Stability theorems for positively graded domains

We consider a positively (and non-trivially) graded integral domain $R=\bigoplus_{i\ge 0}R_i$ of dimension $d\geq 2$, where $\dim(R_0)\geq 1$. Let $A=S^{-1}R$ be the localization of $R$ with respect to a multiplicatively closed set $S\subset R$. In this article, we establish that any unimodular row of length $d+1$ in $A$ can be completed to the first row of an invertible matrix $\alpha$ such that $\alpha$ is homotopic to the identity matrix. We also prove that the injective stability of $\text{K}_1(R)$ is $d+1$. Furthermore, we show that if $I\subset A$ is an ideal such that $\mu(I/I^2)=d$ and $\text{ht}(I)=\text{dim}(A)$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $\mu(-)$ represents the minimal number of generators. As a consequence, every projective $A$-module of rank $d$ with trivial determinant splits off a free summand of rank one. Finally, for a projective $R$-module $P$ of rank $d$, we establish that if the Quillen ideal $J(R_0,P)$ of $P$ is non-zero, then $P$ is cancellative.Comment: 21 pages. Comments are welcom

Efficient generation, unimodular element in a geometric subring of a polynomial ring

Let $R$ be a commutative Noetherian ring of dimension $d$. First, we define the "geometric subring" $A$ of a polynomial ring $R[T]$ of dimension $d+1$ (the definition of geometric subring is more general, see (1.2)). Then we prove that every locally complete intersection ideal of height $d+1$ is a complete intersection ideal. Thus improving the general bound of Mohan Kumar \cite{NMK78} for an arbitrary ring of dimension $d+1$. Afterward, we deduce that every finitely generated projective $A$-module of rank $d+1$ splits off a free summand of rank one. This improves the general bound of Serre \cite{Serre58} for an arbitrary ring. Finally, applications are given to a set-theoretic generation of an ideal in the geometric ring $A$ and its polynomial extension $A[X]$.Comment: 12 page

INFLUENCE OF CANCER SEVERITY AND FUNCTIONAL STATUS OF CANCER ON CARDIAC PARASYMPATHETIC INDICATORS

 Objective: To investigate the influence of cancer severity and functional status of cancer patients on cardiac parasympathetic indicators. Methods: A total of 267 patients with a fresh clinical diagnosis of solid malignant tumor not yet put on cancer therapy and 250 controls matched for age, sex of study subjects were included. Severity of cancer was defined based on the American Joint Committee on Cancer staging. Accordingly, study subjects were subdivided into early stage (Stage I and II combined) and advanced stage (Stage III and IV combined). In cancer patients, the Eastern Cooperative Oncology Group (ECOG) performance score and the Faces Pain Scale score (FPS) was noted. Two indicators of vagal function, expiratory:inspiratory ratio (E:I ratio) and root mean square of successive N-N interval difference (r-MSSD) were included. E:I ratio during deep breathing at six respiratory cycles/minute and r-MSSD at rest was obtained from 1 minute lead II electrocardiogram. Data were analyzed by applying suitable statistical tests. p≤0.05 was considered significant. Results: R-MSSD and E:I ratio was significantly reduced in the early and advanced stage of cancer compared to controls (p≤0.0001). r-MSSD and E:I ratio was significantly reduced in advanced stage compared to the early stage of cancer (p≤0.0001). r-MSSD and E:I ratio was significantly different in subgroups of stages of cancer and controls (p≤0.0001). In cancer patients, r-MSSD was negatively correlated with ECOG and FPS score (p≤0.0278, p≤0.0100). Conclusion: Severity of cancer affects vagal function. However, r-MSSD alone was associated with functional status (ECOG, FPS) of cancer patients

Polycystic ovary syndrome, blood group & diet: A correlative study in South Indian females

Aim: To find out the co-relation between polycystic ovary syndrome (PCOS) with blood group & diet in South Indian females, between the age-group of (20-30) years. Objectives: Correlative analysis of ABO & Rh system, dietary habits & alcohol consumption with PCOS. Materials & Methods: 100 patients between (20-30) years, diagnosed with PCOS were selected. A standard PCOS questionnaire was given. Blood group & dietary status data were collected. Patients were grouped according to ABO & Rh system considering their diet & alcohol intake (p≤0.05 significant). Result: Our data revealed that the highest risk of PCOS was observed in females with blood group ‘O’ positive followed by ‘B’ positive who were on mixed diet & used to consume alcohol. Our study also suggests that Rh negative individuals didn’t show any association with PCOS. Conclusion: The results of our study suggest that ‘O’ positive females, are more prone to PCOS. Though the relative frequency of B positive individuals are more in India, females with blood group O positive are more susceptible to PCOS, contributing factors being mixed diet & alcohol intake. So, early screening of ‘O’ positive &‘B’ positive females of reproductive age-group in South-India, could be used as a measure for timely diagnosis of PCOS, better management &also prevention of complications. However, further research should be done to investigate the multifaceted mechanisms triggering these effects

