Sequential definitions of connectedness

Applied Mathematics Letters 25 (2012) 461–465

Contents lists available at SciVerse ScienceDirect

Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Sequential definitions of connectedness Hüseyin Çakallı ∗ Department of Mathematics, Faculty of Science and Letters, Maltepe University, Marmara Eğitim Köyü, TR 34857, Maltepe, İstanbul, Turkey

article

info

Article history: Received 2 June 2011 Accepted 15 September 2011 Keywords: Sequences Summability G-sequential closure G-sequential continuity G-sequential connectedness

abstract We investigate the impact of changing the definition of convergence of sequences on the structure of the set of connected subsets of a topological group, X . A non-empty subset A of X is called G-sequentially connected if there are no non-empty  and disjoint G-sequentially closed subsets U and V , both meeting A, such that A ⊆ U V . Sequential connectedness in a topological group is a special case of this generalization when G = lim. © 2011 Published by Elsevier Ltd

1. Introduction Connectedness and other related concepts play a very important role not only in pure mathematics but also in other branches of science involving mathematics, especially in geographic information systems, population modelling, and motion planning in robotics. Connor and Grosse-Erdmann [1] have investigated the impact of changing the definition of the convergence of sequences on the structure of sequential continuity of real functions. Çakallı [2] extended this concept to the topological group setting, introducing the concept of G-sequential compactness. He reported further results on G-sequential compactness and G-sequential continuity in [3]. One is often relieved to find that the standard closed-set definition of connectedness for metric spaces can be replaced by a sequential definition of connectedness. That many of the properties of connectedness of sets can be easily derived using sequential arguments has also been, no doubt, a source of relief to the interested mathematics instructor. The aim of this paper is to introduce G-sequential connectedness and to investigate the concept in metrisable topological groups. 2. Preliminaries Before we begin, we will state some definitions and notation. Throughout this paper, N will denote the set of all positive integers. Although some of the definitions that follow make sense for an arbitrary topological group, we prefer using neighbourhoods instead of metrics. In this paper, X will always denote a topological Hausdorff group, written additively, that satisfies the first axiom of countability. We will use boldface letters x, y, z, . . . for sequences x = (xn ), y = (yn ), z = (zn ), . . . of terms of X . The notations s(X ) and c (X ) denote the set of all X -valued sequences and the set of all X -valued convergent sequences of points in X , respectively. Following the idea given in a 1946 American Mathematical Monthly problem [4], a number of authors, Posner [5], Iwinski [6], Srinivasan [7], Antoni [8], Antoni and Salat [9], and Spigel and Krupnik [10], have studied A-continuity defined

∗

Tel.: +90 216 6261050x2248; fax: +90 216 6261113. E-mail addresses: [email protected], [email protected], [email protected].

0893-9659/$ – see front matter © 2011 Published by Elsevier Ltd doi:10.1016/j.aml.2011.09.036

462

H. Çakallı / Applied Mathematics Letters 25 (2012) 461–465

by a regular summability matrix A. Some authors, Öztürk [11], Savaş [12], Savaş and Das [13], and Borsik and Salat [14] have studied A-continuity for methods of almost convergence or for related methods. See also [15] for an introduction to summability matrices. The idea of statistical convergence was formerly described under the name ‘‘almost convergence’’ by Zygmund in the first edition of his celebrated monograph published in Warsaw in 1935 [16]. The concept was formally introduced by Fast [17] and was later reintroduced by Schoenberg [18] and also, independently, by Buck [19]. A sequence (xk ) of points in X is said to be statistically convergent to an element ℓ of X if for each neighbourhood U of 0, 1

|{k ≤ n : xk − ℓ ̸∈ U }| = 0, n and this is denoted by st − limn→∞ xn = ℓ. The statistical limit is an additive function on the group of statistically convergent sequences of points in X . (See [20] for the real case and [21–25] for the topological group setting.) A sequence (xk ) of points in a topological group is called lacunary statistically convergent to an element ℓ of X if lim

