2-step Groups Research Articles

Long-term safety of once-daily lixisenatide in Japanese patients with type 2 diabetes mellitus: GetGoal-Mono-Japan

AimsThis 76-week, open-label, parallel-group study assessed the long-term safety of once-daily lixisenatide monotherapy in Japanese patients with type 2 diabetes mellitus. MethodsPatients were randomized to receive lixisenatide in a 2-step or a 1-step dose-increase regimen. The primary objective was to assess the safety of lixisenatide at week 24 by a descriptive comparison of the 2- and 1-step groups. ResultsAs expected with treatment with a glucagon-like peptide-1 agonist, nausea was the most common treatment-emergent adverse event (2-step group: n=12/33 [36.4%] vs 1-step group: n=18/36 [50.0%] up to week 24). In total, 5/33 patients (15.2%; 2-step group) and 2/36 patients (5.6%; 1-step group) prematurely discontinued treatment up to week 24, mainly due to adverse events. Serious treatment-emergent adverse events occurred in 2/33 patients (6.1%; 2-step group) versus 0/36 patients (0%; 1-step group) up to week 24. Symptomatic hypoglycemia occurred in 2/33 patients (6.1%; 2-step group) versus 1/36 patients (2.8%; 1-step group) up to week 24, with no severe events reported. Glycated hemoglobin, fasting plasma glucose, and body weight were reduced from baseline at weeks 24 and 76. ConclusionIn Japanese patients with type 2 diabetes mellitus, once-daily lixisenatide monotherapy was well tolerated, with less nausea with the 2-step regimen.

Coronary Computed Tomography Angiography Alone Versus Confirmatory Functional Testing for Guiding Treatment Strategy for Patients With Intermediate Coronary Artery Stenosis

Intermediate coronary artery stenosis (≥50% and <90%) on coronary computed tomography angiography (CTA) is usually considered as a significant lesion. However, anatomical diagnosis is not well correlated with the functional significance of myocardial ischemia. We investigated whether functional testing in addition to coronary CTA improves outcomes of patients with intermediate stenosis, compared with the 1-step CTA-alone-based strategy. From 2006 to 2011, we consecutively enrolled 335 patients with chest pain with intermediate stenosis detected by an initially performed coronary CTA. Of these, 159 patients followed the 1-step strategy, whereas 176 followed the 2-step strategy with confirmatory functional tests. One-year follow-up data were obtained for all patients. The primary end point was a composite of cardiovascular death, nonfatal myocardial infarction, and repeated or delayed revascularization (major adverse cardiac event) within a year. Baseline clinical parameters were comparable between patients of the 2 different strategies. The rate of invasive catheterization or percutaneous intervention was 75.5% in the 1-step group and 35.2% in the 2-step group (p <0.001). Consequently, more patients in the 2-step group were medically treated without unnecessary revascularization compared with patients in the 1-step group (71.0% vs 40.9%, p <0.001). Only 2.5% of the patients who received medical treatment in the 2-step group finally received delayed revascularization, whereas 14% in the 1-step group did. Overall, the primary end point occurred in 11.3% in the 1-step group and 4.0% in the 2-step group (p = 0.011). In conclusion, confirmatory functional testing reduces invasive catheterization and coronary intervention and improves clinical outcomes in patients with intermediate stenosis on coronary CTA.

Spectral multipliers on Heisenberg–Reiter and related groups

Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < \infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s > d/2$ is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by M\"uller and Stein and by Hebisch, and for the free group $N_{3,2}$ by M\"uller and the author.

Norms of inner derivations for multiplier algebras of [formula omitted]-algebras and group [formula omitted]-algebras

The derivation constant K ( A ) ⩾ 1 2 has been previously studied for unital non-commutative C ⁎ -algebras A. This paper begins the study of K ( M ( A ) ) where M ( A ) is the multiplier algebra of a non-unital C ⁎ -algebra A. Two results are obtained giving separate conditions on A which imply that K ( M ( A ) ) ⩽ 1 . These results are applied to A = C ⁎ ( G ) for a number of locally compact groups G including SL ( 2 , R ) , SL ( 2 , C ) and several 2-step solvable groups. In these cases, K ( M ( A ) ) = 1 . On the other hand, if G is a (non-abelian) amenable [SIN]-group then K ( M ( A ) ) = 1 2 .

