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Manual:$wgUseTeX - MediaWiki
TeX: $wgUseTeX
Enables the use of <math> (TeX) tags.
1.18.0 (Gerrit change 615882; git #729ec629)
4.1 Check for latex
4.2 Check for the dvips
4.3 Check for the gs
4.4 Check for the convert
This is obsolete in MediaWiki 1.18 and later, replaced by breaking out the feature to the Math extension.
In MediaWiki 1.17 and earlier, enables the use of <math> (TeX ) tags.
$wgUseTeX is variable in the file LocalSetting.php that needs to be assigned value true in order to allow formulas at the MediaWiki sites. Such a setting has syntax
$wgUseTeX = "true";
Without such a setting, the tags <math> and </math> are not recognized as commands.
Such a setting of this variable is necessary, but not sufficient. A lot of other stuff should be loaded, installed, set-up and adjusted to the custom server by a qualified programmer. This stuff is called extensions. The description of this set-up can be found at Extension:WikiTex .
The status of the setting required for the $wgUseTeX can be checked with typing in the MediaWiki article any mathematical formulas with the math tags; for example, the following short code:
<math>O</math>
Such a code is supposed to produce the italics character O. At the current set of this wiki, the result looks as:
{\displaystyle O}
If the variable $wgUseTeX is not set to true, the line appears as it is typed. If the variable is assigned, then the result depends on the setting of other variables and correctness of other files, in particular, those outside the wiki directory.
The errors of the installation of the required extensions are reported with the red messages
(1) Failed to parse (Cannot write to or create math temp directory)
(2) Failed to parse (Missing texvc executable; please see math/README to configure.)
(3) Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert):
In the case (1), it may help to make the math directory writable with the command
chmod 777 math
In the case (2), it may help to set variable $wgTexvc in the LocalSetting.php in the following way:
$wgTexvc = '/math/texvc';
In the case (3), one should verify the existence of the commands latex, dvips, gs, and convert at the server, check that they work properely, and assign the corresponding variables in the LocalSetting.php the appropriate values. This is the most difficult and painful part of the installation of MediaWiki.
Check for latexEdit
The existing of latex at the server can be detected with the simple code, for example, let the file 0.tex contains the four lines below:
Then, at the command line, the command
should produce the file 0.dvi
Check for the dvipsEdit
If the dvips is installed, then, after the test above, the command
dvips -o 0.ps 0.dvi
should produce the file 0.ps
Check for the gsEdit
If the check for latex and the check for dvips above did not cause the error messages, then one may test them together with the gs command typing in the command line
gs 0.ps
If at the console, this should produce a frame with image, showing the simple formula, indicating the square of variable x.
Check for the convertEdit
After the tests above, the program convert and be checked with the command
convert 0.ps 0.png
At the correct installation of the convert (if the tests above passed well), the file 0.png should appear, and, in principle, this file already can be glued into the html document (although the formula occupies very small part of the areas of the frame). However, the software is supposed to do such operations automatically, with correct setting, providing the beautiful formula in the displayed text. (Up to year 2011, the vector mode of displaying of mathematical formulas is not realized, so, for wiki, the formulas are stored as png images, and such images are supported by the most of browsers.)
For the use of MediaWiki, the commands mentioned above should be located, id est, the programmed should know the full path to each of them. This can be realized in the command line with commands
Each of this commands is supposed to type at the screen the full path to each of the executables required. Let the corresponding outputs are p_1, p_2,p_3 and p_4.
CorrectionEdit
Yet, there is no robot to perform the automatic installing of the software. Assume that all the directories required have the mode access 777, and the only problem is that MediaWiki cannot find the support for the conversion of formulas to the images. The programmer should find the corresponding variables in the MediaWiki package and assign them the required values. In some cases, this can be done, editing the LocalSetting.php
The names of variable for latex is already known; so, the corresponding line may have format
$wgLatexCommand = "p_1";
however, the p_1 should be substituted to value returned by the which latex command; for example, for some servers, it may have form
$wgLatexCommand = "/usr/texbin/latex";
Unfortunately, the names of other variables that need to be assigned are not yet reported. There is hypothesis, that the correct name for the convert program is
$wgImageMagickConvertCommand = "p_4";
where p_4 should be substituted to the value specific for the custom server; doe example, for some servers, it may have form
The confirmation of the name of variable for the path of the program convert and search for the names of variables for the programs dvips and gs may be matter for the future research. The assignment of appropriate values to these variables is expected to make the installation of mediaviki usable.
This text is expected to be updated according to the recommendations of more successful wiki-specialists.
The manual about the use of the variable $wgUseTeX and all the requirement accessories is expected to appear at Manual:$wgUseTeX ; currently there is only a stab there.
math/README in SVN
Retrieved from "https://www.mediawiki.org/w/index.php?title=Manual:$wgUseTeX&oldid=5040022"
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Superellipsoid - Maple Help
Home : Support : Online Help : Math Apps : Calculus : Multivariate : Superellipsoid
A superellipsoid is a 3-dimensional solid whose horizontal cross-sections are superellipses with the same exponent r, and whose vertical cross-sections through the centre are superellipses with the same exponent t. The general implicit equation for a superellipsoid is
{\left({|\frac{x}{A}|}^{r}+{|\frac{y}{B}|}^{r}\right)}^{\frac{t}{r}}+{|\frac{z}{C}|}^{t}\le 1
The parameters t and r are positive real numbers which control the amount of flattening at the tips and equator of the solid. Factors A, B, and C scale the basic shape along each axis and are called the semi-diameters of the solid.
t = r
, the equation for a superellipsoid becomes a special case of the superquadric equation.
Interesting Solids
r = 2
, the horizontal cross sections of the solid are circles, so the superellipsoid is a solid of revolution: it can be obtained by rotating a superellipse of exponent t around the vertical axis.
t = r =2
, the solid is an ordinary ellipsoid. In particular, if
A = B = C
, the solid is a sphere of radius A.
A = B = 3, C = 4, t = 2.5
r = 2
, the superellipsoid is a special solid known as Piet Hein's "superegg".
The following graph shows a superellipsoid. Use the sliders to adjust the semi-diameters and parameters to see what solids you can make.
Parameter, t
Parameter, r
Semi-Diameter, C
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The city planning commission has set aside funds to build a bridge going north and south across the Newton River. To save money, they have agreed to build the shortest bridge possible. A map for the
8
-mile wide city is shown at right. The equations below represent the riverbanks where x is the number of miles east of City Hall and y is the number of miles north of City Hall. Determine the location for the shortest bridge that will span the Newton River.
North Bank:
y = \cos( \frac { \pi x } { 2 } )+6
South Bank:
y = \ln(9 - x)
At any give value of
x
, the distance between the North and South Banks is:
y=\left ( \text{cos}\left ( \frac{\pi (x)}{2}\right )+6 \right )-(\text{ln}(9-x))
|
Georg Friedrich Bernhard Riemann (1826-1866) is the person who formulated the modern definition of an integral. He decided that it was not absolutely necessary that each rectangle have the same width. They do not even need to be the same type (i.e. they all do not need to be endpoint or midpoint rectangles).
For example, examine the rectangles at right to estimate the area under
f(x)
. Will they still give a good estimate of area even though the rectangles do not have the same width?
While the rectangles in the diagram will give a good estimate, discuss some of the advantages using rectangles that have the same width?
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Point reflection - Wikipedia
(Redirected from Central inversion)
Geometric symmetry operation
Find sources: "Point reflection" – news · newspapers · books · scholar · JSTOR (April 2010) (Learn how and when to remove this template message)
Not to be confused with Reflection point.
Dual tetrahedra that are centrally symmetric to each other
Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center.
4 Point reflection as a special case of uniform scaling or homothety
5 Point reflection group
6 Point reflections in mathematics
7 Point reflection in analytic geometry
9 Inversion centers in crystallography
10 Inversion with respect to the origin
10.4 Clifford algebras and spin groups
The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections. More narrowly, a reflection refers to a reflection in a hyperplane (
{\displaystyle n-1}
dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where
{\displaystyle 1\leq k\leq n-1}
) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity
{\displaystyle n-1}
on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n).
The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle.
{\displaystyle \mathrm {Ref} _{\mathbf {p} }(\mathbf {a} )=2\mathbf {p} -\mathbf {a} .}
x* = 2a − x
Point reflection as a special case of uniform scaling or homothety[edit]
When the inversion point P coincides with the origin, point reflection is equivalent to a special case of uniform scaling: uniform scaling with scale factor equal to −1. This is an example of linear transformation.
When P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation: homothety with homothetic center coinciding with P, and scale factor −1. (This is an example of non-linear affine transformation.)
Point reflection group[edit]
The composition of two point reflections is a translation. Specifically, point reflection at p followed by point reflection at q is translation by the vector 2(q − p).
Point reflections in mathematics[edit]
Point reflection in analytic geometry[edit]
{\displaystyle P(x,y)}
and its reflection
{\displaystyle P'(x',y')}
with respect to the point
{\displaystyle C(x_{c},y_{c})}
, the latter is the midpoint of the segment
{\displaystyle {\overline {PP'}}}
{\displaystyle {\begin{cases}x_{c}={\frac {x+x'}{2}}\\y_{c}={\frac {y+y'}{2}}\end{cases}}}
Hence, the equations to find the coordinates of the reflected point are
{\displaystyle {\begin{cases}x'=2x_{c}-x\\y'=2y_{c}-y\end{cases}}}
Particular is the case in which the point C has coordinates
{\displaystyle (0,0)}
(see the paragraph below)
{\displaystyle {\begin{cases}x'=-x\\y'=-y\end{cases}}}
Geometrically in 3D it amounts to rotation about an axis through P by an angle of 180°, combined with reflection in the plane through P which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are
{\displaystyle {\overline {1}}}
, Ci, S2, and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry, see cyclic symmetries with n = 1.
Inversion centers in crystallography[edit]
Molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In crystallography, the presence of inversion centers distinguishes between centrosymmetric and noncentrosymmetric compounds. Crystal structures are composed of various polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as tetrahedra, while five-coordinate environments can be square pyramidal or trigonal bipyramidal depending on the bonding angles. All crystalline compounds come from a repetition of an atomic building block known as a unit cell, and these unit cells define which polyhedra form and in what order. These polyhedra link together via corner-, edge- or face sharing, depending on which atoms share common bonds. Polyhedra containing inversion centers are known as centrosymmetric, while those without are noncentrosymmetric. Six-coordinate octahedra are an example of centrosymmetric polyhedra, as the central atom acts as an inversion center through which the six bonded atoms retain symmetry. Tetrahedra, on the other hand, are noncentrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. It is important to note that bonding geometries with odd coordination numbers must be noncentrosymmetric, because these polyhedra will not contain inversion centers.
Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry. Common irregularities found in crystallography include distortions and disorder. Distortion involves the warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic attraction between heteroatoms. For instance, a titanium center will likely bond evenly to six oxygens in an octahedra, but distortion would occur if one of the oxygens were replaced with a more electronegative fluorine. Distortions will not change the inherent geometry of the polyhedra—a distorted octahedron is still classified as an octahedron, but strong enough distortions can have an effect on the centrosymmetry of a compound. Disorder involves a split occupancy over two or more sites, in which an atom will occupy one crystallographic position in a certain percentage of polyhedra and the other in the remaining positions. Disorder can influence the centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center.
Centrosymmetry applies to the crystal structure as a whole, as well. Crystals are classified into thirty-two crystallographic point groups which describe how the different polyhedra arrange themselves in space in the bulk structure. Of these thirty-two point groups, eleven are centrosymmetric. The presence of noncentrosymmetric polyhedra does not guarantee that the point group will be the same—two noncentrosymmetric shapes can be oriented in space in a manner which contains an inversion center between the two. Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair. The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group.
Noncentrosymmetric compounds can be useful for application in nonlinear optics. The lack of symmetry via inversion centers can allow for areas of the crystal to interact differently with incoming light. The wavelength, frequency and intensity of light is subject to change as the electromagnetic radiation interacts with different energy states throughout the structure. Potassium titanyl phosphate, KTiOPO4 (KTP) crystalizes in the noncentrosymmetric, orthorhombic Pna21 space group, and is a useful non-linear crystal. KTP is used for frequency-doubling neodymium-doped lasers, utilizing a nonlinear optical property known as second-harmonic generation. The applications for nonlinear materials are still being researched, but these properties stem from the presence of (or lack thereof) an inversion center.
Inversion with respect to the origin[edit]
In mathematics, reflection through the origin refers to the point reflection of Euclidean space Rn across the origin of the Cartesian coordinate system. Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by
{\displaystyle -1}
, and can also be written as
{\displaystyle -I}
{\displaystyle I}
is the identity matrix. In three dimensions, this sends
{\displaystyle (x,y,z)\mapsto (-x,-y,-z)}
As a scalar matrix, it is represented in every basis by a matrix with
{\displaystyle -1}
on the diagonal, and, together with the identity, is the center of the orthogonal group
{\displaystyle O(n)}
In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension
{\displaystyle 2n}
, it is rotation by 180 degrees in n orthogonal planes;[note 1] note again that rotations in orthogonal planes commute.
It has determinant
{\displaystyle (-1)^{n}}
(from the representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the special orthogonal group SO(2n), and it is orientation-reversing in odd dimension, thus not an element of SO(2n + 1) and instead providing a splitting of the map
{\displaystyle O(2n+1)\to \pm 1}
{\displaystyle O(2n+1)=SO(2n+1)\times \{\pm I\}}
as an internal direct product.
It preserves every quadratic form, meaning
{\displaystyle Q(-v)=Q(v)}
, and thus is an element of every indefinite orthogonal group as well.
In SO(2r), reflection through the origin is the farthest point from the identity element with respect to the usual metric. In O(2r + 1), reflection through the origin is not in SO(2r+1) (it is in the non-identity component), and there is no natural sense in which it is a "farther point" than any other point in the non-identity component, but it does provide a base point in the other component.
Clifford algebras and spin groups[edit]
It should not be confused with the element
{\displaystyle -1\in \mathrm {Spin} (n)}
in the spin group. This is particularly confusing for even spin groups, as
{\displaystyle -I\in SO(2n)}
, and thus in
{\displaystyle \operatorname {Spin} (n)}
there is both
{\displaystyle -1}
and 2 lifts of
{\displaystyle -I}
Estermann measure
Kovner–Besicovitch measure
^ "Orthogonal planes" meaning all elements are orthogonal and the planes intersect at 0 only, not that they intersect in a line and have dihedral angle 90°.
^ This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the spectral theorem, for instance.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Point_reflection&oldid=1075013244"
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J. Nyéki, A. Phillis, B. Cowan, J. Saunders
Recent torsional oscillator measurements on the second layer of
{4}^{}
{\mathrm{He\quad }}^{}
{}^{}
adsorbed on graphite have identified an anomalous superfluid response over a coverage range near third-layer promotion, with four distinct coverage regimes. Here, we present details of the superfluid response in the coverage regime immediately below third-layer promotion. A scaling analysis of the inferred superfluid fraction shows the characteristic temperature governing the superfluid response to decrease, approaching zero near the coverage at which simulations predict the second layer to form a conventional incommensurate solid.
J. Low Temp. Phys. 187(2017) 475
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\mathrm{CB}
R
X
R
A
A
The compressed sparse column form of an
m
A
k
\mathrm{CB}
R
X
k
A
X
k
A
R
k
\mathrm{CB}
n+1
{\mathrm{CB}}_{i}
X
R
i
{\mathrm{CB}}_{1}=1
{\mathrm{CB}}_{n+1}=k+1
i
{\mathrm{CB}}_{i}
{\mathrm{CB}}_{i+1}
\mathrm{cbbase}
{\mathrm{CB}}_{1}=\mathrm{cbbase}
{\mathrm{CB}}_{n+1}=k+\mathrm{cbbase}
A
X
A
R
k
{\mathrm{CB}}_{i}
X
R
i
A
\mathrm{rtable_option}\left(X,\mathrm{storage}\right)
\mathrm{sparse}
{\mathrm{sparse}}_{\mathrm{upper}}
{\mathrm{sparse}}_{\mathrm{lower}}
A
\mathrm{rtable_indfns}\left(A\right)
\mathrm{shape}=\mathrm{symmetric}
\mathrm{storage}={\mathrm{sparse}}_{\mathrm{upper}}
A
A
\mathrm{sfloat},\mathrm{complex}\left(\mathrm{sfloat}\right),{\mathrm{integer}}_{1},{\mathrm{integer}}_{2},{\mathrm{integer}}_{4},{\mathrm{integer}}_{8},{\mathrm{float}}_{4},{\mathrm{float}}_{8},{\mathrm{complex}}_{8}
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
m≔\mathrm{Matrix}\left(6,6,{\left(1,2\right)=-81,\left(2,3\right)=-55,\left(2,4\right)=-15,\left(3,1\right)=-46,\left(3,3\right)=-17,\left(3,4\right)=99,\left(3,5\right)=-61,\left(4,2\right)=18,\left(4,5\right)=-78,\left(5,6\right)=22},\mathrm{datatype}=\mathrm{integer}[4]\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{CompressedSparseForm}\left(m\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{11}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{CompressedSparseForm}\left(m,'\mathrm{form}=\mathrm{row}'\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{11}\\ \textcolor[rgb]{0,0,1}{11}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{6}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{CompressedSparseForm}\left(m,'\mathrm{cbbase}'=0\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{7}\\ \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{10}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{CompressedSparseForm}\left(m,'\mathrm{rbase}'=0\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{11}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{m2}≔\mathrm{Matrix}\left([[0,1,0],[2,0,0],[3,0,4],[5,6,7]],'\mathrm{datatype}=\mathrm{float}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{m2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{5.}& \textcolor[rgb]{0,0,1}{6.}& \textcolor[rgb]{0,0,1}{7.}\end{array}]
\mathrm{cb},r,x≔\mathrm{CompressedSparseForm}\left(\mathrm{m2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3.}\\ \textcolor[rgb]{0,0,1}{5.}\\ \textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{6.}\\ \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{7.}\end{array}]
r
\mathrm{column}≔2
\textcolor[rgb]{0,0,1}{\mathrm{column}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2}
r[\mathrm{cb}[\mathrm{column}]..\mathrm{cb}[\mathrm{column}+1]-1]
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\end{array}]
x
x[\mathrm{cb}[\mathrm{column}]..\mathrm{cb}[\mathrm{column}+1]-1]
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{6.}\end{array}]
\mathrm{m3}≔\mathrm{Matrix}\left(3,3,\left(i,j\right)↦i-j\right)
\textcolor[rgb]{0,0,1}{\mathrm{m3}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{CompressedSparseForm}\left(\mathrm{m3}\right)
\mathrm{m3}
\mathrm{datatype}={\mathrm{integer}}_{4}
\mathrm{m3}≔\mathrm{Matrix}\left(\mathrm{m3},\mathrm{datatype}=\mathrm{integer}[4]\right)
\textcolor[rgb]{0,0,1}{\mathrm{m3}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{-2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{cb},r,x≔\mathrm{CompressedSparseForm}\left(\mathrm{m3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{7}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{-2}\\ \textcolor[rgb]{0,0,1}{-1}\end{array}]
\mathrm{m4}≔\mathrm{Matrix}\left(6,\mathrm{datatype}=\mathrm{integer}[4],\mathrm{shape}=\mathrm{antisymmetric},\mathrm{storage}=\mathrm{sparse}[\mathrm{lower}],\left(i,j\right)↦\mathrm{`if`}\left(\mathrm{irem}\left(i+j,2\right)=1,i-j,0\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{m4}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-5}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{CompressedSparseForm}\left(\mathrm{m4}\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{10}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{6}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{1}\end{array}]
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Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter - MATLAB predict - MathWorks Switzerland
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]
x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]
y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k-1|k-1\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right]
|
g(x)
below and determine if it is differentiable at
x = 2
g ( x ) = \left\{ \begin{array} { c c } { ( x - 1 ) ^ { 2 } } & { \text { for } x < 2 } \\ { 2 \operatorname { sin } ( x - 2 ) + 1 } & { \text { for } x \geq 2 } \end{array} \right.
Differentiability implies continuity. If you are checking for differentiability, you also need to check for continuity.
\lim\limits_{x\rightarrow 2^{-}}g(x)=\lim\limits_{x\rightarrow 2^{+}}g(x)
\lim\limits_{x\rightarrow 2}g(x)=g(2)
\lim\limits_{x\rightarrow 2^{-}}g(x)=\lim\limits_{x\rightarrow 2^{-}}(x-1)^{2}=1
\lim\limits_{x\rightarrow 2^{+}}g(x)=\lim\limits_{x\rightarrow 2^{+}}2\text{sin}(x-2)+1=1
g(2) = 2(\operatorname{sin}(2-2)) + 1 =1
\lim\limits_{x\rightarrow 2^{+}}g(x)=g(2)
g(x)
x = 2
To check for differentiability:
\lim\limits_{x\rightarrow 2^{-}}g'(x)=\lim\limits_{x\rightarrow 2^{+}}g'(x)=g'(2)
\lim\limits_{x\rightarrow 2^{-}}g'(x)=\lim\limits_{x\rightarrow 2^{-}}(2x-2)=2
\lim\limits_{x\rightarrow 2^{+}}g'(x)=\lim\limits_{x\rightarrow 2^{+}}2\text{cos}(x-2)=2
g^\prime(2) = 2\operatorname{cos}(2-2) = 2
g(x)
g^\prime(x)
are continuous at
x = 2
g(x)
x = 2
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Applied Sciences | Special Issue : Automation Control and Robotics in Human-Machine Cooperation
Automation Control and Robotics in Human-Machine Cooperation
Special Issue "Automation Control and Robotics in Human-Machine Cooperation"
Dr. Wojciech Kaczmarek
Faculty of Mechatronics, Armament and Aerospace, Military University of Technology, Kaliskiego 2 Street, 00-908 Warsaw, Poland
Interests: human–robot collaboration; digital tweens; manufacturing simulation; robotics; pick and place processes; production planning and control
Dr. Jarosław Panasiuk
Interests: human–robot collaboration; digital tweens; manufacturing simulation; robotics; pick and place processes; production planning and control; vision systems; machine learning
Interests: industrial informatics; induction heating; electromagnetic field; numerical simulation; optimization; electromagnetic fields; alloys; electromagnetics; computational electromagnetics; electromagnetic compatibility; electromagnetic engineering
Special Issue in Applied Sciences: Advanced Applications of Industrial Informatic Technologies
Special Issue in Electronics: Recent Advanced Applications of Virtual Industrial Informatics and Robotics
Topics: Advances in Energy Market and Power System Modelling and Optimization
I would like to interest you in this Special Issue, entitled “Automation Control and Robotics in Human-Machine Cooperation,” in Applied Sciences and cordially invite you to submit your articles. The purpose of this Special Issue is to compile studies on knowledge, research practice, and forecast development trends in the field of automation control, robotics, and human-machine cooperation.
Every day, people use devices and machines that are more or less automated. Thanks to advanced interfaces, we often do not think about the degree of automation. However, we should be aware of the fact that cooperation between humans and machines is developing very fast both in everyday life and on the production floor. Close cooperation between humans and machines is possible thanks to the rapid development of sensor technologies, data processing, and data acquisition systems. On the one hand, we are trying to increase the functionality of everyday devices by automating them; on the other hand, we are trying to find a place for operators working on production lines by reducing their distance to the machines. In both cases, the main goal is to combine human skill and innovation with machine efficiency and precision. This is particularly evident in modern production lines, where the cooperation of operators with robots in a common workspace is becoming an everyday reality. Nowadays, virtual environments for designing and programming robots and machines enable fast and reliable preparation of technological processes. They also allow simulation studies to determine the performance of processes, which corresponds to the idea of digital tweens (full representation of a real process station by a virtual model).
We welcome the submission of papers on the topics including but not limited to the following:
Human–robot collaboration for manufacturing processes
Sensing and control in robotics
Modelling and simulation in robotics and automation—digital tweens
Advanced environment for modelling and controlling robotics applications
Robots and Industry 4.0 concepts
Industrial robots in research applications
Safety in industrial robot applications
Design and development of robots and robot end-effectors
Study and analysis of robotic processes
Techniques for online, offline robots programming
Sensors in control and steering of the system
modelling and simulation of robotic systems
research robot application
This paper proposes a path correction scheduling strategy for the underground mining boom roadheader by ably combining a back propagation (BP) neural network and state estimation. First, a pose deviation-based tracking model is designed for the roadheader, and it is then further studied [...] Read more.
This paper proposes a path correction scheduling strategy for the underground mining boom roadheader by ably combining a back propagation (BP) neural network and state estimation. First, a pose deviation-based tracking model is designed for the roadheader, and it is then further studied and optimized by incorporating the benefits of BP neural networks into the model adaptation. Considering the fact that there is skidding between tracks on the ground and errors during the instant pose detection of the roadheader underground, singular value decomposition (SVD)–Unscented Kalman filtering is applied to estimate the real pose deviation, based on the summarized distribution regularities of the track skidding ratios and the pose detection errors, instead of complicated analysis mechanisms. The BP neural network and states estimation are well combined in structure, enabling this scheduling strategy to update the control law and revise the control instruction simultaneously in the procedure. The proposed path tracking model for the roadheader is simple and clear, without adding extra devices or massive algorithms, which is attractive in terms of industrial use. The path tracking simulations show that this proposed strategy achieves path tracking well in different scenarios and is of high adaptability when facing complex trajectory while still giving stable control instructions for the roadheader. Full article
Self-Balancing Power Amplifier with a Minimal DC Offset for Launcher Automation Control Circuits of a Surface-to-Air Missile System
Witold Bużantowicz
This paper discusses the design of a new self-balancing amplifier of an AC component power characterized by a minimal output DC offset. The design of the amplifier is based on semiconductor technology and intended for application in low-frequency analog signal processing paths, particularly [...] Read more.
This paper discusses the design of a new self-balancing amplifier of an AC component power characterized by a minimal output DC offset. The design of the amplifier is based on semiconductor technology and intended for application in low-frequency analog signal processing paths, particularly in surface-to-air missile system launcher automation circuits. The proposed solution has several design and technical-implementation advantages, whereas the primary novelty compared to the commonly used ones is the ability for self-generating a near-zero DC component value of output signal. The design features and technical parameters of the developed amplifier make it suitable for use in a wide range of devices that must ensure the stable, prolonged operation of a low-frequency power amplifier under variable weather conditions and a minimal DC offset of output signal. Full article
Junyu Yao
Qijie Lan
This paper addresses a smoother fixed-time obstacle-avoidance trajectory planning based on double-stranded ribonucleic acid (dsRNA) splicing evolutionary algorithm for a dual-arm free-floating space robot, the smoothness of large joint angular velocity is improved by
15.61%
on average compared with the current trajectory [...] Read more.
15.61%
on average compared with the current trajectory planning strategy based on pose feedback, and the convergence performance is improved by
76.44%
compared with the existing optimal trajectory planning strategy without pose feedback. Firstly, according to the idea of pose feedback, a novel trajectory planning strategy with low joint angular velocity input is proposed to make the pose errors of the end-effector and base converge asymptotically within fixed time. Secondly, a novel evolutionary algorithm based on the gene splicing idea of dsRNA virus is proposed to optimize the parameter of the fixed-time error response function and obstacle-avoidance algorithm, which can make joint angular velocity trajectory is planned smooth. In the end, the optimized trajectory planning strategy is applied into the dual-arm space robot system so that the robotic arm can smoothly, fast and accurately complete the tracking task. The proposed novel algorithm achieved 7.56–30.40% comprehensive performance improvement over the benchmark methods, experiment and simulation verify the effectiveness of the proposed method. Full article
This paper presents research on 3D scanning by taking advantage of a camera array consisting of up to five adjacent cameras. Such an array makes it possible to make a disparity map with a higher precision than a stereo camera, however it preserves [...] Read more.
This paper presents research on 3D scanning by taking advantage of a camera array consisting of up to five adjacent cameras. Such an array makes it possible to make a disparity map with a higher precision than a stereo camera, however it preserves the advantages of a stereo camera such as a possibility to operate in wide range of distances and in highly illuminated areas. In an outdoor environment, the array is a competitive alternative to other 3D imaging equipment such as Structured-light 3D scanners or Light Detection and Ranging (LIDAR). The considered kinds of arrays are called Equal Baseline Camera Array (EBCA). This paper presents a novel approach to calibrating the array based on the use of self-calibration methods. This paper also introduces a testbed which makes it possible to develop new algorithms for obtaining 3D data from images taken by the array. The testbed was released under open-source. Moreover, this paper shows new results of using these arrays with different stereo matching algorithms including an algorithm based on a convolutional neural network and deep learning technology. Full article
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Revision as of 11:42, 12 August 2021 by Kleins1 (talk | contribs) (→450.1 Asphalt Binder)
8 450.8 Asphalt Over Rubblized Pavement
9 450.9 Safety EdgeSM
The low temperature number is to remain as indicated in the above table. The high temperature numbers have been set for the traffic loads and operation speeds normally associated with these corridors. The high temperature numbers are recommended minimums and are not to be reduced; however, may be raised to a PG 70-22 or PG 76-22, when deemed necessary, to accommodate actual or anticipated traffic conditions. Typically, the high temperature number should be raised one increment (6 degrees) when traffic speeds are expected to be in the range of 12 to 45 mph and raised two increments (12 degrees) for extremely slow traffic of < 12 mph. Typical candidates for these high temperature number raises are roadways with AADT greater than 3500 that are in a highly congested, urbanized area, have frequent stop and go traffic, or have steep grades with significantly slow traffic speeds. Use of any other PG asphalt binder must be approved by the State Construction and Materials Engineer.
{\displaystyle \approx }
{\displaystyle \approx }
{\displaystyle \approx }
Minor Routes AADT > 3500 and Total Average 24 Hour Commercial Truck Traffic < 600 BP-1 (Sec 401)
Minor Routes AADT < 3500 and Total Average 24 Hour Commercial Truck Traffic < 600 BP-1 or BP-2(3)
Min. Placement Depths and Min. %AC by Volume for Asphalt Treatment Options
3 in. 2 in. 1¾ in. 1½ in. 1 in.
1 When used as the riding surface. When used as a leveling course below the top lift, the minimum lift thickness should be in accordance with EPG 450.4 Level Course.
3 in. 1¾ in 1½ in. 1 in.
The above minimum lift thicknesses are recommended to ensure adequate field density can be achieved and a quality product can be provided. The minimum lift thicknesses do not account for surface irregularities in the roadway. Additional asphalt quantities should be included with single lift asphalt overlays to account for the surface irregularities. The additional asphalt quantities added should be in accordance with the average rutting depth measurements from the roadway as provided in Table 402.1.2.1.
Central Office Construction and Materials may recommend thicknesses greater than the above minimums but projects should not be designed for thicknesses less than the above minimums. If using the minimum lift thickness criteria results in the project exceeding its budget, then a finer gradation asphalt mixture should be used and designed for its minimum lift thickness or greater (for example, if 1.75 in. of BP-1 exceeds project budget, use 1.5 in. or greater of BP-2).
Existing surfaces, both asphaltic and portland cement concrete, should be prepared either by milling or leveling course before the first full-thickness lift of asphalt is placed. Milling is the preferred method of leveling, however PMBP or SP asphaltic concrete may be used for spot wedging and for leveling course as per the standard specifications, providing the minimum thickness is not less than the following:
1½ in. 1 in. ¾ in.
Payment for asphaltic concrete by the square yard should only be used for full depth asphaltic concrete pavements on alternate bid projects. All other payment for asphaltic concrete should be by the ton. For preliminary design, estimate factors may be used to compute quantities for bases, flexible type pavements, seal coats, etc. For final design, the designer should request estimate factors for anticipated rock formations from the District Construction and Materials Engineer to obtain more reliable results. When payment is by the ton, estimate factors should be shown on the plans with a note "For Information Purposes Only." When payment is by the square yard, estimate factors should not be shown on the plans.
450.8 Asphalt Over Rubblized Pavement
Rubblization is the in-place breaking of an existing Portland Cement Concrete (PCC) pavement into an aggregate base for new hot mix asphalt. When major asphalt rehabilitation over existing concrete pavement is considered the existing pavement is rubblized prior to the asphalt overlay. If the existing pavement consists of asphalt over concrete the existing asphalt should be removed prior to rubblization.
When rubblization is used, the asphalt is paid for by the ton of mix necessary to obtain the thickness indicated by Construction and Materials. The plan quantity should be established with consideration for thickness in excess of that specified to account for irregularities in the existing pavement. Similar to unbonded overlay the contractor is required to establish the existing roadway profile and set the final overlay profile. For this reason the bid item for contractor staking should be included in the contract.
There must be adequate subgrade support under old rigid pavements. Dynamic cone penetrometer (DCP) testing of the unbound layers under the PCC pavement, performed during the condition surveys, must indicate that adequate support exists for rubblization to occur.
Shaping the edge of pavement to a 30-degree bevel is an effective strategy for mitigating the negative effects associated with vertical edge drop-off. Refer to Std. Plan 401.00.
As with conventional paving, the graded material adjacent to the Safety EdgeSM should be brought flush with the top of the pavement following paving. If this material should settle or be otherwise displaced in the future, however, the Safety EdgeSM would continue to provide a durable height transition to smoothly conduct vehicles back to the paved road.
Safety EdgeSM must be included in all single-lift overlay projects when the finished surface will be 21 ft. wide or greater. This includes shoulder paving unless the paved width of each shoulder exceeds 4 ft. When using the Safety EdgeSM, the quantity of asphalt mix will increase by approximately 2 percent.
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Collateral - cypher docs
The synthetic assets (cAssets) minted by users via the protocol are backed by stablecoins, starting with USDC. We plan to expand our collateral capabilities to include more stablecoins in the future. Collateralizing cAssets with stablecoins allows for more stability since the collateral is not subject to high volatility the protocol can require lower collateralization ratios (C-ratios) and the protocol will be more capital efficient. The C-ratio of cypher contracts will initially be set at 400% and will allow for a dip to 200% before a minter runs the possibility of being liquidated. Minters incur a debt to the market they provide liquidity to. In order to receive the collateral they initially staked, minters will need to pay back debt by burning cAssets.
Minting Collaterization Ratio
C-ratio for minters is as follows
C_{mint}=\frac{\textrm{capital locked in}}{\textrm{value of minted contracts}}
Currently the optimal C-ratio is 400%. To incentivize minters to maintain the optimal ratio a minter will not be able to collect their pro-rata fee portion if a minted position’s collateral falls below 400% until they add enough capital to meet the desired ratio. There will be a buffer to ensure that small fluctuations in contract value does not impact a minter’s ability to collect the fees. To adjust their ratio, a minter can add capital or burn cAssets to bring the C-ratio back up to the optimum level. A minter could also choose to withdraw capital or mint additional cAssets to bring the C-ratio down to the optimum level.
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Convergence of Solutions to a Certain Vector Differential Equation of Third Order
Cemil Tunç, Melek Gözen, "Convergence of Solutions to a Certain Vector Differential Equation of Third Order", Abstract and Applied Analysis, vol. 2014, Article ID 424512, 6 pages, 2014. https://doi.org/10.1155/2014/424512
Cemil Tunç1 and Melek Gözen1
1Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey
We give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature.
This paper is concerned with the following nonlinear vector differential equation of third order: where and and are continuous functions in their respective arguments.
It should be noted that, in 2005, Afuwape and Omeike [1] considered the following nonlinear vector differential equation of third order: where is real symmetric -matrix. The author established a new result on the convergence of solutions of (2) under different conditions on the function . For some related papers on the convergence of solutions to certain vector differential equations of third order, the readers can referee to the papers of Afuwape [2], Afuwape and Omeike [3], and Olutimo [4]. Further, it is worth mentioning that in a sequence of results Afuwape [2, 5, 6], Afuwape and Omeike [3], Afuwape and Ukpera [7], Ezeilo [8], Ezeilo and Tejumola [9, 10], Meng [11], Olutimo [4], Reissig et al. [12], Tiryaki [13], Tunç [14–16], Tunç and Ateş [17], C. Tunç and E. Tunç [18], and Tunç and Karakas [19] investigated the qualitative behaviors of solutions, stability, boundedness, uniform boundedness and existence of periodic solutions, and so on, except convergence of solutions, for some kind of vector differential equations of third order.
The Lyapunov direct method was used with the aid of suitable differentiable auxiliary functions throughout the mentioned papers. However, to the best of our knowledge, till now, the convergence of the solutions to (1) has not been discussed in the literature. Thus, it is worthwhile to study the topic for (1). It should be noted that the result to be established here is different from that in Afuwape [2], Afuwape and Omeike [1, 3], Olutimo [4], and the above mentioned papers. This paper is an extension and generalization of the result of Afuwape and Omeike [3]. It may be useful for the researchers working on the qualitative behaviors of solutions (see, also, Tunç and Gözen [20]).
It should be noted that throughout the paper will denote the real Euclidean space of -vectors and will denote the norm of the vector in .
Definition 1. Any two solutions , of (1) in will be said to converge to each other if as .
Theorem 2. We assume that there are positive constants , , , , , , and such that the following conditions hold: (i)the Jacobian matrices , , and exist and are symmetric and their eigenvalues satisfy for all , , in ;(ii) satisfies for any , , , , in .
If then any two solutions , of (1) necessarily converge, where , , , are some positive constants with and ,
Remark 3. The mentioned theorem itself still holds valid with (5) replaced by the much weaker condition for arbitrary any , , , , in , where it is assumed that for .
The following lemma is needed in our later analysis.
Lemma 4. Let be a real symmetric -matrix and where and are constants.
Proof (see Afuwape [5]). Our main tool in the proof of our result is the continuous function defined for any triple vectors , , in , by This function can be rearranged as where and
The following result is immediate from the estimate (11).
Lemma 5. Assume that all the conditions on the vectors , , and in the theorem hold. Then, there exist positive constants and such that for arbitrary in .
Proof. Let Then the proof can be easily completed by using Lemma 4. Therefore, we omit the details of the proof.
Proof of the Theorem. Let in be any solution of (1). For such a solution, let and be denoted, respectively, by and . Then, we can rewrite (1) in the following equivalent system form:
Let in be any solution of (1), define by where is the function defined in (11) with , , replaced by , and , respectively.
By Lemma 5, it follows that there exist and such that
When we differentiate the function with respect to along the system (15), it follows, after simplification, that where
Note that the existence of the following estimates is clear (see Afuwape and Omeike [1]): where , , , , , .
Subject to the assumptions, it can be easily obtained that
In view of the assumptions of the theorem, it is also clear that Hence, Using the estimate , it follows that Then if with .
Further, since then where .
Moreover, it is obvious that where .
Hence, Using the assumption (5), we get so that There exists a constant such that provided that , where is a sufficiently small positive constant.
In view of (17), the last estimate implies that for some positive constant .
The conclusion of the theorem is immediate if, provided that , on integrating in (33) between and , we have which implies that By (17), this shows that This completes the proof of the theorem.
Example 6. Let us consider (1), with where are bounded continuous functions on .
Then, it can be easily seen that Thus, , , , , , and .
Let us choose Then, Since , then all the conditions of Theorem 2 hold. Therefore, all solutions of the equation considered converge (see, also, [1]).
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n
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C. Tunç, “On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order,” Applied Mathematics and Mechanics, vol. 20, no. 2, pp. 163–170, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Tunç, “On the boundedness of solutions of certain nonlinear vector differential equations of third order,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 49, no. 3, pp. 291–300, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. Tunç, “On the stability and boundedness of solutions of nonlinear vector differential equations of third order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2232–2236, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Tunç and M. Ateş, “Stability and boundedness results for solutions of certain third order nonlinear vector differential equations,” Nonlinear Dynamics, vol. 45, no. 3-4, pp. 273–281, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. Tunç and E. Tunç, “New ultimate boundedness and periodicity results for certain third-order nonlinear vector differential equations,” Mathematical Journal of Okayama University, vol. 48, pp. 159–172, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
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C. Tunç and M. Gözen, “Stability and uniform boundedness in multidelay functional differential equations of third order,” Abstract and Applied Analysis, vol. 2013, Article ID 248717, 7 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright © 2014 Cemil Tunç and Melek Gözen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Triple product property - WikiMili, The Free Encyclopedia
Triple product property
In abstract algebra, the triple product property is an identity satisfied in some groups.
{\displaystyle G}
be a non-trivial group. Three nonempty subsets
{\displaystyle S,T,U\subset G}
are said to have the triple product property in
{\displaystyle G}
if for all elements
{\displaystyle s,s'\in S}
{\displaystyle t,t'\in T}
{\displaystyle u,u'\in U}
it is the case that
{\displaystyle s's^{-1}t't^{-1}u'u^{-1}=1\Rightarrow s'=s,t'=t,u'=u}
{\displaystyle 1}
{\displaystyle G}
It plays a role in research of fast matrix multiplication algorithms.
In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
In mathematics convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted.
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. As a result, it manages to reduce the complexity of computing the DFT from , which arises if one simply applies the definition of DFT, to , where is the data size. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.
In mathematics, a Lie group is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separated—this makes Lie groups differentiable manifolds. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
Linear algebra is the branch of mathematics concerning linear equations such as
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring. The matrix product is designed for representing the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. In more detail, if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a column of B and summed to produce an entry of AB. When two linear maps are represented by matrices, then the matrix product represents the composition of the two maps.
In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity, just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure, which consists of a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, and satisfies the axioms implied by "vector space" and "bilinear".
In linear algebra, an n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges. Therefore, is a subset of , where is the power set of . The size of vertex set is called the order of the hypergraph, and the size of edges set is the size of the hypergraph.
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang (1935). Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra.
In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by .
In linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, was the asymptotically fastest known matrix multiplication algorithm from 1990 until 2010. It can multiply two matrices in time. This is an improvement over the naïve time algorithm and the time Strassen algorithm. Algorithms with better asymptotic running time than the Strassen algorithm are rarely used in practice, because the large constant factors in their running times make them impractical. It is possible to improve the exponent further; however, the exponent must be at least 2.
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets.
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such as counting the paths through a graph. Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel and distributed systems, where the computational work is spread over multiple processors.
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. arXiv : math.GR/0307321. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.
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Measure Theory Wiki - Optimal Transport Wiki
Welcome to the Measure Theory Wiki!
Here are some New MT Article Ideas
Real Analysis Background
Key Topics from Undergraduate Analysis
Approximation by Open/Compact Sets
{\displaystyle \sigma }
Banach-Tarski Paradox <-- this would be a great article for someone to edit; while it is very well written, it lacks references.
Intersections of Open Sets and Unions of Closed Sets (
{\displaystyle {\mathcal {G}}^{\delta }}
{\displaystyle {\mathcal {F}}^{\sigma }}
sets)
Beppo-Levi Theorem
Egerov's Theorem/Bounded Convergence Theorem
Isomorphism of Measure Spaces
{\displaystyle \sigma }
{\displaystyle {\mathcal {B}}_{\mathbb {R} }}
Vitali's Theorem and non-existence of a measure
Lebesgue-Stieljes Measures
201A Distraction Recommendations
Retrieved from "http://34.106.105.83/wiki/index.php?title=Measure_Theory_Wiki&oldid=2201"
This page was last edited 22:33, 19 December 2020 by Junk. Based on work by Ricky Lee, Andy Sun, Aditya Mohan, Daniel Apsley, Jimin Lin, Pranav Arrepu, Jorge Murillo, Dylan Adams, Xi Gong, Andrew Gracyk, Benedict Lee, Marcos Reyes, Daniel Guo, Blaine Quackenbush, Luis E. Valdivia, Connor FitzGerald, Qiqi Li, ChienHsun Lin, Jose Zavala-Fonseca, Katy Craig, mengrui zhang, Joel E. Pion, Jihye Lee, Arthur Diep-Nguyen, Tomas, Troy Kling, Alexander Louis J. Sabater and Charles Kulick.
