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Mathematical models and dynamic contact analysis of involute/noninvolute beveloid gears
This study investigates an approach for parametric modeling and dynamic contact analysis of involute/noninvolute beveloid gears. Firstly, the mathematical models of involute/noninvolute beveloid gear
pairs are derived based on the theory of gearing and the generation mechanism. Then the parametric modeling programs of involute/noninvolute beveloid gears are developed to automatically generate
exact model via a Matlab code. Subsequently, a numerical example of intersecting axes beveloid gears is presented to evaluate the dynamic stress distribution and dynamic transmission error. Finally,
the dynamic contact characteristics of involute and noninvolute beveloid gears are calculated by three-dimensional dynamic contact finite element method, respectively. The results show that the
noninvolute beveloid gear pairs can relieve the high dynamic stress and contact shock problem of intersecting axes beveloid gear pairs.
1. Introduction
In modern gearing application, spiral bevel gears, hypoid gears, worm gears and face gears can easily realize the non-parallel axes power transmission, and they represent good meshing characteristic
at large shaft angles, close to 90°. However, it is not practical to use them when the shaft angle is less than 45°, due to design and manufacturing difficulties. Beveloid gears, by contrast, also
known as conical gears with perfect meshing characteristics, are suitable for power transmission not only between parallel axes but between non-parallel axes [1, 2].
Recently, a number of studies have been performed on the beveloid gears. Brauer derived the parametric equations for a straight conical involute gear tooth surface [3], and these formulas were used
to create a finite element model [4], then reported on a theoretical study of transmission errors in involute conical gear transmissions [5]. Liu, Ye and Chen established the mathematical models of
variable thickness involute gears and the contact characteristic was also studied [6-8]. Chen solved the tooth face equations of helical involute beveloid gear, and the precise tooth surface was
generated utilizing Matlab [9]. Wu proposed an approach for geometrical design and contact stress analysis of skew conical involute gear drives in approximate line contact [10]. Li investigated the
tooth profile equation of noninvolute beveloid gears and also calculated the tooth profile errors and axial errors [11]. In spite of the above-mentioned studies focusing on meshing theory and
simulation of involute or noninvolute beveloid gears, there are very few published works aimed to solve the engagement equation and tooth profile equation of noninvolute beveloid gears between
intersecting axes by means of tooth face equation of the imaginary helical rack cutter. Furthermore, none also made a comparison between dynamic contact characteristics of involute beveloid gear pair
and noninvolute beveloid gear pair. These gaps will be the emphasis of this paper.
2. Mathematical model of the involute beveloid gear pair
2.1. Transverse tooth profile equation of helical rack cutter
Considering the meshing relationship between imaginary rack cutter and gear blank, we can obtain the tooth profile equation of involute beveloid gear basing on the tooth profile equation of the rack
cutter in transverse cross section. Therefore, the parameters of rack cutter in the transverse cross section should be calculated first.
The relative position relationship between rack cutter and gear blank is shown in Fig. 1. Here, ${r}^{"}$ means gear pitch radius, ${r}^{"}\omega$ is linear velocity of gear and the pitch plane of
imaginary rack cutter is set to form an inclination angle $\delta$ with respect to the pitch plane of gear.
Fig. 1Relative position relationship
Fig. 2 describes the geometry of normal tooth profile of rack cutter. ${S}_{n}\left({X}_{n},{Y}_{n},{Z}_{n}\right)$ means fixed coordinate system linked with normal cross section of rack cutter. The
geometry of normal tooth profile of rack cutter, which mainly contains straight edge $BC$ and tool fillet curve $AB$, is represented. The $BC$ and $AB$ generate the working tooth surface and fillet
surfaces of involute beveloid gear, respectively.
Fig. 2Normal tooth profile of rack cutter
The equation of $BC$ in coordinate system ${S}_{n}$ can be described as:
${\mathbf{R}}_{n}^{z}\left(l\right)=\left[\begin{array}{c}lcos{\alpha }_{n}-{h}_{an}^{*}{m}_{n}\\ ±\left(-l\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{n}+{h}_{an}^{*}{m}_{n}\mathrm{t}\mathrm{a}\mathrm
{n}{\alpha }_{n}+\frac{\pi {m}_{n}}{4}\right)\\ 0\\ 1\end{array}\right],$
where ${\mathbf{R}}_{n}^{z}$ is position vector of arbitrary point on $BC$ in coordinate system ${S}_{n}$; ${\alpha }_{n}$ is the normal pressure angle; ${m}_{n}$ is normal module; ${h}_{an}^{*}$
represents the addendum coefficient; $l$ is the distance between moving point on $BC$ and point $B$; the upper and lower signs of “±” are used to describe the left and right straight edge of rack
The equation of $AB$ in coordinate system ${S}_{n}$ can be described as:
${\mathbf{R}}_{n}^{j}\left(\theta \right)=\left[\begin{array}{c}-{h}_{an}^{*}{m}_{n}-\rho cos\theta +\rho sin{\alpha }_{n}\\ ±\left({h}_{an}^{*}{m}_{n}\mathrm{t}\mathrm{a}\mathrm{n}{\alpha }_{n}+\
frac{\pi {m}_{n}}{4}+\rho \mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{n}-\rho \mathrm{s}\mathrm{i}\mathrm{n}\theta \right)\\ 0\\ 1\end{array}\right],$
where ${\mathbf{R}}_{n}^{j}$ is position vector of arbitrary point on $AB$ in coordinate system ${S}_{n}$; $\theta$ denotes the central angle between moving point on $AB$ and point $A$; $\rho$
represents the radius of fillet.
Using the coordinate transformation from ${S}_{n}\left({X}_{n},{Y}_{n},{Z}_{n}\right)$ to ${S}_{t}\left({X}_{t},{Y}_{t},{Z}_{t}\right)$ which obtains from rotating ${S}_{n}\left({X}_{n},{Y}_{n},{Z}_
{n}\right)$ around the ${X}_{n}$ axis with an angle $\beta$, the coordinate system ${S}_{d}\left({X}_{d},{Y}_{d},{Z}_{d}\right)$ will be got though rotating ${S}_{t}$ around the ${Y}_{t}$ axis with
an angle $\delta$, and they are illustrated in Fig. 2. Combining Eqs. (1) and (2), we can gain the rack cutter equations of straight edge $BC$ and tool fillet curve $AB$ in the ${X}_{t}{O}_{t}{Z}_{t}
$ plane of ${S}_{t}$ as follows:
$\left\{\begin{array}{l}{\mathbf{R}}_{t}^{z}\left(l\right)={\left[\begin{array}{cccc}{R}_{nx}^{z}& \frac{{R}_{ny}^{z}}{\mathrm{c}\mathrm{o}\mathrm{s}\beta }& 0& 1\end{array}\right]}^{T},\\ {\mathbf
{R}}_{t}^{j}\left(\theta \right)={\left[\begin{array}{cccc}{R}_{nx}^{j}& \frac{{R}_{ny}^{j}}{\mathrm{c}\mathrm{o}\mathrm{s}\beta }& 0& 1\end{array}\right]}^{T}.\end{array}\right\$
Then the rack cutter equations of straight edge $BC$ and tool fillet curve $AB$ in the ${X}_{d}{O}_{d}{Z}_{d}$ plane of coordinate system ${S}_{d}$ are as follows:
$\left\{\begin{array}{l}{\mathbf{R}}_{d}^{z}\left(l\right)={\left[\begin{array}{cccc}\frac{{R}_{nx}^{z}}{\mathrm{c}\mathrm{o}\mathrm{s}\delta }& \frac{{R}_{ny}^{z}}{\mathrm{c}\mathrm{o}\mathrm{s}\
beta }+{R}_{nx}^{z}\mathrm{t}\mathrm{a}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\delta & 0& 1\end{array}\right]}^{T},\\ {\mathbf{R}}_{d}^{j}\left(\theta \right)={\left[\begin{array}{cccc}\frac
{{R}_{nx}^{j}}{\mathrm{c}\mathrm{o}\mathrm{s}\delta }& \frac{{R}_{ny}^{j}}{\mathrm{c}\mathrm{o}\mathrm{s}\beta }+{R}_{nx}^{j}\mathrm{t}\mathrm{a}\mathrm{n}\beta \mathrm{t}\mathrm{a}\mathrm{n}\delta &
0& 1\end{array}\right]}^{T}.\end{array}\right\$
2.2. Coordinate transformation of rack cutter
Fig. 3 displays the coordinate systems of rack cutter. Herein, the coordinate system ${S}_{d}$ moving a given translational distance $u$ along negative direction of ${Z}_{d}$ axis and rotating it
around the ${X}_{d}$ axis with an angle $\beta$, the coordinate system ${S}_{c}\left({X}_{c},{Y}_{c},{Z}_{c}\right)$ will be got by rotating ${S}_{p}$ around the ${Y}_{p}$ axis with $\delta$.
Fig. 3Coordinate systems of rack cutter
In the coordinate system ${S}_{c}$, the tooth surface equation of rack cutter can be expressed as:
where ${\mathbf{R}}_{c}$ and ${\mathbf{R}}_{d}$ are position vector of rack cutter in coordinate system ${S}_{c}$ and ${S}_{d}$, respectively; ${\mathbf{M}}_{pd}$ and ${\mathbf{M}}_{cp}$ are
coordinate transformation matrix.
Combining Eqs. (4) and (5), we can obtain the position vector of $AB$ and $AB$ in coordinate system ${S}_{c}$ as follows:
${\mathbf{R}}_{c}^{z}=\left[\begin{array}{c}{R}_{dx}^{z}+ucos\beta sin\delta \\ {R}_{dy}^{z}+usin\beta \\ ucos\beta cos\delta \\ 1\end{array}\right],{\mathbf{R}}_{c}^{j}=\left[\begin{array}{c}{R}_
{dx}^{j}+ucos\beta sin\delta \\ {R}_{dy}^{j}+usin\beta \\ ucos\beta cos\delta \\ 1\end{array}\right].$
2.3. Meshing equation
In coordinate system ${S}_{c}$, equations of arbitrary point on $AB$ and $AB$ are the function which depends on the variables $l/u$ and $\theta /u$, respectively. The unit normal vectors of the rack
cutter can be obtained as:
$\frac{\frac{\partial {\mathbf{R}}_{c}^{z}}{\partial l}×\frac{\partial {\mathbf{R}}_{c}^{z}}{\partial u}}{\left|\frac{\partial {\mathbf{R}}_{c}^{z}}{\partial l}×\frac{\partial {\mathbf{R}}_{c}^{z}}{\
partial u}\right|}=\left[\begin{array}{c}{n}_{cx}^{z}\\ {n}_{cy}^{z}\\ {n}_{cz}^{z}\end{array}\right],\frac{\frac{\partial {\mathbf{R}}_{c}^{j}}{\partial \theta }×\frac{\partial {\mathbf{R}}_{c}^{j}}
{\partial u}}{\left|\frac{\partial {\mathbf{R}}_{c}^{j}}{\partial \theta }×\frac{\partial {\mathbf{R}}_{c}^{j}}{\partial u}\right|}=\left[\begin{array}{c}{n}_{cx}^{j}\\ {n}_{cy}^{j}\\ {n}_{cz}^{j}\
Coordinate relationship between the rack cutter and involute beveloid gear is illustrated in Fig. 4. ${S}_{b}\left({X}_{b},{Y}_{b},{Z}_{b}\right)$ is a spatial fixed coordinate system. ${S}_{j}\left
({X}_{j},{Y}_{j},{Z}_{j}\right)$ is a moving coordinate system attached to gear blank (involute beveloid gear), which forms an angle ${\phi }_{1}$ with respect to the coordinate system ${S}_{b}\left
Fig. 4Coordinate relationship between rack cutter and involute beveloid gear
The angular velocity vector revolving around the ${Z}_{j}$ axis of involute beveloid gear can be represented as:
${\mathbf{\omega }}_{1}=-{\omega }_{1}{\mathbf{k}}_{j}.$
Assuming that the position vector of one point $K$ on rack cutter is described as:
$\stackrel{\to }{{O}_{c}K}={\mathbf{r}}_{c}={x}_{c}{\mathbf{i}}_{c}+{y}_{\text{c}}{\mathbf{j}}_{c}+{z}_{\text{c}}{\mathbf{k}}_{c}.$
Then we can obtain that:
$\stackrel{\to }{PK}=\stackrel{\to }{P{O}_{c}}+\stackrel{\to }{{O}_{c}K}={x}_{c}{\mathbf{i}}_{c}+\left({y}_{c}-{r}^{"}{\phi }_{1}\right){\mathbf{j}}_{c}+{z}_{c}{\mathbf{k}}_{c}.$
Combining Eqs. (8) and (10), the relative velocity vector of one point $K$ on rack cutter is:
${\mathbf{v}}_{12}^{K}={\mathbf{\omega }}_{\text{1}}×\stackrel{\to }{PK}={\omega }_{\text{1}}\left[\left({y}_{c}-{r}^{"}{\phi }_{1}\right){\mathbf{i}}_{c}+{x}_{c}{\mathbf{j}}_{c}\right].$
According to differential geometry and the kinematics of gear geometry [12], the continuous tangency is detected by the equation of meshing which is formulated as:
${\mathbf{n}}^{K}\cdot {\mathbf{v}}_{12}^{K}=0,$
where the unit normal vectors of rack cutter ${\mathbf{n}}^{K}$ is ${\mathbf{n}}^{K}={\left(\begin{array}{ccc}{n}_{x}^{K}& {n}_{y}^{K}& {n}_{z}^{K}\end{array}\right)}^{T}$.
Then the motion parameter (rotation angle) for involute beveloid gear generation is:
${\phi }_{1}=\frac{{y}_{c}}{{r}^{"}}+\frac{{n}_{y}^{K}{x}_{c}}{{n}_{x}^{K}{r}^{\text{'}}}.$
Substituting Eqs. (6) and (7) into Eq. (13) enables us to solve the rotation angle of straight edge $BC$ and tool fillet curve $AB$ for rack cutter:
${\phi }_{1}^{z}=\frac{{R}_{cy}^{z}}{{r}^{"}}+{n}_{cy}^{z}\cdot \frac{{R}_{cx}^{z}}{{n}_{cx}^{z}\cdot {r}^{\text{'}}},\mathrm{}{\phi }_{1}^{j}=\frac{{R}_{cy}^{j}}{{r}^{"}}+{n}_{cy}^{j}\cdot \frac{{R}
_{cx}^{j}}{{n}_{cx}^{j}\cdot {r}^{\text{'}}}.$
2.4. Tooth profile equation of involute beveloid gear
In coordinate system ${S}_{j}$, the tooth surface equation of involute beveloid gear is expressed as:
$\left[\begin{array}{c}{R}_{jx}^{z}\\ {R}_{jy}^{z}\\ {R}_{jz}^{z}\\ 1\end{array}\right]={\mathbf{M}}_{jc}\left[\begin{array}{c}{R}_{cx}^{z}\\ {R}_{cy}^{z}\\ {R}_{cz}^{z}\\ 1\end{array}\right]=\left[\
begin{array}{c}\left({R}_{cx}^{z}+{r}^{"}\right)cos{\phi }_{1}^{z}+\left({r}^{"}{\phi }_{1}^{z}-{R}_{cy}^{z}\right)sin{\phi }_{1}^{z}\\ \left({R}_{cx}^{z}+{r}^{"}\right)sin{\phi }_{1}^{z}-\left({r}^
{"}{\phi }_{1}^{z}-{R}_{cy}^{z}\right)cos{\phi }_{1}^{z}\\ {R}_{cz}^{z}\\ 1\end{array}\right],$
where ${\mathbf{M}}_{jc}$ is coordinate transformation matrix.
Similarly, we can also obtain the fillet equation of involute beveloid gear:
$\left[\begin{array}{c}{R}_{jx}^{j}\\ {R}_{jy}^{j}\\ {R}_{jz}^{j}\\ 1\end{array}\right]=\left[\begin{array}{c}\left({R}_{cx}^{j}+{r}^{"}\right)cos{\phi }_{1}^{j}+\left({r}^{"}{\phi }_{1}^{j}-{R}_{cy}
^{j}\right)sin{\phi }_{1}^{j}\\ \left({R}_{cx}^{j}+{r}^{"}\right)sin{\phi }_{1}^{j}-\left({r}^{"}{\phi }_{1}^{j}-{R}_{cy}^{j}\right)cos{\phi }_{1}^{j}\\ {R}_{cz}^{j}\\ 1\end{array}\right].$
Hence, the mathematical models of the involute beveloid gear have been derived.
3. Mathematical model of noninvolute beveloid gear pair
3.1. Coordinate transformation
According to the differential geometry and the kinematics of gear geometry, the tooth profile equation and meshing equation of noninvolute beveloid gear can be mathematically generated from a
mutually conjugate involute beveloid gear. Fig. 5 describes the relationship between involute beveloid gear cutter and gear blank (noninvolute beveloid gear), including the involute beveloid gear
Gear_1, the noninvolute beveloid gear Gear_2 and the intersection angle $\mathrm{\Sigma }$.
The coordinate system ${S}_{3}\left({X}_{3},{Y}_{3},{Z}_{3}\right)$ and ${S}_{4}\left({X}_{4},{Y}_{4},{Z}_{4}\right)$ are fixed coordinate systems linked with Gear_1 and Gear_2, respectively, and the
coordinate system ${S}_{1}\left({X}_{1},{Y}_{1},{Z}_{1}\right)$ and ${S}_{2}\left({X}_{2},{Y}_{2},{Z}_{2}\right)$ are motion coordinate systems joined to Gear_1 and Gear_2, respectively as
illustrated in Fig. 6.
Fig. 5Position relationship between the involute beveloid gear cutter and gear blank
Under the initial position, the coordinate system ${S}_{1}$ overlaps with ${S}_{3}$ and the coordinate system ${S}_{2}$ coincides with ${S}_{4}$. The Gear_2 rotates clockwise around the ${Z}_{4}$
axis with an angle ${\phi }_{2}$, while the Gear_1 is revolving clockwise around the ${Z}_{3}$ axis with an angle ${\phi }_{1}$.
Fig. 6Coordinate systems of Gear_1 and Gear_2
Consequently, coordinate transformation matrix ${M}_{21}\text{,}$ which is utilized to perform the kinematical relationship between the involute beveloid gear cutter and the generated gear
(noninvolute beveloid gear), can be expressed as ${M}_{21}={M}_{24}{M}_{43}{M}_{31}$, where:
${M}_{24}=\left[\begin{array}{cccc}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}& -\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}& 0& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}& \mathrm{c}\mathrm{o}\mathrm
{s}{\phi }_{2}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],{M}_{43}=\left[\begin{array}{cccc}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }& 0& -\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }&
0\\ 0& 1& 0& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }& 0& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }& 0\\ 0& 0& 0& 1\end{array}\right],$${M}_{31}=\left[\begin{array}{cccc}\mathrm{c}\
mathrm{o}\mathrm{s}{\phi }_{1}& -\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}& 0& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}& \mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1
3.2. Relative velocity and unit normal vectors of involute beveloid gear cutter
Assuming that angular velocity vector clockwise revolving around the axis of Gear_1 is ${\omega }_{1}$ and the angular velocity vector anticlockwise revolving around the axis of Gear_2 is ${\omega }_
{2}$. Consequently, the angular velocity vector can be represented based on three components of unit vector in coordinate system ${S}_{3}\left({X}_{3},{Y}_{3},{Z}_{3}\right)$:
${\mathbf{\omega }}_{1}={\omega }_{1}{\mathbf{k}}_{3},\mathrm{}\mathrm{}\mathrm{}\mathrm{}{\mathbf{\omega }}_{2}=-{\omega }_{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }{\mathbf{k}}_{3}-{\omega }
_{2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }{\mathbf{i}}_{3}.$
The position vector of one contact point $M$ is described as:
Then the relative velocity at the point $M$ can be expressed as:
${\mathbf{v}}_{12}=\left({\mathbf{\omega }}_{1}-{\mathbf{\omega }}_{2}\right)×{\mathbf{r}}_{3}.$
Substituting Eqs. (18) and (19) into Eq. (20) enables us to obtain the relative velocity:
${\mathbf{v}}_{12}^{3}=\left(\begin{array}{ccc}{\mathbf{i}}_{3}& {\mathbf{j}}_{3}& {\mathbf{k}}_{3}\\ {\omega }_{2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }& 0& {\omega }_{1}+{\omega }_{2}\
mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\\ {X}_{3}& {Y}_{3}& {Z}_{3}\end{array}\right)=\left(\begin{array}{c}\left[-{Y}_{3}\left({\omega }_{1}+{\omega }_{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm
{\Sigma }\right)\right]{\mathbf{i}}_{3}\\ \left[-{Z}_{3}{\omega }_{2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }+{X}_{3}\left({\omega }_{1}+{\omega }_{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\
Sigma }\right)\right]{\mathbf{j}}_{3}\\ \left[{Y}_{3}{\omega }_{2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\right]{\mathbf{k}}_{3}\end{array}\right).$
Substituting ${i}_{21}={\omega }_{2}/{\omega }_{1}$ into Eq. (21), ${\mathbf{v}}_{12}^{3}$ can be further written as:
${\mathbf{v}}_{12}^{3}={\omega }_{1}\left(\begin{array}{c}\left[-{Y}_{3}\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\right)\right]{\mathbf{i}}_{3}\\ \left[-{Z}_{3}{i}_{21}\mathrm{s}
\mathrm{i}\mathrm{n}\mathrm{\Sigma }+{X}_{3}\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\right)\right]{\mathbf{j}}_{3}\\ \left[{Y}_{3}{i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\
Sigma }\right]{\mathbf{k}}_{3}\end{array}\right).$
As a consequence, the relative velocity in coordinate system ${S}_{1}$ can be obtained as follows:
${\mathbf{v}}_{12}^{1}={\mathbf{M}}_{13}{\mathbf{v}}_{12}^{3}={\omega }_{1}\left(\begin{array}{c}\left[-{Y}_{1}\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\right)-{Z}_{1}{i}_{21}\
mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right]{\mathbf{i}}_{1}\\ \left[{X}_{1}\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\right)-{Z}_
{1}{i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\right]{\mathbf{j}}_{1}\\ \left[\left({Y}_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}+{X}_{1}\mathrm
{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right){i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\right]{\mathbf{k}}_{1}\end{array}\right),$
where matrix ${\mathbf{M}}_{13}$ describes coordinate transformation from ${S}_{3}$ to ${S}_{1}\text{;}$$\left({X}_{1},{Y}_{1},{Z}_{1}\right)$ means coordinate value of contact point $M$ in
coordinate system ${S}_{1}$, and $\left({\mathbf{i}}_{1},{\mathbf{j}}_{1},{\mathbf{k}}_{1}\right)$ represents three components of unit vector in coordinate system ${S}_{1}$.
According to the position relationship between the involute beveloid gear cutter and noninvolute beveloid gear as shown in Fig. 5, the tooth surface of involute beveloid gear cutter in coordinate
system ${S}_{1}$ can be derived as follows:
$\left[\begin{array}{c}{R}_{1x}\\ {R}_{1y}\\ {R}_{1z}\\ 1\end{array}\right]=\left[\begin{array}{c}{R}_{jx}^{z}\\ {R}_{jy}^{z}\\ {R}_{jz}^{z}+L\\ 1\end{array}\right],$
where $L$ is coordinate value on the ${Z}_{1}$ axis.
Hence, the unit normal vector of Gear_1 (involute beveloid gear) is represented as:
$\frac{\frac{\partial {R}_{1}}{\partial l}×\frac{\partial {R}_{1}}{\partial u}}{\left|\frac{\partial {R}_{1}}{\partial l}×\frac{\partial {R}_{1}}{\partial u}\right|}=\left[\begin{array}{c}\frac{{n}_
{1x}}{\sqrt{{n}_{1x}^{2}+{n}_{1y}^{2}+{n}_{1z}^{2}}}\\ \frac{{n}_{1y}}{\sqrt{{n}_{1x}^{2}+{n}_{1y}^{2}+{n}_{1z}^{2}}}\\ \frac{{n}_{1z}}{\sqrt{{n}_{1x}^{2}+{n}_{1y}^{2}+{n}_{1z}^{2}}}\end{array}\
right]=\left[\begin{array}{c}{n}_{1x}^{"}\\ {n}_{1y}^{\text{'}}\\ {n}_{1z}^{\text{'}}\end{array}\right].$
3.3. Meshing equation and tooth surface equation of noninvolute beveloid gear
According to the kinematics of gear geometry, the equation of meshing is defined as:
$\mathbf{n}\cdot {\mathbf{v}}_{12}=\text{0.}$
Combining Eqs. (23), (25) and (26), we can obtain the meshing equation:
$\left[\begin{array}{c}{n}_{1x}^{"}\\ {n}_{1y}^{\text{'}}\\ {n}_{1z}^{\text{'}}\end{array}\right]\cdot {\omega }_{1}\left(\begin{array}{c}\left[-{Y}_{1}\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\
mathrm{\Sigma }\right)-{Z}_{1}{i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right]{\mathbf{i}}_{1}\\ \left[{X}_{1}\left(1+{i}_{21}\mathrm{c}\mathrm
{o}\mathrm{s}\mathrm{\Sigma }\right)-{Z}_{1}{i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\right]{\mathbf{j}}_{1}\\ \left[\left({Y}_{1}\mathrm{c}\
mathrm{o}\mathrm{s}{\phi }_{1}+{X}_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\right){i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\right]{\mathbf{k}}_{1}\end{array}\right)=0,$
where ${\phi }_{1}$ is angle parameter, which can be derived as:
${\phi }_{1}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\frac{\left(ca-\sqrt{{a}^{2}{b}^{2}+{b}^{4}-{b}^{2}{c}^{2}}\right)a}{\left({a}^{2}+{b}^{2}\right)b}-\frac{c}{b}}{-
$a={i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\left({Y}_{1}{n}_{1z}-{Z}_{1}{n}_{1y}\right),b={i}_{21}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\left({X}_{1}{n}_{1z}-{Z}_{1}{n}_{1x}\
$c=\left(1+{i}_{21}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }\right)\left({X}_{1}{n}_{1y}-{Y}_{1}{n}_{1x}\right).$
Then the tooth profile equation of noninvolute beveloid gear can be obtained by coordinate transformation from ${S}_{1}$ to ${S}_{2}$:
$\left[\begin{array}{c}{R}_{2x}\\ {R}_{2y}\\ {R}_{2z}\\ 1\end{array}\right]={M}_{21}\left[\begin{array}{c}{R}_{1x}\\ {R}_{1y}\\ {R}_{1z}\\ 1\end{array}\right].$
As a consequence:
$\left\{\begin{array}{l}{R}_{2x}=\left(\begin{array}{c}\left(\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }-\mathrm
{s}\mathrm{i}\mathrm{n}{\phi }_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}\right){R}_{1x}\\ +\left(-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\mathrm{c}\
mathrm{o}\mathrm{s}\mathrm{\Sigma }-\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}\right){R}_{1y}+\left(-\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\mathrm{s}\mathrm
{i}\mathrm{n}\mathrm{\Sigma }\right){R}_{1z}\end{array}\right),\\ {R}_{2y}=\left(\begin{array}{c}\left(\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{2}\mathrm{c}\
mathrm{o}\mathrm{s}\mathrm{\Sigma }+\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\right){R}_{1x}\\ +\left(-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\mathrm{s}\
mathrm{i}\mathrm{n}{\phi }_{2}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }+\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{2}\right){R}_{1y}+\left(-\mathrm{s}\mathrm
{i}\mathrm{n}{\phi }_{2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }\right){R}_{1z}\end{array}\right),\\ {R}_{2z}=\mathrm{c}\mathrm{o}\mathrm{s}{\phi }_{1}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\
Sigma }{R}_{1x}+-\mathrm{s}\mathrm{i}\mathrm{n}{\phi }_{1}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{\Sigma }{R}_{1y}+\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Sigma }{R}_{1z}.\end{array}\right\$
Hence, the mathematical model of involute beveloid gear have been derived.
4. Modeling of involute and noninvolute beveloid gear pair
The main design parameters of involute beveloid gear pairs (pinion and wheel) are: number of teeth ${Z}_{1}/{Z}_{2}=$26/51, normal pressure angle ${\alpha }_{n1}/{\alpha }_{n2}=$ 17.5°, normal module
${m}_{n1}/{m}_{n2}=$ 5 mm, tooth width ${b}_{1}/{b}_{2}=$ 94 mm/89 mm, helical angle $\beta =$8°, shaft intersection angle $\delta =$10°; the main design parameters of noninvolute beveloid gear pairs
(pinion and wheel) are: number of teeth ${Z}_{1}^{"}/{Z}_{2}=$ 26/51, the normal module ${m}_{n1}/{m}_{n2}=$ 5 mm, the tooth width ${b}_{1}/{b}_{2}=$ 94 mm/89 mm and shaft intersection angle $\delta
The parametric modeling program is developed via a Matlab code to generate exact three-dimensional (3-D) discrete points on the tooth profiles. The Fig. 7(a) shows the discrete points on tooth
surfaces of involute beveloid gear (wheel). Then the tooth surfaces of involute beveloid gear pair are established after we imported point set into Imageware software. Subsequently, the solid model
will be created by importing tooth surfaces into UG to sew up. In this way, we can obtain the accurate three-dimensional solid model as shown in Fig. 7(b).
Applying the assembly relationship together with the mathematical models of beveloid gear pairs, the assembly models of involute and noninvolute beveloid gear pairs are generated as illustrated in
Fig. 8.
Fig. 7Models of involute beveloid gears
a) Discrete points on tooth surface
Fig. 8Assembly model of involute and noninvolute beveloid gears
5. Dynamic contact analysis of involute and noninvolute beveloid gear pair
5.1. Finite element model
Dynamic contact characteristics of the involute and noninvolute beveloid gear pair will be analyzed by LS-DYNA software. The pinion is rotating at 1600 rpm with a driving torque of 1194 N·m in
numerical simulation analysis. The finite element meshes of gear pair are generated using solid element Solid164. In order to conveniently load the speed and torque [13], internal surfaces of gear
pair are considered as rigid regions utilizing the Shell163 element. Afterwards, degrees of freedom of all nodes in rigid region are coupled to the centroid of rigid body, and apart from axial
rotation, all translational and rotational degrees of freedom of rigid shell elements in pinion and wheel are constrained. The total number of elements is 82160 with 97608 nodes for the finite
element model of involute beveloid helical gear pair, shown as Fig. 9.
Fig. 9Finite element model of involute beveloid helical gear pairs
5.2. Simulation results and discussion
Fig. 10 and Fig. 11 describe the dynamic contact stress contour and single tooth contact force of involute and noninvolute beveloid gear during the mesh cycle.
The results show that the contact area on flank of noninvolute beveloid gear is evidently larger than that of involute beveloid gear; the dynamic contact stress distribution of noninvolute beveloid
gear is more homogeneous and the contact stress reduces significantly; the contact ratio of noninvolute beveloid gear is 3.74, greater than that of involute beveloid gear, which is 3.30; compared
with the single tooth contact force of involute beveloid gear, the noninvolute beveloid gear has smaller single tooth contact force and smoother meshing impact.
Fig. 10Dynamic contact stress contour (Unit: Pa)
a) Involute beveloid gears
b) Noninvolute beveloid gears
Fig. 11Contact force of each mating teeth pair
a) Involute beveloid gears
b) Noninvolute beveloid gears
c) Comparison of single tooth contact force
Fig. 12 and Fig. 13 illustrate the dynamic fillet stress of involute and noninvolute beveloid gear, respectively. The results indicate that the dynamic fillet stress of involute beveloid gear mainly
concentrate on the mid tooth and toe, by contrast, the heel almost do not bear fillet stress; the dynamic fillet stress distribution of noninvolute beveloid gear is more homogeneous on mid tooth, toe
and heel.
Forecasting the dynamic transmission error (DTE) which is an important factor to cause the vibration, is another objective of dynamic contact analysis. The dynamic transmission error is usually
expressed as:
where ${n}_{1}$ and ${n}_{2}$ is the driving and driven shaft’s rotational speeds with $i$ being the transmission ratio of the gear set, respectively.
Fig. 12Dynamic fillet stress curves of involute beveloid gear pairs
Fig. 13Dynamic fillet stress curves of noninvolute beveloid gear pairs
Fig. 14 illustrates the DTE of involute and noninvolute beveloid gear. As observed in Fig. 14, the predicted DTE is approximate sine shape and the trend of peak-peak value of DTE for involute
beveloid gear becomes higher than that of noninvolute beveloid gear due to smaller contact area for involute beveloid gear.
Fig. 14DTE of involute and noninvolute beveloid gear
6. Conclusions
1) The mathematical models of involute and noninvolute beveloid gears are derived and the parametric modeling programs of involute and noninvolute beveloid gears are developed to automatically
generate exact model via Matlab code; then a numerical example of intersecting axes beveloid gear pair is presented to analyze the dynamic contact characteristics.
2) The contact area on flank of noninvolute beveloid gear is evidently larger than that of involute beveloid gear, and the dynamic contact stress distribution of noninvolute beveloid gear is more
homogeneous and the contact stress reduces significantly.
3) The dynamic fillet stress of involute beveloid gear mainly concentrate on the mid tooth and toe, by contrast, the heel almost do not bear fillet stress, the dynamic fillet stress distribution of
noninvolute beveloid gear is more homogeneous on mid tooth, toe and heel.
4) The DTE is approximate sine shape and the trend of peak-peak value for involute beveloid gear becomes higher than that of noninvolute beveloid gear due to smaller contact area for involute
beveloid gear and the noninvolute beveloid gear has gentler meshing impact.
5) Hopefully, noninvolute beveloid gear pairs would relieve the high dynamic contact shock problem of intersecting axes beveloid gear pairs and improve range of application.
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About this article
30 September 2014
involute/noninvolute beveloid gear pairs
mathematical models
contact analysis
The authors are grateful for the financial support provided by the National Natural Science Foundation of China under Contract No. 51175524 and National Science and Technology Support Project under
Contract No. 2013BAF01B04.
Copyright © 2014 JVE International Ltd.
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BS in Actuarial Science | TCU Department of Mathematics
Main Content
BS in Actuarial Science
The Bachelor of Science in Actuarial Science prepares students for careers in insurance, risk analysis, pension management, financial planning, and other related areas. It provides deeper training in
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exams in financial mathematics and probability and also introduces students to advanced statistical modeling.
(Click here if you declared Bachelor of Science in Actuarial Science as your major prior to Spring 2022.)
This degree requires 46 semester hours of courses in mathematics.
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SSC CGL Tier 2 Set 12
Submitted by Atanu Chaudhuri on Tue, 29/05/2018 - 22:41
Trigonometry questions with solutions: SSC CGL Tier 2 Set 12
Trigonometry Questions with Solutions for SSC CGL Tier 2 Set 12 explains how 10 difficult trigonometry questions can be solved in 12 minutes' time.
For best results take the test first at,
12th SSC CGL Tier II level question set and 3rd on Trigonometry.
Solutions to 10 Selected Trigonometry Questions for SSC CGL Tier 2 Set 12 - testing time was 12 mins
Problem 1.
If $2abcos \theta + (a^2-b^2)sin \theta=a^2+b^2$ then the value of $tan \theta$ is,
a. $\displaystyle\frac{1}{2ab}(a^2+b^2)$
b. $\displaystyle\frac{1}{2}(a^2-b^2)$
c. $\displaystyle\frac{1}{2}(a^2+b^2)$
d. $\displaystyle\frac{1}{2ab}(a^2-b^2)$
Solution 1. Problem analysis: identification of promising path
We look at the target $tan \theta$ and chalk out a possible solution path—if we find out $sec \theta - tan \theta$, we should be able to find $sec \theta + tan \theta$ also by just reversing the
value. With these two values in the pocket, finding $tan \theta$ won't pose any problem. There is rarely an easier path to find $tan \theta$.
This decision is based on the rich concept of friendly trigonometric function pairs. and the presence of $sin \theta$ and $cos \theta$ in the given expression.
Rich Concept of friendly trigonometric function pairs
Let us explain this with the first example pair of functions,
$sec\theta$ and $tan\theta$.
We have,
$sec\theta + tan\theta = \displaystyle\frac{(sec\theta +tan\theta)(sec\theta - tan\theta)}{sec\theta - tan\theta}$
$\hspace{25mm}=\displaystyle\frac{1}{sec\theta -tan\theta}$, because $sec^2\theta - tan^2\theta=1$.
The result is somewhat similar to surd rationalization.
In the same way, in the case of the second friendly function pair of $cosec\theta$ and $cot\theta$, we get,
$cosec\theta + cot\theta =\displaystyle\frac{1}{cosec\theta - cot\theta}$.
The inherent friendship mechanism of the third friendly function pair, $sin\theta$ and $cos\theta$ is well known and is used very frequently,
$sin^2 \theta + cos^2 \theta=1$.
Most promising course of action from Initial problem analysis
Objective is to transform the expression in $sec \theta$ and $tan \theta$. This is what we call promising course of action. We may not be absoluterly sure of all the steps to the final solution, but
we choose the most promising course of action at a critical point in the solution process. This promise we know from our Initial problem analysis and domain knowledge.
Usually the most promising course action straightaway leads to the solution, at least in the case of Competitive math problem solving.
In real life complex problems, we take up the most promising course of action and evaluate results to tune the action further.
We are solving again..
But how to get $sec \theta + tan \theta$ or $\sec \theta -tan\theta$ from the given expression!
The first step with this penultimate goal in mind would surely be converting the $sin \theta$ and $cos \theta$ to $tan \theta$ and $sec \theta$.
Solution 1. Problem solving execution
The given equation,
$2abcos \theta + (a^2-b^2)sin \theta=a^2+b^2$,
Dividing by $cos \theta$ and collecting terms with common $a^2$ and $b^2$,
$2ab=a^2(sec \theta - tan \theta) + b^2(sec \theta + tan \theta)$.
Dividing by $ab$ to increase harmony in the expression further,
$\displaystyle\frac{a}{b}(sec \theta -tan \theta)+\displaystyle\frac{b}{a}\times{\displaystyle\frac{1}{sec \theta - tan \theta}}=2$.
This is familiar ground of algebraic simplification, but greatly aided by the most potent trigonometric relation,
$sec \theta + tan \theta=\displaystyle\frac{1}{sec \theta - tan \theta}$.
We will now simplify the equation to an one variable equation by Substituting $x$ in place of the repeating expression $\displaystyle\frac{a}{b}(sec \theta - tan \theta)$. This is a powerful
algebraic problem solving technique in reducing the complexity of the expression being simplified. This also results in reduction of number of variables as well as number of terms.
The last stage is in the form of inverses,
$x + \displaystyle\frac{1}{x}=2$, where, $x=\displaystyle\frac{a}{b}(sec \theta - tan \theta)$,
This is a quadratic equation in $x$,
$x^2 - 2x + 1=(x-1)^2=0$.
$x=\displaystyle\frac{a}{b}(sec \theta - tan \theta) = 1$.
As we have thought at the very beginning, we have obtained the value of $sec \theta - tan \theta=\displaystyle\frac{b}{a}$,
$sec \theta + tan \theta=\displaystyle\frac{a}{b}$.
Subtracting the first from the second and dividing by 2,
Or, $tan \theta =\displaystyle\frac{1}{2ab}(a^2-b^2)$.
Answer: Option d: $\displaystyle\frac{1}{2ab}(a^2-b^2)$.
Key concepts and techniques used: End state analysis approach -- Use of most potent friendly trigonometric function pair -- Working backwards approach -- Initial possible approach -- Most promising
course of action -- Initial problem analysis -- basic trigonometry concepts -- Input transformation to more promising form of relation -- Algebraic simplification techniques -- Substitution
technique -- Variable reduction technique -- roots of a quadratic equation -- Analytical approach example.
Though the problem looked difficult, most of the solution steps could be done in mind.
This is a good example of analytical approach to problem solving.
Problem 2.
$\displaystyle\frac{\sin^2 \theta}{\cos^2 \theta}+\displaystyle\frac{\cos^2 \theta}{\sin^2 \theta}$ is equal to,
a. $\displaystyle\frac{1}{\sin^2 {\theta}\cos^2 \theta}$
b. $\displaystyle\frac{1}{\sin^2 {\theta}\cos^2 \theta} -2$
c. $\displaystyle\frac{1}{\tan^2 \theta - \cot^2 \theta}$
d. $\displaystyle\frac{\sin^2 \theta}{\cot \theta - \sec \theta}$
Solution 2. Problem solving execution
As direct addition would result in $sin^4 \theta + cos^4 \theta$ in the numerator, leaving this uncomfortable path, we would transform to $tan^2 \theta$ and $cot^2 \theta$ and break up each term
further in terms of $sec^2 \theta$ and $cosec^2 \theta$,
$\displaystyle\frac{\sin^2 \theta}{\cos^2 \theta}+\displaystyle\frac{\cos^2 \theta}{\sin^2 \theta}$
$=tan^2 \theta + cot^2 \theta$
$=sec^2 \theta + cosec^2 \theta - 2$
$=\displaystyle\frac{1}{\sin^2 {\theta}\cos^2 \theta} -2$, as $sin^2 \theta + cos^2 \theta=1$ in the numerator after taking inverse and then adding the two terms.
Answer: Option b: $\displaystyle\frac{1}{\sin^2 \theta.cos^2 \theta} -2$.
We have first got rid of fraction terms. effectively reducing number of terms from 4 to 2 considering the 2 numerators and denominators. Transforming again using more potent relations $sec^2 \theta
-1= tan^2 \theta$ and $cosec^2 \theta-1=cot^2 \theta$, we reach very near to solution.
Key concepts and techniques used: Useful pattern identification and exploitation -- basic trigonometry concepts -- friendly trigonometric function pairs concept -- input transformation technique --
efficient simplification.
All steps could easily be done in mind.
Problem 3.
If $cos \theta + sec \theta = 2$, then the value of $cos^5 \theta + sec^5 \theta$ is,
a. $-2$
b. $2$
c. $1$
d. $-1$
Solution 3. Problem analysis
Noticing the inverse relation of the form, $x + \displaystyle\frac{1}{x}=2$ and the target expression also in higher power of inverses, for a moment we considered using the algebraic path of
evaluating inverses.
Then noticing the given expression closely we detected the possibility for finding the value of $cos \theta$ directly.
Solution 3. Problem solving execution
Given expression,
$cos \theta + sec \theta = 2$,
Or, $cos^2 \theta - 2cos \theta + 1=0$
Or, $(cos \theta - 1)^2=0$
Or, $cos \theta = sec \theta =1$
$cos^5 \theta + sec^5 \theta=2$
Answer: Option b: $2$.
Key concepts and techniques used: Pattern identification technique -- End state analysis approach -- Most promising course of action -- Basic algebraic concepts.
All the processing could easily be done mind. The main task was to identify the useful pattern.
Problem 4.
$sin(\alpha + \beta -\gamma)=cos(\beta + \gamma -\alpha)=\displaystyle\frac{1}{2}$ and $tan(\gamma + \alpha -\beta)=1$. If $\alpha$, $\beta$ and $\gamma$ are positive acute angles, value of $2\alpha
+ \beta$ is,
a. $105^0$
b. $110^0$
c. $115^0$
d. $120^0$
Solution 4. Problem analysis
From acute angle conditions we would get actual values of the three angle expressions within brackets. With three variables and three linear equations it is always possible to solve for any
expression in these three angles.
Solution 4. Problem solving execution
First relation,
$sin(\alpha + \beta -\gamma)=\displaystyle\frac{1}{2}$
$\alpha + \beta -\gamma=30^0$.
Second relation,
$cos(\beta + \gamma -\alpha)=\displaystyle\frac{1}{2}$,
$\beta + \gamma -\alpha=60^0$.
The third relation,
$tan(\gamma + \alpha -\beta)=1$,
$\gamma + \alpha -\beta=45^0$.
Adding the first two equations and dividing by 2 we get,
$\beta=45^0$, and adding the first and third equations we get,
$2\alpha + \beta=120^0$
Answer: Option d: $120^0$.
Key concepts and techniques used: Initial problem analysis -- Trigonometric ratio values -- linear algebraic equations -- Analytical approach example.
Problem 5.
If $sin \theta + sin^2 \theta=1$, then the value of $cos^{12} \theta + 3cos^{10} \theta + 3cos^{8} \theta + cos^6 \theta - 1$ is,
a. $1$
b. $0$
c. $2$
d. $3$
Solution 5. Problem analysis and input transformation
As the target expression is more complex, it is apparent that we have to use the input expression to simplify it. But in the present form no apparent promising course of action could be found. The
three terms caused the basic problem. If only we could have two terms in the equation, we might be able to replace one term by the other in the target expression.
On closer inspection the input equation could be transformed in the desired manner,
$sin \theta + sin^2 \theta =1$,
Or, $sin \theta = 1 - sin^2 \theta=cos^2 \theta$.
Solution 5. Problem solving execution
Now the task is cut out. We need to express the target expression in terms of $cos^2 \theta$.
The two coefficients of value 3 could give the help to convert $cos$ expression to a cube of sum,
$E=cos^{12} \theta + 3cos^{10} \theta + 3cos^{8} \theta + cos^6 \theta - 1$
$=(cos^4 \theta + cos^2 \theta)^3 - 1$
$=(sin^2 \theta+cos^2 \theta)^3-1$
Answer: Option b: $0$.
Key concepts and techniques used: Input transformation -- Term reduction technique -- Useful pattern identification -- Substitution -- Friendly trigonometric function pair -- Analytical approach
Problem 6.
The value of $sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta}{1+sin \theta}\right) - 2tan^2 \theta$ is,
a. 4
b. 0
c. 2
d. 1
Solution 6. Problem analysis
We mark the symmetry and possibility of transforming the expression in terms of sum of $sec \theta$ and $tan \theta$. The sum and subtraction expressions of $sec$ and $tan$ (and also $cosec$ and
$cot$) we call golden trigonometric function pairs because of their great potential for simplification.
Solution 6. Problem solving execution
The given expression is,
$E=sec \theta\left(\displaystyle\frac{1+sin \theta}{cos \theta}+\displaystyle\frac{cos \theta}{1+sin \theta}\right) - 2tan^2 \theta$.
Multiply the first and the second terms in the brackets, both numerator and denominator, by $sec \theta$,
$E=sec \theta\left(sec \theta+tan \theta+\displaystyle\frac{1}{sec \theta+tan \theta}\right) - 2tan^2 \theta$
Now substitute inverse of $\displaystyle\frac{1}{sec \theta + tan \theta}=sec \theta - tan \theta$ to the second term and simplify,
$E=sec \theta.2sec \theta - 2tan^2 \theta$
$=2(sec^2 \theta - tan ^2 \theta)$
Answer: Option c: 2.
Quick, elegant and all in mind.
Key concepts and techniques used: Basic trigonometry copncepts -- Trigonometric function transformation -- friendly trigonometric function pairs.
Problem 7.
If $4cos^2 \theta - 4\sqrt{3}cos \theta + 3=0$ and $0^0 \leq \theta \leq 90^0$, then the value of $\theta$ is,
a. $60^0$
b. $90^0$
c. $30^0$
d. $45^0$
Solution 7. Problem analysis and solving execution
In this problem everything hinges on finding the exact value of $\theta$ directly from the quadratic equation in $cos \theta$.
On closer inspection we could indeed express the given expression as a square of sum,
Given expression,
$4cos^2 \theta - 4\sqrt{3}cos \theta + 3=0$,
Or, $(2cos \theta - \sqrt{3})^2=0$,
Or, $cos \theta =\displaystyle\frac{\sqrt{3}}{2}$.
Answer: Option c: $30^0$.
Key concepts used: Pattern identification technique -- input transformation -- roots of qudratic equation -- Basic trigonometry concepts.
Problem 8.
If $sin \theta + cos \theta=\sqrt{2}sin(90^0 - \theta)$ then the value of $cot \theta$ is,
a. $\sqrt{2}-1$
b. $\sqrt{2}+1$
c. $-\sqrt{2}+1$
d. $-\sqrt{2}-1$
Solution 8. Problem solving execution
We will use first complementary trigonometric function concept, $sin (90^0 - \theta)=cos \theta$,
Given expression,
$sin \theta + cos \theta=\sqrt{2}sin(90^0 - \theta)$,
Or, $sin \theta + cos \theta=\sqrt{2}cos \theta$,
Dividing throughout by $cos \theta$,
Or, $tan \theta + 1=\sqrt{2}$,
Or, $tan \theta=\sqrt{2}-1$,
Or, $cot \theta=\displaystyle\frac{1}{\sqrt{2}-1}$,
Or, $cot \theta = \sqrt{2}+ 1$, by surd rationalization, multipltying both numerator and denominator by $\sqrt{2}+1$.
Answer: Option b: $\sqrt{2}+1$.
Key concepts and techniques used: Basic trigonometry concepts -- complementary angle concepts -- Surd rationalization.
Problem 9.
If $x=asin \theta-bcos \theta$ and $y=acos \theta + bsin \theta$, then which of the following is true?
a. $x^2+y^2=a^2+b^2$
b. $\displaystyle\frac{x^2}{a^2}+ \displaystyle\frac{y^2}{b^2}=1$
c. $x^2+y^2=a^2-b^2$
d. $\displaystyle\frac{x^2}{y^2}+ \displaystyle\frac{a^2}{b^2}=1$
Solution 9. Problem analysis
As all the choice values have squares, we will first form the squares of $x$ and $y$.
Solution 9. Problem solving execution
Given equations,
$x=asin \theta-bcos \theta$
$x^2=a^2sin^2 \theta - 2absin \theta.cos \theta + b^2cos^2 \theta$,
Similarly squaring the second equation,
$y^2=a^2cos^2 \theta + 2absin \theta.cos \theta + b^2sin^2 \theta$.
Adding the two and collecting the terms with coefficients $a^2$ and $b^2$,
$x^2 + y^2=a^2 + b^2$, as factor $sin^2 \theta+ cos^2 \theta=1$
Answer: Option a: $x^2 + y^2=a^2 + b^2$.
Key concepts and techniques used: End state analysis approach -- friendly trigonometric function pairs -- target driven simplification.
All in mind and in quick time.
Problem 10.
If $tan \theta=\displaystyle\frac{a}{b}$, then the value of $\displaystyle\frac{asin^3 \theta - bcos^3 \theta}{asin^3 \theta + bcos^3 \theta}$ is,
a. $\displaystyle\frac{a^4-b^4}{a^4+b^4}$
b. $\displaystyle\frac{a^3+b^3}{a^3-b^3}$
c. $\displaystyle\frac{a^3-b^3}{a^3+b^3}$
d. $\displaystyle\frac{a^4+b^4}{a^4-b^4}$
Solution 10. Problem analysis
The target being in the form perfectly suitable for applying Componendo dividendo technique $\left(\text{in the form }\displaystyle\frac{x-y}{x+y}\right)$, the very first step we would take is to
apply the technique on the target expression. After the result is simplified we would substitute the value of $tan \theta$ and again reapply the Componendo dividendo technique to get the target
The method is clear and once we show it, you will easily understand the mechanism. But yes, you have to know componendo dividendo technique.
Rich algebraic technique of Componendo and dividendo
Problem: Simplify, $\displaystyle\frac{x - y}{x + y} = \displaystyle\frac{1}{3}$.
First we add 1 to both sides of the equation,
$1+\displaystyle\frac{x - y}{x + y} = 1+\displaystyle\frac{1}{3}$,
Or, $\displaystyle\frac{2x}{x+ y} = \displaystyle\frac{4}{3}$.
Next we subtract both sides of the equation from 1,
$1-\displaystyle\frac{x - y}{x + y} = 1-\displaystyle\frac{1}{3}$,
Or, $\displaystyle\frac{2y}{x + y} = \displaystyle\frac{2}{3}$.
Now we divide the first result by the second,
$\displaystyle\frac{x}{y} = 2$, a greatly simplified expression with two terms in the numerator and denominator reduced to single terms.
This is a powerful algebraic technique frequently applied whenever we encounter the special form of given expression.
Solution 10. Problem solving execution
Let target expression,
$\displaystyle\frac{asin^3 \theta - bcos^3 \theta}{asin^3 \theta + bcos^3 \theta}=x$.
Applying componendo dividendo technique,
$\displaystyle\frac{1+x}{1-x}=\frac{asin^3 \theta}{bcos^3 \theta}=\displaystyle\frac{a}{b}tan^3 \theta$
Reapplying the componendo dividendo technique (this time first subtract 1 from the equation to get numerator $2x$, next add 1 to the equation to get numerator $2$ and then take ratio),
Answer: Option a: $\displaystyle\frac{a^4-b^4}{a^4+b^4}$.
Key concepts and techniques used: Useful pattern recognition -- basic trigonometry concepts -- componendo dividendo technique -- Analytical approach example.
Note: You will observe that in many of the Trigonometric problems basic and rich algebraic concepts and techniques are to be used. In fact that is the norm. Algebraic concepts are frequently used for
quick solutions of Trigonometric problems.
Guided help on Trigonometry in Suresolv
To get the best results out of the extensive range of articles of tutorials, questions and solutions on Trigonometry in Suresolv, follow the guide,
Reading and Practice Guide on Trigonometry in Suresolv for SSC CHSL, SSC CGL, SSC CGL Tier II Other Competitive exams.
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KC Sinha Maths Solution Class 10
The Mathematics of Class 10 is one of the toughest, and an important subject for the students to make a solid base of the concepts. By making your foundation strong with the help of NCERT Maths
textbook and then moving ahead with the reference Maths books like KC Sinha, scoring in the subject will become easy and quick for the students. Moreover, with consistent practice sessions, you will
fall in love with the subject and will be able to score excellent marks in the exam.
To assist you in the preparation of the Class 10 board exams, NCERTBooks.Guru is providing you with KC Sinha Maths Solutions for Class 10. These solutions are drafted by an experienced and qualified
team of subject mentors and are provided in the chapter-wise format in a well-structured and organized way.
Kc Sinha Maths Solutions for Class 10 Chapter 1
Kc Sinha Maths Solutions for Class 10 Chapter 2
Kc Sinha Maths Solutions for Class 10 Chapter 3
Kc Sinha Maths Solutions for Class 10 Chapter 4
Kc Sinha Maths Solutions for Class 10 Chapter 5
Kc Sinha Maths Solutions for Class 10 Chapter 6
Kc Sinha Maths Solutions for Class 10 Chapter 7
Kc Sinha Maths Solutions for Class 10 Chapter 8
Kc Sinha Maths Solutions for Class 10 Chapter 9
Kc Sinha Maths Solutions for Class 10 Chapter 10
Kc Sinha Maths Solutions for Class 10 Chapter 11
Kc Sinha Maths Solutions for Class 10 Chapter 12
Kc Sinha Maths Solutions for Class 10 Chapter 13
Kc Sinha Maths Solutions for Class 10 Chapter 14
Kc Sinha Maths Solutions for Class 10 Chapter 15
The KC Sinha Maths Solutions for Class 10 drafted by NCERTBooks.Guru is prepared in complete sync with the latest official syllabus of the exam approved by CBSE. Further, covering the complete
syllabus of the exam, you will find these solutions in easy to understand manner, which leads to scoring higher marks in the subject.
Benefits of Using KC Sinha Maths Solutions for Class 10 prepared by NCERTBooks.Guru
Our solutions for Maths subject for Class 10 will help you in your exam preparation study plan in a constructive way. Here are important benefits of these solutions, which are provided below:
• The experienced team of NCERTBooks.Guru has prepared the KC Sinha Mathematics Solutions to aid students in the best way possible.
• These solutions for Class 10 are prepared in sync with the official syllabus of the CBSE.
• The Maths solution is error free and you won’t find any mistakes in it.
• While preparing for the exam, get the best answers to all the questions asked in the textbook from the team of skilled mentors.
• The KC Sinha Maths Solution prepared by NCERTBooks.Guru team is free of cost and the access for the same is easy.
Which is the best KC Sinha Solutions for Class 10th Mathematics?
You will get to know the important topics and questions of the subject in this KC Sinha Maths Solutions for Class 10th, which is prepared by NCERTBooks.Guru. All the answers provided in the solutions
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keeping in mind the latest and official syllabus prescribed by CBSE. Scoring good marks will become easy for you if you prepare for the Maths exam from the KC Sinha Maths Solutions for Class 10.
Is KC Sinha Maths Solution for Class 10 Maths enough to prepare for the exam?
Yes, with the help of KC Sinha Maths Solution for 10th Class, preparing for the exam will become easy and smooth for the students. All the answers provided in these solutions are prepared on the
basis of the official latest syllabus approved by the CBSE.
It is suggested to start your preparation for the exam from the NCERT textbook and then moves ahead with the reference books. KC Sinha is one of the best ‘extra’ book for preparing for the exam and
with the help of KC Sinha Solutions, clear your concepts and gain the ability to solve the questions of the textbook effectively.
Will I get all the answers of all the chapters in NCERTBooks.Guru KC Sinha Solution for Class 10 Maths?
Yes, for sure. These solutions for Class 10th Mathematics covers all the chapters of the subject. The solutions are complete in all respects and not even a single topic is being left by the team of
NCERTBooks.Guru. Also, the solutions are error free and easy to understand for the students and are provided in a well-organized manner.
Is the KC Sinha Solutions for Class 10th Maths based on the latest syllabus?
The simple answer is yes. All the answers provided in KC Sinha Maths Solutions for Class 10 are based on the latest official syllabus of CBSE. There will be no solution or the explanation in the
solutions from outside the syllabus of Class 10 of Mathematics. Moreover, these solutions also cover the important topics and the questions of the exam which will help you in scoring good marks in
the board examination. | {"url":"https://www.ncertbooks.guru/kc-sinha-maths-solution-class-10/","timestamp":"2024-11-03T15:52:54Z","content_type":"text/html","content_length":"86316","record_id":"<urn:uuid:b8dd68dd-a9a8-4c9d-b9b6-3c7ae6125680>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00479.warc.gz"} |
On the 2-independence subdivision number of graphs
[1] M. Atapour, S.M. Sheikholeslami, A. Hansberg, L. Volkmann, and A. Khodkar, 2-domination subdivision number of graphs, AKCE Int. J. Graphs. Combin. 5 (2008), no. 2, 165–173.
[2] M. Atapour, S.M. Sheikholeslami, and A. Khodkar, Roman domination subdivision number of graphs, Aequationes Math. 78 (2009), no. 3, 237–245.
[3] M. Chellali, O. Favaron, A. Hansberg, and L. Volkmann, k-domination and kindependence in graphs: A survey, Graphs Combin. 28 (2012), no. 1, 1–55.
[4] M. Chellali, O. Favaron, T.W. Haynes, and D. Raber, Ratios of some domination parameters in trees, Discrete Math. 308 (2008), no. 17, 3879–3887.
[5] J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, Graph Theory with Applications to Algorithms and Computer Science, John Wiley and Sons. New York, 1985, pp.
[6] T.W. Haynes, S.M. Hedetniemi, and S.T. Hedetniemi, Domination and independence subdivision numbers of graphs, Discuss. Math. Graph Theory 20 (2000), no. 2, 271–280.
[7] T.W. Haynes, S.T. Hedetniemi, and L.C. van des Merwe, Total domination subdivision numbers, J. Combin. Math. Combin. Comput. 44 (2003), no. 3, 115–128. | {"url":"https://comb-opt.azaruniv.ac.ir/article_14224.html","timestamp":"2024-11-09T00:19:04Z","content_type":"text/html","content_length":"47819","record_id":"<urn:uuid:f95008cb-0c60-4b7d-8b8f-aa760c2f12c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00242.warc.gz"} |
Online t-test for independent samples
T Test in SPSS: Output. Your output will include: The Levine’s test for equal variance (the first section of the Independent Samples Test box). If the significance level is larger than .05, you
should use the first line in the output table, Equal variances assumed. If the value is .05 or lower, use the second row of results.
The Independent Samples t Test compares the means of two independent groups in order to determine whether there is statistical evidence that the associated population means are significantly
different. The Independent Samples t Test is a parametric test. This test is also known as: Independent t Test; Independent Measures t Test; Independent Two-sample t Test Mathcracker.com provides
t-test for one and two samples, and for independent and paired samples. Also, you will be able to find calculators of critical value as a well as calculator of probabilities and graphing regions of
the t-distribution. Independent-Samples T Test The Independent-Samples T Test procedure compares means for two groups of cases. Ideally, for this test, the subjects should be randomly assigned to two
groups, so that any difference in response is due to the treatment (or lack of treatment) and not to other factors. The logic and computational details of two-sample t-tests are described in Chapters
9-12 of the online text Concepts & Applications of Inferential Statistics. For the independent-samples t-test, this unit will perform both the "usual" t-test, which assumes that the two samples have
equal variances, and the alternative t-test, which assumes that the two samples have unequal variances. The independent samples t test (also called the unpaired samples t test) is the most common
form of the T test. It helps you to compare the means of two sets of data. For example, you could run a t test to see if the average test scores of males and females are different; the test answers
the question,
Two sample and one sample t-test calculator with step by step explanation. Have Equal Variance (default). Groups Have Unequal Variance (Welch t-test)
The null hypothesis (H0) of a t-test (independent samples) states that both samples come from the same population. I'm interpreting this as Sections 1–12 in Online Stat Book: Logic of Hypothesis
Testing An independent samples t-test is used when you have means from two separate groups that Independent Two-Sample T-Test Calculator, Student T-Test, Welch T-Test, F test for equal variance.
This procedure computes the two-sample t-test and several other two-sample tests of the distribution of differences between independent sample means. 30 Jan 2019 A ttest for dependent samples (often
called a paired t test) uses each What are some simple steps I can take to protect my privacy online? 14 Aug 2018 ISSN: 2575-5919 (Print); ISSN: 2575-5927 (Online). A Brief Review of Independent,
Dependent and One Sample t-test. Banda Gerald. 25 Apr 2019 Since we are conducting a two-sample t-test here, the df formula is slightly different (as we have two samples instead of one). Here it is:.
A t test compares the means of two groups. For example, compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups. Don't
confuse t tests with correlation and regression. The t test compares one variable (perhaps blood pressure) between two groups.
25 Apr 2019 Since we are conducting a two-sample t-test here, the df formula is slightly different (as we have two samples instead of one). Here it is:. The independent-samples t-test (or independent
t-test, for short) compares the means between two unrelated groups on the same continuous, dependent if espresso per latte and a standard deviation of .11 oz. Use alpha = .05 and run an independent
samples t-test to determine if there is a significant difference The first part covers z-tests, single sample t-tests, and dependent t-tests. t-test is appropriate when you want to compare two
independent samples, so two Test to see if the sample mean is significantly different from 65 at the .05 level. an independent groups t test or a correlated t test (test of dependent means)?
relevant Compute a one-sample t-test on this column (with the L values for each
30 Jan 2019 A ttest for dependent samples (often called a paired t test) uses each What are some simple steps I can take to protect my privacy online?
Two-sample t-tests for a difference in mean involve independent samples ( unpaired samples) or paired The t-test uses an approximation to the sampling distribution of the difference in For the 40
independent samples, we plug the sample variances into the Two sample and one sample t-test calculator with step by step explanation. Have Equal Variance (default). Groups Have Unequal Variance
(Welch t-test)
The independent samples t-test compares two independent groups of observations or measurements on a single characteristic. The independent samples t-test is the between-subjects analog to the
dependent samples t-test, which is used when the study involves a repeated measurement (e.g., pretest vs. posttest) or matched observations (e.g., older vs. younger siblings).
T Test in SPSS: Output. Your output will include: The Levine’s test for equal variance (the first section of the Independent Samples Test box). If the significance level is larger than .05, you
should use the first line in the output table, Equal variances assumed. If the value is .05 or lower, use the second row of results. Independent-Samples T Test. The Independent-Samples T Test
procedure compares means for two groups of cases. Ideally, for this test, the subjects should be randomly assigned to two groups, so that any difference in response is due to the treatment (or lack
of treatment) and not to other factors.
30 Jan 2019 A ttest for dependent samples (often called a paired t test) uses each What are some simple steps I can take to protect my privacy online? 14 Aug 2018 ISSN: 2575-5919 (Print); ISSN:
2575-5927 (Online). A Brief Review of Independent, Dependent and One Sample t-test. Banda Gerald. 25 Apr 2019 Since we are conducting a two-sample t-test here, the df formula is slightly different
(as we have two samples instead of one). Here it is:. The independent-samples t-test (or independent t-test, for short) compares the means between two unrelated groups on the same continuous,
dependent if espresso per latte and a standard deviation of .11 oz. Use alpha = .05 and run an independent samples t-test to determine if there is a significant difference The first part covers
z-tests, single sample t-tests, and dependent t-tests. t-test is appropriate when you want to compare two independent samples, so two | {"url":"https://optionedpkmi.netlify.app/pentecost87925pe/online-t-test-for-independent-samples-mi","timestamp":"2024-11-13T18:15:55Z","content_type":"text/html","content_length":"33851","record_id":"<urn:uuid:d42ddec3-1ef6-4836-95f6-71b58b3524f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00165.warc.gz"} |
Properties of the Gaussian
Next: The Wiener-Khinchin Theorem Up: Signals in Radio Astronomy Previous: Introduction Contents
The general statement of gaussianity is that we look at the joint distribution of
Q is a quadratic expression which clearly has to increase to
does the job and has one parameter, the ``Variance''
Such a distribution can be visualised as a cloud of points in 1.2).
The following basic properties are worth noting (and even checking!).
1. We need
2. The constant in front is
3. The average values of inverse of the matrix of a's. For example,
4. By time stationarity,
The extra information about the correlation between 1.8
Next: The Wiener-Khinchin Theorem Up: Signals in Radio Astronomy Previous: Introduction Contents NCRA-TIFR | {"url":"https://www.gmrt.ncra.tifr.res.in/doc/WEBLF/LFRA/node6.html","timestamp":"2024-11-10T17:34:03Z","content_type":"text/html","content_length":"11015","record_id":"<urn:uuid:f8b016b8-a6bc-49f9-9690-d08bd60c3853>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00877.warc.gz"} |
Star98 Educational Dataset
Star98 Educational Dataset¶
This data is on the California education policy and outcomes (STAR program results for 1998. The data measured standardized testing by the California Department of Education that required evaluation
of 2nd - 11th grade students by the the Stanford 9 test on a variety of subjects. This dataset is at the level of the unified school district and consists of 303 cases. The binary response variable
represents the number of 9th graders scoring over the national median value on the mathematics exam.
The data used in this example is only a subset of the original source.
Number of Observations - 303 (counties in California).
Number of Variables - 13 and 8 interaction terms.
Definition of variables names::
NABOVE - Total number of students above the national median for the
math section.
NBELOW - Total number of students below the national median for the
math section.
LOWINC - Percentage of low income students
PERASIAN - Percentage of Asian student
PERBLACK - Percentage of black students
PERHISP - Percentage of Hispanic students
PERMINTE - Percentage of minority teachers
AVYRSEXP - Sum of teachers' years in educational service divided by the
number of teachers.
AVSALK - Total salary budget including benefits divided by the number
of full-time teachers (in thousands)
PERSPENK - Per-pupil spending (in thousands)
PTRATIO - Pupil-teacher ratio.
PCTAF - Percentage of students taking UC/CSU prep courses
PCTCHRT - Percentage of charter schools
PCTYRRND - Percentage of year-round schools
The below variables are interaction terms of the variables defined
Jeff Gill’s Generalized Linear Models: A Unified Approach
Used with express permission from the original author, who retains all rights.
Last update: Oct 29, 2024 | {"url":"https://www.statsmodels.org/devel/datasets/generated/star98.html","timestamp":"2024-11-11T08:08:05Z","content_type":"text/html","content_length":"46320","record_id":"<urn:uuid:300b8cc4-5bf9-4e22-b2c5-2833603ef644>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00458.warc.gz"} |
Thinking Maths Activity lessons
6 Thinking Maths Activity lessons
Each Thinking Maths activity is a stand-alone unit intended for mainstream classes in the 11-14 age range. The example below, TM1 Algebra,shows the unit structure.
Each lesson is presented in two or more episodes; each episode is structured in three ‘acts.
Act 1 Concrete Preparation: introduces the mathematical context at a level all pupils can process, so they can focus on the work to come.
Act 2 Collaborative Learning: small group work, with the intention that each group will have something to contribute in the next act.
Act 3 Collaborative Learning: whole class sharing, where each group shows their ideas or expresses their difficulties to the whole class, enabling the other groups to contribute to and benefit from
the discussion.
The mathematical context and content of the lesson, links to other areas of the mathematics curriculum, resources needed
A summary of the episodes of the lesson | {"url":"https://community.letsthink.org.uk/came/chapter/5/","timestamp":"2024-11-03T18:39:19Z","content_type":"text/html","content_length":"69261","record_id":"<urn:uuid:113d3314-30ba-45ec-a9fa-74ebf309b4f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00523.warc.gz"} |
Network Flow Theory Assignment Help | Expert Solutions
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Inverse differentiation?
• Enter the function y(t)=t^3-2*t+1 for -2<=t<2 and display the function only.
• Create a screenshot of your display.
• Now display the derivative and save the derivative values to a file.
• Open the derivative values file, reading in the derivative as a new function.
• Calculate the integral of this function, using an initial value of t=0, and display the integral only
• Create a second screenshot
Now answer the following:
1. How are the two graphs in the screenshots different?
2. Why are they different?
3. Can you change the setup values and the initial value for t and replot the graphs so as to be as similar as possible?
#differentiation #integration #inverse
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Is it safe to pay someone to do my C++ programming homework? | Hire Someone To Do Programming Assignment
Is it safe to pay someone to do my C++ programming homework?
Is it safe to pay someone to do my C++ programming homework? I can pay the programmer with the right balance of money and time, and they can spend that money for other tasks. (If they want something
they could donate it to a charity that does it for them) It is smart for the business to create some special educational resources for their students, but I would like if I would focus on one C++
project (C++ Mvc for my students by student library B and if a student can do research for a certain class) or on a third project (this is a mixed class) such as creating a library to export one of
my projects for a student library to C++ to be reused as a project. A couple of questions before I commit my degree in design and have my second year of finals completed: What are your plans for 2011
/12/16? Did you have the same math grades for the F12 finals with Math Labs, Core Graphics, Stereos and your Maths Projects made for students I have worked on since 2003? What are your plans for 2011
/12/16? What are your plans for 2011/12/17? Who is your student library that you are working on? A lot of people do the same thing. I am considering building my library (Tower) and storing it for
every student project I am working on including my previous math experiments and projects. If I cut it, how can I do it this way? I have created some very unique stuff to test and set up in 2012. I
am looking for some ways that the library has to be stored. How can I do something like this? How will I use it? A close, but no, way will I be able to do some high-achievement so other libraries
would not only fit me and the program, but also I think I can take a few of the projects I have already done for you (I need grades): (Is it safe to pay someone to do my C++ programming homework? I
want to see what can be done by C++ thinking about C++. It doesn’t need C++ or C++ code and I’m not sure if I will be able to see all the things I can think about, but I’m sure it will be enough. I
just saw some of the pictures on the “dynamic-coercise” on a stackoverflow thread and they all look great. (My apologies, I don’t know where to start because I have been doing lots of work, and then
thought that I could just go to this forum for a little. I had questions, especially after being suggested that I should have just posted about it, but I think there might be a bit more to it because
I am nearly finished building all my own code at once!) The other thing I would really like is to look at “an extension” (or something!) in C++ and apply it to a class in C. Although I don’t know
where exactly this is, I am trying to understand how that works in C++ and it’s in a way I don’t know anything about or even saw, but I’m hopeful anyway. I wrote a piece of code which is somewhere in
the C++ language so I should include it. Basically this compiles with the help of the module and will appear in the top level “libraries page”. Otherwise, I’d be able to see how everything works.
Also, the code (and libraries) currently listed here are going to only be included in the kernel so what I am waiting too for is to add it in to the main class, in C++. (You can find more info and
read through the most efficient way to do this here.) I have a different project that I should follow, maybe including it, because it’s something other teams might like and maybe I would even
encourage them to look at it. First, IIs it safe to pay someone to do my C++ programming homework? A couple days ago, while working on my job, I faced a large error. When I changed my environment to
C/C++, my code would stop working as expected, and not like others had imagined.
Someone Who Grades Test
And I don’t; it sounds like that’s going to lead to confusion when doing C and C++ programming tasks that require multiple C# projects in the same working directory. I’ve tried the error in other
people’s code that involve “C++” and “PHP”, have also managed to get the C/C++ library to work perfectly. I guess someone missed that point when the code was written for PHP because it was supposed
to crash on startup? ~~~ lunari > I haven’t attempted this task. Actually, my very first attempt at it was to not create any project > with a C++ solution: Why would you fail to create a C++ project,
and then simply check the directory and try to create a “building” folder? If you still don’t create your project with your existing directory, why is your directory not created with the code you
just wrote? ~~~ tjowid There’s a difference between creating a project and creating a site… It contains data, so presumably the first website created is a project that has a link to your code you are
writing, but it has to create a website. In Linux you choose an SPS server that has all the necessary tools for helping you to create a site. It sounds like the second site created with your program
is the best “server” that has the best tool to do that. Edit: In the case of the third site, that’s an alternative and probably the only solution. The site creates a home-site that can be installed
at important source time. ~~~ lunari I would love to get this working. You don’t need a subdomain now, right? ~~~ bob242424 you will need multiple subdomains in the hosting address anyway The word
subdomains is where I put the ideas I made for building. —— rtd I run an install of Windows along with an open-source JT plugin that can change directories in Windows Registry 2007 as needed. My
problem is getting a command to run with an old IDE and running vino instantly, after the vino installs (but before the JT is setup). To work with this, I made a’scripts’ folder. With that module, I
added a lot of code that only the code that was in the cmd-edit repository needed in the build file, and built my project with the same syntax. I also imported the file of.net as the modules. I
loaded and run the js scripts | {"url":"https://progassignments.com/is-it-safe-to-pay-someone-to-do-my-c-programming-homework","timestamp":"2024-11-09T04:36:54Z","content_type":"text/html","content_length":"110291","record_id":"<urn:uuid:d601fc12-bb82-41d5-8e4d-7112a0935846>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00658.warc.gz"} |
how to control this rule take effect in both situations
Hi Jiger,
> my intention is if one index of the two PD is the same as the index of
> A, it should be replace by ruAz. I don't know if the rule is right, or
> how to express such an idea with a rule?
> SortCovDsStart[PD];
> ruAz = MakeRule[{PD[-b][PD[-a][A[a]]] , mg[-b, -e] mg[a, c] PD[-c][PD[-a][A[e]]]}];
This rule is syntactically correct and given that you have used
SortCovDsStart[PD] I see that you expect xTensor to infer that the PDs
can be reordered in the pattern matching. But MakeRule is not that
intelligent. MakeRule is a way to construct the simplest rules, which
are also the most frequent, but if you need something with more
complicated pattern constructs, then you have to construct the rule
yourself. Using the output of MakeRule is a good starting point:
HoldPattern[PD[-(b_Symbol)][PD[-(a_Symbol)][A[a_Symbol]]]] :>
Module[{t$1, t$2, t$3}, mg[-b, -t$3]*mg[t$1, t$2]*PD[-t$2][PD[-t
The problem is that the left hand side is only valid for what it says:
the derivative index matching the derivative of A is the most internal
one. We can simply add the two needed alternatives:
HoldPattern[PD[-(b_Symbol)][PD[-(a_Symbol)][A[a_Symbol]]] | PD[-
(a_Symbol)][PD[-(b_Symbol)][A[a_Symbol]]]] :>
Module[{t$1, t$2, t$3}, mg[-b, -t$3]*mg[t$1, t$2]*PD[-t$2][PD[-t
Now this rule works in the two cases you need. You could also replace
the rule by a list of two rules, having the two needed left-hand-
sides, and the same right-hand-side.
I think that MakeRule should be able to see this equivalent case. I'll
add it in the future. | {"url":"https://groups.google.com/g/xact/c/t41EJWWWs_E","timestamp":"2024-11-14T02:33:01Z","content_type":"text/html","content_length":"700337","record_id":"<urn:uuid:6a01e7f6-3199-4d9e-93c8-95b3d93f4b95>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00044.warc.gz"} |
Mathematical Analysis, Modelling, and Applications
The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of
these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The
present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in
this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming
from the real world, from industrial applications, and from complex systems. | {"url":"https://www.math.sissa.it/taxonomy/term/2?page=275","timestamp":"2024-11-08T22:28:04Z","content_type":"application/xhtml+xml","content_length":"45097","record_id":"<urn:uuid:f2ac17bd-441f-40ad-9d1b-06dd8a3158c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00694.warc.gz"} |
The Metropolis Hastings Algorithm 2
Last updated: 2022-04-26
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In this vignette we follow up on the original algorithm with a couple of points: numerical issues that were glossed over, and a useful plot.
Avoiding numerical issues in acceptance probability
The key to the MH algorithm is computing the acceptance probability, which recall is given by \[A= \min \left( 1, \frac{\pi(y)Q(x_t | y)}{\pi(x_t)Q(y | x_t)} \right).\]
In practice both terms in the fraction may be very close to 0, so on a computer you should do this computation on the logarithmic scale before exponentiating; something like this: \[\frac{\pi(y)Q(x_t
| y)}{\pi(x_t)Q(y | x_t)} = \exp\left[ \log\pi(y) + \log Q(x_t | y) - \log \pi(x_t) - \log Q(y | x_t) \right]\]
Furthermore, the log values in this expression should be computed directly, and not by computing them and then taking the log. For example, in our previous example we sampled from a target
distribution that was the exponential: \[\pi(x) = \exp(-x) \qquad (x>0)\] we we can directly compute \(\log \pi(x) = -x\). The code we had in that previous example can therefore be written as
log_target = function(x){
x = rep(0,10000)
x[1] = 100 #initialize; For purposes of illustration I set this to 100
for(i in 2:10000){
current_x = x[i-1]
proposed_x = current_x + rnorm(1,mean=0,sd=1)
A = exp(log_target(proposed_x) - log_target(current_x) )
x[i] = proposed_x # accept move with probabily min(1,A)
} else {
x[i] = current_x # otherwise "reject" move, and stay where we are
An important and useful plot
In practice MCMC is usually performed in high dimensional space. It can therefore be really hard to visualize directly the values of the chain. A simple 1-d summary that is always available is the
log-target density, \(\log \pi(x)\). So you should usually plot a trace of this whenever you run an MCMC scheme.
plot(log_target(x), main = "log-target value")
Here the plot shows “typical” behaviour of MCMC scheme: because the starting point is chosen not to be close to the optimal of \(\pi\), the chain initially takes some iterations to find a part of the
space where \(\pi(x)\) is ``large". Once it finds that part of the space it starts to explore around the region where \(\pi(x)\) is large.
