The prefactor 1/3 takes into account the saturation by the 3 counter-propagating laser beam pairs. - The magnetic field gradient is included by giving a spa- tial dependence of the detuning δ in the expression (2). - An atom will be trapped if its velocity changes of sign within a distance shorter that the beam waist. In figures 3-6 the results of the model are compared to the experimental data. The agreement between the model and the experimental data is correct except at large fre- quency deviation (figures 3 and 4) or at low detuning (fig- ure 5). In those cases the spectrum has some blue detuned components. As mentionned before, this is a complex sit- uation where the assumptions of the simple model do not hold anymore. Fortunately those cases do not have any practical interest because they do not correspond to the optimum transfer efficiency. At the optimum, the model suggests that the transfer is limited by the beam waist (see caption of figures (3-6)). Moreover for all the situation explored in figures 3-5, the magnetic field gradient is strong enough (b = 1G/cm) to have an impact on the capture process, as suggested by the inequality (1). However it is not the transfer limiting factor because the Zeeman shift is easily compensated by a larger frequency excursion or by a larger detuning. Increasing the beam waist would definitely improve the transfer efficiency as showed in figure 7. If the saturation parameter would remain large for all values of beam waist, more than 90% of the atoms would be transferred for a 2 cm beam waist. 25mW of power per beam should be sufficient to achieve this goal. In our experimental set-up, the power is limited to 3mW per beam. So the satura- tion parameter is necessarily reduced once the waist is increased. To take this into account and get a more realis- tic estimation of the efficiency for larger beams, we replace the previous acceleration by the expression ams/(1 + 3s), with s = I/Is the saturation parameter per beam. In this case, the transfer efficiency becomes maximum at 70% for a beam waist of 1.5 cm. 3.3 Temperature Cooling with a broadband FM spectrum on the intercom- biaison line decreases the temperature by three orders of magnitude in comparison with the blue MOT: from 3mK (σb = 0.6m/s) to 2.5µK (see figure 8). For small detuning, the temperature is strongly increasing when the spectrum has some blue detuned components (∆ν > |δ|). Indeed the cooling force and heating rate are strongly modified at the vicinity of zero detuning. This effect is illustrated in figure 8. On the other side at large detuning (δ < −1.5MHz), the temperature becomes constant. This regime corresponds to a detuning independent steady state, as also observed in single frequency cooling (see ref. [12] and section 4). 4 Single frequency cooling About half of the atoms initially in the 461 nm MOT are recaptured in the red one using a broadband laser. The final temperature is 2.5µK i.e. 5 times larger than the photon recoil temperature Tr = 460 nK. To further de- crease the temperature one has to switch to single fre- quency cooling (for time sequences: see figure 1). As we will see in this section, the minimum temperature is now about 600 nK close to the expected 0.8Tr in an 1D mo- lasses [5]. Moreover, one has to note that, under proper conditions described in reference [12], the transfer between the broadband and the single frequency red MOT can be almost lossless. In the steady state regime of the single frequency red MOT, one has kσv ≈ ωr ≈ Γ . Thus, there is no net sepa- ration of different time scales as in MOTs operated with a broad transition where ωr << kσv << Γ . However, here the saturation parameter s always remains high. It cor- responds to the so-called regimes (I) and (II) presented in reference [12]. Thus ωr << Γ 1 + s and the semiclas- sical Doppler theory describes properly the encountered experimental situations. To insure an efficient trapping, the parameter’s val- ues of the single frequency red MOT are different from a usual broad transition MOT: the magnetic field gradi- ent is higher, typically 1000Γ/cm. Moreover the gravity is not negligible anymore by comparison with the typical radiation pressure. Those features lead to an unusual be- havior of the red MOT as we will explain in this section. We will first independently analyze the MOT properties along the vertical dimension (section 4.1) then in the hor- izontal plane (section 4.2), to finally compare those two situations (section 4.3). 4.1 Vertical direction In the regime (I) i.e. at large negative detuning and high saturation (see examples on figure 9a) the temperature is indeed constant. As explained in reference [12], this be- havior is due to the balance between the gravity and the radiation pressure force of the upward laser. At large neg- ative detuning, the downward laser is too far detuned to give a significant contribution. In the semiclassical regime, an atom undergoes a net force of Fz = h̄k 1 + sT + 4(δ − geµBbz − kvz)2/Γ 2 −mg (3) Considering the velocity dependence of the force, the first order term is: T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 5 Fz ≈ −γzvz (4) γz = −4 h̄k2δeff (1 + sT + 4δ /Γ 2)2 where the effective detuning δeff = δ − geµBb < z > is define such as 1 + sT + 4δ = mg (6) sT is the total saturation parameter including all the beams. < z > is the mean vertical position of the cold cloud. Hence δeff is independent of the laser detuning δ and the vertical temperature at larger detuning depends only on the intensity as shown in figures 9a and 9b. The spatial properties of the cloud are also related to the effective detuning δeff which is independent of δ. The mean vertical position depends linearly on the detuning, so that one has : d < z > geµBb The predicted vertical displacement is compared to the experimental data in figure 10a. The agreement is excel- lent (the only adjustable parameter is the unknown origin of the vertical axe). Because the radiation pressure force for an atom at rest does not depend on the laser detuning δ, the vertical rms size should be also δ-independent. This point is also verified experimentally (see figure 10b). 4.2 x− y horizontal plane Let us now study the behavior of the cold cloud in the x−y plane at large laser detuning. As explained in section 4.1, the position of the cloud is vertically shifted downward with respect to the center of the magnetic field quadrupole (see figure 11). The dynamic in the x−y plane occurs thus in the presence of a high bias magnetic field. To derive the expression of the semiclassical force in this unusual situation one has first to project the circular polarizations states of the horizontal lasers on the eigenstates. We define the quantification axis along the magnetic field, one gets: e+x = 1 + sinα cosα√ 1− sinα e−x = 1− sinα cosα√ 1+ sinα where e−i , πi and e i represent respectively the left-handed, linear and right-handed polarisations along the i axis. The angle α between the vertical axis and the local magnetic field is shown on figure 11. For large detuning, α is al- ways small (α ≪ 1 ) and we write α ≈ −x/ < z > considering only the dynamics along the x dimension. For simplicity the magnetic field gradient b is considered as spatially isotropic with b > 0 as sketched on figure 11b. The expression of the radiation pressure force is then: Fx = h̄k ×(10) s(1− sinα)2/4 1 + sT + 4(δ − geµBb < z > (1− tanα)− kvx)2/Γ 2 s(1 + sinα)2/4 1 + sT + 4(δ − geµBb < z > (1− tanα) + kvx)2/Γ 2 Note that this expression is not restricted to the small α values. We expect six terms in the expression (11): three terms for each laser corresponding to the three e− and e+ polarisation eigenstates. However only two terms, corresponding to the e+ state, are close to resonance and thus have a dominant contribution. As for the vertical dimension, the off resonant terms are removed from the expression (11). One has also to note that the effective detuning δeff = δ − geµBb < z > is actually the same as the one along the vertical dimension. The first order expansion of (11) in α and kvx/Γ gives the expression of the horizontal radiation pressure force: Fx ≈ −καα− γxvx = −κxx− γxvx (11) κα = − < z > κx = h̄k 1 + sT + 4δ = mg (12) = −2 h̄k 2δeff (1 + sT + 4δ /Γ 2)2 As for the vertical dimension (equation (6)), the force depends on δeff but at the position of the MOT does not depend on the laser detuning δ. Hence, at large detuning, the horizontal temperature depends only on the intensity as observed in figures 9a and 9b. To understand the trapping mechanisms in the x − y plane, we now consider an atom at rest located at a po- sition x 6= 0 (corresponding to α 6= 0), i.e. not in the center of the MOT. The transition rate of two counter- propagating laser beam is not balanced anymore. This is due to the opposite sign in the α dependency of the prefac- tor in expression (11). This mechanism leads to a restoring force in the x−y plane at the origin of the spatial confine- ment (equation 11). Applying the equipartition theorem one gets the horizontal rms size of the cloud: x2rms = = −< z > kBT Without any free adjusting parameter, the agreement with experimental data is very good as shown in figure 10b. On the other hand there’s no displacement of the center of mass in the x − y plane whatever is the detuning δ as long as the equilibrium of the counter-propagating beams intensities is preserved (figure 10a). 6 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 4.3 Comparing the temperatures along horizontal and vertical axes As seen in sections 4.1 and 4.2, gravity has a dominant impact on cooling in a MOT operated on the intercom- bination line not only along the vertical axe but also in the horizontal plane. Even so we expect different behav- iors along this directions essentially because the gravity renders the trapping potential anisotropic. This is indeed the case for the spatial distribution (figures 10a and 10b) whereas the temperatures are surprisingly the same (fig- ures 9a and 9b). We will now give few simple arguments to physically explain this last point. In the semiclassical approximation, the temperature is defined as the ratio between the friction and the diffusion term: kBTi = Dabsi +D with i = x, y, z (15) Dabs and Dspo correspond to the diffusion coefficients in- duced by absorption and spontaneous emission events re- spectively. The friction coefficients has been already de- rived (equation 13): γz = 2γx,y (16) Indeed cooling along an axe in the x − y plane results in the action of two counter-propagating beams four times less coupled than the single upward laser beam. The same argument holds for the absorption term of the diffusion coefficient: Dabsz = 2D x,y (17) The spontaneous emission contribution in the diffusion co- efficient can be derived from the differential cross-section dσ/dΩ of the emitting dipole [20]. With a strong biased magnetic field along the vertical direction, this calcula- tion is particularly simple as e+z is the only quasi resonant state. Hence dσ/dΩ ∝ (1 + cosφ2) (18) φ is the angle between the vertical axe and the direction of observation. After a straightforward integration, one finds a contribution again two times larger along the vertical Dspoz = 2D x,y (19) From those considerations, the temperature is expected to be isotropic as observed experimentally (see figures 9a and 9b). In the so-called regime (I), the minimum temperature is given by the semiclassical Doppler theory: T = NR s (20) Where NR is a numerical factor which should be close to two [12]. This solution is represented in figure 9 by a dashed line nicely matching the experimental data for s > 8 but with NR = 1.2. Similar results, i.e. with unex- pected low NR values, have been found in [12]. For s ≤ 8 we observed a plateau in the final temperature slightly higher than the low saturation theoretical prediction [5]. We cannot explain why the temperature does not decrease further down as reported in [12]. For quantitative compar- ison with the theory, more detailed studies in a horizontal 1D molasses are required. 4.4 Conclusions Cooling of Strontium atoms using the intercombination line is an efficient technique to reach the recoil temper- ature in three dimensions by optical methods. Unfortu- nately loading from a thermal beam cannot be done di- rectly with a single frequency laser because of the nar- row velocity capture range. We have shown experimentally that more than 50% of the atoms initially in a blue MOT on the dipole-allowed transition are recaptured in the red MOT using a frequency-broadened spectrum. Using a sim- ple model, we conclude that the transfer is limited by the size of the laser beam. If the total power of the beams at 689 nm was higher, transfer rates up to 90% could be expected by tripling our laser beam size. The final tem- perature in the broadband regime is found to be as low as 2.5µK, i.e. only 5 times larger than the photon recoil temperature. The gain in temperature by comparison to the blue MOT (1−10mK) is appreciable. So in absence of strong requirements on the temperature, broadband cool- ing is very efficient and reasonably fast (less than 100ms). The requirements for the frequency noise of the laser are also much less stringent than for single frequency cooling. Using a subsequent single frequency cooling stage, it is possible to reduce the temperature down to 600 nK, slightly above the photon recoil temperature. Analyzing the large detuning regime, we particularly focus our stud- ies on the comparison between vertical and horizontal di- rections. We show how gravity indirectly influences the horizontal parameters of the steady state MOT and find that the trapping potential remains harmonic along all directions, but with an anisotropy. Gravity has a major impact on the MOT as it coun- terbalances the laser pressure of the upward laser (making the steady state independent of the detuning). We show that gravity thus affects the final temperature, which re- mains isotropic, despite different cooling dynamics along the vertical and horizontal directions. 5 Acknowledgments The authors wish to thank J.-C. Bernard and J.-C. Bery for valuable technical assistances. This research is finan- cially supported by the CNRS (Centre National de la Recherche Scientifique) and the former BNM (Bureau Na- tional de Métrologie) actually LNE (Laboratoire national de métrologie et d’essais) contract N◦ 03 3 005. T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 7 References 1. F. Ruschewitz, J. L. Peng, H. Hinderthr, N. Schaffrath, K. Sengstock, and W. Ertmer, Phys. Rev. Lett. 80, 3173 (1998); G. Ferrari, P. Cancio, R. Drullinger, G. Giusfredi, N. Poli, M. Prevedelli, C. Toninelli, and G. M. Tino Phys. Rev. Lett. 91, 243002 (2003); M. Yasuda and H. Katori Phys. Rev. Lett. 92, 153004 (2004); T. Ido, T. H. Loftus, M. M. Boyd, A. D. Lud- low, K. W. Holman, and J. Ye Phys. Rev. Lett. 94, 153001 (2005); R. Le Targat, X. Baillard, M. Fouch, A. Brusch, O. Tcherbakoff, G. D. Rovera, and P. Lemonde Phys. Rev. Lett. 97, 130801 (2006). 2. J. Weiner, V. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. 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Wiley and sons, third edition New York, 1999). Blue MOT Laser Red MOT Laser Red MOT Laser Magnetic field gradient 70 G/cm 1-10 G/cm Du~k vbD 70 ms40 ms 80 ms Fig. 1. Time sequence and cooling stages of Strontium with the dipole-allowed transition and with the intercombination line. 0 5 10 15 20 25 30 35 40 45 Modulation frequency (kHz) Fig. 2. Transfer rate as a function of the modulation frequency. The other parameters are fixed: P = 3mW, δ = −1000 kHz, b = 1G/cm and ∆ν = 1000 kHz http://arxiv.org/abs/quant-ph/0609111 8 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 0 500 1000 1500 2000 2500 3000 Frequency deviation (kHz) Fig. 3. Transfer rate as a function of the frequency deviation (squares). The other parameters are fixed: P = 3mW, δ = −1000 kHz, b = 1G/cm and νm = 25 kHz. The dash and solid line correspond to a simple model prediction (see text). The transfer rate is limited by the frequency deviation of the broad laser spectrum for the dash line and by the waist of the MOT beam for the solid line. 0 500 1000 1500 2000 2500 3000 Frequency deviation (kHz) Frequency deviation (kHz) 0 500 1000 1500 2000 2500 3000 (a) (b) Fig. 4. Transfer rate as a function of the frequency deviation (squares). δ = −1500 kHz and δ = −2000 kHz for (a) and (b) respectively, the other parameters and the definitions are the same than for figure 3. 0 500 1000 1500 2000 2500 Detuning (kHz) Fig. 5. Transfer rate as a function of the detuning (squares). The other parameters are fixed: P = 3mW, ∆ν = 1000 kHz, b = 1G/cm and νm = 25 kHz. The dashed and solid lines have the same signification than in figure 3. 0 1 2 3 4 5 6 7 8 9 10 b (G/cm) Fig. 6. Transfer rate as a function of the magnetic gradient (squares). The other parameters are fixed: P = 3mW, δ = −1000 kHz, ∆ν = 1000 kHz and νm = 25 kHz. The transfer rate is limited by the waist of the MOT beam for all values. The dotted lines represent the case where the magnetic field gradient do not affect the deceleration. 0 1 2 3 4 5 Beam waist (cm) Fig. 7. Transfer rate as a function of the beam waist. The solid lines correspond to a high saturation parameter where as the dash line correspond to a constant power of P = 3mW. The other parameters are fixed: δ = −1000 kHz, ∆ν = 1000 kHz and b = 0.1G/cm. -2000 -1500 -1000 -500 δ (kHz) Fig. 8. Measured temperature as a function of the detuning for a FM spectrum. The other parameters are fixed: P = 3mW, b = 1G/cm, ∆ν = 1000 kHz and νm = 25 kHz T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 9 10 100 1000 -80 -60 -40 -20 0 Detuning (kHz) Fig. 9. Measured temperature as a function of the detuning (a) with I = 4Is or I = 15Is and as a function of the intensity (b) with δ = −100 kHz of single frequency cooling. The circles (respectively stars) correspond to temperature along one of the horizontal (respectively vertical) axis. The magnetic field gradient is b = 2.5G/cm -700 -600 -500 -400 -300 -200 -100 0 Detuning (kHz) -700 -600 -500 -400 -300 -200 -100 0 Detuning (kHz) Fig. 10. Displacement (a) and rms radius (b) of the cold cloud in single frequency cooling along the z axis (star) and in the x−y plane (circle). The intensity per beam is I = 20Is and the magnetic gradient b = 2.5G/cm along the strong axis in the x− y plane. The linear displacement prediction correspond to the plain line (graph a). In graph b, the plain curve correspond to the rms radius prediction based on the equipartition theorem. 10 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line d=-1000kHzd=-100kHz Cloud Quantization axe Fig. 11. (a) Images of the cold cloud in the red MOT. The cloud position for δ = −100 kHz coincides roughly with the center of the MOT whereas it is shifted downward for δ = −1000 kHz. The spatial position of the resonance correspond dot circle. (b) Sketch representing the large detuning case. The coupling efficiency of the MOT lasers is encoded in the size of the empty arrow. The laser form below has maximum efficiency whereas the one pointing downward is absent because is too detuned. Along a horizontal axe, the lasers are less coupled because they do not have the correct polarization. The α angle is the angular position of an atom M with respect to O, the center of the MOT. Introduction Experimental set-up Broadband loading of the red MOT Single frequency cooling Acknowledgments ABSTRACT The intercombination line of Strontium at 689nm is successfully used in laser cooling to reach the photon recoil limit with Doppler cooling in a magneto-optical traps (MOT). In this paper we present a systematic study of the loading efficiency of such a MOT. Comparing the experimental results to a simple model allows us to discuss the actual limitation of our apparatus. We also study in detail the final MOT regime emphasizing the role of gravity on the position, size and temperature along the vertical and horizontal directions. At large laser detuning, one finds an unusual situation where cooling and trapping occur in the presence of a high bias magnetic field. <|endoftext|><|startoftext|> Approximate Selection Rule for Orbital Angular Momentum in Atomic Radiative Transitions I.B. Khriplovich and D.V. Matvienko Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia, and Novosibirsk University Abstract We demonstrate that radiative transitions with ∆l = −1 are strongly dominating for all values of n and l, except small region where l ≪ n. It is well-known that the selection rule for the orbital angular momentum l in electro- magnetic dipole transitions, dominating in atoms, is ∆l = ±1, i. e. in these transitions the angular momentum can both increase and decrease by unity. Meanwhile, the classical radiation of a charge in the Coulomb field is always accompanied by the loss of angular momentum. Thus, at least in the semiclassical limit, the probability of dipole transitions with ∆ l = − 1 is higher. Here we discuss the question how strongly and under what exactly conditions the transitions with ∆l = −1 dominate in atoms. (To simplify the presentation, we mean always, here and below, the radiation of a photon, i. e. transitions with ∆n < 0. Obviously, in the case of photon absorption, i. e. for ∆n > 0, the angular momentum predominantly increases.) The analysis of numerical values for the transition probabilities in hydrogen presented in [1] has demonstrated that even for n and l, comparable with unity, i. e. in a nonclassical situation, radiation with ∆l = −1 can be much more probable than that with ∆l = 1. Later, the relation between the probabilities of transitions with ∆l = −1 and ∆l = 1 was investigated in [2] by analyzing the corresponding matrix elements in the semiclassical approximation. The conclusion made therein is also that the transitions with ∆l = −1 dominate, and the dominance is especially strong when l > n2/3. Here we present a simple solution of the problem using the classical electrodynamics and, of course, the correspondence principle. Our results describe the situation not only in the semiclassical situation. Remarkably enough, they agree, at least qualitatively, with the results of [1], although the latter refer to transitions with |∆n| ∼ n ∼ 1 and l ∼ 1, which are not classical at all. We start our analysis with a purely classical problem. Let a particle with a mass m and charge − e moves in an attractive Coulomb field, created by a charge e, along an ellipse with large semi-axis a and eccentricity ε. It is known [3] that the radiation intensity at a given harmonic ν is here 4e2ω4 ξ2ν + η ; (1) J ′ν(νε), ην = 1− ε2 Jν(νε). (2) In expressions (2), Jν(νε) is the Bessel function, and J ν(νε) is its derivative. We use the Fourier transformation in the following form: x(t) = a iνω0t = 2a ξν cos νω0t, http://arxiv.org/abs/0704.0856v1 y(t) = a iνω0t = 2a ην sin νω0t, where all dimensionless Fourier components ξν and ην are real, and ξ−ν = ξν , η−ν = − ην . We note that the Cartesian coordinates x and y are related here to the polar coordinates r and φ as follows: x = r cos φ, y = r sin φ, where φ increases with time. Thus, the angular momentum is directed along the z axis (but not in the opposite direction). We note also that, since 0 ≤ ε ≤ 1, both Jν(νε) and J ′ν(νε) are reasonably well approximated by the first term of their series expansion in the argument. Therefore, all the Fourier components ξν and ην are positive. In the quantum problem (where ν = |∆n|), the probability of transition in the unit time is h̄ω0ν 4e2ω3 3c3h̄ ξ2ν + η , ω0 = h̄2n3 . (3) Now, the loss of angular momentum with radiation is [3] r× r... . Going over here to the Fourier components, we obtain Ṁν = − 4e2ω2 rν × ṙν , or (with our choice of the direction of coordinate axes, and with the angular momentum measured in the units of h̄) Ṁν = − 4e2ω3 3c3h̄ 2ξνην . (4) Obviously, the last expression is nothing but the difference between the probabilities of transitions with ∆l = 1 and ∆l = −1 in the unit time: Ṁν = W ν −W−ν . (5) Of course, the total probability (3) can be written as Wν = W ν . (6) From explicit expressions (3) and (4) it is clear that inequality W+ν ≪ W−ν holds if 2ξνην ≈ ξ2ν + η2ν , or ην ≈ ξν . The last relation is valid for ε ≪ 1, i. e. for orbits close to circular ones. (The simplest way to check it, is to use in formulae (2) the explicit expression for the Bessel function at small argument: Jν(νε) = (νε) ν/(2νν !).) This conclusion looks quite natural from the quantum point of view. Indeed, it is the state with the orbital quantum number l equal to n − 1 (i. e. with the maximum possible value for given n) which corresponds to the circular orbit. In result of radiation n decreases, and therefore l should decrease as well. The surprising fact is, however, that in fact the probabilities W−ν of transitions with ∆l = −1 dominate numerically everywhere, except small vicinity of the maximum possible eccentricity ε = 1. For instance, if ε ≃ 0.9 (which is much more close to 1 than to 0 !), then at ν = 1 the discussed probability ratio is very large, it constitutes ≃ 12 . The change with ε of the ratio of W+ν to W ν for two values of ν is illustrated in Fig. 1. The curves therein demonstrate in particular that with the increase of ν, the region 0.2 0.4 0.6 0.8 1.0 Ε �������� ���� Fig. 1 where W−ν and W ν are comparable, gets more and more narrow, i. e. when ν grows, the corresponding curves tend more and more to a right angle. Let us go over now to the quantum problem. In the semiclassical limit, the classical expression for the eccentricity is rewritten with usual relations E = −me4/(2h̄2n2) and M = h̄l as . (8) In fact, the exact expression for ε, valid for arbitrary l and n, is [3]: l(l + 1) + 1 . (9) Clearly, in the semiclassical approximation the eccentricity is close to unity only under condition l ≪ n. If this condition does not hold, one may expect that in the semiclas- sical limit the transitions with ∆l = −1 dominate. In other words, as long as l ≪ n, the probabilities of transitions with decrease and increase of the angular momentum are comparable. But if the angular momentum is not small, it is being lost predominantly in radiation. This situation looks quite natural. The next point is that with the increase of |∆n| = ν, the region where W−ν and W+ν are comparable, gets more and more narrow in agreement with the observation made in However, we do not see any hint at some special role (advocated in [2]) of the condition l > n2/3 for the dominance of transitions with ∆l = −1. As mentioned already, the analysis of the numerical values of transition probabilities [1] demonstrates that even for n and l comparable with unity and |∆n| ≃ n, i. e. in the absolutely nonclassical regime, the transitions with ∆l = −1 are still much more probable than those with ∆l = 1. The results of this analysis for the ratio W−/W+ in some transitions are presented in Table 3.1 (first line). Then we indicate in Table 3.1 (last line) W4p→3s W4p→3d W5p→4s W5p→4d W5d→4p W5d→4f W6f→5d W6f→5g W5p→3s W5p→3d W6p→3s W6p→3d exact value 10 3.75 28 72 10.67 13.7 ε̄ 0.87 0.92 0.81 0.75 0.90 0.92 ν = |∆n| 1 1 1 1 2 3 semiclassical value 17.6 8.7 34 58 17.2 15.7 Table 3.1 the values of these ratios obtained in the näıve (semi)classical approximation. Here for the eccentricity ε̄ we use the value of expression (9), calculated with l corresponding to the initial state; as to n, we take its value average for the initial and final states. The table starts with the smallest possible quantum numbers where the transitions, which differ by the sign of ∆l, occur, i. e. with the ratio W4p→3s/W4p→3d. This table demonstrates that the ratio of the classical results to the exact quantum-mechanical ones remains everywhere within a factor of about two. In fact, if one uses as ε̄ expression (8), calculated in the analogous way, the numbers in the last line change considerably. It is clear, however, that the classical approximation describes here, at least qualitatively, the real situation. References [1] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957; §63. [2] N.B. Delone and V.P. Krainov, FIAN Preprint No. 18, 1979; J. Phys. B 27, (1994) 4403. [3] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Nauka, 1973; §70, problem 2 to §72. [4] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Nauka, 1974; §36. ABSTRACT We demonstrate that radiative transitions with \Delta l = - 1 are strongly dominating for all values of n and l, except small region where l << n. <|endoftext|><|startoftext|> Introduction Being a ‘close relative’ of General Relativity (GR), Absolute Parallelism (AP) has many interesting features: larger symmetry group of equations; field irreducibility with respect to this group; vast list of compatible second order equations (discovered by Einstein and Mayer [1]) not restricted to Lagrangian ones. There is the only variant of Absolute Parallelism which solutions are free of arising singularities, if D=5 (there is no room for changes; this variant of AP does not have a Lagrangian, nor match GR); in this case AP has topological features of nonlinear sigma-model. In order to give clear presentation and full picture of the theory’ scope, many items should be sketched: instability of trivial solution and expanding O4-symmetrical ones; tensor Tµν (positive energy, but only three polarizations of 15 carry (and angular) momentum; how to quantize such a stuff ?) and PN-effects; topological classification of symmetric 5D field configurations (alighting on evident parallels with Standard Model’ particle combinatorics) and ‘quantum phenomenology on expanding classical background’ (coexistence); ‘plain’ R2-gravity on very thick brane and change in the Newton’s Law: 1 goes to 1 with distance (not with acceleration – as it is in MOND [2]). At last, an experiment with single photon interference is discussed as the other way to observe very-very long (and very undeveloped) the extra dimension. 2 Unique 5D equation of AP (free of singularities in solutions) There is one unique variant of AP (non-Lagrangian, with the unique D; D=5) which solutions of general position seem to be free of arising singularities. The formal integrability test [3] can be ∗E-mail: zhogin at inp.nsk.su; http://zhogin.narod.ru http://arxiv.org/abs/0704.0857v1 http://zhogin.narod.ru extended to the cases of degeneration of either co-frame matrix, haµ, (co-singularities) or contra- variant frame (or contra-frame density of some weight), serving as the local and covariant (no coordinate choice) test for singularities of solutions. In AP this test singles out the next equation (and D=5, see [4]; ηab = diag(−1, 1, . . . , 1), then h = det haµ = Eaµ = Laµν;ν − 13(faµ + LaµνΦν) = 0 , (1) where (see [4] for more detailed introduction to AP and explanation of notations used) Laµν = La[µν] = Λaµν − Saµν − 23ha[µΦν], Λaµν = 2ha[µ,ν], Sµνλ = 3Λ[µνλ], Φµ = Λaaµ, fµν = 2Φ[µ,ν] = 2Φ[µ;ν]. (2) Coma ”,” and semicolon ”;” denote partial derivative and usual covariant differentiation with symmetric Levi-Civita connection, respectively. One should retain the identities [which follow from the definitions (2)]: Λa[µν;λ] ≡ 0 , haλΛabc;λ ≡ fcb (= fµνhµchνb ), f[µν;λ] ≡ 0. (3) The equation Eaµ;µ = 0 gives ‘Maxwell-like equation’ (we prefer to omit g µν (ηab) in contrac- tions that not to keep redundant information – when covariant differentiation is in use only): (faµ + LaµνΦν);µ = 0, or fµν;ν = (SµνλΦλ);ν (= −12Sµνλfνλ, see below) . (4) Actually the Eq. (4) follows from the symmetric part of equation, E(ab), because skewsymmetric one gives just the identity: 2E[νµ] = Sµνλ;λ = 0, E[µν];ν ≡ 0; note also that the trace part becomes irregular (the principal derivatives vanish) if D = 4 (this number of dimension is forbidden, and the next number, D = 5, is the most preferable): Eµµ = Eaµh ab = 4−D Φµ;µ + (Λ 2) = 0. The system (1) remains compatible under adding fµν = 0, see (4); this is not the case for another covariant, S,Φ, or (some irreducible part of the) Riemannian curvature, which relates to Λ as usually: Raµνλ = 2haµ;[ν;λ]; haµhaν;λ = Sµνλ − Λλµν . 3 Tensor Tµν (despite Lagrangian absence) and PN-effects One might rearrange E(µν)=0 that to pick out (into LHS) the Einstein tensor, Gµν =Rµν− 12gµνR, but the rest terms are not proper energy-momentum tensor: they contain linear terms Φ(µ;ν) (no positive energy ( !); another presentation of ‘Maxwell equation’ (4) is possible instead – as divergence of symmetrical tensor). However, the prolonged equation E(µν);λ;λ = 0 can be written as ‘plain’ (no R-term) R 2-gravity: (−h−1 δ(hRµνGµν)/δgµν=) Gµν;λ;λ +Gǫτ(2Rǫµτν − 12gµνRǫτ ) = Tµν(Λ ′2, . . .), Tµν;ν = 0; (5) up to quadratic terms, Tµν = 2 − fµλfνλ) + Aµǫντ (Λ2);(ǫ;τ) + (Λ2Λ′,Λ4); tensor A has symmetries of Riemann tensor, so the term A′′ adds nothing to momentum and angular momentum. It is worth noting that: (a) the theory does not match GR, but shows ‘plain’ R2-gravity (sure, (5) does not contain all the theory); (b) only f -component (three transverse polarizations in D=5) carries D-momentum and an- gular momentum (‘powerful’ waves); other 12 polarizations are ‘powerless’, or ‘weightless’ (this is a very unusual feature – impossible in the Lagrangian tradition; how to quantize ? let us not to try this, leaving the theory ‘as is’); (c) f -component feels only metric and S-field (‘contorsion’, not ‘torsion’ Λ – to label somehow), see (4), but S has effect only on polarization of f : S[µνλ] does not enter eikonal equation, and f moves along usual Riemannian geodesic (if background has f=0); one may think that all ‘quantum fields’ (phenomenological quantized fields accounting for topological (quasi)charges and carrying some ‘power’; see further) inherit this property; (d) the trace Tµµ = fµνfµν can be non-zero if f 2 6= 0 and this seemingly depends on S- component [which enters the current in (4)]; in other words, ‘mass distribution’ is to depend on distribution of f - and S-component; (e) it should be stressed and underlined that the f -component is not usual (quantum) EM- field – just important covariant responsible for energy-momentum (suffice it to say that there is no gradient invariance for f). 4 Linear domain: instability of trivial solution (with powerless waves) Another strange feature is the instability of trivial solution: some ‘powerless’ polarizations grow linearly with time in presence of ‘powerful’ f -polarizations. Really, from the linearized Eq. (1) and the identity (3) one can write (the following equations should be understood as linearized): Φa,a = 0 (D 6= 4), 3Λabd,d = Φa,b − 2Φb,a, Λa[bc,d],d ≡ 0 ⇒ 3Λabc,dd = −2fbc,a . The last ‘D‘Alembert equation’ has the ‘source’ in its right hand side. Some components of Λ (most symmetrical irreducible parts) do not grow (as well as curvature), because (again, linearized equations are implied below) Sabc,dd = 0, Φa,dd = 0, fab,dd = 0, Rabcd,ee = 0, but the least symmetrical components of the tensor Λ do grow up with time (due to terms ∼ t e−iωt; three growing polarizations which are ‘imponderable’, or powerless) if the ‘ponderable’ waves (three f -polarizations) do not vanish (and this should be the case for solutions of ‘general position’). 5 Expanding O4-symmetrical (single wave) solutions and cosmology The unique symmetry of AP equations gives scope for symmetrical solutions. In contrast to GR, this variant of AP has non-stationary spherically (O4-) symmetric solutions. The O4-symmetric frame field can be generally written as follows [4]: haµ(t, x a bni cni eninj + d∆ij ; i, j = (1, 2, 3, 4), ni = . (6) Here a, . . . , e are functions of time, t = x0, and radius r, ∆ij = δij −ninj, r2 = xixi. As functions of radius, b, c are odd, while the others are even; other boundary conditions: e = d at r = 0, and haµ → δ aµ as r → ∞. Placing in (6) b = 0, e = d (the other interesting choice is b=c=0) and making integrations one can arrive to the next system (resembling dynamics of Chaplygin gas; dot and prime denote derivation on time and radius, resp.; A = a/e = e1/2, B = −c/e): A· = AB′ −BA′ + 3AB/r , B · = AA′ − BB′ − 2B2/r . (7) This system (does not suffer of gradient catastrophe and) has non-stationary solutions; a single- wave solution of proper ‘amplitude’ might serve as a suitable cosmological (expanding) background. The condition fµν=0 is a must for solutions with such a high symmetry (as well as Sµνλ=0); so, these O4-solutions carry no energy, that is, weight nothing (some lack of gravity ! in this theory the universe expansion seemingly has little common with gravity, GR and its dark energy [5]). More realistic cosmological model might look like a single O4-wave (or a sequence of such waves) moving along the radius and being filled with chaos, or stochastic waves, both powerful (weak, ∆h≪ 1) and powerless (∆h < 1, but intense enough that to lead to non-linear fluctuations with ∆h ∼ 1), which form statistical ensemble(s) having a few characteristic parameters (after ‘thermalization’). The development and examination of stability of such a model is an interesting problem. The metric variation in cosmological O4-wave can serve as a time-dependent ‘shallow dielectric guide’ for that weak noise waves. The ponderable waves (which slightly ‘decelerate’ the O4-wave) should have wave-vectors almost tangent to the S 3-sphere of wave-front that to be trapped inside this (‘shallow’) wave-guide; the imponderable waves can grow up, and partly escape from the wave-guide, and their wave-vectors can be less tangent to the S3-sphere. The waveguide thickness can be small for an observer in the center of O4-symmetry, but in co- moving coordinates it can be very large (due to relativistic effect), however still small with respect to the radius of sphere, L≪ R. It seems that the radial dimension has to be very ‘undeveloped’; that is, there are no other characteristic scales, smaller than L, along this extra-dimension. 6 Non-linear domain: topological charges and quasi-charges Let AP-space is of trivial topology: no worm-holes, no compactified space dimensions, no singu- larities. One can continuously deform frame field h(x) to a field of rotation matrices (metric can be diagonalized and ‘square-rooted’) haµ(x) → saµ(x) ∈ SO(1, d); m=D−1. Further deformation can remove boosts too, and so, for any space-like (Cauchy) surface, this gives a (pointed) map, s : Rm ∪∞ = Sm → SOm; ∞ 7→ 1m ∈ SOm. The set of such maps consists of homotopy classes forming the group of topological charge, Π(m): Π(m) = πm(SOm); Π(3) = Z, Π(4) = Z2 + Z2. (8) Here Z is the infinite cyclic group, and Z2 is the cyclic group of order two. It is important that deformation to s-field can keep symmetry of field configuration. Definition: localized field (pointed map) s(x) : Rm → SO(m), s(∞) = 1m, is G-symmetric if, in some coordinates, s(σx) = σs(x)σ−1 ∀ σ ∈ G ⊂ O(m) . (9) The set of such fields C(m)G generally consists of separate, disconnected components – homotopy classes forming the ‘topological quasi-charge group’ denoted here as Π(G;m) ≡ π0(C(m)G ). These QC-groups classify symmetrical localized configurations of frame field. Since field equation does not break symmetry, quasi-charge conserves; if symmetry is not exact (because of distant regions), quasi-charge is not exactly conserving value, and quasi-particle (of zero topological charge) can annihilate (or be created) during colliding with another quasi-particle. The other problem. Let G1 ⊃ G2, such that there is a mapping (embedding) i : C(m)G1 → C which induces the homomorphism of QC-groups: i∗ : Π(G1;m) → Π(G2;m), so one has to describe this morphism. Let us consider the simple (discreet) symmetry group P1 with a plane of reflection symmetry: P1 = {1, p(1)}, where p(1) = diag(−1, 1, . . . , 1) = p−1(1). It is necessary to set field s(x) on the half-space 1 Rm = {x1 ≥ 0}, with additional condition imposed on the surface Rm−1 = {x1 = 0} (stationary points of P1 group) where s has to commute with the symmetry [see (9)]: p(1)x = x ⇒ s(x) = p(1)sp(1) ⇒ s ∈ 1× SOm−1. Hence, accounting for the localization requirement, we have a diad map (relative spheroid; here Dm is anm-ball and Sm−1 its surface) (Dm;Sm−1) → (SOm;SOm−1), and topological classification of such maps leads to the relative (or diad) homotopy group ([6]; the last equality below follows due to fibration SOm/SOm−1 = S m−1): Π(P1;m) = πm(SOm;SOm−1) = πm(S m−1). Similar considerations (of group orbits and stationary points) lead to the following result: Π(Ol;m) = πm−l+1(SOm−l+1;SOm−l) = πm−l+1(S m−l). If l > 3, there is the equality: Π(SOl;m) = Π(Ol;m), while for l = 2, 3 one can find: Π(SO3;m) = πm−2(SO2 × SOm−2;SOm−3) = πm−2(S1 × Sm−3), Π(SO2;m) = πm−1(SOm;SOm−2 × SO2) = πm−1(RG+(m, 2)). The set of quaternions with absolute value one, H1 = {f, |f| = 1}, forms a group under quaternion multiplication, H1 ∼= SU2 = S3, and any s ∈ SO4 can be represented as a pair of such quaternions [6], (f , g) ∈ S3(l) × S3(r), |f | = |g| = 1: x∗ = sx ⇔ x∗ = f x g−1 = f x ḡ ; |x| = |x∗|. The pairs (f,g) and (–f, –g) correspond to the same rotation s, that is, SO4 = S (l) × S3(r)/±. Note that the symmetry condition (9) also splits into two parts: f(axb−1) = af(x)a−1, g(axb−1) = bg(x)b−1 ∀(a,b) ∈ G ⊂ SO4. (10) 7 Example of SO2-symmetric quaternion field Let’s consider an example of SO2{2, 3}−symmetric f–field configuration (g=1), which carries both charge and SO2-quasi-charge (left, of course), f(x): H = R 4 → H1; f(∞) = 1. The symmetry condition (10) reads f(eiφ/2xe−iφ/2) = eiφ/2f(x)e−iφ/2. (11) We’ll switch to ‘double-axial’ coordinates: x = aeiϕ + beiψj. Let us use imaginary quaternions q as stereogrphic coordinates on H1, and take symmetrical field q(x) consistent with Eq. (11): q(x) = x i x̄+ i = −q̄, f(x) = − 1 + q . (12) It is easy to find the ‘center of quasi-soliton’ (1-submanifold, S1) S1 = f−1(−1) = q−1(0) = {a = 0, b = 1} = {x0(ψ) = eiψj} and the ‘vector equipment’ on this circle: dx|x0 = da eiϕ + (db+ i dψ)eiψj, 14df = idb− k ei (ϕ+ψ)da ; i-vector all time looks along the radius b (parallel translation along the circle S1; this is a ‘trivial‘, or ‘flavor’-vector). Two others (’phase’-vectors) make 2π−rotation along the circle. In fact, the field (12) has also symmetry SO2{1, 4}, and this feature restricts possible directions of ‘flavor’-vector (two ‘flavors’ are possible, ±; the P2{1, 4}−symmetry (this is the π-rotation of x1, x4) gives the same effect). The other interesting observation is that the equipped circle can be located also at the stationary points of SO2−symmetry (this increases the number of ‘flavors’). 