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Create Yagi-Uda array antenna - MATLAB - MathWorks France
NumDirectors
DirectorLength
ReflectorLength
Create and View Yagi-Uda Array Antenna
Radiation Pattern of Yagi-Uda Array Antenna
Create Yagi-Uda array antenna
The yagiUda class creates a classic Yagi-Uda array comprised of an exciter, reflector, and N- directors along the z-axis. The reflector and directors create a traveling wave structure that results in a directional radiation pattern.
The exciter, reflector, and directors have equal widths and are related to the diameter of an equivalent cylindrical structure by the equation
w=2d=4r
For a given cylinder radius, use the cylinder2strip utility function to calculate the equivalent width. A typical Yagi-Uda antenna array uses folded dipole as an exciter, due to its high impedance. The Yagi-Uda is center-fed and the feed point coincides with the origin. In place of a folded dipole, you can also use a planar dipole as an exciter.
yu = yagiUda
yu = yagiUda(Name,Value)
yu = yagiUda creates a half-wavelength Yagi-Uda array antenna along the Z-axis. The default Yagi-Uda uses folded dipole as three directors, one reflector, and a folded dipole as an exciter. By default, the dimensions are chosen for an operating frequency of 300 MHz.
yu = yagiUda(Name,Value) creates a half-wavelength Yagi-Uda array antenna, with additional properties specified by one or more name-value pair arguments. Name is the property name and Value is the corresponding value. You can specify several name-value pair arguments in any order as Name1, Value1, ..., NameN, ValueN. Properties not specified retain default values.
dipoleFolded (default) | object
Antenna Type used as exciter, specified as the comma-separated pair consisting of 'Exciter' and an object.
NumDirectors — Total number of director elements
Total number of director elements, specified as a scalar.
Number of director elements should be less than or equal to 20.
Example: 'NumDirectors',13
DirectorLength — Director length
Director length, specified as a scalar or vector in meters.
Example: 'DirectorLength',[0.4 0.5]
DirectorSpacing — Spacing between directors
Spacing between directors, specified as a scalar or vector in meters.
Example: 'DirectorSpacing',[0.4 0.5]
ReflectorLength — Reflector length
Reflector length, specified as a scalar in meters.
Example: 'ReflectorLength',0.3
ReflectorSpacing — Spacing between exciter and reflector
Spacing between exciter and reflector, specified as a scalar in meters.
Example: 'ReflectorSpacing', 0.4
Example: yu.Load = lumpedElement('Impedance',75)
Create and view a Yagi-Uda array antenna with 13 directors.
y = yagiUda('NumDirectors',13);
Plot the radiation pattern of a Yagi-Uda array antenna at a frequency of 300 MHz.
pattern(y,300e6)
dipole | dipoleFolded | slot | cylinder2strip
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Error Analysis of Radial Motion Measurement of Ultra-Precision Spindle
Risheng Zhang1, Jialin Yang1*, Erwei Shang2, Yanqiu Chen2, Yu Liu2
1Institute of Machinery Manufacturing Technology, China Academy of Engineering Physics, Mianyang, China.
2School of Mechanical Engineering, Jiangnan University, Wuxi, China.
This paper gives an error analysis of radial motion measurement of ultra-precision spindle including nonlinearity error of capacitive displacement probes, misalignment error of probes, eccentric error of artifact ball and error induced by different error separating methods. Firstly, nonlinearity of a capacitive displacement probe targeting a spherical surface is investigated through experiment and the phenomena of fake displacement induced by lateral offset of the probe relative to an artifact ball are discussed. It is shown that the error motion in radial and axial direction and eccentric rotation of artifact ball will both induce lateral offset which causes a fake output of probes. Moreover, measurement error induced by angular positioning error for three famous error separating methods is detailed.
Error Motion, Spindle Metrology, Ultra-Precision Spindle, Error Analysis
Zhang, R. , Yang, J. , Shang, E. , Chen, Y. and Liu, Y. (2018) Error Analysis of Radial Motion Measurement of Ultra-Precision Spindle. World Journal of Engineering and Technology, 6, 567-574. doi: 10.4236/wjet.2018.63034.
Ultra-precision spindle or rotating table usually working on aerostatic or hydrostatic principle plays an important role in ultra-precision machine tools. The rotational accuracy of spindle is a main factor influencing the machining accuracy of ultra-precision machine tool [1] . Traditional method no longer applies to error motion measurement for ultra-precision axis because of artifact form error. As a result, several error separating methods have been developed. The most well-known methods such as Donaldson reversal, multi-steps and multi-probe have been demonstrated to approach uncertainty on order of nanometers [2] [3] [4] . Grejda proposed a modified reversal method to eliminate mounting error induced by misalignment of artifact and probe by relocating the spindle stator using a rotary table instead [5] . However, none of the above researches made a comprehensive investigation into error analysis considering all the factors such as alignment error, error separating methods, probe nonlinearity. Moreover, nonlinearity of capacitive probe when targeting spherical artifact is not taken into consideration in detail. Especially, nonlinearity when a probe moving laterally relative to a spherical surface has not been investigated ever before. Although R. Ryan Vallance studied nonlinearity when a probe moving axially relative to a spherical surface [6] .
2.1. Capacitive Probe Nonlinearity Targeting Ball Surface
In order to study nonlinearity of a capacitive probe moving laterally relative to a spherical surface, an experiment is conducted shown in Figure 1(a) where the probe moves laterally every time by some micrometers while a same probe is used to measure the lateral displacement. The data is shown in Figure 1(b) and a quadratic curve is used to fit the experimental data. It is obvious that the nonlinearity exists when the probe moves laterally above the spherical surface. This leads to two problems when measuring radial error of axis of rotation. One is whether the linear gain will change when the probe targeting spherical surface at different lateral offsets. The other is the lateral component of eccentric movement of artifact ball may lead to additional reading error of a capacitive probe which will be discussed later in section 2.2. To investigate the former problem one experiment is made in which readings of a probe approaching the artifact ball at different lateral offsets and the results are shown in Figure 1(c). It is concluded that the linear gain remains constant when lateral displacement varying between 0 and 40 μm. When measure error motion, the lateral displacement is always keep minimum by adjusting the probe in lateral direction to approach the highest point of the artifact ball and this will ensure the linear gain is constant.
2.2. Eccentricity Induced Lateral Misalignment
A misalignment between the artifact and axis of rotation leads to eccentric error in the probe signals. Two primary methods exist to eliminate this effect, such as the least quadratic circle and the Fourier analysis to remove the fundamental frequency. However, little attention is given to the fact that lateral component of eccentric movement vector of artifact ball may lead to additional reading error of capacitive probe. Set the eccentric error to be e. At angular position
\theta
, the lateral and the radial components of eccentric error are
e\ast \mathrm{cos}\theta
e\ast \mathrm{sin}\theta
respectively. Assuming the initial lateral offset of the probe e0 relative to the ball is shown in Figure 2.
According to 2.1, output of the probe
{e}_{\text{lateral_effect}}
caused by lateral offset
\chi
can be presented by
Figure 1. Nonlinearity of a capacitive sensor induced by lateral offset relative to a spherical artifact. (a) Experiment setup; (b) Capacitive probe output when moving laterally at different offset distances. y1 and y2 are quadratic and cubic fitting equations respectively; (c) Experiment data to identify the output characteristic of capacitive probe at different lateral offset distances.
Figure 2. Eccentric error induced lateral offset.
{e}_{\text{lateral_effect}}=a{\chi }^{2}=a{\left({e}_{0}+e\mathrm{cos}\theta \right)}^{2}
\alpha
-the identified coefficient and in this paper a = 0.00082 μm−2.
The total contribution to the probe output caused by eccentric error is expressed by
\begin{array}{l}{e}_{\text{eccentric_effect}}\\ ={e}_{\text{lateral_effect}}+e\mathrm{sin}\theta \\ =\alpha {\left({e}_{0}+e\mathrm{cos}\theta \right)}^{2}+e\mathrm{sin}\theta \\ =\alpha {e}_{0}^{2}+\frac{1}{2}\alpha {e}^{2}+\frac{1}{2}\alpha {e}^{2}\mathrm{cos}2\theta +2\alpha {e}_{0}e\mathrm{cos}\theta +e\mathrm{sin}\theta \end{array}
From this formula, second order and first order errors will be included in the probe output and when the eccentric error e = 5 μm the second order error will be up to 10 nm which will be an unacceptable error and be impossible to be eliminated by mathematical method. The last two 1st order components in this formula can be removed by Fourier analysis to remove the fundamental frequency.
2.3. Misalignment Error of Probe: Tilt Error
When considering radial error motion, one of the important error sources is attributed to misalignment between the capacitive probe and the artifact ball as is shown in Figure 3. The probe output is affected by four error motion components of which two are in the error sensitive direction and the other two in the error insensitive direction. The components have the following relationship:
{e}_{a}={e}_{an}+{e}_{at}
{e}_{r}={e}_{rn}+{e}_{rt}
{e}_{at}
{e}_{rt}
are error motion components in the error sensitive direcion,
{e}_{rn}
{e}_{an}
in the error insensitive direction, and
{e}_{a}
is the axial error motion and
{e}_{r}
the radial error motion. Accordingly, the output
{m}_{1}
of probe can be expressed as the combination effects of two parts, namely
{m}_{1}={S}_{x}+E
{S}_{x}
and E are radial error motion in X direction and the error induced by misalignment, respectively. We have
E={e}_{rn}+{e}_{at}+f\left({e}_{an}+{e}_{rt}\right)-{e}_{r}
{e}_{rn}={e}_{r}\mathrm{cos}\phi
{e}_{at}={e}_{a}\mathrm{sin}\phi
in to (6) yields
E={e}_{r}\left(\mathrm{cos}\phi -1\right)+{e}_{a}\mathrm{sin}\phi +f\left({e}_{an}+{e}_{rt}\right)
f\left(\cdot \right)
corresponds to the lateral offset effects which is detailed in section 2.1 and
\phi
is the tilt angle. Considering the lateral offset
{e}_{an}
{e}_{rt}
are much smaller relative to the initial distance
{e}_{0}
f\left({e}_{an}+{e}_{rt}\right)=a{\left({e}_{0}+{e}_{an}+{e}_{rt}\right)}^{2}-a{e}_{0}^{2}
f\left({e}_{an}+{e}_{rt}\right)\approx 2a{e}_{0}\left({e}_{an}+{e}_{rt}\right)
Figure 3. General misalignment between the probe and the artifact.
\phi
is small enough, we have
E\approx {e}_{a}\phi +2a{e}_{0}\left({e}_{an}+{e}_{rt}\right)
It can be concluded from (2) and (10) that the error motion in radial and axial direction and eccentric rotation of artifact ball will both induce lateral offset which causes a fake output of probes. When axial error motion is 0.4 μm and the initial lateral offset e0 is 20 μm, the maximum error due to lateral offset effects is up to 13 nm, which is a large measurement error in calibration of an ultra-precision spindle.
3. Positioning Error of Different Error Separations Methods
3.1. Donaldson Reversal Method
Let the angular positioning error of artifact after reversal be
\phi
which is shown in Figure 4(a), then the measurement error induced by angular error after reversing the artifact is derived as
E\left(\theta \right)=\frac{R\left(\theta \right)-R\left(\theta +\phi \right)}{2}\approx -\frac{\phi }{2}{R}^{\prime }\left(\theta \right)
R\left(\theta \right)
is roundness of the artifact.
The Donaldson reversal method needs to rotate the probe by 180 degrees relative to the rotor of the spindle measured at the same time. Angular position error of the probe will be introduced into the measurement signal. This kind of error is illustrated in Figure 4(b). The corresponding measurement error is derived as
{M}_{2}\left(\theta \right)=R\left(\theta -\phi \right)-{S}_{x}\left(\theta \right)\mathrm{cos}\phi +{S}_{y}\left(\theta \right)\mathrm{sin}\phi
E\left(\theta \right)=\frac{{M}_{1}\left(\theta \right)-{M}_{2}\left(\theta \right)}{2}-{S}_{x}\left(\theta \right)
E\left(\theta \right)=\frac{1}{2}\left[R\left(\theta \right)-R\left(\theta -\phi \right)+{S}_{x}\left(\mathrm{cos}\phi -1\right)-{S}_{y}\mathrm{sin}\phi \right]
Figure 4. Illustration of positioning error of Donaldson reversal method. (a) Positioning error of artifact ball; (b) Positioning error of probe.
E\left(\theta \right)\approx \frac{1}{2}\left[\phi \left({R}^{\prime }\left(\theta \right)-{S}_{y}\right)-\frac{1}{2}{S}_{x}\left(\theta \right){\phi }^{2}+ο\left(\phi \right)+ο\left({\phi }^{4}\right)\right]
{S}_{x}
{S}_{y}
are error motion components in X and Y directions respectively. If
\phi
is efficiently small and the measurement error will be simplified as
E\left(\theta \right)\approx \frac{1}{2}\left[\phi \left({R}^{\prime }\left(\theta \right)-{S}_{y}\right)+ο\left(\phi \right)\right]
3.2. Multi-Position Method
When using multi-position method to separate roundness of the artifact and rotating the artifact by a constant angle
\phi
, an angular error
{\Delta }_{i}
exists, as is shown in Figure 5(a). According to the principle introduced in [4] , the measurement error induced by angular error
{\Delta }_{k}
is derived as
\begin{array}{c}E\left(\theta \right)=\frac{1}{N}{\sum }_{k=0}^{N-1}R\left(\theta +k\phi +{\Delta }_{k}\right)\\ \approx \frac{1}{N}{\sum }_{k=0}^{N-1}\left[R\left(\theta +k\phi \right)+{R}^{\prime }\left(\theta +k\phi \right){\Delta }_{k}+ο\left({\Delta }_{k}\right)\right]\end{array}
As roundness of the artifact can be expressed as Fourier series and when N is an even integer, we have
{\sum }_{k=0}^{N-1}R\left(\theta +k\phi \right)=0
{\Delta }_{k}
E\left(\theta \right)\approx \frac{1}{N}{\sum }_{k=0}^{N-1}{R}^{\prime }\left(\theta +k\phi \right){\Delta }_{k}
3.3. Multi-Probe Method
Three-probe method is detailed in [5] , here gives only the formulas. Define
M\left(\theta \right)
as linear combination of
{m}_{1}\left(\theta \right)
{m}_{2}\left(\theta \right)
{m}_{3}\left(\theta \right)
with coefficients a, b and c respectively, namely
\left\{\begin{array}{l}M\left(\theta \right)={m}_{A}\left(\theta \right)+b{m}_{B}\left(\theta \right)+c{m}_{C}\left(\theta \right)\\ b=-\frac{\mathrm{sin}\beta }{\left[\mathrm{sin}\left(\beta -\alpha \right)\right]}\\ c=\frac{\mathrm{sin}\alpha }{\left[\mathrm{sin}\left(\beta -\alpha \right)\right]}\end{array}
{m}_{1}\left(\theta \right)
{m}_{2}\left(\theta \right)
{m}_{3}\left(\theta \right)
are outputs of sensors, and
Figure 5. Angular positioning error of multi-position (a) and three-probe methods (b).
b=-\text{sin}\beta /\left[\mathrm{sin}\left(\beta -\alpha \right)\right]
c=\text{sin}\alpha /\left[\mathrm{sin}\left(\beta -\alpha \right)\right]
. Applying discrete Fourier transformation (DFT) to formula (19) yields
M\left(k\right)=\left(1+b{\text{e}}^{-jk\alpha }+c{\text{e}}^{-jk\beta }\right)R\left(k\right)
when three-probe method is used, let angular position errors of probes at positions
\alpha
\beta
\delta \alpha
\delta \beta
respectively, as is shown in Figure 5(b). We have the measurement error of roundness of artifact in frequent domain
E\left(k\right)=\left(\frac{1}{{W}_{1}}-\frac{1}{W}\right)M\left(k\right)-\frac{1}{{W}_{1}}\left[{C}_{1}{S}_{x}\left(k\right)+{C}_{2}{S}_{y}\left(k\right)\right]
{W}_{1}\left(k\right)=1+b{e}^{-jk\left(\alpha +\delta \alpha \right)}+c{e}^{-jk\left(\beta +\delta \beta \right)}
W\left(k\right)=1+b{e}^{-jk\alpha }+c{e}^{-jk\beta }
{C}_{1}=1+\text{bcos}\left(\alpha +\delta \alpha \right)+\text{ccos}\left(\beta +\delta \beta \right)
{C}_{2}=\text{bsin}\left(\alpha +\delta \alpha \right)+\text{csin}\left(\beta +\delta \beta \right)
By inverse Fouries transformation we have the measurement error
e\left(\theta \right)=IDFT\left(E\left(k\right)\right)
Factors influencing measurement error of radial error motion are discussed in detail. Nonlinearity of a capacitive displacement probe targeting a spherical surface is investigated through experiment and the phenomena of fake displacement induced by lateral offset of the probe relative to an artifact ball are discussed. It is shown that the error motion in radial and axial direction and eccentric rotation of artifact ball will both induce lateral offset which causes a fake output of probes.
*Supported by National Science and Technology Major Project of High-level Numerical Controlling Machine Tools and Basic Manufacturing Equipment with the granted No. 2017ZX04011001.
[1] Tauhiduzzaman, M. (2015) Form Error in Diamond Turning. Precision Engineering, 42, 22-36.
[2] Evans, C.J., Hocken, R.J. and Estler, W.T. (1996) Self-Calibration: Reversal, Redundancy, Error Separation, and Absolute Testing. CIRP Annals-Manufacturing Technology, 45, 617-634.
[3] Zhang, G.X., Zhang, Y.H., Yang, S.M., et al. (1997) A Multipoint Method for Spindle Error Motion Measurement. CIRP Annals Manufacturing Technology, 46, 441-445.
[4] Marsh, E., Couey, J. and Vallance, R. (2006) Nanometer-Level Comparison of Three Spindle Error Motion Separation Techniques. Journal of Manufacturing Science and Engineering, 128, 180-187.
[5] Grejda, R., Marsh, E. and Vallance, R. (2005) Techniques for Calibrating Spindles with Nanometer Error Motion. Precision Engineering, 29, 113-123.
[6] Vallance, R.R., Marsh, E.R. and Smith, P.T. (2004) Effects of Spherical Targets on Capacitive Displacement Measurements. Journal of Manufacturing Science & Engineering, 126, 822-829.
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Lending - cypher docs
The lending protocol for the platform will be similar to that of Compound, except it is specifically for trading purposes on the platform. Lending pools will be peer-to-peer unlike a traditional brokerage firm, with smart contracts and collateralization ratios mitigating counterparty risk while allowing participants to maintain self-custody of assets. Each oAsset lending pool will be represented by its own smart contract that controls interest rates paid by borrowers. The utilization ratio will properly adjust interest rates based on supply and demand, which is defined as:
U_{ratio}=\frac{Loans}{Asset\:Value+Loans}
A utilization ratio (U_ratio) will be calculated independently for each lending pool, which can be either a stablecoin or an oAsset.
Interest rates are modeled by a set of two functions, one model for U_ratio≤U_optimal, where interest rates will gradually increase and another for U_ratio > U_optimal, where rates should rapidly increase.
R_v=\begin{Bmatrix} \frac{U}{U_{optimal}}.R_{multiple1}+R_{base} & ,U\leq U_{optimal}\\ \frac{U-U_{optimal}}{1-U_{optimal}}.R_{multiple2}+R_{base} & ,U\gt U_{optimal} \end{Bmatrix}
The variable rate is designed to incentivize borrowing and lending and keep the utilization near Uoptimal. In the future, this variable interest rate could be modeled by an exponential function with a positive linear overlay, but initially we will stick with what peer-to-peer lending giants have shown to be successful.
Variable interest rate model is from Aave’s whitepaper: https://github.com/aave/aave-protocol/blob/master/docs/Aave_Protocol_Whitepaper_v1_0.pdf
Fees will be captured from the lending pool in order to build a reserve pool that will be leveraged as an insurance pool for the protocol. This pool could be utilized for preventing cascading defaults and preventing LPs (minters) from having to cover the total cost of oAssets if there are not enough long/short positions available for payout.
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Bruce R. Southey, Pan Zhang, Marissa R. Keever, Haley E. Rymut, ... Sandra L. Rodriguez-Zas
Broad and dynamic neurochemical alterations in the brain of alcoholic rats
Li Luo, Xiao-Li Min, Xiang He, Fei-Fei Shang
The functional hierarchy of the task-positive networks indicates a core control system of top-down regulation in visual attention
Ping Zhao, Ren-Shu Yu, Yuan Liu, Zheng-Hao Liu, ... Xiao-Tong Wen
“Olfactory three-needle” acupuncture enhances synaptic function in Aβ1-42-induced Alzheimer’s disease via activating PI3K/AKT/GSK-3β signaling pathway
Yuan Wang, Ani Zheng, Huan Yang, Qiang Wang, ... Zhi-Bin Liu
Geniposide protection against Aβ1-42 toxicity correlates with mTOR inhibition and enhancement of autophagy
Dong-Xing Liu, Di Zhang, Wei-Min Hu, Yan-Fang Chang, ... Lin Li
Estrogen receptor 2 mediates intraspecific aggressive behaviors of the female Cricetulus barabensis in the estrous cycle
Jin-Hui Xu, Hui-Liang Xue, Zhe Wang, Chao Fan, ... Lai-Xiang Xu
Punicalagin effect on total sleep deprivation memory deficit in male Wistar rats
Mohammad Nasehi, Mohammad-Hossein 11 Mohammadi-Mahdiabadi-Hasani, Mohammad-Reza Zarrindast, Shahram Zarrabian
Thyrotroph embryonic factor polymorphism predicts faster progression of Parkinson’s disease in a longitudinal study
Ping Hua, Can Cui, Yong Chen, Yan Yao, ... Wei-Guo Liu
J. Integr. Neurosci. 2021, 20(1), 95–101; https://doi.org/10.31083/j.jin.2021.01.366
Vessel territory mapping after cerebral revascularization surgery using selective angiographic flat detector perfusion imaging
Thijs van der Zijden, Annelies Mondelaers, Maurits Voormolen, Laetitia Yperzeele, Tomas Menovsky
The neuroendocrine effects of dehydroepiandrosterone and 17
\beta
-estradiol in the in vitro preterm hyperoxia infant model
Stephanie Hübner, Donna E. Sunny, Christine Zädow, Johanna Ruhnau, ... Matthias Heckmann
The prediction of need of using ankle-foot orthoses in stroke patients based on findings of a transcranial magnetic stimulation study
Yoo Jin Choo, Jang-Hwan Kim, Min Cheol Chang
Mannotriose induced differentiation of mesenchymal stem cells into neuron-like cells
Xiao-Lan Lian, Li-Ming Ji, Li-Na Zhang
Prognostic value of serum pentraxin 3 for intracerebral hemorrhage mortality
Pu-Heng Hao, Chao Fu, Tao Ma, Shi-Ming He, Zhuo-Peng Jia
c.3G
\mathbf{>}
A mutation in the CRYAB gene that causes fatal infantile hypertonic myofibrillar myopathy in the Chinese population
Xin-Guo Lu, Uet Yu, Chun-Xi Han, Jia-Hui Mai, ... Yan-Qi Hou
A patient with acute myeloid leukemia presented with a superior sagittal sinus thrombosis as the first manifestation of Trousseau syndrome
Yi Bu, Na Wei, Yan Liu, Jing-Zhe Han
\delta
The role of mast cells in the gut and brain
Vitamin D, Epstein-Barr virus, and endogenous retroviruses in multiple sclerosis - facts and hypotheses
Christine Brütting, Gabriele I. Stangl, Martin S. Staege
The emerging role of non-coding RNAs from extracellular vesicles in Alzheimer’s disease
Yu-Zhong Xu, Ming-Gang Cheng, Xiong Wang, Yu Hu
The action of aripiprazole and brexpiprazole at the receptor level in singultus
Eman Alefishat, Lujain Aloum, Ovidiu C Baltatu, Georg A Petroianu
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TET3-mediated accumulation of DNA hydroxymethylation contributes to the activity-dependent gene expression of Rab3a in post-mitotic neurons
Kun Luo, Lesheng Wang, Yulong Shen, Shuhua She, ... Kui Liu
Selective executive impairments as neuroimmunological manifestations of the human immunodeficiency virus
Eleni Konstantinopoulou, Panagiotis Ioannidis, Grigorios Kiosseoglou, Eleni Aretouli
Fiona Harding, Mathieu Seynaeve, Johanna Keeler, Hubertus Himmerich, ... Carol Kan
The anti-aging effect of velvet antler polypeptide is dependent on modulation of the gut microbiota and regulation of the PPAR
\alpha
/APOE4 pathway
Xiaoran Liu, Qing Yang, Hui Li, Xingcheng Lan, ... Na Li
Suppressors of cytokine signaling 1 protein in a regenerative model of the Gekko japonicus spinal cord
Bingqiang He, Wenjuan Wang, Chunshuai Sun, Ting Yang, ... Yingjie Wang
The impact of CRMP4 SUMOylation on the Cav1.2 interaction, neurite outgrowth and thermal pain sensitivity
Shangdao Lai, Meiyuan Pan, Hongxing Liao, Jiayu Chen, ... Yalan Li
Radiation exposure during computerized tomography-based neuroimaging for acute ischemic stroke: a case-control study
Subhash C Kapur, Jeevesh Kapur, Vijay K Sharma
Prenatal sevoflurane exposure causes abnormal development of the entorhinal cortex in rat offspring
Ying Gao, Tianyun Zhao, Yanxin Chen, Zhixiang Sun, ... Xingrong Song
Distinguishing brain abscess from necrotic glioblastoma using MRI-based intranodular radiomic features and peritumoral edema/tumor volume ratio
Dongdong Xiao, Jiajing Wang, Xuan Wang, Peng Fu, ... Xiaobing Jiang
Changes in depressive-like behaviors induced by spinal cord injury based on the hypothalamic-pituitary-adrenal axis and hippocampal neurogenesis
Chang-Hong Liu, Bo-Lun Zhao, Wen-Tao Li, Xiao-Hua Zhou, ... Li-Bin An
Altered production of reproductive neuropeptides in rats subjected to chronic intermittent hypoxia
Antonela Romina Terrizzi, María Pilar Martinez, Javier Fernandez-Solari
Dementia risk among Mongolian population with type 2 diabetes: a matched case-control study
Munkh-Undral Munkhsukh, Darisuren Namjil, Puntsagdulam Byambajav, Enkh-Amgalan Angarag, ... Oyuntugs Byambasukh
{}_{2}
Tyler W. LeBaron, Jason Kharman, Michael L. McCullough
Relationships among language ability, the arcuate fasciculus and lesion volume in patients with putaminal hemorrhage: a diffusion tensor imaging study
Min Jye Cho, Sung Ho Jang
Comparison of the effectiveness of pulsed radiofrequency of the suprascapular nerve and intra-articular corticosteroid injection for hemiplegic shoulder pain management
Tae Hoon Kim, Min Cheol Chang
Human locomotion-control brain networks detected with independent component analysis
Pengxu Wei, Tong Zou, Zeping Lv, Yubo Fan
Brain multimodality monitoring in patients suffering from acute aneurysmal subarachnoid hemorrhage: clinical value and complications
Martin Kieninger, Katharina Meichelböck, Sylvia Bele, Elisabeth Bründl, ... Karl-Michael Schebesch
Alteration of Oxidative stress and apoptotic markers alterations in the rat prefrontal cortex influence behavioral response induced by cisplatin and N-acetylcysteine in the tail suspension test
Rade Vukovic, Dragica Selakovic, Jelena S. Katanic Stankovic, Igor Kumburovic, ... Gvozden Rosic
Clinical manifestation and imaging characteristics of transient global amnesia: patent foramen ovale as an underlying factor
Sang-Mi Noh, Hyun Goo Kang
Molecular pathophysiological mechanisms of ischemia/reperfusion injuries after recanalization therapy for acute ischemic stroke
Anamaria Jurcau, Ioana Adriana Ardelean
Neurofeedback therapy for the management of multiple sclerosis symptoms: current knowledge and future perspectives
Samar S. Ayache, Benjamin Bardel, Jean-Pascal Lefaucheur, Moussa A. Chalah
Functional role of peripheral vasoconstriction: not only thermoregulation but much more
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F4_(mathematics) Knowpia
F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.
The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.
There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.
The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8.
In older books and papers, F4 is sometimes denoted by E4.
The Dynkin diagram for F4 is:
Weyl/Coxeter groupEdit
Its Weyl/Coxeter group
{\displaystyle G=W\left(\mathrm {F} _{4}\right)}
is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree
{\displaystyle \mu (G)=24}
[1] which is realized by the action on the 24-cell.
{\displaystyle \left[{\begin{array}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{array}}\right]}
F4 latticeEdit
The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin.
Roots of F4Edit
The 24 vertices of 24-cell (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F4 in this Coxeter plane projection
The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations, representing the vertices of a disphenoidal 288-cell if the edge lengths of the 24-cells are equal:
24-cell vertices:
24 roots by (±1,±1,0,0), permuting coordinate positions
Dual 24-cell vertices:
8 roots by (±1, 0, 0, 0), permuting coordinate positions
16 roots by (±½, ±½, ±½, ±½).
Hasse diagram of F4 root poset with edge labels identifying added simple root position
One choice of simple roots for F4,
, is given by the rows of the following matrix:
{\displaystyle {\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}}
F4 polynomial invariantEdit
Just as O(n) is the group of automorphisms which keep the quadratic polynomials x2 + y2 + ... invariant, F4 is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).
{\displaystyle C_{1}=x+y+z}
{\displaystyle C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}}
{\displaystyle C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}}
Where x, y, z are real valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix:
{\displaystyle M={\begin{bmatrix}x&{\overline {Z}}&Y\\Z&y&{\overline {X}}\\{\overline {Y}}&X&z\end{bmatrix}}}
The set of polynomials defines a 24 dimensional compact surface.
1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912…
The 52-dimensional representation is the adjoint representation, and the 26-dimensional one is the trace-free part of the action of F4 on the exceptional Albert algebra of dimension 27.
There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The fundamental representations are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the Dynkin diagram in the order such that the double arrow points from the second to the third).
^ Saunders, Neil (2014). "Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups". arXiv:0812.0182 [math.GR].
Adams, J. Frank (1996). Lectures on exceptional Lie groups. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0-226-00526-3. MR 1428422.
John Baez, The Octonions, Section 4.2: F4, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at http://math.ucr.edu/home/baez/octonions/node15.html.
Chevalley C, Schafer RD (February 1950). "The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl. Acad. Sci. U.S.A. 36 (2): 137–41. Bibcode:1950PNAS...36..137C. doi:10.1073/pnas.36.2.137. PMC 1063148. PMID 16588959.
Jacobson, Nathan (1971-06-01). Exceptional Lie Algebras (1st ed.). CRC Press. ISBN 0-8247-1326-5.
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Introduction to Chemical Engineering Processes/Notation - Wikibooks, open books for an open world
1 A Note on Notation
2 Base Notation (in alphabetical order)
6 Units Section/Dimensional Analysis
Base Notation (in alphabetical order)Edit
{\displaystyle [i]_{n}}
: Molarity of species i in stream n
a, b, c, d: Stoichiometric coefficients.
C: Molar concentration (mol/L)
K: Equilibrium coefficient
MW: Molecular Weight (Molar Mass)
n: Number of data points (in statistics section)
N: Number of components
x: Mole fraction in the liquid phase OR Mass fraction [1]
X: (molar) extent of reaction
y: Mole fraction in the gas phase
z: Overall composition
↑ Unless specified explicitly, assume that a given percent composition is in terms of the overall flowrate. So if you're given a flowrate in terms of kg/s and a compositoin of 30%, assume that the 30% is a mass fraction. If a given equation requires one or the other, it will explicitly be stated near the equation which is necessary.
{\displaystyle \rho }
{\displaystyle \Sigma }
SubscriptsEdit
If a particular component (rather than an arbitrary one) is considered, a specific letter is assigned to it:
[A] is the molarity of A
{\displaystyle x_{A}}
is the mass fraction of A
Similarly, referring to a specific stream (rather than any old stream you want), each is given a different number.
{\displaystyle {\dot {n}}_{1}}
is the molar flowrate in stream 1.
{\displaystyle {\dot {n}}_{A1}}
is the molar flow rate of component A in stream 1.
Special subscripts:
If A is some value denoting a property of an arbitrary component stream, the letter i signifies the arbitrary component and the letter n signifies an arbitrary stream, i.e.
{\displaystyle A_{n}}
is a property of stream n. Note
{\displaystyle {\dot {n}}_{n}}
is the molar flow rate of stream n.
{\displaystyle A_{i}}
is a property of component i.
The subscript "gen" signifies generation of something inside the system.
The subscripts "in" and "out" signify flows into and out of the system.
If A is some value denoting a property then:
{\displaystyle {\bar {A}}_{n}}
denotes the average property in stream n
{\displaystyle {\dot {A}}_{n}}
denotes a total flow rate in steam n
{\displaystyle {\dot {A}}_{in}}
denotes the flow rate of component i in stream n.
{\displaystyle {\hat {A}}}
indicates a data point in a set.
{\displaystyle A_{i}^{*}}
is a property of pure component i in a mixture.
Units Section/Dimensional AnalysisEdit
In the units section, the generic variables L, t, m, s, and A are used to demonstrate dimensional analysis. In order to avoid confusing dimensions with units (for example the unit m, meters, is a unit of length, not mass), if this notation is to be used, use the unit equivalence character
{\displaystyle {\dot {=}}}
rather than a standard equal sign.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Notation&oldid=3325789"
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\mathrm{CB}
R
X
A
A
A
R
X
X
\mathrm{CB}
A
The compressed sparse column form of an
m
A
k
\mathrm{CB}
R
X
k
A
X
k
A
R
k
\mathrm{CB}
n+1
{\mathrm{CB}}_{i}
X
R
i
{\mathrm{CB}}_{n+1}={\mathrm{CB}}_{1}+k
i
{\mathrm{CB}}_{i}
{\mathrm{CB}}_{i+1}
A
X
A
R
k
{\mathrm{CB}}_{i}
X
R
i
A
X
\mathrm{sfloat},\mathrm{complex}\left(\mathrm{sfloat}\right),{\mathrm{integer}}_{1},{\mathrm{integer}}_{2},{\mathrm{integer}}_{4},{\mathrm{integer}}_{8},{\mathrm{float}}_{4},{\mathrm{float}}_{8},{\mathrm{complex}}_{8}
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
m≔\mathrm{Matrix}\left(5,6,{\left(1,2\right)=-81,\left(2,3\right)=-55,\left(2,4\right)=-15,\left(3,1\right)=-46,\left(3,3\right)=-17,\left(3,4\right)=99,\left(3,5\right)=-61,\left(4,2\right)=18,\left(4,5\right)=-78,\left(5,6\right)=22},\mathrm{datatype}=\mathrm{integer}[4]\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{cb},r,x≔\mathrm{CompressedSparseForm}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{11}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb},r,x\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{cb},r,x≔\mathrm{CompressedSparseForm}\left(m,'\mathrm{form}=\mathrm{row}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{11}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{6}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb},r,x,'\mathrm{form}=\mathrm{row}'\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb},r,x\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{cb},r,x≔\mathrm{CompressedSparseForm}\left(m,'\mathrm{cbbase}'=3,'\mathrm{rbase}'=-2\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{10}\\ \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{13}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-2}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-46}\\ \textcolor[rgb]{0,0,1}{-81}\\ \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{-55}\\ \textcolor[rgb]{0,0,1}{-17}\\ \textcolor[rgb]{0,0,1}{-15}\\ \textcolor[rgb]{0,0,1}{99}\\ \textcolor[rgb]{0,0,1}{-61}\\ \textcolor[rgb]{0,0,1}{-78}\\ \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb},r,x,'\mathrm{rbase}'=-2\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-81}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-55}& \textcolor[rgb]{0,0,1}{-15}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-17}& \textcolor[rgb]{0,0,1}{99}& \textcolor[rgb]{0,0,1}{-61}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{18}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-78}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{22}\end{array}]
\mathrm{m1}≔\mathrm{Matrix}\left([[0,1,0],[2,0,0],[3,0,4]],'\mathrm{datatype}=\mathrm{float}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{m1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{m2}≔\mathrm{Matrix}\left([[0,1,0],[2,0,0],[3,0,4],[0,0,0]],'\mathrm{datatype}=\mathrm{float}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{m2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\end{array}]
\mathrm{cb1},\mathrm{r1},\mathrm{x1}≔\mathrm{CompressedSparseForm}\left(\mathrm{m1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{r1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3.}\\ \textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{cb2},\mathrm{r2},\mathrm{x2}≔\mathrm{CompressedSparseForm}\left(\mathrm{m2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cb2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{r2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{x2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3.}\\ \textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb2},\mathrm{r2},\mathrm{x2}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\end{array}]
\mathrm{FromCompressedSparseForm}\left(\mathrm{cb2},\mathrm{r2},\mathrm{x2},'\mathrm{otherdimension}'=4\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{3.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{4.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\end{array}]
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COIN-OR Knowpia
Computational Infrastructure for Operations Research (COIN-OR), is a project that aims to "create for mathematical software what the open literature is for mathematical theory." The open literature (e.g., a research journal) provides the operations research (OR) community with a peer-review process and an archive. Papers in operations research journals on mathematical theory often contain supporting numerical results from computational studies. The software implementations, models, and data used to produce the numerical results are typically not published. The status quo impeded researchers needing to reproduce computational results, make fair comparisons, and extend the state of the art.
The success of Linux, Apache, and other projects popularized the open-source model of software development and distribution. A group at IBM Research proposed open source as an analogous yet viable means to publish software, models, and data. COIN-OR was conceived as an initiative to promote open source in the computational operations research community and to provide the on-line resources and hosting services required to enable others to run their own open-source software projects.
The COIN-OR website was launched as an experiment in 2000, in conjunction with 17th International Symposium on Math Programming in Atlanta, Georgia. In 2007, COIN-OR had 25 application projects,[1] including tools for linear programming (e.g., COIN-OR CLP), nonlinear programming (e.g., IPOPT), integer programming (e.g., CBC, Bcp and COIN-OR SYMPHONY), algebraic modeling languages (e.g., Coopr) and more. By 2011, this had grown to 48 projects.[2] COIN-OR is hosted by the Institute for Operations Research and the Management Sciences, INFORMS, and run by the educational, non-profit COIN-OR Foundation.
CLPEdit
COIN-OR LP (CLP or Clp) is an open-source linear programming solver written in C++. It is published under the Common Public License so it can be used in proprietary software with none of the restrictions of the GNU General Public License. CLP is primarily meant to be used as a callable library, although a stand-alone executable version can be built. It is designed to be as reliable as any commercial solver, although several times slower,[3] and to be able to tackle very large problems.
CLP is designed to solve linear programming problems such as :
{\displaystyle c_{1}x_{1}+c_{2}x_{2}\,}
subject to problem constraints of the following form
{\displaystyle a_{11}x_{1}+a_{12}x_{2}\leq b_{1}}
{\displaystyle a_{21}x_{1}+a_{22}x_{2}\leq b_{2}}
{\displaystyle a_{31}x_{1}+a_{32}x_{2}\leq b_{3}}
and non-negative variables
{\displaystyle x_{1}\geq 0}
{\displaystyle x_{2}\geq 0}
with up to millions of variables and/or constraints. Its main algorithm is the simplex algorithm.
CLP is used in other COIN-OR projects such as SYMPHONY, Branch Cut and Price (BCP), COIN-OR Branch and Cut (CBC), and others.
CBCEdit
COIN-OR branch and cut (CBC or Cbc) is an open-source mixed integer programming solver written in C++. It can be used as both a stand-alone executable and as a callable library (through A Mathematical Programming Language (AMPL) [natively], General Algebraic Modeling System (GAMS) [using the links provided by the COIN-OR Optimization Services (OS) and GAMSlinks projects], MPL [through the CoinMP project], AIMMS [through the AIMMSlinks project], PuLP, CMPL, OpenSolver for Excel, JuMP, or MiniZinc). Although it has been a popular choice of open source MIP solver for many years, its performance is now significantly inferior to HiGHS.[4]
SYMPHONYEdit
Single- or multi-process optimization over networks (SYMPHONY) is an open source branch and cut framework for solving mixed integer programs (MIPs) over heterogeneous networks.[5] It can use CLP, CPLEX, XPRESS or other linear programming solvers to solve the underlying linear programs.
SYMPHONY is a callable library which implements both sequential and parallel versions of branch, cut and price to solve MILPs. A branch, cut and price algorithm is similar to a branch and bound algorithm but additionally includes cutting-plane methods and pricing algorithms. The user of the library can customize the algorithm in any number of ways by supplying application-specific subroutines for reading in custom data files, generating application-specific cutting planes, or applying custom branching rules, resulting in a customized branch and cut algorithm. Most components of the algorithm, e.g., search tree management, management of linear programming solution, cut pool management, and communication management, are internal to the library and need not be touched by the user. The executables can be built in any number of configurations ranging from completely sequential to fully parallel with independently functioning cut generators, cut pools, and LP solvers. The distributed version currently runs in any environment supported by the PVM message passing protocol. The same source code can also be compiled for shared-memory architectures using any OpenMP compliant compiler.
SYMPHONY reads MPS (through the COIN-OR MPS reader) and GNU MathProg files. SYMPHONY does not have an LP-Solver of its own, but can be used with solvers like Clp, Cplex, Xpress through the Osi-interface. Cuts are generated using COIN's cut generation library: CGL. SYMPHONY also has structure specific implementations for problems like the traveling salesman problem, vehicle routing problem, set partitioning problem, mixed postman problem, etc. SYMPHONY also has an interactive shell where the user can enter commands to execute and control the program.
PuLPEdit
PuLP is an LP/IP modeler written in Python.[6] It can generate MPS or LP files and call GLPK, CLP/CBC, and CPLEX, to solve linear problems. PuLP is the default optimization tool in SolverStudio for Excel.
SMIEdit
SMI is a stochastic programming modeler and solver written in C++.[7] It can read Stochastic MPS and offers direct interfaces for constructing stochastic programs. It generates the deterministic equivalent linear program, solves it, and provides interfaces to access the scenario solutions.
COIN-OR solvers are available in the AIMMS, AMPL and GAMS modeling systems, and in the FortSP solver. They can also be used from within Excel via the OpenSolver and SolverStudio add-ins.
^ COIN-OR Annual Report, 2007
^ http://plato.asu.edu/ftp/lpsimp.html
^ http://plato.asu.edu/ftp/milp.html
^ SYMPHONY
^ PuLP
^ "SMI". Archived from the original on 2014-10-15. Retrieved 2014-01-03.
