L∞ Norm
Maranda
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in an Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
Some authors include non-negativity as part of the definition of "norm", although this is not necessary.Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";[8] these definitions are not equivalent.
The absolute value x \displaystyle is a norm on the vector space formed by the real or complex numbers. The complex numbers form a one-dimensional vector space over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.
The Euclidean norm is by far the most commonly used norm on R n , \displaystyle \mathbb R ^n, [11] but there are other norms on this vector space as will be shown below.However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.
The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane R 2 . \displaystyle \mathbb R ^2. This identification of the complex number x + i y \displaystyle x+iy as a vector in the Euclidean plane, makes the quantity x 2 + y 2 \textstyle \sqrt x^2+y^2 (as first suggested by Euler) the Euclidean norm associated with the complex number. For z = x + i y \displaystyle z=x+iy , the norm can also be written as z z \displaystyle \sqrt \bar zz where z \displaystyle \bar z is the complex conjugate of z . \displaystyle z\,.
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory.In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero.However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness.When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.
There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional of a centrally-symmetric convex body in R n \displaystyle \mathbb R ^n (centered at zero) defines a norm on R n \displaystyle \mathbb R ^n (see Classification of seminorms: absolutely convex absorbing sets below).
Every norm is a seminorm and thus satisfies all properties of the latter. In turn, every seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function.
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic.
Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a family ( p ) \displaystyle (p) of seminorms p \displaystyle p that separates points: the collection of all finite intersections of sets { p
Norm type, specified as 2 (default), a positive real scalar, Inf, or -Inf. The valid values of p and what they return depend on whether the first input to norm is a matrix or vector, as shown in the table.
As an instance of the rv_continuous class, norm object inherits from ita collection of generic methods (see below for the full list),and completes them with details specific for this particular distribution.
If axis is an integer, it specifies the axis of x along which tocompute the vector norms. If axis is a 2-tuple, it specifies theaxes that hold 2-D matrices, and the matrix norms of these matricesare computed. If axis is None then either a vector norm (when xis 1-D) or a matrix norm (when x is 2-D) is returned. The defaultis None.
All minerals and raw materials contain radionuclides of natural origin. The most important for the purposes of radiation protection are the radionuclides in the U-238 and Th-232 decay series. For most human activities involving minerals and raw materials, the levels of exposure to these radionuclides are not significantly greater than normal background levels and are not of concern for radiation protection. However, certain work activities can give rise to significantly enhanced exposures that may need to be controlled by regulation. Material giving rise to these enhanced exposures has become known as naturally occurring radioactive material (NORM).
NORM potentially includes all radioactive elements found in the environment. However, the term is used more specifically for all naturally occurring radioactive materials where human activities have increased the potential for exposure compared with the unaltered situation. Concentrations of actual radionuclides may or may not have been increased; if they have, the term technologically-enhanced NORM (TENORM) may be used.
The acronym TENORM, or technologically enhanced NORM, is often used to refer to those materials where the amount of radioactivity has actually been increased or concentrated as a result of industrial processes. This paper addresses some of these industrial sources, and for simplicity the term NORM will be used throughout.
Another NORM issue relates to radon exposure in homes, particularly those built on granitic ground. Occupational health issues include the exposure of flight crew to higher levels of cosmic radiation, the exposure of tour guides to radon in caves, exposure of miners to radon underground, and exposure of workers in the oil & gas and mineral sands industries to elevated radiation levels in the materials they handle.
* The first four columns represent four of the 14 nuclides in the uranium decay series, the next two represent two of 10 in the thorium series. (For total activity in any coal, assume these are in serial equilibrium, hence multiply U-238 by 14 and Th-232 by 10, then add K-40.)
The amounts of radionuclides involved are noteworthy. US, Australian, Indian and UK coals contain up to about 4 ppm uranium, those in Germany up to 13 ppm, and those from Brazil and China range up to 20 ppm uranium. Thorium concentrations are often about three times those of uranium.
During combustion the radionuclides are retained and concentrated in the flyash and bottom ash, with a greater concentration to be found in the flyash. The concentration of uranium and thorium in bottom and flyash can be up to ten times greater than for the burnt coal, while other radionuclides such as Pb-210 and K-40 can concentrate to an even greater degree in the flyash. Some 99% of flyash is typically retained in a modern power station (90% in some older ones). While much flyash is buried in an ash dam, a lot is used in building construction. Table 3 gives some published figures for the radioactivity of ash. There are obvious implications for the use of flyash in concrete.
In 2017 Australia exported 372 million tonnes of coal. With an average of 0.9 ppm uranium and 2.6 ppm thorium, at least 330 tonnes of uranium per year and 970 tonnes of thorium could conceivably be added to published export figures.
In the USA, 858 million tonnes of coal was used in 2013 for electricity production. With an average content of 1.3 ppm uranium and 3.2 ppm thorium, US coal-fired electricity generation in that year gave rise to 1100 tonnes of uranium and 2700 tonnes of thorium in coal ash. In Victoria, Australia, some 65 million tonnes of brown coal is burned annually for electricity production. This contains about 1.6 ppm uranium and 3.0-3.5 ppm thorium, hence about 100 tonnes of uranium and 200 tonnes of thorium is buried in landfill each year in the Latrobe Valley.
It is evident that even at 1 part per million (ppm) U in coal, there is more energy in the contained uranium (if it were to be used in a fast neutron reactor) than in the coal itself. If coal had 25 ppm uranium and that uranium was used simply in a conventional reactor, it would yield half as much thermal energy as the coal.
In 2007, China National Nuclear Corp (CNNC) commissioned Sparton Resources of Canada with the Beijing No.5 Testing Institute to undertake advanced trials on leaching uranium from coal ash in central Yunnan. In early 2007, Sparton signed an agreement with the Xiaolongtang Guodian Power Company of Yunnan for a program to test and possibly commercialize the extraction of uranium from waste coal ash. Some 250 km southwest of Kunming, the Xiaolongtang, Dalongtang and the Kaiyuan power stations, all located within 20 km of each other burn coal from a centrally located open pit lignite mine with high ash content (20-30%) and very high uranium content. The coal uranium content varies from about 20 to 315 ppm and averages about 65 ppm. The ash averages about 210 ppm U (0.021%U) - above the cut-off level for some uranium mines. The power station ash heap contains over 1000 tU, with annual arisings of 190 tU. (Recovery of this by acid leaching is about 70%.)
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