n→∞

lim

r →∞

1 hr

|{k ∈ Ir : xk − ℓ ̸∈ U }| = 0

for every neighbourhood U of 0 where Ir = (kr −1 , kr ] and k0 = 0, hr : kr − kr −1 → ∞ as r → ∞, and θ = (kr ) is an increasing sequence of positive integers. For a constant lacunary sequence, θ = (kr ), the lacunary statistically convergent sequences in a topological group form a subgroup of the group of all X -valued sequences, and the lacunary statistical limit is an additive function on this space. (See [21] for the topological group setting, and see [26,27] for the real case.) Throughout this paper, we assume that lim infr k kr > 1. r −1 By a method of sequential convergence, or briefly by a method, we mean an additive function G defined on a subgroup cG (X ) of s(X ) into X [2]. A sequence x = (xn ) is said to be G-convergent to ℓ if x ∈ cG (X ) and G(x) = ℓ. In particular, lim denotes the limit function lim x = limn xn on the group c (X ). A method G is called regular if every convergent sequence x = (xn ) is G-convergent with G(x) = lim x. Clearly, if f is G-sequentially continuous on X , then it is G-sequentially continuous on every subset Z of X , but the converse is not necessarily true, as in the latter case, the sequences x are restricted to Z . This distinction was demonstrated by an example in [1] for a real function. We define the sum of two methods of sequential convergence, G1 and G2 , as

(G1 + G2 )(x) = G1 (x) + G2 (x), where cG1 +G2 (X ) = cG1 (X ) ∩ cG2 (X ) [3]. The notion of regularity introduced above coincides with the classical notion of regularity for summability matrices and with regularity in a topological group for limitation methods. See [15] for an introduction to regular summability matrices of real numbers and complex numbers, [28] for an introduction to regular limitation (summability) methods in topological groups, and [29] for a general view of sequences of real or complex numbers. First of all, we recall the definition of G-sequential closure of a subset of X [2,3]. Let A ⊂ X and ℓ ∈ X . Then ℓ is in the G-sequential closure of A (called the G-hull of A in [1]) if there is a sequence x = (xn ) of points in A such that G(x) = ℓ. We G

denote the G-sequential closure of a set A by A . We say that a subset A is G-sequentially closed if it contains all of the points G

in its G-closure, i.e., a subset A of X is G-sequentially closed if A ⊂ A. The null set φ and the whole space X are G-sequentially closed. G G G It is clear that φ = φ and X = X for a regular method G. If G is a regular method, then A ⊂ A ⊂ A , and hence A is G

G

G

G

G-sequentially closed if and only if A = A. Even for regular methods, it is not always true that (A ) = A . Even for regular methods, the union of any two G-sequentially closed subsets of X need not be a G-sequentially closed subset of X , as is seen by considering Counterexample 1 given after Theorem 4 in [3]. Çakallı introduced the concept of G-sequential compactness and proved that a G-sequentially continuous image of any G-sequentially compact subset of X is also G-sequentially compact (see Theorem 7 in [2]). He investigated G-sequential continuity and obtained further results in [3] (See also [30–37] for some other types of continuities that cannot be given by any sequential method). Recently, Mucuk and Sahan [38] investigated further properties of G-sequential closed subsets of X . They modified the definition of the open subset to the G-sequential case in the sense that a subset A of X is G-sequentially open if its complement G

is G-sequentially closed, i.e., X \ A ⊆ X \ A, and obtained that the union of any G-sequentially open subsets of X are G-sequentially open. From the fact that the G-sequential closure of a subset of X includes the set itself for a regular sequential G

method G, we see that a subset A is G-sequentially open if and only if X \ A = X \ A for a regular sequential method G. A function f : X → X is G-sequentially continuous at a point u if, given a sequence (xn ) of points in X , G(x) = u implies that G(f (x)) = f (u). If a function f is G-sequentially continuous on X , then the inverse image f −1 (K ) of any G-sequentially closed subset K of X is G-sequentially closed [2]. If a function f is G-sequentially continuous on X , then the inverse image f −1 (U ) of any G-sequentially open subset U of X is G-sequentially open [38]. For a regular method G, the function fa : X → X , x → a + x is G-sequentially continuous, G-sequentially closed, and G-sequentially open [38]. If A and B are G-sequentially open, then so is the sum A + B [38]. Mucuk and Sahan also obtained that a subset A of X is G-sequentially open if and only if each a ∈ A has a G-sequentially open neighbourhood Ua such that Ua ⊆ A.