J.T. Linderoth and T.K. Ralphs: Noncommercial Software for Mixed-Integer Linear Programming. In: Integer Programming: Theory and Practice, John Karlof (ed.), CRC Press Operations Research Series, 2005, 253-303. (Working paper version)
T. Ralphs: An Introduction to the COIN-OR Optimization Suite: Open Source Tools for Building and Solving Optimization Models. Optimization Days, Montreal, May 7, 2013. (Presentation slides)
Official website COIN-OR, Computational Infrastructure for Operations Research
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Introduction to Chemical Engineering Processes/Significant figures - Wikibooks, open books for an open world
1 Importance of Significant Figures
1.2 Counting Significant Figures
1.3 The Parable of the Cement Block
1.4 Mathematical Operations and Significant Figures
Importance of Significant FiguresEdit
Significant figures (also called significant digits) are an important part of scientific and mathematical calculations, and deals with the accuracy and precision of numbers. It is important to estimate uncertainty in the final result, and this is where significant figures become very important.
Precision and AccuracyEdit
Before discussing how to deal with significant figures one should discuss what precision and accuracy in relation to chemical experiments and engineering are. Precision refers to the reproducibility of results and measurements in an experiment, while accuracy refers to how close the value is to the actual or true value. Results can be both precise and accurate, neither precise nor accurate, precise and not accurate, or vice versa. The validity of the results increases as they are more accurate and precise.
A useful analogy that helps distinguish the difference between accuracy and precision is the use of a target. The bullseye of the target represents the true value, while the holes made by each shot (each trial) represents the validity.
High precision, but unfortunately low accuracy
High accuracy, but not very good precision
As the above images show, the first has a lot of holes (black spots) covering a small area. The small area represents a precise experiment, yet it seems that there is a faultiness within the experiment, most likely due to systematic error, rather than random error. The second image represents an accurate though imprecise experiment. The holes are near the bullseye, even "touching" or within, though the problem is that they are spread out. This could be due to random error, systematic error, or not being careful in measuring.
Counting Significant FiguresEdit
There are three preliminary rules to counting significant. They deal with non-zero numbers, zeros, and exact numbers.
1) Non-zero numbers - all non-zero numbers are considered significant figures
2) Zeros - there are three different types of zeros
leading zeros - zeros that precede digits - do not count as significant figures (example: .0002 has one significant figure)
captive zeros - zeros that are "caught" between two digits - do count as significant figures (example: 101.205 has six significant figures)
trailing zeros - zeros that are at the end of a string of numbers and zeros - only count if there is a decimal place (example: 100 has one significant figure, while 1.00, as well as 100., has three)
3) Exact numbers - these are numbers not obtained by measurements, and are determined by counting. An example of this is if one counted the number of millimetres in a centimetre (10 - it is the definition of a millimetre), but another example would be if you have 3 apples.
How many significant figures do the following numbers have? Assume none of them are exact numbers.
a) 4.2362 - all numbers, so five
b) 2.0 - zeros after a decimal point count, so two
c) 9900 - only two in this case, because there is no decimal point
d) .44205 - there is a "captive zero," which means it counts, so five
e) .05 - only the five counts, so one
f) 3.9400E9 - tricky one, but scientific notation helps make the zeros at the end noticeable; there are five
The Parable of the Cement BlockEdit
People new to the field often question the importance of significant figures, but they have great practical importance, for they are a quick way to tell how precise a number is. Including too many can not only make your numbers harder to read, it can also have serious negative consequences.
As an anecdote, consider two engineers who work for a construction company. They need to order cement bricks for a certain project. They have to build a wall that is 10 feet wide, and plan to lay the base with 30 bricks. The first engineer does not consider the importance of significant figures and calculates that the bricks need to be 0.3333 feet wide and the second does and reports the number as 0.33, figuring that a precision of
{\displaystyle \pm 0.01ft}
(0.1 inches) would be precise enough for the work she was doing.
Now, when the cement company received the orders from the first engineer, they had a great deal of trouble. Their machines were precise but not so precise that they could consistently cut to within 0.0001 feet. However, after a good deal of trial and error and testing, and some waste from products that did not meet the specification, they finally machined all of the bricks that were needed. The other engineer's orders were much easier, and generated minimal waste.
When the engineers received the bills, they compared the bill for the services, and the first one was shocked at how expensive hers was. When they consulted with the company, the company explained the situation: they needed such a high precision for the first order that they required significant extra labor to meet the specification, as well as some extra material. Therefore it was much more costly to produce.
What is the point of this story? Significant figures matter. It is important to have a reasonable gauge of how precise a number is so that you know not only what the number is but how much you can trust it and how limited it is. The engineer will have to make decisions about how precisely he or she needs to specify design specifications, and how precise measurement instruments (and control systems!) have to be. If you do not need 99.9999% purity then you probably don't need an expensive assay to detect generic impurities at a 0.0001% level (though the lab technicians will probably have to still test for heavy metals and such), and likewise you will not have to design nearly as large of a distillation column to achieve the separations necessary for such a high purity.
Mathematical Operations and Significant FiguresEdit
Most likely at one point, the numbers obtained in one's measurements will be used within mathematical operations. What does one do if each number has a different amount of significant figures? If one adds 2.0 litres of liquid with 1.000252 litres, how much does one have afterwards? What would 2.45 times 223.5 get?
For addition and subtraction, the result has the same number of decimal places as the least precise measurement use in the calculation. This means that 112.420020 + 5.2105231 + 1.4 would have have a single decimal place but there can be any amount of numbers to the left of the decimal point (in this case the answer is 119.0).
For multiplication and division, the number that is the least precise measurement, or the number of digits. This means that 2.499 is more precise than 2.7, since the former has four digits while the latter has two. This means that 5.000 divided by 2.5 (both being measurements of some kind) would lead to an answer of 2.0.
So now you know how to pick which numbers to drop if there is a question about significant figures, but one also has to take into account rounding. Once one has decided which digit should be the last digit kept, one must decide whether to round up or down.
If the number is greater than five (6 to 9), one rounds up - 1.36 becomes 1.4
If the number is less than five (1 to 4), one rounds down - 1.34 becomes 1.3
What does one do when there is a five? There is a special case that deals with the number five, since, if you have not noticed, it is in the middle (between 1 and 9). Often in primary school one learns to just round up, but engineers tend to do something different, called unbiased rounding.
If the number before the five is even, then one rounds down - 1.45 becomes 1.4
If the number before the five is odd, then one rounds up - 1.55 becomes 1.6
Another case is this: 1.4501, where the numbers after five are greater than zero, so one would round to 1.5
Note: Remember that rounding is generally done at the end of calculations, not before the calculations are made.
Why is this done? Engineers make many calculations that often matter, since time, money, etc. are being taken into account, it is best to make sure that the final results are not synthetic or untrue to what the actual value should be. This relates back to accuracy and precision.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Significant_figures&oldid=3740219"
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The table below shows the frequency of children of age x years in a hospital:
Use the table to answer the question below:
The figure below is a Venn diagram showing the elements arranged within sets A,B,C,ε.
43. What is the loci of a distance 4cm from a given point P?
A. A straight line of length 4cm
B. a circle of radius 4cm
C. perpendicular to point P at 4cm
D. a circle of diameter 4cm
44. Given that Sin (5\(_x\) − 28)\(^o\) = Cos(3\(_x\) − 50)\(^o\), O\(_x\) < 90\(^o\)
D. 39\(^o\)
48. Calculate 243\(_{six}\) – 243\(_{five}\) expressing your answer in base 10
49. Evaluate ∫\(^2_1\) \(\frac{5}{x}\) dx
50. Tossing a coin and rolling a die are two separate events. What is the probability of obtaining a tail on the coin and an even number on the die?
\frac{1/}{4}
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\int _ { - \pi } ^ { \pi } \operatorname { cos } x d x
To make this problem quicker to solve, remember that
y =\operatorname{cos}x
=2\int_{0}^{\pi}\text{cos}xdx=2\left ( \text{sin}x\left|\begin{matrix} \pi \\ 0 \end{matrix}\right. \right )=2[\text{sin}\pi -\text{sin}0]=0
\int ( 5 \sqrt { y } - \operatorname { sin } y ) d y
Notice that this is an indefinite integral. Don't forget the
+C
\int _ { 1 } ^ { 2.7183 } \frac { 1 } { x } d x
2.7183 ≈ e
(Euler's number) and the derivative of ln
(x)
x^{−1}
\int ( \operatorname { sin } ^ { 2 } x + \operatorname { cos } ^ { 2 } x ) d x
Think! Trig identity. Simplify the integrand before you integrate.
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Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : ParametricSystemTools Subpackage : Specialize
specialize a list of regular chains at a point
Specialize(pt, lrc, R)
point with coordinates in rational number field or a finite field
The command Specialize(pt, lrc, R) returns a list of regular chains obtained from those of lrc by specialization at the point pt.
The point pt is given by a list of rational numbers or a list of elements in a finite field; moreover, the number of coordinates in pt must be less than or equal to the number of variables of R.
All polynomials in each regular chain of lrc are evaluated at the last
\mathrm{nops}\left(\mathrm{pt}\right)
variables of R using the corresponding coordinates of pt.
Regular chains in lrc must specialize well at pt, otherwise an error message displays.
This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form Specialize(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][Specialize](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
\mathrm{with}\left(\mathrm{ConstructibleSetTools}\right):
\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):
The following example shows how to analyze the output of a comprehensive triangular decomposition.
R≔\mathrm{PolynomialRing}\left([x,y,s]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
F≔[s-\left(y+1\right)x,s-\left(x+1\right)y]
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}]
\mathrm{pctd},\mathrm{cells}≔\mathrm{ComprehensiveTriangularize}\left(F,1,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{pctd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{cells}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}]]]
The first part is a list of regular chains which form a pre-comprehensive triangular decomposition of F. The second part is a partition of the projection image of V(F) to the last coordinate. Each constructible set is associated with indices of regular chains in the first part.
\mathrm{lcs}≔[\mathrm{seq}\left(\mathrm{cells}[i][1],i=1..\mathrm{nops}\left(\mathrm{cells}\right)\right)]
\textcolor[rgb]{0,0,1}{\mathrm{lcs}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{constructible_set}}]
Consider a specialization point
\mathrm{pt}\left(s=4\right)
\mathrm{pt}≔[4]
\textcolor[rgb]{0,0,1}{\mathrm{pt}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{4}]
Try to figure out to which partition pt belongs.
\mathrm{li}≔\mathrm{BelongsTo}\left(\mathrm{pt},\mathrm{lcs},R\right);
i≔\mathrm{li}[1]
\textcolor[rgb]{0,0,1}{\mathrm{li}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{2}]
\textcolor[rgb]{0,0,1}{i}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2}
Then retrieve the indices of regular chains that specialize well at pt.
\mathrm{ind}≔\mathrm{cells}[i][2]
\textcolor[rgb]{0,0,1}{\mathrm{ind}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}]
\mathrm{lrc_ind}≔\mathrm{map}\left(i↦\mathrm{pctd}[i],\mathrm{ind}\right)
\textcolor[rgb]{0,0,1}{\mathrm{lrc_ind}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]
\mathrm{map}\left(\mathrm{Info},\mathrm{lrc_ind},R\right)
[[\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{s}]]
Thus you know that the regular chains in lrc_ind all specialize well at the point pt. Then you can do simple substitutions.
\mathrm{lrc_sp}≔\mathrm{Specialize}\left(\mathrm{pt},\mathrm{lrc_ind},R\right)
\textcolor[rgb]{0,0,1}{\mathrm{lrc_sp}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]
Regular chains of
\mathrm{lrc_sp}
form a triangular decomposition of F after specialization at pt.
\mathrm{map}\left(\mathrm{Info},\mathrm{lrc_sp},R\right)
[[\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}]]
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Indexed family — Wikipedia Republished // WIKI 2
Not to be confused with Family of sets.
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects one real number for each integer (possibly the same).
More formally, an indexed family is a mathematical function together with its domain I and image X. Often the elements of the set X are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I is called the index set of the family, and X is the indexed set.
Proof and Problem Solving - Family of Sets and Indexed Family of Sets
Indexed Families of Sets
1 Mathematical statement
2.1 Indexed vectors
2.2 Matrices
3 Operations on indexed families
4 Indexed subfamily
5 Usage in category theory
Definition. Let I and X be sets and f a function such that
{\displaystyle {\begin{aligned}f\colon I&\to X\\f\colon i&\mapsto x_{i}=f(i),\end{aligned}}}
{\displaystyle i}s an element of I and the image
{\displaystyle f(i)}
{\displaystyle i}
under the function f is denoted by
{\displaystyle x_{i}}
{\displaystyle f(3)}
{\displaystyle x_{3}}
{\displaystyle x_{i}}
{\displaystyle x_{i}}
is the element of X indexed by
{\displaystyle i\in I}
. The function f thus establishes a family of elements in X indexed by I, which is denoted by
{\displaystyle (x_{i})_{i\in I}}
, or simply (xi) if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.
Any set X gives rise to a family (xx)x∈X, where X is indexed by itself (meaning that
{\displaystyle f}
is the identity function). However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects. A family contains any element exactly once if and only if the corresponding function is injective.
An indexed family
{\displaystyle (x_{i})_{i\in I}}
defines a set
{\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\}}
, that is, the image of I under f. Since the mapping f is not required to be injective, there may exist
{\displaystyle i,j\in I}
with i ≠ j such that xi = xj. Thus,
{\displaystyle |{\mathcal {X}}|\leq |I|}
, where |A| denotes the cardinality of the set A. For example, the sequence
{\displaystyle \left((-1)^{i}\right)_{i\in \mathbb {N} }}
indexed by the natural numbers
{\displaystyle \mathbb {N} =\{1,2,3,\dots \}}
has image set
{\displaystyle \{(-1)^{i}:i\in \mathbb {N} \}=\{-1,1\}}
. In addition, the set
{\displaystyle \{x_{i}:i\in I\}}
does not carry information about any structures on I. Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set.
Indexed vectors
The vectors v1, ..., vn are linearly independent.
Here (vi)i ∈ {1, ..., n} denotes a family of vectors. The i-th vector vi only makes sense with respect to this family, as sets are unordered so there is no i-th vector of a set. Furthermore, linear independence is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider n = 2 and v1 = v2 = (1, 0) as the same vector, then the set of them consists of only one element (as a set is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent).
Suppose a text states the following:
A square matrix A is invertible, if and only if the rows of A are linearly independent.
{\displaystyle A={\begin{bmatrix}1&1\\1&1\end{bmatrix}}.}
Let n be the finite set {1, 2, ..., n}, where n is a positive integer.
An ordered pair (2-tuple) is a family indexed by the set of two elements, 2 = {1, 2}; each element of the ordered pair is indexed by each element of the set 2.
An n-tuple is a family indexed by the set n.
An infinite sequence is a family indexed by the natural numbers.
A list is an n-tuple for an unspecified n, or an infinite sequence.
An n×m matrix is a family indexed by the Cartesian product n×m which elements are ordered pairs, e.g., (2, 5) indexing the matrix element at the 2nd row and the 5th column.
A net is a family indexed by a directed set.
Operations on indexed families
{\displaystyle \sum _{i\in I}a_{i}.}
When (Ai)i∈I is a family of sets, the union of all those sets is denoted by
{\displaystyle \bigcup _{i\in I}A_{i}.}
Likewise for intersections and Cartesian products.
Indexed subfamily
An indexed family (Bi)i∈J is a subfamily of an indexed family (Ai)i∈I, if and only if J is a subset of I and Bi = Ai holds for all i in J.
Usage in category theory
Main article: Diagram (category theory)
The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.
Array data type
Diagram (category theory)
Family of sets
Net (mathematics)
Tagged union
Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).
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YYiki: Book: Strong Towns: A Bottom-Up Revolution to Rebuild American Prosperity
https://www.amazon.com/Strong-Towns-Bottom-Up-Revolution-Prosperity-ebook/dp/B07YGC4K4V/
Edges. Wall hugging. Thigmotaxis. Jane Jacobs’ The Death and Life of Great American Cities. Stay to the sides. Also Christopher Alexander’s “pattern language”
The emergent design of Pompeii. Human-oriented.
The abundance of resource destroys the need for adaptation. The post war abundance led to lots of issues. Traffic congestion? Build more lanes.
building a neighborhood all at once, instead of incrementally, merely ensures that all the inevitable pressures of decline will occur simultaneously across the entire neighborhood. … as the signs of decline start to become apparent, the more affluent in the neighborhood will move. … all possible future converge on the sole remaining outcome – stagnation or decline – and there is no easy path back once the cycle of decline is established.
Improvement to land value ration is roughly maintained through a cycle of land value rising -> redevelopment
A key characteristic of traditional cities, especially those that reached a level of maturity, was stability. These were human habitats designed to endure, a refinement achieved through thousands of years of trial and error experimentation. … the more mature the city, however, the more entrenched the wealth.
Traditionally, the private sector leads the growth and public sector catches up by building the infrastructure.
“Good party” where each guest bring more food than they eat vs. “bad party” where it’s reverse. The upfront public investment may lead to the “bad party”.
The cities are playing an “infinite game”. It should make profits; otherwise it will not do its job properly.
Most infrastructure project simply cannot pay for itself.
for me, the evidence was pointing to a conclusion I found difficult to believe, yet impossible to ignore: the more our cities build, the poorer they become.
Infrastructure project has a long cycle. It doesn’t cost much until it reaches the end of the maintenance cycle. Growth can temporarily cover the big maintenance cost spike, but it’s like a Ponzi scheme and cannot be sustained. The problem is that it creates the illusion of wealth short-term and growth, which look good short-term, in which many elected officials may operate.
Detroit is not some strange anomaly. It’s just early. It’s just a couple of decades ahead of everyplace else.
The infrastructure cult
The last thing that our citites need is more infrastructure.
Only, that’s not how infrastructure works. The generally accepted accounting practices for municipalities counts infrastructure as an asset, not a liability. There is no accounting of the tax base or the revenue from the community’s wealth; it’s simply ignored.
humans value their time differently than Infrastructure Cult models suggest they should. The best example is the I-65 bridge in Louisville connecting Kentucky to Indiana. The states spent
1.3 billion on a congestion-reduction project including the construction of the 1-65 bridge, which is now tolled up to
4 per crossing. Not only is post-construction traffic not meeting projections, it’s been cut nearly in half from pre-construction levels.” In addition, drivers willing to avoid the toll are now detouring through a longer, slower route to use a nearby, untolled bridge. Traffic is up 75% on the free-of-charge-but-slower bridge, despite the pre-construction claims that saved time is the same as saved money.
Growth or stability
This locked in a pattern of growth, stagnation, and decline that would become one of the defining features of the current American development pattern.
growth once served us, but now we serve growth. The constraint of modern America is that we must experience annual growth in our economy. Without growth, we can’t service our debts, pay our retirements and pensions, and keep up with the rising costs of healthcare and education.
Keynes identified the Paradox of Thrift, the damage done to the national economy when individuals and organizations save instead of spend during an economic downturn, but what about the opposite? What about a Paradox of Avarice, where individuals and organizations don’t save but spend all they have? And more. What are the impacts of such a condition?
Forget strategic investments in growth; Ferguson today is so indebted that it can’t maintain its basic infrastructure systems. In the year Brown was killed, the city spent over
800,000 making interest payments on their debts while allocating only
25,000 to the maintenance of sidewalks. There are good reasons for Ferguson residents to walk in the streets; their sidewalks are falling apart.
If we’re going to have broad American prosperity, if we are to experience the comfort and stability of being truly strong and successful, Americans must again embrace a chaotic but smart approach to evolving our cities. To harmonize competing interests in a successful human habitat, our response to these stresses needs to emerge from within, not be imposed from the outside.
… density, at best, is a byproduct of success, but never the cause. … Instead of density, the math we must focus on is the relationship between private investment and public investment. … When we build or take over a community obligation, do we have enough private wealth to financially sustain that commitment?
Old & Blighted vs. New & Shiny. The old & blighted block hosts 11 small businesses with much more tax revenue. It also means 11 owners who are vested in the city instead of a franchise owner who doesn’t really care about the local community.
Let me summarize: In exchange for 26 years of tax relief, the community was able to get an out-of-town franchise restaurant to abandon their old building and move three blocks up the street where they tore town a block of buildings and replaced them with a development that is 44% less valuable than the development pattern of what was removed. By any financial measure, this is a bad investment, yet cities everywhere routinely do this exact kind of transaction.
Urban3 and John Minicozzi.
The team at Urban3 has modeled hundreds of cities around North America. This massive dataset has revealed a near-universal set of trends, …
Older neighborhoods … outperform …
Blight is not an indicator of financial productivity. …
The more reliant on the automobile a development pattern is, the less financially productive it tends to be.
Americans express a preference for single-family homes on large lots along cul-de-sacs because that’s the lifestyle we subsidize. We’ve been willing to bankrupt our cities, and draw down the wealth prior generations built, in order to provide that subsidy. It can’t go on indefinitely.
Making strong investments
The low-risk side of our public investment portfolio is as obvious as it is boring: Local governments must prioritize basic, routine maintenance in neighborhoods with high financial productivity (high value per acre). … Instead of prioritizing maintenance based on condition or age, cities must prioritize based on financial productivity.
Tatical urbanism by Mike Lydon and Better Block Foundation by Jason Roberts.
Missing middle housing and zoning regulations.
Place-oriented government
How a city’s priorities must shift to build wealth. e.g., City Engineer department should rethink “moving automobiles quickly” to “providing an abundance of mobility options within the city”.
The principle of subsidiarity states that the decision making power should be given to the smallest possible unit. For instance, who should regulate whether I can have chickens in my backyard? A reasonable answer is that whatever the smallest possible (affected) unit of body. More freedome should be given to the local municipalities.
« Book/Strong Towns »
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Crossover frequencies for sector bound - MATLAB getSectorCrossover - MathWorks España
Find Sector Crossover Frequency
Crossover frequencies for sector bound
wc = getSectorCrossover(H,Q) returns the frequencies at which the following matrix M(ω) is singular:
M\left(\omega \right)=H{\left(j\omega \right)}^{H}Q\text{\hspace{0.17em}}H\left(j\omega \right).
When a frequency-domain sector plot exists, these frequencies are the frequencies at which the relative sector index (R-index) for H and Q equals 1. See About Sector Bounds and Sector Indices for details.
Find the crossover frequencies for the dynamic system
G\left(s\right)=\left(s+2\right)/\left(s+1\right)
and the sector defined by:
S=\left\{\left(y,u\right):a{u}^{2}<uy<b{u}^{2}\right\},
for various values of a and b.
In U/Y space, this sector is the shaded region of the following diagram (for a, b > 0).
Q=\left[\begin{array}{cc}1& -\left(a+b\right)/2\\ -\left(a+b\right)/2& ab\end{array}\right];\phantom{\rule{1em}{0ex}}a=0.1,\phantom{\rule{0.2777777777777778em}{0ex}}b=10.
getSectorCrossover finds the frequencies at which
H\left(s{\right)}^{H}QH\left(s\right)
is singular, for
H\left(s\right)=\left[G\left(s\right);I\right]
. For instance, find these frequencies for the sector defined by Q with a = 0.1 and b = 10.
The empty result means that there are no such frequencies.
Now find the frequencies at which
{H}^{H}QH
is singular for a narrower sector, with a = 0.5 and b = 1.5.
Here the resulting frequency is where the R-index for H and Q2 is equal to 1, as shown in the sector plot.
Thus, when a sector plot exists for a system H and sector Q, getSectorCrossover finds the frequencies at which the R-index is 1.
Model to analyze against sector bounds, specified as a dynamic system model such as a tf, ss, or genss model. H can be continuous or discrete. If H is a generalized model with tunable or uncertain blocks, getSectorCrossover analyzes the current, nominal value of H.
To get the frequencies at which the I/O trajectories (u,y) of a linear system G lie in a particular sector, use H = [G;I], where I = eyes(nu), and nu is the number of inputs of G.
wc — Sector crossover frequencies
Sector crossover frequencies, returned as a vector. The frequencies are expressed in rad/TimeUnit, relative to the TimeUnit property of H. If the trajectories of H never cross the boundary, wc = [].
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Margin - cypher docs
Once a wallet is connected, the protocol will generate a margin account for traders to deposit initial funds (USDC). Traders are allowed to leverage their account value up to 2x, where account value is defined as portfolio net of borrowed value. Leverage multiple is defined as:
L_{multiple}=\frac{Borrows}{Account\: Value}=\frac{\sum_{b=b_1}^{b_i}loan(b)}{\sum_{a=a_1}^{a_i}positions(a)-\sum_{b=b_1}^{b_i}loan(b)}
position(a): position value for a given asset a a_i: asset in portfolio loan(b): loan value for a given asset b b_i: borrowed asset
The protocol stipulates that a trader must post an initial collateralization of 150% and allows for a maintenance collateralization of 125%. Having a maintenance collateralization threshold allows for some price movement in assets before a trader runs the risk of being liquidated. If a trader is liquidated the system or a liquidator will take all funds and positions in a margin account.
A trader’s margin collateral ratio is slightly different from their leverage multiple and is defined as:
C_{margin}=\frac{Positions}{Borrows}=\frac{\sum_{a=a_1}^{a_i}position(a)}{\sum_{b=b_1}^{b_i}loan(b)}
where variables are defined above.
Margin account structure was gleaned from mango markets: https://docs.mango.markets/tutorials/trade-on-mango.markets#how-to-trade-with-leverage
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A Detailed and Reduced Reaction Mechanism of Biomass-Based Syngas Fuels | J. Eng. Gas Turbines Power | ASME Digital Collection
e-mail: marina.braun-unkhoff@dlr.de
Nadezhda Slavinskaya,
Nadezhda Slavinskaya
Braun-Unkhoff, M., Slavinskaya, N., and Aigner, M. (June 10, 2010). "A Detailed and Reduced Reaction Mechanism of Biomass-Based Syngas Fuels." ASME. J. Eng. Gas Turbines Power. September 2010; 132(9): 091401. https://doi.org/10.1115/1.4000589
In the present work, the elaboration of a reduced kinetic reaction mechanism is described, which predicts reliably fundamental characteristic combustion properties of two biogenic gas mixtures consisting mainly of hydrogen, methane, and carbon monoxide, with small amounts of higher hydrocarbons (ethane and propane) in different proportions. From the in-house detailed chemical kinetic reaction mechanism with about 55 species and 460 reactions, a reduced kinetic reaction mechanism was constructed consisting of 27 species and 130 reactions. Their predictive capability concerning laminar flame speed (measured at
T0=323 K
, 373 K, and 453 K, at
p=1 bar
, 3 bars, and 6 bars for equivalence ratios
φ
between 0.6 and 2.2) and auto ignition data (measured in a shock tube between 1035 K and 1365 K at pressures around 16 bars for
φ=0.5
and 1.0) are discussed in detail. Good agreement was found between experimental and calculated values within the investigated parameter range.
biofuel, combustion, flames, gas mixtures, gas turbines, laminar flow, reaction kinetics, mechanism reduction, heat release, syngas, flame speed
Flames, Ignition delay, Fuels, Temperature, Combustion, Ignition, Biomass, Syngas
Global Energy Market—Past, Present and Future
Evaluation of Plants and Processes for the Use of Biomass Gasification in Combined Heat and Power Plants
Final Report on Project Study PGBV
, on behalf of the Federal Ministry of Consumer Protection, Food and Agriculture (BMVEL); (FNR, FKZ: 22020700), Germany.
Analysis and Evaluation of Biomass Gasification Plants
Proceedings of the Second World Conference on Biomass for Energy, Industry and Climate Protection
Numerical Simulation of a Lab Scale Syngas Burner Using Complex Chemistry
Proceedings of the GT 2006 Conference on Modelling Fluid Flow (CMFF) ‘06
Validation of Advanced Computational Methods for Determining Flame Transfer Functions in Gas Turbine Combustion Systems
K. -U.
Impact of Radiation on the Wall Heat Load at a Test Bench Gas Turbine Combustion Chamber: Measurements and CFD Simulation
Reduced Chemical Model for High Pressure Methane Combustion With PAH Formation
Skeletal Mechanism for C2H4 Combustion With PAH Formation
Gas Production From Gasification of Biomass for Power Generation
,” Project No. A205/04, funded by Stiftung Energieforschung, Baden-Württemberg, Germany.
Investigations of the Combustion Properties of the Product Gas Gained From the Gasification of Wet Biomass in Supercritical Water
Skeletal Mechanism Production for n-decane
Proceedings of the Second International Workshop on Model Reduction in Reacting Flow
, University of Rome “La Sapienza,” Rome, Italy, Sept. 3–5.
A Modeling Study of Aromatic Soot Precursors in Laminar Methane and Ethene Flames
Kinetic Modeling for High Pressure, Fuel-Rich Methane/Oxygen Combustion
Ber. Bunsenges. Phys. Chem
Experimental and Numerical Determination of Laminar Flame Speeds of Methane/(Ar, N2, CO2)-Air Mixtures as Function of Stoichiometry, Pressure, and Flame Temperature
Laminar Flame Speeds and Extinction Strain Rates of Mixture of Carbon Monoxide With Hydrogen, Methane and Air
The Use of Carbon Monoxide/Hydrogen Burning Velocities to Examine the Rate of the CO+OH Reaction
Shock-Tube Study of the Ignition of Methane/Ethane/Hydrogen Mixtures With Hydrogen Contents From 0% to 100% at Different Pressures
Shock Tube Study of the Ignition of Lean CO/H2 Fuel Blends at Intermediate Temperatures and High Pressure
Gasification of Wet Biomass in Supercritical Water—Results of Pilot Plant Experiments
Proceedings of the 14th European Biomass Conference and Exhibition Biomass for Energy, Industry and Climate Protection
, Paris, France, Oct. 17–21, pp.
Determination of Burning Velocities: A Critical Review
,” http://www.me.berkeley.edu/gri_mechhttp://www.me.berkeley.edu/gri_mech
CHEMKIN-II: A FORTRAN Chemical Kinetics Package for the Analysis of Gas Phase Chemical Kinetics
,” Sandia National Laboratories Report No. SAND89-8009B.
,” Sandia National Laboratories Report No. SAND87-8215.
Uncertainty Analysis of Updated Hydrogen and Carbon Monoxide Oxidation Mechanism
Evaluated Kinetic Data for Combustion Modelling: Supplement II
Evaluated Kinetic Data for Combustion Modelling Supplement I
Chemical Kinetic Database for Combustion Chemistry. Part I. Methane and Related Compounds
Ignition and Oxidation of Lean CO/H2 Fuel Blends in Air
A Compilation of Experimental Data on Laminar Burning Velocities
ANSYS CFX-5
,” http://www.ansys.com/http://www.ansys.com/
,” www.fluent.comwww.fluent.com
KINALC: A CHEMKIN Base Program for Kinetic Analyses
,” available at http://www.chem.leeds.ac.uk/Combustion/kinalc.htmhttp://www.chem.leeds.ac.uk/Combustion/kinalc.htm
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Decode encoded data - MATLAB - MathWorks Italia
Decode Encoded Data For New Images
Decode encoded data
Y = decode(autoenc,Z)
Y = decode(autoenc,Z)returns the decoded data Y, using the autoencoder autoenc.
Trained autoencoder, returned by the trainAutoencoder function as an object of the Autoencoder class.
Decoded data, returned as a matrix or a cell array of image data.
If the autoencoder autoenc was trained on a cell array of image data, then Y is also a cell array of images.
If the autoencoder autoenc was trained on a matrix, then Y is also a matrix, where each column of Y corresponds to one sample or observation.
Train an autoencoder using the training data with a hidden size of 15.
autoenc = trainAutoencoder(X,hiddenSize);
Extract the encoded data for new images using the autoencoder.
features = encode(autoenc,Xnew);
Decode the encoded data from the autoencoder.
Y = decode(autoenc,features);
Y is a 1-by-5000 cell array, where each cell contains a 28-by-28 matrix representing a synthetic image of a handwritten digit.
x\in {ℝ}^{{D}_{x}}
z\in {ℝ}^{{D}^{\left(1\right)}}
z={h}^{{}^{\left(1\right)}}\left({W}^{\left(1\right)}x+{b}^{\left(1\right)}\right),
{h}^{\left(1\right)}:{ℝ}^{{D}^{\left(1\right)}}\to {ℝ}^{{D}^{\left(1\right)}}
{W}^{\left(1\right)}\in {ℝ}^{{D}^{\left(1\right)}×{D}_{{}^{x}}}
{b}^{\left(1\right)}\in {ℝ}^{{D}^{\left(1\right)}}
\stackrel{^}{x}={h}^{{}^{\left(2\right)}}\left({W}^{\left(2\right)}z+{b}^{\left(2\right)}\right),
{h}^{\left(2\right)}:{ℝ}^{{D}_{x}}\to {ℝ}^{{D}_{x}}
{W}^{\left(1\right)}\in {ℝ}^{{D}_{{}^{x}}×{D}^{\left(1\right)}}
{b}^{\left(2\right)}\in {ℝ}^{{D}_{x}}
encode | trainAutoencoder
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Onshore to Offshore Ground‐Surface and Seabed Rupture of the Jordan–Kekerengu–Needles Fault Network during the 2016 Mw 7.8 Kaikōura Earthquake, New ZealandOnshore to Offshore Ground‐Surface and Seabed Rupture of the Jordan–Kekerengu–Needles Fault Network | Bulletin of the Seismological Society of America | GeoScienceWorld
Onshore to Offshore Ground‐Surface and Seabed Rupture of the Jordan–Kekerengu–Needles Fault Network during the 2016 Mw 7.8 Kaikōura Earthquake, New Zealand
School of Geography, Environment and Earth Sciences, Victoria University of Wellington, P.O. Box 600, Wellington 6040, New Zealand, Jesse@kearse.co.nz
GNS Science, 1 Fairway Drive, Avalon 5010, PO Box 30‐368, Lower Hutt 5040, New Zealand
National Institute of Water and Atmospheric Research (NIWA), 301 Evans Bay Parade, Greta Point, Wellington, New Zealand
Robert Langridge;
Joshu Mountjoy;
Will Ries;
Matthew Hill;
Humbolt State University, Geology Faculty, 1 Harpst Street, Arcata, California 95521
Jesse Kearse, Timothy A. Little, Russ J. Van Dissen, Philip M. Barnes, Robert Langridge, Joshu Mountjoy, Will Ries, Pilar Villamor, Kate J. Clark, Adrian Benson, Geoffroy Lamarche, Matthew Hill, Mark Hemphill‐Haley; Onshore to Offshore Ground‐Surface and Seabed Rupture of the Jordan–Kekerengu–Needles Fault Network during the 2016
Mw
7.8 Kaikōura Earthquake, New Zealand. Bulletin of the Seismological Society of America 2018;; 108 (3B): 1573–1595. doi: https://doi.org/10.1785/0120170304
During the 2016
Mw
7.8 Kaikōura earthquake, the Kekerengu fault ruptured the ground surface and produced a maximum of
∼12 m
of net displacement (dextral slip with minor reverse slip), one of the largest five coseismic surface‐rupture displacements so far observed globally. This study presents the first combined onshore to offshore dataset of coseismic ground‐surface and vertical seabed displacements along a near‐continuous
∼83
‐km‐long strike‐slip dominated earthquake surface rupture of large slip magnitude. Onshore on the Kekerengu, Jordan thrust, Upper Kowhai, and Manakau faults, we measured the displacement of 117 cultural and natural markers in the field and using airborne light detection and ranging (lidar) data. Offshore on the dextral‐reverse Needles fault, multibeam bathymetric and high‐resolution seismic reflection data image a throw of the seabed of up to
3.5±0.2 m
. Mean net slip on the total
∼83 km
rupture was
5.5±1 m
, this is an unusually large mean slip for the rupture length compared to global strike‐slip surface ruptures. Surveyed linear features that extend across the entire surface‐rupture zone show that it varies in width from 13 to 122 m. These cultural features also reveal the across‐strike distribution of lateral displacement, 80% of which is, on average, concentrated within the central 43% of the rupture zone. Combining the near‐field measurements of fault offset with published, far‐field Interferometric Synthetic Aperture Radar (InSAR), continuous Global Positioning System (GPS), and coastal deformation data suggests partitioning of oblique plate convergence, with a significant portion of coseismic contractional deformation (and uplift) being accommodated off‐fault in the hanging‐wall crust to the northwest of the main rupturing faults.
Needles Fault
Multiple‐Fault, Slow Rupture of the 2016
Mw
7.8 Kaikōura, New Zealand, Earthquake: Complementary Insights from Teleseismic and Geodetic Data
Surface Rupture of the Hundalee Fault during the 2016
Mw
7.8 Kaikōura Earthquake
Preliminary Geometry, Displacement, and Kinematics of Fault Ruptures in the Epicentral Region of the 2016
Mw
Ground Surface Deformation Caused by the
Mw
5.8 Early Strong Aftershock following the 13 November 2016
Mw
7.8 Kaikōura Mainshock
Seismo‐Ionospheric Observations, Modeling, and Backprojection of the 2016 Kaikōura Earthquake
Hypothetical Real‐Time GNSS Modeling of the 2016
Mw
7.8 Kaikōura Earthquake: Perspectives from Ground Motion and Tsunami Inundation Prediction
Hosgri fault zone, offshore Santa Maria Basin, California
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Triple Oxygen Isotope Systematics in the Hydrologic Cycle | Reviews in Mineralogy and Geochemistry | GeoScienceWorld
Triple Oxygen Isotope Systematics in the Hydrologic Cycle
Jakub Surma;
Institute of Geology and Mineralogy, University of Cologne, Zülpicher Straße 49b, Cologne, 50674, Germany
Presently at: Geoscience Center, Georg August University, Goldschmidtstraße 1 Göttingen, 37077, Germany
* E-mail: jakub.surma@uni-goettingen.de
Sergey Assonov;
Formerly at: Institute of Geology and Mineralogy, University of Cologne, Zülpicher Straße 49b Cologne, 50674, Germany
Michael Staubwasser
Institute of Geology and Mineralogy, University of Cologne, Zülpicher Straße 49b Cologne, 50674, Germany
Jakub Surma, Sergey Assonov, Michael Staubwasser; Triple Oxygen Isotope Systematics in the Hydrologic Cycle. Reviews in Mineralogy and Geochemistry 2021;; 86 (1): 401–428. doi: https://doi.org/10.2138/rmg.2021.86.12
The analysis of hydrogen (δD) and oxygen (δ18O) isotope ratios of H2O are widely used tools for studies of the hydrological cycle (Friedman 1953; Dansgaard 1954; Gonfiantini 1986; Gat 1996; Araguás-Araguás et al. 2000; Gat et al. 2000) and climate reconstruction (Dansgaard 1964; Johnsen et al. 1989; Petit et al. 1999). Natural variations of δD and δ18O in precipitation are well correlated and fall on a common global trend, the Global Meteoric Water Line (GMWL, Craig 1961):
δD=8·...
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Introduction to Chemical Engineering Processes/Basic Statistics and Data Analysis - Wikibooks, open books for an open world
2.1 Example of linear regression
2.2 How to tell how good your regression is
3.2 Power Law
4.2 Limitations of Linear Interpolation
Mean and Standard DeviationEdit
A lot of the time, when you're conducting an experiment, you will run it more than once, especially if it is inexpensive. Scientists run experiments more than once so that the random errors that result from taking measurements, such as having to guess a length between two hash marks on a ruler, cancel themselves out and leave them with a more precise measurement. However, the question remains: how should you consolidate all of the data into something that's more managable to use?
Suppose you have n data points taken under the same conditions and you wish to consolidate them to as few as feasibly possible. One thing which could help is is to use some centralized value, which is in some way "between" all of the original data points. This, in fact, is called the mean of the data set.
There are many ways of computing the mean of a data set depending on how it is believed to be distributed. One of the most common methods is to use the arithmetic mean, which is defined as:
{\displaystyle {\bar {x}}={\frac {\Sigma {\hat {x}}_{k}}{n}}}
Other types of mean include the w:Geometric mean, which should be used when the data are very widely distributed (ex. an exponential distribution) and the "log-mean" which occurs often in transport phenomena.
Having a value for the mean tells you what value the data points "cluster" around but it does not tell you how spread out they are from the center. A second statistical variable called the standard deviation is used for that. The standard deviation is essentially the average distance between the data points and their mean. The distance is expressed as a squared distance in order to prevent negative deviations from lessoning the effect of positive deviations.
The mathematical formulation for the standard deviation
{\displaystyle \sigma }
{\displaystyle \sigma ^{2}={\frac {\Sigma ({\hat {x}}_{k}-{\bar {x}})^{2}}{n-1}}}
The denominator is n-1 instead of n because statisticians found that it gives better results for small numbers of experiments; see w:Standard deviation for a more thorough explanation of this.
The standard deviation of a data set measured under constant conditions is a measure of how precise the data set is. Because this is true, the standard deviation of a data set is often used in conjunction with the mean in order to report experimental results. Typically, results are reported as:
{\displaystyle {\bar {x}}\pm \sigma }
If a distribution is assumed, knowing both the mean and standard deviation can help us to estimate the probability that the actual value of the variable is within a certain range, if there is no systematic bias in the data. If there is (such as use of broken equipment, negligence, and so on) then no statistics could predict the effects of that.
Linear RegressionEdit
Suppose you have a set of data points (
{\displaystyle {\hat {x}}_{k},{\hat {y}}_{k}}
) taken under differing conditions which you suspect, from a graph, can be reasonably estimated by drawing a line through the points. Any line that you could draw will have (or can be written in) the following form:
{\displaystyle y=mx+b}
We seek the best line that we could possibly use to estimate the pattern of the data. This line will be most useful for both interpolating between points that we know, and extrapolating to unknown values (as long as they're close to measured values). In the most usual measure, how "good" the fit is is determined by the vertical distance between the line and the data points (
{\displaystyle R_{k}}
), which is called the residual:
{\displaystyle R_{k}=(m{\hat {x}}_{k}+b)-{\hat {y}}_{k}}
In order to normalize the residuals so that they don't cancel when one's positive and one's negative (and thus helping to avoid statistical bias), we are usually concerned with the square of
{\displaystyle R_{k}}
when doing least-squares regression. We use squared terms and not absolute values so that the function is differentiable, don't worry about this if you haven't taken calculus yet.
In order to take into account all of the data points, we simply seek to minimize the sum of the squared residuals:
{\displaystyle {\mbox{ minimize }}\Sigma {R_{k}}^{2}}
Using calculus, we can take the derivative of this with respect to m and with respect to b and solve the equations to come up with the values of m and b that minimize the sum of squares (hence the alternate name of this technique: least-squares regression. The formulas are as follows, where n is the total number of data points you are regressing[1]:
{\displaystyle m^{*}={\frac {n*\Sigma ({\hat {x}}_{k}*{\hat {y}}_{k})-\Sigma ({\hat {x}}_{k})*\Sigma ({\hat {y}}_{k})}{n*\Sigma ({\hat {x_{k}}}^{2})-(\Sigma ({\hat {x}}_{k}))^{2}}}}
{\displaystyle b^{*}={\frac {\Sigma ({\hat {y}}_{k})-m^{*}*\Sigma ({\hat {x}}_{k})}{n}}}
Example of linear regressionEdit
Suppose you wanted to measure how fast you got to school by a less direct route than looking at the speedometer of your car. Instead, you look at a map and read the distances between each intersection, and then you measure how long it takes to go each distance. Suppose the results were as shown in the table below. How far from home did you start, and what is the best estimate for your average speed?
