The Na(+)-translocating F-ATPase of the thermoalkaliphilic bacterium Clostridium paradoxum harbors an oligomeric ring of c subunits that resists dissociation by sodium dodecyl sulfate. The c ring has been isolated and crystallized in two dimensions. From electron microscopy of these c-ring crystals, a projection map was calculated to 7 A resolution. In the projection map, each c ring consists o...

Given a λ-ring A and a formally etale morphism f : A→ B of commutative rings, one may ask: What are the possible λ-ring strutures on B such that f is a map of λ-rings? We give the answer: Such a lifted λ-ring structure on B is determined uniquely by a compatible choice of lifts of the Adams operations ψ from A to B for all primes p which satisfy Frobenius congruences. In other words, to extend ...

1.1 (a) Rings (resp. σ-rings) are closed under finite (resp. countable) intersections. (b) If R is a ring (resp. σ-ring) then R is an algebra (resp. σ-algebra) iff X ∈ R. (c) If R is a (nonempty) σ-ring then {E ⊂ X : E ∈ R or E ∈ R} is a σ-algebra. (d) If R is a σ-ring, then {E ⊂ X : E ∩ F ∈ R for all F ∈ R} is a σ-algebra. Solution: (a) If R is a ring and E1, E2 ∈ R, then since R is closed und...

LetR = k[x1, . . . , xd] be the polynomial ring in d independent variables, where k is a field of characteristic p > 0. Let DR be the ring of k-linear differential operators of R and let f be a polynomial in R. In this work we prove that the localization R[ 1 f ] obtained from R by inverting f is generated as a DR-module by 1 f . This is an amazing fact considering that the corresponding charac...

The F -thresholds are characteristic p analogs of the jumping coefficients for multiplier ideals in characteristic zero. We give an alternative description of the F -thresholds of an ideal in a regular and F -finite ring R, which enables us to settle two open questions posed in [MTW]. Namely, we show that if the ring is, in addition, essentially of finite type over a field, then the F -threshol...

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant and Hasse-Witt invariant can be written in the Witt ring as the s...

Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z 2(M)-inverse split provided f^(-1)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct...

A ring of 8-15 identical c-subunits is essential for ion-translocation by the rotary electromotor of the ubiquitous F(O)F(1)-ATPase. Here we present the crystal structure at 3.4Å resolution of the c-ring from chloroplasts of a higher plant (Pisum sativum), determined using a native preparation. The crystal structure was found to resemble that of an (ancestral) cyanobacterium. Using elastic netw...

Since the time of the Voyager flybys of Saturn in 1980-1981, Saturn's eccentric F ring has been known to be accompanied on either side by faint strands of material. New Cassini observations show that these strands, initially interpreted as concentric ring segments, are in fact connected and form a single one-arm trailing spiral winding at least three times around Saturn. The spiral rotates arou...

In the title benzonitrile compound, C(12)H(11)F(3)N(2)O, an intra-molecular C-H⋯F hydrogen bond generates an S(7) ring motif. The trifluoro-methyl group is disordered over two orientations with a refined occupancy ratio of 0.549 (16):0.451 (16). The morpholine ring adopts a chair conformation. The benzene ring and mean plane of the morpholine ring make a dihedral angle of 58.04 (10)° with each ...