The plot in the previous section immediately shows that there is an initial period of time where the Markov Chain is unduly influenced by its starting position. In other words, during those
iterations the Markov Chain has not “converged” and those samples should not be considered to be samples from \(\pi\). To address this it is common to discard the first set of iterations of any
chain; the iterations that are discarded are often called “burn-in”.
Here, based on the plot we might discard the first 1000 iterations or so as burn-in. Here are comparisons of the samples with and without burnin discarded:
hist(x, main="without burn-in discarded")
hist(x[-(1:1000)], main="with burn-in discarded")
R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.2
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] Rcpp_1.0.7 whisker_0.4 knitr_1.36 magrittr_2.0.2
[5] workflowr_1.7.0 R6_2.5.1 rlang_0.4.12 fastmap_1.1.0
[9] fansi_0.5.0 highr_0.9 stringr_1.4.0 tools_4.1.0
[13] xfun_0.28 utf8_1.2.2 git2r_0.29.0 jquerylib_0.1.4
[17] htmltools_0.5.2 ellipsis_0.3.2 rprojroot_2.0.2 yaml_2.2.1
[21] digest_0.6.28 tibble_3.1.6 lifecycle_1.0.1 crayon_1.4.2
[25] later_1.3.0 sass_0.4.1 vctrs_0.3.8 fs_1.5.0
[29] promises_1.2.0.1 glue_1.5.0 evaluate_0.14 rmarkdown_2.11
[33] stringi_1.7.5 bslib_0.3.1 compiler_4.1.0 pillar_1.6.4
[37] jsonlite_1.7.2 httpuv_1.6.3 pkgconfig_2.0.3
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A329563 - OEIS
That is, there are 5 primes, counted with multiplicity, among the 10 pairwise sums of any 5 consecutive terms.
Conjectured to be a permutation of the positive integers.
This sequence is quite different from the restriction of the "nonnegative" variant
to positive indices: it seems that the two have no common terms beyond a(6) = 8, except for the accidental a(22) = 15 and maybe some later coincidences of this type. There also appears to be no other
simple relation between the terms of these sequences, in contrast to, e.g.,
For n = 1, we consider pairwise sums among the first 5 terms chosen as small as possible, a(1..5) = (1, 2, 3, 4, 5). We see that we have indeed 5 primes among the sums 1+2, 1+3, 1+4, 1+5, 2+3, 2+4,
2+5, 3+4, 3+5, 4+5.
Then, to get a(6), consider first the pairwise sums among terms a(2..5), (2+3, 2+4, 2+5; 3+4, 3+5; 4+5), among which there are 3 primes, counted with multiplicity (i.e., the prime 7 is there two
times). So the new term a(6) must give exactly two more prime sums with the terms a(2..5). We find that 6 or 7 would give just one more (5+6 resp. 4+7), but a(6) = 8 gives exactly two more, 3+8 and
(PARI) {
(n, show=1, o=1, N=5, M=4, p=[], u=o, U)=for(n=o, n-1, show>0&& print1(o", "); show<0&& listput(L, o); U+=1<<(o-u); U>>=-u+u+=valuation(U+1, 2); p=concat(if(#p>=M, p[^1], p), o); my(c=N-sum(i=2, #p,
sum(j=1, i-1, isprime(p[i]+p[j])))); if(#p<M&&sum(i=1, #p, isprime(p[i]+u))<=c, o=u)|| for(k=u, oo, bittest(U, k-u)|| sum(i=1, #p, isprime(p[i]+k))!=c|| [o=k, break])); show&&print([u]); o} \\
optional args: show=1: print a(o..n-1), show=-1: append them on global list L, in both cases print [least unused number] at the end. See the wiki page for a function S() which returns a vector: a
(0..n-1) = S(5, 5; 1).
(6 primes using 5 consecutive terms),
(6 primes using 6 consecutive terms).
(4 primes using 4 consecutive terms),
(4 primes using 5 consecutive terms).
(3 primes using 4 consecutive terms),
(3 primes using 5 consecutive terms).
(2 primes using 3 consecutive terms),
(2 primes using 4 consecutive terms),
(2 primes using 5 consecutive terms).
(1 (odd) prime using 3 terms),
(1 prime using 2 terms);
(0 primes using 2 terms),
(0 primes using 3 terms),
ff: other variants. | {"url":"https://oeis.org/A329563","timestamp":"2024-11-13T11:45:11Z","content_type":"text/html","content_length":"19140","record_id":"<urn:uuid:0b59d626-7db3-403c-8bf8-a631112b882b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00820.warc.gz"} |
Lesson 19
Ways to Divide Larger Numbers
Warm-up: True or False: Ones, Tens, Twenties (10 minutes)
The purpose of this True or False is to reinforce the relationship between tens and ones (that 1 ten is equal to 10 ones, or 1 group of 10 is 10 groups of 1). This will be helpful when students use
base-ten blocks to represent division and decompose tens into ones to facilitate the process of dividing. It also allows students to practice finding the product of a one-digit whole number and a
multiple of 10.
• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time
• Share and record answers and strategy.
• Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
• \(4 \times 10 = 40 \times 1\)
• \(4 \times 20 = 4 \times 2 \times 10\)
• \(8 \times 20 = 8 \times 2 \times 1\)
• \(8 \times 20 = 16 \times 10\)
Activity Synthesis
• “How can you justify your answer without finding the value of both sides?”
• Consider asking:
□ “Who can restate _____’s reasoning in a different way?”
□ “Does anyone want to add on to _____’s reasoning?”
Activity 1: Divide with Base-Ten Blocks (20 minutes)
The purpose of this activity is for students to use strategies based on place value to find quotients greater than 10. Students use base-ten blocks to represent quotients with single-digit divisors,
for which it is intuitive to think of the divisor as the number of groups. In a later activity, students will be reminded that the divisor can also be interpreted as the size of each group.
Working with base-ten blocks encourages students to divide out the tens and then the ones, and to see that sometimes it is necessary to decompose one or more tens to finish putting the dividend into
equal groups. When students represent a quotient using base-ten blocks they reason abstractly and quantitatively (MP2).
MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important when deciding how to divide up the blocks. Display the sentence frame: “The next
time I use base-ten blocks to divide, I will look for/pay attention to . . . .“
Supports accessibility for: Visual-Spatial Processing, Attention
• Groups of 2
• Give each group base-ten blocks.
• “Use base-ten blocks to represent \(39 \div 3\).”
• 1–2 minutes: independent work time
• Select a student who divided the blocks into 3 groups of 13 to share their final representation, such as:
• “Why are there 3 groups?” (We are dividing by 3.)
• “How could the blocks have been divided to end up like this?” (The tens were put into 3 groups and then ones placed one by one into 3 groups until none were left.)
• Highlight that the blocks could have been divided up by the tens and then the ones.
• “Work with your partner on the first problem.”
• 5 minutes: partner work time
• Pause for a discussion.
• “What was different about using the blocks to find \(45 \div 3\) and using them to find \(55 \div 5\)?” (For \(45 \div 3\), it was necessary to decompose 1 ten to finish putting 45 into 3 equal
groups. That’s not necessary for \(55 \div 5\) because there was already the right number of tens and ones to make the 5 groups.)
• “Now, work independently to find the value of quotients in the second problem.”
• 6–8 minutes: independent work time
Student Facing
1. Use base-ten blocks to represent each expression. Then, find its value.
1. \(55 \div 5\)
2. \(45 \div 3\)
2. Find the value of each expression. Use base-ten blocks if you find them helpful.
1. \(63 \div 3\)
2. \(84 \div 7\)
3. \(100 \div 5\)
Activity Synthesis
• Invite students to share their responses and reasoning for the last set of quotients.
• Ask students who used base-ten blocks or drew diagrams: “Was it necessary to decompose any of the tens into ones to divide?” (It wasn’t necessary for \(63 \div 3\) because there was already the
right number of tens and ones to put into 3 groups. It wasn’t necessary for \(100 \div 5\) because I started with 10 tens and there was already the right number of tens to put into 5 groups.)
• “Why was it necessary or helpful to decompose the tens in 84?” (After putting 7 tens in 7 groups, there’s still 1 ten and 4 ones. The 1 ten couldn’t be split into 7 groups.)
Activity 2: Different Ways to Show Division (15 minutes)
The purpose of this activity is to show that the two meanings of division still apply when dividing larger numbers and that, in some cases, one interpretation may be more helpful than the other.
Students first analyze two ways of using base-ten blocks to represent \(65 \div 5\) and see that the divisor, 5, can be interpreted to mean the number of groups or the size of one group. They then
consider how they might interpret and represent the divisor in other quotients. The reasoning here prepares students to reason more strategically as they divide larger numbers.
• Groups or 2–4
• Give base-ten blocks to each group.
• Ask students to keep their materials closed.
• “Use base-ten blocks to find the value of \(60 \div 5\).”
• 1–2 minutes: independent work time
• “Now take a look at Jada and Han's work in the activity. Which of them represented the division the same way you did?”
• “Work with your partner to make sense of Jada’s and Han’s work and complete the first problem.”
• Pause for a brief discussion.
• “How was Jada’s and Han’s representation different? How did each of them interpret \(60 \div 5\)?” (Jada saw the 5 as the number of groups. Han saw the 5 as the number in each group.)
• Poll the class on how they interpreted \(60 \div 5\) when they represented it during the launch.
• “Now, work independently on the second set of problems.”
• 5 minutes: independent work time
Student Facing
Jada and Han used base-ten blocks to represent \(60 \div 5\).
Here is Jada’s work:
Here’s Han’s work:
1. Make sense of Jada’s and Han’s work.
1. What did they do differently?
2. Where do we see the value of \(60 \div 5\) in each person’s work?
2. How would you use base-ten blocks so you could represent these expressions and find their value? Be prepared to explain your reasoning.
1. \(64 \div 4\): Would you make 4 groups or groups of 4?
2. \(72 \div 6\): Would you make 6 groups or groups of 6?
3. \(75 \div 15\): Would you make 15 groups or groups of 15?
Activity Synthesis
• Invite students to share their responses and reasoning for the last set of problems.
• “How did you decide whether the divisor, the number we’re dividing by, is the number of groups or the amount in each group?” (It depends on the number. In the first two problems, the divisor was
4 and 6, so it was easier to think about 4 groups and 6 groups. In the last problem, the divisor was 15. It was easier to think about how many groups of 15 are in 75 than to think about making 15
groups from 75.)
Lesson Synthesis
“Today, we recalled that the divisor in a division expression can be seen as the number of groups or the size of each group.”
Display: \(96 \div 8\)
“If you are representing this quotient with base-ten blocks, would you put 9 tens and 6 ones in 8 groups, or would you put them into groups of 8?” (I would put them into 8 groups. Eight of the tens
can go into 8 groups easily. The 1 remaining ten and 6 ones make 16 ones, so 2 ones go in each group. I would put them into groups of 8. I know 10 groups of 8 is 80, so that takes care of the 8 tens.
The 1 remaining ten and 6 ones make 16, which is 2 groups of 8.)
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Vehicle state and parameter estimation based on adaptive robust unscented particle filter
In order to solve the problem that the measured values of key state parameters such as the lateral velocity and yaw rate of the vehicle are easily interfered by random errors, a filter estimation
method of vehicle state is proposed based on the principle of robust filtering and the unscented particle filter algorithm. Based on the establishment of a 3-DOF non-linear dynamic model and the
Dugoff tire model of the vehicle, the adaptive robust unscented particle filter(ARUPF) is used to filter and estimate the parameters of the vehicle state, and to realize the longitudinal and lateral
speed as well as the yaw rate of the vehicle during the driving process. The simulation and the real vehicle test results show that based on the adaptive robust unscented particle filter algorithm,
the vehicle driving state estimation can be realized, the measurement parameters can be effectively filtered, and the estimation accuracy is high.
• The adaptive robust unscented particle filter algorithm can realize the vehicle driving state estimation.
• The adaptive robust unscented particle filter algorithm can effectively filter the measurement parameters.
• The adaptive robust unscented particle filter algorithm has higher estimation accuracy.
1. Introduction
When analyzing the active safety of a vehicle, it is particularly important to obtain the state parameters during the driving process of the vehicle. The algorithms used to estimate the key state
parameters of automobiles mainly include Kalman filter algorithm, particle filter algorithm, sliding mode observer algorithm, robust observer algorithm and Romberg observer algorithm. The sliding
mode observer algorithm depends on the accuracy and performance of the sensor, otherwise it is prone to chattering. The robust observer algorithm is prone to underestimation bias in some cases, which
causes the algorithm to diverge. The calculation process of the Romberg observer algorithm is complex, and it is difficult to meet the real-time requirements of the vehicle estimation algorithm.
A vehicle is a strong nonlinear system, but the Kalman filter can only process linear systems. Based on the standard Kalman filter algorithm, the Extended Kalman Filter (EKF) and Unscented Kalman
Filter (UKF) are developed to handle nonlinear systems. The extended Kalman filter algorithm linearizes the nonlinear system by performing Taylor expansion of the mathematical model of the nonlinear
system at the optimal point, and solving the Jacobian matrix of the nonlinear function. When this algorithm is used to linearize the nonlinear system, only the first-order system is retained, and the
second-order or higher-order components are discarded, so there is a certain estimation bias. At the same time, when the estimated target system has strong nonlinearity, its calculation amount is too
large and the Jacobian matrix is complicated to solve, so the divergence is easy to occur. For each sigma point obtained, its mean and variance are ensured the same as the original data, and brought
into the nonlinear system for unscented transformation making it close to Gaussian distribution through weighted summation of samples. Based on the characteristics of Gaussian distribution, the
algorithm can be accurate to the third-order mean and covariance, and the operation is simple, and the obtained system is stable.
The problem of vehicle state estimation has been widely studied. A brief review is presented in what follows.
Lenzo et al.^[1] proposed an Extended Kalman Filter (EKF) method to estimate the vehicle states as well as tyre-road coefficient of friction. Li et al. [2] proposed a new situation-sensitive method
to improve the vehicle detection at nighttime. In order to estimate vehicle states and parameter with high precision, Zhu et al. [3] presented a modified particle filter. Wang et al. [4] proposed a
novel adaptive fault-tolerant extended Kalman filter to estimate vehicle state in case of partial loss of sensor data. In order to improve the driving dynamics and the driving safety, Henning et al.^
[5] presented an integrated lateral dynamics control concept for a over-actuated vehicle. Zhao et al. [6] presented a state estimation method to enable fully autonomous flight in outdoor
environments. Alatorre et al. [7] proposed an algorithm that merges the concepts of least squares method and sliding mode observer for the estimation of the vehicle mass. Heidfeld et al. [8] proposed
an Unscented KALMAN Filter (UKF) for simultaneous state and parameter estimation. Badini et al. [9] presented a simple parameter independent speed estimation algorithm for vector-controlled permanent
magnet synchronous motor (PMSM) drive. Kulikov et al. [10] resolved the lack of square-root implementations by means of hyperbolic QR transforms applied for yielding J-orthogonal square roots.
Malikov [11] solved the problem of differential equations with periodic using the quadratic Lyapunov function. Takikawa1 et al. [12] used a global navigation satellite system (GNSS) Doppler for
accurate vehicular trajectory estimation. Gao et al. [13, 14] presented a new methodology of distributed state fusion for multisensory nonlinear systems by using the sparse-grid quadrature filter.
In the literatures proposed above, some researchers did not implement the algorithms in onboard real-time systems. And some researchers missed the diverse traffic conditions in daytime and nighttime
when estimating states of the vehicle. At the same time, more vehicle parameters including the yaw inertia of the vehicle hadn’t been estimated in a lot of literatures. And the estimation error
covariance as well as the closed-loop operation of the state estimator with a driving dynamic hadn’t been investigated. For some researchers, the usability of the proposed model should be improved.
In the current research, most of the observation noise covariance matrix is set to a fixed value. However, in the actual driving process of the vehicle, the process noise and observation noise are
randomly generated. The unscented particle filter (UPF) algorithm uses the unscented Kalman filter method to generate the proposal density function, so that the prior probability peak value and the
likelihood function peak value have good agreement reducing particle degradation. However, the accuracy is affected by the uncertainty of the system noise. And the lack of an adaptive adjustment
mechanism makes it impossible to adjust the filter gain and related parameters in real time. In order to better estimate the state parameters of the vehicle in real-time and effectively, the adaptive
robust unscented particle filter (ARUPF) method, which can adjust filter parameters in real time and has better adaptability to interference noise is proposed.
The ARUPF algorithm absorbs the advantages of robust estimation, robust adaptive filtering and particle filtering fully. It combines robust estimation principle and the UPF algorithm through
equivalent weight function and adaptive factor. The state model information and measurement model information can be controlled by selecting appropriate weight function and adaptive factor,
suppressing the influence of abnormal interference. The ARUPF algorithm overcomes the shortcomings of using only a single filtering algorithm. The algorithm uses the important density function
selected from the three important steps of UT transformation, equivalent weight and adaptive factor adjustment to perform importance sampling, reducing the degree of particle degradation, and having
high filtering accuracy.
2. Mathematical model of vehicle dynamics
2.1. 3-DOF vehicle model
The vehicle state estimation model is established based on a 3-DOF vehicle model. The 3-DOF vehicle model is shown in Fig. 1. $xoy$ is the vehicle coordinate system, and the origin of the vehicle
coordinate system coincides with the center of mass of the vehicle.
Fig. 13-DOF vehicle model
The paper adopts a simplified estimation model, and establishes a nonlinear 3-DOF vehicle model including longitudinal and lateral as well as yaw motion. And it is assumed that:
(1) The center of mass of the vehicle model coincides with the origin of the vehicle coordinate system;
(2) The suspension has no effect on the vertical movement of the vehicle;
(3) The vehicle has no degree of freedom in the pitch and roll directions;
(4) The impact of the longitudinal rolling resistance is ignored for the state parameter estimation.
In Fig. 1, $a$ and $b$ are the distances of front and rear axles from the center of gravity respectively; ${t}_{f}$ and ${t}_{r}$ are the tracks of the front and rear wheels respectively; ${\alpha }_
{fl,fr}$ are the side slip angles of the left and right front wheels; ${\alpha }_{rl,rr}$ are the side slip angles of the left and right rear wheels; ${{F}_{x_}}_{fl,fr,rl,rr}$ are the longitudinal
forces of the left front, right front, left rear, and right rear wheels; ${{F}_{y_}}_{fl,fr,rl,rr}$ are the lateral forces of the left front, right front, left rear, and right rear wheels; ${\delta }
_{fl,fr}$ are the wheel angles of the left and right front wheels.
The dynamic equation of the 3-DOF vehicle model is as follows:
$\left\{\begin{array}{l}{a}_{x}=\stackrel{˙}{u}-vr,\\ {a}_{y}=\stackrel{˙}{v}+ur,\\ {I}_{z}\stackrel{˙}{r}=\sum M,\end{array}\right\$
where, $u$ and $v$ are the longitudinal and the lateral speed; $r$ is the yaw rate; ${a}_{x}$ and ${a}_{y}$ the longitudinal and lateral acceleration; $M$ is the yaw moment; ${I}_{z}$ is the moment
of inertia around the z axis of the vehicle.
The side slip angle of the center of mass is:
$\beta =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{v}{u}\right).$
According to the kinetic equation, the calculation formulas for other parameters are as follows:
$\begin{array}{l}M=a\left({F}_{x_fl}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fl}+{F}_{y_fl}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{fl}\right)-\frac{{t}_{f}}{2}\left({F}_{x_fl}\mathrm{c}\mathrm{o}\
mathrm{s}{\delta }_{fl}-{F}_{y_fl}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fl}\right)\\ +a\left({F}_{x_fr}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fr}+{F}_{y_fr}\mathrm{c}\mathrm{o}\mathrm{s}{\delta
}_{fl}\right)+\frac{{t}_{f}}{2}\left({F}_{x_fr}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{fr}-{F}_{y_fr}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fr}\right)\\ -b{F}_{x_rl}-\frac{{t}_{f}}{2}{F}_{x_rl}-b
${a}_{x}=\frac{1}{m}\left({F}_{{x}_{fl}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{fl}-{F}_{{y}_{fl}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fl}+{F}_{{x}_{fr}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_
{fr}-{F}_{{y}_{fr}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fr}+{F}_{{x}_{rl}}+{F}_{{x}_{rr}}\right),$
${a}_{y}=\frac{1}{m}\left({F}_{{x}_{fl}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{fl}+{F}_{{y}_{fl}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{fl}+{F}_{{x}_{fr}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_
{fr}+{F}_{{y}_{fr}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{fr}+{F}_{{x}_{rl}}+{F}_{{x}_{rr}}\right),$
$\left\{\begin{array}{l}{\alpha }_{fl,fr}={\delta }_{fl,fr}-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{g}\frac{v+ar}{u±\frac{{t}_{f}}{2}r},\\ {\alpha }_{rl,rr}=-\mathrm{a}\mathrm{r}\mathrm{c}\
$\left\{\begin{array}{l}{v}_{fl,fr}=\sqrt{{\left(u\mp \frac{{t}_{f}}{2}r\right)}^{2}+\left(v+ar{\right)}^{2}},\\ {v}_{rl,rr}=\sqrt{{\left(u\mp \frac{{t}_{r}}{2}r\right)}^{2}+\left(v-br{\right)}^{2}},
$\left\{\begin{array}{l}{{F}_{z_}}_{fl,fr}=\left(\frac{1}{2}mg±m{a}_{y}\frac{h}{{t}_{f}}\right)b-\frac{1}{2}m{a}_{x}h,\\ {{F}_{z_}}_{rl,rr}=\left(\frac{1}{2}mg±m{a}_{y}\frac{h}{{t}_{r}}\right)b+\frac
where, $m$ is the vehicle mass; $h$ is the height of the center of mass; ${R}_{e}$ is the rolling radius of the wheels; $l$ is the wheelbase; ${v}_{fl,fr}$ is the center speed of the left and right
front wheels; ${v}_{rl,rr}$ is the center speed of the left and right rear wheels; ${{F}_{z_}}_{fl,fr}$ is the vertical load of the left and right front wheels; ${{F}_{z_}}_{rl,rr}$ is the vertical
load of the left and right rear wheels.
2.2. Tire model
The expression of the Dugoff tire model is relatively simple, with fewer unknown parameters, and can accurately describe the nonlinear characteristics of tire friction. Therefore, the tire model
selected in this article is the Dugoff tire model [15]:
${F}_{xi}={C}_{\sigma }\frac{{\sigma }_{i}}{1+{\sigma }_{i}}f\left({\lambda }_{i}\right),$
${F}_{yi}={C}_{\alpha }\frac{\mathrm{t}\mathrm{a}\mathrm{n}{\alpha }_{i}}{1+{\alpha }_{i}}f\left({\lambda }_{i}\right),$
$f\left({\lambda }_{i}\right)=\left\{\begin{array}{l}\left(2-{\lambda }_{i}\right){\lambda }_{i},{\lambda }_{i}\le 1,\\ 1,{\lambda }_{i}>1,\end{array}\right\$
${\lambda }_{i}=\frac{\mu {F}_{zi}\left(1+{\sigma }_{i}\right)}{2\sqrt{{C}_{\sigma }^{2}\cdot {\sigma }_{i}^{2}+{C}_{\alpha }^{2}\mathrm{t}\mathrm{a}{\mathrm{n}}^{2}{\alpha }_{i}}},$
where ${C}_{\sigma }$ and ${C}_{\alpha }$ are the longitudinal and cornering stiffness of the tires respectively; ${\sigma }_{i}$ is the longitudinal slip ratio; ${\alpha }_{i}$ is the slip angle of
the tire; $i$ indicates $fl$, $fr$, $rl$ and $rr$; $\mu$ is the road friction coefficient; when ${\lambda }_{i}>1$, the wheel is in the linear state area; when ${\lambda }_{i}\le 1$, the wheel is in
the non-linear state region.
2.3. Nonlinear vehicle system containing noise
The state vector of the nonlinear vehicle system is set as $\mathbf{x}=\left[{v}_{x},u,r{\right]}^{T}$, the system input is $\mathbf{u}=\left[{a}_{x}{\right]}^{T}$, and the observation vector is $\
3. The ARUPF Algorithm
The traditional particle filter algorithm has the defect of particle degradation in the iterative process, resulting in waste of computing resources and low accuracy of estimation results. In order
to solve the above problems, the filtering algorithm is often optimized by increasing the number of particles, re-sampling, and selecting a reasonable proposal density function. Increasing the number
of particles can effectively alleviate the particle degradation, but it increases the computational workload of the system. The re-sampling method can increase the diversity of particles and avoid
particle degradation. The adaptive robust unscented particle filter algorithm uses the unscented transformation algorithm to calculate the mean and covariance of each particle and establish a
reasonable proposal density function combining the robust filter estimation algorithm to automatically adjust the gain matrix and system variance making the distribution of sample points fit better
with the maximum likelihood function. Unscented particle filter algorithm is easy to implement in engineering and can effectively reduce the workload of the system. The specific methods are as
1) Initialization, $k=$0.
Drawing initial state particles from the prior distribution:
$\left\{\begin{array}{l}{\stackrel{-}{x}}_{0}^{\left(i\right)}=E\left[{x}_{0}^{\left(i\right)}\right],\\ {p}_{0}^{\left(i\right)}=E\left[\left({x}_{0}^{\left(i\right)}-{\stackrel{-}{x}}_{0}^{\left(i\
right]=\left[\begin{array}{lll}{p}_{0}^{\left(i\right)}& 0& 0\\ 0& Q& 0\\ 0& 0& R\end{array}\right],\end{array}\right\$
where ${\stackrel{-}{x}}_{0}^{\left(i\right)}$ and ${p}_{0}^{\left(i\right)}$ are the mathematical expectation and variance of the initial particle respectively, ${\stackrel{-}{x}}_{0}^{\left(i\
right)a}$ and ${p}_{0}^{\left(i\right)a}$ are the mathematical expectation and variance of the initial Sigma point, respectively; $Q$ and $R$ are the covariance matrix and the observation covariance
matrix of the system respectively.
2) Importance sampling.
Calculating the mean and variance using the unscented Kalman algorithm.
(1) Extracting the set of Sigma points:
${x}_{k-1}^{\left(i\right)a}=\left[{\stackrel{-}{x}}_{k-1}^{\left(i\right)a}{\stackrel{-}{x}}_{k-1}^{\left(i\right)a}-\sqrt{\left({n}_{a}+\lambda \right){p}_{k-1}^{\left(i\right)a}}{\stackrel{-}{x}}_
{k-1}^{\left(i\right)a}+\sqrt{\left({n}_{a}+\lambda \right){p}_{k-1}^{\left(i\right)a}}\right],$
where ${x}_{k-1}^{\left(i\right)a}$ and ${p}_{k-1}^{\left(i\right)a}$ are the mathematical expectation and variance of the extracted particles respectively; ${n}_{a}$ and $\lambda$ are the state
dimension and scaling factor respectively.
(2) One-step prediction for the Sigma point set:
$\left\{\begin{array}{l}{x}_{k\left|k-1\right}^{\left(i\right)a}=f\left({x}_{k-1}^{\left(i\right)x},k-1\right),\\ {\stackrel{-}{x}}_{k\left|k-1\right}^{\left(i\right)}=\sum _{j=0}^{2{n}_{a}}{W}_{j}^
{m}{x}_{j,k\left|k-1\right}^{\left(i\right)x},\\ {p}_{k\left|k-1\right}^{\left(i\right)}=\sum _{j=0}^{2{n}_{a}}{W}_{j}^{c}\left[{x}_{j,k\left|k-1\right}^{\left(i\right)x}-{\stackrel{-}{x}}_{k\left|
$\left\{\begin{array}{l}{Z}_{k\left|k-1\right}^{\left(i\right)}=h\left({x}_{k\left|k-1\right}^{\left(i\right)x},{x}_{k\left|k-1\right}^{in}\right),\\ {\stackrel{-}{Z}}_{k\left|k-1\right}^{\left(i\
right)}=\sum _{j=0}^{2{n}_{a}}{W}_{j}^{c}{Z}_{j,k\left|k-1\right}^{\left(i\right)},\end{array}\right\$
where ${x}_{k\left|k-1\right}^{\left(i\right)a}$, ${\stackrel{-}{x}}_{k\left|k-1\right}^{\left(i\right)}$ and ${p}_{k\left|k-1\right}^{\left(i\right)}$ are state value, mathematical expectation and
variance of the Sigma particle after one-step prediction respectively; ${Z}_{k\left|k-1\right}^{\left(i\right)}$ and ${\stackrel{-}{Z}}_{k\left|k-1\right}^{\left(i\right)}$ are the observed value and
the observed mean value obtained by inputting the observation equation for the Sigma point after one-step prediction respectively; ${W}_{j}^{m}$ and ${W}_{j}^{c}$ are the calculation weights of the
mean and the covariance corresponding to Sigma respectively.
(3) Integrating the observation data to update the mean, Kalman gain and covariance of the Sigma point set:
$\left\{\begin{array}{l}{P}_{{\stackrel{-}{Z}}_{k},{\stackrel{-}{Z}}_{k}}=\sum _{j=0}^{2{n}_{a}}{W}_{j}^{\left(c\right)}\left[{Z}_{j,k\left|k-1\right}^{\left(i\right)}-{Z}_{k\left|k-1\right}^{\left(i
\right)}\right]\left[{Z}_{j,k\left|k-1\right}^{\left(i\right)}-{Z}_{k\left|k-1\right}^{\left(i\right)}{\right]}^{T},\\ {\stackrel{^}{P}}_{K}^{\left(i\right)}={P}_{k\left|k-1\right}^{\left(i\right)}-
{K}_{k}{P}_{{\stackrel{-}{Z}}_{k},{\stackrel{-}{Z}}_{k}}{K}_{k}^{T},\\ {P}_{{x}_{k},{Z}_{k}}=\sum _{j=0}^{2{n}_{a}}{W}_{j}^{\left(c\right)}\left[{x}_{j,k\left|k-1\right}^{\left(i\right)}-{\stackrel
{-}{x}}_{k\left|k-1\right}^{\left(i\right)}\right]\left[{x}_{j,k\left|k-1\right}^{\left(i\right)}-{\stackrel{-}{x}}_{k\left|k-1\right}^{\left(i\right)}{\right]}^{T},\\ K={P}_{{\stackrel{-}{Z}}_{k},{\
stackrel{-}{Z}}_{k}}{P}_{{x}_{k},{Z}_{k}},\\ {\stackrel{-}{x}}_{k}^{\left(i\right)}={\stackrel{-}{x}}_{k\left|k-1\right}^{\left(i\right)}+{K}_{k}{Z}_{k}-{\stackrel{-}{Z}}_{k\left|k-1\right}^{\left(i\
where ${P}_{{\stackrel{-}{Z}}_{k},{\stackrel{-}{Z}}_{k}}$ and ${P}_{{x}_{k},{Z}_{k}}$ are the observed covariance and system variance obtained by weighted calculation respectively; ${K}_{k}$, ${\
stackrel{-}{x}}_{k}^{\left(i\right)}$ and ${\stackrel{^}{P}}_{K}^{\left(i\right)}$ are the system gain matrix, state value and variance after state update respectively.
3) ARUPF algorithm.
The ARUPF algorithm is based on robust estimation filtering theory, which controls the abnormal situation of the observed value of the dynamic model, and constructs an adaptive factor to control the
error of the dynamic model. If ${\stackrel{-}{P}}_{i}$ is set as the weight matrix of the state matrix ${\stackrel{-}{x}}_{k}^{\left(i\right)}$, the equivalent weight matrix is $\stackrel{-}{P}=\
mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\stackrel{-}{P}}_{1},{\stackrel{-}{P}}_{2},\cdots ,{\stackrel{-}{P}}_{k}\right)$. The principle for using the IGG (Institute of Geodesy & Geophysics)
method to generate the equivalent weight function is as follows:
${\stackrel{-}{P}}_{K}=\left\{\begin{array}{l}{P}_{K},\left(\left|{V}_{K}\right|\le {K}_{0}\right),\\ {P}_{K}\frac{{K}_{0}}{\left|{V}_{K}\right|}{\left(\frac{{K}_{g}-\left|{V}_{K}\right|}{{K}_{g}-{K}
_{0}}\right)}^{2},\left({K}_{0}\le \left|{V}_{K}\right|\le {K}_{g}\right),\\ 0,\left(\left|{V}_{K}\right|\ge {K}_{g}\right),\end{array}\right\$
where ${V}_{K}={A}_{k}{\stackrel{-}{x}}_{k}^{\left(i\right)}-{\stackrel{-}{Z}}_{k\left|k-1\right}^{\left(i\right)}$ is the detect residual values for sensors; the value range of the adjustment factor
are ${K}_{0}\in \left(1,1.5\right)$ and ${K}_{g}\in \left(3,8\right)$.
It is set that the sensor perception matrix is ${A}_{k}$. The system state vector is updated according to the weight matrix obtaining the system state solution vector of the adaptive robust Kalman
$x{r}_{k}=\left({\alpha }_{k}{P}_{{\stackrel{-}{x}}_{i}}+{A}_{k}^{T}{\stackrel{-}{P}}_{k}{A}_{k}{\right)}^{-1}\left({\alpha }_{k}{P}_{{\stackrel{-}{x}}_{i}}{\stackrel{-}{x}}_{k}^{\left(i\right)}+{A}_
{k}^{T}{\stackrel{-}{P}}_{k}{Z}_{k\left|k-1\right}^{\left(i\right)}\right),$${\alpha }_{k}=\left\{\begin{array}{l}1,\left(\left|\mathrm{\Delta }{\stackrel{-}{x}}_{k}\right|\le {c}_{0}\right),\\ \frac
{{c}_{0}}{\mathrm{\Delta }{\stackrel{-}{x}}_{k}}\left(\frac{{c}_{1}-\left|\mathrm{\Delta }{\stackrel{-}{x}}_{k}\right|}{{c}_{1}-{c}_{0}}{\right)}^{2}\left({c}_{0}\le \left|\mathrm{\Delta }{\stackrel
{-}{x}}_{k}\right|\le {c}_{1}\right),\\ 0,\left(\left|\mathrm{\Delta }{\stackrel{-}{x}}_{k}\right|\ge {c}_{1}\right),\\ \mathrm{\Delta }{\stackrel{-}{x}}_{k}=‖x{r}_{k}-{\stackrel{-}{x}}_{k}^{\left(i\
right)}‖/\sqrt{tr\left(\sum {\stackrel{-}{x}}_{k}^{\left(i\right)}\right)}\left(\left|\mathrm{\Delta }{\stackrel{-}{x}}_{k}\right|\right),\end{array}\right\$
where ${\alpha }_{k}$ is the adaptive factor; $tr$ is the matrix trace operator; the ranges of the regulators ${c}_{0}$ and ${c}_{1}$ are ${c}_{0}\in \left(1,1.5\right)$, ${c}_{1}\in \left(3,8\right)
In the above formula, the weight matrix ${\stackrel{-}{P}}_{k}$ is obtained by judging the residual error; the adaptive factor ${\alpha }_{k}$ is obtained by the difference operation between the
state estimated value and the predicted value. The two parameters are used to adjust the Kalman gain, the sampled particle mean and the particle weight at the same time and update the particle and
normalize weights:
$\left\{\begin{array}{l}{K}_{k}^{*}={\stackrel{^}{P}}_{k}^{\left(i\right)}{\stackrel{-}{P}}_{k}^{-1},\\ {\stackrel{-}{x}}_{k}^{\left(i\right)*}={\stackrel{-}{x}}_{k\left|k-1\right}^{\left(i\right)}+
{K}_{k}^{*}\left({Z}_{k}-{\stackrel{-}{Z}}_{k\left|k-1\right}^{\left(i\right)}\right),\\ {\stackrel{^}{P}}_{K}^{\left(i\right)*}={\alpha }_{k}{P}_{k\left|k-1\right}^{\left(i\right)}-{K}_{k}^{*}{P}_
{{\stackrel{-}{Z}}_{k},{\stackrel{-}{Z}}_{k}}{K}_{k}^{T*},\\ {W}_{k}^{\left(i\right)*}\propto \frac{p\left({Z}_{k}\left|{\stackrel{^}{X}}_{k}^{\left(i\right)*}\right\right)p\left({\stackrel{^}{X}}_
where ${K}_{k}^{*}$ is the Kalman gain computed by the robustness algorithm; ${\stackrel{-}{x}}_{k}^{\left(i\right)*}$ is the State sample mean; ${\stackrel{^}{P}}_{K}^{\left(i\right)*}$ is the
sample variance; ${W}_{k}^{\left(i\right)*}$ is the updated particle weight value.
Using the resampling algorithm, the particle set is eliminated and copied based on the normalized weight, and the weight is reset to the new particle. When the prediction model has excessive abnormal
interference, the adaptive factor ${\alpha }_{k}$ is decreased, which can weaken the influence of the interference. When there is a large disturbance in the observation model, the abnormal influence
caused by the disturbance can be reduced by adjusting the weight matrix ${\stackrel{-}{P}}_{k}$. The adaptive robust unscented particle filter algorithm can effectively solve the problem of gross
error and abnormal state of the system observation, and establish a reasonable particle filter which effectively solves the problem of particle degradation.
4. Numerical simulations and experimental verification
4.1. Numerical simulations
A Volkswagen vehicle is verified by a simulation test on the virtual prototype software ADAMS. And the real vehicle used for test is the same with the virtual vehicle model in Adams. That is to say,
the parameters of the vehicle dynamics model in ADAMS are the same as our test vehicle shown in Table 1.
ADAMS/Car adopts a top-down modeling sequence. That is to say, the vehicle model and the system assembly model are built on the basis of each subsystem model. And each subsystem model needs to be
established by templates, so the establishment of each template is the key steps to establish the vehicle model. The template establishment process is as follows:
(1) Simplification of the physical model: According to the relative motion relationship between the various parts of the subsystem, defining the “Topological Structure” of the parts, integrating the
parts, and defining the parts without motion relationship as a “Gene-ralPart”.
(2) Determining “Hard Point”: The hard point is the geometric positioning point at the key connection point of the part, and the determination of the hard point is to determine the geometric
coordinates of the key point of the part. To determine the hard point is to determine the geometric coordinates of the key points of the part in the subsystem coordinate system, and the hard point
can be modified in the vehicle model and template state.
Table 1Simulation parameters
Parameter Value
$m$ (kg) 1558
${I}_{z}$ (kg∙m^2) 2538
$a$ (m) 1.48
$b$ (m) 1.08
$h$ (m) 0.432
${t}_{f}$ (m) 1.51
${t}_{r}$ (m) 1.55
${r}_{e}$ (m) 0.32
(3) Determining the parameters of the part: Calculating or measuring the mass of the integrated part, the position of the center of mass, and the moment of inertia around the three axes of the center
of mass coordinate system. It should be noted that the directions of the three coordinate axes of the center of mass coordinate system must be parallel to the directions of the three axes of the
system coordinate system.
(4) Creating the “Geometry” of the part: Building the geometric model of the part on the basis of the hard point. Since the dynamic parameters of the part have been determined, the shape of the
geometric model has no influence on the dynamic simulation results. But in the kinematic analysis, the outer contour of the part is directly related to the motion check of the part. And the
intuitiveness of the model is considered. And considering the intuitiveness of the model, the geometry of the part should be as close to the actual structure as possible.
(5) Defining “Constrain”: Defining the type of constraint according to the motion relationship between parts. The components are connected by constraints to form the subsystem structure model.
Defining constraints is the key to correct modeling and is directly related to the rationality of the degrees of freedom of the system.
(6) Defining “Mount”: Defining assembly commands at the connections between subsystems and subsystems or external models.
(7) Defining “Subsystem”: Transferring the model built in Template to the standard mode and defining each subsystem model preparing for assembling the vehicle model.
Fig. 2Vehicle model in ADAMS
(8) Defining “Assembly”: In the standard mode, each subsystem is assembled into a complete vehicle model. So that the establishment process of the physical model is completed in the ADAMS/Car module.
By adding attribute files, the simulation analysis of the vehicle under different working conditions can be carried out to obtain the required results.
The vehicle model can be obtained as shown in Fig. 2.
4.1.1. Sine delay test
The sine delay test is carried out on a dry, flat and clean cement test site. The test vehicle drives at a constant speed of (80±2) km/h. The virtual test value, the experimental measurements, the
estimation results of the ARUPF and the UPF algorithm are shown in Fig. 3. The results show that the ARUPF algorithm achieves better estimation results for the longitudinal velocity and the yaw rate,
while the estimation results of the UPF algorithm have a certain amplitude fluctuation in the steering wheel holding stage. For the lateral velocity, the ARUPF There is a small deviation between the
estimated value of the ARUPF algorithm and the virtual test value at the peak. But the overall estimated result meets the engineering needs. And the estimated result of the UPF algorithm is different
from the virtual test value during the steering wheel rotation process. There is a certain magnitude of fluctuation in the steering wheel holding process.
Fig. 3Comparison of the state variables found with the different algorithms (ARUPF and UPF) for a sine delay test road
b) Absolute error of longitudinal velocity
d) Absolute error of lateral velocity
f) Absolute error of yaw rate
The ARUPF algorithm has the advantages of robust estimation and unscented particle filtering. After comprehensively selecting the important density function by using unscented transformation,
equivalent weight function and adaptive adjustment factor, the importance sampling of particles is carried out. And the sensor perception vector is used reasonably. It effectively suppresses the
pollution problem of state parameters and the abnormal disturbance of sensor perception values, reducing the degree of particle degradation in the process of vehicle state estimation and overcoming
the shortcomings of a single filtering algorithm. The reasons for this error are as follows: since the mathematical model ignores the suspension characteristics and tire rolling resistance, the tire
slip angle and the lateral load transfer of the sprung mass have a certain relationship with the tire slip angle and tire vertical load in ADAMS. However, the observer can effectively estimate the
vehicle state such as the longitudinal and lateral velocity, as well as the yaw rate, etc.
4.1.2. Double lane change test
A double lane change test is carried out on a dry, flat and clean cement test site. The test speed is 60 km/h. The test driver make the vehicle pass through the double lane change channel without
touching the stakes. The virtual test value, the experimental measurements, the estimation results of the ARUPF and the UPF algorithm are shown in Fig. 4.
Fig. 4Comparison of the state variables found with the different algorithms (ARUPF and UPF) for a double lane change test road
b) Absolute error of longitudinal velocity
d) Absolute error of lateral velocity
f) Absolute error of yaw rate
This test condition is used to verify the accuracy of the algorithm for estimating the longitudinal and lateral velocity as well as the yaw rate of the vehicle when the vehicle states change rapidly.
The results show that the ARUPF algorithm has achieved good results in the estimation of the longitudinal velocity and the yaw rate. But the UPF algorithm has a small fluctuation in the estimation of
the longitudinal velocity and the yaw rate. For the lateral velocity, there is a certain magnitude of deviation between the estimated results of the ARUPF algorithm and the virtual test value during
the steering wheel rotation, while the estimated results of the UPF algorithm have large fluctuations.
The average absolute error (MAE) and root mean square error (RMSE) are given to verify the estimation accuracy of the proposed algorithm.
Table 2The MAE and RMSE indicators of the two algorithms
Evaluation index State value UPF ARUPF
$v$ (m/s) 0.316 0.140
MAE $v$ (m/s) 0.181 0.0475
$r$ (rad/s) 0.316 0.0180
$v$ (m/s) 0.345 0.141
RMSE $v$ (m/s) 0.243 0.0522
$r$ (rad/s) 0.411 0.0221
From Table 2, it can be seen more intuitively that the estimation accuracy of the ARUPF algorithm is significantly higher than the UPF method.
The ARUPF algorithm has the advantages of robust estimation and unscented particle filtering. After comprehensively selecting the important density function by using unscented transformation,
equivalent weight function and adaptive adjustment factor, the importance sampling of particles is carried out. And the sensor perception vector is used reasonably. It effectively suppresses the
pollution problem of state parameters and the abnormal disturbance of sensor perception values, reducing the degree of particle degradation in the process of vehicle state estimation and overcoming
the shortcomings of a single filtering algorithm. The reasons for this error are as follows: since the mathematical model ignores the suspension characteristics and tire rolling resistance, the tire
slip angle and the lateral load transfer of the sprung mass have a certain relationship with the tire slip angle and tire vertical load in ADAMS. However, the observer can effectively estimate the
vehicle state such as the longitudinal and lateral velocity, as well as the yaw rate, etc.
4.1.3. Slop input test
A slope input test is carried out on a dry, flat and clean cement test site, the vehicle speed is maintained at 80 km/h. The virtual test value, the experimental measurements, the estimation results
of the ARUPF and the UPF algorithm are shown in Fig. 5. The test can make the vehicle gradually enter the limit working condition as the steering wheel angle increases, and the tire gradually
transitions from the linear working area to the nonlinear working area, which leads to the increase of the error of the model of the vehicle.
Fig. 5Comparison of the state variables found with the different algorithms (ARUPF and UPF) for a slop input test road
b) Absolute error of longitudinal velocity
d) Absolute error of lateral velocity
f) Absolute error of yaw rate
It can be seen from Fig. 5 that when the system noise increases, the estimation deviation of the UPF algorithm for the longitudinal lateral speed gradually increases. For the estimation of yaw rate,
both the UPF algorithm and the ARUPF algorithm maintain a high estimation accuracy.
The ARUPF algorithm has the advantages of robust estimation and unscented particle filtering. After comprehensively selecting the important density function by using unscented transformation,
equivalent weight function and adaptive adjustment factor, the importance sampling of particles is carried out. And the sensor perception vector is used reasonably. It effectively suppresses the
pollution problem of state parameters and the abnormal disturbance of sensor perception values, reducing the degree of particle degradation in the process of vehicle state estimation and overcoming
the shortcomings of a single filtering algorithm. The reasons for this error are as follows: since the mathematical model ignores the suspension characteristics and tire rolling resistance, the tire
slip angle and the lateral load transfer of the sprung mass have a certain relationship with the tire slip angle and tire vertical load in ADAMS. However, the observer can effectively estimate the
vehicle state such as the longitudinal and lateral velocity, as well as the yaw rate, etc.
4.2. Experimental verification
According to ISO/TR3888-2004, the real vehicle test on double lane change road is carried out. And the test speed is set as 80 km/h (±3 km/h). A gyroscope is installed on the vehicle to collect the
yaw rate and lateral acceleration of the vehicle in real time. The non-contact speed sensor is used to measure the longitudinal and lateral speed of the vehicle. In addition, the steering wheel angle
tester is used to measure the steering wheel angle. Fig. 6 is the comparison between the estimated value of ARUPF method of three state variables and the real vehicle test value.
Fig. 6Comparison of the estimated and test values
From Fig. 6, it can be seen that although there is a certain error, the estimated value is basically consistent with the experimental value in trend. There is a large difference between the
experimental value and the estimated value of the longitudinal velocity, because the estimation of the longitudinal velocity in the model is affected by many parameters such as longitudinal
acceleration, lateral velocity and yaw rate. And the tire model used in this paper still has some deviation when simulating the mechanical characteristics of real vehicle tire. In addition, the
measurement error and installation position of the sensor are also important reasons for the deviation between the estimated and the test value.
5. Conclusions
Based on the Dugoff tire model, a 3-DOF dynamic model of the vehicle with front wheel steering is established to estimate the longitudinal and lateral speed and the yaw rate of the vehicle. The
equivalent weight function is generated by using the IGG method. By adaptively adjusting the weight matrix, the random error caused by nonlinear factors in the process of vehicle sensor detection can
be effectively suppressed. And the influence of data distortion caused by interference can be reduced, and the accuracy of vehicle state estimation is improved. Based on the principle of adaptive
robust filter and unscented particle filter algorithm, a new state estimation method of vehicle is proposed. The ARUPF method has the advantages of good noise filtering effect and high precision. A
simulation platform is used to simulate, analyze and verify the estimation effect of the proposed algorithm. The simulation results show that the vehicle state estimation based on the ARUPF algorithm
has the characteristics of high precision, strong anti-interference ability and good stability. It has the advantages of low cost, easy engineering implementation providing a new idea for vehicle
state estimation.
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About this article
Vibration in transportation engineering
automotive engineering
vehicle state estimation
adaptive robust unscented particle filter
vehicle handling dynamics
This research was supported by the Science and Technology Program Foundation of Weifang under Grant 2015GX007. And also, this research was financially supported by the Open Research Fund from the
State Key Laboratory of Rolling and Automation, Northeastern University, under Grant 2021RALKFKT008. The first author gratefully acknowledges the support agency.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Conflict of interest
The authors declare that they have no conflict of interest.
Copyright © 2022 Yingjie Liu, et al.
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. | {"url":"https://www.extrica.com/article/22788","timestamp":"2024-11-07T16:18:29Z","content_type":"text/html","content_length":"206344","record_id":"<urn:uuid:fbc33a9b-4865-4bbe-ad36-dd6a017d9333>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00097.warc.gz"} |
Point of view of dynamic theory, systems, which was firstthe complete rotor method may be | bet-bromodomain.com
Point of view of dynamic theory, systems, which was firstthe complete rotor method may be regardedperspective of dynamic the harm evolution of proposed by Chelidze [31]. From the as a high-order
method composed of a “fast time” scale plus a “slow time” scale. Its definition is as follows:xf x, g x, , t,t(7)In Equation (7), x Rn may be the “fast time” scale variable, which is usually measured
directly.R m is the “slow time” scale variable, which can directly reflect the damageMachines 2021, 9,10 oftheory, the harm evolution with the complete rotor program is often regarded as a high-order
technique composed of a “fast time” scale and a “slow time” scale. Its definition is as follows: x = f [ x,), t] . = g( x, , t).(7)In Equation (7), x Rn is the “fast time” scale variable, which may
be measured directly. Rm would be the “slow time” scale variable, which can straight reflect the harm state in the entire system but can’t be measured directly. f ( and g( are the “fast time” and
“slow time” scaling functions, respectively. is actually a function from the variable , t represents time, and (0 1) is usually a constant parameter that defines the harm price from the method. The
response states on the bearing system in the initial time point t0 and after operating to get a specific period tp could be expressed as x0 = F [ x0 , 0), t0 ] xt = F x p , t), t p (eight)Suppose
that the bearing system always maintains its initial state without having experiencing any change or harm, then the response from the bearing method might be 5-Methylcytidine manufacturer calculated
making use of Equation (9). x R t p = F x p , 0), t p (9) Thus, with the initial operating state from the bearing method as the reference state, there’s t0 = t R . Then, the damage state of the
bearing program (or harm tracking) may be expressed as follows: e = F x p , ( P), t p – F x p , ( R), t p (10)In Ref. [31], immediately after performing the Taylor expansion, the bearing system’s
harm state is often finally expressed as e= F p – R O p p – R O(11)where O( represents higher-order infinitesimal. In this paper, the raw acceleration signal from the bearing is regarded as an
observable “fast time” scale variable, whilst the harm state of the bearing method is regarded as a “slow time” scale variable. Generally, to calculate the damage state of the bearing, the phase
space reconstruction theory according to the Takens embedding theorem would be introduced, and also the damage state on the bearing would be quantified on this basis. The phase space reconstruction
is mathematically expressed as follows: y R (n) = [ x R (n), x R (n ), . . . , x R (n ( D – 1))]T n = 1, . . . , N ( D – 1) (12)exactly where y R R D could be the reference initial state from the
phase space of the bearing program, n will be the number of vectors within the phase space, and N would be the total number of observation data points. and D represent the time delay and embedding
dimension of the phase space, which can be calculated employing the mutual facts [32] process and Cao’s approach [33]. Theoretically, the unknown mapping involving the reconstruction vector y R (n)
within the reference phase space and that of your next step y R (n 1) around the “slow time” scale–that is, under the reference state from the bearing R –can be expressed as y R ( n 1) = P [ y R (
n); R ] (13)Machines 2021, 9,11 ofIn engineering, linear regression is the most straightforward and generic model for Oprozomib manufacturer establishing the mapping among reconstruction vectors y R
(n) and y R (n 1) inside the reference phase space, as shown by Equation (14). y R ( n 1) = A n y ( n) (14)exactly where An Rdd1) is.
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Java Sorting Exercises
For the Java Exercise, you need to implement the following sorting methods:
• Insertion sort
• Cocktail Shaker sort
• A variation of Quick Sort
• External sort
You also need to implement an efficient, linear time, algorithm that takes two sorted arrays and returns an array of the elements that occur in both arrays.
You may not use any built-in sorting methods (or in-built classes from the Collections framework such as ArrayList, LinkedList, HashMap etc.) for this Exercise.
Implementation Details
The starter code has been provided to you. Fill in code in class SortingImplementation that implements the following SortingInterface (it's important that you do not modify the signatures of any
public interface SortingInterface {
void insertionSort(Comparable[] array, int lowindex, int highindex, boolean reversed); void shakerSort(Comparable[] array, int lowindex, int highindex, boolean reversed); void modifiedQuickSort
(Comparable[] array, int lowindex, int highindex);
void externalSort(String inputFile, String outputFile, int k, int m);
You will also need to implement the method intersectionOfSortedArrays that takes two sorted arrays and returns an array of elements that occur in both arrays.
You should not use any instance variables for this exercise apart from constants; use only local method variables. You may write private helper methods.
We describe each of the sorting algorithms below:
1. Insertion Sort
public void insertionSort(Comparable[] array, int lowindex, int highindex, boolean reversed);
Modify the code of the insertion sort so that:
• It sorts all elements in the array with indices in the range from lowindex to highindex
(inclusive). Youshouldnotchangeelementsoutsidetherangelowindextohighindex.
• If reversed is false, the list should be sorted in ascending order. If the reversed flag is true, the list should be sorted in descending order.
• Can sort the array of any Comparable objects, not just integers.
2. Cocktail Shaker Sort
public void shakerSort(Comparable[] array, int lowindex, int highindex, boolean reversed);
This is a variation of the bubble sort that sorts in both directions (bubbles the largest element to the end, and bubbles the smallest element to the front on each pass). It has the same asymptotic
running time as bubble sort.
The first pass consists of two parts: we first iterate over the array from left to right and bubble the largest element to the end of the list. Then we take the leftward pass (iterate from the
(last-1) element to the first element) and bubble the smallest element to the front of the array.
The second pass will first bubble the second largest element to the end of the list (to index last-1) and then shift the second smallest element to the correct position at index 1.
After each pass, we reduce the size of the list that needs to be sorted by two elements.
Example (assume we want to sort the list in ascending order, and lowindex =0, and highindex = array.length-1): 4, 10, 6, 9, 2, 3, 8, 4
After the first part of pass 1 we get: 4, 6, 9, 2, 3, 8, 4, 10 (the largest element is in the last position).
After the second part of pass 2 (iterating from right to left): 2, 4, 6, 9, 3, 4, 8, 10. The smallest element is in the first position. Now we need to sort the list from index 1 to index last -1.
After the first part of pass 2, the list will look like this: 2, 4, 6, 3, 4, 8, 9, 10.
After the second part of pass 2 we get: 2, 3, 4, 6, 4, 8, 9, 10
We will now sort from index 2 to index 5 (inclusive):
After the first part of pass 3 we get: 2, 3, 4, 4, 6, 8, 9, 10. The list won't change after the second part of pass 3. It would now need to sort the list from index 3 to index 4. The list won't
change, and we are done. Note: your code should work for any lowindex, highindex and in both ascending and descending order.
3. Modified Quick Sort
public void modifiedQuickSort(Comparable[] array, int lowindex, int highindex);
Change the code of the quick sort so that:
- It sorts the sub-list of the original list (from lowindex to highindex)
- At each pass, it picks three elements of the sub-list:
- the first element in the sublist (at index lowindex),
- the middle element of the sublist (in the middle between lowindex and highindex)
- the last element of the sublist (at index highindex),
and uses the median of these three elements as the pivot. How do you compute a median of three values? If we "sort" three elements, the median is the element in the middle (for instance, if the three
elements are 5, 2, 19, the median is 5); note that it is different from mean (average)! Do not use the mean.
Example: Consider the following array 5, 2, 9, 12, 6, 8, 3, 1 and assume we want to sort the array from lowindex = 2 to highindex = 7 (so from elements 9 to 1). We first find three elements we are
considering for our first pivot: 9, 6 and 1 (9 is the element at lowindex, 1 is the element at highindex, and 6 is the middle element in the sublist). We order them and get 1, 6, 9. And take the
median one, the one in the middle, 6. This is our pivot for the first pass of quick sort.
We then run quicksort as usual:
5, 2, 9, 12, 1, 8, 3, 6 (swap the pivot with the element at highindex).
i starts at index lowindex=2, so points at 9 which is larger than the pivot. j points at 3 which is smaller than the pivot. Swap them:
5, 2, 3, 12, 1, 8, 9, 6
i now points at 12, j moves until it points to 1 (an element less than the pivot. We swap 12 and 1 and increment i, decrement j:
5, 2, 3, 1, 12, 8, 9, 6
i and j now overlap, so we are done with the first pass, we just need to swap the pivot with the element at i and get:
5, 2, 3, 1, 6, 8, 9, 12
So the first pass split the sublist into elements < 6, 6 and elements > 6. We now need to recursively run quicksort on sublists 3, 1 and 8, 9, 12. Why didn't we include 5, 2? Because we were given a
lowindex = 2, so we do not look at the first two elements. For each sublist, we would again pick potential pivot elements and choose a median as a pivot. If the sublist contains only two elements,
randomly pick one of the two as the pivot. Finish this example before you start coding this version of quick sort.
4. External Sort
public void externalSort(String inputFile, String outputFile, int k, int m);
What if we need to sort a very large list that does not fit into memory all at once? Then you need to use external sort that stores partial results in files on the disk. Assume we can only fit k
integers into memory at a time, and we have a text file that contains N integers (one per line, to keep it simple). We can read k integers from the file at a time, store them in a list and sort the
list using some existing efficient sorting algorithm such as quicksort. Then we can write the result to a temporary file. We can repeat this process for another chunk of the original file. We would
need to do it m (number of chunks) = ceiling(N/k) times, until all the partial results are stored in temporary files (the provided test expects you to call them "temp0.txt", "temp1.txt", ...,
"temp99.txt"). In this method, k (how many integers can fit in an array) and m (the number of chunks) are passed as parameters to the method. Please note that N is not passed as a parameter (but it
can be determined from the number of lines in a file).
We need to merge the sorted sublists stored in the temp files into a single sorted "list" stored in the output file. You can use the algorithm similar to the merge step of the mergesort, except that
you would read data from the temp files (all of them need to be open at the same time, you may use an array of BufferedReaders for that) and keep writing the numbers to another file as your algorithm
proceeds with the merge. You will read the first number from each temporary file, find the minimum of those numbers, and write it to the output file. Then read another number from the file that
contained the minimum and again find the minimum of the currently read set of numbers and write it to the output file. For this problem, the list in the output file should be sorted in ascending
Consider the following example (to keep the example short, let's assume the input file has 6 numbers, and our tiny memory is only able to fit k = 3 integers at a time):
The external sort algorithm will read m = 2 chunks from the input file: [8, 4, 10] and [3, 7, 5], sort them with quicksort and save the sorted sublists in two temporary files:
"temp0.txt" 4
"temp1.txt" 3
It will then open both temp files and merge them as following: it will first read 4 and 3 (the first numbers in each temp file), save them into the temporary array, find the minimum (3) and write it
to the output file (assume it is called "output"):
"output" 3
Then it would read another number from "temp1" since it's the file that contained the minimum element. Now the array of elements is [4 , 5], the minimum is a 4 and we write it to the output file"
We read another number from "temp0", 8, and the array is [5, 8]. The minimum is a 5, we write it to the output file:
"output" 3
We read another number from "temp1", a 7, the array is now [8, 7], the minimum is 7 and the output file is:
"output" 3
We continue as before until we write all the elements to the output file. The temporary files can then be deleted (although you are not required to delete them in your program).
To test your external sort, create a large file of integers (also done in the test file, provided in the zip file).
5. Find "Intersection" of two sorted lists
Implement the method intersectionOfSortedArrays that takes two sorted arrays and returns an array of elements that occur in both arrays.
For instance, if we pass the following two sorted arrays to the method:
int[] arr1= {2, 10, 12, 34, 90};
int[] arr2 = {4, 6, 8, 10, 11, 12, 14, 20, 26, 30, 34, 48, 50};
then the method should return: {10, 12, 34}.
Note: your method should be general and work for any two sorted arrays of integers. It is required that it runs in linear time: Theta(n1+n2), where n1 and n2 are the sizes of the two lists and takes
advantages of the fact that the input lists are sorted.
The zip file provides some basic tests for testing several methods of the project, but they simply check if the list is sorted after running your algorithm. Passing the tests does not guarantee that
your code is correct; you should also do your own testing and are encouraged to write your own tests.
Code Style
Your code needs to adhere to the code style described in the StyleGuidelines.pdf document. You may receive a deduction if your code does not follow these guidelines.
package sorting;
/** A class that implements SortingInterface. Has various methods
* to sort a list of elements. */
public class SortingImplementation implements SortingInterface {
* Sorts the sublist of the given list (from lowindex to highindex)
* using insertion sort
* @param array array of Comparable-s
* @param lowindex the beginning index of a sublist
* @param highindex the end index of a sublist
* @param reversed if true, the list should be sorted in descending order
*/ @Override
public void insertionSort(Comparable[] array, int lowindex, int highindex, boolean reversed) {
// FILL ON CODE
* Sorts the sublist of the given list (from lowindex to highindex)
* using the shaker sort (see pdf for description)
* @param array array of Comparable-s
* @param lowindex the beginning index of a sublist
* @param highindex the end index of a sublist
* @param reversed if true, the list should be sorted in descending order
public void shakerSort(Comparable[] array, int lowindex, int highindex, boolean reversed) {
* Sorts the sublist of the given list (from lowindex to highindex)
* using the following modification of quick sort:
* At each pass, it picks three elements of the sub-list:
* - the first element in the sublist (at index lowindex),
* - the middle element of the sublist (in the middle between lowindex and highindex)
* - the last element of the sublist (at index highindex)
* and chooses the median of these three elements as the pivot.
* How do you compute a median of three values?
* If we "sort" three elements, the median is the element in the middle
* (for instance, if the three elements are 5, 2, 19, the median is 5);
* note that it is different from mean! Do NOT take the average of the 3 numbers, pick the median instead.
* @param array array to sort
* @param lowindex the beginning index of a sublist
* @param highindex the end index of a sublist
*/ @Override
public void modifiedQuickSort(Comparable[] array, int lowindex, int highindex) {
// FILL ON CODE
* Implements external sort method
* @param inputFile The file that contains the input list
* @param outputFile The file where to output the sorted list
* @param k number of elements that fit into memory at once
* @param m number of chunks
public void externalSort(String inputFile, String outputFile, int k, int m) {
// FILL IN CODE
* Takes two sorted arrays and returns an array of all the elements that occur in both arrays.
* For instance, if we pass the following two sorted arrays to the method:
* int[] arr1= {2, 10, 12, 34, 90};
* int[] arr2 = {4, 6, 8, 10, 11, 12, 14, 20, 26, 30, 34, 48, 50};
* then the method should return: {10, 12, 34}
* Note: your method should be general and work for any two sorted arrays of integers.
* It is required that it runs in linear time: Theta(n1+n2),
* where n1 and n2 are the sizes of the two lists and takes advantages of the fact that the input lists are sorted.
* Hint: you can modify the merge helper method we wrote in class to solve this problem.
* @param arr1 sorted array 1
* @param arr2 sorted array 2
* @return array of common elements in arr1 and arr2
public int[] intersectionOfSortedArrays(int[] arr1, int[] arr2) {
// FILL IN CODE
int[] res = new int[arr1.length]; // you can computer the actual number of common elements as you go,
// and later return the array of the correct length using copyOf method in class Arrays
return res;
package sorting;
/** An interface that describes several algorithms for sorting a list */ public interface SortingInterface {
void insertionSort(Comparable[] array, int lowindex, int highindex, boolean reversed);
void shakerSort(Comparable[] array, int lowindex, int highindex, boolean reversed); void modifiedQuickSort(Comparable[] array, int lowindex, int highindex);
void externalSort(String inputFile, String outputFile, int k, int m);
Need a custom answer at your budget?
This assignment has been answered 4 times in private sessions. | {"url":"https://codifytutor.com/marketplace/java-sorting-exercises-a6e5499c-fce5-439d-817c-2147d42ed330","timestamp":"2024-11-02T11:27:29Z","content_type":"text/html","content_length":"36461","record_id":"<urn:uuid:f75c8ac0-98e8-4fb9-9a82-1dac4e621ad4>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00361.warc.gz"} |
ECE 299: Statistical Learning Theory (Spring 2011)
There is no required textbook for this class; however, the following two books are useful and have been placed on reserve at the Perkins Library: Here are some additional survey papers that I
recommend (more will be added as the class progresses):
Concentration inequalities
Binary classification | {"url":"http://maxim.ece.illinois.edu/teaching/spring11/reading.html","timestamp":"2024-11-10T01:34:47Z","content_type":"text/html","content_length":"2866","record_id":"<urn:uuid:f4b60405-75c6-4107-8d1b-444cc0efd49e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00379.warc.gz"} |
The Stacks project
Lemma 29.29.3. Let $f : X \to Y$, $g : Y \to Z$ be locally of finite type. If $f$ has relative dimension $\leq d$ and $g$ has relative dimension $\leq e$ then $g \circ f$ has relative dimension $\leq
d + e$. If
1. $f$ has relative dimension $d$,
2. $g$ has relative dimension $e$, and
3. $f$ is flat,
then $g \circ f$ has relative dimension $d + e$.
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distance to momentum calculator
Distance To Momentum Calculator
In the realm of physics, calculating momentum is a fundamental concept that helps us understand the motion of objects. Momentum, denoted by p, is the product of an object’s mass and its velocity.
This article introduces a simple yet effective calculator to compute momentum effortlessly.
How to Use
To utilize the Distance to Momentum Calculator, follow these steps:
1. Enter the mass m of the object in kilograms.
2. Input the velocity v of the object in meters per second.
3. Click the “Calculate” button to obtain the momentum p.
The formula to calculate momentum is given by: p=mv Where:
• p is momentum in kilogram meters per second (kg m/s).
• m is the mass of the object in kilograms (kg).
• v is the velocity of the object in meters per second (m/s).
Example Solve
Let’s say we have an object with a mass of 5 kilograms and a velocity of 10 meters per second. Using the formula mentioned above:
p=5×10=50kg m/s
Hence, the momentum of the object is 50 kg m/s.
Q: Can momentum be negative?
A: Yes, momentum can indeed be negative, indicating motion in the opposite direction of the chosen positive direction.
Q: How does momentum relate to Newton’s laws of motion?
A: Momentum is a crucial concept in Newton’s laws of motion, particularly the second law, which states that the rate of change of momentum of an object is directly proportional to the force acting on
Q: Is momentum conserved in all situations?
A: According to the law of conservation of momentum, in a closed system where no external forces act, momentum is conserved.
Q: What are the units of momentum?
A: Momentum is measured in kilogram meters per second (kg m/s).
The Distance to Momentum Calculator simplifies the process of computing momentum, a vital parameter in understanding the behavior of objects in motion. By employing this tool, users can swiftly
derive momentum values and gain insights into the dynamics of physical systems. | {"url":"https://calculatordoc.com/distance-to-momentum-calculator/","timestamp":"2024-11-13T21:26:31Z","content_type":"text/html","content_length":"84405","record_id":"<urn:uuid:3f76cfc6-2bb8-4f10-adfb-aea2f91f24b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00300.warc.gz"} |
Excel Formula: Count Cells with Specific Text using COUNTIF
In this tutorial, we will learn how to use the COUNTIF function in Excel to count the number of cells in a specific range that contain a specific text. The COUNTIF function is a powerful tool that
allows you to perform calculations based on specific criteria. By using this function, you can easily determine the number of cells that meet a certain condition. In our case, we want to count the
cells in a range that contain the text 'Baskin, David'. Let's dive into the details of how to write this formula in Excel.
To count the number of cells in a range that contain a specific text, we can use the COUNTIF function. The syntax of the COUNTIF function is as follows: =COUNTIF(range, criteria). The 'range'
argument specifies the range of cells to be evaluated, and the 'criteria' argument specifies the text that needs to be matched.
For our example, we want to count the cells in the range A1:A40 that contain the text 'Baskin, David'. To do this, we can write the following formula in Excel: =COUNTIF(A1:A40, "Baskin, David"). This
formula will count the number of cells in the range A1:A40 that contain the exact text 'Baskin, David'.
Let's consider an example to better understand how this formula works. Suppose we have the following data in column A:
Baskin, David
Smith, John
Baskin, David
Doe, Jane
Baskin, David
If we use the formula =COUNTIF(A1:A40, "Baskin, David"), it will return the value 3. This is because there are three cells in the range A1:A40 that contain the exact text 'Baskin, David'.
In conclusion, the COUNTIF function in Excel is a useful tool for counting the number of cells in a range that meet specific criteria. By using this function, you can easily perform calculations
based on specific text or conditions. We hope this tutorial has been helpful in understanding how to use the COUNTIF function to count cells with specific text in Excel.
An Excel formula
=COUNTIF(A1:A40, "Baskin, David")
Formula Explanation
The formula uses the COUNTIF function to count the number of cells in the range A1:A40 that contain the exact text "Baskin, David".
Step-by-step explanation
1. The COUNTIF function takes two arguments: the range of cells to be evaluated and the criteria to be matched.
2. In this case, the range of cells is A1:A40, which means it will count the cells in column A from row 1 to row 40.
3. The criteria is "Baskin, David", which is the exact text that needs to be matched in the cells.
4. The COUNTIF function counts the number of cells in the range that meet the specified criteria.
For example, if we have the following data in column A:
| A |
| |
|Baskin, David|
|Smith, John |
|Baskin, David|
|Doe, Jane |
|Baskin, David|
The formula =COUNTIF(A1:A40, "Baskin, David") would return the value 3, because there are three cells in the range A1:A40 that contain the exact text "Baskin, David". | {"url":"https://codepal.ai/excel-formula-generator/query/1Ntma83N/excel-formula-countif-cell-range","timestamp":"2024-11-04T15:13:13Z","content_type":"text/html","content_length":"92941","record_id":"<urn:uuid:534e6572-ee4b-45b4-8a70-a53ea66a16aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00288.warc.gz"} |
Back to basics - Racehead Engineering
WHY IS THIS SO?
To the readers who are experts in engine design theory this article could be ‘old hat’ so I apologize for boring them and wasting their time.
To others perhaps less theoretically agile, this should be welcome as it will compare design concepts with numbers rather than argument.
We hope you enjoy the following information to better understand the math myths behind engine improvements and how RACEHEAD ENGINEERING improves performance.
The Basics
An engine is a device with a number of cylinders each with a cylinder bore (B) and a stroke (S). This gives the engine a swept volume (Vsv) for each cylinder and a swept volume for the entire engine
(Vtsv). The engine will have a bore to stroke ratio (Kbs). The calculation of this basic data is shown in equations 1-4.
A further important mechanical design parameter is the mean piston speed (Cp), For racing engines, this limit parameter has hardly increased in numeric value in fifty years and that fact reflects the
gradual improvement of cylinder design and lubrication technology since the 1950s.
Then, an air-cooled and iron linered Norton Manx cylinder had a mean piston speed of 20 m/s at its engine speed for peak power. Today, a MotoGP engine at peak power with its liquid-cooled and
silicon-carbide plated cylinder running on a synthetic lube oil has a mean piston speed of 25m/s. One would be hard put to call that a technological breakthrough.
“Racing engine mean piston speed has hardly increased in numeric value in fifty years”
A firing engine produces a turning moment at the crankshaft, the TORQUE. Depending on the speed of rotation of the crankshaft (N) the engine produces a power output (POWER). The TORQUE is measured
with the engine on a dynamometer.
The computation of the POWER output is shown in equations 5-8 There the unit is POWERkW (kW or kilowatts) but if you want horsepower (bhp) instead then divide POWERkw by 0.7457 to get POWERbhp.
Alternatively, one can compute the POWER using the brake mean effective pressure (BMEP). However, and much more likely, having used the measured TORQUE to get the power output from the dynamometer
data, you can calculate the value of the BMEP by back-calculation using a re-arranged Eqn.6, because all other parameters are known data values for the engine.
Somewhat like mean piston speed, this is another parameter which, for the naturally aspirated, gasoline racing engine, has barely changed over the last fifty years. That Norton Manx racing motorcycle
of 1955 attained a BMEP value of almost 14 bar.
Today’s MotoGP engine also has a BMEP value of about 14 bar. Some progress! However, that 800 cc MotoGP engine does it at 17,000 rpm whereas the 500 Norton did at 7000 rpm so the power differential
is huge. As the BMEP potential and the piston speed are such common parameters between engines, the Eqns.7 and 8 become very useful ready-reckoners as to the possible power performance of any engine.
The highest mean piston speed (Cp) I have heard of at the time of writing this article is 26.5 m/s but these boundaries are increasing continually with the development of high strength lightweight
materials so in particular the RPM of large displacement drag race engines increases every year! However the maximum BMEP potential of the simple naturally aspirated, gasoline four-stroke racing
engine at high piston speed is some 15 bar, this limit has not changed much at all.
Nevertheless, Eqns.7 and 8 still permit you to make quick design decisions as to what performance is potentially possible from any engine. It also permits you to numerically trip up those PR-based
engine developers who grossly exaggerate their engine’s power output as evidence of the application of their genius!
“Those engine developers who grossly exaggerate their engine’s power as evidence of their genius!”
Ok so far we have covered some basic principles of engine physics from now on it gets a little more complicated.
Work is defined as the distance (dX) moved by a force (F). In the context of a piston in a cylinder, as seen in Fig.3, the force (F) on the piston is the product of the pressure (P) on it when
applied over the piston area (A).
Hence, the work done on, or by, the piston as it moves is the product of the pressure (P) and the cylinder volume change (dV) as it occurs.
On the power stroke, as the volume increases that work is positive.
On the compression stroke, as the volume decreases that work is negative, i.e., supplied by the engine to the piston.
During the power phase, from bottom dead centre (bdc) to the next bottom dead centre (bdc), or one turn of the crankshaft, the pressure-volume diagram of the in-cylinder events is shown in Fig.4.
It is sketched from data for the MotoGP engine. The net work (POWER WORK) on the piston during this process is the summation (integration) of all of the pressure volume increments over the period and
is shown as the area coloured yellow in the diagram.
If there were no other losses in the system, that would be the work delivered to the crankshaft; but there are.
The yellow area can be represented by the equivalent rectangular area shown in blue, which area has a height of IMEP and a width of the cylinder swept volume (Vsv).
The value of IMEP is known at the indicated mean effective pressure. It is called ‘indicated’ as it is derived from the pressure transducer signal as measured in the cylinder head of the engine.
During the pumping phase that follows, from bottom dead centre (bdc) to the next bottom dead centre (bdc), or the next turn of the crankshaft, the pressure volume diagram is shown at the right of
Here, the work computation would elicit a negative value for the PUMP WORK, the yellow area on that diagram, as the opening (higher) line is of compression (negative dV) during the exhaust stroke.
This yellow area can be equally represented by the equivalent rectangle of height PMEP and width Vsv labelled as the pumping mean effective pressure. Its negative numerical value indicates that the
pumping work is supplied by the piston from the crankshaft; in short, it is lost work.
The rest of the engine work losses are lumped together as ‘frictional’ losses and can be expressed as a FMEP value, the friction mean effective pressure, again officially a negative number. The
upshot of this part of the discussion can be seen in Eqns.9-14.
The net work per cylinder per cycle is shown in Eqn.9 where the brake mean effective pressure (BMEP) is observed to be the result of subtracting the (positive values of) the pumping mean effective
pressure (PMEP) and the friction mean effective pressure (FMEP) from the indicated mean effective
pressure (IMEP). AS IMEP and PMEP data can only be determined from an analysis of measured cylinder pressure diagrams, but BMEP can be calculated from measured dynamometer data through Eqns.6-7, then
one method of determining FMEP is through the re-arrangement of Eqn.9.
The ratio of BMEP to IMEP is known as the ‘mechanical efficiency’ of the engine and is normally in the 75 to 85% range for most racing engines. The MotoGP engine [2], designed for a BMEP of 14 bar at
16,100 rpm, produced exactly that. It had a IMEP value of 18.52 bar, a PMEP value of 1.26 bar (see Fig.4) and a FMEP value of 3.26 bar; the mechanical efficiency was 75.6%.
In Fig.5, Eqn.11, it can be seen that IMEP is the POWER WORK divided by the cylinder swept volume. Alternatively, that in-cylinder work is directly proportional to the heat released (Q) by combustion
of the fuel trapped in the cylinder.
In Eqn.12, this argument is advanced to relate the heat released (Q) to the mass (M) of fuel trapped in the cylinder. However, as air-fuel ratios for racing engines on gasoline are almost fixed,
Eqn.12 reduces to showing that IMEP is directly proportional to the mass of air (Mair) trapped in the cylinder.
It is but a short logical step in Eqn.14 to relate the BMEP to IMEP and the BMEP to the specific mass airflow rate into the engine, i.e., delivery ratio (DR). An even shorter logical step is found by
linking Eqns. 13 and 14 to relate the engine TORQUE output per cylinder to BMEP and DR.
“The MotoGP engine designed for a BMEP of 14 bar at 16,100rpm had a mechanical efficiency of 75.6%”
In short, as BMEP and DR have only minor variations from one racing engine to another, BMEP and DR are far more useful numbers with which to compare the development level of differing engines than is
the output TORQUE, because this number also incorporates the total swept volume of an engine.
The bottom line, design-wise, is that brake mean effective pressure (BMEP) is inextricably linked with the specific mass airflow rate ratio, delivery ratio (DR).
In this discussion, you will note that I have not used or defined the term ‘volumetric efficiency’, which is a volume based specific airflow rate parameter ignoring air mass/volume variations such
temperature, atmospheric and altitude conditions.