Bacterial-sized particle inflow through sutured clear corneal incisions in a laboratory human model

To determine the effectiveness of a single radial suture placement in 2 clear corneal incision (CCI) configurations in preventing inflow of bacterial-sized particles. Wilmer Eye Institute, Baltimore, Maryland, USA. Experimental study. Ten human globes were used. Two 25-gauge needles connected to a saline solution bag and to a digital manometer were inserted through the limbus 120 degrees apart. Four incision-suture combinations were evaluated sequentially in each cornea. All incisions had the same dimensions (2.75 mm width, 3.00 mm length). Two incisions were single plane, and 2 were biplane. One incision from each configuration was left unsutured, and the other was sutured (10-0 nylon). With a preset 10 mm Hg intraocular pressure (IOP), India ink was applied to the incision site and a sudden IOP fluctuation was induced. Inflow was outlined and measured by planimetry. There was a significant increase in area and linear distance of India-ink inflow after pressure challenge in all groups (P < .05), but with important differences among them. When the sutured and unsutured groups from each CCI were compared, there was a significantly smaller area of inflow in the 2-step unsutured group (P < .05). The linear inflow was higher in both sutured groups; however, the difference was significant for the 2-step incision configuration only (P < .05). A single radial suture reduced the area of inflow of bacterial-sized particles but increased the linear distance of inflow in single-plane 3.0 mm incisions. Suture placement in a stepped incision increased inflow of bacterial-sized particles. No author has a financial or proprietary interest in any material or method mentioned.

Editor's Comment and Q&A: Randomized clinical trial comparing control of maxillary anchorage with 2 retraction techniques

Introduction The objective of this pilot randomized clinical trial was to investigate the relative effectiveness of anchorage conservation of en-masse and 2-step retraction techniques during maximum anchorage treatment in patients with Angle Class I and Class II malocclusions. Methods Sixty-four growing subjects (25 boys, 39 girls; 10.2-15.9 years old) who required maximum anchorage were randomized to 2 treatment techniques: en-masse retraction (n = 32) and 2-step retraction (n = 32); the groups were stratified by sex and starting age. Each patient was treated by a full-time clinic instructor experienced in the use of both retraction techniques at the orthodontic clinic of Peking University School of Stomatology in China. All patients used headgear, and most had transpalatal appliances. Lateral cephalograms taken before treatment and at the end of treatment were used to evaluate treatment-associated changes. Differences in maxillary molar mesial displacement and maxillary incisor retraction were measured with the before and after treatment tracings superimposed on the anatomic best fit of the palatal structures. Differences in mesial displacement of the maxillary first molar were compared between the 2 treatment techniques, between sexes, and between different starting-age groups. Results Average mesial displacement of the maxillary first molar was slightly less in the en-masse group than in the 2-step group (mean, –0.36 mm; 95% CI, –1.42 to 0.71 mm). The average mesial displacement of the maxillary first molar for both treatment groups pooled (n = 63, because 1 patient was lost to follow-up) was 4.3 ± 2.1 mm (mean ± standard deviation). Boys had significantly more mesial displacement than girls (mean difference, 1.3 mm; P <0.03). Younger adolescents had significantly more mesial displacement than older adolescents (mean difference, 1.3 mm; P <0.02). Conclusions Average mesial displacement of the maxillary first molar with 2-step retraction was slightly greater than that for en-masse retraction, but the difference did not reach statistical significance. This finding appears to contradict the belief of many clinicians that 2-step canine retraction is more effective than en-masse retraction in preventing clinically meaningful anchorage loss. The objective of this pilot randomized clinical trial was to investigate the relative effectiveness of anchorage conservation of en-masse and 2-step retraction techniques during maximum anchorage treatment in patients with Angle Class I and Class II malocclusions. Sixty-four growing subjects (25 boys, 39 girls; 10.2-15.9 years old) who required maximum anchorage were randomized to 2 treatment techniques: en-masse retraction (n = 32) and 2-step retraction (n = 32); the groups were stratified by sex and starting age. Each patient was treated by a full-time clinic instructor experienced in the use of both retraction techniques at the orthodontic clinic of Peking University School of Stomatology in China. All patients used headgear, and most had transpalatal appliances. Lateral cephalograms taken before treatment and at the end of treatment were used to evaluate treatment-associated changes. Differences in maxillary molar mesial displacement and maxillary incisor retraction were measured with the before and after treatment tracings superimposed on the anatomic best fit of the palatal structures. Differences in mesial displacement of the maxillary first molar were compared between the 2 treatment techniques, between sexes, and between different starting-age groups. Average mesial displacement of the maxillary first molar was slightly less in the en-masse group than in the 2-step group (mean, –0.36 mm; 95% CI, –1.42 to 0.71 mm). The average mesial displacement of the maxillary first molar for both treatment groups pooled (n = 63, because 1 patient was lost to follow-up) was 4.3 ± 2.1 mm (mean ± standard deviation). Boys had significantly more mesial displacement than girls (mean difference, 1.3 mm; P <0.03). Younger adolescents had significantly more mesial displacement than older adolescents (mean difference, 1.3 mm; P <0.02). Average mesial displacement of the maxillary first molar with 2-step retraction was slightly greater than that for en-masse retraction, but the difference did not reach statistical significance. This finding appears to contradict the belief of many clinicians that 2-step canine retraction is more effective than en-masse retraction in preventing clinically meaningful anchorage loss.

