When are the natural embeddings of classical invariant rings pure?

Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl's book: for the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings, and the Pl\"ucker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of $S$, equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is linearly reductive.Comment: 40 pages; comments welcome! Corrected references for the orthogonal group in characteristic two; results and proofs remain the sam

Deep Gaussian Processes with Convolutional Kernels

Deep Gaussian processes (DGPs) provide a Bayesian non-parametric alternative to standard parametric deep learning models. A DGP is formed by stacking multiple GPs resulting in a well-regularized composition of functions. The Bayesian framework that equips the model with attractive properties, such as implicit capacity control and predictive uncertainty, makes it at the same time challenging to combine with a convolutional structure. This has hindered the application of DGPs in computer vision tasks, an area where deep parametric models (i.e. CNNs) have made breakthroughs. Standard kernels used in DGPs such as radial basis functions (RBFs) are insufficient for handling pixel variability in raw images. In this paper, we build on the recent convolutional GP to develop Convolutional DGP (CDGP) models which effectively capture image level features through the use of convolution kernels, therefore opening up the way for applying DGPs to computer vision tasks. Our model learns local spatial influence and outperforms strong GP based baselines on multi-class image classification. We also consider various constructions of convolution kernel over the image patches, analyze the computational trade-offs and provide an efficient framework for convolutional DGP models. The experimental results on image data such as MNIST, rectangles-image, CIFAR10 and Caltech101 demonstrate the effectiveness of the proposed approaches

When are the natural embeddings of classical invariant rings pure?

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as inWeyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with ⊆ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring is a pure subring of S, equivalently, is a direct summand of S as an -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion ⊆ is pure. It turns out that if ⊆ is pure, then either the invariant ring is regular or the group G is linearly reductive

ANTIPYRETIC ACTIVITY OF BLUMEA LACERA LEAVES IN ALBINO RATS

The present study was carried out to determine the antipyretic activity of the methanolic extract of Blumea lacera leaves (BLE) on albino rats. Acute toxicity study of BLE was performed by limit test as per OECD guideline. Antipyretic activity was studied by inducing pyrexia with Brewer's yeast. Acute oral LD50 of BLE in female rats was more than 2000 mg/kg. Three ascending doses of 100 (1/20 LD50), 200 (1/10 LD50) and 400 mg/kg (1/5 LD50) were selected for studying the antipyretic activity of BLE in rats. BLE at dose rates of 200 and 400 mg/kg reduced brewer's yeast induced pyrexia in rats suggesting antipyretic effect of BLE

Convolutional Deep Gaussian Processes

Deep Gaussian processes (DGPs) provide a Bayesian non-parametric alternative to standard parametric deep learning models. A DGP is formed by stacking multiple GPs resulting in a well-regularized composition of functions. The Bayesian framework that equips the model with attractive properties, such as implicit capacity control and predictive uncertainty, makes it at the same time challenging to combine with a convolutional structure. This has hindered the application of DGPs in computer vision tasks, an area where deep parametric models (i.e. CNNs) have made breakthroughs. Standard kernels used in DGPs such as radial basis functions (RBFs) are insufficient for handling pixel variability in raw images. In this paper, we build on the recent convolutional GP to develop Convolutional DGP (CDGP) models which effectively capture image level features through the use of convolution kernels, therefore opening up the way for applying DGPs to computer vision tasks. Our model learns local spatial influence and outperforms strong GP based baselines on multi-class image classification. We also consider various constructions of convolution kernel over the image patches, analyze the computational trade-offs and provide an efficient framework for convolutional DGP models. The experimental results on image data such as MNIST, rectangles-image, CIFAR10, Convex-sets and Caltech101 demonstrate the effectiveness of the proposed approaches. We also propose a method to reduce the computational complexity of the model. We sub-sample the number of patches and show the efficiency of the approach on caltech101 dataset

Fresh versus frozen embryo transfer: a retrospective cohort study

Background: Elective frozen embryo transfer (FET), has recently increased significantly with improvements in cryopreservation techniques. Observational studies and randomized controlled trials suggested that the endometrium in stimulated cycles is not optimally prepared for implantation; risk of ovarian hyperstimulation syndrome reduced and pregnancy rates increased following FET and perinatal outcomes are less affected after FET. However, the evidence is not unequivocal and recent randomised control trials challenge the use of elective FET for the general IVF population. Pregnancy rates were analysed in a cohort of patients undergoing embryo transfers.Methods: This was a retrospective cohort study of patients who underwent embryo transfers from April 2018 to March 2019 at study centre in Surat.175 cycles of embryo transfers (119 fresh and 56 frozen) were included in the study. Outcomes measured were positive pregnancy, clinical pregnancy and ongoing clinical pregnancy rates achieved in the IVF-ET cycles.Results: There were no statistically significant differences between positive pregnancy rate (54.6% versus 60.7%, Odds ratio (OR) 0.78; 95% Confidence Interval (CI) 0.41-1.49), clinical pregnancy rate (48.73% versus 57.14%, OR 0.52; 95% CI 0.1- 2.64) and ongoing clinical pregnancy rate  (45.38% versus 51.78% OR 1.4; 95% CI 0.29 - 6.67) in fresh ET and FET cycles, respectively, p < 0.05 was considered statistically significant for all measures.Conclusions: Despite the observed higher rates of positive biochemical, clinical and ongoing clinical pregnancy per transfer in the FET cohort, these did not reach statistical significance. Thus, both transfer strategies are reasonable options, although there is a trend favouring the freeze-all strategy