An engine inhales air through its intake valve(s) and exhales through its exhaust valve(s). The aperture area (At) through which this flow takes place at any valve lift (Lv) is shown in Fig.6.
Also shown is the basic geometry of a valve seat, a valve seat angle, a valve stem, an inner port, and a duct size at the manifold.
The physical dimensions are labelled as the valve seat angle (As), the diameters at the seat (Dis and Dos) and at the inner port (Dip), and at the manifold (D2).
The manifold diameter (D2) may connect to a number of valves (nv) so, if so, the total aperture area for flow is obviously a multiple (nv times At) of that illustrated for one valve.
The aperture flow area (At) is considered as being the side area of a frustum of a cone and that cone shape changes position with lift.Iif all analyses are to be accurate,
it must be persistently used to the point of pedantry in absolutely every aspect of the design process from the experimental determination of discharge coefficients (Cd), valve flow time-areas,
through to implementation within a theoretical engine simulation. The theoretical computation of the airflow rate is conducted through complex equations.
“It must be persistently used to the point of pedantry in absolutely every aspect of the race engine
design process”
At any one step, the effective area of the aperture is the product of the discharge mass flow coefficient (Cd) and the area (At).
The volume flow is found by multiplying that value by the particle velocity, and the mass flow rate by multiplying that product by the prevailing gas density (rho).
The summation is conducted over the main part of the intake stroke from tdc to bdc. This is the complex step by step integration that proceeds incrementally.
However, this computational approach for the delivery ratio (DR),will never be executed on your pocket calculator
The main variables vary dramatically during the summation process from tdc to bdc. Above shows the variation of the particle velocity (c) at the manifold diameter (D2) for both the exhaust and the
intake processes; the value is plotted as Mach number which is particle velocity (c) divided by the local acoustic velocity.
For the intake flow, from tdc to bdc, the particle velocity rises from near zero to a Mach number of about 0.5 However, while these variations are significant and would inhibit the ‘pocket
calculator’ solution for DR the pattern of all these variations from engine to engine is really quite similar.
Hence, citing these similarities, we can solve for what is known as the specific time area (STA) shown by the graphical result
it is the area coloured blue in the diagram which is the integration of the intake aperture area from tdc to bdc.
The entire intake valve period extends from opening (IVO) to closing (IVC), but the STAip data refers only to the main intake pumping period from tdc to bdc.
The result of the equivalent calculation for the exhaust pumping period, the exhaust stroke from bdc to tdc for the exhaust valve and is labelled as STAep.
By definition, any air mass induced into the engine inevitably becomes the exhaust mass post-combustion (plus the added fuel mass) and which requires to be expelled from the engine.
Therefore, there is an obvious proportionality connection between the STAip and STAep values.
In a racing engine with tuned intake and exhaust systems, scavenging of the trapped exhaust gas during the valve overlap period from the small space that is the clearance volume is a vital part of
effective engine design.
This permits a through-draught of fresh charge to scavenge the cylinder of its exhaust gas and fill it with fresh charge.
If carried out effectively say, in an engine with a compression ratio of 11, it means that the induction process can begin with a delivery ratio of some 10% before the downward piston motion even
begins to suck in air, 10% extra DR can be 10% extra BMEP.
The scavenge process will only be successful, always assuming that the intake and exhaust tuning is well organized, if the phasing of the opening intake valve and the closing exhaust valve apertures
is effective.
This phasing is well-expressed pictorially by the specific time areas for the overlap valve periods for the exhaust valve(s) (STAeo) and the intake valve(s) (STAio) as the red and blue coloured
areas, respectively.
If either value, or both, is numerically deficient then, even with perfect pressure wave tuning, the throughflow scavenge process will be impaired, as will the engine POWER.
In a two-door room with both doors shut, there are no draughts on a windy day.
Another valve area segment to be considered is the period from the opening of the exhaust valve(s) to the bdc position, i.e., the exhaust blowdown period. The specific time area for this period
(STAeb) is shown, coloured red
If this value is numerically inadequate then the cylinder pressure at bdc will be high as a sufficient mass of exhaust gas has not been bled from the cylinder. Hence, the ensuing exhaust pumping
process from bdc to tdc will be conducted with higher than normal cylinder pressures giving increased pumping losses (PMEP) and may even promote excessive exhaust gas backflow up the intake tract as
the intake valve opens.
If this latter situation occurs, even a well-designed scavenge process could be negated because it would be conducted with backflow exhaust gas and not fresh intake charge; a situation guaranteed to
invisibly and inexplicably reduce power output, raise the trapped charge temperature and encourage detonation.
The final valve area segment to be considered is the period from the bdc position on the intake stroke to intake valve closure at IVC, i.e., the intake ramming period. The specific time
area for this period (STAir) is shown, coloured blue.
The higher is the desired delivery ratio (DR), the higher required BMEP, then so too must be the need for effective intake ramming which requires sufficient valve aperture and time at any engine
That a well-designed and phased intake system will give the correct direction of pressure differential to encourage a ramming action can be observed from bdc to IVC.
Many engines, ranging from high performance racing engines to lawnmowers, all at their engine speed for peak horsepower, have indeed a logical numerical connection between their individual STA values
and the BMEP attained by them.
There was, naturally, scatter in this plethora of data from so many sources, but the trends were very clear. A theoretical connection between STA and BMEPcan be established and reduced to equations
and one can match the six individual STA values as closely as possible to their target values for a required BMEP.
What is important is not to have any one STA value seriously deficient of its target value as that will make the design ‘unmatched’ and the engine will breathe badly.
Although the equations can be solved on your pocket calculator, it is too complicated to produce the actual STA values for a given engine. This requires the numerical integration of the six segments
of the two valve lift curves and their aperture areas.
Today, the likes of 4stHEAD software the STA analysis for the empirical design of a new engine, or analysis of an existing one, is conducted within a dedicated computer program. In such software,
the STA-BMEP equations have been enhanced and extended to cope with both spark-ignition and compression-ignition engines; with the use of gasoline, kerosene, methanol and ethanol fuels; with the
employment of differing compression ratios; and the use of supercharging or turbocharging.
The high cylinder pressure during exhaust blowdown, and the low cylinder pressure during the induction stroke, creates compression waves and expansion (suction) waves in their respective ducts. It is
the reflection of these waves at the exhaust pipe end (or mid-section expansions at a branch or collector), or at the bellmouth of the intake, which provides the pressure differential characteristics
to conduct cylinder scavenging during the exhaust overlap period. In the case of the intake, that tuning length also needs to be set correctly to aid the ramming process.
Apart from designing in the correct lengths, the size of the ducts at the manifold is a most important design consideration and one which is rarely, if ever, emphasized in published empirical theory.
If the ducts are too large then the pressure waves will be weak except at the very highest speeds and if too small they will yield waves of excessive amplitude except at the lower engine speeds.
Upon pipe end reflection, weak waves give less effective pressure differentials for the scavenge or ramming processes and waves of excessive amplitude friction-scrub themselves along the pipe walls
with an inevitable reduction in their strength giving the same outcome.
Analysis of many engines, both empirically from their physical geometry and theoretically using complete engine simulations, yields empirical design criteria for the optimum size of these ducts. The
empirical criteria relate the manifold duct size to their valve apertures and the number of valves providing them.
The specific time areas (STA) are incorporated into a design for a piston speed (Cp) of 25 m/s and a BMEP of 14 bar with the exhaust and intake duct sizes. It’s not possible to meet all STA target
values precisely, nor is it vital as empiricism is not an exact science! The reason one cannot precisely mesh actual and target STA values is that one must work within the confines of real valve lift
profiles that must also survive without failure.
This data can be presented to an accurate engine simulation and run over a specified speed range with differing sizes of exhaust and intake ducts. The results for POWER and airflow rate (DR) rate can
be noted. The largest duct pairing gives the highest power and airflow at the higher engine speeds but loses out at lower rpm.
However, the larger duct pairing may exceed the designed power output of (14 bar BMEP at given rpm) So at the expense of a ‘peaky’ power curve and an even ‘peakier’ airflow curve; the latter may not
provide the actual power estimated in the real world and provide on-track difficulties in fuelling smoothly.
The duct sizes should match the design power criterion exactly. The applied Km criteria reveal that the duct size variations from the designed value by about +/- 1 mm for ‘acceptability’ and that
‘acceptability’ is well-nigh proven as these Km criteria exhibit a very narrow dimensional tolerance.
This evidence should provide a cautionary tale for those who may somewhat arbitrarily size their engine ducts and are even now pondering the reasons behind either ‘peaky’ power curves or ‘inadequate’
peak power curves when, by their design lights, they ought to have been ‘perfect’. In short, a well-executed design using the STA-BMEP parameters can be negated to some extent by an incorrect sizing
of the intake and the exhaust ducting.
“A well-executed design can be negated to some extent by an incorrect sizing of the intake and the exhaust ducting”
I regret to say that, while this is a design criterion for the dimensions of an intake valve and is doubtless helpful in that regard, further assistance is not forthcoming for the rest of an engine
However, if one replaces the mean piston speed with the maximum piston speed (Cp) in Eqn.26, i.e., the above-used number 25 would virtually double to about 50 m/s, the Mean Gas Velocities then double
to 168.8 and 152 m/s, respectively, the first of which is not a million miles/hour away from the Mach number optimum of 0.5 (170 m/s) debated above.
This reasonable correlation, between Mach number and a Kl value based on maximum piston speed, lends theoretical credence to its usefulness as a basic method to size an intake valve.
Moreover, if one extends Mean Gas Velocity thinking to the exhaust valves of an engine, where the speed of sound in the elevated temperatures of exhaust gas is some 600 m/s, there the Mach number
criterion of 0.5 translates to (if computed at maximum piston speed) a Mean Gas Velocity of 300 m/s. For the exemplar MotoGP engine, using 50 m/s as the maximum piston speed and 22 mm which is the
inner port diameter (Dip) of each exhaust valve, the exhaust valves area ratio (Kev) is 0.177 from Eqn.25, and an exhaust-based Mean Gas Velocity becomes 284 m/s from Eqn.26; and that is a pretty
good match for the supposedly required value of 300 m/s. Hence, it seems feasible to extend the Mean Gas Velocity concept to the exhaust valves as well; this is important as the relative sizing of
the exhaust and intake valves is a critical design factor which has been previously discussed.
However, while the basic sizing of the valves in any given design may well be guided by using the Mean Gas Velocity for the intake valve(s) and also by this extension for the exhaust valve(s), it
falls short of telling us what to do with either of them to tailor a required engine power characteristic.
Firstly, there is no information as to the valve lift profile which should accompany a Mean Gas Velocity; such as how high should the valve(s) be lifted?; such as the required duration or the angular
positions of valve opening or closing or maximum lift?; or what happens if I employ a more or a less aggressive valve lift profile?
Secondly, in the absence of an extension of the Mean Gas Velocity concept to dimension the exhaust valves, we would not know the required size of the exhaust valve(s) which should accompany the
intake valve(s); or how high and for how long, and when, they should operate, etc., etc?
The good thing about the Mean Gas Velocity (Kl) concept is that it can be easily derived on a hand calculator but as a design tool, even with the above-proposed extension for sizing exhaust valves,
it is much too simplistic to be universally useful.
Technical journalists et al should consider quoting Kl data with the relevant caveats and not as Holy Grail. It was, as I understand it, the late Brian Lovell of Weslake who conceived Mean Gas
Velocity with respect to intake valves.
Before I get literally savaged by some technical journalist who feels that I have demeaned the memory of a great design engineer, I should point out that Brian Lovell proposed Mean Gas Velocity as a
means of comparing engines for which precious little data was available in a design era populated with slide rules and not computers; more complex calculations were definitely not on the menu.
If I have thoroughly bored my expert reader I can only but repeat my earlier apology; the fault is yours, you should have stopped reading back at the first page!
To those to whom this paper is a refresher course from their university days, then that is no bad thing. Although I very much doubt that your undergraduate university course ever extended to unsteady
gas dynamics, pressure waves and engine tuning, not to speak of specific time areas, to be reminded of the fundamentals and see them extended into effective design techniques is, as has been said
before, no bad thing.
To those who find even this level of math somewhat daunting, but yet have a basic understanding of engine tuning, rest easy because all the mathematics of unsteady gas dynamics, valve lift profile
design, valvetrain dynamic analysis, cylinder pressure analysis, discharge coefficient analysis, and specific time area calculations are packaged nowadays into computer software that you can
effectively use for design and thereby gain total understanding of the theoretical concepts which are discussed here.
Why is this empiricism so important if all I have to do is buy a complete engine simulation, like I use here, and just keep stuffing the input data numbers of the engine and duct geometry into it
until I come up with the required engine design?
This is especially the question as some of these engine simulations come with built-in automatic performance optimisers [7].
The answer is that you can keep stuffing numbers as input data into an engine simulation, where the data involved number in the hundreds if not thousands, but you may never attain a design as well
optimised as the exemplar MotoGP engine [2].
The reason is that it was initially created in the 4stHEAD software using the above empiricism to reach a ‘matched’ design which employed real valve lift profiles that not only provided valvetrain
dynamic stability but also a satisfactory cam design and manufacture potential.
It was only when all such design considerations were satisfied that it was run through the engine simulation to check that, as shown in Fig.21, (a) the design target was achieved and, (b) an
effective power and torque characteristic extended over the usable speed range.
All readers, be they experts or tyros, must conclude that does constitute a design process. In short, it is through an understanding of the basics that we get the guidance to efficiently use today’s
sophisticated computational tools for engine design.
[1]G.P. Blair, sidebar contribution on combustion in diesel engines, Race Engine Technology, Volume 5, Issue 2, May
2007 [2]G.P. Blair, “Steel Coils versus Gas”, Race Engine Technology, Volume 5, Issue 3, June/July 2007 [3]G.P. Blair, “Design and Simulation of Four-Stroke Engines”, Society of Automotive Engineers,
1998, SAE reference
R-186. [4]G.P. Blair, “Design and Simulation of Two-Stroke Engines”, Society of Automotive Engineers, 1996, SAE reference
R-161. [5]4stHEAD design software, Prof. Blair and Associates, Belfast, Northern Ireland [6]G.P. Blair, W.M. Cahoon, “Life at the Limit”, Race Engine Technology, Volume 1, Issue 004, Spring 2004. [7]
Virtual 4-Stroke engine simulation, Optimum Power Technology, | {"url":"https://racehead.com.au/designing-performance/back-to-basiscs/","timestamp":"2024-11-08T12:22:31Z","content_type":"text/html","content_length":"104681","record_id":"<urn:uuid:aa973f23-f43f-4342-81b8-ba43ea2eef80>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00830.warc.gz"} |
The Mathematics of BlurXTerminator
BlurXTerminator takes a different approach to deconvolution. Not only does it use a machine learning algorithm, but it was also developed using an updated mathematical formulation of deconvolution
itself. Implicit in the classical formulation of deconvolution is the goal of perfect recovery of an original signal. By recognizing that this represents an unobtainable absolute, and by instead
embracing the fundamental limitations of deconvolution, a new formulation results that is workable and provides a practically achievable goal. This approach also results in a formulation that is
well-suited to training a machine learning algorithm.
The classical description of deconvolution begins with describing its inverse operation, convolution:
where $f$ is the original signal (e.g., image), $g$ is a point spread function (PSF), and $h$ is the image produced by the convolution of $f$ with $g$. Deconvolution is defined as inverting this
process: recovering $f$ given $h$ and $g$.
Real images contain noise, and the above is usually modified to express this explicitly:
Because of the noise, $n$, and also because of imperfect knowledge of the PSF, $g$, amongst other factors such as finite numerical precision, there is no such thing as perfect deconvolution in
practice. All deconvolution algorithms produce an estimate of the original image, $f$. In other words, real deconvolution algorithms, run on real image data, cannot completely deconvolve an image to
a zero-width PSF. Their results can instead be described as the original image, $f$, convolved with a new (hopefully smaller) PSF, $g’$:
where $h’$ is the output of the deconvolution algorithm. It can be easily recognized that in the limit of taking $g’$ to an ideal Dirac delta function, this formulation becomes identical to the
classical description of deconvolution.
Stated another way, and in the context of astronomical images, an ideal algorithm in the classical sense would transform every star in the image into a perfect point source. This is never observed in
practice. It is an “upper bound” that can be approached but never reached. Perfect deconvolution, therefore, exists only as an idealized concept. What is observed in practice is that the existing
PSF, $g$, is in effect replaced by a smaller PSF, $g’$.
Real deconvolution algorithms also introduce artifacts. The most familiar example is ringing. This causes dark halos around stars and the appearance of false structure in the areas of an image
dominated by noise. While ringing by itself could technically be expressed as a PSF that has side lobes with negative values, we can also simply lump it and any other errors produced by a given
algorithm into a single error term:
The error, $e$, may be related to $f$, $g’$, and $n$ in complex ways. The point of collecting all errors into a single term is the following:
$e = f*g’+n-h’$
Defining $\mathcal{F}[x,\pmb{W}]$ as the operation performed by a machine learning deconvolution algorithm on an input, $x$, using a set a trainable weights, $\pmb{W}$, this becomes:
where $e$ is the training loss function. During training, $f$, $g$, $g’$, and $n$ are perfectly known: start with an ideal input image, convolve it with a chosen PSF, and add noise to produce the
input training image, $h=f*g+n$. The corresponding ground truth image, the ideal output of the algorithm, is $f*g’+n$. This is simply the same ideal image convolved with a new (smaller, not
aberrated) PSF, $g’$, plus noise.
Formulating deconvolution in the above way also provides control over the amount of deconvolution. By making the output PSF, $g’$, a parameterized quantity, we can specify exactly how much the
algorithm should reduce the diameter of the PSF, or even how its other characteristics should be modified. This is in stark contrast to the classical algorithms which not only have the unachievable
end goal of a zero-width PSF, but also have no control over whatever final PSF is obtained. They also usually have no clear stopping criteria for their iterations. We run them for as many iterations
as we can before artifacts appear, and we get whatever final PSF we get.
Using the above approach, during training the machine learning algorithm produces an output, $\mathcal{F}[f*g+n,\pmb{W}]$, with error, $e$. A suitable optimization algorithm determines the gradient
of the error with respect to the weights, $\nabla_{\pmb{W}}e$, and updates the weights iteratively to eventually minimize the error. If the distributions of the features in the ideal images, the
chosen PSFs, and the noise are sufficiently broad and representative of real-world scenes and image acquisition systems, an algorithm is obtained that generalizes well to deconvolving real images
with high accuracy.
Separating stellar and non-stellar image components
The above formulation of deconvolution can be taken a step further by decomposing the original image, $f$, into stellar and non-stellar components:
This is physical: some features in an astronomical scene originate directly from stars. Other features originate from non-stellar objects – clouds of gas and dust that reflect, emit, and/or absorb
light. The actual scene in terms of received light is the arithmetic sum of these components.
Decomposing the original scene in this way, presuming that it is mathematically tractable to do so in practice, and given that the output PSF is a parameterized quantity, an additional degree of
freedom in the deconvolution operation results: the stellar and non-stellar features can be deconvolved by different amounts. Stating this formally, while the captured image,
is convolved with a single PSF, $g$, the deconvolved image, $h’$, could, if we so choose, have different PSFs applied to each component:
The motivation for treating stellar and non-stellar features differently in this way is of course purely aesthetic and has no scientific value. Non-stellar features have much lower contrast than
stellar features, and can usually be deconvolved more before artifacts appear, an observation that is as true for the classical algorithms as it is for the present algorithm. Equal treatment of all
features is still an option if $g’_s$ and $g’_{ns}$ are identical.
Proceeding with the above development, any and all errors in the deconvolution estimate have again been lumped into a single term, $e$, so that the training loss function becomes:
Expanding this as before,
This is in fact precisely how BlurXTerminator’s convolutional network is trained.
Non-stationary point spread functions
The PSF of an image captured with real optics is rarely stationary, which is to say uniform for all positions in the image. Various off-axis aberrations such as coma, astigmatism, and field curvature
cause the PSF to vary across the image – features in the corners of an image are rarely as sharp as those in the center. In reality the PSF varies smoothly across an image, and an ideal deconvolution
algorithm would use a different PSF estimate for every position in the image.
An approximation to this ideal can be made by turning a limitation of convolutional networks into an advantage.
Efficient implementation of the computations in a convolutional network on modern hardware requires constructing a directed graph that parallelizes these computations as much as possible. The
construction of this graph represents a significant amount of setup time, but subsequent computations on input data of the same dimensions can be performed using the same constructed graph. The
expensive setup time can thus be amortized over a number of input data sets: an image can be broken up into same-sized chunks, or tiles.
In the case of a deconvolution network, each tile can be deconvolved with an estimated local PSF. Assuming that the PSF does not significantly vary within a tile, and assuming that a reasonable
estimate of the local PSF for each tile is available, this can produce a reasonable step-wise approximation to the ideal.
BlurXTerminator is a blind deconvolution algorithm: a PSF estimate is not supplied as an input. The PSF for each tile is instead inferred from the stars present in that tile, stars being excellent
approximations of point sources and therefore copies of the PSF with varying brightness and color. While this approach can fail on occasional tiles that contain no stars, in practice this is not
frequently observed. A number of measures, such as overlapping the input tiles, are taken to further reduce the probability of not having a reasonable local PSF estimate for every part of an image. | {"url":"https://www.rc-astro.com/the-mathematics-of-blurxterminator/","timestamp":"2024-11-04T17:05:06Z","content_type":"text/html","content_length":"88026","record_id":"<urn:uuid:ff39720e-e034-48ba-91c7-e9a7169eee79>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00704.warc.gz"} |
Daniel Liang
Apr 20, 2022
Abstract:Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro
and Datta (2020) studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of
circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known
to be efficiently PAC learnable (Rocchetto 2018). Here we provide a negative result, showing that proper learning of CNOT circuits is hard for classical learners unless $\textsf{RP} = \textsf{NP}$.
As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if $\textsf{RP} = \textsf{NP}$ then
there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only
if $\textsf{NP} \subseteq \textsf{RQP}$.
* 25 pages, 2 figures | {"url":"https://www.catalyzex.com/author/Daniel%20Liang","timestamp":"2024-11-11T08:53:51Z","content_type":"text/html","content_length":"128862","record_id":"<urn:uuid:a54543f8-24a6-4157-84b4-50d822475f8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00631.warc.gz"} |
Cohomology I
In this part we will go through the main idea of cohomology in algebraic topology.
Detour on Homology
Recall that a homology group comes with two process:
1. Define the chain complex $\{C_*,\partial_*\}$, with a boundary map such that $\partial^2=0$.
$\cdots\rightarrow C_n \overset{\partial}{\rightarrow} C_{n-1} \overset{\partial}{\rightarrow}\cdots$
2. Determine the boundary $B_n\subset Z_n\subset C_n$ and thus define the homology group to be
After a somehow abstract definition, we introduce the notion of homology to CW-complex, from “simplicial chain” of a topological space, to the “singular chain” of a topological space defined by free
Abelian group generated by continuous maps
$\sigma:\Delta^q\rightarrow X$
where if $X$ is a CW-complex, then the two homology are equivalent. Unfortunately (at least for Relue), the definition above cannot be used to compute the homology group of any concrete spaces, even
after we have realized the fact that “Homology only depends on the homotopy type of the space” .
Thus, we ought to extract some theorems/tools/techniques from the definition to do “computation”. Some important theorems must be listed here.
1. Exact sequence: Reveal the basic relation between $X$ and its quotient $X/A$
$\cdots\rightarrow H_n(A)\rightarrow H_n(X)\rightarrow H_n(X,A)\overset{\partial}{\rightarrow} H_{n-1}(A)\rightarrow\cdots$
2. Mayer-Vietoris theorem: Acts as a “van-Kampen theorem” in homotopy theory
$\cdots\rightarrow H_n(A\cap B)\overset{\Phi}{\rightarrow}H_n(A)\oplus H_n(B)\overset{\Psi}{\rightarrow} H_n(X)\overset{\partial}{\rightarrow} H_{n-1}(A\cap B)\rightarrow \cdots$
3. Excision: Elementary characteristic of exact sequence. If $Z\subset A\subset X$ such that $\bar{Z}\subset A^\circ$, then the inclusion $i:(X-Z,A-Z)\rightarrow (X,A)$ induces an isomorphism
$H_n(X-Z,A-Z)\approx H_n(X,A),\qquad \forall n\in\N.$
The above are computations on homology. In cohomology theory, we also wish to derive theories like that – from definitions to techniques that are simple for computation.
The Rise of Cohomology
Given a chain complex $\{C_*,\partial\}$ of free Abelian groups. Define its dualization
to obtain a cochain complex, with a coboundary map
\begin{aligned} \delta:C^n&\rightarrow C^{n+1}\\ \alpha&\mapsto \alpha\circ\partial \end{aligned}
Then the cohomology group with coefficients in $G$ is defined by
[Fact] $\mathrm{Hom}(\cdot,G)$ is right exact. | {"url":"https://mathyuan.com/2024/03/11/Cohomology-I/index.html","timestamp":"2024-11-06T15:18:22Z","content_type":"text/html","content_length":"71017","record_id":"<urn:uuid:3dd16776-0689-46d0-bfb9-338a0a20db36>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00069.warc.gz"} |
Institut für Angewandte Mathematik und Statistik
Permanent URI for this collection
Recent Submissions
• Gut microbiota patterns predicting long-term weight loss success in individuals with obesity undergoing nonsurgical therapy
(2022) Bischoff, Stephan C.; Nguyen, Nguyen K.; Seethaler, Benjamin; Beisner, Julia; Kügler, Philipp; Stefan, Thorsten
The long-term success of nonsurgical weight reduction programs is variable; thus, predictors of outcome are of major interest. We hypothesized that the intestinal microbiota known to be linked
with diet and obesity contain such predictive elements. Methods: Metagenome analysis by shotgun sequencing of stool DNA was performed in a cohort of 15 adults with obesity (mean body mass index
43.1 kg/m2) who underwent a one-year multidisciplinary weight loss program and another year of follow-up. Eight individuals were persistently successful (mean relative weight loss 18.2%), and
seven individuals were not successful (0.2%). The relationship between relative abundancies of bacterial genera/species and changes in relative weight loss or body mass index was studied using
three different statistical modeling methods. Results: When combining the predictor variables selected by the applied statistical modeling, we identified seven bacterial genera and eight
bacterial species as candidates for predicting success of weight loss. By classification of relative weight-loss predictions for each patient using 2–5 term models, 13 or 14 out of 15 individuals
were predicted correctly. Conclusions: Our data strongly suggest that gut microbiota patterns allow individual prediction of long-term weight loss success. Prediction accuracy seems to be high
but needs confirmation by larger prospective trials.
• Test for the model selection from two competing distribution classes
(2016) Chen, Hong; Jensen, Uwe
One of the main tasks in statistics is to allocate an appropriate distribution function to a given set of data. Often the underlying distribution of the data can be approximated by a distribution
function from a parametric distribution model class. This thesis deals with model selection from two given competing parametric model classes. To this end statistical hypothesis tests are
proposed in different settings and their asymptotic behaviour for an increasing data size is analysed. This thesis is part of a DFG-project investigating the lifetime distribution of
mechatronical systems such as DC-motors, which has been conducted in cooperation with engineers of the University of Stuttgart. The considered mechatronical systems are characterised by so-called
covariates, which can influence the lifetime distribution. For DC-motors such covariates could be the electric current, the working load or the operation voltage. For instance, the lifetime
distributions could be modelled by means of the Weibull distribution class or the log-normal distribution class with parameters depending linearly on the covariates. For a given data set an
estimator for the unknown parameter in a model class can be obtained according to the maximum likelihood method. Under suitable conditions, the consistency of the estimator follows from the
maximum likelihood theory for an increasing data size. In this thesis we consider two cases: First we handle the case with a fixed number of covariate values and the number of observations at
each covariate value tending to infinity. After that, we consider the situation the other way round. The distance between the underlying distribution function and the competing model classes is
defined based on the limit value of the maximum likelihood estimator and Cramér-von Mises distance. The reasons for the chosen distance measure are on the one hand the popularity of the maximum
likelihood estimator and on the other hand the simple interpretability of the Cramér-von Mises distance with respect to our intention to approximate the lifetime distribution function. The null
hypothesis is that both models provide an equally well fit. While the test statistic is defined by the estimated difference of the distances. Under suitable conditions, we show the asymptotic
normality of the test statistic. Moreover, it is shown that the asymptotic variance can be estimated consistently by a plug-in estimator. With quantiles of the standard normal distribution for a
given significance level the test decision rules are formulated. For the case with a fixed number of observations at each covariate and an increasing number of covariate values, the limit of the
maximum likelihood estimator is defined analogously. The distance is adjusted accordingly and in the test statistic the empirical distribution is replaced by the Nadaraya-Watson kernel estimator.
For one dimensional covariates we show similar results as in the first case. However, it cannot be extended to the multidimensional case in general. Thus, a one-sided test is proposed. Further,
the consistency of the test is also proven. The results are extended to the case with right random censoring, whereby the Kaplan-Meier and the Beran estimator for distribution functions are used.
At the end of the thesis the applicability of the proposed hypothesis tests is evaluated by means of simulations and a case study.
• SIA matrices and non-negative stationary subdivision
(2012) Li, Xianjun; Jetter, Kurt
This dissertation is concerned with SIA matrices and non-negative stationary subdivision, and is organized as follows: After an introducing chapter where some basic notation is given we describe,
in Chapter 3, how non-negative subdivision is connected to a corresponding non-homogenous Markov process. The family of matrices A, built from the mask of the subdivision scheme, is introduced.
Among other results, Lemma 3.1 and Lemma 3.2 relate the coefficients of the iterated masks to matrix products from the family A, and in the limiting case the values of the basic limit function
are found from the entries in an infinite product of matrices. Chapter 4 and Chapter 5 are the core of this dissertation. In Chapter 4, we first review some spectral and graph properties of
row-stochastic matrices and, in particular, of SIA matrices. We point to the notion of scrambling power, introduced by Hajnal [16], and of the related coefficient of ergodicity. We also consider
the directed graph of such matrices, and we improve upon a condition given by Ren and Beard in [30]. Then we study finite families of SIA matrices, the properties of their indicator matrices and
the connectivity of their directed graphs. We consider this chapter to be an important contribution to the theory of non-negative subdivision, since it explains the background in order to apply
the convergence result of Anthonisse and Tijms [2], which we reprove in Section 4.6, to rank one convergence of infinite products of row stochastic matrices. It does not use the notion of joint
spectral radius but the (equivalent) coefficient of ergodicity. Properties equivalent to SIA are listed in Lemma 4.7 and in the subsequent Lemma 4.8; they connect the SIA property to equivalent
conditions (scrambling property, positive column property) as they appear in the existing literature dealing with convergence of non-negative subdivision. The fifth chapter of the dissertation
contains the full proof of the characterization of uniform convergence for non-negative subdivision, for the univariate and bivariate case, the latter one being a representative for multivariate
aspects. It uses the pointwise definition of the limit function at dyadic points - refering to the dyadic expansion of real vectors from the unit cube - using the Anthonisse-Tijms pointwise
convergence result, and employs the proper extension of the Micchelli-Prautzsch compatibility condition to the multivariate case, taking care of the ambiguity of representation of dyadic points.
As a consequence, the Hölder exponent of the basic limit function can be expressed in terms of the coefficient of ergodicity of the family A. Our convergence theorems, in Theorem 5.1 and Theorem
5.8, include the existing characterizations of uniform convergence for non-negative univariate and bivariate subdivision from the literature, except for the GCD condition, which seems to be a
condition applicable to univariate subdivision only. Chapter 5 also reports on some further attempts where we have tried to extend conditions from univariate subdivision, which are sufficient for
convergence, to the bivariate case. We could find a bivariate analogue of Melkman's univariate string condition, which we call - in the bivariate case - a rectangular string condition. The
chapter concludes with stating the fact that uniform convergence of non-negative stationary subdivision is a property of the support of the mask alone, modulo some apparent necessary conditions
such as the sum rules. A typical application of this support property characterizes uniform convergence in the case where the mask is a convex combination of other masks. The dissertation ends
with two short chapters on tensor product and box spline subdivision, and an appendix where some definitions and useful lemmas and theorems about matrix and graph theory are stated without
• Cox-Type regression and transformation models with change-points based on covariate thresholds
(2007) Lütkebohmert-Marhenke, Constanze; Jensen, Uwe
In this thesis we consider Cox-type regression models and transformation models for right-censored survival time data with bent-line change-points in the underlying regression functions according
to covariate thresholds. We establish the usual asymptotic properties of the estimates such as √(n) consistency and asymptotic normality. Furthermore, we applied the Cox regression model with
change-points to different data sets. | {"url":"https://hohpublica.uni-hohenheim.de/collections/b0948e33-d6a0-4184-8837-3127fdd0facf","timestamp":"2024-11-10T11:31:44Z","content_type":"text/html","content_length":"556706","record_id":"<urn:uuid:99c6fdf8-f269-4247-9a8f-c9ef5739b7c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00454.warc.gz"} |
#22063: Forall-types should be of a TYPE kind · Issues · Glasgow Haskell Compiler / GHC · GitLab
Forall-types should be of a TYPE kind
This is the main ticket for a bunch of related tickets, relating to the kind of (forall a. ty). Specifically
With or without impredicativity, we usually intend a forall-quantified type to be inhabited by values, i.e. that forall a. {- ... -} :: TYPE r for some r. Apparently this is never checked in GHC and
we have roughly:
Gamma, a |- t :: k
Gamma |- (forall a. t) :: k
This implies that all kinds (even closed) are lifted: forall a. a :: k. Of course in presence of TypeFamilies+UndecidableInstances all kinds are already lifted, but this seems somehow worse.
Furthermore such a type can be used to trigger a core lint error, the wording of which ("Non-*-like kind when *-like expected") suggests that at least on some level it is expected that forall a. ...
:: *. Applying a type constructor (other than ->) to a type like forall a. a requires ImpredicativeTypes, but (and this might be an unrelated issue) apparently the type can appear on the left side of
an application even without ImpredicativeTypes.
Steps to reproduce
GHCi says:
> :kind forall a. a
forall a. a :: k
The following are accepted by GHC 9.2.3, 8.6.5 and HEAD:
{-# LANGUAGE RankNTypes, DataKinds, KindSignatures #-}
module M where
type F = (forall (f :: * -> *). f) ()
f :: F
f = f
{-# LANGUAGE RankNTypes, DataKinds, KindSignatures, ImpredicativeTypes #-}
module M where
type B = forall (a :: Bool). a
b :: proxy B
b = b
but with -dcore-lint they explode:
*** Core Lint errors : in result of Desugar (before optimization) ***
a.hs:5:1: warning:
Non-Type-like kind when Type-like expected: * -> *
when checking the body of forall: f_agw
In the type of a binder: f
In the type ‘F’
Substitution: [TCvSubst
In scope: InScope {f_agw}
Type env: [agw :-> f_agw]
Co env: []]
*** Offending Program ***
Rec {
$trModule :: Module
$trModule = Module (TrNameS "main"#) (TrNameS "M"#)
f :: F
f = f
end Rec }
*** End of Offense ***
*** Core Lint errors : in result of Desugar (before optimization) ***
b.hs:5:1: warning:
Non-Type-like kind when Type-like expected: Bool
when checking the body of forall: a_auh
In the type of a binder: b
In the type ‘forall (proxy :: Bool -> *). proxy B’
Substitution: [TCvSubst
In scope: InScope {a_auh proxy_aui}
Type env: [auh :-> a_auh, aui :-> proxy_aui]
Co env: []]
*** Offending Program ***
Rec {
$trModule :: Module
$trModule = Module (TrNameS "main"#) (TrNameS "M"#)
b :: forall (proxy :: Bool -> *). proxy B
b = \ (@(proxy_awC :: Bool -> *)) -> b @proxy_awC
end Rec }
*** End of Offense ***
GHC 8.4.4 will fail the first program with an impredicativity error, but will still accept it with ImpredicativeTypes enabled, and will still crash with a core lint.
Expected behavior
forall a. e should constrain e :: TYPE r, meaning GHCi should report forall a. a :: * or forall a. a :: TYPE r. Both programs should be rejected with a type error for the above reason. The first
program (or programs like it) should raise an "Illegal impredicative type".
• GHC version used: 8.4.4, 8.6.5, 9.2.3, HEAD
• Operating System: GNU/Linux
• System Architecture: x86_64
To upload designs, you'll need to enable LFS and have an admin enable hashed storage. | {"url":"https://gitlab.haskell.org/ghc/ghc/-/issues/22063","timestamp":"2024-11-08T09:19:53Z","content_type":"text/html","content_length":"118876","record_id":"<urn:uuid:4239a593-3829-4bcf-b481-58d84473e7c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00388.warc.gz"} |
is two-photon absorption
What is two-photon absorption cross section?
What is two-photon absorption cross section?
Two-photon absorption (TPA or 2PA) or two-photon excitation or non-linear absorption is the simultaneous absorption of two photons of identical or different frequencies in order to excite a molecule
from one state (usually the ground state) to a higher energy, most commonly an excited electronic state.
How is cross section absorption measured?
This is obtained using, dx/dt = c. σ has units of cm2 and is called the absorption cross-section. The total change in the photon density over a distance, l, is obtained by integration. Transmission
is defined as the intensity of light leaving the sample, divided by the intensity of light entering the sample.
What is meant by absorption cross section?
Absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain
particle-particle interaction, e.g., scattering, electromagnetic absorption, etc.
What is single photon absorption?
Single-photon absorption. Single-photon absorption (SPA or 1PA) is a linear absorption process whereby one photon excites an atom, ion or molecule from a lower energy level to a higher energy level,
for example, from the ground state to the first excited state.
Why is two-photon absorption a third order process?
Two-photon absorption (TPA) is a third order nonlinear optical phenomenon in which a molecule absorbs two photons at the same time. The transition energy for this process is equal to the sum of the
energies of the two photons absorbed.
What is two-photon absorption coefficient?
The two‐photon absorption (TPA) coefficient has been measured for a single‐mode GaAs/AlGaAs quantum well laser at 0.86 μm, near the lasing wavelength of 0.83 μm. Picosecond laser pulses were employed
to resolve the ultrafast TPA from long‐lived carrier‐dependent effects.
How do you measure absorbance?
Absorbance is measured using a spectrophotometer or microplate reader, which is an instrument that shines light of a specified wavelength through a sample and measures the amount of light that the
sample absorbs.
How do you calculate absorption length?
The absorption length arises from the imaginary part of the atomic scattering factor, f2. It is closely related to the absorption cross-section, and the mass absorption coefficient. Specifically, the
atomic photoabsorption cross-section can be computed via: σ = 2 r e λ f 2 {\displaystyle \sigma =2r_{e}\lambda f_{2}}
How do you convert absorbance to absorption?
You can calculate the absorption coefficient using this formula: α=2.303*A/d, where d is thickness, A is absorption and α is the absorption coefficient, respectively.
What happens during photon absorption?
Photon absorption by an atomic electron occurs in the photoelectric effect process, in which the photon loses its entire energy to an atomic electron which is in turn liberated from the atom. This
process requires the incident photon to have an energy greater than the binding energy of an orbital electron.
How do you calculate the absorption of a photon?
Step 3: Use the formula f=Eh f = E h to find the frequency of the photon absorbed, where h=6.63×10−34 J⋅ Hz−1 h = 6.63 × 10 − 34 J ⋅ Hz − 1 is Planck’s constant. The frequency of the photon absorbed
is about 3.15×1015 Hz 3.15 × 10 15 Hz .
What is the difference between 2nd harmonic generation and two-photon absorption?
Second-harmonic generation (SHG) and two-photon absorption (2PA) are nonlinear optical processes. SHG is the second-order nonlinear process, while 2PA is the third-order nonlinear process. The
second-order nonlinear processes occur in the non-centrosymmetric (crystal) nonlinear optical materials.
Why do we measure absorbance?
Why measure absorbance? In biology and chemistry, the principle of absorbance is used to quantify absorbing molecules in solution. Many biomolecules are absorbing at specific wavelengths themselves.
What is the purpose of measuring the absorbance of the sample?
Spectrophotometry is a method to measure how much a substance absorbs light by measuring the intensity of light as a beam of light passes through sample solution. The basic principle is that each
compound absorbs or transmits light over a certain range of wavelength.
On what factors mass absorption coefficient depend?
4- The linear absorption coefficient depends on the density of the absorbed material. When considering the mass of the material, we are talking about the mass attenuation coefficient and we will
arrive at more similar values for the attenuation coefficients of this material.
How do you find the intensity of absorbance of light?
The detector measures the intensity of the light that travels through the sample….Absorbance Measurements – the Quick Way to Determine Sample Concentration
1. Transmission or transmittance (T) = I/I0
2. Absorbance (A) = log (I0/I)
3. Absorbance (A) = C x L x Ɛ => Concentration (C) = A/(L x Ɛ)
How is absorption measured?
How do you calculate absorption?
To find out the absorption rate in real estate, divide the total number of homes sold in a specific period of time by the total number of homes available in that market.
What is the two-photon absorption cross section of metal ions (tpacs)?
Up to now, however, the two-photon absorption cross section (TPACS) of metal ions, which characterizes the capability of TPA, still remains unknown because of the difficulty to measure it, since the
existing methods for studying TPA are not applicable to metal ions.
What are the units of molecular two-photon absorption cross-section?
The molecular two-photon absorption cross-section is usually quoted in the units of Goeppert-Mayer ( GM) (after its discoverer, Nobel laureate Maria Goeppert-Mayer ), where 1 GM is 10 −50 cm 4 s
photon −1.
What is the two-photon absorption coefficient?
The two-photon absorption coefficient is defined by the relation . is the two-photon absorption cross section (cm 4 s/molecule).
What is non-degenerate two-photon absorption?
Absorption of two photons with different frequencies is called non-degenerate two-photon absorption. Since TPA depends on the simultaneous absorption of two photons, the probability of TPA is
proportional to the square of the light intensity, thus it is a nonlinear optical process. | {"url":"https://mystylit.com/essay-writing-tips/what-is-two-photon-absorption-cross-section/","timestamp":"2024-11-05T23:25:17Z","content_type":"text/html","content_length":"54702","record_id":"<urn:uuid:a41e1387-5add-47db-9a0a-f325ed116120>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00776.warc.gz"} |
Number Line Printable Positive And Negative
Number Line Printable Positive And Negative - Web number line for positive and negative fractions and decimals created by rise over run a tall vertical number line is an. Web the last worksheet
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Related Post: | {"url":"https://tineopprinnelse.tine.no/en/number-line-printable-positive-and-negative.html","timestamp":"2024-11-10T11:51:19Z","content_type":"text/html","content_length":"31868","record_id":"<urn:uuid:b5e22b7a-44b6-4f0a-b8f3-7e49a036fa7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00548.warc.gz"} |
pub trait KeyOwnerProofSystem<Key> {
type Proof: Codec;
type IdentificationTuple: Codec;
// Required methods
fn prove(key: Key) -> Option<Self::Proof>;
fn check_proof(
key: Key,
proof: Self::Proof
) -> Option<Self::IdentificationTuple>;
Expand description
Something which can compute and check proofs of a historical key owner and return full identification data of that key owner.
Required Associated Types§
The proof of membership itself.
The full identification of a key owner and the stash account.
Required Methods§
Prove membership of a key owner in the current block-state.
This should typically only be called off-chain, since it may be computationally heavy.
Returns Some iff the key owner referred to by the given key is a member of the current set.
Check a proof of membership on-chain. Return Some iff the proof is valid and recent enough to check.
Object Safety§
Implementations on Foreign Types§ | {"url":"http://git.phron.ai/phronesis_runtime/trait.KeyOwnerProofSystem.html","timestamp":"2024-11-03T17:01:10Z","content_type":"text/html","content_length":"18812","record_id":"<urn:uuid:0292e77b-d4c8-4479-bb5a-f2603b4b752e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00472.warc.gz"} |
Prove that the Goldbach conjecture that every even integer
Elementary Number Theory Problems 3.3 Solution (David M. Burton's 7th Edition) - Q6
My Solution for "Prove that the Goldbach conjecture that every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of
three primes."
All theorems, corollaries, and definitions listed in the book's order:
Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)
All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 3)
I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.
Prove that the Goldbach conjecture that every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of three primes.
[Hint: If $2n - 2 =p_{1}+ p_{2}$, then $2n =p_{1}+ p_{2} + 2$ and $2n + 1 =p_{1}+ p_{2} + 3$.]
Method 1: Using the Hint
Let $m > 5$ be an integer.
If $m$ is odd:
By Theorem 2.1 Division Algorithm, we can write $m = 2n + 1$ for some integer $n$. We know $2n + 1 > 5$, so $2n - 2 > 2$ and $2n - 2$ is an even integer. Thus we can write $2n - 2 = p_{1} + p_{2}$
where $p_{1}$ and $p_{2}$ are two primes. Then $m = 2n + 1 = p_{1} + p_{2} + 3$, which is the sum of three primes.
If $m$ is even:
By Theorem 2.1 Division Algorithm, we can write $m = 2k$ for some integer $k$. We know $2k > 5$, so $2k - 2 > 2$ and $2k - 2$ is an even integer. Then $2k - 2 = p_{1} + p_{2}$. Then $2k = p_{1} + p_
{2} + 2$, which is the sum of three primes.
Let $m > 2$ be an even integer. By Theorem 2.1 Division Algorithm, we can write $m = 2k$ for some integer $k$. We know $2k + 2 > 5$, then $2k + 2 = p_{1} + p_{2} + p_{3}$. Then $2k = p_{1} + p_{2} +
p_{3} - 2$. One of $p_{1}$, $p_{2}$, and $p_{3}$ must be $2$; otherwise, the righthand side will be odd, and the lefthand side will be even. WLOG, let $p_{3} = 2$. Thus $m = 2k = p_{1} + p_{2} + 2 -
2 = p_{1} + p_{2}$.
Therefore, every even integer greater than $2$ is the sum of two primes is equivalent to the statement that every integer greater than $5$ is the sum of three primes.
Method 2: A Shorter Version
The rest is for Premium Members only
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Elementary Number Theory Problems 4.3 Solution (David M. Burton's 7th Edition) - Q10 Paid Members Public
My Solution for "Prove that no integer whose digits add up to $15$ can be a square or a cube. [Hint: For any $a$, $a^{3} \equiv 0$, $1$, or $8$ $\pmod 9$.]"
Elementary Number Theory Problems 4.3 Solution (David M. Burton's 7th Edition) - Q9 Paid Members Public
My Solution for "Find the remainder when $4444^{4444}$ is divided by $9$. [Hint: Observe that $2^{3} \equiv -1 \pmod {9}$.]" | {"url":"https://www.ranblog.com/blog/elementary-number-theory-problems-3-3-solution-david-m-burtons-7th-edition-q6/","timestamp":"2024-11-03T03:22:02Z","content_type":"text/html","content_length":"105349","record_id":"<urn:uuid:a5839cdd-b91f-4137-9a76-7bc28e580aa2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00789.warc.gz"} |
LPeg is a pattern-matching library for Lua, based on Parsing Expression Grammars (PEGs).
For a more formal treatment of LPeg, as well as some discussion about its implementation, see:
http://www.inf.puc-rio.br/~roberto/docs/peg.pdf (A Text Pattern-Matching Tool based on Parsing Expression Grammars.)
Following the Snobol tradition, LPeg defines patterns as first-class objects. That is, patterns are regular Lua values (represented by userdata). The library offers several functions to create and
compose patterns. With the use of metamethods, several of these functions are provided as infix or prefix operators. On the one hand, the result is usually much more verbose than the typical encoding
of patterns using the so called regular expressions (which typically are not regular expressions in the formal sense). On the other hand, first-class patterns allow much better documentation (as it
is easy to comment the code, to use auxiliary variables to break complex definitions, etc.) and are extensible, as we can define new functions to create and compose patterns.
For a quick glance of the library, the following table summarizes its basic operations for creating patterns:
Operator Description
lpeg.P(string) : Matches string literally
lpeg.P(number) : Matches exactly number characters
lpeg.S(string) : Matches any character in string (Set)
lpeg.R("xy") : Matches any character between x and y (Range)
patt^n : Matches at least n repetitions of patt
patt^-n : Matches at most n repetitions of patt
patt1 * patt2 : Matches patt1 followed by patt2
patt1 + patt2 : Matches patt1 or patt2 (ordered choice)
patt1 - patt2 : Matches patt1 if patt2 does not match
-patt : Equivalent to ("" - patt)
#patt : Matches patt but consumes no input
As a very simple example, lpeg.R("09")^1 creates a pattern that matches a non-empty sequence of digits. As a not so simple example, -lpeg.P(1) (which can be written as lpeg.P(-1) or simply -1 for
operations expecting a pattern) matches an empty string only if it cannot match a single character; so, it succeeds only at the subject's end.
In addition to the functions documented here, some arithmetic operators have special effects on patterns:
Returns a pattern that matches only if the input string matches patt, but without consuming any input, independently of success or failure. (This pattern is equivalent to &patt in the original PEG
When it succeeds, #patt produces all captures produced by patt.
Returns a pattern that matches only if the input string does not match patt. It does not consume any input, independently of success or failure. (This pattern is equivalent to !patt in the original
PEG notation.)
As an example, the pattern -lpeg.P(1) matches only the end of string.
This pattern never produces any captures, because either patt fails or -patt fails. (A failing pattern never produces captures.)
patt1 + patt2
Returns a pattern equivalent to an ordered choice of patt1 and patt2. (This is denoted by patt1 / patt2 in the original PEG notation, not to be confused with the / operation in LPeg.) It matches
either patt1 or patt2, with no backtracking once one of them succeeds. The identity element for this operation is the pattern lpeg.P(false), which always fails.
If both patt1 and patt2 are character sets, this operation is equivalent to set union.
patt1 - patt2
Returns a pattern equivalent to !patt2 patt1. This pattern asserts that the input does not match patt2 and then matches patt1.
If both patt1 and patt2 are character sets, this operation is equivalent to set difference. Note that -patt is equivalent to "" - patt (or 0 - patt). If patt is a character set, 1 - patt is its
patt1 * patt2
Returns a pattern that matches patt1 and then matches patt2, starting where patt1 finished. The identity element for this operation is the pattern lpeg.P(true), which always succeeds.
(LPeg uses the * operator [instead of the more obvious ..] both because it has the right priority and because in formal languages it is common to use a dot for denoting concatenation.)
If n is nonnegative, this pattern is equivalent to pattn patt*. It matches at least n occurrences of patt.
Otherwise, when n is negative, this pattern is equivalent to (patt?)-n. That is, it matches at most -n occurrences of patt.
In particular, patt^0 is equivalent to patt*, patt^1 is equivalent to patt+, and patt^-1 is equivalent to patt? in the original PEG notation.
In all cases, the resulting pattern is greedy with no backtracking (also called a possessive repetition). That is, it matches only the longest possible sequence of matches for patt.
patt / string
Creates a string capture. It creates a capture string based on string. The captured value is a copy of string, except that the character % works as an escape character: any sequence in string of the
form %n, with n between 1 and 9, stands for the match of the n-th capture in patt. The sequence %0 stands for the whole match. The sequence %% stands for a single %.
patt / table
Creates a query capture. It indexes the given table using as key the first value captured by patt, or the whole match if patt produced no value. The value at that index is the final value of the
capture. If the table does not have that key, there is no captured value.
patt / function
Creates a function capture. It calls the given function passing all captures made by patt as arguments, or the whole match if patt made no capture. The values returned by the function are the final
values of the capture. In particular, if function returns no value, there is no captured value.
With the use of Lua variables, it is possible to define patterns incrementally, with each new pattern using previously defined ones. However, this technique does not allow the definition of recursive
patterns. For recursive patterns, we need real grammars.
LPeg represents grammars with tables, where each entry is a rule.
The call lpeg.V(v) creates a pattern that represents the nonterminal (or variable) with index v in a grammar. Because the grammar still does not exist when this function is evaluated, the result is
an open reference to the respective rule.
A table is fixed when it is converted to a pattern (either by calling lpeg.P or by using it wherein a pattern is expected). Then every open reference created by lpeg.V(v) is corrected to refer to the
rule indexed by v in the table.
When a table is fixed, the result is a pattern that matches its initial rule. The entry with index 1 in the table defines its initial rule. If that entry is a string, it is assumed to be the name of
the initial rule. Otherwise, LPeg assumes that the entry 1 itself is the initial rule.
As an example, the following grammar matches strings of a's and b's that have the same number of a's and b's:
equalcount = lpeg.P{
"S"; -- initial rule name
S = "a" * lpeg.V"B" + "b" * lpeg.V"A" + "",
A = "a" * lpeg.V"S" + "b" * lpeg.V"A" * lpeg.V"A",
B = "b" * lpeg.V"S" + "a" * lpeg.V"B" * lpeg.V"B",
} * -1
Lua functions
lpeg.B - Matches patt n characters behind the current position, consuming no input
lpeg.C - Creates a simple capture
lpeg.Carg - Creates an argument capture
lpeg.Cb - Creates a back capture
lpeg.Cc - Creates a constant capture
lpeg.Cf - Creates a fold capture
lpeg.Cg - Creates a group capture
lpeg.Cmt - Creates a match-time capture
lpeg.Cp - Creates a position capture
lpeg.Cs - Creates a substitution capture
lpeg.Ct - Creates a table capture
lpeg.locale - Returns a table of patterns matching the current locale
lpeg.match - Matches a pattern against a string
lpeg.P - Converts a value into a pattern
lpeg.print - Outputs debugging information to stdout
lpeg.R - Returns a pattern that matches a range of characters
lpeg.S - Returns a pattern that matches a set of characters
lpeg.setmaxstack - Sets the maximum size for the backtrack stack
lpeg.type - Tests if a value is a pattern
lpeg.V - Creates a non-terminal variable for a grammar
lpeg.version - Returns the LPeg version
Lua bc (big number) functions
Lua bit manipulation functions
Lua package functions
Lua PCRE regular expression functions
Lua script extensions
Lua string functions
Lua syntax
Lua table functions
Lua utilities
Regular Expressions
Scripting callbacks - plugins
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How do you solve a/(9a-2)=1/8? | HIX Tutor
How do you solve #a/(9a-2)=1/8#?
Answer 1
See the entire solution process below:
First, multiply each side of the equation by #color(blue)(8)color(red)((9a - 2))# to eliminate the fractions while keeping the equation balanced. #color(blue)(8)color(red)((9a - 2))# is the Lowest
Common Denominator of the two fractions:
#color(blue)(8)color(red)((9a - 2)) * a/(9a - 2) = color(blue)(8)color(red)((9a - 2)) * 1/8#
#color(blue)(8)cancel(color(red)((9a - 2))) * a/color(red)(cancel(color(black)((9a - 2)))) = cancel(color(blue)(8))color(red)((9a - 2)) * 1/color(blue)(cancel(color(black)(8)))#
#8a = 9a - 2#
Next, subtract #color(red)(9a)# from each side of the equation to isolate the #a# term while keeping the equation balanced:
#-color(red)(9a) + 8a = -color(red)(9a) + 9a - 2#
#(-color(red)(9) + 8)a = 0 - 2#
#-1a = -2#
Now, multiply each side of the equation by #color(red)(-1)# to solve for #a# while keeping the equation balanced:
#color(red)(-1) * -1a = color(red)(-1)* -2#
#1a = 2#
#a = 2#
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Answer 2
To solve the equation ( \frac{a}{9a - 2} = \frac{1}{8} ), cross multiply to get ( 8a = 9a - 2 ). Then, solve for ( a ) by subtracting ( 9a ) from both sides to get ( -a = -2 ), and finally, divide
both sides by ( -1 ) to get ( a = 2 ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Understanding Mathematical Functions: How To Find Function Value Calcu
Introduction: The Importance of Understanding Mathematical Functions
Mathematical functions play a crucial role in various aspects of our lives, from everyday activities to complex calculations in academic and professional settings. Understanding how functions work
and how to effectively utilize function value calculators can significantly simplify mathematical tasks and improve problem-solving abilities.
A Overview of mathematical functions in daily life and various academic fields
Mathematical functions are prevalent in our daily lives, often without us realizing it. From calculating the cost per unit in grocery shopping to predicting future trends in stock markets, functions
are essential tools for making sense of the world around us.
In academic fields such as mathematics, physics, engineering, and economics, functions are used to model real-world phenomena and solve complex problems. They provide a framework for understanding
relationships between variables and are instrumental in making predictions and optimizing solutions.
B The role of function value calculators in simplifying complex calculations
Function value calculators are powerful tools that can handle complex mathematical operations and provide accurate results in a fraction of the time it would take to solve manually. These calculators
are especially useful when dealing with functions that involve multiple variables, exponents, logarithms, and trigonometric functions.
By utilizing function value calculators, individuals can focus on the conceptual aspects of the problem-solving process, rather than getting bogged down in tedious arithmetic calculations. This
allows for a deeper understanding of the underlying principles and promotes more efficient problem-solving strategies.
C Setting the stage for the exploration of how to effectively use function value calculators
With the increasing reliance on technology and computational tools, it is essential to understand how to leverage function value calculators to their full potential. In the following chapters, we
will delve into the various features and functionalities of these calculators, providing practical examples and step-by-step instructions on how to find function values effectively.
Key Takeaways
• Understanding mathematical functions
• Importance of finding function values
• How to use a function value calculator
• Examples of finding function values
• Practice exercises for mastering function values
Key Concepts: Defining Functions and Function Values
Understanding mathematical functions is essential in various fields, including mathematics, physics, engineering, and computer science. In this chapter, we will delve into the key concepts of
defining functions and function values, and how to find function value calculator.
A The definition of a mathematical function and its components (domain, range, etc)
A mathematical function is a relation between a set of inputs (the domain) and a set of possible outputs (the range), where each input is related to exactly one output. The components of a function
• Domain: The set of all possible input values for the function.
• Range: The set of all possible output values that result from applying the function to the domain.
• Mapping: The correspondence between each element in the domain and its corresponding element in the range.
It's important to note that a function must satisfy the condition that each input in the domain maps to exactly one output in the range, and no input can be left unmapped.
B Explanation of what a function value represents
The function value represents the output of the function when a specific input from the domain is used. In other words, it is the result obtained by applying the function to a particular input. The
function value is crucial in understanding how the function behaves and how it relates to different inputs.
For example, in the function f(x) = 2x + 3, if we want to find the value of the function when x = 4, we substitute 4 for x and solve for f(4), which gives us the function value at x = 4.
C The significance of understanding function notation and evaluation
Function notation, typically represented as f(x), g(x), or h(x), is a way to denote the relationship between the input and output of a function. It provides a concise and standardized way to refer to
functions and their values.
Understanding function evaluation is crucial for analyzing and solving problems involving functions. It allows us to determine the output of a function for a given input, which is essential in
various mathematical and real-world applications.
Tools of the Trade: Introduction to Function Value Calculators
Function value calculators are essential tools for anyone working with mathematical functions. These calculators can quickly and accurately determine the value of a function at a given input. In this
chapter, we will explore the different types of function value calculators, key features to look for in a reliable calculator, and the advantages of using calculators over manual calculations.
A Overview of different types of function value calculators
Function value calculators come in various forms, including online calculators, software applications, and handheld devices. Each type has its own set of advantages and limitations.
• Online calculators: These are web-based tools that can be accessed through a web browser. They are convenient and accessible from any device with an internet connection. Many online calculators
also offer additional features such as graphing and equation solving.
• Software applications: There are numerous software applications available for function value calculations, ranging from basic calculators to advanced mathematical software. These applications
often provide a wide range of functions and capabilities for complex mathematical operations.
• Handheld calculators: These are physical devices designed specifically for mathematical calculations. They are portable and can be used in various settings, making them a popular choice for
students and professionals.
B Key features to look for in a reliable function value calculator
When choosing a function value calculator, it is important to consider the following key features to ensure its reliability and usefulness:
• Accuracy: The calculator should provide precise and accurate results for a wide range of functions and inputs.
• User-friendly interface: A well-designed interface with intuitive controls and clear display can enhance the user experience and make calculations more efficient.
• Function support: Look for a calculator that supports a variety of mathematical functions, including trigonometric, logarithmic, and exponential functions.
• Graphing capabilities: Some calculators offer graphing features, allowing users to visualize functions and analyze their behavior.
• Customization options: The ability to customize settings and preferences can make the calculator more adaptable to individual needs.
C Advantages of using calculators over manual calculations
Using function value calculators offers several advantages over manual calculations, including:
• Speed and efficiency: Calculators can perform complex calculations in a fraction of the time it would take to do them manually, saving valuable time and effort.
• Reduced errors: Manual calculations are prone to human errors, while calculators provide accurate results consistently.
• Accessibility: With online calculators and software applications, users can access function value calculators from anywhere, eliminating the need for carrying physical calculators.
• Advanced features: Many calculators offer advanced features such as equation solving, graphing, and data analysis, expanding their utility beyond basic function value calculations.
Step-by-Step Guide: Finding Function Values Using a Calculator
When it comes to understanding mathematical functions, one of the key tasks is finding function values. This can be done manually, but using a calculator can make the process much quicker and more
accurate. Here's a step-by-step guide on how to find function values using a calculator.
A. Inputting the function into the calculator correctly
Before you can find the value of a function using a calculator, you need to input the function correctly. Most scientific calculators have a specific function input mode where you can enter the
entire function, including any constants or variables. Make sure to use the appropriate symbols for operations such as addition (+), subtraction (-), multiplication (*), division (/), and
exponentiation (^).
For example, if you want to find the value of the function f(x) = 2x^2 + 3x - 5, you would input '2*x^2 + 3*x - 5' into the calculator, making sure to use the correct syntax for exponentiation and
B. Setting the correct domain values if applicable
If the function has a specific domain over which it is defined, you may need to set the correct domain values in the calculator. This is especially important for functions that have restrictions on
the input values, such as square roots or logarithmic functions. Many calculators have a feature that allows you to specify the domain for the function.
For example, if you want to find the value of the function g(x) = √(x + 1), you would need to set the domain to x ≥ -1 to ensure that the calculator only evaluates the function for valid input
C. Interpreting the output from the calculator
Once you have input the function and, if necessary, set the domain values, you can proceed to calculate the function value. After entering the input value for the independent variable (x), the
calculator will provide the output value of the function.
It's important to interpret the output correctly, especially if the function has specific units or context. Make sure to double-check the input and output values to ensure accuracy.
By following these steps, you can effectively find function values using a calculator, saving time and minimizing errors in the process.
Using Function Value Calculators in Practical Scenarios
Function value calculators are powerful tools that can be used in various practical scenarios, from educational settings to real-world applications in businesses. Let's explore how these calculators
are utilized in different contexts.
A Case study: Using a calculator in an educational setting for learning purposes
In an educational setting, function value calculators play a crucial role in helping students understand and visualize mathematical functions. These calculators allow students to input different
values for variables and instantly see the corresponding function values. This hands-on approach helps students grasp the concept of functions more effectively and reinforces their understanding of
mathematical principles.
Furthermore, function value calculators can be used to plot graphs of functions, enabling students to visualize the behavior of different functions and how they change with varying input values. This
visual representation enhances the learning experience and makes abstract mathematical concepts more tangible.
Real-world application: How businesses use function calculators for financial projections
Businesses utilize function value calculators for financial projections and analysis. These calculators enable financial analysts to model various scenarios by inputting different variables such as
sales figures, expenses, and growth rates. By using function value calculators, businesses can forecast future financial outcomes and make informed decisions based on the projected results.
Additionally, function value calculators can be used to perform sensitivity analysis, allowing businesses to assess the impact of changes in key variables on their financial projections. This
capability is invaluable for risk management and strategic planning, as it helps businesses anticipate potential outcomes under different circumstances.
Troubleshooting common errors, such as misinterpreting the calculator's syntax
While function value calculators are powerful tools, users may encounter common errors when using them. One of the most prevalent issues is misinterpreting the calculator's syntax, which can lead to
incorrect function values. To troubleshoot this error, users should carefully review the input format and ensure that the calculator is interpreting the variables and functions correctly.
Another common error is inputting incorrect values for variables, resulting in inaccurate function values. Users should double-check their input data to avoid this error and verify that the
calculator is processing the correct information.
By understanding these common errors and how to troubleshoot them, users can make the most of function value calculators and leverage their capabilities effectively.
Maximizing Efficiency: Tips and Tricks for Using Function Value Calculators
Function value calculators are powerful tools for quickly and accurately evaluating mathematical functions. To make the most of these calculators, it's important to understand how to customize
settings for repetitive tasks, utilize shortcuts and hotkeys, and avoid common pitfalls.
A. Customizing calculator settings for repetitive tasks
• Save custom functions: Many function value calculators allow users to save custom functions for easy access. Take advantage of this feature to avoid re-entering the same function repeatedly.
• Adjust display settings: Customize the display settings to show the specific information you need, such as decimal places or scientific notation.
• Use memory functions: Some calculators have memory functions that allow you to store and recall specific values, making it easier to work with repetitive calculations.
B. Shortcuts and hotkeys that can save time during use
• Learn common shortcuts: Familiarize yourself with common shortcuts for functions such as square root, exponentiation, and trigonometric functions. This can significantly speed up the calculation
• Customize hotkeys: If your calculator allows for custom hotkeys, consider setting up shortcuts for the functions you use most frequently.
• Utilize history functions: Take advantage of the calculator's history feature to quickly recall previous calculations without re-entering the entire function.
C. Common pitfalls to avoid when using function value calculators
• Input errors: Double-check your input to ensure that the function is entered correctly, including parentheses, operators, and variables.
• Avoid rounding errors: Be mindful of rounding errors, especially when working with functions that involve repeated calculations or very large or small numbers.
• Understand calculator limitations: While function value calculators are powerful tools, they may have limitations in terms of the types of functions they can handle or the precision of their
calculations. Be aware of these limitations and seek alternative methods if necessary.
Conclusion & Best Practices: Ensuring Accuracy and Understanding
As we conclude our discussion on understanding mathematical functions and how to find function values using calculators, it is important to emphasize the significance of accuracy and understanding in
this process. By following best practices and continuously learning about advancements in calculator technology, individuals can ensure that they are obtaining accurate function values and utilizing
these tools effectively.
A Recap of the importance of knowing how to find function values accurately
Accuracy is paramount when it comes to mathematical functions. The values obtained from function calculators are used in various fields such as engineering, finance, and science, where precision is
crucial. Understanding how to find function values accurately ensures that the results obtained are reliable and can be used with confidence in real-world applications.
Summary of best practices when using function value calculators
• Double-checking results: It is always a good practice to double-check the results obtained from function value calculators. This can help in identifying any potential errors or discrepancies in
the calculations.
• Understanding limitations of the tools: Function value calculators have their limitations, and it is important to be aware of these. For example, certain calculators may have restrictions on the
range of values they can handle or the types of functions they can evaluate. Understanding these limitations can help in using the calculators more effectively.
Final thoughts on continuous learning and staying updated with advancements in calculator technology
Continuous learning is essential in the field of mathematics and technology. Staying updated with advancements in calculator technology allows individuals to leverage the latest tools and features
for finding function values. By keeping abreast of new developments, one can enhance their skills and efficiency in using function value calculators. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-find-function-value-calculator","timestamp":"2024-11-11T18:06:01Z","content_type":"text/html","content_length":"228540","record_id":"<urn:uuid:67557e54-2c3e-4243-982b-71995626a0b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00470.warc.gz"} |
Tutorial on boolean algebra
Algebra Tutorials! Saturday 2nd of November
tutorial on boolean algebra
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Rotating a Parabola variable | free lessons in intermediate algebra | free math programs grade 9 | sat maths printable worksheets special triangles | solved examples of sales tax
Multiplying Fractions
Finding Factors
Miscellaneous Equations Author Message
Mixed Numbers and
Improper Fractions nakc Posted: Thursday 28th of Dec 19:41
Systems of Equations in Can someone help me? I am in deep difficulty. It’s about tutorial on boolean algebra. I tried to find somebody in my neighborhood who can
Two Variables help me with multiplying matrices, rational inequalities and subtracting exponents. But I failed . I also know that it will be hard for me
Literal Numbers to meet the cost . My exams are coming up shortly. What should I do? Someone out there who can assist me?
Adding and Subtracting
Polynomials Registered:
Subtracting Integers 15.10.2004
Simplifying Complex From:
Decimals and Fractions
Multiplying Integers
Logarithmic Functions Vofj Timidrov Posted: Saturday 30th of Dec 08:18
Multiplying Monomials Dear Friend , don't get panties in a bunch. Check out https://gre-test-prep.com/graphing-exponential-functions.html, https://
Mixed gre-test-prep.com/numbers.html and https://gre-test-prep.com/solving-quadratic-equations.html. There is a utility by name Algebrator
The Square of a Binomial available at all the three websites . This tool would provide all the details that you would require on the title Remedial Algebra. But,
Factoring Trinomials ensure that you read through all the lessons intently .
The Pythagorean Theorem Registered:
Solving Radical 06.07.2001
Equations in One From: Bulgaria
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers Ashe Posted: Saturday 30th of Dec 14:11
Solving Quadratic I have tried out several software. I would confidently say that Algebrator has assisted me to come to grips with my problems on adding
Equations Using the functions, gcf and leading coefficient. All I did was to just key in the problem. The answer showed up almost immediately showing all the
Quadratic Formula steps to the solution . It was quite simple to follow. I have relied on this for my algebra classes to figure out College Algebra and Basic
Solving Quadratic Math. I would highly suggest you to try out Algebrator.
Equations Registered:
Algebra 08.07.2001
Order of Operations From:
Dividing Complex Numbers
The Appearance of a
Polynomial Equation thicxolmed01 Posted: Monday 01st of Jan 11:20
Standard Form of a Line Algebrator is a very user friendly software and is surely worth a try. You will also find several interesting stuff there. I use it as
Positive Integral reference software for my math problems and can swear that it has made learning math more fun .
Dividing Fractions
Solving Linear Systems Registered:
of Equations by 16.05.2004
Elimination From: Welly, NZ
Multiplying and Dividing
Square Roots
Functions and Graphs zerokevnz Posted: Wednesday 03rd of Jan 09:42
Dividing Polynomials Oh really? Marvelous. You mean it’s that uncomplicated ? I must definitely try it. Please tell me where I can get hold of this program?
Solving Rational
Use of Parentheses or Registered:
Brackets (The 23.04.2002
Distributive Law) From:
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing Dnexiam Posted: Thursday 04th of Jan 12:30
Multiplying Decimals Yes I’m sure. This is tried and tested. Here: https://gre-test-prep.com/polynomials-1.html. Try to make use of it. You’ll be improving your
Use of Parentheses or solving abilities way faster than just by reading tutorials.
Brackets (The
Distributive Law)
Simplifying Complex Registered:
Fractions 1 25.01.2003
Adding Fractions From: City 17
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
tutorial on boolean algebra
Related topics:
Home free simple elementary algebra worksheets | ninth grade math study books | solving quadratic equations by factoring calculator | calculating the value of a
Rotating a Parabola variable | free lessons in intermediate algebra | free math programs grade 9 | sat maths printable worksheets special triangles | solved examples of sales tax
Multiplying Fractions
Finding Factors
Miscellaneous Equations Author Message
Mixed Numbers and
Improper Fractions nakc Posted: Thursday 28th of Dec 19:41
Systems of Equations in Can someone help me? I am in deep difficulty. It’s about tutorial on boolean algebra. I tried to find somebody in my neighborhood who can
Two Variables help me with multiplying matrices, rational inequalities and subtracting exponents. But I failed . I also know that it will be hard for me
Literal Numbers to meet the cost . My exams are coming up shortly. What should I do? Someone out there who can assist me?
Adding and Subtracting
Polynomials Registered:
Subtracting Integers 15.10.2004
Simplifying Complex From:
Decimals and Fractions
Multiplying Integers
Logarithmic Functions Vofj Timidrov Posted: Saturday 30th of Dec 08:18
Multiplying Monomials Dear Friend , don't get panties in a bunch. Check out https://gre-test-prep.com/graphing-exponential-functions.html, https://
Mixed gre-test-prep.com/numbers.html and https://gre-test-prep.com/solving-quadratic-equations.html. There is a utility by name Algebrator
The Square of a Binomial available at all the three websites . This tool would provide all the details that you would require on the title Remedial Algebra. But,
Factoring Trinomials ensure that you read through all the lessons intently .
The Pythagorean Theorem Registered:
Solving Radical 06.07.2001
Equations in One From: Bulgaria
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers Ashe Posted: Saturday 30th of Dec 14:11
Solving Quadratic I have tried out several software. I would confidently say that Algebrator has assisted me to come to grips with my problems on adding
Equations Using the functions, gcf and leading coefficient. All I did was to just key in the problem. The answer showed up almost immediately showing all the
Quadratic Formula steps to the solution . It was quite simple to follow. I have relied on this for my algebra classes to figure out College Algebra and Basic
Solving Quadratic Math. I would highly suggest you to try out Algebrator.
Equations Registered:
Algebra 08.07.2001
Order of Operations From:
Dividing Complex Numbers
The Appearance of a
Polynomial Equation thicxolmed01 Posted: Monday 01st of Jan 11:20
Standard Form of a Line Algebrator is a very user friendly software and is surely worth a try. You will also find several interesting stuff there. I use it as
Positive Integral reference software for my math problems and can swear that it has made learning math more fun .
Dividing Fractions
Solving Linear Systems Registered:
of Equations by 16.05.2004
Elimination From: Welly, NZ
Multiplying and Dividing
Square Roots
Functions and Graphs zerokevnz Posted: Wednesday 03rd of Jan 09:42
Dividing Polynomials Oh really? Marvelous. You mean it’s that uncomplicated ? I must definitely try it. Please tell me where I can get hold of this program?
Solving Rational
Use of Parentheses or Registered:
Brackets (The 23.04.2002
Distributive Law) From:
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing Dnexiam Posted: Thursday 04th of Jan 12:30
Multiplying Decimals Yes I’m sure. This is tried and tested. Here: https://gre-test-prep.com/polynomials-1.html. Try to make use of it. You’ll be improving your
Use of Parentheses or solving abilities way faster than just by reading tutorials.
Brackets (The
Distributive Law)
Simplifying Complex Registered:
Fractions 1 25.01.2003
Adding Fractions From: City 17
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and
Improper Fractions
Systems of Equations in
Two Variables
Literal Numbers
Adding and Subtracting
Subtracting Integers
Simplifying Complex
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical
Equations in One
Multiplying Binomials
Using the FOIL Method
Imaginary Numbers
Solving Quadratic
Equations Using the
Quadratic Formula
Solving Quadratic
Order of Operations
Dividing Complex Numbers
The Appearance of a
Polynomial Equation
Standard Form of a Line
Positive Integral
Dividing Fractions
Solving Linear Systems
of Equations by
Multiplying and Dividing
Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational
Use of Parentheses or
Brackets (The
Distributive Law)
Multiplying and Dividing
by Monomials
Solving Quadratic
Equations by Graphing
Multiplying Decimals
Use of Parentheses or
Brackets (The
Distributive Law)
Simplifying Complex
Fractions 1
Adding Fractions
Simplifying Complex
Solutions to Linear
Equations in Two
Quadratic Expressions
Completing Squares
Dividing Radical
Rise and Run
Graphing Exponential
Multiplying by a
The Cartesian Coordinate
Writing the Terms of a
Polynomial in Descending
Quadratic Expressions
Solving Inequalities
Solving Rational
Inequalities with a Sign
Solving Linear Equations
Solving an Equation with
Two Radical Terms
Simplifying Rational
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step
Solving Quadratic
Equations Graphically
and Algebraically
Collecting Like Terms
Solving Equations with
Radicals and Exponents
Percent of Change
Powers of ten
(Scientific Notation)
Comparing Integers on a
Number Line
Solving Systems of
Equations Using
Factoring Out the
Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing
Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting
Rational Expressions
with Different
Expressions with
Variables as Exponents
The Quadratic Formula
Writing a Quadratic with
Given Solutions
Simplifying Square Roots
Adding and Subtracting
Square Roots
Adding and Subtracting
Rational Expressions
Combining Like Radical
Solving Systems of
Equations Using
Dividing Polynomials
Graphing Functions
Product of a Sum and a
Solving First Degree
Solving Equations with
Radicals and Exponents
Roots and Powers
Multiplying Numbers
tutorial on boolean algebra
Related topics:
free simple elementary algebra worksheets | ninth grade math study books | solving quadratic equations by factoring calculator | calculating the value of a variable | free lessons in
intermediate algebra | free math programs grade 9 | sat maths printable worksheets special triangles | solved examples of sales tax
Author Message
nakc Posted: Thursday 28th of Dec 19:41
Can someone help me? I am in deep difficulty. It’s about tutorial on boolean algebra. I tried to find somebody in my neighborhood who can help me with multiplying
matrices, rational inequalities and subtracting exponents. But I failed . I also know that it will be hard for me to meet the cost . My exams are coming up shortly.
What should I do? Someone out there who can assist me?
Vofj Timidrov Posted: Saturday 30th of Dec 08:18
Dear Friend , don't get panties in a bunch. Check out https://gre-test-prep.com/graphing-exponential-functions.html, https://gre-test-prep.com/numbers.html and https:/
/gre-test-prep.com/solving-quadratic-equations.html. There is a utility by name Algebrator available at all the three websites . This tool would provide all the
details that you would require on the title Remedial Algebra. But, ensure that you read through all the lessons intently .
From: Bulgaria
Ashe Posted: Saturday 30th of Dec 14:11
I have tried out several software. I would confidently say that Algebrator has assisted me to come to grips with my problems on adding functions, gcf and leading
coefficient. All I did was to just key in the problem. The answer showed up almost immediately showing all the steps to the solution . It was quite simple to follow. I
have relied on this for my algebra classes to figure out College Algebra and Basic Math. I would highly suggest you to try out Algebrator.
thicxolmed01 Posted: Monday 01st of Jan 11:20
Algebrator is a very user friendly software and is surely worth a try. You will also find several interesting stuff there. I use it as reference software for my math
problems and can swear that it has made learning math more fun .
From: Welly, NZ
zerokevnz Posted: Wednesday 03rd of Jan 09:42
Oh really? Marvelous. You mean it’s that uncomplicated ? I must definitely try it. Please tell me where I can get hold of this program?
