Smooth Action Research Articles

Crossed Products of Continuous-Trace C ∗ -Algebras by Smooth Actions

Crossed Products of Continuous-Trace C ∗ -Algebras by Smooth Actions

Smooth conjugacy of algebraic actions on affine varieties

The motivation for this result is the linearity conjecture which asserts that any algebraic action of a reductive algebraic group on affine n-space is conjugate to a linear action. This topic has been popularized by H. Bass and H. Kraft. See for example [4], [2] and [S] for the current status of the conjecture. In addition to being interesting, the conjecture is tough. It is not even known whether the set of algebraic conjugacy classes of algebraic actions of a reductive group of affine n-space is countable, as implied by the linearity conjecture. Theorem 1 is a version of this countability question. We shall see in the course of the proof of Theorem 1 that there are an uncountable number of conjugacy classes of smooth actions of a nontrivial compact Lie group on real n-space. Theorem 1 shows that only countably many of these are conjugate to algebraic actions on real n-space. The main tools in the proof are the compactification theorem of Corollary 4 and the Palais Rigidity Theorem [lo]. Whether this rigidity theorem has an algebraic analog is an interesting question whose consequences we briefly explore at the end. Before giving the proof, we introduce some notation. In this discussion varieties are affine. The ground field is the reals [w or the complex numbers @. An algebraic map between affine spaces (over [w or C) is a map whose coordinate functions are polynomials. An algebraic map between affine varieties is a map which extends to an algebraic map of affine spaces containing the varieties as subvarieties. Let W be a nonsingular variety resp. a smooth manifold and Aut( W, c) for c=a or s denote the group of algebraic resp. smooth automorphisms of W with the C” topology. Let Hom(G, Aut( W, c)) denote the space of c actions of G on W. A point d of this space is a homomorphism d:BG+Aut( W, c) such that the induced map from G x W to W is a c-map. Topologize Hom(G, Aut( W, c)) as the subspace of the space of all smooth maps of G x W to Win the C” topology. The group Aut( W,c) acts by conjugation on the space of c-actions and the orbit space is denoted by Hom(G, Aut( W, c))/Aut( W, c). In the case W is a nonsingular variety V, weakening structure defines a continuous function from the space of a-actions to the space of s-actions and induces a map of corresponding orbit spaces. We are interested in the cardinality of the image. The idea is to factor this map through a countable set. In the case V is simply connected at infinity (i.e. given any compact set C in Vthere is a compact set D containing C

Circle actions on spin manifolds and characteristic numbers

II+ 1970 Atiyah and Hirzebruch [3] proved that the existence of a non-trivial smooth action of the circle group S’ on a closed connected Spin manifold M implies that the A-genus of M vanishes. It is our purpose to explore the vanishing of further Pontrjagin numbers of Spin manifolds admitting smooth S’ actions of odd type; assuming that MS’ # 4, and viewing Z, = ( k 1) c S’, these are the actions for which all components of the fixed set MzL have codimensions congruent to 2 (mod 4) (see [2,3]). We have one general result for such actions; the rest of our results require that the action also be semifree, i.e. that it be free on the complement of the fixed set MS’. The following result was proved during conversations with S. Weinberger.

Deviations from a power-function near miss and their relation to loudness functions

In a previous paper [W. S. Hellman and R. P. Hellman, J. Acoust. Soc. Am. Suppl. 1 81, S53–S54 (1987)], the neural-count function N(I) was derived from the intensity-jnd function J(I) through the relation N(I)1/2 = (h/2)∫dI/[IJ(I)] + a. Loudness functions were then generated by the prescription L = k[N(I) − N]. Using a power-function near miss for J(I), good agreement between the measured and calculated loudness values for pure tones was obtained. While a power function yields a simple evaluation of the integral for N(I)1/2, the results of many recent jnd studies do not easily conform to a power-function fit. In order to determine how the departures from a power-function near miss affect the form of the loudness function within the model, the integration was performed over the segmented intensity jnd functions observed for gated and continuous tones. The calculated loudness functions and their respective input intensity jnd functions are shown for frequencies of 250 and 1000 Hz. In spite of the segmented, and in some instances, nonmonotonic intensity jnd functions, the results reveal that the smoothing action of the integration over J(I) produces loudness functions consistent with experimental results. [Partially supported by the Rehabilitation Research and Development Service of the VA.]