Central nervous system tuberculomata presenting as internuclear ophthalmoplegia

Central nervous system (CNS) tuberculoma can have variable presentation depending upon the site and number of tuberculomata. We are reporting a rare case of a 15 years old girl who presented to our hospital with binocular diplopia on right gaze. Clinical examination revealed left sided internuclear ophthalmoplegia and dysdiadochokinesia and ataxia on left side. Magnetic Resonance Imaging (MRI) of brain revealed multiple tuberculomata in both cerebral hemispheres, cerebellum, left half of medulla and pons. This case highlights the need for a high degree of suspicion for CNS tuberculosis in patients presenting with internuclear ophthalmoplegia.KEYWORDS: Brainstem; Tuberculoma; Internuclear; Ophthalmoplegia; SyndromeontInternet Journal of Medical Update 2012 January;7(1):59-6

Intrauterine platelet rich plasma versus injection G-CSF for treatment of thin endometrium in infertility

Background: For success in IVF treatment, it is essential that the patient has a responsive endometrium together with many other factors. Inspite of numerous treatments available today for growth of endometrium, there is lack of any ideal drug or protocol for increasing endometrial thickness. The study is an attempt to evaluate the role of two drugs for increasing endometrial growth before embryos transfers.Methods: This study is a retrospective cohort study including 50 patients with previously diagnosed as thin endometrium patients who may or may not have failed previous cycles of IVF. Patients were randomly divided into two groups. First group-Group A (n=25) are patients randomly selected to undergo intrauterine PRP instillation for increase in endometrial thickness before embryo transfer. Second group-Group B (n=25) are randomly selected from a retrospective cohort of thin endometrium to take injection GCSF  as intrauterine infusion (total dose 300mcg) on day of trigger or day 11 of cycle followed by 60 units subcutaneously after embryo transfer. The difference in endometrial thickness during transfer and the pregnancy outcomes were compared.Results: Injection GCSF was found to be more effective than intrauterine PRP in improving endometrial thickness in patients with thin endometrium with a p-value of &lt;0.0001.  It was found that the chemical and clinical pregnancy rates were comparable as the p values were 0.77 and 0.37 respectively and hence statistically not significant. Although patients given injection GCSF had a slightly higher clinical pregnancy rate (44%) as compared to patients given intrauterine PRP which was 28%. All other variables were comparable.Conclusions: In the study it was proven that injection GCSF, is more effective for the treatment of thin endometrium patients as compared to intrauterine PRP infusion. Though the clinical and chemical pregnancy rates were comparable, a higher percentage of women were clinically pregnant in the group given injection GCSF.  Intrauterine PRP can also be a good alternative for thin endometrium.  More studies and RCTS are needed for comparison to prove the effectiveness of these drugs for treatment of thin endometrium

A power law solution for FRLW Universe with observational constraints

This paper examines a power law solution under $f(R,T)$ gravity for an isotropic and homogeneous universe by considering its functional form as $f(R,T) = R + \xi RT$, where $\xi$ is a positive constant. In $f(R,T)$ gravity, we have built the field equation for homogeneous and isotropic spacetime. The developed model's solution is $a = \alpha t^{\beta}$. We have used the redshift in the range $0 \leq z \leq 1.965$ and obtained the model parameters $\alpha$, $\beta$, $H_0$ by using the Markov Chain Monte Carlo (MCMC) method. The constrained values of the model parameter are as follows: $H_0 = 67.098^{+2.148}_{-1.792}$ km s$^{-1}$ Mpc$^{-1}$, $H_0 = 67.588^{+2.229}_{-2.170}$ km s$^{-1}$ Mpc$^{-1}$, $H_0 = 66.270^{+2.215}_{-2.181}$ km s$^{-1}$ Mpc$^{-1}$, $H_0 = 65.960^{+2.380}_{-1.834}$ km s$^{-1}$ Mpc$^{-1}$, $H_0 = 66.274^{+2.015}_{-1.864}$ km s$^{-1}$ Mpc$^{-1}$ which have been achieved by bounding the model with the Hubble parameter ($H(z)$) dataset, Baryon Acoustic Oscillations (BAO) dataset, Pantheon dataset, joint $H(z)$ + Pantheon dataset and collective $H(z)$ + BAO + Pantheon dataset, respectively. These computed $H_o$ observational values agree well with the outcomes from the Plank collaboration group. Through an analysis of the energy conditions' behaviour on our obtained solution, the model has been examined and analysed. Using the Om diagnostic as the state finder diagnostic tool and the jerk parameter, we have also investigated the model's validity. Our results show that, within a certain range of restrictions, the proposed model agrees with the observed signatures