Dnexiam Posted: Thursday 04th of Jan 12:30
Yes I’m sure. This is tried and tested. Here: https://gre-test-prep.com/polynomials-1.html. Try to make use of it. You’ll be improving your solving abilities way
faster than just by reading tutorials.
From: City 17
Author Message
nakc Posted: Thursday 28th of Dec 19:41
Can someone help me? I am in deep difficulty. It’s about tutorial on boolean algebra. I tried to find somebody in my neighborhood who can help me with multiplying matrices,
rational inequalities and subtracting exponents. But I failed . I also know that it will be hard for me to meet the cost . My exams are coming up shortly. What should I do?
Someone out there who can assist me?
Vofj Timidrov Posted: Saturday 30th of Dec 08:18
Dear Friend , don't get panties in a bunch. Check out https://gre-test-prep.com/graphing-exponential-functions.html, https://gre-test-prep.com/numbers.html and https://
gre-test-prep.com/solving-quadratic-equations.html. There is a utility by name Algebrator available at all the three websites . This tool would provide all the details that you
would require on the title Remedial Algebra. But, ensure that you read through all the lessons intently .
From: Bulgaria
Ashe Posted: Saturday 30th of Dec 14:11
I have tried out several software. I would confidently say that Algebrator has assisted me to come to grips with my problems on adding functions, gcf and leading coefficient.
All I did was to just key in the problem. The answer showed up almost immediately showing all the steps to the solution . It was quite simple to follow. I have relied on this
for my algebra classes to figure out College Algebra and Basic Math. I would highly suggest you to try out Algebrator.
thicxolmed01 Posted: Monday 01st of Jan 11:20
Algebrator is a very user friendly software and is surely worth a try. You will also find several interesting stuff there. I use it as reference software for my math problems
and can swear that it has made learning math more fun .
From: Welly, NZ
zerokevnz Posted: Wednesday 03rd of Jan 09:42
Oh really? Marvelous. You mean it’s that uncomplicated ? I must definitely try it. Please tell me where I can get hold of this program?
Dnexiam Posted: Thursday 04th of Jan 12:30
Yes I’m sure. This is tried and tested. Here: https://gre-test-prep.com/polynomials-1.html. Try to make use of it. You’ll be improving your solving abilities way faster than
just by reading tutorials.
From: City 17
Posted: Thursday 28th of Dec 19:41
Can someone help me? I am in deep difficulty. It’s about tutorial on boolean algebra. I tried to find somebody in my neighborhood who can help me with multiplying matrices, rational inequalities and
subtracting exponents. But I failed . I also know that it will be hard for me to meet the cost . My exams are coming up shortly. What should I do? Someone out there who can assist me?
Posted: Saturday 30th of Dec 08:18
Dear Friend , don't get panties in a bunch. Check out https://gre-test-prep.com/graphing-exponential-functions.html, https://gre-test-prep.com/numbers.html and https://gre-test-prep.com/
solving-quadratic-equations.html. There is a utility by name Algebrator available at all the three websites . This tool would provide all the details that you would require on the title Remedial
Algebra. But, ensure that you read through all the lessons intently .
Posted: Saturday 30th of Dec 14:11
I have tried out several software. I would confidently say that Algebrator has assisted me to come to grips with my problems on adding functions, gcf and leading coefficient. All I did was to just
key in the problem. The answer showed up almost immediately showing all the steps to the solution . It was quite simple to follow. I have relied on this for my algebra classes to figure out College
Algebra and Basic Math. I would highly suggest you to try out Algebrator.
Posted: Monday 01st of Jan 11:20
Algebrator is a very user friendly software and is surely worth a try. You will also find several interesting stuff there. I use it as reference software for my math problems and can swear that it
has made learning math more fun .
Posted: Wednesday 03rd of Jan 09:42
Oh really? Marvelous. You mean it’s that uncomplicated ? I must definitely try it. Please tell me where I can get hold of this program?
Posted: Thursday 04th of Jan 12:30
Yes I’m sure. This is tried and tested. Here: https://gre-test-prep.com/polynomials-1.html. Try to make use of it. You’ll be improving your solving abilities way faster than just by reading | {"url":"https://gre-test-prep.com/algebra-1-practice-test/exponent-rules/tutorial-on-boolean-algebra.html","timestamp":"2024-11-02T20:25:54Z","content_type":"text/html","content_length":"117390","record_id":"<urn:uuid:5b27e349-95b9-412a-a758-55e9b2dd58a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00744.warc.gz"} |
Vol. 45, No. 2, 2019
HOUSTON JOURNAL OF
Electronic Edition Vol. 45, No. 2, 2019
Editors: D. Bao (San Francisco, SFSU), D. Blecher (Houston), B. G. Bodmann (Houston), H. Brezis (Paris and Rutgers), B. Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke
(LIAFA, Paris7), C. Hagopian (Sacramento), A. Haynes (Houston), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), W. B. Johnson (College Station), M. Rojas (College Station), Min Ru (Houston), S.W.
Semmes (Rice).
Managing Editors: B. G. Bodmann and K. Kaiser (Houston)
Houston Journal of Mathematics
Juan Climent Vidal, Universitat de València, Departament de Lògica i Filosofia de la Ciència, Av. de Blasco Ibáñez, 30-7a, 46010, València, Spain. (Juan.B.Climent@uv.es) and Enric Cosme Llópez,
Université de Lyon, CNRS, ENS de Lyon, UCB Lyon 1, Laboratoire de L'Informatique du Parallélisme, 46 allée d'Italie, 69364, Lyon, France. (Enric.Cosme-Llopez@ens-lyon.fr).
Eilenberg theorems for many-sorted formations, pp. 321-369.
ABSTRACT. A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after
defining, for a fixed set of sorts S and a fixed S-sorted signature Σ, the concepts of formation of congruences with respect to Σ and of formation of Σ-algebras, we prove that the algebraic lattices
of all Σ-congruence formations and of all Σ-algebra formations are isomorphic, which is an Eilenberg’s type theorem. Moreover, under a suitable condition on the free Σ-algebras and after defining the
concepts of formation of congruences of finite index with respect to Σ, of formation of finite Σ-algebras, and of formation of regular languages with respect to Σ, we prove that the algebraic
lattices of all Σ-finite index congruence formations, of all Σ-finite algebra formations, and of all Σ-regular language formations are isomorphic, which is also an Eilenberg’s type theorem.
Asir, T., Department of Mathematics, DDE, Madurai Kamaraj University, Madurai 625 021, Tamil Nadu, India (asirjacob75@gmail.com), Maimani, H. R., Mathematics Section, Department of Basic Sciences,
Shahid Rajaee Teacher Training University, P.O. Box 16785-163, Tehran, Iran (maimani@ipm.ir), Pournaki, M. R., Department of Mathematical Sciences, Sharif University of Technology, P.O. Box
11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran (pournaki@ipm.ir), and Tamizh Chelvam, T., Department of
Mathematics, Manonmaniam Sundaranar University, Tirunelveli 627 012, Tamil Nadu, India (tamche59@gmail.com).
Some bounds for the genus of a class of graphs arising from rings, pp. 371-384.
ABSTRACT. Let R be a commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is R \ J(R), and two distinct
vertices x and y are adjacent if and only if 1-xy is not a unit in R. In this paper, some bounds for the genus of Jacobson graphs are obtained. As an application, all commutative Artinian rings with
nonzero identity whose Jacobson graphs are toroidal is classified up to isomorphism by a similar result for finite case. Finally, we obtain an isomorphism relation between two Jacobson graphs.
Nunokawa, Mamoru, University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan (mamoru_nuno@doctor.nifty.jp), Sokół, Janusz, University of Rzeszów, Faculty of Mathematics and Natural
Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland (jsokol@ur.edu.pl), and Trąbka-Więcław, Katarzyna, Lublin University of Technology, Mechanical Engineering Faculty, ul. Nadbystrzycka 36, 20-618
Lublin, Poland (k.trabka@pollub.pl).
New sufficient conditions for strong starlikeness, pp. 385-393.
ABSTRACT. This paper determines new sufficient conditions for strong starlikeness and some related properties. The proof rests on several genetralizations and corollaries from Nunokawa's lemma, On
the Order of Strongly Starlikeness of Strongly Convex Functions, Proc. Japan Acad. 69, Ser. A (1993) 234-237.
Nan Wu, Department of Mathematics, School of Science, China University of Mining and Technology (Beijing), Beijing,100083, People's Republic of China (wunan2007@163.com).
Deviations and spreads of holomorphic curves of finite lower order, pp. 395-411.
ABSTRACT. In this paper, we consider the relation between the number of maximum modulus points, spread and growth of a holomorphic curve. We use the method of I. I. Marchenko and E. Ciechanowicz to
generalize their results of meromorphic functions to holomorphic curves.
Taylor, Michael, University of North Carolina, Chapel Hill NC 27599 (met@math.unc.edu).
The Weierstrass ℘-function as a distribution on a complex torus, and its Fourier Series, pp. 413-429.
ABSTRACT. We treat the Weierstrass ℘-function associated to a lattice in the complex plane as a principal value distribution on the quotient torus and compute its Fourier coefficients. The
computation of these coefficients for nonzero frequencies is straightforward, but quite pretty. The "constant term" is more mysterious. It leads to a non-absolutely convergent doubly infinite series.
This can be regarded as a version of an Eisenstein series, though as we discuss in Section 4 it differs from the "Eisenstein summation" of the series, as treated in Weil's monograph on elliptic
functions. Material from Section 3 on the Fourier series of elliptic functions arising from the Weierstrass zeta function leads to a formula connecting our sum with the Eisenstein series treated in
Weil's text, and thereby yields a rapidly convergent approximation to the constant term.
Ping Li, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P. R. China (pli@ustc.edu.cn), Wei-Ran Lü, College of Science, China University of
Petroleum, Qingdao, Shandong, 266580, P. R. China (luwr@upc.edu.cn), and Chung-Chun Yang, Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, P. R. China (maccyang@163.com).
Entire solutions of certain types of nonlinear differential equations, pp. 431-437.
ABSTRACT. By utilizing classical Laguerre's theorem, we can resolve the entire solutions of nonlinear differential equations of the form: f(z)f''(z)=(p[1](z) exp(α[1]z)+p[2](z) exp(α[2]z))^2, where p
[1](z) and p[2](z) are nonzero polynomials, and α[1], α[2] are distinct nonzero constants. The results of this paper extend or generalize a result of Titchmarsh. An example is provided to show that
the results in this paper, in a sense, are best possible.
Jianjun Zhang, Mathematics and Information Technology School, Jiangsu Second Normal University, Beijing West Road 77, Nanjing, 210013, P.R.China, (zhangjianjun1982@163.com), Xiaoqing Lu
(corresponding author), Mathematics and Information Technology School, Jiangsu Second Normal University, Beijing West Road 77, Nanjing, 210013, P.R.China (luxiaoqing1984@126.com), and Liangwen Liao,
Department of Mathematics, Nanjing University, Hankou Road 22, Nanjing, 210093, P.R.China (maliao@nju.edu.cn).
On exact transcendental meromorphic solutions of nonlinear complex differential equations, pp. 439-453.
ABSTRACT. In this paper, we will deal with the existence and the form of transcendental meromorphic solutions of nonlinear differential equation f^n+Q[d](z,f)=p[1](z)e^α[1](z)+p[2](z)e^α[2](z),where
n≥ 4 is an integer, Q[d](z,f) is a special differential polynomial in f of degree d = n-1 with rational functions as its coefficients, p[1], p[2] are non-vanishing rational functions and α[1],α[2]
are nonconstant polynomials.In particular, we can show that Conjecture 1 (P. Li and C. C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations. J. Math.
Anal. Appl. 320 (2006), 827-835.) is true when Q[d](z,f) has a special form.
Brendan Guilfoyle, School of Science, Technology, Engineering and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland (brendan.guilfoyle@ittralee.ie) and Wilhelm
Klingenberg, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom (wilhelm.klingenberg@durham.ac.uk).
A global version of a classical result of Joachimsthal, pp. 455-467.
ABSTRACT. A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is
also a line of curvature of the other surface. In this note we prove the following global analogue of this result. Suppose that two closed convex surfaces intersect with constant angle along a curve
that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both non-orientable. We prove this by characterizing
the constant angle intersection of two surfaces in Euclidean 3-space as the intersection of a Lagrangian surface and a foliated hypersurface in the space of oriented lines, endowed with its canonical
neutral Kähler structure. This establishes a relationship between the principal directions of the two surfaces along the intersection curve in Euclidean space. A winding number argument yields the
result. The method of proof is motivated by topology and, in particular, the slice problem for curves in the boundary of a 4-manifold.
Sun, Zonghan, Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China (sun-zh13@mails.tsinghua.edu.cn) , (bipfiic2008xj@163.com), and Zhang, Guangyuan, Department of
Mathematical Sciences, Tsinghua University, Beijing, P. R. China (gyzhang@mail.tsinghua.edu.cn).
The properties of extremal surfaces in Ahlfors' theory of covering surfaces, pp. 469-495.
ABSTRACT. Ahlfors' second fundamental theorem in Ahlfors' theory of covering surfaces claims that for each fixed set E[q] of q (q>2) extended complex numbers (called the special points), the
spherical area of a covering surface is dominated by the number of times when this surface assumes the special points, and a constant h multiple of the spherical perimeter of this surface. The
optimal value of the constant h is called Ahlfors' constant H[0](E[q]), which depends on E[q] in a rather complicated way. One may only consider the surfaces which don't assume special points. In
this case, the optimal value of the constant h is denoted by h[0](E[q]), which also depends on E[q] in a rather complicated way. Few properties of H[0](E[q]) and h[0](E[q]) are known yet, and Zhang
(2012) have determined h[0]({-1,0,1})=4.034.... In order to determine H[0](E[q]) and h[0](E[q]) relatively conveniently, among all surfaces with a fixed bound of perimeters, we should consider the
surfaces with the maximal area, which are called the extremal surfaces. In this paper, we prove that extremal surfaces have many good properties. For example, the boundary of an extremal surface
could always be partitioned into finitely many convex circular arcs, with a common geodesic curvature (simply called the boundary geodesic curvature), and the end points of each of these circular
arcs must be in E[q]. The main result of this paper is an equality that H[0](E[q]) or h[0](E[q]) is exactly (q-2) divided by the limit of the boundary geodesic curvatures of extremal surfaces. This
relation also gives a fast numerical algorithm to compute h[0]({-1,0,1})=4.034...
Hamed Najafi, Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran (hamednajafi20@gmail.com).
A converse of the characterization of operator geometric means, pp. 497-508.
ABSTRACT. In this paper, we used the closed convex hull of unitary orbit of positive operators and completely positive linear maps to investigate reverse of the characterization of operator geometric
means by positive block matrices.
Rim Abid, University of Tunis-El Manar, 2092-El Manar, Tunisia, and Karim Boulabiar, University of Tunis-El Manar, 2092-El Manar, Tunisia (karim.boulabiar@ipest.rnu.tn).
Algebras of disjointness preserving operatcors on Banach lattices, pp. 509-524.
ABSTRACT. Let A be an algebra of disjointness preserving operators on a Banach lattice X. We shall characterize the set of all nilpotent operators in A and we will deduce that if A is semiprime
(i.e., with no nonzero nilpotent elements) then A is automatically commutative. This will lead us to show that if A is semiprime then A has an isometrically-isomorphic copy in the center of some
Banach lattice.
Stammeier, Nicolai, Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0216 Oslo, Norway (nicolsta@math.uio.no).
Topological freeness for *-commuting covering maps, pp. 525-551.
ABSTRACT. We prove a close connection between *-commutativity and independence of group endomorphisms as considered by Cuntz-Vershik. This motivates the study of a family of *-commuting surjective
local homeomorphisms of a compact Hausdorff space. Inspired by Ledrappier's shift, we describe interesting new examples related to cellular automata. To every such family we associate a universal
C*-algebra that we then identify as the Cuntz-Nica-Pimsner algebra of a product system of Hilbert bimodules. This allows us to extend a result of Meier-Carlsen and Silvestrov which yields an
application for irreversible algebraic dynamical systems.
Beanland, Kevin, Washington and Lee University, Lexington, VA 23220 (beanlandk@wlu.edu) and Kania, Tomasz, University of Warwick, Coventry, UK (tomasz.marcin.kania@gmail.com) and Laustsen, Niels
Jakob, Fylde College, Lancaster University, Lancaster, UK (n.laustsen@lancaster.ac.uk).
The algebras of bounded operators on the Tsirelson and Baernstein spaces are not Grothendieck spaces, pp. 553-566.
ABSTRACT. We present two new examples of reflexive Banach spaces X for which the associated Banach algebra B(X) of bounded operators on X is not a Grothendieck space, namely X = T (the Tsirelson
space) and X = B[p] (the pth Baernstein space) for 1<p< infinity.
Timothy Ferguson, Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487 (tjferguson1@ua.edu).
Bounds on integral means of Bergman projections and their derivatives, pp. 567-588.
ABSTRACT. We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman
space norms of derivatives of Bergman projections in terms of weighted L^p norms of certain derivatives of the original function in the &theta direction. These results easily imply the well known
result that the Bergman projection is bounded from the Sobolev space W^k,p into itself for 1 < p < ∞. We also apply our results to derive certain regularity results involving extremal problems in
Bergman spaces. Lastly, we construct a function that approaches 0 uniformly at the boundary of the unit disc but whose Bergman projection is not in H^2.
Mendoza, José M., Universidad Federal de São Carlos, São Carlos, Brazil (josearanda@dm.ufscar.br).
Existence of solutions for a nonhomogeneous semilinear fractional Laplacian problems, pp. 589-599.
ABSTRACT. In this paper we give existence results for a nonhomogeneous semilinear fractional laplacian problems with local coercivity in euclidean bounded domains using variational methods.
Włodzimierz J. Charatonik and Sahika Sahan, Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 West 12th St., Rolla, MO, 65409-0020 (wjcharat@mst.edu},
Zero-dimensional spaces homeomorphic to their Cartesian squares, pp. 601-608.
ABSTRACT. We show that there exists uncountably many zero-dimensional compact metric spaces homeomorphic to their cartesian squares as well as their n-fold symmetric products.
Roshan Adikari and Wayne Lewis, Department of Mathematics and Statistics; Texas Tech University, Lubbock, Texas 79409 (roshan.adikari@ttu.edu), (wayne.lewis@ttu.edu).
Endpoints of nondegenerate hereditarily decomposable chainable continua, pp. 609-624.
ABSTRACT. We show that a nondegenerate hereditarily decomposable chainable continuum must have a pair of opposite endpoints and use this result to investigate more on endpoints of such continua.
Liang-Xue Peng (Corresponding author), Beijing University of Technology, Beijing 100124, China (pengliangxue@bjut.edu.cn) and Pei Zhang, Beijing University of Technology, Beijing 100124, China (
Some properties of bounded sets in certain topological spaces, pp. 625-646.
ABSTRACT. In the first part of this article we give some sufficient conditions under which a bounded set in a topological space (paratopological group) X is strongly bounded in X (p-bounded in X for
every p∈ω^∗). We show that if X is a first-countable topological space, then every bounded subset of X is strongly bounded in X. If G is a paratopological group and for every U∈Ν(e) there exists a
continuous homomorphism p[U]:G→H[U] onto a first-countable Hausdorff paratopological group H[U] such that p[U]^-1(Cl(V[U]))⊂ Cl(U) for some neighborhood V[U] of the neutral element of H[U], then any
bounded set in G is strongly bounded in G. We also give some sufficient conditions under which a semitopological group G satisfies that for any U∈Ν(e) there exists a continuous homomorphism p[U]:G→H
[U] onto a first-countable Hausdorff semitopological group H[U] such that p[U]^-1(Cl(V[U]))⊂Cl(U) for some neighborhood V[U] of the neutral element of H[U]. In the last part of this note, we give
some equivalent conditions for a bounded set in a Tychonoff topological space. We finally show that every pseudocompact submetacompact space is weakly Lindelöf. | {"url":"https://www.math.uh.edu/~hjm/Vol45-2.html","timestamp":"2024-11-09T00:32:57Z","content_type":"text/html","content_length":"23553","record_id":"<urn:uuid:f0a514e1-0197-4c08-a569-cbe13a833b23>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00413.warc.gz"} |
and R
Pose and Resectioning¶
Theia contains efficient and robust implementations of the following pose and resectioning algorithms. We attempted to make each method as general as possible so that users were not tied to Theia
data structures to use the methods. The interface for all pose methods uses Eigen types for feature positions, 3D positions, and pose rotations and translations.
Perspective Three Point (P3P)¶
bool PoseFromThreePoints(const Eigen::Vector2d feature_position[3], const Eigen::Vector3d world_point[3], std::vector<Eigen::Matrix3d> *solution_rotations, std::vector<Eigen::Vector3d> *
Computes camera pose using the three point algorithm and returns all possible solutions (up to 4). Follows steps from the paper “A Novel Parameterization of the Perspective-Three-Point
Problem for a direct computation of Absolute Camera position and Orientation” by [Kneip]. This algorithm has been proven to be up to an order of magnitude faster than other methods. The
output rotation and translation define world-to-camera transformation.
feature_position: Image points corresponding to model points. These should be calibrated image points as opposed to pixel values.
world_point: 3D location of features.
solution_rotations: the rotation matrix of the candidate solutions
solution_translation: the translation of the candidate solutions
returns: Whether the pose was computed successfully, along with the output parameters rotation and translation filled with the valid poses.
Five Point Relative Pose¶
bool FivePointRelativePose(const Eigen::Vector2d image1_points[5], const Eigen::Vector2d image2_points[5], std::vector<Eigen::Matrix3d> *rotation, std::vector<Eigen::Vector3d> *translation)¶
Computes the relative pose between two cameras using 5 corresponding points. Algorithm is implemented based on “Recent Developments on Direct Relative Orientation” by [Stewenius5pt]. This
algorithm is known to be more numerically stable while only slightly slower than the [Nister] method. The rotation and translation returned are defined such that \(E=[t]_{\times} * R\) and \
(y^\top * E * x = 0\) where \(y\) are points from image2 and \(x\) are points from image1.
image1_points: Location of features on the image plane of image 1.
image2_points: Location of features on the image plane of image 2.
returns: Output the number of poses computed as well as the relative rotation and translation.
Four Point Algorithm for Homography¶
bool FourPointHomography(const std::vector<Eigen::Vector2d> &image_1_points, const std::vector<Eigen::Vector2d> &image_2_points, Eigen::Matrix3d *homography)¶
Computes the 2D homography mapping points in image 1 to image 2 such that: \(x' = Hx\) where \(x\) is a point in image 1 and \(x'\) is a point in image 2. The algorithm implemented is the DLT
algorithm based on algorithm 4.2 in [HartleyZisserman].
image_1_points: Image points from image 1. At least 4 points must be passed in.
image_2_points: Image points from image 2. At least 4 points must be passed in.
homography: The computed 3x3 homography matrix.
Eight Point Algorithm for Fundamental Matrix¶
bool EightPointFundamentalMatrix(const std::vector<Eigen::Vector2d> &image_1_points, const std::vector<Eigen::Vector2d> &image_2_points, Eigen::Matrix3d *fundamental_matrix)¶
Computes the fundamental matrix relating image points between two images such that \(x' F x = 0\) for all correspondences \(x\) and \(x'\) in images 1 and 2 respectively. The normalized eight
point algorithm is a speedy estimation of the fundamental matrix (Alg 11.1 in [HartleyZisserman]) that minimizes an algebraic error.
image_1_points: Image points from image 1. At least 8 points must be passed in.
image_2_points: Image points from image 2. At least 8 points must be passed in.
fundamental_matrix: The computed fundamental matrix.
returns: true on success, false on failure.
Perspective N-Point¶
void DlsPnp(const std::vector<Eigen::Vector2d> &feature_position, const std::vector<Eigen::Vector3d> &world_point, std::vector<Eigen::Quaterniond> *solution_rotation, std::vector<Eigen::Vector3d>
Computes the camera pose using the Perspective N-point method from “A Direct Least-Squares (DLS) Method for PnP” by [Hesch] and Stergios Roumeliotis. This method is extremely scalable and
highly accurate for the PnP problem. A minimum of 4 points are required, but there is no maximum number of points allowed as this is a least-squared approach. Theoretically, up to 27
solutions may be returned, but in practice only 4 real solutions arise and in almost all cases where n >= 6 there is only one solution which places the observed points in front of the camera.
The returned rotation and translations are world-to-camera transformations.
feature_position: Normalized image rays corresponding to model points. Must contain at least 4 points.
points_3d: 3D location of features. Must correspond to the image_ray of the same index. Must contain the same number of points as image_ray, and at least 4.
solution_rotation: the rotation quaternion of the candidate solutions
solution_translation: the translation of the candidate solutions
Four Point Focal Length¶
int FourPointPoseAndFocalLength(const std::vector<Eigen::Vector2d> &feature_positions, const std::vector<Eigen::Vector3d> &world_points, std::vector<Eigen::Matrix<double, 3, 4>> *
Computes the camera pose and unknown focal length of an image given four 2D-3D correspondences, following the method of [Bujnak]. This method involves computing a grobner basis from a
modified constraint of the focal length and pose projection.
feature_position: Normalized image rays corresponding to model points. Must contain at least 4 points.
points_3d: 3D location of features. Must correspond to the image_ray of the same index. Must contain the same number of points as image_ray, and at least 4.
projection_matrices: The solution world-to-camera projection matrices, inclusive of the unknown focal length. For a focal length f and a camera calibration matrix \(K=diag(f, f, 1)\), the
projection matrices returned are of the form \(P = K * [R | t]\).
Five Point Focal Length and Radial Distortion¶
bool FivePointFocalLengthRadialDistortion(const std::vector<Eigen::Vector2d> &feature_positions, const std::vector<Eigen::Vector3d> &world_points, const int num_radial_distortion_params,
std::vector<Eigen::Matrix<double, 3, 4>> *projection_matrices, std::vector<std::vector<double>> *radial_distortions)¶
Compute the absolute pose, focal length, and radial distortion of a camera using five 3D-to-2D correspondences [Kukelova]. The method solves for the projection matrix (up to scale) by using a
cross product constraint on the standard projection equation. This allows for simple solution to the first two rows of the projection matrix, and the third row (which contains the focal
length and distortion parameters) can then be solved with SVD on the remaining constraint equations from the first row of the projection matrix. See the paper for more details.
feature_positions: the 2D location of image features. Exactly five features must be passed in.
world_points: 3D world points corresponding to the features observed. Exactly five points must be passed in.
num_radial_distortion_params: The number of radial distortion paramters to
solve for. Must be 1, 2, or 3.
projection_matrices: Camera projection matrices (that encapsulate focal
length). These solutions are only valid up to scale.
radial_distortions: Each entry of this vector contains a vector with the radial distortion parameters (up to 3, but however many were specified in num_radial_distortion_params).
return: true if successful, false if not.
Three Point Relative Pose with a Partially Known Rotation¶
void ThreePointRelativePosePartialRotation(const Eigen::Vector3d &rotation_axis, const Eigen::Vector3d image_1_rays[3], const Eigen::Vector3d image_2_rays[3], std::vector<Eigen::Quaterniond> *
soln_rotations, std::vector<Eigen::Vector3d> *soln_translations)¶
Computes the relative pose between two cameras using 3 correspondences and a known vertical direction as a Quadratic Eigenvalue Problem [SweeneyQEP]. Up to 6 solutions are returned such that
\(x_2 = R * x_1 + t\) for rays \(x_1\) in image 1 and rays \(x_2\) in image 2. The axis that is passed in as a known axis of rotation (when considering rotations as an angle axis). This is
equivalent to aligning the two cameras to a common direction such as the vertical direction, which can be done using IMU data.
Four Point Relative Pose with a Partially Known Rotation¶
void FourPointRelativePosePartialRotation(const Eigen::Vector3d &rotation_axis, const Eigen::Vector3d image_1_origins[4], const Eigen::Vector3d image_1_rays[4], const Eigen::Vector3d
image_2_origins[4], const Eigen::Vector3d image_2_rays[4], std::vector<Eigen::Quaterniond> *soln_rotations, std::vector<Eigen::Vector3d> *soln_translations)¶
Computes the relative pose between two generalized cameras using 4 correspondences and a known vertical direction as a Quadratic Eigenvalue Problem [SweeneyQEP]. A generalized camera is a
camera setup with multiple cameras such that the cameras do not have the same center of projection (e.g., a multi-camera rig mounted on a car). Up to 8 solutions are returned such that \(x_2
= R * x_1 + t\) for rays \(x_1\) in image 1 and rays \(x_2\) in image 2. The axis that is passed in as a known axis of rotation (when considering rotations as an angle axis). This is
equivalent to aligning the two cameras to a common direction such as the vertical direction, which can be done using IMU data.
Two Point Absolute Pose with a Partially Known Rotation¶
int TwoPointPosePartialRotation(const Eigen::Vector3d &axis, const Eigen::Vector3d &model_point_1, const Eigen::Vector3d &model_point_2, const Eigen::Vector3d &image_ray_1, const Eigen::Vector3d
&image_ray_2, Eigen::Quaterniond soln_rotations[2], Eigen::Vector3d soln_translations[2])¶
Solves for the limited pose of a camera from two 3D points to image ray correspondences. The pose is limited in that while it solves for the three translation components, it only solves for a
single rotation around a passed axis.
This is intended for use with camera phones that have accelerometers, so that the ‘up’ vector is known, meaning the other two rotations are known. The effect of the other rotations should be
removed before using this function.
This implementation is intended to form the core of a RANSAC routine, and as such has an optimized interface for this use case.
Computes the limited pose between the 3D model points and the (unit-norm) image rays. Places the rotation and translation solutions in soln_rotations and soln_translations. There are at most
2 solutions, and the number of solutions is returned.
The rotations and translation are defined such that model points are transformed according to \(image_point = Q * model_point + t\)
This function computes the rotation and translation such that the model points, after transformation, lie along the corresponding image_rays. The axis referred to is the axis of rotation
between the camera coordinate system and world (3D point) coordinate system. For most users, this axis will be (0, 1, 0) i.e., the up direction. This requires that the input image rays have
been rotated such that the up direction of the camera coordinate system is indeed equal to (0, 1, 0).
When using this algorithm please cite the paper [SweeneyISMAR2015]. | {"url":"http://theia-sfm.org/pose.html","timestamp":"2024-11-13T05:08:54Z","content_type":"text/html","content_length":"35197","record_id":"<urn:uuid:6a7b2440-4bd3-4ff4-ad18-a7143294743f>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00152.warc.gz"} |
What is an example of a one step equation?
One step equation is an equation that requires only one step to be solved. You only perform a single operation in order to solve or isolate a variable. Examples of one step equations include: 5 + x =
12, x – 3 = 10, 4 + x = -10 etc. To solve this equation, divide both sides of the equation by 3.
What are the 3 steps to solving one step equations?
How to Solve One Step Equations
• 1 Adding or Subtracting to Solve.
• 2 Dividing or Multiplying to Solve.
• 3 Completing Sample Problems.
What is a 2 step equation?
An equation that needs two steps to solve. Example: 5x − 6 = 9. Step 1: Add 6 to both sides: 5x = 15. Step 2: Divide both sides by 5: x = 3. Linear Equations often need two steps to solve.
What is a 1 step problem?
A one-step equation is an algebraic equation you can solve in only one step. You’ve solved the equation when you get the variable by itself, with no numbers in front of it, on one side of the equal
What are the steps in solving the equation?
Solving A Linear Equation: Five Steps To Success. Step 1: Perform any distribution; look for ( ). Step 2: Combine like terms on each side of = sign. Step 3: Add or subtract variable terms to get all
variables on the same side of the = sign.
What are the rules for solving equations?
Solve equations and simplify expressions. In algebra 1 we are taught that the two rules for solving equations are the addition rule and the multiplication/division rule. The addition rule for
equations tells us that the same quantity can be added to both sides of an equation without changing the solution set of the equation.
What is an example of an one step equation?
One step equation is an equation that requires only one step to be solved. You only perform a single operation in order to solve or isolate a variable. Examples of one step equations include: 5 + x =
12, x – 3 = 10, 4 + x = -10 etc. For instance, to solve 5 + x = 12,
What are one step equations?
A one-step equation is an equation that can be solved in one step. There are four types of one-step equations: addition, subtraction, multiplication, and division. | {"url":"https://bookriff.com/what-is-an-example-of-a-one-step-equation/","timestamp":"2024-11-02T11:43:41Z","content_type":"text/html","content_length":"36135","record_id":"<urn:uuid:2689a579-93f8-46db-8fc6-757e18165770>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00706.warc.gz"} |
predator-prey | mages' blog
This article illustrates how ordinary differential equations and multivariate observations can be modelled and fitted with the brms package (Bürkner (2017)) in R1. As an example I will use the well
known Lotka-Volterra model (Lotka (1925), Volterra (1926)) that describes the predator-prey behaviour of lynxes and hares. Bob Carpenter published a detailed tutorial to implement and analyse this
model in Stan and so did Richard McElreath in Statistical Rethinking 2nd Edition (McElreath (2020)).
This evening I will talk about Dynamical systems in R with simecol at the LondonR meeting. Thanks to the work by Thomas Petzoldt, Karsten Rinke, Karline Soetaert and R. Woodrow Setzer it is really
straight forward to model and analyse dynamical systems in R with their deSolve and simecol packages. I will give a brief overview of the functionality using a predator-prey model as an example. This
is of course a repeat of my presentation given at the Köln R user group meeting in March. | {"url":"https://magesblog.com/tags/predator-prey/","timestamp":"2024-11-07T17:26:24Z","content_type":"text/html","content_length":"11727","record_id":"<urn:uuid:1d49549c-dcda-45f2-b1d9-75ed54e70800>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00232.warc.gz"} |
Notes on Global Stress and Hyper-Stress Theories
The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional
counterpart of statics of systems having finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the
manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear
functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space, implies that force functionals may
be represented by objects that generalize the stresses of traditional continuum mechanics.
Publication series
Name Advances in Mechanics and Mathematics
Volume 43
ISSN (Print) 1571-8689
ISSN (Electronic) 1876-9896
• Continuum mechanics
• Differentiable manifold
• Stress
• Hyper-stress
• Global analysis
• Manifold of mappings
• de Rham currents
Dive into the research topics of 'Notes on Global Stress and Hyper-Stress Theories'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/notes-on-global-stress-and-hyper-stress-theories","timestamp":"2024-11-12T06:49:13Z","content_type":"text/html","content_length":"55101","record_id":"<urn:uuid:074f5b1c-9a41-4548-b915-a906e2445c78>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00601.warc.gz"} |
How To Lookup A Number In Between A Range Using Map and Makecontinuous Command - Splunk on Big Data
Home Tips & Tricks How To Lookup A Number In Between A Range Using Map and...
How To Lookup A Number In Between A Range Using Map and Makecontinuous Command
How To Lookup A Number In Between A Range Using Map and Makecontinuous Command
Hi Guys!!!
Today we have come with an interesting trick with two commands i.e. map and makecontinuous to lookup a number between a range.
For showing you this requirement we will use one csv file which we have uploaded as a lookup file named “info.csv” in splunk and the sample data which we have indexed in “test_index” index.
Please, see the below image to see the content of the lookup file named info.csv.
In this lookup file, we have three fields, “start_number”, “end_number” and “location”.
Please, see the below image to see the indexed data.
Here, we are using index “test_index” and sourcetype “test”, where we have our sample data.
We have used the “table” command to show the values of the “number” field in tabular form.
Example: 1
Now, using the “map” command, we will get the values of the “number” field which will fall between the ranges (in between the values “start_number” and “end_number” field) of the lookup file “
info.csv” and also we will get the values of additional field i.e. “location” from the lookup file.
So, let’s see the below query to get the output,
index=test_index sourcetype="test"
| table number
| map search="| inputlookup info.csv | where "end_number" >= $number$ AND "start_number" <= $number$ | eval "number"="$number$" | table location,number,start_number,end_number"
| eval Range=start_number."-".end_number
| fields - start_number,end_number
| stats count, list(number) as number, list(Range) as range by location
First, we have used index “test_index” and sourcetype “test”, where we have our sample data.
We have used the “table” command to show the values of the “number” field in tabular form.
Now we have used,
| map search=”| inputlookup info.csv | where “end_number” >= $number$ AND “start_number” <= $number$ | eval “number”=”$number$” | table location,number,start_number,end_number”
The “map” command works as a looping operator that runs a search repeatedly for each of the input events or results.
map search=”<string>” [This is the syntax of map command]
In the place of string, we have to write the query which we want to run as an ad hoc search to run for each input of the resultset.
Here we have used, “| inputlookup info.csv | where “end_number” >= $number$ AND “start_number” <= $number$ | eval “number”=”$number$” | table location,number,start_number,end_number”
| inputlookup infor.csv -> To get the content of the lookup file “info.csv”
| where “end_number” >= $number$ AND “start_number” <= $number$ | eval “number”=”$number$” -> This will only give the values of number field which is greater than the values of “end_number” field and
less than the value of “start_number” field. Now to get these values in a field named “number” we have used, | eval “number”=”$number$”. And, then we have just “table” command to get the “location”,
“number”, “start_number”, “end_number” fileds’ values in tabular form.
| eval Range=start_number.”-“.end_number -> This will create a new field named Range where we can see the values of the “start_number” field and “end_number” field concatenated with “-”
| fields – start_number,end_number -> This will exclude the “start_number” field and “end_number” [as, we don’t need to show these two fields in the result].
| stats count,list(number) as number,list(Range) as range by location -> This will derive the count and the list of values of the number field and Range field grouped by the field location.
As, you can see, we are getting only the values in the “number” field which fall between the ranges in the “Range” field created from the lookup file (info.csv).
Example: 2
Here, we will show you how to achieve the same thing using “makecontinuous” command. So, let’s start,
index="test_index" sourcetype="test"
| table number
| eval aa="aa"
| append
[| inputlookup info.csv
| table start_number,end_number,location
| eval Range=start_number."-".end_number
| makecontinuous start=0 end=50000 start_number
| filldown end_number,location,Range
| where start_number<=end_number
| rename start_number as number ]
| stats values(*) as * by number
| where isnotnull(aa)
| fields - end_number,aa
| where isnotnull(location)
First, we have used index “test_index” and sourcetype “test”, where we have our sample data.
We have used the “table” command to show the values of the “number” field in tabular form.
Then, we have created a field named “aa” which will contain the value(string) “aa” using the “eval” command.
Then we have used,
“| append
[| inputlookup info.csv
| table start_number,end_number,location
| eval Range=start_number.”-“.end_number
| makecontinuous start=0 end=50000 start_number
| filldown end_number,location,Range
| where start_number<=end_number
| rename start_number as number]”
“append” command we used to append the two queries together. So, let’s see how we inside this query we have used, “makecontinuous” command to get the requirement.
| inputlookup info.csv -> This we used to get the content of the lookup file named “info.csv”.
| table start_number,end_number,location -> This will show the values of “start_number”, “end_number” and “location” field in tabular format.
| eval Range=start_number.”-“.end_number -> This will create a new field named “Range” where we can see the values of the “start_number” field and “end_number” field concatenated with “-”
| makecontinuous start=0 end=50000 start_number -> “makecontinuous” command will fill in the gaps we have in the start_number field by continuing the numbers by incrementing.
[start, end ], set the range or minimum and maximum extents for the values of the field we will use with the command. The data which falls outside of the [start, end] range will be discarded.
| filldown end_number,location,Range -> This will fill the null values in “end_number”, “location”, “Range” fields with the last not null value for corresponding fields.
| where start_number<=end_number -> This will show the results only where the values of the “start_number” field are less than or equal to the field values of the “end_number” field.
| rename start_number as number -> This will rename the “start_number” field as “number”.
| stats values(*) as * by number -> Then by the “stats” command we have sorted two-three fields by the field “number”. So that, the rows which have null values for “end_number”, “location”, “Range”,
“aa” will be discarded.
| where isnotnull(aa) -> This is to discard all the rows where the value of the “aa” field is null.
| fields – end_number,aa -> This is to discard the “end_number” and “aa” fields.
| where isnotnull(location) -> This is to discard all the rows where the value of the “location” field is null.
As you can see, we are getting only the values in the number field which falls between the ranges in the “Range” field created from the lookup file (info.csv) and we are getting the corresponding “
location” field values.
Example: 2.1
Now, if you want to get the results or values of “number” field which does not fall between any range of the “Range” field, you can use the below query,
index="test_index" sourcetype="test"
| table number
| eval aa="aa"
| append
[| inputlookup info.csv
| table start_number,end_number,location
| eval Range=start_number."-".end_number
| makecontinuous start=0 end=50000 start_number
| filldown end_number,location,Range
| where start_number<=end_number
| rename start_number as number ]
| stats values(*) as * by number
| where isnotnull(aa)
| fields - end_number,aa
| where isnull(location)
The explanation of this query is the same as Example: 2, only one difference is there i.e. the last command “| where isnull(location)”, which means we only want to see the rows which contain “null”
values for the “location” field.
If you see the above image, you will understand we are only getting the values of the “number” field which don’t fall between any range of the “Range” field.
Happy Splunking !!
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Wave Power :: Dave Rogers —
I was an ocean engineering major at the Naval Academy. I wasn't an outstanding student by any measure, but I got the degree.
There are things that stuck with me, one of them is "wave power varies as the cube of the height." That is, a wave that is twice as tall has eight times as much power.
Energy is the capacity to do work. Power is the rate at which work is done.
Most ocean waves are wind-generated, and wave height varies linearly with wind velocity. That is, a 10% increase in wind speed yields 10% higher average wave heights.
Wind speed is a function of energy in the atmosphere, temperature and pressure differences in various regions of the atmosphere. A warmer atmosphere has more energy, and so larger differences can
occur, although on a global scale one might think that the average atmospheric temperature is equally greater, so the differences that drive wind would be relatively the same. But, it's complicated
and they're not.
So this report came as little surprise to me.
The analysis revealed that in the era beginning after 1970, California's average winter wave height has increased by 13% or about 0.3 meters (one foot) compared to average winter wave height
between 1931 and 1969. Bromirski also found that between 1996 and 2016 there were about twice as many storm events that produced waves greater than four meters (13 feet) in height along the
California coast compared to the two decades spanning 1949 to 1969.
So (average winter Pacific northwest) wave heights have increased 13% since 1970. That means average wave power has increased 1.13^3, or ~1.44. That's 44%!
That's 44% more power to erode shorelines, move sand. In a rising ocean.
Shoreline erosion is taking place at a much faster rate than it has in the past. We can quibble about whether it's 20%, 30%, or 44%, but no matter what number you want to put on it, you have less
time to "manage" shoreline erosion.
We have to move faster.
Originally posted at Notes From the Underground 09:55 Monday, 7 August 2023 | {"url":"https://b-banzai.micro.blog/2023/08/07/wave-power.html","timestamp":"2024-11-14T20:06:27Z","content_type":"text/html","content_length":"10552","record_id":"<urn:uuid:fb0b1719-0dd9-4795-ada0-6a3b58bf8fc0>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00686.warc.gz"} |
Reasoning and Proof (Justification)
Submitted by Patricia Croskrey on
How many of us ask students to solve a problem and then stop when the student produces a correct answer? Give students a good rich task but don’t ask how they arrived at the answer or how they know
it is correct? This is where reasoning and proof comes in.
I have heard so many teachers (in fact just about all teachers) say, “Students have no number sense!” Well, my question is why not and where do they think number sense comes from (hmmm...a little
self-reflection might be in order)? There is much written about how students come into school with a sense of numbers and problem solving ability and we teach it out of them! We teach process and
procedure without context, without asking students to explain their thinking or how they arrived at an answer, or without asking how they know their answer is correct and we, by doing nothing else,
are teaching number sense right out of our students.
Let’s look at what some of the research says about reasoning and proof. In Elementary and Middle School Mathematics, John Van de Walle says that, “If problem solving is the focus of mathematics,
reasoning is the logical thinking that helps us decide if and why our answers make sense. Students need to develop the habit of providing an argument or rationale as an integral part of every answer.
Justifying answers is a process that enhances conceptual understanding. The habit of providing reasons can begin in kindergarten. However, it is never too late for students to learn the value of
defending ideas through logical argument.” In Adding it Up, researchers call reasoning, “...the glue that holds everything together, the lodestar that guides learning.” Both publications very clearly
and explicitly state that students who disagree about a mathematical answer shouldn’t have to run to the teacher to sort it out, but should be able to use reasoning and justification (proof) to
determine the validity of their solution. Adding it Up also states that it is not sufficient to justify a procedure just once. The development of proficiency occurs over an extended period of time.
Students need to use new concepts and procedures for some time and to explain and justify them by relating them to concepts and procedures that they already understand. For example, it is not
enough for students to do only practice problems on adding fractions after the procedure has been developed. If students are to understand the algorithm, they also need experience in explaining and
justifying it themselves with many different problems and in many different contexts.
I am going to use another fraction example to highlight how the lack of reasoning and proof in our classrooms affects our students. I often see students who are having difficulty comparing and
ordering fractions and what I am about to tell you is a true anecdote. I was working with 4 fifth grade boys on comparing and ordering fractions. I put four fractions up on my white board, 1/4, 2/5,
5/6, and 1/2 and I asked them what they could tell me about the fractions just by looking at them. They could tell me very little. Of course they wanted to cross-multiply (ugh!) and when I told them
that they could only use that process if they could explain the math behind it, that strategy was out. They knew that sixths were the smallest pieces and the halves were the largest pieces, but
beyond that they had nothing. So we spent two or three (maybe even four) 30-minute sessions reasoning through fractions and then proving their sizes with models and number lines. In the end they
could reason through and justify to each other (and me) the order of the fractions from smallest to greatest without performing a paper-pencil “process”. They could explain, for example, that while
sixths were the smallest pieces, 5/6 was the largest fraction because 6/6 would be one whole and 5/6 was only 1/6 away from the whole, a much smaller piece away from its whole than the other three
fractions were away from their wholes. Now that’s what I’m talking about!
Reasoning and proof (or justification) is not only an integral part of problem solving, you can see how it is the “glue” that holds our math together and is also the part of our mathematics that
develops and solidifies our number sense! Please, for the sake of number sense everywhere (and so that teachers will quit asking why their students don’t have any number sense), teach your students
to reason through problems and justify their answers! | {"url":"https://mathplc.com/MathPLCBlogs/reasoning-and-proof-justification","timestamp":"2024-11-05T03:55:34Z","content_type":"text/html","content_length":"31895","record_id":"<urn:uuid:79b5f0e4-7ab0-44e6-be08-222bd6706f03>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00360.warc.gz"} |
History revisited
Climate change? A reality… Both continuous and discrete in geological time
One cannot truthfully claim that climate change on planet earth is just a myth. Fossil and geological evidence in various places have revealed earlier Mediterranean climates as far north as Sweden
and massive glacial ice floes as far south as North Dakota. The causes of such colossal transformations on the earth’s surface have been attributed to various solar and planetary changes. These have
included increased solar activity, decreased volcanic activity, alterations to ocean circulation and even perturbations in the earth’s orbital parameters. At this juncture, much research has
highlighted temperature and precipitation fluctuations as evidence of another impending global climate phenomenon. The heightened rhetoric points in the direction of human intervention into natural
processes as one of its major contributors.
However, recent geological and paleontological research into the origins and the history of the Sahara Desert (the hottest and driest region on the planet) have unearthed some interesting theories
and facts about climate change which go well beyond human influence and impact on the environment.
See: “When the Sahara Desert was Green - Science Documentary 2017” (published September 15, 2017 by Documentary film 2017)
Universal Order or Chaos?
Gottfried Leibniz (1646 - 1716) was a German philosopher, physicist and mathematician. Even as a self-taught mathematician, he is famous for being one of the founders of Calculus, the discoverer of
the binary numbers and the invention of an early version of the calculator, the latter two being precursors to the modern computer. However, it was his personal philosophy that led him to probe into
the dynamics of the universe around him. Leibniz believed that we were living in a deterministic universe whose rhythm was derived from a ‘pre-established harmony.’ This led him to focus on a
formulation of what he called ‘vis viva,’ that unseen ‘life force’ of a moving body, as being a product of its mass times the square of the velocity at which it was moving and that vis viva was
subject to some kind of law of conservation or state of equilibrium. In mathematical terms, Leibniz considered a body’s life force to be = mv^2. This is so reminiscent of Einstein’s relativity
equation between mass and energy which was introduced to the scientific community some two centuries later. E = mc^2 where E represents the energy ( ‘life force’) equivalent to a moving body of mass
m somewhere in the universe and c is the speed of light. In fact, there is a law of conservation built into this equation. It states that the ratio of a body’s energy over its mass is a constant,
implying that energy and mass are in a constant state of equilibrium.
For more on the legacy of Leibniz’s idea of ‘vis viva’ or ‘life force,’ watch the documentary called: Order and Disorder
Note: Here the word ‘chaos’ is defined as the infinity of space or formless matter supposed to have preceded the existence of the ordered universe.
John Forbes Nash Jr.
Mathematician, Economist and Nobel Prize Laureate 1994
It is not often that mathematicians win Nobel Prizes because there is no Nobel Prize in Mathematics. This is the autobiography of one who did. He has 'A Beautiful Mind', the central premise of the
2001 Oscar award winning film of the same name.
Of note is that the Field's Medal, sometimes referred to as the 'Nobel Prize of mathematics' and considered by many to be the most prestigious award a mathematician can receive, was founded as a
legacy of a Canadian mathematician, John Charles Fields.
Resurrecting Napier's Bones
Watch the presentation...
John Napier was a Scottish mathematician and inventor who lived in the latter part of the sixteenth and the early seventeenth centuries. Some of his innovative ideas and devices caused a great stir
at the time.
A Winning Lottery Combination: Mathematics, Psychology and Politics
Speaking of thinking differently, some enterprising MIT students apparently used mathematical thinking to win the Massachusetts’ state lottery consistently from 2005 to 2011. Why were they able to
“beat the System” for such a long period? A clever combination of number theory and geometry – and the ever-pervasive forces of human psychology and politics – may have been the answers to their
How Not to Be Wrong: The Power of Mathematical Thinking (published on June 24, 2015 by The Royal Institution) | {"url":"https://www.athabascau.ca/math-site/historical-perspectives/index.html","timestamp":"2024-11-09T11:19:14Z","content_type":"application/xhtml+xml","content_length":"142055","record_id":"<urn:uuid:07d70781-a441-4ba0-a8cd-e16c0d06a6c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00896.warc.gz"} |
Zipf's Law, ArtEnt Blog Hits
As I look at the hit statistics for the last quarter, I cannot help but wonder how well they fit Zipf’s law (a.k.a. Power Laws, Zipf–Mandelbrot law, discrete Pareto distribution). Zipf’s law states
that the distribution of many ranked things like city populations, country populations, blog hits, word frequency distribution, probability distribution of questions for Alicebot, Wikipedia Hits,
terrorist attacks, the response time of famous scientists, … look like a line when plotted on a log-log diagram. So here are the numbers for my blog hits and, below that, a plot of log(blog hits) vs
log(rank) :
“Deep Support Vector Machines for Regression Problems” 400
Simpson’s paradox and Judea Pearl’s Causal Calculus 223
Standard Deviation of Sample Median 220
100 Most useful Theorems and Ideas in Mathematics 204
Computer Evaluation of the best Historical Chess Players 181
Notes on “A Few Useful Things to Know about Machine Learning” 178
Comet ISON, Perihelion, Mars, and the rule of 13.3 167
Dropout – What happens when you randomly drop half the features? 139
The Exact Standard Deviation of the Sample Median 101
Bengio LeCun Deep Learning Video 99
Category Theory ? 92
“Machine Learning Techniques for Stock Prediction” 89
Approximation of KL distance between mixtures of Gaussians 75
“A Neuro-evolution Approach to General Atari Game Playing” 74
The 20 most striking papers, workshops, and presentations from NIPS 2012 65
Matlab code and a Tutorial on DIRECT Optimization 61
About 51
Not too linear. Hmmm.
(Though Zipf’s “law” has been known for a long time, this post is at least partly inspired by Tarence Tao’s wonderful post “Benford’s law, Zipf’s law, and the Pareto distribution“.)
Related Posts via Categories | {"url":"http://artent.net/2013/09/27/zipfs-law-artent-blog-hits/","timestamp":"2024-11-10T04:35:50Z","content_type":"text/html","content_length":"27990","record_id":"<urn:uuid:0d1f10f2-e1b3-4402-9270-707d4bbfcdbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00823.warc.gz"} |
How do you convert a sparse matrix to dense?
You can use either todense() or toarray() function to convert a CSR matrix to a dense matrix.
How do you convert a sparse matrix into a dense matrix in python?
Sparse Matrices in Python A dense matrix stored in a NumPy array can be converted into a sparse matrix using the CSR representation by calling the csr_matrix() function.
How does sparse matrix work in Python?
Sparse matrices in Python
1. import numpy as np.
2. from scipy. sparse import csr_matrix.
4. # create a 2-D representation of the matrix.
5. A = np. array([[1, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 1],\
6. [0, 0, 0, 2, 0, 0]])
7. print(“Dense matrix representation: \n”, A)
How do you write a sparse matrix?
Description. S = sparse( A ) converts a full matrix into sparse form by squeezing out any zero elements. If a matrix contains many zeros, converting the matrix to sparse storage saves memory. S =
sparse( m,n ) generates an m -by- n all zero sparse matrix.
How do I use Toarray in Python?
“function to change toarray() python” Code Answer
1. import numpy as np.
2. my_list = [2,4,6,8,10]
3. my_array = np. array(my_list)
4. # printing my_array.
5. print my_array.
6. # printing the type of my_array.
7. print type(my_array)
How do you make a matrix dense in Python?
A dense matrix is created by calling the function matrix . The arguments specify the values of the coefficients, the dimensions, and the type (integer, double, or complex) of the matrix. size is a
tuple of length two with the matrix dimensions. The number of rows and/or the number of columns can be zero.
How do you use Linalg in Python?
For example, scipy. linalg. eig can take a second matrix argument for solving generalized eigenvalue problems….Solving equations and inverting matrices.
linalg.solve (a, b) Solve a linear matrix equation, or system of linear scalar equations.
linalg.tensorinv (a[, ind]) Compute the ‘inverse’ of an N-dimensional array.
How do you convert a sparse matrix in python?
1. Create an empty list which will represent the sparse matrix list.
2. Iterate through the 2D matrix to find non zero elements.
3. If an element is non zero, create a temporary empty list.
4. Append the row value, column value, and the non zero element itself into the temporary list.
How do you convert data to sparse matrix in python?
What are the advantages of sparse matrix?
Using sparse matrices to store data that contains a large number of zero-valued elements can both save a significant amount of memory and speed up the processing of that data. sparse is an attribute
that you can assign to any two-dimensional MATLAB® matrix that is composed of double or logical elements.
What does Todense do Python?
Return a dense matrix representation of this matrix. A NumPy matrix object with the same shape and containing the same data represented by the sparse matrix, with the requested memory order. If out
was passed and was an array (rather than a numpy.
What is Tolist () in Python?
tolist(), used to convert the data elements of an array into a list. This function returns the array as an a. ndim- levels deep nested list of Python scalars. The elements are converted to the
nearest compatible built-in Python type through the item function.
What is a sparse matrix in Python?
Sparse Matrix. A sparse matrix is a matrix that is comprised of mostly zero values. Sparse matrices are distinct from matrices with mostly non-zero values, which are referred to as dense matrices.
What is the default dtype for sparse data in Python?
Sparse data should have the same dtype as its dense representation. Currently, float64, int64 and booldtypes are supported. Depending on the original dtype, fill_value default changes −
What is sparsescipy?
SciPy provides tools for creating sparse matrices using multiple data structures, as well as tools for converting a dense matrix to a sparse matrix. Many linear algebra NumPy and SciPy functions that
operate on NumPy arrays can transparently operate on SciPy sparse arrays.
Is it possible to use NumPy sparse array for machine learning?
Further, machine learning libraries that use NumPy data structures can also operate transparently on SciPy sparse arrays, such as scikit-learn for general machine learning and Keras for deep
learning. A dense matrix stored in a NumPy array can be converted into a sparse matrix using the CSR representation by calling the csr_matrix () function. | {"url":"https://ru-facts.com/how-do-you-convert-a-sparse-matrix-to-dense/","timestamp":"2024-11-12T12:50:41Z","content_type":"text/html","content_length":"54639","record_id":"<urn:uuid:7c7eda14-aa05-4d7b-9096-6a2fd07d25cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00493.warc.gz"} |
Diagonal of a Rectangle Calculator - Calculatorway
Diagonal of a Rectangle Calculator
Free online Diagonal of a rectangle calculator – Enter the length and width of the rectangle with different length units then click the calculate button.
The diagonal of a rectangle
The diagonal of a rectangle is a line segment that connects two opposite corners or vertices of the rectangle.
A diagonal is a measure from a vertex to the opposite side, passing through the opposite corner.
Diagonal of Rectangle formula
The formula for calculating the diagonal (d) of a rectangle
d = √(a²+b²)
• d = Diagonal of a rectangle
• a = width of a rectangle
• b = Length of a rectangle
How to calculate the diagonal of a rectangle?
Here’s a step-by-step guide:
• First, Determine the length of a rectangle
• Next, Determine the width of a rectangle
• Next, Square each of the side lengths: a² and b²
• Next, Add the squared values together: a²+b²
• finally, Take the square root of the sum to find the length of the diagonal:d = √(a²+b²)
Solved Example.
Example. 1: The length of the rectangle is 30 metres and the breadth is 40 metres then find the diagonal of the rectangle.
Given values, length = 30 mt, breadth = 40 mt
Diagonal of a rectangle = √(length² + breadth²) = √(40² + 30²) = 50 metres
Example. 2: The length of the rectangle is 20 metres and the breadth is 70 metres then find the diagonal of the rectangle.
Diagonal of a rectangle = √(length² + breadth²) = √(20² + 70²) = 72.801 metres
More Calculator | {"url":"https://www.calculatorway.com/diagonal-of-a-rectangle-calculator/","timestamp":"2024-11-06T23:41:20Z","content_type":"text/html","content_length":"192778","record_id":"<urn:uuid:99e2c91f-d300-4ea8-a72a-ca9f11128769>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00739.warc.gz"} |
Experimental Study on the Mechanics and Impact Resistance of Multiphase Lightweight Aggregate Concrete
School of Civil Engineering, Architecture and Environment, Hubei University of Technology, Wuhan 430068, China
China Construction Third Bureau First Engineering Co., Ltd., Wuhan 430040, China
China Construction Shenzhen Decoration Co., Ltd., Wuhan 430068, China
Author to whom correspondence should be addressed.
Submission received: 21 July 2022 / Revised: 2 August 2022 / Accepted: 3 August 2022 / Published: 4 August 2022
Multiphase lightweight aggregate concrete (MLAC) is a green composite building material prepared by replacing part of the crushed stone in concrete with other coarse aggregates to save construction
ore resources. For the best MLAC performance in this paper, four kinds of coarse aggregate—coal gangue ceramsite, fly ash ceramsite, pumice and coral—were used in different dosages (10%, 20%, 30% and
40%) of the total coarse aggregate replacement. Mechanical property and impact resistance tests on each MLAC group showed that, when coal gangue ceramsite was 20%, the mechanical properties and
impact resistance of concrete were the best. The compressive, flexural and splitting tensile strength and impact energy dissipation increased by 29.25, 19.93, 13.89 and 8.2%, respectively, compared
with benchmark concrete. The impact loss evolution equation established by the two-parameter Weibull distribution model effectively describes the damage evolution process of MLAC under dynamic
loading. The results of a comprehensive performance evaluation of four multiphase light aggregate concretes are coal gangue ceramsite concrete (CGC) > fly ash ceramsite concrete (FAC) > coral
aggregate concrete (CC) > pumice aggregate concrete (PC).
1. Introduction
Due to technological innovation and the impact on resources, the construction industry must become sustainable [
]. As the world’s major construction materials, concrete is widely used in buildings, roads, bridges, and ocean engineering. The need to produce huge amounts of concrete every year entails a large
amount of sand and stone mining, which causes environmental damage and an aggregate shortage. Coal gangue is the waste product of coal mining and washing, and the world produces 350 million tons
every year [
]. At the same time, the world annually produces 78 million tons of industrial fly ash waste [
]. Using this constantly generated waste to replace coarse aggregates to produce multiphase lightweight aggregate concrete (MLAC) maximizes the use of recycled materials, meets the need for
sustainable construction [
] and reduces costs [
]. MLAC provides a sustainable method for waste recycling and utilization and plays an important role in sustainability [
Gangue can be ground into powder as cementitious material or additive, or it can be prepared as aggregates. Ashfaq et al. [
] showed that coal gangue has good earthwork performance, and its use in earthwork projects reduces carbon emissions. Gao et al. [
] studied coal gangue as a coarse aggregate in the production of green concrete and found that increasing the replacement rate reduced the mechanical properties of coal gangue concrete, so it is not
advisable to use a too-high or too-low concrete design grade. Kockal et al. [
] studied the mechanical properties and durability of fly ash light aggregate concrete. As a complete replacement for normal weight coarse aggregate, it resulted in a reduction in compressive and
splitting tensile strength from 62.9 MPa and 5.1 MPa to 42.3 MPa and 3.7 MPa, respectively. Jayabharath and Kesavan et al. [
] found that the impact resistance of concrete produced from conventional aggregates was higher than that of concrete produced from fly ash aggregates. Dahim et al. [
] found that after reducing the size of fly ash from the micron to the nanoscale by ball milling, the concrete had a higher compressive strength compared to concrete containing conventional micro fly
ash. Hemalatha et al. [
] showed that fly ash has been extensively studied as a replacement for cement since the end of the 20th century, and Hasan et al. [
] investigated the impact resistance of concrete produced from fly ash. The results showed that different types of cold consolidated fly ash aggregate slightly reduce compressive and splitting
tensile strength and impact resistance. However, replacing 8–16 mm aggregates with cold consolidated fly ash improved all aspects of concrete performance.
Pumice, a natural material of volcanic origin, has been used as an aggregate in the production of lightweight concrete in many countries. Worldwide, pumice is relatively abundant. Turkey alone
accounts for about 40% of the world’s known pumice reserves (18 billion m
) [
]. Hatice et al. [
] used acidic pumice to make pervious concrete and found that its strength compared to crushed stone decreased because of the fragility of the pumice. Hossain et al. [
] studied the mechanical properties and durability of lightweight volcanic pumice concrete, but experimental results showed that it had low compatibility as an alternative to common coarse aggregate.
Kurt et al. [
] investigated the effect of fly ash on the self-compactness of pumice light aggregate concrete. The experimental results of 20, 40, 60, 80 and 100% pumice as a natural aggregate decreased the flow
diameter of SCC concrete and material by 5, 6, 9 and 19%, respectively. Amel et al. [
] found that concrete with a dry density of 1430–1690 kg/m
could be obtained by using pumice as a coarse aggregate. Bakis [
] studied the effect of dune sand and pumice on the mechanical properties of lightweight concrete, and the results showed that pumice powder can be used as a binder for road pavement in an optimum
binder ratio of 30% pumice and 20% lime.
As the exploitation of marine resources has become more and more important, the demand for concrete for island construction projects has become stronger. If the island is far away from the mainland,
the construction cost is serious. Without damaging the environment, using coral to replace part of the coarse aggregate can effectively reduce the cost and shorten the construction time [
]. Therefore, it is feasible to use coral as an aggregate for mixing concrete. Many researchers have studied the properties of coral concrete. Arumugam et al. [
] found that the strength of normal concrete is higher than that of all-coral aggregate concrete at the same water–cement ratio. Kakooei et al. [
] found that the compressive strength of coral concrete increased with an increase in polypropylene fiber admixture, and the compressive strength was maximized when the admixture was 2 kg/m
. Niu et al. [
] found that the incorporation of 0.05% basalt fibers improved the mechanical properties of coral concrete the most by 9.87% and 1.36% in compressive and splitting compressive strength, respectively,
at 28 days. Rao et al. [
] found that PVA fibers effectively enhanced the mechanical properties of coral concrete with an optimum admixture rate of 2–3 kg/m
. Cheng [
] found that coral sand concrete had better carbonation depth and capillary water absorption, compared to river sand concrete.
In the current construction industry, most aggregates need to be mined, which may lead to severe damage to mountains and forests [
]. Therefore, the use of industrial waste and abandoned coral to replace part of the coarse aggregate not only reduces industrial waste, but also reduces the damage caused by aggregate extraction [
]. Impact resistance is one of the important characteristics of concrete. Many concrete members are subjected to dynamic loads, such as pile foundations, bridge structures, dams and offshore
platforms. These human and natural factors have a serious impact on the safe use of concrete structures, so how to ensure the safety of concrete members under dynamic load is something researchers
need to solve. Based on these problems, four kinds of lightweight aggregate—coal gangue ceramsite, fly ash ceramsite, pumice and coral—were selected to replace part of the coarse aggregate. The
change in concrete performance after their addition was comprehensively evaluated by mechanical tests and impact resistance.
2. Materials and Methods
2.1. Test Materials and Mix Design
The cement was P.O42.5 ordinary silicate produced by China Huaxin Cement Joint Stock Company, and the fly ash was Grade I produced by Henan Hengyuan New Material Corporation. As shown in
Figure 1
, the coarse aggregate was continuous grading crushed stone, coal gangue and fly ash ceramsite, and pumice and coral aggregate. The particle size was controlled at 5–15 mm. The fine aggregate was
natural river sand with a fineness modulus of 2.92 mm, water content 2.51% and water absorption 7.58%. The water reducing agent was a polycarboxylic acid high-performance water-reducing agent with a
water reduction rate of 35%. The physical properties of lightweight aggregate are shown in
Table 1
The reference strength of concrete is C40. To study the influence of different lightweight aggregates on the mechanical properties and durability of concrete, the coarse aggregate was replaced with
coal gangue ceramsite, fly ash ceramsite, pumice and coral, and the replacement rates were 10%, 20%, 30% and 40%. The mix design of concrete is shown in
Table 2
2.2. Test Methods
Because of the porosity and water absorption of lightweight aggregate, the water absorption and water return characteristics occurred during the preparation of the concrete, so the coal gangue
ceramsite, fly ash ceramsite, pumice and coral had to be pre-wetted. After the specimens were soaked in water for 24 h, rinsed and dried, they were prepared in a HJS-60 double-horizontal axis
concrete mixer. The compressive strength test specimen specification was 100 × 100 × 100 mm; the flexural strength test specimen specification was 100 × 100 × 400 mm; the splitting tensile strength
test specimen specification was 100 × 100 × 100 mm; and the drop hammer impact test specimen specification was a Φ150 × 63 mm cylinder. All specimens had to be maintained for 28 days in an
environment at a temperature of 20 ± 2 °C and relative humidity of more than 95% before the studies could be carried out. The design strength of concrete in this test was C40, so according to the
specification, the loading speed in the compressive test was 0.5 MPa/s. In the flexural and splitting tests, the loading speed was 0.05 MPa/s. The relevant parameters were set; the test was
conducted, and the relevant data were recorded.
The impact test was performed using the falling hammer method recommended by the American Concrete Institute: ACI544 standard [
]. It has the advantages of simple operation and low test requirements. The apparatus was a CECS13-2009 concrete falling hammer impact testing machine. Before the test, the cylinder specimen was
placed in the specified test area, and a steel ball was placed in the center of the specimen. The drop hammer was aligned with the steel ball through an infrared device. An electromagnetic relay was
used to carry out the test and record the number of drop hammer impacts. The first visible crack is regarded as the initial crack state and was recorded as initial crack number N
. When the specimen contacted any three of the four baffles, it was regarded as the damage state and recorded as final crack number N
3. Experimental Design and Results
3.1. Cubic Compressive Strength Test
As shown in
Figure 2
, when coal gangue ceramsite content increased, the compressive strength of CGC increased then decreased. When the content of coal gangue was 20%, the compressive strength of CGC reached the optimal
value, and the compressive strength increased by 29.25%. With increased fly ash ceramsite content, the FAC compressive strength increased, decreased, increased again and finally decreased. When the
content of fly ash ceramsite was 10%, the FAC compressive strength increased by 7.06%. When the content was 40%, the FAC compressive strength decreased by 13.02%. When the coral aggregate content was
10%, the CC compressive strength increased by 18.68%. When the content was 40%, the CC compressive strength decreased by 13.11%. With increased pumice aggregate, the CC compressive strength
decreased. When the pumice aggregate was 40%, the CC compressive strength decreased by 36.07%.
3.2. Flexural Strength Test
As shown in
Figure 3
, with increased coal gangue ceramsite, the flexural strength of CGC increased then decreased. When coal gangue ceramsite was 20%, the CGC flexural strength reached optimal value, and the flexural
strength increased by 19.93%. When the coal gangue ceramsite was greater than 20%, the CGC flexural strength began to decrease. When it was 40%, the CGC flexural strength of decreased by 6.41%. With
increased fly ash ceramsite, the FAC increased, decreased, and finally increased again. When the content was 30%, the FAC flexural strength was the strongest, and flexural strength increased by
14.96%. When fly ash ceramsite was 40%, the FAC flexural strength decreased by 10.58%. With the increased coral aggregate content, the CC increased then decreased. When the dosage was 10%, the
flexural strength of CC increased by 13.35%. The PC decreased with the increased pumice aggregate. When the pumice content was 40%, the PC decreased by 19.57%.
3.3. Splitting Tensile Strength
As shown in
Figure 4
, with increased coal gangue ceramsite, the CGC splitting tensile strength increased then decreased. When it was 20%, the CGC splitting tensile strength was the largest, increasing by 13.89%. With
increased fly ash ceramsite, the FAC splitting tensile strength increased, decreased, increased again and finally decreased. When fly ash ceramsite was 30%, the splitting tensile strength reached
maximum value, which was an increase of 5.56%. With increased coral aggregate, the CC increased then decreased. When it was 10%, the CC splitting tensile strength was the largest, with an increase of
5.56%. With the increase of pumice aggregate, PC splitting tensile strength decreased. When the pumice aggregate was 40%, the PC splitting tensile strength decreased by 27.78%.
3.4. Internal Mechanism Analysis of MLAC
The micropore structure inside the ceramsite functions like a micropump and reservoir. Under capillary action, ceramsite migrates water in the concrete, which gives it the characteristics of
absorbing and returning water. It can effectively improve the performance of the interfacial transition zone (ITZ), increase the bonding force between the ceramsite and mortar interface, and make the
structure of concrete interfacial zone closer. This change led to the improved mechanical properties of CGC concrete. The fly ash ceramsite has lower strength and higher water absorption than coal
gangue ceramsite, which makes the internal structure of the concrete more complex. Although there is a role of micro pump in the fly ash ceramsite, the gap in the cement stone is reduced. This change
improves the FAC performance to a certain extent, but the improvement in mechanical properties is lower than that of coal gangue ceramsite. Because coral is a light aggregate, it absorbs water,
giving a fuller internal hydration, which effectively reduces the gap in the interfacial transition zone and improves the concrete performance. However, too much coral aggregate decreases the
mechanical properties when its content increases due to its low strength. The low strength and the higher water absorption of pumice reduce the mechanical properties of PC. When the dosage was 40%,
the compressive, flexural and splitting tensile strength of PC decreased by 36.07%, 19.57% and 27.78%, respectively.
4. Impact Resistance Test
4.1. Impact Specimen Damage Pattern
The impact damage morphology presented by each group of test blocks under the impact load is shown in
Figure 5
. Under continuous impact hammering, the first crack appeared on the surface and then penetrated the whole plain concrete specimen. The damage patterns of CCG, PC, FAC, and CC specimens were similar
with “cracking as destruction” showing obvious brittle damage. In essence, the concrete was unable to prevent the expansion of the internal fine cracks, leading to rapid extension to the surface of
the brittle damage. As shown in
Figure 5
, a small portion of these cracks occurred in the combined zone of cement mortar and coarse aggregate stones, and the majority crossed the cement mortar and light aggregate with a more complex damage
trend, while the natural coarse aggregate was rarely damaged. This indicated that the damage was clearly influenced by the type of aggregate. This phenomenon can be explained by the variability in
the mechanical properties of different aggregates. Several of the test blocks showed three trigonal and star-shaped cracks under impact loading, which divided the test blocks into three parts.
4.2. Impact Performance
The impact test records the number of initial cracks N
and the number of final cracks
for each group of 6 test blocks. The average value was used to calculate the impact energy consumption, which is calculated by the formula
is the impact energy consumption
$N 2$
is the number of impacts when the specimen is damaged;
is the quality of the impact drop hammer, kg (4.5 kg);
is gravitational acceleration (9.81 m/s
; and
is the height of impact hammer drop (0.5 m).
Table 3
, it can be seen that the number of impact resistance varies widely for each group of the six test blocks, indicating a relatively large dispersion of the initial cracking number N
and final cracking number N
because of the highly discrete type of concrete. In
Figure 6
, the variation pattern of the impact number of each group of multi-phase light aggregate concrete can be seen more intuitively. The data analysis and processing of the impact resistance index of
each group of specimens leads to
Table 3
Table 4
, using coal gangue ceramsite, pumice aggregate, fly ash ceramsite and coral aggregate to replace part of the coarse aggregate did not obviously improve ductility, and the specimen showed obvious
brittle failure. After adding the pumice and coral aggregate, the number of initial and final cracks of PC and CC decreased, and the average impact energy consumption was lower than that for the
reference concrete. When the dosage was 40%, the impact energy consumption of PC and CC decreased by 32.89% and 22.34%, respectively. These damaged cracks generally occurred in the cement mortar and
lightweight aggregate areas, where the coarse aggregate crushed stone was rarely damaged, indicating that the destruction of concrete was due to the lower strength of pumice and coral aggregates
compared to that of coarse aggregate crushed stone.
When adding 20% gangue ceramsite, the impact energy consumption of the specimen increased compared to the reference concrete, and the impact number increased by 8.2%. This was because a certain
content of coal gangue ceramsite promoted hydration so that the cement mortar and aggregate matrix were closely combined, and hydration reaction products filled the gap between aggregates, thereby
improving concrete performance. When the coal gangue ceramsite exceeded 20%, the impact energy consumption decreased gradually. When it was 40%, the impact energy consumption decreased by 10.66%,
demonstrating that excessive coal gangue ceramsite reduced the impact energy consumption of concrete. When the contents of fly ash ceramsite were 10% and 30%, the impact energy consumption increased
slightly, which was similar to that of coal gangue ceramsite. Therefore, an appropriate dosage of coal gangue and fly ash ceramsite improved the impact resistance of concrete and reduced cracking
4.3. Impact Resistance Analysis of MLAC Based on Two-Parameter Weibull Distribution Model
4.3.1. Parameter Determination of Two-Parameter Weibull Distribution Model
There are inevitably some tiny cracks during the setting and hardening of cement-based materials. Under the impact load, these tiny cracks continue to expand until the concrete is finally destroyed.
Therefore, concrete is essentially in a process of random cumulative fatigue damage. Researchers tried to use various statistical tools to summarize the changes in experimental results. Normal
distribution was one of them, but the fitting of shock test results using normal distribution was poor [
]. Some scholars found that the impact life of concrete follows the two-parameter Weibull distribution [
]. As a life prediction model, it is widely used in the failure assessment of brittle materials. Therefore, with the final cracking number,
$N 2$
of each group of multi-phase lightweight aggregate concrete as the initial data, this model was selected to conduct an in-depth study on the impact life of each group of multi-phase lightweight
aggregate concrete under different failure probabilities.
The model is a two-parameter function composed of shape parameter
, which determines the shape of the function, and proportional parameter
, which determines the scaling of the function. According to the two-parameter Weibull distribution model, assuming that
$F ( N )$
is the probability density function of the impact life
of the multiphase lightweight aggregate concrete,
$F ( N )$
can be expressed by Equation (2).
$F N = β η N η β − 1 exp − N η β N ≥ 0$
By integrating Equation (2), the corresponding cumulative distribution function
$f ( N )$
can be obtained, as shown in Equation (3):
The function value corresponding to the cumulative distribution function
$f ( N )$
is also called cumulative failure probability
$P 1$
. The survival probability function value
$P 2$
can then be expressed as
Taking the natural logarithm twice for both sides of Equation (4), Equation (5) is obtained:
$ln ln 1 P 2 = β ln 1 η + β ln N$
$Y = ln ln 1 / P 2 , X = ln N$
, then the above equation can be rewritten as
The proportional parameter
in Equation (5) can be expressed by Equation (7):
Equation (5) can be used to verify whether the impact life of MLAC is subject to the two-parameter Weibull distribution. Through linear regression analysis, the values of parameters $a$, $b$ and the
correlation coefficient $R 2$ can be obtained. If there is a good linear relationship between $Y$ and $X$ in the linear fitting results, the two-parameter Weibull distribution model can reasonably
predict and analyze the impact life of MLAC under different failure probabilities. If there is no linear relationship between $Y$ and $X$, the model is not suitable for an impact life analysis.
To test whether the impact life of MLAC conforms to the two-parameter Weibull distribution and whether there is a good linear relationship between
, the survival probability needs to be calculated first. The calculation method is shown in Formula (8):
is the order of each group of experimental data, and
is the total number of samples per group. Through Formulas (5), (6) and (8), with
as the abscissa and
as the ordinate, linear regression analysis was carried out on the experimental data. The values of regression parameters A, B and correlation coefficient
are shown in
Table 5
4.3.2. Impact Life Analysis of MLAC under Multiple Factors
Rahmani et al. [
] pointed out that
$R 2 ≥ 0.7$
could establish a reasonable reliability model at that time. It can be seen from
Table 5
that the minimum regression coefficient
is 0.798, the maximum is 0.978, and both are greater than 0.7. The linear regression fit is good, and the test results are consistent with the distribution law of the Weibull probability density
function, which means that Equation (8) holds. According to Formulas (5)–(8), Formula (9) can be obtained and used to obtain the impact life of an MLAC under different failure probabilities.
$N = exp ln ln 1 / ( 1 − P 1 ) − a b$
can be obtained from
Table 4
. The impact life of each group of MLAC specimens under different failure probabilities can be obtained through Equation (9), as shown in
Table 6
In the optimization of experimental design, it is difficult to directly determine the optimal value of the factors due to the interaction of multiple factors, so the response surface method came into
being. It uses graphic technology to directly reflect the functional relationship between multiple variables such as aggregate replacement rate and failure probability and core elements (impact
times) in the system. In the previous section, the ultimate impact times of each MLAC group under different failure probabilities were obtained through the Weibull distribution model. To study the
influence of different types and percentages of lightweight aggregate and the failure probability on the impact times of MLACs, three-dimensional response diagrams of four different lightweight
aggregate materials under different percentages and different failure probabilities were established.