8 Quasi-charges and their morphisms (in 5D, ie m = 4) If G ⊂ SO4, the QC-group has two isomorphous parts, left and right: Π(G) = Π(l)(G) + Π(r)(G). The Table below describes quasi-charge groups for G ⊂ G0 = (O3 × P4) ∩ SO4 (P4 is spatial inversion, the 4-th coordinate is the extra dimension of G0-symmetric expanding cosmological background). Table. QC-groups Π(l)(G) and their morphisms to the preceding group; G ⊂ G0. G Πl(G) → Πl(G∗) ‘label’ SO{1, 2} Z(e) e→ Z2 e SO{1, 2} × P{3, 4} Z(ν) + Z(H) i,m2→ Z(e) ν0; H0 → e + e SO{1, 2} × P{2, 3} Z(W ) 0→ Z(e) W → e + ν0 SO{1, 2} × P{2, 4} Z(Z) 0→ Z(e) Z0 → e+ e SO{1, 2} × P{3, 4} × Z(γ) 0→ Z(H) γ0 → H0 +H0 ×P{2, 3} 0→ Z(W ) →W +W ‘Quasi-particles’, which symmetry includes P4, seem to be true neutral (neutrinos, Higgs particles, photon). One can assume further that an hadron bag is a specific place where G0−symmetry does not work, and the bag’s symmetry is isomorphous to O4. This assumption can lead to another classification of quasi-solitons (some doubling the above scheme), where self-dual and anti-self- dual one-parameter groups take place of SO2−group. The total set of quasi-particle parameters (parameters of equipped 1-manifold (loop) plus parameters of group) for (anti)self-dual groups, G(4, 2)×RP 2, is larger than the analogous set for groups SO2 ⊂ G0, which is just O3×G(3, 1) = RP 2 . If the number of ‘flavor’-parameters (which are not degenerate and have some preferable particular values; this should be sensitive to discreet part of G – at least photons have the same flavor) is the same as in the case of ‘white’ quasi-particles, the remaining parameters (degenerate, or ‘phase’) can give room for ‘color’ (in addition to spin). So, perhaps one might think about ‘color neutrinos’ (in the context of pomeron, and baryon spin puzzle), ‘color W, Z, and Higgs’ (another context – B-mesons), and so on. Note that in this picture the very notion of quasi-particle depends on the background symmetry (also to note: there are no ’quanta of torsion’ per se). On the other hand, large clusters of quasi-particles (matter) can disturb the background, and waves of such small disturbances (with wavelength larger than the thickness L, perhaps) can be generated as well (but these waves do not carry (quasi)charges, that is, are not quantized). 9 Coexistence: phenomenological ‘quantum fields’ on classical back- ground The non-linear, particle-like field configurations with quasi-charges (quasi-particles) should be very elongated along the extra-dimension (all of the same size L), while being small sized along usual dimensions, λ≪ L. The motion of such a spaghetti-like quasi-particle should be very complicated and stochastic due to ‘strong’ imponderable noise, such that different parts of spaghetti are coming their own paths. At the same time, quasi-particle can acquire ‘its own’ energy–momentum – due to scattering of ponderable waves (which wave-vectors are almost tangent to usual 3D (sub)space); so, it seems that scattering amplitudes1 of those spaghetti’s parts which have the same 3D– coordinates can be summarized providing an auxiliary, secondary field. So, the imponderable waves provides stochasticity (of motion of spaghetti’s parts), while the ponderable waves ensure superposition (with secondary fields). Phenomenology of secondary fields could be of Lagrangian type, with positive energy acquired by quasi-particles, – that to ensure the stability (of all the waveguide with its infill – with respect to quasi-particle production; the least action principle has deep concerns with Lyapunov stability and is deducible, in principle, from the path integral approach). 10 ‘Plain’ R2 gravity on very thick brane and change in the Newton’s Law of Gravitation Let us start with 4d (from 5D) bi-Laplace equation with a δ-source [as weak field, non-relativistic (stationary) approximation (it is assumed that ‘mass is possible’) for R2-gravity (5)] and its solution (R is 4d distance, radius): ∆2ϕ = − a δ(R); ϕ(R2) = lnR2 − b (+ c , but c does not matter); (13) the attracting force between two point masses is Fpoint = , a, b should be proportional to both masses. Now let us suppose that all masses are distributed along the extra dimension with a ‘universal function’, µ(p), µ(p) dp = 1. Then the attracting (gravitation) force takes the next form [see 1 These amplitudes can depend on additional vector-parameters (‘equipment vectors’) relating to differential of field mapping at a ‘quasi-particle center’ – where quasi-charge density is largest (if it has covariant sense). 0 1 2 3 4 5 6 Fig. 1. Deviation δF = F − 1/r2 for different µ(p), see Eq. (14) and text below. (13); r is usual 3d distance]: F (r) = ϕ(r2 + (p− q)2)µ(p)µ(q) dp dq = V − b V ′, V (r) = µ(p)µ(q) dp dq r2 + (p− q)2 . (14) (Note that V (r) can be restored if F (r) is measured.) Taking µ1(p) = π −1/(1 + p2) (typical scale along the extra dimension is taken as unit, L = 1; it seems that L should be greater than ten AU), one can find rV1(r) = 1/(2 + r) and F (r) = 8 + 4r 2b(1 + r) r2(2 + r)2 ; or (now L 6= 1) F (r) = 2L(2L+ r)2 , where a = b = 2/L2. Fig. 1, curve (a) shows δF = F − 1/r2 (deviation from the Newton’s Law; a/b is chosen that δF (0)=0); two other curves, (b) & (c), correspond to µ2 = 2π −1/(1 + p2)2, µ3 = 2π −1p2/(1 + p2)2 (also δF (0)=0; residues help to find rV2 = (10 + 6r + r 2)/(2 + r)3, rV3 = (2 + 2r + r 2)/(2 + r)3). We see that in principle this theory can explain galaxy rotation curves, v2(r)∝ rF r→∞−→ const, without need for Dark Matter (or MOND [2]; about rotation curves and DM see [7]; they are looking for DM in Solar system too, [8]). Q: Can the ‘coherence of mass’ along the extra dimension be disturbed ? (the flyby anomaly, the Pioneer anomaly [9]); can µ(p) be negative in some domains of p ? 11 How to register ‘powerless’ waves This section is added perhaps for some funny recreation (or still not ? who knows). We have learnt that S-waves do not carry momentum and angular momentum, so they can not perform any work or spin flip. But let us conceive that these waves can effect a flip-flop of two neighbor spins. So, a ‘detector’ could be a media with two sorts of spins, A and B. Let sA = sB = 1/2 but gA 6= gB, and let the initial state is prepared as follows: {,}(0) = {1/2,−1/2}. Then the process of spin relaxation starts; turning on appropriate magnetic field Hz (and alternating fields of proper frequencies) one can measure the detector’s state and find the time of spin relaxation. The next step. Skilled experimenters try to generate S-waves and to register an effect of these waves on spin relaxation. The generation of intense ‘coherent’ S-waves could be proceeded perhaps with a similar spin system subjected to alternating polarization. 12 Single photon experiment (that to feel huge extra dimension), and Conclusion Today, many laboratories have sources of single (heralded) photons, or entangled bi-photons (say, for Bell-type experiments [10]); some students can perform laboratory works with single photons, having convinced on their own experience that light is quantized (the Grangier experiment)[11]. It is being suggested a minor modification of the single (polarized) photon interference exper- iment, say, in a Mach-Zehnder fiber interferometer with ‘long’ (the fibers may be rolled) enough arms. The only new element is a fast-acting shutter placed at the beginning of (one of) the inter- ferometer’s arms (the closing-opening time of the shutter should be smaller than the flight time in the arms). For example, a fast electro-optical modulator in combination with polarizer (or a number of such combinations) can be used with polarized photons. Both Quantum mechanics (no particle’s ontology) and Bohmian mechanics (wave-particle dou- ble ontology)[12] exclude any change in the interference figure as a result of separating activity of such a fast shutter (while the photon’s ‘halves’ are making their ways to the place of a meet- ing). However, if a photon has non-local spaghetti-like ontology (along the extra dimension) and fragments of this spaghetti are moving along both arms at once, then the shutter should tear up this spaghetti (mainly without photon absorption), tear out its fragments (which will dissolve in ‘zero-point oscillations’). Hence, if the absorption factor of the shutter (the extinction ratio of polarizer) is large enough, the 50/50-proportion (between the photon’s amplitudes in the arms) will be changed and a significant decrease of the interference visibility should be observed. QM is everywhere (where we can see, of course), and, so, non-linear 5D-field fluctuations, looking like spaghetti-anti-spaghetti loops, should exist everywhere. (This omnipresence can be related to the universality of ‘low-level heat death’, restricted by the presence of topological quasi- solitons – some as the 2D computer experiment by Fermi, Pasta, and Ulam, where the process of thermalization was restricted by the existence of solitons. See also the sections 5–8 (and [4]) for arguments in favor of phenomenological (quantized) ‘secondary fields’ accounting for topological (quasi)charges and obeying superposition, path integral and so on.) AP, at least at the level of its symmetry, seems to be able to cure the gap between the two branches of physics – General Relativity (with coordinate diffeomorphisms) and Quantum Mechanics (with Lorentz invariance).2 Most people give all the rights of fundamentality to quanta, and so, they are trying to quantize gravity, and the very space-time (probing loops, and strings, and branes; see also the warning polemic by Schroer [14]). The other possibility is that quanta have the specific phenomenological origin relating to topological (quasi)charges. 2Rovelli writes[13]: In spite of their empirical success, GR and QM offer a schizophrenic and confused under- standing of the physical world. References [1] A. Einstein and W. Mayer, Sitzungsber. preuss. Akad. Wiss. Kl 257–265 (1931). [2] M. Milgrom, The modified dynamics – a status review, arXiv: astro-ph/9810302. [3] J. F. Pommaret, Systems of Partial Differentiation Equations and Lie Pseudogroups (Math. and its Applications, Vol. 14, New York 1978). [4] I. L. Zhogin, Topological charges and quasi-charges in AP, arXiv: gr-qc/0610076; spherical symmetry: gr-qc/0412130; 3-linear equations (contra-singularities): gr-qc/0203008. [5] S.M. Carroll, Why is the Universe Accelerating ? arXiv: astro-ph/0310342 [6] B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry – Methods and Applica- tions, Springer-Verlag, 1984. [7] M.E. Peskin, Dark Matter: What is it ? Where is it ? Can we make it in the lab ? http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf; M. Battaglia, M.E. Peskin, The Role of the ILC in the Study of Cosmic Dark Matter, hep-ph/0509135 [8] L. Iorio, Solar System planetary orbital motions and dark matter, arXiv: gr-qc/0602095; I.B. Khriplovich, Density of dark matter in Solar system and perihelion precession of planets, astro-ph/0702260. [9] C. Lämmerzahl, O. Preuss, and H. Dittus, Is the physics within the Solar system really understood ? arXiv: gr-qc/0604052; A. Unzicker, Why do we Still Believe in Newton’s Law ? Facts, Myths and Methods in Gravitational Physics, gr-qc/0702009. [10] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998); quant-ph/9810080; W. Tittel, G. Weihs, Photonic Entanglement for Fundamental Tests and Quantum Communication, quant-ph/0107156. [11] See the next links: departments.colgate.edu/physics/research/Photon/root/ , marcus.whitman.edu/ beckmk/QM . [12] H. Nikolić, Quantum mechanics: Myths and facts, arXiv: quant-ph/0609163 . [13] C. Rovelli, Unfinished revolution, gr-qc/0604045 . [14] B. Schroer, String theory and the crisis in particle physics (a Samisdat on particle physics), arXiv: physics/0603112; the other sources of contra-string polemic are seemingly the books: P. Woit, Not even wrong; L. Smolin, The Trouble with Physics (and the blog math.columbia.edu/∼woit/wordpress). http://arxiv.org/abs/astro-ph/9810302 http://arxiv.org/astro-ph/9810302 http://arxiv.org/abs/gr-qc/0610076 http://arXiv.org/gr-qc/0610076 http://arXiv.org/gr-qc/0412130 http://arXiv.org/gr-qc/0203008 http://arxiv.org/abs/astro-ph/0310342 http://arxiv.org/astro-ph/0310342 http://www.slac.stanford.edu/grp/th/mpeskin/Yale1.pdf http://arxiv.org/hep-ph/0509135 http://arxiv.org/abs/gr-qc/0602095 http://arxiv.org/gr-qc/0602095 http://arxiv.org/astro-ph/0702260 http://arxiv.org/abs/gr-qc/0604052 http://arxiv.org/gr-qc/0604052 http://arxiv.org/gr-qc/0702009 http://arXiv.org/quant-ph/9810080 http://arXiv.org/quant-ph/0107156 http://departments.colgate.edu/%physics/research/Photon/root/photon_quantum_mechanics.htm http://marcus.whitman.edu/~beckmk/QM/ http://arxiv.org/abs/quant-ph/0609163 http://arXiv.org/gr-qc/0604045 http://arxiv.org/abs/physics/0603112 http://arXiv.org/physics/0603112 http://www.math.columbia.edu/~woit/wordpress/ Introduction Unique 5D equation of AP (free of singularities in solutions) Tensor T (despite Lagrangian absence) and PN-effects Linear domain: instability of trivial solution (with powerless waves) Expanding O4-symmetrical (single wave) solutions and cosmology Non-linear domain: topological charges and quasi-charges Example of SO2-symmetric quaternion field Quasi-charges and their morphisms (in 5D, ie m=4) Coexistence: phenomenological `quantum fields' on classical background `Plain' R2 gravity on very thick brane and change in the Newton's Law of Gravitation How to register `powerless' waves Single photon experiment (that to feel huge extra dimension), and Conclusion ABSTRACT Galactic rotation curves and lack of direct observations of Dark Matter may indicate that General Relativity is not valid (on galactic scale) and should be replaced with another theory. There is the only variant of Absolute Parallelism which solutions are free of arising singularities, if D=5 (there is no room for changes). This variant does not have a Lagrangian, nor match GR: an equation of `plain' R^2-gravity (ie without R-term) is in sight instead. Arranging an expanding O_4-symmetrical solution as the basis of 5D cosmological model, and probing a universal_function of mass distribution (along very-very long the extra dimension) to place into bi-Laplace equation (R^2 gravity), one can derive the Law of Gravitation: 1/r^2 transforms to 1/r with distance (not with acceleration). <|endoftext|><|startoftext|> Introduction During the last decade, several initiatives have been developed to monitor and collect real world data about malicious activities on the Internet, e.g., the Internet Motion Sensor project [1], CAIDA [2] and Dshield [3]. The CADHo project [4] in which we are involved is complementary to these initiatives and is aimed at: • deploying a distributed platform of honeypots [5] that gathers data suitable to analyze the attack processes targeting a large number of machines on the Internet; • validating the usefulness of this platform by carrying out various analyses, based on the collected data, to characterize the observed attacks and model their impact on security. A honeypot is a machine connected to a network but that no one is supposed to use. If a connection occurs, it must be, at best an accidental error or, more likely, an attempt to attack the machine. The first stage of the project focused on the deployment of a data collection environment (called Leurré.com [6]) based on low-interaction honeypots. As of today, around 40 honeypot platforms have been deployed at various sites from academia and industry in almost 30 different countries over the five continents. Several analyses and interesting conclusions have been derived based on the collected data as detailed e.g., in [4,5,7-9]. Nevertheless, with such honeypots, hackers can only scan ports and send requests to fake servers without ever succeeding in taking control over them. The second stage of our project is aimed at setting up and deploying high-interaction honeypots to allow us to analyze and model the behavior of malicious attackers once they have managed to compromise and get access to a new host, under strict control and monitoring. We are mainly interested in observing the progress of real attack processes and the activities carried out by the attackers in a controlled environment. In this paper, we describe the lessons learned from the development and deployment of such a honeypot. The main contributions are threefold. First, we do confirm the findings discussed in [9] showing that different sets of compromised machines are used to carry out the various stages of planned attacks. Second, we do outline the fact that, despite this apparent sophistication, the actors behind those actions do not seem to be extremely skillful, to say the least. Last, the geographical location of the machines involved in the last step of the attacks and the link with some phishing activities shed a geopolitical and socio-economical light on the results of our analysis. The paper is organized as follows. Section 2 presents the architecture of our high-interaction honeypot and the design rationales for our solution. The lessons learned from the attacks observed over a period of almost 4.5 months are discussed in Section 3. Finally, Section 4 concludes and discusses future work. An extended version of this paper detailing the context of this work and the related state-of-the art is available in [10]. 2. Architecture of our honeypot In our implementation, we decided to use VMware [11] and to install virtual operating system upon it. Compared to solutions based on physical machines, virtual honeypots provide a cost effective and flexible solution that is well suited for running experiments to observe attacks. The objective of our experiment is to analyze the behavior of the attackers who succeed in breaking into a machine. The vulnerability that they exploit is not as crucial as the activity they carry out once they have broken into the host. That's why we chose to use a simple vulnerability: weak passwords for ssh user accounts. Our honeypot is not particularly hardened for two reasons. First, we are interested in analyzing the behavior of the attackers even when they exploit a buffer overflow and become root. So, if we use some kernel patch such as Pax [12], our system will be more secure but it will be impossible to observe some behavior. Secondly, if the system is too hardened, the intruders may suspect something abnormal and then give up. In our setup, only ssh connections to the virtual host are authorized so that the attacker can exploit this vulnerability. A firewall blocks all connection attempts from the Internet, but those to port 22 (ssh). Also, any connection from the virtual host to the outside is blocked Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 to avoid that intruders attack remote machines from the honeypot. This does not prevent the intruder from downloading code, using the ssh connection1. Our honeypot is a standard Gnu/Linux installation, with kernel 2.6, with the usual binary tools. No additional software was installed except the http apache server. This kernel was modified as explained in the next subsection. The real host executing VMware uses the same Gnu/Linux distribution and is isolated from outside. In order to log what the intruders do on the honeypot, we modified some drivers functions (tty_read and tty_write), as well as the exec system call in the Linux kernel. The modifications of tty_read and tty_write enable us to intercept the activity on all the terminals of the system. The modification of the exec system call enables us to record the system calls used by the intruder. These functions are modified in such a way that the captured information is logged directly into a buffer of the kernel memory of the honeypot itself. Moreover, in order to record all the logins and passwords tried by the attackers to break into the honeypot we added a new system call into the kernel of the virtual operating system and we modified the source code of the ssh server so that it uses this new system call. The logins and passwords are logged in the kernel memory, in the same buffer as the information related to the commands used by the attackers. The activities of the intruder logged by the honeypot are preprocessed and then stored into an SQL database. The raw data are automatically processed to extract relevant information for further analyses, mainly: i) the IP address of the attacking machine, ii) the login and the password tested, iii) the date of the connection, iv) the terminal associated (tty) to each connection, and v) each command used by the attacker. 3. Experimental results This section presents the results of our experiments. First, we give global statistics in order to give an overview of the activities observed on the honeypot, then we characterize the various intrusion processes. Finally, we analyze in detail the behavior of the attackers once they break into the honeypot. In this paper, an intrusion corresponds to the activities carried out by an intruder who has succeeded to break into the system. 3.1. Global statistics The high-interaction honeypot has been deployed on the Internet and has been running for 131 days during which 480 IP addresses have tried to contact its ssh port. It is worth comparing this value to the amount of hits observed against port 22, considering all the other low- interaction honeypot platforms we have deployed in the rest of the world (40 platforms). In the average, each platform has received hits on port 22 from around approximately 100 different IPs during the same period of time. Only four platforms have been contacted by more 1 We have sometimes authorized http connections for a short time, by checking that the attackers were not trying to attack other remote hosts. than 300 different IP addresses on that port and only one was hit by more visitors than our high interaction honeypot. Even better, the low-interaction platform maintained in the same subnet as the high-interaction honeypot experimented only 298 visits, i.e. less than two thirds of what the high-interaction did see. This very simple and first observation confirms the fact already described in [9] that some attacks are driven by the fact that attackers know in advance, thanks to scans done by other machines, where potentially vulnerable services are running. The existence of such a service on a machine will trigger more attacks against it. This is what we observe here: the low interaction machines do not have the ssh service open, as opposed to the high interaction one, and, therefore get less attacked than the one where some target has been identified. The number of ssh connection attempts to the honeypot we have recorded is 248717 (we do not consider here the scans on the ssh port). This represents about 1900 connection attempts a day. Among these 248717 connection attempts, only 344 were successful. Table 1 represents the user accounts that were mostly tried (the top ten) as well as the number of different passwords that have been tested by the attackers. It is noteworthy that many user accounts corresponding to usual first names have also regularly been tested on our honeypot. The total number of accounts tested is 41530. Account Number of connection attempts Percentage of connection attempts Number of passwords tested root 34251 13.77% 12027 admin 4007 1.61% 1425 test 3109 1.25% 561 user 1247 0.50% 267 guest 1128 0.45% 201 info 886 0.36% 203 mysql 870 0.35% 211 oracle 857 0.34% 226 postgres 834 0.33% 194 webmaster 728 0.29% 170 Table 1: ssh connection attempts and number of passwords tested Before the real beginning of the experiment (approximately one and a half month), we had deployed a machine with a ssh server correctly configured, offering no weak account and password. We have taken advantage of this observation period to determine which accounts were mostly tried by automated scripts. Using this acquired knowledge, we have created 17 user accounts and we have started looking for successful intrusions. Some of the created accounts were among the most attacked ones and others not. As we already explained in the paper, we have deliberately created user accounts with weak passwords (except for the root account). Then, we have measured the time between the creation of the account and the first successful connection to this account, then the duration between the first successful connection and the first real intrusion (as explained in section 3.2, the first successful connection is very seldom a real intrusion but rather an automatic script which tests passwords). Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 Table 2 summarizes these durations (UAi means User Account i). User Account Duration between creation and first successful connection Duration between first successful connection and first intrusion UA1 1 day 4 days UA2 Half a day 4 minutes UA3 15 days 1 day UA4 5 days 10 days UA5 5 days null UA6 1 day 4 days UA7 5 days 8 days UA8 1 day 9 days UA9 1 day 12 days UA10 3 days 2 minutes UA11 7 days 4 days UA12 1 day 8 days UA13 5 days 17 days UA14 5 days 13 days UA15 9 days 7 days UA16 1 day 14 days UA17 1 day 12 days Table 2: History of breaking accounts The second column indicates that there is usually a gap of several days between the time when a weak password is found and the time when someone logs into the system with this password to issue some commands on the now compromised host. This is a somehow a surprising fact and is described with some more details here below. The particular case of the UA5 account is explained as follows: an intruder succeeded in breaking the UA4 account. This intruder looked at the contents of the /etc/passwd file in order to see the list of user accounts for this machine. He immediately decided to try to break the UA5 account and he was successful. Thus, for this account, the first successful connection is also the first intrusion. 3.2. Intrusion process In the section, we present the conclusions of our analyses regarding the process to exploit the weak password vulnerability of our honeypot. The observed attack activities can be grouped into three main categories: 1) dictionary attacks, 2) interactive intrusions, 3) other activities such as scanning, etc. Figure 3: Classification of observed IP addresses As illustrated in figure 3, among the 480 IP addresses that were seen on the honeypot, 197 performed dictionary attacks and 35 performed real intrusions on the honeypot (see below for details). The 248 IP addresses left were used for scanning activity or activity that we did not clearly identified. Among the 197 IP addresses that made dictionary attacks, 18 succeeded in finding passwords. The others (179) did not find the passwords either because their dictionary did not include the accounts we created or because the corresponding weak password had already been changed by a previous intruder. We have also represented in Figure 3 the corresponding number of IP addresses that were also seen on the low-interaction honeypot deployed in the context of the project in the same network (between brackets). Whereas most of the IP addresses seen on the high interaction honeypot are also observed on the low interaction honeypot, none of the 35 IPs used to really log into our machine to launch commands have ever been observed on any of the low interaction honeypots that we do control in the whole world! This striking result is discussed hereafter. 3.2.1. Dictionary attack. The preliminary step of the intrusion consists in dictionary attacks2. In general, it takes only a couple of days for newly created accounts to be compromised. As shown in Figure 3, these attacks have been launched from 197 IP addresses. By analysing more precisely the duration between the different ssh connection attempts from the same attacking machine, we can say that these dictionary attacks are executed by automatic scripts. As a matter of fact, we have noted that these attacking machines try several hundreds, even several thousands of accounts in a very short time. We have made then further analyses regarding the machines that succeed in finding passwords, i.e., the 18 IP addresses. By searching the leurré.com database containing information about the activities of these addresses against the other low interaction honeypots we found four important elements of information. First, we note that none of our low interaction honeypot has an ssh server running, none of them replies to requests sent to port 22. These machines are thus scanning machines without any prior knowledge on their open ports. Second, we found evidences that these IPs were scanning in a simple sequential way all addresses to be found in a block of addresses. Moreover, the comparison of the fingerprints left on our low interaction honeypots highlights the fact that these machines are running tools behaving the same way, not to say the same tool. Third, these machines are only interested in port 22, they have never been seen connecting to other ports. Fourth, there is no apparent correlation as far as their geographical location is concerned: they are located all over the world. In other words, it comes from this analysis that these IPs are used to run a well known program. The detailed analysis of this specific tool is outside the scope of the paper but, nevertheless, it is worth mentioning that the activities linked to that tool, as observed in our Leurré.com database, indicate that it is unlikely to be a worm but rather an easy to use and widely spread tool. 3.2.2. Interactive attack: intrusion. The second step of the attack consists in the real intrusion. We have noted that, several days after the guessing of a weak 2 We consider as “dictionary attack” any attack that tries more than 10 different accounts and passwords. Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 password, an interactive ssh connection is executed on our honeypot to issue several commands. We believe that, in those situations, a real human being, as opposed to an automated script, is connected to our machine. This is explained and justified in Section 4.3. As shown in Figure 3, these intrusions come from 35 IP addresses never observed on any of the low-interaction honeypots. Whereas the geographic localisation of the machines performing dictionary attacks is very blur, the machines that are used by a human being for the interactive ssh connection are, most of the time, clearly identified. We have a precise idea of their country, geographic address, the responsible of the corresponding domain. Surprisingly, these machines, for half of them, come from the same country, an European country not usually seen as one of the most attacking ones as reported, for instance, by the www.leurrecom.org web site. We then made analyses in order to see if these IP addresses had tried to connect to other ports of our honeypot except for these interactive connections; and the answer is no. Furthermore, the machines that make interactive ssh connections on our honeypot do not make any other kind of connections on this honeypot, i.e, no scan or dictionary attack. Further analyses, using the data collected from the low-interaction honeypots deployed in the CADHo project, revealed that none of the 35 IP addresses have ever been observed on any of our platforms deployed in the word. This is interesting because it shows that these machines are totally dedicated to this kind of attack (they only targeted our high- interaction honeypot and only when they knew at least one login and password on this machine). We can conclude for these analyses that we face two groups of attacking machines. The first group is composed of machines that are specifically in charge of making dictionary attacks. Then the results of these dictionary attacks are published somewhere. Then, another group of machines, which has no intersection with the first group, comes to exploit the weak passwords discovered by the first group. This second group of machines is, as far as we can see, clearly geographically identified and commands are executed by a human being. A similar two steps process was already observed in the CADHo project when analyzing the data collected from the low-interaction honeypots (see [9] for more details). 3.3. Behavior of attackers This section is dedicated to the analysis of the behavior of the intruders. We first characterize the intruders, i.e. we try to know if they are humans or programs. Then, we present in more details the various actions they have carried out on the honeypot. Finally, we try to figure out what their skill level seems to be. We concentrate the analyses on the last three months of our experiment. During this period, some intruders have visited our honeypot only once, others have visited it several times, for a total of 38 ssh intrusions. These intrusions were initiated from 16 IP addresses and 7 accounts were used. Table 3 presents the number of intrusions per account, IP addresses and passwords used for these intrusions. It is of course difficult to be sure that all the intrusions for a same account are initiated by the same person. Nevertheless, in our case, we noted that: • most of the time, after his first login, the attacker changes the weak password into a strong which, from there on, remains unchanged. • when two different IP addresses access the same account (with the same password), they are very close and belong to the same country or company. These two remarks lead us to believe that there is in general only one person associated to the intrusions for a particular account. Account Number of intrusions Number of passwords Number of IP addresses UA2 1 1 1 UA4 13 2 2 UA5 1 1 1 UA8 1 1 1 UA10 9 2 2 UA13 6 1 5 UA16 5 1 3 UA17 2 1 1 Table 3: Number of intrusions per account 3.3.1. Type of the attackers: humans or programs. Before analyzing what intruders do when connected, we can try to identify who they are. They can be of two different natures. Either they are humans, or they are programs which reproduce simple behaviors. For all intrusions but 12, intruders have made mistakes when typing commands. Mistakes are identified when the intruder uses the backspace to erase a previously entered character. So, it is very likely that such activities are carried out by a human, rather than programs. When an intruder did not make any mistake, we analyzed how the data were transmitted from the attacker machine to the honeypot. We can note that, for ssh communications, data transmission between the client and the server is asynchronous. Most of the time, the ssh client implementation uses the function select() to get user input. So, when the user presses a key, this function ends and the program sends the corresponding value to the server. In the case of a copy and a paste into the terminal running the client, the select() function also ends, but the program sends all the values contained in the buffer used for the paste into the server. We can assume that, when tty_read() returns more than one character, these values have been sent after a copy and a paste. If all the activities during a connection are due to a copy and a paste, we can strongly assume that it is due to an automatic script. Otherwise, this is quite likely a human being who uses shortcuts from time to time (such as CTRL-V to paste commands into its ssh session). For 7 out of the last 12 activities without mistakes, intruders have entered several commands on a character-by- character basis. This, once again, seems to indicate that a human being is entering the commands. For the 5 others, their activities are not significant enough to conclude: they have only launched a single command, like w, which is not long enough to highlight a copy and a paste. Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 3.3.2. Attacker activities. The first significant remark is that all of the intruders change the password of the hacked account. The second remark is that most of them start by downloading some files. In all cases, but one, the attackers tried to download some malware to the compromised machines. In a single case, the attacker has first tried to download an innocuous, yet large, file to the machine (the binary for a driver coming from a known web site). This is probably a simple way to assess the connectivity quality of the compromised host. The command used by the intruders to download the software is wget. To be more precise, 21 intrusions upon 38 include the wget command. These 21 intrusions concern all the hacked accounts. As mentioned in section 2, outgoing http connections are forbidden by the firewall. Nevertheless, the intruders still have the possibility to download files through the ssh connection using sftp command (instead of wget). Surprisingly, we noted that only 30% of the intruders did use this ssh connection. 70% of the attackers were unable to download their malware due to the absence of http connectivity! Three explanations can be envisaged at this stage. First, they follow some simplistic cookbook and do not even known the other methods at their disposal to upload a file. Second, the machines where the malware resides do not support sftp. Third, the lack of http connectivity made the attacker suspicious and he decided to leave our system. Surprisingly, the first explanation seems to be the right one in our case as we noticed that the attackers leave after an unsuccessful wget and come back a few hours or days later, trying the same command again as if they were hoping it to work at that time. Some of them have been seen trying this several times. It can be concluded that: i) they are apparently unable to understand why the command fails, ii) they are not afraid to come back to the machine despite the lack of http connectivity, iii) applying such brute force attack reveals that they are not aware of any other method to upload the file. Once the attackers manage to download their malware using sftp, they try to install it (by decompressing or extracting files for example). 75% of the intrusions that installed software did not install it on the hacked account but rather on standard directories such as /tmp, /var/tmp or /dev/shm (which are directories with write access for everybody). This makes the hacker activity more difficult to identify because these directories are regularly used by the operating system itself and shared by all the users. Additionally, we have identified four main activities of the intruders. The first one is launching ssh scans on other networks but these scans have never tested local machines. Their idea is to use the targeted machine to scan other networks, so that it is more difficult for the administrator of the targeted network to localize them. The program used by most intruders, which is easy to find on the Internet, is pscan.c. The second type of activity consists in launching irc clients, e.g., emech [13] and psyBNC. Names of binary files have regularly been changed by intruders, probably in order to hide them. For example, the binary files of emech have been changed to crond or inetd, which are well known Unix binary file names and processes. The third type of activity is trying to become root. Surprisingly, such attempts have been observed for 3 intrusions only. Two rootkits were used. The first one exploits two vulnerabilities: a vulnerability which concerns the Linux kernel memory management code of the mremap system call [14] and a vulnerability which concerns the internal kernel function used to manage process's memory heap [15]. This exploit could not succeed because the kernel version of our honeypot does not correspond to the version of the exploit. The intruder should have realized this because he checked the version of the kernel of the honeypot (uname -a). However, he launched this rootkit anyway and failed. The other rootkit used by intruders exploits a vulnerability in the program ld. Thanks to this exploit, three intruders became root but the buffer overflow succeeded only partially. Even if they apparently became root, they could not launch all desired programs (removing files for example caused access control errors). The last activity observed in the honeypot is related to phishing activities. It is difficult to make precise conclusions because only one intruder has attempted to launch such an attack. He downloaded a forged email and tried to send it through the local smtp agent. But, as far as we could understand, it looked like a preliminary step of the attack because the list of recipient emails was very short. It seems that is was just a preliminary test before the real deployment of the attack. 3.3.3. Attackers skill. Intruders can roughly speaking be classified into two main categories. The most important one is relative to script kiddies. They are inexperienced hackers who use programs found on the Internet without really understanding how they work. The next category represents intruders who are more dangerous. They are named “black hat”. They can make serious damage on systems because they are expert in security and they know how to exploit vulnerabilities on various systems. As already presented in §3.3.2. (use of wget and sftp), we have observed that intruders are not as clever as expected. For example, for two hacked accounts, the intruders don't seem to really understand the Unix file access rights (it's very obvious for example when they try to erase some files whereas they don't have the required privileges). For these two same accounts, the intruders also try to kill the processes of other users. Many intruders do not try to delete the file containing the history of their commands or do not try to deactivate this history function (this file depends on the login shell used, it is .bash_history for example for the bash). Among the 38 intrusions, only 14 were cleaned by the intruders (11 have deactivated the history function and 3 have deleted the.bash_history file). This means that 24 intrusions left behind them a perfectly readable summary of their activity within the honeypot. The IP address of the honeypot is private and we have started another honeypot on this network. This second honeypot is not directly accessible from the outside, it is only accessible from the first honeypot. We have modified the /etc/motd file of the first honeypot (which is automatically printed on the screen during the login Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 process) and added the following message: “In order to use the software XXX, please connect to A.B.C.D”. In spite of this message, only one intruder has tried to connect to the second honeypot. We could expect that an experienced hacker will try to use this information. In a more general way, we have very seldom seen an intruder looking for other active machines on the same network. One important thing to note is relative to fingerprinting activity. No intruder has tried to check the presence of VMware software. For three hacked accounts, the intruders have read the contents of the file /proc/cpuinfo but that's all. None of the methods discussed on Internet was tested to identify the presence of VMware software [16,17]. This probably means that the intruders are not experienced hackers. 4. Conclusion In this paper, we have presented the results of an experiment carried out over a period of 6 months during which we have observed the various steps that lead an attacker to successfully break into a vulnerable machine and his behavior once he has managed to take control over the machine. The findings are somehow consistent with the informal know how shared by security experts. The contributions of the paper reside in performing an experiment and rigorous analyses that confirm some of these informal assumptions. Also, the precise analysis of the observed attacks reveals several interesting facts. First of all, the complementarity between high and low interaction honeypots is highlighted as some explanations can be found by combining information coming from both set ups. Second, it appears that most of the observed attacks against port 22 were only partially automated and carried out by script kiddies. This is very different from what can be observed against other ports, such as 445, 139 and others, where worms have been designed to completely carry out the tasks required for the infection and propagation. Last, honeypot fingerprinting does not seem to be a high priority for attackers as none of them has tried the known techniques to check if they were under observation. It is also worth mentioning a couple of important missing observations. First, we did not observe scanners detecting the presence of the open ssh port and providing this information to other machines in charge of running the dictionary attack. This is different from previous observations reported in [9]. Second, as most of the attacks follow very simple and repetitive patterns, we did not observe anything that could be used to derive sophisticated scenarios of attacks that could be analyzed by intrusion detection correlation engine. Of course, at this stage it is too early to derive definite conclusions from this observation. Therefore, it would be interesting to keep doing this experiment over a longer period of time to see if things do change, for instance if a more efficient automation takes place. We would have to solve the problem of weak passwords being replaced by strong ones though, in order to see more people succeeding in breaking into the system. Also, it would be worth running the same experiment by opening another vulnerability into the system and verifying if the identified steps remain the same, if the types of attackers are similar. Could it be, at the contrary, that some ports are preferably chosen by script kiddies while others are reserved to some more elite attackers? This is something that we are in the process of assessing. Acknowledgement. This work has been partially supported by: 1) CADHo, a research action funded by the French ACI “Securité & Informatique” (www.cadho.org), 2) the CRUTIAL IST-027513 project (crutial.cesiricerca.it), and 3) the ReSIST IST- 026764 project (www.resist-noe.org). 