Randomized clinical trial comparing control of maxillary anchorage with 2 retraction techniques

The objective of this pilot randomized clinical trial was to investigate the relative effectiveness of anchorage conservation of en-masse and 2-step retraction techniques during maximum anchorage treatment in patients with Angle Class I and Class II malocclusions. Sixty-four growing subjects (25 boys, 39 girls; 10.2-15.9 years old) who required maximum anchorage were randomized to 2 treatment techniques: en-masse retraction (n = 32) and 2-step retraction (n = 32); the groups were stratified by sex and starting age. Each patient was treated by a full-time clinic instructor experienced in the use of both retraction techniques at the orthodontic clinic of Peking University School of Stomatology in China. All patients used headgear, and most had transpalatal appliances. Lateral cephalograms taken before treatment and at the end of treatment were used to evaluate treatment-associated changes. Differences in maxillary molar mesial displacement and maxillary incisor retraction were measured with the before and after treatment tracings superimposed on the anatomic best fit of the palatal structures. Differences in mesial displacement of the maxillary first molar were compared between the 2 treatment techniques, between sexes, and between different starting-age groups. Average mesial displacement of the maxillary first molar was slightly less in the en-masse group than in the 2-step group (mean, -0.36 mm; 95% CI, -1.42 to 0.71 mm). The average mesial displacement of the maxillary first molar for both treatment groups pooled (n = 63, because 1 patient was lost to follow-up) was 4.3 ± 2.1 mm (mean ± standard deviation). Boys had significantly more mesial displacement than girls (mean difference, 1.3 mm; P <0.03). Younger adolescents had significantly more mesial displacement than older adolescents (mean difference, 1.3 mm; P <0.02). Average mesial displacement of the maxillary first molar with 2-step retraction was slightly greater than that for en-masse retraction, but the difference did not reach statistical significance. This finding appears to contradict the belief of many clinicians that 2-step canine retraction is more effective than en-masse retraction in preventing clinically meaningful anchorage loss.

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.

The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set \( {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) of K-orbits in the dual \( \mathfrak{n}* \) of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence \( \Delta {\left( {K,N} \right)} \leftrightarrow {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Δ(K, N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.

Analysis of clear corneal incision integrity in an ex vivo model

To determine the most favorable sutureless incision configuration to minimize extraocular fluid inflow after cataract surgery. The Wilmer Eye Institute, Baltimore, Maryland, USA. Five fresh human eyes were used in the study. Two 27-gauge needles connected to a saline solution bag and a digital manometer were inserted through the limbus 180 degrees from each other. Intraocular pressure (IOP) was maintained at 15 to 20 mm Hg. Three incisions were performed in different quadrants of each cornea: uniplanar 1.0 mm and 3.0 mm tunnel lengths and 2-step 3.0 mm tunnel length. India ink was applied to the incision site, and IOP fluctuation was induced by applying pressure to the limbal area of the opposite quadrant using an ophthalmodynamometer. Imaging was performed before and after pressure application. The linear distance of India ink inflow after pressure application was higher than the prepressure measurements in the 1.0 mm and 3.0 mm incision groups (P = .039 and P = .023, respectively). The maximum mean of inflow after pressure application was not higher than the prepressure measurement in the 2-step incision group (P = .105). The total ink area measured before and after pressure applications in the incisions of the 3 groups was not significantly different (P = .285). Intraocular pressure fluctuations may promote entry of bacteria-size particles into the eye when 1.0 mm and 3.0 mm single-plane incisions are performed. Stepped incisions seem to be more resistant to inflow in the presence of IOP fluctuation.