D (yards)
The first thing we should do with any data like this is to graph it and see if a linear fit would be reasonable. Plotting this data, we can see by inspection that a linear fit appears to be reasonable.
Now we need to compute all of the values in our regression formulas, and to do this (by hand) we set up a table:
1 1.1 1.21 559.5 313040 615.45
2 1.9 3.61 759.5 576840 1443.05
3 3.0 9.00 898.2 806763 2694.6
4 3.8 14.44 1116.3 1246126 4241.94
TOTAL 15.1 56.35 4642.2 4655464 15931.15
Now that we have this data we can plug it into our linear regression equation:
{\displaystyle m^{*}={\frac {n*\Sigma ({\hat {x}}_{k}*{\hat {y}}_{k})-\Sigma ({\hat {x}}_{k})*\Sigma ({\hat {y}}_{k})}{n*\Sigma ({\hat {x_{k}}}^{2})-(\Sigma ({\hat {x}}_{k}))^{2}}}}
{\displaystyle ={\frac {5*15931.13-15.1*4642.2}{5*56.35-(15.1)^{2}}}}
{\displaystyle =177.9{\mbox{ }}{\frac {yard}{min}}}
{\displaystyle b={\frac {\Sigma ({\hat {y}}_{k})-m^{*}*\Sigma ({\hat {x}}_{k})}{n}}}
{\displaystyle ={\frac {4642.2-177.9*15.1}{5}}=391.2{\mbox{ yards}}}
Hence the equation for the line of best fit is:
{\displaystyle D=177.9*t+391.2}
The graph of this plotted against the data looks like this:
How to tell how good your regression isEdit
In the previous example, we visually determined if it would be reasonable to perform a linear fit, but it is certainly possible to have a less clear-cut case! If there is some slight curve to the data, is it still "close enough" to be useful? Though it will always come down to your own judgment after seeing the fit line graph against the data, there is a mathematical tool to help you called a correlation coefficient, r, which can be defined in several different ways. One of them is as follows [1]:
{\displaystyle r={\frac {n*\Sigma ({\hat {x}}_{k}*{\hat {y}}_{k})-\Sigma ({\hat {x}}_{k})*\Sigma ({\hat {y}}_{k})}{{\sqrt {n*\Sigma ({{\hat {x}}_{k}}^{2})-(\Sigma {\hat {x}}_{k})^{2}}}*{\sqrt {n*\Sigma ({{\hat {y}}_{k}}^{2})-(\Sigma {\hat {y}}_{k})^{2}}}}}}
It can be shown that this value always lies between -1 and 1. The closer it is to 1 (or -1), the more reasonable the linear fit. In general, the more data points you have, the smaller r needs to be before it's a good fit, but a good rule of thumb is to look for high (higher than 0.85 or 0.9) values and then graph to see if the graph makes sense. Sometimes it will, sometimes it won't, the method is not foolproof.
{\displaystyle r={\frac {5*15931.13-15.1*4642.2}{{\sqrt {5*56.35-(15.1)^{2}}}*{\sqrt {5*4655464-(4642.2)^{2}}}}}}
{\displaystyle r=0.992}
Hence the data correlates very well with a linear model.
Whenever you have to fit a parameter or multiple parameters to data, it is a good idea to try to linearize the function first, because linear regression is much less intensive and more accurate than nonlinear regression. The goal with any linearization is to reduce the function to the form:
{\displaystyle Variable{\mbox{ 1}}=constant+constant*{\mbox{ Variable 2}}}
The difference between this and "standard" linear regression is that Variable 1 and Variable 2 can be any functions of x and y, as long as they are not combined in any way (i.e. you can't have
{\displaystyle ln(x+y)}
as one variable). The technique can be extended to more than two variables using a method called w:multiple linear regression but as that's more difficult to perform, this section will focus on two-dimensional regression.
Power LawEdit
To see some of the power of linearization, let's suppose that we have two variables, x and y, related by a power law:
{\displaystyle y=A*x^{b}}
where A and b are constants. If we have data connecting changes in y to changes in x, we would like to know the values of a and b. This is difficult to do if the equation is in its current form but we can change it into a linear-type function!
The trick here is we need to get rid of the exponent b, so in order to do that we take the natural log of both sides:
{\displaystyle ln{\mbox{ y}}=ln{\mbox{ (A*x}}^{b})}
Using laws of logarithms we can simplify the right-hand side to obtain the following:
{\displaystyle y=A*x^{b}<->ln{\mbox{ y}}=ln{\mbox{ A}}+b*ln{\mbox{ x}}}
The beauty of this equation is that it is, in a sense, linear. If we graph ln(y) vs. ln(x) obtain a straight line with slope b and y-intercept ln(A).
Another common use of linearization is with exponentials, where x and y are related by an expression of the form:
{\displaystyle y=A*b^{x}}
This works for any base but the most common base encountered in practice is Euler's constant, e. Again, we take the natural log of both sides, in order to get rid of the exponent:
{\displaystyle y=A*b^{x}<->ln{\mbox{ y}}=\ln {A}+x*\ln {b}}
This time, Graph ln y vs. x to obtain a line with slope ln(b) and y-intercept ln(A).
Linear InterpolationEdit
Often, when you look up properties on a chart, you will be looking them up at conditions in between two charted conditions. For example, if you were looking up the specific enthalpy of steam at 10 MPa and 430oC you would look in the steam tables and see something like this: [2]
{\displaystyle ({\frac {kJ}{kg}})}
How can you figure out the intermediate value for this? We can't exactly but we can assume that H(T) is a linear function. If we assume that it is linear, then we can easily find the intermediate value. First, we set up a table, including the unknown value like this:
{\displaystyle ({\frac {kJ}{kg}})}
Then since we're assuming the relationship between T and H is linear, and the slope of a line is a constant the slope between points 3 and 2 has to equal the slope between points 3 and 1.
Therefore, we can write that:
{\displaystyle {\frac {2943.4-x}{450-430}}={\frac {2943.4-2832.4}{450-400}}}
Solving gives x = 2899 kJ/kg
The same method can be used to find an unknown T for a given H between two tabulated values.
General formulaEdit
To derive a more general formula (though I always derive it from scratch anyways, it's nice to have a formula), lets replace the numbers by variables ad give them more generic symbols:
{\displaystyle x_{1}}
{\displaystyle y_{1}}
{\displaystyle x^{*}}
{\displaystyle y^{*}}
{\displaystyle x_{2}}
{\displaystyle y_{2}}
Setting the slope between points 3 and 2 equal to that between 3 and 1 yields:
{\displaystyle {\frac {y2-y1}{x2-x1}}={\frac {y2-y^{*}}{x2-x^{*}}}}
This equation can then be solved for x* or y* as appropriate.
Limitations of Linear InterpolationEdit
It is important to remember that linear interpolation is not exact. How inexact it is depends on two major factors:
What the real relationship between x and y is (the more curved it is, the worse the linear approximation)
The difference between consecutive x values on the table (the smaller the distance, the closer almost any function will resemble a line)
Therefore, it is not recommended to use linear interpolation if the spaces are very widely separated. However, if no other method of approximation is available, linear interpolation is often the only option, or other forms of interpolation (which may be just as inaccurate, depending on what the actual function is).
See also w:interpolation.
[1]: Smith, Karl J. The Nature of Mathematics. Pacific Grove, California: Brooks/Cole Publishing company, 6e, p. 683
[2]: Sandler, Stanley I. Chemical, Biochemical, and Engineering Thermodynamics. University of Deleware: John Wiley and Sons inc., 4e, p. 923
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Basic_Statistics_and_Data_Analysis&oldid=3325774"
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Stake liquidity pool - Bright Union
Provide liquidity for trading BRIGHT and ETH pair on UniSwap
Uniswap is a decentralized exchange on the Ethereum Network. It allows anyone to create a pool with a pair of tokens which can be exchanged against each other. Once the pool is created liquidity can be added - which means that sufficient tokens of each pair need to be added - so that the community can start trading. Whoever added liquidity to the pool will earn a small transfer fee for every transaction in the pool. As the community adds more liquidity, the pool becomes 'deeper' and price volatility of the token pair becomes less.
Providing liquidity to a Uniswap pool doesn't come without risks and therefore we only suggest advanced DeFi users to engage in this.
Why do I earn rewards for staking BRIGHT/ETHUniV2 tokens?
Everyone trading BRIGHT tokens (buyers and sellers) benefit from a deeper pool with less price volatility. It means that your transaction will have less impact on the market price of BRIGHT which especially impacts larger transaction amounts.
Bright Union would like to reward the community for providing BRIGHT and ETH tokens to the Uniswap pool. Since the user provided BRIGHT and ETH (with the same value) the APY is double the APY of staking BRIGHT with Bright Union.
How to provide liquidity and stake BRIGHT/ETHUniV2
Staking in Uniswap pools comes with a number of risks which need to be well understood before proceeding. The Bright Union team recommends only experienced DeFi users to stake liquidity pools
Buy BRIGHT tokens (link to guide)
Stake BRIGHT and ETH tokens on Uniswap and receive BRIGHT/ETHUniV2 Pool Tokens (see warning above)
Go to Stake BRIGHT in the application (link to page)
Connect to your Metamask wallet (link to guide)
Press "Stake" in the Stake BRIGHT/ETHUniV2 section of the interface and select an amount
Approve spending of BRIGHT token in your Metamask wallet (~??K GigaWei gas cost)
Approve the transaction in your Metamask wallet (~89K GigaWei gas cost)
Congratulations! You are now earning a transfer fee on Uniswap for providing liquidity and rewards from Bright Union.
How is the APY and my staking return calculated?
The Annual Percentage Yield (APY) is the rewards the user receives for staking BRIGHT/ETHUniV2 tokens for an entire year as a percentage of the amount staked. The APY is a dynamic number which can increase or decrease based on the total amount being staked by all users together.
Users staking BRIGHT/ETHUniV2 are not only earning rewards in the Bright Union application, but also earning a trading fee (~0.5% of trading volume * % of pool they own) on Uniswap
Tokens released per block per year: Bright Union will release 0.25 BRIGHT token for each block released on the Ethereum Blockchain (~every 12 seconds). This amount can be adjusted through votes of the DAO. Blocks Per Year: There are roughly 2.3M blocks released per year. This implies that nearly 600K BRIGHT tokens are released per year. This is roughly ~0.5% of the total supply per year (which is available after 3 years). Total Liquidity Staked: The total amount of UNIBRIGHT tokens staked by all users together denominated in BRIGHT tokens BRIGHT/ETHUniV2Value : The value of UNIBRIGHT denominated in BRIGHT tokens.
APY =TokensReleasedPerBlock * BlocksPerYear / TotalLiquidityStaked
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A Generic Bioheat Transfer Thermal Model for a Perfused Tissue | J. Biomech Eng. | ASME Digital Collection
e-mail: dev@cmrr.umn.edu
Shrivastava, D., and Vaughan, J. T. (June 5, 2009). "A Generic Bioheat Transfer Thermal Model for a Perfused Tissue." ASME. J Biomech Eng. July 2009; 131(7): 074506. https://doi.org/10.1115/1.3127260
A thermal model was needed to predict temperatures in a perfused tissue, which satisfied the following three criteria. One, the model satisfied conservation of energy. Two, the heat transfer rate from blood vessels to tissue was modeled without following a vessel path. Three, the model applied to any unheated and heated tissue. To meet these criteria, a generic bioheat transfer model (BHTM) was derived here by conserving thermal energy in a heated vascularized finite tissue and by making a few simplifying assumptions. Two linear coupled differential equations were obtained with the following two variables: tissue volume averaged temperature and blood volume averaged temperature. The generic model was compared with the widely employed empirical Pennes’ BHTM. The comparison showed that the Pennes’ perfusion term
wCp(1−ε)
should be interpreted as a local vasculature dependent heat transfer coefficient term. Suggestions are presented for further adaptations of the general BHTM for specific tissues using imaging techniques and numerical simulations.
biological tissues, biothermics, blood vessels, differential equations, haemorheology, heat transfer
Biological tissues, Blood, Temperature, Vessels, Heat transfer, Bioheat transfer, Blood vessels
Towards Patient Specific Thermal Modelling of the Prostate
Radiofrequency Heating at 9.4T: In Vivo Temperature Measurement Results in Swine
Temperature and SAR calculations for a Human Head Within Volume and Surface Coils at 64 and 300 MHz
The Effects of Residual Temperature Rise on Ultrasound Heating
Discretizing Large Traceable Vessels and Using DE-MRI Perfusion Maps Yields Numerical Temperature Contours that Match the MR Noninvasive Measurements
3D Numerical Reconstruction of the Hyperthermia Induced Temperature Distribution in Human Sarcomas Using DE-MRI Measured Tissue Perfusion: Validation Against Non-Invasive MR Temperature Measurements
Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm. 1948
Thermal Modeling of the Normal Woman’s Breast
A Hybrid Equation for Simulation of Perfused Tissue During Thermal Treatment
The Energy Conservation Equation for Living Tissue
Heat Transfer in Perfused Biological Tissue. I. General Theory
“Heat Transfer in Perfused Biological Tissue–II. The ‘Macroscopic’ Temperature Distribution in Tissue
Effect of Effective Tissue Conductivity on Thermal Dose Distributions of Living Tissue With Directional Blood Flow During Thermal Therapy
Experimental Measurements of the Temperature Variation Along Artery-Vein Pairs From 200 to 1000 Microns Diameter in Rat Hind Limb
Large Blood Vessel Cooling in Heated Tissues: A Numerical Study
Hybrid Finite Element-Finite Difference Method for Thermal Analysis of Blood Vessels
An Analysis of Peripheral Heat Transfer in Man
Magnetic Resonance Angiography of Collateral Vessel Growth in a Rabbit Femoral Artery Ligation Model
Mathematical Models of Bioheat Transfer
Recent Developments in Modeling Heat Transfer in Blood Perfused Tissues
Bioheat Equation for the Human Thermal System
An Analytical Derivation of Source Term Dependent, 2-D ‘Generalized Poisson Conduction Shape Factors’
An Analytical Study of Heat Transfer in a Finite Tissue Region With Two Blood Vessels and General Dirichlet Boundary Conditions
An Analytical Study of ‘Poisson Conduction Shape Factors’ for Two Thermally Significant Vessels in a Finite, Heated Tissue
Microvascular Architecture Within the Pig Kidney Cortex
Biotransport-Heat and Mass Transfer in Living Systems (Annals of the New York Academy of Sciences)
Quantification of Cerebral Blood Flow and Oxygen Metabolism With 3-Dimensional PET and 15O: Validation by Comparison With 2-Dimensional PET
Absolute Quantification of Cerebral Blood Flow in Normal Volunteers: Correlation Between Xe-133 SPECT and Dynamic Susceptibility Contrast MRI
Mataigne
Whole Brain Quantitative CBF, CBV, and MTT Measurements Using MRI Bolus Tracking: Implementation and Application to Data Acquired From Hyperacute Stroke Patients
Validating Volume Flow Measurements From a Novel Semiautomated Four-Dimensional Doppler Ultrasound Scanner
Experimental Evaluation of Two Simple Thermal Models Using Hyperthermia in Muscle In Vivo
Experimental Evaluation of Two Simple Thermal Models Using Transient Temperature Analysis
Use of Vascular and Non-Vascular Models for the Assessment of Temperature Distribution During Induced Hyperthermia
Evaluation of Tissue Convective Energy Balance Equation in Unheated Tissue With a Realistic Vessel Network
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Introduction to Chemical Engineering Processes/What is a mass balance? - Wikibooks, open books for an open world
1 The "Black Box" approach to problem-solving
1.2 Common assumptions on the conservation equation
The "Black Box" approach to problem-solvingEdit
In this book, all the problems you'll solve will be "black-box" problems. This means that we take a look at a unit operation from the outside, looking at what goes into the system and what leaves, and extrapolating data about the properties of the entrance and exit streams from this. This type of analysis is important because it does not depend on the specific type of unit operation that is performed. When doing a black-box analysis, we don't care about how the unit operation is designed, only what the net result is. Let's look at an example:
Suppose that you pour 1L of water into the top end of a funnel, and that funnel leads into a large flask, and you measure that the entire liter of water enters the flask. If the funnel had no water in it to begin with, how much is left over after the process is completed?
Solution The answer, of course, is 0, because you only put 1L of water in, and 1L of water came out the other end. The answer to this does not depend on the how large the funnel is, the slope of the sides, or any other design aspect of the funnel, which is why it is a black-box problem.
Conservation equationsEdit
The formal mathematical way of describing the black-box approach is with conservation equations which explicitly state that what goes into the system must either come out of the system somewhere else, get used up or generated by the system, or remain in the system and accumulate. The relationship between these is simple:
The streams entering the system cause an increase of the substance (mass, energy, momentum, etc.) in the system.
The streams leaving the system decrease the amount of the substance in the system.
Generating or consuming mechanisms (such as chemical reactions) can either increase or decrease the amount of substance in the system.
What's left over is the amount of the substance in the system.
With these four statements we can state the following very important general principle:
{\displaystyle Accumulation=In-Out+Generation-Consumption}
Its so important, in fact, that you'll see it a million times or so, including a few in this book, and it is used to derive a variety of forms of conservation equations.
Common assumptions on the conservation equationEdit
The conservation equation is very general and applies to any property a system can have. However, it can also lead to complicated equations, and so in order to simplify calculations when appropriate, it is useful to apply assumptions to the problem.
Closed system: A closed system is one which does not have flows in or out of the substance. Almost always, when one refers to a closed system, it is implied that the system is closed to mass flow but not to other flows such as energy or momentum. The equation for a closed system is:
{\displaystyle Accumulation=Generation}
The opposite of a closed system is an open system in which the substance is allowed to enter and/or leave the system. The funnel in the example was an open system because mass flowed in and out of it.
No generation: Certain quantities are always conserved in the strict sense that they are never created or destroyed. These are the most useful quantities to do balances on because then the model does not need to include a generation term:
{\displaystyle Accumulation=In-Out}
The most commonly-used conserved quantities in this class are mass and energy (other conserved quantities include momentum and electric charge). However, it is important to note that though the total mass and total energy in a system are conserved, the mass of a single species is not (since it may be changed into something else in a reaction). Neither is the "heat" in a system if a so-called "heat-balance" is performed (since it may be transformed into other forms of energy. Therefore, one must be careful when deciding whether to discard the generation term).
Steady State: A system which does not accumulate a substance is said to be at steady-state. Often times, this allows the engineer to avoid having to solve differential equations and instead use algebra.
{\displaystyle In-Out+Generation-Consumption=0}
All problems in this text assume steady state but it is not always a valid assumption. It is mostly valid after a process has been running in a controlled manner for long enough that all the flow rates, temperatures, pressures, and other system parameters have reached reasonably constant values. It is not valid when a process is first warming up (or an operating condition is changed) and the system properties change significantly over time. How they change, and how long it takes to become close enough to steady state, is a subject for another course.
TOTAL mass is a conserved quantity (except in nuclear reactions, let's not go there), as is the mass of any individual species if there is no chemical reaction occurring in the system. Let us write the conservation equation at steady state for such a case (with no reaction):
{\displaystyle In-Out=0}
Now, there are two major ways in which mass can enter or leave a system: diffusion and convection. However, if the velocity entering the unit operations is fairly large and the concentration gradient is fairly small, diffusion can be neglected and the only mass entering or leaving the system is due to convective flow:
{\displaystyle Mass_{in}={\dot {m}}_{in}=\rho *v*A}
A similar equation apply for the mass out.
In this book, we generally use the symbol
{\displaystyle {\dot {m}}}
to signify a convective mass flow rate, in units of
{\displaystyle mass/time}
. Since the total flow in is the sum of individual flows, and the same with the flow out, the following steady state mass balance is obtained for the overall mass in the system:
{\displaystyle \sum {{\dot {m}}_{out}}-\sum {{\dot {m}}_{in}}=0}
If it is a batch system, or if we're looking at how much has entered and left in a given period of time (rather than instantaneously), we can apply the same mass balance without the time component. In this book, a value without the dot signifies a value without a time component:
{\displaystyle \sum {m_{out}}-\sum {m_{in}}=0}
Let's work out the previous example (the funnel), but explicitly state the mass balance. We're given the following information:
{\displaystyle {m_{in}}=1L}
{\displaystyle {m_{out}}=1L}
From the general balance equation,
{\displaystyle In-Out=Accumulation}
{\displaystyle Accumulation=1L-1L=0}
Since the accumulation is 0, the system is at steady state.
This is a fairly trivial example, but it gets the concepts of "in", "out", and "accumulation" on a physical basis, which is important for setting up problems. In the next section, it will be shown how to apply the mass balance to solve more complex problems with only one component.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/What_is_a_mass_balance%3F&oldid=3498075"
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Linear separability - Wikipedia
The existence of a line separating the two types of points means that the data is linearly separable
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are linearly separable if there exists at least one line in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if the line is replaced by a hyperplane.
The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises in several areas. In statistics and machine learning, classifying certain types of data is a problem for which good algorithms exist that are based on this concept.
3 Linear separability of Boolean functions in n variables
{\displaystyle X_{0}}
{\displaystyle X_{1}}
be two sets of points in an n-dimensional Euclidean space. Then
{\displaystyle X_{0}}
{\displaystyle X_{1}}
are linearly separable if there exist n + 1 real numbers
{\displaystyle w_{1},w_{2},..,w_{n},k}
, such that every point
{\displaystyle x\in X_{0}}
{\displaystyle \sum _{i=1}^{n}w_{i}x_{i}>k}
and every point
{\displaystyle x\in X_{1}}
{\displaystyle \sum _{i=1}^{n}w_{i}x_{i}<k}
{\displaystyle x_{i}}
{\displaystyle i}
{\displaystyle x}
Equivalently, two sets are linearly separable precisely when their respective convex hulls are disjoint (colloquially, do not overlap).[citation needed]
In simple 2D, it can also be imagined that the set of points under a linear transformation collapses into a line, on which there exists a value, k, greater than which one set of points will fall into, and lesser than which the other set of points fall.
Three non-collinear points in two classes ('+' and '-') are always linearly separable in two dimensions. This is illustrated by the three examples in the following figure (the all '+' case is not shown, but is similar to the all '-' case):
However, not all sets of four points, no three collinear, are linearly separable in two dimensions. The following example would need two straight lines and thus is not linearly separable:
Notice that three points which are collinear and of the form "+ ⋅⋅⋅ — ⋅⋅⋅ +" are also not linearly separable.
Linear separability of Boolean functions in n variables[edit]
A Boolean function in n variables can be thought of as an assignment of 0 or 1 to each vertex of a Boolean hypercube in n dimensions. This gives a natural division of the vertices into two sets. The Boolean function is said to be linearly separable provided these two sets of points are linearly separable. The number of distinct Boolean functions is
{\displaystyle 2^{2^{n}}}
where n is the number of variables passed into the function.[1]
Number of linearly separable Boolean functions in each dimension[2] (sequence A000609 in the OEIS)
6 18446744073709552000 15028134
7 3.402823669 ×10^38 8378070864
8 1.157920892 ×10^77 17561539552946
9 1.340780792 ×10^154 144130531453121108
H1 does not separate the sets. H2 does, but only with a small margin. H3 separates them with the maximum margin.
Classifying data is a common task in machine learning. Suppose some data points, each belonging to one of two sets, are given and we wish to create a model that will decide which set a new data point will be in. In the case of support vector machines, a data point is viewed as a p-dimensional vector (a list of p numbers), and we want to know whether we can separate such points with a (p − 1)-dimensional hyperplane. This is called a linear classifier. There are many hyperplanes that might classify (separate) the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two sets. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum margin classifier.
More formally, given some training data
{\displaystyle {\mathcal {D}}}
{\displaystyle {\mathcal {D}}=\left\{(\mathbf {x} _{i},y_{i})\mid \mathbf {x} _{i}\in \mathbb {R} ^{p},\,y_{i}\in \{-1,1\}\right\}_{i=1}^{n}}
where the yi is either 1 or −1, indicating the set to which the point
{\displaystyle \mathbf {x} _{i}}
{\displaystyle \mathbf {x} _{i}}
is a p-dimensional real vector. We want to find the maximum-margin hyperplane that divides the points having
{\displaystyle y_{i}=1}
{\displaystyle y_{i}=-1}
{\displaystyle \mathbf {x} }
{\displaystyle \mathbf {w} \cdot \mathbf {x} -b=0,}
{\displaystyle \cdot }
denotes the dot product and
{\displaystyle {\mathbf {w} }}
the (not necessarily normalized) normal vector to the hyperplane. The parameter
{\displaystyle {\tfrac {b}{\|\mathbf {w} \|}}}
{\displaystyle {\mathbf {w} }}
If the training data are linearly separable, we can select two hyperplanes in such a way that they separate the data and there are no points between them, and then try to maximize their distance.
Kirchberger's theorem
^ Russell, Stuart J. (2016). Artificial intelligence a modern approach. Norvig, Peter 1956- (Third ed.). Boston. p. 766. ISBN 978-1292153964. OCLC 945899984.
^ Gruzling, Nicolle (2006). "Linear separability of the vertices of an n-dimensional hypercube. M.Sc Thesis". University of Northern British Columbia. {{cite journal}}: Cite journal requires |journal= (help)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Linear_separability&oldid=1076354375"
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On the Calculation of Stress Intensity Factors and
J
-Integrals Using the Submodeling Technique
Marenić, E., Skozrit, I., and Tonković, Z. (August 5, 2010). "On the Calculation of Stress Intensity Factors and
J
-Integrals Using the Submodeling Technique." ASME. J. Pressure Vessel Technol. August 2010; 132(4): 041203. https://doi.org/10.1115/1.4001267
In the present paper, calculations of the stress intensity factor (SIF) in the linear-elastic range and the
J
-integral in the elastoplastic domain of cracked structural components are performed by using the shell-to-solid submodeling technique to improve both the computational efficiency and accuracy. In order to validate the submodeling technique, several numerical examples are analyzed. The influence of the choice of the submodel size on the SIF and the
J
-integral results is investigated. Detailed finite element solutions for elastic and fully plastic
J
-integral values are obtained for an axially cracked thick-walled pipe under internal pressure. These values are then combined, using the General Electric/Electric Power Research Institute method and the reference stress method, to obtain approximate values of the
J
-integral at all load levels up to the limit load. The newly developed analytical approximation of the reference pressure for thick-walled pipes with external axial surface cracks is applicable to a wide range of crack dimensions.
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The graph of a function
f(x)
is given at right. Use the graph to evaluate the following limits.
\lim\limits_ { x \rightarrow - 1 } f ( x )
The limit does not exist, but
y→+∞
\lim\limits_ { x \rightarrow 2 } f ( x )
A limit is a predicted value (which sometimes differs from the actual function value). The prediction must agree from the left and the right.
\lim\limits_ { x \rightarrow 2 ^ { - } } f ( x )
What is the prediction from the left?
\lim\limits_ { x \rightarrow 2 ^ { + } } f ( x )
What is the prediction from the right?
\lim\limits_ { x \rightarrow 5 } f ( x )
Most of the time, the limit and the function value agree.
\lim\limits_ { x \rightarrow \infty } f ( x )
\operatorname{lim }x→∞
\operatorname{lim }x→−∞
will reveal the equation of a horizontal asymptote, if there is one.
Where (if anywhere) does the derivative of
f(x)
not exist?
Look for cusps, endpoints, jumps, holes, and vertical asymptotes.
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Clopen_set Knowpia
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!"[1] emphasizing that the meaning of "open"/"closed" for doors is unrelated to their meaning for sets (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.
A graph with several clopen sets. Each of the three large pieces (that is, components) is a clopen set, as is the union of any two or all three.
{\displaystyle X,}
the empty set and the whole space
{\displaystyle X}
are both clopen.[2][3]
Now consider the space
{\displaystyle X}
which consists of the union of the two open intervals
{\displaystyle (0,1)}
{\displaystyle (2,3)}
{\displaystyle \mathbb {R} .}
{\displaystyle X}
is inherited as the subspace topology from the ordinary topology on the real line
{\displaystyle \mathbb {R} .}
{\displaystyle X,}
{\displaystyle (0,1)}
is clopen, as is the set
{\displaystyle (2,3).}
This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.
{\displaystyle X}
be an infinite set under the discrete metric – that is, two points
{\displaystyle p,q\in X}
have distance 1 if they're not the same point, and 0 otherwise. Under the resulting metric space, any singleton set is open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.
As a less trivial example, consider the space
{\displaystyle \mathbb {Q} }
of all rational numbers with their ordinary topology, and the set
{\displaystyle A}
of all positive rational numbers whose square is bigger than 2. Using the fact that
{\displaystyle {\sqrt {2}}}
{\displaystyle \mathbb {Q} ,}
one can show quite easily that
{\displaystyle A}
is a clopen subset of
{\displaystyle \mathbb {Q} .}
{\displaystyle A}
is not a clopen subset of the real line
{\displaystyle \mathbb {R} }
; it is neither open nor closed in
{\displaystyle \mathbb {R} .}
{\displaystyle X}
is connected if and only if the only clopen sets are the empty set and
{\displaystyle X.}
A set is clopen if and only if its boundary is empty.[4]
Any clopen set is a union of (possibly infinitely many) connected components.
If all connected components of
{\displaystyle X}
are open (for instance, if
{\displaystyle X}
has only finitely many components, or if
{\displaystyle X}
is locally connected), then a set is clopen in
{\displaystyle X}
if and only if it is a union of connected components.
{\displaystyle X}
is discrete if and only if all of its subsets are clopen.
Using the union and intersection as operations, the clopen subsets of a given topological space
{\displaystyle X}
form a Boolean algebra. Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
^ Munkres 2000, p. 91.
^ Bartle, Robert G.; Sherbert, Donald R. (1992) [1982]. Introduction to Real Analysis (2nd ed.). John Wiley & Sons, Inc. p. 348. (regarding the real numbers and the empty set in R)
^ Hocking, John G.; Young, Gail S. (1961). Topology. NY: Dover Publications, Inc. p. 56. (regarding topological spaces)
^ Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 87. ISBN 0-486-66352-3. Let
{\displaystyle A}
be a subset of a topological space. Prove that
{\displaystyle \operatorname {Bdry} (A)=\varnothing }
{\displaystyle A}
is open and closed. (Given as Exercise 7)
Morris, Sidney A. "Topology Without Tears". Archived from the original on 19 April 2013.
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Express 0.0000407, correct to 2 significant figures
2. If x varies inversely as y and y varies directly as z, what is the relationship between x and z?
A. x \(\alpha\) z
B. x \(\alpha\) \(\frac{1}{z}\)
C. a \(\alpha\) z\(^2\)
D. x \(\alpha\) \(\frac{1}{z^2}\)
3. Evaluate \(\frac{3\frac{1}{4} \times 1\frac{3}{5}}{11\frac{1}{3} – 5 \frac{1}{3}}\)
4. The ages of Tunde and Ola are in the ratio 1:2. If the ratio of Ola’s age to Musa’s age is 4:5, what is the ratio of Tunde’s age to Musa’s age?
5. If M = {x : 3 \le
x < 8} and N = {x : 8 < x \le
12}, which of the following is true?
i. 8 ∈ M ∩ N
ii. 8 ∈ M
\cup
iii. M ∩ N = ∅
6. Given that a = log 7 and b =
\mathrm{log}
2, express log 35 in terms of a and b.
A. a + b + 1
B. ab – 1
C. a – b + 1
D. b – a + 1
7. If x = 2/3 and y = – 6, evaluate xy – y/x
8. Solve the equation: 1/5x + 1/x= 3
9. A sum of N18,100 was shared among 5 boys and 4 girls with each boy taking N20.00 more than each girl. Find a boy’s share.
10. One factor of 7x²+3x−10 is
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As an arrow flies through the air, the distance it has traveled in feet at time t is
s ( t ) = 4 \sqrt { t }
. Without your calculator, find the velocity,
s^\prime(t)
t = 1, 4,
16
Three big concepts to choose from are limits, derivatives and integrals.
As a train travels through a station, its velocity, measured in miles per hour, is
v(t) = 9t + 32
. If the train is in the station when
t = 0
, determine the position of the train at
t=1
hour. Explain what concepts of calculus you applied in order to solve this problem.
Both (a) and (b) involve distance and velocity. However, each required a different method or approach. Describe the relationship between distance and velocity, as well as the derivative and area under a curve.
Think back to the Freeway Fatality problem 1-1.
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Burnett equations for simulation of transitional flows | Appl. Mech. Rev. | ASME Digital Collection
Ramesh K Agarwal,
Aerospace Research and Engineering Center, Washington University, St Louis MO 63130
Keon-Young Yun
National Institute for Aviation Research, Wichita State University, Wichita KS 67260-0093
Agarwal , R. K., and Yun , K. (June 10, 2002). "Burnett equations for simulation of transitional flows ." ASME. Appl. Mech. Rev. May 2002; 55(3): 219–240. https://doi.org/10.1115/1.1459080
Hypersonic flows about space vehicles in low earth orbits and flows in microchannels of microelectromechanical devices produce local Knudsen numbers which lie in the continuum-transition regime. The Navier-Stokes equations cannot model these flows adequately since they are based on the assumption of small deviation from local thermodynamic equilibrium. A number of extended hydrodynamics (E-H) or generalized hydrodynamics (G-H) models as well as the Direct Simulation Monte Carlo (DSMC) approach have been proposed to model the flows in the continuum-transition regime over the past 50 years. One of these models is the Burnett equations which are obtained from the Chapman-Enskog expansion of the Boltzmann equation (with Knudsen number (Kn) as a small parameter) to
OKn2.
With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with this model. Several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the Augmented Burnett Equations and the BGK-Burnett Equations. This review article traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and Grad’s 13 moment equations as shown by Struchtrup by employing the Maxwell-Truesdell-Green iteration is also presented. Also, the recent work of Jin and Slemrod on regularization of the Burnett equations via viscoelastic relaxation that ensures positive entropy production and eliminates the instability paradox is discussed. Numerical solutions in 1D, 2D, and 3D are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum-transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier-Stokes calculations, Burnett solutions of other investigators, and DSMC solutions wherever possible. This review article cites 56 references.
hypersonic flow, Knudsen flow, kinetic theory, Boltzmann equation, flow simulation, numerical analysis
Boundary-value problems, Density, Flow (Dynamics), Navier-Stokes equations, Microchannels, Shock (Mechanics), Temperature, Hypersonic flow, Simulation, Computation
Computational hypersonic rarefied flows
The fluid mechanics of microdevices
Bird GA (1994), Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Science Publ, New York NY.
Unique determination of solutions to the Burnett equations
Comeaux KA, Chapman DR, and MacCormack RW (1995), An analysis of the Burnett equations based in the second law of thermodynamics, AIAA Paper No. 95-0415, Reno NV.
Existence of kinetic theory solutions to the shock structure problem
Comments on existence of kinetic theory solutions to the shock structure problem
Moment closure Hierarchies for kinetic theory
The Gaussian moment closure for gas dynamics
Groth CPT, Roe PL, Gombosi TI, and Brown SL (1995), On the nonstationary wave structure of 35-moment closure for rarefied gas dynamics, AIAA Paper No 95-2312, San Diego CA.
Brown S (1996), Approximate Riemann solvers for moment models of dilute gases, PhD Thesis, Univ of Michigan, Ann Arbor MI.
Myong R (1999), A new hydrodynamic approach to computational hypersonic rarefied gas dynamics, AIAA Paper No 99-3578, Norfolk VA.
Eu BC (1992), Kinetic Theory and Irreversible Thermodynamics, John Wiley & Sons, New York, NY.
Cybyk
Direct simulation Monte Carlo: Recent advances and application
Hybrid Euler/particle approach for continuum/rarefied flows
Predicting failure of the continuum fluid equations in transitional hypersonic flows
Struchtrup H (2000), Some remarks on the equations of Burnett and Grad, Proc of the Workshop on Mathematical Models for Simulation of High Knudsen Number Flows, Inst for Math and Applications, Univ of Minnesota, Minneapolis MN.
Jin S and Slemrod M (2000), Regularization of the Burnett equations via relaxation, J. Stat. Phys. (to appear).
The distribution of velocities and mean motion in a slight non-uniform gas
Chapman S and Cowling TG (1970), The Mathematical Theory of Non-Uniform Gases, Cambridge Univ Press, New York NY.
Fiscko KA and Chapman DR (1988), Comparison of Burnett, super-Burnett and Monte-Carlo solutions for hypersonic shock structure, Proc of 16th Int Symp on Rarefied Gas Dynamics, Pasadena CA, 374–395.
Zhong X (1991), Development and computation of continuum higher order constitutive relations for high-altitude hypersonic flow, PhD Thesis, Stanford Univ, Stanford CA.
Welder WT, Chapman DR, and MacCormack RW (1993), Evaluation of various forms of the Burnett equations, AIAA Paper No 93-3094, Orlando FL.
Balakrishnan R and Agarwal RK (1996), Entropy consistent formulation and numerical simulation of the BGK-Burnett equations for hypersonic flows in the continuum-transition regime, Proc of Int Conf on Numerical Methods in Fluid Dynamics, Springer-Verlag, Monterey CA.
Numerical simulation of the BGK-Burnett for hypersonic flows
A model for collision process in gas
Balakrishnan R, Agarwal RK, and Yun K-Y (1997), Higher-order distribution functions, BGK-Burnett equations and Boltzmann’s H-theorem, AIAA Paper No 97-2552, Atlanta GA.
Numerical simulation of three-dimensional augmented Burnett equations for hypersonic flow
Augmented Burnett and Bhatnagar-Gross-Krook-Burnett for hypersonic flow
Burnett’s equations from a 13+9N-field theory
Ferziger JH and Kaper HG (1972), Mathematical Theory of Transport Processes in Gases, Amsterdam, Holland.
Truesdell C and Muncaster RG (1980), Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York NY.
On the higher order hydrodynamic theories of shock structure
Acta Phys. Austriaca, Suppl.
Agarwal RK, Yun K-Y, and Balakrishnan R (1999), Beyond Navier-Stokes: Burnett equations for flow simulations in continuum-transition regime, AIAA Paper No 99-3580, Norfolk VA.
The Chapman-Enskog and Grad methods for solving the Boltzmann equation
On the method of derivation of closed systems for macroparameters of distribution function for Small Knudsen number
On the domain space for constitutive laws in linear viscoelasticity
Biscari P, Cercignani C, and Slemrod M (2000), Time derivatives and frame indifference beyond Newtonian fluids, C. R. Acad. Sci. Paris, to appear.
Joseph DD (1990), Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, New York NY.
Hyperbolic conservation laws with stiff relaxation terms and entropy
Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics
Asymptotic theory of the Boltzmann equation
Veder Warmeleitung in verdumteu Gasen
Annalen der Physik and Chemie
Yun K-Y (1999), Numerical simulation of 3-D augmented Burnett equations for hypersonic flow in continuum-transition regime, PhD thesis, Wichita State Univ, Wichita KS.
Flux vector splitting of the inviscid gas dynamics equations with application to finite-difference methods
A second-order description of shock structure
Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam
Augmented Burnett equation solutions over axisymmetric blunt bodies in hypersonic flow
Vogenitz
Monte Carlo study of blunt body hypersonic viscous shock layers
Moss JN and Bird GA (1984), Direct simulation of transitional flow for hypersonic reentry conditions, AIAA Paper No 84-0223, Reno NV.
Development of particle methods for computing MEMS gas flows
A model for flows in channels, pipes, and ducts at micro and nano scales
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Some Properties of the -Extension of the -Adic Gamma Function
Hamza Menken, Adviye Körükçü, "Some Properties of the -Extension of the -Adic Gamma Function", Abstract and Applied Analysis, vol. 2013, Article ID 176470, 4 pages, 2013. https://doi.org/10.1155/2013/176470
Hamza Menken 1 and Adviye Körükçü2
1Mathematics Department, Science and Arts Faculty, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey
2Mathematics Graduate Program, Institute of Science, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey
We study the -extension of the -adic gamma function . We give a new identity for the -extension of the -adic gamma in the case . Also, we derive some properties and new representations of the -extension of the -adic gamma in general case.
Let be a prime number and let , and denote the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of , respectively. It is well known that the analogous of the classical gamma function in -adic context depends on modifying the factorial function [1]. The factorial function in is defined as The -adic gamma function is defined by Morita [2] as the continuous extension to of the function . That is, is defined by the formula for , where approaches through positive integers. The -adic gamma function had been studied by Diamond [3], Barsky [4], and others. The relationship between some special functions and the -adic gamma function were investigated by Gross and Koblitz [5], Cohen and Friedman [6]. and Shapiro [7].
The -extension of the -adic gamma function is defined by Koblitz as follows.
Definition 1 (see [8]). Let . The -extension of the -adic gamma function is defined by formula for , where approaches through positive integers. We recall that .
The -extension of the -adic gamma function was studied by Koblitz [8, 9], Nakazato [10], Kim et al. [11], and Kim [12].
In the present work, we give a new identity for the -extension of the -adic gamma function in special case . Also, we derive some properties and representations for the -extension of the -adic gamma function .
Theorem 2. If , then for all where is defined by the formula
Proof. Let and . From Proposition 3 in [12] we known that Here, is the greatest integer function. Taking in place of , the relation becomes Now, let in base 2. If , then in base 2 and Thus, we get If , then Hence, Thus, we have and thus, we obtain
We recall that the -factorial is defined in [13] by the formula for , where Note that for , we can define .
We use the following theorem to prove our results.
Theorem 3 (see [12]). Let . Then, where is the greatest integer function. In particular,
Theorem 4. Let and let be the sum of the digits of in base . Then (a)(b) .
Proof. From the Theorem 3 we know that By taking instead of , respectively, we get the relations By multiplying of the equalities above, we can easily obtain Therefore, we get the relation (a) Therefore, we get the relation (b)
Theorem 5. Let and let . Then
Proof. From Theorem 3 it follows that Taking of instead of , respectively, we have the equalities By multiplying of the equalities above, we can easily obtain Thus,
Lemma 6. Let , , and let be a prime number. Then, for
Proof. For For it follows that
Theorem 7. Let and let be the sum of the digits of in base . Then
Proof. This theorem can be proved by using Theorem 4 and Lemma 6.
This work is supported by Mersin University and the Scientific and Technological Research Council of Turkey (TÜBİTAK). The authors would like to thank the reviewers for their useful comments and suggestions.