Vascular muscle cell depolarization and activation in renal arteries on elevation of transmural pressure

The goal of this study was to define some of the cellular and ionic mechanisms of smooth muscle cell activation in dog renal arteries exposed to physiological levels of transmural pressure. Isolated interlobular arteries were cannulated and connected to a pressure reservoir to allow manipulation of transmural pressure in 20-mmHg increment steps from 20 to 120 mmHg. As transmural pressure was increased, vascular smooth muscle exhibited a linear depolarization from an average resting potential of -57 +/- 2 mV at 20 mmHg to -38 +/- 2.4 mV at 120 mmHg. Spontaneous action potentials could often be recorded at pressures greater than 80 mmHg. These appeared to occur primarily at bifurcation points of branching arteries. Vascular smooth muscle depolarization and action potentials occurring in response to increases in transmural pressure were associated with a maintenance of internal diameter of the vessel segments despite increases in transmural pressure in the range between 60 and 120 mmHg. The "pressure-induced" activation of vascular smooth muscle contraction and spontaneous action potentials of small renal arteries at higher transmural pressures were blocked on Ca2+ channel inhibition with verapamil (10(-6) M). These data document a membrane ionic mechanism (probably increased Ca2+ influx) for pressure-induced myogenic activation of isolated renal arteries. It is interesting that the contraction of these vessels occurs over the pressure range in which autoregulation of renal blood flow normally occurs. The physiological significance of these responses needs to be determined.

Actions of neuropeptide Y on innervated and denervated rat tail arteries.

1. Neuropeptide Y caused a dose-dependent contraction and depolarization of the smooth muscle of the rat tail artery. 2. 30 nM-neuropeptide Y increased the contraction caused by either nerve-released noradrenaline or smooth muscle action potentials. 3. 30 nM-neuropeptide Y did not change the amplitude or rate of rise of the smooth muscle action potential. It did not change the amplitude of small excitatory junction potentials, suggesting that it did not affect neurotransmitter release. 4. 30 nM-neuropeptide Y increased the contraction caused by exogenous noradrenaline, 5-hydroxytryptamine and K in concentrations that gave submaximal contractions. It did not affect the response to higher concentrations that gave maximal or near-maximal contractions.

Rigid cyclic group actions on cohomology complex projective spaces

In this paper we discuss smooth and locally smooth cyclic actions on cohomology complex projective spaces. We shall show that in low dimensions the presence of a codimension-two fixed-point component forces many algebraic invariants to behave as they do in the linear examples.

Structure of lymphatic valves in the spinotrapezius muscle of the rat.

Lymphatic valves assure the forward propulsion of fluid along the lymphatic vessels. A description of valve function in skeletal muscle must be based on a knowledge of the valve morphology. To this end, histological sections of valves from lymphatic microvessels of the rat spinotrapezius muscle were examined with light microscopy. All of the approximately 50 valves studied from 20 rats had a bileaflet structure, with a buttress formed at each side of the valve by the fusion of opposing leaflets. This valve structure would allow the valve to close without inversion. There is no evidence for active smooth muscle action to open and close the valve. Since the Reynolds number of lymph flow is very small (about 0.0025), only pressure and viscous forces are available for valve closure. A particular mechanism based on the actual lymphatic valve structure is proposed.