5. References [1] M. Bailey, E. Cooke, F. Jahanian, J. Nazario, The Internet motion sensor - a distributed blackhole monitoring system. Network and Distributed Systems Security Symp. (NDSS 2005), San Diego, USA, 2005. [2] CAIDA Project. Home Page of the CAIDA Project, http://www.caida.org. [3] http://www.dshield.org. Home page of the DShield.org Distributed Intrusion Detection System. [4] E. Alata, M. Dacier, Y. Deswarte, M. Kaaniche, K. Kortchinsky, V. Nicomette, V. Hau Pham, and F. Pouget, Collection and analysis of attack data based on honeypots deployed on the Internet. QOP 2005, 1st Workshop on Quality of Protection (co-located with ESORICS and METRICS), Sept. 15, Milan, Italy, 2005. [5] F. Pouget, M. Dacier, V. Hau Pham. Leurre.com: on the advantages of deploying a large scale distributed honeypot platform. In Proc. of ECCE'05, E-Crime and Computer Conference, Monaco, 2005. [6] Home page of Leurré.com: http://www.leurre.org. [7] Project Leurré.com. Publications web page: http://www.leurrecom.org/paper.htm. [8] M. Dacier, F. Pouget, H. Debar. Honeypots: practical means to validate malicious fault assumptions. 10th IEEE Pacific Rim Int. Symp., pp. 383--388, Tahiti, 2004. [9] F. Pouget, M. Dacier, V. Hau Pham, “Understanding threats: a prerequisite to enhance survivability of computing systems”, Int. Infrastructure Survivability Workshop IISW'04, (25th IEEE Int. Real-Time Systems Symp. (RTSS 04)), Lisboa, Portugal, 2004. [10] E. Alata, V. Nicomette, M. Kaaniche, M. Dacier, M. Herrb, Lessons learned from the deployment of a high- interaction honeypot: Extended version. LAAS Report, July 2006. [11] Inc. VMware. Available on: http://www.vmware.com [12] The PaX Team. Available on: http://pax.grsecurity.net. [13] EnergyMech team. Energymech. Available on: http://www.energymech.net. [14] US-CERT. Linux kernel mremap(2) system call does not properly check return value from do_munmap() function. Available on: http://www.kb.cert.org/vuls/id/981222. [15] US-CERT. Linux kernel do_brk() function contains integer overflow. http://www.kb.cert.org/vuls/id/981222. [16] J. Corey, Advanced honeypot identification and exploitation. Phrack, N 63, Available on: http://www.phrack.org/fakes/p63/p63-0x09.txt. [17] T. Holz and F. Raynal, Detecting honeypots and other suspicious environments. In Systems, Man and Cybernetics (SMC) Information Assurance Workshop. Proc. from the Sixth Annual IEEE, pages 29--36, 2005. Proceedings of the Sixth European Dependable Computing Conference (EDCC'06) 0-7695-2648-9/06 $20.00 © 2006 ABSTRACT This paper presents an experimental study and the lessons learned from the observation of the attackers when logged on a compromised machine. The results are based on a six months period during which a controlled experiment has been run with a high interaction honeypot. We correlate our findings with those obtained with a worldwide distributed system of lowinteraction honeypots. <|endoftext|><|startoftext|> Introduction The idea behind abstract (linear) potential theory, as developed by Choquet [4], Fuglede [9] and Ohtsuka [15], is to replace the Euclidian space Rd by some locally compact space X and the well-known Newto- nian kernel by some other kernel function k : X×X → R∪{+∞}, and ∗ This work was started during the 3rd Summerschool on Potential Theory, 2004, hosted by the College of Kecskemét, Faculty of Mechanical Engineering and Automa- tion (GAMF). Both authors would like to express their gratitude for the hospitality and the support received during their stay in Kecskemét. † The second named author was supported by the Hungarian Scientific Research Fund; OTKA 49448 http://arxiv.org/abs/0704.0859v1 to look at which “potential theoretic” assertions remain true in this gen- erality (see the monograph of Landkof [12]). This approach facilitates general understanding of certain potential theoretic phenomena and allows also the exploration of fundamental principles like Frostman’s maximum principle. Although there is a vast work done considering energy integrals and different notions of energies, the familiar notions of transfinite diame- ter and Chebyshev constants in this abstract setting are sporadically found, sometimes indeed inaccessible, in the literature, see Choquet [4] or Ohtsuka [17]. In [4] Choquet defines transfinite diameter and proves its equality with the Wiener energy in a rather general situation, which of course covers the classical case of the logarithmic kernel on C. We give a slightly different definition for the transfinite diameter that, for infinite sets, turns out to be equivalent with the one of Choquet. The primary aim of this note is to revisit the above mentioned notions and related results and also to partly complement the theory. We already remark here that Zaharjuta’s generalisation of transfi- nite diameter and Chebyshev constant to Cn is completely different in nature, see [24], whereas some elementary parts of weighted potential theory (see, e.g., Mhaskar, Saff [13] and Saff, Totik [20]) could fit in this framework. The power of the abstract potential analytic tools is well illustrated by the notion of the average distance number from metric analysis, see Gross [11], Stadje [21]. The surprising phenomenon noticed by Gross is the following: If (X, d) is a compact connected metric space, there al- ways exists a unique number r(X) (called the average distance number or the rendezvous number of X), with the property that for any finite point system x1, . . . , xn ∈ X there is another point x ∈ X with average distance d(xj , x) = r(X). Stadje generalised this to arbitrary continuous, symmetric functions replacing d. Actually, it turned out, see the series of papers [6, 5, 7] and the references therein, that many of the known results concerning av- erage distance numbers (existence, uniqueness, various generalisations, calculation techniques etc.), can be proved in a unified way using the works of Fuglede and Ohtsuka. We mention for example that Frost- man’s Equilibrium Theorem is to be accounted for the existence for certain invariant measures (see Section 5 below). In these investigations the two variable versions of Chebyshev constants and energies and even their minimax duals had been needed, and were also partly available due to the works of Fuglede [10] and Ohtsuka [16, 17], see also [6]. Another occurrence of abstract Chebyshev constants is in the study of polarisation constants of normed spaces, see Anagnostopoulos, Ré- vész [1] and Révész, Sarantopoulos [19]. Let us settle now our general framework. A kernel in the sense of Fuglede is a lower semicontinuous function k : X × X → R ∪ {+∞} [9, p. 149]. In this paper we will sometimes need that the kernel is symmetric, i.e., k(x, y) = k(y, x). This is for example essential when defining potential and Chebyshev constant, otherwise there would be a left- and right-potential and the like. Another assumption, however a bit of technical flavour, is the pos- itivity of the kernel. This we need, because we would like to avoid technicalities when integrating not necessarily positive functions. This assumption is nevertheless not very restrictive. Since we usually con- sider compact sets of X ×X, where by lower semicontinuity k is nec- essarily bounded from below, we can assume that k ≥ 0. Indeed, as we will see, energy, nth diameter and nth Chebyshev constant are linear in constants added to k. Denote the set of compactly supported Radon measures on X by M(X), that is M(X) := {µ : µ is a regular Borel measure on X, µ has compact support, ‖µ‖ < +∞}. Further, let M1(X) be the set of positive unit measures from M(X), M1(X) := {µ ∈ M(X) : µ ≥ 0, µ(X) = 1}. We say that µ ∈ M1(X) is supported on H if supp µ, which is a compact subset of X, is in H. The set of (probability) measures supported on H are denoted by M(H) (M1(H)). Before recalling the relevant potential theoretic notions from [9] (see also [15]), let us spend a few words on integrals (see [2, Ch. III-IV.]). Let µ be a positive Radon measure on X. Then the integral of a compactly supported continuous function with respect to µ is the usual integral. The upper integral of a positive l.s.c. function f is defined as f dµ := sup 0 ≤ h ≤ f h ∈ Cc(X) h dµ. This definition works well, because by standard arguments (see, e.g., [2, Ch. IV., Lemma 1]) one has k(x, y) = sup 0 ≤ h ≤ k h ∈ Cc(X ×X) h(x, y), where, because of the symmetry assumption, it suffices to take only symmetric functions h in the supremum. What should be here noted, is that this notion of integral has all useful properties that we are used to in case of Lebesgue integrals (note also the necessity of the positivity assumptions). The usual topology onM is the so-called vague topology which is a lo- cally convex topology defined by the family {µ 7→ X f dµ : f ∈ Cc(X)} of seminorms. We will only encounter this topology in connection with families M of measures supported on subsets of the same compact set K ⊂ X. In this case, the weak∗-topology (determined by C(K)) and the vague topology coincide on M, Fuglede [9]. For a potential theoretic kernel k : X ×X → R+ ∪ {0} Fuglede [9] and Ohtsuka [15] define the potential and the energy of a measure µ Uµ(x) := k(x, y) dµ(y) , W (µ) := k(x, y) dµ(y) dµ(x). The integrals exist in the above sense, although may attain +∞ as well. For a given set H ⊂ X its Wiener energy is w(H) := inf µ∈M1(H) W (µ), (1) see [9, (2) on p. 153]. One also encounters the quantities (see [9, p. 153]) U(µ) := sup Uµ(x), V (µ) := sup x∈ supp µ Uµ(x). Accordingly one defines the following energy functions u(H) := inf µ∈M1(H) U(µ), v(H) := inf µ∈M1(H) V (µ). In general, one has the relation w ≤ v ≤ u ≤ +∞, where in all places strict inequality may occur. Nevertheless, under our assumptions we have the equality of the energies v and w, being gen- erally different, see [9, p. 159]. More importantly, our set of conditions suffices to have a general version of Frostman’s equilibrium theorem, see Theorem 9. In fact, at a certain point (in §4), we will also assume Frostman’s maximum principle, which will trivially guarantee even u = v, that is, the equivalence of all three energies treated by Fuglede. Definition. The kernel k satisfies the maximum principle, if for every measure µ ∈ M1 U(µ) = V (µ). As our examples show in §5, this is essential also for the equivalence of the Chebyshev constant and the transfinite diameter. Carleson [3, Ch. III.] gives a class of examples satisfying the maximum principle: Let Φ(r), r = |x|, x ∈ Rd be the fundamental solution of the Laplace equation, i.e., Φ(|x−y|) the Newtonian potential on Rd. For a positive, continuous, increasing, convex function H assume also that H(Φ(r))rd−2 dr < +∞. Then H ◦Φ satisfies the maximum principle; see [3, Ch. III.] and also Fuglede [9] for further examples. Let us now turn to the systematic treatment of the Chebyshev constant and the transfinite diameter. We call a function g : X → R log-polynomial, if there exist w1, . . . , wn ∈ X such that g(x) = j=1 k(x,wj) for all x ∈ X. Accordingly, we will call the wjs and n the zeros and the degree of g(x), respectively. Obviously the sum of two log-polynomials is a log-polynomial again. The terminology here is motivated by the case of the logarithmic kernel k(x, y) = − log |x− y|, where the log-polynomials correspond to negative logarithms of alge- braic polynomials. Log-polynomials give access to the definition of transfinite diameter and the Chebyshev constant, see Carleson [3], Choquet [4], Fekete [8], Ohtsuka [17] and Pólya, Szegő [18]. First we start with the “degree n” versions, whose convergence will be proved later. Definition. Let H ⊂ X be fixed. We define the nth diameter of H as Dn(H) := inf w1,...,wn∈H (n− 1)n 1≤j 6=l≤n k(wj , wl) ; (2) or, if the kernel is symmetric Dn(H) = inf w1,...,wn∈H (n− 1)n 1≤i m, there is always a point from the diagonal ∆ = {(x, x) : x ∈ H} in the definition of Dn(H). This possibility is completely excluded by Choquet in [4], thus allowing only infinite sets. Definition. For an arbitrary H ⊂ X the nth Chebyshev constant of H is defined as Mn(H) := sup w1,...,wn∈H k(x,wk) We are going to show that both nth diameters and nth Chebyshev constants converge from below to some number (or +∞), which are respectively called the transfinite diameter D(H) and the Chebyshev constant M(H). The aim of this paper is to relate these quantities as well as the Wiener energy of a set. 2. Chebyshev constant and transfinite diameter We define the Chebyshev constant and the transfinite diameter of a set H ⊂ X and proceed analogously to the classical case. It turns out, though not very surprisingly, that in general the equality of these two quantities does not hold. First, we prove the convergence of nth diameters and nth Chebyshev constants. This is for both cases classical, we give the proof only for the sake of completeness, see, e.g., Carleson [3], Choquet [4], Fekete [8], Ohtsuka [17] and Pólya, Szegő [18]. PROPOSITION 1. The sequence of nth diameters is monotonically increasing. Proof. Choose x1, . . . , xn ∈ H arbitrarily. If we leave out any index i = 1, 2, . . . , n, then for the remaining n − 1 points we obtain by the definition of Dn−1(H) that (n− 1)(n − 2) 1 ≤ j 6= l ≤ n j 6= i, l 6= i k(xj , xl) ≥ Dn−1(H). After summing up for i = 1, 2, . . . , n this yields 1≤j 6=l≤n k(xj , xl) ≥ n ·Dn−1(H), for each term k(xj , xl) occurs exactly n − 2 times. Now taking the infimum for all possible x1, . . . , xn ∈ H, we obtain n · Dn(H) ≥ n · Dn−1(H), hence the assertion. The limit D(H) := limn→∞Dn(H) is the transfinite diameter of H. Similarly, the nth Chebyshev constants converge, too. PROPOSITION 2. For any H ⊂ X, the Chebyshev constants Mn(H) converge in the extended sense. Proof. The sum of two log-polynomials, p(z) = i=1 k(z, xi) with de- gree n and q(z) = j=1 k(z, yj) with degree m, is also a log-polynomial with degree n+m. Therefore (n+m)Mn+m ≥ nMn +mMm (3) for all n,m follows at once. Should Mn(H) be infinity for some n, then all succeeding terms Mn′(H), n ′ ≥ n are infinity as well, hence the convergence is obvious. We assume now that Mn(H) is a finite sequence. At this point, for the sake of completeness, we can repeat the classical argument of Fekete [8]. Namely, let m,n be fixed integers. Then there exist l = l(n,m) and r = r(n,m), 0 ≤ r < m nonnegative integers such that n = l ·m + r. Iterating the previous inequality (3) we get n ·Mn ≥ l + rMr = nMm + r(Mr −Mm). Fixing now the value of m, the possible values of r remain bounded by m, and the finitely many values of Mr −Mm’s are finite, too. Hence dividing both sides by n, and taking lim infn→∞, we are led to lim inf Mn ≥ lim inf Mr −Mm = Mm . This holds for any fixed m ∈ N, so taking lim supm→∞ on the right hand side we obtain lim inf Mn ≥ lim sup that is, the limit exists. M(H) := limn→∞Mn(H) is called the Chebyshev constant of H. In the following, we investigate the connection between the Chebyshev constant M(H) and the transfinite diameter D(H). THEOREM 3. Let k be a positive, symmetric kernel. For any n ∈ N and H ⊂ X we have Dn(H) ≤ Mn(H), thus also D(H) ≤ M(H). Proof. If Mn(H) = +∞, then the assertion is trivial. So assume Mn(H) < +∞. By the quasi-monotonicity (see (3)) we have that for all m ≤ n also Mm(H) is finite. We use this fact to recursively find w1, . . . wn ∈ H such that k(wi, wj) < +∞ for all i < j ≤ n. At the end we arrive at 1≤i 0, we take, as we may, an “approximate n-Fekete point system” w1, . . . , wn (n− 1)n 1≤i 6=j≤n k(wi, wj) < Dn + ε. (4) For any x ∈ H the points x,w1, . . . , wn form a point system of n + 1 points, so by the definition of Dn+1 we have k(x,wi) + 1≤i 6=j≤n k(wi, wj) ≥ n(n+ 1)Dn+1 ≥ n(n+ 1)Dn, using also the monotonicity of the sequence Dn. This together with (4) lead to pn(x) := k(x,wi) ≥ n(n+ 1) n(n− 1) Dn + ε Taking infimum of the left hand side for x ∈ H we obtain pn(x) ≥ nDn − n(n− 1)ε By the very definition of the nth Chebyshev constant, n · Mn ≥ infx∈H pn(x) holds, hence Mn ≥ Dn − (n− 1)ε/2 follows. As this holds for all ε > 0, we conclude Mn ≥ Dn. Later we will show that, unlike the classical case of C, the strict inequality D < M is well possible. 3. Transfinite diameter and energy We study the connection between the energy w and the transfinite diameter D. Without assuming the maximum principle we can prove the equivalence of these two quantities for compact sets. This result can actually be found in a note of Choquet [4]. There is however a slight difference to the definitions of Choquet in [4]. There the diagonal was completely excluded from the definition of D, that is the infimum in (2) is taken over wi 6= wj, i 6= j and not for systems of arbitrary wj’s . This means, among others, that in [4] the transfinite diameter is only defined for infinite sets. The other assumption of Choquet is that the kernel is infinite on the diagonal. This is completely the contrary to what we assume in Theorem 8. Indeed, with our definitions of the transfinite diameter one can even prove equality for arbitrary sets if the kernel is finite-valued. THEOREM 4. Let k be an arbitrary kernel and H ⊂ X be any set. Then D(H) ≤ w(H). Proof. Let µ ∈ M1(H) be arbitrary, and define ν := j=1 µ the product measure on the product space Xn. We can assume that the kernel is positive because supp µ, and hence supp ν, is compact so we can add a constant to k such that it will be positive on these supports. Consider the following lower semicontinuous functions g and h on Xn g : (x1, . . . , xn) 7→ Dn(H) := inf (w1,...,wn)∈Xn n(n−1) 1≤i 6=j≤n k(wi, wj) h : (x1, . . . , xn) 7→ n(n−1) 1≤i 6=j≤n k(xi, xj). Since 0 ≤ g ≤ h, by the definition of the upper integral the following holds true Dn(H) ≤ n(n− 1) 1≤i 6=j≤n k(xi, xj) dν(x1, . . . , xn) n(n− 1) 1≤i 6=j≤n k(xi, xj) dµ(xi) dµ(xj) = W (µ). Taking infimum in µ yields Dn(H) ≤ w(H), hence also D(H) ≤ w(H). To establish the converse inequality we need a compactness as- sumption. With the slightly different terminology, Choquet proves the following for kernels being +∞ on the diagonal ∆. The arguments there are very similar, except that the diagonal doesn’t have to be taken care of in [4]. We give a detailed proof. PROPOSITION 5 (Choquet [4]). For an arbitrary kernel function k the inequality D(K) ≥ w(K) holds for all K ⊆ X compact sets. Proof. First of all the l.s.c. function k attains its infimum on the compact set K × K. So by shifting k up we can assume that it is positive, and the validity of the desired inequality is not influenced by this. If D(K) = +∞, then by Theorem 4 we have w(K) = +∞, thus the assertion follows. Assume therefore D(K) < +∞, and let n ∈ N, ε > 0 be fixed. Let us choose a Fekete point system w1, . . . , wn from K. Put µ := µn := 1/n i=1 δwi where δwi are the Dirac measures at the points wi, i = 1, . . . , n. For a continuous function 0 ≤ h ≤ k with compact support, we have h dµ dµ = i,j=1 h(wi, wj) h(wi, wi) + i,j=1 h(wi, wj) h(wi, wi) + i,j=1 k(wi, wj) i,j=1 k(wi, wj) Dn(K) ≤ +D(K) using, in the last step, also the monotonicity of the sequenceDn (Propo- sition 1). In fact, we obtain for n ≥ N = N(‖h‖, ε) the inequality h dµ dµ ≤ D + ε. (5) It is known, essentially by the Banach-Alaoglu Theorem, that for a compact set K the measures of M1(K) form a weak ∗-compact subset of M, hence there is a cluster point ν ∈ M1(K) of the set MN := {µn : n ≥ N} ⊂ M1(K). Let {να}α∈I ⊆ MN be a net converging to ν. Recall that να⊗να weak ∗-converges to ν⊗ν. We give the proof. For a function g ∈ C(K ×K), g(x, y) = g1(x) · g2(y) it is obvious that g dνα dνα → g dν dν. (6) The set A of such product-decomposable functions g(x, y) = g1(x)g2(y) is a subalgebra of C(K ×K), which also separates X ×X, since it is already coordinatewise separating. By the Stone–Weierstraß theorem A is dense in C(K ×K). From this, using also that the family MN of measures is norm-bounded, we immediately get the weak∗-convergence (6). All these imply h dν dν ≤ D(K) + ε, w(K) ≤ W (ν) := kdνdν = sup 0 ≤ h ≤ k h ∈ Cc(X ×X) hdνdν ≤ D(K)+ε, for all ε > 0. This shows w(K) ≤ D(K). COROLLARY 6 (Choquet [4]). For arbitrary kernel k and compact set K ⊂ X, the equality D(K) = w(K) holds. Proof. By compactness we can shift k up and therefore assume it is positive. Then we apply Theorem 4 and Proposition 5. The assumptions of Choquet [4] are the compactness of the set plus the property that the kernel is +∞ on the diagonal (besides it is continuous in the extended sense). This ensures, loosely speaking, that for a set K of finite energy an energy minimising measure µ (i.e., for whichW (µ) = w(K)) is necessarily non-atomic, moreover µ ⊗ µ is not concentrated on the diagonal. Therefore to show equality of w with D, one has to exclude the diagonal completely from the definition of the transfinite diameter. We however allow a larger set of choices for the point system in the definition of D. Indeed, we allow Fekete points to coincide, and this also makes it possible to define the transfinite diameter of finite sets. With this setup the inequality D ≤ w is only simpler than in the case handled by Choquet. Whereas, however surprisingly, the equality D(K) = w(K) is still true for compact sets K but without the assumption on the diagonal values of the kernel. We will see in §5 Example 13 that even assuming the maximum prin- ciple but lacking the compactness allows the strict inequality D < w. This phenomena however may exist only in case of unbounded kernels, as we will see below. In fact, we show that if the kernel is finite on the diagonal, thenD = w holds for arbitrary sets. For this purpose, we need the following technical lemma, which shows certain inner regularity properties of D and is also interesting in itself. LEMMA 7. Assume that the kernel k is positive and finite on the diagonal, i.e., k(x, x) < +∞ for all x ∈ X. Then for an arbitrary H ⊂ X we have D(H) = inf K ⊂ H K compact D(K) = inf W ⊂ H #W < ∞ D(W ). (7) Proof. The inequality infD(K) ≤ infD(W ) is clear. For H ⊇ K the inequality D(H) ≤ D(K) is obvious, so we can assume D(H) < +∞. For ε > 0 let W = {w1, . . . , wn} be an approximate n-Fekete point set of H satisfying (4). Then D(W ) = lim Dmn(W ) ≤ lim mn(mn− 1) 1≤i′ 6=j′≤mn k(wi′ , wj′), where wi′ := . . . ′ = i+ rn, r = 0, . . . ,m− 1 . . . Set C := max{k(x, x) : x ∈ W}. So we find D(W ) ≤ lim mn(mn−1) 1≤i 6=j≤n k(wi, wj) + mn(mn−1) 1≤i≤n k(wi, wi) 1≤i 6=j≤n k(wi, wj) lim mn(mn−1) + Cn lim mn(mn−1) 1≤i 6=j≤n k(wi, wj) ≤ (Dn(H) + ε) ≤ D(H) + ε. This being true for all ε > 0, taking infimum we finally obtain W ⊂ H #W < ∞ D(W ) ≤ D(H). Clearly, if k(x, x) = +∞ for all x ∈ W with a finite set #W = n, then for all m > n we have Dm(W ) = +∞. Thus in particular for kernels with k : ∆ → {+∞}, the above can not hold in general, at least as regards the last part with finite subsets. Now, completely contrary to Choquet [4] we assume that the kernel is finite on the diagonal and prove D = w for any set. Hence an example of D < w (see §5 Example 13) must assume k(x, x) = +∞ at least for some point x. THEOREM 8. Assume that the kernel k is positive and is finite on the diagonal, that is k(x, x) < +∞ for all x ∈ X. Then for arbitrary sets H ⊂ X, the equality D(H) = w(H) holds. Proof. By Theorem 4 we have D(H) ≤ w(H). Hence there is nothing to prove, if D(H) = +∞. Assume D(H) < +∞, and let ε > 0 be arbitrary. By Lemma 7 we have for some n ∈ N a finite set W = {w1, w2 . . . , wn} with D(H) + ε ≥ D(W ). In view of Proposition 5 we have D(W ) ≥ w(W ), and by monotonicity also w(W ) ≥ w(H). It follows that D(H) + ε ≥ w(H) for all ε > 0, hence also the “≥” part of the assertion follows. 4. Energy and Chebyshev constant To investigate the relationship between the energy and the Cheby- shev constant the following general version of Frostman’s Equilibrium Theorem [9, Theorem 2.4] is fundamental for us. THEOREM 9 (Fuglede). Let k be a positive, symmetric kernel and K ⊂ X be a compact set such that w(K) < +∞. Every µ which has minimal energy (µ ∈ M1(K),W (µ) = w(K)) satisfy the following properties Uµ(x) ≥ w(K) for nearly every1 x ∈ K, Uµ(x) ≤ w(K) for every x ∈ supp µ, Uµ(x) = w(K) for µ-almost every x ∈ X. Moreover, if the kernel is continuous, then Uµ(x) ≥ w(K) for every x ∈ K. THEOREM 10. Let H ⊂ X be arbitrary. Assume that the kernel k is positive, symmetric and satisfies the maximum principle. Then we have Mn(H) ≤ w(H) for all n ∈ N, whence also M(H) ≤ w(H) holds true. Proof. Let n ∈ N be arbitrary. First let K be any compact set. We can assume w(K) < +∞, since otherwise the inequality holds irrespective of the value of Mn(K). Consider now an energy-minimising measure νK of K, whose existence is assured by the lower semicontinu- ity of µ 7→ k dµ dµ and the compactness of M1(K), see [9, Theorem 2.3]. By the Frostman-Fuglede theorem (Theorem 9) we have UνK (x) ≤ w(K) for all x ∈ supp νK , so V (νK) ≤ w(K), and by the maximum principle even UνK (x) ≤ w(K) for all x ∈ X. 1 The set A of exceptional points is small in the sense w(A) = +∞. Then for all w1, . . . , wn ∈ K k(x,wj) ≤ k(x,wj) dνK(x) ≤ w(K) . Taking supremum for w1, . . . , wn ∈ K, we obtain w1,...,wn∈K k(x,wj) ≤ w(K). So Mn(K) ≤ w(K) for all n ∈ N. Next let H ⊂ X be arbitrary. In view of the last form of (1), for all ε > 0 there exists a measure µ ∈ M1(H), compactly supported in H, with w(µ) ≤ w(H) + ε. Let W = {w1, . . . , wn} ⊂ H be arbitrary and define pW (x) := i k(x,wi). Consider the compact set K := W ∪ supp µ ⊂ H. By definition of the energy, supp µ ⊂ K implies w(K) ≤ w(µ), hence w(K) ≤ w(H) + ε. Combining this with the above, we come to Mn(K) ≤ w(H) + ε. Since W ⊂ K, by definition of Mn(K) we also have pW (x) ≤ Mn(K). (8) The left hand side does not increase, if we extend the inf over the whole of H, and the right hand side is already estimated from above by w(H) + ε. Thus (8) leads to pW (x) ≤ w(H) + ε. This holds for all possible choices of W = {w1, . . . , wn} ⊂ H, hence is true also for the sup of the left hand side. By definition of Mn(H) this gives exactly Mn(H) ≤ w(H) + ε, which shows even Mn(H) ≤ w(H). Remark. In [6] it is proved that M(H) = q(H), where q(H) = inf µ∈M1(H) Uµ(x). The idea behind is a minimax theorem, see also [16, 17]. Trivially w(H) ≤ q(H) ≤ u(H). So the maximum principle implies M(H) = w(H) = q(H) = u(H). 5. Summary of the Results. Examples In this section, we put together the previous results, thus proving the equality of the three quantities being studied, under the assumption of the maximum principle for the kernel. Further, via several instruc- tive examples we investigate the necessity of our assumptions and the sharpness of the results. THEOREM 11. Assume that the kernel k is positive, symmetric and satisfies the maximum principle. Let K ⊂ X be any compact set. Then the transfinite diameter, the Chebyshev constant and the energy of K coincide: D(K) = M(K) = w(K). Proof. We presented a cyclic proof above, consisting of M ≥ D (Theorem 3), D ≥ w (Proposition 5) and finally w ≥ M (Theorem 10). THEOREM 12. Assume that the kernel k is positive, finite and sat- isfies the maximum principle. For an arbitrary subset H ⊂ X the transfinite diameter, the Chebyshev constant and the energy of H co- incide: D(H) = M(H) = w(H). Proof. By finiteness D = w, due to Theorem 8. This with D ≤ M and M ≤ w (Theorems 3 and 10) proves the assertion. Remark. In the above theorem, logically it would suffice to assume that the kernel be finite only on the diagonal. But if this was the case, the maximum principle would then immediately imply the finiteness of the kernel everywhere. Let us now discuss how sharp the results of the preceding sections are. In the first example we show that, if we drop the assumption of compactness the assertions of Theorem 3, Theorem 4 and Theorem 10 are in general the strongest possible. Example 13. Let X = N ∪ {0} endowed with discrete topology and the kernel k(n,m) := +∞ if n = m, 0 if 0 6= n 6= m 6= 0, 1 otherwise. The kernel is symmetric, l.s.c. and has the maximum principle. This latter can be seen by noticing that for a probability measure µ ∈ M1(X) the potential is +∞ on the support of µ. Indeed, since X is countable, all measures µ ∈ M1(X) are necessarily atomic, and if for some point ℓ ∈ X we have µ({ℓ}) > 0, then by definition X k(x, y) dµ(y) = +∞. We calculate the studied quantities of the set H = X (also as in all the examples below). Since the kernel is positive, Dn ≥ 0. On the other hand, choosing w1 := 1, . . . , wn := n, all the values k(wi, wj) will be exactly 0, so it follows that Dn = 0, n = 1, 2, . . ., and hence D = 0. The Chebyshev constant can be estimated from below, if we compute the infimum of a suitably chosen log-polynomial. Consider the log- polynomial p(x) with all zeros placed at 0, that is with w1 = . . . = wn = 0. Then the log-polynomial p(x) is j k(x,wj) = n · k(x, 0). If x 6= 0, we have p(x) = n, which gives M ≥ 1. The upper estimate of M is also easy: suppose that in the system w1, . . . wn there are exactly m points being equal to 0 (say the first m). Then p(x) = +∞ x = w1, . . . , wn, n x = 0, x 6= w1, . . . , wn (if m = 0) m x 6= 0, x 6= w1, . . . , wn This shows for the corresponding log-polynomial inf p(x) = m, so Mn ≤ 1, whence M = 1. The energy is computed easily. Using the above reasoning on the maximum principle, we see W (µ) = +∞ for any µ ∈ M1(X), hence w(X) = +∞. Thus we have an example of +∞ = w > M > D = 0. The above example completes the case of the kernel with maximum principle. Let us now drop this assumption and look at what can happen. Example 14. Let X := {−1, 0, 1} be endowed with the discrete topol- ogy. We define the kernel by k(x, y) := 2 if 0 ≤ |x− y| < 2, 0 if 2 = |x− y|. Then k is continuous and bounded on X×X. This, in any case, implies D = w by Theorem 8. Note that k does not satisfy the maximum principle. To see this, consider, e.g., the measure µ = 1 δ1. Then for the potential Uµ one has Uµ(1) = Uµ(−1) = 1 and Uµ(0) = 2, which shows the failure of the maximum principle. To estimate the nth diameter from above, let us consider the point system {wi} of n = 2m points with m points falling at −1 and m points falling at 1, while no points being placed at 0. Then by definition of Dn := Dn(X) one can write n(n− 1) Dn ≤ 2 · 2 +m2 · 0 = Applying this estimate for all even n = 2m as n → ∞, it follows that D = lim Dn ≤ 1. (9) Next we estimate the Chebyshev constants from below by computing the infimum of some special log-polynomials. For pn(x) = k(x, 0) one has pn(x) ≡ 2 = inf pn. We thus find Mn ≥ 2 and M ≥ 2, showing M > D, as desired. Example 15. Let X := N with the discrete topology. Then X is a locally compact Hausdorff space, and all functions are continuous, hence l.s.c. on X. Let k : X ×X → [0,+∞] be defined as k(n,m) := +∞ if n = m, 2−n−m if n 6= m. Clearly k is an admissible kernel function. For the energy we have again w(X) = +∞, see Example 13. On the other hand let n ∈ N be any fixed number, and compute the nth diameter Dn(X). Clearly if we choose wj := m+ j, with m a given (large) number to be chosen, then we get Dn(H) ≤ (n− 1)n 1≤i 6=j≤n 2−i−j−2m ≤ (n− 1)n ≤ 2−2m , hence we find that the nth diameter is Dn(X) = 0, so D(X) = 0, too. For any log-polynomial p(x) we have inf p(x) = limx→∞ p(x) = 0, hence M(X) = 0. That is we have D(X) = M(X) = 0 < w(X) = +∞. The example shows how important the diagonal, excluded in the definition of D but taken into account in w, may become for particular cases. We can even modify the above example to get finite energy. Example 16. Let X := (0, 1], equipped with the usual topology, and let xn = 1/n. We take now k(x, y) := +∞ if x = y, 2−n−m if x = xn and y = xm (xn 6= xm), − log |x− y| otherwise Compared to the l.s.c. logarithmic kernel, this k assumes different, smaller values at the relatively closed set of points {(xn, xm) : n 6= m} ⊂ X ×X only, hence it is also l.s.c. and thus admissible as kernel. If a measure µ ∈ M1(X) has any atom, say if for some point z ∈ X we have µ({z}) > 0, then by definition X k(x, y) dµ(y) = +∞, hence also w(µ) = +∞. Since for all µ ∈ M1(X) with any atomic component w(µ) = +∞, we find that for the set H := X we have w(H) := inf µ∈M1(H) w(µ) = inf µ∈M1(H) µ not atomic w(µ). But for measures without atoms, the countable set of the points xn are just of measure zero, hence the energy equals to the energy with respect to the logarithmic kernel. Thus we conclude w(H) = e−cap(H) = e−1/4, as cap((0, 1]) = 1/4 is well-known. On the other hand if n ∈ N is any fixed number, we can compute the nth diameter Dn(H) exactly as above in Example 15. Hence it is easy to see that Dn(H) = 0, whence also D(H) = 0. Similarly, we find M(H) = 0, too. This example shows that even in case w(H) < +∞ we can have w(H) > D(H) = M(H). 6. Average distance number and the maximum principle In the previous section, we showed the equality of the Chebyshev con- stant M and the transfinite diameter D, using essentially elementary inequalities and the only theoretically deeper ingredient, the assump- tion of the maximum principle. We have also seen examples showing that the lack of the maximum principle for the kernel allows strict inequality between M and D. These observations certify to the rel- evance of this principle in our investigations. Indeed, in this section we show the necessity of the maximum principle in case of continuous kernels for having M(K) = D(K) for all compact sets K. We need some preparation first. Recall from the introduction the notion of the average distance (or rendezvous) number. Actuyally, a more general assertion than there can be stated, see Stadje [21] or [6]. For a compact connected set K and a continuous, symmetric kernel k, the average distance number r(K) is the uniquely existing number with the property that for all probability measures supported in K there is a point x ∈ K with Uµ(x) = k(x, y) dµ(y) = r(K). This can be even further generalised by dropping the connectedness, see Thomassen [22] and [6]. Even for not necessarily connected but compact spacesK with symmetric, continuous kernel k there is a unique number r(K) with the property that whenever a probability measure on K and a positive ε are given, there are points x1, x2 ∈ K such that Uµ(x1)− ε ≤ r(K) ≤ U µ(x2) + ε. This number is called the (weak) average distance number, and is par- ticularly easy to calculate, when a probability measure with constant potential is available. Such a measure µ is called then an invariant measure. In this case the average distance number r(K) is trivially just the constant value of the potential Uµ, see Morris, Nicholas [14] or [7]. It was proved in [7] that one always has M(K) = r(K), so once we have an invariant measure, then the Chebyshev constant is again easy to determine. Also the Wiener energy w(K) has connection to invariant measures, as shown by the following result, which is a simplified version of a more general statement from [7], see also Wolf [23]. THEOREM 17. Let ∅ 6= K ⊂ X be a compact set and k be a continu- ous, symmetric kernel. Then we have r(K) ≥ w(K). Furthermore, if r(K) = w(K), then there exists an invariant measure in M1(K). As mentioned above, we have r(K) = M(K), so the inequality r(K) ≥ w(K) in the first assertion of the above theorem is also the conse- quence of Theorems 3 and 8. For the proof of the second assertion one can use the Frostman-Fuglede Equilibrium Theorem 9 with the obvious observation that “nearly every” in this context means indeed “every”. Actually any probability measure µ ∈ M1(K) which minimises ν 7→ supK U ν is an invariant measure and its potential is constant M(K), see [7, Thm. 5.2] (such measures undoubtedly exist because of compactness of M1(K)). Henceforth we will indifferently use the terms energy minimising or invariant for expressing this property of measures. THEOREM 18. Suppose that the kernel k is symmetric and continu- ous. If M(K) = D(K) for all compact sets K ⊆ X, then the kernel has the maximum principle. Proof. Recall from Corollary 6 that D(K) = w(K) for all K ⊆ X compact. So we can use Theorem 17 all over in the following arguments. We first prove the assertion in the case when X is a finite set. The proof is by induction on n = #X. For n = 1 the assertion is trivial. Let now #X = 2, X = {a, b}. Assume without loss of generality that k(a, a) ≤ k(b, b). Then we only have to prove that for µ = δa the maximum principle, i.e., the inequality k(a, b) ≤ k(a, a) holds. To see this we calculate M(X) and D(X). We certainly have D(X) ≤ k(a, a). On the other hand for an energy minimising probability measure νp := pδa + (1 − p)δb on X we know that its potential is constant over X, hence pk(a, a) + (1− p)k(b, a) = pk(a, b) + (1− p)k(b, b) = M(X) = D(X) ≤ k(a, a). Here if p = 1, then k(a, a) = k(a, b). If p < 1, then we can write (1− p)k(b, a) ≤ (1− p)k(a, a), hence k(b, a) ≤ k(a, a), so the maximum principle holds. Assume now that the assertion is true for all sets with at most n elements and for all kernels, and let #X = n + 1. For a probability measure µ on X we have to prove supx∈X U µ(x) = supx∈ supp µ U µ(x). If supp µ = X, then there is nothing to prove. Similarly, if there are two distinct points x1 6= x2, x1, x2 ∈ X \ supp µ, then by the induction hypothesis we have x∈X\{x1} Uµ(x) = sup x∈ supp µ Uµ(x) = sup x∈X\{x2} Uµ(x). So for a probability measure µ defying the maximum principle we must have # supp µ = n, say supp µ = X \ {xn+1}; let µ be such a measure. Set K = supp µ and let µ′ be an invariant measure on K. We claim that all such measures µ′ are also violating the maximum principle. If µ = µ′, we are done. Assume µ 6= µ′ and consider the linear combinations µt := tµ+(1− t)µ ′. There is a τ > 1, for which µτ is still a probability measure and supp µτ ( supp µ. By the inductive hypothesis (as # supp µτ < n) we have U µτ (xn+1) ≤ U µτ (a) for some a ∈ supp µτ . We also know that U µ(xn+1) = U µ1(xn+1) > U µ1(a). Hence for the linear function Φ(t) := Uµt(xn+1) − U µt(a) we have Φ(1) > 0 and also Φ(τ) ≤ 0 (τ > 1). This yields Φ(0) > 0, i.e., (xn+1) = U µ0(xn+1) > U µ0(a) = Uµ (y) for all y ∈ K. We have therefore shown that all energy minimising (invariant) measures on K must defy the maximum principle. Let now ν be an invariant measure on X. We have M(X) = Uν(y) = sup Uν(x) = D(X) ≤ D(K) = sup (x) = Uµ (z) < Uµ (xn+1) for all y ∈ X, z ∈ K. Thus we can conclude Uν(y) ≤ Uµ (y) for all y ∈ X and even “<” for y = xn+1. Integrating with respect to ν would yield k dν dν = M(X) < k dµ′ dν = k dν dµ′ = M(X), hence a contradiction, unless ν({xn+1}) = 0. If ν({xn+1}) = 0 held, then ν would be an energy minimising measure on K. This is because obviously supp ν ⊆ K holds, and the potential of ν is constant M(X) over K, so M(X) = k dν dµ′ = k dµ′ dν = M(K) holds. As we saw above, then ν would not satisfy the maximum principle, a contradiction again, since the potential of ν is constant on X. The proof of the case of finite X is complete. We turn now to the general case of X being a locally compact space with continuous kernel. Let µ be a compactly supported probability measure on X and y 6∈ supp µ. Set K = supp µ and note that both M1(K) ∋ ν 7→ supK U ν and ν 7→ Uν(y) are continuous mappings with respect to the weak∗-topology on M1(K). If supK U µ < Uµ(y) were true, we could therefore find, by a standard approximation argument, see for example [6, Lemma 3.8], a finitely supported probability measure µ′ on K for which x∈ supp µ′ (x) ≤ sup (x) < Uµ This is nevertheless impossible by the first part of the proof, thus the assertion of the theorem follows. Acknowledgement The authors are deeply indebted to Szilárd Révész for his insightful suggestions and for the motivating discussions. References 1. Anagnostopoulos, V. and Sz. Gy. Révész: 2006, ‘Polarization constants for products of linear functionals over R2 and C2 and Chebyshev constants of the unit sphere’. Publ. Math. Debrecen 68(1–2), 75–83. 2. 