Anisotropic quaternion Carnot groups: Geometric analysis and Green function

We construct examples of 2-step Carnot groups related to quaternions and study their fine structure and geometric properties. This involves the Hamiltonian formalism, which is used to obtain explicit equations for geodesics and the computation of the number of geodesics joining two different points on these groups. We are able to find the explicit lengths of geodesics. We present the fundamental solutions of the Heat and sub-Laplace equations for these anisotropic groups and obtain some estimates for them, which may be useful.

Safety and effect of two-step portal vein embolization in treatment of hepatic fibrosis: experiment with rabbits

To evaluate the safety and feasibility of two-step portal vein embolization in treatment of hepatic fibrosis. Twenty rabbit underwent intraperitoneal injection of carbon tetrachloride to establish models of hepatic fibrosis, and then were randomly divided into 2 equal groups: one-step group in which ileocolic vein was exposed, portal angiography was conducted, a catheter was inserted into the upper end of the main trunk of portal vein beyond the branches of right lower lobe of liver, a mixture of absolute ethanol- lipiodol was infused into the portal vein branches supplying 70% - 90% of the liver via a catheter and then a coil was put at the proximal side of hepatic portal, and two-step group in which the portal vein branches were embolized by using a mixture of absolute ethanol-lipiodol first at the first step and then by using polyvinyl alcohol particles 4 weeks later during the second operation. Follow-up was conducted for 7 weeks. The serum levels of alanine transaminase (ALT), aspartate aminotransferase (AST), total bilirubin, and albumin were examined before and 4, 7, and 14 days after the portal vein embolization. Computed tomography was performed before and after the embolization to observe the changes of the liver volume and compare the volumes of the embolized and non-embolized lever segments. After the rabbits were sacrificed, pathological examination of the liver was conducted. Successful portal vein embolization was achieved in both groups. One rabbit of the one-step group died during the operation and 2 rabbits of the 2-step group died of bleeding or infection. The levels of ALT and AST increased after the procedure, peaked at the fourth day, and returned to normal one week later. Pathology showed that after the embolization the right lower lobe of liver was hypertrophic to a certain degree and the other part of the liver became atrophic. The hypertrophy percentage of non-embolized liver was higher by 7.0 +/- 1.3% in the one-step group and by 15.4% +/- 3.0% in the two-step group (both P < 0.05). A more great degree of hypertrophy in the non-embolized liver segment can be achieved by two-step portal vein embolization.

Rehabilitation of the severely atrophied maxilla by horseshoe Le Fort I osteotomy (HLFO)

Objective The purpose of this study was to determine the long-term outcome of the horseshoe Le Fort I osteotomy (HLFO) as a preprosthetic operation technique for implant insertion in the extremely atrophied maxilla. Study design 36 patients (8 male, 28 female, average age 57.6 years) underwent HLFO combined with iliac crest bone grafting. They were divided into 2 groups: group A with 12 patients who simultaneously received 100 implants; group B with 24 patients where 176 implants were inserted in 18 patients in a second-stage procedure. Clinical and radiographic outcome with regard to implant osseointegration, alveolar bone height in the canine and molar regions, peri-implant bone loss and satisfaction of patients (esthetics, masticatory function, overall treatment) were investigated in all cases. Results The overall 2-year failure-free fraction of implants was 95.5%; the 5-year failure-free fraction was 89.3%. In the 1-step group the 2-year and 5-year failure-free fractions were 95.9% and 86.9%, respectively, in the 2-step group 95.0% and 91.3% (log rank test P = .57). A total of 27 implants were lost during the entire follow-up: 14 in 6 patients of the 1-stage and 13 in 9 patients of the 2-stage group. The mean loss of alveolar bone after augmentation in the canine and molar regions was almost equal in both groups (overall means for the 2 regions 3.67 ± 2.77 and 4.42 ± 2.72 mm, respectively). The relationship between the jaws and thereby the esthetic profile could be improved in all cases. All patients were satisfied with the dental rehabilitation and the achieved new esthetic appearance. Conclusions HLFO combined with iliac bone grafting is a feasible preprosthetic technique prior to implant insertion in cases of severe atrophy of the maxillary alveolar ridge, leading to satisfying implant survival and rehabilitation of function.