H. Çakallı / Applied Mathematics Letters 25 (2012) 461–465

463

3. G-sequential connectedness In the literature, sequentially connected spaces are defined in various ways (see, for example, [39,40]). Connectedness is much more useful for the covering spaces of topological groups. For the non-connected case, see, for example, [41]. We define a G-sequentially connected space as follows. Definition 1. A non-empty subset A of a topological group X is called  G-sequentially connected if there are no non-empty and disjoint G-sequentially closed subsets U and V such that A ⊆ U V , and A ∩ U and A ∩ V are non-empty. In particular, X is called G-sequentially connected if there are no non-empty, disjoint G-sequentially closed subsets of X whose union is X .  We give the following definition before presenting some characterisation of connectedness of a subset. Definition 2. Let A be a subset of X . A subset F of A is called G-sequentially closed in A if there exists a G-sequentially closed subset U of X such that F = U ∩ A. We say that a subset V of A is G-sequentially open in A if A \ V is G-sequentially closed in A.  Here, we note that a subset B of A is G-sequentially open in A if and only if there exists a G-sequentially open subset V of X such that B = A ∩ V . Now, we give the following lemma. Lemma 1. For a subset A of X , the following are equivalent: i: ii: iii: iv:

A is G-sequentially connected; A cannot be written as a union of non-empty disjoint G-sequentially closed subsets in A; A cannot be written as a union of non-empty disjoint G-sequentially open subsets in A; There is no G-sequentially open and closed proper subset in A.

The proof is straightforward by Definitions 1 and 2 and is therefore omitted. In [2], it was proved that a G-sequentially continuous image of any G-sequentially compact subset of X is also G-sequentially compact. Now, we prove that a G-sequentially continuous image of any G-sequentially connected subset of X is also G-sequentially connected. Theorem 1. A G-sequentially continuous image of any G-sequentially connected subset of X is G-sequentially connected. Proof. Suppose that f (A) is not G-sequentially connected so that f (A) can be covered as a union U ∪ V of non-empty, disjoint G-sequentially closed subsets U and V of X , both meeting f (A). Because the inverse image of a G-sequentially closed subset of X is G-sequentially closed for the G-sequentially continuous function f , f −1 (U ) and f −1 (V ) are non-empty, disjoint G-sequentially closed subsets of X and cover A. This statement implies that A is not G-sequentially connected. This contradiction completes the proof of the theorem.  Definition 3. Let G be a sequential method, B ⊆ A ⊆ X , and let a ∈ A. Then we say a is in the G-sequential closure of B in A G

if there is a sequence x = (xn ) of points in B such that G(x) = a. We denote the G-sequential closure of B in A by BA . G

G

Lemma 2. Let G be a regular sequential method. If B ⊆ A ⊆ X , then BA = B ∩ A. Proof. The proof is straightforward and is therefore omitted.



Lemma 3. Let G be a regular sequential method, and let B ⊆ A ⊆ X . If A is G-sequentially closed in X , and B is G-sequentially closed in A, then B is G-sequentially closed in X . Proof. Let ℓ be a point in the G-sequential closure of B in X . Then there exists a sequence x = (xn ) of points in B such that G

G

G

G(x) = ℓ ∈ X . A ⊆ B implies B ⊆ A , and thus ℓ ∈ A . Because A is G-sequentially closed in X , we have ℓ ∈ A. Therefore, ℓ is a point in the G-sequential closure of B in A. Because B is closed in A, we have that ℓ ∈ B. This conclusion completes the proof of the lemma.  Lemma 4. Let A be a G-sequentially connected subset of X . If U and V are non-empty disjoint G-sequentially closed subsets of X such that A ⊆ U ∪ V , then either A ⊆ U or A ⊆ V . Proof. Suppose that A ̸⊆ U and A ̸⊆ V . A ̸⊆ U implies that there exists an x ∈ A such that x ̸∈ U. Because A ⊆ U ∪ V , x ∈ V and so A ∩ V is not empty. Similarly, A ̸⊆ V implies that A ∩ U is not empty. This contradiction completes the proof. Lemma 5. Let G be a sequential method, A ⊆ X , and U a G-sequentially open and G-sequentially closed subset of X . If A is G-sequentially connected, then either A ⊆ U or A ⊆ X \ U. Proof. If U = ∅ or U = X , the proof is obvious. If U ̸= ∅ and U ̸= X , then A ⊆ U ∪ (X \ U ), and so by Lemma 4, either A ⊆ U or A ⊆ X \ U.  G