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P.: 1975, ‘Transfinite diameter, Chebishev constants, and capacity for compacta in Cn’. Math. USSR Sbornik 25(3), 350–364. ABSTRACT We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the Chebyshev constant of any compact set coincides with its transfinite diameter, the kernel must satisfy the maximum principle. An abundance of examples is provided to show the sharpness of the results. <|endoftext|><|startoftext|> Introduction Event logs have been widely used to analyze the error/failure behavior of computer-based systems and to estimate their dependability. Event logs include a large amount of information about the occurrence of various types of events that are collected concurrently with normal system operation, and as such reflect actual workload and usage. Some of the events are informational and are issued from the normal activity of the target systems, whereas others are recorded when errors and failures affect local or distributed resources, or are related to system shutdown and start-up. The latter events are particularly useful for dependability analysis. Computer system dependability analysis based on event logs has been the focus of several published papers [1, 2, 4, 5, 7, 8, 9]. Various types of systems have been studied (Tandem, VAX/VMS, Unix, Windows NT, Windows 2000, etc.) including mainframes and largely deployed commercial systems. The issues addressed in these studies cover a large spectrum, including the development of techniques and methodologies for the extraction of relevant information from the event logs, the identification of error patterns, their causes and their effects, and the statistical assessment of dependability measures such as failure and recovery rates, reliability and availability. It is widely recognized that such event log based dependability analyses provide useful feedback to software and system designers. Nevertheless, it is important to note that the results obtained are intimately related to the quality and the accuracy of the data recorded in the logs. The study reported in [1] points out various problems that might affect the data included in the event logs and make incorrect conclusions likely, considering as an example the VAX/VMS system. Thus, extreme care is needed to identify deficiencies in the data and to avoid that they lead to incorrect conclusions. In this paper, we show that similar problems can be observed in the event logs maintained by the SunOS/Solaris Unix operating system, and we present a novel approach that is aimed to address such problems and to improve the dependability estimates based on such event logs. These results are illustrated using field data collected during a 4-year period from 373 SunOS/Solaris Unix workstations and servers interconnected through a LAN. The data corresponds to event logs recorded via the syslog daemon. In particular, we use var/adm/messages log files. We focus on the evaluation of machine uptimes, downtimes and availability based on the identification of failures that caused a total service interruption of the machine. In this study, we show that the consideration of the information recorded in the var/adm/messages log files only may lead to dependability estimations that do not faithfully reflect reality due to incomplete or imperfect data recorded in the corresponding logs. For the estimation of these measures, we start with the assumption that machine failures can be identified by the last events recorded in the event log before the machine goes down and then is rebooted. This assumption was considered in the study reported in [3]. However, the validity of this assumption is questionable in the following situations: 1) the machine has a real activity between the last event logged and the reboot without generating events in the logs, 2) the time when the failure occurs is earlier than the timestamp of the last event logged on the machine. To address these problems and to obtain more realistic estimations, we propose a solution based on utilization of additional information obtained from wtmpx Unix files, as well as data characterizing the state of the machines included in the data collection that are recorded at a regular basis during the data collection procedure. The results clearly show that the combined use of this additional information and syslogd log files have a significant impact on the estimations. To our knowledge, the approach discussed in this paper and the corresponding results have not been addressed in the previous studies published on the exploitation of syslogd log files for the dependability analysis of Unix based systems, including our paper The rest of the paper is structured into 5 sections. Section 2 describes the event logging mechanism in Unix and the data collection procedure that we have used in our study. Section 3 presents the dependability measures that we have considered and discusses different approaches and assumptions to estimate them from the collected data. Section 4 presents some results illustrating the benefits of the proposed approach, as well as various statistics characterizing the dependability of the Unix systems considered in our study. 2. Event logging and data collection For the Unix operating system, the event logging mechanism is implemented by the syslog daemon (denoted as syslogd). Running as a background process, this daemon listens for the events generated by different sources: kernel, system components (disk, memory, network interfaces), daemons and applications that are configured to communicate with syslogd. These events inform about the normal activity of the system as well as its behavior under the occurrence of errors and failures including reboot and shutdown events. The configuration file /etc/syslog.conf specifies the destination of each event received by syslogd, depending on its severity level and its origin. The destination could be one or several log files, the administration console or the operator. The events that are relevant to our study are generally stored in the /var/adm/messages log file. Each message stored in a log file refers to an event that occurred on the system due to the local activity or its interaction with other systems on the network. It contains the following information: the date and time of the event, the machine name on which the event is logged and a description of the message. An example of an event recorded in the log file is given below: Mar 2 10:45:12 elgar automountd[124]: server mahler not responding The SunOS/Solaris Unix operating system limits the size of the log files. Generally, only the log files corresponding to the last 5 weeks of activity are kept. It is necessary to set up a data collection strategy in order to archive a large amount of data. This is essential to obtain representative results for the dependability measures characterizing the monitored systems. In our study, we have included all the SunOS/Solaris machines connected through the LAAS local area network, excluding those used for experimental testbeds or maintenance activities. We have developed a data collection strategy to automatically collect the /var/adm/messages log files stored on these machines. This strategy takes into account the frequent evolution of the network configuration during the observation period in terms of variation of the number of connected systems, updates or changes of the operating system versions, modification of software configurations, etc. A shell script executed each week via the cron mechanism implements the strategy and remotely copies the log files from each system included in the study and archives them on a dedicated machine. After each data collection campaign, a text file (named DCSummary) containing a summary of the data collection campaign is created. This summary indicates the status of each machine included in the campaign and how the collection of the corresponding log file has been done. For each machine, the status information reported in the summary is one of the following: • alive_OK: the machine is alive and the copy of its log file succeeded; • alive_KO: the machine is alive but the copy of its log file failed. For this case, a description of the failure symptom and cause is also included: shell problem, connection ended by tiers, etc. • no_answer: the machine did not answer to a ping request before expiration of the default timeout period. The information included in the DCSummary file is used to verify each data collection campaign and solve the problems that may appear during the collection. It is also useful to improve the accuracy of dependability measures estimation (see Section 3.2). More detailed information about the syslogd mechanism and the data collection strategy are reported in [6]. 3. Dependability measures estimation and assumptions Various types of dependability analyses can be carried out based on the information contained in the log files and several quantitative measures can be considered to characterize the dependability of the target machines: machine uptimes and downtimes, reliability, availability, failure and recovery rates, etc. In order to evaluate these measures, it is necessary to identify from the log files the failure occurrences and the corresponding service degradation durations. Such task is tedious and requires the development of heuristics and predefined failure criteria. An example of such analysis is reported in [7]. In our study, we have focused on the availability analysis of the individual machines included in the data collection. In this context, we have considered machine failures leading to a total interruption of the service delivered to the users, followed by a reboot. The time between the failure occurrence and the end of the reboot corresponds to the total service interruption period of the system. Apart from these periods, the system is considered to be in the normal functioning state where it delivers an appropriate service to the users. In order to evaluate the availability of the machines included in the study, we need to estimate for each machine the corresponding uptimes (denoted as UTi) and downtimes (DTi), based on the information recorded in the event logs. Each downtime value DTi corresponds to the total service interruption period associated to the i failure. It is composed by the service degradation period due to the failure occurrence and the reboot period. Each uptime value corresponds to the period between two successive downtimes. Using the uptime and downtime estimates for each machine j, we can evaluate the corresponding availability (noted Aj) and the unavailability (noted UAj). These measures are computed with the following formulas: UAj =� UTi ⁄ �(UTi +DTi) and UAj = 1 - UAj (1) 3.1. Machine uptimes and downtimes estimation The estimation of machine uptimes and downtimes is carried out in two steps: 1) Identification of machine reboots and their duration. 2) Identification of failures associated to each reboot and of the corresponding service interruption period. To identify the occurrence of machine reboots and their duration, we have developed an algorithm based on the sequential parsing and matching of each event recorded in the system log files to specific patterns or sequences of patterns characterizing the occurrence of reboots. Indeed, whereas some reboots can be explicitly identified by a “reboot” or a “shutdown” event, many others can be detected only by identifying the sequence of the initialization events that are generated by the system when it is restarted. The algorithm is described in [4, 6]. It gives, for each reboot i identified in the event logs and for each machine, the timestamp of the reboot start (dateSBi), the timestamp of the reboot end (dateEBi) and the associated service interruption duration. The identification of the timestamp of the failure associated to each reboot and the corresponding service interruption period is more problematic. In the study reported in [3], it was assumed that the timestamp of the last event recorded before the reboot (denoted as dateEBRi) identifies the failure occurrence time. With this assumption, each uptime UTi and downtime DTi can be evaluated as follows: UTi = dateEBRi – dateEBi-1 and DTi = dateEBi - dateEBRi (2) where i is the index of the current reboot, i-1 the index of the previous reboot. The consideration of EBR for the estimation of UTi and DTi parameters may not be realistic in the following situations (denoted as S1 and S2): S1) The system could be in a normal functioning state during a period of time between EBR and the following reboot although it does not generate any event into the log files during that period. S2) The beginning of the service interruption period for the users could be prior to the timestamp of the EBR event. This happens for instance when a critical failure affects the machine in such a way that it becomes completely unusable to the users, without preventing the event logging mechanisms from recording some messages into the log files. A careful analysis of the data collected during our study revealed that the above situations are common. To address this problem and to improve downtime and uptime estimation accuracy, it is necessary to use auxiliary data that provides complementary information on the activity of the target machines. In this paper, we present a solution based on the correlation of data collected from the /var/adm/messages log files, with data issued from wtmpx files also maintained by the SunOS/Solaris operating system. We also use the information recorded in the DCSummary file (see Section 2). The following section presents the method developed to extract the data from the wtmpx file and how we used this data to adjust the estimation of machine uptimes and downtimes. 3.2. Uptime and downtime estimations refinement 3.2.1. wtmpx files. The SunOS/Solaris Unix operating system records into the /var/adm/wtmpx binary file information identifying the users login/logout. Through the pseudo-user reboot it also records information on the system reboots. The wtmpx file is organized into records (named also entries) with a fixed size. Each record has the format of a data structure with the following fields: • the user login name: “user”; • the id associated to the current record in the /etc/inittab file: “init_id”; • the device name (console, lnxx): “device”; • the process id: “pid”; • the record type: “proc_type”; • the exit status for a process marked as DEAD_PROCESS: “exit_status” and “term_status”; • the timestamp of the record: “date”; • the session id: “session_id”; • the length of the machine’s name: “length”; • the machine’s name used by the user to connect, if it is a remote one: “host”. We developed a specific algorithm that collects the wtmpx file of each machine included in the study on a regular basis and processes the binary file to extract the information that is relevant to our study. The results of the algorithm are kept in a separate file for each machine. Figure 1 presents examples of records obtained for a machine of our network. The first two records show that the root user connected to the local system from the system named cubitus on November 6, 2001 at 16h 37mn 41s, using the rlogin command. The next records inform about the occurrence of a reboot event about 3 minutes later. The third record shows that this reboot was done via a shutdown command executed probably by the root user. The sequence of records corresponding to a reboot event is much longer than this example. The whole sequence is not presented in Figure 1, the aim of the illustration is to show some examples of records as extracted from wtmpx files by our algorithm. In the following, we outline the approach that we developed to use the information extracted from the wtmpx files together with the information from the DCSummary files in order to refine the uptime and downtime estimations, considering situations S1 and S2 discussed in Section 3.1. 2001 Nov 6 16:37:41 user=.rlogin host= length=0 init_id=r100 device=/dev/pts/1 pid=25220 proc_type=6 term_status=0 2001 Nov 6 16:37:41 user=root host=cubitus length=8 init_id=r100 device=/dev/pts/1 pid=25220 proc_type=7 term_status=0 2001 Nov 6 16:40:35 user=shutdown host= length=0 init_id= device=~ pid=0 proc_type=0 term_status=0 exit_status=0 2001 Nov 6 16:41:39 user= host= length=0 init_id= device=system boot pid=0 proc_type=2 term_status=0 2001 Nov 6 16:42:09 user= host= length=0 init_id= device=run–level 3 pid=0 proc_type=1 term_status=0 Figure 1. Examples of records from /var/adm/wtmpx obtained with our algorithm 3.2.2. Situation S1: an operational activity exists between EBR and SB events. The detailed analysis of the collected data from the log files and comparison with the information extracted from wtmpx files showed that the situation where a real activity exists between the last event recorded before a reboot (EBR) and the event identifying the start of the following reboot (SB event) recorded in the /var/adm/messages log files appears quite often. This situation occurs when the machine functions normally but its activity doesn’t produce any message into the log file maintained by the syslogd daemon. The cause could be that the applications or services run by the users aren’t configured to communicate with the syslogd daemon. To better understand this case, Figure 2 gives an example of a sequence of events characterizing the state of the corresponding system, taking into account the information extracted from the /var/adm/messages, wtmpx and DCSummary files. For each event, we indicate the timestamp when it is logged, a short description and the source file from which the event is extracted. For wtmpx events, we present only the fields which are useful to identify the system activity, the other fields are not significant for this analysis. For this example, the events recorded in the /var/adm/messages log file let us believe that the system had no activity between December 8 at 18:06 (EBR event) and December 9 at 15:30, the timestamp of the reboot start. However, the analysis of the DCSummary and wtmpx files shows that the system had a real activity between EBR and SB events. In fact, we see that the data collection campaign was successfully carried out on December 9 at 6:43. Event # Event date Event description File where the event is logged .................. 2002 Dec 8 18:06:08 2002 Dec 9 06:43:34 2002 Dec 9 13:18:45 2002 Dec 9 13:35:21 2002 Dec 9 13:47:57 2002 Dec 9 13:48:48 2002 Dec 9 15:18:46 2002 Dec 9 15:29:20 2002 Dec 9 15:29:25 2002 Dec 9 15:29:25 2002 Dec 9 15:29:27 .................. 2002 Dec 9 15:29:52 2002 Dec 9 15:30:52 2002 Dec 9 15:30:52 2002 Dec 9 15:30:52 2002 Dec 9 15:30:52 2002 Dec 9 15:30:53 last event before reboot alive_ok user=UserC; device=pts/0; pid=2362; proc_type=7 user=UserB; device=pts/1; pid=2379; proc_type=7 user=UserB; device=pts/1; pid=2379; proc_type=8 user=UserA; device=pts/1; pid=2434; proc_type=7 user=UserA; device=pts/1; pid=2434; proc_type=8 user=UserB; device=console; pid=2644; proc_type=7 user=UserB; device=console; pid=338; proc_type=8 user=UserB; device=console; pid=2644; proc_type=8 user=LOGIN; device=console; pid=2742; proc_type=6 .................. user=troot; device=console; pid=334; proc_type=7 user=sac; device=; pid=333; proc_type=8 user=troot; device=console; pid=334; proc_type=8 user=; device=run-level 6; pid=0; proc_type=1 user=rc6; device=; pid=2899; proc_type=5 reboot start var/adm/messages log file DCSummary wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx .................. wtmpx wtmpx wtmpx wtmpx wtmpx var/adm/messages log file Figure 2. Example illustrating situation S1 Moreover, the records from wtmpx file show, for example, that UserA used the system on December 9 between 13:48 (information given by the proc_type field value equal to 7, that is the process with pid=2434 started at the time of this record) and 15:18 (proc_type=8, the same process ended at the time of this record), corresponding to an utilization period of the system of nearly one hour and a half. In this situation, the EBR event as defined earlier doesn’t correspond to the beginning of the total service interruption period. Thus, the estimated value of the downtime parameter using the assumption discussed in Section 3.1, does not faithfully reflect the real value of the service interruption period. Based on the correlation of the information provided by the three data source files, a refined and more accurate estimation of machine downtimes and uptimes could be obtained. The refinement consists in associating the failure occurrence time to the timestamps of the last event recorded before the reboot based on the information contained in /var/adm/messages, wtmpx and DCSummary files. 3.2.3. Situation S2: the service interruption period starts before the EBR event. This situation occurs when critical failures affect the system in such a way that it becomes completely unusable, without preventing the event logging mechanisms from recording some messages into the log files. During the recovery phase, the actions performed by the system administrators may include several unsuccessful reboot attempts that are not recorded in the /var/adm/messages log file, but some events referring to them are written in the wtmpx file. Using this information, just like in the previous case, we can refine the downtime and uptime estimations by associating the failure occurrence time to the timestamps of the events recorded in the wtmpx file that better reflects the start of the service interruption. An example of a sequence of events illustrating this case is given in Figure 3. Event # Event date Event description File where the event is logged 2003 Jan 9 10:18:59 2003 Jan 9 10:21:39 2003 Jan 9 10:21:39 2003 Jan 9 10:21:39 2003 Jan 9 10:21:39 2003 Jan 9 10:21:48 2003 Jan 9 10:21:48 2003 Jan 9 10:22:05 2003 Jan 9 10:22:13 2003 Jan 9 10:22:13 2003 Jan 9 10:22:16 user=root; device=console; pid=2370; proc_type=7 user=sac; device=; pid=425; proc_type=8 user=root; device=console; pid=2370; proc_type=8 user=; device=run-level 5; pid=0; proc_type=1 user=rc5; device=; pid=25952; proc_type=5 user=UserC; device=pts/3; pid=11584; proc_type=8 user=UserC; device=pts/1; pid=11359; proc_type=8 last event before reboot user=rc5; device=; pid=25953; proc_type=8 user=uadmin;device=; pid=26121; proc_type=5 reboot start wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx wtmpx var/adm/messages log file wtmpx wtmpx var/adm/messages log file Figure 3. Example illustrating situation S2 We can identify the events extracted from the wtmpx file informing upon the stop of the system: event # 2 with user field “sac” and proc_type “ 8” (dead process) followed by events #3, #4, and #5 notifying the system run-level change to run-level 5 (this one is used to properly stop the system). This example shows that the start of the service interruption period is prior to the EBR event recorded in the /var/adm/messages log file. The refinement of the uptime and downtime estimations corresponding to such situations consists in associating the failure occurrence time to the timestamps of the last event recorded in the wtmpx file before the start of the reboot sequence. 4. Experimental results The analyses presented in this Section are based on /var/adm/messages log file data collected during 45 months (October 1999 – July 2003) from 418 SunOS/Solaris Unix workstations and servers interconnected through the LAAS local area computing network. As shown in Figure 4, the data collection period differed significantly from one machine to another due to the dynamic evolution of the network. For more than 70 % of the machines, the data collection period was longer than 21 months. On the other hand, it can be noticed that some machines have a quite short data collection period. In order to have significant statistical analysis results, we excluded from the analysis the machines for which the data collection period was shorter than 2000 hours (about 3 months). Consequently, the results presented in the following concern 373 Unix machines. Among these machines, 17 correspond to major servers for the entire network or a sub-set of users: WWW, NIS+, NFS, FTP, SMTP, file servers, printing servers, etc. Figure 4. Examples of records from /var/adm/wtmpx obtained with our algorithm The application of the reboot identification algorithm on the collected data allowed us to identify 12805 reboots for the 373 machines, only 476 reboots concern the 17 servers. Based on the information provided by the reboot identification algorithm, we evaluated for each machine the associated uptimes UTi and downtimes DTi, and the availability measure. The collection of wtmpx files started later than the /var/adm/messages log files. For this reason, we were able to analyze the impact of uptimes and downtimes estimation refinement algorithms only on a subset of UTi and DTi values associated with the reboots identified from the log files. Among the 12805 reboots, this analysis concerned 6163 reboots (48.13%). For the remaining 6642 reboots, the corresponding data from the wtmpx files was not available. In the following, we first present in Section 4.1 the results of machine uptimes and downtimes estimation based on the processing of the set of 6163 reboots focusing on the impact of the estimation refinement algorithms. Then, global results taking into account the whole data collected during our study are presented in Section 4.2 in order to give an overall picture on the availability and the rate of occurrence of reboots characterizing the Unix machines included in our study. 4.1. Machine uptimes and downtimes estimation and refinement The correlation of the information contained in the /var/adm/messages log files, the wtmpx files, and the DCSummary files, revealed that both situations S1 and S2 discussed in Section 3.2 are common: • Situation S1 was observed for 79.35% of the analyzed reboots; • Situation S2 was observed for 10.77% of the analyzed reboots; For the 9.88% remaining reboots, the assumption that the EBR recorded in /var/adm/messages file identifies the last event recorded on the machine before the reboot was consistent with the information available in the wtmpx and the DCSummary files. In order to analyze the impact of the estimation refinement algorithms on the results, Table 1 gives the Mean, Median and Standard Deviation of uptime and downtime values, before and after the application of our estimation refinement algorithms discussed in Section 3. Considering the median of the downtime values, it can be seen that the refinement algorithms have a significant impact on the results. The median estimated after the refinement is 66 times lower than the value obtained without the refinement. The refinement algorithms have also an impact on the uptimes estimation, but as expected the improvement factor is lower than the one observed for the downtime values (1.8 compared to 66). Table 1. Machine uptimes and downtimes estimates before and after refinement Uptimes UTi Downtimes DTi before refinement after refinement before refinement after refinement Mean 28.3days 1.1 month 5.9 days 1.9 days Median 6.1 days 10.8 days 8.9 hours 8.1 min 1.7 months 1.8 months 24.1 days 21.1 days The impact of the estimation refinement algorithms on availability is summarized in Table 2. It can be seen the estimated average unavailability after the refinement is three times lower than the value estimated based only on the information in the /var/adm/messages log files. Clearly, the difference is significant and cannot be ignored. Table 2. Impact of the estimation refinement algorithms on Availability and Unavailability before refinement after refinement A 89.3% 96.3 % UA 39.0 days/year 13.7 days/year 4.2. Availability and reboot rates estimated from the whole data set This section presents some results concerning the reboot rates and the availability of the 373 SunOS/Solaris Unix machines included in our study taking into account the whole set of 12805 reboots identified from the /var/adm/messages files. When the wtmpx files were not available (this concerned 6642 reboots), the estimation of the UTi, DTi, availability and reboot rates was based only on the information in the /var/adm/messages files, using the assumption discussed in Section 3.1. In the other case (i.e., for the 6163 reboots), we applied the estimation refinement algorithms presented in Section 3.2. Figure 5 plots the reboot rates estimated for each machine as a function of the data collection period. The estimated reboot rate for each machine corresponds to the average number of reboots recorded during the corresponding observation. It can be seen that the reboot rates are uniformly distributed between 10 /hour and 10 /hour. Figure 5. Unix machines reboot rates as a function of the data collection period As indicated in Table 3, the mean value of machine reboot rates is 1.3 10 /hour, when considering all Unix machines including workstations and servers. If we take into account only the servers, the mean reboot rate is 1.5 times lower (7.7 10 /hour) corresponding to one reboot every two months. Table 3. Reboot rate statistics Mean Median Std. Dev. SunOS/Solaris machines (Workstations + Servers) 1.3 10 /h 1.0 10 /h 1.3 10 Servers only 7.7 10 /h 6.4 10 /h 5.6 10 The results illustrating the availability and unavailability of the Unix machines including workstations and servers are given in Figure 6 and Table 4. The mean availability is 97.81 % corresponding to an average unavailability of 8 days per year. Detailed analysis shows that only 15 among the 373 Unix machines included in the study have an availability lower than 90%. When considering only the servers, the estimated availability varies between 99.36% and 99.1% with an average unavailability of 12 hours per year. Figure 6. SunOS/Solaris Unix machines availability distribution Table 4. Availability and Unavailability statistics Mean Median Std. Dev. A 97.81 % 98.79 % 3.07 % UA 7.99 day/year 4.41 day/year 11.20 day/year 6. Conclusion Dependability analyses based on event logs provide useful feedback to software and system designers. Nevertheless, the results obtained are intimately related to the quality and the completeness of the information recorded in the logs. As the information contained in such event logs could be incomplete or imperfect, it is important to use additional sources of information to ensure that the conclusions derived from such analyses faithfully reflect reality. The approach investigated in this paper is aimed to fulfill this objective considering SunOS/Solaris Unix systems as an example. In particular, we have shown that the combined us of the data contained in the syslogd files and the information recorded in the wtmpx files or through the monitoring of systems state during the data collection campaigns provides uptime and downtime estimations that are closer to reality than the estimations obtained based on syslogd files only. This result is illustrated based on a large set of field data collected from 373 machines during a 45 month observation period. In our future work, we will investigate the applicability of the approach proposed in this paper to other operating systems such as Linux, Windows 2K and Mac OS X. References [1] M. F. Buckley, D. P. Siewiorek, “VAX/VMS Event Monitoring and Analysis”, 25th IEEE Int. Symp. on Fault-Tolerant Computing (FTCS-25), (Pasadena, CA, USA), pp. 414-423, IEEE Computer Society, 1995. [2] R. K. Iyer, D. Tang, “Experimental Analysis of Computer System Dependability”, in Fault-Tolerant Computer System Design, D. K. Pradhan, Ed., Prentice Hall PTR, 1996, pp. 282-392. [3] M. Kalyanakrishnam, Z. Kalbarczyk, R. K. Iyer, “Failure Data Analysis of a LAN of Windows NT Based Computers”, 18th IEEE Symp. on Reliable Distributed Systems (SRDS-18), (Lausanne, Switzerland), pp. 178-187, 1999. [4] C. Simache, M. Kaâniche, “Measurement-based Availability Analysis of Unix Systems in a Distributed Environment”, The 12th Int. Symp. on Software Reliability Engineering (ISSRE-2001), (Hong Kong, China), pp. 346-355, IEEE Computer Society, 2001. [5] C. Simache, M. Kaâniche, “Event Log based Dependability Analysis of Windows NT and 2K Systems”, 2002 Pacific Rim Int. Symposium on Dependable Computing (PRDC-2002), (Tsukuba, Japan), pp. 311-315, IEEE Computer Society, 2002. [6] C. Simache, “Dependability evaluation of Unix and Windows Systems based on operational data: A Method and Application”, PhD Thesis, LAAS Report N°04333, 2004 (in French). [7] A. Thakur, R. K. Iyer, “Analyze-NOW — An Environment for Collection & Analysis of Failures in a Network of Workstations”, IEEE Transactions on Reliability, vol. 45, pp. 561-570, 1996. [8] M. Tsao, D. P. Siewiorek, “Trend Analysis on System Error Files”, 13th IEEE Int. Symp. on Fault-Tolerant Computing (FTCS-13), (Milano, Italy), pp. 116-119, IEEE Computer Society, 1983. [9] J. Xu, Z. Kalbarczyk, R. K. Iyer, “Networked Windows NT System Field Failure Data Analysis”, Proc. 1999 IEEE Pacific Rim Int. Symp. on Dependable Computing (PRDC-1999), (Los Alamitos, CA), pp. 178-185, 1999 /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles false /AutoRotatePages /None /Binding /Left /CalGrayProfile (None) /CalRGBProfile (None) /CalCMYKProfile (None) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.3 /CompressObjects /Off /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJDFFile false /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.1000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails true /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize true /OPM 0 /ParseDSCComments false /ParseDSCCommentsForDocInfo false /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo false /PreserveFlatness true /PreserveHalftoneInfo true /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Remove /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true /NeverEmbed [ true /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 150 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 2.00333 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages false /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] /ColorImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] /GrayImageDict << /QFactor 0.76 /HSamples [2 1 1 2] /VSamples [2 1 1 2] /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 15 /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 /AllowPSXObjects false /CheckCompliance [ /None /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /False /Description << /JPN /DEU /FRA /PTB /DAN /NLD /ESP /SUO /ITA /NOR /SVE /ENU >> setdistillerparams /HWResolution [600 600] /PageSize [612.000 792.000] >> setpagedevice ABSTRACT This paper presents a measurement-based availability assessment study using field data collected during a 4-year period from 373 SunOS/Solaris Unix workstations and servers interconnected through a local area network. We focus on the estimation of machine uptimes, downtimes and availability based on the identification of failures that caused total service loss. Data corresponds to syslogd event logs that contain a large amount of information about the normal activity of the studied systems as well as their behavior in the presence of failures. It is widely recognized that the information contained in such event logs might be incomplete or imperfect. The solution investigated in this paper to address this problem is based on the use of auxiliary sources of data obtained from wtmpx files maintained by the SunOS/Solaris Unix operating system. The results obtained suggest that the combined use of wtmpx and syslogd log files provides more complete information on the state of the target systems that is useful to provide availability estimations that better reflect reality. <|endoftext|><|startoftext|> Introduction Several initiatives have been developed during the last decade to monitor malicious threats and activities on the Internet, including viruses, worms, denial of service attacks, etc. Among them, we can mention the Internet Motion Sensor project [1], CAIDA [2], DShield [3], and CADHo [4]. These projects provide valuable information on security threats and the potential damage that they might cause to Internet users. Analysis and modeling methodologies are necessary to extract the most relevant information from the large set of data collected from such monitoring activities that can be useful for system security administrators and designers to support decision making. The designers are mainly interested in having representative and realistic assumptions about the kind of threats and vulnerabilities that their system will have to cope with once it is used in operation. Knowing who are the enemies and how they proceed to defeat the security of target systems is an important step to be able to build systems that can be resilient with respect to the corresponding threats. From the system security administrators’ perspective, the collected data should be used to support the development of efficient early warning and intrusion detection systems that will enable them to better react to the attacks targeting their systems. As of today, there is still a lack of methodologies and significant results to fulfill the objectives described above, although some progress has been achieved recently in this field. The CADHo project “Collection and Analysis of Data from Honeypots” [4], an ongoing research action started in September 2004, is aimed at contributing to filling such a gap by carrying out the following activities: 1) deploying a distributed platform of honeypots [5] that gathers data suitable to analyze the attack processes targeting a large number of machines connected to the Internet; 2) developing analysis methodologies and modeling approaches to validate the usefulness of this platform by carrying out various analyses, based on the collected data, to characterize the observed attacks and model their impact on security. A honeypot is a machine connected to a network but that no one is supposed to use. In theory, no connection to or from that machine should be observed. If a connection occurs, it must be, at best an accidental error or, more likely, an attempt to attack the machine. The Leurré.com data collection environment [5], set up in the context of the CADHo project, has deployed, as of to date, thirty five honeypot platforms at various locations from academia and industry, in twenty five countries over the five continents. Several analyses carried out based on the data collected so far from these honeypots have revealed that very interesting observations and conclusions can be derived with respect to the attack activities observed on the Internet [4, 6-9]. In addition, several automatic data analyses and clustering techniques have been developed to facilitate the extraction of relevant information from the collected data. A list of papers detailing the methodologies used and the results of these analyses is available in [6]. This paper focuses on modeling-related activities based on the data collected from the honeypots. We first discuss the objectives of such activities and the challenges that need to be addressed. Then we present some examples of models obtained from the data. The paper is organized as follows. Section 2 presents the data collection environment. Section 3 focuses on the modeling of attacks based on the data collected from the honeypots deployed. Modeling examples are presented in Section 4. Finally, Section 5 discusses future work. 2. The data collection environment The data collection environment (called Leurré.com [5]) deployed in the context of the CADHo project is based on low-interaction honeypots using the freely available software called honeyd [10]. Since September 2004, 35 honeypot platforms have been progressively deployed on the Internet at various geographical locations. Each platform emulates three computers running Linux RedHat, Windows 98 and Windows NT, respectively, and various services such as ftp, web, etc. A firewall ensures that connections cannot be initiated from the computers, only replies to external solicitations are allowed. All the honeypot platforms are centrally managed to ensure that they have exactly the same configuration. The data gathered by each platform are securely uploaded to a centralized database with the complete content, including payload of all packets sent to or from these honeypots, and additional information to facilitate its analysis, such as the IP geographical localization of packets’ source addresses, the OS of the attacking machine, the local time of the source, etc. 3. Modeling objectives Modeling involves three main steps: 1) The definition of the objectives of the modeling activities and the quantitative measures to be evaluated. 2) The development of one (or several) models that are suitable to achieve the specified objectives. 3) The processing of the models and the analysis of the results to support system design or operation activities. The data collected from the honeypots can be processed in various ways to characterize the attack processes and perform predictive analyses. In particular, modeling activities can be used to: • Identify the probability distributions that best characterize the occurrence of attacks and their propagation through the Internet. • Analyze whether the data collected from different platforms exhibit similar or different malicious attack activities. • Model the time relationships that may exist between attacks coming from different sources (or to different destinations). • Predict the occurrence of new waves of attacks on a given platform based on the history of attacks observed on this platform as well as on the other platforms. For the sake of illustration, we present in the following sections simple preliminary models based on the data collected from our honeypots that are aimed at fulfilling such objectives. 4. Examples The examples presented in the following address: 1) The analysis of the time evolution of the number of attacks taking into account the geographic location of the attacking machine. 2) The characterization and statistical modeling of the times between attacks. 3) The analysis of the propagation of attacks throughout the honeypot platforms. The data considered for the examples has been collected from January 1st, 2004 to April 17, 2005, corresponding to a data collection period of 320 days. We take into account the attacks observed on 14 honeypot platforms among those deployed so far. The selected honeypots correspond to those that have been active for almost the whole considered period. The total number of attacks observed on these honeypots is 816476. These attacks are not uniformly distributed among the platforms. In particular, the data collected from three platforms represent more than fifty percent of the total attack activity. 4.1 Attack occurrence and geographic distribution The preliminary models presented in this sub-section address: i) the time-evolution modeling of the number of attacks observed on different honeypot platforms, and ii) the analysis of potential correlations for the attack processes observed on the different platforms taking into account the geographic location of the attacking machines and the proportion of attacks observed on each platform, wrt. to the global attack activity. Let us denote by: − Y(t) the function describing the evolution of the number of attacks per unit of time observed on all the honeypots during the observation period, − Xj(t) the function describing the evolution of the number of attacks per unit of time observed on all the honeypots during the observation period for which the IP address of the attacking machine is located in country j . In a first stage, we have plotted, for various time periods, Y(t) and the curves Xj(t) corresponding to different countries j. Visual inspection showed surprising similarities between Y(t) and some Xj(t). To confirm such empirical observations, we have then decided to rigorously analyze this phenomenon using mathematical linear regression models. Considering a linear regression model, we have investigated if Y(t) can be estimated from the combination of the attacks described by Xj(t), taking into account a limited number of countries j. Let us denote by Y*(t) the estimated model. Formally, Y*(t) is defined as follows: Y*(t) = Σαj Xj(t) + β j= 1, 2, .. k (1) Constants αj and β correspond to the parameters of the linear model that provide the best fit with the observed data, and k is the number of countries considered in the regression. The quality of fit of the model is measured by the statistics R2 defined by: R2 = Σ(Y*(i) – Yav) 2/ Σ(Y (i) – Yav) 2 (2) Y (i) and Y*(i) correspond to the observed and estimated number of attacks for unit of time i, respectively. Yav is the average number of attacks per unit of time, taking into account the whole observation period. Indeed, R is the correlation factor between the estimated model and the observed values. The closer the R2 value is to 1, the better the estimated model fits the collected data. We have applied this model considering linear regressions involving one, two or more countries. Surprisingly, the results reveal that a good fit can be obtained by considering the attacks from one country only. For example, the models providing the best fit taking into account the total number of attacks from all the platforms are obtained by considering the attacks issued from either UK, USA, Russia or Germany only. The corresponding R2 values are of the same order of magnitude (0.944 for UK, 0.939 for USA, 0.930 for Russia and 0.920 for Germany), denoting a very good fit of the estimated models to the collected data. For example, the estimated model obtained when considering the attacks from Russia only is defined by equation (3): Y*(t) = 44.568 X1(t) + 1555.67 (3) X1(t) represents the evolution of the number of attacks from Russia. Figure 1 plots the evolution of the observed and estimated number of attacks per unit of time during the data collection period considered in this example. The unit of time corresponds to 4 days. It is noteworthy that, similar conclusions are obtained if we consider another granularity for the unit of time, for example one day, or one week. These results are even more surprising that the attacks from Russia and UK represent only a small proportion of the total number of attacks (1.9% and 3.7% respectively). Concerning the USA, although the proportion is higher (about 18%), it is not sufficient to explain the linear model. Figure 1- Evolution of the number of attacks per unit of time observed on all the platforms and estimated model considering attacks from Russia only We have applied similar analyses by respectively considering each honeypot platform in order to investigate if similar conclusions can be derived by comparing their attack activities per source country to their global attack activities. The results are summarized in Table 1. The second column identifies the source country that provides the best fit. The corresponding R2 value is given in the third column. Finally, the last three columns give the R2 values obtained when considering UK, USA, or Russia in the regression model. It can be noticed that the quality of the regressions measured when considering attacks from Russia only is generally low for all platforms (R2 less than 0.80). This indicates that the property observed at the global level is not visible when looking at the local activities observed on each platform. However, for the majority of the platforms, the best regression models often involve one of the three following countries: USA, Germany or UK, which also provide the best regressions when analyzing the global attack activity considering all the platforms together. Two exceptions are found with P6 and P8 for which the observed attack activities exhibit different characteristics with respect to the origin of the attacks (Taiwan, China), compared to the other platforms. The trends discussed above have been also observed when considering a different granularity for the unit of time (e.g., 1 day or 1 week) as well as different data observation periods. Platform Country providing the best model Best model Russia P1 Germany 0.895 0.873 0.858 0.687 P2 USA 0.733 0.464 0.733 0.260 P4 Germany 0.722 0.197 0.373 0.161 P5 Germany 0.874 0.869 0.872 0.608 P6 UK 0.861 0.861 0.699 0.656 P8 Taiwan 0.796 0.249 0.425 0.212 P9 Germany 0.754 0.630 0.624 0.631 P11 China 0.746 0.303 0.664 0.097 P13 Germany 0.738 0.574 0.412 0.389 P14 Germany 0.708 0.510 0.546 0.087 P20 USA 0.912 0.787 0.912 0.774 P21 SPAIN 0.791 0.620 0.727 0.720 P22 USA 0.870 0.176 0.870 0.111 P23 USA 0.874 0.659 0.874 0.517 Global UK 0.944 0.944 0.939 0.930 Table 1 – Estimated models for each platform: correlation factors for the countries providing the best fit and for UK, USA and Russia To summarize, two main findings can be derived from the results presented above: 1) Some trends exhibited at the global level considering the attack processes on all the platforms together are not observed when analyzing each platform individually (this is the case, for example, of attacks from Russia). On the other hand, we have observed the other situation where the trends observed globally are also visible locally on the majority of the platforms (this is the case, for example, of attacks from USA, UK and Germany). 