Donor source affects the outcome of ocular surface reconstruction in chemical or thermal burns of the cornea

Purpose To study the association between surgical approach and postoperative results in chemical and thermal burns of the cornea. Design Retrospective, interventional, noncomparable case series. Participants Thirty-two eyes of 32 patients with chemical (n = 27) or thermal (n = 5) burns of the cornea that were associated with total limbal dysfunction. Eight eyes had a history of previous keratoplasty. Intervention Patients were treated by amniotic membrane transplantation combined with either conjunctivolimbal autograft transplantation (autograft group, n = 11) or keratolimbal allograft transplantation (allograft group, n = 21). Fifteen eyes had simultaneous penetrating keratoplasty (simultaneous group), and 6 had keratoplasty several months after ocular surface reconstruction (2-step group). Main outcome measures Reconstruction of the corneal surface by corneal epithelium, clarity of the cornea, and incidence of postoperative complications. The outcome was compared between the autograft and allograft groups and also between the simultaneous and 2-step groups. Results At final examination, 17 eyes (53.1%) showed stable corneal epithelialization. Preoperative conditions were similar in the autograft and allograft groups and also in the simultaneous and 2-step groups. The autograft group showed significantly better results than the allograft group in both corneal epithelialization (Kaplan–Meier analysis, P = 0.003) and clear cornea ( P = 0.010). Although the incidences of corneal epithelialization and clear corneas did not significantly differ between the simultaneous and 2-step groups, the former had a higher rate of endothelial rejection in the central graft ( P = 0.019). Conclusions In chemical or thermal burns of the cornea with monocular involvement, autografting should be considered as a first choice of surgery. Even in eyes with opaque corneal stroma, it may be safer to perform ocular surface reconstruction first, followed by keratoplasty as a secondary procedure.

A Blow-up Theorem for regular hypersurfaces on nilpotent groups

We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate for the Hausdorff dimension of the characteristic set relative to C1,1 hypersurfaces in 2-step groups and we prove that it has finite Q–2 Hausdorff measure, where Q is the homogeneous dimension of the group.

Geodesics and isometries of Carnot groups

We will investigate geodesics in 2-step Carnot groups. And, by applying this, we will prove that every isometry of 2-step Carnot group $N$ fixing the identity is an automorphism of Lie group $N$. Our argument will be elementary. Finally, we will see that the same conclusion holds for general Carnot groups.

Studies on effective PCR screening strategies for white spot syndrome virus (WSSV) detection in Penaeus monodon brooders.

We re-tested stored (frozen) DNA samples in 5 independent polymerase chain reaction (PCR) replicates and confirmed that equivocal test results from a previous study on white spot syndrome virus (WSSV) in brooders and their offspring arose because amounts of WSSV DNA in the test samples were near the sensitivity limits of the detection method. Since spawning stress may trigger WSSV replication, we also captured a fresh batch of 45 brooders for WSSV PCR testing before and after spawning. Replicates of their spawned egg batches were also WSSV PCR tested. For these 45 brooders, WSSV prevalence before spawning was 67% (15/45 1-step PCR positive, 15/45 2-step PCR positive and 15/45 2-step PCR negative). Only 27 (60%) spawned successfully. Of the successful spawners, 56% were WSSV PCR positive before spawning and 74% after. Brooders (15) that were heavily infected (i.e. 1-step PCR positive) when captured mostly died within 1 to 4 d, but 3 (20%) did manage to spawn. All their egg batch sub-samples were 1-step PCR positive and many failed to hatch. The remaining 30 shrimp were divided into a lightly infected group (21) and a 2-step PCR negative group (9) based on replicate PCR tests. The spawning rates for these 2 groups were high (81 and 78%, respectively). None of the negative spawners (7) became WSSV positive after spawning and none gave egg samples positive for WSSV. In the lightly infected group (21), 6 brooders were 2-step WSSV PCR negative and 15 were 2-step WSSV PCR positive upon capture. However, all of them were WSSV PCR positive in replicate tests and after spawning or death. Four died without spawning. The remaining 17 spawned but only 2 gave egg samples PCR negative for WSSV. The other 15 gave PCR positive egg samples, but they could be divided into 2 spawner groups: those (7) that became heavily infected (i.e. 1-step PCR positive) after spawning and those (8) that remained lightly infected (i.e. became or remained 2-step PCR positive only). Of the brooders that became heavily infected after spawning, almost all egg sample replicates (91 %) tested 2-step PCR positive. One brooder even gave heavily infected (i.e. 1-step PCR positive) egg samples. For the brooders that remained lightly infected after spawning, only 27% of the egg sample replicates were 2-step PCR positive. Based on these results, we recommend that to avoid false negatives in WSSV PCR brooder tests screening tests should be delayed until after spawning. We also recommend, with our PCR detection system, discarding all egg batches from brooders that are 1-step PCR positive after spawning. On the other hand, it may be possible with appropriate monitoring to use eggs from 2-step PCR positive brooders for production of WSSV-free or lightly infected postlarvae. These may be used to stock shrimp ponds under low-stress rearing conditions.

