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Killing form - formulasearchengine
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. The Killing form was essentially introduced into Lie algebra theory by Template:Harvs in his thesis; although Killing had previously made a passing mention of it, he made no serious use of it.
3 Matrix elements
4 Connection with real forms
Consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad(x) (also written as adx) of g with the help of the Lie bracket, as
{\displaystyle \mathrm {ad} (x)(y)=[x,y].}
Now, supposing g is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form
{\displaystyle B(x,y)=\mathrm {trace} (\mathrm {ad} (x)\mathrm {ad} (y)),}
with values in K, the Killing form on g.
The Killing form B is bilinear and symmetric.
The Killing form is an invariant form, in the sense that it has the 'associativity' property
{\displaystyle B([x,y],z)=B(x,[y,z]),}
where [ , ] is the Lie bracket.
If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a scalar multiple of the Killing form.
The Killing form is also invariant under automorphisms s of the algebra g, that is,
{\displaystyle B(s(x),s(y))=B(x,y)}
for s in Aut(g).
The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
The Killing form of a nilpotent Lie algebra is identically zero.
If I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.
The orthogonal complement with respect to B of an ideal is again an ideal.[1]
If a given Lie algebra g is a direct sum of its ideals I1,...,In, then the Killing form of g is the direct sum of the Killing forms of the individual summands.
Given a basis ei of the Lie algebra g, the matrix elements of the Killing form are given by
{\displaystyle B^{ij}=\mathrm {tr} (\mathrm {ad} (e^{i})\circ \mathrm {ad} (e^{j}))/I_{ad}}
where Iad is the Dynkin index of the adjoint representation of g. Here
{\displaystyle \left({\textrm {ad}}(e^{i})\circ {\textrm {ad}}(e^{j})\right)(e^{k})=[e^{i},[e^{j},e^{k}]]=[e^{i},{c^{jk}}_{m}e^{m}]={c^{im}}_{n}{c^{jk}}_{m}e^{n}}
in Einstein summation notation, where the cijk are the structure coefficients of the Lie algebra. The index k functions as column index and the index n as row index in the matrix ad(ei)ad(ej). Taking the trace amounts to putting k = n and summing, and so we can write
{\displaystyle B^{ij}={\frac {1}{I_{ad}}}{c^{im}}_{n}{c^{jn}}_{m}}
The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indices (co- and contra-variant indices). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for g such that the structure constants with all upper indices are completely antisymmetric.
The Killing form for some Lie algebras g are (for X, Y in g):
gl(n, R) 2n tr(XY) − 2 tr(X)tr(Y)
sl(n, R) 2n tr(XY)
su(n) 2n tr(XY)
so(n, R) (n−2) tr(XY)
so(n) (n−2) tr(XY)
sp(n, R) (2n+2) tr(XY)
sp(n, C) (2n+2) tr(XY)
Connection with real forms
Suppose that g is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries ±1. By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the index of the Lie algebra g. This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra. In particular, a real Lie algebra g is called compact if the Killing form is negative definite. It is known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.
If gC is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is gC, which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form g. The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.
For example, the complex special linear algebra sl(2, C) has two real forms, the real special linear algebra, denoted sl(2, R), and the special unitary algebra, denoted su(2). The first one is noncompact, the so-called split real form, and its Killing form has signature (2, 1). The second one is the compact real form and its Killing form is negative definite, i.e. has signature (0 ,3). The corresponding Lie groups are the noncompact group SL(2, R) of 2 × 2 real matrices with the unit determinant and the special unitary group SU(2), which is compact.
↑ Template:Fulton-Harris See page 207.
Daniel Bump, Lie Groups (2004), Graduate Texts In Mathematics, 225, Springer-Verlag. ISBN 978-0-387-21154-1
Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992) Cambridge University Press. ISBN 0-521-48412-X
Template:Fulton-Harris
Retrieved from "https://en.formulasearchengine.com/index.php?title=Killing_form&oldid=235552"
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Find a vector that has a specified correlation with another vector » SAS博客列表
Math, simulation, Statistical Programming Add comments
Do you know that you can create a vector that has a specific correlation with another vector? That is, given a vector, x, and a correlation coefficient, ρ, you can find a vector, y, such that corr(x, y) = ρ. The vectors x and y can have an arbitrary number of elements, n > 2. One application of this technique is to create a scatter plot that shows correlated data for any correlation in the interval (-1, 1). For example, you can create a scatter plot with n points for which the correlation is exactly a specified value, as shown at the end of this article.
The algorithm combines a mixture of statistics and basic linear algebra. The following facts are useful:
Statistical correlation is based on centered and normalized vectors. When you center a vector, it usually changes the direction of the vector. Therefore, the calculations use centered vectors.
Correlation is related to the angle between the centered vectors. If the angle is θ, the correlation between the vectors is cos(θ).
Projection is the key to finding a vector that has a specified correlation. In linear algebra, the projection of a vector w onto a unit vector u is given by the expression (w`u)*u.
Affine transformations do not affect correlation. For any real number, α, and for any β > 0, the vector α + β y has the same correlation with x as y does. For simplicity, the SAS program in this article returns a centered unit vector. You can scale and translate the vector to obtain other solutions.
The geometry of a correlated vector
Given a centered vector, u, there are infinitely-many vectors that have correlation ρ with u. Geometrically, you can choose any vector on a positive cone in the same direction as u, where the cone has angle θ and cos(θ)=ρ. This is shown graphically in the figure below. The plane marked \(\mathbf{u}^{\perp}\)
\mathbf{u}^{\perp}
is the orthogonal complement to the vector u. If you extend the cone through the plane, you obtain the cone of vectors that are negatively correlated with x
One way to obtain a correlated vector is to start with a guess, z. The vector z can be uniquely represented as the sum \(\mathbf{y} = \mathbf{w} + \mathbf{w}^{\perp}\)
\mathbf{y} = \mathbf{w} + \mathbf{w}^{\perp}
, where w is the projection of z onto the span of u, and \(\mathbf{w}^{\perp}\)
\mathbf{w}^{\perp}
is the projection of z onto the orthogonal complement.
The following figure shows the geometry of the right triangle with angle θ such that cos(θ) = ρ. If you want the vector y to be unit length, you can read off the formula for y from the figure. The formula is
\(\mathbf{y} = \rho \mathbf{w} / \lVert\mathbf{w}\rVert + \sqrt{1 - \rho^2} \mathbf{w}^\perp / \lVert\mathbf{w}^\perp\rVert \)
\mathbf{y} = \rho \mathbf{w} / \lVert\mathbf{w}\rVert + \sqrt{1 - \rho^2} \mathbf{w}^\perp / \lVert\mathbf{w}^\perp\rVert
In the figure, \(\mathbf{v}_1 = \mathbf{w} / \lVert\mathbf{w}\rVert\)
\mathbf{v}_1 = \mathbf{w} / \lVert\mathbf{w}\rVert
and \(\mathbf{v}_2 = \mathbf{w}^\perp / \lVert\mathbf{w}^\perp\rVert\)
\mathbf{v}_2 = \mathbf{w}^\perp / \lVert\mathbf{w}^\perp\rVert
Compute a correlated vector
It is straightforward to implement this projection in a matrix-vector language such as SAS/IML. The following program defines two helper functions (Center and UnitVec) and uses them to implement the projection algorithm. The function CorrVec1 takes three arguments: the vector x, a correlation coefficient ρ, and an initial guess. The function centers and scales the vectors into the vectors u and z. The vector z is projected onto the span of u. Finally, the function uses trigonometry and the fact that cos(θ) = ρ to return a unit vector that has the required correlation with x.
/* Given a vector, x, and a correlation, rho, find y such that corr(x,y) = rho */
/* center a column vector by subtracting its mean */
start Center(v);
return ( v - mean(v) );
/* create a unit vector in the direction of a column vector */
start UnitVec(v);
return ( v / norm(v) );
/* Find a vector, y, such that corr(x,y) = rho. The initial guess can be almost
any vector that is not in span(x), orthog to span(x), and not in span(1) */
start CorrVec1(x, rho, guess);
/* 1. Center the x and z vectors. Scale them to unit length. */
u = UnitVec( Center(x) );
z = UnitVec( Center(guess) );
/* 2. Project z onto the span(u) and the orthog complement of span(u) */
w = (z`*u) * u;
wPerp = z - w;
/* 3. The requirement that cos(theta)=rho results in a right triangle
where y (the hypotenuse) has unit length and the legs
have lengths rho and sqrt(1-rho^2), respectively */
v1 = rho * UnitVec(w);
v2 = sqrt(1 - rho**2) * UnitVec(wPerp);
y = v1 + v2;
/* 4. Check the sign of y`*u. Flip the sign of y, if necessary */
if sign(y`*u) ^= sign(rho) then
return ( y );
The purpose of the function is to project the guess onto the green cone in the figure. However, if the guess is in the opposite direction from x, the algorithm will compute a vector, y, that has the opposite correlation. The function detects this case and flips y, if necessary.
The following statements call the function for a vector, x, and requests a unit vector that has correlation ρ = 0.543 with x:
/* Example: Call the CorrVec1 function */
x = {1,2,3};
guess = {0, 1, -1};
y = CorrVec1(x, rho, guess);
corr = corr(x||y);
print x y, corr;
As requested, the correlation coefficient between x and y is 0.543. This process will work provided that the guess satisfies a few mild assumptions. Specifically, the guess cannot be in the span of x or in the orthogonal complement of x. The guess also cannot be a multiple of the 1 vector. Otherwise, the process will work for positive and negative correlations.
The function returns a vector that has unit length and 0 mean. However, you can translate the vector and scale it by any positive quantity without changing its correlation with x, as shown by the following example:
/* because correlation is a relationship between standardized vectors,
you can translate and scale Y any way you want */
y2 = 100 + 23*y; /* rescale and translate */
corr = corr(x||y2); /* the correlation will not change */
print corr;
When y is a centered unit vector, the vector β*y has L2 norm β. If you want to create a vector whose standard deviation is β, use β*sqrt(n-1)*y, where n is the number of elements in y.
Random vectors with a given correlation
One application of this technique is to create a random vector that has a specified correlation with a given vector, x. For example, in the following program, the x vector contains the heights of 19 students in the Sashelp.Class data set. The program generates a random guess from the standard normal distribution and passes that guess to the CorrVec1 function and requests a vector that has the correlation 0.678 with x. The result is a centered unit vector.
use sashelp.class;
read all var {"Height"} into X;
guess = randfun(nrow(x), "Normal");
mean = 100;
std = 23*sqrt(nrow(x)-1);
v = mean + std*y;
title "Correlation = 0.678";
title2 "Random Normal Vector";
call scatter(X, v) grid={x y};
The graph shows a scatter plot between x and the random vector, v. The correlation in the scatter plot is 0.678. The sample mean of the vector v is 100. The sample standard deviation is 23.
If you make a second call to the RANDFUN function, you can get another random vector that has the same properties. Or you can repeat the process for a range of ρ values to visualize data that have a range of correlations. For example, the following graph shows a panel of scatter plots for ρ = -0.75, -0.25, 0.25, and 0.75. The X variable is the same for each plot. The Y variable is a random vector that was rescaled to have mean 100 and standard deviation 23, as above.
The random guess does not need to be from the normal distribution. You can use any distribution.
This article shows how to create a vector that has a specified correlation with a given vector. That is, given a vector, x, and a correlation coefficient, ρ, find a vector, y, such that corr(x, y) = ρ. The algorithm in this article produces a centered vector that has unit length. You can multiply the vector by β > 0 to obtain a vector whose norm is β. You can multiply the vector by β*sqrt(n-1) to obtain a vector whose standard deviation is β.
There are infinitely-many vectors that have correlation ρ with x. The algorithm uses a guess to produce a particular vector for y. You can use a random guess to obtain a random vector that has a specified correlation with x.
The post Find a vector that has a specified correlation with another vector appeared first on The DO Loop.
Intransitive dice Create a response variable that has a specified R-square value
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Intelligent glove for suppression of resting tremor in Parkinson’s disease | JVE Journals
Adibah M. Zulkefli1 , Asan G. A. Muthalif2 , Diyana N. H. Nordin3 , Thaer M. I. Syam4
1, 3Smart Structures, Systems and Control Research Lab (S3CRL) Department of Mechatronics Engineering, Kulliyyah of Engineering, International Islamic University, Kuala Lumpur, Malaysia
Received 1 October 2019; accepted 8 October 2019; published 28 November 2019
Copyright © 2019 Adibah M. Zulkefli, 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.
One of the significant symptoms in Parkinson’s disease is resting tremor. Resting tremor occurs when the muscle is relaxed, causing the limb to shake. Rhythmic muscle movement of the patients commonly happens within the range of 4 Hz to 6 Hz. Thus, reducing this type of tremor will help improve patients’ quality of life. In this paper, to suppress resting tremors, an intelligent glove was designed utilizing the concepts of vibrations and gyro effect. A rotating brass disc attached to the glove creates a gyroscopic effect of the smart glove. Therefore, the disc will do their utmost to stay upright and counter any input forces instantaneously by providing the counterforce. A reduction of more than 50 % with the intelligent glove is also shown.
The article recommends intelligent glove for suppression of vibrations for resting tremors of Parkinson’s disease.
A prototype model is fabricated to validate performance of the glove.
Experimental study on the prototype has shown up to 90 % reduction for fixed sine signal between 3 Hz-7 Hz.
Keywords: intelligent glove, vibration suppression, gyro effect, acceleration amplitude reduction, frequency response function (FRF).
Parkinson’s disease is a neurodegenerative disorder that affects human limb movements [1]. Dr. James Parkinson first discussed it in “An Essay on the Shaking Palsy”, which was published in 1817 [2]. Primary symptoms of Parkinson’s disease include shaking and trembling. Parkinson’s disease tremor can range from 4-5 Hz for low frequency and 8-10 Hz for high frequency [3]. Patients suffering from tremors might face difficulties in their daily activities as muscle movements are involuntary. There are two types of tremors, namely resting tremor (RT) and action tremor (AT) with resting tremor being recognized as the primary symptom of Parkinson’s disease [4, 5]. Resting tremor occurs at a frequency of 4-6 Hz when the limbs are at rest and disappears with voluntary movements [4, 6]. On the other hand, action tremor occurs during voluntary movements [5].
Currently, no cure has been found to treat Parkinson’s disease [6]. Nevertheless, medical approaches such as medications and surgeries can slow down the progress rate towards disability. However, patients are prone to the side effects of these treatments, and the dosage of the medicine has to be gradually increased as the patients’ body get accustomed to the regular dosage [7].
Though several assistive devices have been designed for Parkinson’s disease patients, they are, however expensive and are usually function-oriented. For example, [8, 9] designed a spoon to assist patients for self-feeding. On the other hand, [10, 11] developed a mobility device to improve patients’ walking gait. Furthermore, [12] developed a monitoring system to use in a diverse Parkinson’s disease population for at least two weeks. [13] determined the influence of dopamine agonist treatment on the prevalence of Parkinson’s disease disturbances. This paper aims at eliminating vibrations so that the patients for daily activities need only one device. In this work, a wearable glove using gyroscopic technology is designed to suppress vibrations due to resting tremors experienced by the Parkinson’s patients.
2. Vibration suppression of rest tremor utilizing an intelligent glove
The intelligent glove designed for vibration suppression of patients’ hand consists of a gyroscope, a motor, an aluminium frame as well as a Velcro hook and loop fastening strap. It utilizes gyroscopic force, which works based on the principle of angular momentum. The spinning discs located inside the gyroscope counters any input force instantaneously. Fig. 1(a) shows the front view of the intelligent glove, which is modelled in a spring-mass mechanism as in Fig. 1(b).
Fig. 1. The intelligent glove, worn on a mannequin hand
2.1. Mathematical modelling of the intelligent glove
The wheel of the gyroscope can be used to derive the mathematical modelling of the gyroscope, which is attached to the intelligent glove. Deriving mathematical modelling of a system is important to know the behaviour of the system.
Fig. 2. Reference diagram for the wheel analysis [14]
Based on the local
xyz
axis as well as using trigonometry’s equations, the summation of moments about point
G
as in,
\sum {M}_{G}
of the gyroscope wheel is written as [15]:
\sum {M}_{G}^{x}={I}_{G}^{x}{\alpha }^{x}-\left({I}_{G}^{y}-{I}_{G}^{z}\right){\omega }^{y}{\omega }^{z},
\sum {M}_{G}^{y}={I}_{G}^{y}{\alpha }^{y}-\left({I}_{G}^{z}-{I}_{G}^{x}\right){\omega }^{z}{\omega }^{x},
\sum {M}_{G}^{z}={I}_{G}^{z}{\alpha }^{z}-\left({I}_{G}^{x}-{I}_{G}^{y}\right){\omega }^{x}{\omega }^{y},
I
is the mass moment of inertia,
\alpha
is the angular acceleration and
\omega
represents angular velocity. However, no tangential acceleration exists since point
G
is travelling in a horizontal circle, resulting in
{\alpha }^{y}= {\alpha }^{z}=0
. Thus, substituting this information into Eqs. (2, 3) will make
{M}_{G}^{y}={M}_{G}^{z}=0
. Also, the mass moment of inertia about point
G
x
y
z
directions can be expressed as:
{I}_{G}^{x}={I}_{G}^{z}=\frac{1}{4}{m}_{w}{r}^{2},
{I}_{G}^{y}=\frac{1}{2}{m}_{w}{r}^{2},
r
m
is the radius and mass of the wheel, respectively. In terms of angular velocity, the angular acceleration term in
x
-axis can be expressed as:
{\alpha }^{x}=-{\omega }_{s}{\omega }_{p}\mathrm{s}\mathrm{i}\mathrm{n}\theta .
Here, subscripts
p
refer to the spin rate of the wheel and precession of the wheel. Thus, substituting Eq. (4-6) into Eq. (1), the equation can finally be written as in Eq. (7). This expression represents the necessary amount of moment needed to counter the resting force from the patients’ hand:
{M}_{G}^{x}=-\frac{1}{4}{m}_{w}{r}^{2}{\omega }_{s}{\omega }_{p}\mathrm{s}\mathrm{i}\mathrm{n}\theta -\frac{1}{4}{m}_{w}{r}^{2}\left({\omega }_{s}+{\omega }_{p}cos\theta \right){\omega }_{p}\mathrm{s}\mathrm{i}\mathrm{n}\theta .
3. Experimental studies of the intelligent glove
The experimental setup, as shown in Fig. 3, consists of HP 3570A dynamic signal analyser (DSA), an accelerometer, a power amplifier, an inertial shaker and a mannequin hand wearing the glove. The accelerometer was used to capture the signals and was directly connected to the dynamic system analyser so that the output can be analysed and saved. The input excitations were also provided by the DSA, which was fed to the inertial shaker through the power amplifier to mimic resting tremor. In this work, the result is displayed in the form of a transfer function, which is the ratio of the accelerometer readings to the input voltage (g/V). Experiments were conducted with and without the intelligent glove to study the performance of the proposed device, in two modes, i.e. swept sine and fixed sine.
It is assumed that the highest frequency of the resting tremor is at 7 Hz. The parameters used in designing the gyroscope are listed in Table 1.
Table 1. Parameters of the gyroscope and the mannequin hand
Mass of mannequin hand
Length of mannequin hand
Mass of gyroscope
3.2.1. Performance of the intelligent glove subjected to swept sine signal
The experiment was initially fed with a swept sine signal within the range of Parkinson’s disease frequency, which is 4 Hz
<f<
7 Hz. In this range, as seen from Fig. 4, a huge difference between the uncontrolled and controlled system can be observed. Using the intelligent glove, a reduction of more than 50 % of the acceleration can be seen.
Fig. 4. Controlled and uncontrolled performances, when swept from 4 Hz to 7 Hz
3.2.2. Performance of the intelligent glove subjected to fixed sine signal
In this section, a fixed harmonic signal (at 4 Hz, 5 Hz, 6 Hz and 7 Hz, respectively) was applied to the system. The performance of the system, with and without the intelligent glove, was studied.
The performance of the intelligent glove in term of normalized frequency response function (FRF) at 4 Hz is shown in Fig. 5(a). At the frequency of 4 Hz, a reduction of about 50 % with the intelligent glove is seen. Fig. 5(b) shows the plots when subjected to 5 Hz of fixed sine signal. The mannequin hand experienced 6.8 times of reduction in vibrations when wearing the intelligent glove.
Fig. 5. Controlled and uncontrolled performances at a) 4 Hz and b) 5 Hz
At 6 Hz as it can be observed in Fig. 6(a) without the glove, the hand experienced 18.8 mg/V of acceleration. However, the intelligent glove managed to suppress vibrations to 2.8 mg/V. Thus, a reduction of 85 % is seen on the graph. Finally, in Fig. 6(b), about 90 % of vibration reduction is seen for 7 Hz.
From the results presented in this section and by looking at the normalized FRF by which the ratio of output acceleration/input voltage is plotted against the frequency, it can be seen that the proposed intelligent glove can cause a significant reduction in the acceleration/voltage amplitude so that it can be used to reduce vibrations due to resting tremors, as experienced by Parkinson’s disease patients. A minimum of 50 % vibration suppression was recorded during the experiment.
The intelligent glove is appropriate for reduction of vibrations due to resting tremors for Parkinson’s disease patients that occurs at a frequency of 3 Hz-7 Hz. Results showed that a reduction of more than 50 % for a swept sine excitation. Also, it showed a minimum of 50 % and a maximum of 90 % reduction in for a fixed sine signal at all frequencies ranges between 3 Hz-7 Hz. The proposed glove is suitable for suppression of vibrations for resting tremors of Parkinson’s disease, and it only uses one device instead of using several devices so that patients can be more comfortable.
Bjorklund L. M., Sánchez Pernaute R., Chung S., Andersson T., Chen I. Y. C., St Mcnaught K. P., Brownell A., Jenkins B. G., Wahlestedt C., Kim K., Isacson O. Embryonic stem cells develop into functional dopaminergic neurons after transplantation in a Parkinson rat model. Proceedings of the National Academy of Sciences of the United States of America, Vol. 99, Issue 4, 2002, p. 2344-2349. [Publisher]
Palacios Sánchez L., Torres Nupan M., Botero Meneses J.-S. James Parkinson and his essay on ‘shaking palsy’, two hundred years later. Arquivos de Neuro-Psiquiatria, Vol. 75, Issue 9, 2017, p. 671-672. [Publisher]
Helmich R. C., Hallett M., Deuschl G., Toni I., Bloem B. R. Cerebral causes and consequences of parkinsonian resting tremor: a tale of two circuits? Brain, Vol. 135, Issue 11, 2012, p. 3206-3226. [Publisher]
Rana Q. A., Siddiqui I., Mosabbir A. A., Qureshi A. R. M., Fattah A., Awan N. Is action tremor in Parkinson’s disease related to resting tremor? Neurological Research, Vol. 36, Issue 2, 2014, p. 107-111. [Publisher]
Bhatia et al. K. P. Consensus Statement on the classification of tremors. from the task force on tremor of the International Parkinson and Movement Disorder Society. Movement Disorders, Vol. 33, Issue 1, 2018, p. 75-87. [Publisher]
Camara C., Isasi P., Warwick K., Ruiz V., Aziz T., Stein J., Bakˇstein E. Resting tremor classification and detection in Parkinson’s disease patients. Biomedical Signal Processing and Control, Vol. 16, 2015, p. 88-97. [Publisher]
Levine C. B., Fahrbach K. R., Siderowf A. D., Estok R. P., Ludensky V. M., Ross S. D. Diagnosis and treatment of Parkinson’s disease: a systematic review of the literature: summary. Evidence Report/Technology Assessment, Vol. 57, 2003, p. 03-E039. [Publisher]
Hermann R. P., Phalangas A. C., Mahoney R. M., Alexander M. A. Powered feeding devices: an evaluation of three models. Archives of Physical Medicine and Rehabilitation, Vol. 80, Issue 10, 1999, p. 1237-1242. [Publisher]
Thinh N. T., Thang L. H., Thanh T. T. Design strategies to improve self-feeding device – FeedBot for Parkinson patients. International Conference on System Science and Engineering (ICSSE), 2017. [Search CrossRef]
Bächlin M., Plotnik M., Roggen D., Maidan I., Hausdorff J. M., Giladi N., Tröster G. Wearable assistant for Parkinsons disease patients with the freezing of gait symptom. IEEE Transactions on Information Technology in Biomedicine, Vol. 14, Issue 2, 2010, p. 436-446. [Publisher]
Paulo J., et al. An innovative robotic walker for mobility assistance and lower limbs rehabilitation. IEEE 5th Portuguese Meeting on Bioengineering (ENBENG), 2017. [Search CrossRef]
Heijmans M., et al. Monitoring Parkinson’s disease symptoms during daily life: a feasibility study. Nature, Vol. 9, 2019, p. 21. [Search CrossRef]
Vargas A. P., Vaz L. S., Reuter A., Couto C. M., Costa Cardoso F. E. Impulse control symptoms in patients with Parkinson’s disease: the influence of dopaminergic agonist. Parkinsonism and Related Disorders, Vol. 68, 2019, p. 17-21. [Publisher]
Gyroscope Physics, https://www.real-world-physics-problems.com/gyroscope-physics.html. [Search CrossRef]
Shigley J. E., John Joseph Uicker J. Theory of Machines and Mechanisms. 2nd Ed., McGraw-Hill, New York, 1995. [Search CrossRef]
Naheed Rihan, Asan G. A. Muthalif, Thaer M. I. Syam, N. H. Diyana Nordin
Eka Mistiko Rini, Endi Sailul Haq
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Structure and Interpretation of Classical Mechanics: Chapter 7
[Go to first, previous, next section; contents; index]
V. I. Arnold, Mathematical Methods of Classical Mechanics [5], footnote, p. 246
There has been a remarkable revival of interest in classical mechanics in recent years. We now know that there is much more to classical mechanics than previously suspected. The behavior of classical systems is surprisingly rich; derivation of the equations of motion, the focus of traditional presentations of mechanics, is just the beginning. Classical systems display a complicated array of phenomena such as nonlinear resonances, chaotic behavior, and transitions to chaos.
Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical systems. We concentrate on developing general methods for studying the behavior of systems, whether or not they have a symbolic solution. Typical systems exhibit behavior that is qualitatively different from the solvable systems and surprisingly complicated. We focus on the phenomena of motion, and we make extensive use of computer simulation to explore this motion.
Even when a system is not symbolically tractable, the tools of modern dynamics allow one to extract a qualitative understanding. Rather than concentrating on symbolic descriptions, we concentrate on geometric features of the set of possible trajectories. Such tools provide a basis for the systematic analysis of numerical or experimental data.
Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context.1 For example, a fundamental result of mechanics is the Lagrange equations. In traditional notation the Lagrange equations are written
The Lagrangian L must be interpreted as a function of the position and velocity components qi and , so that the partial derivatives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations.2 An introduction to our functional notation is in an appendix.
Computation also enters into the presentation of the mathematical ideas underlying mechanics. We require that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer. As a consequence of this requirement the formulas and equations that appear in the text stand on their own. They have clear meaning, independent of the informal context. For example, we write Lagrange’s equations in functional notation as follows:3
D(∂2L ∘ Γ[q]) − ∂1L ∘ Γ[q] = 0.
The Lagrangian L is a real-valued function of time t, coordinates x, and velocities v; the value is L(t, x, v). Partial derivatives are indicated as derivatives of functions with respect to particular argument positions; ∂2L indicates the function obtained by taking the partial derivative of the Lagrangian function L with respect to the velocity argument position. The traditional partial derivative notation, which employs a derivative with respect to a “variable,” depends on context and can lead to ambiguity.4 The partial derivatives of the Lagrangian are then explicitly evaluated along a path function q. The time derivative is taken and the Lagrange equations formed. Each step is explicit; there are no implicit substitutions.
Active exploration on the part of the student is an essential part of the learning experience. Our focus is on understanding the motion of systems; to learn about motion the student must actively explore the motion of systems through simulation and experiment. The exercises and projects are an integral part of the presentation.
That the mathematics is precise enough to be interpreted automatically allows active exploration to be extended to it. The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether the expression is correctly formulated. Experience demonstrates that interaction with the computer in this way uncovers and corrects many deficiencies in understanding.
In this book we express computational methods in Scheme, a dialect of the Lisp family of programming languages that we also use in our introductory computer science subject at MIT. There are many good expositions of Scheme. We provide a short introduction to Scheme in an appendix.
Even in the introductory computer science class we never formally teach the language, because we do not have to. We just use it, and students pick it up in a few days. This is one great advantage of Lisp-like languages: They have very few ways of forming compound expressions, and almost no syntactic structure. All of the formal properties can be covered in an hour, like the rules of chess. After a short time we forget about the syntactic details of the language (because there are none) and get on with the real issues—figuring out what we want to compute.
The advantage of Scheme over other languages for the exposition of classical mechanics is that the manipulation of procedures that implement mathematical functions is easier and more natural in Scheme than in other computer languages. Indeed, many theorems of mechanics are directly representable as Scheme programs.
The version of Scheme that we use in this book is MIT/GNU Scheme, augmented with a large library of software called Scmutils that extends the Scheme operators to be generic over a variety of mathematical objects, including symbolic expressions. The Scmutils library also provides support for the numerical methods we use in this book, such as quadrature, integration of systems of differential equations, and multivariate minimization.
The Scheme system, augmented with the Scmutils library, is free software. We provide this system, complete with documentation and source code, in a form that can be used with the GNU/Linux operating system, on the Internet at mitpress.mit.edu/classical mech.
This book presents classical mechanics from an unusual perspective. It focuses on understanding motion rather than deriving equations of motion. It weaves recent discoveries in nonlinear dynamics throughout the presentation, rather than presenting them as an afterthought. It uses functional mathematical notation that allows precise understanding of fundamental properties of classical mechanics. It uses computation to constrain notation, to capture and formalize methods, for simulation, and for symbolic analysis.
This book is the result of teaching classical mechanics at MIT. The contents of our class began with ideas from a class on nonlinear dynamics and solar system dynamics by Wisdom and ideas about how computation can be used to formulate methodology developed in an introductory computer science class by Abelson and Sussman. When we started we expected that using this approach to formulate mechanics would be easy. We quickly learned that many things we thought we understood we did not in fact understand. Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning. The resulting struggle to make the mathematics precise, yet clear and computationally effective, lasted far longer than we anticipated. We learned a great deal about both mechanics and computation by this process. We hope others, especially our competitors, will adopt these methods, which enhance understanding while slowing research.
We have taught classical mechanics using this text every year at MIT since the first edition was published. We have learned a great deal about what difficulties students encountered with the material. We have found that some of our explanations needed improvement. This edition is the result of our new understanding.
Our software support has improved substantially over the years, and we have exploited it to provide algebraic proofs of more generality than could be supplied in the first edition. This advantage permeates most of the new edition.
In the first chapter we now go more directly to the coordinate representation of the action, without compromising the importance of the coordinate independence of the action. We also added a simple derivation of the Euler–Lagrange equations from the Principle of Stationary Action, supplementing the more formal derivation of the first edition.
In the chapter on rigid-body motion we now provide an algebraic derivation of the existence of the angular-velocity vector. Our new derivation is in harmony with the development of generalized coordinates for a rigid body as parameters of the transformation from a reference orientation to the actual orientation. We also provide a new section on quaternions as a way of avoiding singularities in the analysis of the motion of rigid bodies.
A canonical transformation is a transformation of phase-space coordinates and an associated transformation of the Hamiltonian that maintains a one-to-one correspondence between trajectories. We allow time-dependent systems and transformations, complicating the treatment of canonical transformations. The chapter on canonical transformations has been extensively revised to clarify the relationship of canonical transformations to symplectic transformations. We split off the treatment of canonical transformations that arise from evolution, including Lie transforms, into a new chapter.
We fixed myriad minor mistakes throughout. We hope that we have not introduced more than we have removed.
1In his book on mathematical pedagogy [17], Hans Freudenthal argues that the reliance on ambiguous, unstated notational conventions in such expressions as f(x) and df(x)/dx makes mathematics, and especially introductory calculus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation.
2In his beautiful book Calculus on Manifolds [40], Michael Spivak uses functional notation. On p. 44 he discusses some of the problems with classical notation. We excerpt a particularly juicy passage:
The mere statement of [the chain rule] in classical notation requires the introduction of irrelevant letters. The usual evaluation for D1(f ∘ (g, h)) runs as follows:
If f(u, v) is a function and u = g(x, y) and v = h(x, y) then
\frac{\partial f\left(g\left(x,y\right),h\left(x,y\right)}{\partial x}=\frac{\partial f\left(u,v\right)}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f\left(u,v\right)}{\partial v}\frac{\partial v}{\partial x}
[The symbol ∂u/∂x means ∂/∂x g(x, y), and ∂/∂u f(u, v) means D1f(u, v) = D1f(g(x, y), h(x, y)).] This equation is often written simply
\frac{\partial f}{\partial x}=\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x}.
Note that f means something different on the two sides of the equation!
3This is presented here without explanation, to give the flavor of the notation. The text gives a full explanation.
4“It is necessary to use the apparatus of partial derivatives, in which even the notation is ambiguous.” V.I. Arnold, Mathematical Methods of Classical Mechanics [5], Section 47, p. 258. See also the footnote on that page.
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Climate Change/Science/Sun-Earth System - Wikibooks, open books for an open world
Climate Change/Science/Sun-Earth System
In Sun's Influence on Earth, a simplified view of the Sun-Earth system has been used. The results are correct, and show how simple physical principles directly inform us about some of the most important aspects ofthis view is inadequate; this will become especially clear in the discussion of paleoclimate (ancient climate) and ice ages. Here let us review a few important aspects of the system to set the stage for later discussions.
Perhaps the most profound influence on the Earth-Sun system is the geometry involved. The most basic part of this geometry is the orbit of Earth around the sun, which is governed by the gravitational attraction between the two bodies. Kepler[1] showed that orbits are ellipses, rather than perfect circles. Earth's orbit is slightly elliptical, with an eccentricity of just 0.01671 [2]; even though this value is small, it has important consequences. The perihelion, or smallest distance from the sun, occurs during the northern hemisphere's winter and is about
{\displaystyle 1.47\times 10^{11}m}
, and aphelion, the most distant point from the sun, is during the northern hemisphere's summer and is about
{\displaystyle 152\times 10^{11}m.}
This difference does not cause Earth's seasons, but can influence the severity of seasons (discussed in the paleoclimate section) and does introduce small variations to the annual incoming solar radiation ("insolation") as there are very slow variations in the eccentricity.
A second important effect to consider is the tilt of Earth's spin axis with respect to the ecliptic plane, which is basically the average plane of Earth's orbit around the sun. The angle between the spin axis and the perpendicular to the ecliptic plane is called Earth's obliquity, and is currently about 23.4 degrees. This angle is the primary reason for seasons on Earth, for as the planet traverses its orbit, the amount of insolation at points on the surface slowly change, with winter occurring when the pole faces away from the sun and summer when the pole faces toward the sun. Seasons are more extreme with larger obliquity, and high latitudes (e.g. Antarctica) experience more extreme changes in insolation than the tropics, leading to more pronounced season. Earth's obliquity slowly changes in time, which has important consequences for very long-term climate change.
A third important part of the Earth-sun geometry is called precession, and is actually the combination of parameters. Precession is the slow variation of direction of the spin axis, and is affected by both a turning of the spin axis and a slow change in the shape of Earth's orbit. For the contemporary climate, the precession only matters because it determines the relative position of the poles to the sun during Earth's orbit. There are important consequences for long-term climate change, though, which will be discussed later.
The geometry of the Earth-sun system is a large part of the astronomical basis for Earth's climate. Other astronomical factors that are important include the evolution of the solar system and the sun itself, as well as electromagnetic phenomena (e.g. the solar wind). These topics are well worth studying, even in the context of climate, but they are beyond the scope of this book, as they bear little relevance on contemporary climate change.
^ Eccentricity is defined for all conic sections, and is a relationship between the semimajor (a) and semiminor (b) axes. It can be determined by
{\displaystyle \epsilon \equiv {\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.}
For a perfect circle a = b, so the eccentricity is zero, for an ellipse a > b, and the eccentricity is bounded
{\displaystyle 0<\epsilon <1.}
See also [Wolfram MathWorld].
^ Johannes_Kepler
Retrieved from "https://en.wikibooks.org/w/index.php?title=Climate_Change/Science/Sun-Earth_System&oldid=1999531"
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Revision as of 09:41, 15 May 2015 by MathAdmin (talk | contribs) (Created page with "<span class="exam"> Find the antiderivative of <math style="vertical-align: -50%">\int \frac{1}{3x+2}\,dx.</math> {| class="mw-collapsible mw-collapsed" style = "text-align:...")
{\displaystyle \int {\frac {1}{3x+2}}\,dx.}
Integration by substitution (U - sub): I{\displaystyle f}
{\displaystyle g}
{\displaystyle (f\circ g)'(x)=f'(g(x))\cdot g'(x).}
The Product Rule: I{\displaystyle f}
{\displaystyle g}
{\displaystyle (fg)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).}
The Quotient Rule: I{\displaystyle f}
{\displaystyle g}
{\displaystyle g(x)\neq 0}
{\displaystyle \left({\frac {f}{g}}\right)'(x)={\frac {f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}.}
{\displaystyle \left(x^{n}\right)'\,=\,nx^{n-1},}
{\displaystyle n\neq 0}
{\displaystyle \left(\ln x\right)'\,=\,{\frac {1}{x}}.}
{\displaystyle u=3x+2.}
{\displaystyle du=3dx}
, and after substitution we have
{\displaystyle \int {\frac {1}{3x+2}}=\int {\frac {1}{3u}}du}
{\displaystyle \int {\frac {1}{3u}}du={\frac {\log(u)}{3}}}
Since this integral is an indefinite integral we have to remember to add C at the end.
{\displaystyle \int {\frac {1}{3x+2}}dx={\frac {\ln(3x+2)}{3}}}
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(Not recommended) Generalized solver for discrete-time algebraic Riccati equation - MATLAB gdare - MathWorks Benelux
gdare not recommended
(Not recommended) Generalized solver for discrete-time algebraic Riccati equation
gdare is not recommended. Use idare instead. For more information, see Compatibility Considerations.
[X,L,report] = gdare(H,J,ns)
[X1,X2,D,L] = gdare(H,J,NS,'factor')
[X,L,report] = gdare(H,J,ns) computes the unique stabilizing solution X of the discrete-time algebraic Riccati equation associated with a Symplectic pencil of the form
H-tJ=\left[\begin{array}{ccc}A& F& B\\ -Q& E\prime & -S\\ S\prime & 0& R\end{array}\right]-\left[\begin{array}{ccc}E& 0& 0\\ 0& A\prime & 0\\ 0& B\prime & 0\end{array}\right]
The third input ns is the row size of the A matrix.
Optionally, gdare returns the vector L of closed-loop eigenvalues and a diagnosis report with value:
-1 if the Symplectic pencil has eigenvalues on the unit circle
[X1,X2,D,L] = gdare(H,J,NS,'factor') returns two matrices X1, X2 and a diagonal scaling matrix D such that X = D*(X2/X1)*D. The vector L contains the closed-loop eigenvalues. All outputs are empty when the Symplectic pencil has eigenvalues on the unit circle.
R2019a: gdare not recommended
Starting in R2019a, use the idare command to solve discrete-time Riccati equations. This approach has improved accuracy through better scaling and the computation of K is more accurate when R is ill-conditioned relative to gdare. Furthermore, idare includes an optional info structure to gather the implicit solution data of the Riccati equation.
The following table shows some typical uses of gdare and how to update your code to use idare instead.
[X,L] = gdare(H,J,NS)
There are no plans to remove gdare at this time.
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Non-standard model of arithmetic - formulasearchengine
1.1 From the compactness theorem
1.2 From the incompleteness theorems
1.2.1 Arithmetic unsoundness for models with ~G true
1.3 From an ultraproduct
2 Structure of countable non-standard models
From the compactness theorem
From the incompleteness theorems
Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic. The incompleteness theorems show that a particular sentence G, the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic. By the completeness theorem, this means that G is false in some model of Peano arithmetic. However, G is true in the standard model of arithmetic, and therefore any model in which G is false must be a non-standard model. Thus satisfying ~G is a sufficient condition for a model to be nonstandard. It is not a necessary condition, however; for any Gödel sentence G, there are models of arithmetic with G true of all cardinalities.
Arithmetic unsoundness for models with ~G true
Assuming that arithmetic is consistent, arithmetic with ~G is also consistent. However since ~G means that arithmetic is inconsistent, the result will not be ω-consistent (because ~G is false and this violates ω-consistency).
From an ultraproduct
{\displaystyle \mathbb {N} ^{\mathbb {N} }}
. Identify two sequences if they agree for a set of indices that is a member of a fixed non-principal ultrafilter. The resulting ring is a non-standard model of arithmetic. It can be identified with the hypernatural numbers.
Structure of countable non-standard models
Any countable non-standard model of arithmetic has order type ω + (ω* + ω) · η, where ω is the order type of the standard natural numbers, ω* is the dual order (an infinite decreasing sequence) and η is the order type of the rational numbers. In other words, a countable non-standard model begins with an infinite increasing sequence (the standard elements of the model). This is followed by a collection of "blocks," each of order type ω* + ω, the order type of the integers. These blocks are in turn densely ordered with the order type of the rationals. The result follows fairly easily because it is easy to see that the non-standard numbers have to be dense and linearly ordered without endpoints, and the order type of the rationals is the only countable dense linear order without endpoints.[1][2][3]
It is easy to see that the arithmetical structure differs from ω + (ω* + ω) · η. For instance if u is in the model, then so is m*u for any m, n in the initial segment N, yet u2 is larger than m*u for any standard finite m.