The data obtained in the previous section were imported into the Origin software to draw the corresponding three-dimensional response diagram, as shown in
Figure 7
. The
axis is the failure probability, and the
axis is the replacement ratio of the aggregate. It can be seen from
Figure 7
that with an increased failure probability and aggregate replacement rate of concrete specimens, the color in the figure gradually changes from purple to dark red. In
Figure 7
, the contour density of the
axis is greater than that of the
axis, indicating that the influence of the aggregate replacement rate on the impact performance of MLAC was higher than the probability of failure or the failure of the concrete specimens. When the
failure probability of MLAC was higher, the corresponding number of the impact resistance was higher. It can be seen in
Figure 7
a that when the content of coal gangue ceramsite in GCC was between 15% and 25%, the impact resistance of GCG had the best value. In
Figure 7
b, when the content of pumice aggregate in PC was between 0% and 20%, the impact resistance of PC had the best value. In
Figure 7
c, when the content of fly ash ceramsite was 0–15% and 25–35%, the impact resistance of FAC had the best value.
Figure 7
d shows that when the content of coral aggregate in CC was between 5% and 15%, the impact resistance of CC had the best value.
4.3.3. Impact Damage Analysis of MLAC
In the previous section, the two-parameter Weibull distribution model was used to analyze the impact life of each MLAC group under different failure probabilities, but the damage process of MLAC
after repeated dynamic loading was not systematically studied. Damage caused by repeated drop hammer impacts on MLAC structures was essentially the result of accumulated damage within the concrete
structure after multiple drop hammer impacts.
With the gradual increase in the number of falling hammer impacts, the probability of failure damage also increased, so the probability of failure damage and damage variables of the concrete
structure were considered a simultaneous superposition process during the whole impact breakage process. It can be considered that the concrete damage degree and failure probability accumulated
simultaneously during the drop hammer impact. When the concrete fails after
times, the failure probability of concrete is
$P 1 ( N ) = 1$
, and the damage degree is
$D ( N ) = 1$
. It can be seen that the failure probability and damage degree of concrete can be treated equivalently, that is,
$P 1 ( N ) = D ( N )$
. In summary, the impact damage model of MLAC based on the two-parameter Weibull distribution can be expressed by Equation (10):
The data in
Table 5
are substituted into Equation (7) to obtain
. The calculated values are substituted into Equation (10) to obtain the evolution equation of impact damage resistance for each group of multiphase light aggregate concrete under repeated falling
hammer impact loads. To improve accuracy, six specimens were used in each group. The fitting formula was obtained by taking the average value of the experimental data, as shown in
Table 7
According to the impact damage evolution equation in
Table 7
, the damage degree curve of each group of lightweight aggregate concrete specimens can be drawn, as shown in
Figure 8
. From the damage degree change curve, it can be seen that during the drop hammer impact tests, the damage degree change of the concrete was not obvious at the initial stage because the internal
structure of concrete tended to be stable, and the degree of crack extension was low. The concrete with 20% coal gangue ceramsite had the maximum limit impact resistance times, which showed that the
CGC with this content had the best impact resistance. With the increase in impact times, the damage degree of concrete increased sharply. When the damage degree was 1, the concrete appeared through
the cracks and was completely destroyed. The Weibull distribution probability damage model and the experimental data had high similarity, which showed that the model described the damage variation of
concrete specimens during impact resistance well.
5. Conclusions
The following conclusions can be drawn from the mechanical and impact test results of MLAC:
• With the increase in coal gangue ceramsite, the mechanical properties of CGC first increased and then decreased. With the increase in fly ash ceramsite, the mechanical properties of FAC
increased, decreased, increased again and finally decreased. With the increase in coral aggregate content, the CC increased then decreased. With the increase in pumice aggregate, the PC
decreased. The comprehensive performance was CGC > FAC > CC > PC.
• When coal gangue ceramsite was 20%, the mechanical properties and impact resistance of concrete were the best. The compressive, flexural and splitting tensile strength and the impact energy
consumption increased by 29.25%, 19.93%, 13.89% and 8.2%, respectively, compared with the reference concrete.
• The impact test results of MLAC obeyed the distribution law of the two-parameter Weibull distribution model, which can be used to predict and describe the impact life of multi-phase lightweight
aggregate concrete under different failure probabilities.
• The impact resistance of MLAC under multiple factors was analyzed in depth. The analysis showed that the influence of the aggregate replacement rate on the impact resistance of multi-phase
lightweight aggregate concrete was higher than the probability of failure or the failure of the concrete specimens.
• Through the establishment of the impact damage evolution equation, the damage degradation of each specimen under drop hammer impact was studied in depth. The variation law of the data derived
from the equation was highly consistent with the experimental results. The damage degradation of MLAC under dynamic load can be reasonably described by the equation.
6. Prospect
In this experiment, the effects of four lightweight aggregates on the mechanical properties and impact resistance of MLAC were studied, and the durability of MLAC was further studied in the follow-up
work. Whether the addition of fibers or other cementitious materials can improve the mechanical properties and durability of concrete can be considered.
Author Contributions
Conceptualization, J.M. and A.Z.; methodology, Z.X. and J.M.; software, J.M.; validation, J.M., A.Z. and Z.X.; formal analysis, Z.X., J.M. and Z.L.; investigation, J.M., Z.L. and S.C.; resources,
J.M.; data curation, J.M. and C.W.; writing—original draft preparation, J.M.; writing—review and editing, J.M.; visualization, J.M. and B.Z.; supervision, J.M. and A.Z.; project administration, J.M.
and A.Z.; funding acquisition, J.M. and A.Z. All authors have read and agreed to the published version of the manuscript.
This research was funded by An Zhou, grant number BSQD12155.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data presented in this study are available on request from the corresponding author.
Conflicts of Interest
The author declares no conflict of interest.
Figure 1. (a) Coal gangue ceramsite; (b) pumice aggregate; (c) fly ash ceramsite; (d) coral aggregate.
Figure 6. Relationship between impact resistance times of MLAC and aggregate content (a) CGC; (b) PC; (c) FAC; (d) CC.
No. Bulk Density Apparent Density Water Absorption (%) Tube Compressive Strength (MPa)
(kg/m^3) (kg/m^3) 1 h 24 h
Coal gangue ceramsite 975 1730 5.07 7.43 6.8
Pumice aggregate 690 1593 16.44 17.32 2.98
Fly ash ceramsite 650 1323 12.51 12.98 6.5
Coral aggregate 915 1841 8.5 11.0 3.1
No. Substitution Rate (kg/m^3) Water (kg/m^3) Cement (kg/m^3) Fly Ash (kg/m^3) Gravel (kg/m^3) Lightweight Aggregate (kg/m^3) Sand (kg/m^3) Water
Reducing Agent (kg/m^3)
BC0 0 180 366.4 91.6 1070 0 656 2
CGC1 10 180 366.4 91.6 963 107 656 2
CGC2 20 180 366.4 91.6 856 214 656 2
CGC3 30 180 366.4 91.6 749 321 656 2
CGC4 40 180 366.4 91.6 642 428 656 2
PC1 10 180 366.4 91.6 963 107 656 2
PC2 20 180 366.4 91.6 856 214 656 2
PC3 30 180 366.4 91.6 749 321 656 2
PC4 40 180 366.4 91.6 642 428 656 2
FAC1 10 180 366.4 91.6 963 107 656 2
FAC2 20 180 366.4 91.6 856 214 656 2
FAC3 30 180 366.4 91.6 749 321 656 2
FAC4 40 180 366.4 91.6 642 428 656 2
CC1 10 180 366.4 91.6 963 107 656 2
CC2 20 180 366.4 91.6 856 214 656 2
CC3 30 180 366.4 91.6 749 321 656 2
CC4 40 180 366.4 91.6 642 428 656 2
No. N[1]/N[2]
BC0 853/854 1003/1004 1065/1066 1046/1047 923/924 986/987
CGC1 861/863 927/928 902/903 952/953 763/765 894/895
CGC2 1274/1276 1167/1169 976/979 1102/1104 1077/1079 953/955
CGC3 796/797 853/854 965/967 1023/1024 1058/1059 871/872
CGC4 684/685 1034/1034 749/749 852/852 957/957 932/933
PC1 755/756 975/976 959/961 783/784 732/733 910/911
PC2 692/693 728/729 831/832 921/922 925/927 744/745
PC3 635/635 838/839 764/764 822/822 637/637 706/707
PC4 732/733 580/580 613/613 669/669 706/706 636/636
FAC1 954/955 891/892 933/935 967/968 1131/1132 1035/1036
FAC2 740/741 713/714 948/949 885/886 821/822 869/870
FAC3 1107/1108 1065/1066 1009/1010 853/854 864/865 965/966
FAC4 715/715 736/737 834/834 872/872 928/928 800/800
CC1 823/825 835/836 891/892 1084/1085 1102/1103 1054/1055
CC2 742/744 756/757 872/873 971/972 915/916 946/947
CC3 685/686 718/719 783/784 813/814 868/869 914/915
CC4 670/671 754/754 836/837 765/766 825/825 703/703
No. Average of the Number of Impacts Impact Energy Consumption/(w/J)
N[1] N[2] N[2] − N[1]
BC0 976 977 1 21,564.833
CGC1 913 914 1 20,174.265
CGC2 1055 1057 2 23,330.633
CGC3 928 929 1 20,505.353
CGC4 873 873 0 19,269.293
PC1 852 853 1 18,827.843
PC2 807 808 1 17,834.58
PC3 734 734 0 16,201.215
PC4 656 656 0 14,479.56
FAC1 985 986 1 21,763.485
FAC2 829 930 1 20,527.425
FAC3 977 978 1 21,586.905
FAC4 814 814 0 17,967.015
CC1 965 966 1 21,322.035
CC2 867 868 1 19,158.93
CC3 797 798 1 17,613.855
CC4 759 759 0 16,753.028
No. Regression Parameters Correlation Coefficient
$a$ $b$ R^2
BC0 10.949 −75.854 0.973
CGC1 11.356 −77.494 0.899
CGC2 8.208 −57.866 0.927
CGC3 8.018 −55.224 0.943
CGC4 5.803 −39.674 0.976
PC1 6.807 −46.364 0.863
PC2 7.011 −47.357 0.880
PC3 7.21 −47.997 0.898
PC4 10.342 −67.519 0.978
FAC1 10.151 −70.422 0.839
FAC2 8.243 −55.833 0.956
FAC3 8.322 −57.731 0.93
FAC4 9.088 −61.345 0.956
CC1 6.249 −43.373 0.798
CC2 7.699 −52.519 0.902
CC3 8.279 −55.746 0.973
CC4 11.148 −74.181 0.809
No. $P 1 = 0.1$ $P 1 = 0.3$ $P 1 = 0.5$ $P 1 = 0.7$ $P 1 = 0.9$
BC0 831 929 987 1038 1101
CGC1 754 840 890 935 990
CGC2 876 1017 1102 1179 1276
CGC3 740 862 936 1003 1087
CGC4 632 780 874 962 1075
PC1 652 780 860 933 1026
PC2 622 741 814 881 966
PC3 570 675 740 799 874
PC4 551 620 661 697 742
FAC1 825 931 994 1049 1118
FAC2 665 771 836 894 967
FAC3 786 910 985 1053 1138
FAC4 667 763 820 872 936
CC1 721 876 975 1065 1181
CC2 685 802 875 940 1022
CC3 640 742 804 859 929
CC4 634 707 751 789 836
$B C : D N = 1 − exp − n 1020 . 356 10 . 949$ $C G C 1 : D N = 1 − exp − n 919 . 547 11 . 356$ $C G C 2 : D N = 1 − exp − n 1152 . 275 8 . 209$
$C G C 3 : D N = 1 − exp − n 979 . 671 8 . 018$ $C G C 4 : D N = 1 − exp − n 932 . 017 5 . 803$ $P C 1 : D N = 1 − exp − n 908 . 235 6 . 807$
$P C 2 : D N = 1 − exp − n 858 . 317 7 . 011$ $P C 3 : D N = 1 − exp − n 778 . 508 7 . 21$ $P C 4 : D N = 1 − exp − n 684 . 397 10 . 342$
$F A C 1 : D N = 1 − exp − n 1029 . 858 10 . 151$ $F A C 2 : D N = 1 − exp − n 874 . 61 8 . 243$ $F A C 3 : D N = 1 − exp − n 1029 . 962 8 . 322$
$F A C 4 : D N = 1 − exp − n 853 . 94 9 . 088$ $C C 1 : D N = 1 − exp − n 1033 . 501 6 . 249$ $C C 2 : D N = 1 − exp − n 917 . 754 7 . 699$
$C C 3 : D N = 1 − exp − n 840 . 127 8 . 279$ $C C 4 : D N = 1 − exp − n 776 . 011 11 . 148$ —
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Unveiling the Inventory Carrying Cost Formula: The Key to Calculating Procurement Costs - oboloo
Unveiling the Inventory Carrying Cost Formula: The Key to Calculating Procurement Costs
Unveiling the Inventory Carrying Cost Formula: The Key to Calculating Procurement Costs
Are you tired of constantly overspending on inventory procurement? Do you struggle to find ways to minimize your expenses and maximize your profits? Look no further! The answer lies in the Inventory
Carrying Cost Formula. This powerful tool allows businesses to accurately calculate their procurement costs, giving them the ability to make more informed decisions about their inventory management.
In this article, we will delve into what exactly the Inventory Carrying Cost Formula is, how it works, and the many benefits it can bring to your business. So buckle up and get ready for a
game-changing discovery that will revolutionize the way you manage your inventory!
What is the Inventory Carrying Cost Formula?
Inventory carrying cost is the expense that an organization bears to store and maintain its inventory. It includes expenses such as rent, utilities, insurance, taxes, depreciation, obsolescence and
more. The Inventory Carrying Cost Formula is a method of calculating these costs so that businesses can determine the true cost of holding inventory.
The formula takes into account various factors including the annual inventory holding cost percentage rate (which varies based on industry), average inventory value during a specific period of time
and the length of time for which the inventory is carried. By using this formula businesses are able to calculate how much it costs them to keep their products in stock.
While it may seem like just another accounting calculation, knowing your Inventory Carrying Cost can be vital for procurement professionals who need to make strategic decisions about when they should
order new stock or reduce existing stock levels. By accurately identifying these costs organizations can also identify areas where they may be able to cut back or optimize their spending.
Understanding what the Inventory Carrying Cost Formula is and how it works can help organizations make informed decisions regarding their procurement strategy while keeping profitability at top
How to Use the Inventory Carrying Cost Formula
Once you have an understanding of what the Inventory Carrying Cost Formula is and why it’s important, you need to know how to use it. The formula itself is relatively simple: Total Inventory Carrying
Cost = (Average Inventory Level x Cost per Unit) x (Carrying Cost Percentage/100).
Firstly, you’ll want to calculate your average inventory level. This involves taking a sum of your inventory levels over a specific period of time, such as a year or quarter, and dividing that number
by the total number of periods.
Next, determine your cost per unit. This can include factors such as production costs, storage fees, transportation expenses and insurance premiums.
Figure out your carrying cost percentage – this is essentially the percentage rate at which you’re charged for holding onto inventory over a certain period of time. It may encompass factors like
interest rates on loans used for purchasing stock or rent paid on warehouse space.
By plugging all these numbers into the equation above and doing some basic arithmetic, you’ll be able to calculate exactly how much money it’s costing your company to hold onto excess stock. Armed
with this knowledge, procurement managers can make informed decisions about when to order new inventory and in what quantities – ultimately saving their organization valuable time and resources in
the process.
The Benefits of Using the Inventory Carrying Cost Formula
The Inventory Carrying Cost Formula is a valuable tool for businesses that want to optimize their procurement costs. By calculating the cost of carrying inventory, companies can make informed
decisions about how much inventory to order and when.
One of the main benefits of using this formula is that it helps businesses avoid overstocking. When a company has too much inventory on hand, they risk incurring additional carrying costs such as
storage fees, insurance premiums and other expenses associated with storing items for long periods. Overstocking also ties up cash flow that could be used elsewhere in the business.
Another benefit of using the Inventory Carrying Cost Formula is that it allows companies to identify which products are costing them the most money to carry. Armed with this information, businesses
can adjust their ordering patterns or even discontinue certain products altogether if they are not profitable.
In addition, by accurately measuring inventory carrying costs, businesses can negotiate better deals with suppliers by having more precise data on how much it actually costs them to hold each item in
stock. This empowers businesses to save money through strategic purchasing decisions.
Utilizing the Inventory Carrying Cost Formula provides significant advantages for companies looking to streamline their procurement processes and maximize profits.
To sum up, the inventory carrying cost formula is an essential tool for calculating procurement costs. By taking into account all of the expenses associated with holding inventory, businesses can
make informed decisions about their purchasing and supply chain management strategies.
By using this formula, companies can identify areas where they can reduce costs and optimize their cash flow. This will ultimately lead to a more efficient and profitable business operation.
Understanding how to calculate your inventory carrying costs is crucial for any business that wants to succeed in today’s competitive marketplace. So why not take some time to evaluate your own
company’s inventory carrying costs today? You might be surprised at what you learn!
The Complete Guide to Online Procurement Systems
October 17, 2024 | Uncategorized
Analyzing Procurement Spend: Unveiling Cost-Saving Opportunities
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How To Average Grades Using Points
Averaging grades using the total point system can be relatively simple, provided you keep track of the points so you can calculate your grades. Usually the points are tracked for you in an online
system so you can access them at any time. The basic formula for averaging the grades is to take the number of points earned and divide it by the total number of points possible. Multiply the answer
by 100 for a percentage grade.
Step 1
Collect all the assignments done for the course in question. Make sure not to leave any out because the grade average will not be accurate. If you want to keep track of your points earned by
yourself, making an Excel spreadsheet or using graph paper can be helpful. List the assignment names as titles and list the points earned below the appropriate title.
Step 2
Add up the points earned from the assignments. This total reflects the amount of points awarded to the student, not the amount of points that could have been earned. It is a good idea to recheck the
addition in case an error occurs, such as punching in a wrong number into the calculator. Repeat this task for adding up the number of points that could have been awarded for each assignment, making
sure to check the math. Now there should be two sets numbers: one for the points awarded, and one for the points that were possible.
Step 3
Divide the number of points awarded from the assignments by the number of points that could have been awarded if you received a perfect score (number of points possible). Multiply the answer by 100
to convert the grade into a percentage. A good formula to follow when doing this calculation is: [(number of points awarded)/(number of points possible at this point in the semester)]*100.
TL;DR (Too Long; Didn't Read)
It is important to only divide by the number of possible points earned at that point in the semester. If you want to know your average halfway through the course and you divide the points awarded so
far by the points possible at the end of the semester, the average may show you are failing when in fact you may be doing fine in the class.
Cite This Article
Spear, Marcia. "How To Average Grades Using Points" sciencing.com, https://www.sciencing.com/average-grades-using-points-8760708/. 24 April 2017.
Spear, Marcia. (2017, April 24). How To Average Grades Using Points. sciencing.com. Retrieved from https://www.sciencing.com/average-grades-using-points-8760708/
Spear, Marcia. How To Average Grades Using Points last modified March 24, 2022. https://www.sciencing.com/average-grades-using-points-8760708/ | {"url":"https://www.sciencing.com:443/average-grades-using-points-8760708/","timestamp":"2024-11-09T16:17:25Z","content_type":"application/xhtml+xml","content_length":"70761","record_id":"<urn:uuid:64a1a113-defa-4c81-bebb-0ab24097648f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00110.warc.gz"} |
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CS 371 - Introduction to Artificial Intelligence
Homework #3
Due at the beginning of class 8
Note: Work is to be done in pairs
Stochastic Local Optimization
Important: To avoid loss of points, remove/comment-out unspecified printing from your code before submission. You would not believe how those best/current energy lines in SLS really add up in the log
1. Rook Jumping Maze Generation:
Rook Jumping Maze Instructions: Starting at the circled cell in the upper-left corner, find a path to the goal cell marked “G”. From each numbered cell, one may move that exact number of cells
horizontally or vertically in a straight line. How many moves does the shortest path have?
For this exercise, you will generate random n-by-n Rook Jumping Mazes (RJMs) (5 ≤ n ≤ 10) where there is a legal move (jump) from each non-goal state. You will also use stochastic local search to
optimize for (1) solvability and (2) maximum length of the shortest maze solution path.
Let array row and column indices both be numbered (0, ..., n - 1). The RJM is represented by a 2D-array of jump numbers. A cell's jump number is the number of array cells one must move in a straight
line horizontally or vertically from that cell. The start cell is located at (0, 0). For the goal cell, located at (n - 1, n - 1), let the jump number be 0. For all non-goal cells, the randomly
generated jump number must allow a legal move. In the example 5-by-5 maze above, legal jump numbers for the start cell are {1, 2, 3, 4}, whereas legal jump numbers for the center cell are {1, 2}. In
general, the minimum legal jump number for a non-goal cell is 1, and the maximum legal jump number for a non-goal cell (r, c) is the maximum of n - 1 - r, r, n - 1 - c, and c. This defines a directed
graph where vertices are cells, edges are legal jumps, and the outdegree is 0 for the goal cell vertex and positive otherwise.
There are many features of a good RJM. One obvious feature is that the maze has a solution, i.e. one can reach the goal from the start. One simple measure of maze quality is the minimum number of
moves from the start to the goal. For this exercise, we will limit our attention to these two measures of maze quality.
Using breadth-first search, or some other suitable graph algorithm, compute the minimum distance (i.e. depth, number of moves) to each cell from the start cell. Create an objective function (a.k.a.
energy function) that returns the negated distance from start to goal, or a large positive number (use 1,000,000) if no path from start to goal exists. Then the task of maze generation can be
reformulated as a search through changes in the maze configuration so as to minimize this objective function.
Implement stochastic local search state RookJumpingMaze according to this specification. Note that the specification of implemented interface methods is listed towards the top of the RookJumpingMaze
Once you have RookJumpingMaze implemented, complete the implementation of RJMGenerator.java. For a size 5 RJM, you should be able to implement and tune the parameters of a stochastic local search
algorithm of your choice to achieve a median energy of less than or equal to -18.0 in 5000 iterations of search or less. That is, your median-energy maze generated should require a minimum of 18
moves to solve.
Hint: When computing the energy() method, you should not use your previous uniformed search code. Instead, coding a simple breadth-first search within the RookJumpingMaze class will allow you to
easily do repeated-state elimination. Initialize the depth of the initial location to 0, and all others to a negative constant representing "unreached" status. Replace "unreached" values with search
depths (one greater than current depth) as child locations are being added to the queue, and not adding children to the queue that have already been visited (have non-negative depth).
2. Distant Sampling: In many different problem, it is useful to be able to select a biased sample from a data set such that the items are very different from each other. Examples include
initialization of clustering algorithms, choosing a color palette where colors are well distinguished from one another, and choosing sites to geographically distribute resources.
For this problem, you will create a stochastic local search State implemented according to this specification. Note that the specification of implemented interface methods is listed towards the top
of the DistantSamplerState documentation. To summarize, the state is constructed with a 2D array where each row is an n-dimensional double point/vector, and a given number of distant samples from
among these is the objective. Thus, to compute the energy, one computes the inverse squared Euclidean distance for each pair of points and sums these to get the "energy" of the State. To compute the
Euclidean distance, compute the sum of the squared differences in each dimension between two points/vectors and then take the square root of this sum. To get the inverse squared Euclidean distance,
you divide one by the distance squared. (Alternatively, don't take the square root when computing the distance and directly compute the inverse (i.e. reciprocal) without squaring. The square root and
squaring are inverse operations and cancel.) It is as if we're simulating electrostatic rupulsion forces between our selected sample data points.
A stochastic local search step here is switching from one unique sample choice (no repetitions are allowed) to a different unique sample choice. It is convenient to internally represent our sample
choices as a list of row indices from the original given data. More details to more methods are given in this specification.
Once you have DistantSamplerState implemented, complete the implementation of DistantSampler.java. For the default dataset given therein, you should be able to implement and tune the parameters of a
stochastic local search algorithm of your choice to achieve a median energy of less than 132.0 in 10000 iterations of search or less.
Common problems and solutions:
• Problem: Relying on classes HillDescender or SimulatedAnnealer which aren't submitted/assumed. Solution: Embed your favorite SLS algorithm within your optimizing method.
• Problem: Your best energy is always equal to your current energy and is nonmonotonic (goes up and down in a single run). Solution: Do a completely deep clone rather than a shallow/partially deep
• Problem: Your program seems to work, but when testing gives a different result. Solution: The JUnit test harness performs multiple tests of the code in sequence. Make sure that successive test
runs do not corrupt the returned result of the previous test runs.
Hopefully, in looking at these problems, you'll see the wide applicability of such optimization algorithms. Often, the most challenging aspect of such optimization is devising a good, efficient next
state generator that gives beneficial structure to the state space, rather than merely bouncing randomly from one state to another. Good next state generators both allow one to traverse the entire
state space, and immediately sample states that are similar in quality to the current state. | {"url":"http://cs.gettysburg.edu/~tneller/cs371/hw3.html","timestamp":"2024-11-04T13:47:34Z","content_type":"text/html","content_length":"9937","record_id":"<urn:uuid:04342856-38e4-4c8f-a83f-e04bd2489f69>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00266.warc.gz"} |
The same sort of technique can be used with a multi-dimensional array, with start, stop, and (optionally) step specified along each dimension, with the dimensions separated by a comma. The syntax
would be:
my2Darray[start1:stop1:step1, start2:stop2:step2]
With the same rules as above. You can also combine slicing with fixed indices to get some or all elements from a single row or column of your array.
For example, array b created above is a 3x3 array with the values 1-9 stored in it. We can do several different things:
b[0,:] # get the first row
b[:,2] # get the third column
b[1,::2] # get every other element of the first row, starting at element 0
b[:2,:2] # get a square array containing the first two elements along each dimension
b[-2:,-2:] # get a square array containing the last two elements along each dimension
b[::2,::2] # get a square array of every other element along each dimension
b[-1::-1,-1::-1] # original sized array, but reversed along both dimensions | {"url":"https://notebook.community/ComputationalModeling/spring-2017-danielak/past-semesters/fall_2016/day-by-day/day23-agent-based-modeling-day1/Numpy_2D_array_tutorial","timestamp":"2024-11-06T20:51:42Z","content_type":"text/html","content_length":"70861","record_id":"<urn:uuid:ff4b5888-72a8-40f7-a0d5-ef0cff07bf02>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00436.warc.gz"} |
A* Pathfinding
In a 2D top down game I am making I intend to have a fairly decent AI that can navigate around the world in a realistic, efficient and competitive manor. This scenario operates on a grid based world,
but the scenario that I intend to use my A* algorithm on is a much higher resolution, so I plan to make a second version of this with a pixel based resolution and rather than using a grid I will use
probably triangles that are the correct shape and size.
Source is available under a GPL 3.0 licence or later.
My current implementaiton has a "nodeMap" object on each grid tile that stores an object reference to each of it's adjenct nodeMaps or null (eg at the edges of the world). The actual searching
algorithm isn't particularly advanced compared to most A* programmes, but the point is that I made it all by my self and it is a base to work off from when I try to make it work in higher res worlds.
Basically what it does is on the start node it creates a "nodeSearch" object that stores the cumulative cost to get there, the nodes parent (or null for the start node), and the F score. I then
explore each of it's neighbouring nodes that are not null or untraversable (black/wall). Out of these I only acutally explore them if they have not already been explored (in the closed list), or if
we have already found them I only explore them if our cumulative cost is less than what we have already found (sometimes you will see the numbers on the nodes change, that is when the pathfinder has
found a shorter route to the same node.) So the algorithm keeps on exploring, trying out the nodes closest to the finish first (because of the heuristic) until it findes a path. This path can then be
reconstructed by examining the parent node of each nodeSearch object in turn until you reach the start again. I have made it so that the algrithm only expands one node per an act so you can use the
speed bar at the bottom to watch it happen in real time if you want.
Currently I support 4 and 8 way movement (8 way in this example) and heuristics of:
- 0 (Dijkstra's algorithm)
- Manhattan
- Manhattan with straight line vector tie breaker (slightly (*0.001) prefers nodes that are directly in line between the start and the finish. Although it doesn't make a difference in the example as
it is 8-way, it does in 4-way though.
==========WHAT THE NUMBERS MEAN============
BLUE: The blue number in the top left is the cumulative cost of the fastest route of getting to that node. (i.e it costs 1 for each node you travel up, down, left, or right and slightly more
GREEN: The green number in the top right is the heuristic. It is basically an estimate of how much further we have to travel to get to the finish node assuming there is nothing in the way. That is
what makes the program try the nodes that are closes first.
ORANGE: The orange number in the bottom left is the F Score, this is simply the blue number + the green number and it tells the pathfinder which node to try next, the ones with the lowest F score are
tried first.
================HOW TO PLAY=================
To place the Start (green) node press "s" and then click where you would like it to be.
To place the Finish (red) node press "f" and then click where you would like it to be.
To make a node untreversable (make a wall) just click on it.
ONCE YOU ARE DONE PRESS THE [SPACE] BAR TO START THE SEARCH :)
Don't press space more than once! (I will fix it....)
Request For Comments
Looks good. I like the stepping through process. I implemented the A* algorithm here http://greenfootgallery.org/scenarios/1259 and have used it for a few of my games. But I like how your program
steps and displays what's going on, though it is difficult to make out the numbers.
Cool, I looked at your source and generally the idea is the same, obviously not quite the same implementation, but it has the same features like open/closed list (or variable in the object itself in
your case) and checking to see if we have already found the node and if we do have a lower cost to that node then making ourself the parent node. One big difference is that as I say in the
description I have an object on each grid square that stores it's cost (*1 for ver/hoz movement and *sqrt(2) for diagonal) and it stores object references to each of the neighbouring nodes whereas
yours just relies on counting grid positions to find it's neighbouring nodes.. As for the text, I made each grid tile 32x32 pixels so I had to make the font height 9px to fit it in, I guess I could
have made the grid tiles a bit bigger instead... Anyway, all will be revealed when I release the source code soon, I just need to neaten things up a bit and add a couple extra comments :) (Oh yeah,
yours was the first source code I've read in the gallery that has comments in it!)
Pretty cool.
A new version of this scenario was uploaded on Thu Sep 29 18:29:00 UTC 2011 As promised, the source code is now here. Licensed under the GPL v3 or Later. I didn't manage to make many changes, just
added a couple more comments and fixed the bug if you press space twice (well in a way, I guess I could have cleared the open and closed list so you could watch it again. Yeah shoulda done that.) So
now that you have the source you can see the 4 and 8 directional movement and try out all my different heuristic functions. :)
Here's the source code, as promised!
Very nice. Will look over the source code :)
Want to leave a comment? You must first log in. | {"url":"https://www.greenfoot.org/scenarios/3459","timestamp":"2024-11-02T01:26:08Z","content_type":"text/html","content_length":"22761","record_id":"<urn:uuid:96600540-1f18-48f4-a6e3-e04ddb902db2>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00456.warc.gz"} |
Reconstruction modeling of crustal thickness and paleotopography of western North America since 36 Ma
The complex deformation history of the western United States since 36 Ma involves a dramatic transition from a subduction-dominated to a transform-dominated margin. This transition involved
widespread extension, collapse of topography, and development of the interior Basin and Range region. The topographic collapse resulted in significant exhumation of deep crustal rocks exposed in
metamorphic core complexes of the southwestern Cordillera. We use calculated position estimate changes for the western United States from previous work based on a comprehensive compilation of
geological and structural information, and incorporation of constraints from Pacific plate motion history to determine lithospheric strain rates through time, and integrate these strain rates, to
provide quantitative models of crustal thickness and surface elevation, along with formal standard errors, since 36 Ma. Our crustal thickness model at 36 Ma is consistent with a significant crustal
welt, with an average thickness of ∼56.5 ± 2.5 km, in eastern portions of northern to southern Nevada as well as parts of eastern California and through Arizona. Our final integrated topography model
shows a Nevadaplano of ∼3.95 ± 0.3 km average elevation in central, eastern, and southern Nevada, western Utah, and parts of easternmost California. A belt of high topography also trends through
northwestern, central, and southeastern Arizona at 36 Ma (Mogollon Highlands). Our model shows little to no elevation change for the Colorado Plateau and the northern Sierra Nevada (north of 36°N)
since at least 36 Ma, and that between 36 and 5 Ma, the Sierra Nevada was located at the Pacific Ocean margin, with a shoreline on the eastern edge of the present-day Great Valley.
Crustal thickness and topography are a direct result of interactions among mantle convection, continental dynamics, and climatic and erosional processes. Hence, the topographic evolution of mountain
belts and continental interiors reflects directly upon the coupling between mantle and surface processes. The Basin and Range province of the western United States, located between the Sierra Nevada
and the Colorado Plateau, is a unique modern intracontinental extensional province (Dickinson, 2002, 2006) (Fig. 1). This extension follows a protracted phase (155–60 Ma) of crustal shortening and
mountain building associated with a history of subduction and terrane accretion (Hamilton, 1969; Saleeby, 1983; Burchfiel et al., 1992; Saleeby et al., 2003; Liu et al., 2008; Sigloch and Mihalynuk
2013). Since ca. 40 Ma, the western North American plate tectonic history has involved a complex transition from early shallow- to flat-subduction of the east-dipping Farallon slab along the western
margin to its present transtensional environment associated with the evolution of the San Andreas fault system (Bird, 1988, 1998; Liu, 2001; Dickinson, 2002, 2003, 2006; Humphreys and Coblentz, 2007;
Humphreys, 2009). Through this evolution of changing boundary forces, the lithosphere underwent a profound transformation (Atwater, 1970; Coney and Harms, 1984), leading up to the present-day setting
involving both shear and extension. Over time, as the boundary conditions evolved, topography, along with crustal thicknesses, were dramatically altered from probable high elevations of orogenic
plateaus, with corresponding thick crustal welts (Coney and Harms, 1984; Wernicke, 2011), to the current Basin and Range–style topography and thin crustal structure (Wernicke et al., 1987; Atwater
and Stock, 1998; McQuarrie and Wernicke, 2005; Roy et al., 2009).
While the Basin and Range landscape has increased in area by a factor of two over the past ∼20 m.y. (Hamilton and Myers, 1966; Wernicke et al., 1988; Burchfiel et al., 1992), the southern part of the
Great Basin has undergone 200% west-east extension and the northern portion has undergone 50% since the middle Miocene (Wernicke et al., 1988; McQuarrie and Wernicke, 2005). The average present-day
continental crust thickness within the Basin and Range area of the western U.S. is ∼30 km (Shen and Ritzwoller, 2016). Although it is recognized that the widespread extension across the Basin and
Range impacted the regional climatic, faunal, floral, and mammal evolution across the North American southwest (Spencer et al., 2008; Badgley, 2010; Badgley et al., 2014), considerable controversy
surrounds the mechanisms responsible for such extension (Sonder and Jones, 1999; Flesch et al., 2000, 2007; Humphreys and Coblentz, 2007; Ghosh and Holt, 2012; Ghosh et al., 2013a, 2013b).
The eastern and southern margins of the Great Basin contain belts of metamorphic core complexes. These unique features are thought to be a result of the extreme thinning of crust within the Basin and
Range province (Davis, 1980). In the crustal thinning process, deep crustal levels are pulled from underneath an area of extensional detachment and exposed at the surface (Wernicke and Burchfiel,
1982; Davis, 1988; Dickinson, 2004). Coney and Harms (1984) argued that the extreme extension occurred above zones of crustal welts, which resulted from pre-extensional crustal horizontal shortening.
We reconstruct the crustal thickness and paleotopographic evolution of the Great Basin and southern Basin and Range regions using time-dependent crustal strain rate estimates with formal
uncertainties obtained from position estimates of geologic reconstruction by McQuarrie and Wernicke (2005). Quantitative models of crustal thickness evolution between 36 Ma and the present are
achieved by integrating the strain rates back through time. We then adopt a compensation model in order to calculate a paleoelevation. The compensation model involves a variable upper mantle density
with a depth of compensation of 100 km below sea level for the present-day western U.S. To produce the quantitative models of surface elevation evolution from 36 Ma to present, we track the positions
of this upper mantle density field through time in the same way that we track the positions of crustal thickness. We discuss the methodology and the propagation of errors for the strain rates,
crustal thicknesses, and surface elevations. We also discuss the methodology for testing the effect of thermal perturbations as a result of slab rollback and the migration of volcanism on our upper
mantle density and surface elevation models beneath the Basin and Range area.
The Basin and Range province is an ideal locality to test the validity of our assumptions for calculating topography and lithosphere thickness through time due to: (1) the widespread and detailed
geologic mapping that has been completed, resulting in a fairly complete palinspastic reconstruction of Tertiary extension (Coney and Harms, 1984; McQuarrie and Wernicke, 2005), as well as some tests
of paleoelevation from paleothermometers like stable isotopes, and clumped isotopes in lacustrine carbonates (e.g., Huntington et al., 2010; Lechler et al., 2013); (2) Earthscope USArray seismic
network experiments that provide valuable new constraints on crustal and mantle structures (Schmandt et al., 2015; Porter et al., 2016; Shen and Ritzwoller, 2016); and (3) studies directed at
lithosphere evolution through the Cenozoic using thermochronology, geochronology, and geochemistry (e.g., Flowers et al., 2008; Bidgoli et al., 2015).
Our crustal thickness and topography models for the western U.S. will provide constraints for a high-standing topography prior to Basin and Range extension. The estimates of paleotopography and
crustal thickness evolution, with formal uncertainties through time, will provide constraints for quantifying the magnitude and distribution of lithospheric body force distributions and their
influence on collapse and development of the Basin and Range within the Pacific–North American plate boundary zone. Our topography model also may help by providing constraints for studies that
explore linkages between topographic evolution, landscape evolution, and floral and faunal changes. Furthermore, our time-dependent crustal thickness model provides a regional context for
understanding the spatial distribution of exhumation histories, reflected in the vast and growing number of thermochronological studies.
Computation of a Velocity Gradient Tensor Field through Time within the Basin and Range
McQuarrie and Wernicke (2005) provided position estimate changes with uncertainties based on a comprehensive compilation of structural information, synthesis of east-west profiles, and incorporation
of constraints from Pacific plate motion history. For a number of points, they provided the direction and magnitude of displacements, which occurred between specified time intervals. Although this
deformation is accommodated by a large number of fault-bounded blocks (resulting in the Basin and Range province), we model the field through time as a continuum. Our continuum approach is justified
due to a number of practical limitations in our knowledge of the true field of finite fault slip history. For example, whereas the slip rate of an individual fault may, at a specific point in time,
possess high uncertainties, the integrated offset across several faults over a finite time interval will be more precisely known. Thus, the estimate of the average strain rate within a finite volume
is going to be more precise than a point estimate of strain rate at a specific fault location. We argue that the strain rates within finite volumes of the field provided by McQuarrie and Wernicke
(2005) can be reliably quantified, and a continuum treatment is the simplest approach for modeling the strain history in the western U.S.
To reconstruct the paleotopography of the western U.S. since 36 Ma, it is necessary to obtain estimates of the kinematics of the lithosphere through time. To this end, we use position estimates from
the fault displacement compilation of McQuarrie and Wernicke (2005) for 857 sites in the western U.S. to calculate displacement rates (Fig. 2). McQuarrie and Wernicke (2005) provided values in 2 m.y.
bins for periods between the present time to 18 Ma, and in 6 m.y. bins for periods between 18 and 36 Ma. Using these position estimates, we obtain crustal velocities, with errors, at 857 sites at 1,
3, 5, 7, 9, 11, 13, 15, 17, 21, 27, and 33 Ma. We take the midpoint between the latitude and longitude of beginning and end points of each site for a specified time interval, along with the total
magnitude and direction of displacement, to obtain a velocity vector for that location. This vector represents the average velocity of the deforming lithosphere for that location during the specified
interval of time. We define a grid where we can use all velocity estimates to determine a continuous model of a velocity gradient tensor field for the time periods above (Fig. 2). Because locations
of the stable blocks like the Colorado Plateau and Great Plains are the firmest constraints for delineating important boundaries between relatively undeformed and deformed lithosphere during the time
interval of interest (0–36 Ma) (McQuarrie and Wernicke, 2005), we put a grid of points (0.5° × 0.5°) with zero velocities and 1 mm/yr standard errors as additional constraints for the strain rate
model within these stable blocks (Fig. 2). During the time period that the Rio Grande rift was opening, based on McQuarrie and Wernicke’s (2005) model, we add the constraint of velocities on the
Colorado Plateau that favor little to no internal deformation and that are consistent with the direction of opening of the rift. We also add a dense set of points (1.0° × 1.0°) with 2 mm/yr standard
errors on the Pacific plate as constraints based on the direction of motion of the Pacific plate relative to stable North America between 16 Ma and the present (Atwater and Stock, 1998; DeMets and
Dixon, 1999; McQuarrie and Wernicke, 2005) (Fig. 2). Having constraints for the Pacific plate through time provides the added benefit of resulting in a correct prediction of the timing of the coastal
fragment of California being transferred to the Pacific plate. Additionally, these constraints help inform strain rate and crustal thickness models during the time interval that Baja California is
transferred to the Pacific plate and the Gulf of California opens up.
The continuous approach we adopt enables us to quantify formal errors in strain rates and rotation rates averaged within the areas in Figure 2, given a set of velocity values with uncertainties. The
smoothed velocity gradient tensor field solution is obtained using a damped least-squares inversion method (Beavan and Haines, 2001; Holt et al., 2000). In this method, the horizontal velocity field
within a specified frame of reference is expressed as u(r) = W(r) × r, where W(r) is a continuous rotation vector function, and r is the radial position vector, passing through the Earth’s center and
defining a point on the Earth’s surface. In the least-squares inversion, the values of W(r) are determined at the knotpoints of the grid, the grid line intersections (2475 points on the grid in Fig.
2). A continuous field of velocities throughout the domain is provided by bicubic spline interpolation of W(r) parameters (Beavan and Haines, 2001). Spatial derivatives of these continuous functions,
W(r), provide strain and rotation rates (Haines and Holt, 1993). The parameterization thus gives us a continuous model field of horizontal velocities, strain rates , and rotation rates about the
vertical axis (Beavan and Haines, 2001; Holt et al., 2000) throughout the domain for each given time step on the surface of the sphere (Fig. 3) (see Supplemental Animation S1^^1).
In the inversion procedure, the objective functional that is minimized is the following:
where σ
and σ
are the formal standard errors in velocity for north and east directions, respectively; here,
is the observed velocity in north direction,
is the observed velocity in east direction, and velocities superscripted with “mod” represent the model velocity for that component. Positive directions are east and north, respectively. In the
second part of the functional, are the model values of strain rates, and υ is a weighting factor. In practice, the second part of the functional, involving the second invariant of only model strain
rates, is achieved by defining zero values for observed strain rates within all areas (
Beavan and Haines, 2001
). Note that the weighting factor υ consists of probabilistic constraints on these zero values of observed strain rates within areas, where:
Here, ) is the variance for zero-valued observed strain rates; S[i] is the area of the i^th cell on the grid. Thus, 1/υ is an adjustable strain rate variance parameter controlling the degree to which
the continuous parameters can fit the input velocities (point values) or the zero strain rate constraints (area averages) (Beavan and Haines, 2001). The value of υ is a single adjustable parameter,
assigned as a constant for all areas.
Large variances in strain rates impose small weighting on the second part of the functional in Equation 1. Small variances in strain rates impose larger weighting on the second part of Equation 1.
Minimization of Equation 1 provides the minimum weighted sum of squares misfit to the velocities, under the constraint that there is a global minimum for the second invariant of the model strain
rates, weighted by υ. Minimization of the functional of Equation 1 with small weighting factor υ implies that large model strain rates could be produced in order to minimize the weighted sum of
squares misfit to the velocities.
The shape of the continuous model strain rate field is controlled by how close we want to match the velocities. If the observed and modeled velocities are matched closely, then the field might be
fairly irregular; however, if larger misfits are allowed with the observed velocities, the model field will be smoother. Hence, choosing a proper value for υ is of high importance.
Beavan and Haines (2001)
argued that an optimal value of υ was one that gave the sum of the squares misfit of the velocity components,
, equal to the number of degrees of freedom (
). Because there are two degrees of freedom associated with each horizontal velocity estimate,
= 1714 for 857 velocity observation sites. However, we settle on an optimum value for υ that provides sum of the squares divided by the number of degrees of freedom approximately equal to 1.5. This
allows for the fact that the velocity errors may be underestimated, and results in a smoother model solution. We quantify the misfit using the
a posteriori
formal standard error of unit weight (SEUW) (
Beavan and Haines, 2001
), which is:
We explore three different levels of smoothing for solutions within all time bins, with SEUW equal to 1.0 (underdamped), 1.3 (intermediate), and 1.7 (overdamped) (Fig. 4). If the uncertainties in
velocities account for all unknowns in position and timing for accumulated fault motions over a given time interval, then what we call the underdamped solution would be optimal. Our solution of
choice (SEUW = 1.3) provides results that are slightly more smoothed than the one where SEUW = 1.0, and accounts for the possibility that average uncertainties for all velocities are ∼30% higher than
the estimates used.
The formal uncertainties in model strain rates are controlled by the spatial density of velocity observations, the velocity errors, and the degree of formal smoothing in the velocity field
interpolation (Fig. 5). Note from Figure 5, which shows the formal standard errors for vertical strain rates (), the greater the degree of smoothing (with lower resolution), the smaller the formal
errors (see Supplemental Animation S2 [^footnote 1]).
Calculating Crustal Thickness and Position Changes through Time
Given the solutions for the different time bins, we time-integrate the instantaneous flow field estimates to obtain coordinate changes and crustal thickness changes through time. In our analysis, we
assume that the lithospheric deformation is vertically coherent, or that vertical variations in horizontal velocity within the lithosphere are small in comparison with horizontal gradients of
horizontal velocity. We also approximate zero volume change, ∇ u = 0, and thus the vertical strain rates . Furthermore, we also ignore effects of erosion and igneous crustal addition. Given these
assumptions, the vertical strain rates are used to calculate crustal thickness evolution throughout the southwestern U.S.
Later we will discuss the possibility of lower crustal flow within regions that have experienced extreme crustal extension and mid-crustal exhumation (metamorphic core complexes). We argue here that
the assumption of vertical coherence is a reasonable first approximation even in the presence of lower crustal flow as long as (1) this lower crustal flow is not pervasive, but rather is isolated
beneath zones of high topography and thick crust that underwent extreme extension, and (2) horizontal strain rates, averaged over 200 km length scales, do not differ by more than a factor of two
between upper and lower crust. Not accounting for a factor of two difference between upper and lower crust, for extensional strain rate magnitudes found in this paper, results in only a ∼10% error in
final crustal thickness estimates. Yet such a difference in strain rate, when integrated over a geological time scale (10 m.y.), can result in substantial differential offsets, much like that
observed in the vicinity of metamorphic core complexes.
Calculating Crustal Thickness
The calculation of crustal thickness evolution involves (1) tracking coordinate changes through time, based on the time-dependent velocity field, and (2) tracking the crustal thickness changes of
those corresponding coordinates.
The parameterization of a continuous velocity field enables us to track position changes using relatively small time steps (0.5 m.y.). We use a fixed grid and velocities in North American frame, and
we assume that the strain rates do not change temporally within each time interval. There is no advection of fault source structures during each time interval (2 m.y. intervals between 0 and 18 Ma,
and 6 m.y. intervals between 18 and 36 Ma), but the advection of source structures from one time interval to the next time interval is accounted for. Using the horizontal velocity gradient tensor
field for each specific time interval, we integrate points back in time to obtain past position coordinates for a dense grid of points (47,616 points, 0.1° × 0.1° spacing).
In general, the relationship between a vector
in a deforming region at time
= 0, and time
, is:
) is the deformation gradient tensor (
McKenzie and Jackson, 1983
). For our specific case,
would be a vertical vector representing crustal thickness,
The time dependence of
is the velocity gradient tensor (
McKenzie and Jackson, 1983
). We are interested in the component F
for a strain rate field that is considered vertically coherent. Using this vertical coherency for the vertical strain rates (
), we have from
Equation 5
Solving this first-order differential equation, we have , where Δ
is the incremental time step (0.5 m.y.). From
Equation 5
, the time-dependent crustal thickness,
, is
is the crustal thickness at the beginning of the time step, and the vertical strain rate (
) is appropriate for a specific time interval and coordinate position within the generally spatially varying (latitude and longitude only) strain rate field (
Fig. 6
We start with a dense set of coordinates of 0.1° spacing (longitude = ϕ, latitude = θ) and calculate a shift in position (Δϕ, Δθ) relative to the stable North American frame, using a time increment
of Δ
= −0.5 m.y. and the instantaneous velocity gradient tensor field solution. For the
– 1 time interval, the position shift at the
time interval is Δϕ = ϕ
– ϕ
Δθ = θ
– θ
, where ϕ
= ϕ
+ Δϕ, and θ
= θ
+ Δθ. The crustal thickness (
) for the
time interval is:
where is the vertical strain rate for the
velocity gradient tensor field solution, where
= 1–72 and
= 1–12. There are 72 time steps (
) because the interval over which we have strain rate constraints is 36 m.y. and the time increment is Δ
= −0.5 m.y. There are 12 velocity gradient tensor field solutions (
), which cover the time periods of 0–2, 2–4, 4–6, 6–8, 8–10, 10–12, 12–14, 14–16, 16–18, 18–24, 24–30, and 30–36 Ma. Our starting crustal thickness data set,
; θ
), is from
Shen and Ritzwoller (2016)
, and all model solutions for crustal thickness and position at a given time in the past can only be obtained by starting at
= 1,
= 1 (present-day), and integrating backward in order. Because the
Shen and Ritzwoller (2016)
crustal thickness model covers only the entire U.S. and not Mexico (southern Basin and Range area) and offshore (oceanic plate), we merge the crustal thickness data set for Mexico and offshore from
the CRUST 1.0 global crustal model (
Laske et al., 2013
) with the crustal thickness data set for western U.S. from
Shen and Ritzwoller (2016)
Fig. 7
We make the simplifying assumption that the strain rate field is stationary during the time interval covering the particular velocity gradient tensor field solution (j). Because strain rates are
spatially averaged and smoothed, the change in coordinates (Δϕ, Δθ) for any single time increment k (Δt = −0.5 m.y.) is small in comparison to distances over which there are substantial changes in
strain rate values. Thus, the inaccuracies are small relative to the exact solution, which would involve tracking strain rate values in an advecting field containing fault source terms. With a change
to a new velocity gradient tensor field solution (j = j + 1), the spatial shift in the strain rate field properly accounts for the advection of source faults (McQuarrie and Wernicke, 2005) relative
to the North American frame.
Crustal extension and exhumation of the middle crust, as exemplified in core complex exposures, likely involved some flow of weak lower crust (e.g., Block and Royden, 1990). Such a flow of a weak and
hot lower crust, enhanced by magmatic activity (Armstrong and Ward, 1991), can be a result of tectonic denudation and isostatic adjustment to crustal stretching, topographic forces, and sedimentary
loading by erosion (Wernicke, 1992). Yet today, the Moho is relatively flat below regions containing core complexes (Snow and Wernicke, 2000). A present-day flat Moho suggests that the paleo–crustal
root must have flattened as middle crustal rocks were exhumed. Moreover, a weak lower crust would have likely facilitated a broadening of the crustal roots prior to and during the extension, and
consequently a broadening of the topography. As discussed above, our treatment using vertical coherence cannot account for lower crustal flow. We have also argued that not accounting for lower
crustal flow is unlikely to lead to errors of more than 10% for paleo–crustal thickness. Nevertheless, we approximate the influence of the lower crustal flow in the following way and present crustal
thickness models that incorporate this approximation in supplementary for comparison. We approximate the effect of lower crustal flow by applying additional spatial smoothing of the model strain
rates. This smoothing has the influence of spreading the strain rates out spatially, but preserving the total integral of those strain rates (total extension). We will show that in comparison with
the unsmoothed strain rates, the smoothed solution produces broader crustal welts and slightly lower elevations.
Integration back to 36 Ma shows a substantial crustal welt with an average crustal thickness of ∼56.5 ± 2.5 km within eastern Nevada, eastern California, and northwestern, central, and southeastern
Arizona (Fig. 7). The formal estimates of standard errors from the model strain rate fields enable us to calculate error propagation for crustal thickness H (ϕ, θ, t) at all time steps (see the
Appendix) (Fig. 8) (see Supplemental Animations S3, S4 [^footnote 1]). The smoothed solution that approximates the effect of lower crustal flow shows a slightly wider zone of crustal welt that with
an average of ∼54.3 ± 2.5 km, which, as expected, is wider and not as thick as the unsmoothed solution (see Supplemental Fig. S1^^2, and Supplemental Animation S5 [^footnote 1]). We also analyzed the
CRUST 1.0 model (Laske et al., 2013) for the western U.S. for comparison with results obtained using the model of Shen and Ritzwoller (2016) (see Supplemental Figs. S2, S3 [^footnote 2]).
Calculating Upper Mantle Density and Surface Elevation
To create topographic maps of the western U.S. since 36 Ma, it is necessary to have a compensation model. Becker at al. (2014) argued for significant contributions from dynamic topography in the
presence of a thinned lithosphere-asthenosphere boundary below the Great Basin and southern Basin and Range. Time-dependent topographic variations owing to possible temporal variations in the
convecting regime are beyond the scope of the study, and such a treatment would require input from time-dependent models (e.g., Liu et al., 2008; Moucha et al., 2008). Because predictions for dynamic
topography in the region do not exceed ∼1 km since 36 Ma (Moucha et al., 2008; Liu et al., 2010), our overall findings are not compromised because such a signal from dynamic topography would
constitute only 25% of the total paleoelevation.
However, we adopt a simple model that assumes pressure equilibrium at an upper mantle depth of 100 km, consistent with the inference of negligible elevation variations owing to dynamic topography
within the western U.S. (
Levandowski et al., 2014
). We start by investigating the present-day topography and crustal thickness estimates. We plot topography (ETOPO5 elevation data;
) versus crustal thickness (
Fig. 9A
). If all topography points were compensated at 100 km depth relative to a reference column with a non-variable upper mantle density (e.g., 3300 kg/m
), then all points in a crustal thickness versus elevation plot would lie on a line defined by:
Here H[A] is the theoretical crustal thickness, H[SL] is the line intercept (crustal thickness at sea level), h[e] is elevation, and [ρ[m] / (ρ[m] – ρ[c])] is the slope of the line, where ρ[c] is
crustal density and ρ[m] is the reference upper mantle density. For this perfect compensation model (Airy compensation), both ρ[c] and ρ[m] do not vary laterally.
If points do not lie on the line, then the crustal density, the mantle density, or both do not match with the reference model. While it is recognized that there are lateral variations in the average
crustal density within the western U.S. (
Levandowski et al., 2014
), in our simple approach, we assume that all pressure variations are due to upper mantle densities relative to the reference model, and that the crustal density is constant. For example, if
topography points lie above the best-fit line (
Fig. 9
), then they are supported by mantle densities that are greater than the reference density; points below the line are supported by mantle densities less than that of the reference model. Using an
average crustal density of 2700 kg/m
for the western U.S. (
Mooney and Kaban, 2010
Levandowski et al., 2014
), the slope of the best-fit line in
Figure 9A
yields a best-fit reference upper mantle density of 3150 kg/m
, and the intercept defines
of 27 km (thickness at sea level). As part of the assumption that topography is compensated at 100 km, we solve for the upper mantle density for each point. Allowing for a variable upper mantle
density (ρ
′), the revised relationship between actual crustal thickness (
) (
Shen and Ritzwoller, 2016
), crustal density (ρ
), elevation (
), reference upper mantle density (ρ
), and upper mantle density (ρ
′) is:
is the distance from base of crust to the compensation depth of 100 km below sea level (
Fig. 9B
). By subtracting
Equation 9
Equation 10
, we solve for the variable upper mantle density (ρ
Solving for ρ
′ yields a coherent pattern that shows lower densities in the Great Basin and southern Basin and Range provinces, normal densities in the Colorado Plateau, and higher densities in the Great Plains (
Fig. 10
) (see Supplemental Animation S6 [
^footnote 1
]). The patterns in this upper mantle density model are consistent with patterns of seismic velocity structure for uppermost mantle at depths of 90–100 km in the western U.S. (
Yang et al., 2008
Schmandt and Humphreys, 2010
Moschetti et al., 2010
Obrebski et al., 2011
Schmandt et al., 2015
Porter et al., 2016
Shen and Ritzwoller, 2016
). That is, comparing with these seismic velocity models, the present-day density model (
Fig. 10
) shows values lower than the reference value in areas corresponding to slower-than-average upper mantle S- and P-wave velocities (Great Basin, southern Basin and Range, southern margin of the
Colorado Plateau). Likewise, higher densities correspond to areas where upper mantle P- and S-wave velocities are generally higher (northern Colorado Plateau, western edge of the Great Plains). Our
present-day density model is also in agreement with upper mantle temperature models (
Lowry and Pérez-Gussinyé, 2011
Afonso et al., 2016
). Having the crustal thickness and upper mantle density, using
Equation 10
, and because
= 100 km –
, we solve for the elevation (
) as:
As a first approach, to calculate the paleoelevation, we make the simplifying assumption that ρ
′ does not change throughout time. We later discuss the effect of thermal perturbation on a new time-dependent upper mantle density model, with the same depth of compensation, and recalculate the
paleoelevation based on the variation of ρ
′ through time. We track the position of the ρ
′(ϕ, θ) values (
Fig. 10
) in the same way that we track all crustal thickness points through time. Then, by substituting the time-dependent crustal thickness from
Equation 8
Equation 12
, we arrive at the elevation (
) for the
time interval (
Fig. 11
where again the integration must start at
= 1,
= 1, and proceed in order. Using the formal estimates of standard errors for crustal thickness enables us to calculate formal standard errors for surface elevation at all time steps (see the
Appendix) (
Fig. 12
) (see Supplemental Animations S7, S8 [
^footnote 1
]). To test the sensitivity to compensation depth, we solve for ρ
′ and surface elevation assuming a depth of compensation of 150 km. This yields a mean value of 3170 kg/m
for ρ
′ and a slightly narrower range (3120–3220 kg/m
), and a nearly identical prediction for surface elevation through time. We also analyzed the CRUST 1.0 model (
Laske et al., 2013
) for comparison with results obtained using the model of
Shen and Ritzwoller (2016)
(see Supplemental Figs. S4, S5, S6, S7 [
^footnote 2
The final integrated result leads to a high Nevadaplano with an average elevation of ∼3.95 ± 0.3 km at 36 Ma for the optimal solution (SEUW = 1.3). This result of a high Nevadaplano at 36 Ma appears
to be robust, as it is also obtained for both the underdamped (4.67 ± 0.4 km) and overdamped (3.75 ± 0.2 km) strain solutions (Fig. 13) (see Supplemental Fig. S8 [^footnote 2]).
By taking into account the fact that lower crustal flow was likely a factor, the smoothed surface elevation models yield a broader plateau and at slightly lower elevation of ∼3.57 ± 0.3 km (see
Supplemental Fig. S9 [^footnote 2] and Supplemental Animation S9 [^footnote 1]).
Influence of Thermal Perturbations on Western U.S. Upper Mantle Densities
Major thermal perturbations occurred in the upper mantle during the Oligocene and Miocene under the Basin and Range, as indicated by the mid-Tertiary ignimbrite flareup and the initiation of Basin
and Range extension (Armstrong and Ward, 1991; Humphreys, 1995; Best et al., 2013, 2016). The thermal perturbation associated with this magmatic history likely impacted the density field of the upper
mantle through time, as the present-day mantle densities beneath the Great Basin reflect warmer upper mantle that both postdates and likely results from the Farallon slab rollback during the
Cenozoic. Therefore, to address the effect of thermal perturbation on the upper mantle density variation and compensated topography, we derive a thermal model for the upper mantle density of the
western U.S. using the North American Volcanic and Intrusive Rock Database (NAVDAT; http://www.navdat.org) for magmatism that occurred in the western U.S. from 36 Ma to present. To this end, we use a
finite element package to solve the steady-state conductive heat flow equation in COMSOL Multiphysics software (version 5.2) with imposed sources for temperature perturbation linked to the temporal
and spatial history of magmatism from the NAVDAT data set (see the Appendix).
Our model for the thermal perturbation history for the upper mantle is an approximation. It is a superposition of four steady-state conductive heat distribution models. The four models divide the 36
m.y. time histories into four roughly equal parts (see Supplemental Fig. S10 [^footnote 2], and Supplemental Animation S10 [^footnote 1]). Our result can only be an approximation because we are
obtaining solutions to the steady-state heat flow problem and not the time-dependent heat flow equations. Moreover, our model history between 36 Ma and 0 Ma is not constrained to match present-day
heat flow. Instead, our approach is designed to test the sensitivity of topography models to the lithosphere’s upper mantle density changes arising from the estimates of thermal perturbations
associated with the temporal and spatial history of magmatic activity (NAVDAT data set). The areas with densities of ∼3100 kg/m^3 for this new upper mantle model at 36 Ma would represent density
values following slab rollback, considering that the bulk of the north-to-south slab rollback was complete across southern Nevada (north of 36°N) by 34 Ma (Humphreys, 1995; Dickinson, 2003, 2006; Liu
et al., 2010) (see Supplemental Fig. S11 [^footnote 2], and Supplemental Animations S11, S12 [^footnote 1]).
This new variable upper mantle density model yields a new surface elevation model showing slight differences from the original upper mantle density model, which lacked density changes associated with
temperature variations. The main differences are slight increases in Nevadaplano elevation (by ∼1 km) as temperatures increase there (by 150 °C) owing to magmatic activity between 36 and 30 Ma.
Following 30 Ma, the effect of topographic collapse dominates over temperature increase, and elevations for the Nevadaplano gradually decrease. There are also slight increases in elevation of the
Rocky Mountains and even the Colorado Plateau (by <300 m), owing to volcanic activity in the Rockies and some conductive heat increase in the mantle lithosphere below the Colorado Plateau. One
important outcome of our investigation using the NAVDAT data set is that the collapse of topography correlates in space and time with the distribution of active volcanism during the Oligocene and
Miocene within the Basin and Range of the western U.S. (see Supplemental Fig. S12 [^footnote 2], and Supplemental Animation S13 [^footnote 1]). This collapse of topography was presumably facilitated,
in part, by the thermal perturbation in the upper mantle associated with the magmatism (see Supplemental Digital Files^^3).
The timing and extent of the strain history embedded in our models are linked directly with McQuarrie and Wernicke’s (2005) model, who carefully compiled the existing structural data and estimates of
net movements through time, along with estimates of uncertainties. Our contribution is to use this information to provide quantitative models of crustal thickness and surface elevation with
associated formal uncertainties. Our formal uncertainties do not account for erosion and volume addition from igneous activity. Erosion reduces crustal thickness if the sediments leave the system.
Therefore, starting with the present-day crustal thicknesses and integrating them back to estimate original thicknesses provides minimum estimates of pre-erosion crustal thicknesses. However, the
basins of the Basin and Range province have significant thicknesses of locally derived sediments, which are part of the present-day crustal thickness distribution estimated through seismology and
used in this study (Shen and Ritzwoller, 2016). Therefore, if most of the sediments are locally deposited, we do not expect to grossly underestimate original crustal thicknesses. On the other hand,
igneous addition adds material to crustal thicknesses over time. Thus, not accounting for igneous addition causes an overestimation for crustal thickness through time. Because these two unknown
factors have opposite signs, they may tend to cancel one another if the mass of sediments that have left the system through erosion is roughly equivalent to the mass of crustal igneous addition.
Crustal Thickness Model of the Western U.S.
Our time-dependent crustal thickness model at 36 Ma is consistent with a significant crustal welt at that time, looking remarkably similar to the welt proposed by Coney and Harms (1984) from northern
to southern Nevada (Nevadaplano), eastern California, and Arizona. However, our model does not predict the 50–55 km welt north of 38°N within eastern California predicted by Coney and Harms (1984) (
Fig. 14).
A roughly 55-km-thick crust in Nevada is also consistent with high Sr/Y ratios of Jurassic–Cretaceous intermediate continental calc-alkaline magmatic rocks (Chapman et al., 2015). Decreasing Sr/Y
ratios suggest mid-Eocene to Oligocene extension and decreased crustal thickness to 30–40 km by the Miocene as a result of extension (Chapman et al., 2015).
Such a thick crustal welt at around 36 Ma could have driven extension, crustal thinning, and collapse of topography, resulting ultimately in the present-day geometry of the Great Basin and Basin and
Range provinces. Because this crustal welt would have been present long before the initiation of collapse (Sonder et al., 1987; DeCelles, 2004; Liu and Gurnis, 2010), such a collapse was probably
initiated by reduction of viscosity by a mantle-derived heating event (Liu and Shen, 1998; Liu, 2001), thermal relaxation of the overthickened crust (Sonder et al., 1987), or collapse and steepening
of a previously shallowly dipping Laramide Benioff zone, which may have reduced the regional stress and possibly started extension (Coney, 1980; Armstrong, 1982; Spencer, 1984; Humphreys, 1995; Jones
et al., 1996; Dickinson, 2002, 2003).
Armstrong (1982), Coney and Harms (1984), and Parrish et al. (1988) suggested that metamorphic core complexes were extensional in origin and mainly Tertiary in age (65–2 Ma). Coney and Harms (1984)
argued that the metamorphic core complexes formed above sites of extreme crustal thickening. Tracking the positions of core complexes back to 36 Ma locates them above the thickest crust and the
Nevadaplano (Fig. 14) (see Supplemental Animations S14, S15 [^footnote 1]), confirming the findings of Coney and Harms (1984). However, Coney and Harms (1984) predicted a large crustal welt in
eastern California (north of 38°N) where no core complexes are observed. By contrast, our model predicts only a modest crustal welt there.
Paleotopography Model of the Western U.S.
Our final integrated topography model shows a highland with an average elevation of ∼3.95 ± 0.3 km in central, eastern, and southern Nevada, western Utah, parts of easternmost California, and for
northwestern Arizona (Nevada plano). The Mogollon Highlands (Cooley and Davidson, 1963; Elston and Young, 1991) are also present within central and southeastern Arizona at 36 Ma (Fig. 11). This
topographic belt stretching from northern Nevada to southeastern Arizona and northern Mexico results from a significant crustal welt that was likely a consequence of Sevier-Laramide convergence (
Dickinson, 2002; DeCelles, 2004; Sigloch and Mihalynuk, 2013).
The calculated surface elevation estimates at ca. 36 Ma represent elevations following slab rollback, where the crustal root supporting much of that elevation was inherited prior to 36 Ma from
Sevier-Laramide convergence (DeCelles et al., 1995; Dickinson, 2002; DeCelles, 2004). Our surface elevation models show that the crustal topography changes of the Nevadaplano, owing to the transition
in upper mantle density associated with slab rollback and thermal perturbation, constitutes only ∼25% of the total elevation of the Nevadaplano. This heating of the upper mantle lithosphere
accompanied the migration of volcanism following slab rollback and is represented as a heating event in our models (temperature increase of 150 °C in upper mantle lithosphere) between 36 Ma and 30
Ma. This temperature increase results in a reduction of 60 kg/m^3 in upper mantle density, which yields an elevation increase of ∼1 km for Nevadaplano. Whereas 1 km is a substantial uplift, the bulk
of the elevation of the Nevadaplano (average 3.95 ± 0.3 km) results from the crustal welt as opposed to the upper mantle density changes associated with slab rollback and thermal perturbation (e.g.,
Mix el al., 2011).
Our topography model also shows that between 36 and 5 Ma, the Sierra Nevada was located adjacent to paleo–sea level, with a shoreline on the eastern edge of the present-day Great Valley (Fig. 11).
The model for the northern Sierra Nevada (north of 36°N) shows little to no elevation change throughout the deformation period since at least 36 Ma (Fig. 11), consistent with idea that the majority
of uplift occurred prior to 36 Ma, in the Late Cretaceous to early Cenozoic (Cassel et al., 2009; Cecil et al., 2010). However, by contrast, our model for the southern Basin and Range at 14 Ma
involves a steep gradient in crustal thickness from ∼30 km in eastern California to ∼50–55 km just east of the California–southern Nevada border region (Fig. 7). The topography model for this time
period shows variable elevation in eastern California ranging from 0.5 to 1.5 km and reaching high elevations of ∼3.5–4 km in the area of the 50–55-km-thick crust in southern Nevada (Fig. 11).
Our paleotopography estimates can be compared to a variety of proxies that have been employed by many studies to estimate paleotopography and paleoaltitude across the western U.S. Because the
isotopic composition of precipitation somewhat directly scales with elevation, it can be used to reconstruct topographic histories of mountain belts (Mix et al., 2011), though the basins in which the
carbonates that can be analyzed are found clearly represent a minimum elevation. Maps of the δ^18O of precipitation for different time bins during the Cenozoic highlight the major isotopic shifts
observed within individual basins along the Cordillera (Mix et al., 2011; Horton et al., 2004; Kent-Corson et al., 2006). Using spatial and temporal patterns of δ^18O of precipitation, Mix et al.
(2011) determined paleoelevations for Elko Basin in northeastern Nevada (36–28 Ma, ∼3.4 km), Copper Basin in northeastern Nevada (37 Ma, ∼3.4 km), and the Sage Creek Basin in southwestern Montana
(38.8–32.0 Ma, ∼3.7 km). They also show an Eocene–Oligocene highland of ∼3.4 km in eastern Nevada and western Utah. The paleobotanical study by Wolfe et al. (1998) also suggests the presence of an
Eocene–Oligocene highland of ∼4 km or higher in the Copper Basin region (northeastern Nevada).
Paleobotanical results for the late Eocene House Range flora in the Sevier Desert (west-central Utah) yield a paleoelevation of ∼3–4 km at ca. 31 Ma (Gregory-Wodzicki, 1997). Based on δD measurements
in hydrated glass from ignimbrites, Cassel et al. (2014) indicated the presence of a high, broad orogen that stretched across northern to southern Nevada during the Eocene to Oligocene, with the
highest elevations of 3.5 km in the late Oligocene. Gébelin et al. (2012) results based on δ^18O and δD values calculated from muscovite from Snake Range Mountain (northwestern Nevada) yield an
elevation of ∼3850 ± 650 m between 27 and 20 Ma. These are in agreement with our paleotopography models for 36–20 Ma, showing a Nevadaplano with an average elevation of ∼3.95 ± 0.3 km in central,
eastern, and southern Nevada, western Utah, and parts of easternmost California (Fig. 11).
Paleobotanical analyses obtained by Wolfe et al. (1997) for several mid-Miocene floras in eastern Nevada suggest a paleoaltitude of ∼3 km at ca. 15–16 Ma. Their assumption that the highland had
collapsed by ca. 13 Ma is in agreement with our topography model showing the altitudes similar to the present-day in eastern Nevada at 13 Ma and afterward (Fig. 11).
Mix et al. (2011) suggested that slab rollback may be the primary source of a wave of uplift of the Nevadaplano that swept from north to south during the Oligocene. They argued that the southernmost
measurement achieved an elevation change of ∼2.5 km late in the time between 36 and 28 Ma, possibly when slab rollback was occurring beneath the region. As we have explained above, we estimate a 60
kg/m^3 change in upper mantle density beneath a ∼55-km-thick crust associated with thermal heating, and this produces only ∼1 km of uplift. Although Mix et al. (2011) may have indeed resolved a
component of uplift associated with slab rollback, we show that elevations are expected to have still been high (in excess of 3 km) above the substantial crustal welt that was present just prior to
the slab rollback. Another potential source of uplift prior to 36 Ma may have resulted from hydration of the lithosphere from the Farallon slab during the flat slab subduction phase (Humphreys et
al., 2003; Jones et al., 2015; Porter et al., 2017).
Cassel et al. (2009) also showed that northern Sierra Nevada elevation (100 km east of the paleo-coastline) in the Oligocene was similar to that of the present day (∼2800 km). This is consistent with
results of Chamberlain and Poage (2000) who measured little change in δ^18O of smectites from the northern Sierra Nevada since 16 Ma, suggesting that the Sierra Nevada has been approximately at the
same elevation since 16 Ma. This is in agreement with an elevation of ∼2.5 km shown in our model from 36 Ma to the present day. Our model also shows a Sierra Nevada adjacent to the paleo–sea level
shoreline before 5 Ma. This is consistent with the idea that volcanic materials originating in northern and central Nevada were deposited during the Paleogene into the Pacific Ocean within what is
now the Great Valley (Faulds et al., 2005; Garside et al., 2005; Henry, 2009).
Clumped isotope (Δ47) thermometry of lacustrine carbonates (∼17–24 °C) from the central Basin and Range and the southern Sierra Nevada Bena Basin indicate that middle Miocene paleoelevations in the
Death Valley region were <1.5 km (Lechler et al., 2013), consistent with our estimates. However, our results do not agree with the 1.5 km elevations in the middle Miocene for the Spring Mountains
region obtained by Lechler et al. (2013). They acknowledged that the 1.5 km elevation was exceptionally low for the expected pre-extensional crustal thicknesses there (>52 km), and that final
post-extensional elevations can only be explained by igneous volume addition. Supporting evidence for the topography in our model lies in the agreement we see for the exhumation of core complexes
along the belt near the border region between eastern California and southernmost Nevada. Possible discrepancies may be explicable by the sampling of lacustrine carbonates within low-lying basins
that are not resolvable with our modeling technique, which addresses the smoothed components of the deformation field and thus smoothed topography. Huntington et al.’s (2010) paleoelevation estimates
from apatite (U-Th)/He data of the Grand Canyon reveal that most of the Colorado Plateau’s lithospheric buoyancy and uplift occurred at ca. 80–60 Ma. This is in agreement with our surface elevation
model showing almost no elevation change for Colorado Plateau at least during the past 36 m.y.
Our paleotopography reconstruction models provide a framework for considering the timing of the topographic development of the Sierra Nevada and Basin and Range province. Future models can be refined
through the incorporation of results from isotopic and thermochronological studies.
We use the compilation of McQuarrie and Wernicke (2005) to determine a horizontal velocity gradient tensor field solution for the lithosphere through time. Using present-day crustal thickness
estimates for initial conditions, and assuming volume conservation, we integrate these solutions to produce models of crustal thickness and surface elevation, along with formal uncertainties, through
the past 36 m.y. Our crustal thickness model at 36 Ma is consistent with a significant crustal welt, with an average crustal thickness of 56.5 ± 2.5 km, looking similar to the welt proposed by Coney
and Harms (1984) from northern to southern Nevada and through central and southeast Arizona, which was likely a consequence of Sevier-Laramide convergence. Such a thick crustal welt at ca. 36 Ma
could have played a major role in driving extension, crustal thinning, and collapse of topography, resulting ultimately in the present-day geometry of the Great Basin and Basin and Range provinces.
Our final integrated topography model shows a Nevadaplano of ∼3.95 ± 0.3 km average elevation in central, eastern, and southern Nevada, western Utah, parts of easternmost California, and northwestern
Arizona. Highlands of significant elevation (∼3–3.5 km) are also present along a belt in central and southeastern Arizona at 36 Ma (Mongollon Highlands). Our model, between 36–5 Ma, shows a Sierra
Nevada adjacent to paleo–sea level, with a shoreline on the eastern edge of the present-day Great Valley. Moreover, based on our model, the Colorado Plateau and the northern Sierra Nevada (north of
36°N) show little to no elevation change since at least 36 Ma.
We would like to express our gratitude to reviewers Brian Wernicke, an anonymous reviewer, and Science Editor Raymond Russo for their careful reading of our manuscript and their many insightful
comments and valuable suggestions, which greatly improved the quality of the manuscript. We thank Yuanyuan Liu and Rubin Smith for assistance with early calculations during the initial stages of this
work. We are very grateful to Weisen Shen for kindly providing his most recent database for crustal structure of the United States. Grateful thanks are also expressed to Nadine McQuarrie, Daniel
Davis, Catherine Badgley, Tara Smiley, Ryan Porter, and Gavin Piccione for their general interest, and fruitful discussions, which greatly assisted in the development and refinement of this study.
This research was supported by the National Science Foundation under grant numbers EAR-1246971 and EAR-1052989, and NASA ESI under grant number NNX16AL18G, as well as Southern California Earthquake
Center under grant numbers 16291 and 14226. Some data products used in this study were made possible through EarthScope (www.earthscope.org; EAR-0323309), supported by the National Science
Foundation. All figures provided in this manuscript are created using GMT (Wessel et al., 2013).
To calculate the formal standard error for crustal thickness and surface elevation through time, we start with present-day crustal thickness of the western U.S. (Shen and Ritzwoller, 2016; Laske et
al., 2013), and then we calculate the crustal thickness and surface elevation variations back in time to 36 Ma for every 0.5 m.y. time increment.
Computation of Formal Standard Error for Crustal Thickness through Time
As mentioned in the paper, strain rates and positions are tracked in temporal order, starting at the present day. The vertical strain rates for the
position and time (ϕ
= ϕ
+ Δϕ; θ
= θ
+ Δθ, where ϕ is longitude, θ is latitude), and the
strain solution, are denoted as , where
= 1–12,
= 1–72, and time increment Δ
= −0.5 m.y. When solutions are tracked in order, starting at
= 1, the crustal thickness at the
time interval (
) is:
The formal variances in strain rates that are used in our error propagation calculation involve the spatially averaged quantities within each 1° × 1° area on the grid. As mentioned in the paper, this
sized area is reasonable as it reflects our confidence in spatially averaged quantities and also represents the current resolution in crustal thickness determination inferred from seismology.