W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis, vol. 4, Cambridge University Press, Cambridge, UK, 1984. View at: Zentralblatt MATH | MathSciNet
Y. Morita, “A
p
-adic analogue of the
\mathrm{\Gamma }
-function,” Journal of the Faculty of Science, vol. 22, no. 2, pp. 255–266, 1975. View at: Google Scholar | MathSciNet
J. Diamond, “The
p
-adic log gamma function and
p
-adic Euler constants,” Transactions of the American Mathematical Society, vol. 233, pp. 321–337, 1977. View at: Google Scholar | MathSciNet
D. Barsky, “On Morita's
p
-adic gamma function,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 89, no. 1, pp. 23–27, 1981. View at: Publisher Site | Google Scholar | MathSciNet
B. H. Gross and N. Koblitz, “Gauss sums and the
p
\mathrm{\Gamma }
-function,” Annals of Mathematics, vol. 109, no. 3, pp. 569–581, 1979. View at: Publisher Site | Google Scholar | MathSciNet
H. Cohen and E. Friedman, “Raabe's formula for
p
-adic gamma and zeta functions,” Annales de l'Institut Fourier, vol. 58, no. 1, pp. 363–376, 2008. View at: Publisher Site | Google Scholar | MathSciNet
I. Shapiro, “Frobenius map and the
p
-adic gamma function,” Journal of Number Theory, vol. 132, no. 8, pp. 1770–1779, 2012. View at: Publisher Site | Google Scholar | MathSciNet
N. Koblitz, “q-extension of the p-adic gamma function,” Transactions of the American Mathematical Society, vol. 260, no. 2, 1980. View at: Google Scholar
N. Koblitz, “
q
p
-adic gamma function. II,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 111–129, 1982. View at: Publisher Site | Google Scholar | MathSciNet
H. Nakazato, “The
q
-analogue of the
p
-adic gamma function,” Kodai Mathematical Journal, vol. 11, no. 1, pp. 141–153, 1988. View at: Publisher Site | Google Scholar | MathSciNet
T. K. Kim et al., “A note on analogue of gamma functions,” in Proceedings of the 5th Transcendental Number Theory, pp. 111–118, Gakushin University, Tokyo, Japan, 1997. View at: Google Scholar
Y. S. Kim, “
q
-analogues of
p
-adic gamma functions and
p
-adic Euler constants,” Korean Mathematical Society, vol. 13, no. 4, pp. 735–741, 1998. View at: Google Scholar | MathSciNet
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q
q
-beta functions,” Applicable Analysis, vol. 8, no. 2, pp. 125–141, 1978/79. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2013 Hamza Menken and Adviye Körükçü. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Convert gain and phase variation into disk-based gain variation - MATLAB getDGM - MathWorks América Latina
getDGM
Smallest Disk-Based Gain Margin Corresponding to Gain and Phase Margins
Balanced Disk-Based Gain Margin from Gain and Phase Margins
Disk-Based Gain Margin for Specified Gain-Variation Range
Disk-Based Gain and Phase Ranges for Multiple Target Gain and Phase Variations
Convert gain and phase variation into disk-based gain variation
DGM = getDGM(GM,PM,'tight')
[DGM,DPM] = getDGM(___)
In disk margin analysis, gain and phase variations are modeled as a factor F(s) multiplying the open loop response L(s). This factor takes values in a disk D centered on the real axis with real-axis intercepts gmin and gmax. The disk margin determines the largest disk size [gmin,gmax] for which the feedback loop remains stable. This provides a gain margin of at least DGM = [gmin,gmax] and also some phase margin DPM determined by the disk geometry.
Conversely, getDGM takes desired gain and phase margins GM and PM and computes the smallest disk D that delivers both. This disk is characterized by its real-axis intercepts gmin, gmax and the corresponding disk-based gain margin DGM = [gmin,gmax] and phase margin DPM meet or exceed GM and PM.
For more information about the disk model of gain and phase variation, see Algorithms.
DGM = getDGM(GM,PM,'tight') computes the smallest disk that captures the target gain and phase variations specified by GM and PM.
If GM and PM are scalars, then the disk captures gain that can increase or decrease by a factor of GM, and phase that can increase or decrease by PM.
If GM and PM are vectors of the form [glo,ghi] and [pmin,pmax] then the disk captures relative gain and phase variations in these ranges.
If either GM or PM is [], that removes the corresponding constraint on the disk size.
The output is of the form DGM = [gmin,gmax], and describes a disk that represents absolute gain variations within that range. For instance, DGM = [0.8,1.8] models gain that can vary from 0.8 times the nominal value to 1.8 times the nominal value, and phase variations determined by the disk geometry. This disk might have non-zero skew (see Algorithms). Use DGM to create a umargin block that models these gain and phase variations.
DGM = getDGM(GM,PM,'balanced') computes the smallest disk that represents a symmetric gain variation, that is, DGM = [gmin,gmax] where gmin = 1/gmax. This disk has zero skew (see Algorithms).
[DGM,DPM] = getDGM(___) also returns the disk-based phase range DPM modeled by the disk that DGM describes. You can use this output argument with any of the previous syntaxes.
Find the smallest disk-based gain margin that represents relative a gain variation of ±6 dB relative to the nominal value and phase variation of ±40°. Convert the gain variation into absolute units.
GM = db2mag(6)
DGM describes a disk that models both gain and phase variations. The values in DGM represent the range of gain variation in the absence of phase variation. Note that the DGM range is slightly larger than the specified [1/GM,GM] range as the phase margin requirement is more stringent and determines the disk size. Visualize the full range of gain and phase variations represented by DGM.
The 'tight' constraint computes the smallest disk that delivers both target gain and phase variations, which does not necessarily represent a symmetric gain range. In this case, the disk represents gain that can decrease somewhat more than it can increase. Examine the disk of uncertainty defined by this particular DGM.
To enforce symmetric gain variation, use the 'balanced' option.
Determine the disk-based gain margin that delivers symmetric gain variation of
±
5 dB and phase variation of
±
30 degrees.
The 'balanced' constraint models a disk of uncertainty that is symmetric around the nominal value. The function returns a symmetric disk-based gain margin DGM = [gmin,gmax], with gmin=1/gmax.
In this case, DPM slightly exceeds the target phase variation and DGM is equal to the target gain variation.
Determine the disk-based gain margin corresponding to gain variations between 90% and 160% of the nominal value, and phase variations from -15 to +15 degrees.
gainRange = [0.9,1.6];
phaseRange = [-15,15];
DGMt = getDGM(gainRange,phaseRange,'tight')
DGMt = 1×2
The 'tight' constraint models the smallest disk that delivers target gain and phase variations. This disk is modeled with gain variation that skews toward gain increase.
Alternatively, you can use the 'balanced' option to constrain the disk-based gain margin to a symmetrical range of the form gmin = 1/gmax. This means that the gain can increase or decrease by equal amount.
DGMb = getDGM(gainRange,phaseRange,'balanced')
DGMb = 1×2
Visualize the range of simultaneous gain and phase variations corresponding to both gain ranges.
diskmarginplot([DGMt;DGMb])
The balanced range DGMb models a larger, symmetric gain range (gmin = 1/gmax) and larger phase variations than the ones you specify. If you are confident that gain varies more in one direction than the other in your system, then this balanced model might be overly conservative.
Determine the balanced disk-based gain margin ranges that delivers gain variations of ±4 dB, ±6 dB, and ±12 dB and phase variation of ±30°. You can get all the disk-based gain ranges at once by stacking the desired target ranges into a column vector.
GM = db2mag([4;6;12]);
Each row in the matrix DGM gives the disk-based gain variation for the corresponding entry in GM. For instance, the smallest balanced (symmetric) disk that captures gain variation of ±4 dB and phase variation of ±30° is specified by DGM(1,:) = [0.58 1.73].
This disk represents somewhat more than the target ±4 dB, in order to capture the full target gain variation of ±30°. For the targets ±6 dB and ±12 dB, the disk meets the target gain variation exactly, but the corresponding disks describe larger phase variations.
GM — Target amount of relative gain variation
scalar | vector | two-column matrix
Target range of relative gain variation, specified as a scalar, vector, or two-column matrix.
If GM is a scalar, then the disk captures gain that can increase or decrease by a factor of GM. For instance, if GM = 2, then the output DGM represents gain that can decrease or increase by a factor of 2.
If GM is a vectors of the form [glo,ghi] then the disk captures relative gain variations in this range. For instance, if GM = [0.8,1.9], then DGM represents gain that can vary between 0.8 and 1.9 times the nominal value.
If GM [], then getDGM returns a disk that captures the phase variation specified by PM, and the corresponding gain variation determined by the disk model.
Multiple Ranges at Once
To get DGM corresponding to multiple target gain ranges at once, specify GM as a column vector [GM1;...;GMn] or a matrix [glo1,ghi1;...;gloN,ghiN].
PM — Target amount of phase variation
Target phase variation, specified as a scalar, vector, or two-column matrix.
If PM is a scalar, then the disk captures phase that can increase or decrease by PM. For instance, if PM = 20, then the output DGM represents phase that can vary by ±20°.
If PM is a vector of the form [pmin,pmax] with pmin < 0 and pmax > 0, then the disk captures phase that can vary by ±min(abs(pmin),pmax). For instance, if [pmin,pmax] = [-20,40] then the disk captures phase variation in the range [-40,40].
If PM [], then getDGM returns a disk that captures the relative gain variation specified by GM, and the corresponding phase variation determined by the disk model.
To get DGM corresponding to multiple target phase ranges at once, specify PM as a column vector [PM1;...;PMn] or a matrix of the form [-pm1,pm1;...;-pmN,pmN].
DGM — Modeled range of relative gain variation
Modeled range of relative gain variation, returned as a two-element vector of the form [gmin,gmax], where gmin < 1 and gmax > 1. For instance, DGM = [0.8 1.5] represents a gain that can vary between 80% and 150% of its nominal value (that is, change by a factor between 0.8 and 1.5). gmin can be negative, defining a range of relative gain variation that includes a change in sign. When you use the 'balanced' option, the gain change is symmetric, that is, the gain can increase or decrease by the same amount (gmin = 1/gmax).
The range [gmin,gmax] describes a disk of gain and phase uncertainty where the gain can vary by [gmin,gmax] and the phase can vary by an amount determined by the disk geometry. For instance, the following plot shows a disk characterized by DGM = [0.5,2] (For more information about the disk-based uncertainty model, see Algorithms). The corresponding phase variation (returned in DPM) is ±30°.
In general, DGM or the corresponding DPM might capture larger ranges of variation than those you specify with the inputs GM and PM. The disk always captures at least the specified variations.
If GM is a column vector or matrix representing multiple target ranges of gain variation, DGM is a two-column matrix of the form [gmin1,gmax1; ...;gminN,gmaxN], where each row is a corresponding disk-based gain range.
Disk-based phase margin, returned as a two-element vector of the form [-pm,pm]. The amount of phase variation is determined by the geometry of the disk described by DGM (see Algorithms).
If PM is a column vector or matrix representing multiple target ranges of phase variation, DPM is a two-column matrix of the form [-pm1,pm1; ...;-pmN,pmN], where each row is a corresponding disk-based gain range.
umargin and diskmargin model gain and phase variations in an individual feedback channel as a frequency-dependent multiplicative factor F(s) multiplying the nominal open-loop response L(s), such that the perturbed response is L(s)F(s). The factor F(s) is parameterized by:
F\left(s\right)=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta \left(s\right)}{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta \left(s\right)}.
The factor F takes values in a disk centered on the real axis and containing the nominal value F = 1. The disk is characterized by its intercept DGM = [gmin,gmax] with the real axis. gmin < 1 and gmin > 1 are the minimum and maximum relative changes in gain modeled by F, at nominal phase. The phase uncertainty modeled by F is the range DPM = [-pm,pm] of phase values at the nominal gain (|F| = 1). For instance, in the following plot, the right side shows the disk F that intersects the real axis in the interval [0.71,1.4]. The left side shows that this disk models a gain variation of ±3 dB and a phase variation of ±19°.
DGM = [0.71,1.4]
getDGM converts the target gain and phase variations that you want to model into the disk-based gain-variation range DGM. This range fully characterizes the disk F. The corresponding phase range DPM is thus determined by DGM and the disk model.
For further details about the uncertainty model for gain and phase variations, see Stability Analysis Using Disk Margins.
diskmargin | diskmarginplot | getDPM | umargin
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Sketcher BSplinePoleWeight - FreeCAD Documentation
Sketcher BSplinePoleWeight
Previous: Show/hide B-spline knot multiplicity
Next: Convert geometry to B-spline
Sketch → Sketcher B-spline tools → Show/hide B-spline control point weight
3 Weight Explanation
4 Changing Weights
Shows or hides the display of the weights for the control points of a B-spline curve (see below for an explanation of weights).
B-spline with control point weights displayed in brackets
Select a B-spline and use the toolbar button Show/hide B-spline control point weight.
Weight Explanation
The Bézier curve is calculated using this formula:
{\displaystyle \quad {\textrm {Bezier}}(n,t)=\sum _{i=0}^{n}\underbrace {\binom {n}{i}} _{\text{polynomial term}}\underbrace {\left(1-t\right)^{n-i}t^{i}} _{\text{polynomial term}}\;\underbrace {P_{i}} _{\text{point coordinate}}}
n is hereby the degree of the curve. So a Bézier curve of degree n is a polygon with order n. The factors
{\displaystyle P_{i}}
are hereby in fact the coordinates of the Bézier curves' control points. For a visualization see this page.
The term weight in FreeCAD is a bit misleading because in literature the factors
{\displaystyle P_{i}}
are often called weights as well. FreeCAD's weights are something different. The idea of these weights is to modify the spline so that the different control points are "weighted". The idea is that a point with weight 2 should have twice as much influence than a point with weight 1. This is achieved by using this different formula to calculate the spline:
{\displaystyle \quad \mathrm {Rational\ Bezier} (n,t)={\cfrac {\sum _{i=0}^{n}{\binom {n}{i}}\left(1-t\right)^{n-i}t^{i}w_{i}P_{i}}{\sum _{i=0}^{n}{\binom {n}{i}}\left(1-t\right)^{n-i}t^{i}w_{i}\;\;\;\,}}}
{\displaystyle w_{i}}
is the weight for the point
{\displaystyle P_{i}}
This is a new class of Bézier curves because despite the points are indeed weighted as desired, the curve is no longer a polynomial but a fractional polynomial. Therefore these curves are called rational Bézier curves and the B-splines is then called rational B-splines.
The consequence is that you gain more flexibility in defining the spline shape. If all weights are equal, the shape of the spline does not change. So the weights relative to each other is important, not the value alone. For example this spline has exactly the same shape as the one in the first image:
Same B-spline as in first image but with different absolute weight values
A weight of zero would be a singularity in the equation to calculate the rational Bézier curves, therefore FreeCAD assures that it cannot become zero. Nevertheless, small values have the same effect as if the control point would almost not exist:
Same B-spline with an almost zero weight control point
How to change weights is described in this Wiki page.
Retrieved from "http://wiki.freecadweb.org/index.php?title=Sketcher_BSplinePoleWeight&oldid=1129084"
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Table 3 Two states at a fully informative for one parent locus built from the previous state
\stackrel{\to }{v}
〈0, 1〉 〈1, 1〉 〈1, 1〉 〈0, 0〉 〈1, 1〉 0
State hap 〈a, g〉 〈a, a〉 〈g, a〉 〈a, a〉 〈a, a〉 a, a〉 〈a, a〉
\stackrel{\to }{v}
〈1, 1〉 〈0, 1〉 〈0, 1〉 〈 0, 0〉 〈0, 1〉 4
State hap 〈g, a〉 〈a, a〉 〈g, a〉 〈a, a〉 〈a, a〉 〈 a, a〉 〈a, a〉
\stackrel{\to }{v}
An example locus with one heterozygous and one homozygous parent that shows one state at the previous locus and the two states Hapi builds based on this previous state. This example is from the real dataset discussed in Results. The rows labeled
\stackrel{\to }{v}
show the states' inheritance vectors and the rows labeled hap give haplotype assignments of the alleles. Hapi copies the inheritance vector values corresponding to the homozygous parent from the previous state to states a and b. Recombinations result from differing inheritance vector values from the previous state; these differences appear in bold and the states' total number of recombinations appear in the right-most column. Note that the heterozygous parent's inheritance vector values in the two states are exactly opposite each other and are therefore equivalently labeled.
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{\displaystyle n}
{\displaystyle n}
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle (n-1)^{\textrm {th}}}
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value tha{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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11. Solve: – \(\frac{1}{4}\) < \(\frac{3}{4}\) (3x – 2) < \(\frac{1}{2}\)
A. -\(\frac{5}{9}\) < x <\(\frac{8}{9}\)
B. -\(\frac{8}{9}\) < x <\(\frac{7}{9}\)
C. -\(\frac{8}{9}\) < x <\(\frac{5}{9}\)
D. -\(\frac{7}{9}\) < x <\(\frac{8}{9}\)
12. Simplify; 3x – (p – x) – (r – p)
A. 2x – r
B. 2x + r
C. 4x – r
D. 2x – 2p – r
13. An arc of a circle of radius 7.5cm is 7.5cm long. Find, correct to the nearest degree, the angle which the arc subtends at the centre of the circle. [Take π=22/7]
14. Water flows out of a pipe at a rate of 40πcm2
\pi c{m}^{2}
per seconds into an empty cylinder container of base radius 4cm. Find the height of water in the container after 4 seconds.
15. The dimensions of water tank are 13cm, 10cm and 70cm. If it is half-filled with water, calculate the volume of water in litres
16. If the total surface area of a solid hemisphere is equal to its volume, find the radius
17. Which of the following is true about parallelogram?
B. opposite angles are complementary
C. opposite angles are equal
D. opposite angles are reflex angles
18. Calculate the gradient (slope) of the joining points (-1, 1) and (2, -2)
19. If P(2,3) and Q)2, 5) are points on a graph, calculate the length PQ
20. A bearing of 320º expressed as a compass bearing is
A. N 50º W
B. N 40º W
C. N 50º E
D. N 40o E
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Divine battlemage potion - OSRS Wiki
(Redirected from Divine battlemage potion(4))
1 dose of divine battlemage potion.
Divine battlemage potion(1)Divine battlemage potion(2)Divine battlemage potion(3)Divine battlemage potion(4)? (edit)1 dose2 dose3 dose4 dose? (edit)? (edit) ? (edit)File:Divine battlemage potion(1).pngFile:Divine battlemage potion(2).pngFile:Divine battlemage potion(3).pngFile:Divine battlemage potion(4).png30 April 2020 (Update)? (edit)YesNoYes? (edit)? (edit)NoNoYes? (edit)Drink, Empty? (edit)Drop? (edit)1 dose of divine battlemage potion.2 doses of divine battlemage potion.3 doses of divine battlemage potion.4 doses of divine battlemage potion.? (edit)180360540720? (edit)180 coins360 coins540 coins720 coinstrue? (edit)108 coins216 coins324 coins432 coins? (edit)72 coins144 coins216 coins288 coins? (edit)108216324432? (edit)0.0300.0300.0300.135? (edit)0.03 kg0.03 kg0.03 kg0.135 kg? (edit)trueDivine battlemage potion(1)Divine battlemage potion(2)Divine battlemage potion(3)Divine battlemage potion(4)330772548894146253,307 coins (info)7,254 coins (info)8,894 coins (info)14,625 coins (info)infobox-cell-shown2,00020007451,018
SMW Subobject for (3)High Alchemy value: 324Buy limit: 2000Examine: 3 doses of divine battlemage potion.Is members only: trueIs variant of: Divine battlemage potionImage: File:Divine battlemage potion(3).pngUses infobox: ItemWeight: 0.030Value: 540Version anchor: 3 doseRelease date: 30 April 2020Item ID: 24626
SMW Subobject for (1)High Alchemy value: 108Buy limit: 2000Examine: 1 dose of divine battlemage potion.Is members only: trueIs variant of: Divine battlemage potionImage: File:Divine battlemage potion(1).pngUses infobox: ItemWeight: 0.030Value: 180Version anchor: 1 doseRelease date: 30 April 2020Item ID: 24632
The divine battlemage potion is a stat-boosting potion that increases the player's Magic level by 4, and the player's Defence level by 5 + 15% of the player's Defence level for 5 minutes. During these 5 minutes, the player's Magic and Defence levels will not drain below the maximum boost of the potion. Although natural stat drain is disabled, a player's stats can still be decreased or increased by the use of other spells and potions.
It is created by using crystal dust on a battlemage potion, requiring 86 Herblore.
Mixing divine battlemage potions yields 0.5 experience per dose.
1 Defence boost by level
Defence boost by level[edit | edit source]
Defence boost is calculated with:
{\displaystyle \lfloor DefenceLevel\times {\frac {15}{100}}\rfloor +5}
Battlemage potion(1) 1 2,369
Divine battlemage potion(1) 1 3,307
Battlemage potion(4) 1 10,654
Divine battlemage potion(4) 1 14,625
Divine battlemage potion(1) 3,307
The inventory icon of the divine battlemage potion has been graphically updated to match the look of normal potions.
Retrieved from ‘https://oldschool.runescape.wiki/w/Divine_battlemage_potion?oldid=14189485#4_dose’
Pumpkin (Construction)
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Rapid haplotype inference for nuclear families | Genome Biology | Full Text
Table 5 States at a fully informative for one parent locus built from a state with ambiguous values
From: Rapid haplotype inference for nuclear families
\stackrel{\to }{v}
〈0, 1〉 〈1, 1〉? 〈1, 1〉 〈0, 0〉? 〈1, 1〉 0
State hap 〈a, g〉 〈a, a〉 〈g, a〉 〈a, a〉 〈a, a〉 〈a, a〉 〈a, a〉 4
\stackrel{\to }{v}
〈1, 1〉 〈0, 0〉 〈0, 1〉 〈0, 0〉 〈0, 1〉
State hap 〈g, a〉 〈a, a〉 〈g, a〉 〈a, a〉 〈a, a〉 〈a, a〉 〈a, a〉 1
\stackrel{\to }{v}
An example, modified from Table 3 and not from real data, showing a state with ambiguous inheritance values (marked by ?) at the previous locus, and the two states Hapi builds based on it. For unambiguous children's inheritance vector values, the system copies the bits corresponding to the homozygous parent from the previous state. For ambiguous children, two opposite inheritance values are valid for the previous state, and the system uses the homozygous parent bit from the inheritance value that matches the heterozygous parent's bit in the state being built. Both of the two inheritance values are necessarily represented, one in each of the resulting states. As the underlined values show, the inheritance values for the homozygous parent differ across the two outputs. As such, the states are not equivalent, and Hapi cannot eliminate either. Bold values indicate recombinations.
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Development of GaSb Photoreceiver Arrays for Solar Thermophotovoltaic Systems | J. Sol. Energy Eng. | ASME Digital Collection
GaSb
Photoreceiver Arrays for Solar Thermophotovoltaic Systems
, E.T.S.I de Telecomunicación, Avda. Complutense 38, 28040 Madrid, Spain; and Centro de Estudios Superiores “Felipe II”,
, Capitán 39, 28300 Aranjuez. Madrid, Spain
e-mail: dmartin@cesfelipesegundo.com
Carlos Algora,
Victoria Corregidor,
Victoria Corregidor
, E.T.S.I de Telecomunicación, Avda. Complutense 38, 28040 Madrid, Spain
Martín, D., Algora, C., Corregidor, V., and Datas, A. (April 7, 2006). "Development of
GaSb
Photoreceiver Arrays for Solar Thermophotovoltaic Systems." ASME. J. Sol. Energy Eng. August 2007; 129(3): 283–290. https://doi.org/10.1115/1.2734567
In comparison to conventional solar photovoltaics, where sun radiation is converted into electricity directly by solar cells, solar thermophotovoltaic (STPV) conversion has some specific advantages. These advantages come from the fact that in thermophotovoltaics the photon radiator is always inside the conversion system and near the photovoltaic cells. For these reasons we are developing small prototypes with sun heated emitters and photoreceiver arrays to be installed inside complete STPV systems. In order to achieve these complete STPV systems, the first step is to determine the optimum way of packaging the TPV cells into STPV arrays, choosing the best series/parallel configurations depending on the TPV cell band gap, the size of arrays, and the materials. This is the goal of this paper. To carry out the calculations, 18 and 24 cell arrays have been connected following different series and parallel configurations, using the PSPICE commercial circuit-simulation software. Each TPV cell is simulated as a block consisting of the well-known photogenerated current source, two dark diodes of ideality factors equal to one and two, and two resistances, one in parallel and the other in series. As a result, recommendations about the size and front grid design of the
GaSb
cells are obtained. When the optimally designed cells are connected to be included in two specific systems, recommendations about the best parallel/series connection are achieved. Evaluation on the performance of the arrays at nonuniform illumination is also carried out. The first photoreceiver arrays are being constructed and implemented in real STPV systems following these recommendations.
thermophotovoltaic cells, SPICE, solar cells, solar radiation, electrical engineering computing, gallium compounds, III-V semiconductors, solar thermophotovoltaics, concentrators, III-V semiconductors
Design, Optimization, Photons, Radiation (Physics), Semiconductors (Materials), Simulation, Solar cells, Solar energy, Temperature, Tungsten, Energy gap, Electricity (Physics)
Thermophotovoltaic Energy Conversion
Proc. 20th Annual Power Sources Conference
Analysis, Optimization and Assessment of Radioisotope Thermophotovoltaic System Desing for an Illustrative Space Mission
Proc. of the 1st NREL Conference on Thermophotovoltaic Generation of Electricity
, Copper Mountain, Colorado. July 24–27, AIP Conference Proceedings 321, pp.
Operation and Component Testing of a Solar Thermophotovoltaic Power System
Proceedings 25th PVSC Conference
Solar Thermophotovoltaic Using Al2O3∕Er3Al5O12 Eutectic Composite Selective Emitter
Proc. 28th IEEE PVSC Conference
, Anchorage, Alaska, September 15–22, pp.
Proc. of the 15th Annual Power Sources Conference
, Atlantic, New Jersey, June 20–22, pp.
Drubka
Analysis of Solar Thermophotovoltaic Test Data from Experiments Performed at McDonnell Douglas
, Copper Mountain, Colorado, July 24–27, AIP Conference Proceedings 321, pp.
Testing and Modeling of a Solar Thermophotovoltaic Power System
Proc. of the 2nd NREL Conference on Thermophotovoltaic Generation of Electricity
, Colorado Springs, Colorado, July 17–19, AIP Conference Proceedings 358, pp.
Solar TPV: Brief Review and a New Look
Theoretical Limits of Photovoltaic Conversion
Handbook of Photovoltaic Science and Technology
A Thermophotovoltaic Electric Generator using GaSb Cells With a Hydrocarbon Burner
Proc. of the 1st World Conference on Photovoltaic Solar Energy Conversion
, Wailoloa, Hawaii, December 5–9, pp.
Development of a Small Air-Cooled “Midnight Sun” Thermophotovoltaic Electric Generator
Efficiency and Power Density Potential of Thermophotovoltaic Systems Using Low Bandgap Photovoltaic Cells
Khovostikov
Portable TPV Generator Based on Metallic Emitter and 1.5-Amp GaSb Cells
Modelling and Manufacturing GaSb TPV Converters
GaSb Thermophotovoltaic Converters Manufactured by Single Zinc Difussion Using Spin-On Masking Layers
, Munich, Germany, October 22–26, pp.
Key Issues for an Accurate Modelling of GaSb TPV Converters
Theoretical Comparison Between GaSb Diffused and Epitaxial TPV Cells
Proceedings of the 6th International Conference on Thermophotovoltaic Generation of Electricity
, Freiburg, Germany, June 14–16, AIP Conference Proceedings 738.
Performance of Antireflecting Coating-AlGaAs Window Layer Coupling for Terrestrial Concentrator GaAs Solar Cells
Modelling of Resistive Losses due to the Bus-Bar and External Connections in III–V High-Concentrator Solar Cells
Temperature-dependent GaSb Material Parameters for Reliable Thermophotovoltaic Cell Modeling
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These are some WIP notes on credit derivatives.
Discount processes
Risky Annuties
\mathcal F
Z(t,T)
discount curve
r(t)
short rate process
D(t)
discount process
\Gamma(t)
annuity process
A(t)
annuity valuation process
\tau
Q(t,T)
survival curve
\lambda(t)
hazard rate process
\hat Z (t,T)
risky discount curve
\hat A (t)
risky annuity valuation process
Assume that all stochastic processes, unless otherwise specified, are adapted to some common filtration
\mathcal F
Consider a hypothetical bond maturing at time
t
(the maturity or tenor) having the following properties:
Unit-Notional
the bond holder will be paid one unit of currency. This payment is the notional.
No payments to the bond holder will be made before time
T
(otherwise, such payments are called coupons).
The notional is guaranteed to be paid in full and on time.
Denote the fair value of this bond ascertained at time
s
Z(s,t)
. The bitemporal function
Z
is called the (risk-free) discount curve. It represents the market demand for money.
Some assumed propeties for all real values
t\geq s
h
s
s\mapsto Z(s,t)
is a stochastic process with respect to the filtration
\mathcal{F}_s
Z(s,t) > 0
Z(s,t+h)=Z(s,t)\mathbb{E}[Z(t,t+h)|\mathcal F_s]
We can deduce from the above properties that
Z(t,t)=1
. This is because, applying the second property:
Z(t,t)=Z(t,t+0)=Z(t,t)Z(t,t+0)=Z(t,t)^2
Let us additionally assume that
t \mapsto Z(s,t)
Z'
Z
with respect to its second argument. That is,
Z'(smt)=\frac{\partial Z(s,t)}{\partial t}
\begin{aligned} Z'(s,t) &= \lim_{h\to 0} \frac{Z(s,t+h)-Z(s,t)}{h} \\ &= \lim_{h\to 0} \frac{Z(s,t)\mathbb{E}[Z(t,t+h)|\mathcal{F}_s]-Z(s,t)}{h} \\ &= Z(s,t) \mathbb{E}\left[\lim_{h\to 0} \frac{Z(t,t+h)-1}{h}\middle\vert\mathcal{F}_s\right] \\ &= Z(s,t)Z'(t,t) \end{aligned}
\begin{aligned} Z(s,t) &= \mathbb{E}\left[ \exp\left(\int_s^t Z'(\tau,\tau)d\tau \right) \middle\vert \mathcal{F}_s \right] \\ &= \mathbb{E}\left[ \exp\left(-\int_s^t r(\tau)d\tau \right) \middle\vert \mathcal{F}_s \right]. \end{aligned}
r(t)=-Z'(t,t)
is the short rate process. It can be thought of as the "instanteous" interest rate at time
t
That is, the interest rate of a zero coupon bond during the period
(s,t)
\frac{1}{Z(s,t)}\frac{1-Z(s,t)}{t-s}.
s\to t
would then yield
r(t)
The discount process corresponding to the short rate process
r(t)
D(t) = \exp \int_0^t r(\tau)d\tau.
With this notation, the discount curve may be written as
Z(s,t)=\frac{\mathbb E [ D(t) | \mathcal F_s]}{D(s)}.
An annuity
\Gamma
is a schedule of future payments. Denote
\Gamma (t)
as the total payments up to time
t
\Gamma
may be modeled as either a deterministic function or a stochastic process.
Assuming zero risk, the (forward-looking) fair value of this annuity at time
t
is defined as the following Riemann-Stieltjes integral:
A(t) = \int_t^\infty Z(t,s) d\Gamma (s).
Consider an annuity consisting of discrete payments
\gamma_1,\ldots,\gamma_N
t_1,\ldots,t_N
A(t) = \sum_{n=1}^{N} \gamma_n Z(t,t_n)I(t>t_n),
I
is the indicator variable.
\Gamma
is smooth. And define
\gamma(t) = \Gamma' (t)
as the annuity rate. Then
A(t) = \int_t^\infty Z(t,s)\gamma (s) ds.
A credit event is a contractually-obligated event wherein a debt security (e.g. a bond) has been determined to not be fully honored.
Mathematically, a credit event
\tau
is a stopping time. That is, a random variable representing some point in time.
It's distribution can be described the the survival curve:
Q(t,T) = \text{Pr}(\tau > T|\mathcal F_t).
This is the probability ascertained at time
that a credit event will not occur before or during time
T
Consider an annuity
\Lambda (t)
that is risky. That is, scheduled payment occuring at or after a credit event
\tau
are unrealized. The forward-looking fair value of this annuity is given by the risky value process:
\hat A(t) = \int_t^\infty Z(t,s) Q(t,s) d\Gamma(s)
The hazard rate
\lambda (t)
is the instanteous likelihood that a credit event will occur at time
t
. Mathematically, it is a stochastic process
\lambda
\lambda(t) = \left. -\frac{\partial Q(t,s)}{\partial s} \right\vert_{s=t}.
\lambda(t) dt
is the likelihood that a credit event will occur between times
t
t+dt
Equivalently, the survival curve may be derived from an a priori defined hazard rate:
Q(t,T)= \mathbb E \left[ \exp \left( -\int_t^T \lambda(s)ds \right) \middle\vert \mathcal F_t \right].
\lambda
Q
is analogous to that of
r
Z
Consider a payment of one unit of currency paid out at time
\tau
\tau < T
for some tenor
T
. This is a form of insurance. And it's value at time
t
\int_{s=t}^{s=T} Z(t,s) dQ(t,s) = \int_t^T Z(t,s) \lambda(s) ds.
The Riemann-Stieltjes integral on the left side of the above equation is with respect to the parameter
s
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Use your calculator to approximate the following limits to two decimal places.
\lim\limits_ { x \rightarrow 1 } \frac { 2 ^ { x } - 2 } { 3 ^ { x } - 3 }
≈ 0.42
\lim\limits_ { x \rightarrow 0 } \frac { 1 - \operatorname { cos } x } { x ^ { 2 } }
It is important to recognize what skills you have and what skills you do not have. Think about why you are not expected to compute these limits without a calculator.
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106.3.2.32 TM-32, Determination of Purity of Sodium Chloride - Engineering_Policy_Guide
This method determines the percent of sodium chloride in commercial grades of sodium chloride.
1 106.3.2.32.1 Reagents and Apparatus
2 106.3.2.32.2 Preparation of Standard Solution
3 106.3.2.32.3 Procedure
4 106.3.2.32.4 Calculation and Report
106.3.2.32.1 Reagents and Apparatus
a) Millivolt meter equipped with a combination chloride electrode.
b) Nitric Acid (HNO3), chloride free, 1.42 specific gravity.
c) Sodium Chloride (NaCl), Reagent Grade, dried at 105-110°C for 1 hour prior to use.
d) Silver Nitrate (AgNO3), Reagent Grade.
e) A source of chloride-free distilled or deionized water.
106.3.2.32.2 Preparation of Standard Solution
Standard Sodium Chloride Solution (0.0100N). Weigh 0.5844 g dried NaCl, dissolve in distilled water and dilute to 1L.
Standard Silver Nitrate Solution (0.01N). Weigh 1.699 g AgNO3, dissolve in distilled water and dilute to 1L. Standardize to the nearest 0.0001 N against 0.0100 N NaCl.
Using a suitable sample splitting technique, divide the salt, as received, to obtain a representative sample of about 75 g. Dry at least 3 hours at 105-110C and cool to room temperature in a dessicator. Weigh the sample to the nearest 0.1 mg and transfer to a 1000-mL beaker. Add approximately 500 mL of distilled water and 20 mL of HNO3. Heat near boiling 1 hour, occasionally stirring and crushing any insoluble matter against the bottom of the beaker with the flattened end of a stirring rod. Cool to room temperature, quantitatively transfer to a 1000-mL volumetric flask and dilute to volume with distilled water. Transfer a 10-mL aliquot to a 1000-mL volumetric flask and dilute to volume with distilled water. Transfer a 10-mL aliquot to a 250-mL beaker, adding 90 mL of distilled water and 1 mL of HNO3. Determine the chloride concentration by potentiometric titration with the standard silver nitrate solution.
106.3.2.32.4 Calculation and Report
Report percent sodium chloride to the nearest 0.1% as follows:
% Sodium Chloride (NaCl) =
{\displaystyle 100\times {\frac {A\times C\times 584.5}{D}}}
A = Milliliters of AgNO3 solution to titrate sample
C = Normality of AgNO3 solution
D = Grams of sample
Retrieved from "https://epg.modot.org/index.php?title=106.3.2.32_TM-32,_Determination_of_Purity_of_Sodium_Chloride&oldid=23408"
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Revision as of 12:03, 4 March 2008 by Smithk (talk | contribs) (article construction)
{\displaystyle \mu \,}
Bolts and nuts specified to meet the requirements of ASTM A 307 shall be accompanied by a manufacturer's certification statement that the bolts and nuts were manufactured to comply with the requirements of ASTM A 307 and, if required by the specifications, galvanized to comply with the requirements of AASHTO M 232, Class C or were mechanically galvanized in accordance with the requirements of AASHTO M 298, Class 55. High strength bolts, nuts and washers shall be accompanied by a manufacturer's inspection test report for each production lot or shipping lot furnished and certifying that the bolts furnished conform to the requirements specified. All bolts, nuts and washers are to be identifiable as to type and manufacturer. Bolts, nuts and washers manufactured to meet ASTM A 307 will normally be identified on the packaging since no special markings are required on the item. The specified AASHTO is to be consulted for the required identification marks on high strength bolts and nuts. Dimensions are to be as shown on the plans or as specified. Weight of zinc coating, when specified, is to be determined by magnetic gauge in the same manner as described in the previous paragraph except that a smaller number of single-spot tests will be sufficient. Samples for Central Laboratory testing are only required when requested by the State Construction and Materials Engineer or when field inspection indicates questionable compliance. When samples are taken, they are to be taken at the frequency and of the size shown in Table 2, Sampling Requirements.
Reports shall indicate acceptance, qualified acceptance or rejection. Appropriate remarks as described in 106.9 Reporting Test Results are to be included in the report to clarify conditions of acceptance or rejection. Distribution of reports or materials purchased under a MoDOT purchase order is to be as described in 1101 Materials Purchased by a MoDOT Purchase Order.
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Iteratively reweighted ℓ1-penalized robust regression
2021 Iteratively reweighted
{\ell }_{1}
-penalized robust regression
Xiaoou Pan, Qiang Sun, Wen-Xin Zhou
Xiaoou Pan,1 Qiang Sun,2 Wen-Xin Zhou1
1Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
2Department of Statistical Sciences, University of Toronto, Toronto, ON M5S 3G3, Canada
This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear models have only bounded second moments, we show that iteratively reweighted
{\ell }_{1}
-penalized adaptive Huber regression estimator satisfies exponential deviation bounds and oracle properties, including the oracle convergence rate and variable selection consistency, under a weak beta-min condition. Computationally, we need as many as
\mathcal{O}\left(logs+loglogd\right)
iterations to reach such an oracle estimator, where s and d denote the sparsity and ambient dimension, respectively. Extension to a general class of robust loss functions is also considered. Numerical studies lend strong support to our methodology and theory.
Sun was supported in part by the NSERC Grant RGPIN-2018-06484. Zhou acknowledges the support from NSF Grant DMS-1811376 and UCSD General Campus Research Grant.
The authors are very grateful to the Editor and two anonymous referees for their careful reading of the manuscript, and many valuable remarks and suggestions.
Xiaoou Pan. Qiang Sun. Wen-Xin Zhou. "Iteratively reweighted
{\ell }_{1}
-penalized robust regression." Electron. J. Statist. 15 (1) 3287 - 3348, 2021. https://doi.org/10.1214/21-EJS1862
Keywords: Adaptive Huber regression , convex relaxation , heavy-tailed noise , nonconvex regularization , optimization error , oracle property , oracle rate
Xiaoou Pan, Qiang Sun, Wen-Xin Zhou "Iteratively reweighted
{\ell }_{1}
-penalized robust regression," Electronic Journal of Statistics, Electron. J. Statist. 15(1), 3287-3348, (2021)
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18.3: Statistical Power - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F18%253A_Quantifying_Effects_and_Desiging_Studies%2F18.03%253A_Statistical_Power
Remember from the previous chapter that under the Neyman-Pearson hypothesis testing approach, we have to specify our level of tolerance for two kinds of errors: False positives (which they called Type I error) and false negatives (which they called Type II error). People often focus heavily on Type I error, because making a false positive claim is generally viewed as a very bad thing; for example, the now discredited claims by Wakefield (1999) that autism was associated with vaccination led to anti-vaccine sentiment that has resulted in substantial increases in childhood diseases such as measles. Similarly, we don’t want to claim that a drug cures a disease if it really doesn’t. That’s why the tolerance for Type I errors is generally set fairly low, usually at
\alpha = 0.05
. But what about Type II errors?
power = 1 - \beta
Another important aspect of the Neyman-Pearson model that we didn’t discuss above is the fact that in addition to specifying the acceptable levels of Type I and Type II errors, we also have to describe a specific alternative hypothesis – that is, what is the size of the effect that we wish to detect? Otherwise, we can’t interpret
\beta
– the likelihood of finding a large effect is always going to be higher than finding a small effect, so
\beta
will be different depending on the size of effect we are trying to detect.
There are three factors that can affect power:
This simulation shows us that even with a sample size of 96, we will have relatively little power to find a small effect (
d = 0.2
\alpha = 0.005
. This means that a study designed to do this would be futile – that is, it is almost guaranteed to find nothing even if a true effect of that size exists.
There are at least two important reasons to care about statistical power, one of which we discuss here and the other of which we will return to in Chapter 32. If you are a researcher, you probably don’t want to spend your time doing futile experiments. Running an underpowered study is essentially futile, because it means that there is a very low likelihood that one will find an effect, even if it exists.
Let’s say that we are interested in running a study of how a particular personality trait differs between users of iOS versus Android devices. Our plan is collect two groups of individuals and measure them on the personality trait, and then compare the two groups using a t-test. In order to determine the necessary sample size, we can use the pwr.t.test() function from the pwr library:
However, it’s rare in science to be doing an experiment where we expect to find such a large effect – just as we don’t need statistics to tell us that 16-year-olds are taller than than 6-year-olds. When we run a power analysis, we need to specify an effect size that is plausible for our study, which would usually come from previous research. However, in Chapter 32 we will discuss a phenomenon known as the “winner’s curse” that likely results in published effect sizes being larger than the true effect size, so this should also be kept in mind.
18.3: Statistical Power is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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28.2: Comparing Two Means - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F28%253A_Comparing_Means%2F28.02%253A_Comparing_Two_Means
A more common question that often arises in statistics is whether there is a difference between the means of two different groups. Let’s say that we would like to know whether regular marijuana smokers watch more television. We can ask this question using the NHANES dataset; let’s take a sample of 200 individuals from the dataset and test whether the number of hours of television watching per day is related to regular marijuana use. The left panel of Figure 28.1 shows these data using a violin plot.
t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}
\bar{X}_1
\bar{X}_2
S
S
are the variances for each of the groups, and
n_1
n_2
are the sizes of the two groups. Under the null hypothesis of no difference between means, this statistic is distributed according to a t distribution with n-2 degrees of freedom (since we have computed two parameter estimates, namely the means of the two groups). We can compute the t-test in R using the t.test() function. In this case, we started with the specific hypothesis that smoking marijuana is associated with greater TV watching, so we will use a one-tailed test. Since the t.test function orders the conditions alphabetically, the “No” group comes first, and thus we need to test the alternative hypothesis of whether the first group is less than the second (“Yes”) group; for this reason, we specify ‘less’ as our alternative.
## t = -3, df = 198, p-value = 0.004
28.2: Comparing Two Means is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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a
f(x)
x = 1
f ( x ) = \left\{ \begin{array} { l l } { ( x + 2 ) ^ { 2 } - 3 } & { \text { for } x < 1 } \\ { a \operatorname { sin } ( x - 1 ) + 6 } & { \text { for } x \geq 1 } \end{array} \right.
f(x)
x = 1
, you must demonstrate that the derivatives agree from the left and right.
But do not forget that 'differentiablity implies continuity'. In other words, if the function is not continuous at
x = 1
, then it CANNOT be differentiable (even if the derivatives agree). So use the Three Conditions of Continuity to demonstrate that
f(x)
x = 1
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Revision as of 11:36, 9 April 2017 by MathAdmin (talk | contribs) (Created page with "<span class="exam">The displacement from equilibrium of an object in harmonic motion on the end of a spring is: ::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\...")