Also you can define "square roots" such as the least v such that v2 > 2*u. It is easy to see that these can't be within a standard finite number of any rational multiple of u. By analogous methods to Non-standard analysis you can also use PA to define close approximations to irrational multiples of a non-standard number u such as the least v with v > π*u (these can be defined in PA using non-standard finite rational approximations of π even though pi itself can't be). Once more, v - (m/n)*u/n has to be larger than any standard finite number for any standard finite m,n. {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}
This shows that the arithmetical structure of a countable non-standard model is more complex than the structure of the rationals. There is more to it than that though.
Tennenbaum's theorem shows that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as (standard) natural numbers such that either the addition or multiplication operation of the model is a computable on the codes. This result was first obtained by Stanley Tennenbaum in 1959.
Boolos, G., and Jeffrey, R. 1974. Computability and Logic, Cambridge University Press. ISBN 0-521-38923-2
Skolem, Th. (1934) Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23, 150–161.
↑ Andrey Bovykin and Richard Kaye Order-types of models of Peano arithmetic: a short survey June 14, 2001
↑ Andrey Bovykin On order-types of models of arithmetic thesis submitted to the University of Birmingham for the degree of Ph. D. in the Faculty of Science 13th April 2000
↑ - LINEAR ORDERS, DISCRETE, DENSE, AND CONTINUOUS - includes proof that Q is the only countable dense linear order.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Non-standard_model_of_arithmetic&oldid=249948"
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Free vibration analysis of simply supported rectangular plates | JVE Journals
Ganesh Naik Guguloth1 , Baij Nath Singh2 , Vinayak Ranjan3
2Indian Institute of Technology, Dhanbad, India
Copyright © 2019 Ganesh Naik Guguloth, 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.
In this paper, numerical analysis for free vibration of simply supported thin rectangular plates has been simulated using Ansys. The vibration of plate follows Kirchhoff plate theory. The natural frequency calculated using Ansys has been compared with Levy type solution results available in the literature. It is observed that the natural frequencies of the simply supported rectangular plate is in close agreement with exact solution results as reported in literature.
Keywords: Kirchhoff plate theory, Levy solution, simply supported rectangular plate.
A thin plate is a solid structural body where the thickeness of the plate is small as compared to other two dimensions.
In modern science explorations, rectangular thin plate structures are used in aerospace industries, mechanical, automotive sectors, civil and marine industries etc. In the actual engineering environment, there are many external forces which cause vibrations in the thin plate structures. These forced vibrations are the cause of failure and damage to the engineering structures. To investigate the dynamic response due to forced vibration of the plate structure, natural frequencies and their mode shape estimations are very important. So, this paper is about the analysis of free vibrations of rectangular thin plates in simply supported condition using finite element with Ansys software.
Tanaka et al. [2] solved the free vibrations of elastic plates using integral equation process. Karunasena and Kitipornchai et al. [3] determined the free vibration analysis of deformable triangular element. Wu and Liu et al. [4] have been studied the new numerical solution technique called as differential cubature method for free vibration analysis of arbitrary shaped plates. Moon and Choi et al. [5] have formulated the transfer dynamic stiffness coefficient method for vibration analysis of frame structures. Myung [6] has developed the finite element transfer stiffness coefficient method for free vibration analysis of plate structures. The approach is based on the combination of the modelling techniques in FEM and the transfer technique of the stiffness coefficient in the transfer stiffness coefficient method. Kumar et al. [7] use dynamic stiffness method to extract natural frequency and mode shapes of thin plate. Piyush et al. [8] used Rayleigh-Ritz method to compute the natural frequencies of thin plate.
The Kirchhoff plate theory is a two dimensional mathematical model that is used to determine the stresses and deformation in thin plates when they are subjected to force and moments.
The assumptions made in Kirchhoff plate theory are:
1) Straight lines normal to the mid surface remains straight after deformation.
2) The thickness of the plate does not change during a deformation.
3) Straight lines normal to the mid surface remain normal to the mid surface after deformation.
This theory is used to model the transverse vibration of the plate. According to Kirchhoff plate theory, the equations for the transverse vibration of plate is given by:
D\left[\frac{{\partial }^{2}\omega }{\partial {x}^{4}}+2\frac{{\partial }^{2}\omega }{\partial {x}^{2}\partial {y}^{2}}+\frac{{\partial }^{4}\omega }{\partial {y}^{4}}\right]=0,
D
is flexural rigidity:
D=\frac{E{h}^{3}}{12\left(1-{\vartheta }^{2}\right)}.
The finite element method is used to calculate the natural frequency. The general equation of finite element for the transverse deflection of thin plate is given by:
\left[M\right]\left\{\stackrel{¨}{q}\right\}+\left[K\right]\left\{q\right\}=0,
M
] is the mass matrix, [
K
] is stiffness matrix,
\left\{\stackrel{¨}{q}\right\}
is the nodal acceleration vector and
\left\{q\right\}
The analytical calculation of natural frequencies are also carried out using Eq. (4):
{\omega }_{mn}=\sqrt{\frac{D}{\rho h}}\left[{\left(\frac{m\pi }{a}\right)}^{2}+{\left(\frac{n\pi }{b}\right)}^{2}\right],
{\omega }_{mn}
– natural frequencies,
\rho
– density,
a
b
– length and width of rectangular plate,
h
– thickness of plate.
The material properties and dimensions that are considered for the analysis of simply supported rectangular plate are referred from Ramu et al. [1].
The material properties of rectangular plate are:
Material = Aluminium;
E=
70×109 N/m2;
\rho =
2700 kg/m3;
Poisons ratio
\vartheta =
The dimensions of plate taken are:
Length of plate
a=
600 mm;
Width of plate
b=
h=
6.25 mm, 12.5 mm, 25 mm, 50 mm.
Fig. 1. Dimensions of the rectangular plate
Finite element analysis formulation for rectangular element have four nodes. There are three degrees of freedom at each node, one is along the thickness of the plate and the other two are rotations along
X
Y
directions. The solid rectangular plate is modelled in Ansys software and the size of the mesh is 1.e-002 m. The number of elements taken are 2400 and the number of nodes are 5002. The face meshing of the rectangular plate gives the solutions which are close agreement to the theoretical values.
The geometry of plate 600 mm×400 mm with varying thickness.
In this section, the first six natural frequencies and mode shapes of simply supported rectangular plates are estimated using Finite Element Method. The present natural frequencies results are compared with those available in literature and with exact solutions.
Tables 1-4 shows that the first six natural frequencies of thin rectangular plate. These results are nearly same as Ramu [1] and analytical solutions under different thickness of the plate.
Fig. 3. First six mode shapes of simply supported rectangular plate
d) 4th mode shape
e) 5th mode shape
f) 6th mode shape
Table 1. Comparison of the natural frequency parameter with analytical solution and Ramu [1] for simply supported rectangular plate with the thickeness of 6.25 mm
Ramu [1]
Present (FEA)
Table 2. Comparison of the natural frequency parameter with analytical solution and Ramu [1] for simply supported rectangular plate with the thickness of 12.5 mm
Table 3. Comparison of the natural frequency parameter with analytical solution and Ramu [1] for simply supported rectangular plate with the thickness of 25 mm
In this paper, numerical analysis for free vibration analysis of thin rectangular solid plate is carried out using Finite Element Method under simply supported condition with varying the thickness. It is found that those natural frequency results are quite close to the reported literatures. As the thickness of the rectangular plate increases, the values are not so close to the analytical values.
Ramu I., Mohanty S. C. Study on free vibration analysis of rectangular plate structures using finite element method. Procedia Engineering, Vol. 38, 2012, p. 2758-2766. [Publisher]
Tanaka M., Yamagiwa K., Miyazaki K., Ueda T. Free vibration analysis of elastic plate structures by boundary element method. Engineering Analysis, Vol. 5, Issue 4, 1988, p. 182-188. [Publisher]
Karunasena W., Kitipornchai S. Free vibrations of shear deformable general triangular plates. Journal of Sound and Vibration, Vol. 199, Issue 5, 1997, p. 595613. [Search CrossRef]
Wu L., Liu J. Free vibration analysis of arbitrary shaped thick plates by differential cubature method. International Journal of Mechanical Sciences, Vol. 47, 2005, p. 6381. [Publisher]
Moon D. H., Choi M. S. Vibration analysis for frame structures using transfer of dynamic stiffness coefficient. Journal of Sound and Vibration, Vol. 234, Issue 5, 2000, p. 725736. [Publisher]
Myung Soo Choi Free vibration analysis of plate structures using finite element-transfer stiffness coefficient method. KSME International Journal, Vol. 17, Issue 6, 2003, p. 805-815. [Publisher]
Pratap Singh P., Azam M. S., Ranjan Vinayak Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on Winkler-Pasternak foundation under different combinations of boundary conditions. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2018. [Search CrossRef]
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Nilpotent_group Knowpia
In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, its central series is of finite length or its lower central series terminates with {1}.
Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.[1]
Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.
Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.
The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G:
G has a central series of finite length. That is, a series of normal subgroups
{\displaystyle \{1\}=G_{0}\triangleleft G_{1}\triangleleft \dots \triangleleft G_{n}=G}
{\displaystyle G_{i+1}/G_{i}\leq Z(G/G_{i})}
{\displaystyle [G,G_{i+1}]\leq G_{i}}
G has a lower central series terminating in the trivial subgroup after finitely many steps. That is, a series of normal subgroups
{\displaystyle G=G_{0}\triangleright G_{1}\triangleright \dots \triangleright G_{n}=\{1\}}
{\displaystyle G_{i+1}=[G_{i},G]}
G has an upper central series terminating in the whole group after finitely many steps. That is, a series of normal subgroups
{\displaystyle \{1\}=Z_{0}\triangleleft Z_{1}\triangleleft \dots \triangleleft Z_{n}=G}
{\displaystyle Z_{1}=Z(G)}
{\displaystyle Z_{i+1}}
is the subgroup such that
{\displaystyle Z_{i+1}/Z_{i}=Z(G/Z_{i})}
For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. (By definition, the length is n if there are
{\displaystyle n+1}
different subgroups in the series, including the trivial subgroup and the whole group.)
Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most n, then it is sometimes called a nil-n group.
It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups.[2][3]
A portion of the Cayley graph of the discrete Heisenberg group, a well-known nilpotent group.
As noted above, every abelian group is nilpotent.[2][4]
For a small non-abelian example, consider the quaternion group Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q8; so it is nilpotent of class 2.
The direct product of two nilpotent groups is nilpotent.[5]
All finite p-groups are in fact nilpotent (proof). The maximal class of a group of order pn is n (for example, any group of order 2 is nilpotent of class 1). The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
Furthermore, every finite nilpotent group is the direct product of p-groups.[5]
The multiplicative group of upper unitriangular n × n matrices over any field F is a nilpotent group of nilpotency class n − 1. In particular, taking n = 3 yields the Heisenberg group H, an example of a non-abelian[6] infinite nilpotent group.[7] It has nilpotency class 2 with central series 1, Z(H), H.
The multiplicative group of invertible upper triangular n × n matrices over a field F is not in general nilpotent, but is solvable.
Any nonabelian group G such that G/Z(G) is abelian has nilpotency class 2, with central series {1}, Z(G), G.
Explanation of termEdit
Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group
{\displaystyle G}
of nilpotence degree
{\displaystyle n}
{\displaystyle g}
{\displaystyle \operatorname {ad} _{g}\colon G\to G}
{\displaystyle \operatorname {ad} _{g}(x):=[g,x]}
{\displaystyle [g,x]=g^{-1}x^{-1}gx}
{\displaystyle g}
{\displaystyle x}
) is nilpotent in the sense that the
{\displaystyle n}
th iteration of the function is trivial:
{\displaystyle \left(\operatorname {ad} _{g}\right)^{n}(x)=e}
{\displaystyle x}
{\displaystyle G}
This is not a defining characteristic of nilpotent groups: groups for which
{\displaystyle \operatorname {ad} _{g}}
is nilpotent of degree
{\displaystyle n}
(in the sense above) are called
{\displaystyle n}
-Engel groups,[8] and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.
An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).
Since each successive factor group Zi+1/Zi in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.
Every subgroup of a nilpotent group of class n is nilpotent of class at most n;[9] in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent[9] of class at most n.
The following statements are equivalent for finite groups,[10] revealing some useful properties of nilpotency:
G is a nilpotent group.
If H is a proper subgroup of G, then H is a proper normal subgroup of NG(H) (the normalizer of H in G). This is called the normalizer property and can be phrased simply as "normalizers grow".
Every Sylow subgroup of G is normal.
G is the direct product of its Sylow subgroups.
If d divides the order of G, then G has a normal subgroup of order d.
(a)→(b)
By induction on |G|. If G is abelian, then for any H, NG(H) = G. If not, if Z(G) is not contained in H, then hZHZ−1h−1 = h'H'h−1 = H, so H·Z(G) normalizers H. If Z(G) is contained in H, then H/Z(G) is contained in G/Z(G). Note, G/Z(G) is a nilpotent group. Thus, there exists an subgroup of G/Z(G) which normalizers H/Z(G) and H/Z(G) is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in G and it normalizes H. (This proof is the same argument as for p-groups – the only fact we needed was if G is nilpotent then so is G/Z(G) – so the details are omitted.)
(b)→(c)
Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G), 1 ≤ i ≤ s. Let P = Pi for some i and let N = NG(P). Since P is a normal subgroup of N, P is characteristic in N. Since P char N and N is a normal subgroup of NG(N), we get that P is a normal subgroup of NG(N). This means NG(N) is a subgroup of N and hence NG(N) = N. By (b) we must therefore have N = G, which gives (c).
(c)→(d)
Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G), 1 ≤ i ≤ s. For any t, 1 ≤ t ≤ s we show inductively that P1P2···Pt is isomorphic to P1×P2×···×Pt.
Note first that each Pi is normal in G so P1P2···Pt is a subgroup of G. Let H be the product P1P2···Pt−1 and let K = Pt, so by induction H is isomorphic to P1×P2×···×Pt−1. In particular,|H| = |P1|⋅|P2|⋅···⋅|Pt−1|. Since |K| = |Pt|, the orders of H and K are relatively prime. Lagrange's Theorem implies the intersection of H and K is equal to 1. By definition,P1P2···Pt = HK, hence HK is isomorphic to H×K which is equal to P1×P2×···×Pt. This completes the induction. Now take t = s to obtain (d).
(d)→(e)
Note that a P-group of order pk has a normal subgroup of order pm for all 1≤m≤k. Since G is a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups, G has a normal subgroup of order d for every divisor d of |G|.
(e)→(a)
For any prime p dividing |G|, the Sylow p-subgroup is normal. Thus we can apply (c) (since we already proved (c)→(e)).
Statement (d) can be extended to infinite groups: if G is a nilpotent group, then every Sylow subgroup Gp of G is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).
Many properties of nilpotent groups are shared by hypercentral groups.
^ Dixon, M. R.; Kirichenko, V. V.; Kurdachenko, L. A.; Otal, J.; Semko, N. N.; Shemetkov, L. A.; Subbotin, I. Ya. (2012). "S. N. Chernikov and the development of infinite group theory". Algebra and Discrete Mathematics. 13 (2): 169–208.
^ a b Suprunenko (1976). Matrix Groups. p. 205.
^ Tabachnikova & Smith (2000). Topics in Group Theory (Springer Undergraduate Mathematics Series). p. 169.
^ Hungerford (1974). Algebra. p. 100.
^ a b Zassenhaus (1999). The theory of groups. p. 143.
^ Haeseler (2002). Automatic Sequences (De Gruyter Expositions in Mathematics, 36). p. 15.
^ Palmer (2001). Banach algebras and the general theory of *-algebras. p. 1283.
^ For the term, compare Engel's theorem, also on nilpotency.
^ a b Bechtell (1971), p. 51, Theorem 5.1.3
^ Isaacs (2008), Thm. 1.26
Von Haeseler, Friedrich (2002). Automatic Sequences. De Gruyter Expositions in Mathematics. Vol. 36. Berlin: Walter de Gruyter. ISBN 3-11-015629-6.
Hungerford, Thomas W. (1974). Algebra. Springer-Verlag. ISBN 0-387-90518-9.
Palmer, Theodore W. (1994). Banach Algebras and the General Theory of *-algebras. Cambridge University Press. ISBN 0-521-36638-0.
Stammbach, Urs (1973). Homology in Group Theory. Lecture Notes in Mathematics. Vol. 359. Springer-Verlag. review
Tabachnikova, Olga; Smith, Geoff (2000). Topics in Group Theory. Springer Undergraduate Mathematics Series. Springer. ISBN 1-85233-235-2.
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Talk:QB/a18ElectricChargeField findE - Wikiversity
Talk:QB/a18ElectricChargeField findE
{\displaystyle \varepsilon _{0}=}
{\displaystyle k_{e}={\tfrac {1}{4\pi \varepsilon _{0}}}=}
{\displaystyle {\vec {F}}=Q{\vec {E}}}
{\displaystyle {\vec {E}}={\tfrac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\tfrac {q_{i}}{r_{Pi}^{2}}}{\hat {r}}_{Pi}}
{\displaystyle {\vec {E}}=\int {{\tfrac {dq}{{r}^{2}}}{\hat {r}}}}
{\displaystyle dq=\lambda d\ell =\sigma da=\rho dV}
{\displaystyle E={\tfrac {\sigma }{2\varepsilon _{0}}}}
{\displaystyle {\vec {E}}({\vec {r}})={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {{\widehat {\mathcal {R}}}_{i}Q_{i}}{|{\mathcal {\vec {R}}}_{i}|^{2}}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {{\vec {\mathcal {R}}}_{i}Q_{i}}{|{\mathcal {\vec {R}}}_{i}|^{3}}}}
{\displaystyle {\vec {r}}}
{\displaystyle {\vec {r}}_{i}}
{\displaystyle {\vec {\mathcal {R}}}_{i}={\vec {r}}-{\vec {r}}_{i},}
Retrieved from "https://en.wikiversity.org/w/index.php?title=Talk:QB/a18ElectricChargeField_findE&oldid=1947835"
Return to "QB/a18ElectricChargeField findE" page.
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Conic Sections - Maple Help
Home : Support : Online Help : Math Apps : Algebra and Geometry : Geometry : Conic Sections
The conic sections are the curves formed by intersecting a cone with a plane. The four non-degenerate conics are the circle, the ellipse, the parabola, and the hyperbola:
The degenerate conics occur when the plane passes through the apex of the cone. These consist of the following types: a single point, a line, and the intersection of two lines.
Visualization: Intersection of a cone and with a plane
Use the sliders to manipulate the plane. See how the intersection with the cone changes to form a circle, ellipse, parabola, or a hyperbola.
distance from origin =
Visualization: General Form
The general form of a conic is:
{\mathrm{Ax}}^{2 }+\mathrm{Bxy} +{\mathrm{Cy}}^{2 }+\mathrm{Dx} +\mathrm{Ey}+F = 0
where A, B, C, D, E, F are real-valued parameters.
The classification of conics can be expressed using the following discriminants:
{B}^{2}-4 A C
\mathrm{Δ}=4ACF-A{E}^{2}+BE\mathrm{D}-{B}^{2}F-C{\mathrm{D}}^{2}
B =0
A =C
C \mathrm{Δ} > 0
{B}^{2}-4 \mathrm{AC} <0
C \mathrm{\Delta } > 0
B ≠0
A ≠C
{B}^{2}-4 \mathrm{AC} = 0
\mathrm{Δ}≠0
{B}^{2 }-4 \mathrm{AC} > 0
\mathrm{\Delta }≠0
Line(s), Point
\mathrm{Δ}=0
Use the sliders to modify coefficients of the general equation of a conic and see how it affects the conic .
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Difference between revisions of "Analytic continuation" - SEG Wiki
Difference between revisions of "Analytic continuation"
In [[Dictionary:analytic Function|complex analysis]] we may consider extending the domain of a given function <math> f(z) </math> which is [[Dictionary:analytic Function|analytic]] in a region <math> \mathcal R </math>
by finding another function <math> g(z) </math> analytic in a region <math> {\mathcal R}_2 </math>, if a region <math> {\mathcal R}_3 </math> exists such
that <math> {\mathcal R}_3 = {\mathcal R} \cap {\mathcal R}_2 </math>. We say that <math> g(z) </math> is ''the analytic continuation'' of <math> f(z) </math>
that <math> {\mathcal R}_3 = {\mathcal R} \cap {\mathcal R}_2 </math> and if <math> f(z) = g(z) </math> for all <math> z \in {\mathcal R}_3 </math> then we say that <math> g(z) </math> is ''the analytic continuation'' of <math> f(z) </math
into <math> {\mathcal R}_2 </math>.
In complex analysis we may consider extending the domain of a given function
{\displaystyle f(z)}
which is analytic in a region
{\displaystyle {\mathcal {R}}}
by finding another function
{\displaystyle g(z)}
analytic in a region
{\displaystyle {\mathcal {R}}_{2}}
, if a region
{\displaystyle {\mathcal {R}}_{3}}
{\displaystyle {\mathcal {R}}_{3}={\mathcal {R}}\cap {\mathcal {R}}_{2}}
{\displaystyle f(z)=g(z)}
{\displaystyle z\in {\mathcal {R}}_{3}}
{\displaystyle g(z)}
is the analytic continuation of
{\displaystyle f(z)</mathinto<math>{\mathcal {R}}_{2}}
Retrieved from "https://wiki.seg.org/index.php?title=Analytic_continuation&oldid=32246"
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Big O notation - Simple English Wikipedia, the free encyclopedia
In mathematics and computer science, Big O notation is a way of comparing rates of growth of different functions. It is often used to compare the efficiency of different algorithms, which is done by calculating how much memory is needed, and how much time it takes to complete.
The Big O notation is often used in identifying how complex a problem is, also known as the problem's complexity class. The mathematician Paul Bachmann (1837-1920) was the first to use this notation, in the second edition of his book "Analytische Zahlentheorie", in 1896. Edmund Landau (1877-1938) made the notation popular. For this reason, when people talk about a Landau symbols, they refer to this notation.
Big O notation is named after the term "order of the function", which refers to the growth of functions. Big O notation is used to find the upper bound (the highest possible amount) of the function's growth rate, meaning it works out the longest time it will take to turn the input into the output. This means an algorithm can be grouped by how long it can take in a worst-case scenario, where the longest route will be taken every time.
More specifically, given two positive functions
{\displaystyle f(x)}
{\displaystyle g(x)}
{\displaystyle f}
is in the big O of
{\displaystyle g}
(written
{\displaystyle f\in O(g)}
) if for large enough
{\displaystyle x}
{\displaystyle f(x)\leq k\cdot g(x)}
{\displaystyle k}
Big O is an expression that finds worst-case scenario run-time, showing how efficient an algorithm is without having to run the program on a computer. This is also useful due to the fact that different computers may have different hardware, and therefore need different amounts of time to complete it. Since Big O always assumes the worst-case, it can show a consistent measurement of speed: regardless of the hardware,
{\displaystyle O(1)}
is always going to complete faster than
{\displaystyle O(n!)}
, because they have different levels of efficiency.
2 Little-o notation
The following examples all use code written in Python. Note that this is not a complete list of Big O types.
Constant[change | change source]
{\displaystyle O(1)}
always takes the same amount of time regardless of input. For example, take a function that accepts an integer (called x) and returns double its value:
return x * 2 #Return the value of x times 2
After accepting the input, this function will always take one step to return an output. It is constant because it will always take the same amount of time, so it is
{\displaystyle O(1)}
Linear[change | change source]
{\displaystyle O(n)}
increases according to the size of the input, represented by
{\displaystyle n}
. For example, for the following function that accepts n and returns every number from 1 to n:
i = 1 #Create a counter called "i" with a value of 1
while i <= n: #While i is less-than or equal to n
print(i) #Print the value of i
i = i + 1 #Redefine i as "the value of i + 1"
If we were to input the value of 5, then this would output
{\displaystyle 1,2,3,4,5}
, requiring 5 loops to complete. Similarly, if we input 100, then it would output
{\displaystyle 1,2,3...98,99,100}
, requiring 100 loops to complete. If the input is
{\displaystyle n}
, then the algorithm's run time is exactly
{\displaystyle n}
loops every time, therefore it is
{\displaystyle O(n)}
Factorial[change | change source]
{\displaystyle O(n!)}
increases in factorial amounts, meaning the time taken increases massively with the input. For example, say we wish to visit five cities around the world and want to see every possible ordering (permutation). An algorithm we could write using Python's itertools library is as follows:
import itertools #Import the itertools library
cities = ['London', 'Paris', 'Berlin', 'Amsterdam', 'Rome'] #An array of our chosen cities
def permutations(cities): #Taking an array of cities as input:
for i in itertools.permutations(cities): #For each permutation of our items (assigned to variable "i")
print(i) #Output i
This algorithm will calculate each unique permutation of our cities and then output it. Examples of output will include:
('London', 'Paris', 'Berlin', 'Amsterdam', 'Rome')
('London', 'Paris', 'Berlin', 'Rome', 'Amsterdam')
('London', 'Paris', 'Amsterdam', 'Berlin', 'Rome')
('Rome', 'Amsterdam', 'Paris', 'Berlin', 'London')
('Rome', 'Amsterdam', 'Berlin', 'London', 'Paris')
('Rome', 'Amsterdam', 'Berlin', 'Paris', 'London')
Here, our input list is 5 items long, and for every selection our remaining options decreases by 1. In other words, our 5 inputs choose
{\displaystyle 5\times 4\times 3\times 2\times 1}
items (or
{\displaystyle 5!}
). If our input is
{\displaystyle n}
cities long, then the number of outputs is
{\displaystyle n!}
; in general, assuming that we go through every permutation, then we will require
{\displaystyle O(n!)}
loops to complete it.
Little-o notation[change | change source]
A related concept to big-O notation is little-o notation. Big-O is used to say that a function does not grow faster than another function, while little-o is used to say that a function grows more slowly than another function. If two functions grow at the same rate, big-O can be used but little-o cannot. The difference between big-O and little-o is similar to the difference between
{\displaystyle \leq }
{\displaystyle <}
{\displaystyle f(x)}
grows more slowly than
{\displaystyle g(x)}
{\displaystyle f(x)\in O(g(x))}
{\displaystyle f(x)\in o(g(x))}
{\displaystyle f(x)}
{\displaystyle g(x)}
{\displaystyle f(x)\in O(g(x))}
{\displaystyle f(x)\not \in o(g(x))}
{\displaystyle f(x)}
grows at the faster than
{\displaystyle g(x)}
{\displaystyle f(x)\not \in O(g(x))}
{\displaystyle f(x)\not \in o(g(x))}
↑ "Big-O notation (article) | Algorithms". Khan Academy. Retrieved 2020-09-22.
↑ "Big O Notation | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-09-22.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Big_O_notation&oldid=8004976"
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Circular law - Wikipedia
In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an n × n random matrix with independent and identically distributed entries in the limit n → ∞.
It asserts that for any sequence of random n × n matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to 1/n, the limiting spectral distribution is the uniform distribution over the unit disc.
Plot of the real and imaginary parts (scaled by sqrt(1000)) of the eigenvalues of a 1000x1000 matrix with independent, standard normal entries.
Precise statement[edit]
{\displaystyle (X_{n})_{n=1}^{\infty }}
be a sequence of n × n matrix ensembles whose entries are i.i.d. copies of a complex random variable x with mean 0 and variance 1. Let
{\displaystyle \lambda _{1},\ldots ,\lambda _{n},1\leq j\leq n}
denote the eigenvalues of
{\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}
. Define the empirical spectral measure of
{\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}
{\displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}(A)=n^{-1}\#\{j\leq n:\lambda _{j}\in A\}~,\quad A\in {\mathcal {B}}(\mathbb {C} ).}
With these definitions in mind, the circular law asserts that almost surely (i.e. with probability one), the sequence of measures
{\displaystyle \displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}}
converges in distribution to the uniform measure on the unit disk.
For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre.[1] In the 1980s, Vyacheslav Girko introduced[2] an approach which allowed to establish the circular law for more general distributions. Further progress was made[3] by Zhidong Bai, who established the circular law under certain smoothness assumptions on the distribution.
The assumptions were further relaxed in the works of Terence Tao and Van H. Vu,[4] Guangming Pan and Wang Zhou,[5] and Friedrich Götze and Alexander Tikhomirov.[6] Finally, in 2010 Tao and Vu proved[7] the circular law under the minimal assumptions stated above.
The circular law result was extended in 1988 by Sommers, Crisanti, Sompolinsky and Stein to an elliptical law for ensembles of matrices with arbitrary correlations.[8] The elliptic and circular laws were further generalized by Aceituno, Rogers and Schomerus to the hypotrochoid law which includes higher order correlations.[9]
^ Ginibre, Jean (1965). "Statistical ensembles of complex, quaternion, and real matrices". J. Math. Phys. 6: 440–449. Bibcode:1965JMP.....6..440G. doi:10.1063/1.1704292. MR 0173726.
^ Girko, V.L. (1984). "The circular law". Teoriya Veroyatnostei i ee Primeneniya. 29 (4): 669–679.
^ Bai, Z.D. (1997). "Circular law". Annals of Probability. 25 (1): 494–529. doi:10.1214/aop/1024404298. MR 1428519.
^ Tao, T.; Vu, V.H. (2008). "Random matrices: the circular law". Commun. Contemp. Math. 10 (2): 261–307. arXiv:0708.2895. doi:10.1142/s0219199708002788. MR 2409368.
^ Pan, G.; Zhou, W. (2010). "Circular law, extreme singular values and potential theory". J. Multivariate Anal. 101 (3): 645–656. arXiv:0705.3773. doi:10.1016/j.jmva.2009.08.005.
^ Götze, F.; Tikhomirov, A. (2010). "The circular law for random matrices". Annals of Probability. 38 (4): 1444–1491. arXiv:0709.3995. doi:10.1214/09-aop522. MR 2663633.
^ Tao, Terence; Vu, Van (2010). appendix by Manjunath Krishnapur. "Random matrices: Universality of ESD and the Circular Law". Annals of Probability. 38 (5): 2023–2065. arXiv:0807.4898. doi:10.1214/10-AOP534. MR 2722794.
^ Sommers, H.J.; Crisanti, A.; Sompolinsky, H.; Stein, Y. (1988). "Spectrum of Large Asymmetric Matrices". Physical Review Letters. 60 (19): 1895–1898.
^ Aceituno, P.V.; Rogers, T.; Schomerus, H. (2019). "Universal hypotrochoidic law for random matrices with cyclic correlations". Physical Review E. 100 (1): 010302.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Circular_law&oldid=1015007282"
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Zhan, S. , Shu, Z. and Jiang, H. (2018) Research on Two-Echelon Green Supply Chain Decision under Government Subsidy. American Journal of Industrial and Business Management, 8, 487-495. doi: 10.4236/ajibm.2018.83032.
D=a-bp+\alpha \theta
\xi {\theta }^{2}/2
{\pi }_{r}=\left(p-w\right)D=\left(p-w\right)\left(a-bp+\alpha \theta \right)
{\pi }_{m}=\left(w-c+r\theta \right)\left(a-bp+\alpha \theta \right)-\beta {\theta }^{2}
{p}_{d}^{\ast }=\left(a+\alpha \theta +bw\right)/2b
{w}_{d}^{\ast }=\frac{bc+a}{2b}+\frac{2b\left(br-\alpha \right)\left(ar-\alpha c\right)-{\left(br-\alpha \right)}^{2}\left(bc+a\right)}{2b{\left(br-\alpha \right)}^{2}+8{b}^{2}\left(\alpha r-\beta \right)}
{\theta }_{d}^{\ast }=\left[\left(br-\alpha \right)\left(bc+a\right)+2b\left(\alpha c-ar\right)\right]/\left[{\left(br-\alpha \right)}^{2}+4b\left(\alpha r-\beta \right)\right]
{p}_{d}^{\ast }=\frac{bc+3a}{4b}+\frac{\left(3\alpha -br\right)\left(br-\alpha \right)\left(bc+a\right)+\left(2{b}^{2}r-6\alpha b\right)\left(ar-\alpha c\right)}{4b{\left(br-\alpha \right)}^{2}+16{b}^{2}\left(\alpha r-\beta \right)}
{\pi }_{sc}=\left(p-c+r\theta \right)D-\beta {\theta }^{2}/2=\left(p-c+r\theta \right)\left(a-bp+\alpha \theta \right)-\beta {\theta }^{2}/2
{p}_{c}^{\ast }=\frac{a+bc}{2b}-\frac{{\left(br-\alpha \right)}^{2}\left(a+bc\right)+2b\left(br-\alpha \right)\left(\alpha c-ar\right)}{2b{\left(br-\alpha \right)}^{2}+4{b}^{2}\left(2\alpha r-\beta \right)}
{\theta }_{c}^{\ast }=\left[\left(br-\alpha \right)\left(a+bc\right)+2b\left(\alpha c-ar\right)\right]/\left[{\left(br-\alpha \right)}^{2}+2b\left(2\alpha r-\beta \right)\right]
{\pi }_{r}^{c}=\left(p-w\right)\left(a-bp+\alpha \theta \right)-F
{\pi }_{m}^{c}=\left(w-c+r\theta \right)\left(a-bp+\alpha \theta \right)-\beta {\theta }^{2}/2+F
{w}^{c}=\left[bcr\alpha -2bc\beta +2b{r}^{2}+c{\alpha }^{2}+a\alpha r\right]/\left[{\left(br-\alpha \right)}^{2}+2b\left(2\alpha r-\beta \right)\right]
{w}^{c}\le {w}_{d}^{\ast }\le {p}_{d}^{\ast }
{\pi }_{m}^{0}
{\pi }_{r}^{0}
{\pi }_{m}^{c}\ge {\pi }_{m}^{0}
F\ge {F}_{1}={\pi }_{m}^{0}-\left({w}^{C}-c+r{\theta }_{C}^{*}\right)\left(a-b{p}_{C}^{*}+\alpha {\theta }_{C}^{*}\right)-\beta {\theta }_{C}^{*}{}^{2}/2
{\pi }_{r}^{c}\ge {\pi }_{r}^{0}
F\le {F}_{2}=\left({p}_{C}^{*}-{w}^{C}\right)\left(a-b{p}_{C}^{*}+\alpha {\theta }_{C}^{*}\right)-{\pi }_{r}^{0}
[1] Menges, R. (2003) Supporting Renewable Energy on Liberalized Markets: Green Electricity between Additionality and Consumer Sovereignty. Energy Policy, 7, 583-596.
[2] Bowen, F.E., Cousins, P.D., Lamming, R.C., et al. (2001) The Role of Supply Management Capabilities in Green Supply. Production and Operations Management, 10, 174-189.
[3] Li, Y.N. and Ye, F. (2011) Institutional Pressures, Environmental Innovation Practices and Firm Performance—An Institutional Theory and Ecological Modernization Theory Perspective. Studies in Science of Science, 29, 1884-1894.
[4] Eriksson, C. (2004) Can Green Consumerism Replace Environmental Regulation? A Differentiated-Products Example. Resource and Energy Economics, 3, 281-293.
[5] Zhu, Q.H. and Dou, Y.J. (2007) An Evolutionary Model between Governments and Core-Enterprises in Green Supply Chains. Systems Engineering Theory and Practice, 12, 85-89.
[6] Cohen, M.C., Lobel, R. and Perakis, G. (2015) The Impact of Demand Uncertainty on Consumer Subsidies for Green Technology Adoption. Management Science, 9, 1-24.
[7] Conrad, K. (2005) Price Competition and Product Differentiation When Consumers Care for the Environment. Environment and Resource Economics, 31, 1-19.
[8] Liu, Z., Anderson, T.D. and Cruz, J.M. (2012) Consumer Environmental Awareness and Competition in Two-Stage Supply Chains. European Journal of Operational Research, 218, 602-613.
[9] Zhang, L.H., Wang, J.G. and You, J.X. (2015) Consumer Environmental Awareness and Channel Coordination with Two Substitutable Products. European Journal of Operational Research, 241, 63-73.
[10] Mitra, S. and Webster, S. (2008) Competition in Remanufacturing and the Effect of Government Subsidies. International Journal of Production Economics, 111, 287-298.
[11] Cohen, M.C., Perakis, G. and Thraves, C. (2015) Competition and Externalities in Green Technology Adoption. Working Paper. Massachusetts Institute of Technology, Cambridge.
[12] Ling, L.Y., Dong, H.X. and Liang, L. (2012) Analysis of Monopoly Market for Green Products from the Perspective of Government Subsidies. Operations Research and Management Science, 21, 139-144.
[13] Savaskan, R.C. and Van Wassenhove, L.N. (2006) Reverse Channel Design: The Case of Competing Retailers. Management Science, 52, 1-14.
[14] Swami, S. and Shah, J. (2013) Channel Coordination in Green Supply Chain Management. Journal of the Operational Research Society, 64, 336-351.
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Mighty Max started the wrestling season weighing
135
pounds. By the time the season ended, he weighed
128
pounds. What was the percent decrease in his weight? That is, what was his decrease in weight as a percent of his starting weight?
Find the value of the decrease as a percent of the original weight.
135 − 128 = 7 \text{ pounds}
7 = \left(x\right)\left(135\right)
5\%
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The Hask Category / Benjamin R. Bray
math category-theory functional-programming haskell
The Category Hask
(Hask is Not a Category)
(Towards a Solution)
(Truth from Lies)
(Hask is Not Cartesian Closed)
Endofunctors in Hask
Endofunctors, Categorically
Expressing Functors in Haskell
Aside: Endofunctors Enriched in Hask
Aside: Desugaring Functor
Natural Transformations in Hask
Examples of Natural Transformations in Hask
Yoneda Lemma in Hask
Monads in Hask
Exponentials in Hask
(note: this page is not original research, but rather a synthesis of the best explanations I could find online for each of these topics individually)
The Haskell Wikibook is written from the perspective of someone who knows a bit of Haskell and wants to know how it connects to category theory. Instead, I'm someone who knows a bit of category theory and wants to know how it connects to Haskell.
Functional programming concepts like Functor and Monad do not always correspond in an obvious way to their category theory counterparts. These notes aim to clarify the relationship between Haskell syntax and category theory.
Haskellers like to imagine that there is a category Hask such that:
objects in Hask are concrete data types of kind *.
morphisms in Hask are Haskell functions (which are values). For two concrete types A and B, the hom-set Hom(A,B) is the set of functions with signature A -> B.
function composition is given by f . g
the [[polymorphic]] function id provides an identity morphism id :: A -> A for every data type A
The rest of this section explains why Hask is not actually a category.
We follow the explanation by user K. A. Buhr on StackOverflow. According to HaskellWiki/Hask, the magically-strict seqfunctionallows us to construct morphisms which violate the category laws.
Consider their monomorphic specializations to the type () -> (),
undef1, undef2 :: () -> ()
undef1 = undefined
It turns out that undef1 . id = undef2, so if the morphisms Hask are to be Haskell functions, the category laws mandate that undef1 and undef2 should refer to the same Haskell function. How can we check?
In Haskell, primitives like Integer can be directly compared. For compound types, we normally think of two values as being different if we can construct an expression that witnesses a difference in directly comparable values:
-- 4 is a witness to the fact that sqrt =/= id, allowing
-- us to conclude `sqrt` and `id` are different values
id 4 = 4
Similarly, seq witnesses the difference between undef1 and undef2:
seq undef1 ()
-- use defn of undef1
= seq undefined ()
-- seq semantics: WHNF of undefined is _|_, so value is _|_
= _|_
= seq (\_ -> undefined) ()
-- seq semantics: (\_ -> undefined) is already in WHNF
-- and is not _|_, so value is second arg ()
Since undef1 . id = undef2, but undef1 ≠ undef2, our category law is violated.
So, the wiki proposes instead that morphisms in Hask should be equivalence classes of Haskell functions, where f and g equivalent iff f x = g x for all inputs x.
However, Andrej Bauer points out that due to the lack of a formal [[operational-semantics]] for Haskell, there is no rigorous way to determine if two expressions (therefore two morphisms) are equal. This idea is explained further by user K. A. Buhr on StackOverflow.
The Haskell Wikibook proposes a different solution, which involves redefining function composition to be strict:
We can define a new strict composition function, f .! g = ((.) $! f) $! g, that makes Hask a category. We proceed by using the normal (.), though, and attribute any discrepancies to the fact that seq breaks an awful lot of the nice language properties anyway.
A reddit comment in response to Andrej Bauer's post proposes this:
Hask is this category:
The objects are types of kind *.
For objects A, B, the hom-set hom(A,B) is the quotient (A -> B) / equal3{A,B}, where A -> B is the set of values of that type.
For object A, the identity morphism is the equal3{A,A} equivalence class containing (\a -> a) :: A -> A.
For objects A, B, C, and morphisms F ∈ hom(A,B), G ∈ hom(B,C), the composition G . F is the equal3{A,C} equivalence class containing \a -> g (f a), where f ∈ F, g ∈ G.
For the above reasons, among others, we cannot confidently say that Hask is a genuine category. Nevertheless, many constructs in Haskell are inspired by category theory, and treating Hask like a genuine category can lend some clarity to their definitions.
Moreover, fast and loose reasoning is morally correct:
@[danielsson2006_fast-loose-morally-correct]
Putting aside that Hask isn't even a category, the wiki points out a number of technical limitations prevent Hask from being a [[cartesian-closed-category]] as well.
it does not have sums, products, or an initial object
() is not a terminal object
the monad identities fail for almost all instances of Monad
Because of these difficulties, Haskell developers tend to think in some subset of Haskell (PlatonicHask) where types do not have [[bottom]] values.