Group actions and equivariant Lipschitz analysis

The idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by Singer. The concrete realization of this, in the case of rational Pontrjagin classes, was the result of works by Sullivan, who showed that topological manifolds have essentially unique Lipschitz structures, and Teleman, who showed that Lipschitz manifolds are the type of object for which one can analyze the signature operator. (A refinement of the same method proves the KO[l/2] orientability of topological bundles.) Much of this can be done for manifolds with group actions. The main implication is that for certain sorts of topological group actions (including all smooth and PL actions) one can construct an signature operator whose class in equivariant if-homology is a topological invariant. This class satisfies the G-signature theorem. The topological method has implications for various precise classification problems (e.g. semifree topological actions on the sphere), but here we shall present only the consequences of these Lipschitzian ideas: (A) Topologically locally linear G-manifolds for odd-order G have KOG[1/2] orientations (Madsen-Rothenberg). For all finite G there are Lipschitz structures, but the signature class is a zero divisor for even-order groups. The same method produces, upon application of the localization theorem, Atiyah-Singer classes for some actions with nonmanifold fixed set (Weinberger, extending earlier joint work with Cappell). (B) For odd-order groups, topological and linear conjugacy of linear representations are equivalent (Hsiang-Pardon, Madsen-Rothenberg). This is a consequence of (A). Alternatively one can return to the original argument of Atiyah-Bott-Milnor for semifree representations. A stronger result is true: the Atiyah-Bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations. This distinguishes many pairs of representations of even-order groups. (C) R Top(G) = R Lip(G) for all finite G, where R Cat is the Grothendieck group of representations up to Cat conjugacy (Weinberger, after inverting 2). This implies that there is no analytic definition of Reidemeister torsion that works well in the Lipschitz setting. The reason is that, following

Modulation of smooth muscle responses by serine proteases and related enzymes.

Standard isolated smooth muscle tissue preparations were used to screen the musculotropic actions of serine proteases and other substances that are generated during hemostatic activation. Sera obtained from human, rabbit, and guinea pigs produced a dose-dependent contraction of these isolated tissue preparations. These sera also augmented the actions of epinephrine and other agonists on different tissue preparations. Both contractile and relaxant actions of proteases were observed with various serine proteases and their complexes. Thrombins (alpha, beta, and gamma) produced varying degrees of contractile actions of rabbit aortic strips; however, Factor Xa produced a marked relaxation of this preparation. Trypsin and kallikrein did not produce any action on the aortic strip and thrombin, however, produced contraction of the guinea pig ileum. Protease complexes produced varying musculotropic actions on the isolated tissue preparations, which may be related to their composition. The musculotropic actions of serum and thrombin were not blocked by conventional pharmacologic antagonists. However, protease inhibitors, such as hirudin, D-Phe-Pro-Arginal, and other derivatives, produced varying effects of the musculotropic actions of serum and thrombins. These results indicate that thrombin and related proteases generated during the activation of the hemostatic system are capable of producing direct musculotropic actions of various smooth muscles. Various serine proteases and serine protease complexes were found to produce strong hemodynamic effects in a rabbit model for the study of blood plasma. The smooth muscle modulant actions of serine proteases should be taken into account to clarify the vascular pathologic changes in relation to spasmogenic and spasmolytic syndromes observed during hemostasis disorders.

Effects of cold-restraint stress on gastric ulceration and motility in rats

The effects of stress by restraint at 4°C for 2 hr and of drug treatment on gastric lesion formation and motor activities (contraction frequency, amplitude and tone) were studied in rats. Restraint at room temperature (22°C) produced a small ulcer index in the controls and did not significantly affect gastric motor activities; atropine and verapamil reduced but bethanechol increased gastric contractions under the same experimental conditions. Restraint at 4°C markedly elevated the ulcer index. The frequency of gastric contractions was significantly increased in the first hr but the amplitude was depressed during the whole 2-hr observation period. Gastric tone initially fell but rose in the second hr of cold-restraint stress. Atropine and verapamil pretreatment prevented stress-induced ulcer formation and suppressed the frequency and amplitude of gastric contraction. Bethanechol stimulated both frequency and amplitude without significantly influencing stress ulcer size. It is unlikely that gastric hypermotility plays a major role in stress ulceration; the stomach smooth muscle-relaxing action of atropine and verapamil may contribute only partly to their antiulcer effects.

