Regularity theory of Kolmogorov operator revisited

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

The European Reception of Livy (1469–1600)

Chapter 1 explores Livy’s early reception and translation in Renaissance Europe, examining the first, key decades in which the history reached a wider audience through its publication on the continent. The chapter first examines the literary fame enjoyed by Livy in Europe towards the end of the fifteenth century as well as the attempts of his earliest editors in print to impose some kind of critical order onto this monolithic work. The focus then moves to the first vernacular translations of Livy to have appeared in Europe, including the first renderings of the history into French and Italian. The final section considers the various translation styles at work in early-modern England and how these manifest themselves in each of the sixteenth-century translations of Livy.

Reduction of Medicine of PD’s Patients to Reduce Side Effects of Medicine and Brain Disorder Effects Using Crashing of Precedence Relationship Process of Operational Research (Preprint)

BACKGROUND Case of a Parkinson’s disease (PD’s). PD’s is caused by a decreased production of dopamine. It is progressive which means it develops gradually over a long period of time. It’s a neurodegenerative condition that’s caused by a loss of nerve cells in a certain part of the brain. People suffering from Parkinson’s disease can try something new, make a list of small tasks and complete them can help their body produce the dopamine it needs, and that reduce some symptoms of PD’s. There is no cure, so treatment focuses on managing symptoms. So while eating the right food and considering a lifestyle certainly cannot offset the effects of Parkinson’s disease entirely, a recommended diet and lifestyle can support their body’s ability to produce dopamine and combat symptoms of the disease. OBJECTIVE This study was undertaken with aim to investigate the effects of Crashing. The objective of study is to reduce the quantity of medicine while minimizing effect of neurodegenerative brain disorder. METHODS The concept of crashing is the method for minimizing the side effects by reducing one or more critical order of medicine to decrease their normal quantity of medicine. Critical order of medicine is an order of medicine in which the sequence of order of medicine which add up to longest overall quantity of medicine. This determines the less quantity of medicine possible to reduce the neurodegenerative disorder effect. RESULTS A medicine consists of interrelated relationship which is to be related in a certain order before the entire order of medicine is completed. The medicine is represented in a form of interrelated logical sequence which is known as precedence relationship diagram for the purpose of analytical treatment to get the solution for reducing the amount of medicine and controlling the side effects of distinct medicine. When we speedup the activity, the crash effect of medicine will be higher when compare to normal effect of medicine to find the optimal crashed quantity of medicine to reduce movement disorder. CONCLUSIONS When speedup the activity, the normal quantity of medicine can be reduced to crash quantity of medicine. People suffering from Parkinson’s disease can do the following activity to increase their dopamine levels: Try something new, Make a list of small tasks and complete them, Avoid foods with excess fat and sugar, try to maintain a balanced diet that contains adequate protein, vitamins and minerals, Listen to music you enjoy, get enough sleep , do yoga and Exercise regularly can help their body produce the dopamine it needs, and that reduce some symptoms of Parkinson’s disease. No food or diet can serve as a cure for Parkinson’s disease. But eating certain foods can help to minimize symptoms and help them to get the most out of their medication.