PlatonicHask only includes functions that terminate, and typically only finite values
the corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads
A [[functor]] is a map between categories that preserves categorical structure. In Haskell, we care most about [[endofunctors]], from the category Hask to itself.
An endofunctor F : Hask -> Hask in the category Hask assigns
to each type A in Hask a type F A in Hask.
to each Haskell function f : A -> B a function F f : F A -> F B
such that F (id :: A) = (id :: F A)
such that F ( g . f ) = (Fg) . (Ff) for f : x -> y, g : y -> z
paraphrased from leftaroundabout on StackOverflow
Mathematically speaking, a functor is a [[dependent]] pair, where
the object map Type -> Type lives in Haskell's type-level world
the morphism map (a -> b) -> f a -> f b lives in the value world
Haskell doesn't have a way of specifying such dependent pairs. The Functor typeclass tricks its way around this limitation by allowing the use of type constructors as the Type -> Type object mapping.
This helps, because type constructors are unique. Every one of them can be assigned a well-defined morphism-mapping through Haskell's typeclass mechanism.
However, the image of a Functor is always the proper (full?) subcategory of types in the image of the type constructor, which prevents us from specifying arbitrary object mappings, as we might with a type synonym.
IThe takeaway is that there are some (endo)functors in Hask which cannot be captured by a Functor instance.
Example. (Identity Functor) For example, the identityFunctor cannot map a type directly to itself. Instead, it sends each type to a wrapped, isomorphic copy:
newtype Identity a = Identity { unwrap :: a }
So, Identity isn't, strictly speaking, the identity functor. Instead, the accessor unwrap :: Identity a -> a is a natural isomorphism from Identity to the true, category-theoretic identity functor, which is expressible in Haskell as a type alias:
Type aliases in Haskell can't be [[typeclass]] instances, so we have to work with the naturally isomorphic Identity functor instead.
Remark. Haskell does have a TypeSynonymInstances extension, which might not work as expected without also enabling FlexibleInstances and OverlappingInstances. See an answer by mb14 on StackOverflow.
Example. (Const Functor) Similarly, a [[const-functor]] is naturally isomorphic to the constant functor we normally think of in category theory.
type CF m a = m
-- we have a constant functor
-- with object mapping CF m
-- with morphism mapping id @m (type application syntax)
-- but no way to pair them together!
Due to Haskell's limitations, though, we end up in a situation where Const Int Char and Const Int Bool are technically speaking different, albeit isomorphic, types.
Every Functor in Haskell is an endofunctors from Hask to Hask.
If you're wondering why we specify an fmap for morphisms but apparently no corresponding omap for objects, you're not alone!
SO, "Why Functor class has no return function?"
SO, "Why is pure required for Applicative and not for Functor?"
SO, "Why does the Functor class in Haskell not include a function on objects? Is pure that function?"
SO, "How are functors in Haskell related to functors in category theory?"
Here are two example instances of Functor:
fmap [] = []
fmap f (x:xs) = (f x) : (fmap xs)
-- Maybe sends type T to (Maybe T)
Notice that the type constructor Maybe sends each type T to a new type Maybe T. Hence the object map for the ******************Maybe******************** functor is the type constructor itself!**
-- this isn't a real signature
Maybe :: T -> Maybe T
From an answer by Eduardo Pareja Tobes on StackOverflow:
One important point about this is that what you really want is functors [[enriched]] in Hask, not just plain old functors. Hask is cartesian closed (not really, but it tries hard to be onesuch), and so it is naturally enriched in itself.
Now, [[enriched-endofunctors]] give you a way of restricting to those implementable within the language:
an enriched functor Hask -> Hask is a function at the level of objects (types) f a and for each pair of objects a, b a morphism in Hask going f : ***********Hask************(a,b) -> ************Hask************(fa,fb)*. Of course, this is just fmap :: (a -> b) -> f a -> f b
We can desugar the typeclass syntax according to the rules described in @[peyton-jones2011_classes-not-as-we-know-them], giving
-- to make a functor instance, we just need an fmap
data Functor f
= MkFunctor ((a -> b) -> f a -> f b)
-- given a Functor, selects the correct fmap function
fmap :: Num a -> ((a -> b) -> f a -> f b)
fmap (MkFunctor m) = m
-- an instance declaration
dFunctorList :: Functor []
dFunctorList = MkFunctor map where
map [] = []
map f (x:xs) = (f x) : (fmap xs)
Following Milewski, "CTfP, Natural Transformations"
@[fong2020_cats4progs] Chapter 3
Milewski, "Parametricity: Money for Nothing and Theorems for Free"
Given functors
F
G
, a natural transformation
\alpha : F \Rightarrow G
assigns
to each datatype
x \in \Hask
a Haskell funcion
\alpha_x : Fx \rightarrow Gx
\alpha_y \circ Ff = Gf \circ \alpha_x
for all functions
f : x \rightarrow y
In Haskell, a [[natural transformation]] between functors f and g is just a [[polymorphic]] function:
type Nat f g = forall a. f a -> g a
Every polymorphic function already gives a map between datatypes. As a consequence of [[parametric polymorphism]], a Haskell function of the type below automatically obeys the natural transformation law!
-- a family of functions parameterized by `a`
alpha :: forall a. F a -> G a
To see why, recall that the action of a Haskell Functor F on a function f is implemented with fmap :: (a -> b) -> (F a -> F b). So, the natural transformation law above can be rewritten as
-- given functors F, G from Hask => Hask
-- for any f :: x -> y, we require that
alpha . (fmap f) = (fmap f) . alpha
-- (both sides of the equation have type Fx -> Gy)
Where parametric polymorphism allows us to omit subscripts.
[[TODO]]: Finish this explanation of why parametrically polymorphic functions are automatically natural transformations. See @[reynolds1972-definitional-interpreters-higher-order-pl], @[wadler1989-theorems-for-free], and a post by Milewski.
Q: is parametric polymorphism related to a universal property? see e.g. @[ghani2015_parametric-polymorphism-universally]
Example. The parametrically polymorphic function safeHead is a natural transformation from the List functor to the Maybe functor.
-- naturality condition
fmap f . safeHead = safeHead . fmap f
Example. A natural transformation from or to a [[const-functor]] looks just like a function that's either polymorphic in its return type or in its argument type.
newtype Const a b = Const a
fmap :: (a -> b) -> Const a -> Const b
fmap _ (Const a) = Const (f b)
see [[yoneda-lemma]] for examples of Yoneda and coYoneda in Hask
Following the Haskell Wikibook, a [[monad]] in the category
C
is an endofunctor
M : C \rightarrow C
together with two natural transformations
\begin{aligned} \mathrm{unit}^M &: 1_C \rightarrow M \\ \mathrm{join}^M &: M \circ M \rightarrow M \end{aligned}
This translates to the following Haskell typeclass:
Hask is an endofunctor together with two morphisms for each
see [[monoid-in-the-category-of-endofunctors]]
An [[exponential]] is a universal function object.
see Milewski, "CTfP: Function Types"
coends?
Category Hask
Haskell/CategoryTheory
Haskell Wiki, "Hask"
Andrej Bauer 2016, "Hask is not a Category"
StackOverflow, Is Hask Even a Category?
Makoto Hamana 2007, "What is the Category for Haskell?"
Fong, "C4P: Is Haskell a Category?"
StackOverflow, "Do all Type Classes in Haskell Have a Category-Theoretic Analogue?"
StackOverflow, "Where to values fit in Category of Hask?"
talks about a different category where values are objects and functions are either morphisms or functors
[my question] StackOverflow, "Is Haskell'sConstFunctor analogous to the constant functor from category theory?"
StackOverflow, "Why Functor class has no return function?"
StackOverflow, "What is a natural transformation in Haskell?"
Wikipedia, "Monad (Category Theory)"
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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Constructors : Copy
construct a copy of a Matrix or Vector
Copy(MV)
Matrix or Vector to copy
The Copy(MV) command returns an identical copy of its argument, MV. Because this copy is not simply another reference to MV, changes to MV or to the copy do not affect the other.
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
v≔\mathrm{Vector}[\mathrm{row}]\left([1,2,3],\mathrm{datatype}=\mathrm{float}[8],\mathrm{attributes}=[\mathrm{blue}]\right)
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}\end{array}]
w≔\mathrm{Copy}\left(v\right)
\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}\end{array}]
\mathrm{attributes}\left(w\right)
\textcolor[rgb]{0,0,1}{\mathrm{blue}}
w[1]≔10:
v[1]
\textcolor[rgb]{0,0,1}{1.}
M≔\mathrm{Matrix}\left([[1,2,3],[4,5],[6]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper}]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
N≔\mathrm{Matrix}\left(M\right)
\textcolor[rgb]{0,0,1}{N}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{MatrixOptions}\left(M\right)
\textcolor[rgb]{0,0,1}{\mathrm{shape}}\textcolor[rgb]{0,0,1}{=}[{\textcolor[rgb]{0,0,1}{\mathrm{triangular}}}_{\textcolor[rgb]{0,0,1}{\mathrm{upper}}}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{datatype}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{storage}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{triangular}}}_{\textcolor[rgb]{0,0,1}{\mathrm{upper}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{order}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}
\mathrm{MatrixOptions}\left(N\right)
\textcolor[rgb]{0,0,1}{\mathrm{shape}}\textcolor[rgb]{0,0,1}{=}[]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{datatype}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{storage}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{rectangular}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{order}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}
P≔\mathrm{Copy}\left(M\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{MatrixOptions}\left(P\right)
\textcolor[rgb]{0,0,1}{\mathrm{shape}}\textcolor[rgb]{0,0,1}{=}[{\textcolor[rgb]{0,0,1}{\mathrm{triangular}}}_{\textcolor[rgb]{0,0,1}{\mathrm{upper}}}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{datatype}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{storage}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathrm{triangular}}}_{\textcolor[rgb]{0,0,1}{\mathrm{upper}}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{order}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}
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Potential temperature - Wikipedia
The potential temperature of a parcel of fluid at pressure
{\displaystyle P}
is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure
{\displaystyle P_{0}}
, usually 1,000 hPa (1,000 mb). The potential temperature is denoted
{\displaystyle \theta }
and, for a gas well-approximated as ideal, is given by
{\displaystyle \theta =T\left({\frac {P_{0}}{P}}\right)^{R/c_{p}},}
{\displaystyle T}
is the current absolute temperature (in K) of the parcel,
{\displaystyle R}
is the gas constant of air, and
{\displaystyle c_{p}}
is the specific heat capacity at a constant pressure.
{\displaystyle R/c_{p}=0.286}
for air (meteorology).
3 Potential temperature perturbations
5 Potential virtual temperature
The concept of potential temperature applies to any stratified fluid. It is most frequently used in the atmospheric sciences and oceanography.[1] The reason that it is used in both fields is that changes in pressure can result in warmer fluid residing under colder fluid – examples being dropping air temperature with altitude and increasing water temperature with depth in very deep ocean trenches and within the ocean mixed layer. When potential temperature is used instead, these apparently unstable conditions vanish as a parcel of fluid is invariant along its isolines.
Potential temperature is a more dynamically important quantity than the actual temperature. This is because it is not affected by the physical lifting or sinking associated with flow over obstacles or large-scale atmospheric turbulence. A parcel of air moving over a small mountain will expand and cool as it ascends the slope, then compress and warm as it descends on the other side- but the potential temperature will not change in the absence of heating, cooling, evaporation, or condensation (processes that exclude these effects are referred to as dry adiabatic). Since parcels with the same potential temperature can be exchanged without work or heating being required, lines of constant potential temperature are natural flow pathways.
Under almost all circumstances, potential temperature increases upwards in the atmosphere, unlike actual temperature which may increase or decrease. Potential temperature is conserved for all dry adiabatic processes, and as such is an important quantity in the planetary boundary layer (which is often very close to being dry adiabatic).
Potential temperature and hydrostatic stability
Potential temperature is a useful measure of the static stability of the unsaturated atmosphere. Under normal, stably stratified conditions, the potential temperature increases with height,[2]
{\displaystyle {\frac {\partial \theta }{\partial z}}>0}
and vertical motions are suppressed. If the potential temperature decreases with height,[2]
{\displaystyle {\frac {\partial \theta }{\partial z}}<0}
the atmosphere is unstable to vertical motions, and convection is likely. Since convection acts to quickly mix the atmosphere and return to a stably stratified state, observations of decreasing potential temperature with height are uncommon, except while vigorous convection is underway or during periods of strong insolation. Situations in which the equivalent potential temperature decreases with height, indicating instability in saturated air, are much more common.
Since potential temperature is conserved under adiabatic or isentropic air motions, in steady, adiabatic flow lines or surfaces of constant potential temperature act as streamlines or flow surfaces, respectively. This fact is used in isentropic analysis, a form of synoptic analysis which allows visualization of air motions and in particular analysis of large-scale vertical motion.[2]
Potential temperature perturbationsEdit
The atmospheric boundary layer (ABL) potential temperature perturbation is defined as the difference between the potential temperature of the ABL and the potential temperature of the free atmosphere above the ABL. This value is called the potential temperature deficit in the case of a katabatic flow, because the surface will always be colder than the free atmosphere and the PT perturbation will be negative.
The enthalpy form of the first law of thermodynamics can be written as:
{\displaystyle dh=T\,ds+v\,dp,}
{\displaystyle dh}
denotes the enthalpy change,
{\displaystyle T}
the temperature,
{\displaystyle ds}
the change in entropy,
{\displaystyle v}
the specific volume, and
{\displaystyle p}
For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to:
{\displaystyle dh=v\,dp.}
For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state,
{\displaystyle pv=RT}
can be substituted into the 1st law yielding, after some rearrangement:
{\displaystyle {\frac {dp}{p}}={{\frac {c_{p}}{R}}{\frac {dT}{T}}},}
{\displaystyle dh=c_{p}dT}
was used and both terms were divided by the product
{\displaystyle pv}
{\displaystyle \left({\frac {p_{1}}{p_{0}}}\right)^{R/c_{p}}={\frac {T_{1}}{T_{0}}},}
{\displaystyle T_{0}}
, the temperature a parcel would acquire if moved adiabatically to the pressure level
{\displaystyle p_{0}}
{\displaystyle T_{0}=T_{1}\left({\frac {p_{0}}{p_{1}}}\right)^{R/c_{p}}\equiv \theta .}
Potential virtual temperatureEdit
The potential virtual temperature
{\displaystyle \theta _{v}}
{\displaystyle \theta _{v}=\theta \left(1+0.61r-r_{L}\right),}
is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. It is used as a practical substitute for density in buoyancy calculations. In this definition
{\displaystyle \theta }
is the potential temperature,
{\displaystyle r}
is the mixing ratio of water vapor, and
{\displaystyle r_{L}}
is the mixing ratio of liquid water in the air.
Related quantitiesEdit
The Brunt–Väisälä frequency is a closely related quantity that uses potential temperature and is used extensively in investigations of atmospheric stability.
^ Stewart, Robert H. (September 2008). "6.5: Density, Potential Temperature, and Neutral Density". Introduction To Physical Oceanography (pdf). Academia. pp. 83–88. Retrieved March 8, 2017.
^ a b c Dr. James T. Moore (Saint Louis University Dept. of Earth & Atmospheric Sciences) (August 5, 1999). "Isentropic Analysis Techniques: Basic Concepts" (pdf). COMET COMAP. Retrieved March 8, 2017.
M K Yau and R.R. Rogers, Short Course in Cloud Physics, Third Edition, published by Butterworth-Heinemann, January 1, 1989, 304 pages. ISBN 9780750632157 ISBN 0-7506-3215-1
Eric Weisstein's World of Physics at Wolfram Research
Retrieved from "https://en.wikipedia.org/w/index.php?title=Potential_temperature&oldid=1066375919"
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Dynamic load computational modelling of containers placed on a flat wagon at railroad ferry transportation | JVE Journals
The article presents result of computational modelling of container dynamic load during transportation as a part of trains of intermodal transport on a railway ferry. The computational models were developed which account the movement of the container with regard to the frame of the flat wagon while moving of the railway ferry. It was assumed that there is no movement of the flat wagon with regard to the deck, since these movements were limited by fastening means. The obtained acceleration rates, as components of the dynamic loads acting on the container, were accounted while determining the containers stability coefficient with regard to the flat wagons. The railway ferry heeling angles which ensure the stability of the containers were determined. The researches will ensure safety of transportation of containers as a part of trains of intermodal transport on a railway ferry, as well as increase the efficiency of operation of intermodal transport in the international transport.
Keywords: freight wagon, bearing structure, container, dynamic load, strength, railroad ferry.
The development of international economic activity between Eurasian countries enables the commissioning of competitive transport systems. To date the most priority among them is the container transportation. Container transportation as a part of intermodal trains was expanded to reduce the time of cargo delivery and transportation costs. These trains can run not only in relation to main-line railways, but also in international rail and water transportations involving railroad ferries. It is important to study the load of intermodal trains to ensure the safety of their transportation on railroad ferries, as conditions for the container transportation by sea are significantly different from the conditions of their operation relative to the main-line railways. Therefore, the article focuses on the issue of researching the dynamic load of containers during transportation as a part of intermodal trains on railroad ferries and determining the permissible heeling angles at which the stability of containers is ensured in relation to flat wagons.
The peculiarities of using simplified methods for measuring the stress-strain state of the body-container of variable volume are given in [1]. The scheme of load and test methods of body-containers for transverse and longitudinal swash is proposed in the work.
The peculiarities of invention a container for the transportation of fruit and vegetables are covered in [2]. The requirements for the body-container are given, its construction is proposed and the calculation of the strength using the finite element method is made in the article. It is important to note that the study of the dynamic load of containers is not carried out in these works, and the determination of strength indicators is carried out taking into account the regulatory values of loads.
An overview of the structure and properties of nanomaterials obtained by isostatic compression is carried out in [3]. Possibilities of the implementation of this material in bearing structures of vehicles for ensuring strength under operational load conditions is not carried out in the work.
The peculiarities of testing units of rolling stock on roller stand to determine their dynamic properties under operating conditions is carried out in [4]. The work does not specify the possibility of using this equipment to determine the dynamic load of containers during transportation on railroad ferries.
Measures concerning the improvement of the automatic coupling draw gear in order to reduce the dynamic load of wagons are highlighted in [5, 6]. The results obtained by mathematical modelling are confirmed by computer simulation of the dynamic load of wagons.
Identification of the substantiation peculiarities of the open wagon service life prolongation that have exhausted their normative resource is given in [7]. The study of the open wagon dynamic load and strength was carried out taking into account the actual wear values of the bearing structure elements of the open wagon in operation. The task of studying the dynamic load of bearing structures of containers under operating conditions in these works is not stated.
Peculiarities of the improvement of the open wagon bearing structure to ensure the reliability of its fastening on the deck of the railroad ferry are given in [8, 9]. The proposed technical solutions are confirmed by calculations on strength; the results are given in the article. The problems of determining the dynamic loading of containers during transportation as a part of intermodal trains on railroad ferries are not considered in the work.
Other advanced methods for dynamic analysis of mechanical structures, including examples of their application, are described in [10, 11].
3. Computational model of a container placed on a flat wagon
The aim of the research was to create a computational model describing the peculiarities on dynamic load of containers placed on a flat wagon during transportation by a railroad ferry. To achieve this goal, the following tasks were defined:
– to develop a mathematical model for determining the dynamic load of a container placed on a flat wagon during transportation by a railroad ferry;
– to determine the stability coefficient of the container placed on a flat wagon during transportation by railroad ferry.
The calculation scheme for researching the dynamic load of a container placed on a flat wagon during fluctuations of a railroad ferry is shown in Fig. 1.
Fig. 1. Calculating scheme for researching the dynamic load of a container placed on a flat wagon during fluctuations of a railroad ferry
The computational model was developed to determine the dynamic load of the container during transportation as a part of intermodal train on the railroad ferry. This model accounts the angular movements of the elements of the system (“railroad ferry – container”) around the longitudinal axis (roll), as a case of the greatest load of the bearing structure of the container during transportation by railroad ferry, as well as ensuring its stability with respect to the frame of the flat wagon. The model does account the movements of the flat wagon. That is, it is assumed that they are limited by fastening means relative to the deck.
Equations of motion for researching the dynamic load of a container placed on a flat wagon during fluctuations of a railroad ferry [12] can be written in form:
\left(\frac{D}{12g}\left({B}^{2}+4{z}_{g}^{2}\right)\right){\stackrel{¨}{\theta }}_{1}+\left({\mathrm{\Lambda }}_{\theta }\frac{B}{2}\right){\stackrel{˙}{\theta }}_{1}={p}_{FB}^{\text{'}}\frac{h}{2}+{\mathrm{\Lambda }}_{\theta }\frac{B}{2}F\left(t\right),
{I}_{K}^{\theta }{\stackrel{¨}{\theta }}_{2}={p}_{K}^{\text{'}}\frac{{h}_{K}}{2}+{M}_{K}^{D},
{\theta }_{1}
is a generalized coordinate which corresponds to the angular displacement around the longitudinal axis of a railroad ferry,
{\theta }_{2}
is a generalized coordinate which corresponds to the angular displacement around the longitudinal axis of the container. The coordinate system origin is located in the railroad ferry centre of mass.
For railroad ferry
D
is the weight displacement,
B
h
is the height of the board,
{\mathrm{\Lambda }}_{\theta }
is coefficient of resistance to fluctuations,
{z}_{g}
is coordinate of the gravity centre,
{p}_{FB}^{\text{'}}
is the wind load on the uncovered projection,
F\left(t\right)
is the time course of force that moves a railroad ferry with wagons located on its decks.
For the container
{I}_{K}^{\theta }
is the moment of inertia of the container,
{h}_{k}
is the height of the side surface of the container,
{p}_{K}^{\text{'}}
is wind load on the side of the container,
{M}_{K}^{D}
is force moment that occurs between the container and the deck at angular movements relative to the longitudinal axis.
The research has been conducted relating to the ferry “Geroi Shipki”, which moves along the Black Sea water area. The model 13-4012 was chosen as the basic model of the flat wagon and the model of 1СС with a gross mass of 24 tons was chosen as a basic model of the container. The hydrometeorological characteristics of the Black Sea area are determined according to the data given in [14]. The speed of the railroad ferry was assumed to be equal to its operating speed of 18.6 knots (9.57 m/s) and was accounted as permanent when moving along the sea area.
Differential equations of motion were solved applying the programming environment Mathcad [15, 16] by using the Runge-Kutta method.
It was established that the maximum acceleration value acting on the container is about 1.5 m/s² and occurs at the wave-to-course angles of 60 deg and 120 deg in relation to the hull of the railroad ferry. The total value of the acceleration acting on the container is 3.57 m/s² (0.36 g) taking into account the horizontal component of acceleration of free fall, caused by the angle of the railway train ferry. At the same time, the estimated heeling angle of the railroad ferry at a static wind action on the uncovered projection was 12.2 deg.
Studies of the equilibrium stability coefficient were performed at angular movements of the railroad ferry relative to the longitudinal axis in order to assess the stability of containers in relation to the frame of the flat wagon.
The accelerations were taken into account in determining the overturning moment. These accelerations were calculated using mathematical modelling and are the components of the dynamic load acting on the container.
To ensure the stability of the container’s equilibrium with regard to the frame of the flat wagon, the following condition must be fulfilled:
{k}_{c}=\frac{{M}_{rest}}{{M}_{ov}}\ge 1,
where the magnitude of the restoring moment
{M}_{rest}
is the magnitude of the overturning moment
{M}_{ov}
are determined by equations:
{M}_{ov}={p}_{K}^{\text{'}}\frac{{h}_{K}}{2}+{M}_{g}\left(g\mathrm{sin}{\theta }_{1}+{\stackrel{¨}{\theta }}_{2}\right)\frac{{h}_{K}}{2},
{M}_{rest}={P}_{g}\frac{{B}_{K}}{2}\mathrm{cos}{\theta }_{1}+{n}_{f}\left({M}_{g}\left(g\mathrm{sin}{\theta }_{1}+{\stackrel{¨}{\theta }}_{2}\right)\frac{{h}_{f}}{2},
{M}_{g}
{P}_{g}
{B}_{K}
is the width of the container,
{n}_{f}
is the number of fitting pieces which support the container at angular movements relative to the longitudinal axis and hf is height of the fitting piece.
The results of the calculation in the form of characteristic curve are shown in Fig. 2.
In this case, the threshold of stability is established when the magnitudes of the restoring and overturning moments are equal. The stability curve is indicated by blue, the trend line (black colour) is described by the equation:
{k}_{c}=0.0055{{\theta }_{1}}^{2}-0.4006{\theta }_{1}+8.014.
The conducted studies allowed us to conclude that the stability of the container under this calculation scheme was provided at the corners of the railroad ferry up to 30 deg taking into account the possible movements of container fittings with respect to the fitting of the flat wagon.
Fig. 2. Dependence of the stability coefficient of the container placed on a flat wagon on the heeling angle of the railroad ferry
The computational model for determining the dynamic load of a container placed on a flat wagon during transportation by a railroad ferry was developed. It was stated that the maximum acceleration value acting on the container is about 1.5 m/s2 and occurs at the wave-to-course angles in relation to the hull of the railroad ferry of 60 deg and 120 deg. The total value of the acceleration acting on the container, taking into account the horizontal component of acceleration of free fall is 3.57 m/s² (0.36 g).
The stability coefficient of the container placed on the flat wagon during transportation by a railroad ferry was determined. Taking into account possible movements of container fittings with respect to fittings of the flat wagon, the container stability is provided at the railroad ferry heeling angle up to 30 deg.
The conducted research will provide the development of recommendations on the safety of container transportation on railroad ferries by sea, clarification of existing regulatory documentation on the design and calculation of vehicles for transportation on railroad ferries [17-19], as well as increase the efficiency of intermodal transport operations.
Mishuta D. V., et al. Simplified methods for measuring the stress-strain state of a container body of variable volume. Devices and Measurement Methods, Vol. 2, Issue 5, 2012, p. 100-103, (in Russian). [Search CrossRef]
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Fomin O., et al. Research of the strength of the bearing structure of the flat wagon body from round pipes during transportation on the railway ferry. MATEC Web of Conferences, Vol. 235, 2018, p. 00003. [Publisher]
Tkachenko V., et al. Research of resistance to the motion of vehicles related to the direction by railway. Eastern-European Journal of Enterprise Technologies, Vol. 5, Issues 7-89, 2017, p. 65-72. [Publisher]
Okorokov A. M, et al. Research into a possibility to prolong the time of operation of universal semi-wagon bodies that have exhausted their standard resource. Eastern-European Journal of Enterprise Technologies, Vol. 3/7, Issues 93, 2018, p. 20-26. [Publisher]
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Lovska A. Simulation of loads on the carrying structure of an articulated flat car in combined transportation. International Journal of Engineering and Technology, Vol. 7, Issues 4, 2018, p. 140-146. [Publisher]
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Roman Varbanets, Oleksij Fomin, Václav Píštěk, Valentyn Klymenko, Dmytro Minchev, Alexander Khrulev, Vitalii Zalozh, Pavel Kučera
Oleksandr Plakhtii, Volodymyr Nerubatskyi, Denys Hordiienko, Yakiv Scherbak, Svitlana Podnebenna, Andriy Synyavskyi
Yakiv Scherbak, Oleksandr Plakhtii, Volodymyr Nerubatskyi, Denys Hordiienko, Dmytro Shelest, Yuryi Semenenko
Determination of residual resource of flat wagons load-bearing structures with a 25-year service life
O Fomin, G Vatulia, M Horbunov, A Lovska, V. Píštěk, P. Kučera
Improvements in passenger car body for higher stability of train ferry
Oleksij Fomin, Alyona Lovska
Dynamic load of the carrying structure of an articulated wagon with new draft gear concepts
Dynamic Load Modelling within Combined Transport Trains during Transportation on a Railway Ferry
Jan Voparil, Ales Prokop, Kamil Rehak
Calculation of Loads on Carrying Structures of Articulated Circular-Tube Wagons Equipped with New Draft Gear Concepts
Alyona Lovska, Oleksij Fomin, Pavel Kučera, Václav Píštěk
Lubomir Drapal, Jozef Dlugos, Jan Voparil
The dynamics and strength of the carrying structure of a flat wagon while conducting fire from it
O Fomin, A Lovska, Ya Kichuk, N Urum
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Advances in Aging Research > Vol.7 No.3, May 2018
Establishing the Reliability and Validity of Health in Motion© Automated Falls Screening Tool ()
1School of Occupational Therapy, University of Indianapolis, Indianapolis, IN, USA.
2Blue Marble Health Company, Altadena, CA, USA.
Introduction: Blue Marble Health Company has created a digital fall risk screening tool (Health in Motion©) that can be used by means of self-report (touch/mouse) or by means of motion capture (Microsoft Kinect Sensor). Health in Motion© consists of automated versions of the Fall Risk Question-naire, 30-Second Chair Stand Test, and the One Leg Stance Test. Methods: We compared the three methods (self-report, sensor, and clinical standard measurement) using stopwatch and observation in 15 community-dwelling older adults, aged 63 - 80 years old. Each version was completed three times each in random order, for a total of nine trials. Results: Health in Motion© falls screening tool accessible via self-report and sensor is a valid and reliable automated at-home self-assessment for falls risk. Conclusion: Results support the use of Health in Motion© falls screening tools as viable alternatives to standard falls risk assessments for use by older adults at home.
Falls Prevention, Falls Screening, Self-Assessment, Mobile Health Technology, Reliability, Validity
\text{ICC}\left(3,1\right)=\frac{\text{BMS}-\text{EMS}}{\text{BMS}+\left(k-1\right)\text{EMS}}
\text{SEM}=\sqrt{\text{MSE}}
{\text{MDC}}_{95}=\text{SEM}\times \sqrt{2}\times 1.96
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On Separable Higher Gauss Maps
August 2019 On Separable Higher Gauss Maps
Michigan Math. J. 68(3): 483-503 (August 2019). DOI: 10.1307/mmj/1555574416
m
th Gauss map in the sense of F. L. Zak of a projective variety
X\subset {\mathbb{P}}^{N}
over an algebraically closed field in any characteristic. For all integers
m
n:=dim\left(X\right)⩽m<N
, we show that the contact locus on
X
of a general tangent
m
-plane is a linear variety if the
m
th Gauss map is separable. We also show that for smooth
X
n<N-2
\left(n+1\right)
th Gauss map is birational if it is separable, unless
X
is the Segre embedding
{\mathbb{P}}^{1}×{\mathbb{P}}^{n}\subset {\mathbb{P}}^{2n-1}
. This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.
Katsuhisa Furukawa. Atsushi Ito. "On Separable Higher Gauss Maps." Michigan Math. J. 68 (3) 483 - 503, August 2019. https://doi.org/10.1307/mmj/1555574416
Received: 24 June 2017; Revised: 30 October 2017; Published: August 2019
Digital Object Identifier: 10.1307/mmj/1555574416
Katsuhisa Furukawa, Atsushi Ito "On Separable Higher Gauss Maps," Michigan Mathematical Journal, Michigan Math. J. 68(3), 483-503, (August 2019)
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The command RegularizeInitial(f, rc, R) returns a list of pairs
[{f}_{i},{\mathrm{rc}}_{i}]
{\mathrm{rc}}_{i}
is a regular chain of R and each
{f}_{i}
is a polynomial of R.
The set of all the regular chains
{\mathrm{rc}}_{i}
form a decomposition of in_rc in the sense of Kalkbrener.
Each polynomial
{f}_{i}
is either constant or its initial is regular modulo
{\mathrm{rc}}_{i}
{f}_{i}
{\mathrm{rc}}_{i}
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
R≔\mathrm{PolynomialRing}\left([x,y,z]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
T≔\mathrm{Empty}\left(R\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
T≔\mathrm{Chain}\left([\left(z+1\right)\left(z+2\right),{y}^{2}+z,\left(x-z\right)\left(x-y\right)],T,R\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}
\mathrm{rtl}≔\mathrm{RegularizeInitial}\left(\left(z+1\right)\left({x}^{3}+5\right),T,R\right)
\textcolor[rgb]{0,0,1}{\mathrm{rtl}}\textcolor[rgb]{0,0,1}{≔}[[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{regular_chain}}]]
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{nops}\left(\mathrm{rtl}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{Equations}\left(\mathrm{rtl}[i][2],R\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{IsRegular}\left(\mathrm{rtl}[i][1],\mathrm{rtl}[i][2],R\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Monotone Iterative Technique and Symmetric Positive Solutions to Fourth-Order Boundary Value Problem with Integral Boundary Conditions
Huihui Pang, Chen Cai, "Monotone Iterative Technique and Symmetric Positive Solutions to Fourth-Order Boundary Value Problem with Integral Boundary Conditions", Discrete Dynamics in Nature and Society, vol. 2014, Article ID 583875, 7 pages, 2014. https://doi.org/10.1155/2014/583875
Huihui Pang1 and Chen Cai1
1College of Science, China Agricultural University, Beijing 100083, China
The purpose of this paper is to investigate the existence of symmetric positive solutions for a class of fourth-order boundary value problem: , , , , where , . By using a monotone iterative technique, we prove that the above boundary value problem has symmetric positive solutions under certain conditions. In particular, these solutions are obtained via the iteration procedures.
The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourth-order ordinary differential equation BVP (short for boundary value problem). At present, two-point situation of fourth-order BVP has been studied by many authors, generally using the nonlinear alternatives of Leray-Schauder, the fixed point index theory, and the method of upper and lower solutions, monotone iteration; see [1–6].
Recently, problems with integral boundary value conditions arise naturally in thermal conduction problems [7], semiconductor problems [8], and hydrodynamic problems [9]. Hence, the existence results of positive solutions to this kind of problems have received a great deal of attentions. We refer the readers to [10–15].
In [13], Ma studied the following problem: where and and are continuous. The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones.
In [14], authors study the existence and nonexistence of symmetric positive solutions of the following fourth-order BVP: The argument was based on the fixed point theory in cones.
For fourth-order differential equation subject to boundary value conditions (2), author in [15] established the existence of positive solutions by the use of the Krasnoseliis fixed point theorem in cone.
The existing literature indicates that researches of fourth-order two point BVPs are excellent and methods are developed to be various. However, as to fourth-order BVPs with integral boundary value conditions, methods applied are relatively limited. Most of results are obtained by the use of fixed point theory in the cone or the fix point index theorem.
In this paper, we will apply the monotone iterative technique to the following fourth-order BVP with integral boundary conditions: We do not assume that the upper and lower solutions to the boundary value problem should exist but construct the specific form of the symmetric upper and lower solutions. And we will construct successive iterative schemes for approximating solutions. In addition, it is worth stating that the first term of our iterative scheme is a simple function or a constant function. Therefore, the iterative scheme is feasible. Under the appropriate assumptions on nonlinear term, a new and general result to the existence of symmetric positive solution of BVP (5) and (6) is obtained.
We assume that the following conditions hold throughout the paper:(S1); (S2); (S3), , , .
Given , let and , . Denoted by , , the Green’s function of the following problem: Then, careful calculation yields
Lemma 1 (see [15]). Suppose that hold. Then, for any , solves the problem if and only if , where
During the process of getting the above solution, we can also know
Lemma 2. If is satisfied, the following results are true:(1), for , ;(2), for , .
Denote As , it is easy to check that and , for . Hence, from the symmetry and concavity of , we have In addition, for , the following results hold: Further, and therefore
We consider Banach space equipped with the norm , where . In this paper, a symmetric positive solution of (5) means a function which is symmetric and positive on and satisfies (5) as well as the boundary conditions (6).
In this paper, we always suppose that the following assumptions hold:(H1) for , , ;(H2), for , ;(H3), for .
Denote It is easy to see that is a cone in .
We define the operator as follows: By the above argument, we know that, for any , and
Lemma 3. If are satisfied, is completely continuous; that is, is continuous and compact.
Proof. For any , from (21) and (22), combining Lemma 2 and (), we know that and for . We now prove that is symmetric about .
For , So, . The continuity of is obvious. We now prove that is compact. Let be a bounded set. Then, there exists such that For any , we have Therefore, from (17) and (18), we have and from (19), we have So, is uniformly bounded. Next we prove that is equicontinuous.
For , we have where and According to the Lagrange mean value theorem, we obtain that Similarly, we have Hence, there exists a positive constant such that And the similar results can be obtained for and .
The Arzelà-Ascoli theorem guarantees that is relatively compact which means that is compact.
3. Existence and Iterative of Solutions for BVP (5) and (6)
Theorem 4. Assume that hold. If there exists two positive numbers such that where and satisfy Then, problem (5) and (6) has concave symmetric positive solution with where where
Proof. We denote . In what follows, we first prove that .
Let ; then , .
By assumption and (33), for , we have
For any , by Lemma 3, we know that . According to (17), (18), and (33), we get and from (19) and (33), we get
Hence, . Thus, we get . Let , for ; then and . Let ; then . We denote
From the definition of , (16), (18), and (38), it follows that On the other hand, from (15), (18), and (38), we have From , it follows that By induction,
Since , we have , . From Lemma 3, is completely continuous. We assert that has a convergent subsequence and there exists such that .
Let , ; then . Let ; then ; we denote
Similarly to , we assert that has a convergent subsequence and there exists , such that .
Since , we have Hence, By induction, , , , (). Hence, we assert that , .
If , , then the zero function is not the solution of BVP (5) and (6). Thus, ; we have
It is well known that the fixed point of operator is the solution of BVP (5) and (6). Therefore, and are two positive, concave, and symmetric solutions of BVP (5) and (6).
Example 5. Consider the following fourth-order boundary value problem with integral boundary conditions: where
The calculation yields It is easy to check that assumptions hold. Set , . Then we can verify that conditions and and (33) are satisfied. Then applying Theorem 4, BVP (50) has two concave symmetric positive solutions with where where
This research is supported by the Beijing Higher Education Young Elite Teacher Project (Project no. YETP0322) and Chinese Universities Scientific Fund (Project no. 2013QJ004).