{\displaystyle y={\frac {1}{3}}\cos(12t)-{\frac {1}{4}}\sin(12t)}
{\displaystyle y}
{\displaystyle t}
{\displaystyle t={\frac {\pi }{8}}.}
What is the relationship between position
{\displaystyle s(t)}
{\displaystyle v(t)}
{\displaystyle v(t)=s'(t)}
{\displaystyle t={\frac {\pi }{8}},}
{\displaystyle t={\frac {\pi }{8}}}
{\displaystyle y.}
{\displaystyle {\begin{array}{rcl}\displaystyle {y{\bigg (}{\frac {\pi }{8}}{\bigg )}}&=&\displaystyle {{\frac {1}{3}}\cos {\bigg (}{\frac {12\pi }{8}}{\bigg )}-{\frac {1}{4}}\sin {\bigg (}{\frac {12\pi }{8}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {1}{3}}\cos {\bigg (}{\frac {3\pi }{2}}{\bigg )}-{\frac {1}{4}}\sin {\bigg (}{\frac {3\pi }{2}}{\bigg )}}\\&&\\&=&\displaystyle {0-{\frac {1}{4}}(-1)}\\&&\\&=&\displaystyle {{\frac {1}{4}}{\text{ foot}}.}\end{array}}}
{\displaystyle {\begin{array}{rcl}\displaystyle {v(t)}&=&\displaystyle {y'}\\&&\\&=&\displaystyle {{\frac {-1}{3}}\sin(12t)(12)-{\frac {1}{4}}\cos(12t)(12)}\\&&\\&=&\displaystyle {-4\sin(12t)-3\cos(12t).}\end{array}}}
{\displaystyle t={\frac {\pi }{8}}}
{\displaystyle {\begin{array}{rcl}\displaystyle {v{\bigg (}{\frac {\pi }{8}}{\bigg )}}&=&\displaystyle {-4\sin {\bigg (}{\frac {3\pi }{2}}{\bigg )}-3\cos {\bigg (}{\frac {3\pi }{2}}{\bigg )}}\\&&\\&=&\displaystyle {-4(-1)+0}\\&&\\&=&\displaystyle {4{\text{ feet/second}}.}\end{array}}}
{\displaystyle {\frac {1}{4}}{\text{ foot}}.}
{\displaystyle 4{\text{ feet/second}}.}
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Compute or plot passivity index as function of frequency - MATLAB passiveplot - MathWorks España
G=\left(s+2\right)/\left(s+1\right)
\left(G\left(s\right)+G{\left(s\right)}^{H}\right)/2
\left(G{\left(s\right)}^{-1}+G{\left(s\right)}^{-H}\right)/2
\text{G}\left(s\right)+\text{G}{\left(s\right)}^{H}>2\tau \left(\omega \right)\left(I+\text{G}{\left(s\right)}^{H}\text{G}\left(s\right)\right),
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The annual cycle of daily high temperatures in Cabanaville is shown in the graph below from January 1st to December 31st. The
x
-axis is marked in segments of
30
days and the
y
-axis represents temperature in °F.
Approximately when is Cabanaville at its hottest? The coldest? How can you tell?
This is a temperature graph. Change in temperature is a rate. How is rate shown on a temperature graph? How about the 'fastest' rate?
When is the temperature changing the fastest? What is this called?
This is a temperature graph. During what month was temperature the highest?
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Work Energy And Power, Popular Questions: Karnataka Class 11-science PHYSICS, Physics Puc I (part 1) - Meritnation
How i cap into i cap (dot product ) is equal to 1??
\mu
Q. The velocity of a particle is expressed as v = kx2 where v is the velocity and x is displacement along X axis . The particle is at origin ( 0,0) initially (t = 0 ) . The kinetic energy of particle after 2 second is
(A) 5J (B) 2 J (C) 0 (D) none of these
Find the angle between the vectors A=i^-2j^-5k^ and B=2i^+j^-4k^
A stone of mass m is to be thrown to a height h
a) With what minimum velocity should it be thrown?
b)At what height does KE and PEbecome equal
c)Find the velocity at that height.
mass of Bob of a simple pendulum is ten gram it makes 10 oscillation in 15 second passes the mean position with a speed 4 cm per second work done by tension in the suspension thread in oscillation is:
Spring of weighing machine is compressed by 1cm when a sand bag of mass 0.1 kg is dropped on it from a height 0.25 m. From what height should the sand bag be dropped to cause a compression of 4cm ?
A uniform ladder of mass 5kg and length 2m carries a block of mass 2.5 kg attached to one end. When the ladder is moved from orizontal position to veritical position calculate the change in its gravitational potential energy.
Tharun D asked a question
A 2000 kg car moves on a level road at a constant speed of 36 km/h. The driver switches off the engine. Initially for a distance of 250 m the air resistance is 150 N after which the air resistance reduces to 50 N. How far does the car travel after the engine is switched off? Ignore friction from the road.
a bullet emerges out from a wooden plank wit 75%of its initial kinetic energy. the number of additional planks required to stop the bullet is? (ans: 3)
{h}_{0}
{h}_{1}
{h}_{2}
{h}_{0}>{h}_{1}>{h}_{2}
{h}_{0}={h}_{1}={h}_{2}
{h}_{0}>{h}_{1}={h}_{2}
{h}_{0}={h}_{2}>{h}_{1}
Q45. A block starts from rest from top of incline plane whose last
\frac{1}{3}
rd part is rough. If block comes to rest at bottom, then value of coefficient of friction between block and incline plane is
\left(1\right) \mathrm{\mu } = \mathrm{tan} \mathrm{\theta }\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(2\right) \mathrm{\mu } = \frac{1}{3} \mathrm{tan} \mathrm{\theta }\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(3\right) \mathrm{\mu } =\frac{1}{2} \mathrm{tan} \mathrm{\theta }\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(4\right) \mathrm{\mu } =3 \mathrm{tan} \mathrm{\theta }
\sqrt[20]{3} \text{m/s}
\sqrt[20]{2} \text{m/s}
Aditya Phalwal asked a question
A block A having a mass ‘mA ’ is released from rest at the position P shown and slides freely down the smooth inclined ramp. When it reaches the bottom of the ramp it slides horizontally onto the surface of a cart of mass mc for which the coefficient of friction between the cart and the box is ‘µ’. If ‘h’ be the initial height of A, determine the final velocity of the cart once the block comes to rest in it. Also determine the position ‘x’ of the box on the cart after it comes to rest relative to cart. (The cart moves on smooth horizontal surface.)
a stone oof mass m is thrown to a height h.at what height does the kinetic energy and potential energy be equal
Shiksha Priya asked a question
A 5 kg block is kept on a horizontal platform at rest . At t=0 , the platform starts moving with a constant acceleration of 1 m/s^2 . The coefficient of friction between block and the platform is 0.2 . The work done force of friction on the block in reference frame fixed with ground in 10 sec is
In the ques.. if coefficient of friction becomes 0.02 , the work done . (same conditions applied)
calculate the work done in lifting body of mass 10kg to a height 10km above the ground....
Q.8. A boy is holding a book of mass 500 gm by pressing his two hands against the two opposite vertical faces of the book. If coefficient of friction between his hand and the surface of the book is 0.2, and if he exerts equal forces in opposite directions by his two hands as shown in the figure, the minimum force (F) he has to exert, in order to stop the book from slipping down is
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Revision as of 22:39, 12 April 2015 by MathAdmin (talk | contribs) (Created page with "right|350px <span class="exam">A 15-foot ladder is leaning against a house. The base of the ladder is pulled away from the house at a rate of 2 feet p...")
{\displaystyle x^{2}+y^{2}\,\,=\,\,15^{2}\,\,=\,\,225,}
{\displaystyle x}
{\displaystyle y}
{\displaystyle 2x{\frac {dx}{dt}}+2y{\frac {dy}{dt}}\,\,=\,\,0,}
{\displaystyle {\frac {dy}{dt}}\,\,=\,\,-{\frac {x}{y}}\cdot {\frac {dx}{dt}},}
{\displaystyle x=9,\quad y=12,\quad dx/dt=2.}
{\displaystyle {\frac {dy}{dt}}\,\,=\,\,-{\frac {x}{y}}\cdot {\frac {dx}{dt}}\,\,=\,\,-{\frac {9}{12}}\cdot 2\,\,=\,\,-{\frac {3}{2}}}
{\displaystyle -3/2}
{\displaystyle -3/2}
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Introduction to Chemical Engineering Processes/Converting Information into Mass Flows - Wikibooks, open books for an open world
1 Converting Information into Mass Flows - Introduction
2 Volumetric Flow rates
2.1 Why they're useful
2.3 How to convert volumetric flow rates to mass flow rates
3.3 How to convert velocity into mass flow rate
4 Molar Flow Rates
4.3 How to Change from Molar Flow Rate to Mass Flow Rate
Converting Information into Mass Flows - IntroductionEdit
In any system there will be certain parameters that are easier (often considerably) to measure and/or control than others. When you are solving any problem and trying to use a mass balance or any other equation, it is important to recognize what pieces of information can be interconverted. The purpose of this section is to show some of the more common alternative ways that mass flow rates are expressed, mostly because it is easier to, for example, measure a velocity than it is to measure a mass flow rate directly.
Volumetric Flow ratesEdit
A volumetric flow rate is a relation of how much volume of a gas or liquid solution passes through a fixed point in a system (typically the entrance or exit point of a process) in a given amount of time. It is denoted as:
{\displaystyle {\dot {V}}_{n}{\dot {=}}{\frac {Volume}{time}}}
in stream n
Why they're usefulEdit
Volumetric flow rates can be measured directly using flow meters. They are especially useful for gases since the volume of a gas is one of the four properties that are needed in order to use an equation of state (discussed later in the book) to calculate the molar flow rate. Of the other three, two (pressure, and temperature) can be specified by the reactor design and control systems, while one (compressibility) is strictly a function of temperature and pressure for any gas or gaseous mixture.
Volumetric Flowrates are Not Conserved. We can write a balance on volume like anything else, but the "volume generation" term would be a complex function of system properties. Therefore if we are given a volumetric flow rate we should change it into a mass (or mole) flow rate before applying the balance equations.
Volumetric flowrates also do not lend themselves to splitting into components, since when we speak of volumes in practical terms we generally think of the total solution volume, not the partial volume of each component (the latter is a useful tool for thermodynamics, but that's another course entirely). There are some things that are measured in volume fractions, but this is relatively uncommon.
How to convert volumetric flow rates to mass flow ratesEdit
Volumetric flowrates are related to mass flow rates by a relatively easy-to-measure physical property. Since
{\displaystyle {\dot {m}}{\dot {=}}mass/time}
{\displaystyle {\dot {V}}{\dot {=}}volume/time}
, we need a property with units of
{\displaystyle mass/volume}
in order to convert them. The density serves this purpose nicely!
{\displaystyle {\dot {V}}_{n}*{\rho }_{i}={\dot {m}}_{n}}
The "i" indicates that we're talking about one particular flow stream here, since each flow may have a different density, mass flow rate, or volumetric flow rate.
VelocitiesEdit
The velocity of a bulk fluid is how much lateral distance along the system (usually a pipe) it passes per unit time. The velocity of a bulk fluid, like any other, has units of:
{\displaystyle v_{n}={\frac {distance}{time}}}
By definition, the bulk velocity of a fluid is related to the volumetric flow rate by:
{\displaystyle {v}_{n}={\frac {{\dot {V}}_{n}}{A_{n}}}}
This distinguishes it from the velocity of the fluid at a certain point (since fluids flow faster in the center of a pipe). The bulk velocity is about the same as the instantaneous velocity for relatively fast flow, or especially for flow of gasses.
For purposes of this class, all velocities given will be bulk velocities, not instantaneous velocities.
(Bulk) Velocities are useful because, like volumetric flow rates, they are relatively easy to measure. They are especially useful for liquids since they have constant density (and therefore a constant pressure drop at steady state) as they pass through the orifice or other similar instruments. This is a necessary prerequisite to use the design equations for these instruments.
Like volumetric flowrates, velocity is not conserved. Like volumetric flowrate, velocity changes with temperature and pressure of a gas, though for a liquid, velocity is generally constant along the length of a pipe with constant cross-sectional area.
Also, velocities can't be split into the flows of individual components, since all of the components will generally flow at the same speed. They need to be converted into something that can be split (mass flow rate, molar flow rate, or pressure for a gas) before concentrations can be applied.
How to convert velocity into mass flow rateEdit
In order to convert the velocity of a fluid stream into a mass flow rate, you need two pieces of information:
The cross sectional area of the pipe.
In order to convert, first use the definition of bulk velocity to convert it into a volumetric flow rate:
{\displaystyle {\dot {V}}_{n}=v_{n}*A_{n}}
Then use the density to convert the volumetric flow rate into a mass flow rate.
{\displaystyle {\dot {m}}_{n}={\dot {V}}_{n}*{\rho }_{n}}
The combination of these two equations is useful:
{\displaystyle {\dot {m}}_{n}=v_{n}*{\rho }_{n}*A_{n}}
Molar Flow RatesEdit
The concept of a molar flow rate is similar to that of a mass flow rate, it is the number of moles of a solution (or mixture) that pass a fixed point per unit time:
{\displaystyle {\dot {n}}_{n}{\dot {=}}{\frac {moles}{time}}}
Molar flow rates are mostly useful because using moles instead of mass allows you to write material balances in terms of reaction conversion and stoichiometry. In other words, there are a lot fewer unknowns when you use a mole balance, since the stoichiometry allows you to consolidate all of the changes in the reactant and product concentrations in terms of one variable.
Unlike mass, total moles are not conserved. Total mass flow rate is conserved whether there is a reaction or not, but the same is not true for the number of moles. For example, consider the reaction between hydrogen and oxygen gasses to form water:
{\displaystyle H_{2}+{\frac {1}{2}}O_{2}\rightarrow H_{2}O}
This reaction consumes 1.5 moles of reactants for every mole of products produced, and therefore the total number of moles entering the reactor will be more than the number leaving it.
However, since neither mass nor moles of individual components is conserved in a reacting system, it's better to use moles so that the stoichiometry can be exploited, as described later.
The molar flows are also somewhat less practical than mass flow rates, since you can't measure moles directly but you can measure the mass of something, and then convert it to moles using the molar flow rate.
How to Change from Molar Flow Rate to Mass Flow RateEdit
Molar flow rates and mass flow rates are related by the molecular weight (also known as the molar mass) of the solution. In order to convert the mass and molar flow rates of the entire solution, we need to know the average molecular weight of the solution. This can be calculated from the molecular weights and mole fractions of the components using the formula:
{\displaystyle {\bar {MW}}_{n}=[\Sigma ({MW}_{i}*y_{i})]_{n}}
where i is an index of components and n is the stream number.
{\displaystyle y_{i}}
signifies mole fraction of each component (this will all be defined and derived later).
Once this is known it can be used as you would use a molar mass for a single component to find the total molar flow rate.
{\displaystyle {\dot {m}}_{n}={\dot {n}}_{n}*{\bar {MW}}_{n}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Converting_Information_into_Mass_Flows&oldid=3512585"
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N Refine - Esolang
N Refine is an OISC esolang by User:PythonshellDebugwindow.
2 Example: infinite loop
Each N Refine program is a series of unbounded space-separated integers. Each command takes up two of these integers; let's call these integers A and B. If A is 0 (notwithstanding number redefinition), then the program jumps to position
{\displaystyle 2B}
, as each command takes up two integers. If A is nonzero, however, a random number is then picked from
{\displaystyle \{0,1\}}
: if the number is 0, then it redefines the number A to be the number B; if the number is 1, then it redefines the number B to be the number A.
Unknown; numbers can be used for storage, but storage is probabilistic, and all jumps are absolute.
An interpreter on GitHub
Retrieved from "https://esolangs.org/w/index.php?title=N_Refine&oldid=78327"
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Simplified physical explanations of lift on an airfoilEdit
Explanation based on flow deflection and Newton's lawsEdit
Controversy regarding the Coandă effectEdit
Explanations based on an increase in flow speed and Bernoulli's principleEdit
False explanation based on equal transit-timeEdit
Obstruction of the airflowEdit
Issues common to both versions of the Bernoulli-based explanationEdit
Basic attributes of liftEdit
Pressure differencesEdit
{\displaystyle {\frac {\operatorname {d} p}{\operatorname {d} R}}=\rho {\frac {v^{2}}{R}}}
Angle of attackEdit
Airfoil shapeEdit
Flow conditionsEdit
Air speed and densityEdit
Boundary layer and profile dragEdit
StallingEdit
Bluff bodiesEdit
A more comprehensive physical explanationEdit
Lift at the airfoil surfaceEdit
The wider flow around the airfoilEdit
Mutual interaction of pressure differences and changes in flow velocityEdit
How simpler explanations fall shortEdit
Quantifying liftEdit
Pressure integrationEdit
{\displaystyle L=\oint p\mathbf {n} \cdot \mathbf {k} \;\mathrm {d} S,}
{\displaystyle {\begin{aligned}D_{p}&=\oint p\mathbf {n} \cdot \mathbf {i} \;\mathrm {d} S,\\Y&=\oint p\mathbf {n} \cdot \mathbf {j} \;\mathrm {d} S.\end{aligned}}}
Lift coefficientEdit
{\displaystyle C_{L}}
{\displaystyle C_{L}}
{\displaystyle L={\tfrac {1}{2}}\rho v^{2}SC_{L}}
{\displaystyle L}
{\displaystyle \rho }
{\displaystyle v}
{\displaystyle S}
{\displaystyle C_{L}}
Mathematical theories of liftEdit
Navier–Stokes (NS) equationsEdit
Reynolds-averaged Navier–Stokes (RANS) equationsEdit
Inviscid-flow equations (Euler or potential)Edit
Linearized potential flowEdit
Circulation and the Kutta–Joukowski theoremEdit
Three-dimensional flowEdit
Wing tips and spanwise distributionEdit
Horseshoe vortex systemEdit
Manifestations of lift in the farfieldEdit
Integrated force/momentum balance in lifting flowsEdit
{\displaystyle -L'}
{\displaystyle -L'}
Lift reacted by overpressure on the ground under an airplaneEdit
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Inductor Ferrites » Electronics Notes - Miscellaneous - 2022
Ferrites are one of the main core materials used in inductors and transformers.
Inductor ferrite is used to provide an increase in the permeability of the medium around the coil to increase the inductance of the inductor.
Ferrites are widely used within inductor technology to improve the performance of the inductor.
Ferrites are basically iron based magnetic material in the form of a ceramic.
Ferrites are made from a powder and can therefore the ferrite cores used in inductors and other applications can be manufactured in a variety of shapes according to the requirements.
Ferrites, or as they are also known ferromagnetic materials can be classified into two categories based on their magnetic coercivity, or persistence of internal magnetism:
Soft ferrites: Soft ferrites are ferrite materials that are able to easily reverse their polarity of their magnetisation without a significant amount of energy being needed to reverse the magnetic polarity. This means that there is only a relatively small loss of energy.
Soft ferrites also have a high electrical resistance and therefore, when used in inductors and transformers eddy current losses are also low.
Soft ferrites are often made from a blend of iron, nickel, zinc or manganese oxides. Manganese-zinc and nickel-zinc magnets are the most common of the soft ferrite magnets. As a result of their high resistance, soft ferrites are widely used in the cores of inductors or transformers because they result in minimal energy loss.
Generally soft ferrites are accepted as those having a coercivity of less than 1 kA.m.
Hard ferrites: Hard ferrites may also be called permanent magnets. They retain the polarity of their magnetisation once the magnetising field has been removed, i.e. they have a high remanance level.
Hard ferrite magnets are typically made of barium, iron or strontium oxides. They are cheap to produce and are the magnets that are used in a wide number of applications, but may be most commonly seen in such applications as standard household magnets (e.g., kitchen magnets).
Generally hard ferrites are considered to be those with coercivity levels of greater than 10 kA/m.
Ferrites are generally chemically inert ceramic iron-based materials. They generally have the chemical structure of the format XFe2O4, where X is a transition material.
Transition Metals used in Ferrites
To manufacture the ferrites used in inductors and other applications, the powers of the metals are mixed in proportions and then milled to give the required grain size and then pressed into shape.
Sintering entails heating the material to between about 1150°C and 1300°C.
Sintering is a process where a powdered ceramic material is held in a mold to give it the required shape and then heated to a temperature which is below the material melting point. It is found that the atoms in the powder particles diffuse across the particle boundaries, so that the particles become fused together. In this way a single solid item is created.
The sintered core of the inductor ferrite may still require further finishing - it may be ground to provide a very flat surface for situations where mating halves of a core are required. Here flat surfaces are essential to ensure that air gaps in inductors or transformers, etc., are as small as possible.
The finished ferrite material contains thousands of small crystals or grains. Typically these are around 10µm across. Within each grain or crystal there are many more smaller magnetic domains that can will have a random orientation after heating. With the application of an external field, these domains will tend to orientate in the same direction.
Ferrite permeability
There are many parameters that are of importance when a ferrite is used within an inductor. However the chief parameter for an inductor ferrite is the permeability. The level of permeability of the inductor ferrite enables the inductor to have a much greater inductance than it would if only an air core were used.
The permeability of ferrites used within inductors varies considerably between different types of ferrite. They can have permeability levels that may range between 20 to more than 15,000, although some very specialised ones may be higher.
Inductor ferrite core losses
One major parameter of interest to the electronic engineer using ferrites in inductors is the core losses they exhibit and their frequency dependence.
The core losses of a ferrite core can be expressed in the following manner:
{P}_{c} ={P}_{h} + {P}_{e} + {P}_{r}
Pc = total core loss
Ph = hysteresis loss
Pr = residual loss
It is found that he hysteresis loss increases linearly with increasing frequency and flux. The eddy current loss increases exponentially with increasing frequency and flux. However it is found that the hysteresis loss is the dominant core loss up to a frequency determined by the performance of the core. Above this the eddy current loss predominates.
To improve high frequency performance the grain size used in the preparation of the ferrite used for the inductor must be small, and also the mixture used must be free from impurities.
Watch the video: Transformer winding and calculation in detail (May 2022).
In-Circuit Test ICT Design for Test Guidelines
Copyright 2022 \ Inductor Ferrites ...
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Classic Tutorial - vantage6
In this section the basic steps for creating an algorithm for horizontal partitioned data are explained.
The final code of this tutorial is published on Github. The algorithm is also published in our Docker registry: harbor2.vantage6.ai/demo/average
It is assumed that it is mathematically possible to create a federated version of the algorithm you want to use. In the following sections we create a federated algorithm to compute the average of a distributed dataset. An overview of the steps that we are going through:
Mathematically decompose the model
Federated implementation and local testing
Vantage6 algorithm wrapper
Dockerize and push to a registry
This tutorial shows you how to create a federated mean algorithm.
Mathematical decomposition
Q = [q_1, q_2 ... q_n]
Q_{mean} = \frac{1}{n} \sum \limits_{i=1}^{n} {q_i} = \frac{q_1 + q_2 + ... + q_n}{n}
When dataset
Q
is horizontally partitioned in dataset
A = [a_1, a_2 ... a_j] = [q_1, q_2 ... q_j]
B = [b_{1}, b_{2} ... b_k] = [q_{j+1}, q_{j+2}...q_{n}]
, we would like to compute
Q_{mean}
from dataset A and B. This could be computed as:
Q_{mean} = \frac{(a_1+a_2+...+a_j) + (b_1+b_2+...+b_k)}{j+k} = \frac{\sum A + \sum B }{j+k}
Both the number of samples in each dataset and the total sum of each dataset is needed. Then we can compute the global average of dataset
A
B
We cannot simply compute the average on each node and combine them, as this would be mathematically incorrect. This would only work if dataset A and B contain the exact same number of samples.
Federated implementation
In this example we use python, however you are free to use any language. The only requirements are: 1) It has to be able to create HTTP-requests, and 2) has to be able to read and write to files.
However, if you use a different language you are not able to use our wrapper. Reach out to us on Discord to discuss how this works.
A federated algorithm consist of two parts:
A federated part of the algorithm which is responsible for creating the partial results. In our case this would be computing (1) the sum of the observations, and (2) the number of observations.
A central part of the algorithm which is responsible for combining the partial results from the nodes. In the case of the federated mean that would be dividing the total sum of the observations by the total number of observations.
The central part of the algorithm can either be run on the machine of the researcher himself or in a master container which runs on a node. The latter is the preferred method.
In case the researcher runs this part, he/she needs to have a proper setup to do so (i.e. Python 3.5+ and the necessary dependencies). This can be useful when developing new algorithms.
1 Federated part
The node that runs this part contains a CSV-file with one column (specified by the argument column_name) which we want to use to compute the global mean. We assume that this column has no NaN values.
def federated_part(path, column_name="numbers"):
"""Compute the sum and number of observations of a column"""
# extract the column numbers from the CSV
numbers = pandas.read_csv(path)[column_name]
# compute the sum, and count number of rows
local_sum = numbers.sum()
local_count = len(numbers)
# return the values as a dict
"sum": local_sum,
"count": local_count
2 Central part
The central algorithm receives the sums and counts from all sites and combines these to a global mean. This could be from one or more sites.
def central_part(node_outputs):
"""Combine the partial results to a global average"""
global_sum = 0
for output in node_outputs:
global_sum += output["sum"]
global_count += output["count"]
return {"average": global_sum / global_count}
To test, simply create two datasets A and B, both having a numerical column numbers. Then run the following:
federated_part("path/to/dataset/A"),
federated_part("path/to/dataset/B")
Q_average = central_part(outputs)["average"]
print(f"global average = {Q_average}.")
Vantage6 integration
A good starting point would be to use the boilerplate code from our Github. This section outlines the steps needed to get to this boilerplate but also provides some background information.
In this example we use a csv-file. It is also possible to use other types of data-sources. This tutorial makes use of our algorithm wrapper which is currently only available for csv and SPARQL.
Other wrappers like SQL, OMOP, etc. are under consideration. Let us now if you want to use one of these or other data-sources.
Now that we have a federated implementation of our algorithm we need to make it compatible with the vantage6 infrastructure. The infrastructure handles the communication with the server and provides data access to the algorithm.
The algorithm consumes a file containing the input. This contains both the method name to be triggered as well as the arguments provided to the method. The algorithm also has access to a CSV file (in the future this could also be a database) on which the algorithm can run. When the algorithm is finished, it writes back the output to a different file.
The central part of the algorithm has to be able to create (sub)tasks. These subtasks are responsible for executing the federated part of the algorithm. The central part of the algorithm can either be executed on one of the nodes in the vantage6 network or on the machine of a researcher. In this example we only show the case in which one of the nodes executes the central part of the algorithm. The node provides the algorithm with a JWT token so that the central part of the algorithm has access to the server to post these subtasks.
📂Algorithm Structure
The algorithm needs to be structured as a Python package. This way the algorithm can be installed within the Docker image. The minimal file-structure would be:
└── algorithm_pkg
We also recommend adding a README.md, LICENSE and requirements.txt to the project_folder.
Contains the setup method to create a package from your algorithm code. Here you specify some details about your package and the dependencies it requires.
# we're using a README.md, if you do not have this in your folder, simply
# replace this with a string.
# Here you specify the meta-data of your package. The `name` argument is
# needed in some other steps.
name='v6-average-py',
description='vantage6 average',
url='https://github.com/IKNL/v6-average-py',
'vantage6-client',
# list your dependencies here:
# pandas, ...
The setup.py above is sufficient in most cases. However if you want to do more advanced stuff (like adding static data, or a CLI) you can use the extra arguments from setup.
Contains the recipe for building the Docker image. Typically you only have to change the argument PKG_NAME to the name of you package. This name should be the same as as the name you specified in the setup.py. In our case that would be v6-average-py.
# This specifies our base image. This base image contains some commonly used
# dependancies and an install from all vantage6 packages. You can specify a
# different image here (e.g. python:3). In that case it is important that
# `vantage6-client` is a dependancy of you project as this contains the wrapper
# we are using in this example.
FROM harbor.vantage6.ai/algorithms/algorithm-base
# Change this to the package name of your project. This needs to be the same
# as what you specified for the name in the `setup.py`.
ARG PKG_NAME="v6-average-py"
# This will install your algorithm into this image.
# This will run your algorithm when the Docker container is started. The
# wrapper takes care of the IO handling (communication between node and
# algorithm). You dont need to change anything here.
ENV PKG_NAME=${PKG_NAME}
CMD python -c "from vantage6.tools.docker_wrapper import docker_wrapper; docker_wrapper('${PKG_NAME}')"
This contains the code for your algorithm. It is possible to split this into multiple files, however the methods that should be available to the researcher should be in this file. You can do that by simply importing them into this file (e.g. from .average import my_nested_method)
We can distinguish two types of methods that a user can trigger:
Central part of the algorithm. Receives a client as argument which provides an interface to the central server. This way the master can create tasks and collect their results.
(client, data, *args, **kwargs)
Consumes the data at the node to compute the partial.
RPC_
(data, *args, **kwargs)
The client the master method receives is a ContainerClient which is different than the client you use as a user.
Everything that is returned by thereturn statement is sent back to the central vantage6-server. This should never contain any privacy-sensitive information.
For our average algorithm the implementation will look as follows:
from vantage6.tools.util import info
def master(client, data, column_name):
"""Combine partials to global model
First we collect the parties that participate in the collaboration.
Then we send a task to all the parties to compute their partial (the
row count and the column sum). Then we wait for the results to be
ready. Finally when the results are ready, we combine them to a
Note that the master method also receives the (local) data of the
node. In most usecases this data argument is not used.
The client, provided in the first argument, gives an interface to
the central server. This is needed to create tasks (for the partial
results) and collect their results later on. Note that this client
is a different client than the client you use as a user.
# Info messages can help you when an algorithm crashes. These info
# messages are stored in a log file which is send to the server when
# either a task finished or crashes.
info('Collecting participating organizations')
# Collect all organization that participate in this collaboration.
# These organizations will receive the task to compute the partial.
organizations = client.get_organizations_in_my_collaboration()
ids = [organization.get("id") for organization in organizations]
# Request all participating parties to compute their partial. This
# will create a new task at the central server for them to pick up.
# We've used a kwarg but is is also possible to use `args`. Although
# we prefer kwargs as it is clearer.
info('Requesting partial computation')
task = client.create_new_task(
input_={
'method': 'average_partial',
'column_name': column_name
organization_ids=ids
# Now we need to wait untill all organizations(/nodes) finished
# their partial. We do this by polling the server for results. It is
# also possible to subscribe to a websocket channel to get status
info("Waiting for results")
task_id = task.get("id")
task = client.get_task(task_id)
while not task.get("complete"):
# Once we now the partials are complete, we can collect them.
info("Obtaining results")
results = client.get_results(task_id=task.get("id"))
# Now we can combine the partials to a global average.
global_sum += result["sum"]
global_count += result["count"]
def RPC_average_partial(data, column_name):
"""Compute the average partial
The data argument contains a pandas-dataframe containing the local
data from the node.
# extract the column_name from the dataframe.
info(f'Extracting column {column_name}')
numbers = data[column_name]
info('Computing partials')
Now that we have a vantage6 implementation of the algorithm it is time to test it. Before we run it in a vantage6 setup we can test it locally by using the ClientMockProtocol which simulates the communication with the central server.
Before we can locally test it we need to (editable) install the algorithm package so that the Mock client can use it. Simply go to the root directory of your algorithm package (with the setup.py file) and run the following:
Then create a script to test the algorithm:
from vantage6.tools.mock_client import ClientMockProtocol
# Initialize the mock server. The datasets simulate the local datasets from
# the node. In this case we have two parties having two different datasets:
# a.csv and b.csv. The module name needs to be the name of your algorithm
# package. This is the name you specified in `setup.py`, in our case that
# would be v6-average-py.
client = ClientMockProtocol(
datasets=["local/a.csv", "local/b.csv"],
module="v6-average-py"
# to inspect which organization are in your mock client, you can run the
org_ids = ids = [organization["id"] for organization in organizations]
# we can either test a RPC method or the master method (which will trigger the
# RPC methods also). Lets start by triggering an RPC method and see if that
# works. Note that we do *not* specify the RPC_ prefix for the method! In this
# example we assume that both a.csv and b.csv contain a numerical column `age`.
average_partial_task = client.create_new_task(
'method':'average_partial',
'column_name': 'age'
organization_ids=org_ids
# You can directly obtain the result (we dont have to wait for nodes to
# complete the tasks)
results = client.get_results(average_partial_task.get("id"))
# To trigger the master method you also need to supply the `master`-flag
# to the input. Also note that we only supply the task to a single organization
# as we only want to execute the central part of the algorithm once. The master
# task takes care of the distribution to the other parties.
average_task = client.create_new_task(
'method':'master',
organization_ids=[org_ids[0]]
results = client.get_results(average_task.get("id"))
Now that we have a fully tested algorithm for the vantage6 infrastructure. We need to package it so that it can be distributed to the data-stations/nodes. Algorithms are delivered in Docker images. So that's where we need the Dockerfile for. To build an image from our algorithm (make sure you have docker installed and it's running) you can run the following command from the root directory of your algorithm project.
docker build -t harbor2.vantage6.ai/demo/average .
The option -t specifies the (unique) identifier used by the researcher to use this algorithm. Usually this includes the registry address (harbor2.vantage6.ai) and the project name (demo).
In case you are using docker hub as registry, you do not have to specify the registry or project as these are set by default to the Docker hub and your docker hub username.
docker push harbor2.vantage6.ai/demo/average
Reach out to us on Discord if you want to use our registries (harbor.vantage6.ai and harbor2.vantage6.ai).
Cross-language serialization
It is possible that a vantage6 algorithm is developed in one programming language, but you would like to run the task from another language. For these use-cases, the Python algorithm wrapper and client support cross-language serialization. By default, input to the algorithms and output back to the client are serialized using pickle. However, it is possible to define a different serialization format.
Input and output serialization can be specified as follows:
client.post_task(
name='mytask',
image='harbor2.vantage6.ai/testing/v6-test-py',
collaboration_id=COLLABORATION_ID,
organization_ids=ORGANIZATION_IDS,
data_format='json', # Specify input format to the algorithm
'method': 'column_names',
'kwargs': {'data_format': 'json'}, # Specify output format
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EVII (or E7(-25)), which has maximal compact subgroup SO(2)·E6/(center), infinite cyclic fundamental group and outer automorphism group of order 2.
u = (1, φ, 0, -1, φ, 0,0)
v = (φ, 0, 1, φ, 0, -1,0)
w = (0, 1, φ, 0, -1, φ,0)
1) 2 points at the origin
2) 2 icosahedrons
3) 1 icosadodecahedron
4) 2 dodecahedrons
for a total of 126 vertices.
The roots are all the 8×7 permutations of (1,−1,0,0,0,0,0,0) and all the
{\displaystyle {\begin{pmatrix}8\\4\end{pmatrix}}}
permutations of (½,½,½,½,−½,−½,−½,−½)
They are listed so that their corresponding nodes in the Dynkin diagram are ordered from left to right (in the diagram depicted above) with the side node last.
An alternative descriptionEdit
{\displaystyle 4\times {\begin{pmatrix}6\\2\end{pmatrix}}}
permutations of (±1,±1,0,0,0,0,0) preserving the zero at the last entry, all of the following roots with an even number of +½
{\displaystyle \left(\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over 2},\pm {1 \over {\sqrt {2}}}\right)}
{\displaystyle \left(0,0,0,0,0,0,\pm {\sqrt {2}}\right).}
Thus the generators consist of a 66-dimensional so(12) subalgebra as well as 64 generators that transform as two self-conjugate Weyl spinors of spin(12) of opposite chirality, and their chirality generator, and two other generators of chiralities
{\displaystyle \pm {\sqrt {2}}}
{\displaystyle {\begin{bmatrix}1&-1&0&0&0&0&0\\0&1&-1&0&0&0&0\\0&0&1&-1&0&0&0\\0&0&0&1&-1&0&0\\0&0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}\\0&0&0&0&1&-1&0\\\end{bmatrix}}.}
{\displaystyle {\begin{bmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&0\\0&0&-1&2&-1&0&-1\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&0\\0&0&0&-1&0&0&2\end{bmatrix}}.}
Important subalgebras and representationsEdit
E7 Polynomial InvariantsEdit
{\displaystyle C_{1}=pq-qp+Tr[PQ]-Tr[QP]}
{\displaystyle C_{2}=(pq+Tr[P\circ Q])^{2}+pTr[Q\circ {\tilde {Q}}]+qTr[P\circ {\tilde {P}}]+Tr[{\tilde {P}}\circ {\tilde {Q}}]}
{\displaystyle {\tilde {P}}\equiv \det(P)P^{-1}}
and the binary circle operator is defined by
{\displaystyle A\circ B=(AB+BA)/2}
{\displaystyle C_{2}=Tr[(XY)^{2}]-{\dfrac {1}{4}}Tr[XY]^{2}+{\frac {1}{96}}\epsilon _{ijklmnop}\left(X^{ij}X^{kl}X^{mn}X^{op}+Y^{ij}Y^{kl}Y^{mn}Y^{op}\right)}
{\displaystyle {\frac {1}{\mathrm {gcd} (2,q-1)}}q^{63}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^{8}-1)(q^{6}-1)(q^{2}-1)}
Importance in physicsEdit
In string theory, E7 appears as a part of the gauge group of one of the (unstable and non-supersymmetric) versions of the heterotic string. It can also appear in the unbroken gauge group E8 × E7 in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.
^ Platonov, Vladimir; Rapinchuk, Andrei (1994) [1991], Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Boston, MA: Academic Press, ISBN 978-0-12-558180-6, MR 1278263 (original version: Платонов, Владимир П.; Рапинчук, Андрей С. (1991). Алгебраические группы и теория чисел. Наука. ISBN 5-02-014191-7. ), §2.2.4
^ Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985). Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press. p. 46. ISBN 0-19-853199-0.
^ Carter, Roger W. (1989). Simple Groups of Lie Type. Wiley Classics Library. John Wiley & Sons. ISBN 0-471-50683-4.
^ Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics. Vol. 251. Springer-Verlag. ISBN 1-84800-987-9.
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Dimensional Analysis - Bobbie's Blog
Once one of my engineering friends asked me: “Why would you need all these dimensionless numbers? There are so many of them and they just complicate stuff!”
If something is not useful, it wouldn’t have been invented in science; this is particularly true for dimensional analysis which has made great success across all fields of science.
To put things simple, dimensional analysis has two major fundamental applications:
Simplify equations and lead to new scientific branches/specialties.
Guide scientific experiments with dynamic similitude.
Simplification of the Navier-Stokes equation
Look at the Navier-Stokes equation
At a particular point in a flow, We may find that some term in this equation is orders of magnitude bigger than others. We could simplify the N-S equation by retaining those terms that matter, and eliminating those that don’t.
How do we go about to eliminate the negligible terms? To put the conclusion first, the magnitudes of each term in the N-S equation depend on both the sturcture of the flow and location in the flow – with appropriate flow structure and at appropriate location, one can safely delete certain terms in the N-S equation. This whole process of simplfying the equations, is called dimensional analysis.
There are three steps in performing dimensional analysis.
Identify characteristic scales of a flow.
characteristic length $L$ is related to the size of the boundaries
characteristic velocity $U$ is determined by the particular mechanism driving the flow
characteristic time $T$ is either imposed by external means or simply defined as $T=L/U$.
In the case of unidirectional flow through channel: $U = $ max velocity across channel.
In the case of uniform flow past stationary body: $U = $ velocity of the incident flow, $L =$ diameter of the body.
In the case of forced oscillatory flow: $T = $ period of oscillation.
Nondimensionalization:
Rescale each variable in the equation into a dimensionless variable.
\hat{\mathbf{u}}:=\frac{\mathbf{u}}{U},\qquad \hat{\mathbf{x}}:=\frac{\mathbf{x}}{L},\qquad \hat{t}:=\frac{t}{T},\qquad \hat{p}:=\frac{pL}{\mu U}.
Rewrite the equation into a dimensionless form.
\beta \frac{\partial \hat{\mathbf{u}}}{\partial \hat{t}} + \mbox{Re}\,\hat{\mathbf{u}}\cdot\hat{\nabla}\hat{\mathbf{u}} = -\hat{\nabla} \hat{p} + \hat{\nabla}^2 \hat{\mathbf{u}} + \frac{\mbox{Re}}{\mbox{Fr}^2}\frac{\mathbf{g}}{g}
The multiplication factors of each term are nondimensional groups, called dimensionless numbers
Frequency parameter: $\beta:=\frac{L^2}{\nu T}$, expresses the relative magnitudes of the inertial acceleration force and the viscous force
Reynolds number: $\mbox{Re}:=\frac{UL}{\nu}$, expresses the relative magnitudes of the inertia convective force and the viscous force
Froude number: $\mbox{Fr}:=\frac{U}{\sqrt{gL}}$ expresses the relative magnitudes of the inertial convective force and the body force
The group $\frac{\mbox{Re}}{\mbox{Fr}^2}=\frac{gL^2}{\nu U}$, expresses the relative magnitudes of the body force and the viscous force
Eliminate the dominated terms.
If $\mbox{Re}\ll 1$ and $\beta\ll1$, the dominant terms give Stokes equation
-\nabla p + \mu \nabla^2\mathbf{u}+ \rho\mathbf{g}=\mathbf{0}
where the inertia terms go away, resulting in a quasi-steady fluid.
If $\mbox{Re}\gg 1$ and $\beta\gg1$, and rescale the pressure by $\hat{p}=\frac{p}{\rho U^2}$ instead, the dominant terms give Euler’s equation
\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \rho\mathbf{g}
where the viscous term goes away, making it behave like an inviscid fluid.
Dynamic similitude
Pozrikidis C. (2011). Introduction to Theoretical and Computational Fluid Dynamics (2nd ed.). Oxford University Press.
Posted by 波儿比 Bobbie 08-11-16 16:00 fluid-dynamics, pde
« 写数学 Blog 不要计算 »
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(Redirected from Fundamental interactions)
{\displaystyle {\frac {1}{r}}\ e^{-m_{\rm {W,Z}}\ r}}
{\displaystyle {\sim r}}
{\displaystyle {\frac {1}{r}}}
{\displaystyle {\frac {1}{r}}}
{\displaystyle 4000\ {\mbox{g}}\,{\rm {{H}_{2}{\rm {{O}\cdot {\frac {1\ {\mbox{mol}}\,{\rm {{H}_{2}{\rm {O}}}}}{18\ {\mbox{g}}\,H_{2}O}}\cdot {\frac {10\ {\mbox{mol}}\,e^{-}}{1\ {\mbox{mol}}\,H_{2}O}}\cdot {\frac {96,000\ {\mbox{C}}\,}{1\ {\mbox{mol}}\,e^{-}}}=2.1\times 10^{8}C\ \,\ }}}}}
{\displaystyle {1 \over 4\pi \varepsilon _{0}}{\frac {(2.1\times 10^{8}\mathrm {C} )^{2}}{(1m)^{2}}}=4.1\times 10^{26}\mathrm {N} .}
{\displaystyle {\frac {\lambda _{i}}{\sqrt {2}}}{\bar {\psi }}\phi '\psi ={\frac {m_{i}}{\nu }}{\bar {\psi }}\phi '\psi }
{\displaystyle \lambda _{i}}
{\displaystyle m_{i}}
{\displaystyle V(r)=-{\frac {m_{i}m_{j}}{m_{\rm {H}}^{2}}}{\frac {1}{4\pi r}}e^{-m_{\rm {H}}\,c\,r/\hbar }}
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Default Risk Premium - Course Hero
Introduction to Finance/Interest Rates/Default Risk Premium
Bond securities are created by a company that is issuing the bond as a form of loan. In essence, the investor is loaning the company money when the investor purchases a bond. Assume that an investor is going to lend $100,000. The borrower already has $1 million in the bank and has always paid back any loans. The investor is reasonably assured that the loan will be paid. Thus, the lender charges a small interest rate. Conversely, if the borrower had no money in the bank and had not repaid loans in the past, then the lender would want a higher return to compensate for the associated risk of default. This investor, like all investors, is risk averse, meaning that the investor expects compensation for increased risk. The danger that the borrower will not repay the loan will earn the investor a default risk premium. Default risk premium is an additional fee that a borrower is charged to compensate for the possibility that the borrower might be unable to pay back the loan.