Determining error propagation involves the need for estimating variances for each incremental change in crustal thickness (Δ
) for the
time step and the
solution, where:
Using the general rule for calculating variance for a multivariable nonlinear function (
Snedecor and Cochran, 1994
), the variance for the change in crustal thickness from the
– 1 to the
time interval is:
The partial derivative of function
with respect to:
and the partial derivative of function
with respect to
To calculate , we use the formal variances and covariances for the spatially averaged strain rates within each 1° × 1° area of the grid for the
solution, obtained in the least-squares inversion (
Beavan and Haines, 2001
). Considering the volumetric strain and volume conservation, the variance for vertical strain rate is:
To calculate covariance between and
, we sample the strain rate field and crustal thickness field on a dense set of points such that the covariance within each 1° × 1° area is defined as:
is the total number points in each 1° × 1° area in our western U.S. grid area, is the average of vertical strains in each area on the grid, and is the average of crustal thicknesses in each area at
time interval. The variance in crustal thickness at time
= Δ
involves the sum of variances of incremental changes in crustal thicknesses for each time step, where
is the total number of time steps under consideration. By substituting
Equations A4
Equation A3
, we arrive at the variance for crustal thickness at the
time interval, var(
Finally, the formal standard error of crustal thickness at
time interval [se(
)] is (
Fig. 8
Computation of Formal Standard Error for Surface Elevation through Time
To calculate and propagate the formal standard errors for surface elevations of the western U.S., incremental changes in surface elevation [Δ
, θ
)] for each time step are needed. From
Equation 13
in the main text, this can be written as:
where Δ
, θ
) is surface elevation change, Δ
is crustal thickness changes from the
– 1 to the
time interval, ρ
is the reference upper mantle density, ρ
is the crustal density, ρ
, θ
) is the upper mantle density beneath the western North American plate, and
is the thickness of the crust at sea level.
The variance of function Δ
, θ
) is
At present we do not consider errors in our density models for the calculation of error propagation for elevation. We only consider the influence of errors in crustal thickness. Hence, the partial
derivative of function Δ
, θ
) with respect to crustal thickness changes is:
The variance in surface elevation {var[
, θ
)]} at time
= Δ
involves the sum of variances of incremental changes in surface elevation for each time step, where
is the total number of time steps under consideration:
Thus the formal standard error of surface elevation {se[
, θ
)]} at the
time interval is (
Fig. 12
Calculating Two-Dimensional Steady-State Conductive Thermal Perturbations of Western U.S. Upper Mantle Density
We generate time-dependent two-dimensional (2-D) steady-state conductive solutions of heat distribution for each 0.5 m.y. (k = 72−1) from 36 Ma to present day based on four different patterns (i =
1–4) of magmatism during the past 36 m.y.: 72 < k < 61, i = 1; 60 < k < 37, i = 2; 36 < k < 17, i = 3; and 16 < k < 0, i = 4 (see Supplemental Fig. S10 [^footnote 2]).
Using the Laplace equation and assuming constant thermal conductivity, the steady-state conductive heat distribution with no heat generation (
Eckert and Drake, 1987
) is:
Equation A16
yields the 2-D temperature (
) as a function of the two independent space coordinates (
= ϕ
= θ
). Using the Fourier equations (
Eckert and Drake, 1987
), then the heat flow (
) in the ϕ
and θ
directions is calculated as:
is thermal conductivity (W/m·°C),
is the surface area (m
), and
is the temperature variation based on four different patterns of magmatism during the past 36 m.y. and the boundary condition. We use 3.5 W/m·°C as thermal conductivity of upper mantle (
Robertson, 1988
). The heat flow at any point on our 2-D grid area is the sum of the solutions for A17 and A18.
We use 150 °C as the differential temperature for the areas affected by magmatism and ignimbrite flareups. This choice of a thermal perturbation linked to active magmatism comes from constraints for
present-day upper mantle lithosphere obtained by Schutt et al. (2012). Their study, inferred from present-day seismic and heat flow constraints, shows an average of ∼150 °C temperature variation of
upper mantle lithosphere between undeformed blocks like the Great Plains and Colorado Plateau and magmatically and tectonically active regions such as Yellowstone, the margin of the Colorado Plateau,
and the western margin of the Great Basin. To compute density variations owing to an increase in lithosphere upper mantle temperature of this amount, we only need to consider solutions for a relative
temperature increase. Thus, we define boundary conditions of ΔT = 0 °C around our grid area, and also for the continental margin of the western U.S., as well as ΔT = 150 °C for the heat sources,
based on the NAVDAT data set of magmatic source locations (see Supplemental Fig. S10 [^footnote 2]). The temperature distribution models for upper mantle are produced in COMSOL. We then advect the
heat distribution model for each solution time interval (i = 1–4), and produce time-dependent temperature models for the upper mantle density.
In general, changing either the pressure or the temperature can change the density of the upper mantle. We ignore the pressure dependence for the density of the upper mantle lithosphere, and only
consider density changes owing to temperature changes within the upper mantle lithosphere. Hence, we calculate a new time- and temperature-dependent upper mantle density model [ρ
, θ
)] for each 0.5 m.y. time interval from 36 Ma to the present day (
= 72−1) based on thermal expansion of the upper mantle at constant pressure and differential temperatures as:
where ρ
, θ
) is the upper mantle density at the
time interval, α is the thermal expansion coefficient of the upper mantle, and Δ
, θ
) is the differential temperature between heat distribution solutions for the upper mantle (e.g., Δ
= T
– T
, 4 <
< 1). Here we use 4 × 10
as the thermal expansion coefficient of the upper mantle (
Suzuki, 1975
Robertson, 1988
Omitting details, we constrained the starting mantle density values prior to 36 Ma to be such that the thermal perturbation history gives back the present-day density values for the upper mantle,
obtained through compensation of present-day topography. That is, the starting density values prior to 36 Ma are higher and the thermal perturbations through time cause density reductions over time,
particularly in the vicinity of magmatic activities (Supplemental Fig. S11 [^footnote 2], and Supplemental Animations S11, S12 [^footnote 1]). Based on the differential temperature solutions for each
0.5 m.y. [ΔT(ϕ[k], θ[k])] and the thermal coefficient of expansion for the upper mantle, and using Equation A19, we calculate a new time-dependent upper mantle density [ρ[T](ϕ[k], θ[k])] for the
western U.S. from 36 Ma to the present day.
Supplemental Animations 1–15. Time-slice animations of western U.S. contoured dilatation strain rates, crustal thickness, paleotopography, and formal standard errors for all calculations along with
two-dimensional steady-state conductive heat distribution model of upper mantle and upper mantle density for every 0.5 m.y. from 36 Ma to present. Please visit
or the full-text article on
to view the Supplemental Animations.
Supplemental Figures S1–S12. Time-slice figures of western U.S. crustal thickness, upper mantle density, paleotopography, and formal standard errors for all models using the CRUST 1.0 dataset (
Laske et al., 2013
) for 7 m.y. time intervals from 36 Ma to present. Please visit
or the full-text article on
to view the Supplemental Figures.
Supplemental Digital Files. Calculated velocity, strain rate, crustal thickness, and paleotopography fields for every 0.5 m.y. from 36 Ma to present. Please visit
or the full-text article on
to view the Supplemental Digital Files.
Science Editor: Raymond M. Russo | {"url":"https://pubs.geoscienceworld.org/gsa/geosphere/article/14/3/1207/530582/reconstruction-modeling-of-crustal-thickness-and","timestamp":"2024-11-15T04:17:51Z","content_type":"text/html","content_length":"375661","record_id":"<urn:uuid:274dc82a-3b5f-4ae3-8463-1dd9f90b59db>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00297.warc.gz"} |
Differentiation Formulas: Rules, Formula List PDF
Differentiation Formulas PDF: Differentiation is one of the most important topics for Class 11 and 12 students. Therefore, every student studying in the Science stream must have a thorough knowledge
of differentiation formulas and examples at their fingertips. We have provided a list of differentiation formulas for students’ reference so that they can use it to solve problems based on
differential equations.
In this article, we have provided you with the list of complete differentiation formulas along with trigonometric formulas, formulas for logarithmic, polynomial, inverse trigonometric, and hyperbolic
functions. These derivative formulas will help you solve various problems related to differentiation.
Differentiation Formulas PDF: What Is Differentiation?
Differentiation is a process of calculating a function that represents the rate of change of one variable with respect to another. Differentiation and derivatives have immense application not only in
our day-to-day life but also in higher mathematics.
Differentiation Definition: Let’s say y is a function of x and is expressed as \(y=f(x)\). Then, the rate of change of “y” per unit change in “x” is given by \(\frac{dy}{dx} \).
Here, \(\frac{dy}{dx} \) is known as differentiation of y with respect to x. It is also denoted as \({f}'(x)\).
In general, if the function f(x) undergoes infinitesimal change h near to any point x, then the derivative of the function is depicted as:
\(\underset{h\to \infty }{\mathop{\lim }}\,\frac{f(x+h)-f(x)}{h}\)
Rules Of Differentiation: Differentiation Formulas PDF
There are mainly 7 types of differentiation rules that are widely used to solve problems relate to differentiation:
1. Power Rule: When we need to find the derivative of an exponential function, the power rule states that:
\(\frac{d}{dx}{{x}^{n}}=n\times {{x}^{n-1}}\)
2. Product Rule: When \(f(x)\) is the product of two functions, \(a(x)\) and \(b(x)\), then the product rule states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ a(x)\times b(x) \right]=b(x)\times \frac{d}{dx}a(x)+a(x)\times \frac{d}{dx}b(x)\)
3. Quotient Rule: When \(f(x)\) is of the form \(\frac{a(x)}{b(x)}\), then the quotient rule states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ \frac{a(x)}{b(x)} \right]=\frac{b(x)\times \frac{d}{dx}a(x)-a(x)\times \frac{d}{dx}b(x)}{{{\left[ b(x) \right]}^{2}}}\)
4. Sum or Difference Rule: When a function \(f(x)\) is the sum or difference of two functions \(a(x)\) and \(b(x)\), then the sum or difference formula states that:
\(\frac{d}{dx}f(x)=\frac{d}{dx}\left[ a(x)\pm b(x) \right]=\frac{d}{dx}a(x)\pm \frac{d}{dx}b(x)\)
5. Derivative of a Constant: Derivative of a constant is always zero.
Suppose \(f(x)=c\), where c is a constant. We have,
6. Derivative of a Constant Multiplied with a Function \(f\): When we need to find out the derivative of a constant multiplied with a function, we apply this rule:
\(\frac{d}{dx}\left[ c\times f(x) \right]=c\times \frac{d}{dx}f(x)\)
7. Chain Rule: The chain rule of differentiation states that:
\(\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\)
Differentiation Formulas List
The table below provides the derivatives of basic functions, constant, a constant multiplied with a function, power rule, sum and difference rule, product and quotient rule, etc. Differentiation
formulas of basic logarithmic and polynomial functions are also provided.
(i) \(\frac{d}{dx} (k)= 0\)
(ii) \(\frac{d}{dx} (ku)= k\frac{du}{dx}\)
(iii) \(\frac{d}{dx} (u±v)= \frac{du}{dx}±\frac{dv}{dx}\)
(iv) \(\frac{d}{dx} (uv)= u\frac{dv}{dx}+v\frac{du}{dx}\)
(v) \(\frac{d}{dx} (u/v)= \frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\)
(vi) \(\frac{dy}{dx}.\frac{dx}{dy}= 1\)
(vii) \(\frac{d}{dx} (x^n)= nx^{n-1}\)
(viii) \(\frac{d}{dx} (e^x)= e^x\)
(ix) \(\frac{d}{dx} (a^x)= a^x\log a\)
(x) \(\frac{d}{dx} (\log x)= \frac{1}{x}\)
(xi) \(\frac{d}{dx} \displaystyle \log _{a}x= \frac{1}{x}\displaystyle \log _{a}e\)
(xii) \(\frac{d^n}{dx^n} (ax+b)^n= n!a^n\)
Let us now look into the differentiation formulas for different types of functions.
Differentiation Formulas For Trigonometric Functions
Sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec), and cotangent (cot) are the six commonly used trigonometric functions each of which represents the ratio of two sides of a
triangle. The derivatives of trigonometric functions are as under:
(i) \(\frac{d}{dx} (\sin x)= \cos x\)
(ii) \(\frac{d}{dx} (\cos x)= -\sin x\)
(iii) \(\frac{d}{dx} (\tan x)= \sec^2 x\)
(iv) \(\frac{d}{dx} (\cot x)= – cosec^2 x\)
(v) \(\frac{d}{dx} (\sec x)= \sec x \tan x\)
(vi) \(\frac{d}{dx} (cosec x)= – cosec x \cot x\)
(vii) \(\frac{d}{dx} (\sin u)= \cos u \frac{du}{dx}\)
(viii) \(\frac{d}{dx} (\cos u)= -\sin u \frac{du}{dx}\)
(ix) \(\frac{d}{dx} (\tan u)= \sec^2 u \frac{du}{dx}\)
(x) \(\frac{d}{dx} (\cot u)= – cosec^2 u \frac{du}{dx}\)
(xi) \(\frac{d}{dx} (\sec u)= \sec u \tan u \frac{du}{dx}\)
(xii) \(\frac{d}{dx} (cosec u)= – cosec u \cot u \frac{du}{dx}\)
Differentiation Formulas For Inverse Trigonometric Functions
Inverse trigonometric functions like (\(\sin^{-1}~ x)\) , (\(\cos^{-1}~ x)\) , and (\(\tan^{-1}~ x)\) represents the unknown measure of an angle (of a right angled triangle) when lengths of the two
sides are known. The derivatives of inverse trigonometric functions are as under:
(i) \(\frac{d}{dx}(\sin^{-1}~ x)\) = \(\frac{1}{\sqrt{1-x^2}}\)
(ii) \(\frac{d}{dx}(\cos^{-1}~ x)\) = -\(\frac{1}{\sqrt{1-x^2}}\)
(iii) \(\frac{d}{dx}(\tan^{-1}~ x)\) = \(\frac{1}{{1+x^2}}\)
(iv) \(\frac{d}{dx}(\cot^{-1}~ x)\) = -\(\frac{1}{{1+x^2}}\)
(v) \(\frac{d}{dx}(\sec^{-1}~ x)\) = \(\frac{1}{x\sqrt{x^2-1}}\)
(vi) \(\frac{d}{dx}(coses^{-1}~ x)\) = -\(\frac{1}{x\sqrt{x^2-1}}\)
(vii) \(\frac{d}{dx}(\sin^{-1}~ u)\) = \(\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}\)
(viii) \(\frac{d}{dx}(\cos^{-1}~ u)\) = -\(\frac{1}{\sqrt{1-u^2}}\frac{du}{dx}\)
(ix) \(\frac{d}{dx}(\tan^{-1}~ u)\) = \(\frac{1}{{1+u^2}}\frac{du}{dx}\)
Formulas for Hyperbolic Functions Differentiation
The hyperbolic function of an angle is expressed as a relationship between the distances from a point on a hyperbola to the origin and to the coordinate axes. The derivatives of hyperbolic functions
are as under:
(i) \(\frac{d}{dx} (\sinh~ x)= \cosh x\)
(ii) \(\frac{d}{dx} (\cosh~ x) = \sinh x\)
(iii) \(\frac{d}{dx} (\tanh ~x)= sech^{2} x\)
(iv) \(\frac{d}{dx} (\coth~ x)=-cosech^{2} x\)
(v) \(\frac{d}{dx} (sech~ x)= -sech x \tanh x\)
(vi) \(\frac{d}{dx} (cosech~ x ) = -cosech x \coth x\)
(vii) \(\frac{d}{dx}(\sinh^{-1} ~ x)\) = \(\frac{1}{\sqrt{x^2+1}}\)
(viii) \(\frac{d}{dx}(\cosh^{-1} ~ x)\) = \(\frac{1}{\sqrt{x^2-1}}\)
(ix) \(\frac{d}{dx}(\tanh^{-1} ~ x)\) = \(\frac{1}{{1-x^2}}\)
(x) \(\frac{d}{dx}(\coth^{-1} ~ x)\) = -\(\frac{1}{{1-x^2}}\)
(xi) \(\frac{d}{dx}(\sec h^{-1} ~ x)\) = -\(\frac{1}{x\sqrt{1-x^2}}\)
(xii) \(\frac{d}{dx}(cos h^{-1} ~ x)\) = -\(\frac{1}{x\sqrt{1+x^2}}\)
So, now you are aware of the differentiation formulas, i.e. derivatives of popular trigonometric, polynomial, inverse trigonometric, logarithmic, and hyperbolic functions. You can download
Differentiation Formulas cheat sheet and Pdf on Embibe.
Check other important Maths articles:
FAQs On Differentiation Formulas Class 12
You can find the important FAQs answered by our experts below:
Q1: What are the differentiation formulae?
Ans: When you calculate a function that represents the rate of change of one variable with respect to another, differentiation is used and the associated formulas are differentiation formulas.
Q2: How do I memorize all the integration and differentiation formulas for trigonometry?
Ans: The best way to memorize the complex integration and differentiation formulas is by solving questions. Start with the topics and then consistently move towards the end of the chapter. Do keep
referring to these formulas whenever you get stuck on a question. With passing time, you will improve and not require the formula sheet anymore.
Q3: Is there any website where I can practice differentiation formulas for?
Ans: You can practice differential calculus questions at Embibe.
Q4. What are the common formulas of differentiation?
Ans: The common formulas of differentiation include: Derivatives of basic functions, Derivatives of Logarithmic and Exponential functions, Derivatives of Trigonometric functions, Derivatives of
Inverse trigonometric functions, Differentiation rules.
Q5. What are some of the basic rules of differentiation?
Ans: Some of the basic rules of differentiation are:
Power Rule: (d/dx) (x^n ) = nx^{n-1}
Product Rule: (d/dx) (fg)= fg’ + gf’
Quotient Rule: (d/dx) (f/g) = [(gf’ – fg’)/g^2]
Sum Rule: (d/dx) (f ± g) = f’ ± g’
Q6. What are some of the commonly used derivatives of trigonometric functions?
Ans: The commonly used derivatives of six trigonometric functions are:
(d/dx) sin x = cos x
(d/dx) cos x = -sin x
(d/dx) tan x = sec^2 x
(d/dx) sec x = sec x tan x
(d/dx) cot x = -cosec^2 x
(d/dx) cosec x = -cosec x cot x
Q7. What is d/dx?
Ans: d/dx is the general representation of the derivative function. d/dx denotes the differentiation with respect to the variable x.
Q8. What is a UV formula?
Ans: (d/dx)(uv) = v(du/dx) + u(dv/dx)
This formula is used to find the derivative of the product of two functions.
Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.
Practice Questions and Mock Tests for Maths (Class 8 to 12)
We hope that this complete list of differentiation formulas helps you. If you have any questions, feel to ask in the comment section below. We will get back to you at the earliest.
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If sectheta = a and sintheta < 0, find the exact valaues of sintheta? | HIX Tutor
If #sectheta = a# and #sintheta < 0#, find the exact valaues of #sintheta#?
I think the answer is like $- \sqrt{{a}^{2} - 1}$
Thanks in advance!!!
Answer 1
#sectheta=a# and #sin theta>0#
#sin theta=+-sqrt(1-1/a^2)=+-sqrt(a^2-1)/a#
However #sintheta <0#, then #sintheta=-sqrt(a^2-1)/a#
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Answer 2
If sec(θ) = a and sin(θ) < 0, we can use the given information to find the exact value of sin(θ).
From the definition of secant and sine, we know that:
sec(θ) = 1/cos(θ) = a
From this, we can find the value of cos(θ):
cos(θ) = 1/sec(θ) = 1/a
Since sin(θ) < 0, θ must lie in the third or fourth quadrant, where sin is negative.
In the third or fourth quadrant, the cosine is negative. Since cos(θ) = 1/a is positive, θ must be in the fourth quadrant. Therefore, we can write:
cos(θ) = -√(1 - sin²(θ)) = 1/a
Solving for sin(θ):
-√(1 - sin²(θ)) = 1/a 1 - sin²(θ) = 1/a² sin²(θ) = 1 - 1/a² sin²(θ) = (a² - 1)/a² sin(θ) = ±√((a² - 1)/a²)
Since sin(θ) < 0, sin(θ) = -√((a² - 1)/a²).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Working with Big Numbers
I have a script that asks you to type in a number. If you type in a big number it puts it in scientific notation or something. It looks like this:
In my script I have to get how many digits are in the number and then go through a loop that adds the digits together like:
-- theNumber equals 1.8975126541345E+13
set theLength to length of (theNumber as string)
set tempNumber to 0
repeat with i from 1 to theLength
set tempNumber to (character i of (theNumber as string) as number) + tempNumber
end repeat
This doesn’t work because it has that decimal and E+13 thing going on. How do you get applescript to stop doing this and get my code to work?
According to the documentation, the largest value that can be expressed as an integer is 2^29-3 although on my computer it is 2^29-1 as you can see in the following script:
set n to (2 ^ 29 - 3) as integer
set n to (n + 1) as integer
on error
beep 2
exit repeat
end try
end repeat
{n as integer, n + 1}
So, if the integer is greater than 2^29-1 and you want to display it as integer, then you need to convert it to text. You can use text manipulation or math to convert it. I don’t know which is
faster, but check this out:
set t to “18975126541345”
display dialog “Enter an integer:” default answer t
set n to (text returned of result) as number
return {n div 10, n mod 10}
Here’s what I got using the div and mod operators:
set t to “18975126541345”
display dialog “Enter an integer:” default answer t
set n to (text returned of result) as number
set temp_n to n
set int_text to {}
repeat until temp_n = 0
set beginning of int_text to (temp_n mod 10) as integer
set temp_n to (temp_n div 10)
end repeat
return int_text as string
Need to error check the text returned from the dialog.
Here is a very robust handler by Nigel Garvey:
Wow, thanks. It works great. I never figured it would be so complicated. | {"url":"https://www.macscripter.net/t/working-with-big-numbers/30453","timestamp":"2024-11-06T13:41:35Z","content_type":"text/html","content_length":"26059","record_id":"<urn:uuid:1da0caca-8a23-4d63-acb5-987b6e612117>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00623.warc.gz"} |
Go to the source code of this file.
subroutine ddisna (JOB, M, N, D, SEP, INFO)
Function/Subroutine Documentation
subroutine ddisna ( character JOB,
integer M,
integer N,
double precision, dimension( * ) D,
double precision, dimension( * ) SEP,
integer INFO
Download DDISNA + dependencies
[TGZ] [ZIP] [TXT]
DDISNA computes the reciprocal condition numbers for the eigenvectors
of a real symmetric or complex Hermitian matrix or for the left or
right singular vectors of a general m-by-n matrix. The reciprocal
condition number is the 'gap' between the corresponding eigenvalue or
singular value and the nearest other one.
The bound on the error, measured by angle in radians, in the I-th
computed vector is given by
DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed
to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
the error bound.
DDISNA may also be used to compute error bounds for eigenvectors of
the generalized symmetric definite eigenproblem.
JOB is CHARACTER*1
Specifies for which problem the reciprocal condition numbers
should be computed:
[in] JOB = 'E': the eigenvectors of a symmetric/Hermitian matrix;
= 'L': the left singular vectors of a general matrix;
= 'R': the right singular vectors of a general matrix.
M is INTEGER
[in] M The number of rows of the matrix. M >= 0.
N is INTEGER
[in] N If JOB = 'L' or 'R', the number of columns of the matrix,
in which case N >= 0. Ignored if JOB = 'E'.
D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
dimension (min(M,N)) if JOB = 'L' or 'R'
[in] D The eigenvalues (if JOB = 'E') or singular values (if JOB =
'L' or 'R') of the matrix, in either increasing or decreasing
order. If singular values, they must be non-negative.
SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
[out] SEP dimension (min(M,N)) if JOB = 'L' or 'R'
The reciprocal condition numbers of the vectors.
INFO is INTEGER
[out] INFO = 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 118 of file ddisna.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/dc/d37/ddisna_8f.html","timestamp":"2024-11-08T14:12:16Z","content_type":"application/xhtml+xml","content_length":"11953","record_id":"<urn:uuid:ceaaa220-1b0b-4158-bf97-ee5e3971eb71>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00170.warc.gz"} |
THIS IS ONE WAY A Penny BECOMES $10 Million AND JUST WHY You Should Care | Michelle Chew
THIS IS ONE WAY A Penny BECOMES $10 Million AND JUST WHY You Should Care
1 million right now or offer you a magical penny that will double every day for 31 times, which would you select? If you’re like my “old” personal, then I bet you’d take the million and run. That
seems enticing and who would like a scent that doubles? Well, check out why the magical scent might be considered a good choice! Have you heard about the magic penny before? Well, before we go on to
talk about this magical cent, let me ask you a straightforward question and you are wished by me to answer it honestly. 1 million or one penny doubled every day for 31 days? I’m going to bet that 95%
of you are going to pick the million dollars.
Heck, that’s quite attractive isn’t it. To have one million dollars in cash just seated there waiting for you. Oh, I am given by it goosebumps. The other 5% of you’ll pick the penny doubled and I
applaud you. Because it’s the right answer. You’re considering with a long-term view on the immediate gratification view. I used to be an area of the 95%, but I’ve since transformed my thought
processes since I acquired out of consumer debt ever.
Some of you are probably shaking your mind thinking that a cent is worthless. I believe someone called it a pocket weight at one point in time. I hate getting, and having around pennies, each day for
31 days but easily can get one to double in value, then I’d love me some pennies.
Before you mind over to the “x” button in your browser, let me show you some simple mathematics which makes the penny so enticing. I’m heading to show you the simple idea that has transformed my
brain about trading and conserving my money. I’m going showing you what compound interest is focused on.
Some have called it the eighth wonder of the world. I would contemplate it as well after doing the simple mathematics around it. Chemical substance interest really off will rock and roll my socks!
Month 10 million in just one, I thought it was crazy. My head was spinning just trying to figure it out.
The mathematics didn’t work in my mind. Now, I found out about this trick a long time ago but never talked about it on this website. I’ve spoken about compound interest before, which explains why
investing and saving are so worth while. It’s the only real way I am able to retire. It’s the only path for many people.
Compound interest is helping me retire people! 0.Every day for 31 days 01 doubled? 0.01 doubled every day for 31 days? It wasn’t until I actually put this little mathematics conundrum down on paper
(OK, it was Excel) that I truly observe how it works. Here is the background. 0.01 every day for one month, what can you get? You’re probably considering just like me and saying “not much!” It’s hard
to comprehend this little equation. 0.Month 01 everyday for one. 1 million to make it even more enticing.
• 31 Jan 2015 – A virtual tour of the ABIM Foundation’s condominium is published
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• Commercial Properties Diversify The Risks
• 4 years ago from Syracuse, NY
• State and local income taxes
• The base rate applicable for the tenure for which the deposit has been in force with the Bank
• Work out which debt charges you the most
1 million and run with it. It’s just too tempting, and it’s an amount most people couldn’t fathom having in their ownership. A few years ago, I would’ve done the same thing. 0.Month 01 everyday for
one? Do you there see that up? 0.A month 01 and dual it everyday for.
Wow, that’s the true power of compounding interest. This is actually the same thing that forces investments and savings generally. Now, I want to first speak some truth. This little mathematics trick
is to show you the true power behind the compound interest just. You won’t find any investment or savings account that delivers you with 100% returns. | {"url":"https://michellechew.com/3891-this-is-one-way-a-penny-becomes-10-million-and-just-why-you-should-care-36/","timestamp":"2024-11-13T14:38:40Z","content_type":"text/html","content_length":"145787","record_id":"<urn:uuid:de5bfb35-b9d2-421e-8fa5-11f0a06b414c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00548.warc.gz"} |
Department of Mathematics
Mathematics |
The mathematics graduate program offers an opportunity to do research in both pure and applied areas of mathematics. Our program is designed for students who expect to pursue a career in academia as
well as those preparing for careers as mathematicians in industry and government.
The Department of Mathematics offers a Doctor of Philosophy degree in mathematics.
Bourama Toni
Department Chair (202) 806-6830 Email
Henok Mawi
Director of Graduate Studies 202-806-7122 Email
Regin Maxwell
Administrative Coordinator 202-806-7122 Email
• Related Degrees: Ph.D.
• Program Frequency: Full-Time
• Format: In Person
Students admitted into the graduate mathematics program must have at least a bachelor's degree and a GPA of 3.2 in their major undergraduate mathematics courses.
Note: In response to COVID-19, the Graduate School has made temporary changes to the GRE requirement. For the 2021-22 academic year, the GRE requirement has been waived for all programs in the
Graduate School. Applicants will be evaluated holistically: GPA, letters of recommendation, statement of academic interests and professional goals, and an autobiographical statement that foregrounds
your research interests. For more information, contact the Department of Mathematics' Director of Graduate Studies, Dr. Henok Mawi
Degree Requirements
The Expository Writing Requirement
Howard University mandates that all entering graduate students pass an expository writing requirement administered by their department of study, unless the student has earned a score of 5.0 or better
on the GRE Analytical Writing test. The expository writing requirement must be met within the student's first year of enrollment.
There are several options mathematics majors can use to satisfy the Department of Mathematics writing requirement:
a. Score a 5.0 or better on the Analytical Writing portion of the GRE;
b. Publish an article in a professional mathematics or education journal;
c. Have written a master's thesis at an accredited institution;
d. Complete the McGraw-Hill Connect adaptive learning module for developing writers. A link to the course is given below:
e. Complete the History of Mathematics course with a B or better.
i. In this course, students are required to write a paper in LaTeX on a mathematical history topic. The paper must include both historical and mathematical content.
ii. Using a rubric developed by a subcommittee of the Department of Mathematics Graduate Committee, the paper will be evaluated according to analysis, language control, grammar, clarity, and logic.
Residence Requirements
At least two semesters of full-time study or the equivalent, shall be undertaken in the Department of Mathematics within the Graduate School of Arts and Science.
Other Requirements
Graduate students shall regularly attend seminars, lecture series, and colloquia sponsored by the Department of Mathematics.
The Ph.D. Degree Program
This degree program requires a minimum of 60 graduate credits beyond the B.S. degree or a minimum of 36 graduate credits beyond the M.S. degree in course work. In addition 12 graduate credits are
required for the Ph.D. dissertation.
Course Requirements
The courses for the Ph.D. degree presented by a candidate must include at most one course from Group 1, all courses from Group 2, at least two courses from Group 3 and a course on topics in History
of Mathematics. Additional courses to cover the areas of qualifying examinations as well as topics courses will be on subjects corresponding to the research interests of the faculty.
Core Course Groups
Group 1
• Introduction to Analysis I (MATH-220 / MATH-195)
• Introduction to Analysis ll (MATH-221 / MATH-196)
• Introduction to Modern Algebra I (MATH-208 / MATH-197)
• Introduction to Modern Algebra ll (MATH-209 / MATH-198)
• Introduction to Complex Analysis (MATH-185)
• Introduction to Differential Geometry (MATH-186)
• Probability and Statistics (MATH-189)
• Introduction to Number Theory (MATH-184)
• Introduction to General Topology (MATH-199)
Group 2
• Algebra I (MATH-210)
• Algebra II (MATH-211)
• Real Analysis I (MATH-222)
• Real Analysis II (MATH-223)
• Topology I (MATH-250)
• Complex Analysis I (MATH-229)
Group 3
• NumberTheory I (MATH-214)
• Applications of Analysis (MATH-224)
• Complex Analysis II (MATH-230)
• Functional Analysis I (MATH-231)
• Algebraic Topology I (MATH-252)
• Algebraic Topology II (MATH-253)
• Differential Geometry I (MATH-259)
• Differential Geometry II (MATH-260)
• Partial Differential Equations II (MATH-237)
Ph.D. Degree: Admission and Examination Requirements
To obtain a Ph.D. degree, a student admitted to the program must:
1. pass two qualifying examinations on subjects, not closely related to each other, chosen from two of the following six groups:
□ Real Analysis or Complex Analysis or Functional Analysis or Harmonic Analysis
□ Algebra or Number Theory
□ Combinatorics
□ Geometry or Topology
□ Dynamical Systems or Ordinary Differential Equations or Partial Differential Equations
□ Probability or Mathematical Statistics.
2. take a third qualifying examination in an area of the student's choice, that may include one from the above six groups, and
3. write a Ph.D. dissertation and defend it satisfactorily.
Financial Support
Financial support from the university is contingent upon the student making satisfactory progress. Students in the Ph.D. degree program are expected to have successfully completed six graduate
courses in the first year in the Ph.D. program and to have passed at least two of the qualifying examinations by the end of their second year in the Ph.D. program in order to obtain continuing
university support.
Language Requirement
Students must exhibit proficiency in one of the following languages: Arabic, Chinese, French, German, Russian. In exceptional cases, other languages may be accepted by the Department. In lieu of a
language from the above list and upon approval of the Chairman of the Department, students may take suitable graduate level courses from one of the following departments or schools: Computer Science,
Sociology, Economics, Biology or Education.
Requirements for Admission to Candidacy for the Ph.D. degree:
1. Candidates must have passed two of the Qualifying Examinations.
2. Candidates must satisfy the language requirement and the writing skills requirement.
Other Requirements
1. A minimum of 18 credits of work toward the Ph.D. degree shall be pursued after admission to candidacy.
2. Doctoral candidates shall participate actively in at least two seminars during their candidacy.
3. Only courses in which students earn grades of "A" or "B" may be counted toward the Ph.D. degree.
4. A student in the Ph.D. program who accumulates more than two courses of grades below "B" shall be dropped from the Mathematics Graduate Program.
Admission to Candidacy
A student should file for admission to candidacy after 12 hours of work has been completed and this student has satisfied the GSAS writing proficiency requirement. Forms provided by the dean should
be filed a semester before graduation and approved by the student's thesis committee and the Executive Committee of the Graduate School of Arts and Sciences.
Residence Requirements
At least four semesters of residence and full-time study or the equivalent, shall be in the Department of Mathematics of Howard University.
Group 1:
• Introduction to Analysis I (MATH-220 / MATH-195)
• Introduction to Analysis ll (MATH-221 / MATH-196)
• Introduction to Modern Algebra I (MATH-208 / MATH-197)
• Introduction to Modern Algebra ll (MATH-209 / MATH-198)
• Introduction to Number Theory (MATH-184)
• Introduction to Complex Analysis (MATH-185)
• Introduction to Differential Geometry (MATH-186)
• Probability and Statistics (MATH-189)
• Introduction to General Topology (MATH-199)
Group 2:
• Algebra I (MATH-210)
• Algebra II (MATH-211)
• Real Analysis I (MATH-222)
• Real Analysis II (MATH-223)
• Complex Analysis I (MATH-229)
• Topology I (MATH-250)
Group 3:
• Number Theory I (MATH-214)
• Applications of Analysis (MATH-224)
• Complex Analysis II (MATH-230)
• Functional Analysis I (MATH-231)
• Partial Differential Equations II (MATH-237)
• Algebraic Topology I (MATH-252)
• Algebraic Topology II (MATH-253)
• Differential Geometry I (MATH-259)
• Differential Geometry II (MATH-260)
• Combinatorics I (MATH-273)
• Combinatorics II (MATH-274) | {"url":"https://mathematics.howard.edu/index.php/academics/graduate-program","timestamp":"2024-11-04T15:32:58Z","content_type":"text/html","content_length":"36308","record_id":"<urn:uuid:44035900-00e9-42ff-8000-c2e279657a78>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00544.warc.gz"} |
Equivalence of two sets of functional dependencies
Two or more than two sets of FD's are called equivalence if the right hand side of one FD can be determined using the second FD set, similarly the right hand side of the second FD set can be
determined using the first FD set.
Given a relation with different FD sets for that relation, we have to find out whether one FD set is subset of other or both are equal.
How to find relationship between two FD's
Let FD1 and FD2 are two FD sets for a relation R. Then
1. If all FD of FD1 can be derived from FD's present in FD2, we can say that FD2 is subset of FD1
2. If all FD of FD2 can be derived from FD's present in FD1, we can say that FD1 is subset of FD2
3. If both above two conditions hold then FD1=FD2
Example: Consider two relational schemas F(ACDEH) and G(ACDEH) with functional dependencies F={ A->C, AC->D, E->AD, E->H} and G={ A->CD,E->AH}. Which of the following is true?
1. F is equivalent to G
2. F is not equivalent to G
3. We can't compare F with G
4. None of these.
i) checking if F is subset of G:
To find this calculate Closure of left side attribute in G Using FDs in F
Therefore F is subset of G ----------------(1)
ii) checking if G is subset of F:
To find this calculate Closure of left side attribute in F Using FDs in G
Therefore G is subset of F ----------------(2)
From (1) and (2) we can say that F is equivalent to G. So option B is correct
0 comments : | {"url":"https://www.tutorialtpoint.net/2022/04/equivalence-of-two-sets-of-functiona-dependencies.html","timestamp":"2024-11-14T01:18:56Z","content_type":"application/xhtml+xml","content_length":"190469","record_id":"<urn:uuid:236068c0-9267-44f1-b9a8-014d78b620be>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00213.warc.gz"} |
Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be :
Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator,
then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be :
1 Answer | {"url":"https://clay6.com/qa/120166/preeti-reached-the-metro-station-and-found-that-the-escalator-was-not-worki","timestamp":"2024-11-11T23:42:31Z","content_type":"text/html","content_length":"18286","record_id":"<urn:uuid:c29b6218-214a-486b-a5f1-5e1ac48daea9>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00455.warc.gz"} |
One-Way ANOVA in R
Master One-Way ANOVA in R: Analyzing Group Differences Made Easy
One-way analysis of variance (ANOVA) is a statistical technique used to compare the means of three or more groups that are independent of each other. It is commonly used in various fields of research
to test the differences between multiple groups, such as comparing the effects of different treatments, assessing the performance of different products, or analyzing the impact of various factors on
a given outcome. In this tutorial, we will explore how to perform a one-way ANOVA and check its assumptions in R, a powerful open-source statistical software, and learn how to interpret the results
of the analysis. By the end of this tutorial, you will have a better understanding of how to use R to conduct one-way ANOVA and how to interpret the results to make meaningful conclusions.
Preparing Data for One-Way ANOVA
The below code creates an example dataset that can be used to explore the weight of apples produced in three different orchards. The dataset is constructed using three numeric vectors, each
representing the weight of apples produced in one of the orchards (A, B, or C). These vectors are combined into a dataframe called apple_data, where the orchard column indicates which orchard the
weight values correspond to and the weight column contains the actual weight values. Additionally, the block variable is added to indicate the block from which each observation was obtained. This
dataset can be used to generate visualizations and conduct statistical analyses to explore the differences in apple weight among the different orchards and blocks.
# Create example data
orchardA <- c(2.3, 1.9, 2.2, 2.5, 2.1)
orchardB <- c(1.8, 1.7, 2.0, 2.2, 2.1)
orchardC <- c(2.1, 1.8, 1.9, 2.0, 1.7)
# Combine data into a data frame
apple_data <- data.frame(orchard = factor(c(rep("A", 5), rep("B", 5), rep("C", 5))),
weight = c(orchardA, orchardB, orchardC))
# Add block variable
apple_data$block <- factor(rep(1:5, times = 3))
# orchard weight block
# 1 A 2.3 1
# 2 A 1.9 2
# 3 A 2.2 3
# 4 A 2.5 4
# 5 A 2.1 5
# 6 B 1.8 1
Checking Assumptions for One-Way ANOVA
Before conducting the one-way ANOVA, it is important to check its assumptions. There are three main assumptions to check:
• Normality
• Homogeneity of variances
• Independence of observations
Assumption of normality
The provided code conducts the Shapiro-Wilk test for normality on the weight values of apples produced in each of the three orchards (A, B, and C) contained in the apple_data dataframe using the
shapiro.test function. The split function is used to split the “weight” variable by orchard, creating a list of weight values for each orchard. The lapply function is then applied to each list
element, which in this case are the weight values of each orchard. The names function is used to label each of the three tests with the corresponding orchard name. Finally, the output of the
Shapiro-Wilk test for each orchard is displayed by calling the shapiro_test object. This test is useful for assessing whether the distribution of the apple weights in each orchard can be assumed to
be normal or not, which is an important assumption for many statistical analyses.
# Check normality assumption with Shapiro-Wilk test
shapiro_test <- lapply(split(apple_data$weight,
names(shapiro_test) <- levels(apple_data$orchard)
# $A
# Shapiro-Wilk normality test
# data: X[[i]]
# W = 0.99929, p-value = 0.9998
# $B
# Shapiro-Wilk normality test
# data: X[[i]]
# W = 0.95235, p-value = 0.754
# $C
# Shapiro-Wilk normality test
# data: X[[i]]
# W = 0.98676, p-value = 0.9672
For orchards A, B, and C, the Shapiro-Wilk tests resulted in W values of 0.99929, 0.95235, and 0.98676, respectively, and all p-values are greater than 0.05 (0.9998, 0.754, and 0.9672). Based on
these results, we cannot reject the null hypothesis that the weight values in each orchard are normally distributed. Therefore, we can assume that the normality assumption is met for this data set,
which is important for performing many statistical analyses.
Assumption of homogeneity of variances
The below code conducts Levene’s test for equality of variances on the weight values of apples produced in each of the three orchards (A, B, and C) contained in the ‘apple_data’ dataframe. The
leveneTest function is used to conduct this test, which tests the null hypothesis that the variances of the weight values are equal across all three orchards. The alternative hypothesis is that at
least one of the variances is different from the others. The “weight” variable is specified as the dependent variable and “orchard” is specified as the independent variable. The output of the
Levene’s test is stored in the levene_test object and can be used to assess whether the equal variance assumption is met for these datasets. This assumption is important for many statistical
analyses, including ANOVA and t-tests.
# Check equal variance assumption with Levene's test
levene_test <- leveneTest(weight ~ orchard, data = apple_data)
# Levene's Test for Homogeneity of Variance (center = median)
# Df F value Pr(>F)
# group 2 0.2105 0.8131
# 12
The output shows the results of Levene’s test for equality of variances with a center of median, which assesses whether the variance of a continuous variable is the same across different groups. In
this case, there are three groups, and the test was conducted on 12 observations. The null hypothesis is that the variances of the groups are equal, and the alternative hypothesis is that at least
one group’s variance is different from the others.
The test statistics are reported as an F-value, which is calculated as the ratio of the mean square deviation among the groups to the mean square deviation within the groups. In this example, the
F-value is 0.2105, and the associated p-value is 0.8131. Since the p-value is greater than 0.05, we cannot reject the null hypothesis. Therefore, we can assume that the variance of the continuous
variable does not significantly differ across the three groups.
Assumption of independence of observations
The assumption of independence of observations is not something that can be directly tested with statistical tests. Rather, it is an assumption that should be evaluated based on the study design and
data collection methods. For example, if the apple weight data were collected by measuring the weights of different apples from different trees in different orchards, and each measurement was taken
independently of the others, then we can assume that the observations are independent.
However, if the data were collected by measuring the weights of apples from the same tree or the same branch, then the observations may not be independent and this assumption may be violated.
Therefore, it is important to carefully consider the study design and data collection methods when assessing the assumption of independence of observations.
Conducting One-Way ANOVA
The below code fits a one-way ANOVA model with weight as the dependent variable and orchard and block as independent variables. It then prints a summary of the results including the number of degrees
of freedom, the sum of squares, the mean squares, the F-value, and the p-value for each of the independent variables (orchard and block) as well as for the residuals. The ANOVA results allow us to
test if there are any statistically significant differences between the mean weights of apples across the three orchards and the five blocks.
# Fit one-way ANOVA model
model <- aov(weight ~ orchard + block, data = apple_data)
# Df Sum Sq Mean Sq F value Pr(>F)
# orchard 2 0.2520 0.12600 5.771 0.0281 *
# block 4 0.2973 0.07433 3.405 0.0660 .
# Residuals 8 0.1747 0.02183
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The ANOVA table shows that the effect of orchard is statistically significant with an F-value of 5.771 and a p-value of 0.0281. This suggests that there are significant differences in the mean weight
of apples produced by the different orchards.
The effect of block is not statistically significant with an F-value of 3.405 and a p-value of 0.0660. This suggests that there is insufficient evidence to conclude that the blocks have different
effects on the mean weight of apples produced.
The model appears to be a reasonable fit as the residuals have a small mean square value of 0.02183. However, it is important to note that the ANOVA results only provide information about the
statistical significance of the model and do not provide information on the size or direction of the effects.
Pairwise difference using Tukey test
The below code performs a Tukey Honest Significant Differences (HSD) test on the model object model using the TukeyHSD function. The TukeyHSD test is used to determine which pairs of group means are
significantly different from each other. The results are stored in the tukey_results object, which is a matrix containing the pairwise differences in means, standard errors, confidence intervals, and
p-values for each comparison. The results can be viewed by calling the ‘tukey_results’ object.
# Perform Tukey test for pairwise differences
tukey_results <- TukeyHSD(model)
# View the results
# Tukey multiple comparisons of means
# 95% family-wise confidence level
# Fit: aov(formula = weight ~ orchard + block, data = apple_data)
# $orchard
# diff lwr upr p adj
# B-A -0.24 -0.5070348 0.02703476 0.0765292
# C-A -0.30 -0.5670348 -0.03296524 0.0298606
# C-B -0.06 -0.3270348 0.20703476 0.8018580
# $block
# diff lwr upr p adj
# 2-1 -0.26666667 -0.68346984 0.1501365 0.2664199
# 3-1 -0.03333333 -0.45013650 0.3834698 0.9984348
# 4-1 0.16666667 -0.25013650 0.5834698 0.6543929
# 5-1 -0.10000000 -0.51680317 0.3168032 0.9142837
# 3-2 0.23333333 -0.18346984 0.6501365 0.3728834
# 4-2 0.43333333 0.01653016 0.8501365 0.0415206
# 5-2 0.16666667 -0.25013650 0.5834698 0.6543929
# 4-3 0.20000000 -0.21680317 0.6168032 0.5052382
# 5-3 -0.06666667 -0.48346984 0.3501365 0.9785138
# 5-4 -0.26666667 -0.68346984 0.1501365 0.2664199
The results of the Tukey multiple comparison test show the differences between the means of each orchard and block.
For the orchard comparison, there is a significant difference between orchard A and C (p = 0.0298606), but no significant difference between orchard A and B (p = 0.0765292) or between orchard B and C
(p = 0.8018580).
For the block comparison, there is a significant difference between blocks 2 and 4 (p = 0.0415206), but no significant difference between blocks 1 and 2, 1 and 3, 1 and 4, 1 and 5, 2 and 3, 2 and 5,
3 and 5, or 4 and 5.
Overall, including the block variable as a factor in the ANOVA model has slightly changed the significance levels of the orchard means compared to the ANOVA model without the block factor. The block
factor itself has only one significant difference between block 2 and block 4.
Visualization of One-Way ANOVA Results
Boxplots for each orchard
Below code creates a boxplot using the ggplot2 package in R. The data used for the plot is contained in a dataframe called apple_data. The plot displays the weight of apples produced in different
orchards, with the orchards on the x-axis and the weight of the apples on the y-axis in grams. The plot is titled “Apple Weight by Orchard”, and the x and y-axes are labeled “Orchard” and “Weight
(grams)”, respectively. The boxplot itself shows the distribution of the apple weights within each orchard. The theme_bw() function is used to set the plot’s background to white and the plot elements
to black to improve its visual clarity.
# Create boxplot
ggplot(apple_data, aes(x = orchard, y = weight)) +
geom_boxplot() + theme_bw() +
labs(title = "Apple Weight by Orchard",
x = "Orchard", y = "Weight (grams)")
The boxplot shows the distribution of weight measurements for each orchard. The boxes represent the interquartile range (IQR), which contains the middle 50% of the data. The horizontal line inside
each box is the median. The whiskers extend to the minimum and maximum values within 1.5 times the IQR, and any points beyond the whiskers are considered outliers. From the boxplot, we can see that
orchard A has the highest median weight and the least variation among measurements, while orchard C has the lowest median weight and the greatest variation among measurements.
Mean weight plot with error bars
The plot displays the mean weight of apples produced in different orchards, with the orchards on the x-axis and the mean weight of the apples on the y-axis. The plot is titled “Mean Weight by
Orchard”, and the x and y-axes are labeled “Orchard” and “Weight”, respectively. The bars in the plot are colored light blue, and error bars are included that represent one standard deviation on
either side of the mean weight for each orchard. The theme_minimal() function is used to set the plot’s background to white and the plot elements to black to improve its visual clarity.
ggplot(apple_data, aes(x = orchard, y = weight)) +
geom_bar(stat = "summary", fun = mean, fill = "lightblue") +
geom_errorbar(stat = "summary", fun.min = function(x) mean(x) - sd(x),
fun.max = function(x) mean(x) + sd(x), width = 0.2) +
labs(x = "Orchard", y = "Weight") +
ggtitle("Mean Weight by Orchard") +
The bar plot with error bars shows the mean weight of apples for each orchard, along with the variability (standard deviation) of the weights within each orchard. Orchard A has the highest mean
weight, followed by orchard B and C. However, the error bars for orchard B overlap with those of orchard A, indicating that there may not be a significant difference between the mean weights of these
two orchards. On the other hand, the error bars for orchard C are clearly separated from those of A and B, suggesting that the mean weight of apples from orchard C is significantly different from the
other two orchards.
Interaction plot
The below code creates an interaction plot using the “ggplot2” library in R. The plot shows the relationship between the weight of apples produced in each of the three orchards (A, B, and C) across
different blocks. The ‘apple_data’ dataframe is used, where the ‘orchard’ variable represents the orchard, the ‘weight’ variable represents the weight of the apples, and the ‘block’ variable
represents the block number. The ggplot function is used to initialize the plot, and the aes function specifies the mapping of the x-axis to the orchard variable, the y-axis to the weight variable,
and the color aesthetic to the block variable. geom_point and geom_line are used to add points and lines to the plot respectively, with the latter grouped by block using the group parameter in aes.
Finally, the plot is labeled with appropriate axis labels, legend titles, and a title using the labs and ggtitle functions.
# Create the interaction plot
ggplot(apple_data, aes(x = orchard, y = weight, color = block)) +
geom_point() +
geom_line(aes(group = block)) +
labs(x = "Orchard", y = "Weight", color = "Block") +
ggtitle("Interaction Plot: Orchard vs. Weight by Block") +
The interaction plot shows the relationship between orchard and weight, stratified by block. The plot suggests that there may be an interaction effect between orchard and block, as the relationship
between orchard and weight appears to be different for each block. For example, the weight of apples appears to be highest for Orchard 3 in Block 1 and Block 2, but is highest for Orchard 2 in Block
3. This suggests that the effect of orchard on weight may depend on the block in which the apples were grown. Further analysis, such as a two-way ANOVA, may be necessary to investigate this
interaction effect further.
Based on the one-way ANOVA model and Tukey’s HSD test, we can conclude that there is a statistically significant difference in mean weight among the three orchards. Specifically, the mean weight of
apples from Orchard 1 is significantly different from the mean weight of apples from Orchard 3. However, there is no significant difference between the mean weight of apples from Orchard 1 and
Orchard 2, or between Orchard 2 and Orchard 3. These findings suggest that Orchard 3 may have an advantage in producing larger apples compared to the other two orchards, but more investigation would
be needed to confirm this.
Download R program — Click_here
Download R studio — Click_here | {"url":"https://www.agroninfo.com/master-one-way-anova-in-r-analyzing-group-differences-made-easy/","timestamp":"2024-11-09T02:56:47Z","content_type":"text/html","content_length":"334663","record_id":"<urn:uuid:5464bb89-bcb4-438b-bb4f-fff65cc97f74>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00171.warc.gz"} |
SAVAE: Leveraging the variational Bayes autoencoder for survival analysis (2024)
Patricia A. Apellániz
Information Processing
and Telecommunications Center
Universidad Politécnica de Madrid
&Juan Parras
Information Processing
and Telecommunications Center
Universidad Politécnica de Madrid
&Santiago Zazo
Information Processing
and Telecommunications Center
Universidad Politécnica de Madrid
As in many fields of medical research, survival analysis has witnessed a growing interest in the application of deep learning techniques to model complex, high-dimensional, heterogeneous, incomplete,
and censored medical data. Current methods often make assumptions about the relations between data that may not be valid in practice. In response, we introduce SAVAE (Survival Analysis Variational
Autoencoder), a novel approach based on Variational Autoencoders. SAVAE contributes significantly to the field by introducing a tailored ELBO formulation for survival analysis, supporting various
parametric distributions for covariates and survival time (as long as the log-likelihood is differentiable). It offers a general method that consistently performs well on various metrics,
demonstrating robustness and stability through different experiments. Our proposal effectively estimates time-to-event, accounting for censoring, covariate interactions, and time-varying risk
associations. We validate our model in diverse datasets, including genomic, clinical, and demographic data, with varying levels of censoring. This approach demonstrates competitive performance
compared to state-of-the-art techniques, as assessed by the Concordance Index and the Integrated Brier Score. SAVAE also offers an interpretable model that parametrically models covariates and time.
Moreover, its generative architecture facilitates further applications such as clustering, data imputation, and the generation of synthetic patient data through latent space inference from survival
KeywordsSurvival Analysis$\cdot$Time to Event$\cdot$Deep Learning$\cdot$Variational Autoencoders
1 Introduction
In recent years, there has been a significant transformation in medical research methodologies towards the adoption of Deep Learning (DL) techniques for predicting critical events, such as disease
development and patient mortality. Despite their potential to handle complex data, practical applications in this domain remain limited, with most studies still relying on traditional statistical
Survival Analysis (SA), or time-to-event analysis, is an essential tool for studying specific events in various disciplines, not only in medicine but also in fields such as recommendation systems [1
], employee retention [2], market modeling [3], and financial risk assessment [4].
According to the existing literature, the Cox proportional hazards model (Cox-PH) [5] is the dominant SA method that offers a semiparametric regression solution to the non-parametric Kaplan-Meier
estimator problem [6]. Unlike the Kaplan-Meier method, which uses a single covariate, Cox-PH incorporates multiple covariates to predict event times and assess their impact on the hazard rate at
specific time points. However, it is crucial to acknowledge that the Cox-PH model is built on certain strong assumptions. One of these is the proportional hazards assumption, which posits that
different individuals have hazard functions that remain constant over time. Furthermore, the model assumes a linear relation between the natural logarithm of the relative hazard (the ratio of the
hazard at time $t$ to the baseline hazard) and the covariates. Furthermore, it assumes the absence of interactions among these covariates. It is worth noting that these assumptions may not hold in
real-world datasets, where complex interactions between covariates and non-linear relations might exist. Other traditional parametric statistical models for SA make specific assumptions about the
distribution of event times. For instance, models like those presented in [7, 8] assume exponential and Weibull distributions, respectively, for event times. However, one drawback of these models is
that they lack flexibility when it comes to changing the assumed distribution for survival times, making them less adaptable to diverse datasets.
In response, researchers have explored Deep Neural Networks (DNNs) to effectively capture the intricate and non-linear relations between predictive variables and a patient’s risk of failure.
Significant emphasis has been placed on improving the Cox PH model, which has been the standard approach in SA.
Recent approaches have introduced Neural Networks (NN) in various configurations, either enhancing the Cox-PH model with neural components or proposing entirely novel architectures. This exploration
of NN applications for SA traces back to 1995 with the work of [9], who initially employed a simple feed-forward NN to replace linear interaction terms while incorporating non-linearities.
Subsequently, the field saw the emergence of DeepSurv [10], a model designed to extract non-linearities from input data, albeit still assuming the proportional hazards assumption. This assumption
persists in other related models like the one proposed by [11]. Beyond addressing non-linearity, some researchers have sought to enhance prediction accuracy and model interpretability by combining
Bayesian networks with the Cox-PH model, as demonstrated by [12]. Additionally, efforts have been made to introduce concepts that facilitate analysis when data availability is limited, as seen in the
work of [13, 14]. However, it is essential to note that all these models still depend on the proportional hazards assumption. As a result, novel architectures such as DeepHit [15] have emerged as
alternatives that do not rely on the proportional hazards assumption. While DeepHit has exhibited superior performance compared to other state-of-the-art models, it operates exclusively in the
discrete-time domain, which comes with certain limitations, notably the requirement for a dataset with a substantial number of observations, a condition that may not be feasible in real-world
In light of the persistent limitations of existing approaches in the realm of SA, this paper introduces a novel, versatile algorithm grounded in DL advances, named SAVAE (Survival Analysis
Variational Autoencoder). SAVAE has been meticulously designed to predict the time distribution that leads to a predefined event and adapts to application in various domains, with a specific focus on
the medical context. Then, our main contributions consist of:
• •
We introduce a generative approach that underpins the development of a flexible tool, SAVAE, based on Variational Autoencoders (VAEs). SAVAE can effectively reproduce the data by analytically
modeling the discrete or continuous time to a specific event. This analytical approach enables the calculation of all necessary statistics with precision, as the output provided by SAVAE are the
parameters of the predicted time distribution
• •
SAVAE is a flexible tool that enables us to use a wide variety of distributions to model the time-to-event and the covariates. This allows us to not assume proportional hazards. By using NN, it
permits modeling complex, non-linear relations between the covariates and the time-to-event too, as opposed to the linearity assumptions in the state of the art. Also, the time-to-event is
trained with standard likelihood techniques, unlike state-of-the-art models like DeepHit, which trains the Concordance Index (C-index). This makes our approach more general and flexible, as any
differentiable distribution could be used to model the time and the covariates.
• •
Furthermore, our proposal can be trained on censored data, effectively leveraging information from patients who have not yet experienced the event of interest.
• •
We have conducted comprehensive time-to-event estimation experiments using datasets characterized by continuous and discrete time-to-event values and varying covariate natures, encompassing both
clinical and genomic data. These experiments involve a comparative analysis with the traditional Cox-PH model and other DL techniques. The results indicate that SAVAE is competitive with these
models in terms of the C-index and the Integrated Brier score (IBS).
2 Background
To establish context, we will define SA and VAEs. SA is a branch of applied statistics that examines random processes related to system failures and mortality. Following this, we will provide an
analytical overview of VAEs before introducing SAVAE.
2.1 Survival Analysis
In a conventional time-to-event or SA setup, N observations are given. Each of these observations is described by $D=(x_{i},t_{i},d_{i})^{N}_{i=1}$ triplets, where $x_{i}=(x_{i}^{1},...,x_{i}^{L})$
is an $L$-dimensional vector where $l=1,2,...,L$ indexes the covariates, $t_{i}$ is the time-to-event, and $d_{i}\in\{0,1\}$ is the censor indicator. When $d_{i}=0$ (censored), the subject has not
experienced an event up to time $t_{i}$, while $d_{i}=1$ indicates the observed events (ground truth). SA models are conditional on covariates: time probability density function $p(t|x)$, hazard rate
function (the instantaneous rate of occurrence of the event at a specific time) $h(t|x)$, or survival function $S(t|x)=P(T>t)=1-F(t|x)$, also known as the probability of a failure occurring after
time $t$, where $F(t|x)$ is the Cumulative Distribution Function (CDF) of the time. From standard definitions of the survival function, the relations between these three characterizations are
formulated as:
$p(t|x)=h(t|x)S(t|x).$ (1)
2.2 Vanilla Variational Autoencoder
In 2013, [16] proposed the original VAE, a powerful approach employing DNNs for Bayesian inference. It addresses the problem of a dataset consisting of $N$ i.i.d. samples $x_{i}$ of a continuous or
discrete variable, where $i\in 1,2,...,N$, $x_{i}$ are generated by the following random process, which is depicted in Figure 1:
1. 1.
A latent variable $z_{i}$ is sampled from a given prior probability distribution $p(z)$. [16] assumes a form $p_{\theta}(z)$, i.e., the prior depends on some parameters $\theta$, but its main
result drops this dependence. Therefore, in this paper, a simple prior $p(z)$ is assumed.
2. 2.
A conditional distribution, $p_{\theta}(x|z)$, with parameters $\theta$ generates the observed values, $x_{i}$. This process is governed by a generative model. Certain assumptions are made,
including the differentiability of probability density functions (pdfs), $p(z)$, and $p_{\theta}(x|z)$, regarding $\theta$ and $z$.
The latent variable $z$ and the parameters $\theta$ are unknown. Without simplifying assumptions, evaluating the true posterior density $p_{\theta}(x)=\int p(z)p_{\theta}(x|z)dz$ is infeasible. This
true posterior density can be defined as Equation 2 using Bayes’ theorem:
$p_{\theta}(z|x)=\frac{p_{\theta}(x|z)p(z)}{p_{\theta}(x)}.$ (2)
Variational methods offer a solution by introducing a variational approximation, $q_{\phi}(z|x)$, to the true posterior. This approximation involves finding the best parameters for a chosen family of
distributions through optimization. The quality of the approximation depends on the expressiveness of this parametric family.
2.2.1 ELBO derivation
Since an optimization problem must be solved, the optimization target needs to be developed. Considering $x_{i}$ are assumed to be i.i.d., the marginal likelihood of a set of points $\{x_{i}\}_{i=1}^
{N}$ can be expressed as
$\log p_{\theta}(x_{1},x_{2},...,x_{N})=\sum_{i=1}^{N}\log p_{\theta}(x_{i}),$ (3)
$\begin{split}p_{\theta}(x)=\int p_{\theta}(x,z)dz=\int p_{\theta}(x,z)\frac{q_%{\phi}(z|x)}{q_{\phi}(z|x)}dz=\mathbb{E}_{q_{\phi}(z|x)}\left[\frac{p_{\theta}%(x,z)}{q_{\phi}(z|x)}\right].\end (4)
Using Jensen’s inequality, we can obtain:
$\begin{split}\log p_{\theta}(x)=\log\Bigg{[}\mathbb{E}_{q_{\phi}(z|x)}\left[%\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}\right]\Bigg{]}\geq\mathbb{E}_{q_{\phi}(z%|x)}\left[\log\frac{p_{\theta}(x,z)} (5)
Rearranging Equation 5, we can express it as follows:
$\begin{split}\mathbb{E}_{q_{\phi}(z|x)}\Bigg{[}\log\left(\frac{p_{\theta}(x,z)%}{q_{\phi}(z|x)}\right)\Bigg{]}=\int q_{\phi}(z|x)\log\frac{p_{\theta}(x|z)p(z%)}{q_{\phi}(z|x)}dz=\int q_{\phi}
(z|x)\log\frac{p(z)}{q_{\phi}(z|x)}dz\\+\int q_{\phi}(z|x)\log p_{\theta}(x|z)dz=-\int q_{\phi}(z|x)\log\frac{q_{\phi%}(z|x)}{p(z)}dz+\int q_{\phi}(z|x)\log p_{\theta}(x|z)dz\\=-D_{KL}(q_{\phi} (6)
(z|x)||p(z))+\mathbb{E}_{q_{\phi}(z|x)}\left[\log p_{\theta}(%x|z)\right]=\pazocal{L}(x,\theta,\phi),\end{split}$
where $D_{KL}(p||q)$ is the Kullback-Leibler divergence between distributions $p$ and $q$, and $\pazocal{L}(x,\theta,\phi)$ is the Evidence Lower BOund (ELBO), whose name comes from Equation 5:
$\begin{split}\log p_{\theta}(x)\geq-D_{KL}(q_{\phi}(z|x)||p(z))+\mathbb{E}_{q_%{\phi}(z|x)}\left[\log p_{\theta}(x|z)\right]=\pazocal{L}(x,\theta,\phi),\end{split}$ (7)
that is, the ELBO is a lower bound for the marginal log-likelihood of the relevant set of points. Thus, by maximizing the ELBO, the log-likelihood of the data is maximized. This would be the
optimization problem to solve.
2.2.2 Implementation
The ELBO derived from Equation 7 can be effectively implemented using a DNN-based architecture. However, computing the gradient of the ELBO concerning $\phi$ presents challenges due to the presence
of $\phi$ in the expectation term (the second part of the ELBO in Equation 7). To address this issue, [16] introduced the reparameterization trick. This method involves modifying the latent space
sampling process to make it differentiable, enabling the use of gradient-based optimization techniques. Rather than sampling directly from the latent space distribution, VAEs sample $\epsilon$ from a
simple distribution, often a standard normal distribution. Subsequently, a deterministic transformation $g_{\phi}$ is applied to $\epsilon$, producing $z=g_{\phi}(x,\epsilon)$ where $z\sim q_{\phi}(z
|x)$ and $\epsilon\sim p(\epsilon)$. In this case, the ELBO can be estimated as follows.
$\begin{split}\pazocal{\hat{L}}(x,\theta,\phi)=\frac{1}{N}\sum_{i=1}^{N}\bigg{(%}-D_{KL}(q_{\phi}(z|x_{i})||p(z))+\log p_{\theta}(x_{i}|g_{\phi}(x_{i},%\epsilon_{i}))\bigg{)}.\end{split}$ (8)
This modification facilitates the calculation of the ELBO gradient concerning $\theta$ and $\phi$, allowing the application of standard gradient optimization methods.
Equation 8 offers a solution using DNNs, with functions parameterized by $\phi$ and $\theta$. Gradients can be conveniently computed using the Backpropagation algorithm, which is automated by various
programming libraries. The term VAE derives from the fact that Equation 8 resembles the architecture of an Autoencoder (AE) [17], as illustrated in Figure 2. Notably, the variational distribution $q_
{\phi}$ can be implemented using a DNN with weights $\phi$, taking an input sample $x$ and outputting parameters for the deterministic transformation $g_{\phi}$. The VAE’s latent space comprises the
distribution of the latent variable $z$, which is a deterministic transformation $g_{\phi}$ of the encoder DNN output and random ancillary noise $\epsilon$. A sampled value $z_{i}$ is drawn from the
latent distribution and used to generate an output sample, where another DNN with weights $\theta$ acts as a decoder, taking $z$ as input and providing parameters of the distribution $p_{\theta}(x|z)
$ as output.
Two key observations emerge.
1. 1.
The ELBO losses in Equation 7 include a regularization term penalizing deviations from the prior in the latent space and a reconstruction error term that enforces similarity between generated
samples from the latent space and inputs.
2. 2.
In contrast to standard AEs, VAEs incorporate intermediate sampling, rendering them non-deterministic. This dual sampling process is retained in applications where the distribution of output
variables is of interest, facilitating the derivation of input value distribution parameters.
3 Materials and Methods
The interest lies in using VAEs to obtain the predictive distribution of time-to-event given covariates. The proposed approach, termed Survival Analysis VAE (SAVAE) and depicted in Figure 3, extends
the Vanilla VAE. SAVAE includes a continuous latent variable $z$, two vectors (an observable covariate vector $x$ and the time-to-event $t$), and generative models $p_{\theta_{1}}(x|z)$ and $p_{\
theta_{2}}(t|z)$, assuming conditional independence, which is a characteristic inherent to VAEs and their ability to effectively model the joint distribution of variables. This means that knowing $z$
, the components of the vector $x$ and $t$ can be generated independently. To define the predictive distribution based on covariates, a single variational distribution estimates the variational
posterior $p(z|x)$. While it is possible to include the effect of time ($p(z|t,x)$), this approach focuses on using only covariates to obtain the latent space, as the time $t$ can be unknown to
predict survival times for test patients and could be censored. SAVAE combines VAEs and survival analysis, offering a flexible framework for modeling complex event data.
3.1 Goal
To achieve the main objective, which is to obtain the predictive distribution for the time to event, variational methods will be used in the following way defined in [18]:
$\begin{split}p\left(t^{*}|x^{*},\left\{x_{i},t_{i}\right\}^{N}_{i=1}\right)=%\int p\left(t^{*}|z,\left\{x_{i},t_{i}\right\}^{N}_{i=1}\right)p\left(z|x^{*},%\left\{x_{i},t_{i}\right\}^{N}_{i=1} (9)
where $x^{*}$ represents the covariates of a certain patient, and its survival time distribution $p\left(t^{*}|z,\left\{x_{i},t_{i}\right\}^{N}_{i=1}\right)$ needs to be estimated.
3.2 ELBO derivation
Considering our main objective and the use of VAE as the architecture on which we base our approach, the ELBO development seen previously can be extended to apply to our case. SAVAE assumes that the
two generative models $p_{\theta_{1}}(x|z)$ and $p_{\theta_{2}}(t|z)$ are conditionally independent. This implies that if $z$ is known, it is possible to generate $x$ or $t$. Furthermore, due to the
VAE architecture, it is assumed that each component of the covariate vector $x$ is also conditionally independent given $z$. Therefore,
$p(x,t,z)=p_{\theta_{1}}(x|z)p_{\theta_{2}}(t|z)p(z)=p_{\theta}(x,t|z)p(z).$ (10)
It also assumes that the distribution families of $p_{\theta_{1}}(x|z)$ and $p_{\theta_{2}}(t|z)$ are known, but not the parameters $\theta_{1}$ and $\theta_{2}$.Taking into account these
assumptions, the ELBO can be computed in a similar way to the case of the Vanilla VAE. First, the conditional likelihood of a set of points $\left\{x_{i},t_{i}\right\}^{N}_{i=1}$ can be expressed as
$\begin{split}\log p_{\theta}(x_{1},x_{2},...,x_{N},t_{1},t_{2},...,t_{N}|z)=%\sum_{i=1}^{N}\log p_{\theta}(x_{i},t_{i}|z)\\=\sum_{i=1}^{N}\left(\log p_{\theta_{2}}(t_{i}|z)+\sum_{l=1}^{L}\log (11)
where the expected conditional likelihood can be expressed as:
$\begin{split}\mathbb{E}_{z}\left[p_{\theta}(x,t|z)\right]=\int p_{\theta}(x,t|%z)p(z)dz=\int\frac{p_{\theta}(x,t,z)}{p(z)}p(z)dz\\=\int p_{\theta}(x,t,z)dz=p_{\theta}(x,t)=\int p_{\theta} (12)
As the interest lies in computing the log-likelihood:
$\begin{split}\log p_{\theta}(x,t)=\log\left[\mathbb{E}_{q_{\phi}(z|x)}\left[%\frac{p_{\theta}(x,t,z)}{q_{\phi}(z|x}\right]\right]\\\geq\mathbb{E}_{q_{\phi}(z|x)}\left[\log\frac{p_{\theta} (13)
where the inequality comes from applying Jensen’s inequality. Then, this could be rearranged as:
$\begin{split}\mathbb{E}_{q_{\phi}(z|x)}\left[\log\left(\frac{p_{\theta}(x,t,z)%}{q_{\phi}(z|x)}\right)\right]=\int q_{\phi}(z|x)\log\frac{p_{\theta_{1}}(x|z)%p_{\theta_{2}}(t|z)p(z)}{q_{\phi}
(z|x)}dz\\=-\int q_{\phi}(z|x)\log\frac{q_{\phi}(z|x)}{p(z)}dz+\int q_{\phi}(z|x)\left(%\log p_{\theta_{1}}(x|z)+\log p_{\theta_{2}}(t|z)\right)dz\\=-D_{KL}(q_{\phi}(z|x)||p(z))+\mathbb{E}_{q_ (14)
{\phi}(z|x)}\left[\log p_{\theta_{%1}}(x|z)+\log p_{\theta_{2}}(t|z)\right]\\=\pazocal{L}(x,\theta_{1},\theta_{2},\phi).\end{split}$
After computing this ELBO, it can be seen that it is similar to the Vanilla VAE’s one (Equation 8). The only difference lies in the reconstruction term, which is expressed differently in order to
explicitly distinguish between the covariates and the time-to-event. By using Equation 11 and the reparameterization trick, the ELBO estimator is obtained, explicitly accounting for each dimension of
the covariates vector:
$\begin{split}\pazocal{\hat{L}}(x,\theta_{1},\theta_{2},\phi)\\=\frac{1}{N}\sum_{i=1}^{N}\Bigg{(}-D_{KL}(q_{\phi}(z|x_{i})||p(z))+\log p_{%\theta_{2}}(t_{i}|g_{\phi}(x_{i},\epsilon_{i}))+\sum_ (15)
{l=1}^{L}\log p_{\theta_{%1}}(x_{i}^{l}|g_{\phi}(x_{i},\epsilon_{i}))\Bigg{)}.\end{split}$
In terms of implementation, three DNNs have been used, as specified in Figure 4. Note that the decoder DNNs output the parameters of each distribution.
3.2.1 Divergence computation
SAVAE assumes that $q_{\phi}(z|x)$ follows a multidimensional Gaussian distribution defined by a vector of means $\mu$, where each element is $\mu_{j}$ and by a diagonal covariance matrix C, where
the main diagonal consists of variances $\sigma^{2}_{j}$. According to [16], it can be stated that:
$-D_{KL}(q_{\phi}(z|x)||p(z))=\frac{1}{2}\sum_{j=1}^{J}(1+\log(\sigma_{j}^{2})-%\mu_{j}^{2}-\sigma_{j}^{2}),$ (16)
where $J$ is the dimension of the latent space $z$. This means that the Kullback-Leibler divergence from the ELBO Equation 15 can be calculated analytically.
3.2.2 Time modeling
One significant challenge in handling survival data is the issue of censorship, which occurs when a patient has not yet experienced the event of interest. In such cases, the true survival time
remains unknown, resulting in partial or incomplete observations. Consequently, SA models must employ specific techniques capable of accommodating censored observations along with uncensored ones to
reliably estimate relevant parameters.
In our case, to account for censoring in survival data, we start from the time $t$ reconstruction term from Equation 15 for a single patient:
$\pazocal{\hat{L}}_{time}(x_{i},\theta_{2},\phi)=\log p_{\theta_{2}}(t_{i}|g_{%\phi}(x_{i},\epsilon_{i})).$ (17)
Taking into account the censoring indicator $d_{i}$:
$d_{i}=\begin{cases}0\;\;\;\text{if censored}\\1\;\;\;\text{if event experienced}\end{cases},$ (18)
we could just use the information given by uncensored patients. However, we would waste information, since we know that the censored patients have not experienced the event until time $t_{i}$. Hence,
considering Equation 1 and following [19], we model the time pdf as:
$p_{\theta_{2}}(t_{i}|g_{\phi}(x_{i},\epsilon_{i}))=h(t_{i}|g_{\phi}(x_{i},%\epsilon_{i}))^{d_{i}}S(t_{i}|g_{\phi}(x_{i},\epsilon_{i})).$ (19)
Therefore, the hazard function term is only taken into account when the event has been experienced, that is, when the data are not censored. This way, SAVAE incorporates information from censored
observations, providing consistent parameter estimates.
Regarding the distribution chosen for the time event, we have followed several publications such as [8] where the Weibull distribution model is used. This distribution is two-parameter, with positive
support, that is, $p(t)=0,\forall t<0$. The two scalar parameters of the distribution are $\lambda$ and $\alpha$, where $\lambda>0$ controls the scale and $\alpha>0$ controls the shape as follows:
exp{\left(-\left(\frac{t}{\lambda}\right)^{%\alpha}\right)}\\h(t;\alpha,\lambda)=\frac{p(t;\alpha,\lambda)}{S(t;\alpha,\lambda)}=\frac{%\alpha}{\lambda}\left(\frac{t}{\lambda}\right)^{\ (20)
Although the Weibull distribution is our primary choice for modeling time-to-event data in SAVAE, it is crucial to highlight that other distributions are feasible, as long as their hazard functions
and CDFs can be analytically calculated. This versatility distinguishes SAVAE from other models. For example, the exponential distribution, a special case of Weibull with $\alpha=1$, can represent
constant hazard functions. Integrating alternative distributions, such as the exponential, into SAVAE is straightforward and only requires adjusting the terms in Equation 19. The ability of SAVAE to
predict the distribution parameters for each patient facilitates the calculation of various statistics, such as means, medians and percentiles, providing flexibility beyond the models customized to a
single distribution.
3.2.3 Marginal log-likelihood computation
Assigning distribution models to patient covariates in the reconstruction term is essential in SAVAE. This choice enables control over the resulting output variable distribution, but it also implies
that the model approximates the chosen distribution even if the actual distribution differs. The third component of the ELBO (15) depends on the log-likelihood of the data, which for some
representative distributions is:
• •
Gaussian distribution: Suitable for real-numbered variables ($x_{i}^{l}\in(-\infty,+\infty)$), it has parameters $\mu\in(-\infty,+\infty)$ and $\sigma\in(0,+\infty)$, known for its symmetric
nature. Its log-likelihood function is:
$\begin{split}\log(p(x_{i}^{l};\mu,\sigma))=-\log(\sigma\sqrt{2\pi})-\frac{1}{2%}\left(\frac{x_{i}^{l}-\mu}{\sigma}\right)^{2}.\end{split}$ (21)
• •
Bernoulli distribution: Applied to binary variables ($x_{i}^{l}\in\{0,1\}$), it has a single parameter $\beta\in[0,1]$, representing the probability of $x_{i}^{l}=1$.Its log-likelihood function
$\log(p(x_{i}^{l};\beta))=x_{i}^{l}\log(\beta)+(1-x_{i}^{l})\log(1-\beta).$ (22)
• •
Categorical distribution: Models discrete variables with $K$ possible values. We can think of $x_{i}^{l}$ as a categorical scalar random variable with $K$ different values. Each possible outcome
is assigned a probability $\theta_{k}$ (note that $\sum_{k=1}^{K}\theta_{k}=1$). The log-likelihood function can be computed based on the Probability Mass Function (PMF) following the expression:
$\log(p(x_{i}^{l}|\theta_{1},\theta_{2},...,\theta_{k}))=\log\left(\prod_{k=1}^%{K}\theta_{k}^{\mathbb{I}(x_{i}^{l}=k)}\right),$ (23)
where the indicator function means:
$\mathbb{I}(x_{i}^{l}=k)=\begin{cases}1\quad x_{i}^{l}=k\\0\quad x_{i}^{l}eq k\end{cases}.$ (24)
Recall that other desired distributions can be implemented in SAVAE, as long as their log-likelihood is differentiable.
4 Results and Discussion
Once SAVAE has been defined, the next step is to proceed with the experimental validation. First, the data used as input to the model will be discussed, followed by the experimental setup (network
architecture and training process). Finally, SAVAE’s performance evaluation will be analyzed. The code can be found in https://github.com/Patricia-A-Apellaniz/savae.