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Copyright © 2014 Huihui Pang and Chen Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Rd Sharma 2018 for Class 10 Math Chapter 2 - Polynomials
Rd Sharma 2018 Solutions for Class 10 Math Chapter 2 Polynomials are provided here with simple step-by-step explanations. These solutions for Polynomials are extremely popular among Class 10 students for Math Polynomials Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2018 Book of Class 10 Math Chapter 2 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2018 Solutions. All Rd Sharma 2018 Solutions for class Class 10 Math are prepared by experts and are 100% accurate.
p\left(x\right)={x}^{2}+2\sqrt{2}x-6
q\left(x\right)=\sqrt{3}{x}^{2}+10x+7\sqrt{3}
f\left(x\right)={x}^{2}-\left(\sqrt{3}+1\right) x+\sqrt{3}
h\left(s\right)=2{s}^{2}-\left(1+2\sqrt{2}\right)s+\sqrt{2}
f\left(v\right)={v}^{2}+4\sqrt{3}v-15
p\left(y\right)={y}^{2}+\frac{3\sqrt{5}}{2}y-5
q\left(y\right)=7{y}^{2}-\frac{11}{3}y-\frac{2}{3}
h\left(t\right) = {t}^{2} - 15\phantom{\rule{0ex}{0ex}}h\left(t\right) = {\left(t\right)}^{2} - {\left(\sqrt{15}\right)}^{2}\phantom{\rule{0ex}{0ex}}h\left(t\right) = \left(t + \sqrt{15}\right) \left(t - \sqrt{15}\right)
h\left(t\right) = 0\phantom{\rule{0ex}{0ex}}\left(t - \sqrt{15}\right) \left(t + \sqrt{15}\right) = 0\phantom{\rule{0ex}{0ex}}\left(t - \sqrt{15}\right) = 0\phantom{\rule{0ex}{0ex}}t = \sqrt{15}\phantom{\rule{0ex}{0ex}}\mathrm{or} \phantom{\rule{0ex}{0ex}}\left(t + \sqrt{15}\right) = 0\phantom{\rule{0ex}{0ex}}t = -\sqrt{15}
\mathrm{Hence}, \mathrm{the} \mathrm{zeros} \mathrm{of} h\left(t\right) \mathrm{are} \alpha = \sqrt{15} \mathrm{and} \beta = - \sqrt{15}.
h\left(s\right)=2{s}^{2}-\left(1+2\sqrt{2}\right)s+\sqrt{2}
h\left(s\right)=2{s}^{2}-s-2\sqrt{2}s+\sqrt{2}\phantom{\rule{0ex}{0ex}}h\left(s\right)=s\left(2s-1\right)-\sqrt{2}\left(2s-1\right)\phantom{\rule{0ex}{0ex}}h\left(s\right)=\left(2s-1\right)\left(s-\sqrt{2}\right)
2{s}^{2}-\left(1+2\sqrt{2}\right)s+\sqrt{2}=0\phantom{\rule{0ex}{0ex}}\left(2s-1\right)\left(s-\sqrt{2}\right)=0\phantom{\rule{0ex}{0ex}}\left(2s-1\right)=0 \mathrm{or} \left(s-\sqrt{2}\right)=0\phantom{\rule{0ex}{0ex}}s=\frac{1}{2} \mathrm{or} s=\sqrt{2}
h\left(s\right)=\left(2s-1\right)\left(s-\sqrt{2}\right) \mathrm{are} \alpha =\frac{1}{2} \mathrm{and} \beta =\sqrt{2}
\alpha +\beta
=\frac{1}{2}+\sqrt{2}
\frac{-\mathrm{Coefficient} \mathrm{of} s}{\mathrm{Coefficient} \mathrm{of} {s}^{2}}\phantom{\rule{0ex}{0ex}}=-\left(\frac{-\left(1+2\sqrt{2}\right)}{2}\right)\phantom{\rule{0ex}{0ex}}=\frac{1+2\sqrt{2}}{2}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}+\sqrt{2}
=\frac{1}{2}×\sqrt{2}=\frac{1}{\sqrt{2}}
\frac{\mathrm{Constant} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {s}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}
f\left(v\right)={v}^{2}+4\sqrt{3}v-15
f\left(v\right)={v}^{2}+5\sqrt{3}v-\sqrt{3}v-15\phantom{\rule{0ex}{0ex}}={v}^{2}-\sqrt{3}v+5\sqrt{3}v-15\phantom{\rule{0ex}{0ex}}=v\left(v-\sqrt{3}\right)+5\sqrt{3}\left(v-\sqrt{3}\right)\phantom{\rule{0ex}{0ex}}=\left(v-\sqrt{3}\right)\left(v+5\sqrt{3}\right)
{v}^{2}+4\sqrt{3}v-15=0\phantom{\rule{0ex}{0ex}}\left(v+5\sqrt{3}\right)\left(v-\sqrt{3}\right)=0\phantom{\rule{0ex}{0ex}}\left(v-\sqrt{3}\right)=0 \mathrm{or} \left(v+5\sqrt{3}\right)=0\phantom{\rule{0ex}{0ex}}v=\sqrt{3} \mathrm{or} v=-5\sqrt{3}
f\left(v\right)=\left(v-\sqrt{3}\right)\left(v+5\sqrt{3}\right) \mathrm{are} \alpha =\sqrt{3} \mathrm{and} \beta =-5\sqrt{3}
\alpha +\beta
=\sqrt{3}-5\sqrt{3}=-4\sqrt{3}
\frac{-\mathrm{Coefficient} \mathrm{of} v}{\mathrm{Coefficient} \mathrm{of} {v}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{-4\sqrt{3}}{1}=-4\sqrt{3}
=\sqrt{3}×\left(-5\sqrt{3}\right)=-15
\frac{\mathrm{Constant} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {v}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{-15}{1}=-15
p\left(y\right)={y}^{2}+\frac{3\sqrt{5}}{2}y-5
p\left(y\right)=\frac{1}{2}\left(2{y}^{2}+4\sqrt{5}y-\sqrt{5}y-10\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\left[2y\left(y+2\sqrt{5}\right)-\sqrt{5}\left(y+2\sqrt{5}\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\left[\left(2y-\sqrt{5}\right)\left(y+2\sqrt{5}\right)\right]
p\left(y\right)=\frac{1}{2}\left(2y-\sqrt{5}\right)\left(y+2\sqrt{5}\right) \mathrm{are} \alpha =\frac{\sqrt{5}}{2}\mathrm{and} \beta =-2\sqrt{5}
\alpha +\beta
=\frac{\sqrt{5}}{2}-2\sqrt{5}=\frac{\sqrt{5}-4\sqrt{5}}{2}=\frac{-3\sqrt{5}}{2}
\frac{-\mathrm{Coefficient} \mathrm{of} y}{\mathrm{Coefficient} \mathrm{of} {y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{\frac{-3\sqrt{5}}{2}}{1}=\frac{-3\sqrt{5}}{2}
=\frac{\sqrt{5}}{2}×-2\sqrt{5}\phantom{\rule{0ex}{0ex}}=-5
\frac{\mathrm{Constant} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{-5}{1}=-5
q\left(y\right)=7{y}^{2}-\frac{11}{3}y-\frac{2}{3}
q\left(y\right)=\frac{1}{3}\left(21{y}^{2}-11y-2\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left(21{y}^{2}-14y+3y-2\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left[7y\left(3y-2\right)+1\left(3y-2\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\left[\left(7y+1\right)\left(3y-2\right)\right]
q\left(y\right)=\frac{1}{3}\left(7y+1\right)\left(3y-2\right) \mathrm{are} \alpha =\frac{-1}{7}\mathrm{and} \beta =\frac{2}{3}
\alpha +\beta
=\frac{-1}{7}+\frac{2}{3}\phantom{\rule{0ex}{0ex}}=\frac{11}{21}
\frac{-\mathrm{Coefficient} \mathrm{of} y}{\mathrm{Coefficient} \mathrm{of} {y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{-\left(-\frac{11}{3}\right)}{7}=\frac{11}{21}
=\frac{-1}{7}×\frac{2}{3}\phantom{\rule{0ex}{0ex}}=\frac{-2}{21}
\frac{\mathrm{Constant} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {y}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{\frac{-2}{3}}{7}\phantom{\rule{0ex}{0ex}}=\frac{-2}{21}
-\frac{8}{3}, \frac{4}{3}
\frac{21}{8}, \frac{5}{16}
-2\sqrt{3}, -9
\frac{-3}{2\sqrt{5}}, -\frac{1}{2}
f\left(x\right)=k\left\{{x}^{2}-\left(\mathrm{Sum} \mathrm{of} \mathrm{zeroes}\right)x+\mathrm{Product} \mathrm{of} \mathrm{zeroes}\right\}
\frac{-8}{3}
\frac{4}{3}
f\left(x\right)=k\left({x}^{2}+\frac{8}{3}x+\frac{4}{3}\right)
f\left(x\right)=k\left({x}^{2}+\frac{8}{3}x+\frac{4}{3}\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(3{x}^{2}+8x+4\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(3{x}^{2}+6x+2x+4\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(3x\left(x+2\right)+2\left(x+2\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(3x+2\right)\left(x+2\right)
x=-\frac{2}{3} \mathrm{and} x=-2
\frac{21}{8}
\frac{5}{16}
f\left(x\right)=k\left({x}^{2}-\frac{21}{8}x+\frac{5}{16}\right)
f\left(x\right)=k\left({x}^{2}-\frac{21}{8}x+\frac{5}{16}\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{16}\left(16{x}^{2}-42x+5\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{16}\left(16{x}^{2}-40x-2x+5\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{16}\left(16{x}^{2}-2x-40x+5\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(2x\left(8x-1\right)-5\left(8x-1\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{3}\left(8x-1\right)\left(2x-5\right)
x=\frac{1}{8} \mathrm{and} x=\frac{5}{2}
-2\sqrt{3}
f\left(x\right)=k\left({x}^{2}+2\sqrt{3}x-9\right)
f\left(x\right)=k\left({x}^{2}+2\sqrt{3}x-9\right)\phantom{\rule{0ex}{0ex}}=k\left({x}^{2}+3\sqrt{3}x-\sqrt{3}x-9\right)\phantom{\rule{0ex}{0ex}}=k\left(x+3\sqrt{3}\right)\left(x-\sqrt{3}\right)
x=-3\sqrt{3} \mathrm{and} x=\sqrt{3}
\frac{-3}{2\sqrt{5}}
\frac{-1}{2}
f\left(x\right)=k\left({x}^{2}+\frac{3}{2\sqrt{5}}x-\frac{1}{2}\right)
f\left(x\right)=\frac{k}{2\sqrt{5}}\left(2\sqrt{5}{x}^{2}+3x-\sqrt{5}\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{2\sqrt{5}}\left(2\sqrt{5}{x}^{2}+5x-2x-\sqrt{5}\right)\phantom{\rule{0ex}{0ex}}=\frac{k}{2\sqrt{5}}\left[\sqrt{5}x\left(2x+\sqrt{5}\right)-1\left(2x+\sqrt{5}x\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{k}{2\sqrt{5}}\left(2x+\sqrt{5}\right)\left(\sqrt{5}x-1\right)
x=\frac{-\sqrt{5}}{2} \mathrm{and} x=\frac{1}{\sqrt{5}}
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-\mathrm{\alpha \beta }
\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}
\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}
\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{a}}+2\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}\right)+3\mathrm{\alpha \beta }
\frac{{\mathrm{\alpha }}^{2}}{{\mathrm{\beta }}^{2}}+\frac{{\mathrm{\beta }}^{2}}{{\mathrm{\alpha }}^{2}}=\frac{{p}^{2}}{{q}^{2}}-\frac{4{p}^{2}}{q}+2
\frac{2\mathrm{\alpha }}{\mathrm{\beta }} \mathrm{and} \frac{2\mathrm{\beta }}{\mathrm{\alpha }}
\frac{1}{2\mathrm{\alpha }+\mathrm{\beta }}\mathrm{and} \frac{1}{2\mathrm{\beta }+\mathrm{\alpha }}
\frac{\mathrm{\alpha }-1}{\mathrm{\alpha }+1}, \frac{\mathrm{\beta }-1}{\mathrm{\beta }+1}
\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }
\frac{1}{a\mathrm{\alpha }+b}+\frac{1}{a\mathrm{\beta }+b}
\frac{\mathrm{\beta }}{a\mathrm{\alpha }+b}+\frac{\mathrm{\alpha }}{a\mathrm{\beta }+b}
a\left(\frac{{\mathrm{\alpha }}^{2}}{\mathrm{\beta }}+\frac{{\mathrm{\beta }}^{2}}{\mathrm{\alpha }}\right)+b\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}\right)
{\alpha }^{4}+{\beta }^{4}={\left[{\left(\frac{-b}{ a}\right)}^{2}-2×\left(\frac{c}{a}\right)\right]}^{2}-2×{\left(\frac{c}{a}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}= {\left[\frac{{b}^{2}}{{a}^{2}}-\frac{2c}{a}\right]}^{2}-2×{\left(\frac{c}{a}\right)}^{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}= {\left[\frac{{b}^{2}-2ac}{{a}^{2}}\right]}^{2}-2×\frac{{c}^{2}}{{a}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}= \frac{{\left({b}^{2}-2ac\right)}^{2}}{{a}^{4}}-2×\frac{{c}^{2}}{{a}^{2}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{{\left({b}^{2}-2ac\right)}^{2}-2{c}^{2}{a}^{2}}{{a}^{4}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Hence} \mathrm{the} \mathrm{value} \mathrm{of} {\alpha }^{4}+{\beta }^{4} \mathrm{is} \frac{{\left({b}^{2}-2ac\right)}^{2}-2{c}^{2}{a}^{2}}{{a}^{4}}.
f\left(x\right)=2{x}^{3}+{x}^{2}-5x+2; \frac{1}{2}, 1, -2
g\left(t\right)={t}^{2}-3, f\left(t\right)=2{t}^{4}+3{t}^{3}-2{t}^{2}-9t-12
g\left(x\right)={x}^{3}-3x+1, f\left(x\right)={x}^{5}-4{x}^{3}+{x}^{2}+3x+1
g\left(x\right)=2{x}^{2}-x+3, f\left(x\right)=6{x}^{5}-{x}^{4}+4{x}^{3}-5{x}^{2}-x-15
-\sqrt{3} \mathrm{and} \sqrt{3}
-\sqrt{\frac{3}{2}} \mathrm{and} \sqrt{\frac{3}{2}}
\sqrt{2} \mathrm{and} -\sqrt{2}
-\sqrt{3} \mathrm{and} \sqrt{3}
-\sqrt{2} \mathrm{and} \sqrt{2}
\sqrt{2}
6{x}^{3} + \sqrt{2}{x}^{2} - 10x - 4\sqrt{2}
\sqrt{2}
6{x}^{3}+\sqrt{2}{x}^{2}-10x-4\sqrt{2}
x-\sqrt{2}
6{x}^{3}+\sqrt{2}{x}^{2}-10x-4\sqrt{2}
x-\sqrt{2}
6{x}^{2}+7\sqrt{2}x+4
f\left(x\right)=g\left(x\right)×q\left(x\right)+r\left(x\right)
f\left(x\right)=\left(x-\sqrt{2}\right)\left(6{x}^{2}+7\sqrt{2}x+4\right)+0\phantom{\rule{0ex}{0ex}}=\left(x-\sqrt{2}\right)\left(\sqrt{2}x+1\right)\left(3\sqrt{2}x+4\right)
-\frac{1}{\sqrt{2}} \mathrm{and} -\frac{4}{3\sqrt{2}}
x - \sqrt{5}
{x}^{3} - 3\sqrt{5}{x}^{2} + 13x - 3\sqrt{5}
x-\sqrt{5}
{x}^{3}-3\sqrt{5}{x}^{2}+13x-3\sqrt{5}
{x}^{3}-3\sqrt{5}{x}^{2}+13x-3\sqrt{5}
x-\sqrt{5}
{x}^{2}-2\sqrt{5}x+3
f\left(x\right)=g\left(x\right)×q\left(x\right)+r\left(x\right)
f\left(x\right)=\left(x-\sqrt{5}\right)\left({x}^{2}-2\sqrt{5}x+3\right)+0\phantom{\rule{0ex}{0ex}}=\left(x-\sqrt{5}\right)\left[x-\left(\sqrt{5}+\sqrt{2}\right)\right]\left[x-\left(\sqrt{5}-\sqrt{2}\right)\right]
\sqrt{5},\sqrt{5}+\sqrt{2} \mathrm{and} \sqrt{5}-\sqrt{2}
-\frac{1}{2}
-\frac{1}{4}
2\sqrt{3}
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}=
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}
\frac{7}{3}
-\frac{7}{3}
\frac{3}{7}
-\frac{3}{7}
\alpha +\beta +\gamma
\alpha +\beta +\gamma =\frac{-\mathrm{Coefficient} \mathrm{of} x}{\mathrm{Coefficient} \mathrm{of} {x}^{2}}\phantom{\rule{0ex}{0ex}} = -\frac{\left(-3k\right)}{2} = \frac{3k}{2}
\alpha +\beta +\gamma
\alpha +\beta +\gamma =\frac{3k}{2}
\frac{1}{\mathrm{\alpha }}\mathrm{and} \frac{1}{\mathrm{\beta }}
\frac{3}{2}
-\frac{3}{2}
\frac{2}{3}
-\frac{2}{3}
\frac{3}{2}
-\frac{3}{2}
\frac{9}{2}
-\frac{9}{2}
\frac{1}{c}
-\frac{1}{c}
\frac{1}{6}
\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}+\frac{1}{\mathrm{\gamma }}=
-\frac{b}{d}
\frac{c}{d}
-\frac{c}{d}
-\frac{c}{a}
\frac{{b}^{2}-ac}{{a}^{2}}
\frac{{b}^{2}-2ac}{a}
\frac{{b}^{2}+2ac}{{b}^{2}}
\frac{{b}^{2}-2ac}{{a}^{2}}
\frac{1}{\mathrm{\alpha \beta }}+\frac{1}{\mathrm{\beta \gamma }}+\frac{1}{\mathrm{\gamma \alpha }}=
\frac{1}{{\mathrm{\alpha }}^{2}}+\frac{1}{{\mathrm{\beta }}^{2}}=
\frac{{b}^{2}-2ac}{{a}^{2}}
\frac{{b}^{2}-2ac}{{c}^{2}}
\frac{{b}^{2}+2ac}{{a}^{2}}
\frac{{b}^{2}+2ac}{{c}^{2}}
\frac{-d}{a}
\frac{c}{a}
\frac{-b}{a}
\frac{b}{a}
\sqrt{5} \mathrm{and} -\sqrt{5}
\sqrt{5} \mathrm{and} -\sqrt{5}
p\left(x\right)=a{\left(x+2\right)}^{n}{\left(x-5\right)}^{m}
\left(k-1\right){x}^{2}+kx+1
-3
f\left(-3\right)=0
\left(k-1\right){\left(-3\right)}^{2}+k\left(-3\right)+1=0\phantom{\rule{0ex}{0ex}}⇒9\left(k-1\right)-3k+1=0\phantom{\rule{0ex}{0ex}}⇒9k-9-3k+1=0\phantom{\rule{0ex}{0ex}}⇒6k-8=0\phantom{\rule{0ex}{0ex}}⇒k=\frac{4}{3}
127×1=127
\frac{c}{a}
-\frac{b}{a}
{x}^{2}+\left(a+1\right)x+b
-3
a = -7, b = -1 \phantom{\rule{0ex}{0ex}}
a = 5 , b = -1
a =2 , b = -6
a = 0 , b = -6
{x}^{2}+\left(a+1\right)x+b=0
\alpha =2
\beta =-3
\mathrm{Sum} \mathrm{of} \mathrm{zeroes}=-\frac{\mathrm{Coefficient} \mathrm{of} x}{\mathrm{Coefficient} \mathrm{of} {x}^{2}}\phantom{\rule{0ex}{0ex}}⇒2+\left(-3\right)=-\frac{\left(a+1\right)}{1}\phantom{\rule{0ex}{0ex}}⇒-1=-a-1\phantom{\rule{0ex}{0ex}}⇒a=0
\mathrm{Product} \mathrm{of} \mathrm{zeroes}=\frac{\mathrm{Constant} \mathrm{term}}{\mathrm{Coefficient} \mathrm{of} {x}^{2}}\phantom{\rule{0ex}{0ex}}⇒2×\left(-3\right)=\frac{b}{1}\phantom{\rule{0ex}{0ex}}⇒b=-6
a{x}^{3} + b{x}^{2} + cx + d
\frac{-c}{a}
\frac{c}{a}
\frac{-b}{a}
p\left(x\right)=a{x}^{3}+b{x}^{2}+cx+d
⇒a{\left(0\right)}^{3}+b{\left(0\right)}^{2}+c\left(0\right)+d=0\phantom{\rule{0ex}{0ex}}⇒d=0
p\left(x\right)=a{x}^{3}+b{x}^{2}+cx=x\left(a{x}^{2}+bx+c\right)
a{x}^{2}+bx+c=0
a{x}^{2}+bx+c=0
\alpha \beta =\frac{c}{a}
a \ne 0,
\frac{c}{a}
-\frac{b}{a}
{x}^{3} + a{x}^{2} + bx + c
-1
b - a + 1
b - a - 1
a - b + 1
a - b - 1
p\left(x\right)={x}^{3}+a{x}^{2}+bx+c
⇒{\left(-1\right)}^{3}+a{\left(-1\right)}^{2}+b\left(-1\right)+c=0\phantom{\rule{0ex}{0ex}}⇒-1+a-b+c=0\phantom{\rule{0ex}{0ex}}⇒a-b+c=1\phantom{\rule{0ex}{0ex}}⇒c=1-a+b
\alpha , \beta , \gamma
a{x}^{3}+b{x}^{2}+cx+d
\alpha \beta \gamma =-\frac{d}{a}
p\left(x\right)={x}^{3}+a{x}^{2}+bx+c
\alpha \beta \left(-1\right)=\frac{-c}{1}=\frac{-\left(1-a+b\right)}{1}\phantom{\rule{0ex}{0ex}}⇒\alpha \beta =1-a+b
a{x}^{3} + b{x}^{2} + cx + d
- \frac{b}{a}
\frac{b}{a}
\frac{c}{a}
-\frac{d}{a}
f\left(x\right)=a{x}^{3}+b{x}^{2}+cx+d
\alpha =0 \mathrm{and} \beta =0
\alpha +\beta +\gamma =\frac{-b}{a}\phantom{\rule{0ex}{0ex}}⇒0+0+\gamma =\frac{-b}{a}\phantom{\rule{0ex}{0ex}}⇒\gamma =\frac{-b}{a}
-
-
⇒{2}^{2}+3×2+k=0\phantom{\rule{0ex}{0ex}}⇒4+6+k=0\phantom{\rule{0ex}{0ex}}⇒k=-10
a{x}^{2}+bx+c
c \ne 0
a{x}^{2}+bx+c
\alpha
\beta
\alpha
\beta
\alpha \beta >0
\alpha \beta =\frac{c}{a}
\frac{c}{a}>0
{x}^{2}+ax+b
f\left(x\right)={x}^{2}+ax+b
\alpha \mathrm{and} -\alpha
\therefore \alpha +\left(-\alpha \right)=\frac{-a}{1}=-a\phantom{\rule{0ex}{0ex}}⇒a=0
f\left(x\right)={x}^{2}+b
\alpha \beta =\frac{b}{1}=b\phantom{\rule{0ex}{0ex}}⇒\alpha \left(-\alpha \right)=b\phantom{\rule{0ex}{0ex}}⇒-{\alpha }^{2}=b
a{x}^{2}+bx+c
y=a{x}^{2}+bx+c
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Why and how to use pen and paper - Designing algorithms with no code - DEV Community
Why and how to use pen and paper - Designing algorithms with no code
#beginners #codenewbie #productivity #codequality
It's a common thing to see programming students staring silently at their code editor, with a blank look in their eyes, not knowing where to start when they are given an assignment. Although most teachers say "pick up pen and paper before programming", there is a reason why some just don't do it: mostly because they don't know why or how.
So, how important is to "pick up pen and paper"?
Your coding skills will always be limited by your problem solving skills
Programming is, by definition, designing a solution. In most cases this means designing an algorithm (sorry, declarative programming).
To be a great programmer, it is more important to develop your ability to solve problems algorithmically and think abstractly about data than to know perfectly the syntax of any language. It will let you deconstruct the problem and solve it with whatever tools you have.
How does it help using pen and paper?
In pseudocode, a flowchart or a drawing, data need no type, there are no syntax errors and you can focus on what you have and what you want, with the level of abstraction you need.
Our minds understand visual information better
With code, it happens like with mathematical language. Our brain knows how to interpret images almost directly, but needs some more resources to interpret code or math.
A \cap B = \{ x : x \in A, x \in B \}
The process your mind takes to understand the image is much shorter than to understand the expression.
That applies to programming too
The same happens when actively creating something. It's easier for you to come up with a solution while using your eyes and hands on the paper than directly trying to express it in code.
Furthermore, when you design before writing, you can debug more easily. The design offers you a panoramic view of the algorithm, which you can contrast with the code in successive iterations, to recognize the nature of the error.
If you're already convinced to pick up that pen and paper, you might find useful some tips for the process.
Some guidelines on how to think when designing an algorithm
There are several ways to do this. In fact, Software Engineering offers a lot of different models on how to design software and abstract architectures to approach a lot of different problems. Here I want to give a very generic method, on which almost any other process is based.
Thinking on the data representation
What information do you have and how you want/need to represent it? Are you going to use predefined types or structures of the language? Do you need to define your own?
Draw some examples of the data and let your brain perceive the gaps or redundancies.
What are the basic operations you can perform on this information? You should be clear on what transformations you can apply to your data.
In many languages the most basic is asignation. Be sure to understand how it works. The rest usually depends on whether it is a number, a string, a collection, an object, etc
You will be able to use these basic operations as the building blocks of your algorithm.
Thinking on the process
These are not sequential steps, but separate advice to develop your solutions.
Before even trying to define the algorithm, it is better to have some examples of input and output. For this you can also specify some preconditions and postconditions: What is true before the algorithm is executed? What must be true after the algorithm is executed?
If the algorithm depends on some arguments, pay special attention to them. Also, what useful information can you extract from them? Write that down too, if you think it will be worhtwhile.
You don't need to think it straight from the first to the last step. Sometimes the last part of the algorithm is clearer than the first one. Go ahead and specify its latter part, and then focus on how to get to that point.
Write or draw the steps you see clearly, try to connect them and fill the missing steps to finish.
Divide the problem into subproblems until you are able to solve them with your basic operations. You can write first a high-level pseudocode or flow-chart version of the solution, and then go through each step, decomposing it into more steps.
If you think you have more than one option, you can even specify various solutions and try them in code, to see if one works better than another.
Different minds deal better with different methods, but in the end, learning and practising are the two only ways to improve.
There is a direct synergy between solving problems and programming, but always remember: the solution can exist without the code; the code can't exist without the solution.
Do you have any advice for writing better code and designing algorithms more efficiently? I'll be glad to read your comments on this topic!
CasyHarward
Nowadays, you kow that many different gadgets can be used instead of pen and paper. Still, I personally think that using pen and paper will give you more benefits then the writing on gadgets as you will remember all the things you write with your hand so it will be an edge, for example if you are writing a research paper. There is some problem statement that occurs in it you can easily tackle it next time the same or similar problem occurs if you have written it by pen and paper.
These are some common steps to use pen and paper to design an algorithm and this assignment help online information is highly suggestive for students that need how to sketch out their story. After learning core concepts you can easily design a program.
I love to write. And this is the only thing in which I consider myself a professional. Writing is a kind of psychotherapy, and of course, writing is work.
Even if you just need to roll data from one database to another, then we make models, add marketing plan writing service and that's it. Because it is not so long and scary if you know how, and anyone who is familiar with the host framework can read and understand the code.
Custom solution for fast navigation in Bash
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BidiagonalForm - Maple Help
Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Solvers : BidiagonalForm
reduce a Matrix to bidiagonal form
BidiagonalForm(A, out, ip, options, outopts)
(optional) equation of the form output = obj where obj is one of 'U', 'B', 'Vt', or 'NAG', or a list containing one or more of these names; selects result objects to compute
(optional) equation(s) of the form outputoptions[o] = list where o is one of 'U', 'B', 'Vt', or 'NAG'; constructor options for the specified result object
The BidiagonalForm(A) function returns a Matrix in bidiagonal form. This routine operates in the floating-point domain. Hence, the entries in Matrix A must necessarily be of type complex(numeric).
A bidiagonal Matrix has nonzero entries only on the main diagonal and either the first super-diagonal or the first subdiagonal.
The original Matrix A and the left and right reduction Matrices U and Vt are related by
A=U·B·\mathrm{Vt}
, where B is the bidiagonal form of A.
If A is real, then U and Vt are orthogonal.
If A is complex, then U and Vt are unitary.
If A is an m x n Matrix and
m<n
, then B is lower bidiagonal. If
m\ge n
, then B is upper bidiagonal.
Depending on what is included in the output option, an expression sequence containing one or more of the factors U (the left reduction Matrix), B (the bidiagonal form), or V (the right reduction Matrix) can be returned. If output is a list, the objects are returned in the same order as specified in the list.
If NAG is included in the output list, then the returned objects are an expression sequence consisting of a Matrix, a Vector, and a Vector. The second and third Vector objects contain additional details of the orthogonal/unitary Matrices U and Vt. The returned objects are encoded in NAG format:
m\ge n
, then the diagonal and first super-diagonal of the first returned object B contain the upper bidiagonal solution. Elements below the diagonal are overwritten by details of the orthogonal Matrix U and elements above the first super-diagonal are overwritten by details of the orthogonal Matrix Vt.
m<n
, then the diagonal and the first subdiagonal of the first returned object B contain the lower bidiagonal solution. Elements below the first subdiagonal are overwritten by details of the orthogonal Matrix U and elements above the diagonal are overwritten by details of the orthogonal Matrix Vt.
The specified output denoted by output='NAG' precludes the output of any other specified objects.
left reducing Matrix
bidiagonal form
right reducing Matrix
This function is part of the LinearAlgebra package, and so it can be used in the form BidiagonalForm(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[BidiagonalForm](..).
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
\mathrm{UseHardwareFloats}≔\mathrm{false}
\textcolor[rgb]{0,0,1}{\mathrm{UseHardwareFloats}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{false}}
A≔\mathrm{RandomMatrix}\left(4,\mathrm{generator}=-10..10\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{-5}& \textcolor[rgb]{0,0,1}{-4}\\ \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{-10}& \textcolor[rgb]{0,0,1}{-4}\\ \textcolor[rgb]{0,0,1}{-10}& \textcolor[rgb]{0,0,1}{-6}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-10}\\ \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{-4}\end{array}]
B≔\mathrm{BidiagonalForm}\left(A\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-17.57839583}& \textcolor[rgb]{0,0,1}{12.45964361}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{1.611363883}& \textcolor[rgb]{0,0,1}{-12.25053573}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{-3.981959872}& \textcolor[rgb]{0,0,1}{-3.846900539}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{-11.06483232}\end{array}]
U,\mathrm{Vt}≔\mathrm{BidiagonalForm}\left(A,\mathrm{output}=['U','\mathrm{Vt}']\right)
\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Vt}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-0.511992112}& \textcolor[rgb]{0,0,1}{-0.1213173437}& \textcolor[rgb]{0,0,1}{-0.8173225584}& \textcolor[rgb]{0,0,1}{0.2347978218}\\ \textcolor[rgb]{0,0,1}{-0.4551040994}& \textcolor[rgb]{0,0,1}{-0.6547009920}& \textcolor[rgb]{0,0,1}{0.2209216328}& \textcolor[rgb]{0,0,1}{-0.5616409017}\\ \textcolor[rgb]{0,0,1}{0.5688801242}& \textcolor[rgb]{0,0,1}{-0.7350840489}& \textcolor[rgb]{0,0,1}{-0.1505236346}& \textcolor[rgb]{0,0,1}{0.3367038477}\\ \textcolor[rgb]{0,0,1}{-0.4551040994}& \textcolor[rgb]{0,0,1}{-0.1276720577}& \textcolor[rgb]{0,0,1}{0.5104117013}& \textcolor[rgb]{0,0,1}{0.7183731618}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.0}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{-0.917722116}& \textcolor[rgb]{0,0,1}{0.3972230054}& \textcolor[rgb]{0,0,1}{1.605182349}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-10}}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{-0.1770984061}& \textcolor[rgb]{0,0,1}{-0.4091583868}& \textcolor[rgb]{0,0,1}{-0.8951120425}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{-0.3555590957}& \textcolor[rgb]{0,0,1}{-0.8214641185}& \textcolor[rgb]{0,0,1}{0.4458412623}\end{array}]
U·B·\mathrm{Vt}
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{9.000000007}& \textcolor[rgb]{0,0,1}{5.000000013}& \textcolor[rgb]{0,0,1}{-5.000000009}& \textcolor[rgb]{0,0,1}{-4.000000009}\\ \textcolor[rgb]{0,0,1}{8.000000003}& \textcolor[rgb]{0,0,1}{3.000000005}& \textcolor[rgb]{0,0,1}{-10.00000000}& \textcolor[rgb]{0,0,1}{-3.999999995}\\ \textcolor[rgb]{0,0,1}{-10.00000000}& \textcolor[rgb]{0,0,1}{-6.000000004}& \textcolor[rgb]{0,0,1}{1.000000003}& \textcolor[rgb]{0,0,1}{-10.00000000}\\ \textcolor[rgb]{0,0,1}{8.000000003}& \textcolor[rgb]{0,0,1}{9.000000002}& \textcolor[rgb]{0,0,1}{6.000000000}& \textcolor[rgb]{0,0,1}{-4.000000002}\end{array}]
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Home : Support : Online Help : MapleSim : MapleSim Component Library : Magnetic : Fundamental Wave : Overview
Overview of the concept of fundamental waves
The exact magnetic field in the air gap of an electric machine is usually determined by an electro-magnetic finite element analysis. The waveform of the magnetic field, e.g., the magnetic potential difference,
{\stackrel{^}{V}}_{m}\left(\mathrm{ϕ}\right)
, consists of a spatial fundamental wave with respect to an equivalent two pole machine and additional harmonic waves of higher order. The fundamental wave is dominant in the air gap of an electric machine.
Fig. 1: Field lines of a four-pole induction machine
In the fundamental wave theory only a pure sinusoidal distribution of magnetic quantities is assumed, other harmonic wave effects are ignored.
Fig. 2: Magnetic potential difference of a four pole machine, where
\mathrm{ϕ}
is the angle of the spatial domain with respect to one pole pair.
The waveforms of the magnetic field quantities, e.g., the magnetic potential difference,
{\stackrel{^}{V}}_{m}\left(\mathrm{ϕ}\right)
, can be represented by a complex phasor,
{\underset{¯}{V}}_{m}
{\underset{¯}{V}}_{m}={V}_{m,r}+j{V}_{m,i}
{V}_{m}\left(\mathrm{ϕ}\right)=ℜ\left({\underset{¯}{V}}_{m}\mathrm{exp}\left(-j\mathrm{ϕ}\right)\right)={V}_{m,r}\mathrm{cos}\left(\mathrm{ϕ}\right)+{V}_{m,i}\mathrm{sin}\left(\mathrm{ϕ}\right)
It is important to note that the magnetic potential used in this library always refers to an equivalent two-pole machine.
Fig. 3: Spatial distribution of the magnetic potential difference (red shade = positive sine wave, blue shade = negative sine wave) including complex phasor representing this spatial distribution
The potential and flow quantities of this library are the complex magnetic potential difference and the complex magnetic flux as defined in the basic magnetic port. Due to the sinusoidal distribution of magnetic potential and flux, such a complex phasor representation can be used. This way, the Fundamental Wave library can be seen as a spatial extension of the standard magnetic library.
The specific arrangement of windings in electric machines with pole pairs gives rise to sinusoidal dominant magnetic potential wave. The spatial period of this wave is determined by one pole pair [Mueller70, Spaeth73].
The main components of an electric machine model based on the Fundamental Wave library are multi-phase and single-phase windings, air gap as well as symmetric or salient cage models. The electric machine models provided in this library are based on symmetrical windings in the stator and equivalent two or three phase windings in squirrel cage rotors. Slip ring induction machines may have different phase numbers in the stator and rotor.
The machine models of the FundamentalWave library are currently based on the following assumptions
The number of stator phases is greater or equal to three [Eckhardt82]
The phase windings are assumed to be symmetrical; an extension to this approach can be considered
Only fundamental wave effects are taken into account
The magnetic potential difference refers to an equivalent two pole machine
There are no restrictions on the waveforms of voltages and currents
All resistances and inductances are modeled as constant quantities; saturation effects, cross coupling effects [Li07], temperature dependency of resistances and deep bar effects could be considered in an extension to this library
Hysteresis losses are currently not considered [Haumer09]
The losses dissipated in the electric machine models are
ohmic heat losses,
eddy current losses in the stator core,
stray load losses,
The term fundamental-wave refers to spatial waves of the electro-magnetic quantities. This library has no limitations with respect to the waveforms of the time-domain signals of any voltages, currents, etc.
Electrical machines and their components
Complex magnetic sensors
Complex magnetic sources
[Beuschel00] M. Beuschel, A uniform approach for modelling electrical machines, Modelica Workshop, pp. 101-108, October 23-24, 2000.
[Eckhardt82] H. Eckhardt, "Grundzüge der elektrischen Maschinen" (in German), B. G. Teubner Verlag, Stuttgart, 1982.
[Haumer09] A. Haumer, and C. Kral, The AdvancedMachines Library: Loss Models for Electric Machines, Modelica Conference, 2009.
[Lang84] W. Lang, "Über die Bemessung verlustarmer Asynchronmotoren mit Käfigläufer für Pulsumrichterspeisung" (in German), Doctoral Thesis, Technical University of Vienna, 1984.
[Laughton02] M.A. Laughton, D.F. Warne, "Electrical Engineer's Reference Book," Butterworth Heinemann, 16th edition, ISBN 978-0750646376, 2002
[Li07] Y. Li, Z. Q. Zhu, D. Howe, and C. M. Bingham, "Modeling of Cross-Coupling Magnetic Saturation in Signal-Injection-Based Sensorless Control of Permanent-Magnet Brushless AC Motors," IEEE Transactions on Magnetics, vol. 43, no. 6, pp. 2552-2554, June 2007.
[Mueller70] G, Müller, "Elektrische Maschinen -- Grundlagen, Aufbau und Wirkungsweise" (in German), VEB Verlag Technik Berlin, 4th edition, 1970.
[Spaeth73] H. Späth, "Elektrische Maschinen -- Eine Einführung in die Theorie des Betriebsverhaltens" (in German), Springer-Verlag, Berlin, Heidelberg, New York, 1973.
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Total bases - Wikipedia
Number of bases a baseball player has gained with hits
{\displaystyle TB=(1\times 1B)+(2\times 2B)+(3\times 3B)+(4\times HR)}
In baseball statistics, total bases is the number of bases a player has gained with hits. It is a weighted sum for which the weight value is 1 for a single, 2 for a double, 3 for a triple and 4 for a home run. For example, three singles is three total bases, a double and a home run is six total bases, and three triples is nine total bases.
Only bases attained from hits count toward this total. Reaching base by other means (such as a base on balls) or advancing further after the hit (such as when a subsequent batter gets a hit) does not increase the player's total bases. In box scores and other statistical summaries, total bases is often denoted by the abbreviation TB.[1][2]
The total bases divided by the number of at bats is the player's slugging percentage.
Hank Aaron (left) and Babe Ruth hold the MLB records for total bases in a career and in a single season, 6,856 and 457, respectively.
Shawn Green (left) and Josh Hamilton hold the records for total bases in a single game for the National League and American League, 19 and 18, respectively.
See also: List of Major League Baseball career total bases leaders
Hank Aaron's 6,856 career total bases make him the all-time MLB record holder.[3] Having spent the majority of his career playing in the National League, he also holds that league's record with 6,591 total bases.[4] Aaron hit for 300 or more total bases in a record 15 different seasons.[5] Aaron regarded this record as his proudest accomplishment, over his career home run record, because he felt it better reflected his performance as a team player.[6] Ty Cobb's 5,854 total bases constitute the American League record.[7] Albert Pujols is the active leader and fourth all-time, with 6,030 TB as of August 23 of the 2021 MLB season.[8][9]
The single season MLB and American League records are held by Babe Ruth, who hit for 457 TB in the 1921 season.[10] The following season saw Rogers Hornsby set the National League record when he hit for 450 total bases.[11]
Shawn Green holds the single game total bases record of 19 TB. Green hit four home runs, a single and a double for the Los Angeles Dodgers against the Milwaukee Brewers on May 23, 2002.[12] The equivalent American League record is held by Josh Hamilton, who hit four home runs and a double (18 TB) for the Texas Rangers in a May 8, 2012, game versus the Baltimore Orioles.[12]
Dustin Pedroia collected the most total bases in a single interleague game during the regular season, with 15. Pedroia hit three home runs, a single and a double for the Boston Red Sox on June 24, 2010, in a game against the Colorado Rockies at Coors Field.[13]
The 2003 Boston Red Sox and 2019 Minnesota Twins jointly hold the American League single season team record with 2,832 total bases; the National League record is held by the 2001 Colorado Rockies (2,748 TB).[14] The Red Sox also have the record for most total bases by a team in one game: they hit for 60 TB in a 29–4 victory over the St. Louis Browns on June 8, 1950.[15]
Among major league pitchers, Phil Niekro gave up the most total bases in a career (7,473),[16] while Robin Roberts (555 TB allowed in 1956) holds the single season record.[17] The record number of total bases allowed in a single game by one pitcher is 42, by Allan Travers of the Detroit Tigers.[18]
Two players have hit for 14 total bases in a postseason game.[19] Albert Pujols is the only player to accomplish this in the World Series, doing so for the St. Louis Cardinals in Game 3 of the 2011 World Series, when he had two singles and three home runs.[20] Bob Robertson also achieved the feat while playing for the Pittsburgh Pirates in Game 2 of the 1971 National League Championship Series, with a double and three home runs.[21] David Freese holds the record for a single postseason, with 50 total bases during the 2011 playoffs for the St. Louis Cardinals, while Derek Jeter has the career postseason record of 302 total bases, all with the New York Yankees.[22]
The Boston Red Sox hit for 45 total bases in their 23–7 victory over the Cleveland Indians in Game 4 of the 1999 American League Division Series, a postseason record. The most total bases by a team in a World Series game is 34, by the Atlanta Braves in Game 5 of the 1991 World Series, when they beat the Minnesota Twins by a score of 14–5.[23]
Ted Williams hit for a record 10 total bases (two singles and two home runs) in the All-Star Game when representing the American League in the 1946 edition.[24][25] The 1954 edition, when the American League had 29 and the National League had 23, produced the most total bases in a single All-Star Game, 52.[26] The most total bases by one team in an All-Star Game is 29, achieved by the American League in both the 1954 and 1992 editions. The National League had a high of 25 total bases in the 1951 game.[27]
^ "Team Batting Game Finder: From 1988 to 2018, Playing for SFG, (requiring TB>=40), sorted by greatest TB". Baseball Reference. Retrieved August 24, 2018.
^ "Giants 13, Braves 4". MLB.com. Retrieved August 24, 2018.
^ "Career Leaders & Records for Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Batting Season & Career Finder: Spanning Multiple Seasons or entire Careers, Playing in the NL, From 1871 to 2018, (requiring TB>=5500), sorted by greatest Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Batting Season & Career Finder: For Single Seasons, From 1871 to 2018, (requiring TB>=300), sorted by greatest Seasons matching criteria". Baseball Reference. Retrieved July 8, 2018.
^ Aaron, Henry; Wheeler, Lonnie (2014). I Had a Hammer (2 ed.). Harper-Collins. p. 202.
^ "Batting Season & Career Finder: Spanning Multiple Seasons or entire Careers, Playing in the AL, From 1871 to 2018, (requiring TB>=5500), sorted by greatest Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Active Leaders & Records for Total Bases". Baseball-Reference.com. Retrieved April 16, 2020.
^ "Career Leaders & Records for Total Bases". Baseball-Reference.com. Retrieved April 16, 2020.
^ "Single-Season Leaders & Records for Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Batting Season & Career Finder: For Single Seasons, From 1871 to 2018, (requiring TB>=425), sorted by greatest Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ a b "Batting Game Finder: From 1908 to 2018, (requiring TB>=17), sorted by greatest TB". Baseball Reference. Retrieved July 8, 2018.
^ "Batting Game Finder: From 1908 to 2018, in Inter-league play, (requiring TB>=13), sorted by greatest TB". Baseball Reference. Retrieved July 8, 2018.
^ "Team Batting Season Finder: For Single Seasons, from 1871 to 2021, Standard stats, requiring Total Bases >= 2700, sorted by greatest Total Bases". Stathead Baseball. Retrieved May 24, 2021.
^ "Team Batting Game Finder: From 1908 to 2018, (requiring TB>=50), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
^ "Pitching Season & Career Finder: Spanning Multiple Seasons or entire Careers, From 1871 to 2018, (requiring TB>=6000), Stats only available back to 1908 and some partially complete., sorted by greatest Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Pitching Season & Career Finder: For Single Seasons, From 1871 to 2018, (requiring TB>=475), Stats only available back to 1908 and some partially complete., sorted by greatest Total Bases". Baseball Reference. Retrieved July 8, 2018.