Smoothness of the action of the gauge transformation group on connections

The NLF–Lie group structure of the group 𝒢 of the gauge transformations, defined as the group of sections of the bundle P[G] associated to the principal bundle P(M,G), is discussed. Other current definitions of the group of gauge transformations are shown to admit a nontrivial smooth structure only in the case of compact G. The space 𝒞 of principal connections, as well, is given the structure of local affine NLF-manifold, after identifications of connections with sections of a convenient vector bundle on M. Finally, the smoothness of the action of 𝒢 on 𝒞 is proved in general. In the case of compact M, the group 𝒢 becomes a tame Fréchet–Lie group and the action a tame smooth action.

Finite group actions on 3-manifolds

If G is a finite group acting smoothly on a closed surface F, it is well known that G leaves invariant some Riemannian metric of constant curvature on F. Thus any action of G on the 2-sphere S 2 is conjugate in Diff(S 2) to an orthogonal action. If G acts on the torus SX• S 1, there is a G-invariant flat metric on S a • S 1, and if G acts on a surface F with negative Euler number, then F admits a G-invariant hyperbolic metric. Recently Thurston, [Th 1, Th 2, Th 3], has described the eight 3-dimensional geometries which provide geometric structures for closed 3-manifolds in the same way that the 2-sphere S 2, the Euclidean plane E 2 and the hyperbolic plane H 2 provide geometric structures for surfaces. See also the survey article by Scott [Sc4]. Thurston also conjectured that if M is a closed 3-manifold with a geometric structure modelled on one of these eight geometries, say X, then any smooth action of a finite group G on M should leave invariant some metric on M inducing the geometry X. We will say that G preserves the geometric structure on M in this case. It should be noted that the restriction to smooth actions of G on M is essential. For Bing [Bi] showed that there are involutions of S 3 whose fixed set is a wild 2-sphere. However, in dimension two, it was proved by Eilenberg [Ei] that any action of a finite group on a surface is conjugate to a smooth action. In this paper, our main result asserts that Thurston's conjecture holds for five of the eight geometries. The result is the following.

Reversal of renal and smooth muscle actions of the thromboxane mimetic U-44069 by diltiazem

U-44069 is a stable prostaglandin (PG) H2 analogue and a potent vasoconstrictor. Its in vivo and in vitro actions mimic those of thromboxane A2. We have studied the effects of the calcium antagonist diltiazem upon the vasoconstriction induced by U-44069 using isolated rat aortic smooth muscle and isolated perfused rat kidney (IPRK). The administration of 10(-6)M U-44069 elicited maximally effective contractions in isolated aortic rings and increased 45Ca uptake from a control value of 285 +/- 6 mumol/kg to 344 +/- 8 mumol/kg. Diltiazem reduced U-44069-induced tension development and 45Ca uptake of isolated aortic smooth muscle 73 +/- 2 and 91 +/- 3%, respectively. The dose dependency of each of these effects of diltiazem was similar (EC50 = 369 nM and 334 nM for tension and 45Ca flux, respectively). When administered to the IPRK, 10(-6) M U-44069 caused a 82 +/- 3% decrease in glomerular filtration rate (GFR) and a 80 +/- 4% decrease in filtration fraction but reduced renal perfusate flow (RPF) only 13 +/- 8% (P less than 0.005). Diltiazem completely reversed the actions of U-44069 on the IPRK (EC50 = 288 nM and 323 nM for GFR and RPF, respectively). Diltiazem thus inhibited U-44069-induced tension development and 45Ca uptake by vascular smooth muscle and increased GFR within identical dose ranges. The contractile response of isolated rat glomeruli was also assessed. U-44069 reduced the volume of isolated glomeruli, but this action was neither prevented nor reversed by diltiazem. These results are consistent with the hypothesis that diltiazem increased GFR by inhibiting U-44069-induced Ca influx at preglomerular vessels.

