Uncertainty dimension

The method of the uncertainty dimension was introduced in 1985 by Grebogi et al. Suppose we are given a fractal that can be covered by the interval [0,1]. Also, suppose that we are provided with a mapping function / that generates the fractal and determines the orbits of seeds xo in [0,1]. Choose e > 0 and N = 1/e random points xj in [0,1]. Now, for every one of the N points determine its lifetime Ij. The lifetime is the number of times the seed xj can be iterated forward with / without ionizing it. Then, for every Xj determine the lifetimes of Xj + e and Xj — e. Call them Ij and lj respectively. For Ij = Ij = lj the point Xj is called e certain. Otherwise, the point Xj is called e uncertain. This procedure measures the fraction of gaps in the fractal on a scale e. Define /(e) as the ratio of e uncertain points and the total number of points N. Then, /(e) is also called the uncertainty fraction. 

The uncertainty categorization should be based on some criteria to ensure consistency. Three categories are used for the uncertainty dimension ... 

The results from the cost-effectiveness study can be presented graphically in a cost-effectiveness uncertainty diagram, for example as introduced in Abrahamsen et al. (2009). An alternative diagram, not including the uncertainty dimension, is presented in Haver et al. (2008). 

Table A2.5.1 shows the Ising exponents for two and tluee dimensions, as well as the classical exponents. The uncertainties are those reported by Guida and Ziim-Justin [27]. These exponent values satisfy the equalities (as they must, considering the scalmg assumption) which are here reprised as functions of p and y ...

In the discussion of Table 4.1, we acknowledged that there might be some uncertainty in the values of the quantities tabulated, but we sidestepped the origin of the uncertainty. In the next section we shall consider one of these areas the effect of crystal dimensions of the value of T j,. 

Effect of Uncertainties in Thermal Design Parameters. The parameters that are used ia the basic siting calculations of a heat exchanger iaclude heat-transfer coefficients tube dimensions, eg, tube diameter and wall thickness and physical properties, eg, thermal conductivity, density, viscosity, and specific heat. Nominal or mean values of these parameters are used ia the basic siting calculations. In reaUty, there are uncertainties ia these nominal values. For example, heat-transfer correlations from which one computes convective heat-transfer coefficients have data spreads around the mean values. Because heat-transfer tubes caimot be produced ia precise dimensions, tube wall thickness varies over a range of the mean value. In addition, the thermal conductivity of tube wall material cannot be measured exactiy, a dding to the uncertainty ia the design and performance calculations. 

Some general quantitative characteristics (orders of magnitudes) of LDA systems are velocity measurement range 1 mm s -100 m s relative measurement uncertainty 0.1-1% rate of accepted data 0.1-10 kHz size of the optical probe 10 p.m-1 mm for each dimension measuring distance 0.1-1 m. 

In experiments related to flow and heat transfer in micro-channels, some parameters, such as the flow rate and channel dimensions are difficult to measure accurately because they are very small. For a single-phase flow in micro-channels the uncertainty of ARe is (Guo and Li 2002,2003)... 

Although the general circulation patterns are fairly well known, it is difficult to quantify the rates of the various flows. Abyssal circulation is generally quite slow and variable on short time scales. The calculation of the rate of formation of abyssal water is also fraught with uncertainty. Probably the most promising means of assigning the time dimension to oceanic processes is through the study of the distribution of radioactive chemical tracers. Difficulties associated with the interpretation of radioactive tracer distributions lie both in the models used, nonconservative interactions, and the difference between the time scale of the physical transport phenomenon and the mean life of the tracer. 

In this line, Hofstede (1980) conducted a transcultural study in which five relevant psychological dimensions were identified empirically in all the cultures that were studied, i.e. power-distance (from small to large), collectivism-individualism, femininity-masculinity, uncertainty avoidance (from weak to strong) and time orientation (from short term to long term). 

The uncertainty in multivariate calibration is characterized with respect to the evaluation functions at Eqs. (6.75) and (6.79). The prediction of a row vector x of dimension n from a row vector y of dimension m results from... 

Quantization puts a serious constraint on five-dimensional motion. If there is an independence on x5, then the particle is freely moving in x5. However, being that dimension compactihed leads to an uncertainty in the position with the size of 2irRc, where Rc is the compactification radius. Thus a Bohr-type quantum condition appears formally ... 

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