4.1 Survival data
Dataset # Samples # Censored # Covariates Event Time (mean, (min - max)) Censoring Time (mean, (min - max))
WHAS 1638 948 (57.88%) 5 1045.42 (1 - 1999) days 1298.92 (371 - 1999) days
SUPPORT 9104 2904 (31.89%) 14 478.45 (3 - 2029) days 1060.22 (344 - 2029) days
GBSG 1546 965 (43.23%) 7 44.49 (0.26 - 87.36) months 65.15 (0.26 - 87.36) months
FLCHAIN 6524 4562 (69.92%) 8 3647.5 (0 - 5166) days 4296.74 (1 - 5166) days
NWTCO 4028 3457 (85.82%) 6 2276.68 (4 - 6209) days 2588.23 (4 - 6209) days
METABRIC 1980 854 (56.18%) 21 2944.81 (3 - 9193) days 3424.81 (21 - 9193) days
PBC 418 257 (61.48%) 17 63.93 (1.37 - 159.8) months 75.22 (17.77 - 159.83) months
STD 877 530 (60.43%) 21 369 (1 - 1519) days 420 (1 - 1519) days
PNEUMON 3470 3397 (97.9%) 13 9.84 (0.5 - 12) months 9.98 (0.5 - 12) months
In SA datasets, each patient contributes information about whether events of interest occurred during a study period, categorizing them as censored or uncensored, along with their respective
follow-up times. To evaluate SAVAE, we trained it in nine diverse disease datasets, including WHAS, SUPPORT, GBSG, FLCHAIN, NWTCO, METABRIC, PBC, STD, and PNEUMON. We followed pre-processing
procedures similar to state-of-the-art models, ensuring a fair evaluation on established benchmarks in SA.
The Worcester Heart Attack Study (WHAS) [20] focuses on patients with acute myocardial infarction (AMI), providing clinical and demographic data. The Study to Understand Prognoses Outcomes and Risks
of Treatment (SUPPORT) [21] investigates seriously ill hospitalized adults and includes information on demographics, comorbidities, and physiological measurements. The Rotterdam & German Breast
Cancer Study Group (GBSG) [22, 23] combines data from node-positive breast cancer patients and a chemotherapy trial. The FLCHAIN [24] dataset studies the relationship between mortality and serum
immunoglobulin Free Light Chains, which are important in hematological disorders. NWTCO [25] studies Wilms tumor in children, Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) [
26] explores breast cancer, PBC focuses on Primary Biliary Cholangitis, STD deals with sexually transmitted diseases, and PNEUMON examines infant pneumonia.
Table 1 offers a more comprehensive view of the temporal aspects and occurrences of events within the various datasets considered. It becomes evident that a deliberate selection of various disease
datasets has been made, each characterized by distinct types and quantities of information. Significantly, the evaluation of the model has been carried out systematically in datasets that show
varying proportions of censored samples and differing time-to-event ranges. This strategic approach aims to provide a broader perspective on how the model might perform when applied to other
real-world datasets.
4.2 Performance metrics
Recalling from Section Survival Analysis, each dataset is described by $D=(x_{i},t_{i},d_{i})^{N}_{i=1}$ triplets, where $x_{i}=x_{i}^{1},...,x_{i}^{L}$ is an L-dimensional vector of covariates, $t_
{i}$ is the time to event and $d_{i}\in\{0,1\}$ is the censoring indicator.
When evaluating an SA model, the literature shows that the most commonly used metric is the C-index, which is the generalization of the ROC curve for all data. It is a measure of the rank correlation
between predicted risk and observed times. The concept arises from the intuition that a higher risk of an event occurring has a complete relation with a short time to the event. Therefore, a high
number of correlating pairs, i.e. pairs of samples that meet this expectation, is decisive to say that the model has good predictive quality.
In this case, the time-dependent C-index described in [27] will be used since the original one [28] cannot reflect the possible changes in risk over time being only computed at the initial time of
observation. This C-index is defined as follows:
$\begin{split}C_{index}=P\Big{(}\hat{F}(t|x_{i})>\hat{F}(t|x_{j})|d_{i}=1,t_{i}%<t_{j},t_{i}\leq t\Big{)},\end{split}$ (25)
where $\hat{F}(t|x_{i})$ is the CDF estimated by the model at the time $t$ given a set of covariates $x_{i}$. The probability is estimated by comparing the relative risks pairwise, as already
Based on the prediction index defined in [29], [30] proposed the second evaluation metric that has been used in this analysis: Brier Score (BS). It is essentially a square prediction error based on
the Inverse Probability of Censoring Weighting (IPCW) [31], a technique designed to recreate an unbiased scenario compensating for censored samples by giving more weight to samples with similar
features that are not censored. So, given a time $t$ the BS can be calculated as follows, with $G(\cdot)$ being the survival function corresponding to censoring ($1/G(t)$ is the IPCW):
$\begin{split}BS(t)=\frac{1}{N}\sum_{i=1}^{N}\Bigg{[}\frac{(S(t|x_{i}))^{2}}{G(%t_{i})}\cdot\mathbb{I}(t_{i}<t,d_{i}=1)\\+\frac{(1-S(t|x_{i}))^{2}}{G(t)}\cdot\mathbb{I}(t_{i}\geq t)\Bigg{]}.\ (26)
Since the C-index does not take into account the actual values of the predicted risk scores, BS can be used to assess calibration, i.e., if a model predicts a 10% risk of experiencing an event at
time t, the observed frequency in the data should match this percentage for a well-calibrated model. On the other hand, it is also a measure of discrimination: whether a model can predict risk scores
that allow us to correctly determine the order of events.
In this case, the evaluation is made using the integral form of BS since it does not depend on the selection of a specific time $t$:
$IBS(t_{max})=\frac{1}{t_{max}}\int_{0}^{t_{max}}BS(t)dt.$ (27)
To statistically assess the performance of each model based on the C-index globally, we propose the Mean Reciprocal Rank (MRR) as the third metric. It measures the effectiveness of a prediction by
considering the rank of the first relevant C-index within a list composed of the C-indices obtained from each model. Formally, the Reciprocal Rank (RR) for a set of results for each model is the
inverse of the position of the first pertinent result. For example, if the first relevant result is in position 1, its RR is 1; if it is in position 2, the RR is 0.5; if it is in position 3, the RR
is approximately 0.33, and so on. Thus, the MRR is the average of the RRs for a set of models:
$MRR=\frac{1}{Q}\sum_{i=1}^{Q}\frac{1}{rank_{i}},$ (28)
where $Q$ is the total number of models that are being compared, and $rank_{i}$ is the position of the first relevant C-index for the $i-th$ model. Higher MRR values indicate that relevant results
tend to appear higher in the list.
Finally, to add more statistical information on the performance of the models, we performed hypothesis testing to compare the mean C-index and IBS values of our model with those of the
state-of-the-art models in multiple folds, since we are using a five-fold cross-validation method. Specifically, we formulated a null hypothesis that assumes that the mean performance metrics of the
state-of-the-art models are greater than our model’s mean performance metrics. To assess the validity of this null hypothesis, we used $p$-values as a statistical measure. We established a
significance threshold of 0.05, a common practice in hypothesis testing. When the obtained $p$-value for each case fell below this threshold, we rejected the null hypothesis. In practical terms, this
indicated that our model exhibited superior performance compared to the other models. On the contrary, if the $p$-value exceeded 0.05, we concluded that there were no statistically significant
differences between our model and the others. It is important to note that this approach considered variations in results across different folds, providing a more comprehensive assessment of model
performance beyond just the average results.
4.3 Experimental setting
To begin with, the implementation of SAVAE was executed using the PyTorch framework [32]. As defined in Section 3.2, three different DNNs were trained, consisting of one encoder and two decoders.
These decoders were designed to infer covariates and time parameters, respectively. The Gaussian encoder exhibits a straightforward architecture, characterized by a single hidden linear layer
featuring a Rectified Linear Unit (ReLU) activation function and an output linear layer with hyperbolic tangent activation. The input to this encoder consists of the covariate vectors from the
training dataset, while the output generates a Gaussian latent space. The dimensionality of this latent space has been fixed to 5. The generated latent space serves as input for both decoders, each
featuring two linear layers. The first layer employs a ReLU activation function and incorporates a dropout rate of 20%. However, the final layer of the decoders employs different activation functions
based on the specified distribution, thereby tailoring the output to the parameters of the respective covariate distribution. Furthermore, the number of neurons in each hidden layer was also fixed at
50. The training process involved 3000 epochs with a batch size of 64 samples while incorporating an Early Stop mechanism in the event of an insufficient reduction in validation loss.
To evaluate the results while ensuring their robustness against data partitioning, we used a five-fold cross-validation technique. This method was applied not only to our model but also to the
state-of-the-art models used for performance comparison and result evaluation, including Cox-PH, DeepHit, and DeepSurv. Moreover, due to the inherent sensitivity of VAE architectures to initial
conditions, we conducted training using up to 10 different random seeds. Subsequently, the C-index was averaged among the three best performing seeds. We consider that the average performance of
three seeds provides a representative and sufficient evaluation. Lastly, note that the three state-of-the-art models have been implemented using the Pycox package [33], as well as the different
metrics used for validation, C-index and IBS. The MRR has been calculated manually, while the $p$-value has been obtained using the SciPy [34] package.
4.4 Results
In this section, we present a comprehensive assessment of the performance of our proposed model, SAVAE, compared to three well-established state-of-the-art models. Cox-PH, DeepSurv, and DeepHit.
Across multiple datasets that encompass a diverse range of medical and clinical scenarios, we conducted extensive experiments to assess the performance of these models. The key focus was on
evaluating their ability to predict survival outcomes, considering censored and uncensored data points.
As the initial set of results, our focus is on comparing the performance and results in terms of the C-index. Table 2 provides a comprehensive view of how our model is completely comparable to the
state-of-the-art models in terms of the average C-index. Additionally, note that all intervals for the minimum and maximum values across various folds overlap, indicating consistent performance
across different data subsets. The results displayed in the table reveal that our model consistently achieves a higher MRR compared to others across multiple datasets, showcasing its superiority in
many cases regarding the average C-index. However, it is essential to acknowledge that the C-index results among the different models are generally similar, highlighting the competitiveness of our
model within the field. Furthermore, it is important to note that the broad intervals are primarily attributed to the limited sample sizes commonly found in medical databases, a characteristic that
poses challenges when assessing model performance. To address this issue, we employed cross-validation as previously mentioned, ensuring that our model’s performance is robust and reliable. In
summary, while our model demonstrates its strength by outperforming other models in terms of MRR and achieving competitive average C-index scores, the overall similarity in C-index results
underscores its robustness and suitability for various medical datasets.
Dataset COXPH DEEPSURV DEEPHIT SAVAE
Avg. C-index (min, max) Avg. C-index (min, max) Avg. C-index (min, max) Avg. C-index (min, max)
WHAS 0.74 (0.66, 0.81) 0.78 (0.57, 0.88) 0.89 (0.82, 0.95) 0.74 (0.67, 0.80)
SUPPORT 0.58 (0.39, 0.78) 0.57 (0.37, 0.82) 0.55 (0.37, 0.73) 0.61 (0.40, 0.86)
GBSG 0.66 (0.61, 0.71) 0.67 (0.58, 0.73) 0.66 (0.58, 0.72) 0.67 (0.62, 0.72)
FLCHAIN 0.69 (0.50, 0.80) 0.67 (0.55, 0.80) 0.78 (0.73, 0.82) 0.79 (0.75, 0.83)
NWTCO 0.71 (0.64, 0.79) 0.70 (0.60, 0.79) 0.72 (0.66, 0.78) 0.71 (0.63, 0.79)
METABRIC 0.59 (0.52, 0.68) 0.61 (0.52, 0.69) 0.56 (0.46, 0.64) 0.61 (0.53, 0.70)
PBC 0.81 (0.64, 0.94) 0.80 (0.65, 0.92) 0.80 (0.62, 0.93) 0.81 (0.62, 0.95)
STD 0.60 (0.47, 0.72) 0.60 (0.49, 0.71) 0.59 (0.50, 0.68) 0.59 (0.46, 0.71)
PNEUMON 0.62 (0.54, 0.70) 0.65 (0.49, 0.80) 0.67 (0.57, 0.77) 0.65 (0.53, 0.77)
MRR 0.56 0.60 0.62 0.76
Bold highlights the best mean. For C-index and MRR, higher is better
Model WHAS SUPPORT GBSG FLCHAIN NWTCO METABRIC PBC STD PNEUMON
COXPH 0.579 0.058 0.0 0.0 0.268 0.003 0.45 0.887 0.003
DEEPSURV 1.0 0.02 0.149 0.0 0.135 0.549 0.28 0.927 0.382
DEEPHIT 1.0 0.0 0.0 0.01 0.644 0.0 0.228 0.727 0.935
Bold Implies a $p$-value below our threshold, 0.05. This means that SAVAE is significantly better than the other models.
In our validation process, we performed a statistical analysis using $p$-values to determine whether our model exhibited superior performance in terms of the C-index. To carry out this analysis, we
compared the average C-index of our model with the mean C-index values obtained from multiple folds for each of the state-of-the-art models. The objective was to determine whether the performance of
our model was statistically better than the alternative models. We established a significance threshold of 0.05, a common practice in hypothesis testing. Our findings in Table 3 reveal several
instances in which our model outperformed the state-of-the-art models, as evidenced by $p$-values below the 0.05 threshold. These results highlight the effectiveness and competitiveness of our
proposed approach. This comprehensive analysis, which considers the diverse C-index values in multiple folds, provides a robust evaluation of the performance of the model, extending beyond simple
average comparisons.
Our validation through IBS values (Tables 4 and 5) yielded conclusions that closely parallel those derived from the C-index analysis. Overall, it is important to note that our model’s IBS results
align closely with those of the state-of-the-art models, demonstrating comparable performance. However, our proposed model consistently demonstrated competitiveness and emerged as the top performer
in the various datasets used in our study. This convergence of results across different evaluation metrics reinforces the robustness and effectiveness of our novel approach. While our model maintains
a competitive edge within the context of the state-of-the-art models, further solidifying its potential and utility in the field of SA, it also stands out as a top-performing solution.
Dataset COXPH DEEPSURV DEEPHIT SAVAE
Avg. IBS (min, max) Avg. IBS (min, max) Avg. IBS (min, max) Avg. IBS (min, max)
WHAS 0.171 (0.109, 0.279) 0.134 (0.067, 0.260) 0.120 (0.067, 0.175) 0.159 (0.114, 0.205)
SUPPORT 0.208 (0.074, 0.374) 0.205 (0.057, 0.363) 0.219 (0.086, 0.370) 0.208 (0.063, 0.385)
GBSG 0.182 (0.142, 0.223) 0.179 (0.137, 0.228) 0.208 (0.168, 0.248) 0.179 (0.139, 0.222)
FLCHAIN 0.137 (0.089, 0.185) 0.142 (0.088, 0.186) 0.121 (0.098, 0.145) 0.102 (0.078, 0.124)
NWTCO 0.107 (0.080, 0.138) 0.109 (0.082, 0.149) 0.111 (0.083, 0.147) 0.127 (0.101, 0.152)
METABRIC 0.186 (0.137, 0.233) 0.191 (0.143, 0.244) 0.214 (0.153, 0.275) 0.180 (0.127, 0.236)
PBC 0.147 (0.043, 0.281) 0.146 (0.046, 0.268) 0.195 (0.087, 0.340) 0.138 (0.034, 0.267)
STD 0.210 (0.121, 0.302) 0.212 (0.123, 0.305) 0.224 (0.142, 0.315) 0.209 (0.121, 0.307)
PNEUMON 0.016 (0.004, 0.031) 0.017 (0.004, 0.034) 0.016 (0.004, 0.031) 0.021 (0.007, 0.037)
MRR 0.55 0.55 0.47 0.71
Bold highlights the best mean. For IBS lower is better and for MRR, higher is better
Model WHAS SUPPORT GBSG FLCHAIN NWTCO METABRIC PBC STD PNEUMON
COXPH 1 0.47 0.998 1 0.0 0.995 0.888 0.575 0.0
DEEPSURV 0.0 0.341 1 0.0 1.0 0.549 0.868 0.746 0.0
DEEPHIT 0.0 0.950 1 1 0.0 1 1.0 0.995 0.0
Bold Implies a $p$-value below our threshold, 0.05. This means that SAVAE is significantly better than the other models.
5 Conclusions
In this paper, we have successfully described an SA model (SAVAE), which stands out for its ability to avoid assumptions that can limit performance in real-world scenarios. It is a model based on
VAEs in charge of estimating continuous or discrete survival times, first, modeling complex non-linear relations among covariates due to the use of highly expressive DNNs, and second, taking
advantage of a combination of loss functions that capture the censoring inherent to survival data. Our model demonstrates efficiency compared to various state-of-the-art models, namely Cox-PH,
DeepSurv, and DeepHit, because of its freedom from assumptions related to linearity and proportional hazards. In contrast to DeepHit, which directly learns the C-Index metric, we train using standard
likelihood techniques. Note that this means that our approach is more flexible, as it allows using many different distributions to model the data, and the performance is competitive, as it performs
well in C-Index and IBS.
Furthermore, the adaptability of our model is a notable strength. While we have assumed specific distributions for both survival times and covariates in our experiments, SAVAE’s versatility extends
to accommodating any other parametric distribution, as long as their CDF and hazard function are differentiable, making it a scalable tool. Notably, our model can efficiently handle censoring to
mitigate bias, introducing a novel improvement in results.
This work raises several attractive lines for the future. An additional advantage lies in our model’s architecture, where time and covariates are reconstructed from latent space information. This
feature opens opportunities for its utility to be expanded to various tasks that have been developed using VAEs, including clustering [35], imputation of missing data [36], and data augmentation [37]
by the generation of synthetic patients. Thus, this tool has great potential and can be exploited in future work to have different functionalities even in the world of Federated Learning [38] [39].
In summary, SAVAE emerges as a versatile and robust model for SA, surpassing state-of-the-art methods while offering extensibility to a broader range of healthcare applications. It presents a
compelling solution for healthcare professionals looking for enhanced performance and adaptability in SA tasks.
This research was supported by GenoMed4All project. GenoMed4All has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 101017549. The
authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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• [36]JohnT. McCoy, Steve Kroon, and Lidia Auret.Variational autoencoders for missing data imputation with application to a simulated milling circuit.IFAC-PapersOnLine, 51(21):141–146, 2018.5th
IFAC Workshop on Mining, Mineral and Metal Processing MMM 2018.
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Engineering, 39:307–320, 01 2021.
• [39]Mirko Polato.Federated variational autoencoder for collaborative filtering.In 2021 International Joint Conference on Neural Networks (IJCNN), pages 1–8, 2021. | {"url":"https://yoitiv.pics/article/savae-leveraging-the-variational-bayes-autoencoder-for-survival-analysis","timestamp":"2024-11-03T10:40:05Z","content_type":"text/html","content_length":"471842","record_id":"<urn:uuid:5a806bc6-cfad-47fc-8c8c-141ad39a5a98>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00256.warc.gz"} |
Formula – Univer
Architecture of Formula Engine
📊📝📽️ Univer General
Chapter outline discussing Univer's formula engine architecture, it is recommended to read the "Overall Architecture" chapter first.
The primary objectives in designing the formula engine are:
1. Support Univer's various document types and related functionalities to connect to formulas
2. Provide a smooth user experience, supporting computation on web worker and server side
3. Support advanced formula capabilities, aligned with features provided by Microsoft Office 365, including but not limited to:
1. Sense formula execution status, support for stopping formula execution, support for loop reference detection and iteration calculation execution limit setting
2. Let / lambda and other custom functions
3. Supporting supertables, formulae, and naming ranges.
Overall Architecture
The architecture diagram is presented as follows:
1. Model: Stores the initial formula data, such as location and formula string.
2. Engine: Responsible for syntactic and semantic analysis of the formula string, analyzing the dependencies between formulas, and more.
3. Service: Provides a formula computation environment, supports registering functions, and offers custom name, supertable, and formula scheduling services.
4. Command and Controller: Control the formula module coordination.
5. Function: Function implementation.
The Engine is the core of the formula engine, providing the following capabilities:
• Dependency analysis, determining the order of execution for a set of formulas.
• Syntax and lexical analysis of each formula string, generating a syntax tree.
• Performing computation through the syntax tree.
• Providing basic operations for functions, including addition, subtraction, multiplication, and division, string concatenation, trigonometric functions, etc.
The Engine architecture is presented in the following figure:
The Lexer is responsible for the lexical analysis of the formula string. It matches tokens defined by the engine and generates nodes based on the rules, constructing a tree of LexerNode nodes. For
example, A1, B10, SUM will all be recognized as LexerNode. The recognition of node types will be handled by the Parser. For instance:
=(sum(sum(A1:B10), E10, 100) + 5) * 6 - 1
The transformed LexerNode tree looks as follows:
After generating the LexerNode tree, the engine will call the conversion method, using postfix notation (opens in a new tab) to replace the original infix expression, eliminating the parentheses in
the computation. For example:
Will be converted to:
The Parser's main responsibilities are to transform the LexerNode tree generated by the Lexer as follows, creating an AstNode tree:
1. Convert node names containing functions like SUM to FunctionNode formula nodes, and references like E10 to ReferenceNode reference nodes, and operators like + to OperatorNode operation nodes.
2. Other node types include:
1. LambdaNode, specific to lambda functions, for parametrization and wrapping as lambda-value-object
2. UnionNode, merging A1:B10 as RangeReference
3. PrefixNode, recognizing - as a negative number and compatibility with @ in older versions
4. SuffixNode, recognizing % as percentage and # as shorthand for dynamic array formula range
5. ValueNode, recognizing text, number, and logical value as the three basic types
3. Convert let to lambda execution
4. Inject parameters for lambda
The illustration of the transformed LexerNode tree to AstNode tree is as follows:
Responsible for executing a single expression, recursively obtaining the return value by invoking the methods of the AST node, with the following main responsibilities:
1. Converting operators to meta functions and executing them, primarily including arithmetic and comparison operators
2. Instantiating characters, numbers, and Boolean values as ValueObject, and instantiating arrays as ArrayValueObject
3. Instantiating references as ReferenceObject, and calling the internal method to convert them to ArrayValueObject
4. Invoking specific functions to start the calculation, with the values received by the function being the base class BaseValueObject, and returning ReferenceObject | BaseValueObject |
5. Asynchronous calculations will be awaited for results in the upper layer and passed to the lower layer, so there is no need to write asynchronous methods in the function, just passing the Promise
as a parameter to AsyncValueObject and returning
6. For INDIRECT, OFFSET, and other reference functions, return ReferenceObject.
BaseValueObject is a crucial operation type in the formula engine computation, with the following inheritance types:
Representing a null value, treated as false or 0 when calculated with other value types. An ErrorValueObject is returned directly if unable to calculate.
Representing an error, similar to Excel's #VALUE!, #NAME!, #REF!, etc. An ErrorValueObject can be returned directly in the function to represent a calculation error.
Comprised of three basic value types: NumberValueObject, StringValueObject, and BooleanValueObject, each implementing their respective numeric calculation methods. The precision of the underlying
computation is handled by big.js (opens in a new tab).
Passing lambda functions as arguments to lower-level functions allows them to be applied in functions such as MAKEARRAY (opens in a new tab) and REDUCE (opens in a new tab).
This is the core of matrix computation and can be computed with any PrimitiveValueObject or another ArrayValueObject. The calculation is demonstrated in the following figure:
It also supports calculations such as sum, average, min, max, std, var, power, and more. Furthermore, it has implemented capabilities like Numpy's slice (opens in a new tab) and filter (opens in a
new tab) for functions such as vlookup, xlookup, match, and others.
export class Vlookup extends BaseFunction {
override calculate(
lookupValue: BaseValueObject,
tableArray: BaseValueObject,
colIndexNum: BaseValueObject,
rangeLookup: BaseValueObject = new PrimitiveValueObject(false)
) {
const colIndexNumValue = this.getIndexNumValue(colIndexNum);
// Extract the first column of the tableArray
const searchArray = (tableArray as ArrayValueObject).slice(0, 1);
// Extract the column specified by colIndexNumValue
const resultArray = (tableArray as ArrayValueObject).slice(colIndexNumValue - 1, colIndexNumValue);
// Use the pick method of the resultArray to filter the matrix based on the lookupValue
// Return the first result value found
return resultArray.pick((searchArray as ArrayValueObject).isEqual(lookupValue) as ArrayValueObject).getFirstCell();
In situations with a high number of formulas, ArrayValueObject implements a reverse index on the list to enhance iteration performance.
Responsible for formula dependency analysis, marking formulas that need to be calculated and outputting a queue of the marked formulas for execution.
As shown in the above image, when the content of cell A1 changes, the formulas in A2, A3, A4, A5 will be marked as dirty, and the marked formulas will undergo dependency analysis, resulting in the
final output order of A2 -> A3 -> A5 -> A4.
If INDIRECT / OFFSET (opens in a new tab) -like reference functions are encountered, the Dependency module will call Lexer and Parser to perform pre-computation. This allows for the advance
calculation of the reference range of these types of functions, which is then used to calculate their dependencies.
Provides various services for the computation process. Here are a few of the key services:
IFormulaCurrentConfigService and IFormulaRuntimeService
Used for loading Univer data and storing temporary data for the formula execution process. The runtime returns all calculation results once the formula execution is complete.
Used for registering functions and their descriptions. Users can also register quick, custom functions through uniscript.
Registers features in the sheet domain, such as pivot tables, conditional formatting, and data validation. For example, pivot tables:
1. Pivot tables can register a dependency range and a getDirtyData method.
2. After the dependency range is marked as dirty, the getDirtyData method is executed, implementing the calculation logic within the pivot table.
3. The getDirtyData method can return a dirty area and temporary data for the dirty area, which are used for further calculation dependent on the pivot table results. The final, correct result is
Registers formulas for doc and slide, which are non-table domains. These formulas are not dependent on sheet formulas, so they do not need to return dirty areas and temporary data.
The core method for triggering formula calculation, which provides the following functions:
1. Circular dependency execution
2. Return of runtime status, including the total number of formulas executed and the number of completed formulas.
3. Secondary marking of array formulas as dirty, and their execution after the results are returned.
4. Uses requestImmediateMacroTask to avoid the 4ms limit of setTimeout, enabling formula execution in macro tasks and supporting termination of formula execution.
5. Formula execution time statistics.
Implemented with matrix calculation, reducing code size and centralizing core logic. This leads to more standardized function implementation and improved accuracy and quality.
The formula engine references Numpy's matrix operation concept, reducing the amount of code needed for function implementation and centralizing numerical computation, trigonometric function
computation, and other core logic in BaseValueObject. This lays the groundwork for standardizing function implementation.
The following is an example of a Sum formula implementation:
export class Sum extends BaseFunction {
override calculate(...variants: BaseValueObject[]) {
// Initialize an accumulator variable for summing
let accumulatorAll: BaseValueObject = new NumberValueObject(0);
// The number of parameters for the Sum function is determined by the user. The following loop obtains and calculates.
for (let i = 0; i < variants.length; i++) {
let variant = variants[i];
// If the input is a reference range like A1:B10, the upper layer will automatically convert it to an ArrayValueObject.
if (variant.isArray()) {
// Call the sum function on ArrayValueObject to sum the value of all ValueObjects
variant = (variant as ArrayValueObject).sum();
// Call the numerical computation function Plus on ValueObject to perform the sum
accumulatorAll = accumulatorAll.plus(variant);
return accumulatorAll; | {"url":"https://univer.ai/guides/sheet/architecture/formula","timestamp":"2024-11-11T01:05:22Z","content_type":"text/html","content_length":"169389","record_id":"<urn:uuid:d9a1b218-f3b6-416f-8e5b-7e2026d09129>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00158.warc.gz"} |
Confidence Interval | Big Data Mining & Machine Learning
Confidence Interval
A confidence interval estimates an interval of a specific parameter that tells us something about the overall data space from which our dataset is a sample. In statistics the overall data space is
called a population and our dataset is typically assumed to be derived from this population or represents a sample of this population. The estimation gives us a range of values (interval) that act as
good estimates of the unknown data space or population parameter. Its usage is often in context of some evaluation of a model (e.g. linear regression, see example below). Especially in huge
population sizes or big data spaces it makes sense to evaluate the model since the more data is available more noise in the data too.
It is provided as a range of values such that with, for example, 95% interval probability, the range will contain the unknown parameter. The range is defined in terms of lower and upper limits that
are computed from the sample of our data. Standard errors can be used to compute confidence intervals while these errors are often obtained during a model fitting process (e.g. linear regression, see
example below).
Confidence Interval R Example
A confidence interval might be used to assess the accuracy of the coefficient estimates of a linear regression model. Please refer to our article ‘R Linear Regression’ in order to understand the
basics of linear regression and a concrete application example that we visualize using R here:
> attach(Boston)
> plot(lstat,medv)
> abline (model.fit ,lwd =3, col =”red “)
Given that our regression model visualized as a red line above is stored in model.fit, we are able to obtain the confidence interval for its coefficient estimates. The following R command can be used
to obtain the intervals:
> confint(model.fit)
2.5 % 97.5 %
(Intercept) 33.448457 35.6592247
lstat -1.026148 -0.8739505
In the case of the Boston data used in the example, the 95% confidence interval for medv is [33.448, 35.659] and the 95% confidence interval for lstat is [-1.026, -0.873]. Therefore, we can conclude
that in the absence of any low socioeconomic percentage in neighborhoud (lstat, 0%), the median house prices will, on average, fall somewhere between 33,448 and 35,659 (unit $1000). Furthermore, for
a decrease in socioeconomic percentage, there will be an average decrease in median house price of between -1.026 and -0.873.
More Information about a Confidence Interval
Please refer to the following short video about the topic:
Follow us on Facebook: | {"url":"http://www.big-data.tips/confidence-interval","timestamp":"2024-11-05T08:48:03Z","content_type":"text/html","content_length":"63085","record_id":"<urn:uuid:104eaf65-9e45-4f81-b174-b97955c06ba1>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00432.warc.gz"} |
Tiger / Rat Tail for Handheld Radios
Handheld radios are getting more and more sophisticated and versatile. The bottleneck for modern handheld radios is often the stock antenna. There is an extremely simple yet very effective add-on
called a “Tiger Tail” or “Rat Tail” to remedy this situation. This article is going to explain how to make your own.
For less than $1 in material, you can significantly increase the receive and transmit performance of pretty much any handheld radio. Not just amateur radio, but practically any radio out there,
including WiFi routers. The following picture shows a Tiger Tail for a 2m band HT.
So if all you need is a bit of wire and a ring terminal, then why bother to write a lengthy article? Well, there are a few caveats and tricks with a Tiger Tail. For instance, some math needs to be
done to get the exact wire length just right. Most articles about the Tiger Tail just mention fixed numbers and completely disregard that the amateur radio bands are not the same around the world.
They also neglect commercial and low-power (Part 15) applications. And to my surprise, many articles do not even bother to mention that a Tiger Tail is a tuned element. A Tiger Tail that may work
perfectly on VHF, may perform pretty bad on UHF. So let’s get started!
The following shows all the tools you will need in some form or another:
You’ll need a ring-terminal appropriate for your wire diameter, some wire (14 AWG / 1.6 mm), wire strippers, a crimping tool and quite possibly a calculator.
Like I said above, the Tiger Tail is a tuned element and needs to be calculated for the specific frequency range of interest. Since I favor metric over imperial units, let’s start with the formula to
use if you like metric:
Length = length of Tiger Tail in cm
f = frequency in MHz
What this formula does is calculate a quarter wavelength for the given frequency + 5%. The same formula rearranged for imperial looks like this:
If you would like to calculate the length in inches, simply divide the result by 2.54. Or use the following formula instead:
Lenn(in) = length of Tiger Tail in inches
f = frequency in MHz
Remember that this Tiger Tail works for a single band ONLY. But there’s a pretty easy trick: if you would like to cover more than one band, like 2m and 70cm at the same time, simply calculate a Tiger
Tail for each band individually and connect them to the radio at the same time.
So after you calculate the correct length, simply crimp a ring-terminal on the wire and — just for good luck — isolate the other end with a piece of heatshrink tubing. That’s it, no black magic at
all. And this is what the final result should look like:
And in case you don’t like to read and you’d like to see some of the math being done for you, here’s a video I made on the same topic. The video also contains a cross-check of the math using a
spectrum analyzer:
Please cite this article as:
Westerhold, S. (2014),
"Tiger / Rat Tail for Handheld Radios".
Baltic Lab High Frequency Projects Blog. ISSN (Online): 2751-8140., https://baltic-lab.com/2014/06/tiger-rat-tail-for-handheld-radios/, (accessed: November 7, 2024).
Funding: If you liked this content, please consider contributing. Any help is greatly appreciated.
2 thoughts on “Tiger / Rat Tail for Handheld Radios”
1. If you were to calculate this for a 5/8 wave antenna, how would that affect your formula above?
2. S: I love your article and how well you explained everything. Thank you!
I’m relatively new to 2m/70cm HT HAM. So, I’m mystified by one item and hoping you or someone else can enlighten me further. Your formula for the length of the tiger tail includes adding 5% to
the length. Why the extra 5%?
I’ve calculated the 1/4 wavelengths for the top, bottom and average of the 2 meter band for phone traffic. I, too, prefer to use the metric system for these measurements, and I get an average for
the 2 meter band phone traffic of 51.2cm. The top of the phone band is 50.7cm and the bottom is 51.9cm. Adding 5% to the average (51.2cm x 1.05 = 53.76cm) which is outside the entire 2 meter band
phone traffic frequencies.
So, like I said, I’m mystified about the 5% extra on the 1/4 wave length tiger tail.
Thanks again for sharing. Your post on this is the best I’ve seen. | {"url":"https://baltic-lab.com/2014/06/tiger-rat-tail-for-handheld-radios/","timestamp":"2024-11-07T10:32:32Z","content_type":"text/html","content_length":"48256","record_id":"<urn:uuid:63f08e5b-1dca-403e-9729-5d63da5ef89c>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00287.warc.gz"} |
Applied Math Seminar | Jemma Shipton, Compatible finite element methods and parallel-in-time schemes for numerical weather prediction | Applied Mathematics
Monday, September 16, 2019 10:30 am - 10:30 am EDT (GMT -04:00)
MC 5417
Jemma Shipton | Imperial College, London
Compatible finite element methods and parallel-in-time schemes for numerical weather prediction
I will describe Gusto, a dynamical core toolkit built on top of the Fire-drake finite element library; present recent results from a range of test cases and outline our plans for future code
development. Gusto uses compatible finite element methods, a form of mixed finite element methods (meaning that different finite element spaces are used for different fields) that allow the exact
representation of the standard vector calculus identities div-curl=0 and curl-grad=0. The popularity of these methods for numerical weather prediction is due to the flexibility to run on
non-orthogonal grid, thus avoiding the communication bottleneck at the poles, while retaining the necessary convergence and wave propagation properties required for accuracy. However, this does not
solve the parallel scalability problem inherent in spatial domain decomposition: we need to find a way to perform parallel calculations in the time domain. While this sounds counterintuitive since we
expect the future state of the atmosphere to depend sequentially on its past state, current research by Prof. Wingate, Dr. Shipton and others demonstrates that schemes based on exponential
integrators offer the potential for large timesteps and parallel computation in evaluating the matrix exponential using a rational approximation. Of particular interest is the parareal method, which
uses an accurate timestepping scheme to iteratively refine, in parallel, the output of a computationally cheap 'coarse propagator' that can take large timesteps. | {"url":"https://uwaterloo.ca/applied-mathematics/events/applied-math-seminar-jemma-shipton-compatible-finite-element","timestamp":"2024-11-11T04:09:17Z","content_type":"text/html","content_length":"111719","record_id":"<urn:uuid:ef2b575a-bded-404a-a805-a1cc5649653d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00145.warc.gz"} |
Books - Linear Algebra
Systems Biology: Linear Algebra for Pathway Modeling
This is an introductory book on linear algebra with some emphasis on systems biology. Print copies can be purchased at Amazon.
This book is meant to be a companion book for a control theory for biologists (2017), and a control theory for bioengineering text (2016). These two new books are currently being developed. Given the
importance of linear algebra in modern systems biology, particularly in relation to stoichiometry and network topology, I decided that rather than include supplementary sections on linear algebra in
the main text books, it would be more efficient to write a shorter draft that readers can refer to as needed.
If readers have any suggestions for improvements, any mistakes or typos in the text, please contact me at
hsauro @ uw dot edu
1. Vectors
2. Matrices
3, Systems of Linear Equations
4. Determinants
5. Vector Spaces
6. Eigenvalues and Eigenvectors
7. Summary
8. Basic Matrix Factorizations
9. Linear Programming
10. Linear Algebra with Python and Julia
Appendix A: Complex Numbers
Appendix B: Further Reading | {"url":"https://books.analogmachine.org/linear-algebra","timestamp":"2024-11-05T13:53:28Z","content_type":"text/html","content_length":"100252","record_id":"<urn:uuid:fb591ad7-40d3-48ee-a5bf-25325fb46edb>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00806.warc.gz"} |
Singh, Vijay P.
Permanent URI for this collection
Recent Submissions
• Quantifying the effect of land use and land cover changes on green water and blue water in northern part of China
(Copernicus Publications on behalf of the European Geosciences Union, 2009-06-12) Liu, X.; Ren, L.; Yuan, F.; Singh, V. P.; Fang, X.; Yu, Z.; Zhang, W.
Changes in land use and land cover (LULC) have been occurring at an accelerated pace in northern parts of China. These changes are significantly impacting the hydrology of these parts, such as
Laohahe Catchment. The hydrological effects of these changes occurring in this catchment were investigated using a semi-distributed hydrological model. The semi-distributed hydrological model was
coupled with a two-source potential evaportranspiration (PET) model for simulating daily runoff. Model parameters were calibrated using hydrometeorological and LULC data for the same period. The
LULC data were available for 1980, 1989, 1996 and 1999. Daily streamflow measurements were available from 1964 to 2005 and were divided into 4 periods: 1964–1979, 1980–1989, 1990–1999 and
2000–2005. These periods represented four different LULC scenarios. Streamflow simulation was conducted for each period under these four LULC scenarios. The results showed that the change in LULC
influenced evapotranspiration (ET) and runoff. The LULC data showed that from 1980 to 1996 grass land and water body had decreased and forest land and crop land had increased. This change caused
the evaporation from vegetation interception and vegetation transpiration to increase, whereas the soil evaporation tended to decrease. Thus during the period of 1964–1979 the green water or ET
increased by 0.95%, but the blue water or runoff decreased by 8.71% in the Laohahe Catchment.
• Kinematic wave model for transient bed profiles in alluvial channels under nonequilibrium conditions
(American Geophysical Union, 2007-12-27) Tayfur, Gokmen; Singh, Vijay P.
Transient bed profiles in alluvial channels are generally modeled using diffusion (or dynamic) waves and assuming equilibrium between detachment and deposition rates. Equilibrium sediment
transport can be considerably affected by an excess (or deficiency) of sediment supply due to mostly flows during flash floods or floods resulting from dam break or dike failure. In such
situations the sediment transport process occurs under nonequilibrium conditions, and extensive changes in alluvial river morphology can take place over a relatively short period of time.
Therefore the study and prediction of these changes are important for sustainable development and use of river water. This study hence developed a mathematical model based on the kinematic wave
theory to model transient bed profiles in alluvial channels under nonequilibrium conditions. The kinematic wave theory employs a functional relation between sediment transport rate and
concentration, the shear-stress approach for flow transport capacity, and a relation between flow velocity and depth. The model satisfactorily simulated transient bed forms observed in laboratory
• Hybrid fuzzy and optimal modeling for water quality evaluation
(American Geological Union, 2007-05-08) Wang, Dong; Singh, Vijay P.; Zhu, Yuansheng
Water quality evaluation entails both randomness and fuzziness. Two hybrid models are developed, based on the principle of maximum entropy (POME) and engineering fuzzy set theory (EFST).
Generalized weighted distances are defined for considering both randomness and fuzziness. The models are applied to 12 lakes and reservoirs in China, and their eutrophic level is determined. The
results show that the proposed models are effective tools for generating a set of realistic and flexible optimal solutions for complicated water quality evaluation issues. In addition, the
proposed models are flexible and adaptable for diagnosing the eutrophic status.
• Kinematic wave model of bed profiles in alluvial channels
(American Geophysical Union, 2006-06-21) Tayfur, Gokmen; Singh, Vijay P.
A mathematical model, based on the kinematic wave (KW) theory, is developed for describing the evolution and movement of bed profiles in alluvial channels. The model employs a functional relation
between sediment transport rate and concentration, a relation between flow velocity and depth and Velikanov's formula relating suspended sediment concentration to flow variables. Laboratory flume
and field data are used to test the model. Transient bed profiles in alluvial channels are also simulated for several hypothetical cases involving different water flow and sediment concentration
characteristics. The model‐simulated bed profiles are found to be in good agreement with what is observed in the laboratory, and they seem theoretically reasonable for hypothetical cases. The
model results reveal that the mean particle velocity and maximum concentration (maximum bed form elevation) strongly affect transient bed profiles.
• Downstream hydraulic geometry relations: 2. Calibration and testing
(American Geophysical Union, 2003-12-04) Singh, Vijay P.; Yang, Chih Ted; Deng, Zhi-Qiang
Using 456 data sets under bank-full conditions obtained from various sources, the geometric relations, derived in part 1 [ Singh et al., 2003 ], are calibrated and verified using the split
sampling approach. The calibration of parameters shows that the change in stream power is not shared equally among hydraulic variables and that the unevenness depends on the boundary conditions
to be satisfied by the channel under consideration. The agreement between the observed values of the hydraulic variables and those predicted by the derived relations is close for the verification
data set and lends credence to the hypotheses employed in this study.
• Downstream hydraulic geometry relations: 1. Theoretical development
(American Geophysical Union, 2003-12-04) Singh, Vijay P.; Yang, Chih Ted; Deng, Z. Q.
In this study, it is hypothesized that (1) the spatial variation of the stream power of a channel for a given discharge is accomplished by the spatial variation in channel form (flow depth and
channel width) and hydraulic variables, including energy slope, flow velocity, and friction, and (2) that the change in stream power is distributed among the changes in flow depth, channel width,
flow velocity, slope, and friction, depending on the constraints (boundary conditions) the channel has to satisfy. The second hypothesis is a result of the principles of maximum entropy and
minimum energy dissipation or its simplified minimum stream power. These two hypotheses lead to four families of downstream hydraulic geometry relations. The conditions under which these families
of relations can occur in field are discussed.
• Solute transport under steady and transient conditions in biodegraded municipal solid waste
(American Geophysical Union, 1999-08) Bendz, David; Singh, Vijay P.
The transport of a conservative tracer (lithium) in a large (3.5 m3) undisturbed municipal solid waste sample has been investigated under steady and fully transient conditions using a simple
model. The model comprises a kinematic wave approximation for water movement, presented in a previous paper, and a strict convective solute flux law. The waste medium is conceptualized as a
three-domain system consisting of a mobile domain (channels), an immobile fast domain, and an immobile slow domain. The mobile domain constitutes only a minor fraction of the medium, and the
access to the major part of medium is constrained by diffusive transport. Thus the system is in a state of physical nonequilibrium. The fast immobile domain is the part of the matrix which
surrounds the channels and forms the boundary between the channels and the matrix. Owing to its exposure to mobile water, which enhances the biodegradation process, this domain is assumed to be
more porous and loose in its structure and therefore to respond faster to a change in solute concentration in the mobile domain compared to the regions deep inside the matrix. The diffusive mass
exchange between the domains is modeled with two first-order mass transfer expressions coupled in series. Under transient conditions the system will also be in a state of hydraulic
nonequilibrium. Hydraulic gradients build up between the channel domain and the matrix in response to the water input events. The gradients will govern a reversible flow and convective transport
between the domains, here represented as a source/sink term in the governing equation. The model has been used to interpret and compare the results from a steady state experiment and an unsteady
state experiment. By solely adjusting the size of the fraction of the immobile fast domain that is active in transferring solute, the model is capable of accurately reproducing the measured
outflow breakthrough curves for both the steady and unsteady state experiments. During transient conditions the fraction of the immobile fast domain that is active in transferring solute is found
to be about 65% larger than that under steady state conditions. It is therefore concluded that the water input pattern governs the size of the fraction of the immobile fast domain which, in turn,
governs the solute residence time in the solid waste. It can be concluded that the contaminant transport process in landfills is likely to be in a state of both physical, hydraulic, and chemical
nonequilibrium. The transport process for a conservative solute is here shown to be dominated by convective transport in the channels and a fast diffusive mass exchange with the surrounding
matrix. This may imply that the observed leachate quality from landfills mainly reflects the biochemical conditions in these regions. The water input pattern is of great importance for the
transport process since it governs the size of the fraction of the immobile fast domain which is active in transferring solute. This may be the reason for leachate quality to be seasonally or
water flux dependent, which has been observed in several investigations. The result also has a significant practical implication for efforts to enhance the biodegradation process in landfills by
recycling of the leachate.
• Kinematic wave model for water movement in municipal solid waste
(American Geophysical Union, 1998-11) Bendz, David; Singh, Vijay P.; Rosqvist, H?�kan; Bengtsson, Lars
The movement of water in a large (3.5 m3) undisturbed sample of 22-year-old municipal solid waste has been modeled using a kinematic wave approximation for unsaturated infiltration and internal
drainage. The model employs a two-parameter power expression as macroscopic flux law. The model parameters were determined and interpreted in terms of the internal geometry of the waste medium by
fitting the model to one set of infiltration and drainage data. The model was validated using another set of data from a sequence of water input events. The results of the validation show that
the model performs satisfactorily, but further development of the model to incorporate spatial variability would increase its capability.
• An entropy-based morphological analysis of river basin networks
(American Geophysical Union, 1993-04) Fiorentino, Mauro; Claps, Pierluigi; Singh, Vijay P.
Under the assumption that the only information available on a drainage basin is its mean elevation, the connection between entropy and potential energy is explored to analyze drainage basins
morphological characteristics. The mean basin elevation is found to be linearly related to the entropy of the drainage basin. This relation leads to a linear relation between the mean elevation
of a subnetwork and the logarithm of its topological diameter. Furthermore, the relation between the fall in elevation from the source to the outlet of the main channel and the entropy of its
drainage basin is found to be linear and so is also the case between the elevation of a node and the logarithm of its distance from the source. When a drainage basin is ordered according to the
Horton-Strahler ordering scheme, a linear relation is found between the drainage basin entropy and the basin order. This relation can be characterized as a measure of the basin network
complexity. The basin entropy is found to be linearly related to the logarithm of the magnitude of the basin network. This relation leads to a nonlinear relation between the network diameter and
magnitude, where the exponent is found to be related to the fractal dimension of the drainage network. Also, the exponent of the power law relating the channel slope to the network magnitude is
found to be related to the fractal dimension of the network. These relationships are verified on three drainage basins in southern Italy, and the results are found to be promising.
• A stochastic model for sediment yield using the Principle of Maximum Entropy
(American Geophysical Union, 1987-05) Singh, V. P.; Krstanovic, P. F.
The principle of maximum entropy was applied to derive a stochastic model for sediment yield from upland watersheds. By maximizing the conditional entropy subject to certain constraints, a
probability distribution of sediment yield conditioned on the probability distribution of direct runoff volume was obtained. This distribution resulted in minimally prejudiced assignment of
probabilities on the basis of given information. The parameters of this distribution were determined from such prior information about the direct runoff volume and sediment yield as their means
and covariance. The stochastic model was verified by using three sets of field data and was compared with a bivariate normal distribution. The model yielded sediment yield reasonably accurately.
• A kinematic model for surface irrigation: Verification by experimental data
(American Geophysical Union, 1983-12) Singh, Vijay P.; Ram, Rama S.
A kinematic model for surface irrigation is verified by experimental data obtained for 31 borders. These borders are of varied characteristics. Calculated values of advance times, water surface
profiles when water reaches the end of the border, and recession times are compared with their observations. The prediction error in most cases remains below 20% for the advance time and below
15% for the recession time. The water surface profiles computed by the model agree with observed profiles reasonably well. For the data analyzed here the kinematic wave model is found to be
sufficiently accurate for modeling the entire irrigation cycle except for the vertical recession.
• A kinematic model for surface irrigation: An extension
(American Geophysical Union, 1982-06) Sherman, Bernard; Singh, Vijay P.
The kinematic model for surface irrigation, reported previously by Sherman and Singh (1978), is extended. Depending upon the duration of irrigation and time variability of infiltration, three
cases are distinguished. Explicit solutions are obtained when infiltration is constant. When infiltration is varying in time, a numerical procedure is developed which is stable and has fast
convergence. A rigorous theoretical justification is developed for computation of the depth of water at and the time history of the front wall of water advancing down an infiltrating plane or
channel. A derivation is given of the continuity and momentum equations when there is lateral inflow and infiltration into the channel bed.
• A distributed converging overland flow model: 3. Application to Natural Watersheds
(American Geophysical Union, 1976-10) Singh, Vijay P.
The proposed distributed converging overland flow model is utilized to predict surface runoff from three natural agricultural watersheds. The Lax-Wendroff scheme is used to obtain numerical
solutions. For determination of the kinematic wave friction relationship parameter a simple relation between the parameter and topographic slope is hypothesized. The simple relation contains two
constants which are optimized for each watershed by the Rosenbrock-Palmer optimization algorithm. The model results are in good agreement with runoff observations from these watersheds. It is
shown that if the model structure is sound, it will suffice to optimize model parameters on hydrograph peak only even for prediction of the entire hydrograph. The model results suggest that a
distributed approach to kinematic wave modeling of watershed surface runoff is potentially promising and warrants further investigation.
• Granulometric characterization of sediments transported by surface runoff generated by moving storms
(Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union, 2008-12-16) de Lima, J. L. M. P.; Souza, C. C. S.; Singh, V. P.
Due to the combined effect of wind and rain, the importance of storm movement to surface flow has long been recognized, at scales ranging from headwater scales to large basins. This study
presents the results of laboratory experiments designed to investigate the influence of moving rainfall storms on the dynamics of sediment transport by surface runoff. Experiments were carried
out, using a rain simulator and a soil flume. The movement of rainfall was generated by moving the rain simulator at a constant speed in the upstream and downstream directions along the flume.
The main objective of the study was to characterize, in laboratory conditions, the distribution of sediment grain-size transported by rainfall-induced overland flow and its temporal evolution.
Grain-size distribution of the eroded material is governed by the capacity of flow that transports sediments. Granulometric curves were constructed using conventional hand sieving and a laser
diffraction particle size analyser (material below 0.250 mm) for overland flow and sediment deliveries collected at the flume outlet. Surface slope was set at 2%, 7% and 14%. Rainstorms were
moved with a constant speed, upslope and downslope, along the flume or were kept static. The results of laboratory experiments show that storm movement, affecting the spatial and temporal
distribution of rainfall, has a marked influence on the grain-size characteristics of sediments transported by overland flow. The downstream-moving rainfall storms have higher stream power than
do other storm types.
• A distributed converging overland flow model: 1. Mathematical solutions
(American Geophysical Union, 1976-10) Sherman, Bernard; Singh, Vijay P.
In models for overland flow based on kinematic wave theory the friction parameter is assumed to be constant. This paper studies a converging geometry and allows continuous spatial variability in
the parameter. Parameter variability results in a completely distributed approach, reduces the need to use a complex network model to simulate watershed surface runoff, and saves much
computational time and effort. This paper is the first in a series of three. It develops analytical solutions for a converging geometry with no infiltration and temporally constant lateral
inflow. Part 2 discusses the effect of infiltration on the runoff process, and part 3 discusses application of the proposed model to natural agricultural watersheds.
• A fractional dispersion model for overland solute transport
(American Geophysical Union, 2006-03-18) Deng, Zhi-Qiang; de Lima, M. Isabel P.; Singh, Vijay P.; de Lima, Jo?�o L. M. P.
Using the kinematic-wave overland flow equation and a fractional dispersion-advection equation, a process-oriented, physically-based model is developed for overland solute transport. Two
scenarios, one consisting of downslope and the other of upslope rainstorm movements, are considered for numerical computations. Under these conditions, the hydrograph displays a long-tailed
distribution due to the variation in flow velocity in both time and distance. The solute transport exhibits a complex behavior. Pollutographs are characterized by a steep rising limb, with a
peak, and a long, stretched receding limb; whereas the solute concentration distributions feature a rapid receding limb followed by a long stretched rising limb. Downslope moving storms cause
much higher peak in both hydrographs and pollutographs than do upslope moving storms. Both hydrographs and the pollutographs predicted by the fractional dispersion model are in good agreement
with the data measured experimentally using a soil flume and a moving rainfall simulator.
• A distributed converging overland flow model: 2. Effect of infiltration
(American Geophysical Union, 1976-10) Sherman, Bernard; Singh, Vijay P.
The overland flow on an infiltrating converging surface is studied. Mathematical solutions are developed to study the effect of infiltration on nonlinear overland flow dynamics. To develop
mathematical solutions, infiltration and rainfall are represented by simple time and space in variant functions. For complex rainfall and infiltration functions, explicit solutions are not | {"url":"https://oaktrust.library.tamu.edu/collections/6a612c7a-97d0-4bf9-8b3d-79cd778baa1b","timestamp":"2024-11-11T15:58:49Z","content_type":"text/html","content_length":"696842","record_id":"<urn:uuid:da85f3fb-4a78-47a5-833f-df88aecb67ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00585.warc.gz"} |
1 Introduction
The Lesser Antilles arc is a zone of convergence between the American plate and the Caribbean plate at a rate of about 2 cm/yr (Lopez et al., 2006). This movement is absorbed by the subduction of the
American plate below the Caribbean plate and deformation of the wedge of the upper plate on a 100-250 km-wide zone, producing an extended system of active crustal faults (Fig. 1 insert, (Feuillet et
al., 2002)). It results in a high seismicity level (about 1000 detected events per year) located on the subduction interface and within the slab with hypocentral depths ranging from 10 km up to 220
km, and within the deformed Caribbean plate with shallow crustal seismicity from 2 km up to 15–20 km in depth. Very shallow earthquakes occurring below/or very close to Guadeloupe archipelago islands
can be felt sometimes with magnitude less than 2.0.
Fig. 1
Since the French volcanological and seismological observatories (OVSG and OVSM) located in the Lesser Antilles are maintaining operational real-time seismic networks, they are responsible for
detecting and informing local authorities and public of any felt earthquake occurrence and main event characteristics: location (epicenter and depth), type (tectonic or volcanic), magnitude, and
maximum reported intensity in Guadeloupe and Martinique islands. Location and magnitude calculation are determined in a systematic way, using hand-picked phase arrivals and hypocenter inversion, and
are available within few tens of minutes after an event, thanks to observatory permanent duty. Macroseismic intensities are determined later, as a result of detailed investigations in the field.
However, in the case of a strongly felt earthquake, the first need of the local authorities is to get practical information on event location and maximum possible effects in the living areas. If this
information can be delivered rapidly, it may be used to evaluate and focus assistance in the most affected zones.
On November 21, 2004, the occurrence of Les Saintes event, $Mw=6.3$ and thousands of aftershocks in few days (Bazin et al., 2010; Beauducel et al., 2005; Bertil et al., 2004; Courboulex et al., 2010;
IPGP, 2004) offered an exceptional new strong-motion database thanks to the French permanent accelerometric network (Pequegnat et al., 2008) installed in 2002–2004. Combined with collected
testimonies and official intensity estimations for largest events, this provided a unique opportunity to establish a first local ground motion model adapted to the observatory needs.
In this article, we present the modeling strategy, dataset, results and applications of our empirical model. This work has been previously described in an internal report (Beauducel et al., 2005),
named $B3$ (from initials of the three original authors), and is presently used in Guadeloupe and Martinique seismological observatories to produce automatic reports.
2 Methodology
Our goal is to produce a predictive model of macroseismic intensities with a final uncertainty of about one intensity level, paying special attention to the maximum values that will be published
after each earthquake. To be usable in an operative way, the model must be applicable to a wide range of magnitudes and hypocentral distances, and, ideally, independently from its tectonic context or
Due to insular configuration of Lesser Antilles, most of epicenters occur offshore: it concerns 95% of $M≥2.5$ detected events (OVSG-IPGP database). Classical macroseismic intensity models cannot be
used because they are based on maximum intensity at epicenter, $I0$ (see for instance Pasolini et al. (2008), Sorensen et al. (2009)), a meaningless parameter for offshore events. Moreover, we do not
have sufficient intensity data to well-constrain a predictive model for intensities. We have then proceeded by combining, first, a ground motion predictive equation (GMPE) constrained by peak ground
accelerations (PGA) local data, and second, applying a forward empirical relation between intensities and accelerations.
Many empirical relations to predict earthquake ground motions have been developed for engineering purposes (see Abrahamson and Shedlock (1997), Bommer et al. (2010), Douglas (2003), Strasser et al.
(2009) for a short review). Due to the necessary high precision for these specific applications (like building damage studies), models are developed using very selected datasets for specific
applicability ranges of site conditions, magnitude and depth. Moreover, none of them is valid for magnitudes lower than 4.
Furthermore, a recent study Douglas et al. (2006), shows that ground motions observed on Guadeloupe and Martinique are poorly estimated by commonly-used GMPE, having smaller and more variable
amplitudes than expected.
In this work, we do not intend to produce a new GMPE for the engineering community; we need a more general model with certainly higher uncertainty, but applicable over a wide range of earthquakes to
be used in an operative way. In the following, we check results and residuals of our obtained PGA model as an intermediate stage, but in order to validate the choices made to produce automatic
reports, we emphasize tests of the final intensity model performance in terms of medians across full range of intensity and distance applicability and beyond.
3 Intermediate PGA predictive equation
3.1 Formulation and dataset
Due to the limited database and model purpose, we use one of the simplest form of GMPEs with only 3 parameters (Berge-Thierry et al., 2003):
$log(PGA)=aM+bR−log(R)+c$ (1)
where $PGA$ is the horizontal acceleration peak (in g), $M$ is the magnitude, $R$ is the hypocentral distance (in km), and $a$, $b$, $c$ are constant parameters.
This functional form implies many hypothesis. In particular, a radial distribution of ground motion around a point source, neglecting geological heterogeneities, tectonic origin, source extension and
radiation pattern. Fukushima (1996) also points out that a linear $log(D)/M$ formulation is not verified for magnitudes $≥6.5$ for which a $M2$ term should be necessary. This concerns magnitudes out
of our study range, but we will keep in mind that accelerations should be underestimated at long distance for large magnitudes.
To inverse the three parameters, we use seismic data recorded at 14 strong-motion permanent stations in Guadeloupe (see Fig. 1), with mixed site conditions, rock and soil (details about the seismic
stations can be found in (Bengoubou-Valérius et al., 2008)), in the period from November 21 to December 28, 2004. The dataset includes about 400 earthquakes associated to 1430 triggers of 3-component
acceleration waveforms. These events correspond to Les Saintes main shock $Mw=6.3$ and mostly the associated aftershocks, but also some regional events that we voluntarily kept in the database.
Locations and magnitudes come from the seismic catalog of the Guadeloupe observatory (OVSG-IPGP). Magnitudes were computed using the classical formula of duration magnitude from Lee et al. (1975) for
events $Md≤4.5$ (Clément et al., 2000; Feuillard, 1985), and we imposed the moment magnitude from worldwide networks for greater events. This allows us to overcome the problem of duration magnitude
saturation for magnitude greater than 4.5. The consistency of magnitude scale ($Md$ versus $Mw$) has been checked by Bengoubou-Valérius et al. (2008).
For each event, a value of PGA is calculated as the maximum amplitude of horizontal acceleration signals, using the modulus of a complex vector defined by the two horizontal and orthogonal components
$x(t)$ and $y(t)$. The PGA dataset is presented in Figs. 2 and 3. Magnitudes range from 1.1 to 6.3, hypocentral distances from 2 to 450 km, and PGA from 16 $μ$g to 0.36 g.
Fig. 2
Fig. 3
3.2 Best model determination and residuals
To calculate the 3 parameters in Eq. (1), we minimized a misfit function using the $L2$-norm. Due to the inhomogeneous dataset (magnitudes follow a power-law and there is more short-distance values),
we applied a simple weighting function by multiplying the misfit by the magnitude and a power of the hypocentral distance. This gave more weight for large magnitudes and long distances.
The inversion scheme yields the following parameters: $a=0.61755$, $b=−0.0030746$, and $c=−3.3968$. It produced an RMS residual on $log(PGA)$ of 0.47 (a factor of 3 in PGA, see Fig. 4). This value is
higher than classical published GMPE results (around 0.3, see Strasser et al. (2009)), and it confirms the observation of Douglas et al. (2006) about abnormal data variability in Lesser Antilles.
However, interestingly, this factor corresponds to the average ratio between rock and soil conditions in the observed PGA (Bengoubou-Valérius et al., 2008). This might also reflect the wide range of
magnitudes and distances in a too simple functional form. In order to follow some of the key considerations used to develop GMPEs (Bommer et al., 2010), we checked medians and sigmas of PGA residuals
(Fig. 4): it shows a very consistent distribution in the full magnitude range (from 2 to 6), while we observe a significant PGA underestimation (median around $+0.5$ so a factor 3 in amplitude) for
$D<15$ km.
Fig. 4
Eq. (1) with the parameters found is represented as an abacus in Fig. 5 showing calculated PGA as a function of hypocentral distance (from 3 to 500 km) and magnitudes 1 to 8.
Fig. 5
Note that we voluntarily limited the minimum hypocentral distance for each magnitude, as we do not take into account the near fault saturation term. It is reasonable to assume that this minimum
hypocentral distance is greater than rupture size. Earthquake magnitude reflects the seismic moment which is proportional to the total displacement averaged over the fault surface (Aki, 1972;
Kanamori, 1977). Many authors propose a simple formula to express the relationship between magnitude and fault length or rupture area (Liebermann and Pomeroy, 1970; Mark, 1977; Wells and Coppersmith,
1994; Wyss, 1979). Here we use Wyss's formula (Wyss, 1979):
where $M$ is the magnitude and $A$ the rupture surface. We decide to restrict the attenuation law of Eq. (1) to the domain $R>L$, where $L≈A12$ is an estimation of the fault characteristic size.
3.3 Examples of predicted and observed PGA
Fig. 6 shows representative events with observed PGA compared to our model predictions. We do not limit examples to the events from the dataset which reflects the previous residual analysis (Fig. 4),
but present events in the period 2004 to 2007 with various depths, in crustal or subduction context, and for which sufficient triggers were available. As seen in Fig. 6, most of PGA values are
predicted within the model uncertainty. Medians of $log(PGA)$ residuals are equal to $+0.15$, $+0.28$, $+0.10$, $+0.19$, $+0.24$, and $−0.01$ for Fig. 6a to f events, respectively. We denote, for
these 6 particular examples, a light tendency for PGA underestimation, which seems independent from magnitude. This is consistent with Fig. 4 residual analysis. The only significant PGA misfit
appears for one soil condition station in the near field ($≈15$ km) for Les Saintes aftershocks (Figs. 6b and c), that is systematically underestimated by a factor of about 10.
Fig. 6
We also compare these results with two published GMPE adapted to shallow crustal events: Sadigh et al. (1997) and Ambraseys (1995). The Sadigh et al. (1997) model is very similar to our PGA model for
magnitudes $≥5.0$ (Figs. 6a, c and f) but has poor fitting for lower magnitudes (Figs. 6b, d and e) with a systematic overestimation. The Ambraseys (1995) model has a globally poor fitting, with an
overestimation of PGA, particularly for $M<5.0$.
4 Macroseismic intensities
4.1 Formulation
Although we know that the spectral frequency content of ground acceleration and peak velocity have important implications on building damage, establishing a direct relation between a single PGA value
and macroseismic intensity has proved its efficiency in many cases (Chiaruttini and Siro, 1981; Margottini et al., 1992; Murphy et al., 1977; Wald et al., 1999). For the Lesser Antilles, we follow
the suggestion of Feuillard (1985) who studied the historical and instrumental seismicity using the simple empirical relation of Gutenberg and Richter (1942):
where $I$ is the mean intensity (MSK scale), PGA is maximum acceleration (in cm.s$−2≈$ mg). Combining Eqs. (1), (2) and (3) made the final empirical model formulation (hereafter called the $B3$
$I=1.85265M−0.0092238R−3log(R)+0.3096R>10M−4.152$ (4)
Note that following the MSK scale, intensity must be an integer value. In this article, we decided arbitrarily to round $I$ to the nearest and smallest integer (e.g., $I=6.0$ to $6.9$ correspond to
intensity of VI).
The resulting model for intensities is presented as right $Y$-axis in Fig. 5. Following Eq. (3), the 0.47 uncertainty on our predicted $log(PGA)$, would imply an uncertainty on $I$ of $±1.4$, on
which we should add the uncertainty of Eq. (3) itself, which is unknown.
4.2 Intensity model residuals
We test our model on a database of 20 recent earthquakes for which we have intensity reports (a total of 254 observations) as well as instrumental magnitudes and hypocenter locations. Events are from
various origins with magnitudes 1.6 to 7.4, distances from 4 to 500 km, and observed intensities from I to VIII. This wide panel of event characteristics allows us to check our model applicability.
We present in Fig. 7 the intensity residuals versus observed intensity and hypocentral distance. Global standard deviation equal 0.8, with a near zero median value. Residuals are also well
distributed over the intensity and distance ranges. Since this database is not statistically sufficient, we will keep uncertainty on intensities deduced from the PGA residuals, i.e., $σ=1.4$
corresponding to 68% confidence interval. We also checked that maximum observed intensity for each event is strictly below this probability level (see Fig. 7 solid circles).
Fig. 7
4.3 Examples of simulated and observed intensities
In Fig. 8, we detail eight examples of the most significant events with observed and predicted intensities (see epicenters in Fig. 1).
Fig. 8
Fig. 8a shows the October 10, 1974 “Antigua” earthquake (McCann et al., 1982; Tomblin and Aspinall, 1975), $Ms=7.4$, a shallow 30 km-depth with normal-fault mechanism, $Ms$ from NEIC USGS, location
and MSK intensities from McCann et al. (1982). Maximum intensities and distance of observations vary from VIII at 45 km in Antigua to II at 400 km in Virgin Islands. All the observations (9 sites)
are within the $B3$ prediction uncertainty limits. The median of intensity residuals equals $−0.6$, sigma is $0.5$. This is an unexpected positive result since the model is extrapolated for
magnitudes larger than Les Saintes ($Mw=6.3$); so this magnitude 7.4 is formally out of our interval of validity. Note also that near-field intensities (at 45 km) seem correctly fitted by the model
while this hypocentral distance is very close to our limit defined by Eq. (2), which gives $L=42$ km.
Fig. 8b shows the March 10, 1976 earthquake, a magnitude $Mb=5.9$, 56 km-depth on subduction interface north of Guadeloupe ($Mb$ from USGS-NEIC, location and MSK intensities from Feuillard (1985)).
Maximum intensities and distances of observations vary from V in Le Moule (Guadeloupe) at 85-km, to II in Martinique at 150 km distance. Most of the 22 observed intensities are underestimated (median
of residuals is $+0.4$) but still within one sigma uncertainty ($RMS$ equals 0.5).
Fig. 8c shows the January 30, 1982 earthquake, a magnitude $Mw=6.0$, 63 km-depth on subduction interface north of Guadeloupe ($Mw$ and location from Global CMT Project, MSK intensities from Feuillard
(1985)). Maximum intensities and distances of observations vary from V in various urban districts of Guadeloupe and Antigua at 90 km distance, to II in Barbuda (130 km). Most of the 34 observed
intensities are within the $B3$ uncertainty limits, with a zero median and $RMS$ on intensity residuals equal to 0.7.
Fig. 8d shows the March 16, 1985 “Redonda” earthquake (Feuillet et al., 2010; Girardin et al., 1991), a magnitude $Mw=6.3$, 10 km-depth normal-fault ($Mw$ and location from Global CMT Project, MSK
intensities from Feuillard (1985)). Maximum intensities and distances of observations vary from VI at 30 km in Montserrat to II at 300 km in Martinique. We added a supposed intensity of VII-VIII
(light gray dashed rectangle) because important cliff collapses have been observed in the Redonda island, at 10 km-distance from epicenter. All the 23 observed intensities are within the $B3$
uncertainty limits ($RMS=0.7$) with zero median. Note a very local amplification effect that occurred in the region of Pointe-à-Pitre (Guadeloupe) with an intensity of V to VI at 120 km from the
Fig. 8e shows the November 21, 2004 Les Saintes main shock earthquake of magnitude $Mw=6.3$, $Mw$ from Global CMT Project, location from Bazin et al. (2010), EMS98 intensities (see definition in
Grunthal et al. (1998)) from an official survey by the BCSF (Cara et al., 2005). Maximum intensities and distances of observations vary from VIII at 20 km in Les Saintes to IV at 140 km in
Martinique, and correspond to detailed studies carried on by BCSF in 33 different urban districts. All the 29 observed intensities are within the $B3$ uncertainty limits ($RMS=0.6$, median = $−0.9$).
Fig. 8f shows the largest Les Saintes aftershock, on February 14, 2005 of magnitude $Mw=5.8$, located south of Terre-de-Haut ($Mw$ and location from Global CMT Project, MSK intensities from
OVSG-IPGP). Maximum intensities and distances of observations vary from VII at 14 km in Les Saintes to IV at 74 km in Anse-Bertrand (Guadeloupe). All the 25 observed intensities are within the $B3$
uncertainty limits ($RMS=0.3$, median = $−1.0$) with a global light overestimation.
Fig. 8g shows one of the numerous Les Saintes aftershocks, on December 22, 2005 of magnitude $Md=4.2$, located north of Terre-de-Bas ($Md$, location and MSK intensities from OVSG-IPGP, unpublished).
Maximum intensities and distances of observations vary from V at 15 km in Basse-Terre to II at 58 km in Saint-François (Guadeloupe). All the 7 observed intensities are within the $B3$ uncertainty
limits ($RMS=0.6$, median = $−0.3$).
Fig. 8h shows the November 29, 2007 Martinique intermediate-depth (152 km) intraslab earthquake of magnitude $Mw=7.4$, $Mw$ and location from Bouin et al. (2010) and Global CMT Project, with EMS98
intensities from an official survey by the BCSF (Schlupp et al., 2008). Maximum intensities and distances of observations vary from VII at 150 km in Martinique to II at 400 km in St-Barthelemy, and
correspond to detailed studies carried on by BCSF in 70 different urban districts in Guadeloupe and Martinique, plus other islands reports. Most of the 74 observed intensities are within the $B3$
uncertainty limits ($RMS=0.83$, median = $−0.1$), but we note three underestimated intensities at long distances: V in Saint-Vincent (250 km) and Trinidad (500 km), and IV in Anguilla (443 km). This
may be due to local site amplifications because of low frequency content of the seismic waves.
These eight examples confirm that $B3$ model seems able to predict average intensities within a global residual of $σ=1.4$ degree in the MSK scale, for events of magnitudes up to 7.4 in Lesser
Antilles context with various hypocentral distances. This value corresponds to 68% of confidence interval and gives a convincing maximum possible intensity even when local site effects are observed.
5 Automatic intensity report
These good results and the apparent robustness of the $B3$ model made us confident of the release of a semi-automatic theoretical intensity report at the Guadeloupe and Martinique observatories. For
each located event, maximum intensity is computed for all towns of Lesser Antilles islands. If at least one location reaches an intensity of II, it means that the event has been potentially felt and
an automatic report is produced, waiting for seismologist validation.
This simulation allows us: (1) to confirm that inhabitants may have (or not) felt the event when intensity interval varies from II to III in a town; and (2) to publish immediately and blindly
(without any testimonies) the information of a possible felt earthquake when the predicted maximum intensity reaches IV, which means a 68% confidence level for an intensity between I-II and IV.
The report (see an example in Fig. 9) includes a synthetic text resuming the date, location and type of event, the maximum intensity prediction value and corresponding town name and distance. To
better take into account potential site effects and increase the precision of the result, the average prediction is given together with the upper limit value ($I+σ=I+1.4$) for potential site effects,
and MSK intensities are indicated in half-unit values, i.e., $I=6.0$ to 6.4 is “VI”, and $I=6.5$ to 6.9 is “VI-VII”. The exhaustive list of urban districts for which theoretical intensity reaches at
least II is given. Note that it includes all islands in the Lesser Antilles, while our model has been mainly checked with Guadeloupe and Martinique intensities. This may constitutes a future
extension of our study.
Fig. 9
The report also includes a location map that presents the islands and towns, earthquake epicenter and theoretical isoseist curves using a shaded color map. A detailed table legend explains the MSK
scale and corresponding name, color, PGA interval, potential damage and human perception.
6 Discussion and conclusions
We propose a simple empirical model for macroseismic intensities prediction for observatory operational purpose. The model is based on intermediate PGA model that has been adjusted using a shallow
crustal normal-fault sequence of events. The functional form is only 3-parameters dependent which implies many assumptions and simplifications, but makes it also extremely robust with an uncertainty
higher than usual GMPE (a factor of 3). This can be explained also by the fact that we do not select specific site conditions in the database, mixing rock and soil stations. The obtained PGA model
has strong potential limits and may not be very useful for engineering purposes, but it exhibits a better fit than previous existing GMPE for Lesser Antilles. Its application domain should be limited
to crustal events, magnitude range up to 6.3, and distance range up to 100–200 km.
The deduced intensity model is tested on a wider range of magnitudes, distances and source types of earthquakes. We suggest that the $B3$ model is able to correctly predict intensities within $±1.4$
($1σ$), for magnitudes up to 7.4 and hypocentral distance up to 300 km. At longer distances, we observe a clear underestimation of intensities. A major result of our work is that the final equation
seems to exhibit a larger applicability range than intermediate PGA predictive equation. In particular, greater magnitudes and other types of earthquakes such as those located in the subduction slab
are well modeled within the given uncertainties.
This model is currently used to produce automatic reports in Guadeloupe (since January 2005) and Martinique (since September 2008) observatories in order to anticipate potentially felt events
immediately after the location and magnitude calculation. On a total amount of about $10,000$ located events in Guadeloupe, a third has been potentially felt (minimal intensity of II) and has
produced an automatic report. Following the observatory convention, only 200 reports were effectively sent as a public communiqué, when the minimum theoretical intensity reached IV or, in case of
lower intensity (II or III), when immediate testimonies were received from inhabitants.
During more than 5 years of continuous seismic monitoring and thanks to inhabitants testimonies, the $B3$ model is daily controlled by observatory team: comparisons between observations and predicted
intensities exhibit an average uncertainty less than $±1$ unit in the MSK scale.
The reports were also used for seismic hazard awareness and education of the public and local authorities. Particularly, explaining the fundamental difference between magnitude and intensity of an
earthquake, the MSK scale, the uncertainty of prediction due to the law's empirical aspect and simplicity, and the potential site condition effects, thus earthquake-resistant construction advice.
FB thanks Pascal Bernard and Nathalie Feuillet for useful discussions, Victor Huerfano and Fabrice Cotton for very constructive comments that helped us to greatly improve the initial manuscript.
Acceleration data come from the French national strong motion permanent network Réseau Accélérométrique Permanent (RAP), available at http://www-rap.obs.ujf-grenoble.fr/. Stations have been installed
and maintained from 2002 to 2004 thanks to the effort of technicians from the observatory of Guadeloupe (OVSG-IPGP): Alberto Tarchini$†$, Christian Lambert, Laurent Mercier, Alejandro Gonzalez, and
Thierry Kitou. Authors warmly thank Guadeloupe inhabitants for their collaboration in collecting testimonies. This is an IPGP contribution #3222. | {"url":"https://comptes-rendus.academie-sciences.fr/geoscience/articles/10.1016/j.crte.2011.09.004/","timestamp":"2024-11-11T21:29:34Z","content_type":"text/html","content_length":"142329","record_id":"<urn:uuid:b623945c-19e1-45b0-85fd-7084c474b5bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00209.warc.gz"} |
what's wrong with this code
I am using the following code to solve two unknown variables (T and delta) with two equations. If I do not use "bounds t>1" AND "optimize" option, it gives me value of T close to zero to all the
observations. However, if I add these two codes, the results for all observations are 1.01 for T. I checked proc model code with simple quadratic equation to get only negative results. It worked
fine. Anyone could please help me? What is wrong with my current code for this more complex equations??? I am in urgent need to solve this. Please help!!!
Additional information: I got the following information when running the code. It seems that the code did not even iterate once to solve the equations and gave error of zero?? That does not seem
NOTE: Optimal.
NOTE: Objective = 0.4636390687.
NOTE: The NLP solver is called.
NOTE: The Interior Point algorithm is used.
Objective Optimality
Iter Value Infeasibility Error
0 0.49897143 0 0
"proc model data=HAVE noprint out=NEED;
bounds t>1;
eq.one = (1-PD) - (probnorm((-log(PD)-(s - 0.5*(delta*delta))*T)/(delta*sqrt(T)))
-PD*exp(s*T)* probnorm((-log(PD)-(s - 0.5*(delta*delta))*T)/(delta*sqrt(T))- delta*sqrt(T)));
eq.two = delta_E*(1-PD) - delta*probnorm((-log(PD)-(s - 0.5*(delta*delta))*T)/(delta*sqrt(T)));
solve delta T /optimize solveprint ;
id companyid date pd s delta_e;
06-25-2018 12:29 AM | {"url":"https://communities.sas.com/t5/Mathematical-Optimization/what-s-wrong-with-this-code/td-p/472865","timestamp":"2024-11-11T17:23:03Z","content_type":"text/html","content_length":"333563","record_id":"<urn:uuid:3c941ee0-91db-4c2b-a823-f205efa39e0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00584.warc.gz"} |
Drgania Swobodne i Wyboczenie Płyty Trójkątnej Wzmocnionej Żebrami
Engineering Transactions, 6, 2, pp. 233-252, 1958
Drgania Swobodne i Wyboczenie Płyty Trójkątnej Wzmocnionej Żebrami
In the first part of this paper, an isotropic triangular plate is considered. The plate is simply supported on the edges and rests on an elastic foundation. The stiffening rib is parallel to one of
the sides adjacent to the right angle. The plate is loaded by longitudinal forces uniformly distributed along the edges. The stiffening bar is loaded by a concentrated force S Fig. 1a. The
interaction between the plate and the rib is replaced by a continuous load p(x). The load p (x, y) of the plate due to the action of the rib, is expressed by a double trigonometric series using the
expression for a triangular plate loaded by the unit force. Starting from the differential equation for plates, the amplitude surface w (x, y) is determined for a compressed plate loaded additionally
by p(x, y), Eq. (1.3). The amplitude curve for the rib v (zc), Eq. (1.4)), is determined from the differential equation for the rib subjected to longitudinal compression, and the additional load p
(zc) expressed by means of a simple trigonometric series. From the compatibility condition between the plate and the rib, Eq. (1.5), the Eq. (1.7) is obtained. Finally, the determinant of a system of
equations is obtained in which the unknown quantity is o the frequency of free vibration of the system (plate and rib). Setting =0, the critical compressive force Scr for the rib or the critical load
Ncr may be found. In the second part, certain particular cases are examined. These concern a triangular plate with one or more ribs loaded by concentrated forces.