^ "Pitching Game Finder: From 1908 to 2018, (requiring TB>=35), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
^ "Batting Game Finder: In the Postseason, From 1903 to 2017, (requiring TB>=12), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
^ "St. Louis Cardinals 16, Texas Rangers 7". Retrosheet. October 22, 2011. Retrieved April 16, 2020.
^ "Pittsburgh Pirates 9, San Francisco Giants 4". Retrosheet. October 3, 1971. Retrieved April 16, 2020.
^ "All-time and Single-Season Postseason Batting Leaders". Baseball Reference. Retrieved August 27, 2018.
^ "Team Batting Game Finder: In the Postseason, From 1903 to 2017, (requiring TB>=32), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
^ "Team Batting Game Finder: In the All-Star Game, From 1933 to 2017, (requiring TB>=8), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
^ "American League 12, National League 0". Retrosheet. July 9, 1946. Retrieved April 16, 2020.
^ "All-Star Game Records: Team All-Star Game Hitting Records". Baseball Almanac. Retrieved April 16, 2020.
^ "Team Batting Game Finder: In the All-Star Game, From 1933 to 2017, (requiring TB>=22), sorted by greatest TB". Baseball Reference. Retrieved July 9, 2018.
Total Bases Records at Baseball Almanac
Retrieved from "https://en.wikipedia.org/w/index.php?title=Total_bases&oldid=1080250341"
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Bound-Constrained Quadratic Programming, Solver-Based - MATLAB & Simulink
Create Boundary Conditions
Create Objective Function Matrices
For a problem-based version of this example, see Bound-Constrained Quadratic Programming, Problem-Based.
{E}_{stretch}
is approximately proportional to the second derivative of the material height, times the height. You can approximate the second derivative by the 5-point finite difference approximation (assume that the finite difference steps are of size 1). Let
\Delta x
\Delta y
represent a shift by 1 in the second coordinate direction.
{E}_{stretch}\left(p\right)=\left(-1\left(x\left(p+{\Delta }_{x}\right)+x\left(p-{\Delta }_{x}\right)+x\left(p+{\Delta }_{y}\right)+x\left(p-{\Delta }_{y}\right)\right)+4x\left(p\right)\right)x\left(p\right).
{E}_{stretch}\left(p\right)
The height matrix defines the lower bounds on the solution x. To restrict the solution to be zero at the boundary, set the upper bound ub to be zero on the boundary.
The quadprog problem formulation is to minimize
\frac{1}{2}{x}^{T}Hx+{f}^{T}x
In this case, the linear term
{f}^{T}x
corresponds to the potential energy of the material height. Therefore, specify f = 1/3000 for each component of x.
Create the finite difference matrix representing
{E}_{stretch}
by using the delsq function. The delsq function returns a sparse matrix with entries of 4 and -1 corresponding to the entries of 4 and -1 in the formula for
{E}_{stretch}\left(p\right)
. Multiply the returned matrix by 2 to have quadprog solve the quadratic program with the energy function as given by
{E}_{stretch}
View the structure of the matrix H. The matrix operates on x(:), which means the matrix x is converted to a vector by linear indexing.
Solve the problem by calling quadprog.
Reshape the solution x to a matrix S. Then plot the solution.
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Pool Contract - Sublime Docs
In the following sections we elaborate the Pool.sol specification.
The Pool.sol contract handles the core logic required to implement pool-based borrowing. It handles important functions such as lending, borrowing, repayments, etc.
gaining ownership from previous admin
can transfer ownership to new admin
can update the pool logic
can update thresholds for relevant parameters
Functions performed by the admin are meant to be transferred to governance over time.
can call functions surrounding the loan - withdrawing amount raised, repayments, responding to margin calls, etc.
can supply liquidity, withdraw repayments, or execute margin calls on the borrower
Borrowing from a Pool
Once the collection period for a pool ends, the borrower can start withdrawing liquidity from the pool. Withdrawing the borrowed amount must be done within the withdrawal period. Having a withdrawal period ensures that the borrower cannot maliciously lock away capital from lenders by never withdrawing assets from the pool.
function withdrawBorrowedAmount()
external override onlyBorrower(msg.sender) nonReentrant {
current time must be greater than loanStartTime and less than the loanWithdrawalDeadline
msg.sender should be the borrower as defined through pool creation
the total amount collected in the pool must be greater than or equal to the minimum borrow amount
the pool's current collateral ratio must be greater than or equal to idealCollateralRatio
None - msg.sender has to be the borrower
Supplying capital to a Pool
Lenders can deposit liquidity into a pool during the collection period. Liquidity deposited cannot be withdrawn unless one of the following conditions are met:
the pool is terminated due to malicious activity
the pool is cancelled by the borrower
the borrower fails to withdraw the loan amount before the loanWithdrawalDeadline
the borrower repays the loan at the end of the loan duration
the pool is liquidated due to late repayments or a margin call by the lender
function lend(
address _lender,
bool _fromSavingsAccount
If the pool creator has specified a lenderVerifier then the lender must be verified by them
the loan must be in the collection period
_lender: address of the lender. Note that the msg.sender need not be _lender
_amount: amount the lender wishes to lend
_fromSavingsAccount: if true, _amount from msg.sender's Sublime Savings Account is withdrawn. If false _amount is directly transferred from msg.sender's wallet address
Liquidating a Pool
An entire pool can be liquidated in case of missed repayments. In case the normal interval deadline is missed, the borrower enters a grace period, during which it is still possible for them to repay any interests due within that period. The borrower further has the possibility of requesting for an extension of the deadline for any given instalment period. However, each pool can only receive a single extension. In case all of the above measures fail, a liquidator can liquidate the pool collateral to receive the pool's collateral and the liquidation reward.
function liquidatePool(
bool _fromSavingsAccount,
bool _toSavingsAccount,
bool _recieveLiquidityShare
the pool must be ACTIVE
the borrower is currently in default
_fromSavingsAccount: if true, liquidity from msg.sender's Sublime Savings Account is used to buy out the collateral. If false, msg.sender directly transfers liquidity from their wallet
_toSavingsAccount: if true, the liquidated collateral is transferred to msg.sender's Sublime Savings Account. If false, the liquidated collateral is transferred to msg.sender's wallet
_receiveLiquidityShare: If true, liquidated collateral is directly transferred as LP tokens. If false, liquidated collateral is first converted to base tokens before transferring to msg.sender
Liquidating a margin call
Lenders can use margin calls to request the borrower to add additional collateral to the pool in case they fall below idealCollateralRatio. If the borrower fails to meet a margin call, a liquidator can liquidate the lender's position in the pool by transferring an amount (denominated in borrowAsset) equal to the value of the collateral backing the lender's position.
function liquidateForLender(
loan status must be ACTIVE
the current time must be greater than or equal to marginCallEndTime
idealCollateralRatio must be greater than the borrower's collateral ratio wrt the lender that executed the margin call
_lender must be a valid lender in the pool
_lender: address of the lender whose position is being liquidated
Withdrawing repayments from pool
As the borrower repays the loan, lenders can start withdrawing them. The amount that a given lender can withdraw depends on the number of pool tokens in their possession, and the amount of interest they've already withdrawn. Repayments are directly transferred to msg.sender's wallet address.
function withdrawRepayment()
external isLender(msg.sender) nonReentrant {
msg.sender must possess valid pool tokens in their wallet
None: msg.sender should be a valid lender (ie, possess pool tokens)
Adding collateral to a pool
Due to price volatilities, the borrower's collateral ratio can often drop. To prevent their collateral ratio from dropping below idealCollateralRatio, borrowers can deposit extra collateral throughout the loan duration.
function depositCollateral(uint256 _amount, bool _transferFromSavingsAccount)
external payable override {
_amount: The amount of collateral msg.sender wishes to deposit
_transferFromSavingsAccount: If true, collateral is directly transferred from msg.sender's Sublime Savings Account
Adding collateral to a margin call
In case of individual margin calls, a borrower might be requested to deposit extra collateral. Note that the collateral only goes to cover the lender that requested the margin call. The borrower only has a set amount of time since the beginning of the margin call for the excess collateral to be added. If they fail to do so, collateral backing the lender might be liquidated.
function addCollateralInMarginCall(
bool _transferFromSavingsAccount
) external payable override nonReentrant {
the current time must be less than or equal to the marginCallEndTime
_lender: address of the lender that has an active margin call
_amount: amount of collateral msg.sender wishes to transfer
_transferFromSavingsAccount: If true, collateral is directly transferred from msg.sender's Sublime Savings Account. If false, collateral is transferred from msg.sender's wallet
At the end of the loan period, a pool is closed automatically when the borrower repays the last instalment. At this point, the borrower's collateral is transferred to the borrower's wallet address.
function closeLoan()
external payable override nonReentrant onlyRepaymentImpl {
only repayment contract can call this functino
the current status of the loan must be ACTIVE
Withdrawing liquidity from a pool
Lenders can withdraw liquidity from a pool if it is CLOSED, CANCELLED, DEFAULTED, or TERMINATED. They receive liquidity proportional to the number of pool tokens they hold.
function withdrawLiquidity()
the pool status must match CLOSED, CANCELLED, DEFAULTED, or TERMINATED
None: msg.sender's balance is checked for the relevant pool tokens
The following describes the typical flow while creating loan requests via pools:
Borrower can create a pool by posting a collateral and specifying the various parameters for the pool.
Lenders can lend borrow tokens to the pool in return for poolTokens till the pool start time
If the total amount lent is higher than the min amount specified by borrower, then pool starts.
If the total amount lent is lower than the min amount specified, then pool is cancelled and borrower can withdraw their collateral without any penalty and lenders can withdraw the lent amount by burning the poolTokens.
If the pool starts, borrower can still cancel the pool within the withdrawal deadline till they withdraw the lent amount. But borrower will have to pay penalty based on the time between pool start and pool cancel times. Penalty received is distributed to the lenders in the proportion of poolTokens they hold.
Pool can liquidated if repayments aren't correctly made or collateral ratio is not maintained above threshold.
In case repayments not made, any one can liquidate the pool in return for a part of collateral as incentive. Liquidator has to supply borrow tokens equivalent to the collateral - (liquidator incentive) while liquidation.
In case of collateral ratio not maintained above threshold, any lender can liquidate collateral proportional to the poolTokens they hold by making margin calls.
Once a margin call is made, borrower has to supply enough collateral specific to the lender who made the margin call, so that borrower's collateral ratio is higher than the threshold. The collateral supplied is specific to the lender and can only be liquidated for that lender.
If all repayments are made as per the pool parameters, the pool will be closed as soon as the principal is repaid and collateral is returned back to the borrower.
PoolConstants
This function is used to query parameters which are set while creating pool and are constant
function poolConstants() returns (
uint256 borrowAmountRequested,
uint256 minborrowAmount,
uint256 loanStartTime,
uint256 loanWithdrawalDeadline,
address borrowAsset,
uint256 idealCollateralRatio,
uint256 noOfRepaymentIntervals,
uint256 repaymentInterval,
address poolSavingsStrategy
borrower : Borrower requesting loan from the pool
borrowAmountRequested : Maximum amount of borrowAsset tokens requested
minborrowAmount : Minimum amount of borrowAsset tokens below which pool will be cancelled
loanStartTime : Timestamp till which borrowAsset tokens can be lent and at which the loan starts
loanWithdrawalDeadline : Timestamp till which borrower can withdraw lent tokens after loan starts
borrowAsset : Address of token which is being borrowed
idealCollateralRatio : Ratio of collateral to the totalDebt (principal + interest) which is to be maintained subject to a volatility threshold that can be queried from poolFactory (link here). This parameter is multiplied by
10^{30}
borrowRate : Rate of interest per annum multiplied by
10^{30}
noOfRepaymentIntervals : Number of intervals in which repayment has to be made
repaymentInterval : Time in seconds per repayment interval
collateralAsset : Address of token which is posted as collateral
poolSavingsStrategy : Address of strategy into which collateral tokens are invested while pool is active
PoolVars
This function is used to query the variables used to maintain the state of the Pool
function poolVars() returns (
uint256 baseLiquidityShares,
uint256 extraLiquidityShares,
LoanStatus loanStatus,
uint256 penalityLiquidityAmount
baseLiquidityShares : Liquidity shares received by investing the collateral in pool's strategy
extraLiquidityShares : Liquidity shares received by investing the collateral added as part of margin calls
loanStatus : Status of the Pool (ref)
penalityLiquidityAmount : Amount of collateral held as penalty for cancelling the pool
This function is used to query the details of lender to the pool
function lenders(address lender) returns (
uint256 principalWithdrawn,
uint256 interestWithdrawn,
uint256 lastVoteTime,
uint256 marginCallEndTime,
uint256 extraLiquidityShares
lender : Lender of the pool
interestWithdrawn : Interest withdrawn by the lender till now
marginCallEndTime : End time of margin call if call is made, otherwise 0
extraLiquidityShares : Liquidity Shares for the extra collateral received from borrower as part of margin calls
InterestTillNow
This function is used to query the interest that is outstanding at the time of call
function interestTillNow() view returns(uint256 interest)
interest : Interest that is outstanding till now
Note: The interest changes with every second, hence every block. So please keep in mind that interest returned by the above function is at the time of query.
calculateCollateralRatio
This function is used to query the collateral ratio of any user, given the balance of the user and total collateral allocated to the user.
function calculateCollateralRatio(uint256 balance, uint256 liquidityShares) returns (uint256 ratio)
balance : Pool Token balance of user to calculate collateral ratio
liquidityShares : Liquidity shares of collateral to calculate collateral ratio
ratio : Collateral ratio calculated
getCurrentCollateralRatio
This function is used to query current collateral ratio of pool
function getCurrentCollateralRatio() returns (uint256 ratio)
RETURNS : Collateral ratio of the pool
Note: Collateral ratio might change every block as prices of the assets might change every block. Hence note that collateral ratio returned is for the block at which query is made.
This function is used to query current collateral ratio of lender
function getCurrentCollateralRatio(address lender) returns (uint256 ratio)
lender : Address of lender
This function is used to query interest per second for a specified principal
function interestPerSecond(uint256 principal) view returns (uint256)
principal : Principal for which interestPerSecond is calculated
RETURNS : Interest per second for the specified principal multiplied by
10^{30}
interestPerPeriod
This function is used to query interest per instalment period for a specified principal
function interestPerPeriod(uint256 principal) view returns (uint256)
principal : Principal for which interestPerPeriod is calculated
RETURNS : Interest per period for the specified principal multiplied by
10^{30}
calculateCurrentPeriod
This function is used to calculate the current instalment period of the pool
function calcualteCurrentPeriod() returns (uint256)
RETURNS : Current Instalment period
calculateRepaymentWithdrawable
This function is used to query the amount of borrow tokens that can be withdrawn as repayment
function calculateRepaymentWithdrawable(address lender) returns (uint256)
lender : Lender to calculate repayments received for
RETURNS : Repayment that lender can withdraw
getMarginCallEndTime
This function is used to query the end time of the margin call for a lender
function getMarginCallEndTime(address lender) view returns (uint256)
lender : Lender to calculate margin call end time for
RETURNS : Timestamp at which margin call will end, if margin call is active otherwise 0
This function is used to query the total amount of tokens lent to the Pool
function totalSupply() view rfunction totalSupply() public view
override(ERC20Upgradeable, IPool) returns (uint256)
RETURNS : Total number of borrow tokens lent to the pool
As borrow tokens cannot be withdrawn or lent while pool is active, this function will returns constant value while pool is active.
getLoanStatus
This function is used to query the status of the Pool
function getLoanStatus() view returns (uint256)
RETURNS : Status of the pool, reference to the returned values is below
LoanStatus
getEquivalentTokens
This function is used to query equivalent tokens for an amount given source and target tokens based on their prices
function getEquivalentTokens(address source, address target, uint256 amount) returns (uint256)
source : Address of the source token
target : Address of the target token in which equivalent tokens are to be calculated
amount : Amount of source tokens to calculate equivalent target tokens for
RETURNS : Equivalent target tokens for the amount of source tokens specified
Note: Equivalent tokens might change every block as prices of the assets might change every block. Hence note that equivalent tokens returned is for the block at which query is made.
This function is used to query borrower of the pool
function borrower() view returns (address)
RETURNS : Address of the borrower of the pool
function/modifier
Not Borrower
OnlyBorrower
Not Lender
Not Repayments contract
onlyRepaymentImpl
can’t deposit 0
Lender can’t deposit collateral
_initialDeposit
Pool not active
addCollateralInMarginCall
Margin call ended
Pool hasn’t started or withdrawal deadline passed
withdrawBorrowedAmount
Minimum borrowable amount not reached
Borrower can’t lend
Lender not verified
Not in collection period
Token transfers paused
can’t transfer to borrower
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Price variance is the actual unit cost of an item less its standard cost, multiplied by the quantity of actual units purchased. The standard cost of an item is its expected or budgeted cost based on engineering or production data. The variance shows that some costs need to be addressed by management because they are exceeding or not meeting the expected costs.
Price variance is the actual unit cost of a purchased item, minus its standard cost, multiplied by the quantity of actual units purchased.
Price variance is a crucial factor in budget preparation.
A price variance shows that some costs need to be addressed by management because they are exceeding or not meeting the expected costs.
How Price Variance Works in Cost Accounting
Price variance is important for budgeting and planning purposes, particularly when companies are deciding what quantities of items to order. The formula for price variance is:
\begin{aligned} &\text{Price Variance} = ( \text{P} - \text{Standard Price} ) \times \text{Q} \\ &\textbf{where:} \\ &\text{P} = \text{Actual Price} \\ &\text{Q} = \text{Actual Quantity} \\ \end{aligned}
Price Variance=(P−Standard Price)×Qwhere:P=Actual PriceQ=Actual Quantity
Based on the equation above, a positive price variance means the actual costs have increased over the standard price, and a negative price variance means the actual costs have decreased over the standard price.
In cost accounting, price variance comes into play when a company is planning its annual budget for the following year. The standard price is the price a company's management team thinks it should pay for an item, which is normally an input for its own product or service. Since the standard price of an item is determined months prior to actually purchasing the item, price variance occurs if the actual price at the time of purchase is higher or lower than the standard price determined in the planning stage of the company's annual budget.
The most common example of price variance occurs when there is a change in the number of units required to be purchased. For example, at the beginning of the year, when a company is planning for Q4, it forecasts it needs 10,000 units of an item at a price of $5.50. Since it is purchasing 10,000 units, it receives a discount of 10%, bringing the per unit cost down to $5. When the company gets to Q4, however, if it only needs 8,000 units of that item, the company will not receive the 10% discount it initially planned, which brings the per unit cost to $5.50 and the price variance to 50 cents per unit.
Achieving a Favorable Price Variance
A company might achieve a favorable price variance by buying goods in bulk or large quantities, but this strategy brings the risk of excess inventory. Buying smaller quantities is also risky because the company may run out of supplies, which can lead to an unfavorable price variance. Businesses must plan carefully using data to effectively its price variances.
Sales price variance is the difference between the price a business expects to sell its products or services for and what it actually sells them for.
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Home : Support : Online Help : Mathematics : Discrete Mathematics : Combinatorics : Iterator : Mixed Radix : Rank
compute the rank of an ordinary mixed radix tuple
Rank(a,m)
{list,rtable}; tuple
Rank computes the rank of an ordinary mixed radix tuple.
The a parameter is the tuple. It is a list or one-dimensional rtable of nonnegative integers. The first element is the low-order element.
\mathrm{with}\left(\mathrm{Iterator}:-\mathrm{MixedRadix}\right):
Print the ranks of mixed radix tuples.
\mathrm{radices}≔[3,1,4,1]:
M≔\mathrm{Iterator}:-\mathrm{MixedRadixTuples}\left(\mathrm{radices}\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{MixedRadixTuples}}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\right)
\mathbf{for}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}M\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{printf}\left("%2d : %d\n",\mathrm{Rank}\left(a,\mathrm{radices}\right),a\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
The Iterator[MixedRadix][Rank] command was introduced in Maple 2016.
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Index Generator - APL Wiki
(Redirected from Index generator)
This page is about generating indices for an array of a given size. See Indexing, Indices, Index of, and Interval Index for other operations named after indices.
Index Generator (⍳) or Interval, often called by the name of its glyph Iota, is a monadic primitive function which returns an array of indices with shape given by the right argument. In the result, the element at each index is that index.
Originally, Iota was defined only on a single number. In APL/360, and all later APLs, ⍳l for a scalar l returns a simple numeric vector of length l counting from the index origin up. In nested APLs, this result may be seen as an array of scalar indices; NARS extended Iota to allow a vector argument by making the result for a non-singleton vector be an array of vector indices. Flat APLs do not use this extension, but may extend to multiple argument elements in different ways. A Dictionary of APL defines Iota to have function rank 0, and SHARP APL gives it rank 1 but requires each row (1-cell) of the argument to have length 1. In A+ and J Iota of a vector returns an array which counts up in ravel order, that is, it matches Iota of its bound, reshaped. A+ refers to this function as Interval while J calls it Integers.
1.1 Vector arguments
1.2 Negative arguments
2 Scalar-vector discrepancy in nested APLs
⍳n returns the first n indices in order:
⍳ 5
This result depends on index origin: if the index origin is set to zero, it starts counting from zero instead:
⎕IO ← 0
An argument of 0 produces the empty vector, Zilde:
⍬ ≡ ⍳ 0
In A+ and J, Iota always returns a simple array. The equivalence ⍳V
{\displaystyle \Leftrightarrow }
V⍴⍳×/V defines the result for non-singleton vectors. If the argument is an empty vector, the result is a scalar: the index origin (which is always 0 in these languages).
⍳ 2 3 4
⍳ 0⍴0
Works in: A+
We might call such a function a "ravel index generator" since the result is an array of scalar indices into its own ravel. In contrast, nested APLs, if they implement Iota for a vector argument (APL2 and APLX, for example, do not), use indices of elements:
⍳ 2 5
│1 1│1 2│1 3│1 4│1 5│
⍳⍬
(⊂⍬) ≡ ⍳⍬
Works in: NARS, NARS2000, Dyalog APL, GNU APL, ngn/apl, dzaima/APL
The result of ⍳⍬ should have shape ⍬, implying it is a scalar, and its only element must be the only possible index into a scalar, the empty vector. This means it must be ⊂⍬. However, Dyalog APL returned the scalar ⎕IO instead prior to version 13.0.
Two languages extend Iota to allow negative arguments. In both cases, the shape of the result is the absolute value of the argument.
In the nested APL NARS2000, negating one number in the argument subtracts that value from each index along the corresponding axis.
⍳2 ¯3
1 ¯2 1 ¯1 1 0
Works in: NARS2000, ⎕FEATURE[⎕IO]←1
In J, a negative number reverses the result along that axis.
i. 2 _3
Scalar-vector discrepancy in nested APLs
In NARS and later nested APLs which share its definition, the result of Iota is simple when the argument is a singleton and nested otherwise. This is because, when the argument is a singleton, each index in the result is represented as a scalar rather than a 1-element vector. dzaima/APL breaks from this definition by using vector indices whenever the argument is a vector, and scalar indices if the argument is a scalar, so that the result is only simple if the argument was scalar. This definition has the property that the shape of each index matches the shape of the argument.
Such an extension breaks compatibility with earlier non-nested APLs: although these APLs required the argument of Iota to be a singleton, they always allowed a scalar or vector, and sometimes any singleton. This is because a shape to be passed to Iota would often naturally be a vector: for example, Shape always returns a vector, and a function such as Tally to return a scalar length did not exist.
NARS defines its extension using the equivalence ⍳R
{\displaystyle \Leftrightarrow }
⊃∘.,/⍳¨R based on an Outer Product reduction. For a singleton vector R, this does yield a simple result, because ∘.,/ leaves a singleton argument unchanged. However, this is arguably due to inconsistent design in Reduce: NARS's creator, Bob Smith, argues that Catenate reduction should yield a DOMAIN ERROR, and the same arguments apply to its Outer Product.[1]
The Greek letter iota, corresponding to the Latin letter i, now usually treated as a mnemonic for "integers" or "index", more likely stood for "identity" in early Iverson notation. Iverson's 1960 paper The Description of Finite Sequential Processes states that "the identity permutation vector
{\displaystyle \iota }
{\displaystyle \iota _{j}=j}
."[2] Sinc{\displaystyle e}
is a more typical letter to use for the identity permutation, it's possible that Iverson's choice was also motivated by the convention of using
{\displaystyle \iota }
for an inclusion map, where iota now indicates "inclusion" or "injection".[3] The term "inclusion" typically indicates that a particular injection is meant to be treated as an identity.
{\displaystyle \iota _{o}(n)}
, "Interval", was used in A Programming Language for the vector of the first
{\displaystyle n}
indices starting at index origin
{\displaystyle o}
. Both arguments were optional, with
{\displaystyle n}
being implied by conformability if omitted. It was adopted with the obvious alterations in APL\360, and extended to a vector argument for nested APLs by NARS and for flat arrays by A+.
The name "iota" has sometimes been used to indicate an increasing sequence of integers even in languages other than APL. In the C++11 standard library, std::iota() fills an iterator with such a sequence, and was named after the APL glyph.[4][3][5] The Go language also uses the name iota as a predeclared identifier which represents increasing integers: each time it is used within a single constant declaration its value will be one higher.[6] The popular Scheme list library SRFI 1 also includes a function called iota in reference to APL, which takes a count as well as optional starting value (default 0) and step size (default 1) arguments.[7] The ArrayFire library for general-purpose computing on graphics processing units has an iota function with a result similar to that of A+ and J when given a vector argument.[8]
BQN (as Range)
↑ Smith, Bob. "Reduction of Singletons". Minnowbrook 2010.
↑ Ken Iverson. The Description of Finite Sequential Processes. Proceedings of a Conference on Information Theory, C. Cherry and W. Jackson, Editors, Imperial College, London, 1960-08-29.
↑ 3.0 3.1 Sean Parent. "#iotashaming". sean-parent.stlab.cc. 2019-01-04.
↑ cppreference.com. std::iota. Retrieved 2020-04-28.
↑ Rob Mayoff. Answer to "What does iota of std::iota stand for?" on Stack Overflow. Retrieved 2020-04-28.
↑ The Go Programming Language Specification. Iota. 2020-01-14.
↑ Olin Shivers. SRFI 1: List Library. Finalized 1999-10-09.
↑ ArrayFire: Functions. iota. Jun 2, 2015.
Retrieved from ‘https://aplwiki.com/index.php?title=Index_Generator&oldid=7101’
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Pool Factory - Sublime Docs
In the following sections we elaborate the PoolFactory.sol specification.
Pool Factory is used to create pools that allows a borrower to borrow from a group of lenders. Pool Factory stores the various global parameters to manage pools. Borrowers need to be verified to be able to create a pool.
Can update logic contract implementation addresses
Can update thresholds for parameters
Users verified by one of the supported Verifiers
Can call the createPool() to create new pools for loan requests
A verified user can create a lending pool to borrow from a group of lenders. The following function is used to create a pool.
uint256 _poolSize,
uint256 _borrowRate,
address _borrowToken,
uint256 _collateralRatio,
uint256 _volatilityThreshold,
uint256 _repaymentInterval,
uint256 _noOfRepaymentIntervals,
address _poolSavingsStrategy,
uint256 _collateralAmount,
bool _transferFromSavingsAccount,
bytes32 _salt,
address _lenderVerifier
msg.sender should approve collateral tokens to the pool to be created, by pre-generating the pool address.
msg.sender: verified users only
msg.value: Only if collateralToken is address(0) (Ether) should be == _collateralAmount
Note: All the parameters might be subject to range specified by admin. We threshold the following parameters:
_collateralRatio
\in
[collateralRatioLimit.min, collateralRatioLimit.max]
_borrowRate
\in
[borrowRateLimit.min, borrowRateLimit.max]
_noOfRepaymentIntervals
\in
[noOfRepaymentIntervalsLimit.min, noOfRepaymentIntervalsLimit.max]
_repaymentInterval
\in
[repaymentIntervalLimit.min, repaymentIntervalLimit.max]
These ranges have been defined purely to better manage risks, especially during the early stages. This thresholding can be removed at a later stage.
_poolSize: loan amount requested
_borrowRate: Interest rate per annum for borrowing. The interest rate fraction should be multiplied by 10**30. e.g. interest rate of 10% should be converted to 10**29
_borrowToken: Borrow Asset requested. If Ether, address(0) to be used
_collateralToken: Asset posted as collateral for the loan. If Ether, address(0) to be used
_collateralRatio: Limit on the ratio of collateral to the debt (principal + interest), which if falls below a threshold, results in liquidation (Refer to Broken link). The _collateralRatio should be multiplied by 10**30
_repaymentInterval: Interval in seconds after which 1 instalment of interest repayment for loan should be repaid
_noOfRepaymentIntervals: Number of instalments in which interest of loan will be repaid
_poolSavingsStrategy: Whitelisted strategy in which the collateral tokens should be invested in Savings Account
_collateralAmount: Amount of asset to be deposited as collateral for the loan
_transferFromSavingsAccount: Flag to determine if collateral will be added from the Savings Account or directly transferred from the borrower's wallet
_salt: random and unique initial seed used in pre-generation of the pool address
_verifier: Verifier that the borrower chooses to represent their identity for this pool. A modifer (onlyBorrower()) is passed this parameter to check whether they've actually been verified by _verifier or not
_lenderVerifier: Verifier that lenders that wish to participate in the pool need to be verified by
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Study on the Cure Kinetic Behavior of Thermosetting Polyurethane Solids and Foams: Effect of Temperature, Density, and Carbon Nanofiber | J. Eng. Mater. Technol. | ASME Digital Collection
Bipul Barua,
Bipul Barua
Saha, M. C., Barua, B., and Mohan, S. (December 3, 2010). "Study on the Cure Kinetic Behavior of Thermosetting Polyurethane Solids and Foams: Effect of Temperature, Density, and Carbon Nanofiber." ASME. J. Eng. Mater. Technol. January 2011; 133(1): 011015. https://doi.org/10.1115/1.4002649
Cure kinetic behavior was studied for both thermosetting polyurethane (PU) solids and foams. The effects of cure temperature, foam density, and carbon nanofiber (CNF) contents were examined. Cure studies were performed experimentally by measuring the evolution of complex shear modulus as a function of time using an advanced polymer analyzer operating in dynamic shear mode. Isothermal cure behavior of PU solid and foams was investigated at four different temperatures, namely,
25°C
45°C
60°C
80°C
and at three different amounts of CNF, namely, 0.01%, 0.05%, and 0.1% by weight. The cure data were analyzed by using an autocatalytic cure kinetic model. The cure behavior of both solid and foam was found to be temperature dependent. Addition of CNF was also found to affect the cure behavior of the PU foam. It was observed that the PU foam with 0.1% CNF shows the highest polymerization reaction compared with the neat foam. It was also observed that the reaction rate constants follow an Arrhenius dependence on temperature, whereas the reaction orders remain fairly constant. A simple predictive model using the reaction orders indicated that the maximum cure reaction rate was occurred at 37.5% conversion.
carbon fibres, curing, nanofabrication, nanofibres, polymer foams, polymerisation, reaction rate constants, shear modulus
Carbon, Density, Foams (Chemistry), Nanofibers, Polymerization, Shear modulus, Solids, Temperature, Urethane elastomers, Reaction rate constants, Hardening (Curing), Polymers
Polyurethane Handbook
Handbook of Polyurethanes
The Thermal Decomposition of Isocyanurates
Br. Polym. J.
Analysis of Rigid Polyurethane Foam as a Shock Mitigator
Proceedings of the SESA Fall Meeting
, Indianapolis, IN, pp.
The Mechanical Properties of Cellular Solids
Effect of Density, Microstructure, and Strain Rate on Compression Behavior of Polymeric Foams
Effect of Strain Rate and Temperature on Compression Properties of Polyurethane Foam
Proceedings of the SAMPE
, Baltimore, MD, p.
Proceedings of the SPE (Automotive Division)
, Troy, MI.
Fabrication, Synthesis and Characterization of Nanoparticles Infused Polyurethane Foams
Effect of Nanoparticles on Mode—I Fracture Toughness of Polurethane Foams
Effect of Ultrasound Sonication in Carbon Nanofiber/Polyurethane Foam Composites
J. -T.
Chemorheological Studies on a Thermoset PU/Clay Nanocomposite System
Dantas Neto
DSC Monitoring of the Cure Kinetics of a Castor-Oil based Polyurethane
M. -K.
A Study of Cure Kinetics by the Use of Dynamic Differential Scanning Calorimetry
Differential Scanning Calorimetry of Epoxy Cure: Isothermal Cure Kinetics
Effect of Filler on Cure Behavior of an Epoxy System: Cure Modeling
Rheological Cure Characterization of Phosphazene-Triazine Polymers
Kulichikhin
Rheokinetics of Curing of Epoxy Resins Near the Glass Transition
Kinetic Studies of Thermoset Cure Reactions: A Review
Kinetics and Energetics of a Fast Polyurethane Cure
Sapochak
FTIR Analysis of Chemical Gradients in Thermally Processed Molded Polyurethane Foam
Monitoring of Polyurethane Foam Cure
J. Cell. Plast.
Material Data Sheet AT-415, Utah Foam Products, Inc., Salt Lake City, UT.
Applied Sciences, Inc., Cedarville, Ohio 45314.
Thermal and Kinetic Characterization of Thermosetting Resins During Cure
,” Ph.D. thesis, McGill University, Montreal, QC, Canada.
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Map (higher-order function) - Wikipedia
For the similarly-titled abstract data type composed of (key,value) pairs, see Associative array.
Find sources: "Map" higher-order function – news · newspapers · books · scholar · JSTOR (November 2012)
In many programming languages, map is the name of a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type. It is often called apply-to-all when considered in functional form.
1 Examples: mapping a list
1.1 Visual example
4 Language comparison
Examples: mapping a list[edit]
Suppose we have a list of integers [1, 2, 3, 4, 5] and would like to calculate the square of each integer. To do this, we first define a function to square a single number (shown here in Haskell):
Afterwards we may call
>>> map square [1, 2, 3, 4, 5]
which yields [1, 4, 9, 16, 25], demonstrating that map has gone through the entire list and applied the function square to each element.
Below, you can see a view of each step of the mapping process for a list of integers X = [0, 5, 8, 3, 2, 1] that we want to map into a new list X' according to the function
{\displaystyle f(x)=x+1}
View of processing steps when applying map function on a list
The map is provided as part of the Haskell's base prelude (i.e. "standard library") and is implemented as:
See also: Functor and Category theory
In Haskell, the polymorphic function map :: (a -> b) -> [a] -> [b] is generalized to a polytypic function fmap :: Functor f => (a -> b) -> f a -> f b, which applies to any type belonging the Functor type class.
The type constructor of lists [] can be defined as an instance of the Functor type class using the map function from the previous example:
Other examples of Functor instances include trees:
-- a simple binary tree
Mapping over a tree yields:
>>> fmap square (Fork (Fork (Leaf 1) (Leaf 2)) (Fork (Leaf 3) (Leaf 4)))
Fork (Fork (Leaf 1) (Leaf 4)) (Fork (Leaf 9) (Leaf 16))
For every instance of the Functor type class, fmap is contractually obliged to obey the functor laws:
fmap id ≡ id -- identity law
fmap (f . g) ≡ fmap f . fmap g -- composition law
where . denotes function composition in Haskell.
Among other uses, this allows defining element-wise operations for various kinds of collections.
Moreover, if F and G are two functors, a natural transformation is a function of polymorphic type
{\displaystyle h:\forall T.F(T)\to G(T)}
which respects fmap:
{\displaystyle h_{Y}\circ \operatorname {fmap} (f)=\operatorname {fmap} (f)\circ h_{X}}
{\displaystyle f:X\to Y}
If the h function is defined by parametric polymorphism as in the type definition above, this specification is always satisfied.
The mathematical basis of maps allow for a number of optimizations. The composition law ensures that both
(map f . map g) list and
map (f . g) list
lead to the same result; that is,
{\displaystyle \operatorname {map} (f)\circ \operatorname {map} (g)=\operatorname {map} (f\circ g)}
. However, the second form is more efficient to compute than the first form, because each map requires rebuilding an entire list from scratch. Therefore, compilers will attempt to transform the first form into the second; this type of optimization is known as map fusion and is the functional analog of loop fusion.[1]
Map functions can be and often are defined in terms of a fold such as foldr, which means one can do a map-fold fusion: foldr f z . map g is equivalent to foldr (f . g) z.
The implementation of map above on singly linked lists is not tail-recursive, so it may build up a lot of frames on the stack when called with a large list. Many languages alternately provide a "reverse map" function, which is equivalent to reversing a mapped list, but is tail-recursive. Here is an implementation which utilizes the fold-left function.
reverseMap f = foldl (\ys x -> f x : ys) []
Since reversing a singly linked list is also tail-recursive, reverse and reverse-map can be composed to perform normal map in a tail-recursive way, though it requires performing two passes over the list.
Language comparison[edit]
The map function originated in functional programming languages.
The language Lisp introduced a map function called maplist[2] in 1959, with slightly different versions already appearing in 1958.[3] This is the original definition for maplist, mapping a function over successive rest lists:
maplist[x;f] = [null[x] -> NIL;T -> cons[f[x];maplist[cdr[x];f]]]
The function maplist is still available in newer Lisps like Common Lisp,[4] though functions like mapcar or the more generic map would be preferred.
Squaring the elements of a list using maplist would be written in S-expression notation like this:
(maplist (lambda (l) (sqr (car l))) '(1 2 3 4 5))
Using the function mapcar, above example would be written like this:
(mapcar (function sqr) '(1 2 3 4 5))
Today mapping functions are supported (or may be defined) in many procedural, object-oriented, and multi-paradigm languages as well: In C++'s Standard Library, it is called std::transform, in C# (3.0)'s LINQ library, it is provided as an extension method called Select. Map is also a frequently used operation in high level languages such as ColdFusion Markup Language (CFML), Perl, Python, and Ruby; the operation is called map in all four of these languages. A collect alias for map is also provided in Ruby (from Smalltalk). Common Lisp provides a family of map-like functions; the one corresponding to the behavior described here is called mapcar (-car indicating access using the CAR operation). There are also languages with syntactic constructs providing the same functionality as the map function.
Map is sometimes generalized to accept dyadic (2-argument) functions that can apply a user-supplied function to corresponding elements from two lists. Some languages use special names for this, such as map2 or zipWith. Languages using explicit variadic functions may have versions of map with variable arity to support variable-arity functions. Map with 2 or more lists encounters the issue of handling when the lists are of different lengths. Various languages differ on this. Some raise an exception. Some stop after the length of the shortest list and ignore extra items on the other lists. Some continue on to the length of the longest list, and for the lists that have already ended, pass some placeholder value to the function indicating no value.
In languages which support first-class functions and currying, map may be partially applied to lift a function that works on only one value to an element-wise equivalent that works on an entire container; for example, map square is a Haskell function which squares each element of a list.
Map in various languages
Map 2 lists
Map n lists
Handling lists of different lengths
APL func list list1 func list2 func/ list1 list2 list3 list4 APL's array processing abilities make operations like map implicit length error if list lengths not equal or 1
Common Lisp (mapcar func list) (mapcar func list1 list2) (mapcar func list1 list2 ...) stops after the length of the shortest list
C++ std::transform(begin, end, result, func) std::transform(begin1, end1, begin2, result, func) in header <algorithm>
begin, end, and result are iterators
result is written starting at result
C# ienum.Select(func)
The select clause ienum1.Zip(ienum2, func) Select is an extension method
ienum is an IEnumerable
Zip is introduced in .NET 4.0
Similarly in all .NET languages stops after the shortest list ends
CFML obj.map(func) Where obj is an array or a structure. func receives as arguments each item's value, its index or key, and a reference to the original object.
Clojure (map func list) (map func list1 list2) (map func list1 list2 ...) stops after the shortest list ends
D list.map!func zip(list1, list2).map!func zip(list1, list2, ...).map!func Specified to zip by StoppingPolicy: shortest, longest, or requireSameLength
Erlang lists:map(Fun, List) lists:zipwith(Fun, List1, List2) zipwith3 also available Lists must be equal length
Elixir Enum.map(list, fun) Enum.zip(list1, list2) |> Enum.map(fun) List.zip([list1, list2, ...]) |> Enum.map(fun) stops after the shortest list ends
F# List.map func list List.map2 func list1 list2 Functions exist for other types (Seq and Array) Throws exception
Groovy list.collect(func) [list1 list2].transpose().collect(func) [list1 list2 ...].transpose().collect(func)
Haskell map func list zipWith func list1 list2 zipWithn func list1 list2 ... n corresponds to the number of lists; predefined up to zipWith7 stops after the shortest list ends
Haxe array.map(func)
list.map(func)
Lambda.map(iterable, func)
J func list list1 func list2 func/ list1, list2, list3 ,: list4 J's array processing abilities make operations like map implicit length error if list lengths not equal
Java 8+ stream.map(func)
ECMAScript 5 array#map(func) List1.map(function (elem1, i) {
return func(elem1, List2[i]); }) List1.map(function (elem1, i) {
return func(elem1, List2[i], List3[i], ...); }) Array#map passes 3 arguments to func: the element, the index of the element, and the array. Unused arguments can be omitted. Stops at the end of List1, extending the shorter arrays with undefined items if needed.
