Algebraic Laws, Congruences and Axiomatizations

Modelling Geometric Measure of Variation About the Population Mean

10.31730/osf.io/en5ph ◽

2021 ◽

Author(s):

Benedict Troon

Keyword(s):

Mean Deviation ◽

Average Deviation ◽

Statistical Tool ◽

Population Mean ◽

Geometric Measure ◽

Initial Dataset ◽

Algebraic Laws ◽

Average Variation ◽

Measures Of Dispersion ◽

The Mean

Measures of dispersion are important statistical tool used to illustrate the distribution of datasets. These measureshave allowed researchers to define the distribution of various datasets especially the measures of dispersion from the mean.Researchers and mathematicians have been able to develop measures of dispersion from the mean such as mean deviation, variance and standard deviation. However, these measures have been determined not to be perfect, for example, variance give average of squared deviation which differ in unit of measurement as the initial dataset, mean deviation gives bigger average deviation than the actual average deviation because it violates the algebraic laws governing absolute numbers, while standarddeviation is affected by outliers and skewed datasets. As a result, there was a need to develop a more efficient measure of variation from the mean that would overcome these weaknesses. The aim of this paper was to model a geometric measure of variation about the population mean which could overcome the weaknesses of the existing measures of variation about the population mean. The study was able to formulate the geometric measure of variation about the population mean that obeyedthe algebraic laws behind absolute numbers, which was capable of further algebraic manipulations as it could be used further to estimate the average variation about the mean for weighted datasets, probability mass functions and probability density functions. Lastly, the measure was not affected by outliers and skewed datasets. This shows that the formulated measure was capable of solving the weaknesses of the existing measures of variation about the mean

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A theory of mixin modules: algebraic laws and reduction semantics

Mathematical Structures in Computer Science ◽

10.1017/s0960129502003687 ◽

2002 ◽

Vol 12 (6) ◽

pp. 701-737 ◽

Cited By ~ 3

Author(s):

DAVIDE ANCONA ◽

ELENA ZUCCA

Keyword(s):

Normal Forms ◽

Preceding Paper ◽

Denotational Semantics ◽

Axiomatic Approach ◽

Algebraic Laws ◽

Small Set ◽

The Given ◽

Rewriting Rules

Mixins are modules that may contain deferred components, that is, components not defined in the module itself; moreover, in contrast to parameterised modules (like ML functors), they can be mutually dependent and allow their definitions to be overridden. In a preceding paper we defined a syntax and denotational semantics of a kernel language of mixin modules. Here, we take instead an axiomatic approach, giving a set of algebraic laws expressing the expected properties of a small set of primitive operators on mixins. Interpreting axioms as rewriting rules, we get a reduction semantics for the language and prove the existence of normal forms. Moreover, we show that the model defined in the earlier paper satisfies the given axiomatisation.

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An Algebraic Model for Bounding Threshold Circuit Depth

DAIMI Report Series ◽

10.7146/dpb.v17i239.7595 ◽

1988 ◽

Vol 17 (239) ◽

Cited By ~ 1

Author(s):

Joan Boyar ◽

Gudmund Skovbjerg Frandsen ◽

Carl Sturtivant

Keyword(s):

Threshold Model ◽

Algebraic Model ◽

Constant Factor ◽

Arbitrary Integer ◽

Arithmetic Circuit ◽

Threshold Circuit ◽

Polynomial Size ◽

Algebraic Laws ◽

Threshold Circuits ◽

Circuit Depth

We define a new structured and general model of computation: circuits using arbitrary fan- in arithmetic gates over the characteristic two finite fields (F_2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold circuits that allow arbitrary integer weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model only differs by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Sum and ½ gates; there is no bound on the number of ''wires'' . We show that these results are provable for any ''reasonable'' correspondance between bit strings of n-bits and elements of F_ 2n. And, we find two distinct characterizations of ''reasonable''. Thus, we have shown that arbitrary fan-in arithmetic computations over F_ 2n constitute a precise abstraction of boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

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ALGEBRAIC LAWS OF SEMANTICS OF FIRST-ORDER PREDICATE LOGIC

Algebraic Methods of Mathematical Logic ◽

10.1016/b978-1-4832-3123-5.50012-4 ◽

1967 ◽

pp. 181-199

Keyword(s):

Predicate Logic ◽

First Order ◽

Algebraic Laws ◽

First Order Predicate Logic

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Using transformations in the implementation of higher-order functions

Journal of Functional Programming ◽

10.1017/s0956796800000204 ◽

1991 ◽

Vol 1 (4) ◽

pp. 459-494 ◽

Cited By ~ 1

Author(s):

Hanne Riis Nielson ◽

Flemming Nielson

Keyword(s):

Code Generation ◽

Partial Evaluation ◽

Current Trend ◽

Higher Order ◽

Functional Languages ◽

Program Transformations ◽

Time Level ◽

Algebraic Laws ◽

Polymorphic Type ◽

Run Time

AbstractTraditional functional languages do not have an explicit distinction between binding times. It arises implicitly, however, as one typically instantiates a higher-order function with the arguments that are known, whereas the unknown arguments remain to be taken as parameters. The distinction between ‘known’ and ‘unknown’ is closely related to the distinction between binding times like ‘compile-time’ and ‘run-time’. This leads to the use of a combination of polymorphic type inference and binding time analysis for obtaining the required information about which arguments are known.Following the current trend in the implementation of functional languages we then transform the run-time level of the program (not the compile-time level) into categorical combinators. At this stage we have a natural distinction between two kinds of program transformations: partial evaluation, which involves the compile-time level of our notation, and algebraic transformations (i.e., the application of algebraic laws), which involves the run-time level of our notation.By reinterpreting the combinators in suitable ways we obtain specifications of abstract interpretations (or data flow analyses). In particular, the use of combinators makes it possible to use a general framework for specifying both forward and backward analyses. The results of these analyses will be used to enable program transformations that are not applicable under all circumstances.Finally, the combinators can also be reinterpreted to specify code generation for various (abstract) machines. To improve the efficiency of the code generated, one may apply abstract interpretations to collect extra information for guiding the code generation. Peephole optimizations may be used to improve the code at the lowest level.

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On the theory of quantum mechanics

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1926.0133 ◽

1926 ◽

Vol 112 (762) ◽

pp. 661-677 ◽

Cited By ~ 474

Keyword(s):

Dynamical System ◽

Differential Equation ◽

Wave Function ◽

Quantum Mechanics ◽

Wave Equation ◽

Atomic System ◽

Hamiltonian Equation ◽

Canonical Variables ◽

New Development ◽

Algebraic Laws

The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e ., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation principle a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H ( q r , P r - W = 0 is the Hamiltonian equation of the system, where the q r , P r are canonical variables, then the wave equation for ψ is {H( q r , ih ∂/∂ q ) - W} ψ = 0.

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