A study on the gear cutting method of spiral bevel gears with a tapered tooth height (Point-contact gear cutting method without cutter-tilt)

Point-contact spiral bevel gears substantially have both lengthwise and profile ease-off to get smooth action in the gearing. This paper is concerned with the gear cutting method of point-contact spiral bevel gears with a tapered tooth height. In the method, the cutter axis is not tilted. The point-contact gears are obtained by the Gleason type cutter in which inside blades are modified to a circular arc. A pair of gears cut by this method meshes with little relative curvature. The main procedure of the method is to determine THE MACHINE SETTING taking account of the gear engagement. The theory is developed assuming that the Gleason hypoid generator is used without cutter-tilt and the generating relationships are unaltered.

Fixed point sets of smooth group actions on disks and Euclidean spaces. A survey

Fixed point sets of smooth group actions on disks and Euclidean spaces. A survey

Pseudofree representations and 2-pseudofree actions on spheres

We characterize 1 1 -pseudofree real representations of finite groups. We apply this to show that the representations at fixed points of a 2 2 -pseudofree smooth action of a finite group on a sphere of dimension ⩾ 5 \geqslant 5 are topologically equivalent. Moreover with one possible exception, the sphere is G G -homeomorphic to a linear representation sphere.

Interaction of 2',3'-di-O-nitro-5'-(N-ethylcarboxamido)adenosine with adenosine receptors in intestinal smooth muscle.

The adenosine derivative, 2'3'-di-O-nitro-(5'-N-ethylcarboxamido)adenosine (DINECA), caused relaxation in several isolated smooth muscle preparations including guinea pig taenia caeci, beef coronary arteries, and rabbit small intestine. In rabbit small intestine the response profile of DINECA action differed from that of established adenosine receptor agonists and, in contrast with the latter, its relaxant effect was only partially reversed by the antagonist 8-p-sulfophenyltheophylline. Concentration-response curves to 5'-(N-ethylcarboxamido)adenosine (NECA), but not those to DINECA, were significantly shifted to the right by 100 microM of 8-sulfophenyltheophylline. Tissues exposed previously to DINECA became refractory to adenosine, an effect not observed with tissues exposed to NECA, suggesting that DINECA became bound to adenosine receptors. Adenylate cyclase from neuroblastoma cells, containing Ra-type adenosine receptors, was stimulated by 2-chloroadenosine and NECA but not by DINECA. The results suggest that most of the smooth muscle relaxant actions of DINECA are not due to interaction with adenosine receptors but are probably due to its function as a nitrate. However, DINECA appears to interact with adenosine receptors, causing long lasting inhibition of adenosine action in rabbit intestine. Such actions may contribute to the overall response to DINECA application in vivo, although lowering of blood pressure due to the high reactivity of the vasculature to nitrates may be the initial and major effect.

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π, X,ρ) be such an action, where ρ: π→ Diff( X) is a homomorphism. We assume that ϱ extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\\ X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π i , X i , ρ i ) be actions as above ( i= 1,2). Then, given an isomorphism f of π 1 onto ϱ 2, there is a diffeomorphism h: X 1→ X 2 such that h( ρ 1( r) x)= ρ 2( f( r) h( x). As an application, we try to decide the structure of affine motions of some euclidean space R n . First we verify the conjecture of [17, § 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A( n) acts properly discontinuously and with compact quotient of R n , then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff( R n ) if they are isomorphic (cf.[8]).

A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Zp action on Sn or Dn, then F is a smooth manifold with a unitary structure. Conversely; most Zp homology disks or spheres with unitary structures are fixed point sets of smooth Zp actions on Dn or Sn for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Zp action on some Z[l/2]-homology disk or sphere. This result is in general the best possible.