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).
S. Timoshenko, Theory of Plates and Shells, 1940.
Z. Kączkowski, Drgania swobodne i wyboczenia płyty trójkątnej, Arch. Mech. Stos. 1 (1956).
W. Nowacki, Stateczność płyt prostokątnych wzmocnionych zebrami, Arch. Mech. Stos. 2 (1954).
W. Nowacki i A. Kacner, Stateczność rusztów wzmocnionych płytą, Arch. Inz. Lad. 1 (1955). | {"url":"https://et.ippt.gov.pl/index.php/et/article/view/2980","timestamp":"2024-11-12T05:59:33Z","content_type":"text/html","content_length":"20398","record_id":"<urn:uuid:6e291e19-4c50-40b6-87e0-fc6ea291cdf6>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00110.warc.gz"} |
Where can I find someone who specializes in matrix partitioning techniques in R programming? | Pay Someone To Take My R Programming Assignment
Where can I find someone who specializes in matrix partitioning techniques in R programming? In particular, what is a “residual component” or “residual block” in R and why is that necessary? Is it
possible for a database software to be constructed that has a “residual block” in it? Or all of the above uses the same database software? A: Given that you have just been introduced to matrix
partitioning, your data does not need any persistence. You can create a matrix exactly like the matrix before. Now, make a matrix. Now, you can use any of the other tricks that you have just
described to locate matrix partitions in R: A matrix will be in your database file so by using matrix partitioning, you can create partitioning by any of your data attributes that you do not need. I
don’t know if you can type a matrix on your processor and use it to calculate the corresponding partitioning of the partition. You could also you can use matrix and set size of matrix. …but you need
the matrix partitioning. In Matrix partitioning just set a bit more field to set for the partition to your first partition. A: Theres an issue with your two questions. I’ll have to set it up some
more. Consider creating a dummy matrix of size 256 by mapping a start1-4 row of data to the end1-5 row of data. This dummy matrix has all dimensions as an nx nx nx 4×4 type matrix and will contain
all m x m rows (or non-m x n x n x 2×2 or n n m x n t x 2×2 or t x 2×2) a knockout post one row. I recommend you simply to add code that you can freely use: part(6, 2) = 64 part(6, 2.6, n = 6.64586,
m = 3) = 1024 if there are M lx(7) that represent m x n x m entries, add lines one, two and three to figure out if m x n x m entries will contain 0 and 1. If not, using the data partitioning would be
easier. I removed the initial vector and just wrote a vector that shows the number 6500 in the first step, so that you can also type it part(6500, 2) = 1 is = 60 add line one add line two I will
state that if this is an existing write function, make sure that in that procedure it is a combination of a function and a set.
My Class And Me
I guess this equation is what needs to be written, but if not, you can try with it too. Edit: thanks to the reviewer here, this code is much faster with the c(v)() expression. So my suggestion
(please copy some other answers) is to just do use them all for calling some other functions.Where can I find someone who specializes in matrix partitioning techniques in R programming? I’m a little
confused as to why I can’t find a reference for this one, since matrices are typically used in other multithreaded programming paradigms. The best I can suggest is to look for source code published
under the Matlab community’s project Let me first explain about matrix partitioning techniques in R. One of the most commonly used techniques is division. A division algorithm assumes that all the
data may be stored in some state machine. However, not every state machine needs to be in some physical state and can be manipulated in several ways. Specifically, since a division algorithm can
incorporate random variables, there are several ways to combine state machines of different state to obtain a state machine in which all the data do not exist. A specific technique for this is to
create a state machine that contains data that you define individually or grouped. This gives us a great deal of information that can be useful for information graph research. Interestingly, the
formula for division algorithm is designed to take the state machine to the local “A” in a unit N rows vector. The initial state machine can be transformed to a particular matrix in the form of a
matrix of the form That means the division algorithm can group the elements of the matrix in some way. For example, if the value each each row or each column of the matrix were to be placed in the
same vertical distance xy to preserve the symmetry of the matrices being moved by the algorithm, then one can move rows or columns to a different position in the computation matrix. This sort of
transformation is called “division” by common terms to distinguish it from other sorts of transformations. Additionally, it can be helpful to look at how division maps to what other-dimensional
vectors to use, e.g. with linear combinations. For example, if x xy describes a way to group several rows out at a very small angle to form a general linear combination, the division algorithm can
tell us if a certain column of matrix has a “weight” that is small or large depending on the amount of data the divisions takes. Essentially, if any element of the division-based matrix is an offset
parameter, this weight allows the division algorithm to do division things better.
No Need To Study Reviews
The division-based matrix is then used to divide by y or z before simply resampling to the appropriate data model. This does work sometimes like being able to transform the division algorithm into a
Matérn instance, but not always as much as you would want. The division algorithm can be implemented using Matlab code as well as an R2-based division library. In this case, you can simply think of
it as grouping all the data as a whole group and replacing data that has some property that identifies it as belonging to the group as a separate column or row. In fact, you can actually do stuff
like this in the division based matrix like mulselater for this aspect of division: The division-based matrix can be usedWhere can I find someone who specializes in matrix partitioning techniques in
R programming? This question is extremely long and it is not necessary for my professor and his team to mention me when I say this. What is your setup for matrix partition? Open-source R code can be
found at http://www.realdata.com/packages/downloads/ Thank you, Mesurys Byrne BAREN Duke University “4 hours to read a column. Read 30 columns of data (big, with up to 45 rows/column). Do up to 4
weeks of programming!” Thanks for the input! dude Rstudio (R) Danish “2 hours to read a column. Read 120 columns of data… 4 weeks of programming!” Thanks also Drede! Adalberto Adalberto Adalberto
Adalberto Cantor, AZ “The quality of our hardware and our development infrastructure should be measured ahead of the early days of the project. This isn’t easy, and each day can be something extra.
Each step by step should be taken care of accomplice of our project.” Adalberto, what are the benefits and drawbacks of your projects? We are currently building projects where people will be using
almost everything because we simply cannot do the things for real time in R. This is as well a solution for teams and non-tactical teams and developers. In the smaller projects (the ones used in our
class), we do not have the usefull project resources that we have in our architecture, and for things that people will really like. We do need to better understand the benefits and disadvantages of R
in the real world.
What Are The Advantages Of Online Exams?
We really like the idea of doing this with our teams and at project level. We are quite new for us and will be going out of this area in years to come. We decided to concentrate on making our
projects cheaper to take care of. It is enough that we have the capabilities to scale up in our company. We can see the application in how well we are developing and how we always operate. On the
other hand, in general, we think what the biggest thing about the R development team is is that that they can be very good at moving code into your way of thinking. We need to not feel as though we
are investing our resources to change more than usual. We are convinced that the experience of learning new stuff will give us more real understanding of what we understand. Then, of course, R really
well is the right application. In the end, it is going to be top notch in terms of learning. Hahaha I think the advantage of developing for real time is that the quality of the code is minimal. Or
rather, it is in a sense “feels good”. None of the old R modules are that good with that. Now I would like to ask if there is any real real benefit about moving synthetic one’s design to R as opposed
to a modern distrib. At this stage, we want to make sure that we take advantage of many types of syntactic noise, namely the syntactic noise from high level tools and old version-mode programmers. As
far as the experience in the design for this program to me seems really obvious, but I do not seem to be helping you find the ones you want to support. R would also solve the issue of what I defined
as “errors with 0”. The problem with those “errors are all syntactic jumbles, make sure you are bookmarked and reference your source code to the book!” is that they are not quite as fast or as
important as they could have been. After installing R for the real world, as you mentioned, your code for some reason feels really different (not even right here than a clean reference file).
However, in general, it helps you figure out what you might want to do with wikipedia reference code.
Pay People To Take Flvs Course For You
From the simple fact that the two processes are alike, and probably related: you are happy to be able to quickly test and debug (or take care of) the code. There are also obvious issues of timing and
/or readability but also other. You will find that our designers and maintainers will know all the nuances. I am glad you joined us. I am a big fan of learning new things. I liked the design because
it was easy to learn. I can no longer do I’ve always felt like learning as many times as I wanted to learn. That is always easier for me. | {"url":"https://rprogrammingassignments.com/where-can-i-find-someone-who-specializes-in-matrix-partitioning-techniques-in-r-programming","timestamp":"2024-11-04T20:41:10Z","content_type":"text/html","content_length":"199542","record_id":"<urn:uuid:ed574ae8-8aa3-4c57-9663-a51b4e3daff4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00822.warc.gz"} |
Finding Unknowns in Matrix Equations
Question Video: Finding Unknowns in Matrix Equations Mathematics • First Year of Secondary School
Given that ๐ ด = [3, 0, โ 2 and โ 2, 1, ๐ ฅ and 0, 3, 7], find ๐ ฅ such that ๐ ดยฒ = [9, โ 6, โ 20 and โ 8, โ 20, โ 52 and โ 6, 24, 28].
Video Transcript
Given that ๐ ด is the three-by-three matrix three, zero, negative two, negative two, one, ๐ ฅ, zero, three, seven, find the value of ๐ ฅ such that ๐ ด squared is equal to the three-by-three
matrix nine, negative six, negative 20, negative eight, negative 20, negative 52, negative six, 24, 28.
In this question, we are given a three-by-three matrix ๐ ด containing an unknown value of ๐ ฅ. We need to find this value of ๐ ฅ. To do this, we are given the matrix ๐ ด squared. We can recall
that we define squaring a matrix in the same way we square a number. We multiply it by itself, so ๐ ด squared is ๐ ด times ๐ ด. We can use this to find ๐ ด squared. We need to multiply ๐ ด by
itself. This gives us the following product for ๐ ด squared. Before we calculate the square of this matrix, we can replace ๐ ด squared in the equation with the given matrix ๐ ด squared.
We now recall that we multiply matrices by finding the sum of the products of each entry in each row of the first matrix with the corresponding elements in the columns of the second matrix. For
instance, we can apply this to the second row of the first matrix and the third column of the second matrix to get the element in the second row and third column of ๐ ด squared. Equating the entry
in the second row and third column of ๐ ด squared with the sum of the products of these entries gives us the following linear equation in ๐ ฅ.
We can then solve this equation for ๐ ฅ. We simplify each side of the equation. We then subtract four from both sides of the equation to obtain negative 56 equals eight ๐ ฅ. Then, we divide through
by eight to get that ๐ ฅ equals negative seven. | {"url":"https://www.nagwa.com/en/videos/746140591746/","timestamp":"2024-11-12T23:58:51Z","content_type":"text/html","content_length":"249623","record_id":"<urn:uuid:6df03837-226b-4848-87ef-02b7b5fa17b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00765.warc.gz"} |
Math Colloquia - Homogeneous dynamics and its application to number theory
Homogeneous dynamics, the theory of flows on homogeneous spaces, has been proved useful for certain problems in Number theory.
In this talk, we will explain what kind of geometry and dynamics we need to solve certain number theoretic questions such as counting matrices of integer entries, or some problems in Diophantine
approximation. The appropriate manifold can often be seen as a space of lattices, and its asymptotic geometry is governed by the smallest length of a non-zero vector in a given lattice, which is also
the backbone of post-quantum cryptography.
We will then explain how (partial) solutions of Oppenheim conjecture and Littlewood conjecture were obtained using homogeneous dynamics. We will also survey some recent results and remaining | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&sort_index=room&order_type=asc&page=10&document_srl=1266013","timestamp":"2024-11-14T18:17:03Z","content_type":"text/html","content_length":"43729","record_id":"<urn:uuid:b9ea1a79-7e3f-4c4e-b93c-42be89736834>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00567.warc.gz"} |
The Stacks project
Situation 77.7.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $u : \mathcal{F} \to \mathcal{G}$ be a homomorphism of quasi-coherent $\mathcal{O}_
X$-modules. For any scheme $T$ over $B$ we will denote $u_ T : \mathcal{F}_ T \to \mathcal{G}_ T$ the base change of $u$ to $T$, in other words, $u_ T$ is the pullback of $u$ via the projection
morphism $X_ T = X \times _ B T \to X$. In this situation we can consider the functor
$$\label{spaces-flat-equation-iso} F_{iso} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets}, \quad T \longrightarrow \left\{ \begin{matrix} \{ *\} & \text{if }u_ T\text{ is an isomorphism}, \\
\emptyset & \text{else.} \end{matrix} \right.$$
There are variants $F_{inj}$, $F_{surj}$, $F_{zero}$ where we ask that $u_ T$ is injective, surjective, or zero.
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Adding And Subtracting Mixed Numbers Like Denominators Worksheets
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6 77 10 6 18 10 1 1 4 2 16 4 3 5 8 4 3 8 or or or 26 or
Welcome to The Adding and Subtracting Two Mixed Fractions with Similar Denominators Mixed Fractions Results and Some Simplifying Fillable A Math Worksheet from the Fractions Worksheets Page at Math
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Subtracting Mixed Numbers Worksheet
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Adding And Subtracting Mixed Fractions A Fractions Worksheet
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Adding And Subtracting Two Mixed Fractions With Similar Denominators
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Welcome to The Adding and Subtracting Two Mixed Fractions with Similar Denominators Mixed Fractions Results and Some Simplifying Fillable A Math Worksheet from the Fractions Worksheets Page at Math
Drills This math worksheet was created or last revised on 2023 09 15 and has been viewed 2 435 times
This page has worksheets on subtracting fractions and mixed numbers Includes like and unlike denominators Worksheets for teaching basic fractions equivalent fractions simplifying fractions comparing
fractions and ordering fractions
Add And Subtract Mixed Numbers With Like Denominators Anchor Fractions Math Fractions 4th
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When Most VNM-Coherent Preference Orderings Have Convergent Instrumental Incentives — LessWrong
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orderings have the same statistical incentives as utility functions; most such orderings will incentivize power-seeking in the settings covered by the power-seeking theorems.
In certain contexts, coherence theorems can have non-trivial implications, in that they provide Bayesian evidence about what the coherent agent will probably do. In the situations where the
power-seeking theorems apply, coherent preferences do suggest some degree of goal-directedness. Somewhat more precisely, VNM-coherence is Bayesian evidence that the agent prefers to stay alive, keep
its options open, etc.
However, VNM-coherence over action-observation histories tells you nothing about what behavior to expect from the coherent agent, because there is no instrumental convergence for generic utility
functions over action-observation histories!
The result follows because the VNM utility theorem lets you consider VNM-coherent preference orderings to be isomorphic to their induced utility functions (with equivalence up to positive affine
transformation), and so these preference orderings will have the same generic incentives as the utility functions themselves.
Let be outcomes, in a sense which depends on the context; outcomes could be world-states, universe-histories, or one of several fruits. Outcome lotteries are probability distributions over outcomes,
and can be represented as elements of the -dimensional probability simplex (ie as element-wise non-negative unit vectors).
A preference ordering is a binary relation on lotteries; it need not be eg complete (defined for all pairs of lotteries). VNM-coherent preference orderings are those which obey the VNM axioms. By the
VNM utility theorem, coherent preference orderings induce consistent utility functions over outcomes, and consistent utility functions conversely imply a coherent preference ordering.
Definition 1: Permuted preference ordering. Let be an outcome permutation, and let be a preference ordering. is the preference ordering such that for any lotteries : if and only if .
EDIT: Thanks to Edouard Harris for pointing out that Definition 1 and Lemma 3 were originally incorrect.
Definition 2: Orbit of a preference ordering. Let be any preference ordering. Its orbit is the set .
The orbits of coherent preference orderings are basically all the preference orderings induced by "relabeling" which outcomes are which. This is made clear by the following result:
Lemma 3: Permuting coherent preferences permutes the induced utility function. Let be a VNM-coherent preference ordering which induces VNM-utility function , and let . Then induces VNM-utility
function , where is any outcome.
Proof. Let be any lotteries.
1. By the definition of a permuted preference ordering, if and only if .
2. By the VNM utility theorem and the fact that is coherent, iff .
3. Since there are finitely many outcomes, we convert to vector representation: .
4. By associativity, .
5. But this is just equivalent to .
As a corollary, this lemma implies that if is VNM-coherent, so is , since it induces a consistent utility function over outcomes.
Consider the orbit of any . By the VNM utility theorem, each preference ordering can be considered isomorphic to its induced utility function (with equivalence up to positive affine transformation).
Then let be any utility function compatible with . By the above lemma, consider the natural bijection between the (preference ordering) orbit of and the (utility function) orbit of , where .
When my theorems on power-seeking are applicable, some proportion of the right-hand side is guaranteed to make (formal) power-seeking optimal. But by the bijection and by the fact that the preference
orderings incentivize the same things (by the VNM theorem in the reverse direction), the (preference ordering) orbit must have the exact same proportion of elements for which (lotteries representing
formal) power-seeking are optimal.
Conversely, if we know that some set A of lotteries tends to be preferred over another set B of lotteries (in the preference order orbit sense), then the same argument shows that A tends to have
greater expected utility than B (in the utility function orbit sense). This holds for all (utility function) orbits, because every utility function corresponds to a VNM-coherent preference ordering.
So: orbit-level instrumental convergence for utility functions is equivalent to orbit-level instrumental convergence for VNM-coherent preference orderings.
• Instrumental convergence does not exist when maximizing expected utility over action observation histories (AOH).
□ Therefore, VNM-coherence over action observation history lotteries tells you nothing about what behavior to expect from the agent.
□ Coherence over AOH tells you nothing because there is no instrumental convergence in that setting!
• In certain contexts, coherence theorems can have non-trivial implications, in that they provide Bayesian evidence about what the coherent agent will probably do.
□ In the situations where the power-seeking theorems apply, coherent preferences do suggest some degree of goal-directedness.
□ Somewhat more precisely, VNM-coherence is Bayesian evidence that the agent prefers to stay alive, keep its options open, etc.
• In some domains, preference specification may be more natural than utility function specification. However, in theory, coherent preferences and utility functions have the exact same statistical
□ In practice, they will differ. For example, suppose we have a choice between specifying a utility function which is linear over state features, or of doing behavioral cloning on elicited
human preferences over world states. These two methods will probably tend to produce different incentives.
The quest for better convergence theorems
Goal-directedness seems to more naturally arise from coherence over resources. (I think the word 'resources' is slightly imprecise here, because resources are only resources in the normal context of
human life; money is useless when alone in Alpha Centauri, but time to live is not. So we want coherence over things-which-are-locally-resources, perhaps.)
In his review of Seeking Power is Often Convergently Instrumental in MDPs, John Wentworth wrote:
in a real-time strategy game, units and buildings and so forth can be created, destroyed, and generally moved around given sufficient time. Over long time scales, the main thing which matters to
the world-state is resources - creating or destroying anything else costs resources. So, even though there's a high-dimensional game-world, it's mainly a few (low-dimensional) resource counts
which impact the long term state space. Any agents hoping to control anything in the long term will therefore compete to control those few resources.
More generally: of all the many "nearby" variables an agent can control, only a handful (or summary) are relevant to anything "far away". Any "nearby" agents trying to control things "far away"
will therefore compete to control the same handful of variables.
Main thing to notice: this intuition talks directly about a feature of the world - i.e. "far away" variables depending only on a handful of "nearby" variables. That, according to me, is the main
feature which makes or breaks instrumental convergence in any given universe. We can talk about that feature entirely independent of agents or agency. Indeed, we could potentially use this
intuition to derive agency, via some kind of coherence theorem; this notion of instrumental convergence is more fundamental than utility functions.
In his review of Coherent decisions imply consistent utilities, John wrote:
"resources" should be a derived notion rather than a fundamental one. My current best guess at a sketch: the agent should make decisions within multiple loosely-coupled contexts, with all the
coupling via some low-dimensional summary information - and that summary information would be the "resources". (This is exactly the kind of setup which leads to instrumental convergence.) By
making pareto-resource-efficient decisions in one context, the agent would leave itself maximum freedom in the other contexts. In some sense, the ultimate "resource" is the agent's action space.
Then, resource trade-offs implicitly tell us how the agent is trading off its degree of control within each context, which we can interpret as something-like-utility.
This seems on-track to me. We now know what instrumental convergence looks like in unstructured environments, and how structural assumptions on utility functions affect the shape and strength of that
instrumental convergence, and this post explains the precise link between "what kinds of instrumental convergence exists?" and "what does VNM-coherence tell us about goal-directedness?". I'd be
excited to see what instrumental convergence looks like in more structured models.
Footnote representative: In terms of instrumental convergence, positive affine transformation never affects the optimality probability of different lottery sets. So for each (preference ordering)
orbit element , it doesn't matter what representative we select from each equivalence class over induced utility functions — so we may as well pick !
Copying over a Slack comment from Abram Demski:
I think this post could be pretty important.
It offers a formal treatment of "goal-directedness" and its relationship to coherence theorems such as VNM, a topic which has seen some past controversy but which has -- till now -- been dealt
with only quite informally. Personally I haven't known how to engage with the whole goal-directedness debate, and I think part of the reason for that is the vagueness of the idea.
Goal-directedness doesn't seem that cruxy for most of my thinking, but some other people seem to really strongly perceive it as a crux for miri-type thought, and sometimes as a crux for AI risk
more generally. (I once made a "tool AI" argument against AI risk myself, although in hindsight I would say that was all motivated cognition, which ignored the idea that even tool AI has to
optimize strongly in order to have high capabilities.)
So, as I see it, there's been something of a stalemate between people who think the "goal-directed AI" vs "non-goal-directed AI" distinction is important for one reason or another, vs people who
don't think that.
Alex Turner seems to give real technical meaning to this distinction, showing that most VNM-coherent preferences are indeed "goal directed" in the sense of acting broadly like we expect agents to
act (that is, behaving in ways consistent with instrumental convergence). However, he also gives a class of VNM-coherent preferences which are not goal-directed in this sense, instead exhibiting
essentially random behavior. This gives us a plausible formal proxy for the "goal directed vs not goal directed" distinction!
I'm not sure how it can/should carry the broader conversation forward, yet, but it seems like something to think about.
No problem! Glad it was helpful. I think your fix makes sense.
I'm not quite sure what the error was in the original proof of Lemma 3; I think it may be how I converted to and interpreted the vector representation.
Yeah, I figured maybe it was because the dummy variable was being used in the EV to sum over outcomes, while the vector was being used to represent the probabilities associated with those outcomes.
Because and are similar it's easy to conflate their meanings, and if you apply to the wrong one by accident that has the same effect as applying to the other one. In any case though, the main result
seems unaffected.
New Comment
4 comments, sorted by Click to highlight new comments since:
Thanks for writing this.
I have one point of confusion about some of the notation that's being used to prove Lemma 3. Apologies for the detail, but the mistake could very well be on my end so I want to make sure I lay out
everything clearly.
First, is being defined here as an outcome permutation. Presumably this means that 1) for some , ; and 2) admits a unique inverse . That makes sense.
We also define lotteries over outcomes, presumably as, e.g., , where is the probability of outcome . Of course we can interpret the geometrically as mutually orthogonal unit vectors, so this lottery
defines a point on the -simplex. So far, so good.
But the thing that's confusing me is what this implies for the definition of . Because is defined as a permutation over outcomes (and not over probabilities of outcomes), we should expect this to be
The problem is that this seems to give a different EV from the lemma:
(Note that I'm using as the dummy variable rather than , but the LHS above should correspond to line 2 of the proof.) Doing the same thing for the lottery gives an analogous result. And then looking
at the inequality that results suggests that lemma 3 should actually be " induces " as opposed to " induces ".
(As a concrete example, suppose we have a lottery with the permutation , , . Then and our EV is
Yet which appears to contradict the lemma as stated.)
Note that even if this analysis is correct, it doesn't invalidate your main claim. You only really care about the existence of a bijection rather than what that bijection is — the fact that your
outcome space is finite ensures that the proportion of orbit elements that incentivize power seeking remains the same either way. (It could have implications if you try to extend this to a metric
space, though.)
Again, it's also possible I've just misunderstood something here — please let me know if that's the case!
Thanks! I think you're right. I think I actually should have defined differently, because writing it out, it isn't what I want. Having written out a small example, intuitively, should hold iff ,
which will also induce as we want.
I'm not quite sure what the error was in the original proof of Lemma 3; I think it may be how I converted to and interpreted the vector representation. Probably it's more natural to represent as ,
which makes your insight obvious.
The post is edited and the issues should now be fixed. | {"url":"https://www.lesswrong.com/s/fSMbebQyR4wheRrvk/p/LYxWrxram2JFBaeaq","timestamp":"2024-11-06T06:04:26Z","content_type":"text/html","content_length":"1048907","record_id":"<urn:uuid:8452ee7f-5b92-4eda-bc24-84ec5a819283>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00255.warc.gz"} |
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Question Text 55
Updated On Jan 10, 2023
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 50
Avg. Video Duration 4 min | {"url":"https://askfilo.com/user-question-answers-mathematics/55-begin-array-l-x-rightarrow-a-frac-cos-x-cos-a-cot-x-cot-a-33373631363536","timestamp":"2024-11-08T14:17:25Z","content_type":"text/html","content_length":"530668","record_id":"<urn:uuid:4423c8cd-e284-4894-8e4e-8c5863c99bf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00644.warc.gz"} |
statistics introduction to probability, Math Homework Help - onlinehomeworkexperts.com
statistics introduction to probability, Math Homework Help
Problems need to include all required steps and answer(s) for full credit. All answers need to be reduced to lowest terms where possible.
Answer the following problems showing your work and explaining (or analyzing) your results.
1. In a poll, respondents were asked if they have traveled to Europe. 68 respondents indicated that they have traveled to Europe and 124 respondents said that they have not traveled to Europe. If
one of these respondents is randomly selected, what is the probability of getting someone who has traveled to Europe?
2. The data set represents the income levels of the members of a golf club. Find the probability that a randomly selected member earns at least $100,000.
INCOME (in thousands of dollars)
1. A poll was taken to determine the birthplace of a class of college students. Below is a chart of the results.
1. What is the probability that a female student was born in Orlando?
2. What is the probability that a male student was born in Miami?
3. What is the probability that a student was born in Jacksonville?
Gender Number of students Location of birth
Male 10 Jacksonville
Female 16 Jacksonville
Male 5 Orlando
Female 12 Orlando
Male 7 Miami
Female 9 Miami
1. Of the 538 people who had an annual check-up at a doctor’s office, 215 had high blood pressure. Estimate the probability that the next person who has a check-up will have high blood pressure.
2. Find the probability of correctly answering the first 4 questions on a multiple choice test using random guessing. Each question has 3 possible answers.
3. Explain the difference between independent and dependent events.
4. Provide an example of experimental probability and explain why it is considered experimental.
5. The measure of how likely an event will occur is probability. Match the following probability with one of the statements. There is only one answer per statement.
0 0.25 0.60 1
a. This event is certain and will happen every time.
b. This event will happen more often than not.
c. This event will never happen.
d. This event is likely and will occur occasionally.
1. Flip a coin 25 times and keep track of the results. What is the experimental probability of landing on tails? What is the theoretical probability of landing on heads or tails?
2. A color candy was chosen randomly out of a bag. Below are the results:
Color Probability
Blue 0.30
Red 0.10
Green 0.15
Yellow 0.20
Orange ???
a. What is the probability of choosing a yellow candy?
b. What is the probability that the candy is blue, red, or green?
c. What is the probability of choosing an orange candy?
The assignment this week is to collect quantitative data for a minimum of 10 days from ONE of your daily activities. Some examples of data collection include:
• The number of minutes you spend studying every day.
• The time it takes to cook meals each day.
• The amount of daily time spent talking on the phone.
• The amount of time you drive each day.
In a paper (1–3 pages), describe the data you are going to collect and how you are going to keep track of the time. Within the paper, incorporate the concepts we are learning in the module, including
(but not limited to) probability theory, independent and dependent variables, and theoretical and experimental probability. Discuss your predictions of what you anticipate the data to look like and
events that can skew the data. Collect data for at least 10 days. Include at least 3-5 days of data with your SLP 1 submission. Continue collecting data for the remaining days for use in SLP 2. Do
you think the data will provide a valid representation of these activities? Explain why or why not.
Submit your paper at the end of Module 1. Future SLP assignments depend on this data and thus this assignment needs to be completed early in the session.
SLP Assignment Expectations
Answer all questions posted in the instructions. Use information from the modular background readings and videos as well as any good-quality resource you can find. Cite all sources in APA style and
include a reference list at the end of your paper.
Note about page length: Your ability to clearly articulate and explain these concepts is being assessed. The page length is a general guideline. A 3- or 4-page paper does not necessarily guarantee a
grade of “A.” An “A” paper would include detailed information and explanations of all the assignment requirements listed above. The letter grade will be based upon demonstrated mastery of the content
and ability to articulate and apply the concepts in the assignment. Keep this in mind while writing your paper.
Do you need a similar assignment written for you from scratch? We have qualified writers to help you. You can rest assured of an A+ quality paper that is plagiarism free. Order now for a FREE first
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https://onlinehomeworkexperts.com/wp-content/uploads/2021/10/logo.png 0 0 James https://onlinehomeworkexperts.com/wp-content/uploads/2021/10/logo.png James2024-10-08 19:35:212024-10-08 19:35:21
statistics introduction to probability, Math Homework Help | {"url":"https://onlinehomeworkexperts.com/statistics-introduction-to-probability-math-homework-help/","timestamp":"2024-11-11T20:57:20Z","content_type":"text/html","content_length":"50851","record_id":"<urn:uuid:e2a59274-c0ef-4566-9bc2-0bb0f9322161>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00743.warc.gz"} |
What is the column index of a table
A peculiar but fascinating problem presented itself to me today. It was one of those; "oh yeah, I'd better look at that" kind of problems that you think you'll solve in a few minutes but eventually
grow to take hours.
Given a cell in a HTML table, what is it's column index?
This seems like it's simple. You just count the cells preceding it. Fine. Works for 99% of cases. But what if the table looks like this?:
│ │cell 2 │cell 3 │ │ │cell 6 │
│ ├───────┼──────┬──────────┤ │cell 5 ├───────┤
│ │cell 7 │ │ │ │ │cell 10│
│cell 1├───────┤ │cell 9 │cell 4├───────┼───────┤
│ │ │cell 8│ │ │cell 12│ │
│ │cell 11│ ├──────────┤ ├───────┤cell 13│
│ │ │ │My index? │ │cell 15│ │
Suddenly things are very complicated again. The last row really only contains two cells and the browser reports cell indexes 0 and 1 on those, respectively. Not the 3 and 4 I was hoping for.
My first thought was simply adding up the number of cells preceding this one which had a greater row index + row span than the row index of my subject cell. This doesn't work because of cells like #4
& #13.
Eventually I simply ended up with a brute force method that does the job. But is unfortunately neither fast nor elegant. But then, what code of mine is?
It's presented here in case anyone else needs to solve the same annoying problem. Or if anyone can suggest a nicer (or preferably a faster) method of doing this. | {"url":"https://borgar.net/s/2008/03/table-cell-column-index/","timestamp":"2024-11-14T19:05:24Z","content_type":"text/html","content_length":"5933","record_id":"<urn:uuid:6f6e260c-53e6-427b-8ec2-4429246ff7ac>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00712.warc.gz"} |
Effect Of Slip On Rotor Parameters : Part 1
Effect of Slip on Rotor Parameters
In case of a transformer, frequency of the induced e.m.f. in the secondary is same as the voltage applied to primary. Now in case of induction motor at start N = 0 and slip s = 1. Under this
condition as long as s = 1, the frequency of induced e.m.f. in rotor is same as the voltage applied to the stator. But as motor gathers speed, induction motor has some slip corresponding to speed N.
In such case, the frequency of induced e.m.f. in rotor is no longer same as that of stator voltage. Slip affects the frequency of rotor induced e.m.f. Due to this some other rotor parameters also get
affected. Let us study the effect of slip on the following rotor parameters.
1. Rotor frequency
2. Magnitude of rotor induced e.m.f.
3. Rotor reactance
4. Rotor power factor
5. Rotor current
1. Effect on rotor frequency
In case of induction motor, the speed of rotating magnetic field is,
N[s] = (120 f )/P ……….(1)
Where f = Frequency of supply in Hz
At start when N = 0, s = 1 and stationary rotor has maximum relative motion with respect to R.M.F. Hence maximum e.m.f. gets induced in the rotor at start. The frequency of this induced e.m.f. at
start is same as that of supply frequency.
As motor actually rotates with speed N, the relative speed of rotor with respect R.M.F. decreases and becomes equal to slip speed of N[s] – N. The induced e.m.f. in rotor depends on rate of cutting
flux i.e. relative speed N[s]
– N. Hence in running condition magnitude of induced e.m.f. decreases so as to its frequency. The rotor is wound for same number of poles as that of stator i.e. P. If f[r ]is the frequency of rotor
induced e.m.f. in running condition at slip speed N[s] – N then there exists a fixed relation between (N[s] – N), f[r] and P similar to equation (1). So we can write for rotor in running condition,
(N[s] – N) = (120 f[r])/P , rotor poles = stator poles = P ……….(2)
Dividing (2) by (1) we get,
(N[s] – N)/N[s] = (120 f[r] / P)/(120 f / P) but (N[s] – N)/N[s] = slip s
s = f[r]/f
f[r] = s f
Thus frequency of rotor induced e.m.f. in running condition (f[r]) is slip times the supply frequency (f).
At start we have s = 1 hence rotor frequency is same as supply frequency. As slip of the induction motor is in the range 0.01 to 0.05, rotor frequency is very small in the running condition.
Example: A 4 pole, 3 phase, 50 Hz induction motor runs at a speed of 1470 r.p.m. speed. Find the frequency of the induced e.m.f in the rotor under this condition.
Solution : The given values are,
P = 4, f = 50 Hz, N = 1470 r.p.m.
N[s] = (120 f )/ P = (120 x 50)/4 = 1500 r.p.m.
s = (N[s] – N)/N[s] = (1500-1470)/1500 = 0.02
f[r] = s f = 0.02 x 50 = 1 Hz
It can be seen that in running condition, frequency of rotor induced e.m.f. is very small. | {"url":"https://electricallive.com/2015/03/effect-of-slip-on-rotor-parameters-part.html","timestamp":"2024-11-02T23:14:14Z","content_type":"text/html","content_length":"58456","record_id":"<urn:uuid:6a3a3ff7-1193-48a8-9301-6ce7e9a988b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00792.warc.gz"} |
My professor was teaching about hypothesis testing in class today.
It reminded me of some blogs by Allen Downey that I bookmarked ages ago.
I read through them in class and this is the framework to takeaway about hypothesis tests.
1. Compute test statistic that measures size of apparent effect. It could be a difference between two groups, absolute difference in means, see more examples here. We call this test statistic 𝛿
2. Define a null hypothesis, which is a model of the world under which the assumption that effect is not real, ex: if you think there is a difference between group A and B, H0 = there is no
difference between A and B.
3. Model of null hypothesis should be stochastic, that is, capable of simulating data similar to original data.
4. Goal: compute p-value (probability of seeing an effect as big as 𝛿 under null hypothesis). You can estimate p-value using simulation: calculate the same test statistic you used on the actual data
for each simulation.
5. Count the fraction of times the test statistic exceeds 𝛿. This fraction approximates p-value. If it's sufficiently small, you can conclude that the apparent effect is unlikely due to chance.
Why simulation?
• analytical methods are slow and expensive, but even as computation gets faster, they are appealing because they are
□ inflexible: using a standard test -> particular test statistic and model, might not be appropriate for problem domain.
□ opaque: real-world scenario has many possible models, based on different assumptions. In standard tests, assumptions are implicit, not easy to know whether model is appropriate.
• simulation on the other hand, are
□ explicit: creating a simulation forces you to think about your modeling decisions, the simulations themselves document those decisions.
□ arbitrarily flexible: can try out several test statistics and models, can choose most appropriate one for the scenario. | {"url":"https://www.bneo.xyz/posts/testing","timestamp":"2024-11-03T22:35:29Z","content_type":"text/html","content_length":"23134","record_id":"<urn:uuid:a90edac5-b125-4e02-b7cd-bbd4f4514e7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00037.warc.gz"} |
Every doubly transitive group action is primitive - Solutions to Linear Algebra Done Right
Every doubly transitive group action is primitive
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.8
A transitive permutation group $G \leq S_A$ acting on $A$ is called doubly transitive if for all $a \in A$, the subgroup $\mathsf{stab}(a)$ is transitive on $A \setminus \{a\}$.
(1) Prove that $S_n$ is doubly transitive on $\{1,2,\ldots,n\}$ for all $n \geq 2$.
(2) Prove that a doubly transitive group is primitive. Deduce that $D_8$ is not doubly transitive in its action on the four vertices of a square.
(1) We know that $S_n$ is transitive on $A = \{1,2,\ldots,n\}$. Now if $n \geq 2$ and $k \in A$, we have a natural isomorphism $\mathsf{stab}(k) \cong S_{A \setminus \{k\}}$; this permutation group
is also transitive in its action on $A \setminus \{k\}$. Thus $S_n$ is doubly transitive.
(2) Let $G \leq S_A$ act transitively on $A$, and suppose further that the action is doubly transitive. Let $B \subseteq A$ be a proper block; then there exist elements $b \in B$ and $a \in A \
setminus B$. By a previous exercise, we have $\mathsf{stab}(b) \leq \mathsf{stab}(B)$. Thus if $\sigma \in \mathsf{stab}(b)$, we have $\sigma[B] = B$. Suppose now that there exists an element $c \in
B$ with $c \neq b$. Because $G$ is doubly transitive on $A$, there exists an element $\tau \in \mathsf{stab}(b)$ such that $\tau(c) = a$. Thus $\tau[B] \neq B$, a contradiction. So no such element
$c$ exists and we have $B = \{b\}$. Now every block is trivial, thus the action of $G$ on $A$ is primitive.
We saw that the action of $D_8$ on the four vertices of a square is not primitive in the previous exercise. Thus this action is not doubly transitive. | {"url":"https://linearalgebras.com/solution-abstract-algebra-exercise-4-1-8.html","timestamp":"2024-11-11T17:11:42Z","content_type":"text/html","content_length":"55319","record_id":"<urn:uuid:dd01c0f6-3f73-4e21-9b36-cdb628ee086e>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00015.warc.gz"} |
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P-value can be interpreted as the observed chance of committing a false positive error. Therefore, if the chance is sufficiently low, we may reject the null hypothesis without being likely to be
wrong. The usual threshold is 0.05.
With today's advanced computing facilities, p-value can often be obtained from a statistical software. However, we need to be clear about the null and the alternative hypotheses before we can
properly draw our conclusions.
Example 1:
What does a p-value of 0.03 mean? Of course it is a statistically significant result at 5% level of significance. But we need to know the null and the alternative hypotheses before we can draw the
correct conclusion. In a t-test on H0: mean blood pressure >= 120 against HA: mean blood pressure < 120 mmHg, a p-value = 0.03 means we have sufficient evidence to demonstrate the mean blood pressure
was below 120 mmHg. However, in a t-test on H0: mean blood pressure <= 120 against HA: mean blood pressure > 120 mmHg, the interpretation will be quite different.
Example 2
What does a p-value of 0.56 mean? Of course, it is statistically insignificant at any reasonably level of significance.
In a Chi-square test for association, it means we do not have sufficient evidence to show the association at say 5% level of significance. Note however we should not say there is no association since
we will almost surely reach a significant result as the sample size becomes large.
bottom of page | {"url":"https://www.biostat.hku.hk/a-z/p-value","timestamp":"2024-11-13T04:17:35Z","content_type":"text/html","content_length":"575328","record_id":"<urn:uuid:26183055-c63e-4b5a-bc46-267accd4f389>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00292.warc.gz"} |
Polynomial LCM Python
Oops, something went wrong. Please try again in a few moments.
from math import gcd
class PolynomialLCM:
Class to handle the polynomial equation and its Least Common Multiple (LCM) operations.
- a, b, c: int
These are the coefficients for the polynomial in the format ax^2 + bx + c.
def __init__(self, a: int, b: int, c: int):
Constructor to instantiate the PolynomialLCM class.
- a: int
Coefficient for x^2 in our polynomial equation.
- b: int
Coefficient for x in our polynomial equation.
- c: int
The constant term in our polynomial equation.
- ValueError:
Throws an error if any of the coefficients are zero since we want non-zero coefficients.
# Verifying that none of the coefficients are zero.
if a == 0 or b == 0 or c == 0:
raise ValueError("Coefficients should not be zero.")
# Assigning the coefficients to the instance variables.
self.a = a # Setting the coefficient for x^2
self.b = b # Setting the coefficient for x
self.c = c # Setting the constant term
def calculate_lcm(self):
Calculates the LCM (Least Common Multiple) of the three coefficients: a, b, and c.
- int:
Gives back the LCM of the coefficients.
- ValueError:
Will raise an error if any coefficient is zero, which would make the LCM indeterminate.
# Calculating the LCM of first two coefficients 'a' and 'b'
lcm_ab = (self.a * self.b) // gcd(self.a, self.b)
# Moving on to get the LCM of 'lcm_ab' (from previous calculation) and 'c'
lcm_abc = (lcm_ab * self.c) // gcd(lcm_ab, self.c)
return lcm_abc
def evaluate_polynomial(self, x: float):
Returns the evaluated value of the polynomial equation at a given 'x'.
- x: float
The specific point/value for which the polynomial is to be computed.
- float:
The final computed value of the polynomial equation at 'x'.
# Evaluate the polynomial equation using the formula: ax^2 + bx + c
result = self.a * (x ** 2) + self.b * x + self.c
# Returning the final computed value.
return result
def main():
# Examples of using the PolynomialLCM class:
# Example 1: Initializing and evaluating the polynomial
polynomial1 = PolynomialLCM(2, 3, 4)
x_value1 = 2
result1 = polynomial1.evaluate_polynomial(x_value1)
print(f"For the polynomial {polynomial1.a}x^2 + {polynomial1.b}x + {polynomial1.c}, the value at x={x_value1} is {result1}.")
# Example 2: Calculating the LCM of coefficients
polynomial2 = PolynomialLCM(1, 2, 3)
lcm2 = polynomial2.calculate_lcm()
print(f"For the polynomial {polynomial2.a}x^2 + {polynomial2.b}x + {polynomial2.c}, the LCM of coefficients is {lcm2}.")
# Example 3: Initializing with zero coefficient (should raise an error)
polynomial3 = PolynomialLCM(1, 0, 3)
result3 = polynomial3.evaluate_polynomial(1)
print(f"For the polynomial {polynomial3.a}x^2 + {polynomial3.b}x + {polynomial3.c}, the value at x=1 is {result3}.")
except ValueError as e:
print(f"Error while initializing polynomial: {e}")
if __name__ == "__main__": | {"url":"https://codepal.ai/code-generator/query/QiF6HVNw/polynomial-lcm-python","timestamp":"2024-11-07T20:10:56Z","content_type":"text/html","content_length":"109880","record_id":"<urn:uuid:cde5e6e1-fcd2-47d4-8e57-edfa42c4a47f>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00020.warc.gz"} |
Programma Dettagliato
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Anno Accademico 2017/2018
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 099301 - COMPUTATIONAL FLUID DYNAMICS OF REACTIVE FLOWS
Cfu 5.00 Tipo insegnamento Monodisciplinare
Docenti: Titolare (Co-titolari) Cuoci Alberto
Corso di Studi Codice Piano di Studio preventivamente Da A Insegnamento
approvato (compreso) (escluso)
Ing Ind - Inf (1 liv.)(ord. 270) - MI (347) INGEGNERIA CHIMICA * A ZZZZ 099301 - COMPUTATIONAL FLUID DYNAMICS OF
REACTIVE FLOWS
081256 - FLUIDODINAMICA DEGLI INCENDI
Ing Ind - Inf (Mag.)(ord. 270) - MI (422) INGEGNERIA DELLA PREVENZIONE E DELLA SICUREZZA * A ZZZZ
NELL'INDUSTRIA DI PROCESSO 099301 - COMPUTATIONAL FLUID DYNAMICS OF
REACTIVE FLOWS
099301 - COMPUTATIONAL FLUID DYNAMICS OF
Ing Ind - Inf (Mag.)(ord. 270) - MI (472) CHEMICAL ENGINEERING - INGEGNERIA CHIMICA * A ZZZZ REACTIVE FLOWS
081256 - FLUIDODINAMICA DEGLI INCENDI
Programma dettagliato e risultati di apprendimento attesi
This course is an introduction to the Computational Fluid Dynamics (CFD) of reacting flows (i.e. flows with chemical reactions), both in laminar and turbulent conditions.
The first part of the course is focused on the fundamentals of Computational Fluid Dynamics: transport equations of mass, momentum, energy and species; spatial discretization and time integration of
transport equations; numerical algorithms for pressure-velocity coupling; numerical methods for parabolic and elliptic equations. Then, the mathematical and numerical modeling of turbulent flows will
be discussed and analyzed: URANS (Unsteady Reynolds Averaged Navier-Stokes) and LES (Large Eddy Simulation) methods.
The second part of the course is devoted to the numerical modeling of reacting flows in a CFD context: kinetic-turbulence interactions; EDC (Eddy Dissipation Concept) models; Transported PDFs;
fundamentals of turbulent combustion modeling; steady-state laminar flamelets.
In the last part of the class special topics are covered: numerical modeling of multiphase flows, verification and validation applied to CFD, large-scale problems and HPC (High Performance
The final aim of this class is to introduce the learner to CFD, to develop their understanding of the theory and operation of CFD, and to develop their competency in the employment of CFD to solve
practical engineering problems. In particular, the specific objectives of the course are:
1. To introduce and develop the main approaches and techniques that constitute the basis of Computational Fluid Dynamics for Chemical Engineers.
2. To familiarize students with the numerical implementation of these techniques and numerical schemes, to provide them with the means to write their own codes and software, and so acquire the
knowledge necessary for the skillful utilization of CFD.
3. To cover a range of modern approaches for CFD, without entering all these topics in detail, but aiming to provide students with a general knowledge and understanding of the subject, including
recommendations for further studies.
This course continues to be a work in progress. New curricular materials are being developed for this course, and feedback from students is always welcome and appreciated during the term. For
example, reviews on specific topics can be provided based on requests from students.
The prerequisite courses include fundamentals of fluid mechanics, principles of transport phenomena, fundamentals of numerical methods, and basic knowledge of computer programming. This is a
relatively advanced level treatment, but in all cases every topic is introduced in a relatively elemantary way. The elementary aspects will, however, be covered quickly so students should have
background in numerical methods and fluid dynamics. Some programming experience, such as with MATLAB(R) or C++, is also essential.
1. Introduction to Computational Fluid Dynamics (CFD); the philosophy behind CFD and its influence on engineering analysis and design; brief history of CFD; commercial and open-source codes
2. Fundamentals of numerical analysis applied to CFD: accuracy, stability, consistency; applications to 2D advection-convection equation and multidimensional boundary value problems (steady-state);
iterative methods for solving linear systems of equations.
3. Tranport equations: mass (continuity), momentum, energy, and species; integral vs differential formulations; constitutive laws: Newton’s, Fick’s, and Fourier’s laws; classification of partial
differential equations (PDE). Special cases: Euler equations, incompressible fluids, Stokes equations. Boundary and initial conditions. Discussion of their physical meaning, and presentation of
forms particularly suited to CFD. Vorticity and derivation of Navier-Stokes equations in vorticity formulation.
4. Spatial discretization of transport equations: meshes, finite difference (FD) and finite-volume (FV) techniques. First and second order discretization schemes; QUICK schemes. High-order
discretization: numerical diffusion and dispersion, the Godunov's theorem, the Godunov's method, flux vector splitting, artificial viscosity, the modern view.
5. Numerical algorithms for pressure-velocity coupling: staggered grids, momentum equations, advection, pressure and viscous terms; the pressure equation.
6. Parabolic equations: one-dimensional problems (explicit, implicit, Crank-Nicolson, accuracy, stability); multi-dimensional problems (Alternating Direction Implicit, approximate factorization,
7. Elliptic equations: examples of elliptic equations, iterative Methods, SOR on vector computers, iteration as time integration, convergence of iterative Methods (basic discussion), multigrid
methods, fast direct method, ADI for elliptic equations, Krylov Methods
8. Navier-Stokes equations: Navier-Stokes equations in primitive variables, colocated grids, high-order in time, other methods (SIMPLE), boundary conditions, all-speed methods
9. Introduction to numerical modeling of turbulent flows: Richardson and Kolmogorov theories, DNS (Direct Numerical Simulation), LES (Large Eddy Simulation), U-RANS (Unsteady Reynolds Averaged
Navier Stokes)
10. Kinetic-turbulence interactions:EDC (Eddy Dissipation Concept) and Transported PDF methods
11. Introduction to turbulent combustion: PVA (Primitive Variable Approach) methods, mixture fraction, SLFM (Steady Laminar Flamelet Model)
12. Validation and Verification: Verification, Method of Manufactured Solutions (MMS), Richardson Extrapolation, Validation, Uncertainty Quantification (basics)
13. Modeling of multiphase flows: general modeling of multiphase flows (Eulerian/Eulerian vs Eulerian/Lagrangian approaches), methods to track moving fluid interfaces, bubbly flows
14. Large-scale problems and HPC: software tools for CFD, large-scale problems, parallelization (shared and distributed)
15. Special topic based on the requests from students
Practical sessions
Most of practical sessions will be based on MATLAB(R) (https://it.mathworks.com/) and the OpenFOAM (https://openfoam.org/) framework, an open-source CFD code for the simulation of multidimensional
reacting flows, in laminar or turbulent conditions, with arbitrarily complex meshes.
Note Sulla Modalità di valutazione
There will be a final project for this class. Students can select the topic of their project in consultation with the instructor. Possible projects include:
1. Comprehensive reviews of material not covered in detail in class, with some numerical examples
2. Specific fluid-related problems or questions that are numerically studied or solved by the applications of approaches, methods or schemes covered in class
The final examination consists of two parts:
1. project presentation to the instructor (max. 2 people per project)
2. individual, oral examination about the topics presented and discussed during the lessons.
Grading will be based on both the quality of the CFD work, the presentation of the results, and the oral examination.
Oran E.S., Boris J.P., Numerical Simulation of Reactive Flow, Editore: Cambridge University Press, Anno edizione: 2001, ISBN: 9780521022361 http://www.cambridge.org/catalogue/catalogue.asp?isbn=
0521022363 Versteeg H.K., Malalasekera W., An Introduction to Computational Fluid Dynamics, Editore: Prentice Hall, Anno edizione: 2009, ISBN: 9780131274983 https://www.pearson.ch/HigherEducation/
PrenticeHall/EAN/9780131274983/An-Introduction-to-Computational-Fluid-Dynamics-The-Finite-Volume-Method Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics, Editore: Springer, Anno
edizione: 2001, ISBN: 9783642560262 http://www.springer.com/gp/book/9783540420743
Nessun software richiesto
Tipo Forma Didattica Ore didattiche
lezione 38.0
esercitazione 18.0
laboratorio informatico 0.0
laboratorio sperimentale 0.0
progetto 0.0
laboratorio di progetto 0.0
Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese | {"url":"https://www11.ceda.polimi.it/schedaincarico/schedaincarico/controller/scheda_pubblica/SchedaPublic.do?&evn_default=evento&c_classe=666793&polij_device_category=DESKTOP&__pj0=0&__pj1=c58983933b9ff74265418999747686d2","timestamp":"2024-11-07T18:46:23Z","content_type":"text/html","content_length":"25880","record_id":"<urn:uuid:a5fea24a-2cfa-4605-83ae-7c5258d4c217>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00636.warc.gz"} |
Patterns again
Back to patterns in Haskell, an unruly puzzle that’s run through the last few years of my life, trying to work out how I want to represent my music. Here’s the current state of my types:
data Pattern a = Sequence {arc :: Range -> [Event a]}
| Signal {at :: Rational -> [a]}
type Event a = (Range, a)
type Range = (Rational, Rational)
A Range is a time range, with a start (onset) and duration. An Event is of some type a, that occurs over a Range. A Pattern can be instantiated either as a Sequence or Signal. These are directly
equivalent to the distinction between digital and analogue, or discrete and continuous. A Sequence is a set of discrete events (with start and duration) occurring within a given range, and a Signal
is a set of values for a given position in time. In other words, both are represented as functions from time to values, but Sequence is for representing a set of events which have beginnings and
ends, and Range is for a continuously varying set of values.
This is a major improvement on my previous version, simply because the types are significantly simpler, which makes the code significantly easier to work with. This simplicity is due to the
structure of patterns being represented entirely with functional composition, so is closer to my (loose) understanding of functional reactive programming..
The Functor definition is straightforward enough:
mapSnd f (x,y) = (x,f y)
instance Functor Pattern where
fmap f (Sequence a) = Sequence $ fmap (fmap (mapSnd f)) a
fmap f (Signal a) = Signal $ fmap (fmap f) a
The Applicative definition allows signals and patterns to be combined in in a fairly reasonable manner too, although I imagine this could be tidied up a fair bit:
instance Applicative Pattern where
pure x = Signal $ const [x]
(Sequence fs) <*> (Sequence xs) =
Sequence $ \r -> concatMap
(\((o,d),x) -> map
(\(r', f) -> (r', f x))
(\((o',d'),_) -> (o' >= o) && (o' < (o+d)))
(fs r)
(xs r)
(Signal fs) <*> (Signal xs) = Signal $ \t -> (fs t) <*> (xs t)
(Signal fs) <*> px@(Sequence _) =
Signal $ \t -> concatMap (\(_, x) -> map (\f -> f x) (fs t)) (at' px t)
(Sequence fs) <*> (Signal xs) =
Sequence $ \r -> concatMap (\((o,d), f) ->
map (\x -> ((o,d), f x)) (xs o)) (fs r)
In the Pattern datatype, time values are represented using Rational numbers, where each whole number represents the start of a metrical cycle, i.e. something like a bar. Therefore, concatenating
patterns involves ‘playing’ one cycle from each pattern within every cycle:
cat :: [Pattern a] -> Pattern a
cat ps = combine $ map (squash l) (zip [0..] ps)
where l = length ps
squash :: Int -> (Int, Pattern a) -> Pattern a
squash n (i, p) = Sequence $ \r -> concatMap doBit (bits r)
where o' = (fromIntegral i)%(fromIntegral n)
d' = 1%(fromIntegral n)
cycle o = (fromIntegral $ floor o)
subR o = ((cycle o) + o', d')
doBit (o,d) = mapFsts scaleOut $ maybe [] ((arc p) . scaleIn) (subRange (o,d) (subR o))
scaleIn (o,d) = (o-o',d* (fromIntegral n))
scaleOut (o,d) = ((cycle o)+o'+ ((o-(cycle o))/(fromIntegral n)), d/ (fromIntegral n))
subRange :: Range -> Range -> Maybe Range
subRange (o,d) (o',d') | d'' > 0 = Just (o'', d'')
| otherwise = Nothing
where o'' = max o (o')
d'' = (min (o+d) (o'+d')) - o''
-- chop range into ranges of unit cycles
bits :: Range -> [Range]
bits (_, 0) = []
bits (o, d) = (o,d'):bits (o+d',d-d')
where d' = min ((fromIntegral $ (floor o) + 1) - o) d
Well this code could definitely be improved..
If anyone is interested the code is on github, but is not really ready for public consumption yet. Now I can get back to making music with it though, more on that elsewhere, soon, maybe under a new
1 Comment
1. I like half your events model, giving each event a duration. Though, it seems odd I could represent events that vary wildly based on the query. | {"url":"https://slab.org/2012/09/18/patterns-again/","timestamp":"2024-11-12T23:09:17Z","content_type":"text/html","content_length":"45856","record_id":"<urn:uuid:e9b1ebdc-ea03-4448-8b3f-1d19aa1ba83e>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00358.warc.gz"} |
Understanding the Quotient Rule: Examples and Explanations
The Quotient Rule is an important topic in mathematics that appears in any level of calculus or higher mathematics. To be able to tackle any higher-level concepts, one must first understand the
fundamentals of the Quotient Rule. In this article, we will define the Quotient Rule and explore the mathematics behind it, as well as looking at some examples and tips to help with understanding and
applying it. We’ll also look at common mistakes and the advantages and disadvantages of using the Quotient Rule, before looking at how to use it in real-world situations.
Definition of the Quotient Rule
The Quotient Rule states that for a function f(x) = (g(x))/(h(x)), the derivative of f(x) will be given by f′(x) = (h(x) * g′(x) – g(x) * h′(x))/(h^2(x)). In other words, the derivative of a function
that is the quotient of two other functions is the difference between the product of the derivatives of each function divided by the square of the denominator.
The Quotient Rule is an important tool for calculus students to understand, as it allows them to calculate the derivatives of more complex functions. It is also useful for finding the rate of change
of a function, which can be used to solve a variety of problems. Knowing how to apply the Quotient Rule can help students to better understand the concepts of calculus and to apply them in real-world
Explaining the Mathematics Behind the Quotient Rule
To explain the mathematics behind the quotient rule, let’s consider a specific example. Suppose we have a function given by f(x) = (5x^2 + 3x – 2)/(x^2 + 1). To find the derivative of f(x), we can
plug this into the Quotient Rule and get f′(x) = (x^2+1*10x + 3 * 2x – 0 – (5x^2 + 3x – 2)*2) / (x^2 + 1)^2, or, more simply, f′(x) = (10x^2 + 12x) / (x^2 + 1)^2. This is the same result we would get
if we found the derivative of each individual function, multiplied them together and divided by the square of the denominator.
The Quotient Rule is a useful tool for finding the derivatives of functions that are expressed as the ratio of two polynomials. It is important to remember that the Quotient Rule is only applicable
when the denominator is a polynomial, and not when the denominator is a constant. Additionally, it is important to remember that the Quotient Rule is only applicable when the numerator and
denominator are both functions of the same variable.
Examples of Applying the Quotient Rule
Now that we’ve discussed the fundamentals, let’s take a look at some examples that illustrate how to apply the Quotient Rule. For example, suppose we have a function given by g(x) = (13x – 5)/(4x +
2). To find its derivative using the Quotient Rule, we can substitute this into the formula to get g′(x) = (4x + 2 * 13 – 13x – 5 * 4)/(4x + 2)^2, which simplifies to g′(x) = -17/(4x + 2)^2.
Tips for Understanding and Applying the Quotient Rule
The Quotient Rule can be a tricky concept to understand, so here are some tips to help with understanding and applying it:
• Focus on understanding the mathematics behind it. Once you understand why and how it works, it will be easier to apply it.
• Write out each step of the calculation as you go along, so that you can keep track of everything that is happening.
• Practice! The more you apply it, the more comfortable you will become with using it.
Common Mistakes to Avoid When Using the Quotient Rule
Using the Quotient Rule can be tricky, so here are some common mistakes to avoid:
• Forgetting to square the denominator when plugging the expression into the formula.
• Mixing up the signs when subtracting terms.
• Not differentiating each function before multiplying them together.
Advantages and Disadvantages of Using the Quotient Rule
The main advantage of using the Quotient Rule is that it can be used to quickly find derivatives when it would take significantly more time to do so by hand. It also allows for derivatives which
would otherwise be extremely complex to calculate. The main disadvantage of using the Quotient Rule is that it can be difficult to understand, which can lead to confusion and errors in calculation if
not properly understood.
How to Use the Quotient Rule in Real-World Situations
The Quotient Rule is most often used in calculus when taking derivatives in order to find things like rates of change and maxima/minima points. It is also used in economic models to calculate
elasticity and in robotics to calculate motion curves. Therefore, understanding how to correctly use the Quotient Rule is essential for effectively utilizing such higher-level mathematics.
Conclusion: Understanding and Applying the Quotient Rule
In conclusion, understanding and applying the Quotient Rule is an important part of any higher-level mathematics or physics course. By following the definition and explanation provided in this
article and taking advantage of the tips and examples given, one should have no difficulty understanding and applying the Quotient Rule in any real-world situation. | {"url":"https://mathemista.com/understanding-the-quotient-rule-examples-and-explanations/","timestamp":"2024-11-07T12:59:43Z","content_type":"text/html","content_length":"58152","record_id":"<urn:uuid:45def034-0072-4454-a656-5a21de32aaa6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00416.warc.gz"} |
Last Call Review of draft-ietf-sfc-proof-of-transit-08
I have reviewed this document as part of the security directorate's ongoing
effort to review all IETF documents being processed by the IESG. These
comments were written primarily for the benefit of the security area directors.
Document editors and WG chairs should treat these comments just like any other
last call comments.
This document proposes a security mechanism to prove that traffic transited through
all specified nodes in a path. The mechanism works by adding a short option to
each packet for which transit shall be verified. The option consists of a random number
set by the originator of the packet, and a sum field to which each transit node
adds a value depending on public parameters, on the random number and on secrets
held by the node. The destination has access to all the secrets held by the nodes
on the path, and can verify whether or not the final sum corresponds to the sum
of expected values. The proposed size of the random number and the sum field is 64 bits.
In the paragraph above, I described the mechanism without mentioning the algorithm
used to compute these 64 bit numbers. The 64 bit size is obviously a concern: for
cryptographic applications, 64 bits is not a large number, and that might be a
weakness whatever the proposed algorithm. The actual algorithm appears to be a
bespoke derivation of Shamir's Secret Sharing algorithm (SSS). In other word, it is
a case of "inventing your own crypto".
SSS relies on the representation of polynomials as a sum of
Lagrange Basis Polynomials. Each of the participating nodes holds a share of the
secret represented by a point on the polynomial curve. A polynomial of degree
K on the field of integers modulo a prime number N can only be revealed if at
list K+1 participants reveal the value of their point. The safety of the
algorithm relies on the size of the number N and on the fact that the
secret shall be revealed only once. But the algorithm does not use SSS
directly, so it deserves its own security analysis instead of relying
simply on Shamir's work.
The proposed algorithm uses two polynomials of degree K for a path containing
K+1 nodes, on a field defined by a prime number N of 64 bits. One of the
polynomial, POLY-1, is secret, and only fully known by the verifying node.
The other, POLY-2 is public, with the constant coefficient set at a random
value RND for each packet.
For each packet, the goal is compute the value of POLY-1 plus POLY-2 at the
point 0 -- that is, the constant coefficient of POLY-3 = POLY-1 + POLY-2.
Without going in too much details, one can observe that the constant
coefficient of POLY-3 is equal to the sum of the constant coefficients
of POLY-1 and POLY-2, and that the constant coefficient of POLY-2 is
the value RND present in each packet. In the example given in section
3.3.2, the numbers are computed modulo 53, the constant coefficient
of POLY-1 is 10, and the value RND is 45. The final sum CML is indeed
10 + 45 = 2 mod 53.
To me, this appears as a serious weakness in the algorithm. If an adversary
can observe the value RND and CML for a first packet, it can retrieve the
constant coefficient of POLY-1, and thus can predict the value of CML for
any other packet. That does not seem very secure.
My recommendation would be to present the problem and ask the CFRG for
algorithm recommendations. | {"url":"https://datatracker.ietf.org/doc/review-ietf-sfc-proof-of-transit-08-secdir-lc-huitema-2021-09-19/00/","timestamp":"2024-11-02T02:28:18Z","content_type":"text/html","content_length":"36147","record_id":"<urn:uuid:d2e05c44-535d-484d-9066-0b0c3a4bef00>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00284.warc.gz"} |
How do I write a VBA for 3 consecutive dates?
I am a VBA newbie and I am reformatting some older data by entering two
consecutive dates on separate rows. For example A1 (1/1/2006) has a date and
I need A2 and A3 to equal the next two days (i.e., 1/2/2006 and 1/3/2006).
I also don’t know how to define cells as dates and how to assign a value
defined by a variable to a cell. The code below makes logical sense but the
syntax is wrong and I don’t know how to fix it. Thank you for your help.
Sub Dates_OfCapture()
Dim Startdate As Date
Dim N1 As Integer
Dim N2 As Integer
Dim x As Integer
x = 10 'starting row
y = 1 'add one day
For N1 = 1 To 100
For N2 = 1 To 2 'two turns of For Loop for every Startdate
z = x + 1
Startdate = Cells(x, 2).Date 'defining the Startdate
Cells(z, 2).Date = DateAdd("dd", y, Startdate) 'adding a day to the
row below the starting date
x = x + 1 'add one to the the row number
y = y + 1 'add one to the number of days beind added to Startdate
Next N2
z = 1 'reset z (number of days added) back to 1
Next N1
End Sub
Sub Dates_OfCapture()
Dim Startdate As Date
Dim N1 As Integer
Dim N2 As Integer
Dim x As Integer
Dim y As Integer 'ADDED
Dim z As Integer 'ADDED
x = 10 'starting row
y = 1 'add one day
For N1 = 1 To 100
For N2 = 1 To 2 'two turns of For Loop for every Startdate
z = x + 1
Startdate = Cells(x, 2) '.Date 'defining the Startdate
Cells(z, 2) = DateAdd("dd", y, Startdate) 'adding a day to the
row below the starting date
x = x + 1 'add one to the the row number
y = y + 1 'add one to the number of days beind added to
Next N2
z = 1 'reset z (number of days added) back to 1
Next N1
End Sub
....is that any better?
WhytheQ said:
Sub Dates_OfCapture()
Dim Startdate As Date
Dim N1 As Integer
Dim N2 As Integer
Dim x As Integer
Dim y As Integer 'ADDED
Dim z As Integer 'ADDED
x = 10 'starting row
y = 1 'add one day
For N1 = 1 To 100
For N2 = 1 To 2 'two turns of For Loop for every Startdate
z = x + 1
Startdate = Cells(x, 2) '.Date 'defining the Startdate
Cells(z, 2) = DateAdd("dd", y, Startdate) 'adding a day to the
row below the starting date
x = x + 1 'add one to the the row number
y = y + 1 'add one to the number of days beind added to
Next N2
z = 1 'reset z (number of days added) back to 1
Next N1
End Sub
....is that any better?
I tried it and I got an "Invalid procedure or argument" notice that highlighted the Cells(z, 2) = DateAdd("dd", y, Startdate) line.
Change the "dd" to just "d"
I just ran it with 01-Jan-06 in the cell B10 and it didn't increment by
just one day at a atime but produced the following. Is this what you
Thank you for your help. Using your suggestions I was able to make it work.
My goal was to fill two consecutive dates following a start date.
I was able to make it work so that it did the following
1/4/1990 (start date)
5/8/1990 (start date)
Bellow is the code I used. Thanks again for your help J. I hope this code
will be useful to someone else.
Sub Dates_OfCapture()
Dim Startdate As Date
Dim N1 As Integer
Dim N2 As Integer
Dim x As Integer
Dim y As Integer
Dim z As Integer
x = 20 'starting row
y = 1 'add one day
Do While Cells(x, 2).Value <> ""
Startdate = Cells(x, 2) 'defining the Startdate
For N2 = 1 To 2 'two turns of For Loop for every Startdate
z = x + 1 'look at the cell bellow the Startdate
Cells(z, 2) = DateAdd("d", y, Startdate) 'adding a day to the row
below the starting date
x = x + 1 'add one to the the row number
y = y + 1 'add one to the number of days being added to Startdate
Next N2
y = 1 'reset the number of days added to Startdate back to 1
x = x + 1 'add one more so that the next Startdate is not the last row but
the following row
End Sub
Glad I could help a bit.
p.s your initial code was pretty accomplished and yet you called
yourself a newbie?! | {"url":"https://www.pcreview.co.uk/threads/how-do-i-write-a-vba-for-3-consecutive-dates.2749054/","timestamp":"2024-11-02T20:27:33Z","content_type":"text/html","content_length":"79412","record_id":"<urn:uuid:bf2b8a87-8226-4e37-8867-84cb68954a87>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00118.warc.gz"} |
ECE 7251: Signal Detection and Estimation
Instructor: Prof. Aaron Lanterman
Office: GCATT 334B
Phone: 404-385-2548
E-mail: lanterma@ece.gatech.edu
Course website: users.ece.gatech.uiuc.edu/~lanterma/ece7251
When and where: MWF, 12:05-12:55,
212 Engineering Science and Mechanics Building
Prerequisites: Knowledge of probability and random processes
at the level of the texts such as Papoulis, Stark and Woods,
or Leon-Garcia, i.e. ECE6601 or equivalent.
Some questions on the take-home
portion of the exams will require some basic programming skills;
although you are welcome to use whatever programming language you like,
competency in MATLAB will be extremely helpful.
Required Texts:
• A.O. Hero, “Statistical Methods for Signal Processing,” 1998-2002.
This is a set of course notes written by Prof. Al Hero of the Univ. of
Michigan, which he plans to eventually turn into a textbook. A link to the
PDF file will be provided, and copies will be made available for purchase
in the campus bookstore.
• H.V. Poor, “An Introduction to Signal Detection and Estimation,”
2nd Edition, Springer, 1994. ($69.95 on amazon.com
Prof. Hero’s notes are well thought out and will provide the main structure
of the course and the inspiration for my lectures. However, the notes are
a bit lacking on textual descriptions at the moment, hence, I thought it best
that everyone have a copy of Vince Poor’s book too, in order to have a
solid reference.
Grade breakdown:
Exam 1 (Feb. 6): 20%
Exam 2 (March 13): 20%
Exam 3 (April 3): 20%
Final Exam: 40%
(If you have a conflict with these any of these times, please let me know
Each exam will have an in-class part and a take-home part.
The in-class part will be open book and open notes, although this is primary
you won’t panic; the in-class section will emphasize your “intuitive”
understanding of the material. If you find yourself spending
most of your time on the in-class portion frantically flipping through your
notes trying to find the answer, you will run out of time.
The take home will consist of more
in-depth problems which couldn’t possibly completed in an in-class
exam, although it will not be intended to be exceptionally time consuming.
You will be given a generous amount of time to complete it.
The take-home portion may involve some highly instructive computational
experiments. You will be allowed to consult with any resources in the
library, posted on web sites, etc., as long as these are properly
cited in solution. You will not, however, be allowed to discuss the take-home
portion in any way with anyone besides myself.
I’m giving three exams, which is more than usual for a grad class, since it
helps reduce the bad day effect. The bad day effect hits
you when you happen to be having a bad day (you know, one of those days
where nothing is going right and you’ve run out of coffee
and you can’t seem to get your brain working)
on the one day you’re taking
the single midterm exam. When there are several midterm exams, you’re not
likely to be having a bad day on all the exam days, so if you do have a
bad day, it can get averaged out.
A note on homework, or the lack thereof:
Notice that no homework will be collected and graded. In lecture, various
problems will be suggested for you to try at your leisure; working through
as many problems as you can will be the best way to prepare for the exams,
particularly the in-class portion.
You are strongly encouraged to work together in groups, preferably with
a lot of coffee. It’s also best
to tackle a problem or two per day, perhaps soons after the lecture while
the material is still fresh in your head,
rather than sit down and try a whole
bunch at once in a marathon session right befor the exam.
Office hours:
If you have an office in GCATT, just go ahead and drop by; I will be there
most afternoons and evenings, although never in the morning. If you are
stationed outside of GCATT (or are having trouble getting a hold of me in
send me an e-mail, and I’ll set up a time to meet with you in Van Leer
(or wherever is most convenient for me.)
I will
generally go to lunch after class; people are welcome to join me for lunch
if they have questions or generally want to chat. I will have more formal
office hours before the exams.
Reserve Books:
The following books have been placed on reserve in the library:
E-mail hours:
As anyone who’s taken a course from me before will attest, I tend to send
out a lot of course-related e-mail. Make sure your e-mail account is working
(and not over quota or something like that) so you don’t miss anything good.
Tentative schedule
• General Structure
□ Sufficient Statistics; Exponential Families
• Parameter Estimation
□ Bayesian Estimation (MAP and MMSE)
□ Orthogonality Principle of MMSE
□ Linear Minimum Mean Squared Error Estimation
□ Maximum-Likelihood
□ Method of Moments
□ Cramer-Rao Bounds (both random and nonrandom)
• Computational Techniques
□ EM Algorithm: Theory
□ EM Algorithm: Examples
□ Markov Chain Monte Carlo algorithms
• Filtering for Discrete-Time Processes
□ Kalman Filter
□ Wiener Filter
• Simple Hypothesis Testing
□ Bayesian Detection
□ Minmax Detection
□ Neyman-Pearson Lemma; ROC Curves
□ Chernoff Bounds
• Composite Hypothesis Testing
□ Uniformly Most Powerful (UMP) Tests
□ Locally Most Powerful (LMP) Tests
□ Generalized Likelihood Ratio Tests (GLRT)
□ Detector Structures for Discrete-Time Data with Gaussian, Laplacian,
and Cauchy Noise
• Model Order Estimation
□ Schwarz’s Bayesian Information Criterion
□ Minimum Description Length Criterion
□ Stochastic Complexity
• Continuous-Time Extensions
□ Karhunen-Loeve Expansions
□ Grenander’s Theorem
□ Detection with Continuous Data
□ Parameter Estimation with Continuous Data | {"url":"https://lanterman.ece.gatech.edu/dude/syl1/","timestamp":"2024-11-10T09:23:16Z","content_type":"text/html","content_length":"51577","record_id":"<urn:uuid:5752ba32-4587-444b-af05-8507b3ebea60>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00047.warc.gz"} |
multiplication Archives
Summer is here and the living is easy.
Multiplication facts, though, are still giving my youngest a hard time.
She’s working on memorizing her times tables so she can sail into third grade math this fall with no trouble.
To make it a little more fun, I thought we’d try this printable pineapple multiplication by four game!
This is a simple and cute way to help kids work on multiplication fluency!
Keep reading to see how to get yours – free! And, for more ways to have fun with math, take a look at our list of outdoor math games to play!
Pineapple Multiplication by Four Game
To use this printable multiplication roll and cover game, you’ll need the following: (This post contains affiliate links. For details, see our Disclosure Policy.)
This product includes two versions of the multiplication roll and cover game:
• One for use with a six-sided die
• And one for use with a ten-sided die
The rules are simple.
Students roll a die and then find the corresponding multiple of four on the game board.
Then they just cover that number with a math counter and keep going!
You can also use buttons, counting bears, pom-poms, or coins in place of counters.
Students can play the game alone or with a partner. If they’re playing alone, challenge them to see how quickly they can roll and cover all the multiples.
If they’re playing with a partner, the first one to cover the last multiple wins!
A line art version of both game boards is also included.
Personally, I love to add a pop of color by printing them on AstroBright paper – as above. It definitely makes the activity a bit more cheerful.
Scroll down to get your Pineapple Multiplication by Four Game!
Check out these other fun math learning ideas!
To get your copy of this Pineapple Multiplication by Four Roll and Cover Game, click the image or the link below to download it to your computer! | {"url":"https://www.lookwerelearning.com/tag/multiplication/","timestamp":"2024-11-05T02:27:07Z","content_type":"text/html","content_length":"265715","record_id":"<urn:uuid:afadfd8c-9cff-4c8a-8ebb-a11a0eff82ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00169.warc.gz"} |
Seeing Fractions Differently
Here is the hook.
Before continuing on, try and make an estimate. What is too small? What is too big? What sections can you easily identify?
Maybe you fit into one of the following strategies (They are not mutually exclusive), or maybe you want to switch after reading them. (I know I did).
1) From @suffolkmaths
The “square” he refers to is one of the smaller 5 squares that make up the plus sign. Using this understanding of 1/5, he is able to place his entire reasoning into a single tweet. Pretty efficient
if you ask me. Because each obtuse, shaded triangle has a base of the entire (smaller) square, and a height of half of a square, the area is one quarter of the square. Four groups of one quarter
(actually of 1/20th of the whole) makes one entire square. Since there are five squares total, the shaded area is 1/5.
2) From @KhatriMath
In response, she classifies her strategy as more complex. At its root, they are doing the same thing–trying to mash the area into a single square representing 1/5. She also employs the area formula
for a triangle. There is a moment that she uses similar triangle notation, but it doesn’t seem to play a huge role in the argument.
She first establishes a congruency between the yellow and orange sections. She is then left with a 2-to-1 relationship between the green and orange to complete the migration to the middle square. At
the end, the middle square is easily observed as 1/5 of the whole. The strategy has a dynamic feel to it.
3) From @HRSBMathematics
This had me immediately curious. After asking for some further clarification, it was a very cool visual interpretation that still had connections to the previous two. There still was an
area calculation going on,
but now there was a distinct division of the entire 3×3 grid into thirds. The idea of fifths emerges when the corners (each half shaded) are subtracted.
There are now 3 sets of 4 triangles that can be envisioned as rotating around the center like a pinwheel. Well, if the one entire group is shaded and there are three total groups, the shaded portion
represents 1/3 of the 9 smaller squares. (Or three smaller squares total). It follows that the corners need to be subtracted to get back to the original shape. Each is half shaded, and there are four
of them. So, you will subtract 2 aggregate squares of shading. If 3 squares were shaded originally, and you lost 2, you re left with one square left. Once again, the square is 1/5 of the original
plus sign shape.
I love this!!
4) From @Simon_Gregg
Here, a different “base shape” is defined. Actually, two of them are used. Simon does such an elegant job of depicting them. They are given names (house and triangle), and values. He then uses
subtraction to calculate the “house-but-not-triangle” area. This term warms my heart.
This sort of unique verbiage signals unique sense-making. A sort of local dialect emerges from the classroom (in this case, twitter). The shape is not dissected into triangles and squares, but
sections of house-but-not-triangles. Beautiful.
It is my hope that Fraction Talks play out exactly like this. Depending on your grade level, you will get differing levels of intricacy. Honour multiplicity, entertain diversity. Allows students to
use formulas, envision pinwheeling, define new terms, and play with the relationships.
Damian Watson built this Geogebra animation of how he visualizes the 1/5th. It is stunningly beautiful.
I would love to further spotlight various classroom levels of reasoning. If you have stories from your Fraction Talks, please share. I believe it is crucial that teachers see productive | {"url":"http://fractiontalks.com/seeing-fractions-differently/","timestamp":"2024-11-12T09:10:56Z","content_type":"text/html","content_length":"48409","record_id":"<urn:uuid:4e149e8e-5afb-463e-abfc-20cad2f13bc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00695.warc.gz"} |