2) The attack processes observed on each platform are very often highly correlated with the attack processes originating from a particular country. The country providing the best regressions locally, does not necessary exhibit high correlations when considering other platforms or at the global level. These trends seem to result from specific factors that govern the attack processes observed on each platform. 4.2 Distribution of times between attacks In this example, we focus on the analysis and the modeling of the times between attacks observed on different honeypot platforms. Let us denote by ti, the time separating the occurrence of attack i and attack (i-1). Each attack is associated to an IP address, and its occurrence time is defined by the time when the first packet is received from the corresponding address at one of the three virtual machines of the honeypot platform. All the packets received from the same IP address within 24 hours are supposed to belong to the same attack session. We have analyzed the distribution of the times between attacks observed on each honeypot platform. Our objective was to find analytical models that faithfully reflect the empirical data collected from each platform. In the following, we summarize the results obtained considering 5 platforms for which we have observed the highest attack activity. 4 .2.1 Empirical analyses Table 2 gives the number of intervals of times between attacks observed at each platform considered in the analysis as well as the corresponding number of IP addresses. As illustrated by Figure 2, most of these addresses have been observed only once at a given platform. Nevertheless, some IP addresses have been observed several times, the maximum number of visits per IP address for the five platforms was 57, 96, 148, 183, and 83 (respectively). Indeed, the curves plotting the number of IP addresses as a function of the number of attacks for each address follow a heavy-tailed power law distribution. It is noteworthy that such distributions have been observed in many performance and dependability related studies in the context of the Internet, e.g., transfer and interarrival times, burst sizes, sizes of files transferred over the web, error rates in web servers, etc. P5 P6 P9 P20 P23 Number of ti 85890 148942 46268 224917 51580 Number of IP addresses 79549 90620 42230 162156 47859 Table 2 - Numbers of intervals of times between attacks (ti) and of different IP addresses observed at each platform Figure 2- Number of IP addresses versus the number of attacks per IP address observed at each platform (log-log scale) 4 .2.2 Modeling Finding tractable analytical models that faithfully reflect the observed times between attacks is useful to characterize the observed attack processes and to find appropriate indicators that can be used for prediction purposes. We have investigated several candidate distributions, including Weibull, Lognormal, Pareto, and the Exponential distribution, which are traditionally used in reliability related studies. The best fit for each platform has been obtained using a mixture model combining a Pareto and an exponential distribution. Let us denote by T the random variable corresponding to the time between the occurrence of two consecutive attacks at a given platform, and t a realization of T. Assuming that the probability density function pdf(t) associated to T is characterized by a mixture distribution combining a Pareto distribution and an exponential distribution, then f(t) is defined as follows. pdf (t) = P (t +1) + (1" P k is the index parameter of the Pareto distribution, λ is the rate associated to the exponential distribution and Pa is a probability. We have used the R statistical package [11] to estimate the parameters that provide the best fit to the collected data. The quality of fit is assessed by applying the Kolmogorov-Smirnov statistical test. The results are presented in Figure 3. It can be noticed that for all the platforms, the mixed distribution provides a good fit to the observed data whereas the exponential distribution is not suitable to describe the observed attack processes. Thus, the traditional assumption considered in hardware reliability evaluation studies assuming that failures occur according to a Poisson process does not seem to be satisfactory when considering the data observed form our honeypots. These results have been also confirmed when considering the data collected during other observation periods. 1 31 61 91 121 151 181 211 241 271 Time between attacks (seconds) Pa = 0.0051 k = 0.173 ! = 0.121/sec. p-value = 0.90 Data Mixture (Pareto, Exp.) Exponential 1 31 61 91 121 151 181 211 241 271 Time between attacks (seconds) Mixture (Pareto, Exp.) Exponential Pa = 0.0115 k = 0.1183 ! = 0.1364/sec. p-value = 0.999 a) P5 b) P6 1 31 61 91 121 151 181 211 241 271 Time between attacks (seconds) Mixture (Pareto, Exp.) Exponential Pa = 0.0019 k = 0.1668 ! = 0.276/sec. p-value = 0.99 1 31 61 91 121 151 181 211 241 271 Time between attacks (seconds) Mixture (Pareto, Exp.) Exponential Pa = 0.0144 k = 0.0183 ! = 0.0136/sec. p-value = 0.90 c) P9 d) P20 1 31 61 91 121 151 181 211 241 271 Time between attacks (seconds) Mixture (Pareto, Exp.) Exponential Pa = 0.0031 k = 0.1240 ! = 0.275/sec. p-value = 0.985 e) P23 Figure 3- Observed and estimated times between attacks probability density functions. 4.3 Propagation of attacks Besides analyzing the attack activities observed at each platform in isolation, it is useful to identify phenomena that reflect propagation of attacks through different platforms. In this section, we analyze simple scenarios where a propagation between two platforms is assumed to occur when the IP address of an attacking machine observed at a given platform is also observed at another platform. Such a situation might occur for example as a result of a scanning activity or might be resulting from the propagation of worms. For the sake of illustration, we restrict the analysis to the five platforms considered in the previous example. For each attacking IP address in the data collected from the five platforms during the period of the study, we identified: 1) all the occurrences with the same source address, 2) the times of each occurrence and 3) the platform on which each occurrence has been reported. A propagation is said to occur for this IP address from platform Pi to platform Pj when the next occurrence of this address is observed on Pj after visiting Pi. Based on this information we build a propagation graph where each node identifies a platform and a transition between two nodes identifies a propagation between the nodes. A probability is associated to each transition to characterize its likelihood of occurrence. Figure 4 presents the propagation graph obtained for the five platforms included in the analysis. Considering platforms P6 and P20, it can be seen that only a few IP addresses that attacked these platforms have been observed on the other platforms. The situation is different when considering platforms P5, P9, and P23. In particular, it can be noticed that propagation between P5 and P9 is highly probable. This is related in particular to the fact that the addresses of the corresponding platforms belong to the same /8 network domain. More thorough and detailed analyses are currently carried out based on the propagation graph in order to take into account timing information for the corresponding transitions and also the types of attacks observed, in order to better explain the propagation phenomena illustrated by the graph. Figure 4- Propagation graph 5. Conclusion This paper presented simple examples and preliminary models illustrating various types of empirical analysis and modeling activities that can be carried out based on the data collected from honeypots in order to characterize attack processes. The honeypot platforms deployed so far in our project belong to the family of so-called “low interaction honeypots”. Thus, hackers can only scan ports and send requests to fake servers without ever succeeding in taking control over them. In our project, we are also interested in running experiments with “high interaction” honeypots where attackers can really compromise the targets. Such honeypots are suitable to collect data that would enable us to study the behaviors of attackers once they have managed to get access to a target and try to progress in the intrusion process to get additional privileges. Future work will be focused on the deployment of such honeypots and the exploitation of the collected data to better characterize attack scenarios and analyze their impact on the security of the target systems. The ultimate objective would be to build representative stochastic models that will enable us to evaluate the ability of computing systems to resist to attacks and to validate them based on real attack data. Acknowledgement. This work has been carried out in the context of the CADHo project, an ongoing research action funded by the French ACI “Securité & Informatique” (www.cadho.org). It is partially supported by the ReSIST European Network of Excellence (www .resist-noe.org). References [1] M. Bailey, E. Cooke, F. Jahanian, J. Nazario, and D. Watson, "The Internet Motion Sensor: A Distributed Blackhole Monitoring System," Proc. 12th Annual Network and Distributed System Security Symposium (NDSS), San Diego, CA, Feb. 2005. [2] Home Page of the CAIDA Project, http://www.caida.org/ [3] DShield Distributed Detection System homepage, http://www.honeynet.org/ [4] E. Alata, M. Dacier, Y. Deswarte, M. Kaâniche, K. Kortchinsky, V. Nicomette, V.H. Pham, F. Pouget, Collection and Analysis of Attack data based on honeypots deployed on the Internet”, 1st Workshop on Quality of Protection, Milano, Italy, September 2005. [5] F. Pouget, M. Dacier, V. H. Pham, “Leurré.com: On the Advantages of Deploying a Large Scale Distributed Honeypot Platform”, Proc. E-Crime and Computer Evidence Conference (ECCE 2005), Monaco, Mars 2005. [6] L. Spitzner, Honeypots: Tracking Hackers, Addison- Wesley, ISBN from-321-10895-7, 2002 [7] Project Leurré.com. Publications web page, http://www.leurrecom.org/paper.htm [8] M. Dacier, F. Pouget, H. Debar, “Honeypots: Practical Means to Validate Malicious Fault Assumptions on the Internet”, Proc. 10th IEEE International Symposium Pacific Rim Dependable Computing (PRDC10), Tahiti, March 2004, pages 383-388. [9] M. Dacier, F. Pouget, H. Debar, “Attack Processes found on the Internet”, Proc. OTAN Symp. on Adaptive Defense in Unclassified Networks, Toulouse, France, April 2004. [10] Honeyd Home page, http://www.citi.umich.edu/u/provos/honeyd/ [11] R statistical package Home page, http://www.r-project.org ABSTRACT Honeypots are more and more used to collect data on malicious activities on the Internet and to better understand the strategies and techniques used by attackers to compromise target systems. Analysis and modeling methodologies are needed to support the characterization of attack processes based on the data collected from the honeypots. This paper presents some empirical analyses based on the data collected from the Leurr{\'e}.com honeypot platforms deployed on the Internet and presents some preliminary modeling studies aimed at fulfilling such objectives. <|endoftext|><|startoftext|> Draft version October 24, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE LOW CO CONTENT OF THE EXTREMELY METAL POOR GALAXY I ZW 18 Adam Leroy , John Cannon , Fabian Walter , Alberto Bolatto , Axel Weiss Draft version October 24, 2018 ABSTRACT We present sensitive molecular line observations of the metal-poor blue compact dwarf I Zw 18 obtained with the IRAM Plateau de Bure interferometer. These data constrain the CO J = 1 → 0 luminosity within our 300 pc (FWHM) beam to be LCO < 1×10 5 K km s−1 pc2 (ICO < 1 K km s −1), an order of magnitude lower than previous limits. Although I Zw 18 is starbursting, it has a CO luminosity similar to or less than nearby low-mass irregulars (e.g. NGC 1569, the SMC, and NGC 6822). There is less CO in I Zw 18 relative to its B-band luminosity, H I mass, or star formation rate than in spiral or dwarf starburst galaxies (including the nearby dwarf starburst IC 10). Comparing the star formation rate to our CO upper limit reveals that unless molecular gas forms stars much more efficiently in I Zw 18 than in our own galaxy, it must have a very low CO-to-H2 ratio, ∼ 10 −2 times the Galactic value. We detect 3mm continuum emission, presumably due to thermal dust and free-free emission, towards the radio peak. Subject headings: galaxies: individual (I Zw 18); galaxies: ISM; galaxies: dwarf, radio lines: ISM 1. INTRODUCTION With the lowest nebular metallicity in the nearby uni- verse (12+ logO/H ≈ 7.2, Skillman & Kennicutt 1993), the blue compact dwarf I Zw 18 plays an important role in our understanding of galaxy evolution. Vigorous ongo- ing star formation implies the presence of molecular gas, but direct evidence has been elusive. Vidal-Madjar et al. (2000) showed that there is not significant diffuse H2, but Cannon et al. (2002) found ∼ 103 M⊙ of dust organized in clumps with sizes 50 – 100 pc. Vidal-Madjar et al. (2000) did not rule out compact, dense molecular clouds, and Cannon et al. (2002) argued that this dust may in- dicate the presence of molecular gas. Observations by Arnault et al. (1988) and Gondhalekar et al. (1998) failed to detect CO J = 1 → 0 emission, the most commonly used tracer of H2. This is not surprising. The low dust abundance and intense radiation fields found in I Zw 18 may have a dramatic impact on the formation of H2 and structure of molecular clouds. A large fraction of the H2 may exist in extended envelopes surrounding relatively compact cold cores. In these envelopes, H2 self-shields while CO is dissociated (Maloney & Black 1988). The result may be that in such galaxies [CII] or FIR emission trace H2 better than CO (Madden et al. 1997; Israel 1997a; Pak et al. 1998). Further, H2 may simply be underabundant, as there is a lack of grains on which to form while photodissociation is enhanced by an intense UV field. Indeed, Bell et al. (2006) found that at Z = Z⊙/100, a molecular cloud may take as long as a Gyr to reach chemical equilibrium. A low CO content in I Zw 18 is then expected, and a stringent upper limit would lend observational support to predictions for molecular cloud structure at low metallic- 1 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany; email: leroy@mpia-hd.mpg.de 2 Astronomy Department, Wesleyan University, Middletown, CT 06459, cannon@astro.wesleyan.edu 3 Radio Astronomy Lab, UC Berkeley, 601 Campbell Hall, Berkeley, CA, 94720 4 MPIfR, Auf dem Hügel 69, 53121, Bonn, Germany ity. However, while the existing upper limits are sensitive in an absolute sense, they do not even show I Zw 18 to have a lower normalized CO content than a spiral galaxy (e.g. less CO per B-band luminosity). The low luminos- ity (MB ≈ −14.7, Gil de Paz et al. 2003) and large dis- tance (d=14 Mpc, Izotov & Thuan 2004) of this system require very sensitive observations to set a meaningful upper limit. In this letter we present observations, obtained with the IRAM Plateau de Bure Interferometer (PdBI)5, that constrain the CO luminosity, LCO, to be equal to or less than that of nearby CO-poor (non-starbursting) dwarf irregulars. 2. OBSERVATIONS I Zw 18 was observed with the IRAM Plateau de Bure Interferometer on 17, 21, and 27 April and 13 May 2004 for a total of 11 hours. The phase calibrators were 0836+710 (Fν(115GHz) ≈ 1.1 Jy), and 0954+556 (Fν(115GHz) ≈ 0.35 Jy). One or more calibrators with known fluxes were also observed during each track. The data were reduced at the IRAM facility in Grenoble us- ing the GILDAS software package; maps were prepared using AIPS. The final CO J = 1 → 0 data cube has beam size 5.59′′ × 3.42′′, and a velocity (frequency) resolution of 6.5 km s−1 (2.5 MHz). The velocity coverage stretches from vLSR ≈ 50 to 1450 km s −1. The data have an RMS noise of 3.77 mJy beam−1 (18 mK; 1 Jy beam−1 = 4.8 K). The 44′′ (FWHM) primary beam completely covers the galaxy. Based on variation of the relative fluxes of the calibrators, we estimate the gain uncertainty to be < 15%. 3. RESULTS 3.1. Upper Limit on CO Emission To search for significant CO emission, we smooth the cube to 20 km s−1 velocity resolution, a typical line width 5 Based on observations carried out with the IRAM Plateau de Bure Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).” http://arxiv.org/abs/0704.0862v1 Fig. 1.— CO 1 → 0 spectra of I Zw 18 towards the radio continuum/Hα peak (left) and the highest significance spectra (right), which is still too faint to classify as more than marginal. The locations of both spectra are shown in Figure 2. Dashed hor- izontal lines show the magnitude of the RMS noise. for CO at our spatial resolution (e.g., Helfer et al. 2003). The noise per channel map in this smoothed cube is σ20 ≈ 0.25 K km s −1. Over the H I velocity range (710 – 810 km s−1, van Zee et al. 1998), there are no regions with ICO,20 > 1 K km s −1 (4σ) within the primary beam. We pick a slightly conservative upper limit for two rea- sons. First, if there were CO emission with this intensity we would be certain of detecting it. Second, the noise in the cube is slightly non-Gaussian, so that the false posi- tive rate for ICO,20 > 1 K km s −1 — estimated from the negatives and the channel maps outside the H I velocity range — is ∼ 0.2%, very close to that of a 3σ deviate. For d = 14 Mpc, the synthesized beam has a FWHM of 300 pc and an area of 1.0 × 105 pc2. Our intensity limit, ICO < 1 K km s −1, therefore translates to a CO luminosity limit of LCO < 1× 10 5 K km s−1 pc2. There is a marginal signal toward the southern knot of Hα emission (9h34m02s.4, 55◦14′23′′.0). This emission has the largest |ICO,20| found over the H I velocity range, corresponding to LCO ∼ 8×10 4 K km s−1 pc2, just below our limit. This same line of sight also shows |ICO| > 2σ over three consecutive channels, a feature seen along only one other line of sight (in negative) over the H I velocity range. The marginal signal is suggestively located in the southeast of I Zw 18, where Cannon et al. (2002) identi- fied several potential sites of molecular gas from regions of relatively high extinction. While tantalizing, the sig- nal is not strong enough to be categorized as a detection. Figure 1 shows CO spectra towards the Hα/radio contin- uum peak (Cannon et al. 2002, 2005; Hunt et al. 2005a, see Figure 2) and this marginal signal. 3.2. Continuum Emission We average the data over all channels and produce a continuum map with noise σ115GHz = 0.35 mJy beam The highest value in the map is I115GHz = 1.06±0.35mJy beam−1 at α2000 = 9 h34m02s.1, δ2000 = +55 ◦ 14′ 27′′.0. This is within a fraction of a beam of the 1.4 GHz peak identified by Cannon et al. (2005, α2000 = 9 h34m02s.1, δ2000 = +55 ◦ 14′ 28′′.06) and Hunt et al. (2005a, α2000 = 9h34m02s, δ2000 = +55 ◦ 14′ 29′′.06). Figure 2 shows the radio continuum peak and 115 GHz continuum contours plotted over Hα emission from I Zw 18 (Cannon et al. 2002). There is only one other region with |I115GHz| > 3σ115GHz within the primary beam and the star-forming extent of I Zw 18 occupies ≈ 10 % of the primary beam. Therefore, we estimate the chance of a false positive co- incident with the galaxy to be only ∼ 10%. 4. DISCUSSION Here we discuss the implications of our CO upper limit and continuum detection. We adopt the following prop- erties for I Zw 18, all scaled to d = 14 Mpc: MB = −14.7 (Gil de Paz et al. 2003), MHI = 1.4 × 10 (van Zee et al. 1998), Hα luminosity log10 Hα = 39.9 erg s−1 (Cannon et al. 2002; Gil de Paz et al. 2003), 1.4 GHz flux F1.4 = 1.79 mJy (Cannon et al. 2005). 4.1. Point Source Luminosity Our upper limit along each line of sight, LCO < 1 × 105 K km s−1 pc2, matches the luminosity of a fairly massive Galactic giant molecular cloud (Blitz 1993). For a Galactic CO-to-H2 conversion factor, 2 × 1020 cm−2 (K km s−1)−1, the corresponding molecular gas mass is MMol ≈ 4.4× 10 5 M⊙, similar to the mass of the Orion-Monoceros complex (e.g. Wilson et al. 2005). 4.2. Comparison With More Luminous Galaxies In galaxies detected by CO surveys, the CO content per unit B-band luminosity is fairly constant. Figure 3 shows the CO luminosity normalized by B-band lumi- nosity, LCO/LB, as a function of absolute B-band mag- nitude (LB is extinction corrected). LCO/LB is nearly constant over two orders of magnitude in LB, though with substantial scatter (much of it due to the extrapo- lation from a single pointing to LCO). Based on these data and assuming that LCO is not a function of the metallicity of the galaxy, we may ex- trapolate to an expected CO luminosity for I Zw 18. For MB,IZw18 ≈ −14.7 the CO luminosity correspond- ing to the median value of LCO/LB (dashed line) in Figure 3 is LCO,IZw18 ≈ 1.7 × 10 6 K km s−1 pc2. The Hα, 1.4 GHz, and H I luminosities lead to simi- lar predictions. Young et al. (1996) found MH2/LHα ≈ 10L⊙/M⊙ for Sd–Irr galaxies, which implies LCO,IZw18 ∼ 4 × 106 K km s−1 pc2. Murgia et al. (2005) measured FCO/F1.4 ≈ 10 Jy km s −1 (mJy)−1 for spirals, that would imply LCO,IZw18 ∼ 10 7 K km s−1. For Sd/Sm galax- ies, MH2/MHI ≈ 0.2 (Young & Scoville 1991), leading to LCO,IZw18 ∼ 5 × 10 6 K km s−1 pc2. Both MH2/LHα and MH2/MHI tend to be even higher in earlier-type spirals. Therefore, surveys would predict LCO,IZw18 & 2 × 106 K km s−1 pc2, very close to the previously established upper limits of 2− 3 × 106 K km s−1pc2 (Arnault et al. 1988; Gondhalekar et al. 1998). With the present obser- vations, we constrain LCO < 1 × 10 5 K km s−1pc2 and thus clearly rule out LCO ∼ 10 6 K km s−1 pc2. This may be seen in Figure 3; even if I Zw 18 has the highest possible CO content, it will still have a lower LCO/LB than 97% of the survey galaxies. 4.3. Comparison With Nearby Metal-Poor Dwarfs The subset of irregular galaxies detected by CO surveys tend to be CO-rich and actively star-forming, resembling scaled-down versions of spiral galaxies (Young et al. 1995, 1996; Leroy et al. 2005). Such galaxies may not be representative of all dwarfs. Because they are nearby, several of the closest dwarf irregulars have been detected Fig. 2.— V -band and Hα (right, Cannon et al. 2002) images of I Zw 18. Overlays on the left image show the size of the synthesized beam and the locations of the spectra shown in Figure 1. Contours on the right image show continuum emission in increments of 0.5σ significance and the location of the radio continuum peak. The primary beam is larger than the area shown. Both optical maps are on linear stretches. V -band data obtained from the MAST Archive, originally observed for GO program 9400, PI: T. Thuan). despite very small LCO. With their low masses and metallicities, they may represent good points of compar- ison for I Zw 18. Table 1 and Figure 3 show CO lumi- nosities and LCO/LB for four nearby dwarfs: NGC 1569, the Small Magellanic Cloud (SMC), NGC 6822, and IC 10. The SMC, NGC 1569, and NGC 6822 have LCO ∼ 10 5 K km s−1 pc2, close to our upper limit, and occupy a region of LCO/LB-LB parameter space similar to I Zw 18. All four of these galaxies have active star for- mation but very low CO content relative to their other properties. We test whether our observations would have detected CO in NGC 1569, the SMC, and IC 10 at the plausible lower limit of 10 Mpc (from H0 = 72 km s −1) or our adopted distance of 14 Mpc. We convolve the integrated intensity maps to resolutions of 210 and 300 pc and mea- sure the peak integrated intensity. The results appear in columns 4 and 5 of Table 1. The PdBI observations of NGC 1569 resolve out most of the flux, so we also apply this test to a distribution with the size and luminosity derived by Greve et al. (1996) from single dish observa- tions. Our observations would detect an analog to IC 10 but not the SMC, with NGC 1569 an intermediate case. With a factor of ∼ 3 better sensitivity (requiring ∼ 10 times more observing time) we would expect to detect all three nearby galaxies. However, achieving such sen- sitivity with present instrumentation will be quite chal- lenging. ALMA will likely be necessary to place stronger constraints on CO in galaxies like I Zw 18. IC 10 may be the nearest blue compact dwarf (Richer et al. 2001), so it may be telling that we would detect it at the distance of I Zw 18. The blue compact galaxies that have been detected in CO have LCO/LB similar to IC 10 (Gondhalekar et al. 1998, the diamonds in Figure 3). Most searches for CO towards BCDs have yielded nondetections, so those detected may not be rep- resentative, but I Zw 18 is clearly not among the “CO- rich” portion of the BCD population. 4.4. Interpretation of the Continuum We measure continuum intensity of F115GHz = 1.06± 0.35 mJy towards the radio continuum peak. The continuum is detected along only one line of sight, so we refer to it here as a point source and com- pare it to integrated values for I Zw 18. F115GHz is expected to be the product of mainly two types of emission: thermal free-free emission and thermal dust emission. At long wavelengths, the integrated ther- mal free-free emission is F1.4GHz(free− free) ≈ 0.52 – 0.75 mJy (Cannon et al. 2005; Hunt et al. 2005a), imply- ing F115GHz(free− free) = 0.36 – 0.51 mJy at 115 GHz (Fν ∝ ν −0.1). The Hα flux predicts a similar value, F115GHz(free− free) = 0.34 mJy (Cannon et al. 2005, Equation 1). Hunt et al. (2005b) placed an upper limit of Fν(850) < 2.5 mJy on dust continuum emission at 850µm; this is consistent with the ∼ 5 × 103 M⊙ esti- mated by Cannon et al. (2002) given almost any reason- able dust properties. Extrapolating this to 2.6 mm as- suming a pure blackbody spectrum, the shallowest plau- sible SED, constrains thermal emission from dust to be < 0.25 mJy at 115 GHz. Based on these data, we would predict F115GHz . 0.75 mJy. Thus our measured F115GHz is consistent with, but somewhat higher than, the ther- mal free-free plus dust emission expected based on opti- cal, centimeter, and submillimeter data. 4.5. Relation to Star Formation I Zw 18 has a star formation rate ∼ 0.06 – 0.1 M⊙ yr−1, based on Hα and cm radio continuum measure- ments (Cannon et al. 2002; Kennicutt 1998a; Hunt et al. 2005a). Our continuum flux suggests a slightly higher value ≈ 0.15 – 0.2 M⊙ yr −1 (following Hunt et al. 2005a; Condon 1992), with the exact value depending on the contribution from thermal dust emission. For any value in this range, the star formation rate per CO luminosity, SFR/LCO is much higher in I Zw 18 than in spirals. For Fig. 3.— CO luminosity normalized by absolute blue mag- nitude for galaxies with Hubble Type Sb or later (black cir- cles, Young et al. 1995; Elfhag et al. 1996; Böker et al. 2003; Leroy et al. 2005). We also plot nearby dwarfs from Ta- ble 1 (crosses) and blue compact galaxies compiled by Gondhalekar et al. (1998, , diamonds). The shaded regions shows our upper limit for I Zw 18, with the range inMB for distances from 10 to 20 Mpc. The dashed line and light shaded region show the median value and 1σ scatter in LCO/LB for spirals and dwarf star- bursts. Methodology: We extrapolate from ICO in central pointings to LCO assuming the CO to have an exponential profile with scale length 0.1 d25 (Young et al. 1995), including only galaxies where the central pointing measures > 20% of LCO. We adopt B mag- nitudes (corrected for internal and Galactic extinction), distances (Tully-Fisher when available, otherwise Virgocentric-flow corrected Hubble flow), and radii from LEDA (Paturel et al. 2003). comparison, our upper limit and the molecular “Schmidt Law” derived by Murgia et al. (2002) predicts a star for- mation rate . 2 × 10−4 M⊙ yr −1. Fits by Young et al. (1996) and Kennicutt (1998b, applied to just the molec- ular limit) yield similar values. Again, I Zw 18 is similar to the SMC and NGC 6822, which have star formation rates of 0.05 M⊙ yr −1 and 0.04 M⊙ yr −1 (Wilke et al. 2004; Israel 1997b) and LCO ∼ 10 5 K km s−1 pc2. 4.6. Variations in XCO Several calibrations of the CO-to-H2 conversion factor, XCO as a function of metallicity exist in the literature. The topic has been controversial and these calibrations range from little or no dependence (e.g. Walter 2003; Rosolowsky et al. 2003) to very steep dependence (e.g., XCO ∝ Z −2.7 Israel 1997a). Comparing the star for- mation rate to our CO upper limit, we may rule out that I Zw 18 has a Galactic XCO unless molecular gas in I Zw 18 forms stars much more efficiently than in the Galaxy. Either the ratio of CO-to-H2 is low in I Zw 18 or molecular gas in this galaxy forms stars with an effi- ciency two orders of magnitude higher than that in spiral galaxies. 5. CONCLUSIONS We present new, sensitive observations of the metal- poor dwarf galaxy I Zw 18 at 3 mm using the Plateau de Bure Interferometer. These data constrain the integrated CO J = 1 → 0 intensity to be ICO < 1 K km s −1 over our 300 pc (FWHM) beam and the luminosity to be LCO < 1× 105 K km s−1 pc2. I Zw 18 has less CO relative to its B-band luminosity, H Imass, or SFR than spiral galaxies or dwarf starbursts, including more metal-rich blue compact galaxies such as IC 10 (ZIC 10 ∼ Z⊙/4, Lee et al. 2003). Because of its small size and large distance, these are the first observa- tions to impose this constraint. We show that I Zw 18 should be grouped with several local analogs — NGC 1569, the SMC, NGC 6822 — as a galaxy with active star formation but a very low CO content relative to its other properties. In these galax- ies, observations suggest that the environment affects the molecular gas and these data suggest that the same is true in I Zw 18. A simple comparison of star formation rate to CO content shows that this must be true at a basic level: either the ratio of CO to H2 is dramatically low in I Zw 18 or molecular gas in this galaxy forms stars with an efficiency two orders of magnitude higher than that in spiral galaxies. We detect 3mm continuum with F115 GHz = 1.06 ± 0.35 mJy coincident with the radio peak identified by Cannon et al. (2005) and Hunt et al. (2005a). This flux is consistent with but somewhat higher than the thermal free-free plus dust emission one would predict based on centimeter, submillimeter, and optical measurements. Finally, we note that improving on this limit with cur- rent instrumentation will be quite challenging. The order of magnitude increase in sensitivity from ALMA will be needed to place stronger constraints on CO in galaxies like I Zw 18. We thank Roberto Neri for his help reducing the data. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr). 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These data constrain the CO J=1-0 luminosity within our 300 pc (FWHM) beam to be L_CO < 1 \times 10^5 K km s^-1 pc^2 (I_CO < 1 K km s^-1), an order of magnitude lower than previous limits. Although I Zw 18 is starbursting, it has a CO luminosity similar to or less than nearby low-mass irregulars (e.g. NGC 1569, the SMC, and NGC 6822). There is less CO in I Zw 18 relative to its B-band luminosity, HI mass, or star formation rate than in spiral or dwarf starburst galaxies (including the nearby dwarf starburst IC 10). Comparing the star formation rate to our CO upper limit reveals that unless molecular gas forms stars much more efficiently in I Zw 18 than in our own galaxy, it must have a very low CO-to-H_2 ratio, \sim 10^-2 times the Galactic value. We detect 3mm continuum emission, presumably due to thermal dust and free-free emission, towards the radio peak. <|endoftext|><|startoftext|> Introduction The Model The BPS code and the formation of hot subdwarfs Monte-Carlo simulation parameters Spectral library Observables from the model Simulations Standard simulation set Simulation sets with varying model parameters Simulation sets for composite stellar populations Results and Discussion Simple stellar populations The model for composite stellar populations Theory versus observations The UV-upturn and metallicity Comparison with previous models Summary and Conclusion REFERENCES ABSTRACT The discovery of a flux excess in the far-ultraviolet (UV) spectrum of elliptical galaxies was a major surprise in 1969. While it is now clear that this UV excess is caused by an old population of hot helium-burning stars without large hydrogen-rich envelopes, rather than young stars, their origin has remained a mystery. Here we show that these stars most likely lost their envelopes because of binary interactions, similar to the hot subdwarf population in our own Galaxy. We have developed an evolutionary population synthesis model for the far-UV excess of elliptical galaxies based on the binary model developed by Han et al (2002, 2003) for the formation of hot subdwarfs in our Galaxy. Despite its simplicity, it successfully reproduces most of the properties of elliptical galaxies with a UV excess: the range of observed UV excesses, both in $(1550-V)$ and $(2000-V)$, and their evolution with redshift. We also present colour-colour diagrams for use as diagnostic tools in the study of elliptical galaxies. The model has major implications for understanding the evolution of the UV excess and of elliptical galaxies in general. In particular, it implies that the UV excess is not a sign of age, as had been postulated previously, and predicts that it should not be strongly dependent on the metallicity of the population, but exists universally from dwarf ellipticals to giant ellipticals. <|endoftext|><|startoftext|> Baltic Astronomy, vol.12, XXX–XXX, 2003. THE REDSHIFT OF LONG GRBS’ Z. Bagoly1 and I. Csabai2 and A. Mészáros3 and P. Mészáros4 and I. Horváth5 and L. G. Balázs6 and R. Vavrek7 1 Lab. for Information Technology, Eötvös University, H-1117 Budapest, Pázmány P. s. 1./A, Hungary 2 Dept. of Physics for Complex Systems, Eötvös University, H-1117 Bu- dapest, Pázmány P. s. 1./A, Hungary 3 Astronomical Institute of the Charles University, V Holešovičkách 2, CZ-180 00 Prague 8, Czech Republic 4 Dept. of Astronomy & Astrophysics, Pennsylvania State University, 525 Davey Lab., University Park, PA 16802, USA 5 Dept. of Physics, Bolyai Military University, H-1456 Budapest, POB 12, Hungary 6 Konkoly Observatory, H-1505 Budapest, POB 67, Hungary 7 Max-Planck-Institut für Astronomie, D-69117 Heidelberg, 17 Königstuhl, Germany Received October 20, 2003 Abstract. The low energy spectra of some gamma-ray bursts’ show ex- cess components beside the power-law dependence. The consequences of such a feature allows to estimate the gamma photometric redshift of the long gamma-ray bursts in the BATSE Catalog. There is good correla- tion between the measured optical and the estimated gamma photometric redshifts. The estimated redshift values for the long bright gamma-ray bursts are up to z = 4, while for the the faint long bursts - which should be up to z = 20 - the redshifts cannot be determined unambiguously with this method. The redshift distribution of all the gamma-ray bursts with known optical redshift agrees quite well with the BATSE based gamma photometric redshift distribution. Key words: Cosmology - Gamma-ray burst 1. INTRODUCTION In this article we present a new method called gamma photometric redshift (GPZ) estimation of the estimation of the redshifts for the http://arxiv.org/abs/0704.0864v1 2 Z.Bagoly et. al long GRBs. We utilize the fact that broadband fluxes change sys- tematically, as characteristic spectral features redshift into, or out of the observational bands. The situation is in some sense similar to the optical observations of galaxies, where for galaxies and quasars the photometric redshift estimation (Csabai et. al (2000), Budavári et. al (2001)) achieved a great success in estimating redshifts from photometry only. We construct our template spectrum that will be used in the GPZ process in the following manner: let the spectrum be a sum of the Band’s function and of a low energy soft excess power-law function, observed in several cases (Preece et. al (2000)). The low energy cross-over is at Ecr = 90 keV, Eo = 500 keV, and the spectral indices are α = 3.2, β = 0.5 and γ = 3.0. Let us introduce the peak flux ratio (PFR hereafter) in the fol- lowing way: PFR = l34 − l12 l34 + l12 where lij is the BATSE DISCSC flux in energy channel Ei < E < Ej , here E1 = 25 keV, E2 = E3 = 55 keV, E4 = 100 keV. 0 2 4 6 8 10 12 14 α=3.2 β=0.5 Ecr=90 keV Fig. 1. The theoretical PFR curves calculated from the template spec- trum using the average detector re- sponse matrix. The spectra are changing quite rapidly with time; the typ- ical timescale for the time vari- ation is ≃ (0.5 − 2.5) s (Ryde & Svensson (1999, 2000)). There- fore, we will consider the spectra in the 320ms time interval cen- tered around the peak-flux. If we redshift the template spec- trum and use the detector re- sponse matrix of the given burst, we can get for any redshift the observed flux and the PFR value. On Fig. 1. we plot the the- oretical PFR curves calculated from the above defined template spectrum using the average detector response matrices for the 8 bursts that have both BATSE data and measured redshifts (Klose (2000)) In the used range of z (i.e. for z< 4) the relation between z and PFR is invertible, hence we can use it to estimate the gamma photometric The Redshift of Long GRBs’ 3 redshift (GPZ) from a measured PFR. For the 7 considered GRBs (leaving out GRB associated with the supernova and GRB having upper redshift limit only) the estimation error between the real z and the GPZ is ∆z =≈ 0.33. 2. ESTIMATION OF THE REDSHIFTS Here restrict ourselves to long and not very faint GRBs with T90 > 10 s and F256 < 0.65 photon/(cm 2s) to avoid the problems with the instrumental threshold (Pendleton et. al (1997), Hakkila et. al, (2000)). Introducing an another cut at F256 > 2.00 photon/(cm 2s) we can investigate roughly the brighter half of this sample. As the soft-excess range redshifts out from the BATSE DISCSC energy channels around z ≈ 4, the theoretical curves converge to a constant value. For higher z it starts to decrease. This means that the method is ambiguous: for the given value of PFR one may have two redshifts - below and above z ≈ 4. Because for the bright GRBs the values above z ≈ 4 are practically excluded, for them the method is usable. Using only the 25 − 55 keV and 55 − 100 keV BATSE energy channels, this method can be used to estimate GPZ only in the redshift range z < 0 1 2 3 4 5 Gamma Photometric Redshift F256>0.65 ph/cm F256>2.0 ph/cm Fig. 1. The distribution of the GPR estimators of the long GRBs having DISCSC data. Let us assume for a moment that all observed long bursts, we have selected above, have z < 4. Then we can simply calculate the zGPZ redshift for any GRB, which has PFR from the DISCSC data. Fig. 2. shows the distribu- tion of the estimated derived red- shifts under the assumption that all GRBs are below z ≈ 4. The dis- tribution has a clear peak value around PFR ≈ 0.2, which corre- sponds to z ≈ (1.5− 2.0). Although there is a problem with the degeneracy (e.g. two possible redshift values) we think that the great majority of values of z obtained for the bright half are correct. This opinion may be supported by the following arguments: the obtained distribution of GRBs in z for the bright half is very similar to the obtained distribution of Schmidt (2001) and Schaefer 4 Z.Bagoly et. al et. al (2001). An another problem for z as it moves into z> 4 regime for the bright GRB is the extremely high GRB luminosities, ≃ 1053ergs/s (Mészáros & Mészáros, 1996). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 17 bursts with known redshift GPZ{ F256>0.65 ph/cm2/s Fig. 3. The redshift distribution of the 17 GRBs’ with known z and the distributions from the GPZ estima- tors. As an additional statistical test we compared the redshift distribution of the 17 GRB with observed redshift with our re- constructed GRB z distributions (limited to the z < 4 range). For the F256 > 0.65 photon/(cm group the KS test suggests a 38% probability, i.e. the observed N(< z) probability distribution agrees quite well with the GPZ reconstructed function. ACKNOWLEDGMENTS The useful remarks with Drs. T. Budavári, S. Klose, D. Reichart, A.S. Szalay are kindly acknowl- edged. This research was supported in part through OTKA grants T024027 (L.G.B.), F029461 (I.H.) and T034549, Czech Research Grant J13/98: 113200004 (A.M.), NASA grant NAG5-9192 (P.M.). REFERENCES ??udavári, T., Csabai, I., Szalay, A.S. et. al, 2001, AJ, 122, 1163 ??sabai, I., Connolly, A.J., Szalay, A.S. et. al, 2000, AJ, 119, 69 ??akkila, J., Haglin, D. J., Pendleton, G. N. et. al, 2000, ApJ, 538, ??lose, S. 2000, Reviews in Modern Astronomy 13, Astronomische Gesellschaft, Hamburg, p.129 ??