Concentration of the Langevin Algorithm's Stationary Distribution

A canonical algorithm for log-concave sampling is the Langevin Algorithm, aka the Langevin Diffusion run with some discretization stepsize $\eta > 0$. This discretization leads the Langevin Algorithm to have a stationary distribution $\pi_{\eta}$ which differs from the stationary distribution $\pi$ of the Langevin Diffusion, and it is an important challenge to understand whether the well-known properties of $\pi$ extend to $\pi_{\eta}$. In particular, while concentration properties such as isoperimetry and rapidly decaying tails are classically known for $\pi$, the analogous properties for $\pi_{\eta}$ are open questions with direct algorithmic implications. This note provides a first step in this direction by establishing concentration results for $\pi_{\eta}$ that mirror classical results for $\pi$. Specifically, we show that for any nontrivial stepsize $\eta > 0$, $\pi_{\eta}$ is sub-exponential (respectively, sub-Gaussian) when the potential is convex (respectively, strongly convex). Moreover, the concentration bounds we show are essentially tight. Key to our analysis is the use of a rotation-invariant moment generating function (aka Bessel function) to study the stationary dynamics of the Langevin Algorithm. This technique may be of independent interest because it enables directly analyzing the discrete-time stationary distribution $\pi_{\eta}$ without going through the continuous-time stationary distribution $\pi$ as an intermediary

Impact of COVID-19 pandemic (wave 2) and associated lockdown on wound care and the resultant increase in the number of amputations

Background: The COVID-19 pandemic has had an unparalleled impact on the socio-economic and healthcare structure of the entire world, including India. The risk of major amputations has increased during the COVID-19 lockdown.Aim of the study was to conduct a single-centered study on the prevalence of major amputations during the pre-pandemic and pandemic period to evaluate the indirect effect of the COVID-19 lockdown on people with lower limb cellulites and wounds. This study also emphasizes on the importance of easy and routine access to foot-care specialist.Methods: The data of patients attending the outpatient and emergency room of general surgery in K. R. Hospital with complaints of cellulites/wounds/ulcers during the months of March to July was evaluated. The number of amputations done during same time period was analyzed.Results: The patients attending outpatient department (OPD) during the lockdown had a fall, while cases and amputations immediately after lockdown had increased.Conclusions: The findings of the present study, reiterate the role of preventive actions in wound care and stress on the importance awareness of complications if such wounds are ignored

Pulmonary arterial hypertension: a current review of pharmacological management

Pulmonary hypertension (PHTN) is a rare and devastating disease characterized by progressive increases in pulmonary arterial pressure and pulmonary vascular resistance, which eventually leads to right ventricular failure and death. At present there is no cure for pulmonary arterial hypertension (PAH); however over the past decade targeted pharmaceutical options have become available for the treatment of PAH. Prior to evaluation for therapeutic options a definitive diagnosis of pulmonary arterial hypertension must be made via comprehensive physical exam and definitive diagnostic testing. Screening test of choice remains echocardiography and gold standard for definitive diagnosis is right heart catheterization. Once the establishment of a diagnosis of PAH is made therapeutic options may be a possibility based on a diagnostic algorithm and disease severity of the PAH patient. There are different classes of medications available with different mechanisms of actions which net a vasodilatory effect and improve exercise tolerance, quality of life as well and survival

Effects on Breathing of Agonists to μ-opioid or GABA\u3csub\u3eA\u3c/sub\u3e Receptors Dialyzed into the Ventral Respiratory Column of Awake and Sleeping Goats

Pulmonary ventilation (V̇I) in awake and sleeping goats does not change when antagonists to several excitatory G protein-coupled receptors are dialyzed unilaterally into the ventral respiratory column (VRC). Concomitant changes in excitatory neuromodulators in the effluent mock cerebral spinal fluid (mCSF) suggest neuromodulatory compensation. Herein, we studied neuromodulatory compensation during dialysis of agonists to inhibitory G protein-coupled or ionotropic receptors into the VRC. Microtubules were implanted into the VRC of goats for dialysis of mCSF mixed with agonists to either μ-opioid (DAMGO) or GABAA (muscimol) receptors. We found: (1) V̇I decreased during unilateral but increased during bilateral dialysis of DAMGO, (2) dialyses of DAMGO destabilized breathing, (3) unilateral dialysis of muscimol increased V̇I, and (4) dialysis of DAMGO decreased GABA in the effluent mCSF. We conclude: (1) neuromodulatory compensation can occur during altered inhibitory neuromodulator receptor activity, and (2) the mechanism of compensation differs between G protein-coupled excitatory and inhibitory receptors and between G protein-coupled and inotropic inhibitory receptors

State-Dependent and -Independent Effects of Dialyzing Excitatory Neuromodulator Receptor Antagonists into the Ventral Respiratory Column

Unilateral dialysis of the broad-spectrum muscarinic receptor antagonist atropine (50 mM) into the ventral respiratory column [(VRC) including the pre-Bötzinger complex region] of awake goats increased pulmonary ventilation (V̇i) and breathing frequency (f), conceivably due to local compensatory increases in serotonin (5-HT) and substance P (SP) measured in effluent mock cerebral spinal fluid (mCSF). In contrast, unilateral dialysis of a triple cocktail of antagonists to muscarinic (atropine; 5 mM), neurokinin-1, and 5-HT receptors does not alter V̇i or f, but increases local SP. Herein, we tested hypotheses that 1) local compensatory 5-HT and SP responses to 50 mM atropine dialyzed into the VRC of goats will not differ between anesthetized and awake states; and 2) bilateral dialysis of the triple cocktail of antagonists into the VRC of awake goats will not alter V̇i or f, but will increase local excitatory neuromodulators. Through microtubules implanted into the VRC of goats, probes were inserted to dialyze mCSF alone (time control), 50 mM atropine, or the triple cocktail of antagonists. We found 1) equivalent increases in local 5-HT and SP with 50 mM atropine dialysis during wakefulness compared with isoflurane anesthesia, but V̇i and f only increased while awake; and 2) dialyses of the triple cocktail of antagonists increased V̇i, f, 5-HT, and SP

Successful pulmonary thromboendarterectomy in a patient with sickle cell disease and associated resolution of a leg ulcer

© 2017 Indian Chest Society | Published by Wolters Kluwer -Medknow. Pulmonary hypertension (PH) is a relatively frequent and severe complication of sickle cell disease (SCD). PH associated with SCD is classified as Group 5 PH. The exact pathogenesis of PH in SCD in not known. There are also very limited treatment options available at this time for such patients with Group 5 PH. Patients with SCD are predisposed to a hypercoagulable state and thus can also suffer from chronic thromboembolism. These patients can have associated chronic thromboembolic pulmonary hypertension (CTEPH), thus being classified as Group 4 PH. We present such a case of a patient with SCD diagnosed with severe PH who was found to have CTEPH and successfully underwent a thromboendarterectomy with resolution of his symptoms such as reduction of his oxygen requirements and healing of chronic leg ulcer. This case illustrates the importance of screening patients with SCD and elevated pulmonary artery pressures for CTEPH as this would offer possible treatment options such as pulmonary thromboendarterectomy and/or riociguat in this subset of patients