Because commercial bonds are generated from the debt of the issuing company, they carry a risk of default as well. These types of bonds are rated by their credit in order to help identify the investment risk. The highest grade that a bond can have is AAA. The lowest grades are C and D. A bond with a low rating can be called a junk bond, or a high-yield bond. A high-yield bond is a bond with a low credit rating for which investors can expect a high return on investment. The risk of default is high and the matching default risk premium is also high. The bonds are rated by an outside agency; Moody's and the S&P 500 are two of the most common rating agencies. Because of the risk of default in low-rated bonds, some investor organizations, such as governmental agencies, are forbidden to invest in them.
Default Risk on Commercial Bonds
The change in default risk will shift the demand for the bonds from D1 to D2, decreasing the price for corporate bonds and increasing the price for Treasury bonds.
How to Calculate Default Risk Premium
Because U.S. Treasury securities, such as Treasury bills, Treasury bonds, and Treasury notes, are assumed to have no default risk, commercial securities are often compared to them by adding the Treasury bond rate to the default risk premium. Default risk is the level of risk investors and lenders may incur if a borrower defaults on a loan. The default risk premium, or the fee a borrower is charged to compensate for the chance they might be unable to pay back the loan, can be calculated by working backward from the risk-free rate. The risk-free interest rate is the expected rate of return with no risks or financial losses, often estimated from U.S. bonds and treasuries. At the risk-free level, Treasury securities and commercial securities will have the same market rate, the rate of interest that a free market can bear.
To calculate the risk-free interest rate, the real rate of interest is added to the inflation premium. The real rate of interest is the rate of interest an investor receives, or expects to receive, after allowing for inflation. The inflation premium is the interest rate that results from lenders compensating for expected inflation. For example, if a 10-year Treasury bond yields two percent, investors would consider two percent to be the risk-free rate of return.
\text{Risk-Free Interest Rate}=\text{Real Rate of Interest}+\text{Inflation Premium}
The commercial market interest rate has the same calculation plus the default risk premium, so the default risk premium can be found by subtracting the two. The default risk premium is the amount over and above the rate for Treasury bonds that the investor would like to earn.
\text{Default Risk Premium}=\text{Market Interest Rate}-\text{Risk-free Interest Rate}
If the market interest rate for a commercial security is 7 percent and the equivalent risk-free security has a rate of 3 percent, then the default risk premium is 4 percent, or
7\%-3\%=4\%
. To get a real default risk premium, subtract the inflation from this number.
Logic dictates that the default risk premium would be solely calculated based on the bond credit rating of the company. Research has shown this concept is difficult to put into practice. The governance of the issuing company seems to have an effect on the bond credit rating and on the bond yield, but a one-to-one formulaic calculation cannot be made to take advantage of this predictive factor. Likewise, bond ratings seem to be linked to negative stock prices, but increases in stock prices do not seem to increase bond ratings. Again, though this is a predictive factor, it is difficult to use it to calculate a default risk premium.
When an investor is presented with two equally attractive investments, the investor will take the one with the least risk. In this way, investors are risk averse. All of the risk premiums discussed are a response to the natural risk aversion of investors. Risk premiums offer a monetary incentive for taking on the risk associated with the investment. Commercial bonds are often compared to Treasury bonds, government-issued bonds with a maturity date over 10 years, because investors are averse to risk. All things being equal, investors would always take Treasury bonds, so premiums are paid on commercial bonds to make them more attractive.
Risk aversion changes when discussing portfolio investments instead of single-security investments. Portfolios are groupings of investments that balance out the risks associated with any single investment. Portfolios are used to balance risk with return. The risk-aversion coefficient was developed to quantify the investor's risk aversion within a portfolio of investments. This coefficient and the aggregate variance of the portfolio components are used to get an overall utility rating for the portfolio that can be compared to the going rates for risk-free securities.
It is important to note that risk aversion is different from risk tolerance. All rational investors are risk averse, but risk tolerance differs from investor to investor. Risk tolerance is essentially the amount of risk that an investor is willing to take on. For example, younger investors with years of investing ahead of them have a higher tolerance for risk than do older investors. Risk tolerance requires that the investor look at the various forms of risk and identify which are acceptable and which are not.
<Maturity in Debt Securities
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Longformer self attention employs self attention on both a “local” context and a “global” context. Most tokens only attend “locally” to each other meaning that each token attends to its
\frac{1}{2} w
previous tokens and
\frac{1}{2} w
succeding tokens with
w
being the window length as defined in config.attention_window. Note that config.attention_window can be of type List to define a different
w
for each layer. A selected few tokens attend “globally” to all other tokens, as it is conventionally done for all tokens in BertSelfAttention.
Note that “locally” and “globally” attending tokens are projected by different query, key and value matrices. Also note that every “locally” attending token not only attends to tokens within its window
, but also to all “globally” attending tokens so that global attention is symmetric.
Using Longformer self attention, the memory and time complexity of the query-key matmul operation, which usually represents the memory and time bottleneck, can be reduced from
\mathcal{O}(n_s \times n_s)
\mathcal{O}(n_s \times w)
n_s
w
being the average window size. It is assumed that the number of “globally” attending tokens is insignificant as compared to the number of “locally” attending tokens.
input_ids = tokenizer.encode('This is a sentence from [MASK] training data', return_tensors='pt')
mlm_labels = tokenizer.encode('This is a sentence from the training data', return_tensors='pt')
class transformers.LongformerConfig < source >
This is the configuration class to store the configuration of a LongformerModel. It is used to instantiate an Longformer model according to the specified arguments, defining the model architecture. Instantiating a configuration with the defaults will yield a similar configuration to that of the RoBERTa roberta-base architecture with a sequence length 4,096.
class transformers.LongformerTokenizer < source >
class transformers.LongformerTokenizerFast < source >
class transformers.models.longformer.modeling_longformer.LongformerBaseModelOutput < source >
class transformers.models.longformer.modeling_longformer.LongformerBaseModelOutputWithPooling < source >
class transformers.models.longformer.modeling_longformer.LongformerMaskedLMOutput < source >
class transformers.models.longformer.modeling_longformer.LongformerQuestionAnsweringModelOutput < source >
class transformers.models.longformer.modeling_longformer.LongformerSequenceClassifierOutput < source >
class transformers.models.longformer.modeling_longformer.LongformerMultipleChoiceModelOutput < source >
class transformers.models.longformer.modeling_longformer.LongformerTokenClassifierOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerBaseModelOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerBaseModelOutputWithPooling < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerMaskedLMOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerQuestionAnsweringModelOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerSequenceClassifierOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerMultipleChoiceModelOutput < source >
class transformers.models.longformer.modeling_tf_longformer.TFLongformerTokenClassifierOutput < source >
class transformers.LongformerModel < source >
( input_ids = None attention_mask = None global_attention_mask = None head_mask = None token_type_ids = None position_ids = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerBaseModelOutputWithPooling or tuple(torch.FloatTensor)
>>> model = LongformerModel.from_pretrained('allenai/longformer-base-4096')
>>> tokenizer = LongformerTokenizer.from_pretrained('allenai/longformer-base-4096')
>>> SAMPLE_TEXT = ' '.join(['Hello world! '] * 1000) # long input document
>>> attention_mask = torch.ones(input_ids.shape, dtype=torch.long, device=input_ids.device) # initialize to local attention
>>> global_attention_mask = torch.zeros(input_ids.shape, dtype=torch.long, device=input_ids.device) # initialize to global attention to be deactivated for all tokens
>>> global_attention_mask[:, [1, 4, 21,]] = 1 # Set global attention to random tokens for the sake of this example
... # Usually, set global attention based on the task. For example,
... # classification: the <s> token
... # QA: question tokens
... # LM: potentially on the beginning of sentences and paragraphs
class transformers.LongformerForMaskedLM < source >
( input_ids = None attention_mask = None global_attention_mask = None head_mask = None token_type_ids = None position_ids = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerMaskedLMOutput or tuple(torch.FloatTensor)
>>> from transformers import LongformerForMaskedLM, LongformerTokenizer
>>> model = LongformerForMaskedLM.from_pretrained('allenai/longformer-base-4096')
>>> attention_mask = None # default is local attention everywhere, which is a good choice for MaskedLM
... # check `LongformerModel.forward` for more details how to set *attention_mask*
>>> outputs = model(input_ids, attention_mask=attention_mask, labels=input_ids)
class transformers.LongformerForSequenceClassification < source >
( input_ids = None attention_mask = None global_attention_mask = None head_mask = None token_type_ids = None position_ids = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerSequenceClassifierOutput or tuple(torch.FloatTensor)
>>> model = LongformerForSequenceClassification.from_pretrained('allenai/longformer-base-4096')
>>> model = LongformerForSequenceClassification.from_pretrained('allenai/longformer-base-4096', problem_type="multi_label_classification")
class transformers.LongformerForMultipleChoice < source >
( input_ids = None token_type_ids = None attention_mask = None global_attention_mask = None head_mask = None labels = None position_ids = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerMultipleChoiceModelOutput or tuple(torch.FloatTensor)
>>> model = LongformerForMultipleChoice.from_pretrained('allenai/longformer-base-4096')
class transformers.LongformerForTokenClassification < source >
( input_ids = None attention_mask = None global_attention_mask = None head_mask = None token_type_ids = None position_ids = None inputs_embeds = None labels = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerTokenClassifierOutput or tuple(torch.FloatTensor)
>>> model = LongformerForTokenClassification.from_pretrained('allenai/longformer-base-4096')
class transformers.LongformerForQuestionAnswering < source >
( input_ids = None attention_mask = None global_attention_mask = None head_mask = None token_type_ids = None position_ids = None inputs_embeds = None start_positions = None end_positions = None output_attentions = None output_hidden_states = None return_dict = None ) → transformers.models.longformer.modeling_longformer.LongformerQuestionAnsweringModelOutput or tuple(torch.FloatTensor)
>>> answer_tokens = all_tokens[torch.argmax(start_logits) :torch.argmax(end_logits)+1]
>>> answer = tokenizer.decode(tokenizer.convert_tokens_to_ids(answer_tokens)) # remove space prepending space token
class transformers.TFLongformerModel < source >
( input_ids = None attention_mask = None head_mask = None global_attention_mask = None token_type_ids = None position_ids = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None training = False **kwargs )
head_mask (tf.Tensor of shape (encoder_layers, encoder_attention_heads), optional) — Mask to nullify selected heads of the attention modules. Mask values selected in [0, 1]:
global_attention_mask (tf.Tensor of shape (batch_size, sequence_length), optional) — Mask to decide the attention given on each token, local attention or global attention. Tokens with global attention attends to all other tokens, and all other tokens attend to them. This is important for task-specific finetuning because it makes the model more flexible at representing the task. For example, for classification, the token should be given global attention. For QA, all question tokens should also have global attention. Please refer to the Longformer paper for more details. Mask values selected in [0, 1]:
position_ids (tf.Tensor of shape (batch_size, sequence_length), optional) — Indices of positions of each input sequence tokens in the position embeddings. Selected in the range [0, config.max_position_embeddings - 1].
class transformers.TFLongformerForMaskedLM < source >
( input_ids = None attention_mask = None head_mask = None global_attention_mask = None token_type_ids = None position_ids = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None labels = None training = False **kwargs ) → transformers.models.longformer.modeling_tf_longformer.TFLongformerMaskedLMOutput or tuple(tf.Tensor)
>>> model = TFLongformerForMaskedLM.from_pretrained('allenai/longformer-base-4096')
class transformers.TFLongformerForQuestionAnswering < source >
( input_ids = None attention_mask = None head_mask = None global_attention_mask = None token_type_ids = None position_ids = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None start_positions = None end_positions = None training = False **kwargs ) → transformers.models.longformer.modeling_tf_longformer.TFLongformerQuestionAnsweringModelOutput or tuple(tf.Tensor)
>>> tokenizer = LongformerTokenizer.from_pretrained('allenai/longformer-large-4096-finetuned-triviaqa')
>>> model = TFLongformerForQuestionAnswering.from_pretrained('allenai/longformer-large-4096-finetuned-triviaqa')
class transformers.TFLongformerForSequenceClassification < source >
( input_ids = None attention_mask = None head_mask = None token_type_ids = None position_ids = None global_attention_mask = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None labels = None training = False **kwargs ) → transformers.models.longformer.modeling_tf_longformer.TFLongformerSequenceClassifierOutput or tuple(tf.Tensor)
>>> model = TFLongformerForSequenceClassification.from_pretrained('allenai/longformer-base-4096')
class transformers.TFLongformerForTokenClassification < source >
( input_ids = None attention_mask = None head_mask = None token_type_ids = None position_ids = None global_attention_mask = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None labels = None training = False **kwargs ) → transformers.models.longformer.modeling_tf_longformer.TFLongformerTokenClassifierOutput or tuple(tf.Tensor)
>>> model = TFLongformerForTokenClassification.from_pretrained('allenai/longformer-base-4096')
class transformers.TFLongformerForMultipleChoice < source >
( input_ids = None attention_mask = None head_mask = None token_type_ids = None position_ids = None global_attention_mask = None inputs_embeds = None output_attentions = None output_hidden_states = None return_dict = None labels = None training = False **kwargs ) → transformers.models.longformer.modeling_tf_longformer.TFLongformerMultipleChoiceModelOutput or tuple(tf.Tensor)
input_ids (tf.Tensor of shape (batch_size, num_choices, sequence_length)) — Indices of input sequence tokens in the vocabulary.
attention_mask (tf.Tensor of shape (batch_size, num_choices, sequence_length), optional) — Mask to avoid performing attention on padding token indices. Mask values selected in [0, 1]:
global_attention_mask (tf.Tensor of shape (batch_size, num_choices, sequence_length), optional) — Mask to decide the attention given on each token, local attention or global attention. Tokens with global attention attends to all other tokens, and all other tokens attend to them. This is important for task-specific finetuning because it makes the model more flexible at representing the task. For example, for classification, the token should be given global attention. For QA, all question tokens should also have global attention. Please refer to the Longformer paper for more details. Mask values selected in [0, 1]:
token_type_ids (tf.Tensor of shape (batch_size, num_choices, sequence_length), optional) — Segment token indices to indicate first and second portions of the inputs. Indices are selected in [0, 1]:
position_ids (tf.Tensor of shape (batch_size, num_choices, sequence_length), optional) — Indices of positions of each input sequence tokens in the position embeddings. Selected in the range [0, config.max_position_embeddings - 1].
>>> model = TFLongformerForMultipleChoice.from_pretrained('allenai/longformer-base-4096')
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Equilibrium constant of a reaction[change | change source]
{\displaystyle \alpha A+\beta B...\rightleftharpoons \sigma S+\tau T...}
{\displaystyle K={\frac {{\{S\}}^{\sigma }{\{T\}}^{\tau }...}{{\{A\}}^{\alpha }{\{B\}}^{\beta }...}}}
{\displaystyle K_{c}={\frac {{[S]}^{\sigma }{[T]}^{\tau }...}{{[A]}^{\alpha }{[B]}^{\beta }...}}}
{\displaystyle \Delta G}
{\displaystyle \Delta G=-RT\ln K}
{\displaystyle R}
{\displaystyle T}
{\displaystyle K_{p}}
{\displaystyle K_{c}}
[change | change source]
{\displaystyle PV=nRT\,}
{\displaystyle {\frac {n}{V}}={\frac {P}{RT}}}
{\displaystyle C={\frac {n}{V}}}
{\displaystyle C={\frac {P}{RT}}}
{\displaystyle P}
{\displaystyle V}
{\displaystyle n}
{\displaystyle R}
{\displaystyle T}
{\displaystyle {\frac {[AB]}{[A][B]}}={\frac {\frac {P_{AB}}{RT}}{{\frac {P_{A}}{RT}}{\frac {P_{B}}{RT}}}}}
{\displaystyle K_{c}={\frac {P_{AB}}{{P_{A}}{P_{B}}}}\times {RT}^{1+1-1}}
{\displaystyle P_{X}}
{\displaystyle X}
{\displaystyle {\frac {P_{AB}}{{P_{A}}{P_{B}}}}=K_{p}}
{\displaystyle K_{c}{(RT)}^{-1-1+1}=K_{p}}
{\displaystyle K_{p}}
{\displaystyle \alpha A+\beta B...\rightleftharpoons \sigma S+\tau T...}
{\displaystyle K_{c}{(RT)}^{-\alpha -\beta ...+\sigma +\tau ...}=K_{p}={\frac {{p_{\mathrm {S} }}^{\sigma }{p_{\mathrm {T} }}^{\tau }...}{{p_{\mathrm {A} }}^{\alpha }{p_{\mathrm {B} }}^{\beta }...}}}
{\displaystyle K_{c}}
{\displaystyle K_{p}}
{\displaystyle K_{c}{(RT)}^{\Delta n}=K_{p}}
{\displaystyle \Delta n}
{\displaystyle K_{eq}}
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Stability radius - Wikipedia
The stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
{\displaystyle {\hat {p}}}
denotes the nominal point,
{\displaystyle P}
denotes the space of all possible values of the object
{\displaystyle p}
, and the shaded area,
{\displaystyle P(s)}
, represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.
1 Abstract definition
3 Relation to Wald's maximin model
4 Info-gap decision theory
6 Stability radius of functions
Abstract definition[edit]
The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful[1][2]
{\displaystyle {\hat {\rho }}({\hat {p}}):=\max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}}
{\displaystyle B(\rho ,{\hat {p}})}
denotes a closed ball of radius
{\displaystyle \rho }
{\displaystyle P}
{\displaystyle {\hat {p}}}
It looks like the concept was invented in the early 1960s.[3][4] In the 1980s it became popular in control theory[5] and optimization.[6] It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest.
Relation to Wald's maximin model[edit]
It was shown[2] that the stability radius model is an instance of Wald's maximin model. That is,
{\displaystyle \max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}\equiv \max _{\rho \geq 0}\min _{p\in B(\rho ,{\hat {p}})}f(\rho ,p)}
{\displaystyle f(\rho ,p)=\left\{{\begin{array}{cc}\rho &,\ p\in P(s)\\-\infty &,\ p\notin P(s)\end{array}}\right.}
The large penalty (
{\displaystyle -\infty }
) is a device to force the
{\displaystyle \max }
player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.
Info-gap decision theory[edit]
Info-gap decision theory is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown[2] that its robustness model, namely
{\displaystyle {\hat {\alpha }}(q,{\tilde {u}}):=\max \ \{\alpha \geq 0:r_{c}\leq R(q,u),\forall u\in U(\alpha ,{\tilde {u}})\}}
is actually a stability radius model characterized by a simple stability requirement of the form
{\displaystyle r_{c}\leq R(q,u)}
{\displaystyle q}
denotes the decision under consideration,
{\displaystyle u}
denotes the parameter of interest,
{\displaystyle {\tilde {u}}}
denotes the estimate of the true value of
{\displaystyle u}
{\displaystyle U(\alpha ,{\tilde {u}})}
denotes a ball of radius
{\displaystyle \alpha }
{\displaystyle {\tilde {u}}}
Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the local robustness of decisions in the neighborhood of the estimate
{\displaystyle {\tilde {u}}}
Sniedovich[2] argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.
There are cases where it is more convenient to define the stability radius slightly different. For example, in many applications in control theory the radius of stability is defined as the size of the smallest destabilizing perturbation in the nominal value of the parameter of interest.[7] The picture is this:
{\displaystyle {\hat {\rho }}(q):=\min _{p\notin P(s)}dist(p,{\hat {p}})}
{\displaystyle dist(p,{\hat {p}})}
denotes the distance of
{\displaystyle p\in P}
{\displaystyle {\hat {p}}}
Stability radius of functions[edit]
The stability radius of a continuous function f (in a functional space F) with respect to an open stability domain D is the distance between f and the set of unstable functions (with respect to D). We say that a function is stable with respect to D if its spectrum is in D. Here, the notion of spectrum is defined on a case by case basis, as explained below.
Formally, if we denote the set of stable functions by S(D) and the stability radius by r(f,D), then:
{\displaystyle r(f,D)=\inf _{g\in C}\{\|g\|:f+g\notin S(D)\},}
where C is a subset of F.
Note that if f is already unstable (with respect to D), then r(f,D)=0 (as long as C contains zero).
The notion of stability radius is generally applied to special functions as polynomials (the spectrum is then the roots) and matrices (the spectrum is the eigenvalues). The case where C is a proper subset of F permits us to consider structured perturbations (e.g. for a matrix, we could only need perturbations on the last row). It is an interesting measure of robustness, for example in control theory.
Let f be a (complex) polynomial of degree n, C=F be the set of polynomials of degree less than (or equal to) n (which we identify here with the set
{\displaystyle \mathbb {C} ^{n+1}}
of coefficients). We take for D the open unit disk, which means we are looking for the distance between a polynomial and the set of Schur stable polynomials. Then:
{\displaystyle r(f,D)=\inf _{z\in \partial D}{\frac {|f(z)|}{\|q(z)\|}},}
where q contains each basis vector (e.g.
{\displaystyle q(z)=(1,z,\ldots ,z^{n})}
when q is the usual power basis). This result means that the stability radius is bound with the minimal value that f reaches on the unit circle.
{\displaystyle f(z)=z^{8}-9/10}
(whose zeros are the 8th-roots of 0.9) has a stability radius of 1/80 if q is the power basis and the norm is the infinity norm. So there must exist a polynomial g with (infinity) norm 1/90 such that f+g has (at least) a root on the unit circle. Such a g is for example
{\displaystyle g(z)=-1/90\sum _{i=0}^{8}z^{i}}
. Indeed, (f+g)(1)=0 and 1 is on the unit circle, which means that f+g is unstable.
^ Zlobec S. (2009). Nondifferentiable optimization: Parametric programming. Pp. 2607-2615, in Encyclopedia of Optimization, Floudas C.A and Pardalos, P.M. editors, Springer.
^ a b c d Sniedovich, M. (2010). A bird's view of info-gap decision theory. Journal of Risk Finance, 11(3), 268-283.
^ Wilf, H.S. (1960). Maximally stable numerical integration. Journal of the Society for Industrial and Applied Mathematics, 8(3),537-540.
^ Milne, W.E., and Reynolds, R.R. (1962). Fifth-order methods for the numerical solution of ordinary differential equations. Journal of the ACM, 9(1), 64-70.
^ Hindrichsen, D. and Pritchard, A.J. (1986). Stability radii of linear systems, Systems and Control Letters, 7, 1-10.
^ Zlobec S. (1988). Characterizing Optimality in Mathematical Programming Models. Acta Applicandae Mathematicae, 12, 113-180.
^ Paice A.D.B. and Wirth, F.R. (1998). Analysis of the Local Robustness of Stability for Flows. Mathematics of Control, Signals, and Systems, 11, 289-302.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Stability_radius&oldid=960898035"
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The Formulas - mimo protocol
This page shows the formulas used in mimo protocol.
In each trade, traders trade certain amount of a particular token for certain amount of another token with the price defined by a formula. There is no orderbook and waiting for fulfillment.
The formula that mimo uses is the famous
x * y = k
that has been widely adopted by AMM based DEX, such as uniswap.
x * y = k
X
is the source token, and
Y
is the destination token. In mimo,
X
Y
could be either IOTX or XRC20 tokens. Let
x
y
be X-token, Y-token in current liquidity pool, respectively.
Based on the famous AMM equation
x * y = k
k
x
y
remains the same before and after trading. For details, please refer to vbuterin's post.
Pricing based on the inputs
Let's further define
d_x
d_y
are how many X-tokens you want to pay, and how many Y-tokens you will get, respectively.
We'd like to know, the price based on
d_x
d_y
. IfgetInputPrice denotes how many Y-Tokens (i.e.
d_y
) can be bought by selling a given
d_x
getInputPrice(x, y, d_x) = \dfrac{y * 997 * dx}{1000 * x + 997 * d_x}
getInputPrice(x, y, dx) = (y * 997 * dx) / (1000 x + 997 dx)
If getOutputPrice denotes how many X-tokens is needed to buy
d_y
Y-tokens,
getOutputPrice(x, y, d_y) = \dfrac{1000 * x * d_y}{(y-d_y)*997} + 1
getOutputPrice(x, y, dy) = 1000 x * dy / ((y - dy) * 997) + 1
where / in above equations denotes divToInteger, which means divide with rounding to floor of the results.
In AMM, the price would change after each trade,
d_x
d_y
x, y
the liquidity in the pool. The price impact is what traders want to know before the trade.
There are two ways of calculating price impact. It can be based on
x, y, d_x
x, y ,d_y
. One is based on input , one is based on output.
PriceImpact(x, y, d_x) = \frac{ (1000*x)^2} { (1000*x + 997*d_x)^2} -1
PriceImpact(x, y, d_y) = \dfrac{(y-d_y)^2}{y^2} -1
price impact(x, y, dx) = (1000*x)^2 / (1000*x + 997*dx)^2 - 1
price impact(x, y, dy) = (y - dy)^2 / y^2 - 1
Note that the price impact is always between -1 and 0.
Cross-Trading Price Impact
If there are no direct trading pairs between two tokens, like in V1 where we only support IOTX/token pairs, traders need to use one token, such as IOTX, as a bridge to trade among two tokens.
In this case, the price impact would be
PriceImpact_{cross} = PI_1 * PI_2 + PI_1 + PI_2
price impact = PI1 * PI2 + PI1 + PI2
PI1 is the price impact of first trading pair, such as x to IOTX
PI2 is the price impact of second trading pair, such as IOTX to y
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Adaptive Cruise Control Using Extremum Seeking Control - MATLAB & Simulink
ACC Using Extremum Seeking Control
Specify Extremum Seeking Control Parameters
Simulate Adaptive Cruise Control System
This example shows how to implement adaptive cruise control using an extremum seeking control (ESC) approach. In this example, the goal is to make an ego car travel at a set velocity while maintaining a safe distance from a lead car by controlling longitudinal acceleration and braking.
Adaptive cruise control (ACC) is a system designed to help vehicles maintain a safe following distance and stay within the speed limit. A vehicle equipped with an ACC system (ego car) uses radar to measure relative distance (
{D}_{rel}
) and relative velocity (
{V}_{rel}
) with respect to the leading vehicle. The ACC system is designed to maintain a desired cruising speed (
{V}_{set}
) or maintain a relative safe distance (
{D}_{safe}
) from the leading car. The switch in the control objective is determined based on the following conditions.
{D}_{rel}>{D}_{safe}
, the ACC system follows the desired reference cruise velocity commanded by the driver.
{D}_{rel}<{D}_{safe}
, the ACC system controls the relative position of the ego car with respect to the lead car.
This example uses the same ego and lead car model as the Adaptive Cruise Control System Using Model Predictive Control (Model Predictive Control Toolbox).
Implement the longitudinal vehicle dynamics as a simple second-order linear model.
G = tf(1,[0.5,1,0]);
Configure the ACC parameters for the example.
Tf = 150; % Duration (s)
Specify the initial position and velocity for both the lead car and ego car.
x0_lead = 50; % Initial lead car position (m)
v0_lead = 25; % Initial lead car velocity (m/s)
x0_ego = 10; % Initial ego car position (m)
v0_ego = 20; % Initial ego car velocity (m/s)
An extremum seeking controller achieves satisfactory control performance by adjusting control parameters to maximize an objective function in real time. For this example, use the following objective function, which depends on relative distance, safe distance, relative velocity, and set velocity.
J=-\int {Q}_{d}{\left({D}_{rel}-{D}_{safe}\right)}^{2}+{Q}_{v}{\left({v}_{rel}-{v}_{set}\right)}^{2}
{\mathit{Q}}_{\mathit{d}}
{\mathit{Q}}_{\mathit{v}}
are objective function weights for the distance error and velocity error terms, respectively.
Qd = 0.5;
Qv = 1;
The extremum seeking controller adapts the following controller gains.
{K}_{xerr}
— Position error gain
{K}_{verr}
— Velocity error gain
{K}_{vrel}
— Relative velocity gain
Specify initial guesses for the gain values.
Kverr = 1; % ACC velocity error gain
Kxerr = 1; % ACC spacing error gain
Kvrel = 0.5; % ACC relative velocity gain
Simulink Control Design software implements the ESC algorithm using the Extremum Seeking Control block. Configure the parameters for this block.
Specify the number of parameters to tune (the three controller gains). The controller uses a separate tuning loop for each parameter.
Specify the initial conditions for the parameter update integrators by scaling the initial gain values with respect to the learning rate for each parameter lr.
lr = 0.02*[2 3 1];
IC = [Kverr,Kxerr,Kvrel];
Configure the demodulation and modulation signals by specifying their frequencies (omega), phases (phi_1 and phi_2), and amplitudes (a and b). Each parameter must use a different forcing frequency. For this example, use the same modulation and demodulation phases and amplitudes for all parameters.
omega = 0.8*[5,7,8]; % Forcing frequency (rad/s)
a = 0.01; % Demodulation amplitude
b = 0.5*lr; % Modulation amplitude
phi_1 = 0; % Demodulation phase (rad)
phi_2 = pi/4; % Modulation phase (rad)
Use a low-pass filter to remove high-frequency noise from the demodulated signal and a high-pass filter to remove bias from the perturbed objective function signal. Specify the cutoff frequencies for these filters.
omega_lpf = 0.04;
omega_hpf = 0.01;
To simulate the ESC adaptive cruise controller, open the ExtremumSeekingControlACC model.
mdl = 'ExtremumSeekingControlACC';
The Plant Dynamics and Objective subsystem contains the ACC models and computes the objective function for the ESC algorithm.
open_system([mdl '/Plant Dynamics and Objective'])
Simulate the model. During the simulation, the lead car velocity varies sinusoidally. Therefore, the ego car must adjust its velocity to compensate.
The following plot shows the relative distance between the lead and ego cars and the safe distance.
The safe distance varies as the ego car velocity changes.
The relative distance between the ego and lead cars occasionally drops slightly below the safe distance. This result is because the ACC system enforces the relative distance using a soft constraint.
open_system([mdl '/Plant Dynamics and Objective/Simulation results/Distance'])
View the velocities of the ego and lead cars along with the ego car set velocity. To maintain a safe distance the ACC system adjusts the ego car velocity as the lead car velocity changes. When the lead car velocity is greater than the set velocity, the ego car stops tracking the lead car velocity and cruises at the set velocity.
open_system([mdl '/Plant Dynamics and Objective/Simulation results/Velocity'])
The next plot shows the cost function that ESC seeks to optimize when searching for optimal control gains.
open_system([mdl '/Plant Dynamics and Objective/Simulation results/Cost'])
View the resulting controller gains, which adapt over the course of the simulation. The top plot is
{K}_{verr}
, the middle plot is
{K}_{xerr}
, and the bottom plot is
{K}_{vrel}
. Fluctuations in the gain values are due to the modulation signals from the Extremum Seeking Control block.
open_system([mdl '/Plant Dynamics and Objective/Gains'])
bdclose('ExtremumSeekingControlACC')
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Hypergeometric_function_of_a_matrix_argument Knowpia
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals.
Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument.
{\displaystyle p\geq 0}
{\displaystyle q\geq 0}
{\displaystyle X}
{\displaystyle m\times m}
complex symmetric matrix. Then the hypergeometric function of a matrix argument
{\displaystyle X}
{\displaystyle \alpha >0}
{\displaystyle _{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot C_{\kappa }^{(\alpha )}(X),}
{\displaystyle \kappa \vdash k}
{\displaystyle \kappa }
{\displaystyle k}
{\displaystyle (a_{i})_{\kappa }^{(\alpha )}}
is the generalized Pochhammer symbol, and
{\displaystyle C_{\kappa }^{(\alpha )}(X)}
is the "C" normalization of the Jack function.
Two matrix argumentsEdit
{\displaystyle X}
{\displaystyle Y}
{\displaystyle m\times m}
complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:
{\displaystyle _{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X,Y)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot {\frac {C_{\kappa }^{(\alpha )}(X)C_{\kappa }^{(\alpha )}(Y)}{C_{\kappa }^{(\alpha )}(I)}},}
{\displaystyle I}
{\displaystyle m}
Not a typical function of a matrix argumentEdit
Unlike other functions of matrix argument, such as the matrix exponential, which are matrix-valued, the hypergeometric function of (one or two) matrix arguments is scalar-valued.
The parameter αEdit
In many publications the parameter
{\displaystyle \alpha }
is omitted. Also, in different publications different values of
{\displaystyle \alpha }
are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984),
{\displaystyle \alpha =2}
whereas in other settings (e.g., in the complex case—see Gross and Richards, 1989),
{\displaystyle \alpha =1}
. To make matters worse, in random matrix theory researchers tend to prefer a parameter called
{\displaystyle \beta }
{\displaystyle \alpha }
which is used in combinatorics.
{\displaystyle \alpha ={\frac {2}{\beta }}.}
Care should be exercised as to whether a particular text is using a parameter
{\displaystyle \alpha }
{\displaystyle \beta }
and which the particular value of that parameter is.
Typically, in settings involving real random matrices,
{\displaystyle \alpha =2}
{\displaystyle \beta =1}
. In settings involving complex random matrices, one has
{\displaystyle \alpha =1}
{\displaystyle \beta =2}
K. I. Gross and D. St. P. Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Theory, 59, no. 2, 224–246, 1989.
J. Kaneko, "Selberg Integrals and hypergeometric functions associated with Jack polynomials", SIAM Journal on Mathematical Analysis, 24, no. 4, 1086-1110, 1993.
Plamen Koev and Alan Edelman, "The efficient evaluation of the hypergeometric function of a matrix argument", Mathematics of Computation, 75, no. 254, 833-846, 2006.
Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
Software for computing the hypergeometric function of a matrix argument by Plamen Koev.
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Home - Past Questions - Mathematics 2019 JAMB Past Question
Make q the subject of the formula in the equation \(\frac{mn}{a^2} – \frac{pq}{b^2} = 1\)
A. \(q = \frac{b^2(mn – a^2)}{a^2 p}\)
B. \(q = \frac{m^2 n – a^2}{p^2}\)
C. \(q = \frac{mn – 2b^2}{a^2}\)
D. \(q = \frac{b^2 (n^2 – ma^2)}{n}\)
2. The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is 78°. Find the height of the tree correct to the nearest whole number.
3. A binary operation ⊗⊗ is defined by m⊗n=mn+m−nm⊗n=mn+m−n on the set of real numbers, for all m, n ∈∈ R. Find the value of 3 ⊗⊗ (2 ⊗⊗ 4).
5. In a class of 50 students, 40 students offered Physics and 30 offered Biology. How many offered both Physics and Biology?
6. Rationalize \(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} – \sqrt{3}}\)
A. \(-5 – 2\sqrt{6}\)
B. \(-5 + 3\sqrt{2}\)
C. \(5 – 2\sqrt{3}\)
D. \(5 + 2\sqrt{6}\)
Find the length of the chord |AB| in the diagram shown above.
8. Given
\mathrm{sin}58°=\mathrm{cos}p°
9. \(\frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} – \frac{1}{3}}\)
10. If \(6x^3 + 2x^2 – 5x + 1\) divides \(x^2 – x – 1\), find the remainder.
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Governments and security companies are coming to rely more heavily on facial recognition software to locate persons of interest.
Consider a hypothetical situation. Suppose that facial recognition software can accurately identify a person
99.9\%
of the time, and suppose the suspect is among
200,000
facial images available to a government agency. When the software makes a positive identification, what is the probability that it is not the suspect?
Use the model at right as a starting point.
Complete it by including the probabilities.
P(suspect)
=
1/200,000, P(ID is correct)
=0.999
P(not the suspect)
=
?, P(ID is not correct)
=
If a person has been identified as the suspect, what is the probability that he or she is not actually the suspect?
\text{Probability} =\frac{\text{# of desired outcomes}}{\text{# of possible outcomes}}
The desired outcome is a person who is not the suspect being incorrectly identified, or situation D.
The possible outcomes are situation D and the suspect being correctly identified, or situation A.
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Axisymmetric Thermal and Structural Analysis of Disc Brake - MATLAB & Simulink - MathWorks Deutschland
Disc Brake Properties and Geometry
Thermal Analysis: Compute Temperature Distribution
Structural Analysis: Compute Thermal Stress
Heat Flux Function
Plot Results Function
This example shows a quasistatic axisymmetric thermal stress analysis workflow by reproducing the results of the simplified disc brake model discussed in [1]. Disc brakes absorb mechanical energy through friction and transform it into thermal energy, which then dissipates. The example uses a simplified model of a disc brake in a single braking process from a constant initial angular speed to a standstill. The workflow has two steps:
Transient thermal analysis to compute the temperature distribution in the disc using the heat flux from brake pads
Quasistatic structural analysis to compute thermal stresses at several solution times using previously obtained temperature distribution to specify thermal loads
The resulting plots show the temperature distribution, radial stress, hoop stress, and von Mises stress for the corresponding solution times.
Based on the assumptions used in [1], the example reduces the analysis domain to a rectangular region corresponding to the axisymmetric section of the annular disc. Because of the geometric and load symmetry of the disc, the example models only half the thickness of the disc and the effect of one pad. In the following figure, the left edge corresponds to the inner radius of the disc
{\mathit{r}}_{\mathit{d}}
. The right edge corresponds to the outer radius of the disc
{\mathit{R}}_{\mathit{d}}
and also coincides with the outer radius of the pad
{\mathit{R}}_{\mathit{p}}
. The disc experiences pressure from the pad, which generates the heat flux. Instead of modeling the pad explicitly, include its effect in the thermal analysis by specifying this heat flux as a boundary condition from the inner radius of the pad
{\mathit{r}}_{\mathit{p}}
to the outer radius of the pad
{\mathit{R}}_{\mathit{p}}
Create a transient axisymmetric thermal model.
modelT = createpde('thermal','transient-axisymmetric');
Create a geometry with two adjacent rectangles. The top edge of the longer rectangle (on the right) represents the disc-pad contact region.
R1 = [3,4, [ 66, 76.5, 76.5, 66, -5.5, -5.5, 0, 0]/1000]';
R2 = [3,4, [76.5, 113.5, 113.5, 76.5, -5.5, -5.5, 0, 0]/1000]';
gdm = [R1 R2];
ns = char('R1','R2');
g = decsg(gdm,'R1 + R2',ns');
Assign the geometry to the thermal model.
Plot the geometry with the edge and face labels.
pdegplot(modelT,'EdgeLabels','on','FaceLabels','on')
Generate a mesh. To match the mesh used in [1], use the linear geometric order instead of the default quadratic order.
generateMesh(modelT,'Hmax',0.5E-04,'GeometricOrder','linear');
Specify the thermal material properties of the disc.
alphad = 1.44E-5; % Diffusivity of disc
cpd = Kd/rhod/alphad;
thermalProperties(modelT,'ThermalConductivity',Kd, ...
'MassDensity',rhod, ...
'SpecificHeat',cpd);
Specify the heat flux boundary condition to account for the pad region. For the definition of the qFcn function, see Heat Flux Function.
thermalBC(modelT,'Edge',6,'HeatFlux',@qFcn);
thermalIC(modelT,20);
Solve the model for the times used in [1].
tlist = [0 0.1 0.2 1.0 2.0 3.0 3.96];
Plot the temperature variation with time at three key radial locations. The resulting plot is comparable to the plot obtained in [1].
iTRd = interpolateTemperature(Rt,[0.1135;0],1:numel(Rt.SolutionTimes));
iTrp = interpolateTemperature(Rt,[0.0765;0],1:numel(Rt.SolutionTimes));
iTrd = interpolateTemperature(Rt,[0.066;0],1:numel(Rt.SolutionTimes));
plot(tlist,iTRd)
plot(tlist,iTrp)
title('Temperature Variation with Time at Key Radial Locations')
legend('R_d','r_p','r_d')
xlabel 't, s'
ylabel 'T,^{\circ}C'
Create an axisymmetric static structural analysis model.
model = createpde('structural','static-axisymmetric');
Assign the geometry and mesh used for the thermal model.
model.Geometry = modelT.Geometry;
model.Mesh = modelT.Mesh;
Specify the structural properties of the disc.
structuralProperties(model,'YoungsModulus',99.97E9, ...
'CTE',1.08E-5);
Constrain the model to prevent rigid motion.
structuralBC(model,'Edge',[3,4],'ZDisplacement',0);
Specify the reference temperature that corresponds to the state of zero thermal stress of the model.
model.ReferenceTemperature = 20;
Specify the thermal load by using the transient thermal results Rt. The solution times are the same as in the thermal model analysis. For each solution time, solve the corresponding static structural analysis problem and plot the temperature distribution, radial stress, hoop stress, and von Mises stress. For the definition of the plotResults function, see Plot Results Function. The results are comparable to figure 5 from [1].
for n = 2:numel(Rt.SolutionTimes)
structuralBodyLoad(model,'Temperature',Rt,'TimeStep',n);
plotResults(model,R,modelT,Rt,n);
This helper function computes the transient value of the heat flux from the pad to the disc. It uses the empirical formula from [1].
function q = qFcn(r,s)
Kd = 51; % Conductivity of disc
rhod = 7100; % Density of disc
cpd = Kd/rhod/alphad; % Specific heat capacity of disc
alphap = 1.46E-5; % Diffusivity of pad
Kp = 34.3; % Conductivity of pad
rhop = 4700; % Density of pad
cpp = Kp/rhop/alphap; % Specific heat capacity of pad
f = 0.5; % Coefficient of friction
omega0 = 88.464; % Initial angular velocity
ts = 3.96; % Stopping time
p0 = 1.47E6*(64.5/360); % Pressure only spans 64.5 deg occupied by pad
omegat = omega0*(1 - s.time/ts); % Angular speed over time
eta = sqrt(Kd*rhod*cpd)/(sqrt(Kd*rhod*cpd) + sqrt(Kp*rhop*cpp));
q = (eta)*f*omegat*r.r*p0;
This helper function plots the temperature distribution, radial stress, hoop stress, and von Mises stress.
function plotResults(model,R,modelT,Rt,tID)
pdeplot(modelT,'XYData',Rt.Temperature(:,tID), ...
'ColorMap','jet','Contour','on')
title({'Temperature'; ...
['max = ' num2str(max(Rt.Temperature(:,tID))) '^{\circ}C']})
xlabel 'r, m'
ylabel 'z, m'
pdeplot(model,'XYData',R.Stress.srr, ...
title({'Radial Stress'; ...