észáros, A., & Mészáros, P. 1996, ApJ, 466, 29 ??reece, R.D., Briggs, M.S., Pendleton, G.N., et. al 1996, ApJ, 473, ??reece, R.D., Briggs, M.S., Mallozzi, et. al, 2000, ApJS, 126, 19 ??yde, F., & Svensson, R. 1999, ApJ, 512, 693 ??yde, F., & Svensson, R. 2000, ApJ, 529, L13 ??chaefer, B. E., Deng, M. & Band, D. L., 2001, ApJ, 563, L123 ??chmidt, M. 2001, ApJ, 552, 36 ABSTRACT The low energy spectra of some gamma-ray bursts' show excess components beside the power-law dependence. The consequences of such a feature allows to estimate the gamma photometric redshift of the long gamma-ray bursts in the BATSE Catalog. There is good correlation between the measured optical and the estimated gamma photometric redshifts. The estimated redshift values for the long bright gamma-ray bursts are up to z=4, while for the the faint long bursts - which should be up to z=20 - the redshifts cannot be determined unambiguously with this method. The redshift distribution of all the gamma-ray bursts with known optical redshift agrees quite well with the BATSE based gamma photometric redshift distribution. <|endoftext|><|startoftext|> Introduction The increasing complexity of software systems raises major concerns in various critical application domains, in particular with respect to the validation and analysis of performance, timing and dependability requirements. Model-driven engineering approaches based on architecture description languages (ADLs) aim at mastering this complexity at the design level. Over the last decade, considerable research has been devoted to ADLs leading to a large number of proposals [1]. In particular, AADL (Architecture Analysis and Design Language) [2] has received an increasing interest from the safety-critical industry (i.e., Honeywell, Rockwell Collins, Lockheed Martin, the European Space Agency, Airbus) during the last years. It has been standardized under the auspices of the International Society of Automotive Engineers (SAE), to support the design and analysis of complex real-time safety-critical applications. AADL provides a standardized textual and graphical notation, for describing architectures with functional interfaces, and for performing various analyses to determine the behavior and performance of the system being modeled. AADL has been designed to be extensible to accommodate analyses that the core language does not support, such as dependability and performance. In critical application domains, one of the challenges faced by the software engineers concerns: 1) the description of the software architecture and its dynamic behavior taking into account the impact of errors and failures, and 2) the evaluation of quantitative measures of relevant dependability properties such as reliability, availability and safety, allowing them to assess the impact of errors and failures on the service. For pragmatic reasons, the designers using an AADL-based engineering approach are interested in using an integrated set of methods and tools to describe specifications and designs, and to perform dependability evaluations. The AADL Error Model Annex [3] has been defined to complement the description capabilities of the AADL core language standard by providing features with precise semantics to be used for describing dependability-related characteristics in AADL models (faults, failure modes and repair assumptions, error propagations, etc.). AADL and the AADL Error Model Annex are supported by the Open Source AADL Tool Environment (OSATE)1. At the current stage, there is a lack of methodologies and guidelines to help the developers, using an AADL based engineering approach, to use the notations defined in the standard for describing complex dependability models reflecting real-life systems with multiple dependencies between components. The objective of this paper is to propose a structured method for AADL dependability model construction. The AADL model is built and validated iteratively, taking into account progressively the dependencies between the components. The approach proposed in this paper is complementary to other research studies focused on the extension of the AADL language capabilities to support formal verifications and analyses (see e.g. [4]). Also, it is intended to be complementary to other studies focused on the integration of formal verification, dependability and performance related activities in the general context of 1 http://lwww.aadl.info/OpenSourceAADLToolEnvironment.html model driven engineering approaches based on ADLs and on UML (see e.g., [5-9]). The remainder of the paper is organized as follows. Section 2 presents the AADL concepts that are necessary for understanding our modeling approach. Section 3 gives an overview of our framework for system dependability modeling and evaluation using AADL. Section 4 presents the iterative approach for building the AADL dependability model. Section 5 illustrates some of the concepts of our approach on a small example and section 6 concludes the paper. 2. AADL concepts The AADL core language allows analyzing the impact of different architecture choices (such as scheduling policy or redundancy scheme) on a system’s properties [10]. An architecture specification in AADL is an hierarchical collection of interacting components (software and compute platform) combined in subsystems. Each AADL component is modeled at two levels: in the component type and in one or more component implementations corresponding to different implementation structures of the component in terms of subcomponents and connections. The AADL core language is designed to describe static architectures with operational modes for their components. However, it can be extended to associate additional information to the architecture. AADL error models are an extension intended to support (qualitative and quantitative) analyses of dependability attributes. The AADL Error Model Annex defines a sub- language to declare reusable error models within an error model annex library. The AADL architecture model serves as a skeleton for error model instances. Error model instances can be associated with components of the system and with the system itself. The component error models describe the behavior of the components with which they are associated, in the presence of internal faults and recovery events, as well as in the presence of external propagations from the component’s environment. Error models have two levels of description: the error model type and the error model implementation. The error model type declares a set of error states, error events (internal to the component) and error propagations2 (events that propagate, from one component to other components, through the connections and bindings between components of the architecture model). Propagations have associated directions (in or out or in out). Error model implementations declare transitions between states, triggered by events and propagations declared in the error model type. Both the type and the implementation can declare Occurrence properties that 2 Error states can also model error free states, error events can also model repair events and error propagations can model all kinds of notifications. specify the arrival rate or the occurrence probability of events and propagations. An out propagation occurs according to a specified Occurrence property when it is named in a transition and the current state is the origin of the transition. If the source state and the destination state of a transition triggered by an out propagation are the same, the propagation is sent out of the component but does not influence the state of the sender component. An in propagation occurs as a consequence of an out propagation from another component. Figure 1 shows an error model example. Error Model Type [simple] error model simple features Error_Free: initial error state; Failed: error state; Fail: error event {Occurrence => Poisson λ}; Recover: error event {Occurrence => Poisson µ}; KO: in out error propagation {Occurrence => fixed p}; end simple; Error Model Implementation [simple.general] error model implementation simple.general transitions Error_Free-[Fail] -> Failed; Error_Free-[in KO] -> Failed; Failed-[Recover] -> Error_Free; Failed-[out KO] -> Failed; end simple.general; Figure 1. Simple error model Error model instances can be customized to fit a particular component through the definition of Guard properties that control and filter propagations by means of Boolean expressions. The system error model is defined as a composition of a set of concurrent finite stochastic automata corresponding to components. In the same way as the entire architecture, the system error model is described hierarchically. The state of a system that contains subcomponents can be specified as a function of its subcomponents’ states (i.e., the system has a derived error model). 3. Overview of the modeling framework For complex systems, the main difficulty for building a dependability model arises from dependencies between the system components. Dependencies can be of several types, identified in [11]: functional, structural or related to the recovery and maintenance strategies. Exchange of data or transfer of intermediate results from one component to another is an example of functional dependency. The fact that a thread runs on a processor induces a structural dependency between the thread and the processor. Sharing a recovery or maintenance facility between several components leads to a recovery or maintenance dependency. Functional and structural dependencies can be grouped into an architecture-based dependency class, as they are triggered by physical or logical connections between the dependent components at architectural level. Instead, recovery and maintenance dependencies are not always visible at architectural level. A structured approach is necessary to model dependencies in a systematic way, to promote model reusability, to avoid errors in the resulting model of the system and to facilitate its validation. In our approach, the AADL dependability-oriented model is built in a progressive and iterative way. More concretely, in a first iteration, we propose to build the model of the system’s components, representing their behavior in the presence of their own faults and recovery events only. The components are thus modeled as if they were isolated from their environment. In the following iterations, dependencies between basic error models are introduced progressively. This approach is part of a complete framework that allows the generation of dependability analysis and evaluation models from AADL models. An overview of this framework is presented in Figure 2. Figure 2. Modeling framework The first step is devoted to the modeling of the application architecture in AADL (in terms of components and operational modes of these components). The AADL architecture model may be available if it has been already built for other purposes. The second step concerns the specification of the application behavior in the presence of faults through AADL error models associated with components of the architecture model. The error model of the application is a composition of the set of component error models. The architecture model and the error model of the application form the dependability-oriented AADL model, referred to as the AADL dependability model. The third step aims at building an analytical dependability evaluation model, from the AADL dependability model, based on model transformation rules. The fourth step is devoted to the dependability evaluation model processing that aims at evaluating quantitative measures characterizing dependability attributes. This step is entirely based on existing processing tools. The iterative approach can be applied to the second step of the modeling framework only or to the second and third steps together. In the latter case, semantic validation based on the analytical model, after each iteration, is helpful to identify specification errors in the AADL dependability model. Due to space limitations, we focus only on the first and second steps in this paper. A transformation from AADL to generalized stochastic Petri nets (GSPN) for dependability evaluation purposes is presented in [12]. 4. AADL dependability model construction To illustrate the proposed approach, the rest of this section presents successively guidelines for modeling an architecture-based dependency (structural or functional) and a recovery and maintenance dependency. More general practical aspects for building the AADL dependability model are given at the end of this section. Note that we illustrate the principles using the graphical notation for AADL composite components (system components). However, they apply to all types of components and connections. 4.1. Architecture-based dependency The dependency is modeled in the error models associated with the dependent components, by specifying respectively outgoing and incoming propagations and their impact on the corresponding error model. An example is shown in Figure 3: Component 1 sends data to Component 2, thus we assume that, at the error model level, the behavior of Component 2 depends on that of Component 1. Figure 3. Architecture-based dependency Instances of the same error model, shown in Figure 1, are associated both with Component 1 and with Component 2. However, the AADL dependability model is asymmetric because of the unidirectional connection between Component 1 and Component 2. Thus, the out propagation KO declared in the error model instance associated with Component 2 is inactive (i.e., even if it occurs, it cannot propagate to Component 1). The out propagation KO from the error model instance of Component 1, together with its Occurrence property and the AADL transition triggered by it form the “sender” part of the dependency. It means that when Component 1 fails, it sends a propagation through the unidirectional connection. The in propagation KO from the error model instance of Component2 together with the AADL transition triggered by it form the “receiver” part of the dependency. Thus, an incoming propagation KO causes the failure of the receiving component. In real applications, architecture-based dependencies usually require using more advanced propagation controlling and filtering through Guard properties. In particular, Boolean expressions can be defined to specify the consequences of a set of propagations occurring in a set of sender components on a receiver component. 4.2. Recovery and maintenance dependency Recovery and maintenance dependencies need to be described when recovery and maintenance facilities are shared between components or when the maintenance activity of some components has to be carried out according to a given order or a specified strategy (i.e., a thread can be restarted only if another thread is running). Components that are not dependent at architectural level may become dependent due to the recovery and maintenance strategy. Thus, the AADL dependability model might need some adjustments to support the description of dependencies related to the maintenance strategy. As error models interact only via propagations through architectural features (i.e., connections, bindings), the recovery and maintenance dependency between components’ error models must be supported by the architecture model. Thus, besides the architecture components, we may need to model (at architectural level) a component allowing to describe the recovery and maintenance strategy. Figure 4-a shows an example of AADL dependability model. In this architecture, Component 3 and Component 4 do not interact at the architecture level. However, if we assume that they share a recovery and maintenance facility, the recovery and maintenance strategy has to be taken into account in the error model of the application. Thus, it is necessary to represent the recovery and maintenance facility at the architectural level, as shown in Figure 4-b in order to model explicitly the dependency between Components 3 and Component 4. Also, the error models of dependent components with regards to the recovery and maintenance strategy might need some adjustments. For example, to represent the fact that Component 3 can only restart if Component 4 is running, one needs to distinguish between a failed state of Component 3 and a failed state where Component 3 is allowed to restart. - a - - b - Figure 4. Maintenance dependency 4.3. Practical aspects The order for modeling dependencies does not impact the final AADL dependability model. However, it may impact the reusability of parts of the model. Thus, the order may be chosen according to the context of the targeted analysis. For example, if the analysis is meant to help the user to choose the best-adapted structure for a system whose functions are completely defined, it may be convenient to introduce first functional dependencies between components and then structural dependencies, as the model corresponding to functional dependencies is to be reused. Generally, recovery and maintenance dependencies are modeled at the end, as one important aim of the dependability evaluation is to find the best- suited recovery and maintenance strategies for an application. Recovery and maintenance dependencies may have an impact on the system’s structure. Not all the details of the architecture model are necessary for the AADL dependability model. Only components that have associated error models and all connections and bindings between them are necessary. This allows a designer to evaluate dependability measures at different stages in the development cycle by moving from a lower fidelity AADL dependability model to a detailed one. In some cases, not all components having associated error models are part of the AADL dependability model. The AADL Error Model Annex offers two useful abstraction options for error models of components composed of subcomponents: − The first option is to declare an abstract error model for a system component. In this case, the corresponding component is seen as a black box (i.e., the detailed subcomponents’ error models are not part of the AADL dependability model). This option is useful to abstract away modeling details in case an architecture model with too detailed error models associated with components does exist for other purposes. Issues linked to the relationship between abstract and concrete error models have been mentioned in [13]. − The second option is to define the state of a system component as a function of its subcomponents’ states. This option can be used to specify state classes for the overall application. These classes are useful in the evaluation of measures. If the user wishes to evaluate reliability or availability, it is necessary to specify the system states that are to be considered as failed states. If in addition, the user wishes to evaluate safety, it is necessary to specify the system states that are considered as catastrophic. 5. Example In this section we illustrate our modeling approach on a small software architecture representing a process whose functional role is to compute a result. The computation is divided in three sub computations, each of them being performed by a thread. The thread Compute2 uses the result obtained by the thread Compute1 and the thread Compute3 uses the result obtained by the thread Compute2 to compute the result expected from the process. The three threads are connected through data connections according to the pipe and filter architectural style [14]. Due to space limitations, we only take into account two dependencies: − An architecture-based dependency between the computing threads: a failure in one of the computing threads may cause the failure of the following thread (with a probability p). In some cases, cascading failures can occur. − A recovery dependency: Compute3 can only recover if Compute1 and Compute2 are error free. We assume that Compute2 can recover if Compute1 is not error free. The AADL dependability model of this application is shown in Figure 5 using the AADL graphical notation. Figure 5. AADL dependability model The AADL dependability model of this application is built in three iterations. The computing threads’ behavior in the presence of their own fault and recovery events is represented in the first iteration. The propagation KO together with corresponding transitions are added in a second iteration to represent the architecture-based dependency. The thread Compute1 can have an impact on Compute2 and Compute2 can have an impact on Compute3. We remind that the opposite is not possible, as the connections between threads are unidirectional. The recovery dependency is modeled in the third iteration. It requires the existence of a Recovery thread in the architecture model (see light grey part of Figure 5). Its role is to send (through the out port to3) a RecoverAuthorize propagation to Compute3 if Compute1 and Compute2 are error free. Figure 6-a shows the error model Comp.general associated with threads Compute1 and Compute2. Figure 6-b shows the error model Comp3.general associated with the threads Compute3. The three iterations are highlighted. Each line tagged with a (+) sign is added to the error model corresponding to the previous iteration while each line tagged with a (-) sign is removed from it during the current iteration. The first and second iterations are the same for all three computing threads. In the third iteration, it is necessary to distinguish between a failed state and a failed state from which Compute3 is authorized to restart. This leads to removing a transition declared in the first iteration, and adding a state (CanRecover) and two transitions linking it to the state machine. Figure 7 shows the Guard_Out property applied to port to3 of the Recovery thread in the third iteration. This property specifies that a RecoverAuthorize propagation is sent to Compute3 through port to3 when OK propagations are received through ports in1 and in2 (meaning that Compute1 and Compute2 are error free). The Recovery thread has an associated error model that is not shown here. It declares in and out propagations used in the Guard_Out property. The main idea of this method is to verify and validate the model at each iteration. If a problem arises during iteration i, only the part of the current AADL dependability model corresponding to iteration i is questioned. Thus, the validation process is facilitated especially in the context of complex systems. 6. Conclusion This paper presented an iterative approach for system dependability modeling using AADL. This approach is meant to ease the task of analyzing dependability characteristics and evaluating dependability measures for the AADL users community. Our approach assists the user in the structured construction of the AADL dependability model (i.e., architecture model and dependability-related information). To support and trace model evolution, this approach proposes that the user builds the model iteratively. Components’ behaviors in the presence of faults are modeled in the first iteration as if they were isolated. Then, each iteration introduces a new dependency between system components. Error models representing the behavior of several types of system components and several types of dependencies may be placed in a library and then instantiated to minimize the modeling effort and maximize the reusability of models. The OSATE toolset is able to support our modeling approach. It also allows choosing component models and error models from libraries. For the sake of illustration, we used simple examples in this paper. We have already applied the iterative modeling approach to a system with multiple dependencies in [12] and we plan to validate it against other complex case studies. Error Model Type [Comp] error model Comp features -- iteration 1 (+) Error_Free: initial error state; (+) Failed: error state; (+) Fail: error event (+) {Occurrence => Poisson λ}; (+) Recover: error event (+) {Occurrence => Poisson µ}; -- iteration 2 (+) KO: in out error propagation (+) {Occurrence => fixed p}; -- iteration 3 (+) OK: out error propagation (+) {Occurrence => fixed 1}; end Comp; Error Model Type [Comp3] error model Comp3 features -- iteration 1 (+) Error_Free: initial error state; (+) Failed: error state; (+) Fail: error event (+) {Occurrence => Poisson λ}; (+) Recover: error event (+) {Occurrence => Poisson µ}; -- iteration 2 (+) KO: in out error propagation (+) {Occurrence => fixed p}; -- iteration 3 (+) CanRecover: error state; (+) OK: in error propagation; end Comp3; Error Model Implementation [Comp.general] error model implementation Comp.general transitions -- iteration 1 (+) Error_Free-[Fail]->Failed; (+) Failed-[Recover]->Error_Free; -- iteration 2 (+) Error_Free-[in KO]->Failed; (+) Failed-[out KO]->Failed; -- iteration 3 (+) Error_Free-[out OK]->Error_Free; end Comp.general; Error Model Implementation [Comp3.general] error model implementation Comp3.general transitions -- iteration 1 (+) Error_Free-[Fail]->Failed; (+) Failed-[Recover]->Error_Free; -- iteration 2 (+) Error_Free-[in KO]->Failed; (+) Failed-[out KO]->Failed; -- iteration 3 (-) Failed-[Recover]->Error_Free; (+) Failed-[RecoverAuthorize]->CanRecover; (+) CanRecover-[Recover]->Error_Free; end Comp3.general; a: Error Model for Compute1 and Compute2 b: Error Model for Compute3 Figure 6. Error model for Compute1 / Compute2 Guard_Out [port Recovery.to3] -- iteration 3 (+) Guard_Out => (+) RecoverAuthorize when (+) (from1[OK]and from2[OK]) (+) mask when others (+) applies to to3; Figure 7. Guard_Out property (port Recovery.to3) Acknowledgements This work is partially supported by 1) the European Commission (European integrated project ASSERT No. IST 004033 and network of excellence ReSIST No. IST 026764). and 2) the European Social Fund. References [1] N. Medvidovic and R. N. Taylor, A classification and comparison framework for Software Architecture Description Languages, IEEE Transactions on Software Engineering, 26, 2000, 70-93. [2] SAE-AS5506, Architecture Analysis and Design Language, Society of Automotive Engineers, 2004. [3] SAE-AS5506/1, Architecture Analysis and Design Language (AADL) Annex Volume 1, Annex E: Error Model Annex, Society of Automotive Engineers, 2006. [4] J.-M. 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ABSTRACT For efficiency reasons, the software system designers' will is to use an integrated set of methods and tools to describe specifications and designs, and also to perform analyses such as dependability, schedulability and performance. AADL (Architecture Analysis and Design Language) has proved to be efficient for software architecture modeling. In addition, AADL was designed to accommodate several types of analyses. This paper presents an iterative dependency-driven approach for dependability modeling using AADL. It is illustrated on a small example. This approach is part of a complete framework that allows the generation of dependability analysis and evaluation models from AADL models to support the analysis of software and system architectures, in critical application domains. <|endoftext|><|startoftext|> Introduction Let X be a compact connected Kähler manifold of dimension n ∈ N∗. Throughout the article ω denotes a smooth closed form of bidegree (1, 1) which is nonnegative and big, i.e. such that ωn > 0. We continue the study started in [GZ 2], [EGZ] of the complex Monge-Ampère equation (MA)µ (ω + dd cϕ)n = µ, where ϕ, the unknown function, is ω-plurisubharmonic: this means that ϕ ∈ L1(X) is upper semi-continuous and ω+ ddcϕ ≥ 0 is a positive current. We let PSH(X,ω) denote the set of all such functions (see [GZ 1] for their basic properties). Here µ is a fixed positive Radon measure of total mass µ(X) = ωn, and d = ∂ + ∂, dc = 1 (∂ − ∂). Following [GZ 2] we say that a ω-plurisubharmonic function ϕ has fi- nite weighted Monge-Ampère energy, ϕ ∈ E(X,ω), when its Monge-Ampère measure (ω+ ddcϕ)n is well defined, and there exists an increasing function χ : R− → R− such that χ(−∞) = −∞ and χ ◦ ϕ ∈ L1((ω + ddcϕ)n). In general χ has very slow growth at infinity, so that ϕ is far from being bounded. The purpose of this article is twofold. First we extend one of the main results of [GZ 2] by showing THEOREM A. There exists ϕ ∈ E(X,ω) such that µ = (ω+ddcϕ)n if and only if µ does not charge pluripolar sets. This results has been established in [GZ 2] when ω is a Kähler form. It is important for applications to complex dynamics and Kähler geometry to consider as well forms ω that are less positive (see [EGZ]). http://arxiv.org/abs/0704.0866v2 2 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI We then look for conditions on the measure µ which insure that the solution ϕ is almost bounded. Following the seminal work of S. Kolodziej [K 2,3], we say that µ is dominated by the Monge-Ampère Capacity Capω if there exists a function F : R+ → R+ such that limt→0+ F (t) = 0 and (†) µ(K) ≤ F (Capω(K)), for all Borel subsets K ⊂ X. Here Capω denotes the global version of the Monge-Ampère capacity intro- duced by E.Bedford and A.Taylor [BT] (see section 2). Observe that µ does not charge pluripolar sets since F (0) = 0. When F (x) . xα vanishes at order α > 1 and ω is Kähler, S. Kolodziej has proved [K 2] that the solution ϕ ∈ PSH(X,ω) of (MA)µ is continuous. The boundedness part of this result was extended in [EGZ] to the case when ω is merely big and nonnegative. If F (x) . xα with 0 < α < 1, two of us have proved in [GZ 2] that the solution ϕ has finite χ−energy, where χ(t) = −(−t)p, p = p(α) > 0. This result was first established by U. Cegrell in a local context [Ce]. Another objective of this article is to fill in the gap inbetween Cegrell’s and Kolodziej’s results, by considering all intermediate dominating functions F. Write Fε(x) = x[ε(− ln(x)/n)] n where ε : R → [0,∞[ is nonincreasing. Our second main result is: THEOREM B. If µ(K) ≤ Fε(Capω(K)) for all Borel subsets K ⊂ X, then µ = (ω + ddcϕ)n where ϕ ∈ PSH(X,ω) satisfies supX ϕ = 0 and Capω(ϕ < −s) ≤ exp(−nH −1(s)). Here H−1 is the reciprocal function of H(x) = e ε(t)dt + s0, where s0 = s0(ε, ω) ≥ 0 only depends on ε and ω. This general statement has several useful consequences: ε(t)dt < +∞, thenH−1(s) = +∞ for s ≥ s∞ := e ε(t)dt+ s0, hence Capω(ϕ < −s) = 0. This means that ϕ is bounded from below by −s∞. This result is due to S. Kolodziej [K 2,3] when ω is Kähler, and [EGZ] when ω ≥ 0 is merely big; • the condition (†) is easy to check for measures with density in Lp, p > 1. Our result thus gives a simple proof (Corollary 3.2), following the seminal approach of S. Kolodziej ([K2]), of the C0-a priori estimate of S.T. Yau [Y], which is crucial for proving the Calabi conjecture (see [T] for an overview); • when ε(t)dt = +∞, the solution ϕ is generally unbounded. The faster ε(t) decreases towards zero, the faster the growth of H−1 at infinity, hence the closer is ϕ from being bounded; • the special case ε ≡ 1 is of particular interest. Here µ(·) ≤ Capω(·), and our result shows that Capω(ϕ < −s) decreases exponentially fast, hence ϕ has “ loglog-singularities”. These are the type of sin- gularities of the metrics used in Arakelov geometry in relation with measures µ = fdV whose density has Poincaré-type singularities (see [Ku], [BKK]). We prove Theorem B in section 3, after establishing Theorem A in section 2.1 and recalling some useful facts from [GZ 2], [EGZ] in section 2.2. We A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 3 then test the sharpness of our estimates in section 4, where we give examples of measures fulfilling our assumptions: these are absolutely continuous with respect to ωn, and their density do not belong to Lp, for any p > 1. 2. Weakly singular quasiplurisubharmonic functions The class E(X,ω) of ω-psh functions with finite weighted Monge-Ampère energy has been introduced and studied in [GZ 2]. It is the largest subclass of PSH(X,ω) on which the complex Monge-Ampère operator (ω+ddc·)n is well-defined and the comparison principle is valid. Recall that ϕ ∈ E(X,ω) if and only if (ω + ddcϕj) n(ϕ ≤ −j) → 0, where ϕj := max(ϕ,−j). 2.1. The range of the Monge-Ampère operator. The range of the operator (ω + ddc·)n acting on E(X,ω) has been characterized in [GZ 2] when ω is a Kähler form. We extend here this result to the case when ω is merely nonnegative and big. Theorem 2.1. Assume ω is a smooth closed nonnegative (1,1) form on X, and µ is a positive Radon measure such that µ(X) = ωn > 0. Then there exists ϕ ∈ E(X,ω) such that µ = (ω + ddcϕ)n if and only if µ does not charge pluripolar sets. Proof. We can assume without loss of generality that µ and ω are normalized so that µ(X) = ωn = 1. Consider, for A > 0, CA(ω) := {ν probability measure / ν(K) ≤ A · Capω(K), for all K ⊂ X}, where Capω denotes the Monge-Ampère capacity introduced by E.Bedford and A.Taylor in [BT] (see [GZ 1] for this compact setting). Recall that Capω(K) := sup (ω + ddcu)n / u ∈ PSH(X,ω), 0 ≤ u ≤ 1 We first show that a measure ν ∈ CA(ω) is the Monge-Ampère of a func- tion ψ ∈ Ep(X,ω), for any 0 < p < 1, where Ep(X,ω) := {ψ ∈ E(X,ω) / ψ ∈ Lp (ω + ddcψ)n Indeed, fix ν ∈ CA(ω), 0 < p < 1, and ωj := ω + εjΩ, where Ω is a kähler form on X, and εj > 0 decreases towards zero. Observe that PSH(X,ω) ⊂ PSH(X,ωj), hence Capω(.) ≤ Capωj(.), so that ν ∈ CA(ωj). It follows from Proposition 3.6 and 2.7 in [GZ 1] that there exists C0 > 0 such that for any v ∈ PSH(X,ωj) normalized by supX v = −1, we have Capωj(v < −t) ≤ , for all t ≥ 1. This yields Ep(X,ωj) ⊂ L p(ν): if v ∈ Ep(X,ωj) with supX v = −1, then (−v)pdν = p · tp−1ν(v < −t)dt ≤ pA · tp−1Capω(v < −t)dt+ Cp + Cp < +∞. 4 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI It follows therefore from Theorem 4.2 in [GZ 2] that there exists ϕj ∈ Ep(X,ωj) with supX ϕj = −1 and (ωj+dd n = cj ·ν, where cj = ωnj ≥ 1 decreases towards 1 as εj decreases towards zero. We can assume without loss of generality that 1 ≤ cj ≤ 2. Observe that the ϕj ’s have uniformly bounded energies, namely (−ϕj) p(ωj + dd n ≤ 2 (−ϕj) pdν ≤ 2 Since supX ϕj = −1, we can assume (after extracting a convergent subse- quence) that ϕj → ϕ in L 1(X), where ϕ ∈ PSH(X,ω), supX ϕ = −1. Set φj := (supl≥j ϕl) ∗. Thus φj ∈ PSH(X,ωj), and φj decreases towards ϕ. Since φj ≥ ϕj , it follows from the “fundamental inequality” (Lemma 2.3 in [GZ 2]) that (−φj) p(ωj + dd n ≤ 2n (−ϕj) p(ωj + dd n ≤ C ′ < +∞. Hence it follows from stability properties of the class Ep(X,ω) that ϕ ∈ Ep(X,ω) (see Proposition 5.6 in [GZ 2]). Moreover (ωj + dd n ≥ inf (ωl + dd n ≥ ν, hence (ω + ddcϕ)n = lim(ωj + dd n ≥ ν. Since ωn = ν(X) = 1, this yields ν = (ω + ddcϕ)n as claimed above. We can now prove the statement of the theorem. One implication is obvious: if µ = (ω+ddcϕ)n, ϕ ∈ E(X,ω), then µ does not charge pluripolar sets, as follows from Theorem 1.3 in [GZ 2]. So we assume now µ that does not charge pluripolar sets. Since C1(ω) is a compact convex set of probability measures which contains all measures (ω + ddcu)n, u ∈ PSH(X,ω), 0 ≤ u ≤ 1, we can project µ onto C1(ω) and get, by a generalization of Radon-Nikodym theorem (see [R], [Ce]), µ = f · ν, ν ∈ C1(ω), 0 ≤ f ∈ L 1(ν). Now ν = (ω + ddcψ)n for some ψ ∈ E1/2(X,ω), ψ ≤ 0, as follows from the discussion above. Replacing ψ by eψ shows that we can actually assume ψ to be bounded (see Lemma 4.5 in [GZ 2]). We can now apply line by line the same proof as that of Theorem 4.6 in [GZ 2] to conclude that µ = (ω+ddcϕ)n for some ϕ ∈ E(X,ω). � 2.2. High energy and capacity estimates. Given χ : R− → R− an increasing function, we consider, following [GZ 2], Eχ(X,ω) := ϕ ∈ E(X,ω) / (−χ)(−|ϕ|) (ω + ddcϕ)n < +∞ Alternatively a function ϕ ≤ 0 belongs to Eχ(X,ω) if and only if (−χ) ◦ ϕj (ω + dd n < +∞, where ϕj := max(ϕ,−j) is the canonical approximation of ϕ by bounded ω-psh functions. When χ(t) = −(−t)p, Eχ(X,ω) is the class E p(X,ω) used in previous section. The properties of classes Eχ(X,ω) are quite different whether the weight χ is convex (slow growth at infinity) or concave. In previous works [GZ 2], A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 5 two of us were mainly interested in weights χ of moderate growth at infinity (at most polynomial). Our main objective in the sequel is to construct solutions ϕ of (MA)µ which are “almost bounded”, i.e. in classes Eχ(X,ω) for concave weights χ of arbitrarily high growth. For this purpose it is useful to relate the property ϕ ∈ Eχ(X,ω) to the speed of decreasing of Capω(ϕ < −t), as t → +∞. We set Êχ(X,ω) := ϕ ∈ PSH(X,ω) / tnχ′(−t)Capω(ϕ < −t)dt < +∞ An important tool in the study of classes Eχ(X,ω) are the “fundamental inequalities” (Lemmas 2.3 and 3.5 in [GZ 2]), which allow to compare the weighted energy of two ω-psh functions ϕ ≤ ψ. These inequalities are only valid for weights of slow growth (at most polynomial), while they become immediate for classes Êχ(X,ω). So are the convexity properties of Êχ(X,ω). We summarize this and compare these classes in the following: Proposition 2.2. The classes Êχ(X,ω) are convex and stable under maxi- mum: if Êχ(X,ω) ∋ ϕ ≤ ψ ∈ PSH(X,ω), then ψ ∈ Êχ(X,ω). One always has Êχ(X,ω) ⊂ Eχ(X,ω), while Eχ̂(X,ω) ⊂ Êχ(X,ω), where χ ′(t− 1) = tnχ̂′(t). Since we are mainly interested in the sequel in weights with (super) fast growth at infinity, the previous proposition shows that Êχ(X,ω) and Eχ(X,ω) are roughly the same: a function ϕ ∈ PSH(X,ω) belongs to one of these classes if and only if Capω(ϕ < −t) decreases fast enough, as t→ +∞. Proof. The convexity of Êχ(X,ω) follows from the following simple observa- tion: if ϕ,ψ ∈ Êχ(X,ω) and 0 ≤ a ≤ 1, then {aϕ+ (1− a)ψ < −t} ⊂ {ϕ < −t} ∪ {ψ < −t} . The stability under maximum is obvious. Assume ϕ ∈ Êχ(X,ω). We can assume without loss of generality ϕ ≤ 0 and χ(0) = 0. Set ϕj := max(ϕ,−j). It follows from Lemma 2.3 below that (−χ) ◦ ϕj (ω + dd χ′(−t)(ω + ddcϕj) n(ϕj < −t)dt χ′(−t)tnCapω(ϕ < −t)dt < +∞, This shows that ϕ ∈ Eχ(X,ω). The other inclusion goes similarly, using the second inequality in Lemma 2.3 below. � If ϕ ∈ Eχ(X,ω) (or Êχ(X,ω)), then the bigger the growth of χ at −∞, the smaller Capω(ϕ < −t) when t → +∞, hence the closer ϕ is from being bounded. Indeed ϕ ∈ PSH(X,ω) is bounded iff it belongs to Eχ(X,ω) for all weights χ, as was observed in [GZ 2], Proposition 3.1. Similarly PSH(X,ω) ∩ L∞(X) = Êχ(X,ω), where the intersection runs over all concave increasing functions χ. We will make constant use of the following result: 6 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI Lemma 2.3. Fix ϕ ∈ E(X,ω). Then for all s > 0 and 0 ≤ t ≤ 1, tnCapω(ϕ < −s− t) ≤ (ϕ<−s) (ω + ddcϕ)n ≤ snCapω(ϕ < −s), where the second inequality is true only for s ≥ 1. The proof is a direct consequence of the comparison principle (see Lemma 2.2 in [EGZ] and [GZ 2]). 3. Measures dominated by capacity From now on µ denotes a positive Radon measure on X whose total mass is V olω(X): this is an obvious necessary condition in order to solve (MA)µ. To simplify numerical computations, we assume in the sequel that µ and ω have been normalized so that µ(X) = V olω(X) = ωn = 1. When µ = ehωn is a smooth volume form and ω is a Kähler form, S.T.Yau has proved [Y] that (MA)µ admits a unique smooth solution ϕ ∈ PSH(X,ω) with supX ϕ = 0. Smooth measures are easily seen to be nicely dominated by the Monge-Ampère capacity (see the proof of Corollary 3.2 below). Measures dominated by the Monge-Ampère capacity have been exten- sively studied by S.Kolodziej in [K 2,3,4]. Following S. Kolodziej ([K3], [K4]) with slightly different notations, fix ε : R → [0,∞[ a continuous decreasing function and set Fε(x) := x[ε(− lnx/n)] n, x > 0. We will consider probability measures µ satisfying the following condition : for all Borel subsets K ⊂ X, µ(K) ≤ Fε(Capω(K)). The main result achieved in [K 2], can be formulated as follows: If ω is a Kähler form and ε(t)dt < +∞ then µ = (ω + ddcϕ)n for some contin- uous function ϕ ∈ PSH(X,ω). The condition ε(t)dt < +∞ means that ε decreases fast enough towards zero at infinity. This gives a quantitative estimate on how fast ε(− lnCapω(K)/n), hence µ(K), decreases towards zero as Capω(K) → 0. ε(t)dt = +∞, it follows from Theorem 2.1 that µ = (ω + ddcϕ)n for some function ϕ ∈ E(X,ω), but ϕ will generally be unbounded. Our second main result measures how far ϕ is from being bounded: Theorem 3.1. Assume for all compact subsets K ⊂ X, (3.1) µ(K) ≤ Fε(Capω(K)). Then µ = (ω + ddcϕ)n where ϕ ∈ E(X,ω) is such that supX ϕ = 0 and Capω(ϕ < −s) ≤ exp(−nH −1(s)), for all s > 0. Here H−1 is the reciprocal function of H(x) = e ε(t)dt + s0, where s0 = s0(ε, ω) ≥ 0 is a constant which only depends on ε and ω. In particular ϕ ∈ Eχ(X,ω) where −χ(−t) = exp(nH −1(t)/2). A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 7 Recall that here, and troughout the article, ω ≥ 0 is merely big. Before proving this result we make a few observations. • It is interesting to consider as well the case when ε(t) increases to- wards +∞. One can then obtain solutions ϕ such that Capω(ϕ < −t) decreases at a polynomial rate. When e.g. ω is Kähler and µ(K) ≤ Capω(K) α, 0 < α < 1, it follows from Proposition 5.3 in [GZ 2] that µ = (ω + ddcϕ)n where ϕ ∈ Ep(X,ω) for some p = pα > 0. Here E p(X,ω) denotes the Cegrell type class Eχ(X,ω), with χ(t) = −(−t)p. • When ε(t) ≡ 1, Fε(x) = x and H(x) ≍ e.x. Thus Theorem 3.1 reads µ ≤ Capω ⇒ µ = (ω + dd cϕ)n, where Capω(ϕ < −s) . exp (−ns/e) . This is precisely the rate of decreasing corresponding to functions which look locally like − log(− log ||z||), in some local chart z ∈ U ⊂ Cn. This class of ω-psh functions with “loglog-singularities” is important for applications (see [Ku], [BKK]). • If ε(t) decreases towards zero, then Capω(ϕ < −t) decreases at a superexponential rate. The faster ε(t) decreases towards zero, the slower the growth of H, hence the faster the growth of H−1 at infin- ity. When ε(t)dt < +∞, the function ε decreases so fast that Capω(ϕ < −t) = 0 for t >> 1, thus ϕ is bounded. This is the case when µ(K) ≤ Capω(K) α for some α > 1 [K 2], [EGZ]. • When ε(t)dt = +∞, the solution ϕmay well be unbounded (see Examples in section 4). At the critical case where µ ≤ Fε(Capω) for all functions ε such that ε(t)dt = +∞, we obtain µ = (ω + ddcϕ)n with ϕ ∈ PSH(X,ω) ∩ L∞(X), as follows from Proposition 3.1 in [GZ 2]. This partially explains the difficulty in describing the range of Monge-Ampère operators on the set of bounded (quasi-)psh functions. Proof. The assumption on µ implies in particular that it vanishes on pluripo- lar sets. It follows from Theorem 2.1 that there exists a function ϕ ∈ E(X,ω) such that µ = (ω + ddcϕ)n and supX ϕ = 0. Set g(s) := − logCapω(ϕ < −s), ∀s > 0. The function g is increasing on [0,+∞] and g(+∞) = +∞, since Capω vanishes on pluripolar sets. Observe also that g(s) ≥ 0 for all s ≥ 0, since g(0) = − logCapω(X) = − log V olω(X) = 0. It follows from Lemma 2.3 and (3.1) that for all s > 0 and 0 ≤ t ≤ 1, tnCapω(ϕ < −s− t) ≤ µ(ϕ < −s) ≤ Fε (Capω(ϕ < −s)) . Therefore for all s > 0 and 0 ≤ t ≤ 1, (3.2) log t− log ε ◦ g(s) + g(s) ≤ g(s + t). We define an increasing sequence (sj)j∈N by induction setting sj+1 = sj + eε ◦ g(sj), for all j ∈ N. 8 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI The choice of s0. Recall that (3.2) is only valid for 0 ≤ t ≤ 1. We choose s0 ≥ 0 large enough so that (3.3) e.