Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from C-n to C-3 n >= 3

In this article we show that for corank 1, quasi-homogeneous and finitely determined map germs f : (C-n, 0)-> (C-3, 0), n >= 3 one can obtain formulae for the polar multiplicities defined on the following stable types of f, f(Delta(f) and f(Sigma(n-2,1)(f), in terms of the weights and degrees of f. As a consequence we show how to compute the Euler obstruction of such stable types, also in terms of the weights and degrees of f.

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0) and g:(X,0)→(C,0). We give applications when g is a generic linear form and when f and g have isolated singularities.

Finite determinacy and Whitney equisingularity of map germs from C(n) to C(2n-1)

We show that a holomorphic map germ f : (C(n), 0) -> (C(2n-1), 0) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n >= 3, we have that mu(D(2)(f)) = 2 mu(D(2)(f)/S(2))+C(f)-1, where D(2)(f) is the lifting of the double point curve in (C(n) x C(n), 0), mu(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f(t)(x)) of f and show that if F is mu-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f(t) is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.

Stable singularities of co-rank one quasi homogeneous map germs from (ℂn+1, 0) to (ℂn, 0), n = 2, 3

In this article, we investigate the geometry of quasi homogeneous corank one finitely determined map germs from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. We give a complete description, in terms of the weights and degrees, of the invariants that are associated to all stable singularities which appear in the discriminant of such map germs. The first class of invariants which we study are the isolated singularities, called 0-stable singularities because they are the 0-dimensional singularities. First, we give a formula to compute the number of An points which appear in any stable deformation of a quasi homogeneous co-rank one map germ from (ℂn+1, 0) to (ℂn, 0) with n = 2, 3. To get such a formula, we apply the Hilbert's syzygy theorem to determine the graded free resolution given by the syzygy modules of the associated iterated Jacobian ideal. Then we show how to obtain the other 0-stable singularities, these isolated singularities are formed by multiple points and here we use the relation among them and the Fitting ideals of the discriminant. For n = 2, there exists only the germ of double points set and for n = 3 there are the triple points, named points A1,1,1 and the normal crossing between a germ of a cuspidal edge and a germ of a plane, named A2,1. For n = 3, there appear also the one-dimensional singularities, which are of two types: germs of cuspidal edges or germs of double points curves. For these singularities, we show how to compute the polar multiplicities and also the local Euler obstruction at the origin in terms of the weights and degrees. © 2013 Pushpa Publishing House.

Krylov subspace approximations for the exponential Euler method: Error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product φ(tA)b where φ(z) = [exp(z) - 1]/z. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where A is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to φ(tA)b, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error.

Going through the obstruction, the Torrens System Assurance Fund and contemporary solutions : a tale weaved from a story of a nun, a romantic triangle and sibling corruption

Private title insurance has been the subject of much debate by law reform bodies and academics. This article adds a new dimension to the discussion by analysing its role against a recent scenario where a nun was betrayed by the actions of her brother, and compensation payable from the assurance fund, after much challenge by the registrar, amounted to in excess of $4 million.We ask whether the slow burning of title insurance into the psyche of Australian home purchasers will see state-based assurance fundings looking to minismise their role in the Torrens system. We also query how the rather more immediate electronic establishment of electronic conveyancing will alter the balance between the assurance fund, private title insurance and the increasing responsibilities on stakeholdes involved in conveyancing.

