Index Of Coincidence

This paper explains thoroughly the Index of Coincidence as well as some basics of probability used. This topic was chosen because it is an interesting topic related to mathematics and we should all read about it. This paper shows how the formula to find the index of coincidence was derived, gives some applications of it and more significant results for specific languages. It also, has some examples for better understanding by anyone

Miro1-dependent Mitochondrial Dynamics in Parvalbumin Interneurons

Parvalbumin interneurons are fast-spiking inhibitory cells that have been implicated in the generation of rhythmic network activity in the γ-frequency band (30-80Hz) by coordinating principal cell activity. In order to sustain their high firing rates, parvalbumin interneurons have a high mitochondrial content reflecting their large energy utilization. Therefore, parvalbumin interneurons are more susceptible to incidents of mitochondrial impairment as mitochondria provide the predominant source of energy. Miro1 is a Ca²⁺-sensing adaptor protein that links mitochondria to the trafficking apparatus, for their microtubule-dependent transport along axons and dendrites, in order to meet the metabolic needs of the cell. Here, we explore the role of Miro1 in parvalbumin interneurons and how changes in the mitochondrial distribution could alter network activity. To investigate mitochondrial dynamics we generated a mouse line where mitochondria are specifically labelled in parvalbumin interneurons. The Cre- recombinase is expressed exclusively in parvalbumin interneurons and excises a termination signal upstream of a fluorescent reporter allowing for the visualisation of mitochondria only in these cells. We further crossed this line with the Miro1(f/f) mouse, generating a transgenic mouse where Miro1 was conditionally knocked-out exclusively in parvalbumin interneurons. Specifically we investigated the conditional removal of Miro1 in mitochondrial dynamics in parvalbumin interneurons. Using live-imaging of ex-vivo organotypic brain cultures, we demonstrated a reduction in mitochondrial trafficking in parvalbumin interneurons in the hippocampus under basal conditions. This lead to accumulation of mitochondria in the soma and their depletion from synaptic terminals. Loss of Miro1 resulted in alterations in axonal but not dendritic branching in parvalbumin interneurons. This was accompanied by altered synaptic transmission and increased frequency of γ-oscillations in hippocampal brain slices. In this study, we show for the first time that Miro1 and Miro1-dependent mitochondrial positioning are essential for correct parvalbumin interneuron function and network activity

Bayes Theorem and its Recent Applications

Individuals who have studied maths to a specific level have come across Bayes’ Theorem or Bayes Formula. Bayes’ Theorem has many applications in areas such as mathematics, medicine, finance, engineering and many other. This paper covers Bayes’ Theorem at a basic level and looks at some of the history of this theorem as well as how the formula was derived. We also, look at some extended forms of the formula and give an explicit example. Lastly, we discuss recent applications of Bayes’ Theorem in valuating depression and predicting water quality conditions. We have chosen this topic for an exposition paper as it is an interesting topic in mathematics and our goal was to explain it in simple words for everyone to understand

Multi-step Fermi normal coordinates

We generalize the concept of Fermi normal coordinates adapted to a geodesic to the case where the tangent space to the manifold at the base point is decomposed into a direct product of an arbitrary number of subspaces, so that we follow several geodesics in turn to find the point with given coordinates. We compute the connection and the metric as integrals of the Riemann tensor. In the case of one subspace (Riemann normal coordinates) or two subspaces, we recover some results previously found by Nesterov, using somewhat different techniques.Comment: 9 pages, 4 figure

A new derivation of singularity theorems with weakened energy hypotheses

The original singularity theorems of Penrose and Hawking were proved for matter obeying the Null Energy Condition or Strong Energy Condition respectively. Various authors have proved versions of these results under weakened hypotheses, by considering the Riccati inequality obtained from Raychaudhuri's equation. Here, we give a different derivation that avoids the Raychaudhuri equation but instead makes use of index form methods. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by Quantum Energy Inequalities. In this last case, we make quantitative estimates of the initial conditions required for our singularity theorems to apply

Energy conditions allow eternal inflation

Eternal inflation requires upward fluctuations of the energy in a Hubble volume, which appear to violate the energy conditions. In particular, a scalar field in an inflating spacetime should obey the averaged null energy condition, which seems to rule out eternal inflation. Here we show how eternal inflation is possible when energy conditions (even the null energy condition) are obeyed. The critical point is that energy conditions restrict the evolution of any single quantum state, while the process of eternal inflation involves repeatedly selecting a subsector of the previous state, so there is no single state where the conditions are violated. </p

Energy conditions in general relativity and quantum field theory

This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on matter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in mathematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are systematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy condition, has recently received increased attention. It is expected to be a universal property of the dynamics of all gravitating physical matter, even in the context of semiclassical or quantum gravity.Comment: 61 pages, 6 figures, Topical review accepted by Classical and Quantum Gravit

Effect of steel hardness on soot wear

Due to incomplete combustion, high levels of soot can accumulate in engine lubricants between drain intervals. This soot can promote wear of engine parts such as timing chains and cam followers. One standard approach to reducing wear is to increase the hardness of the rubbing components used. According to the Archard wear equation, wear rate should be broadly inversely proportional to hardness. To explore this approach for controlling soot wear, wear tests have been conducted in a High Frequency Reciprocating Rig (HFRR) with HFRR steel discs of various hardness against a hard steel ball. Carbon black (soot surrogate) dispersions in model lubricants based on solutions of ZDDP and dispersant in GTL base oils have been studied. Wear volumes have been measured and wear scars and tribofilms analysed using scanning white light interferometry and SEM-EDS. It is found that, while most oils show wear that reduces with increasing hardness, for blends that contain both ZDDP and carbon black, wear rate markedly increases with disc hardness as the latter approaches the hardness of the ball. The results support the prevalence of a corrosive-abrasive wear mechanism when carbon black and ZDDP are both present in a lubricant and suggests that selection of very hard surfaces may not be a useful way to control soot