Julia map(func, list) map(func, list1, list2) map(func, list1, list2, ..., listN) ERROR: DimensionMismatch
Logtalk map(Closure, List) map(Closure, List1, List2) map(Closure, List1, List2, List3, ...) (up to seven lists) Only the Closure argument must be instantiated. Failure
Mathematica func /@ list
Map[func, list] MapThread[func, {list1, list2}] MapThread[func, {list1, list2, ...}] Lists must be same length
Maxima map(f, expr1, ..., exprn)
maplist(f, expr1, ..., exprn) map returns an expression which leading operator is the same as that of the expressions;
maplist returns a list
OCaml List.map func list
Array.map func array List.map2 func list1 list2 raises Invalid_argument exception
PARI/GP apply(func, list) N/A
Perl map block list
map expr, list In block or expr special variable $_ holds each value from list in turn. Helper List::MoreUtils::each_array combines more than one list until the longest one is exhausted, filling the others with undef.
PHP array_map(callable, array) array_map(callable, array1,array2) array_map(callable, array1,array2, ...) The number of parameters for callable
should match the number of arrays. extends the shorter lists with NULL items
Prolog maplist(Cont, List1, List2). maplist(Cont, List1, List2, List3). maplist(Cont, List1, ...). List arguments are input, output or both. Subsumes also zipWith, unzip, all Silent failure (not an error)
Python map(func, list) map(func, list1, list2) map(func, list1, list2, ...) Returns a list in Python 2 and an iterator in Python 3. zip() and map() (3.x) stops after the shortest list ends, whereas map() (2.x) and itertools.zip_longest() (3.x) extends the shorter lists with None items
Ruby enum.collect {block}
enum.map {block} enum1.zip(enum2).map {block} enum1.zip(enum2, ...).map {block}
[enum1, enum2, ...].transpose.map {block} enum is an Enumeration stops at the end of the object it is called on (the first list); if any other list is shorter, it is extended with nil items
Rust list1.into_iter().map(func) list1.into_iter().zip(list2).map(func) the Iterator::map and Iterator::zip methods both take ownership of the original iterator and return a new one; the Iterator::zip method internally calls the IntoIterator::into_iter method on list2 stops after the shorter list ends
S-R lapply(list, func) mapply(func, list1, list2) mapply(func, list1, list2, ...) Shorter lists are cycled
Scala list.map(func) (list1, list2).zipped.map(func) (list1, list2, list3).zipped.map(func) note: more than 3 not possible. stops after the shorter list ends
Scheme (including Guile and Racket) (map func list) (map func list1 list2) (map func list1 list2 ...) lists must all have same length (SRFI-1 extends to take lists of different length)
Smalltalk aCollection collect: aBlock aCollection1 with: aCollection2 collect: aBlock Fails
Standard ML map func list ListPair.map func (list1, list2)
ListPair.mapEq func (list1, list2) For 2-argument map, func takes its arguments in a tuple ListPair.map stops after the shortest list ends, whereas ListPair.mapEq raises UnequalLengths exception
Swift sequence.map(func) zip(sequence1, sequence2).map(func) stops after the shortest list ends
XQuery 3 list ! block
for-each(list, func) for-each-pair(list1, list2, func) In block the context item . holds the current value stops after the shortest list ends
Zipping (computer science) or zip, mapping 'list' over multiple lists
^ "Map fusion: Making Haskell 225% faster"
^ J. McCarthy, K. Maling, S. Russell, N. Rochester, S. Goldberg, J. Slagle. LISP Programmer's Manual. March-April, 1959
^ J. McCarthy: Symbol Manipulating Language - Revisions of the Language. AI Memo No. 4, October 1958
^ Function MAPC, MAPCAR, MAPCAN, MAPL, MAPLIST, MAPCON in ANSI Common Lisp
Retrieved from "https://en.wikipedia.org/w/index.php?title=Map_(higher-order_function)&oldid=1068935723"
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Logical_constant Knowpia
In logic, a logical constant of a language
{\displaystyle {\mathcal {L}}}
is a symbol that has the same semantic value under every interpretation of
{\displaystyle {\mathcal {L}}}
. Two important types of logical constants are logical connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systems of logic.
One of the fundamental questions in the philosophy of logic is "What is a logical constant?";[1] that is, what special feature of certain constants makes them logical in nature?[2]
Some symbols that are commonly treated as logical constants are:
T "true"
F "false"
¬ "not"
∧ "and"
∨ "or"
→ "implies", "if...then"
∀ "for all"
∃ "there exists", "for some"
= "equals"
{\displaystyle \Box }
"necessarily"
{\displaystyle \Diamond }
"possibly"
Many of these logical constants are sometimes denoted by alternate symbols (e.g., the use of the symbol "&" rather than "∧" to denote the logical and).
Defining logical constants is a major part of the work of Gottlob Frege and Bertrand Russell. Russell returned to the subject of logical constants in the preface to the second edition (1937) of The Principles of Mathematics noting that logic becomes linguistic: "If we are to say anything definite about them, [they] must be treated as part of the language, not as part of what the language speaks about."[3] The text of this book uses relations R, their converses and complements as primitive notions, also taken as logical constants in the form aRb.
^ Peacocke, Christopher (May 6, 1976). "What is a Logical Constant?". The Journal of Philosophy. 73 (9): 221–240. doi:10.2307/2025420. Retrieved Jan 12, 2022.
^ Bertrand Russell (1937) Preface to The Principles of Mathematics, pages ix to xi
Stanford Encyclopedia of Philosophy entry on logical constants
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Rd Sharma 2018 for Class 10 Math Chapter 15 - Statistics
Rd Sharma 2018 Solutions for Class 10 Math Chapter 15 Statistics are provided here with simple step-by-step explanations. These solutions for Statistics are extremely popular among Class 10 students for Math Statistics Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2018 Book of Class 10 Math Chapter 15 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2018 Solutions. All Rd Sharma 2018 Solutions for class Class 10 Math are prepared by experts and are 100% accurate.
\left({x}_{i}\right)
\left({f}_{i}\right)
{d}_{i}={x}_{i}-A
{u}_{i}=\frac{1}{h}×{d}_{i}
{f}_{i}{u}_{i}
-
-
-
-
-
-
-
-
-
\sum {f}_{i}=30
\sum {f}_{i}{u}_{i}=4
{d}_{\mathit{i}}={x}_{i}-A\phantom{\rule{0ex}{0ex}} ={x}_{i}-42
{u}_{i}=\frac{{d}_{i}}{h}=\frac{{d}_{i}}{5}
{f}_{i}{u}_{i}
-
-
-
-
-
-
-
\sum _{}^{}{f}_{i}=70
\sum _{}^{}{f}_{i}{u}_{i}=-79
{d}_{i}={x}_{i}-A\phantom{\rule{0ex}{0ex}} ={x}_{i}-12.5
{f}_{i}{d}_{i}
\sum _{}{f}_{i}=64
\sum _{}{f}_{i}{d}_{i}=48
\overline{)X}=A+\frac{\sum _{}{f}_{i}{d}_{i}}{\sum _{}{f}_{i}}
=12.5+\frac{48}{64}\phantom{\rule{0ex}{0ex}}=12.5+0.75\phantom{\rule{0ex}{0ex}}=13.25
=\frac{3620}{140}\phantom{\rule{0ex}{0ex}}=25.857
18=18+2\left(\frac{x-20}{x+44}\right)\phantom{\rule{0ex}{0ex}}⇒2\left(\frac{x-20}{x+44}\right)=0\phantom{\rule{0ex}{0ex}}⇒x-20=0\phantom{\rule{0ex}{0ex}}⇒x=20
{d}_{i}={x}_{i}-A\phantom{\rule{0ex}{0ex}} ={x}_{i}-57
{u}_{i}=\frac{{d}_{i}}{h}=\frac{{d}_{i}}{3}
{f}_{i}{u}_{i}
-
-
-
-
-
\sum _{}^{}{f}_{i}=400
\sum _{}^{}{f}_{i}{u}_{i}=25
\sum _{}{f}_{i}=
\sum _{}{f}_{i}{u}_{i}=
=0.10+0.04\left[\frac{1}{30}×\left(-1\right)\right]\phantom{\rule{0ex}{0ex}}=0.10-\frac{0.04}{30}\phantom{\rule{0ex}{0ex}}=0.10-0.001\phantom{\rule{0ex}{0ex}}=0.099
Number of days: 0−6 6−10 10−14 14−20 20−28 28−38 38−40
Let the assume mean A = 17.
{\mathbit{u}}_{\mathbf{i}}\mathbf{=}\frac{{\mathbit{x}}_{\mathbit{i}}\mathbf{-}\mathbf{55}}{\mathbf{10}}
{\mathbit{f}}_{\mathbit{i}}{\mathbit{u}}_{\mathbit{i}}
\sum _{}{f}_{i}{u}_{i}=-370
N=\sum _{}{f}_{i}=1000, h=10, A=55, \sum _{}{f}_{i}{u}_{i}=-370
\overline{X}=55+10\left[\frac{1}{1000}×\left(-370\right)\right]\phantom{\rule{0ex}{0ex}}=55-3.7\phantom{\rule{0ex}{0ex}}=51.3 \mathrm{years}
{\mathbit{u}}_{\mathbf{i}}\mathbf{=}\frac{{\mathbit{x}}_{\mathbit{i}}\mathbf{-}\mathbf{18}}{\mathbf{2}}
{\mathbit{f}}_{\mathbit{i}}{\mathbit{u}}_{\mathbit{i}}
\sum _{}{f}_{i}{u}_{i}=-8+f
N=\sum _{}{f}_{i}=40+f, h=2, A=18, \sum _{}{f}_{i}{u}_{i}=-8+f
18=18+2\left[\frac{1}{\left(40+f\right)}×\left(-8+f\right)\right]\phantom{\rule{0ex}{0ex}}⇒0=\frac{-16+2f}{40+f}\phantom{\rule{0ex}{0ex}}⇒-16+2f=0\phantom{\rule{0ex}{0ex}}⇒f=8
{\mathbit{u}}_{\mathbit{i}}\mathbf{=}\frac{{\mathbit{x}}_{\mathbit{i}}\mathbf{-}\mathbf{50}}{\mathbf{20}}
{\mathbit{f}}_{\mathbit{i}}{\mathbit{u}}_{\mathbit{i}}
\sum _{}{f}_{i}{u}_{i}=4-{f}_{1}+{f}_{2}
h=20, A=50,
\sum _{}{f}_{i}{u}_{i}=4-{f}_{1}+{f}_{2}
50=50+20\left(\frac{4-{f}_{1}+{f}_{2}}{68+{f}_{1}+{f}_{2}}\right)\phantom{\rule{0ex}{0ex}}⇒{f}_{1}-{f}_{2}=4\phantom{\rule{0ex}{0ex}}⇒{f}_{1}=4+{f}_{2} .....\left(1\right)
{f}_{1}+{f}_{2}= 120-68=52\phantom{\rule{0ex}{0ex}}⇒4+{f}_{2}+{f}_{2}=52 \left[\mathrm{Using} \left(1\right)\right]\phantom{\rule{0ex}{0ex}}⇒{f}_{2}=24\phantom{\rule{0ex}{0ex}}\mathrm{So},\phantom{\rule{0ex}{0ex}}{f}_{1}=4+24=28
{\mathbit{u}}_{\mathbf{i}}\mathbf{=}\frac{{\mathbit{x}}_{\mathbit{i}}\mathbf{-}\mathbf{500}\mathbf{.}\mathbf{5}}{\mathbf{200}}
{\mathbit{f}}_{\mathbit{i}}{\mathbit{u}}_{\mathbit{i}}
\sum _{}{f}_{i}{u}_{i}=-36
N=\sum _{}{f}_{i}=50, h=200, A=500.5, \sum _{}{f}_{i}{u}_{i}=-36
\overline{X}=500.5+200\left[\frac{1}{50}×\left(-36\right)\right]\phantom{\rule{0ex}{0ex}}=500.5-144\phantom{\rule{0ex}{0ex}}=356.5
=94.5+\left(\frac{50-34}{33}\right)×10\phantom{\rule{0ex}{0ex}}=94.5+\frac{160}{33}\phantom{\rule{0ex}{0ex}}=94.5+4.85\phantom{\rule{0ex}{0ex}}=99.35
Rent (in Rs.): 15−25 25−35 35−45 45−55 55−65 65−75 75−85 85−95
No. of Houses: 8 10 15 25 40 20 15 7
Thus, the cumulative frequency just greater than 70 is 98 and the corresponding class is .
=19.5+\left(\frac{178.5-53}{140}\right)×5\phantom{\rule{0ex}{0ex}}=19.5+\frac{125.5}{140}×5\phantom{\rule{0ex}{0ex}}=19.5+\frac{125.5}{28}\phantom{\rule{0ex}{0ex}}=\frac{546+125.5}{28}\phantom{\rule{0ex}{0ex}}=23.98
\sum _{}{f}_{i}{x}_{i}=
\frac{1}{N}\sum _{}^{}{f}_{i}{x}_{i}=\frac{1}{40}×5880=147
=144.5+\left(\frac{20-17}{12}\right)×9\phantom{\rule{0ex}{0ex}}=144.5+\frac{27}{12}\phantom{\rule{0ex}{0ex}}=144.5+2.25\phantom{\rule{0ex}{0ex}}=146.75
l=12, h=6, f=5, F=4+x, N=20
14.4=12+\frac{6×\left(10-4-x\right)}{5}\phantom{\rule{0ex}{0ex}}⇒12=36-6x\phantom{\rule{0ex}{0ex}}⇒6x=24\phantom{\rule{0ex}{0ex}}⇒x=4
⇒x+y=10\phantom{\rule{0ex}{0ex}}⇒y=10-4=6
l=50, h=10, f=20, F=p+40, N=90
50=50+\frac{45-\left(p+40\right)}{20}×10\phantom{\rule{0ex}{0ex}}⇒0=\frac{5-p}{2}\phantom{\rule{0ex}{0ex}}⇒p=5\phantom{\rule{0ex}{0ex}}\mathrm{And}, \phantom{\rule{0ex}{0ex}}p+q+78=90\phantom{\rule{0ex}{0ex}}⇒ p+q=12\phantom{\rule{0ex}{0ex}}⇒q=12-5=7
\frac{\sum _{}fx}{\sum _{}f}=\frac{2830}{80}=35.37
\frac{\sum _{}{f}_{i}{x}_{i}}{\sum _{}{f}_{i}}=\frac{1022.5}{35}=29.2
\sum _{}{f}_{i}{x}_{i}=4225
\therefore \mathrm{Mode}=l+\frac{f-{f}_{1}}{2f-{f}_{1}-{f}_{2}}×h\phantom{\rule{0ex}{0ex}}=40+\frac{20-12}{2×20-12-11}×10\phantom{\rule{0ex}{0ex}}=40+\frac{8}{17}×10\phantom{\rule{0ex}{0ex}}=40+4.7\phantom{\rule{0ex}{0ex}}=44.7
\frac{N}{2}=34
\therefore \mathrm{Mode}=l+\frac{f-{f}_{1}}{2f-{f}_{1}-{f}_{2}}×h\phantom{\rule{0ex}{0ex}}=4000+\frac{18-4}{2×18-4-9}×1000\phantom{\rule{0ex}{0ex}}=4000+\frac{14}{23}×1000\phantom{\rule{0ex}{0ex}}=4000+608.7\phantom{\rule{0ex}{0ex}}=4608.7
\mathrm{Mode}=l+\left(\frac{{f}_{1}-{f}_{0}}{2{f}_{1}-{f}_{0}-{f}_{2}}\right)×h\phantom{\rule{0ex}{0ex}}=5+\left(\frac{80-45}{2×80-45-55}\right)×2\phantom{\rule{0ex}{0ex}}=5+\frac{35}{60}×2\phantom{\rule{0ex}{0ex}}=5+\frac{35}{30}\phantom{\rule{0ex}{0ex}}=6.2
\mathrm{Mode}=l+\left(\frac{{f}_{1}-{f}_{0}}{2{f}_{1}-{f}_{0}-{f}_{2}}\right)×h\phantom{\rule{0ex}{0ex}}=10000+\left(\frac{41-26}{2×41-26-16}\right)×5000\phantom{\rule{0ex}{0ex}}=10000+\frac{15}{40}×5000\phantom{\rule{0ex}{0ex}}=10000+1875\phantom{\rule{0ex}{0ex}}=11875
303+9p=314.88+7.68p\phantom{\rule{0ex}{0ex}}9p-7.68p=314.88-303
66=\frac{10340+55x}{260}
=\frac{2470}{1000}\phantom{\rule{0ex}{0ex}}=2.47
20=\frac{295+5p\left(20+p\right)}{15+5p}
5{p}^{2}+100p+295=300+100p\phantom{\rule{0ex}{0ex}}⇒5{p}^{2}=300-295=5\phantom{\rule{0ex}{0ex}}⇒{p}^{2}=1\phantom{\rule{0ex}{0ex}}⇒p=1
\therefore \frac{N}{2}=33
\therefore \frac{N}{2}=33
\therefore \frac{N}{2}=50
\frac{n+1}{2}
\frac{n-1}{2}
\frac{n}{2}
\frac{n}{2}+1
=\frac{1+2+3+...+n}{n}\phantom{\rule{0ex}{0ex}}=\frac{\frac{n\left(n+1\right)}{2}}{n}\phantom{\rule{0ex}{0ex}}=\frac{n+1}{2}
\overline{)\mathrm{X}}
\overline{)\mathrm{X}}+n
\overline{)\mathrm{X}}+\frac{n}{2}
\overline{)\mathrm{X}}+\frac{n+1}{2}
{x}_{1}, {x}_{2},{x}_{3},...,{x}_{n}
=\overline{)X}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}
⇒{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}=n\overline{)X}
{x}_{1}+1, {x}_{2}+2,{x}_{3}+3,...,{x}_{n}+n
\frac{\left({x}_{1}+1\right)+\left( {x}_{2}+2\right)+\left({x}_{3}+3\right)+...+\left({x}_{n}+n\right)}{n}
=\frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}+\left(1+2+3+...+n\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{n\overline{)X}+\frac{n\left(n+1\right)}{2}}{n}\phantom{\rule{0ex}{0ex}}=\overline{)X}+\frac{n+1}{2}
\frac{\mathrm{\Sigma } {f}_{i}{x}_{i}}{\mathrm{\Sigma } {f}_{i}}
\frac{1}{n}\sum _{i=1}^{n}{f}_{i} {x}_{i}
\frac{\sum _{i=1}^{n} {f}_{i}{x}_{i}}{\sum _{i=1}^{n} {x}_{i}}
\frac{\sum _{i=1}^{n}{f}_{i}{x}_{i}}{\sum _{1=1}^{n}i}
\therefore \sum _{}x=5x+30, n=5, \overline{)X}=10
\frac{3+2+3+4+3+3+p}{7}
⇒7×\left(4-1\right)=17+p\phantom{\rule{0ex}{0ex}}⇒21=17+p\phantom{\rule{0ex}{0ex}}⇒p=4
⇒\frac{6+7+x+8+y+14}{6}=9\phantom{\rule{0ex}{0ex}}⇒35+x+y=54\phantom{\rule{0ex}{0ex}}⇒x+y=54-35=19
\overline{)x}
\overline{)x}+\left(2n+1\right)
\overline{)x}+\frac{n+1}{2}
\overline{)x}+\left(n+1\right)
\overline{)x}-\frac{n+1}{2}
{x}_{1}, {x}_{2},{x}_{3},...,{x}_{n}
=\overline{)x}=\frac{{x}_{1}+{x}_{2}+...+{x}_{n}}{n}
⇒{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}=n\overline{)x}
{x}_{1}+1, {x}_{2}+2,{x}_{3}+3, ... ,{x}_{n}+n
\frac{\left({x}_{1}+1\right)+\left( {x}_{2}+2\right)+\left({x}_{3}+3\right)+...+\left({x}_{n}+n\right)}{n}
=\frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}+\left(1+2+3+...+n\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{n\overline{)x}+\frac{n\left(n+1\right)}{2}}{n}\phantom{\rule{0ex}{0ex}}=\overline{)x}+\frac{n+1}{2}
\frac{5n}{9}
⇒\frac{1+2+3+...+n}{n}=\frac{5n}{9}\phantom{\rule{0ex}{0ex}}⇒\frac{\frac{n\left(n+1\right)}{2}}{n}=\frac{5n}{9}\phantom{\rule{0ex}{0ex}}⇒\frac{n+1}{2}=\frac{5n}{9}\phantom{\rule{0ex}{0ex}}⇒9n+9=10n\phantom{\rule{0ex}{0ex}}⇒n=9
\frac{n+1}{2}
\frac{n}{2}
=\frac{1+3+5+...+\left(2n-1\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{\frac{n}{2}\left(1+2n-1\right)}{n} \left[{S}_{n}=\frac{n}{2}\left(a+l\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{2n}{2}\phantom{\rule{0ex}{0ex}}=n
\frac{{n}^{2}}{81}
=\frac{1+3+5+...+\left(2n-1\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{\frac{n}{2}\left(1+2n-1\right)}{n} \left[{S}_{n}=\frac{n}{2}\left(a+l\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{2n}{2}\phantom{\rule{0ex}{0ex}}=n
\frac{{n}^{2}}{81}
\therefore n=\frac{{n}^{2}}{81}\phantom{\rule{0ex}{0ex}}⇒n=81
\mathrm{Mean}=\frac{7+8+x+11+14}{5}\phantom{\rule{0ex}{0ex}}⇒x=\frac{40+x}{5}\phantom{\rule{0ex}{0ex}}⇒5x=40+x\phantom{\rule{0ex}{0ex}}⇒4x=40\phantom{\rule{0ex}{0ex}}⇒x=10
⇒\frac{1+2+3+...+n}{n}=15\phantom{\rule{0ex}{0ex}}⇒\frac{\frac{n\left(n+1\right)}{2}}{n}=15\phantom{\rule{0ex}{0ex}}⇒\frac{n+1}{2}=15\phantom{\rule{0ex}{0ex}}⇒n+1=30\phantom{\rule{0ex}{0ex}}⇒n=29
{x}_{1}, {x}_{2}, ...., {x}_{n} \mathrm{is} \overline{)x}
a\overline{)x}
\overline{)x}-a
\overline{)x}+a
\frac{\overline{)x}}{a}
{x}_{1}, {x}_{2}, ..., {x}_{n} \mathrm{is} \overline{)x}
\therefore \frac{{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}}{n}=\overline{)x}\phantom{\rule{0ex}{0ex}}⇒{x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}=n\overline{)x}
=\frac{\left({x}_{1}+a\right)+\left({x}_{2}+a\right)+\left({x}_{3}+a\right)+...+\left({x}_{n}+a\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{\left({x}_{1}+{x}_{2}+{x}_{3}+...+{x}_{n}\right)+\left(a+a+a+...+a\right)}{n}\phantom{\rule{0ex}{0ex}}=\frac{n\overline{)x}+na}{n}\phantom{\rule{0ex}{0ex}}=\overline{)x}+a
\overline{)x}
\frac{\overline{)x}}{m}+n
\frac{\overline{)x}}{n}+m
\overline{)x}+\frac{n}{m}
\overline{)x}+\frac{m}{n}
{y}_{1}, {y}_{2}, {y}_{3},..., {y}_{k}
\overline{)x}
⇒\frac{{y}_{1}+{y}_{2}+{y}_{3}+...+{y}_{k}}{k}=\overline{)x}\phantom{\rule{0ex}{0ex}}⇒{y}_{1}+{y}_{2}+{y}_{3}+...+{y}_{k}=k\overline{)x} .....\left(1\right)
\frac{{y}_{1}}{m}+n, \frac{{y}_{2}}{m}+n, \frac{{y}_{3}}{m}+n, ... , \frac{{y}_{k}}{m}+n
=\frac{\left(\frac{{y}_{1}}{m}+n\right)+ \left(\frac{{y}_{2}}{m}+n\right)+...+\left(\frac{{y}_{k}}{m}+n\right)}{k}\phantom{\rule{0ex}{0ex}}=\frac{\left(\frac{{y}_{1}}{m}+\frac{{y}_{2}}{m}+...+\frac{{y}_{k}}{m}\right)+ \left(n+n+...+n\right)}{k}\phantom{\rule{0ex}{0ex}}=\frac{{y}_{1}+{y}_{2}+...+{y}_{k}}{mk}+\frac{nk}{k}\phantom{\rule{0ex}{0ex}}=\frac{k\overline{)x}}{mk}+\frac{nk}{k}\phantom{\rule{0ex}{0ex}}=\frac{\overline{)x}}{m}+n
{u}_{i}=\frac{{x}_{i}-25}{10}, \mathrm{\Sigma }{f}_{i}{u}_{i}=20, \mathrm{\Sigma }{f}_{i}=100, \mathrm{then} \overline{)x}
{u}_{i}=\frac{{x}_{i}-25}{10}, \mathrm{\Sigma }{f}_{i}{u}_{i}=20 \mathrm{and} \mathrm{\Sigma }{f}_{i}=100
\frac{{x}_{i}-25}{10}
\overline{\mathrm{X}} = a + h \left(\frac{1}{N}\Sigma {f}_{i}{u}_{i}\right)
{u}_{i}
\frac{{x}_{i} + a}{h}
h\left({x}_{i} - a\right)
\frac{{x}_{i}- a}{h}
\frac{a -{x}_{i}}{h}
{u}_{i}=\frac{{x}_{i}-A}{h}\phantom{\rule{0ex}{0ex}}\mathrm{Where} {x}_{i} \mathrm{are} \mathrm{the} \mathrm{mid} \mathrm{values}, A \mathrm{is} \mathrm{assumed} \mathrm{mean} \mathrm{and} h \mathrm{is} \mathrm{class} \mathrm{size}.
\therefore \frac{N}{2}=33
\therefore \frac{N}{2}=40
\therefore \frac{N}{2}=33.5
\overline{\mathrm{X}} = \mathrm{a}+\frac{\mathrm{\Sigma } {\mathrm{f}}_{\mathrm{i}}{\mathrm{d}}_{\mathrm{i}}}{\mathrm{\Sigma } {\mathrm{f}}_{\mathrm{i}}}
{{d}_{i}}^{{\text{'}}^{s}}
a
{x}_{i}-a
\frac{N}{2}=\frac{57}{2}=28.5
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Valuation with Missing Data - MATLAB & Simulink
The Capital Asset Pricing Model (CAPM) is a venerable but often maligned tool to characterize comovements between asset and market prices. Although many issues arise in CAPM implementation and interpretation, one problem that practitioners face is to estimate the coefficients of the CAPM with incomplete stock price data.
This example shows how to use the missing data regression functions to estimate the coefficients of the CAPM. You can run the example directly using CAPMdemo.m located at matlabroot/toolbox/finance/findemos.
Given a host of assumptions that can be found in the references (see Sharpe [11], Lintner [6], Jarrow [5], and Sharpe, et. al. [12]), the CAPM concludes that asset returns have a linear relationship with market returns. Specifically, given the return of all stocks that constitute a market denoted as M and the return of a riskless asset denoted as C, the CAPM states that the return of each asset Ri in the market has the expectational form
E\left[{R}_{i}\right]={\alpha }_{i}+C+{\beta }_{i}\left(E\left[M\right]-C\right)
for assets i = 1, ..., n, where βi is a parameter that specifies the degree of comovement between a given asset and the underlying market. In other words, the expected return of each asset is equal to the return on a riskless asset plus a risk-adjusted expected market return net of riskless asset returns. The collection of parameters β1, ..., βn is called asset betas.
The beta of an asset has the form
{\beta }_{i}=\frac{\mathrm{cov}\left({R}_{i},M\right)}{\mathrm{var}\left(M\right)},
which is the ratio of the covariance between asset and market returns divided by the variance of market returns.Beta is the price volatility of a financial instrument relative to the price volatility of a market or index as a whole. Beta is commonly used with respect to equities. A high-beta instrument is riskier than a low-beta instrument. If an asset has a beta = 1, the asset is said to move with the market; if an asset has a beta > 1, the asset is said to be more volatile than the market. Conversely, if an asset has a beta < 1, the asset is said to be less volatile than the market.
The standard CAPM model is a linear model with additional parameters for each asset to characterize residual errors. For each of n assets with m samples of observed asset returns Rk,i, market returns Mk, and riskless asset returns Ck, the estimation model has the form
{R}_{k,i}={\alpha }_{i}+{C}_{k}+{\beta }_{i}\left({M}_{k}-{C}_{k}\right)+{V}_{k,i}
for samples k = 1, ..., m and assets i = 1, ..., n, where αi is a parameter that specifies the nonsystematic return of an asset, βi is the asset beta, and Vk,i is the residual error for each asset with associated random variable Vi.
The collection of parameters α1, ..., αn are called asset alphas. The strict form of the CAPM specifies that alphas must be zero and that deviations from zero are the result of temporary disequilibria. In practice, however, assets may have nonzero alphas, where much of active investment management is devoted to the search for assets with exploitable nonzero alphas.
To allow for the possibility of nonzero alphas, the estimation model generally seeks to estimate alphas and to perform tests to determine if the alphas are statistically equal to zero.
The residual errors Vi are assumed to have moments
E\left[{V}_{i}\right]=0
E\left[{V}_{i}{V}_{j}\right]={S}_{ij}
for assets i,j = 1, ..., n, where the parameters S11, ..., Snn are called residual or nonsystematic variances/covariances.
The square root of the residual variance of each asset, for example, sqrt(Sii) for i = 1, ..., n, is said to be the residual or nonsystematic risk of the asset since it characterizes the residual variation in asset prices that are not explained by variations in market prices.
Although betas can be estimated for companies with sufficiently long histories of asset returns, it is difficult to estimate betas for recent IPOs. However, if a collection of sufficiently observable companies exists that can be expected to have some degree of correlation with the new company's stock price movements, that is, companies within the same industry as the new company, it is possible to obtain imputed estimates for new company betas with the missing-data regression routines.
To illustrate how to use the missing-data regression routines, estimate betas for 12 technology stocks, where a single stock (GOOG) is an IPO.
Load dates, total returns, and ticker symbols for the 12 stocks from the MAT-file CAPMuniverse.
The assets in the model have the following symbols, where the last two series are proxies for the market and the riskless asset:
'AAPL' 'AMZN' 'CSCO' 'DELL' 'EBAY' 'GOOG' 'HPQ'
'IBM' 'INTC' 'MSFT' 'ORCL' 'YHOO' 'MARKET' 'CASH'
The data covers the period from January 1, 2000 to November 7, 2005 with daily total returns. Two stocks in this universe have missing values that are represented by NaNs. One of the two stocks had an IPO during this period and, so, has significantly less data than the other stocks.
Compute separate regressions for each stock, where the stocks with missing data have estimates that reflect their reduced observability.
fprintf(1,'Separate regressions with ');
fprintf(1,'daily total return data from %s to %s ...\n', ...
datestr(StartDate,1),datestr(EndDate,1));
fprintf(1,' %4s %-20s %-20s %-20s\n','','Alpha','Beta','Sigma');
fprintf(1,' ---- -------------------- ');
fprintf(1,'-------------------- --------------------\n');
% Estimate CAPM for each asset separately
% Estimate ideal standard errors for covariance parameters
[StdParam, StdCovar] = ecmmvnrstd(TestData, TestDesign, ...
Covar, 'fisher');
% Estimate sample standard errors for model parameters
% Set up results for output
Alpha = Param(1);
Sigma = sqrt(Covar);
StdAlpha = StdParam(1);
StdBeta = StdParam(2);
StdSigma = sqrt(StdCovar);
% Display estimates
Assets{i},Alpha(1),abs(Alpha(1)/StdAlpha(1)), ...
Beta(1),abs(Beta(1)/StdBeta(1)),Sigma(1),StdSigma(1));
This code fragment generates the following table.
Separate regressions with daily total return data from 03-Jan-2000
to 07-Nov-2005 ...
HPQ 0.0001 ( 0.1747) 1.3745 ( 24.2390) 0.0255 ( 0.0049)
IBM -0.0000 ( 0.0312) 1.0807 ( 28.7576) 0.0169 ( 0.0032)
The Beta column contains beta estimates for each stock that also have t-statistics enclosed in parentheses. For all stocks but GOOG, the hypothesis that the betas are nonzero is accepted at the 99.5% level of significance. It seems, however, that GOOG does not have enough data to obtain a meaningful estimate for beta since its t-statistic would imply rejection of the hypothesis of a nonzero beta.
The Sigma column contains residual standard deviations, that is, estimates for nonsystematic risks. Instead of t-statistics, the associated standard errors for the residual standard deviations are enclosed in parentheses.
To estimate stock betas for all 12 stocks, set up a joint regression model that groups all 12 stocks within a single design. (Since each stock has the same design matrix, this model is actually an example of seemingly unrelated regression.) The routine to estimate model parameters is ecmmvnrmle, and the routine to estimate standard errors is ecmmvnrstd.
Because GOOG has a significant number of missing values, a direct use of the missing data routine ecmmvnrmle takes 482 iterations to converge. This can take a long time to compute. For the sake of brevity, the parameter and covariance estimates after the first 480 iterations are contained in a MAT-file and are used as initial estimates to compute stock betas.
Now estimate the parameters for the collection of 12 stocks.
fprintf(1,'Grouped regression with ');
% Set up grouped asset data and design matrices
% Estimate CAPM for all assets together with initial parameter
[Param, Covar] = ecmmvnrmle(TestData, TestDesign, [], [], [],...
Param0, Covar0);
[StdParam, StdCovar] = ecmmvnrstd(TestData, TestDesign, Covar,...
'fisher');
Grouped regression with daily total return data from 03-Jan-2000
Alpha Beta Sigma
Although the results for complete-data stocks are the same, the beta estimates for AMZN and GOOG (the two stocks with missing values) are different from the estimates derived for each stock separately. Since AMZN has few missing values, the differences in the estimates are small. With GOOG, however, the differences are more pronounced.
The t-statistic for the beta estimate of GOOG is now significant at the 99.5% level of significance. However, the t-statistics for beta estimates are based on standard errors from the sample Hessian which, in contrast to the Fisher information matrix, accounts for the increased uncertainty in an estimate due to missing values. If the t-statistic is obtained from the more optimistic Fisher information matrix, the t-statistic for GOOG is 8.25. Thus, despite the increase in uncertainty due to missing data, GOOG nonetheless has a statistically significant estimate for beta.
Finally, note that the beta estimate for GOOG is 0.62 — a value that may require some explanation. Although the market has been volatile over this period with sideways price movements, GOOG has steadily appreciated in value. So, it is less tightly correlated with the market, implying that it is less volatile than the market (beta < 1).
[1] Caines, Peter E. Linear Stochastic Systems. John Wiley & Sons, Inc., 1988.
[2] Cramér, Harald. Mathematical Methods of Statistics. Princeton University Press, 1946.
[3] Dempster, A.P, N.M. Laird, and D.B Rubin. “Maximum Likelihood from Incomplete Data via the EM Algorithm.”Journal of the Royal Statistical Society, Series B. Vol. 39, No. 1, 1977, pp. 1-37.
[4] Greene, William H. Econometric Analysis. 5th ed., Pearson Education, Inc., 2003.
[5] Jarrow, R.A. Finance Theory. Prentice-Hall, Inc., 1988.
[6] Lintner, J. “The Valuation of Risk Assets and the Selection of Risky Investments in Stocks.” Review of Economics and Statistics. Vol. 14, 1965, pp. 13-37.
[7] Little, Roderick J. A and Donald B. Rubin. Statistical Analysis with Missing Data. 2nd ed., John Wiley & Sons, Inc., 2002.
[8] Meng, Xiao-Li and Donald B. Rubin. “Maximum Likelihood Estimation via the ECM Algorithm.” Biometrika. Vol. 80, No. 2, 1993, pp. 267-278.
[9] Sexton, Joe and Anders Rygh Swensen. “ECM Algorithms that Converge at the Rate of EM.” Biometrika. Vol. 87, No. 3, 2000, pp. 651-662.
[10] Shafer, J. L. Analysis of Incomplete Multivariate Data. Chapman & Hall/CRC, 1997.
[11] Sharpe, W. F. “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” Journal of Finance. Vol. 19, 1964, pp. 425-442.
[12] Sharpe, W. F., G. J. Alexander, and J. V. Bailey. Investments. 6th ed., Prentice-Hall, Inc., 1999.
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Coordination number - Wikipedia
Number of atoms, molecules or ions bonded to a molecule or crystal
In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/molecule/atom is called a ligand. This number is determined somewhat differently for molecules than for crystals.
For molecules and polyatomic ions the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds).[1] For example, [Cr(NH3)2Cl2Br2]− has Cr3+ as its central cation, which has a coordination number of 6 and is described as hexacoordinate. The common coordination numbers are 4, 6 and 8.
1 Molecules, polyatomic ions and coordination complexes
2 Simple and commonplace cases
2.1 Polyhapto ligands
3 Surfaces and reconstruction
5 Usage in quasicrystal, liquid and other disordered systems
Molecules, polyatomic ions and coordination complexes[edit]
Ball-and-stick model of gaseous U(BH4)4, which features 12-coordinate metal centre.[2]
[Co(NH3)6]3+, which features 6-coordinate metal centre with octahedral molecular geometry.
Chloro(triphenylphosphine)gold(I), which features 2-coordinate metal centre.
In chemistry, coordination number (C.N.), defined originally in 1893 by Alfred Werner, is the total number of neighbors of a central atom in a molecule or ion.[1][3] The concept is most commonly applied to coordination complexes.
Simple and commonplace cases[edit]
The most common coordination number for d-block transition metal complexes is 6. The CN does not distinguish the geometry of such complexes, i.e. octahedral vs trigonal prismatic.
For transition metal complexes, coordination numbers range from 2 (e.g., AuI in Ph3PAuCl) to 9 (e.g., ReVII in [ReH9]2−). Metals in the f-block (the lanthanoids and actinoids) can accommodate higher coordination number due to their greater ionic radii and availability of more orbitals for bonding. Coordination numbers of 8 to 12 are commonly observed for f-block elements. For example, with bidentate nitrate ions as ligands, CeIV and ThIV form the 12-coordinate ions [Ce(NO3)6]2− (ceric ammonium nitrate) and [Th(NO3)6]2−. When the surrounding ligands are much smaller than the central atom, even higher coordination numbers may be possible. One computational chemistry study predicted a particularly stable PbHe2+
15 ion composed of a central lead ion coordinated with no fewer than 15 helium atoms.[4] Among the Frank–Kasper phases, the packing of metallic atoms can give coordination numbers of up to 16.[5] At the opposite extreme, steric shielding can give rise to unusually low coordination numbers. An extremely rare instance of a metal adopting a coordination number of 1 occurs in the terphenyl-based arylthallium(I) complex 2,6-Tipp2C6H3Tl, where Tipp is the 2,4,6-triisopropylphenyl group.[6]
Polyhapto ligands[edit]
Coordination numbers become ambiguous when dealing with polyhapto ligands. For π-electron ligands such as the cyclopentadienide ion [C5H5]−, alkenes and the cyclooctatetraenide ion [C8H8]2−, the number of adjacent atoms in the π-electron system that bind to the central atom is termed the hapticity.[7] In ferrocene the hapticity, η, of each cyclopentadienide anion is five, Fe(η5-C5H5)2. Various ways exist for assigning the contribution made to the coordination number of the central iron atom by each cyclopentadienide ligand. The contribution could be assigned as one since there is one ligand, or as five since there are five neighbouring atoms, or as three since there are three electron pairs involved. Normally the count of electron pairs is taken.[8]
Surfaces and reconstruction[edit]
The coordination numbers are well defined for atoms in the interior of a crystal lattice: one counts the nearest neighbors in all directions. The number of neighbors of an interior atom is termed the bulk coordination number. For surfaces, the number of neighbors is more limited, so the surface coordination number is smaller than the bulk coordination number. Often the surface coordination number is unknown or variable.[9] The surface coordination number is also dependent on the Miller indices of the surface. In a body-centered cubic (BCC) crystal, the bulk coordination number is 8, whereas, for the (100) surface, the surface coordination number is 4.[10]
A common way to determine the coordination number of an atom is by X-ray crystallography. Related techniques include neutron or electron diffraction.[11] The coordination number of an atom can be determined straightforwardly by counting nearest neighbors.
α-Aluminium has a regular cubic close packed structure, fcc, where each aluminium atom has 12 nearest neighbors, 6 in the same plane and 3 above and below and the coordination polyhedron is a cuboctahedron. α-Iron has a body centered cubic structure where each iron atom has 8 nearest neighbors situated at the corners of a cube.
A graphite layer, carbon atoms and C–C bonds shown in black.
The two most common allotropes of carbon have different coordination numbers. In diamond, each carbon atom is at the centre of a regular tetrahedron formed by four other carbon atoms, the coordination number is four, as for methane. Graphite is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons; atoms in other layers are further away and are not nearest neighbours, giving a coordination number of 3.[12]
bcc structure
Ions with coordination number six comprise the highly symmetrical "rock salt structure".
For chemical compounds with regular lattices such as sodium chloride and caesium chloride, a count of the nearest neighbors gives a good picture of the environment of the ions. In sodium chloride each sodium ion has 6 chloride ions as nearest neighbours (at 276 pm) at the corners of an octahedron and each chloride ion has 6 sodium atoms (also at 276 pm) at the corners of an octahedron. In caesium chloride each caesium has 8 chloride ions (at 356 pm) situated at the corners of a cube and each chloride has eight caesium ions (also at 356 pm) at the corners of a cube.