Adjustment of apnea-hypopnea index with severity of obstruction events enhances detection of sleep apnea patients with the highest risk of severe health consequences

Introduction Presently, the severity of obstructive sleep apnea (OSA) is estimated based on the apnea-hypopnea index (AHI). Unfortunately, AHI does not provide information on the severity of individual obstruction events. Previously, the severity of individual obstruction events has been suggested to be related to the outcome of the disease. In this study, we incorporate this information into AHI and test whether this novel approach would aid in discriminating patients with the highest risk. We hypothesize that the introduced adjusted AHI parameter provides a valuable supplement to AHI in the diagnosis of the severity of OSA. Methods This hypothesis was tested by means of retrospective follow-up (mean ± sd follow-up time 198.2 ± 24.7 months) of 1,068 men originally referred to night polygraphy due to suspected OSA. After exclusion of the 264 patients using CPAP, the remaining 804 patients were divided into normal (AHI < 5) and OSA (AHI ≥ 5) categories based on conventional AHI and adjusted AHI. For a more detailed analysis, the patients were divided into normal, mild, moderate, and severe OSA categories based on conventional AHI and adjusted AHI. Subsequently, the mortality and cardiovascular morbidity in these groups were determined. Results Use of the severity of individual obstruction events for adjustment of AHI led to a significant rearrangement of patients between severity categories. Due to this rearrangement, the number of deceased patients diagnosed to have OSA was increased when adjusted AHI was used as the diagnostic index. Importantly, risk ratios of all-cause mortality and cardiovascular morbidity were higher in moderate and severe OSA groups formed based on the adjusted AHI parameter than in those formed based on conventional AHI. Conclusions The adjusted AHI parameter was found to give valuable supplementary information to AHI and to potentially improve the recognition of OSA patients with the highest risk of mortality or cardiovascular morbidity.

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multidimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel mode

Lavage treatment of distal intestinal obstruction syndrome in children with cystic fibrosis

The efficacy, adverse reactions, and long-term effects of intestinal lavage treatment with a balanced electrolyte solution (Golytely) was evaluated in patients with cystic fibrosis and distal intestinal obstruction syndrome. Twenty-two patients with cystic fibrosis (mean age 21.8 years, range 14 to 34 years, 15 boys or men) who sough medical attention because of abdominal pain and a mass in the right iliac fossa received Golytely, 5.6 ± 1.9 L (mean ± 1 SD), either orally (n = 14) or via nasogastric tube (n = 8) during 5.6 ± 2.4 hours. No serious side effects occurred. Serum electrolyte values remained within normal limits. Body weight did not change significantly. Minor adverse reactions included bloating (n = 12), nausea (n = 8), vomiting (n = 1), and chills (n = 3). All but one patient reported impressive relief of symptoms and remained pain free for an average of 3 months (range 1 to 19 months). Symptoms of abdominal pain and radiologic signs of fecal impaction assessed before and after lavage both decreased significantly (P < .0001). During follow-up (mean 15.2 months, range 4 to 26 months), 11 patients required a total of 38 (range one to nine) additional doses of Golytely. Seven patients drank the solution at home (21 treatments); only two patients chose a nasogastric tube. In ten patients with symptoms of recurrent distal intestinal obstruction syndrome prior to institution of therapy, duration of hospitalization was significantly reduced by this treatment (5.1 ± 7.6 v 2.3 ± 6.3 hospital days per annum, P < .02). It is concluded that intestinal lavage is a well-accepted, safe, and effective therapy for distal intestinal obstruction syndrome in patients with cystic fibrosis.

Treatment of distal intestinal obstruction syndrome in cystic fibrosis with a balanced intestinal lavage solution

Conventional treatment of distal intestinal obstruction syndrome (DIOS) with high doses of pancreatic enzymes, mucolytic agents, and enemas is neither predictably effective nor rapid in action. In 6 cystic fibrosis patients with DIOS a balanced, non-absorbable intestinal lavage solution produced clinical and radiological improvement and striking improvement in DIOS scores. It is suggested that a balanced intestinal lavage solution should be considered as an alternative treatment for DIOS in patients with cystic fibrosis.

Neonatal intestinal obstruction associated with oral calcium supplementation

A case report of a 920 g infant developing a small intestinal obstruction following therapy for congestive cardiac failure is presented. Although the causation was thought to be milk curd obstruction, subsequent analysis revealed high concentration of calcium and phosphate in the stools. The possible pathogenesis is discussed in relation to the inspissated milk syndrome.