['min = ' num2str(min(R.Stress.srr)/1E6,'%3.2f') ' MPa']; ...
['max = ' num2str(max(R.Stress.srr)/1E6,'%3.2f') ' MPa']})
pdeplot(model,'XYData',R.Stress.sh, ...
title({'Hoop Stress'; ...
['min = ' num2str(min(R.Stress.sh)/1E6,'%3.2f') ' MPa']; ...
['max = ' num2str(max(R.Stress.sh)/1E6,'%3.2f') ' MPa']})
pdeplot(model,'XYData',R.VonMisesStress, ...
title({'Von Mises Stress'; ...
['max = ' num2str(max(R.VonMisesStress)/1E6,'%3.2f') ' MPa']})
sgtitle(['Time = ' num2str(Rt.SolutionTimes(tID)) ' s'])
[1] Adamowicz, Adam. "Axisymmetric FE Model to Analysis of Thermal Stresses in a Brake Disc." Journal of Theoretical and Applied Mechanics 53, issue 2 (April 2015): 357–370. https://doi.org/10.15632/jtam-pl.53.2.357.
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Revision as of 18:38, 31 May 2015 by MathAdmin (talk | contribs) (Created page with "'''Question ''' Consider the following rational function, <center><math>f(x) = \frac{x^2+x-2}{x^2-1}</math></center> <br> a. What is the domain of f?...")
{\displaystyle f(x)={\frac {x^{2}+x-2}{x^{2}-1}}}
{\displaystyle a_{n}=(-a)^{n}}
{\displaystyle y=x^{2}}
{\displaystyle cos^{2}(x)-cos(x)=0}
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10.10: Reversing a Conditional Probability- Bayes’ Rule - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F10%253A_Probability%2F10.10%253A_Reversing_a_Conditional_Probability-_Bayes%25E2%2580%2599_Rule
In many cases, we know
P(A|B)
but we really want to know
P(B|A)
. This commonly occurs in medical screening, where we know
P(\text{positive test result| disease})
but what we want to know is
P(\text{disease|positive test result})
. For example, some doctors recommend that men over the age of 50 undergo screening using a test called prostate specific antigen (PSA) to screen for possible prostate cancer. Before a test is approved for use in medical practice, the manufacturer needs to test two aspects of the test’s performance. First, they need to show how sensitive it is – that is, how likely is it to find the disease when it is present:
\text{sensitivity} = P(\text{positive test| disease})
. They also need to show how specific it is: that is, how likely is it to give a negative result when there is no disease present:
\text{specificity} = P(\text{negative test|no disease})
. For the PSA test, we know that sensitivity is about 80% and specificity is about 70%. However, these don’t answer the question that the physician wants to answer for any particular patient: what is the likelihood that they actually have cancer, given that the test comes back positive? This requires that we reverse the conditional probability that defines sensitivity: instead of
P(positive test| disease)
P(disease|positive test)
P(B|A) = \frac{P(A|B)*P(B)}{P(A)}
If we have only two outcomes, we can express Bayes’ rule in a somewhat clearer way, using the sum rule to redefine
P(A)
P(A) = P(A|B)*P(B) + P(A|\neg B)*P(\neg B)
P(B|A) = \frac{P(A|B)*P(B)}{P(A|B)*P(B) + P(A|\neg B)*P(\neg B)}
We can plug the relevant numbers into this equation to determine the likelihood that an individual with a positive PSA result actually has cancer – but note that in order to do this, we also need to know the overall probability of cancer in the person, which we often refer to as the base rate. Let’s take a 60 year old man, for whom the probability of prostate cancer in the next 10 years is
P(\text{cancer})=0.058
. Using the sensitivity and specificity values that we outlined above, we can compute the individual’s likelihood of having cancer given a positive test:
P(\text{cancer|test}) = \frac{P(\text{test|cancer})*P(\text{cancer})}{P(\text{test|cancer})*P(\text{cancer}) + P(\text{test|}\neg\text{cancer})*P(\neg\text{cancer})}
= \frac{0.8*0.058}{0.8*0.058 +0.3*0.942 } = 0.14
That’s pretty small – do you find that surprising? Many people do, and in fact there is a substantial psychological literature showing that people systematically neglect base rates (i.e. overall prevalence) in their judgments.
10.10: Reversing a Conditional Probability- Bayes’ Rule is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Partial Quark-Lepton Universality and Neutrino CP Violation
Jiajun Liao, D. Marfatia, K. Whisnant, "Partial Quark-Lepton Universality and Neutrino CP Violation", Advances in High Energy Physics, vol. 2015, Article ID 561910, 5 pages, 2015. https://doi.org/10.1155/2015/561910
Jiajun Liao,1,2 D. Marfatia,1 and K. Whisnant2
1Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822, USA
2Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
We study a model with partial quark-lepton universality that can naturally arise in grand unified theories. We find that constraints on the model can be reduced to a single condition on the Dirac CP phase in the neutrino sector. Using our current knowledge of the CKM and PMNS mixing matrices, we predict at .
Our understanding of neutrinos has progressed steadily in the last two decades. After the observation of nonzero by the Daya Bay [1, 2], RENO [3], and Double Chooz [4] experiments, we now know the three mixing angles , , and and the two mass squared differences to good precision. For the normal hierarchy, the current ranges of the three mixing angles from a global three-neutrino oscillation analysis are [5]
The focus of next generation neutrino oscillation experiments is shifted to the Dirac CP phase and the neutrino mass hierarchy. Predictions of many theoretical models designed to explain the observed mixing patterns await verification. Among these models, quark-lepton universality (QLU) [6] is well motivated. It is based on simple relations in grand unified theories (GUT) and connects the mixing matrices of quarks and leptons. Exact quark-lepton universality leads to a symmetric PMNS mixing matrix. However, using the current ranges of the oscillation parameters [5], we find the moduli of the neutrino mixing matrix elements are We see that the exactly symmetric PMNS mixing matrix is disfavored by the current data. This aspect of the PMNS matrix with or has been studied in [7–9].
In this paper, we discuss partial quark-lepton universality [6], which does not require the unitary matrices that diagonalize the upper and lower components of the weak doublets to be the same. We find that partial QLU fits the current data very well and we can make a prediction for the unknown Dirac CP phase.
In Section 2, we review partial quark-lepton universality and discuss renormalization group effects on the model. In Section 3, we discuss the phenomenological results of this model and predict the Dirac CP phase. We conclude in Section 4.
2. Partial Quark-Lepton Universality
Partial quark-lepton universality can be derived from some simple relations in grand unified theories [6]. We start with the SU(5) relation obtainable in lopsided models [10], and where , , and are the mass matrices of the charged-leptons, up-type quarks, and down-type quarks, respectively. If we assume is Hermitian, which can be achieved by imposing left-right symmetry [6]1, then from (3) we find that both the down-type quarks and charged-leptons can be diagonalized by a unitary matrix
Also, from (4), we know that the up-type quarks can be diagonalized by a unitary matrix If the Dirac neutrino matrix and the right-handed Majorana neutrino mass matrix are also diagonalized by (as in some SO(10) models [6]), Then, below the seesaw scale, the light neutrino mass matrix, , is diagonalized by as well. Consider From (5), (6), and (8), we can find that the observable mixing matrices are related to
Note that, for exact quark-lepton universality, we must have , which indicates that and the mixing matrix is symmetric. This is disfavored by the current data. In the next section, we show that partial quark-lepton universality is still allowed by current data. A caveat to partial QLU is that small perturbations to the leading order relations of (9) and (10) are needed to reproduce the measured fermion masses. In [6], it was shown that, with a specific form for the perturbations, the measured fermion masses can be obtained while keeping the mixing matrices unchanged. Consequently, we focus on the connection between the mixing matrices of quarks and leptons.2
The current data that determine the CKM and PMNS mixing matrices are measured at low energies, while the quark-lepton universality relations are realized at the grand unification scale. In order to use the current data to analyze the model, we must consider renormalization group (RG) effects. For the CKM matrix, the RG effects are very small; that is, the next order relative corrections to the CKM matrix are of the order [11, 12], where . The RG effects in the neutrino sector are strongly dependent on the mass spectrum of the light neutrinos. For the inverted and quasi-degenerate mass hierarchies, the effects can be large [13–16]. However, with quark-lepton universality it is more natural to assume that the light neutrinos are very hierarchical with the normal mass spectrum. In this case, RG effects on the three angles are very small [17, 18]; for example, , and in the MSSM with if the lightest neutrino mass is 0.01 eV. Since the current uncertainties in the three angles are larger than the RG effects, we neglect the RG effects in our analysis.
In this section, we introduce a simple approach based on the properties of unitary matrices to reduce the constraints on the model to a single condition, which allows us to easily constrain the Dirac CP phase.
Partial QLU predicts the two observable mixing matrices to have the form of (9) and (10), which can be rewritten as Hence, in order for the model to work, both and should be symmetric. However, the two constraints are not independent. Since (9) implies , (11) follows from (12).
Solutions for and will always exist because if is symmetric, then it can be diagonalized by a unitary matrix ; that is, , where is diagonal. This means that we can find the solution, . Once is known, the solution for can be obtained from (9). Although solutions for and exist, they are not unique. We can always insert a combination of a real rotation matrix into the middle of the right-handed side of (11) or (12). And since , the equation will not change. This can also be seen from (9) and (10). For any real rotation matrix , and are also unitary; hence if we let and , the two observable mixing matrices will remain the same.
Now, if we define then the only constraint from the model is that is symmetric. Since both and are unitary matrices, is also unitary. For a unitary matrix, it can be shown that being symmetric is equivalent to the moduli of being symmetric under phase redefinition [19]. This constraint still imposes three conditions: , , and . However, the conditions are not independent. Since is unitary, . Hence, indicates and vice versa. Similarly, is equivalent to . Therefore, there is only one independent condition that constrains the model. Here, we choose it to be .
The CKM matrix can be written in terms of the Wolfenstein parameters [20] as follows: and the PMNS matrix can be written in the standard form, which is where , denote and , respectively, and Majorana phases are not included. From (13), we see that the condition becomes Note that (16) cannot be satisfied when . Keeping in mind that , the and terms can be neglected since they are of the same order of magnitude as the terms dropped in the Wolfenstein parametrization. Then, we get a simple expression for the cosine of the Dirac CP phase: where . We see that for very small the numerator of the above equation is always larger than the denominator, so that there is no solution for .
Using the currently favored CKM [21] and PMNS [5] parameters with their respective uncertainties and solving the condition numerically without any approximation, we find that the Dirac CP phase in the PMNS matrix lies between and at . The asymmetry around 0 is due to the small CP violation in the CKM matrix, which does not enter the approximate result in (17).
We also find predictions for each mixing angle versus given the best-fit values and allowed regions of the other two mixing angles and the CKM parameters. The results are shown in Figure 1. With the constraints from the other two mixing angles and the CKM parameters, we find that , and at . The partial QLU model is perfectly consistent with the current data, and rather large is strongly favored for the measured solar and atmospheric mixing angles. Note that the relevant neutrino mass squared differences are trivially accommodated.
The allowed regions (shaded bands) in the (, ), (, ), and (, ) planes using measurements (with uncertainties) of the other two neutrino mixing angles and the CKM parameters. The solid curves within the shaded bands are the model predictions for the best-fit values of the other two mixing angles and the CKM parameters. The horizontal solid lines mark the best-fit values and the horizontal dashed lines mark the limits of , , and .
A measurement of by future long baseline neutrino oscillation experiments will provide a stringent test of the viability of the partial quark-lepton universality model.
We studied partial quark-lepton universality, which can naturally arise in grand unified theories. Constraints on the model can be reduced to one simple condition, . Dropping terms of order from this condition, we find a simple expression for the Dirac CP phase in the neutrino sector. We also studied the allowed parameter regions of the model numerically. Our prediction that lies within the range at the level will be tested by future long baseline neutrino experiments.
This research was supported by the U.S. Department of Energy Grant no. DE-SC0010504.
Implementing a Hermitian in a GUT is difficult because SU(5) does not incorporate left-right symmetry, and, in SO(10), the mass matrices arising from the couplings of fermions to Higgs fields in the 10 and 126 representations are complex symmetric (and not Hermitian), while those arising from couplings to 120 are complex antisymmetric.
An example in which (3), (4), (7), and the Hermiticity of naturally arise is an SO(10) scheme with the superpotential terms [6]where , are 16-plet Higgs, is a -plet Higgs, is a charge conjugation matrix in SO(10), and are generation indices, and , , , , and are SO(10) indices. The Lorentz indices and the standard charge conjugation matrix are suppressed. and contain neutral fields with the quantum numbers of and , so that the vacuum expectation value for breaks SO(10) while SU(5) is preserved. We take the contribution to to be zero or subdominant compared to and , so is only generated from . By imposing an additional symmetry, , which leads to , a Hermitian can be obtained.
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Filter disturbances through vector error-correction (VEC) model - MATLAB filter - MathWorks Italia
\begin{array}{rcl}\Delta {y}_{t}& =& c+A{B}^{\prime }{y}_{t-1}+{\Phi }_{1}\Delta {y}_{t-1}+{\epsilon }_{t}\\ & =& \\ & =& \left[\begin{array}{c}-1\\ -3\\ -30\end{array}\right]+\left[\begin{array}{cc}-0.3& 0.3\\ -0.2& 0.1\\ -1& 0\end{array}\right]\left[\begin{array}{ccc}0.1& -0.2& 0.2\\ -0.7& 0.5& 0.2\end{array}\right]{y}_{t-1}+\left[\begin{array}{ccc}0& 0.1& 0.2\\ 0.2& -0.2& 0\\ 0.7& -0.2& 0.3\end{array}\right]\Delta {y}_{t-1}+{\epsilon }_{t}.\end{array}
\Sigma =\left[\begin{array}{ccc}1.3& 0.4& 1.6\\ 0.4& 0.6& 0.7\\ 1.6& 0.7& 5\end{array}\right].
\mu ={\Phi }^{-1}\left(L\right)c.
\Delta {y}_{t}={\stackrel{^}{\Phi }}^{-1}\left(L\right)\left(\stackrel{^}{c}+\stackrel{^}{d}t+\stackrel{^}{A}\stackrel{^}{B}\prime {y}_{t-1}+\stackrel{^}{\beta }{x}_{t}+{e}_{t}\right).
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Capacitor Formulas - Information - 2022
There are many calculations and equations associated with capacitors. The capacitor reactance equations and calculations are common, but there are many more capacitor calculations that may need to be performed.
Capacitor equations and capacitor calculations include many aspects of capacitor operation including the capacitor charge, capacitor voltage capacitor reactance calculations and many more.
Basic capacitance formulae
The very basic capacitor equations link the capacitance with the charge held on the capacitor, and the voltage across the plates.
Q is the charge held on the plates in coulombs
V is the potential difference across the plates in volts
This equation can then be developed to calculate the work required for charging a capacitor, and hence the energy stored in it.
W={\int }_{0}^{Q}V dQ
W={\int }_{0}^{Q}\frac{q}{C} dQ
W=\frac{1}{2}C{V}^{2}
Watch the video: How to select capacitor for single phase motor. Capacitor selection formula (May 2022).
Copyright 2022 \ Capacitor Formulas ...
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1. a) A salt solution is to be concentrated by evaporating the water in a salt pan, with a condensing surface above it to gather the evaporated water. Suppose 1200g of salt solution are emptied into the pan. Once all the water is evaporated, the salt is weighed and found to weigh 100g. What percent of the original solution was water?
b) Now suppose that 0.1 L of the evaporated water was added back to the salt, to bring it to the desired concentration. How much water remains to be used elsewhere?
c) Do you think the salt solution would be safe to drink? Why or why not?
2. a) In a stone quarry, limestone is to be crushed and poured into molds for manufacture of floor tiles. Suppose that a limestone company uses three trucks, each of which is capable of carrying 3000 kg of limestone. The quarry itself is 20 miles away from the processing plant, and the trucks get there at an average speed of 30 miles/hour. Once at the plant, the limestone is ground into fine powder and then melted and poured into the molds. If each of the resulting slabs weighs 2 kg and the plant operates 24 hours a day, how many slabs can the company make in a day?
b) How could this plant become more efficient? Plot the number of slabs the company can make as a function of distance from the quarry and capacity of the trucks. What factors might keep the company from simply moving as close to the quarry as possible and using the largest trucks possible?
3. What is the volumetric flowrate of a solution with density 1.5 kg/m^3 flowing at a velocity of 5 m/s and a mass flow rate of 500 g/min? What is the area of the pipe? If it is circular, what is the radius?
4. Suppose you have a pipe that constricts halfway through from a radius of 0.5 cm to a radius of 0.2 cm. A liquid approaches the constriction at a velocity of 0.5 m/s. What is the velocity of the fluid after the constriction? (Hint: Apply conservation of mass on both sides of the constriction).
Challenge: What kind of energy does the fluid gain? Energy is never created or destroyed, so where does it come from?
5. Suppose that a river with a molar flow rate of 10000 mol/s meets another, larger river flowing at 500000 m^3/s at room temperature. What is the mass flow rate of the river downstream of the intersection if you assume steady state?
b} Evaluate the feasibility of the steady state assumption in this situation. Also qualitatively evaluate the probability that the flowrates are actually constant.
6. Suppose that the population of a certain country n years after year 2000 if there is no emigration can be modeled with the equation:
{\displaystyle P=2.5*10^{8}*e^{0.045*n}}
Also, suppose that in the country, a net emigration of 100,000 people per month actually occurs. What is the total accumulation of people in this country from year 2000 to 2003?
b) What was the population of people in 2002, according to this model?
c) What are some possible problems with this model? For example, what doesn't it take into account? What happens when n is 100? Where did those constants come from? Would they be the same for every country, or for the same country across generations?
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Solve the following equations. Be accurate to three decimal places. Be sure to check your answers.
7 = 4.2^x
\log \left(7\right)\ =\ \log \left(4.2\right)^x
Use the Power Property of Logarithms to rewrite the left side of the equation, then divide both sides of the equation by log (4.2).
\log\left(7\right)\ =\ x\cdot\log\left(4.2\right)
3x^5 = 126
3
x^5 = 42
Take the fifth root of both sides to remove the radical.
You may use your calculator.
x\ =\ \sqrt[5]{42}
14 = 2 · 4^x - 10
Solve the equation to isolate
4^x
4^x = 12
Now follow the steps in part (a).
x ≈ 1.792
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Flow Boiling of R134a in Circular Microtubes—Part II: Study of Critical Heat Flux Condition | J. Heat Transfer | ASME Digital Collection
A companion article has been published: Flow Boiling of R134a in Circular Microtubes—Part I: Study of Heat Transfer Characteristics
Basu, S., Ndao, S., Michna, G. J., Peles, Y., and Jensen, M. K. (February 3, 2011). "Flow Boiling of R134a in Circular Microtubes—Part II: Study of Critical Heat Flux Condition." ASME. J. Heat Transfer. May 2011; 133(5): 051503. https://doi.org/10.1115/1.4003160
A detailed experimental study was carried out on the critical heat flux (CHF) condition for flow boiling of R134a in single circular microtubes. The test sections had inner diameters (ID) of 0.50 mm, 0.96 mm, and 1.60 mm. Experiments were conducted over a large range of mass flux, inlet subcooling, saturation pressure, and vapor quality. CHF occurred under saturated conditions at high qualities and increased with increasing mass fluxes, tube diameters, and inlet subcoolings. CHF generally, but not always, decreases with increasing saturation pressures and vapor qualities. The experimental data were mapped to the flow pattern maps developed by Hasan [2005, “Two-Phase Flow Regime Transitions in Microchannels: A Comparative Experimental Study,” Nanoscale Microscale Thermophys. Eng., 9, pp. 165–182] and Revellin and Thome [2007, “A New Type of Diabatic Flow Pattern Map for Boiling Heat Transfer in Microchannels,” J. Micromech. Microeng., 17, pp. 788–796]. Based on these maps, CHF mainly occurred in the annular flow regime in the larger tubes. The flow pattern for the 0.50 mm ID tube was not conclusively identified. Four correlations—the Bowring correlation, the Katto-Ohno correlation, the Thome correlation, and the Zhang correlation—were used to predict the experimental data. The correlations predicted the correct experimental trend, but the mean absolute error (MAE) was high
(>15%)
A new correlation was developed to fit the experimental data with a MAE of 10%.
critical heat flux, R134a, boiling, microscale, boiling, flow measurement, heat transfer, mass transfer, pattern formation, pipe flow, undercooling
Boiling, Critical heat flux, Flow (Dynamics), Subcooling
Critical Heat Flux (CHF) for Water Flow in Tubes—I. Compilation and Assessment of World CHF Data
Study of the Critical Heat Flux Condition with Water and R-123 During Flow Boiling in Microtubes. Part I: Experimental Results and Discussion of Parametric Effects
Evaporative Heat Transfer, Pressure Drop, and Critical Heat Flux in a Small Vertical Tube With R-113
Low-Flow Critical Heat Flux in Microchannels
Investigation of Saturated Critical Heat Flux in a Single, Uniformly Heated Microchannel
An Improved Version of the Generalized Correlation of Critical Heat Flux for the Forced Convective Boiling in Uniformly Heated Vertical Tubes
Experimental Study of Saturated Flow Boiling Critical Heat Flux in Microchannels
,” ASME meeting, Limerick, Ireland, Jun. 19–21.
Phase Change in Microchannel Heat Sinks With Integrated Temperature Sensors
Proceedings of the International Heat Transfer Conference, IHTC14
, Washington DC, Aug. 8–13.
Two-Phase Flow Regime Transitions in Microchannels: A Comparative Experimental Study
A New Type of Diabatic Flow Pattern Map for Boiling Heat Transfer in Microchannels
A Simple but Accurate Round Tube Uniform Heat Flux Dryout Correlation over the Pressure Range 0.7–17 MN/m2
(100–2500 psia),”
, Winfrith, England, Report AEEW-R789.
Correlation of Critical Heat Flux for Flow Boiling of Water in Mini-Channels
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Finding the Probabilities of Dependent and Independent Events | Texas Gateway
Introduction Determining the Probability of Independent Events Determining the Probability of Dependent Events Determining Probability—“And” versus “Or” Summary
You’ve used probability to describe the likelihood of events occurring. In this lesson, you will investigate how you can calculate the probability of two or more events.
Probability is used to describe how likely something is to happen. For example, you may have heard the weather forecaster describe the chances that it will rain today. She is using probability to make that forecast.
Probability is formally defined as the ratio of the number of desired outcomes (what you want to happen) to the number of total possible outcomes (what could possibly happen). Probability is a ratio, so it can be expressed as a fraction, a decimal, or a percent.
The probability of rain today is 80%. That means that there is an 80% chance that the event of rain will occur.
Out of a dozen dyed eggs, there are 2 blue ones. The probability of randomly choosing a blue egg is
\frac{2}{12}
. Since this is only one event occurring, this is called a simple event.
There is a 7 out of 10 chance that someone will catch the flu if they don't get a flu shot. The probability of getting the flu is 0.7.
The relationship between multiple events that occur is important. Sometimes, when an event occurs, it does not affect the chances of the next event(s) occurring. For example, when you roll a pair of number cubes, the number that lands on the top face of one number cube does not affect the number that lands on the top of the second number cube.
When there is an independent relationship between two or more events, those events are called independent events.
Consider the spinner shown.
Use the image of the spinner to determine the probability of the spinner landing on red, aqua, or yellow.
The probabilities that you just identified are for the simple event of the spinner being spun once. Suppose, however, that the spinner is spun twice. Because two simple events will occur, these events become compound events.
You will also notice that the outcomes of these compound events (the spinner being spun twice) are not related to each other. That is, the results of the second spin do not depend on the results of the first spin. Since that is the case, we can call these events independent events.
To determine the probability of a set of independent events, we must first identify the probabilities of each of the events occurring by themselves.
Three friends are using the spinner to play a board game. Drag the fraction that represents the probability of each of the following events to the space indicated. For example, for the first line, drag the fraction representing the probability of the spinner landing on yellow to the box in the Probability of First Event column. Drag the fraction representing the probability of the spinner landing on red to the box in the Probability of Second Event column.
Now, calculate the probability of both events occurring for each player by multiplying the probability of the first event and second event together.
The principle that allows us to calculate the probability of two or more events occurring is also called the Multiplication Rule and is written as follows:
We are also often interested in the probabilities of events resulting from a drawing. If something is drawn, replaced, and then drawn again, the events are independent. This means that the outcome of the second drawing does not depend on the outcome of the first drawing.
Mr. Aimone wrote the names of eight students on slips of paper. He uses the paper slips to randomly draw a student's name to answer questions in his class. After each drawing, Mr. Aimone replaces the slip so that there are always eight slips in the bag. The names of the eight students are as follows:
Sometimes, when you have a series of compound events, the outcome of the first event does affect the outcomes of the subsequent events. These events are called dependent events since the outcome of the second (or third) event depends on the outcome of the first event.
Suppose a bag contains several color tiles: 6 red tiles, 4 green tiles, 3 yellow tiles, and 2 blue tiles.
Ramona reaches into the bag without looking and randomly pulls out one color tile. Use the bag containing the tiles to determine the probability of each color being drawn.
The probabilities that you just identified are for the simple event of one color tile being drawn. Suppose, however, that Ramona draws two tiles at the same time. Because two simple events will occur, these events become compound events.
However, unlike the spinner in the previous section of the lesson, the outcomes of these compound events (two color tiles being drawn at the same time) are dependent on each other. That is, the results of the second tile that is drawn depend on the results of what happened when the first tile was drawn. Since that is the case, we can call these events dependent events.
To determine the probability of a set of dependent events, we must first identify the probabilities of each of the events occurring by themselves. However, the circumstances describing the second event depend on what happened the first time.
In the case of drawing objects, once the first tile has been selected, there are fewer possible outcomes for the second tile (one less). So, we must reduce the denominator in the probability of the second tile by one.
Ramona will draw two tiles from the bag at the same time. Drag the fraction that represents the probability of each of the following events to the space indicated. For example, for the first line, drag the fraction representing the probability of Ramona drawing a red tile first to the box in the Probability of First Event column. Drag the fraction representing the probability of Ramona drawing a green tile second to the box in the Probability of Second Event column.
Hint: The number of tiles left after the first tile has been selected has decreased by one.
You will also see the words “without replacement” in problems involving the drawing of two objects. This phrase is a clue that you have a situation with dependent events.
Ms. Dawson’s 1st period science class has 18 girls and 12 boys. Ms. Dawson needs to select a pair of students to help her set up a laboratory investigation. She has decided to randomly select 2 students from the entire class.
In the previous two sections, you have investigated situations where two events took place. What happens to the probability if we are only concerned about whether or not a combination of choices could happen? As long as these choices could not happen together, they are called mutually exclusive events. There are shortcuts for determining the probability of these events.
Use the spinner to determine the probability of the spinner landing on red, blue, yellow, or orange.
To determine the probability of a set of mutually exclusive events, we must first identify the probabilities of each of the events occurring by themselves.
Now, calculate the probability of both events occurring for each player by adding the probability of the first event and second event together.
The principle that allows us to calculate the probability of two or more mutually exclusive events occurring is also called the Addition Rule and is written as follows:
Brandy and her sister are playing a card memory game. Brandy wrote the letters shown on index cards.
Brandy will turn the cards over and shuffle them, then randomly draw a card from the pile.
In this lesson, you investigated different ways to compute the probability of two or more events occurring.
Mutually exclusive events are two or more events that cannot both happen at the same time. You can calculate the probability of a set of mutually exclusive events by using the Addition Rule of Probability as follows:
Independent events are two or more events that occur in sequence where the outcome of the first event does not affect the outcome of the events that follow. You can calculate the probability of a series of independent events by using the Multiplication Rule of Probability as follows:
Dependent events are two or more events that occur in sequence where the outcome of the first event does affect the outcome of the events that follow. You can calculate the probability of a series of dependent events by using the Multiplication Rule of Probability as follows:
P(A and B) = P(A) × P(B|A)
The flowchart shown can be used to help you determine how to approach different probability problems.
Distinguishing Between Independent and Dependent Events
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Periodic_sequence Knowpia
In mathematics, a periodic sequence (sometimes called a cycle[citation needed]) is a sequence for which the same terms are repeated over and over:
a1, a2, ..., ap, a1, a2, ..., ap, a1, a2, ..., ap, ...[citation needed]
The number p of repeated terms is called the period (period).[1]
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a1, a2, a3, ... satisfying One size fits all
[2] for all values of n.[1][3][4][5][6] If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.[citation needed] The smallest p for which a periodic sequence is p-periodic is called its least period[1][7] or exact period.[7][verification needed]
Every constant function is 1-periodic.[5]
{\displaystyle 1,2,1,2,1,2\dots }
is periodic with least period 2.[3]
The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:
{\displaystyle {\frac {1}{7}}=0.142857\,142857\,142857\,\ldots }
More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).[8][verification needed]
{\displaystyle -1,1,-1,1,-1,1,\ldots }
More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.[citation needed]
A periodic point for a function f : X → X is a point x whose orbit
{\displaystyle x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots }
is a periodic sequence. Here,
{\displaystyle f^{n}(x)}
means the n-fold composition of f applied to x.[7][verification needed] Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.[citation needed]
{\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n}}
Where k and m<p are natural numbers.[citation needed]
Partial ProductsEdit
{\displaystyle \prod _{n=1}^{kp+m}a_{n}=({\prod _{n=1}^{p}a_{n}})^{k}*\prod _{n=1}^{m}a_{n}}
Periodic 0, 1 sequencesEdit
{\displaystyle \sum _{k=1}^{1}\cos(-\pi {\frac {n(k-1)}{1}})/1=1,1,1,1,1,1,1,1,1...}
{\displaystyle \sum _{k=1}^{2}\cos(2\pi {\frac {n(k-1)}{2}})/2=0,1,0,1,0,1,0,1,0...}
{\displaystyle \sum _{k=1}^{3}\cos(2\pi {\frac {n(k-1)}{3}})/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...}
{\displaystyle ...}
{\displaystyle \sum _{k=1}^{N}\cos(2\pi {\frac {n(k-1)}{N}})/N=0,0,0...,1{\text{ sequence with period }}N}
[citation needed][clarification needed]
A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:
1 / 56 = 0 . 0 1 7 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 ...[citation needed]
A sequence is ultimately periodic if it satisfies the condition
{\displaystyle a_{k+r}=a_{k}}
for some r and sufficiently large k.[1]
A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x1, x2, x3, ... is asymptotically periodic if there exists a periodic sequence a1, a2, a3, ... for which
{\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}
[5][9][10][verification needed]
1 / 3, 2 / 3, 1 / 4, 3 / 4, 1 / 5, 4 / 5, ...
is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....[citation needed]
^ a b c d "Ultimately periodic sequence - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 February 2011. Retrieved 13 August 2021. {{cite web}}: CS1 maint: url-status (link)
^ John cDNA
^ a b Weisstein, Eric W. "Periodic Sequence". mathworld.wolfram.com. Retrieved 2021-08-13.
^ Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). www.math.ru.nl. Retrieved 13 August 2021. {{cite web}}: CS1 maint: url-status (link)
^ a b c Janglajew, Klara; Schmeidel, Ewa (2012-11-14). "Periodicity of solutions of nonhomogeneous linear difference equations". Advances in Difference Equations. 2012 (1): 195. doi:10.1186/1687-1847-2012-195. ISSN 1687-1847. S2CID 122892501.
^ Menezes, Alfred J.; Oorschot, Paul C. van; Vanstone, Scott A. (2018-12-07). Handbook of Applied Cryptography. CRC Press. ISBN 978-0-429-88132-9.
^ a b c Weisstein, Eric W. "Least Period". mathworld.wolfram.com. Retrieved 2021-08-13.
^ Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021. {{cite web}}: CS1 maint: url-status (link)
^ Cheng, SuiSun (2017-09-29). New Developments in Difference Equations and Applications: Proceedings of the Third International Conference on Difference Equations. Routledge. ISBN 978-1-351-42880-4.
^ Shlezinger, Nir; Todros, Koby (2019-01-01). "Performance analysis of LMS filters with non-Gaussian cyclostationary signals" (PDF). Signal Processing. 154: 260–271. arXiv:1708.00635. doi:10.1016/j.sigpro.2018.08.008. ISSN 0165-1684. S2CID 53521677.
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Belite - Wikipedia
This article is about the mineral. For the aircraft manufacturer, see Belite Aircraft.
Belite is an industrial mineral important in Portland cement manufacture. Its main constituent is dicalcium silicate, Ca2SiO4, sometimes formulated as 2 CaO · SiO2 (C2S in cement chemist notation).
The name was given by Törnebohm in 1897 to a crystal identified in microscopic investigation of Portland cement.[1] Belite is a name in common use in the cement industry, but is not a recognised mineral name. It occurs naturally as the mineral larnite, the name being derived from Larne, Northern Ireland, the closest town to Scawt Hill where it was discovered.[2]
Simplified crystal structure of belite
The belite found in Portland cement differs in composition from pure dicalcium silicate. It is a solid solution and contains minor amounts of other oxides besides CaO and SiO2. A typical composition:[3]
Based on this, the formula can be expressed as Ca1.94Mg0.02Na0.01K0.03Fe0.02Al0.07Si0.90P0.01O3.93. In practice, the composition varies with the bulk composition of the clinker, subject to certain limits. Substitution of calcium ions or orthosilicate ions requires that electric charges be kept in balance. For instance, a limited number of orthosilicate (SiO4−
4) ions can be replaced with sulfate (SO2−
4) ions, provided that for each sulfate ion, two aluminate (AlO5−
4) ions are also substituted.
Polymorphs[edit]
Dicalcium silicate is stable, and is readily prepared from reactive CaO and SiO2 at 300 °C. The low temperature form is γ-belite, or lime olivine. This form does not hydrate, and is avoided in cement manufacture.
As the temperature rises, it passes through several polymorphic states:
>1425 α Hexagonal
1160-1425 α'H Orthorhombic
680-1160 α'L Orthorhombic
500-680 β Monoclinic
<500 γ Orthorhombic
Hydration[edit]
Belite is the mineral in Portland cement responsible for development of "late" strength. The other silicate, alite contributes "early" strength, due to its higher reactivity. Belite reacts with water (roughly) to form calcium silicate hydrates (C-S-H) and portlandite (Ca(OH)2) according to the reaction:
{\displaystyle {\ce {\underbrace{{2 Ca2SiO4}}_{{Belite}}+ \underbrace{{4 H2O}}_{{Water}}-> 3 CaO . 2 SiO2. 3 H2O + \underbrace{{Ca(OH)2}}_{{Portlandite}}}}}
This rapid reaction is "chemically analogue" to the slow natural hydration of forsterite (the magnesium end-member of olivine) leading to the formation of serpentine and brucite in nature, although the kinetic of hydration of poorly crystallized artificial belite is much faster than the slow weathering of well crystallized Mg-olivine under natural conditions.
{\displaystyle {\ce {\underbrace{{2 Mg2SiO4}}_{{Forsterite}}+\underbrace{{3H2O}}_{{Water}}-> \underbrace{{Mg3Si2O5(OH)4}}_{{serpentine }}+ \underbrace{{Mg(OH)2}}_{{brucite }}}}}
The hydrate phase, [3 CaO · 2 SiO2 · 3 H2O], is referred to as the "C-S-H" phase. It grows as a mass of interlocking needles that provide the strength of the hydrated cement system. Relatively high belite reactivity is desirable in Portland cement manufacture, and the formation of the unreactive γ-form must be rigorously avoided. This is achieved by rapid cooling, forming crystals that are small, distorted and highly defective. Defects provide sites for initial water attack. Failure to cool the clinker rapidly leads to inversion of belite to the γ-form. The γ-form has a substantially different structure and density, so that inversion leads to degradation of the crystal and its surrounding matrix, and can also trigger decomposition of the neighboring alite. This is observed macroscopically as "dusting": the clinker nodules fall to a fine dust.
Clinker section 0.15 x 0.15 mm
The minerals in Portland cement clinker may be observed and quantified by petrographic microscopy. Clinker nodules are cut and ground to a flat, polished surface. The exposed minerals are made visible and identifiable by etching the surface. The surface can then be observed in reflected light by optical microscopy. In the example, a clinker nodule has been polished and etched with hydrogen fluoride vapour. The alite shows as brown, the belite as blue, and the melt phases as white. Electron microscopy can also be used, in which case the minerals may be identified by microprobe analysis. The preferred method to quantify the minerals accurately is X-ray diffraction on the powdered clinker, using the Rietveld analysis technique. Belite is much harder to grind in a cement mill than alite.
Hydration reaction of forsterite (olivine) in serpentinisation
CCN, cement chemist notation
^ Jean-Pierre Bournazel, Yves Malier, Micheline Moranville Regourd, 1998, Concrete, from Material to Structure RILEM Publications, ISBN 2-912143-04-7.
^ Deer, William Alexander; Howie, R. A; Zussman, J (1997-05-01). "Larnite". Disilicates and ring silicates. pp. 248–249. ISBN 9781897799895.
^ Taylor H.F.W. (1990), Cement Chemistry, Academic Press, 1990, ISBN 0-12-683900-X, pp. 10-11.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Belite&oldid=1054763940"
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Mathematics 2018 Past Questions (Theory) | WAEC
A used car was purchased at N900,000.00. Its value depreciated by 30% in the first year. In each subsequent year, the depreciation was 22% of its value at the beginning of the year. If the car was bought on the 1st of March, 2011, calculate, correct to the nearest hundred naira, the value of the car on the 28th of February, 2015.
{}^{}
2. (a) The graph of
y=2px²−p²x−14
passes through the point (3, 10). Find the values of p.
(b) Two lines,
3y-2x=21
4y+5x=5
intersect at the point Q. Find the coordinates of Q.
3. (a) The diagonals of a rhombus are 10.2 cm and 9.3 cm long. Calculate, correct to one decimal place, the perimeter of the rhombus.
(b) Given that sinx=3/5, 0°< x <90°, find the value of 5cosx−4tanx
(i) The value of x; (ii) <RSQ
(b) If 2N4seven=15Nnine,
5. (a) If the mean of m, n, s, p and q is 12, calculate the mean of (m + 4), (n – 3), (s + 6), (p – 2) and (q + 8).
(b) In a community of 500 people, the 75th percentile age is 65 years while the 25th percentile age is 15 years. How many of the people are between 15 and 65 years?
6. In a road worthiness test on 240 cars, 60% passed. The number that failed had faults in Clutch, Brakes and Steering as follows: Clutch only – 28, Clutch and Steering – 14; Clutch, Steering and Brakes – 8; Clutch and Brakes – 20; Brakes and Steering only – 6. The number of cars with faults in Steering only is twice the number of cars with faults in Brakes only.
(b) How many cars had : (i) Faulty Brakes? (ii) Only one fault?
7. (a) Find the equation of the line passing through the points (2, 5) and (-4, -7).
(b) Three ships P, Q and R are at sea. The bearing of Q from P is 030° and the bearing of P and R is 300°. If |PQ| = 5 km and |PR| = 8 km,
(i) Illustrate the information in a diagram.
(ii) Calculate, correct to three significant figures, the:
(1) distance between Q and R
(2) bearing of R from Q.
8. (a) Lamin bought a book for N300.00 and sold it to Bola at a profit of x%. Bola then sold the same book at a profit of x%. If James paid
N\left(6x+\frac{3}{4}\right)
more for the book than Lamin paid, find the value of x.
(b) Find the range of values of x which satisfies the inequality
3x-2<10+x<2+5x
In the diagram, |PT| = 4 cm, |TS| = 6 cm, |PQ| = 6 cm and < SPR = 30°. Calculate, correct to the nearest whole number:
(a) |SR| ;
(b) area of TQRS.
90°, Find, correct to three significant figures, |PR|.
(b) The length of two ladders, L and M are 10m and 12m respectively. They are placed against a wall such that each ladder makes angle with the horizontal ground. If the foot of L is 8m from the foot of the wall.
(i) Draw a diagram to illustrate this information; (ii) Calculate the height at which M touches the wall.
11. (a) Copy and complete the table of values for y=2x²+x−10 for −5≤x≤4
x -5 -4 -3 -2 -1 0 1 2 3 4
y 5 -9 -10 0
(b) Using scales of 2cm to 1 unit on the x- axis and 2cm to 5 units on the y- axis, Draw the graph of y=2x²+x−1 for −5≤x≤4.
(c) Use the graph to find the solution of :
(i) 2x²+x=10
(ii) 2x²+x−10=2x
Frequency 1 1 x 5 y 1 4 3 1
The frequency distribution shows the marks distribution of a class of 30 students in an examination.
The mean mark of the distribution is 52.
(a) Find the values of x and y.
(b) Construct a group frequency distribution table starting with a lower class limit of 1 and class interval of 10.
(c) Draw a histogram for the distribution
(d) Use the histogram to estimate the mode
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Radical Functions - Course Hero
College Algebra/Rational and Radical Functions/Radical Functions
A radical function is one whose rule can be written using an expression with a variable under a root, such as a square root or cube root:
f(x)=\sqrt[\scriptsize{n}]{p(x)}
There is an inverse relationship with power functions that have whole-number powers for simple radical functions of the form:
f(x)=\sqrt[\scriptsize{n}]{x}
It is a reflection of the graph across the line
y=x
f(x)=x^n
For even functions, the inverse relationship holds only for values of the power function with
x\geq0
Power Functions and Radicals
Even Power Functions and Radicals
Odd Power Functions and Radicals
A function and its inverse are reflections of each other across the line
y=x
. Whether the inverse of a power function of the form
f(x)=x^n
is a function depends on the value of
n
n
is positive and even, the inverse is not a function unless the domain is restricted to
x\geq0
n
is positive and odd, the inverse is a function, and the domain does not need to be restricted.
Radical functions can be difficult to graph by hand. Even when radical functions are graphed using technology, it can be useful to analyze the functions to understand the properties of the graphs.
For some calculators, it may be difficult to enter roots other than square and cube roots. In such cases, it may be necessary to use rational exponents. Taking the
n
th root of an expression is the same as raising it to the power of
\frac{1}{n}
\sqrt[\scriptsize{n}]{x}={x^{\footnotesize{\frac{1}{n}}}}
f(x)=\sqrt[\scriptsize{4}]{3x^2+4x-1}
To graph the function, it may need to be entered as:
(3x^2+4x-1)^{\footnotesize{\frac{1}{4}}}
Graph and then analyze the function:
f(x)=\sqrt{x^2-4x-5}
Look at the expression under the radical. Values of
x
that make this expression negative are not in the domain of
because the square root of a negative value is not a real number.
Graph the given function together with:
g(x)=x^2-4x-5
g
is quadratic and can be factored as:
(x-5)(x+1)
The zeros are –1 and 5. The values are negative for
-1<x<5
. So, the domain is restricted to values outside that interval.
Identify the domain and range of
x\leq-1
x\geq5
The graph shows that all the
y
-values are on or above the
x
-axis, so the range is
y\geq0
Analyze the function. Identify maxima (highest points), minima (lowest points), and zeros. Describe where the function is increasing and decreasing.
The graph shows the minimum value of zero, which occurs at
x=-1
x=5
. There is no maximum.
The zeros of the function are also at
x=-1
x=5
The function is decreasing for
x<-1
x>5
<Rational Functions>Solving Rational and Radical Equations
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