In some compounds the metal-ligand bonds may not all be at the same distance. For example in PbCl2, the coordination number of Pb2+ could be said to be seven or nine, depending on which chlorides are assigned as ligands. Seven chloride ligands have Pb-Cl distances of 280–309 pm. Two chloride ligands are more distant, with a Pb-Cl distances of 370 pm.[13]
In some cases a different definition of coordination number is used that includes atoms at a greater distance than the nearest neighbours. The very broad definition adopted by the International Union of Crystallography, IUCR, states that the coordination number of an atom in a crystalline solid depends on the chemical bonding model and the way in which the coordination number is calculated.[14][15]
Some metals have irregular structures. For example, zinc has a distorted hexagonal close packed structure. Regular hexagonal close packing of spheres would predict that each atom has 12 nearest neighbours and a triangular orthobicupola (also called an anticuboctahedron or twinned cuboctahedron) coordination polyhedron.[12][16] In zinc there are only 6 nearest neighbours at 266 pm in the same close packed plane with six other, next-nearest neighbours, equidistant, three in each of the close packed planes above and below at 291 pm. It is considered to be reasonable to describe the coordination number as 12 rather than 6.[15] Similar considerations can be applied to the regular body centred cube structure where in addition to the 8 nearest neighbors there 6 more, approximately 15% more distant,[12] and in this case the coordination number is often considered to be 14.
NiAs structure
Many chemical compounds have distorted structures. Nickel arsenide, NiAs has a structure where nickel and arsenic atoms are 6-coordinate. Unlike sodium chloride where the chloride ions are cubic close packed, the arsenic anions are hexagonal close packed. The nickel ions are 6-coordinate with a distorted octahedral coordination polyhedron where columns of octahedra share opposite faces. The arsenic ions are not octahedrally coordinated but have a trigonal prismatic coordination polyhedron. A consequence of this arrangement is that the nickel atoms are rather close to each other. Other compounds that share this structure, or a closely related one are some transition metal sulfides such as FeS and CoS, as well as some intermetallics. In cobalt(II) telluride, CoTe, the six tellurium and two cobalt atoms are all equidistant from the central Co atom.[12]
Two other examples of commonly-encountered chemicals are Fe2O3 and TiO2. Fe2O3 has a crystal structure that can be described as having a near close packed array of oxygen atoms with iron atoms filling two thirds of the octahedral holes. However each iron atom has 3 nearest neighbors and 3 others a little further away. The structure is quite complex, the oxygen atoms are coordinated to four iron atoms and the iron atoms in turn share vertices, edges and faces of the distorted octahedra.[12] TiO2 has the rutile structure. The titanium atoms 6-coordinate, 2 atoms at 198.3 pm and 4 at 194.6 pm, in a slightly distorted octahedron. The octahedra around the titanium atoms share edges and vertices to form a 3-D network. The oxide ions are 3-coordinate in a trigonal planar configuration.[17]
Usage in quasicrystal, liquid and other disordered systems[edit]
First coordination number of Lennard-Jones fluid
Second coordination number of Lennard-Jones fluid
The coordination number of systems with disorder cannot be precisely defined.
The first coordination number can be defined using the radial distribution function g(r):[18][19]
{\displaystyle n_{1}=4\pi \int _{r_{0}}^{r_{1}}r^{2}g(r)\rho \,dr,}
where r0 is the rightmost position starting from r = 0 whereon g(r) is approximately zero, r1 is the first minimum. Therefore, it is the area under the first peak of g(r).
The second coordination number is defined similarly:
{\displaystyle n_{2}=4\pi \int _{r_{1}}^{r_{2}}r^{2}g(r)\rho \,dr.}
Alternative definitions for the coordination number can be found in literature, but in essence the main idea is the same. One of those definition are as follows: Denoting the position of the first peak as rp,
{\displaystyle n'_{1}=8\pi \int _{r_{0}}^{r_{p}}r^{2}g(r)\rho \,dr.}
The first coordination shell is the spherical shell with radius between r0 and r1 around the central particle under investigation.[20][21]
^ a b IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "coordination number". doi:10.1351/goldbook.C01331
^ Haaland, Arne; Shorokhov, Dmitry J.; Tutukin, Andrey V.; Volden, Hans Vidar; Swang, Ole; McGrady, G. Sean; Kaltsoyannis, Nikolas; Downs, Anthony J.; Tang, Christina Y.; Turner, John F. C. (2002). "Molecular Structures of Two Metal Tetrakis(tetrahydroborates), Zr(BH4)4 and U(BH4)4: Equilibrium Conformations and Barriers to Internal Rotation of the Triply Bridging BH4 Groups". Inorganic Chemistry. 41 (25): 6646–6655. doi:10.1021/ic020357z. PMID 12470059. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ De, A.K. (2003). A Text Book of Inorganic Chemistry. New Age International Publishers. p. 88. ISBN 978-8122413847.
^ Hermann, Andreas; Lein, Matthias; Schwerdtfeger, Peter (2007). "The Search for the Species with the Highest Coordination Number". Angewandte Chemie International Edition. 46 (14): 2444–7. doi:10.1002/anie.200604148. PMID 17315141.
^ Sinha, Ashok K. (1972). "Topologically close-packed structures of transition metal alloys". Progress in Materials Science. Elsevier BV. 15 (2): 81–185. doi:10.1016/0079-6425(72)90002-3. ISSN 0079-6425.
^ Niemeyer, Mark; Power, Philip P. (1998-05-18). "Synthesis and Solid-State Structure of 2,6-Trip2C6H3Tl (Trip=2,4,6-iPr3C6H2): A Monomeric Arylthallium(I) Compound with a Singly Coordinated Thallium Atom". Angewandte Chemie International Edition. 37 (9): 1277–1279. doi:10.1002/(SICI)1521-3773(19980518)37:9<1277::AID-ANIE1277>3.0.CO;2-1. ISSN 1521-3773. PMID 29711226.
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "hapticity". doi:10.1351/goldbook.H01881
^ Crabtree, Robert H. (2009). The Organometallic Chemistry of the Transition Metals. John Wiley & Sons. ISBN 9780470257623.
^ De Graef, Marc; McHenry, Michael E. (2007). Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry (PDF). Cambridge University Press. p. 515. ISBN 978-0-521-65151-6. Retrieved 15 March 2019.
^ "Closest Packed Structures". Chemistry LibreTexts. 2 October 2013. Retrieved 28 July 2020.
^ Massa, Werner (1999). Crystal Structure Determination (English ed.). Springer. pp. 67–92.
^ a b c d e Wells, A.F. (1984). Structural Inorganic Chemistry (5th ed.). Oxford Science Publications. ISBN 978-0198553700.
^ "II. Coordination of the atoms". Archived from the original on 2012-06-13. Retrieved 2014-11-09.
^ a b Mittemeijer, Eric J. (2010). Fundamentals of Materials Science: The Microstructure–Property Relationship using metals as model systems. Springer. ISBN 9783642105005.
^ Piróth, A.; Sólyom, Jenö (2007). Fundamentals of the Physics of Solids: Volume 1: Structure and Dynamics. Springer. p. 227. ISBN 9783540726005.
^ Diebold, Ulrike (2003). "The surface science of titanium dioxide". Surface Science Reports. 48 (5–8): 53–229. Bibcode:2003SurSR..48...53D. doi:10.1016/S0167-5729(02)00100-0. ISSN 0167-5729.
^ Waseda, Y. (1980). The Structure of Non-crystalline Materials: Liquids and Amorphous Solids. Advanced Book Program. McGraw-Hill International Book Company. ISBN 978-0-07-068426-3. Retrieved 16 October 2020.
^ Vahvaselkä, K. S.; Mangs, J. M. (1988). "X-ray diffraction study of liquid sulfur". Physica Scripta. 38 (5): 737. Bibcode:1988PhyS...38..737V. doi:10.1088/0031-8949/38/5/017.
^ Toofan, Jahansooz (1994). "A Simple Expression between Critical Radius Ratio and Coordination Number". Journal of Chemical Education. 71 (2): 147. Retrieved 3 January 2022.
^ "Errata". Retrieved 3 January 2022. {{cite journal}}: Cite journal requires |journal= (help)
Pentagonal planar
Capped octahedral
Capped trigonal prismatic
Square antiprismatic
Bicapped trigonal prismatic
Retrieved from "https://en.wikipedia.org/w/index.php?title=Coordination_number&oldid=1085187948"
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Variable-area spool orifice in an isothermal system - MATLAB - MathWorks Italia
Spool Orifice (IL)
Flow force output
Variable-area spool orifice in an isothermal system
The Spool Orifice (IL) block models a variable-area orifice between a spool and a sleeve with holes. The sleeve holes can be either a series of round or rectangular holes. The flow rate is based on the total opening area between the sleeve, holes, and spool, which extends or retracts according to the signal received at port S. Multiple Spool Orifice (IL) blocks can be connected for multiple sets of holes along a spool-sleeve pair.
If the spool displacement in your system is supplied by an external source or custom block and you would like the axial flow force to be transmitted to the system, you can use the Spool Orifice Flow Force (IL) block, which applies the same equations for force as the Spool Orifice (IL) block.
F=\frac{-{\stackrel{˙}{m}}_{A}^{2}}{\rho A}\mathrm{cos}\left(\alpha \right)\epsilon ,
{\stackrel{˙}{m}}_{A}
{\alpha }_{jet}=0.3663+0.8373\left(1-{e}^{\frac{-h}{1.848c}}\right),
where c is the Radial clearance, and h is the orifice opening.
ε is the Orifice orientation, which indicates orifice opening that is associated with a positive or negative signal at S.
For variable orifices, setting Orifice orientation to Positive spool displacement opens the orifice indicates that the orifice opens when the control member extends, while Negative spool displacement opens the orifice indicates that the orifice opens when the control member retracts.
The Leakage area, Aleak, is considered a small area open to flow when the orifice is closed, which maintains numerical continuity. Additionally, a nonzero Smoothing factor can provide increased numerical stability when the orifice is in near-closed or near-open position.
Setting Orifice parameterization to Round holes evenly distributes a user-defined number of holes along the sleeve perimeter with the equal diameters and centers aligned in the same plane.
{A}_{orifice}={n}_{0}\frac{{d}_{0}^{2}}{8}\left(\theta -\mathrm{sin}\left(\theta \right)\right)+{A}_{leak}.
The orifice opening angle, θ, is set by the control signal at S:
\theta =2{\mathrm{cos}}^{-1}\left(\theta -\frac{2\Delta S}{{d}_{0}}\right),
Aleak is
{A}_{leak}=c{d}_{0}{n}_{0}.
ΔS is the control member travel distance, ε(S - Smin), where Smin is the Control member position at closed orifice.
{A}_{\mathrm{max}}=\frac{\pi }{4}{d}_{0}^{2}{n}_{0}+{A}_{leak}.
Setting Orifice parameterization to Rectangular slot models one rectangular slot in the tube sleeve.
{A}_{orifice}=w\Delta S+{A}_{leak},
{A}_{\mathrm{max}}=w\Delta {S}_{\mathrm{max}}+{A}_{leak}.
where ΔSmax is the slot orifice Spool travel between closed and open orifice distance.
At the minimum orifice opening area, the leakage area is:
{A}_{leak}=cw.
\Delta \stackrel{^}{S}=\frac{\Delta S}{\Delta {S}_{\mathrm{max}}},
\Delta {\stackrel{^}{S}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\left(\Delta \stackrel{^}{S}\right)}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\Delta \stackrel{^}{S}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.
\Delta {S}_{smoothed}=\Delta {\stackrel{^}{S}}_{smoothed}\Delta {S}_{\mathrm{max}}.
The flow through a spool orifice is calculated by the pressure-flow rate equation:
\stackrel{˙}{m}=\frac{{C}_{d}{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{{A}_{orifice}}{A}\right)}^{2}\right)}}\frac{\Delta p}{{\left[\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right]}^{1/4}},
A is the Cross-sectional area at ports A and B.
\overline{\rho }
Aorifice, is the orifice open area, unless:
The opening is larger than or equal to the area at the Spool travel between closed and open orifice distance. The orifice area is then Amax.
The orifice opening is less than or equal to the minimum opening distance. The orifice area is then Aleak.
\Delta {p}_{crit}=\frac{\pi \overline{\rho }}{8{A}_{orifice}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2}.
Pressure loss describes the reduction of pressure in the orifice due to a decrease in area. PRloss is calculated as:
P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{{A}_{orifice}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}-{C}_{d}\frac{{A}_{orifice}}{{A}_{port}}}{\sqrt{1-{\left(\frac{{A}_{orifice}}{{A}_{port}}\right)}^{2}\left(1-{C}_{d}^{2}\right)}+{C}_{d}\frac{{A}_{orifice}}{{A}_{port}}}.
Pressure recovery describes the positive pressure change in the orifice due to an increase in area. If you do not wish to capture this increase in pressure, set Pressure recovery to Off. In this case, PRloss is 1.
Control member displacement for a variable-area orifice, in m.
Axial flow force, in N.
Orifice parameterization — Type of orifice
Hole geometry in the sleeve. The round holes are spaced evenly about the cross-sectional circumference. There is only one hole in the Rectangular slot setting.
To enable this parameter, set Orifice parametrization to Round holes.
To enable this parameter, set Orifice parametrization to Rectangular slot.
Spool travel between closed and open orifice — Control member maximum stroke
Maximum distance of the control member travel. This value provides an upper limit to calculations so that simulations do not return unphysical values.
Flow force output — Whether to model axial hydraulic force on the spool
Whether to model the axial hydraulic force on the spool. When this parameter is set to On, port F is enabled and outputs the axial force as a physical signal, in N.
To enable this parameter, set Flow force effect to On.
Spool position at closed orifice — Spool offset
Spool offset when the orifice is fully open. A positive, nonzero value indicates a partially closed orifice. A negative, nonzero value indicates an overlapped orifice that remains open for an initial displacement set by the physical signal at port S.
Cross-sectional area at ports A and B — Orifice area at conserving ports
Positive spool displacement opens the orifice (default) | Negative spool displacement opens the orifice
Direction of the area change for variable orifices. A positive opening orientation indicates an increase in the orifice opening. A negative orientation indicates a decrease in the orifice opening. The magnitude is always positive.
Spool Orifice Flow Force (IL) | Variable Overlapping Orifice (IL) | Annular Leakage (IL) | Orifice (IL)
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Local_ring Knowpia
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.
The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe.[1] The English term local ring is due to Zariski.[2]
A ring R is a local ring if it has any one of the following equivalent properties:
R has a unique maximal left ideal.
R has a unique maximal right ideal.
1 ≠ 0 and the sum of any two non-units in R is a non-unit.
1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.
If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,[3] necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals, where two ideals I1, I2 are called coprime if R = I1 + I2.
In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal. Before about 1960 many authors required that a local ring be (left and right) Noetherian, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed.
A local ring that is an integral domain is called a local domain.
All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
is a local ring (p prime, n ≥ 1). The unique maximal ideal consists of all multiples of p.
More generally, a nonzero ring in which every element is either a unit or nilpotent is a local ring.
An important class of local rings are discrete valuation rings, which are local principal ideal domains that are not fields.
{\displaystyle \mathbb {C} [[x]]}
, whose elements are infinite series
{\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}}
where multiplications are given by
{\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}}
{\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}}
, is local. Its unique maximal ideal consists of all elements which are not invertible. In other words, it consists of all elements with constant term zero.
More generally, every ring of formal power series over a local ring is local; the maximal ideal consists of those power series with constant term in the maximal ideal of the base ring.
Similarly, the algebra of dual numbers over any field is local. More generally, if F is a local ring and n is a positive integer, then the quotient ring F[X]/(Xn) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a geometric series to invert all other polynomials modulo Xn. If F is a field, then elements of F[X]/(Xn) are either nilpotent or invertible. (The dual numbers over F correspond to the case n = 2.)
Nonzero quotient rings of local rings are local.
The ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers localized at 2.
More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization; that is, the maximal ideal consists of all elements a/s with a ∈ P and s ∈ R - P.
The ring of polynomials
{\displaystyle K[x]}
{\displaystyle K}
is not local, since
{\displaystyle x}
{\displaystyle 1-x}
are non-units, but their sum is a unit.
Ring of germsEdit
To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open interval around 0 of the real line. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an equivalence relation, and the equivalence classes are what are called the "germs of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.
To see that this ring of germs is local, we need to characterize its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then by continuity there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. (Conversely, if f is invertible, then there is some g such that f(0)g(0) = 1, hence f(0) ≠ 0.)
With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs f with f(0) = 0.
Exactly the same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point. All these rings are therefore local. These examples help to explain why schemes, the generalizations of varieties, are defined as special locally ringed spaces.
Valuation theoryEdit
Local rings play a major role in valuation theory. By definition, a valuation ring of a field K is a subring R such that for every non-zero element x of K, at least one of x and x−1 is in R. Any such subring will be a local ring. For example, the ring of rational numbers with odd denominator (mentioned above) is a valuation ring in
{\displaystyle \mathbb {Q} }
Given a field K, which may or may not be a function field, we may look for local rings in it. If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with
F(P) = G(P) = 0,
is an indeterminate form at P. Considering a simple example, such as
approached along a line
Y = tX,
one sees that the value at P is a concept without a simple definition. It is replaced by using valuations.
Non-commutativeEdit
Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local.
If k is a field of characteristic p > 0 and G is a finite p-group, then the group algebra kG is local.
Some facts and definitionsEdit
Commutative caseEdit
We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ring in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R. If (R, m) is a commutative Noetherian local ring, then
{\displaystyle \bigcap _{i=1}^{\infty }m^{i}=\{0\}}
(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. The theorem is a consequence of the Artin–Rees lemma together with Nakayama's lemma, and, as such, the "Noetherian" assumption is crucial. Indeed, let R be the ring of germs of infinitely differentiable functions at 0 in the real line and m be the maximal ideal
{\displaystyle (x)}
. Then a nonzero function
{\displaystyle e^{-{1 \over x^{2}}}}
{\displaystyle m^{n}}
for any n, since that function divided by
{\displaystyle x^{n}}
is still smooth.
As for any topological ring, one can ask whether (R, m) is complete (as a uniform space); if it is not, one considers its completion, again a local ring. Complete Noetherian local rings are classified by the Cohen structure theorem.
In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field of the local ring or residue field of the point P.
If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f : R → S with the property f(m) ⊆ n.[4] These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S. For example, consider the ring morphism
{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x,y]/(x^{3},x^{2}y,y^{4})}
{\displaystyle x\mapsto x}
. The preimage of
{\displaystyle (x,y)}
{\displaystyle (x)}
. Another example of a local ring morphism is given by
{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x]/(x^{2})}
The Jacobson radical m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.[5]
For an element x of the local ring R, the following are equivalent:
x has a left inverse
x has a right inverse
x is invertible
x is not in m.
If (R, m) is local, then the factor ring R/m is a skew field. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J.
A deep theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated is a simple corollary to Nakayama's lemma. This has an interesting consequence in terms of Morita equivalence. Namely, if P is a finitely generated projective R module, then P is isomorphic to the free module Rn, and hence the ring of endomorphisms
{\displaystyle \mathrm {End} _{R}(P)}
is isomorphic to the full ring of matrices
{\displaystyle \mathrm {M} _{n}(R)}
. Since every ring Morita equivalent to the local ring R is of the form
{\displaystyle \mathrm {End} _{R}(P)}
for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.
^ Krull, Wolfgang (1938). "Dimensionstheorie in Stellenringen". J. Reine Angew. Math. (in German). 1938 (179): 204. doi:10.1515/crll.1938.179.204.
^ Zariski, Oscar (May 1943). "Foundations of a General Theory of Birational Correspondences" (PDF). Trans. Amer. Math. Soc. American Mathematical Society. 53 (3): 490–542 [497]. doi:10.2307/1990215. JSTOR 1990215.
^ Lam (2001), p. 295, Thm. 19.1.
^ "Tag 07BI".
^ The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.
Lam, T.Y. (2001). A first course in noncommutative rings. Graduate Texts in Mathematics (2nd ed.). Springer-Verlag. ISBN 0-387-95183-0.
The philosophy behind local rings
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Revision as of 22:14, 24 March 2018 by AnaCurcio (talk | contribs) (Created page with "El uso del término de 4<sup>th</sup> orden dado por una expansión de Taylor corrige el indeseable efecto de "palo de hockey".")
{\displaystyle t_{x}^{2}=t_{0}^{2}\left[1+\left({\frac {x}{t_{0}V}}\right)^{2}-{\frac {2\eta \left({\frac {x}{t_{0}V}}\right)^{4}}{1+(1+2\eta )\left({\frac {x}{t_{0}V}}\right)^{2}}}\right]}
{\displaystyle \eta ={\frac {\varepsilon -\delta }{1+2\delta }}}
{\displaystyle \varepsilon }
{\displaystyle \delta }
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RegSplit - Maple Help
Home : Support : Online Help : Programming : Names and Strings : StringTools Package : Pattern Matching : RegSplit
split a string on a regular expression
RegSplit( pattern, text )
The RegSplit(pattern, text) command splits a string text at substrings matching the regular expression pattern. The sequence of substrings of text that remain after substrings of text that match pattern have been elided is returned.
Splitting the empty string on any regular expression yields the expression sequence NULL.
If the regular expression pattern matches the empty string, an exception is raised.
Empty strings may result when adjacent matches to pattern occur within text. These can be removed as shown in the examples below.
Use StringTools[Split] to split a string at any of a set of characters. (See the examples, below.) Although this can be accomplished with RegSplit, StringTools[Split] is more efficient for this special case.
8
\mathrm{with}\left(\mathrm{StringTools}\right):
\mathrm{RegSplit}\left("ab","xabyabz"\right)
\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}
\mathrm{RegSplit}\left("ab","xababz"\right)
\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}
\mathrm{RegSplit}\left("\left(a|b\right)","xabyabz"\right)
\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}
\mathrm{RegSplit}\left("\left(\\.|,|;\right)","The Levi-Civita theorem provides a straightforward test for separability; however, because it is only a local characterization, it does not, in general, aid in the determination of separable coordinates."\right)
\textcolor[rgb]{0,0,1}{"The Levi-Civita theorem provides a straightforward test for separability"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"however"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"because it is only a local characterization"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"it does not"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"in general"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"aid in the determination of separable coordinates"}
\mathrm{RegSplit}\left("\left[a-e\right]*","xabyabz"\right)
Error, (in StringTools:-RegSplit) empty string would match
\mathrm{RegMatch}\left("\left[a-e\right]*",""\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{RegSplit}\left("\left[a-e\right]+","xabyabz"\right)
\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}
\mathrm{RegSplit}\left("\left[a-e\right]","xabyabz"\right)
\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}
\mathrm{RegSplit}\left("\left[x-z\right]","xabyabz"\right)
\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ab"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"ab"}
\mathrm{Split}\left("xabyabz","abcde"\right)
[\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{""}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}]
You can remove unwanted empty strings as follows.
\mathrm{remove}\left(\mathrm{type},[\mathrm{RegSplit}]\left("ab","abxabyabz"\right),""\right)
[\textcolor[rgb]{0,0,1}{"x"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"y"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"z"}]
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Turael - The RuneScape Wiki
The Old School RuneScape Wiki also has an article on: osrsw:Turael
Not to be confused with Torrell.
NormalFighting Lucien
Turael? (edit)? (edit) ? (edit)NormalFighting Lucien? (edit)26 January 2005 (Update)26 November 2008 (Update)? (edit)? (edit)NoYes? (edit)
? (edit)BurthorpeChaos Temple (hut)? (edit)? (edit)? (edit)? (edit)? (edit)He looks dangerous!? (edit)no map to displayfalsetrueHuman8273, 8302, 8303, 8461? (edit)8273&&SPLITPOINT&&8302&&SPLITPOINT&&8303&&SPLITPOINT&&8461? (edit)MRND? (edit)Slayer Equipment (Burthorpe)NoMale? (edit)? (edit)Versions: 2
SMW Subobject for NormalNPC ID: 8273, 8302, 8303, 8461Examine: He looks dangerous!Release date: 26 January 2005Is members only: falseIs variant of: Turael
SMW Subobject for Fighting LucienNPC ID: 8273, 8302, 8303, 8461Examine: He looks dangerous!Release date: 26 November 2008Is members only: trueIs variant of: Turael
Turael is the lowest level Slayer Master, along with his daughter Spria. Turael resides in Burthorpe, just north of the lodestone and south of the castle. There are no Slayer or combat level requirement for players to receive Slayer assignments from him.
If asked, Turael can change a player's Slayer task assigned by a different master to one that he will assign. He only does this once per assignment, and only if the first assignment is for a creature that Turael himself would not assign. After completion of Smoking Kills, changing a task in this way will reset the task count back to zero. Turael's tasks also do not award Slayer points upon completion, nor do they count for the five no-point tasks before beginning to gain them.
He is involved in the Animal Magnetism and While Guthix Sleeps quests. During While Guthix Sleeps, he is killed by Lucien and is later replaced by his daughter, Spria.
1.2 Assisting Ava
1.3 Fighting Lucien
Turael excelled in Slayer and combat, and eventually rose to the rank of a Slayer Master. Turael married and had a daughter, Spria, whom he trained even at a young age to fight various creatures. Under Turael's tutelage, Spria excelled in Slayer and combat. Turael aspired to eventually retire and have his daughter replace him as Slayer Master.
Assisting Ava[edit | edit source]
Ava, a budding scientist and young assistant to the Professor of Draynor Manor, moved into Draynor Manor and sought to improve the residence by providing a new bed. Among the materials she needed to create the bed are twigs from an undead tree in the area. Turael, or his daughter Spria if he is replaced, assists the adventurer by offering a blessed hatchet, which can cut down undead trees, in exchange for a mithril hatchet and a holy symbol of Saradomin. With Turael's, or Spria's, help, the adventurer is successfully able to provide all required materials for Ava's new bed. Turael, and also Spria, continue to offer blessed hatchets after Ava's mission is complete.
Fighting Lucien[edit | edit source]
Turael joined a band of eight heroes recruited by the adventurer to fight Lucien. During this time, Spria was formally given the position of Slayer Master and assumed Turael's responsibilities, as Turael was unable to serve his usual functions as a Slayer Master during this mission. These eight heroes jointly attempted to save the adventurer from Lucien after the adventurer's identity was compromised. Turael was one of the six who perished in the ensuing battle. Despite his considerable strength as a human, he was no match for the immensely powerful Lucien, who wielded the Staff of Armadyl at the time. During the battle, he tried to directly attack Lucien with his Steel halberd, but it simply broke without dealing any damage. On Turael's death, the adventurer vowed to avenge him. Turael was succeeded as a Slayer Master by his daughter, Spria. Statues of him were built in front of the Slayer Tower, White Knights' Castle, and Falador Park in his honour, alongside statues of the other Slayer Master who perished in the battle, Duradel.
He planned to retire and take up viticulture and give Bernald some competition after killing Lucien, had he survived the battle, with Spria replacing him. Unfortunately, he never survived to fulfil this dream, though Spria did replace him, as he wished.
Birds 10–20 Chicken 1 (3) Terrorbird 28 (34)
Cave bugs Cave bug 8 (7), 12 (8) None
Cows 10–25 Cow calf 2 (6) Cow 2 (8)
Crawling hands 10–20 Crawling hand 7, 8, 11, 12 (15, 16, 18, 19) Zombie hand, Skeletal hand 90 (100), 80 (100)
Gelatinous abomination None 6 (5) None
Trolls Troll Troll general, Mountain troll, Cliff -
Wolves Adolescent White wolf 3 (6) Big wolf, Ice wolf 73 (74), 96/132 (70/70)
*Many slayer monsters have numerous variants of the monster assigned. This list is not inclusive of all variants. Please check the monster's page for more specific information about the different variants.
'ello, and what are you after then?
After a player has performed several Slayer tasks for Turael, he will occasionally offer a special task in place of a regular Slayer task. The player can decline to take the special task without penalty. If the player accepts the special task, completing the task earns 1,000 Slayer experience (no Slayer experience is earned during the task).
Turael's basement, where his Slayer Challenge takes place.
Turael's special task is for the player to slay, in a single trip, all of the beasts in Turael's basement. The monsters are:
2 x goblin (level 1)
1 x goblin (level 11)
3 x wolf (level 15)
5 x minotaur (level 44)
His basement is found next to his house east of Challenge Mistress Fara, south-east of Turael himself. The ladder is south of the house and not marked on the map.
A statue of Turael at the Morytania Slayer Tower
{\displaystyle {\frac {w}{S}}\times 100\%}
{\displaystyle w}
{\displaystyle S}
{\displaystyle S}
. For Turael, assuming all possible assignments are available,
{\displaystyle S}
Turael speaks in:
Achievement Paths
Turael after 26 January 2005 (update)
Turael's chathead after 26 January 2005 (update)
The 'Let Us Slay' achievement's platform-specific description now refers to either Turael or Spria, according to your current Burthorpe Slayer Master.
If a player has previously completed a Troll Warzone Gelatinous Abomination task, Turael will now skip the step in his Burthorpe path that required the player to defeat Gelatinous Abominations.
Added a missing spotlight to one of the stages of Turael's second combat interface tutorial on mobile.
Clarified and standardised the text for choosing a Path in Turael's dialogue.
Turael no longer has empty dialog lines.
Resolved an issue where one of Turael's combat tutorials didn't appear when using a specific HUD layout.
Dying at the same time as Morningstar will cause his drops to appear in the Burthorpe Troll Cave instance until they are picked up. Otherwise, they can be claimed from Turael after advancing to the next path state.
Monsters killed in Turael/Spria's special slayer assignment without initially tagging them (such as via Corruption Shot/Blast) now count towards task completion.
Fixed in an issue with unrelated audio tracks playing when in Turael's house.
The NPC Contact lunar spell has updated its address book and can now find Turael.
Slayer shops will now correctly restock slayer gem packs at midnight.
When getting your first Slayer task (gelatinous abominations from Turael/Spria), the slayer counter UI is now turned on.
The slayer shop in Burthorpe will now correctly stock slayer gem packs.
Turael/Spria will now correctly point players towards south-west of Lumbridge for their spider slayer assignment.
Some dialogue for Spria and Tureal has been changed so it is not gender specific.
A spelling error on a Slayer tip from Turael has been fixed.
Turael's new hood is a recoloured version of the veteran hood.
The first task everyone gets from him is always Gelatinous abominations.
Retrieved from ‘https://runescape.wiki/w/Turael?oldid=35826638’
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Strip plot - MATLAB strips
Strip Plot of Frequency-Modulated Sinusoid
Strip Plot of Speech Signal
strips(x)
strips(x,n)
strips(x,sd,fs)
strips(x,sd,fs,scale)
strips(x) plots x in horizontal strips of length 250.
strips(x,n) plots x in strips that are each n samples long.
strips(x,sd,fs) plots x in strips of duration sd given the sample rate of fs samples per second.
strips(x,sd,fs,scale) also scales the vertical axes.
Plot two seconds of a frequency-modulated sinusoid in 0.25-second strips. Specify a sample rate of 1 kHz.
x = vco(sin(2*pi*t),[10 490],fs);
strips(x,0.25,fs)
{F}_{s}=7418\phantom{\rule{0.2777777777777778em}{0ex}}Hz
. The file contains the recording of a female voice saying the word "MATLAB®."
Plot the signal in 0.18-second long strips. Scale the vertical axes to 125%.
strips(mtlb,0.18,Fs,1.25)
Input signal, specified as a vector or matrix. If x is a matrix, the strips function plots each column of x as a horizontal strip on the same plot. The function ignores the imaginary part of complex-valued x.
n — Length
Length of strips, specified as a real positive scalar.
sd — Duration
Duration in seconds, specified as a real positive scalar. If sd is specified, then you must also specify fs.
Sample rate, specified as a real positive scalar. fs has units of hertz.
Scale factor, specified as a scalar. The strips function ignores the imaginary part of complex-valued scale.
plot | stem
|
Right angle - Simple English Wikipedia, the free encyclopedia
90° angle (π/2 radians): an angle that bisects the angle formed by two halves of a straight line
A right angle is an angle with a measurement of 90 degrees. When two lines cross each other so that all the angles have the same size, the result is four right angles. The top of the letter T is at right angles to the vertical line. Walls of buildings are usually at right angles to the floor. Two things that are at right angles are called "perpendicular" or "orthogonal". This is expressed using the
{\displaystyle \bot }
symbol (such as in
{\displaystyle \ell _{1}\,\bot \,\ell _{2}}
↑ Weisstein, Eric W. "Perpendicular". mathworld.wolfram.com. Retrieved 2020-09-21.
↑ "Perpendicular - math word definition - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-21.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Right_angle&oldid=7228518"
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Q-less QR decomposition for real-valued matrices - Simulink - MathWorks Switzerland
Q-less QR decomposition for real-valued matrices
The Real Partial-Systolic Q-less QR Decomposition block uses QR decomposition to compute the economy size upper-triangular R factor of the QR decomposition A = QR, where A is a real-valued matrix, without computing Q. The solution to A'Ax = B is x = R\R'\b.
When Regularization parameter is nonzero, the Real Partial-Systolic Q-less QR Decomposition block computes the upper-triangular factor R of the economy size QR decomposition of
\left[\begin{array}{c}\lambda {I}_{n}\\ A\end{array}\right]
where λ is the regularization parameter.
Rows of real matrix A, specified as a vector. A is an m-by-n matrix where m ≥ 2 and n ≥ 2. If A is a fixed-point data type, A must be signed and use binary-point scaling. Slope-bias representation is not supported for fixed-point data types.
Economy size QR decomposition matrix R, returned as a vector or matrix. R is an upper triangular matrix. The output at R has the same data type as the input at A(i,:).
Complex Partial-Systolic Q-less QR Decomposition | Real Partial-Systolic Q-less QR Decomposition with Forgetting Factor | Real Partial-Systolic QR Decomposition | Real Burst Q-less QR Decomposition
|
Untitled - Manim Web
Manim Web: Mathematical Animation Engine, for the web
Having a better CLI
Currently, the CLI is a bit wonky. It can break at any moment for any reason. If you try to compile a scene with a syntax error, it will crash without explaining why it failed. The way it compiles scenes isn't great: instead of getting all the needed data once, the CLI executes multiple times the Dart file. It may be interesting to use the Embedding Dart API. Also, there are way too many options (--gl, --webdev, --html, ...). It's difficult to know which flag is important.
Support other rendering back-ends
Currently, the only supported rendering API is the HTML 2D Canvas. However, using Cairo to render non-interactive animations would be faster than the 2D Canvas API: the scene code could be compiled to a native executable which would run a lot quicker compared to a JavaScript file. I started implementing the Cairo backend, but it's not finished yet: gradients aren't implemented.
3D rendering could be done with OpenGL/WebGL. However, these libraries are made to render triangles or polygons not shapes defined with Bézier curves. So, it could take a while to implement.
Precompiled videos
(suggested by JCosmos)
When rendering complex animations, it would be better to render this animation in the compilation step. Otherwise, it may lag someone's computer when running this animation. It could be done right now but it's better to wait for the Cairo backend integration and the CLI improvement.
I hope it can be done quite easily with something similar to this:
class TestScene extends Scene {
Future construct() async {
await part1();
Future part1() async {
// animations here
@precompiled
One advantage of Dart is dartdoc. It's a tool that'll generate documentation based on comments in the code. For example, a well-documented class is the AbstractDisplay (API Reference page)
AsciiMath integration
AsciiMath is an alternative syntax for
\LaTeX
. For example, when you want to draw this formula:
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
You can use this
\LaTeX
code:\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}. However, the backslash character is reserved for strings and needs to be escaped with another backslash: \\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{\\pi^2}{6}
But, in AsciiMath, it's : sum_(n=1)^oo 1/n^2 = pi^2/6. Also, most
\LaTeX
code is valid AsciiMath code.
GUI for ManimWeb
It's very unlikely this goal will be achieved this year. However, it may be interesting to have suggestions on the way it would work.
This year, it would be awesome to have projects that use ManimWeb. If someone wants to make a project with ManimWeb and has some problems, I'd gladly help.
Also, it could be nice to have a simpler logo for the Discord server and the website.
MIT 2022 © Hugo SALOU.
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Two capacitors C1 =5.2μF ± 0 .1μFandc3= 12 .2μFare joined…
{C}_{1} =5.2\mu F ± 0 .1\mu Fand{c}_{3}= 12 .2\mu Fare
joined (i)I n series (ii) In paraUel Find the net capacitance in these two cases.
A. 2.8%, 1.23%
C. 3.4%, 1.3%
D. 3.9%, 1.15%
Drawing Inference MCQ Manufacturing & Production Engineering MCQ Networking MCQ Statement & Assumption MCQ Payroll MCQ C++ Programming MCQ Algae MCQ Concrete Technology and Design MCQ General Physics MCQ Airport Engineering MCQ Angiosperms (Embryology Life Cycles) MCQ Area MCQ
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A Decentralized, Communication-Free Force Distribution Method With Application to Collective Object Manipulation | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
A Decentralized, Communication-Free Force Distribution Method With Application to Collective Object Manipulation
Shadi Tasdighi Kalat,
e-mail: stasdighikalat@wpi.edu
Siamak G. Faal,
Siamak G. Faal
Soft Robotics Laboratory,
e-mail: sghorbanifaal@wpi.edu
e-mail: cdonal@wpi.edu
Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received December 30, 2016; final manuscript received March 5, 2018; published online April 30, 2018. Editor: Joseph Beaman.
Kalat, S. T., Faal, S. G., and Onal, C. D. (April 30, 2018). "A Decentralized, Communication-Free Force Distribution Method With Application to Collective Object Manipulation." ASME. J. Dyn. Sys., Meas., Control. September 2018; 140(9): 091012. https://doi.org/10.1115/1.4039669
We present a novel approach to achieve decentralized distribution of forces in a multirobot system. In this approach, each robot in the group relies on the behavior of a cooperative virtual teammate that is defined independent of the population and formation of the real team. Consequently, such formulation eliminates the need for interagent communications or leader–follower architectures. In particular, effectiveness of the method is studied in a collective manipulation problem where the objective is to control the position and orientation of a body in time. To experimentally validate the performance of the proposed method, a new swarm agent,
Δρ
(Delta-Rho), is introduced. A multirobot system, consisting of five
Δρ
agents, is then utilized as the experimental setup. The obtained results are also compared with a norm-optimal centralized controller by quantitative metrics. Experimental results prove the performance of the algorithm in different tested scenarios and demonstrate a scalable, versatile, and robust system-level behavior.
Control theory, Mechatronics, Robotics
Algorithms, Control equipment, Errors, Robots, Robotics, Center of mass, Teams, Robustness
Ant-like Task Allocation and Recruitment in Cooperative Robots
Towards Collective Manipulation Without Inter-Agent Communication
, Pisa, Italy, Apr. 4–8, pp.
Cooperative Transport by Ants and Robots
Cooperative Manipulation With Least Number of Robots Via Robust Caging
), Kachsiung, Taiwan, July 11–14, pp.
Cooperative Enclosing and Grasping of an Object by Decentralized Mobile Robots Using Local Observation
Int. J. Soc. Rob.
Collective Transport of Complex Objects by Simple Robots: Theory and Experiments
, St. Paul, MN, May 6–10, pp.
PPSN VIII
), Birmingham, UK, Sept. 18–22, pp.
Kinematic Multi-Robot Manipulation With No Communication Using Force Feedback
), Stockholm, Sweden, May 16–21, pp. 427–432.
Zavlanos
), New Orleans, LA, Dec. 12–14, pp.
Massive Uniform Manipulation: Controlling Large Populations of Simple Robots With a Common Input Signal
), Tokyo, Japan, Nov. 3–7, pp.
), San Diego, CA, June 6–8, pp.
Moore–Penrose Pseudoinverses
Rétornaz
Dash: A Dynamic 16 g Hexapedal Robot
Tribot: A Minimally-Actuated Accessible Holonomic Hexapedal Locomotion Platform
Tasdighikalat
Hierarchical Kinematic Design of Foldable Hexapedal Locomotion Platforms
Ouijabots: Omnidirectional Robots for Cooperative Object Transport With Rotation Control Using No Communication
International Conference Distributed Autonomous Robotics Systems
), London, Nov. 7–9, pp. 117–131.
Trajectory Planning of Unicycle Mobile Robots With a Trapezoidal-Velocity Constraint
Hybrid Swarm System for Time Series Forecasting
VI Encontro Nacional De Inteligência Artif. (ENIA)
, Uberlândia, Brazil.
On the Validation of Models
Phys. Geogr.
Jellins
K-Redundant Trees for Safe and Efficient Multi-Robot Recovery in Complex Environments
Scalable Collective Impedance Control of an Object Via a Decentralized Force Control Method
, “Virtual Coordination in Collective Object Manipulation,” Master's thesis, Worcester Polytechnic Institute, Worcester, MA.
Decentralized Obstacle Avoidance in Collective Object Manipulation
), Pasadena, CA, July 24–27, pp.
GL-Link: A Novel Telerobotics-Based Platform Supporting Distributed Mechatronic Research Via the Internet
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Borel measure - Wikipedia
Measure defined on all open sets of a topological space
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets).[1] Some authors require additional restrictions on the measure, as described below.
2 On the real line
4.1 Lebesgue–Stieltjes integral
4.3 Hausdorff dimension and Frostman's lemma
4.4 Cramér–Wold theorem
{\displaystyle X}
be a locally compact Hausdorff space, and let
{\displaystyle {\mathfrak {B}}(X)}
be the smallest σ-algebra that contains the open sets of
{\displaystyle X}
; this is known as the σ-algebra of Borel sets. A Borel measure is any measure
{\displaystyle \mu }
defined on the σ-algebra of Borel sets.[2] A few authors require in addition that
{\displaystyle \mu }
is locally finite, meaning that
{\displaystyle \mu (C)<\infty }
{\displaystyle C}
. If a Borel measure
{\displaystyle \mu }
is both inner regular and outer regular, it is called a regular Borel measure. If
{\displaystyle \mu }
is both inner regular, outer regular, and locally finite, it is called a Radon measure.
