An exact formulation for the synthesis of broadside Chebyshev arrays of 2N elements with interelement spacingd < λ/2

2006 ◽  
Vol 48 (3) ◽  
pp. 457-463 ◽  
Author(s):  
Derek A. McNamara
Keyword(s):  
Exact Formulation

General Schema Theory for Genetic Programming with Subtree-Swapping Crossover: Part II

2003 ◽  
Vol 11 (2) ◽  
pp. 169-206 ◽  
Author(s):  
Riccardo Poli ◽  
Nicholas Freitag McPhee
Keyword(s):  
Genetic Programming ◽  
Schema Theory ◽  
Exact Formulation ◽  
Expected Number ◽  
Size Evolution ◽  
General Schema ◽  
Definition Of ◽  
Effective Fitness ◽  
Selection Of ◽  
Exact Definition

This paper is the second part of a two-part paper which introduces a general schema theory for genetic programming (GP) with subtree-swapping crossover (Part I (Poli and McPhee, 2003)). Like other recent GP schema theory results, the theory gives an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. The theory is based on a Cartesian node reference system, introduced in Part I, and on the notion of a variable-arity hyperschema, introduced here, which generalises previous definitions of a schema. The theory includes two main theorems describing the propagation of GP schemata: a microscopic and a macroscopic schema theorem. The microscopic version is applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. Therefore, this theorem is applicable to Koza's GP crossover with and without uniform selection of the crossover points, as well as one-point crossover, size-fair crossover, strongly-typed GP crossover, context-preserving crossover and many others. The macroscopic version is applicable to crossover operators in which the probability of selecting any two crossover points in the parents depends only on the parents' size and shape. In the paper we provide examples, we show how the theory can be specialised to specific crossover operators and we illustrate how it can be used to derive other general results. These include an exact definition of effective fitness and a size-evolution equation for GP with subtree-swapping crossover.

scholarly journals On the Use of Approximation Methods for Microcrack Shielding Problems

1992 ◽  
Vol 59 (3) ◽  
pp. 497-501 ◽  
Author(s):  
H. Cai ◽  
K. T. Faber
Keyword(s):  
Fracture Toughness ◽  
Numerical Solution ◽  
Experimental Evidence ◽  
Discrete Model ◽  
Brittle Materials ◽  
Approximation Methods ◽  
Exact Formulation

There is experimental evidence that stress-induced microcracking near a macrocrack tip enhances the fracture toughness of brittle materials. In considering the interaction of the macrocrack with multiple microcracks using a discrete model, it is essential to use approximation methods in order to keep the amount of the computation to a tractable level. However, when crack distances are small, the results of the approximation methods can be significantly different from the numerical solution based upon the exact formulation. The results obtained by these approximation methods will be compared with the numerical solution to show the applicability ranges in which the errors are acceptably small. The use of results obtained by the approximation methods outside applicability ranges in literature is shown to lead to incorrect conclusions concerning microcrack shielding.

Influence of errors in natural mortality estimates in cohort analysis

1997 ◽  
Vol 54 (7) ◽  
pp. 1608-1612 ◽  
Author(s):  
G Mertz ◽  
R A Myers
Keyword(s):  
Mortality Rate ◽  
Similar Pattern ◽  
Cohort Analysis ◽  
Transient Behavior ◽  
Natural Mortality ◽  
Exact Formulation ◽  
Fishing Mortality ◽  
Bias Factor ◽  
Special Case ◽  
Mortality Estimates

The accuracy of the estimation of cohort strength from catch data may be greatly degraded if a poor estimate of the natural mortality rate is entered into the calculation. A straightforward, exact formulation for the error in cohort reconstruction due to a misspecified natural mortality rate is presented. The special case of constant fishing mortality is particularly transparent, allowing the error to be segmented into easily interpreted terms. A change in the fishing mortality may result in a distinct hump in the transient behavior of the bias factor, rather than a simple monotonic adjustment. This implies a similar pattern in estimated cohort strength.

The Rise of Normative Judaism. II. To the Close of the Mishnah

1925 ◽  
Vol 18 (1) ◽  
pp. 1-38
Author(s):  
George Foot Moore
Keyword(s):  
Exact Formulation ◽  
Biblical Exegesis ◽  
Next Generation ◽  
Written Law ◽  
Unwritten Law

The older and younger contemporaries of Gamaliel II and their disciples and successors in the next generation are the fundamental authorities of normative Judaism as we know it in the literature which it has always esteemed authentic. One main division of their learned labors was the definition and exact formulation of the rules of the unwritten law (Halakah), as they had been received through tradition, or were adapted to meet new conditions, or were developed by biblical exegesis or casuistic discussion. Along with this ran the minute study, in course, of the written law in the Pentateuch from Exodus to Deuteronomy, in primary intention a juristic exegesis with constant reference to the Halakah.

scholarly journals Generating dense packings of hard spheres by soft interaction design

2018 ◽  
Vol 4 (6) ◽  
Author(s):  
Thibaud Maimbourg ◽  
Mauro Sellitto ◽  
Guilhem Semerjian ◽  
Francesco Zamponi
Keyword(s):  
Interaction Design ◽  
Optimization Problem ◽  
Hard Spheres ◽  
Hard Core ◽  
Packing Fraction ◽  
Exact Formulation ◽  
Simple Liquids ◽  
Soft Interaction ◽  
Dense Packings ◽  
Number Of Particles

Packing spheres efficiently in large dimension dd is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on the packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be efficiently constructed by this method, up to a packing fraction close to 7 \, d \, 2^{-d}7d2−d. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.

scholarly journals The physics of osmotic pressure

2017 ◽  
Author(s):  
M. G. Bowler
Keyword(s):  
Osmotic Pressure ◽  
Virial Theorem ◽  
Exact Formulation ◽  
Physical Processes ◽  
Intuitive Notion ◽  
Kinetic Treatment

AbstractThe proper application of the virial theorem to the origin of osmotic pressure provides a simple, vivid and complete explanation of the physical processes involved; osmosis is driven by differential solvent pressures. This simple and intuitive notion is widely disregarded, but the virial theorem allows an exact formulation and unifies a number of different treatments. It is closely related to a kinetic treatment devised by Ehrenfest over 100 years ago.

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