Theorem 2. Let G be a regular sequential method, B ⊆ X , and B ⊆ A ⊆ B . If B is G-sequentially connected, then so is A.

464

H. Çakallı / Applied Mathematics Letters 25 (2012) 461–465 G

G

G

G

G

Proof. If B ⊆ A ⊆ B , then A ⊆ B ∩ A = BA . On the other hand, BA ⊆ A. Therefore BA = A, where BA is the G-sequential closure of B in A. Now, conversely, suppose that A is not G-sequentially connected. So there are non-empty and disjoint G-sequentially closed subsets U and V of X such that A ⊆ U ∪ V , and A ∩ U and A ∩ V are non-empty. Because B is connected G

G

G

G

by Lemma 4, either B ⊆ U or B ⊆ V . If B ⊆ U, then B ⊆ U , and so BA ⊆ U ∩ A. Because G is regular and U is G-sequentially G

G

closed in X , we have that U = U. So we have that A = BA ⊆ A ∩ U, which implies that A = A ∩ U. Similarly, if B ⊆ V , then A = A ∩ V . This contradiction completes the proof.  G

Corollary 1. If G is a regular sequential method, and A is a G-sequentially connected subset of X , then so is A . G

We know by Lemma 1 in [3] that for a G-regular method and a subset A, A = A if and only if G is a subsequential method. Here, A denotes the usual closure of A. Thus, we can state the following corollary. Corollary 2. Let G be a regular subsequential method. If X has a G-sequentially connected and dense subset in X , then X is G-sequentially connected. Theorem 3. Let {Ai | i ∈ I } be a class of G-sequentially connected subsets of X . If G-sequentially connected.



i∈I

Ai is non-empty, then



i∈I

Ai is

Proof. Suppose that A is not G-sequentially connected, so that there exist non-empty disjoint G-sequentially closed subsets U and V of X such that A ⊆ U ∪ V . Because each Ai is G-sequentially connected, by Lemma 4, either Ai ⊆ U or Ai ⊆ V . If Ai ⊆ U and Aj ⊆ V for i ̸= j, then Ai ∩ Aj = ∅. Because i∈I Ai is non-empty, for all i ∈ I, either Ai ⊆ U or Ai ⊆ V . Therefore, either A ⊆ U or A ⊆ V . If A ⊆ U, then A = A ∩ U. If A ⊆ V , then A = A ∩ V , which is a contradiction. Thus, A is G-sequentially connected.  Corollary 3. Let {Ai | i ∈ I } be a class of G-sequentially connected subsets of X . Let B ⊆ X be G-sequentially connected such  that B ∩ Ai is non-empty for each i ∈ I. Then B ∪ ( i∈I Ai ) is also G-sequentially connected. Proof. Because B ∩ Ai is non-empty, is G-sequentially connected for each i ∈ I, and  Bi = B ∪ Ai  Therefore, by Theorem 3, the union i∈I Bi = B ∪ ( i∈I Ai ) is G-sequentially connected. 



i∈I

Bi is non-empty.

The following lemma, which is adopted from [3], is very useful for the proof of Theorem 4. Lemma 6. Let G be a regular method, and let A and B be subsets of X . Then the following are satisfied: G

G

i: If A ⊂ B, then A ⊂ B ; G

G

G

ii: A + B ⊂ A + B ; G G iii: −A = −A . G

Theorem 4. Let G be a regular sequential method. If H is a G-sequentially connected (normal) subgroup of X , then so is H . G

Proof. Let H be a G-sequentially connected subgroup of X . Then, by Corollary 1, H is G-sequentially connected because H G

G

G

G

G

G

G

G

is a subgroup H − H ⊆ H. By Lemma 6, H − H ⊆ H − H ⊆ H . Therefore, H − H ⊆ H , and so H is a subgroup of X . Furthermore, if H is normal, a + H − a ⊆ H for each a ∈ X . Thus, by Lemma 6, G

G

G

G

G

{a} + H − {a} ⊆ a + H − a ⊆ H . G

G

G

Because G is regular, {a} ⊆ {a}, and so {a} + H − {a} ⊆ H . Therefore, H is normal.