Tibial nerve function during tibial lengthening. Measurement of nerve conduction and blood flow in rabbits.

We evaluated in 20 Japanese white rabbits the effects of tibial lengthening on tibial nerve conduction and intraneural blood flow at the end of lengthening. Both tibiae were distracted 1 mm per day. The distraction frequency was in 2 steps (0.5 mm/12 h) on the right side and in 120 steps (0.0083 mm/12 min) on the left. The rabbits were separated into 4 subgroups based on the percentage of lengthening: 0 (control), 10, 20, and 30 percent. In the 2-step group, nerve conduction was delayed at 20 and 30 percent lengthening, compared to the control group, while in the 120-step group, it was delayed only at 30 percent lengthening. Intraneural blood flow in the 2-step group was decreased at 10, 20, and 30 percent lengthenings, while in the 120-step group it was reduced at 30 percent lengthening. Our findings indicate that an increase in the frequency of distraction reduces the impairment of nerve function during bone lengthening.

On 2-step solvable groups of finite Morley rank

We prove the following results: Theorem 1. Let G G be a connected, centerless, solvable group of class 2 and of finite Morley rank. Then we can interpret in G G finitely many connected, solvable of class 2 and centerless algebraic groups G ~ 1 , … , G ~ n {\tilde G_1}, \ldots ,{\tilde G_n} over algebraically closed fields K i {K_i} in such a way that G G interpretably imbeds in G ~ = G ~ 1 ⊕ ⋯ ⊕ G ~ n \tilde G = {\tilde G_1} \oplus \cdots \oplus {\tilde G_n} . Furthermore, G ′ = ( G ~ ) ′ G’ = (\tilde G)’ . Let F ( G ) F(G) denote the Fitting subgroup of G G . Theorem 2. Let G , G ~ , G ~ i G,\tilde G,{\tilde G_i} be as in Theorem 1. Then (i) F ( G ) = F ( G ~ ) ∩ G F(G) = F(\tilde G) \cap G . (ii) F ( G ) F(G) has a complement V V in G : G = F ⋊ V G:G = F \rtimes V . (iii) Elements of F ( G ) F(G) are unipotent elements of G G in G ~ \tilde G . (iv) If the characteristic of each base field K i {K_i} of G ~ i {\tilde G_i} is different from 0, then V V is definable and its elements are semi-simple in G ~ \tilde G .

Solvability of Transversally Elliptic Differential Operators on Nilpotent Lie Groups

1. Introduction. In this paper, we study some aspects of solvability of left invariant differential operators on connected, simply connected nilpotent Lie groups. In Part I, we prove a general necessary condition for microlocal solvability and apply it to the case of a transversally elliptic operator on a 2-step nilpotent Lie group. We show in Part II that for a special class of 2-step groups, the criterion of Part I is also sufficient for local solvability; the main argument here involves the division of distributions by analytic functions. Finally, Part III is devoted primarily to a sufficient condition for the existence of a global fundamental solution; this condition is satisfied for the operators discussed in Part II. We now describe these results in more detail. Let G be a (connected, simply connected) 2-step nilpotent Lie group with Lie algebra g. Then g = g1 0 g2 (a vector space direct sum) with g2 = [g, g] = [g1, gl ]. This grading of g induces a grading of the universal enveloping algebra u(g), u(g) = (30=o u,(g). Thus a left invariant differential operator L E u(g) can be written L = ET0 LL, with Li homogeneous of degree j. In the homogeneous case (L = Lm), criteria for local solvability of L have been given by the authors individually ([21], [3], [4]) and severally ([5]), by Levy-Bruhl ([12], [13], [14]), by Rothschild-Tartakoff ([24]), and by others; a survey of some of these results is found in [18]. In Parts I and II of this paper, we use new techniques to obtain results for non-homogeneous operators. We say that L is tranisversally elliptic (or elliptic in the genieraitinlg directions) if for every nontrivial 1-dimensional unitary representation u of G, a(Lm) * 0. (Note that a 0 on g2.) A differential operator P is locally solvable at xo if there is a neighborhood U of x0 such that the equation