ε ◦ g(s0) ≤ 1. This will allow us to use (3.2) with t = tj = sj+1 − sj ∈ [0, 1], since ε ◦ g is decreasing, while sj ≥ s0 is increasing, hence 0 ≤ tj = eε ◦ g(sj) ≤ eε ◦ g(s0) ≤ 1. We must insure that s0 = s0(ε, ω) can chosen to be independent of ϕ. This is a consequence of Proposition 2.7 in [GZ 1]: since supX ϕ = 0, there exists c1(ω) > 0 so that 0 ≤ (−ϕ)ωn ≤ c1(ω), hence g(s) := − logCapω(ϕ < −s) ≥ log s− log(n+ c1(ω)). Therefore g(s0) ≥ ε −1(1/e) for s0 = s0(ε, ω) := (n+ c1(ω)) exp(nε −1(1/e)), which is independent of ϕ. This yields e.ε ◦ g(s0) ≤ 1, as desired. The growth of sj. We can now apply (3.2) and get g(sj) ≥ j + g(s0) ≥ j. Thus lim g(sj) = +∞. There are two cases to be considered. If s∞ = lim sj ∈ R +, then g(s) ≡ +∞ for s > s∞, i.e. Capω(ϕ < −s) = 0, ∀s > s∞. Therefore ϕ is bounded from below by −s∞, in particular ϕ ∈ Eχ(X,ω) for all χ. Assume now (second case) that sj → +∞. For each s > 0, there exists N = Ns ∈ N such that sN ≤ s < sN+1. We can estimate s 7→ Ns: s ≤ sN+1 = (sj+1 − sj) + s0 = e ε ◦ g(sj) + s0 ε(j) + s0 ≤ e.ε(0) + e ε(t)dt+ s0 =: H(N), Therefore H−1(s) ≤ N ≤ g(sN ) ≤ g(s), hence Capω(ϕ < −s) ≤ exp(−nH −1(s)). Set now −χ(−t) = exp(nH−1(t)/2). Then tnχ′(−t)Capω(ϕ < −t)dt ε(H−1(t)) + s̃0 exp(−nH−1(t)/2)dt tn exp(−nt/2)dt < +∞. This shows that ϕ ∈ Eχ(X,ω) where χ(t) = − exp(nH −1(−t)/2). It follows from the proof above that when ε(t)dt < +∞, the solution ϕ is bounded since in this case we have s∞ := lim sj ≤ s0(ε, ω) + e ε(0) + e ε(t)dt < +∞ where s0(ε, ω) is an absolute constant satisfying (3.3) (see above). � A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 9 Let us emphasize that Theorem 3.1 also yields a slightly simplified proof of the following result [K 2], [EGZ]: if µ(K) ≤ Fε(Capω(K)) for some decreas- ing function ε : R → R+ such that ε(t)dt < +∞, then the sequence (sj) above is convergent, hence µ = (ω + dd cϕ)n, where ϕ ∈ PSH(X,ω) is bounded. For the reader’s convenience we indicate a proof of the following important particular case: Corollary 3.2. Let µ = fωn be a measure with density 0 ≤ f ∈ Lp(ωn), where p > 1 and fωn = ωn. Then there exists a unique bounded function ϕ ∈ PSH(X,ω) such that (ω + ddcϕ)n = µ, supX ϕ = 0 and 0 ≤ ||ϕ||L∞(X) ≤ C(p, ω).||f || Lp(ωn) where C(p, ω) > 0 only depends on p and ω. This a priori bound is a crucial step in the proof by S.T.Yau of the Calabi conjecture (see [Ca], [Y], [A], [T], [Bl]). The proof presented here follows Kolodziej’s new and decisive pluripotential approach (see [K2]). Let us stress that the dependence ω 7−→ C(p, ω) is quite explicit, as we shall see in the proof. This is important when considering degenerate situations [EGZ]. Proof. We claim that there exists C1(ω) such that (3.4) µ(K) ≤ C1(ω)||f || Lp(ωn) [Capω(K)] , for all Borel sets K ⊂ X. Assuming this for the moment, we can apply Theorem 3.1 with ε(x) = C1(ω)||f || Lp(ωn) exp(−x), which yields, as observed at the end of the proof of Theorem 3.1 ||ϕ||L∞(X) ≤M(f, ω), whereM(f, ω) := s0(ε, ω)+e ε(0)+e ε(t)dt = s0(ε, ω)+2eC1(ω)||f || Lp(ωn) and s0 = s0(ε, ω) is a large number s0 > 1 satisfying the inequality (3.3). In order to give the precise dependence of the uniform bound M(f, ω) on the Lp−norm of the density f , we need to choose s0 more carefully. Observe that condition (3.3) can be written Capω({ϕ ≤ −s0}) ≤ exp(−nε −1(1/e). Since nε−1(1/e) = log enC1(ω) n‖f‖Lp(ωn) , we must choose s0 > 0 so that (3.5) Capω({ϕ ≤ −s0}) ≤ enC1(ω)n‖f‖Lp(ωn) We claim that for any N ≥ 1 there exists a uniform constant C2(N, p, ω) > 0 such that for any s > 0, (3.6) Capω({ϕ ≤ −s}) ≤ C2(N, p) s −N ‖f‖Lp(ωn). Indeed observe first that by Hölder inequality, (−ϕ)Nωnϕ = (−ϕ)Nfωn ≤ ‖f‖Lp(ωn)‖ϕ‖ LNq (ωn) 10 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI Since ϕ belongs to the compact family {ψ ∈ PSH(X,ω); supX ψ = 0} ([GZ2]), there exists a uniform constant C ′2(N, p, ω) > 0 such that ‖ϕ‖ LNq(ωn) C ′2(N, p, ω), hence (−ϕ)Nωnϕ ≤ C 2(N, p, ω)‖f‖Lp(ωn). Fix u ∈ PSH(X,ω) with −1 ≤ u ≤ 0 and N ≥ 1 to be specified later. If follows from Tchebysheff and energy inequalities ([GZ2]) that {ϕ≤−s} (ω + ddcu)n ≤ s−N (−ϕ)N (ω + ddcu)n ≤ cN s −N max (−ϕ)Nωnϕ, (−u)Nωnu ≤ cN s −N max C ′2(N, p, ω), 1 ‖f‖Lp(ωn). We have used here the fact that ‖f‖Lp(ωn) ≥ 1, which follows from the normalization : 1 = fωn ≤ ‖f‖Lp(ωn). This proves the claim. SetN = 2n, it follows from (3.6) that s0 := C1(ω) nenC2(2n, p, ω)‖f‖ Lp(ωn) satisfies the required condition (3.5), which implies the estimate of the the- orem. We now establish the estimate (3.4). Observe first that Hölder’s inequality yields (3.7) µ(K) ≤ ||f ||Lp(ωn) [V olω(K)] , where 1/p + 1/q = 1. Thus it suffices to estimate the volume V olω(K). Recall the definition of the Alexander-Taylor capacity, Tω(K) := exp(− supX VK,ω), where VK,ω(x) := sup{ψ(x) /ψ ∈ PSH(X,ω), ψ ≤ 0 on K}. This capacity is comparable to the Monge-Ampère capacity, as was observed by H.Alexander and A.Taylor [AT] (see Proposition 7.1 in [GZ 1] for this compact setting): (3.8) Tω(K) ≤ e exp Capω(K) It thus remains to show that V olω(K) is suitably bounded from above by Tω(K). This follows from Skoda’s uniform integrability result: set ν(ω) := sup {ν(ψ, x) /ψ ∈ PSH(X,ω), x ∈ X} , where ν(ψ, x) denotes the Lelong number of ψ at point x. This actually only depends on the cohomology class {ω} ∈ H1,1(X,R). It is a standard fact that goes back to H.Skoda (see [Z]) that there exists C2(ω) > 0 so that ωn ≤ C2(ω), for all functions ψ ∈ PSH(X,ω) normalized by supX ψ = 0. We infer (3.9) V olω(K) ≤ V ∗K,ω ωn ≤ C2(ω)[Tω(K)] 1/ν(ω). A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 11 It now follows from (3.7), (3.8), (3.9), that µ(K) ≤ ||f ||Lp [C2(ω)] 1/qe1/qν(ω) exp qν(ω)Capω(K) The conclusion follows by observing that exp(−1/x1/n) ≤ Cnx 2 for some explicit constant Cn > 0. � 4. Examples 4.1. Measures invariant by rotations. In this section we produce exam- ples of radially invariant functions/measures which show that our previous results are essentially sharp. The first example is due to S.Kolodziej [K 1]. Example 4.1. We work here on the Riemann sphere X = P1(C), with ω = ωFS, the Fubini-Study volume form. Consider µ = fω a measure with density f which is smooth and positive on X \ {p}, and such that f(z) ≃ |z|2(log |z|)2 , c > 0, in a local chart near p = 0. A simple computation yields µ = ω + ddcϕ, where ϕ ∈ PSH(P1, ω) is smooth in P1 \ {p} and ϕ(z) ≃ −c′ log(− log |z|) near p = 0, c′ > 0, hence logCapω(ϕ < −t) ≃ −t, Here a ≃ b means that a/b is bounded away from zero and infinity. This is to be compared to our estimate logCapω(ϕ < −t) . −t/e (Theo- rem 3.1 ) which can be applied, as it was shown by S.Kolodziej in [K 1] that µ . Capω. Thus Theorem 3.1 is essentially sharp when ε ≡ 1. We now generalize this example and show that the estimate provided by Theorem 3.1 is essentially sharp in all cases. Example 4.2. Fix ε as in Theorem 3.1. Consider µ = fω on X = P1(C), where ω = ωFS is the Fubini-Study volume form, f ≥ 0 is continuous on 1 \ {p}, and f(z) ≃ ε(log(− log |z|)) |z|2(log |z|)2 in local coordinates near p = 0. Here ε : R → R+ decreases towards 0 at +∞. We claim that there exists A > 0 such that (4.1) µ(K) ≤ ACapω(K)ε(− logCapω(K)), for all K ⊂ X. This is clear outside a small neighborhood of p = 0 since the measure µ is there dominated by a smooth volume form. So it suffices to establish this estimate when K is included in a local chart near p = 0. Consider K̃ := {r ∈ [0, R] ; K ∩ {|z| = r} 6= ∅}. It is a classical fact (see e.g. [Ra]) that the logarithmic capacity c(K) of K can be estimated from below by the length of K̃, namely l(K̃) ≤ c(K̃) ≤ c(K). 12 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI Using that ε is decreasing, hence 0 ≤ −ε′, we infer µ(K) ≤ 2π ∫ l(K̃) f(r)rdr ∫ l(K̃) ε(log(− log r))− ε′(log− log r) r(log r)2 ε(log(− log l(K̃))) − log l(K̃) ε(log(− log 4c(K))) − log 4c(K) Recall now that the logarithmic capacity c(K) is equivalent to Alexander- Taylor’s capacity T∆(K), which in turn is equivalent to the global Alexander- Taylor capacity Tω(K) (see [GZ 1]): c(K) ≃ T∆(K) ≃ Tω(K). The Alexander- Taylor’s comparison theorem [AT] reads − log 4c(K) ≃ − log Tω(K) ≃ 1/Capω(K), thus µ(K) ≤ ACapω(K)ε(− logCapω(K)). We can therefore apply Theorem 3.1. It guarantees that µ = (ω + ddcϕ), where ϕ ∈ PSH(P1, ω) satisfies logCapω(ϕ < −s) ≃ −nH −1(s), with H(s) = eA ε(t)dt + s0. On the other hand a simple computation shows that ϕ is continuous in P1 \ {p} and ϕ ≃ −H(log(− log |z|)) , near p = 0. The sublevel set (ϕ < −t) therefore coincides with the ball of radius exp(− exp(H−1(t))), hence logCapω(ϕ < −s) ≃ −H −1(s). 4.2. Measures with density. Here we consider the case when µ = fdV is absolutely continuous with respect to a volume form. Proposition 4.3. Assume µ = fωn is a probability measure whose density satisfies f [log(1 + f)]n ∈ L1(ωn). Then µ . Capω. More generally if f [log(1 + f)/ε(log(1 + | log f |))]n ∈ L1(ωn) for some continuous decreasing function ε : R → R+∗ , then for all K ⊂ X, µ(K) ≤ Fε(Capω(K)), where Fε(x) = Ax , A > 0. Proof. With slightly different notations, the proof is identical to that of Lemma 4.2 in [K 4] to which we refer the reader. � We now give examples showing that Proposition 4.3 is almost optimal. Example 4.4. For simplicity we give local examples. The computations to follow can also be performed in a global compact setting. Consider ϕ(z) = − log(− log ||z||), where ||z|| = |z1|2 + . . .+ |zn|2 de- notes the Euclidean norm in Cn. One can check that ϕ is plurisubharmonic in a neighborhood of the origin in Cn, and that there exists cn > 0 so that µ := (ddcϕ)n = f dVeucl, where f(z) = ||z||2n(− log ||z||)n+1 Observe that f [log(1 + f)]n−α ∈ L1, ∀α > 0 but f [log(1 + f)]n 6∈ L1. A PRIORI ESTIMATES FOR SOLUTIONS OF MONGE-AMPÈRE EQUATIONS 13 When n = 1 it was observed by S. Kolodziej [K 1] that µ(K) . Capω(K). Proposition 4.3 yields here µ(K) . Capω(K)(| logCapω(K)|+ 1). For n ≥ 1, it follows from Proposition 4.3 and Theorem 3.1 that logCapω(ϕ < −s) . −nH −1(s). On the other hand, one can directly check that logCapω(ϕ < −s) ≃ −nH −1(s). One can get further examples by considering ϕ(z) = χ ◦ log ||z||, so that (ddcϕ)n = ′ ◦ log ||z||)n−1χ′′(log ||z||) ||z||2n dVeucl. References [AT] H.ALEXANDER & B.A.TAYLOR: Comparison of two capacities in Cn. Math. Zeit, 186 (1984),407-417. [A] T.AUBIN: Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2) 102 (1978), no. 1, 63–95. [BT] E.BEDFORD & B.A.TAYLOR: A new capacity for plurisubharmonic func- tions. Acta Math. 149 (1982), no. 1-2, 1–40. [Bl] Z.BLOCKI: On uniform estimate in Calabi-Yau theorem. Sci. China Ser. A 48 (2005), suppl., 244–247. [BKK] G.BURGOS & J.KRAMER & U.KUHN: Arithmetic characteristic classes of automorphic vector bundles. Doc. Math. 10 (2005), 619–716. [Ca] E.CALABI: On Kähler manifolds with vanishing canonical class. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 78–89. Princeton Univ. Press, Princeton, N. J. (1957). [Ce] U.CEGRELL: Pluricomplex energy. Acta Math. 180 (1998), no. 2, 187–217. [EGZ] P.EYSSIDIEUX & V.GUEDJ & A.ZERIAHI: Singular Kähler-Einstein met- rics. Preprint arxiv math.AG/0603431. [GZ 1] V.GUEDJ & A.ZERIAHI: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15 (2005), no. 4, 607-639. [GZ 2] V.GUEDJ & A.ZERIAHI: The weighted Monge-Ampère energy of quasi- plurisubharmonic functions. J. Funct. Anal. 250 (2007), 442-482. [K 1] S.KOLODZIEJ: The range of the complex Monge-Ampère operator. Indiana Univ. Math. J. 43 (1994), no. 4, 1321–1338. [K 2] S.KOLODZIEJ: The complex Monge-Ampère equation. Acta Math. 180 (1998), no. 1, 69–117. [K 3] S.KOLODZIEJ: The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52 (2003), no. 3, 667–686 [K 4] S.KOLODZIEJ: The complex Monge-Ampère equation and pluripotential theory. Mem. Amer. Math. Soc. 178 (2005), no. 840, x+64 pp. [Ku] U.KUHN: Generalized arithmetic intersection numbers. J. Reine Angew. Math. 534 (2001), 209–236. [R] J.RAINWATER: A note on the preceding paper. Duke Math. J. 36 (1969) 799–800. [Ra] T.RANSFORD: Potential theory in the complex plane. London Mathemati- cal Society Student Texts, 28. Cambridge University Press, Cambridge, 1995. x+232 pp. [T] G.TIAN: Canonical metrics in Kähler geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2000). [Y] S.T.YAU: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. [Z] A.ZERIAHI: Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J. 50 (2001), no. 1, 671–703. http://arxiv.org/abs/math/0603431 14 S.BENELKOURCHI & V.GUEDJ & A.ZERIAHI Slimane BENELKOURCHI & Vincent GUEDJ & Ahmed ZERIAHI Laboratoire Emile Picard UMR 5580, Université Paul Sabatier 118 route de Narbonne 31062 TOULOUSE Cedex 09 (FRANCE) benel@math.ups-tlse.fr guedj@math.ups-tlse.fr zeriahi@math.ups-tlse.fr 1. Introduction 2. Weakly singular quasiplurisubharmonic functions 2.1. The range of the Monge-Ampère operator 2.2. High energy and capacity estimates 3. Measures dominated by capacity 4. Examples 4.1. Measures invariant by rotations 4.2. Measures with density References Bibliography ABSTRACT Let $X$ be a compact K\"ahler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_{\chi}(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Amp\`ere energy. When the weight $\chi$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Amp\`ere capacity, then it belongs to the range of the Monge-Amp\`ere operator on some class ${\mathcal E}_{\chi}(X,\om)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori ${\mathcal C}^0$-estimate. <|endoftext|><|startoftext|> Density oscillation in highly flattened quantum elliptic rings and tunable strong dipole radiation S.P. Situ, Y.Z. He, and C.G. Bao∗ The State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan University, Guangzhou, 510275, P.R. China A narrow elliptic ring containing an electron threaded by a magnetic field B is studied. When the ring is highly flattened, the increase of B would lead to a big energy gap between the ground and excited states, and therefore lead to a strong emission of dipole photons. The photon frequency can be tuned in a wide range by changing B and/or the shape of the ellipse. The particle density is found to oscillate from a pattern of distribution to another pattern back and forth against B. This is a new kind of Aharonov-Bohm oscillation originating from symmetry breaking and is different from the usual oscillation of persistent current. ∗Corresponding author It is recognized that micro-devices are important to micro-techniques. Various kinds of micro-devices, includ- ing the quantum rings,1 have been extensively studied theoretically and experimentally in recent years. Quan- tum rings are different from other devices due to their special geometry. A distinguished phenomenon of the ring is the Aharonov-Bohm (A-B) oscillation of the ground state energy and persistent current2−5. It is be- lieved that geometry would affect the properties of small systems. Therefore, in addition to circular rings, ellip- tic rings or other rings subjected to specific topological transformations deserve to be studied, because new and special properties might be found. There have been a number of literatures devoted to elliptic quantum dots6−9 and rings10−12. It was found that the elliptic rings have two distinguished features. (i) The avoided crossing of the levels and the suppression of the A-B oscillation. (ii) The appearance of localized states which are related to bound states in infinite wires with bends.13 These feature would become more explicit if the eccentricity is larger and the ring is narrower. On the other hand, as a micro-device, the optical prop- erty is obviously essential to its application. It is guessed that very narrow rings with a high eccentricity might have special optical property, this is a point to be clari- fied. This paper is dedicated to this topic. It turns out that the optical properties of a highly flattened narrow ring is greatly different from a circular ring due to having a tunable energy gap, which would lead to strong dipole transitions with wave length tunable in a very broad range (say, from 0.1 to 0.001cm). Besides, a kind of A-B density-oscillation originating from symmetry breaking was found as reported as follows. We consider an electron with an effective mass m∗ con- fined on a one-dimensional elliptic ring with a half major axis rax and an eccentricity ε. Let us introduce an argu- ment θ so that a point (x, y) at the ring is related to θ as x = rax cos θ and y = ray sin θ, where ray = rax 1− ε2 is the half minor axis. A uniform magnetic field B confined inside a cylinder with radius rin vertical to the plane of the ring is applied. The associated vector potential reads A = Br2int/2r, where t is a unit vector normal to the position vector r. Then, the Hamiltonian reads H = G/(1− ε2 cos2 θ)[− d − i2α 1− ε2 (1− ε2 sin2 θ) 1− ε2 cos2 θ 1− ε2 sin2 θ ] (1) where G = ~2/(2m∗r2ax), α = φ/φo, φ = πr inB is the flux, φo = hc/e is the flux quantum. The eigen-states are expanded as Ψj = ∑kmax k=kmin eikθ, where k is an integer ranging from kmin to kmax, and j = 1, 2, · · · denotes the ground state, the second state, and so on. The coefficients C are obtained via the diagonalization of H . In practice, B takes positive values, kmin = −100 and kmax = 10. This range of k assures the numerical results having at least four effective figures. The energy of the j − th state is Ej = 〈H〉j ≡ dθ(1 − ε2 cos2 θ)Ψ∗jHΨj (2) where the eigen-state is normalized as dθ(1− ε2 cos2 θ)Ψ∗jΨj (3) In the follows the units meV, nm, and Tesla are used, m∗ = 0.063me (for InGaAs rings), and rin is fixed at 25. When rax = 50, ε = 0 and 0.4, the evolution of the low-lying spectra with B are given in Fig.1. When ε = 0.4, the effect of eccentricity is still small, the spec- trum is changed only slightly from the case ε = 0, but the avoided crossing of levels can be seen.10,11 In par- ticular, the A-B oscillation exists and the period of φ remains to be φo. However, when ε becomes large, three remarkable changes emerge as shown in Fig.2. (i) The A-B oscillation of the ground state vanishes gradually. (ii) The energy of the second state becomes closer and closer to the ground state. (iii) There is an energy gap lying between the ground state and the third state, the http://arxiv.org/abs/0704.0867v1 0 2 4 6 8 10 E(meV) B(Tesla) ε =0.4rax=50(b) ε =0rax=50(a) FIG. 1: Low-lying spectrum (in meV) of an one-electron sys- tem on an elliptic ring against B. rax = 50nm and ε = 0 (a) and 0.4 (b). The period of the flux φo = hc/e is associated with B = 2.106 Tesla. 0 4 8 12 16 20 B (Tesla) E(meV) rax= 50, ε = 0.8 FIG. 2: Similar to Fig.1 but ε = 0.8. The lowest eight levels are included, where a great energy gap lies between the ground and the third states. gap width increases nearly linearly with B. The exis- tence of the gap is a remarkable feature which has not yet been found before from the rings with a finite width. This feature is crucial to the optical properties as shown later. Fig.3 demonstrates further how the gap varies with ε, rax, and B , where B is from 0 to 30 (or φ from 0 to 14.24φo). One can see that, when ε is large and rax is small, the increase of B would lead to a very large gap. 0 5 10 15 20 25 30 0 5 10 15 20 25 30 (b) ε = 0.8 ε =0.8 B (Tesla) E3-E1 (meV) (a) r ε =0.6 ε=0.4 FIG. 3: Evolution of the energy gap E3 − E1 when rax and ε are given. The A-B oscillation of the ground state energy is given in Fig.4. The change of ε does not affect the period (2.106 Tesla). However, when ε is large, the amplitude of the oscillation would be rapidly suppressed. Thus, for a highly flattened elliptic ring, the A-B oscillation appears only when B is small. 0 2 4 6 8 10 B (Tesla) ε =0, 0.4, 0.8rax=50 FIG. 4: The A-B oscillation of the ground state energy. The solid, dash-dot-dot, and dot lines are for ε = 0, 0.4, and 0.8, respectively. The persistent current of the j − th state reads14 Jj = G/~[Ψ 1− ε2 (1− ε2 sin2 θ) )Ψj + c.c.] (4) The A-B oscillation of Jj is plotted in Fig.5. When ε is small (≤ 0.4), just as in Fig.4, the effect of ε is small as shown in 5a. When ε is large there are three noticeable points: (i) The oscillation of the ground state current would become weaker and weaker when B increases. (ii) The current of the second state has a similar amplitude as the ground state, but in opposite phase. (iii) The third (and higher) state has a much stronger oscillation of current. 0 2 4 6 8 10 B (Tesla) (b) rax=50, ε =0.8 ε=0, 0.4, 0.8(a) r FIG. 5: The A-B oscillation of the persistent current J . (a) is for the ground state with ε = 0 (solid line), 0.4 (dash-dot- dot), and 0.8 (dot). (b) is for the first (ground), second and third states (marked by 1,2, and 3 by the curves) with ε fixed at 0.8. The ordinate is 106 times J/c in nm−1. For elliptic rings, the angular momentum L is not con- served. However, it is useful to define (L)j = 〈−i ∂∂θ 〉j (refer to eq.(2)). This quantity would tend to an integer if ε → 0. It was found that (i) When ε is small (≤ 0.4), (L)1 of the ground state decreases step by step with B, each step by one, just as the case of circular rings. How- ever, when ε is large, (L)1 decreases continuously and nearly linearly. (ii) When ε is small, |(L)i− (L)1| is close (not close) to 1 if 2 ≤ i ≤ 3 (otherwise). Since L would be changed by ±1 under a dipole transition, the ground state would therefore essentially jump to the second and third states. Accordingly, the dipole photon has essen- tially two energies, namely, E2 −E1 and E3 −E1 . How- ever, this is not exactly true when ε is large. There is a relation between the dipole photon energies and the persistent current.15 For ε = 0, the ground state with L = k1 would have the current J1 = G(k1 + α)/π~, while the ground state energy E(k1) = G(k1 + α) 2. Ac- cordingly the second and third states would have L = k1 ± 1, therefore we have |E3 − E2| = |E(k1 + 1)− E(k1 − 1)| = 2hJ1 (5) This relation implies that the current can be accurately measured simply by measuring the energy difference of the photons emitted in dipole transitions. For elliptic rings, this relation holds approximately when ε is small (≤ 0.4), as shown in Fig.6a. However, the deviation is quite large when ε is large as shown in 6c. 0 2 4 6 8 10 (c) ε = 0.7 B (Tesla) (b) ε = 0.5 (a) ε = 0.3 FIG. 6: E3 − E2 and the persistent current of the ground state. The solid line denotes (E3 − E2)/(2hc)10 6, the dash- dot-dot line denotes |J |/c·106 . They overlap nearly if ε < 0.3. The probability of dipole transition from Ψj to Ψj′ reads (ω/c)3|〈x∓ iy〉j′,j |2 (6) where ~ω = Ej′ − Ej is the photon energy, 〈x∓ iy〉j′,j = rax dθ(1 − ε2 cos2 θ)Ψ∗j′ [cos θ ∓ i 1− ε2 sin θ]Ψj (7) The probability of the transition of the ground state to the j′ − th state is shown in Fig.7. When ε is small (≤ 0.4) and B is not very large (≤ 10), the allowed final states essentially Ψ2 and Ψ3, and the oscillation of the probability is similar to the case of circular rings with the same period as shown in 7a and 7b. In particular, P±3,1 is considerably larger than P±2,1 due to having a larger pho- ton energy, thus the third state is particularly important to the optical properties. When ε is large (Fig.7c), the oscillation disappears gradually with B,while the prob- ability increases very rapidly due to the factor (ω/c)3. Since E3 − E1 is nearly proportional to B as shown in Fig.3, the probability is nearly proportional to B3. This leads to a very strong emission (absorption). Further- more, in Fig.7c the black solid curve is much higher than the dash-dot-dot curve, it implies that the final states can be higher than Ψ3, this leads to an even larger prob- ability. 0 2 4 6 8 10 B(Tesla) c) ε =0.8 b) ε =0.4 a) ε =0 FIG. 7: Evolution of the probability of dipole transition of the ground state. The green line is for Ψ1 to Ψ2 transition, red line for Ψ1 to Ψ3, dash-dot-dot line is for the sum of the above two, solid line in black is for the total probability. For circular rings, the particle densities ρ of all the eigen-states are uniform under arbitraryB. However, for elliptic rings, ρ is no more uniform as shown in Fig.8. For the ground state (8a), when φ =0, the non-uniformity is slight and ρ is a little smaller at the two ends of the major axis (θ = 0, π). When φ increases, the density at the two ends of the minor axis (θ = π/2, 3π/2) increases as well. When φ = 4φo the non-uniformity is very strong as shown by the curve 9, where ρ ≈ 0 when θ ≈ 0 or π. The second state has a parity opposite to the ground state, 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Arc length (nm) fourth state third state 8ε =0.8, r ground state FIG. 8: Particle densities ρ as functions of the arc length (the according change of θ is 0 to π). The fluxes are given as φ = (i− 1)φo/2, where i is an integer from 1 to 9 marked by the curves. The first group of curves (in violet) have φ/φo = integer, the second group (in green) have φ/φo = half-integer. When φ increases, the curve of ρ jumps from the first group to the second group and jumps back, and repeatedly. but their densities are similar. For the third state (8b), ρ is peaked not at the ends of the major and minor axes but in between. In particular, when B increases, ρ oscillates from one pattern (say, in violet line) to another pattern (in green line), and repeatedly. The density oscillation would become stronger in higher states (8c). The period of oscillation remains to be φo, thus it is a new type of A-B oscillation without analogue in circular rings (where ρ remains uniform). Incidentally, the density oscillation does not need to be driven by a strong field, instead, a small change of φ from 0 to φo is sufficient. Let us evaluate Ej roughly by using (L)j to replace the operator −i ∂ in eq.(2) Then, Ej ≈ G dθ{[(L)j + α 1− ε2 1− ε2 sin2 θ αε2 sin 2θ 2(1− ε2 sin2 θ) ]2}Ψ∗jΨj (8) There are two terms at the right each is a square of a pair of brackets (for circular rings the second term does not exist). It is reminded that, while α = φ/φo is given positive, (L)j is negative. Thus there is a cancellation inside the first term. Therefore, when ε and α are large, the second term would be more important. It is recalled that both Ψ1 and Ψ2 are mainly distributed around θ = π/2 and 3π/2 (refer to Fig.8a), where the second term is zero due to the factor sin 2θ. Accordingly the energies of Ψ1 and Ψ2 are lower. On the contrary, both Ψ3 and Ψ4 are distributed close to the peaks of the second term (refer to Fig.8b and 8c), this leads to a higher energy. This effect would be greatly amplified by αε2 , this leads to the large energy gap shown in Fig.3. In summary, the optical property of highly flattened elliptic narrow rings was found to be greatly different from circular rings. For the latter, both the energy of the dipole photon and the probability of transition are low, and they are oscillating in small domains. On the contrary, for the former, both the energy and the prob- ability are not limited, the energy (probability) is nearly proportional to B (B3), they are tunable by changing ε, rax and/or B. It implies that a strong source of light with frequency adjustable in a wide domain can be de- signed by using highly flattened, narrow, and small rings. Furthermore, a new type of A-B oscillation, namely, the density oscillation, originating from symmetry breaking, was found. This is a noticeable point because the density oscillation might be popular for the systems with broken symmetry (e.g., with C3 symmetry). Acknowledgment: The support under the grants 10574163 and 90306016 by NSFC is appreciated. References 1, S.Viefers, P. Koskinen, P. Singha Deo, M. Manninen, Physica E 21 , 1 (2004) 2, U.F. Keyser, C. Fühner, S. Borck, R.J. Haug, M. Bichler, G. Abstreiter, and W. Wegscheider, Phys. Rev. Lett. 90, 196601 (2003) 3, D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993) 4, A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Nature (London) 413, 822 (2001) 5, A.E. Hansen, A. Kristensen, S. Pedersen, C.B. Sorensen, and P.E. Lindelof, Physica E (Amsterdam) 12, 770 (2002) 6, M. van den Broek, F.M. Peeters, Physica E,11, 345 (2001) 7, E. Lipparini, L. Serra, A. Puente, European Phys. J. B 27, 409 (2002) 8, J. Even, S. Loualiche, P. Miska, J. of Phys.: Cond. Matt., 15, 8737 (2003) 9, C. Yannouleas, U. Landman, Physica Status Solidi A 203, 1160 (2006) 10, D. Berman, O Entin-Wohlman, and M. Ya. Azbel, Phys. Rev. B 42, 9299 (1990) 11, D. Gridin, A.T.I. Adamou, and R.V. Craster, Phys. Rev. B 69, 155317 (2004) 12, A. Bruno-Alfonso, and A. Latgé, Phys. Rev. B 71, 125312 (2005) 13, J. Goldstone and R.L. Jaffe, Phys. Rev. B 45, 14100 (1992) 14, Eq.(4) originates from a 2-dimensional system via the following steps. (i) the components of the current along X- and Y-axis are firstly obtained from the conser- vation of mass as well known. (ii) Then, the component along the tangent of ellipse jθ can be obtained. (iii) jθ is integrated along the normal of the ellipse under the assumption that the wave function is restricted in a very narrow region along the normal, then it leads to eq.(4). 15, Y.Z. He, C.G. Bao (submitted to PRB) ABSTRACT A narrow elliptic ring containing an electron threaded by a magnetic field B is studied. When the ring is highly flattened, the increase of B would lead to a big energy gap between the ground and excited states, and therefore lead to a strong emission of dipole photons. The photon frequency can be tuned in a wide range by changing B and/or the shape of the ellipse. The particle density is found to oscillate from a pattern of distribution to another pattern back and forth against $B$. This is a new kind of Aharonov-Bohm oscillation originating from symmetry breaking and is different from the usual oscillation of persistent current. <|endoftext|><|startoftext|> Effect of electron-electron interaction on the phonon-mediated spin relaxation in quantum dots Juan I. Climente,1, ∗ Andrea Bertoni,1 Guido Goldoni,1, 2 Massimo Rontani,1 and Elisa Molinari1, 2 1CNR-INFM National Center on nanoStructures and bioSystems at Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy 2Dipartimento di Fisica, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy (Dated: October 21, 2018) We estimate the spin relaxation rate due to spin-orbit coupling and acoustic phonon scattering in weakly-confined quantum dots with up to five interacting electrons. The Full Configuration Interaction approach is used to account for the inter-electron repulsion, and Rashba and Dresselhaus spin-orbit couplings are exactly diagonalized. We show that electron-electron interaction strongly affects spin-orbit admixture in the sample. Consequently, relaxation rates strongly depend on the number of carriers confined in the dot. We identify the mechanisms which may lead to improved spin stability in few electron (> 2) quantum dots as compared to the usual one and two electron devices. Finally, we discuss recent experiments on triplet-singlet transitions in GaAs dots subject to external magnetic fields. Our simulations are in good agreement with the experimental findings, and support the interpretation of the observed spin relaxation as being due to spin-orbit coupling assisted by acoustic phonon emission. PACS numbers: 73.21.La,71.70.Ej,72.10.Di,73.22.Lp I. INTRODUCTION There is currently interest in manipulating electron spins in quantum dots (QDs) for quantum information and quantum computing purposes.1,2,3 A major goal in this research line is to optimize the spin relaxation time (T1), which sets the upper limit of the spin coherence time (T2): T2 ≤ 2T1.4 Therefore, designing two-level spin systems with long spin relaxation times is an im- portant step towards the realization of coherent quantum operations and read-out measuraments. Up to date, spin relaxation has been investigated almost exclusively in single-electron4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 and two- electron20,21,22,23,24,25,26,27,28,29,30 QDs. Spin relaxation in QDs with a larger number of electrons has seldom been considered28,31, even though Coulomb blockade makes it possible to control the exact number of carriers con- fined in a QD.32 Yet, recent theoretical works suggest that Coulomb interaction renders few-electron charge de- grees of freedom more stable than single-electron ones33, which leads to the question of whether similar findings hold for spin degrees of freedom. Moreover, in weakly- confined QDs, acoustic phonon emission assisted by spin- orbit (SO) interaction has been identified as the domi- nant spin relaxation mechanism when cotunneling and nuclei-mediated relaxation are reduced.6,8,31 The com- bined effect of Coulomb interaction and SO coupling has been shown to influence the energy spectrum of few- electron QDs profoundly,34,35,36 but the consequences on the spin relaxation remain largely unexplored.37 In Ref. 28 we investigated the effect of a magnetic field on the triplet-singlet (TS) spin relaxation in two and four-electron QDs with SO coupling, so as to under- stand related experimental works. Motivated by the very different response observed for different number of con- fined particles, in this work we shall focus on the role of electron-electron interaction in spin relaxation processes, extending our analysis to different number of carriers, highlighting, in particular, the different physics involved in even and odd number of confined electrons. Further- more, we will explicitly compare the predictions of our theoretical model with very recent experiments on spin relaxation in two-electron GaAs QDs.29 We study theoretically the energy structure and spin relaxation of N interacting electrons (N = 1 − 5) in parabolic GaAs QDs with SO coupling, subject to axial magnetic fields. Both Rashba38 and Dresselhaus39 SO terms are considered, and the electron-electron repulsion is accounted for via the Full Configuration Interaction method.40,41 By focusing on the two lowest spin states, two different classes of systems are distinguished. For N odd (1,3,5) and weak magnetic fields, the ground state is a doublet and then the two-level system is defined by the Zeeman-split sublevels of the lowest orbital. For N even (2,4), the two-level system is defined by a singlet and a triplet. We analyze these two classes of systems sepa- rately because, as we shall comment below, the physics involved in the spin transition differs. Thus, we compare the phonon-induced spin relaxation of N = 1, 3, 5 elec- trons and that of N = 2, 4 separately. As a general rule, the larger the number of confined carriers, the stronger the SO mixing, owing to the increasing density of elec- tronic states. This would normally yield faster relax- ation rates. However, we note that this is not necessarily the case, and few-electron states may display compara- ble or even slower relaxation than their single-electron and two-electron counterparts. This is due to charac- teristic features of the few-particle energy spectra which tend to weaken the admixture between the initial and final spin states. In N -odd systems, it is the presence of low-energy quadruplets for N > 1 that reduces the admixture between the Zeeman sublevels of the (dou- blet) ground state, hence inhibiting the spin flipping. In N -even systems, electronic correlations partially quench http://arxiv.org/abs/0704.0868v2 phonon emission33, and the relaxation can be further sup- pressed forN > 2 if one selects initial and final spin states differing in more than one quantum of angular momen- tum, which inhibits direct triplet-singlet SO mixing via linear Rashba and Dresselhaus SO terms.28 Noteworthy, all these effects are connected with Coulomb interaction between confined carriers. The paper is organized as follows. In Section II we give details about the theoretical model we use. In Section III we study the energy structure and spin relaxation of a QD with an odd number of electrons (N = 1, 3, 5). In Section IV we do the same for QDs with an even number of electrons (N = 2, 4). In Section V we compare our numerical simulations with experimental data recently reported for N = 2 GaAs QDs. Finally, in Section VI we present the conclusions of this work. II. THEORY We consider weakly-confined GaAs/AlGaAs QDs, which are the kind of samples usually fabricated by different groups to investigate spin relaxation processes.7,8,20,22 In these structures, the dot and the sur- rounding barrier have similar elastic properties, and the lateral confinement (which we approximate as circular) is much weaker than the vertical one. A number of use- ful approximations can be made for such QDs. First, since the weak lateral confinement gives inter-level spac- ings within the range of few meV, only acoustic phonons have significant interaction with bound carriers, while op- tical phonons can be safely neglected. Second, the elasti- cally homogeneous materials are not expected to induce phonon confinement, which allows us to consider three- dimensional bulk phonons. Finally, the different energy scales of vertical and lateral electronic confinement allow us to decouple vertical and lateral motion in the building of single-electron spin-orbitals. Thus, we take a parabolic confinement profile in the in-plane (x, y) direction, with single-particle energy gaps h̄ω0, which yields the Fock- Darwin states.42 In the vertical direction (z) the confine- ment is provided by a rectangular quantum well of width Lz and height determined by the band-offset between the QD and barrier materials (the zero of energy is then the bottom of the conduction band). The quantum well eigenstates are derived numerically. In cylindrical coor- dinates, the single-electron spin-orbitals can be written ψµ(ρ, θ, z; sz) = eimθ Rn,m(ρ) ξ0(z)χsz , (1) where ξ0 is the lowest eigenstate of the quantum well, χsz is the spinor eigenvector of the spin z-component with eigenvalue sz, and Rn,m is the n−th Fock-Darwin orbital with azimuthal angular momentum m, Rn,m(ρ) = (n+ |m|)! 0 L|m|n In the above expression L|m|n denotes a generalized La- guerre polynomial and l0 = h̄/m∗ω0 is the effective length scale, with m∗ standing for the electron effec- tive mass. The energy of the single-particle Fock-Darwin states is given by En,m = (2n + 1 + |m|)h̄Ωc + m2 h̄ωc, where ωc = is the cyclotron frequency and Ωc = + (ωc/2)2 is the total (spatial plus magnetic) con- finement frequency. With regard to Coulomb interaction, we need to go beyond mean field approximations in order to properly include electronic correlations, which play an important role in determining the phonon-induced electron scatter- ing rate.43 Moreover, since we are interested in the re- laxation time of excited states, we need to know both ground and excited states with comparable accuracy. Our method of choice is the Full Configuration Interac- tion approach: the few-electron wave functions are writ- ten as linear combinations |Ψa〉 = cai|Φi〉, where the Slater determinants |Φi〉 = Πµic†µi |0〉 are obtained by filling in the single-electron spin-orbitals µ with the N electrons in all possible ways consistent with symmetry requirements; here c†µ creates an electron in the level µ. The fully interacting Hamiltonian is numerically diago- nalized, exploiting orbital and spin symmetries.40,41 The few-electron states can then be labeled by the total az- imuthal angular momentumM = 0,±1,±2 . . ., total spin S and its z-projection Sz. The inclusion of SO terms is done following a similar scheme to that of Ref. 44, although here we consider not only Rashba but also linear Dresselhaus terms. For a quantum well grown along the [001] direction, these terms read:38,39 HR = α (kysx − kxsy), (3) HD = γc 〈k2z〉(kysy − kxsx), (4) where α and γc are coupling constants, while sj and kj are the j-th Cartesian projections of the electron spin and canonical momentum, respectively, along the main crys- talographic axes (〈k2z〉 = (π/Lz)2 for the lowest eigen- state of the quantum well). The momentum operator includes a magnetic field B applied along the vertical di- rection z. Other SO terms may also be present in the conduction band of a QD, such as the contribution aris- ing from the system inversion asymmetry in the lateral dimension or the cubic Dresselhaus term. However, in GaAs QDs with strong vertical confinement, HR and HD account for most of the SO interaction.36 We rewrite Eqs.(3,4) in terms of ladder operators as: HR = α (k+s− − k−s+), (5) HD = β (k+s+ + k−s−), (6) where k± and s± change m and sz by one quantum, respectively, and β = γc (π/Lz) 2 is the Dresselhaus in- plane coupling constant. It is worth mentioning that when only Rashba (Dresselhaus) coupling is present, the total angular momentum j = m + sz (j = m − sz) is conserved. However, in the general case, when both coupling terms are present and α 6= β, all symmetries are broken. Still, SO interaction in a large-gap semi- conductor such as GaAs is rather weak, and the low- lying states can be safely labelled by their approximate quantum numbers (M,S, Sz) except in the vicinity of the level anticrossings.11,26,45 Since the few-electron M and Sz quantum numbers are given by the algebraic sum of the single-particle states m and sz quantum numbers, it is clear from Eqs. (5,6) that Rashba interaction mixes (M,Sz) states with (M ± 1, Sz ∓ 1) ones, while Dressel- haus interaction mixes (M,Sz) with (M ± 1, Sz ± 1). The SO terms of Eqs. (5,6) can be spanned on a basis of correlated few-electron states.46 The SO matrix elements are then given by sums of single-particle contributions of the form: 〈n′m′ s′z| HR +HD |nmsz〉 = C∗R O+n′m′ nm δm′ m+1 δs′z sz−1+CR O n′m′ nm δm′ m−1 δs′z sz+1+ C∗D O+n′m′ nm δm′ m+1 δs′z sz+1+CD O n′m′ nm δm′ m−1 δs′ sz−1. Here CR = α and CD = −iβ are constans for the Rashba and Dresselhaus interactions respectively, andO± are the form factors: Rnm(t), Rnm(t), with t = ρ2/l20. The above forms factors have analytical expressions which depend on the set of quantum num- bers {n′m′, nm}. The resulting SO-coupled eigenvec- tors are then linear combinations of the correlated states, |ΨSOA 〉 = cAa|Ψa〉. We assume zero temperature, which suffices to capture the main features of one-phonon processes.9,16 Indeed, it is one-phonon processes that account for most of the low- temperature experimental observations in the SO cou- pling regime.2,6,8,28,29,31 We evaluate the relaxation rate between the initial (occupied) and final (empty) states of the SO-coupled few-electron state, B and A, using the Fermi Golden Rule: τ−1B→A = c∗BbcAa c∗bicaj〈Φi|Vνq|Φj〉 δ(EB−EA−h̄ωq), where the electron states |ΨSOK 〉 (K = A,B) have been written explicitly as linear combinations of Slater deter- minants, EK stands for the K electron state energy and h̄ωq represents the phonon energy. Vνq is the interac- tion operator of an electron with an acoustic phonon of momentum q via the mechanism ν, which can be either deformation potential or piezoelectric field interaction. Details about the electron-phonon interaction matrix el- ements can be found elsewhere.33 In this work we study a GaAs/Al0.3Ga0.7As QDs, us- ing the following material parameters:47 electron effective massm∗ = 0.067, band-offset Vc = 243 meV, crystal den- sity d = 5310 kg/m3, acoustic deformation potential con- stant D = 8.6 eV, effective dielectric constant ǫ = 12.9, and piezoelectric constant h14 = 1.41 · 109 V/m. The Landé factor is g = −0.44.5 As for GaAs sound speed, we take cLA = 4.72 · 103 m/s for longitudinal phonon modes and cTA = 3.34 · 103 m/s for transversal modes.48 Unless otherwise stated, a lateral confinement of h̄ω0 = 4 meV and a quantum well width of Lz = 10 nm are assumed for the QD under study, and a Dressehlaus coupling pa- rameter γc = 25.5 eV·Å3 is taken49, so that β ≈ 25 meV·Å. The value of the Rashba coupling constant can be modulated externally e.g. with external electric fields. Here we will investigate systems both with and without Rashba interaction. When present, we shall mostly con- sider α = 50 meV·Å, to represent the case where Rashba effects prevail over Dresselhaus ones. Few-body correlated states (M,S, Sz) are obtained us- ing a basis set composed by the Slater determinants (SDs) which result from all possible combinations of 42 single-electron spin-orbitals (i.e., from the six lowest en- ergy shells of the Fock-Darwin spectrum at B = 0) filled with N electrons. For N = 5, this means that the basis rank may reach ∼ 2 · 105. The SO Hamiltonian is then diagonalized in a basis of up to 56 few-electron states, which grants a spin relaxation convergence error below 2%. Since SO terms break the spin and angular mo- mentum symmetries, the SO-coupled states |ΨSOK 〉 are described by a linear combination of SDs coming from different (M,S, Sz) subspaces. Thus, for N = 5, the states are described by up to ∼ 8.5 · 105 SDs. To evalu- ate the electron-phonon interaction matrix elements, we note that only a small percentage of the huge number of possible pairs of SDs (∼ 7 · 1011 for N = 5) may give non-zero matrix elements, owing to spin-orbital or- thogonalities. We scan all pairs of SDs and filter those which may give non-zero matrix elements writing the de- terminants in binary representation and using efficient bit-per-bit algorithms.40,41 The matrix elements of the remaining pairs (∼ 2 ·106 for N = 5) are evaluated using massive parallel computation. 