Birth and death

I cannot judge whether Truesdell s kind of continuum mechanics is of use to mechanical engineers who have to design structures to withstand specific demands, but the total absence of diagrams causes me to wonder. In any case, I understand (Walters 1998, Tanner and Walters 1998) that rational mechanics was effectively Truesdell s invention and is likely to end with him. The birth and death of would-be disciplines go on all the time. 

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

Equation (3.14) is thus eonsistent with the general population balanee (equation 3.8) when B = D = Q i.e. nueleation oeeurs at zero size and both the birth and death terms due to agglomeration and breakage are negleeted, and the feed is erystal-free. 

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798, the rate of change of the population N of Earth is dN/dt = births — deaths. The numbers of births and deaths are proportional to the population, with proportionality constants b and d. Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below ... 

Number of k-fold Precursor Particles. Dynamic differential equations were written for the concentration of the k-fold precursors to account for birth and death by coagulation, growth by propagation, and the formation of primary precursors by homogeneous nucleation. There... 

MANOLAGAS s c (2000) Birth and death of bone cells basic regulatory mechanisms and implications for the pathogenesis and treatment of osteoporosis. Endocr Rev. 21 (2) 115-37. 

The left-hand sideofEq. (40)isthe accumulationofparticlesofagivensize. The terms on the right-hand side are, in turn, the bulk flow into and out of the control volume, the convective flux along the size axis due to layering and attrition, the birth of new particles due to nucleation, and birth and death of granules due to coalescence. 

The solar body that is the fruit of the Great Work is a vehicle in which the powers of both the physical and astral bodies are fully manifested. The astral body takes on the materiality of the physical, while the physical body inherits the powers of the astral vehicle. The result is a body, freed from the Wheel of Birth and Death, able to materialize at will—physical enough to be touched, to eat, yet subtle enough to raise the vibrations of its atoms at will and so operate upon any level of the... 

Life We celebrate its arrival and bemoan its passing. Between birth and death, we protect life, cling to it, and perhaps prepare for what may come after it. 

This picture changed in the 1886 when an American chemist, Charles Martin Hall (1863— 1914), and a French chemist, Paul Louis-Toussaint Heroult (1863—1914), both discovered, at about the same time, a new process for extracting aluminum from molten aluminum oxide by electrolysis. (It might be noted that both discoverers have the same birth and death dates as well as the same date of discovery.) Hall was inspired by his teacher to find a way to inexpensively produce aluminum metal. He wired together numerous wet cells to form a battery that produced enough electricity to separate the aluminum from the melted aluminum oxide (mixed with the minerals cryolyte or fluorite), by the process known as electrolysis. Hall formed the Pittsburgh Reduction Co., which is now known as the Aluminum Company of America, or Alcoa. His company produced so much aluminum that the price dropped to about sixty cents per kilogram. 

The revelatory power of the new astronomy, especially astronomy associated with the extreme forms of radiation, resides in its capacity to expose previously unknown processes to reason and understanding gamma astronomy, the most violent phenomena in the Universe, such as the rupture and destruction of stars, and infrared astronomy, the gentle events, such as the birth of stars. Optical astronomy fills the relatively calm gap between stellar birth and death, whilst millimetre radioastronomy opens our minds to the formation of molecular structure in great clouds of cold gases and opaque dusts, far from any devastating light. 

Thales s successor, Anaximander—the exact dates of his birth and death are unknown, but he was said to have been 64 years old in 546 B.C.—agreed that there was one primal material. But he didn t think it was ever encountered on Earth in its pure state. According to Anaximander everything in the world was made of apeiron, a substance that was infinite and eternal, and which could take on numerous forms, including those of all the familiar terrestrial materials. It is neither water nor any of the so-called elements, Anaximander said, but a nature different from them and infinite, from which arise all the heavens and the worlds within them. ... 

In this section we shall present a few of the elementary type reactions that have been solved exactly. By elementary we mean unimolecular and bimolecular reactions, and simple extensions of them. In a more classical stochastic context, these reactions may be thought of as birth and death processes, unimolecular reactions being linear birth and death processes and bimolecular being quadratic. These reactions may be described by a finite or infinite set of states, (x), each member of which corresponds to a specified number of some given type of molecule in the system. One then describes a set of transition probabilities of going from state x to x — i, which in unimolecular reactions depend linearly upon x and in bimolecular reactions depend quadratically upon x. The simplest example is that of the unimolecular irreversible decay of A into B, which occurs particularly in radioactive decay processes. This process seems to have been first studied in a chemical context by Bartholomay.6... 

Exercise. In a population of n bacteria, each individual has a probability a per unit time to die and / to give birth to a new individual. Construct the M-equation ( birth and death process , compare chapter VI). 

The one-step or birth-and-death processes are a special class of Markov processes, which occur in many applications and can be analyzed in some detail. 

Many stochastic processes are of a special type called birth-and-death processes or generation-recombination processes . We employ the less loaded name one-step processes . This type is defined as a continuous time Markov process whose range consists of integers n and whose transition matrix W permits only jumps between adjacent sites,... 

The coefficients r and gn derive their names from recombination and generation of charge carriers in semiconductors, and are often denoted by pin and Xn in birth-and-death problems. 

Systems Under Birth and Death Conditions Lotka and Lotka-Volterra Models... 

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