Quality assurance of simultaneous treatment of two targets in pelvic region planned with single isocenter using three dimensional conformal radiotherapy (3DCRT) technique

Purpose: The purpose of this study was to conduct quality assurance of a three dimensional conformal radiotherapy (3DCRT) of two targets in pelvis region planned with single isocenter technique. Methods: A treatment plan was generated with two identical water phantoms with ionization chamber (IC) sleeves (IC-1 & IC-2), simulated as if targets are in pelvis region, simultaneously irradiated with single isocenter technique with a dose prescription of 300 cGy for point dose verification. A two dimensional ion chamber array detector was used for fluence verification.Results: Calculated minimum, mean and maximum dose (in cGy) for IC-1 & IC-2 were 295, 303 and 307 as per dose volume histogram. The global dose maximum was found to be 307.4 cGy. Measured point doses to both lesions were within ±2.5% of the computed dose. A pass percentage of 97% was obtained with the set of criteria 3 mm distance to agreement and 3% dose difference for fluence verification.Conclusion: Treatment execution of two targets simultaneously with single isocenter can reduce positional errors and delivery time

Subrings of polynomial rings and the conjectures of Eisenbud-Evans

Let $R$ be a commutative Noetherian ring of dimension $d$. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring $R[T]$. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of $R[T]$ of dimension $d+1$, containing $R$. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class. Furthermore, for a graded subring $B$ of $R[T]$ containing $R$, we improve some existing stability theorems for projective modules over $B$, generalizing results due to Bass, Serre and Vaser{\v{s}}te{\u{\i}}n.Comment: 25 pages. New results in section 6. Comments are welcom

Cancellation of projective modules in polynomial rings of prime characteristic

Let $A$ be a commutative Noetherian ring of characteristic $p>0$, such that $\dim(A)=d$. Let $P$ be a projective $A[T_1,...,T_n]$-module of rank $d$. We show that $P$ is cancellative if and only if $P/P$ is cancellative. We deduce some applications. In one of the interesting consequences, we show that the Bass-Quillen conjecture has an affirmative answer in dimension three, when $2$ is invertible.Comment: The proof of 2.2 is not correc

Splitting criteria of projective modules on polynomial extensions over various base rings

In this article we prove the following results: 1. A monic inversion principle on polynomial extension of affine $\overline{\mathbb{F}}_p$-algebras. 2. Let $R$ be an affine algebra over an algebraically closed field (of characteristic $\not=2$) of dimension $d\ge 2$ and $P$ be a projective $R[T]$-module of rank $d$. Suppose that $P$ surjects onto a complete intersection ideal of height $d$. Then $P$ splits off a free summand of rank one. Moreover if the ground field is $\overline{\mathbb{F}}_p$ with $(d-1)!\in R^*$, we show this condition is necessary as well. 3. Let $R$ be a finite $\mathbb{Z}$-algebra such that there exists a positive integer $n \in R^*$, other than $1$. Suppose that $P$ is a projective $R[T]-$module of rank $d$, which surjects onto a locally complete intersection ideal of height $d$, which contains a monic polynomial. Then $P$ splits off a free summand of rank one.Comment: 28 pages. Preliminary version. Comments are welcom

Quality assurance of simultaneous treatment of two targets in pelvic region planned with single isocenter using three dimensional conformal radiotherapy (3DCRT) technique

Purpose: The purpose of this study was to conduct quality assurance of a three dimensional conformal radiotherapy (3DCRT) of two targets in pelvis region planned with single isocenter technique. Methods: A treatment plan was generated with two identical water phantoms with ionization chamber (IC) sleeves (IC-1 &amp; IC-2), simulated as if targets are in pelvis region, simultaneously irradiated with single isocenter technique with a dose prescription of 300 cGy for point dose verification. A two dimensional ion chamber array detector was used for fluence verification.Results: Calculated minimum, mean and maximum dose (in cGy) for IC-1 &amp; IC-2 were 295, 303 and 307 as per dose volume histogram. The global dose maximum was found to be 307.4 cGy. Measured point doses to both lesions were within ±2.5% of the computed dose. A pass percentage of 97% was obtained with the set of criteria 3 mm distance to agreement and 3% dose difference for fluence verification.Conclusion: Treatment execution of two targets simultaneously with single isocenter can reduce positional errors and delivery time.</p