0 1 3 B (T) FIG. 1: Low-lying energy levels in a QD with N = 1, 3, 5 interacting electrons, as a function of an axial magnetic field. The SO interaction coefficients are α = 50 meV· Å and β = 25 meV· Å. The dot has h̄ω0 = 4 meV and Lz = 10 nm. Note the increasing size of the SO-induced anticrossing gaps and zero-field splittings with increasing N . III. SPIN RELAXATION IN A QD WITH N ODD A. Energy structure When the number of electrons confined in the QD is odd and the magnetic field is weak enough, the ground and first excited states are usually the Zeeman sz = 1/2 and sz = −1/2 sublevels of a doublet [Fig. 1]. Since the initial and final spin states belong to the same orbital, ∆M = 0 and SO mixing (which requires ∆M = ±1) is only possible with higher-lying states. In addition, the phonon energy (corresponding to the electron tran- sition energy) is typically small (in the µeV scale). In this case, the relaxation rate is determined essentially by the phonon density, the strength and nature of the SO interaction, and the proximity of higher-lying states.9,11 In order to gain some insight on the influence of these factors, in Fig. 1 we compare the energy structure of a QD with N = 1, 3, 5 vs. an axial magnetic field, in the presence of Rashba and Dresselhaus interactions.55 One can see that the increasing number of particles changes the energy magneto-spectrum drastically. This is be- cause the quantum numbers of the low-lying energy levels change, resulting in a different field dependence, and be- cause Coulomb interaction leads to an increased density of electron states, as well as to a more complicated spec- trum. At first sight, the energy spectra of Fig. 1 closely resem- ble those in the absence of SO effects. For instance, the N = 1 spectrum is very similar to the pure Fock-Darwin spectrum.42 Rashba and Dresselhaus interactions were expected to split the degenerate |m| > 0 shells at B = 0, shift the positions of the level crossings and turn them into anticrossings36,52,53,54, but here such signatures are hardly visible because SO interaction is weak in GaAs. In fact, the magnitude of the SO-induced zero-field en- ergy splittings and that of the anticrossing gaps is of very few µeV, and SO effects simply add fine features to the N = 1 spectrum.52 A significantly different picture arises in the N = 3 and N = 5 cases. Here, the increased density of elec- tronic states enhances SO mixing as compared to the single-electron case.56 As a result, the anticrossing gaps can be as large as 30 µeV (N = 3) and 60 µeV (N = 5). Moreover, unlike in the N = 1 case, where the ground state orbital has m = 0, here it has |M | = 1. Therefore, the Zeeman sublevels involved in the fundamental spin transition are subject to SO-induced zero-field splittings. To illustrate this point, in Fig. 2 we zoom in on the energy spectrum of the four lowest states of N = 3 and N = 5 under weak magnetic fields, without (left panels) and with (right panels) Rashba interaction. Clearly, the four- fold degeneracy of |M | = 1 spin-orbitals at B = 0 has been lifted by SO interaction.36 One can also see that the order of the two lowest sublevels at B ∼ 0 changes when Rashba interaction is switched on. Thus, for N = 3 and α = 0, the two lowest sublevels are (M = −1, Sz = 1/2) and (M = −1, Sz = −1/2), but this order is reversed when α = 50 meV·Å. The opposite level order as a func- tion of α is found for N = 5. This behavior constitutes a qualitative difference with respect to the N = 1 case in two aspects. First, the phonon energy (i.e., the energy of the fundamental spin transition) is no longer given by the bare Zeeman splitting. Instead, it has a more compli- cated dependence on the magnetic field, and it is greatly influenced by the particular values of α and β. This is apparent in the N = 5 panels, where the energy splitting between the two lowest states strongly differs depending on the relative value of α and β. Second, it is possible to find situations where the ground state at B ∼ 0 has Sz = −1/2 and the first excited state has Sz = 1/2 (e.g. N = 3 when α > β or N = 5 when α < β). In these cases, the Zeeman splitting leads to a weak anticrossing of the two sublevels (highlighted with dashed circles in Fig. 2) which has no counterpart in single-electron sys- tems. This kind of B-induced (i.e., not phonon-induced) ground state spin mixing, also referred to as “intrinsic spin mixing”, has been previously reported for singlet- triplet transitions in N = 2 QDs.58 Here we show that they may also exist in few-electron QDs with N odd. (−2,1/2)(1,1/2) (−1,1/2) (1,1/2) (1,1/2) (−4,1/2) (−2,1/2) (−4,1/2) (−1,1/2) (−1,1/2) (−1,1/2) (1,1/2) α = 50, β = 25α = 0, β = 25 N = 3 N = 5 127.0 127.5 128.0 128.5 0.5 1.0 0.0 236.0 236.5 237.0 B (T) 0.0 0.5 1.0 B (T) FIG. 2: The four lowest energy levels in a QD with N = 3, 5 interacting electrons, as a function of an axial magnetic field, without (left column) and with (right column) Rashba SO interaction. The approximate quantum numbers (M,S) of the levels are shown, with arrows denoting the spin projection Sz = 1/2 (↑) and Sz = −1/2 (↓). The dashed circles highlight the region of intrinsic spin mixing of the ground state. Figure 1 puts forward yet another qualitative differ- ence between SO coupling in single- and few-electron QDs: while in the former low-energy anticrossings are due to Rashba interaction11,36,52, in few-electron QDs, when S = 3/2 states come into play, both Rashba and Dresselhaus terms may induce anticrossings. For exam- ple, the (M = −1, Sz = 1/2) sublevel couples directly to both (M = −2, Sz = −1/2) and (M = −2, Sz = 3/2) sublevels, via the Dresselhaus and Rashba interaction, respectively. Coupling to S = 3/2 states is a characteris- tic feature of N > 1 systems, which has important effects on the spin relaxation rate, as we will discuss below. B. Spin relaxation between Zeeman sublevels In Fig. 3 we compare the magnetic field dependence of the spin relaxation rate between the two lowest Zeeman sublevels of N = 1, 3, 5. Dashed lines (solid lines) are used for systems without (with) Rashba interaction.59 While for N = 1 the well-known exponential dependence with B is found2,6,9, and the main effect of Rashba cou- pling is to shift the curve upwards (i.e., to accelerate the relaxation), for N = 3 and N = 5 the relaxation rate ex- hibits complicated trends which strongly depend on the values of the SO coupling parameters. α = 50, β = 25 α = 0, β = 25 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 B (T) 0 1 2 3 FIG. 3: Spin relaxation rate in a QD with N = 1, 3, 5 inter- acting electrons as a function of an axial magnetic field. Solid (dashed) lines stand for the system with (without) Rashba interaction. Note the strong influence of the SO interaction in the shape of the relaxation curve for N > 1. To understand this result, one has to bear in mind that in spin relaxation processes two well-distinguished and complementary ingredients are involved, namely SO in- teraction and phonon emission. Phonon emission grants the conservation of energy in the electron relaxation, but phonons have zero spin and therefore cannot cou- ple states with different spin. It is the SO interaction that turns pure spin states into mixed ones, thus enabling the phonon-induced transition. The overall efficiency of the scattering event is then given by the combination of the two phenomena: the phonon emission efficiency modulated by the extent of the SO mixing. The shape of spin relaxation curves shown in Fig. 3 can be directly related to the energy dispersion of the phonon, which cor- responds to the splitting between the two lowest levels of the electron spectrum. Thus, for N = 1, the phonon energy is simply proportional to B through the Zeeman splitting, but for N = 3 and N = 5 it has a non-trivial dependence on B, as shown in Fig. 2. Actually, the relax- ation minima in Fig. 3 are connected with the magnetic field values where the two lowest levels anticross in Fig. 2. In these magnetic field windows, in spite of the fact that SO coupling is strong, the phonon density is so small that the relaxation rate is greatly suppressed.28 Similarly, the relaxation rate fluctuations of N = 3 at B ∼ 3 T are signatures of the anticrossings with high-angular momen- tum states. For larger fields (B > 3 T), the ground state approaches the maximum density droplet configuration and high-spin states are possible.44 In this work, how- ever, we restrict ourselves to the magnetic field regime where the ground state is a doublet. eV∆ (µ ) 10 10 10 10 10 10 10 10 10 0 20 40 60 80 10 FIG. 4: (Color online). Spin relaxation rate in a QD with N = 1, 3, 5 interacting electrons as a function of the energy splitting between the two lowest spin states. Top panel: α = 0, β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25 meV·Å. The relaxation of N = 3 is slower than that of N = 1 for a wide range of ∆12. The irregular data distribution is due to the irregular relaxation rates vs. magnetic field. For example, the strongly deviated points of N = 3 come from the peaks at B ∼ 3 in Fig. 3. For a more direct comparison between the relaxation rates of N = 1, 3, 5, in Fig. 4 we replot the data of Fig. 3 as a function of the energy splitting between the two lowest states, ∆12, without (top panel) and with (bot- tom panel) Rashba interaction. Since the phonon energy is identical for all points with the same ∆12, differences in the relaxation rate arise exclusively from the different strength of SO interaction. ∆12 is also a relevant pa- rameter from the experimental point of view, since it is usually required that it be large enough for the states to be resolvable. In this sense, it is worth noting that, even if the inter-level splittings shown in Fig. 4 are fairly small, a number of experiments have successfully addressed this regime.5,8,21 A most striking feature observed in the figure is that, for most values of ∆12, the N = 3 relaxation rate is clearly slower than the N = 1 one. Likewise, N = 5 shows a similar (or slightly faster) relaxation rate than N = 1. These are interesting results, for they suggest that improved spin stability may be achieved using few- electron QDs instead of the single-electron ones typically employed up to date.8 At first sight the results are sur- prising, because the higher density of states in the few- electron systems implies smaller inter-level spacings, and hence stronger SO mixing, which should translate into enhanced relaxation. It then follows that another physi- cal mechanism must be acting upon the few-electron sys- tems, which reduces the transition probability between the initial and final spin states, and may even make it smaller than for N = 1. Here we propose that such mechanism is the SO admixture with low-lying quadru- plet (S = 3/2) states, which become available for N > 1. By coupling to S = 3/2 levels, the projection of the dou- blet Sz = 1/2 levels onto Sz = −1/2 ones is reduced, and this partly inhibitis the transition between the low- est doublet sublevels. Let us explain this by comparing the spin transition for N = 1 and N = 3. For N = 1, the spin configuration of the initial and final states, in the absence of SO coupling, is |Sz = −1/2〉 and |Sz = +1/2〉, respectively. The tran- sition between these states is spin-forbidden. However, when SO coupling is switched on, the two states become admixed with higher-lying S = 1/2 states fulfilling the ∆Sz = ±1 condition. The transition between the initial and final states can then be represented schematically as: ca |Sz = −1/2〉+cb|Sz = +1/2〉 ⇒ cr |Sz = +1/2〉+cs|Sz = −1/2〉, where ci are the admixture coefficients (in general ca ≫ cb and cr ≫ cs). Clearly now both spin configurations of the initial state have a finite overlap with the final state, and so the transition is possible. Let us next consider the N = 3 case. In the absence of SO coupling, the initial and final states are again the Sz = −1/2 and Sz = +1/2 doublets, respectively, and the transition is spin- forbidden. When we switch on SO coupling, we note that the ∆Sz = ±1 condition allows for mixing not only with Sz = ±1/2 states (either doublets or quadruplets) but also with Sz = ±3/2 quadruplets, so that the transition can be represented as: ca|Sz = −1/2〉+ cb|Sz = +1/2〉+ cc|Sz = −3/2〉 ⇒ cr|Sz = +1/2〉+ cs|Sz = −1/2〉+ ct|Sz = +3/2〉, where, in general, ca ≫ cb, cc, and cr ≫ cs, ct. In this case, |Sz = −3/2〉 has no overlap with the final state con- figurations. Likewise, |Sz = +3/2〉 has no overlap with the initial state configurations. Therefore, these quadru- plet configurations are inactive from the point of view of the transition, and the more important they are (i.e., the stronger the SO coupling with quadruplet states), the less likely the transition is. To prove this argument quantitatively, in Fig. 5 we il- lustrate the spin relaxation of N = 3 calculated by diag- onalization of the SO Hamiltonian including and exclud- ing the low-lying S = 3/2 states from the basis set. As expected, when the quadruplets are not considered, the transition is visibly faster. For N = 5, low-lying S = 3/2 levels are also available, but in this case they barely com- pensate for the large density of electron states, so that the overall scattering rate turns out to be comparable to that of N = 1. eV∆ (µ ) without S=3/2 with S=3/2 10 10 10 10 10 0 20 40 60 80 FIG. 5: (Color online). Spin relaxation rate in a QD with N = 3 interacting electrons as a function of the energy splitting between the two lowest spin states. α = 0 and β = 25 meV·Å. Symbol + (×) stands for SO Hamiltonian diagonalized in a basis which includes (excludes) S = 3/2 states. Clearly, the inclusion of S = 3/2 states slows down the relaxation. To test the robustness of the few-electron spin states stability predicted above, we also compare the relax- ation rate of N = 1 and N = 3 in a QD with dif- ferent confinement, namely h̄ω0 = 6 meV, in Fig. 6. Since the lateral confinement of the dot is now stronger, (M = −1, S = 1/2) is the N = 3 ground state up to large values of the magnetic field (B ∼ 5 T). This allows us to investigate larger Zeeman splittings (i.e., larger ∆12), which may be easier to resolve experimentally. As seen in the figure, the relaxation rate of N = 3 is again slower than that of N = 1 for a wide range of ∆12, the behavior being very similar to that of Fig. 4, albeit extended to- wards larger inter-level spacings. The crossing between N = 3 and N = 1 relaxation rates at large ∆12 val- ues, both in Fig. 4 and Fig. 6, is due to the proximity of high-angular momentum levels coming down in energy for N = 3 when the magnetic field (and hence the Zee- man splitting) is large. Such levels bring about strong SO admixture and thus fast relaxation (see middle panel of Fig. 3 at B ∼ 3 T). IV. SPIN RELAXATION IN A QD WITH N A. Energy structure When the number of electrons confined in the QD is even and the magnetic field is not very strong, the ground and first excited states are usually a singlet (S = 0) and a triplet (S = 1) with three Zeeman sublevels eV∆ (µ ) 10 10 10 10 10 10 10 10 10 10 0 40 80 120 FIG. 6: (Color online). Spin relaxation rate in a QD with N = 1, 3 interacting electrons as a function of the energy splitting between the two lowest spin states. The QD has h̄ω0 = 6 meV. Top panel: α = 0, β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25 meV·Å. As for the weaker- confined dot of Fig. 4, the relaxation of N = 3 is slower than that of N = 1 for a wide range of ∆12. (Sz = +1, 0,−1). Unlike in the previous section, here the initial and final states of the spin transition may have dif- ferent orbital quantum numbers, and the inter-level split- ting ∆12 may be significantly larger (in the meV scale). Under these conditions, the phonon emission efficiency no longer exhibits a simple proportionality with the phonon density, but it further depends on the ratio between the phonon wavelength and the QD dimensions.50,51 More- over, SO interaction is sensitive to the quantum numbers of the initial and final electron states.26,28 Therefore, in this class of spin transitions the details of the energy structure are also relevant to determine the relaxation rate. In Fig. 7 we plot the energy levels vs. magnetic field for a QD with N = 2, 4 in the presence of Rashba and Dres- selhaus interactions. The approximate quantum num- bers (M,S) of the lowest-lying states are written between parenthesis. For N = 2 and weak fields, the ground state is the (M = 0, S = 0) singlet, and the first excited state is the (M = −1, S = 1) triplet. As in the previous sec- tion, SO interaction introduces small zero-field splittings and anticrossings in the energy levels with |M | > 0.36 As a consequence, when α > β, the zero-field ordering of the (M = −1, S = 1) Zeeman sublevels is such that they anticross in the presence of an external magnetic field. This anticrossing is highlighted in the figure by a dashed circle. On the other hand, as B increases the singlet- triplet energy spacing is gradually reduced, and then the singlet experiences a series of weak anticrossings with all (−1,1) (−2,0) (−3,1) (0,1) (0,0) 0 1 2 3 B (T) FIG. 7: Low-lying energy levels in a QD with N = 2, 4 in- teracting electrons as a function of an axial magnetic field. α = 50 meV· Å and β = 25 meV· Å. The approximate quantum numbers (M,S) of the lowest states are shown. The dashed circle in N = 2 highlights the anticrossing between M = −1 Zeeman sublevels. three Zeeman sublevels of the triplet. These anticross- ings are due to the fact that (M = 0, S = 0, Sz = 0) couples to the (M = −1, S = 1, Sz = −1) sublevel via Dresselhaus interaction, to the (M = −1, S = 1, Sz = +1) sublevel via Rashba interaction, and finally to the (M = −1, S = 1, Sz = 0) sublevel indirectly through higher-lying states.26,28 For N = 4, the density of electronic states is larger than for N = 2, which again reflects in a larger magni- tude of the anticrossings gaps due to the enhanced SO interaction. The ground state at B = 0 is a triplet, (M = 0, S = 1), but soon after it anticrosses with a singlet, (M = −2, S = 0). After this, and before the formation of Landau levels, two different branches of the first excited state can be distinguished: when B < 1 T, the first excited state is (M = 0, S = 1), and when B > 1 T it is (M = −3, S = 1). It is worth pointing out that the complexity of the N = 4 spectrum, as compared to the simple N = 2 one, implies a greater flexibility to select initial and final spin states by means of external fields. As we shall discuss below, this degree of freedom has important consequences on the relaxation rate. B. Triplet-singlet spin relaxation In a recent work, we have investigated the magnetic field dependence of the TS relaxation due to SO cou- pling and phonon emission in N = 2 and N = 4 QDs.28. Here we study this kind of transition from a different perspective, namely we compare the spin relaxation of two- and four-electron systems in order to highlight the changes introduced by inter-electron repulsion. Increas- ing the number of electrons confined in the QD has three important consequences on the TS transition. First, it increases the density of electronic states (and then the SO mixing), leading to faster relaxation. Second, as mentioned in the previous section, it introduces a wider choice of orbital quantum numbers for the singlet and triplet states. Third, it increases the strength of elec- tronic correlations. Since now the initial and final spin states have different orbital wave functions, the latter factor effectively reduces phonon scattering, in a similar fashion to charge relaxation processes33 (this effect has been recently pointed out in Ref. 30 as well). To find out the overall combined effect of these three factors, in this section we analyze quantitative simulations of correlated We focus on the magnetic field regions where the ground state is a singlet and the excited state is a triplet. A complete description of the TS transition should then include spin relaxation between the Zeeman-split sub- levels of the triplet. However, for the weak fields we con- sider this relaxation is orders of magnitude slower than the TS one (compare Figs. 3 and 8),60 the reason for this being the small Zeeman energy and the fact that the Zeeman sublevels are not directly coupled by Rashba and Dresselhaus terms, as mentioned in Section III. There- fore, it is a good approximation to assume that all three triplet Zeeman sublevels are equally populated and they relax directly to the singlet.26 α = 50, β = 25 α = 0, β = 25 10 10 10 10 10 10 0.5 1 1.5 2 2.5 3 B (T) FIG. 8: Spin relaxation rate in a QD with N = 2, 4 interact- ing electrons as a function of an axial magnetic field. Solid (dashed) lines stand for the system with (without) Rashba interaction. The relaxation of N = 4 when B < 1 T is slower than that of N = 2. Figure 8 represents the TS relaxation rate in a QD with N = 2, 4, after averaging the relaxation from the three triplet sublevels. Solid (dashed) lines stand for the case with (without) Rashba interaction.59 The main effect of Rashba and Dresselhaus interactions is to accelerate the spin transition by shifting the relaxation curve upwards. This is in contrast to the N -odd case, where these terms may induce drastic changes in the shape of the relaxation rate curve (see Fig. 3). Figure 8 also reveals a different behavior of the N = 2 and N = 4 TS relaxation rates. The former increases gradually with B and then drops in the vicinity of the TS anticrossing, due to the small phonon energies.28,29,30 Conversely, for N = 4 an addi- tional feature is found, namely an abrupt step at B ∼ 1. This is due to the change of angular momentum of the excited triplet. For B < 1 T the triplet has M = 0, and for B > 1 T it has M = −3. Since the ground state is a singlet with M = −2, the M = 0 triplet does not fulfill the ∆M = ±1 condition for linear SO coupling. This inhibits direct spin mixing between initial and final states and reduces the relaxation rate by about one order of magnitude.28 meV∆ ( ) N=4, M=0 N=4, M=−3 10 10 10 10 10 10 10 0 0.4 0.8 1.2 FIG. 9: (Color online). Spin relaxation rate in a QD with N = 2, 4 interacting electrons as a function of the energy spacing between the singlet and the triplet. Here M stands for the angular momentum of the triplet. Top panel: α = 0, β = 25 meV·Å. Bottom panel: α = 50 meV·Å, β = 25 meV·Å. The relaxation of N = 4 is comparable to that of N = 2 when the triplet has M = −3, and it is much smaller when M = 0. Noteworthy, the choice of states differing in more than one quantum of angular momentum is only possible for N > 2 QDs. One may then wonder if it is more conve- nient to use these systems instead of the N = 2 ones dom- inating the experimental literature up to date20,21,29, i.e. if it compensates for the increased density of electronic states. Interestingly, Fig. 8 predicts slower relaxation for the N = 4 QD with M = 0 triplet than for N = 2. To verify that this arises from weakend SO coupling rather than from different phonon energy values, in Fig. 9 we replot the spin relaxation rate of N = 2, 4 as a function of the TS energy splitting. In the figure, the upper and bottom panels represent the situations without and with Rashba interaction, respectively. While N = 4 shows similar relaxation rate to N = 2 when the triplet has M = −3, the relaxation is slower by about one order of magnitude when the triplet has M = 0. This result indicates that the weakening of SO mixing due to the violation of the ∆M = ±1 condition clearly exceeds the strengthening due to the higher density of states, con- firming that N = 4 systems are more attractive than N = 2 ones to obtain long triplet lifetimes. We also point out that, in spite of the different density of states, the relaxation rate of N = 2 and N = 4, M = −3 triplets is quite similar. This can be ascribed to the phonon scat- tering reduction by electronic correlations,33 which may also explain the fact that experimentally resolved TS re- laxation rates of N = 8 QDs and N = 2 QDs be quite similar.20,31 V. COMPARISON WITH N = 2 EXPERIMENTS Whereas, to our knowledge, no experiments have mea- sured transitions between Zeeman-split sublevels in N > 1 systems yet, a number of works have dealt with TS re- laxation in QDs with few interacting electrons. In Ref. 28 we showed that our model correctly predicts the trends observed in experiments with N = 2 and N = 8 QDs subject to axial magnetic fields.20,21,31 In this section, we extend the comparison to new experiments available for N = 2 TS relaxation in QDs,29 which for the first time provide continuous measurements of the average triplet lifetime against axial magnetic fields, from B = 0 to the vicinity of the TS anticrossing. By using a simple model, the authors of the experimental work showed that the measuraments are in clear agreement with the behavior expected from SO coupling plus acoustic phonon scatter- ing. However, in such model: (i) the TS energy splitting was a taken directly from the experimental data, (ii) the SO coupling effect was accounted for by parametrizing the admixture of the lowest singlet and triplet states only, and (iii) the B-dependence of the SO-induced admix- ture was neglected. Approximation (ii) may overlook the correlation-induced reduction of phonon scattering,30,33 that we have shown above to be significant, and which may have an important contribution from higher excited states in weakly-confined QDs. In turn, approximation (iii) may overlook the important influence of SO coupling in the B-dependence of the triplet lifetime, as we had anticipated in Ref. 28. Here we compare with the exper- imental findings using our model, which includes these effects properly. We assume a QD with an effective well width Lz = 30 nm, as expected by Ref. 29 authors, and a lateral confinement parabola of h̄ω0 = 2 meV which, as we shall see next, fits well the position of the TS an- ticrossing. Yet, the comparison is limited by the lack of detailed information about the Rashba and Dresselhaus interaction constants, and because we deal with circular QDs instead of elliptical ones (the latter effect introduces simple deviations from the circular case26). In addition, in the experiment a tilted magnetic field of magnitude B∗, forming an angle of 68◦ with the vertical direction was used. Here we consider the vertical component of the field (B = 0.37B∗), which is the main responsible for the changes in the energy structure, and the effect of the in-plane component enters via the Zeeman splitting only. Figure 10 illustrates the average triplet lifetime for N = 2. The bottom axis shows the vertical magnetic field B value, while the top axis shows the value to be compared with the experiment B∗.59 As can be seen, the triplet lifetime first decreases with the field and then it abruptly increases in the vicinity of the TS anticross- ing, due to the small phonon density.28 This behavior is in clear agreement with the experiment (cf. Fig. 3 of Ref. 29). The position of the anticrossing (B∗ ∼ 2.9 T) is also close to the experimental value (B∗ ∼ 2.8 T), which confirms that that h̄ω0 = 2 meV is similar to the mean confinement frequency of the experimental sample. A departure from the experimental trend appears at weak fields (B < 0.5 T), where we observe a continuous in- crease of T1 with decreasing B, while the experiment re- ports a plateau. This is most likely due to the ellipticity of the experimental sample, which renders the electron states (and consequently the relaxation rate) insensitive to the field in the B∗ = 0 − 0.5 T region (see Fig. 1a in 29). In any case, Fig. 10 clearly confirms the role of phonon-induced relaxation in the experiments, using a realistic model for the description of correlated electron states, SO admixture and phonon scattering. A comment is worth here on the magnitude of the SO coupling terms. In Fig. 10, we obtain good agreement with the experimental relaxation times by using small values of the SO coupling parameters. In particular, a close fit is obtained using β = 1, α = 0.5 meV·Å, which yields a spin-orbit length λSO = 48 µm. This value, which coincides with the experimental guess (λSO ≈ 50 µm), indicates that SO coupling is several times weaker than that reported for other GaAs QDs.8 Typical GaAs parameters are often larger. For instance, measuraments of the Rashba and Dresselhaus constants by analysis of the weak antilocalization in clean GaAs/AlGaAs two- dimensional gases revealed α = 4−5 meV·Å, and γc = 28 eV·Å3 (i.e, β = 3 meV·Å for our quantum well of Lz = 30 nm).61 To be sure, the small SO coupling parameters in the experiment have a major influence on the lifetime scale. Compare e.g. the β = 1 and β = 5 meV·Å curves in Fig. 10. Actually, we note that accurate comparison with the timescale reported for other GaAs samples31 is also possible within our model, but assuming stronger SO coupling constants.28 In Ref. 29, it was suspected that the weak SO coupling inferred from the experimen- tal data could be the result of the exclusion of higher orbitals and the magnetic field dependence of SO ad- α = 0.5 β = 1 β = 5 β = 2 β = 1 α = 0 B (T) 0.2 0.4 0.6 0.8 1 0 0.53 1.58 2.11 2.631.05 B (T) FIG. 10: Average triplet lifetime in a QD with N = 2 elec- trons as a function of an axial magnetic field. Only the field region before the TS anticrossing is shown. α and β are in meV·Å units. B is the applied axial magnetic field, and B∗ is the equivalent tilted magnetic field, for comparison with Ref. 29 experiment. mixture in their model (higher states reduce the effective SO coupling constants by decreasing the phonon-induced scattering30,33). Here we have considered both these ef- fects and still small SO coupling constants are needed to reproduce the experiment. Therefore, understanding the origin of their small value remains as an open question. One possibility could be that the particular direction of the tilted magnetic field used in the experiment corre- sponded to a reduced degree of SO admixture.30 VI. CONCLUSIONS We have investigated theoretically the energy structure and spin relaxation rate of weakly-confined QDs with N = 1 − 5 interacting electrons, subject to axial mag- netic fields, in the presence of linear Rashba and Dressel- haus SO interactions. It has been shown that the num- ber of electrons confined in the dot introduces changes in the energy spectrum which significantly influence the intensity of the SO admixture, and hence the spin re- laxation. In general, the larger the number of confined carriers, the higher the density of electronic states. This decreases the energy splitting between consecutive lev- els and then enhances SO admixture, which should lead to faster spin relaxation. However, we find that this is not necessarily the case, and slower relaxation rate may be found for few-electron QDs as compared to the usual single and two-electron QDs used up to date. The physi- cal mechanisms responsible for this have been identified. For N -odd systems, when the spin transition takes place between Zeeman-split sublevels, it is the presence of low- energy S = 3/2 states for N > 1 that reduces the pro- jection of the doublet Sz = 1/2 sublevels into Sz = −1/2 ones, thus partly inhibiting the spin transition. For N - even systems, when the spin transition takes place be- tween triplet and singlet levels, there are two underlying mechanisms. On the one hand, electronic correlations tend to reduce phonon emission efficiency. 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Fabian, Phys. Rev. B 72, 155410 (2005). 53 O. Voskoboynikov, C.P. Lee, and O. Tretyak, Phys. Rev. B 63, 165306 (2001). 54 W.H. Kuan, and C.S. Tang, J. Appl. Phys. 95, 6368 (2004). 55 The energy magneto-spectrum of GaAs parabolic QDs with SO interaction and up to four interacting electrons was also investigated in Ref. 35, but considering Rashba interaction only. 56 Coulomb-enhanced SO interaction was previously pre- dicted for higher-dimensional structures.57 Here we report it for QDs. 57 G.H. Chen, and M.E. Raikh, Phys. Rev. B 60, 4826 (1999). 58 C.F. Destefani, S.E. Ulloa, and G.E. Marques, Phys. Rev. B 69, 125302 (2004). 59 For simplicity of the discussion, in Figs. 3, 8 and 10, the near vicinity of B = 0 T is not shown. In that range one finds damped phonon-induced relaxation rates due to de- generacies arising from the time-reversal symmetry and the circular symmetry of the confinement we have assumed. We do not expect these features to be observable in exper- iments, because QDs are not perfectly circular and because hyperfine interaction is expected to be the dominant spin relaxation mechanism for very weak fields (see Refs. 18,23). 60 Greatly suppressed TS spin relaxation, comparable to that of inter-Zeeman sublevels at very weak B, may be achieved by means of geometrically or field-induced acoustic phonon emission minima.27,28 61 J.B. Miller, D.M. Zumbühl, C.M. Marcus, Y.B. Lyanda- Geller, D. Goldhaber-Gordon, K. Campman, and A.C. Gossard, Phys. Rev. Lett. 90, 076807 (2003). http://www.s3.infm.it/donrodrigo ABSTRACT We estimate the spin relaxation rate due to spin-orbit coupling and acoustic phonon scattering in weakly-confined quantum dots with up to five interacting electrons. The Full Configuration Interaction approach is used to account for the inter-electron repulsion, and Rashba and Dresselhaus spin-orbit couplings are exactly diagonalized. We show that electron-electron interaction strongly affects spin-orbit admixture in the sample. Consequently, relaxation rates strongly depend on the number of carriers confined in the dot. We identify the mechanisms which may lead to improved spin stability in few electron (>2) quantum dots as compared to the usual one and two electron devices. Finally, we discuss recent experiments on triplet-singlet transitions in GaAs dots subject to external magnetic fields. Our simulations are in good agreement with the experimental findings, and support the interpretation of the observed spin relaxation as being due to spin-orbit coupling assisted by acoustic phonon emission. <|endoftext|><|startoftext|> Introduction The Asymmetric Simple Exclusion Process (ASEP) is a lattice model of parti- cles with hard core interactions. Due to its simplicity, the ASEP appears as a minimal model in many different contexts such as one-dimensional transport phenomena, molecular motors and traffic models. From a theoretical point of view, this model has become a paradigm in the field of non-equilibrium statistical mechanics; many exact results have been derived using various methods, such as continuous limits, Bethe Ansatz and matrix Ansatz (for re- views, see e.g., Spohn 1991, Derrida 1998, Schütz 2001, Golinelli and Mallick 2006). In a recent work (Golinelli and Mallick 2007), we applied the algebraic Bethe Ansatz technique to the Totally Asymmetric Exclusion Process (TASEP). http://arxiv.org/abs/0704.0869v1 Golinelli, Mallick — Connected Operators for TASEP 2 This method allowed us to construct a hierarchy of ‘generalized Hamiltonians’ that contain the Markov matrix and commute with each other. Using the algebraic relations satisfied by the local jump operators, we derived explicit formulae for the transfer matrix and the generalized Hamiltonians, generated from the transfer matrix. We showed that the transfer matrix can be inter- preted as the generator of a discrete time Markov process and we described the actions of the generalized Hamiltonians. These actions are non-local be- cause they involve non-connected bonds of the lattice. However, connected operators are generated by taking the logarithm of the transfer matrix. We conjectured for the connected operators a combinatorial formula that was verified for the first ten connected operators by using a symbolic calculation program. The aim of the present work is to present an analytical calculation of the connected operators and to prove the formula that was proposed in (Golinelli and Mallick 2007). This paper is a sequel of our previous work, however, in section 2, we briefly review the main definitions and results already obtained so that this work can be read in a fairly self-contained manner. In section 3, we derive the general expression of the connected operators. 2 Review of known results We first recall the dynamical rules that define the TASEP with n particles on a periodic 1-d ring with L sites labelled i = 1, . . . , L. The particles move according to the following dynamics: during the time interval [t, t + dt], a particle on a site i jumps with probability dt to the neighboring site i+ 1, if this site is empty. This exclusion rule which forbids to have more than one particle per site, mimics a hard-core interaction between particles. Because the particles can jump only in one direction this process is called totally asymmetric. The total number n of particles is conserved. The TASEP being a continuous-time Markov process, its dynamics is entirely encoded in a 2L × 2L Markov matrix M , that describes the evolution of the probability distribution of the system at time t. The Markov matrix can be written as Mi , (1) where the local jump operator Mi affects only the sites i and i + 1 and represents the contribution to the dynamics of jumps from the site i to i+1. Golinelli, Mallick — Connected Operators for TASEP 3 2.1 The TASEP algebra The local jump operators satisfy a set of algebraic equations : M2i = −Mi, (2) Mi Mi+1 Mi = Mi+1 Mi Mi+1 = 0, (3) [Mi,Mj ] = 0 if |i− j| > 1. (4) These relations can be obtained as a limiting form of the Temperley-Lieb algebra. On the ring we have periodic boundary conditions : Mi+L = Mi. The local jumps matrices define an algebra. Any product of the Mi’s will be called a word. The length of a given word is the minimal number of operators Mi required to write it. A word, that can not be simplified further by using the algebraic rules above, will be called a reduced word. Consider any word W and call I(W ) the set of indices i of the operators Mi that compose it (indices are enumerated without repetitions). We remark that, if W is not annihilated by application of rule (3), the simplification rules (2, 4) do not alter the set I(W ), i.e., these rules do not introduce any new index or suppress any existing index in I(W ). This crucial property is not valid for the algebra associated with the partially asymmetric exclusion process (see Golinelli and Mallick 2006). Using the relation (2) we observe that for any i and any real number λ 6= 1 we have (1 + λMi) −1 = (1 + αMi) with α = . (5) 2.2 Simple words A simple word of length k is defined as a word Mσ(1)Mσ(2) . . .Mσ(k), where σ is a permutation on the set {1, 2, . . . , k}. The commutation rule (4) implies that only the relative position of Mi with respect to Mi±1 matters. A simple word of length k can therefore be written as Wk(s2, s3, . . . , sk) where the boolean variable sj for 2 ≤ j ≤ k is defined as follows : sj = 0 if Mj is on the left of Mj−1 and sj = 1 if Mj is on the right of Mj−1. Equivalently, Wk(s2, s3, . . . , sk) is uniquely defined by the recursion relation Wk(s2, s3, . . . , sk−1, 1) = Wk−1(s2, s3, . . . , sk−1) Mk , (6) Wk(s2, s3, . . . , sk−1, 0) = Mk Wk−1(s2, s3, . . . , sk−1) . (7) The set of the 2k−1simple words of length k will be called Wk. For a simple word Wk, we define u(Wk) to be the number of inversions in Wk, i.e., the Golinelli, Mallick — Connected Operators for TASEP 4 number of times that Mj is on the left of Mj−1 : u(Wk(s2, s3, . . . , sk)) = (1− sj) . (8) We remark that simple words are connected, they cannot be factorized in two (or more) commuting words. 2.3 Ring-ordered product Because of the periodic boundary conditions, products of local jump opera- tors must be ordered adequately. In the following we shall need to use a ring ordered product O () which acts on words of the type W = Mi1Mi2 . . .Mik with 1 ≤ i1 < i2 < . . . < ik ≤ L , (9) by changing the positions of matrices that appear in W according to the following rules : (i) If i1 > 1 or ik < L, we define O (W ) = W . The word W is well- ordered. (ii) If i1 = 1 and ik = L, we first write W as a product of two blocks, W = AB, such that B = MbMb+1 . . .ML is the maximal block of matrices with consecutive indices that contains ML, and A = M1Mi2 . . .Mia , with ia < b− 1, contains the remaining terms. We then define O (W ) = O (AB) = BA = MbMb+1 . . .MLM1Mi2 . . .Mia . (10) (iii) The previous definition makes sense only for k < L. Indeed, when k = L, we have W = M1M2 . . .ML and it is not possible to split W in two different blocks A and B. For this special case, we define O (M1M2 . . .ML) = |1, 1, . . . , 1〉〈1, 1, . . . , 1| , (11) which is the projector on the ‘full’ configuration with all sites occupied. The ring-orderingO () is extended by linearity to the vector space spanned by words of the type described above. 2.4 Transfer matrix and generalized Hamiltonians Hk The algebraic Bethe Ansatz allows to construct a one parameter commuting family of transfer matrices, t(λ), that contains the translation operator T = t(1) and the Markov matrix M = t′(0). For 0 ≤ λ ≤ 1, the operator Golinelli, Mallick — Connected Operators for TASEP 5 t(λ) can be interpreted as a discrete time process with non-local jumps : a hole located on the right of a cluster of p particles can jump a distance k in the backward direction, with probability λk(1 − λ) for 1 ≤ k < p, and with probability λp for k = p. The probability that this hole does not jump at all is 1 − λ. This model is equivalent to the 3-D anisotropic percolation model of Rajesh and Dhar (1998) and to a 2-D five-vertex model. It is also an adaptation on a periodic lattice of the ASEP with a backward- ordered sequential update (Rajewsky et al. 1996, Brankov et al. 2004), and equivalently of an asymmetric fragmentation process (Rákos and Schütz 2005). The operator t(λ) is a polynomial in λ of degree L given by t(λ) = 1 + λkHk , (12) where the generalized HamiltoniansHk are non-local operators that act on the configuration space. [We emphasize that the notation used here is different from that of our previous work : t(λ) was denoted by tg(λ) in (Golinelli and Mallick 2007).] We have H1 = M and more generally, as shown in (Golinelli and Mallick 2007), Hk is a homogeneous sum of words of length k 1≤i1