Lemma 7. Let G be a regular sequential method and U a symmetric neighbourhood of 0. If U is G-sequentially connected, then so is U + U. Proof. If U is G-sequentially connected, then, by Theorem 1, for each  x ∈ U, the set x + U is G-sequentially connected, and as U is symmetric, x + U includes the identity. Because U + U = x∈U x + U by Theorem 3, the set U + U is G-sequentially connected.  Theorem 5. Let G be a regular sequential method, and let H be a subgroup of X . If H is G-sequentially open, then it is G-sequentially closed. Proof. Let H be a G-sequentially open subgroup of X . Then a + H is G-sequentially open for each a ∈ X . On the other hand, because X \H =



a+H

a∈X \H

and the union of G-sequentially open subsets is open, X \ H becomes G-sequentially open. Therefore, H is sequentially closed. 

H. Çakallı / Applied Mathematics Letters 25 (2012) 461–465

465

4. Conclusion The present work introduces a concept of G-sequential connectedness using a G-sequential method via sequences, and it investigates interesting results related to G-sequential connectedness, G-sequential closedness, and G-sequential continuity in first countable Hausdorff topological groups. One may expect this concept to be a useful tool in the field of topology in modelling various problems occurring in many areas of science, geographic information systems, population modelling, and motion planning in robotics. It seems that an investigation of the present work taking ‘‘nets’’ instead of ‘‘sequences’’ could be done, using the properties of ‘‘nets’’ instead of the properties of ‘‘sequences’’. As the vector space operations, namely, vector addition and scalar multiplication, are continuous in a cone normed space, and therefore, cone normed spaces are special topological groups, we see that the results are also valid in cone normed spaces ([42] for the definition of a cone normed space, see also [43]). For further study, we also suggest investigating the present work for the fuzzy case. However, due to the change in settings, the definitions and methods of proofs will not always be analogous to those in the present work (see [44] for the definitions in the fuzzy setting). References [1] J. Connor, K.-G. Grosse-Erdmann, Sequential definitions of continuity for real functions, Rocky Mountain J. Math. 33 (1) (2003) 93–121. MR 2004e:26004. [2] H. Çakallı, Sequential definitions of compactness, Appl. Math. Lett. 21 (6) (2008) 594–598. MR 2009b:40005. [3] H. Çakallı, On G-continuity, Comput. Math. Appl. 61 (2011) 313–318. [4] R.C. Buck, Solution of problem 4216, Amer. Math. Monthly 55 (36) (1948) MR 15 26874. [5] E.C. Posner, Summability preserving functions, Proc. Amer. Math. Soc. 12 (1961) 73–76. MR 2212327. [6] T.B. Iwinski, Some remarks on Toeplitz methods and continuity, Comment. Math. Prace Mat. 17 (1972) 37–43. MR 48759. [7] V.K. Srinivasan, An equivalent condition for the continuity of a function, Tex. J. Sci. 32 (1980) 176–177. MR 81f:26001. [8] J. Antoni, On the A-continuity of real functions II, Math. Slovaca 36 (3) (1986) 283–287. MR 88a:26001. [9] J. Antoni, T. Salat, On the A-continuity of real functions, Acta Math. Univ. Comenian. 39 (1980) 159–164. MR 82h:26004. [10] E. Spigel, N. Krupnik, On the A-continuity of real functions, J. Anal. 2 (1994) 145–155. MR 95h:26004. [11] E. Öztürk, On almost-continuity and almost A-continuity of real functions, Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. 32 (1983) 25–30. MR 86h:26003. [12] E. Savaş, On invariant continuity and invariant A-continuity of real functions, J. Orissa Math. Soc. 3 (1984) 83–88. MR 87m:26005. [13] E. Savaş, G. Das, On the A-continuity of real functions, İstanbul Univ. Fen Fak. Mat Derg. 53 (1994) 61–66. MR 97m:26004. [14] J. Borsik, T. Salat, On F -continuity of real functions, Tatra Mt. Math. Publ. 2 (1993) 37–42. MR 94m:26006. [15] J. Boos, Classical and Modern Methods in Summability, Oxford Univ. Press, Oxford, 2000. [16] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, UK, 1979. [17] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. MR 14:29c. [18] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (5) (1959) 361–375. MR 104946. [19] R.C. Buck, Generalised asymptotic density, Amer. J. Math. 75 (1953) 335–346. MR 14,854f. [20] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. MR 87b:40001. [21] H. Çakallı, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (2) (1995) 113–119. MR 95m:40016. [22] H. Çakallı, On statistical convergence in topological groups, Pure Appl. Math. Sci. 43 (1–2) (1996) 27–31. MR 99b:40006. [23] H. Çakallı, A study on statistical convergence, Funct. Anal. Approx. Comput. 1 (2) (2009) 19–24. MR2662887. [24] G.D. Maio, L.D.R. Kocinac, Statistical convergence in topology, Topology Appl. 156 (2008) 28–45. MR 2009k:54009. [25] H. Çakallı, M.K. Khan, Summability in topological spaces, Appl. Math. Lett. 24 (2011) 348–352. [26] J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1) (1993) 43–51. MR 94j:40014. [27] J.A. Fridy, C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl. 173 (1993) 497–504. MR 95f:40004. [28] H. Çakallı, B. Thorpe, On summability in topological groups and a theorem of D.L. Prullage, Ann Soc. Math. Pol. Comm. Math. Ser. I 29 (1990) 139–148. MR 91g:40010. [29] A. Zygmund, Trigonometric Series, 2nd ed., vol. II, Cambridge Univ. Press, London, New York, 1959, MR 58:29731. [30] H. Çakallı, Forward continuity, J. Comput. Anal. Appl. 13 (2) (2011) 225–230. [31] H. Çakallı, δ -quasi-Cauchy sequences, Math. Comput. Modelling 53 (1–2) (2011) 397–401. [32] H. Çakallı, Slowly oscillating continuity, Abstr. Appl. Anal. 2008 (2008), doi:10.1155/2008/485706. p. 5, Article ID 485706. MR 2009b:26004. [33] M. Dik, I. Canak, New types of continuities, Abstr. Appl. Anal. 2010 (2010), doi:10.1155/2010/258980. p. 6, Article ID 258980. [34] H. Çakallı, İ. Çanak, M. Dik, δ -quasi-slowly oscillating continuity, Appl. Math. Comput. 216 (2010) 2865–2868. [35] H. Çakallı, New kinds of continuities, Comput. Math. Appl. 61 (2011) 960–965. [36] H. Çakallı, Statistical quasi-Cauchy sequences, Math. Comput. Modelling (2011), doi:10.1016/j.mcm.2011.04.037. [37] H. Çakallı, Statistical ward continuity, Appl. Math. Lett. 24 (10) (2011) 1724–1728. [38] O. Mucuk, T. Sahan, On G-sequential continuity, Comput. Math. Appl. (submitted for publication). [39] Q. Huang, S. Lin, Notes on sequentially connected spaces, Acta Math. Hungar. 110 (1–2) (2006) 159–164. [40] A. Fedeli, A.L. Donne, On good connected preimages, Topology Appl. 125 (2002) 489–496. [41] R. Brown, O. Mucuk, Covering groups of nonconnected topological groups revisited, Math. Proc. Cambridge Philos. Soc. 115 (1) (1994) 97–110. MR 95k:22001. [42] A. Sonmez, H. Çakallı, Cone normed space and weighted means, Math. Comput. Modelling 52 (2010) 1660–1666. [43] H. Çakallı, A. Sonmez, Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Appl. Math. Lett. (2011), doi:10.1016/j.aml.2011.09.029. [44] H. Çakalli, Pratulananda Das, Fuzzy compactness via summability, Appl. Math. Lett. 22 (11) (2009) 1665–1669. MR 2010k:54006.