On the real line[edit]
The real line
{\displaystyle \mathbb {R} }
with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case,
{\displaystyle {\mathfrak {B}}(\mathbb {R} )}
is the smallest σ-algebra that contains the open intervals of
{\displaystyle \mathbb {R} }
. While there are many Borel measures μ, the choice of Borel measure that assigns
{\displaystyle \mu ((a,b])=b-a}
for every half-open interval
{\displaystyle (a,b]}
is sometimes called "the" Borel measure on
{\displaystyle \mathbb {R} }
. This measure turns out to be the restriction to the Borel σ-algebra of the Lebesgue measure
{\displaystyle \lambda }
, which is a complete measure and is defined on the Lebesgue σ-algebra. The Lebesgue σ-algebra is actually the completion of the Borel σ-algebra, which means that it is the smallest σ-algebra that contains all the Borel sets and has a complete measure on it. Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e.,
{\displaystyle \lambda (E)=\mu (E)}
for every Borel measurable set, where
{\displaystyle \mu }
is the Borel measure described above).
Product spaces[edit]
If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets
{\displaystyle B(X\times Y)}
of their product coincides with the product of the sets
{\displaystyle B(X)\times B(Y)}
of Borel subsets of X and Y.[3] That is, the Borel functor
{\displaystyle \mathbf {Bor} \colon \mathbf {Top} _{2CHaus}\to \mathbf {Meas} }
from the category of second-countable Hausdorff spaces to the category of measurable spaces preserves finite products.
Lebesgue–Stieltjes integral[edit]
Main article: Lebesgue–Stieltjes integration
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind.[4]
One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral[5]
{\displaystyle ({\mathcal {L}}\mu )(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t).}
An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a distribution function f. In that case, to avoid potential confusion, one often writes
{\displaystyle ({\mathcal {L}}f)(s)=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt}
{\displaystyle \lim _{\varepsilon \downarrow 0}\int _{-\varepsilon }^{\infty }.}
Hausdorff dimension and Frostman's lemma[edit]
Main articles: Hausdorff dimension and Frostman lemma
Given a Borel measure μ on a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dimHaus(X) ≥ s. A partial converse is provided by the Frostman lemma:[6]
Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent:
Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure.
There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such that
{\displaystyle \mu (B(x,r))\leq r^{s}}
holds for all x ∈ Rn and r > 0.
Cramér–Wold theorem[edit]
Main article: Cramér–Wold theorem
The Cramér–Wold theorem in measure theory states that a Borel probability measure on
{\displaystyle \mathbb {R} ^{k}}
is uniquely determined by the totality of its one-dimensional projections.[7] It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
^ D. H. Fremlin, 2000. Measure Theory Archived 2010-11-01 at the Wayback Machine. Torres Fremlin.
^ Alan J. Weir (1974). General integration and measure. Cambridge University Press. pp. 158–184. ISBN 0-521-29715-X.
^ Vladimir I. Bogachev. Measure Theory, Volume 1. Springer Science & Business Media, Jan 15, 2007
^ Halmos, Paul R. (1974), Measure Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90088-9
^ Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.
^ K. Stromberg, 1994. Probability Theory for Analysts. Chapman and Hall.
Gaussian measure, a finite-dimensional Borel measure
Feller, William (1971), An introduction to probability theory and its applications. Vol. II., Second edition, New York: John Wiley & Sons, MR 0270403 .
J. D. Pryce (1973). Basic methods of functional analysis. Hutchinson University Library. Hutchinson. p. 217. ISBN 0-09-113411-0.
Ransford, Thomas (1995). Potential theory in the complex plane. London Mathematical Society Student Texts. Vol. 28. Cambridge: Cambridge University Press. pp. 209–218. ISBN 0-521-46654-7. Zbl 0828.31001.
Wiener's lemma related
Borel measure at Encyclopedia of Mathematics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Borel_measure&oldid=1088367823"
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Modular Log - Maple Help
Home : Support : Online Help : Mathematics : Number Theory : Modular Log
ModularLog(a, b, n, output = o, method = m)
(optional) keyword argument where o is a list of the names
\mathrm{result}
\mathrm{char}
\mathrm{characteristic}
method = m
(optional) keyword argument where m is one of the names
\mathrm{optimal}
\mathrm{shanks}
\mathrm{indexcalculus}
The ModularLog function computes the discrete logarithm of rationals under modular arithmetics. The base b discrete logarithm of a modulo n is a number
y
{b}^{y}=a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n
If a is not an integer, then the denominator of a must be relatively prime to the modulus n.
If the output option is not specified, then the return value is the smallest non-negative
y
, if it exists. An error message is displayed if
y
If the output option is specified, then the return value is a sequence obtained from output by replacing each instance of
\mathrm{result}
y
and each instance of
\mathrm{char}
\mathrm{characteristic}
with a number
such that for every non-negative integer
t
{b}^{st+y}=a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}n
s
is minimal. Similarly, an error message is displayed if
y
When the method option is not specified or set to
\mathrm{optimal}
, then the algorithm used is chosen automatically based on the size of the problem. Set method to
\mathrm{shanks}
to force Shanks' Baby-Step Giant-Step algorithm. Set method to
\mathrm{indexcalculus}
to force the Index Calculus algorithm.
\mathrm{with}\left(\mathrm{NumberTheory}\right):
\mathrm{ModularLog}\left(9,4,11\right)
\textcolor[rgb]{0,0,1}{3}
5
does not have a base
2
logarithm modulo
7
\mathrm{ModularLog}\left(5,2,7\right)
Error, (in NumberTheory:-ModularLog) no solutions exist
\mathrm{ModularLog}\left(17,13,101,\mathrm{output}=[\mathrm{result},\mathrm{char}]\right)
\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{50}
{\mathrm{infolevel}}_{\mathrm{ModularLog}}
2
or greater, userinfo messages will be printed. The ModularLog function prints messages at levels
2
4
\mathrm{infolevel}[\mathrm{ModularLog}]≔4
{\textcolor[rgb]{0,0,1}{\mathrm{infolevel}}}_{\textcolor[rgb]{0,0,1}{\mathrm{ModularLog}}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{4}
\mathrm{ModularLog}\left(1441,5,10007,\mathrm{method}=\mathrm{shanks}\right)
ModularLog: "using Shanks' method to compute log[25](1441) mod 10007"
\textcolor[rgb]{0,0,1}{5000}
\mathrm{ModularLog}\left(1441,5,10007,\mathrm{method}=\mathrm{indexcalculus}\right)
ModularLog: "using the index calculus method to compute log[5](1441) mod 10007"
ModularLog: "found new equation 1 out of 4"
\textcolor[rgb]{0,0,1}{5000}
\mathrm{ModularLog}\left(\frac{2}{3},6,41\right)
\textcolor[rgb]{0,0,1}{11}
The NumberTheory[ModularLog] command was introduced in Maple 2016.
|
Beam elements as approximate models of vibrating mechanical subsystems of freight wagons | JVE Journals
Andrzej Buchacz1
1Silesian University of Technology, Faculty of Mechanical Engineering, Institute of Engineering Processes Automation and Integrated Manufacturing Systems, Konarskiego 18A street, 44-100 Gliwice, Poland
Received 3 November 2016; accepted 10 November 2016; published 30 May 2017
The complex mechanical system which is a subsystem of mechatronic system composed with many continuous mechanical elements having the same length and variable cross section, loaded by the harmonic force is analyzed. The main subject of deliberation was to determine the flexibility of the mechanical subsystem with constant cross section using algorithms of exact and approximate methods. Next, the comparison of methods and the correction of the approximate method has been made. The model of subsystem of complex transverse vibrating mechanical system of freight wagon was considered and after deliberation the approximate method has been chosen. The presented research of subsystems establishes the foundation to complex systems analysis with cascade structure. By analyzing the diagrams of characteristics of confirmed system it has been determined that in case of approximate method the resonance frequencies cover with those which have been determined with the exact method. However, the values of the characteristic in other areas are different. Therefore, there is the mistake within the approximate method, which while of studying the single system does not have any influence because in resonance areas the characteristic values of the system approach to the infinity. However, the difference between values of flexibility within two methods has the great influence on the result of complex systems. That is why it was necessary to correct the results of the approximate method.
Keywords: dynamical flexibility, mechanical or mechatronic complex systems, algorithms of exact and approximate methods of analysis.
The problems concerned with mechatronic systems and problems of electrostriction and piezoelectricity were presented for example in [8-10]. Other, diverse problems have been modelled by different methods [1, 2] and next they were examined and analyzed in (e.g. [3-7]).
2. The algorithm of the exact method for obtaining the solution of transverse vibrating free beam – simple subsystem of complex beam system
In the global case the equation of motion of the beam is considered:
EIy\left(x,t{\right)}_{\text{'}xxxx}+\rho Fy\left(x,t{\right)}_{\text{'}tt}=0,
y\left(x,t\right)
is deflection at the time moment
of the lining beam section within the distance x from the beginning of the system,
E
– Young modulus,
\rho
– mass density of material of the beam,
I
– polar inertia moment of the beam cross section,
F
– area of the beam cross section.
The boundary conditions of beam system are known as: free (
F
), clamped (
C
), pinned (
P
) and sliding (
S
) ends.
The solution of Eq. (1) is the function in form of:
y\left(x,t\right)=X\left(x\right)\mathrm{s}\mathrm{i}\mathrm{n}\omega t.
After transformations own functions are obtained as:
{X}_{n}=f\left(\frac{{z}_{n}}{l}x\right), n=1,2,3.
The problem of determining of own function was presented for discrete-continuous transverse vibrating mechatronic systems in (e.g. [8-10]).
3. The algorithm of the exact method for obtaining the dynamical flexibility of transverse vibrating beam – subsystem of freight wagons
Taking into consideration that mechanical subsystem of transverse vibrating complex system is extorted with harmonic force in form
P\left(t\right)={P}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\omega t
, so one boundary condition in combination
F
C
P
S
Eq. (1) takes form:
EIy\left(x,t{\right)}_{\text{'}xxxx}+\rho Fy\left(x,t{\right)}_{\text{'}tt}=P\left(t\right).
After substituting of derivatives and transformation the deflection of the beam is considered as:
y\left(x,t\right)=f\left(E,I,k,l\right){P}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\omega t.
According to definition of dynamic flexibility it takes form:
Y=f\left(E,\mathrm{ }I,k,l\right).
The graphs of absolute value of dynamical flexibility Eq. (6), obtained for
x=l
, for combinations of boundary conditions: free end – (
F
) and pinned end – (
P
) are presented in Fig. 1 and for clamped end – (
C
) and sliding end – (
S
) – in Fig. 2.
Detailed transformations and final sentences of combinations of boundary conditions have been presented in (E.g. [8-10]).
4. The algorithm of approximate method for obtaining the dynamical flexibility of transverse vibrating beam - subsystem of freight wagons
It has to be considered that if the beam is under the action of moment with continuous factorization threw the beam length with the value
F\left(x\right)\mathrm{s}\mathrm{i}\mathrm{n}\omega t
on the length unit – then the equation of motion of the element with length
{d}_{x}
lining in the point
x
EI{y}_{\mathrm{\text{'}}xxxx}+\rho A{y}_{\mathrm{\text{'}}tt}={P}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\omega t,
{P}_{0}=F/h
Using approximate method – Galerkin’s one, the solution of Eq. (7), after transformations is given in the form:
y\left(x,t\right)={\sum }_{n=1}^{\mathrm{\infty }}{y}_{n}\left(x,t\right)={\sum }_{n=1}^{\mathrm{\infty }}{A}^{\left(n\right)}f\left(k,x\right)\mathrm{s}\mathrm{i}\mathrm{n}\omega t.
After transformation, the equation of dynamical flexibility is:
{Y}_{xl}^{\left(n\right)}=f\left(a,k,x,l,\omega \right).
In general case the dynamical flexibility at the arbitrary point of the beam after transformations gets a form of:
{Y}_{xl}={\sum }_{n=1}^{\mathrm{\infty }}{Y}_{xl}^{\left(n\right)}.
Plots of absolute values of dynamical flexibility defined by Eq. (10) for combinations of boundary conditions (
C
S
F
P
) are presented in Fig. 1 and 2.
Fig. 1. Absolute values of dynamical flexibility of the beam with combination of boundary conditions (
F
P
), for
n=
C
S
n=
The problems presented in this paper are considered to be the base and introduction to the solution of inverse task, meaning the synthesis of transverse vibrating complex mechanical or mechatronic beam systems with assumed frequency spectrum.
This work has been conducted as a part of Research Project PBS2/A6/17/2013 supported by the National Centre for Research and Development in 2013-2016.
Bellert S., Woźniacki H. The Analysis and Synthesis of Electrical Systems by Means of the Method of Structural Numbers. WNT, Warszawa, 1968, (in Polish). [Search CrossRef]
Berge C. Graphs and Hypergraphs. American Elsevier Publishing Co., Inc., New York/North Holland Publishing Co., Amsterdam-London, 1973. [Search CrossRef]
Buchacz A. Modelling, synthesis and analysis of bar systems characterized by a cascade structure represented by graphs. Mechanism and Machine Theory, Vol. 30, Issue 7, 1995, p. 969-986. [Search CrossRef]
Białas K., Buchacz A., Dzitkowski T. Synthesis of Active Mechanical Systems with Dumping in View of Polar Graphs and Structural Numbers. Monograph No. 230. Silesian University of Technology Press, Gliwice, 2009, (in Polish). [Search CrossRef]
Buchacz A., Żurek K. Reverse Task of Dynamic of Active Mechanical Systems by Means the Graphs and Structural Numbers Methods. Monograph No 81. Silesian University of Technology Press, Gliwice, 2005, (in Polish). [Search CrossRef]
Buchacz A. Exact and approximate analysis of mechanical and mechatronic systems. Journal of Achievements in Materials and Manufacturing Engineering, Vol. 33, Issue 1, 2009, p. 47-52. [Search CrossRef]
Buchacz A. Modelling, synthesis, modification, sensitivity and analysis of mechanic and mechatronic systems. Journal of Achievements in Materials and Manufacturing Engineering, International OCOSCO World Press, Vol. 24, Issue 1, 2007, p. 198-207. [Search CrossRef]
Buchacz A. The supply of formal notions to synthesis of the vibrating discrete-continuous mechatronic systems. Journal of Achievements in Materials and Manufacturing Engineering, International OCOSCO World Press, Vol. 44, Issue 2, 2011, p. 168-178. [Search CrossRef]
Callahan H. B. Vibration monitoring of cylindrical shells using piezoelectric sensors. Finite Elements in Analysis and Design, Vol. 23, 1996, p. 303-318. [Search CrossRef]
Kurnik W. Damping of mechanical vibrations utilizing shunted piezoelements. Machine Dynamics Problems, Vol. 28, Issue 4, 2004, p. 15-26. [Search CrossRef]
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Visualize prior and posterior densities of Bayesian linear regression model parameters - MATLAB plot
Plot Prior and Posterior Distributions
Plot Distributions to Separate Figures
Return Default Distribution and Evaluations
Specify Values for Density Evaluation and Plotting
pointsUsed
posteriorDensity
priorDensity
Visualize prior and posterior densities of Bayesian linear regression model parameters
plot(PriorMdl)
plot(PosteriorMdl,PriorMdl)
pointsUsed = plot(___)
[pointsUsed,posteriorDensity,priorDensity] = plot(___)
[pointsUsed,posteriorDensity,priorDensity,FigureHandle] = plot(___)
plot(PosteriorMdl) or plot(PriorMdl) plots the posterior or prior distributions of the parameters in the Bayesian linear regression model PosteriorMdl or PriorMdl, respectively. plot adds subplots for each parameter to one figure and overwrites the same figure when you call plot multiple times.
plot(PosteriorMdl,PriorMdl) plots the posterior and prior distributions in the same subplot. plot uses solid blue lines for posterior densities and dashed red lines for prior densities.
plot(___,Name,Value) uses any of the input argument combinations in the previous syntaxes and additional options specified by one or more name-value pair arguments. For example, you can evaluate the posterior or prior density by supplying values of β and σ2, or choose which parameter distributions to include in the figure.
pointsUsed = plot(___) also returns the values of the parameters that plot uses to evaluate the densities in the subplots.
[pointsUsed,posteriorDensity,priorDensity] = plot(___) also returns the values of the evaluated densities.
If you specify one model, then plot returns the density values in PosteriorDensity. Otherwise, plot returns the posterior density values in PosteriorDensity and the prior density values in PriorDensity.
[pointsUsed,posteriorDensity,priorDensity,FigureHandle] = plot(___) returns the figure handle of the figure containing the distributions.
{\text{GNPR}}_{t}={\beta }_{0}+{\beta }_{1}{\text{IPI}}_{t}+{\beta }_{2}{\text{E}}_{t}+{\beta }_{3}{\text{WR}}_{t}+{\epsilon }_{t}.
t
{\epsilon }_{t}
{\sigma }^{2}
\beta |{\sigma }^{2}\sim {N}_{4}\left(M,{\sigma }^{2}V\right)
M
V
{\sigma }^{2}\sim IG\left(A,B\right)
A
B
PriorMdl is a conjugateblm Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance.
plot plots the marginal prior distributions of the intercept, regression coefficients, and disturbance variance.
Suppose that the mean of the regression coefficients is
\left[-20\phantom{\rule{0.2777777777777778em}{0ex}}4\phantom{\rule{0.2777777777777778em}{0ex}}0.001\phantom{\rule{0.2777777777777778em}{0ex}}2{\right]}^{\prime }
and their scaled covariance matrix is
\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0.001& 0& 0\\ 0& 0& 1e-8& 0\\ 0& 0& 0& 0.1\end{array}\right].
Also, the prior scale of the disturbance variance is 0.01. Specify the prior information using dot notation.
PriorMdl.Mu = [-20; 4; 0.001; 2];
PriorMdl.V = diag([1 0.001 1e-8 0.01]);
PriorMdl.B = 0.01;
Request a new figure and plot the prior distribution.
plot replaces the current distribution figure with a plot of the prior distribution of the disturbance variance.
Load the Nelson-Plosser data set, and create variables for the predictor and response data.
y = DataTable.GNPR;
Estimate the posterior distributions.
\beta
{\sigma }^{2}
plot(PosteriorMdl);
Plot the prior and posterior distributions of the parameters on the same subplots.
plot(PosteriorMdl,PriorMdl);
Consider the regression model in Plot Prior and Posterior Distributions.
Load the Nelson-Plosser data set, create a default conjugate prior model, and then estimate the posterior using the first 75% of the data. Turn off the estimation display.
PosteriorMdlFirst = estimate(PriorMdl,X(1:floor(d*end),:),y(1:floor(d*end)),...
Plot the prior distribution and the posterior distribution of the disturbance variance. Return the figure handle.
[~,~,~,h] = plot(PosteriorMdlFirst,PriorMdl,'VarNames','Sigma2');
h is the figure handle for the distribution plot. If you change the tag name of the figure by changing the Tag property, then the next plot call places all new distribution plots on a different figure.
Change the name of the figure handle to FirstHalfData using dot notation.
h.Tag = 'FirstHalfData';
Estimate the posterior distribution using the rest of the data. Specify the posterior distribution based on the final 25% of the data as the prior distribution.
PosteriorMdl = estimate(PosteriorMdlFirst,X(ceil(d*end):end,:),...
y(ceil(d*end):end),'Display',false);
Plot the posterior of the disturbance variance based on half of the data and all the data to a new figure.
plot(PosteriorMdl,PosteriorMdlFirst,'VarNames','Sigma2');
This type of plot shows the evolution of the posterior distribution when you incorporate new data.
Load the Nelson-Plosser data set and create a default conjugate prior model.
Plot the prior distributions. Request the values of the parameters used to create the plots and their respective densities.
[pointsUsedPrior,priorDensities1] = plot(PriorMdl);
pointsUsedPrior is a 5-by-1 cell array of 1-by-1000 numeric vectors representing the values of the parameters that plot uses to plot the corresponding densities. The first element corresponds to the intercept, the next three elements correspond to the regression coefficients, and the last element corresponds to the disturbance variance. priorDensities1 has the same dimensions as pointsUsed and contains the corresponding density values.
Estimate the posterior distribution. Turn off the estimation display.
Plot the posterior distributions. Request the values of the parameters used to create the plots and their respective densities.
[pointsUsedPost,posteriorDensities1] = plot(PosteriorMdl);
pointsUsedPost and posteriorDensities1 have the same dimensions as pointsUsedPrior. The pointsUsedPost output can be different from pointsUsedPrior. posteriorDensities1 contains the posterior density values.
Plot the prior and posterior distributions. Request the values of the parameters used to create the plots and their respective densities.
[pointsUsedPP,posteriorDensities2,priorDensities2] = plot(PosteriorMdl,PriorMdl);
All output values have the same dimensions as pointsUsedPrior. The posteriorDensities2 output contains the posterior density values. The priorDensities2 output contains the prior density values.
Confirm that pointsUsedPP is equal to pointsUsedPost.
compare = @(a,b)sum(a == b) == numel(a);
cellfun(compare,pointsUsedPost,pointsUsedPP)
The points used are equivalent.
Confirm that the posterior densities are the same, but that the prior densities are not.
cellfun(compare,posteriorDensities1,posteriorDensities2)
cellfun(compare,priorDensities1,priorDensities2)
When plotting only the prior distribution, plot evaluates the prior densities at points that produce a clear plot of the prior distribution. When plotting both a prior and posterior distribution, plot prefers to plot the posterior clearly. Therefore, plot can determine a different set of points to use.
Load the Nelson-Plosser data set and create a default conjugate prior model for the regression coefficients and disturbance variance. Then, estimate the posterior distribution and obtain the estimation summary table from summarize.
summaryTbl = summarize(PosteriorMdl);
summaryTbl = summaryTbl.MarginalDistributions;
summaryTbl is a table containing the statistics that estimate displays at the command line.
For each parameter, determine a set of 50 evenly spaced values within three standard deviations of the mean. Put the values into the cells of a 5-by-1 cell vector following the order of the parameters that comprise the rows of the estimation summary table.
Points = cell(numel(summaryTbl.Mean),1); % Preallocation
for j = 1:numel(summaryTbl.Mean)
Points{j} = linspace(summaryTbl.Mean(j) - 3*summaryTbl.Std(j),...
summaryTbl.Mean(j) + 2*summaryTbl.Std(j),50);
Plot the posterior distributions within their respective intervals.
plot(PosteriorMdl,'Points',Points)
PosteriorMdl — Bayesian linear regression model storing posterior distribution characteristics
conjugateblm model object | empiricalblm model object
Bayesian linear regression model storing posterior distribution characteristics, specified as a conjugateblm or empiricalblm model object returned by estimate.
When you also specify PriorMdl, then PosteriorMdl is the posterior distribution composed of PriorMdl and data. If the NumPredictors and VarNames properties of the two models are not equal, plot issues an error.
PriorMdl — Bayesian linear regression model storing prior distribution characteristics
Bayesian linear regression model storing prior distribution characteristics, specified as a conjugateblm, semiconjugateblm, diffuseblm, empiricalblm, customblm, mixconjugateblm, mixsemiconjugateblm, or lassoblm model object returned by bayeslm.
When you also specify PosteriorMdl, then PriorMdl is the prior distribution that, when combined with the data likelihood, forms PosteriorMdl. If the NumPredictors and VarNames properties of the two models are not equal, plot issues an error.
Example: 'VarNames',["Beta1"; "Beta2"; "Sigma2"] plots the distributions of regression coefficients corresponding to the names Beta1 and Beta2 in the VarNames property of the model object and the disturbance variance Sigma2.
VarNames — Parameter names
cell vector of character vectors | string vector
Parameter names indicating which densities to plot in the figure, specified as the comma-separated pair consisting of 'VarNames' and a cell vector of character vectors or string vector. VarNames must include "Intercept", any name in the VarNames property of PriorMdl or PosteriorMdl, or "Sigma2".
By default, plot chooses "Intercept" (if an intercept exists in the model), all regression coefficients, and "Sigma2". If the model has more than 34 regression coefficients, then plot chooses the first through the 34th only.
VarNames is case insensitive.
If your model contains many variables, then try plotting subsets of the parameters on separate plots for a better view of the distributions.
Example: 'VarNames',["Beta(1)","Beta(2)"]
Points — Parameter values for density evaluation and plotting
numeric vector | cell vector of numeric vectors
Parameter values for density evaluation and plotting, specified as the comma-separated pair consisting of 'Points' and a numPoints-dimensional numeric vector or a numVarNames-dimensional cell vector of numeric vectors. numPoints is the number of parameters values that plot evaluates and plots the density.
If Points is a numeric vector, then plot evaluates and plots the densities of all specified distributions by using its elements (see VarNames).
If Points is a cell vector of numeric vectors, then:
numVarNames must be numel(VarNames), where VarNames is the value of VarNames.
Cells correspond to the elements of VarNames.
For j = 1,…,numVarNames, plot evaluates and plots the density of the parameter named VarNames{j} by using the vector of points in cell Points(j).
By default, plot determines 1000 adequate values at which to evaluate and plot the density for each parameter.
Example: 'Points',{1:0.1:10 10:0.2:25 1:0.01:2}
pointsUsed — Parameter values used for density evaluation and plotting
cell vector of numeric vectors
Parameter values used for density evaluation and plotting, returned as a cell vector of numeric vectors.
Suppose Points and VarNames are the values of Points and VarNames, respectively. If Points is a numeric vector, then PointsUsed is repmat({Points},numel(VarNames)). Otherwise, PointsUsed equals Points. Cells correspond to the names in VarNames.
posteriorDensity — Evaluated and plotted posterior densities
cell vector of numeric row vectors
Evaluated and plotted posterior densities, returned as a numVarNames-by-1 cell vector of numeric row vectors. numVarNames is numel(VarNames), where VarNames is the value of VarNames. Cells correspond to the names in VarNames. posteriorDensity has the same dimensions as priorDensity.
priorDensity — Evaluated and plotted prior densities
Evaluated and plotted prior densities, returned as a numVarNames-by-1 cell vector of numeric row vectors. priorDensity has the same dimensions as posteriorDensity.
FigureHandle — Figure window containing distributions
Figure window containing distributions, returned as a figure object.
plot overwrites the figure window that it produces.
If you rename FigureHandle for a new figure window, or call figure before calling plot, then plot continues to overwrite the current figure. To plot distributions to a different figure window, change the figure identifier of the current figure window by renaming its Tag property. For example, to rename the current figure window called FigureHandle to newFigure, at the command line, enter:
FigureHandle.Tag = newFigure;
Because improper distributions (distributions with densities that do not integrate to 1) are not well defined, plot cannot plot them very well.
\ell \left(\beta ,{\sigma }^{2}|y,x\right)=\prod _{t=1}^{T}\varphi \left({y}_{t};{x}_{t}\beta ,{\sigma }^{2}\right).
forecast | simulate | summarize | summarize
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NAD(P)+ nucleosidase - Wikipedia
NAD+ hydrolase homooctamer, Human
In enzymology, a NAD(P)+ nucleosidase (EC 3.2.2.6) is an enzyme that catalyzes the chemical reaction
NAD(P)+ + H2O
{\displaystyle \rightleftharpoons }
ADP-ribose(P) + nicotinamide
The 3 substrates of this enzyme are NAD+, NADP+, and H2O, whereas its two products are ADP-ribose and nicotinamide.
This enzyme belongs to the family of hydrolases, specifically those glycosylases that hydrolyse N-glycosyl compounds. The systematic name of this enzyme class is NAD(P)+ glycohydrolase. Other names in common use include nicotinamide adenine dinucleotide (phosphate) nucleosidase, triphosphopyridine nucleotidase, NAD(P) nucleosidase, NAD(P)ase, and nicotinamide adenine dinucleotide (phosphate) glycohydrolase. This enzyme participates in nicotinate and nicotinamide metabolism.
ALIVISATOS SG, WOOLLEY DW (1956). "Solubilization and purification of the diphosphopyridine nucleotidase from beef spleen". J. Biol. Chem. 219 (2): 823–32. PMID 13319302.
ZATMAN LJ, KAPLAN NO, COLOWICK SP (1953). "Inhibition of spleen diphosphopyridine nucleotidase by nicotinamide, an exchange reaction". J. Biol. Chem. 200 (1): 197–212. PMID 13034774.
Zatman LJ, Kaplan NO, Colowick SP, Ciotti MM (1953). "Formation of the isonicotinic acid hydrazide analog of DPN". J. Am. Chem. Soc. 75 (13): 3293–3294. doi:10.1021/ja01109a527.
Retrieved from "https://en.wikipedia.org/w/index.php?title=NAD(P)%2B_nucleosidase&oldid=1038379373"
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Development of a Spark Discharge PM Sensor for Measurement of Engine-Out Soot Emissions | J. Eng. Gas Turbines Power | ASME Digital Collection
Greg R. Pucher,
Gardiner, D. P., Pucher, G. R., Allan, W. D., and LaViolette, M. (May 18, 2011). "Development of a Spark Discharge PM Sensor for Measurement of Engine-Out Soot Emissions." ASME. J. Eng. Gas Turbines Power. November 2011; 133(11): 114502. https://doi.org/10.1115/1.4003711
There is interest in measuring the engine-out particulate emissions from diesel engines in real time in order to optimize the engine control strategies such as exhaust gas recirculation (EGR) and monitor the loading of the particulate filter. This paper presents experimental results obtained using a measurement technique that produced a signal proportional to the peak voltage of a spark discharge that was exposed to the exhaust gas. The sensor was tested on a turbocharged diesel engine with exhaust soot levels from
<0.1 FSN
>3.5 FSN
, and compared with reference measurements of filter smoke number (FSN) from an AVL 415S smokemeter. The sensor signal showed a high correlation with the reference FSN measurements. Conversion of the FSN values to mass concentration values (using published techniques for the reference instrument) indicated that the output of the spark discharge soot sensor was nearly linear with mass concentration over a substantial portion of the measuring range. The sensor showed a response time of less than 2 s to step changes in FSN levels.
diesel engines, smoke detectors
Engines, Exhaust systems, Gas discharge (Electric discharge), Sensors, Signals, Soot, Emissions, Smoke, Diesel engines
Ion Probe in the Exhaust Manifold of Diesel Engines
Real-Time Smoke Sensor for Diesel Engines
Diesel Smoke Transient Control Using a Real-Time Smoke Sensor
Particle Sensor for Diesel Combustion Monitoring
Proceedings of the DEER04 Meeting
, San Antonio, TX, Oct. 24–27, SAE Paper No. 2005-01-3792.
,” Society of Automotive Engineers, SAE Paper No. 2004-01-2906.
Smoke Particulate Sensors for OBD and High Precision Measuring
Commercial Vehicle Engineering Congress and Exhibition
, Chicago, IL, Oct. 31–Nov. 2, SAE Paper No. 2006-01-3549.
Smoke Sensing Device for Internal Combustion Engines
W. D. E.
Development of a Smoke Sensor for Diesel Engines
SAE Powertrain Congress
, Pittsburgh, PA, Oct. SAE Paper No. 03-01-3084.
,” ASME Paper No. ICEF2005-1302.
,” ASME Paper No. ICEF2010-35141.
A Theoretical Model for the Correlation of Smoke Number to Dry Particulate Concentration in Diesel Exhaust
AVL 415S Smoke Meter, The New Correlation Curve
,” AVL Application Note, updated Oct. 15.
Spark Ignition: Its Physics and Effect on the Internal Combustion Engine
Fuel Economy in Road Vehicles Powered by Spark Ignition Engines
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Generate Code for quadprog - MATLAB & Simulink
First Steps in quadprog Code Generation
Modify Example for Efficiency
This example shows how to generate code for the quadprog optimization solver. Code generation requires a MATLAB® Coder™ license. For details about code generation requirements, see Code Generation for quadprog Background.
The problem is to minimize the quadratic expression
\frac{1}{2}{x}^{T}Hx+{f}^{T}x
H=\left[\begin{array}{ccc}1& -1& 1\\ -1& 2& -2\\ 1& -2& 4\end{array}\right]
f=\left[\begin{array}{c}2\\ -3\\ 1\end{array}\right]
0\le x\le 1
\sum x=1/2
Create a file named test_quadp.m containing the following code.
function [x,fval] = test_quadp
f = [2;-3;1];
Aeq = ones(1,3);
[x,fval] = quadprog(H,f,[],[],Aeq,beq,lb,ub,x0,opts)
Generate code for the test_quadp file.
codegen -config:mex test_quadp
After some time, codegen creates a MEX file named test_quadp_mex.mexw64 (the file extension varies, depending on your system). Run the resulting C code.
[x,fval] = test_quadp_mex
Following some of the suggestions in the topic Optimization Code Generation for Real-Time Applications, configure the generated code to have fewer checks and to use static memory allocation.
Create a file named test_quadp2.m containing the following code. This code sets a looser optimality tolerance than the default 1e-8.
function [x,fval,eflag,output] = test_quadp2
opts = optimoptions('quadprog','Algorithm','active-set',...
[x,fval,eflag,output] = quadprog(H,f,[],[],Aeq,beq,lb,ub,x0,opts)
Generate code for the test_quadp2 file.
codegen -config cfg test_quadp2
[x,fval,eflag,output] = test_quadp2_mex
Changing the optimality tolerance does not affect the optimization process, because the 'active-set' algorithm does not check this tolerance until it reaches a point where it stops.
Create a third file that limits the number of allowed iterations to 2 to see the effect on the optimization process.
function [x,fval,exitflag,output] = test_quadp3
opts = optimoptions('quadprog','Algorithm','active-set','MaxIterations',2);
[x,fval,exitflag,output] = quadprog(H,f,[],[],Aeq,beq,lb,ub,x0,opts)
To see the effect of these settings on the solver, run test_quadp3 in MATLAB without generating code.
[x,fval,exitflag,output] = test_quadp3
quadprog stopped because it exceeded the iteration limit,
options.MaxIterations = 2.000000e+00.
message: '↵Solver stopped prematurely.↵↵quadprog stopped because it exceeded the iteration limit,↵options.MaxIterations = 2.000000e+00.↵↵'
In this case, the solver reached the solution in fewer steps than the default. Usually, though, limiting the number of iterations does not allow the solver to reach a correct solution.
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Sophomore's dream - Wikipedia
Identity expressing an integral as a sum
In mathematics, the sophomore's dream is the pair of identities (especially the first)
{\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}\\\end{aligned}}}
{\displaystyle {\begin{aligned}\int _{0}^{1}x^{x}\,dx&=\sum _{n=1}^{\infty }(-1)^{n+1}n^{-n}=-\sum _{n=1}^{\infty }(-n)^{-n}\end{aligned}}}
discovered in 1697 by Johann Bernoulli.
The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively.
The name "sophomore's dream"[1] is in contrast to the name "freshman's dream" which is given to the incorrect[note 1] identity (x + y)n = xn + yn. The sophomore's dream has a similar too-good-to-be-true feel, but is true.
1.1 Historical proof
Graph of the functions y = xx (red, lower) and y = x−x (grey, upper) on the interval x ∈ (0, 1].
The proofs of the two identities are completely analogous, so only the proof of the second is presented here. The key ingredients of the proof are:
to write xx = exp(x log x) (using the notation log for the natural logarithm and exp for the exponential function;
to expand exp(x log x) using the power series for exp; and
to integrate termwise, using integration by substitution.
In details, one expands xx as
{\displaystyle x^{x}=\exp(x\log x)=\sum _{n=0}^{\infty }{\frac {x^{n}(\log x)^{n}}{n!}}.}
{\displaystyle \int _{0}^{1}x^{x}\,dx=\int _{0}^{1}\sum _{n=0}^{\infty }{\frac {x^{n}(\log x)^{n}}{n!}}\,dx.}
By uniform convergence of the power series, one may interchange summation and integration to yield
{\displaystyle \int _{0}^{1}x^{x}\,dx=\sum _{n=0}^{\infty }\int _{0}^{1}{\frac {x^{n}(\log x)^{n}}{n!}}\,dx.}
To evaluate the above integrals, one may change the variable in the integral via the substitution
{\displaystyle x=\exp \left(-{\frac {u}{n+1}}\right).}
With this substitution, the bounds of integration are transformed to
{\displaystyle 0<u<\infty ,}
giving the identity
{\displaystyle \int _{0}^{1}x^{n}(\log x)^{n}\,dx=(-1)^{n}(n+1)^{-(n+1)}\int _{0}^{\infty }u^{n}e^{-u}\,du.}
By Euler's integral identity for the Gamma function, one has
{\displaystyle \int _{0}^{\infty }u^{n}e^{-u}\,du=n!,}
{\displaystyle \int _{0}^{1}{\frac {x^{n}(\log x)^{n}}{n!}}\,dx=(-1)^{n}(n+1)^{-(n+1)}.}
Summing these (and changing indexing so it starts at n = 1 instead of n = 0) yields the formula.
Historical proof[edit]
The original proof, given in Bernoulli,[2] and presented in modernized form in Dunham,[3] differs from the one above in how the termwise integral
{\displaystyle \int _{0}^{1}x^{n}(\log x)^{n}\,dx}
is computed, but is otherwise the same, omitting technical details to justify steps (such as termwise integration). Rather than integrating by substitution, yielding the Gamma function (which was not yet known), Bernoulli used integration by parts to iteratively compute these terms.
The integration by parts proceeds as follows, varying the two exponents independently to obtain a recursion. An indefinite integral is computed initially, omitting the constant of integration
{\displaystyle +C}
both because this was done historically, and because it drops out when computing the definite integral. One may integrate
{\displaystyle \int x^{m}(\log x)^{n}\,dx}
by taking u = (log x)n and dv = xm dx, which yields:
{\displaystyle {\begin{aligned}\int x^{m}(\log x)^{n}\,dx&={\frac {x^{m+1}(\log x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m+1}{\frac {(\log x)^{n-1}}{x}}\,dx\qquad {\text{(for }}m\neq -1{\text{)}}\\&={\frac {x^{m+1}}{m+1}}(\log x)^{n}-{\frac {n}{m+1}}\int x^{m}(\log x)^{n-1}\,dx\qquad {\text{(for }}m\neq -1{\text{)}}\end{aligned}}}
(also in the list of integrals of logarithmic functions). This reduces the power on the logarithm in the integrand by 1 (from
{\displaystyle n}
{\displaystyle n-1}
) and thus one can compute the integral inductively, as
{\displaystyle \int x^{m}(\log x)^{n}\,dx={\frac {x^{m+1}}{m+1}}\cdot \sum _{i=0}^{n}(-1)^{i}{\frac {(n)_{i}}{(m+1)^{i}}}(\log x)^{n-i}}
where (n) i denotes the falling factorial; there is a finite sum because the induction stops at 0, since n is an integer.
In this case m = n, and they are integers, so
{\displaystyle \int x^{n}(\log x)^{n}\,dx={\frac {x^{n+1}}{n+1}}\cdot \sum _{i=0}^{n}(-1)^{i}{\frac {(n)_{i}}{(n+1)^{i}}}(\log x)^{n-i}.}
Integrating from 0 to 1, all the terms vanish except the last term at 1,[note 2] which yields:
{\displaystyle \int _{0}^{1}{\frac {x^{n}(\log x)^{n}}{n!}}\,dx={\frac {1}{n!}}{\frac {1^{n+1}}{n+1}}(-1)^{n}{\frac {(n)_{n}}{(n+1)^{n}}}=(-1)^{n}(n+1)^{-(n+1)}.}
This is equivalent to computing Euler's integral identity
{\displaystyle \Gamma (n+1)=n!}
for the Gamma function on a different domain (corresponding to changing variables by substitution), as Euler's identity itself can also be computed via an analogous integration by parts.
^ Incorrect in general, but correct when one is working in a commutative ring of prime characteristic p with n being a power of p. The correct result in a general commutative context is given by the binomial theorem.
^ All the terms vanish at 0 because
{\displaystyle \lim _{x\to 0^{+}}x^{m}(\log x)^{n}=0}
by l'Hôpital's rule (Bernoulli omitted this technicality), and all but the last term vanish at 1 since log 1 = 0.
Bernoulli, Johann (1697). Opera omnia. Vol. 3. pp. 376–381.
Borwein, Jonathan; Bailey, David H.; Girgensohn, Roland (2004). Experimentation in Mathematics: Computational Paths to Discovery. pp. 4, 44. ISBN 9781568811369.
Dunham, William (2005). "Chapter 3: The Bernoullis (Johann and
{\displaystyle x^{x}}
)". The Calculus Gallery, Masterpieces from Newton to Lebesgue. Princeton University Press. pp. 46–51. ISBN 9780691095653.
OEIS, (sequence A083648 in the OEIS) and (sequence A073009 in the OEIS)
Pólya, George; Szegő, Gábor (1998), "Part I, problem 160", Problems and Theorems in Analysis, p. 36, ISBN 9783540636403
Weisstein, Eric W. "Sophomore's Dream". MathWorld.
Max R. P. Grossmann (2017): Sophomore's dream. 1,000,000 digits of the first constant
Literature for x^x and Sophomore's Dream, Tetration Forum, 03/02/2010
The Coupled Exponential, Jay A. Fantini, Gilbert C. Kloepfer, 1998
Sophomore's Dream Function, Jean Jacquelin, 2010, 13 pp.
Lehmer, D. H. (1985). "Numbers associated with Stirling numbers and xx". Rocky Mountain Journal of Mathematics. 15 (2): 461. doi:10.1216/RMJ-1985-15-2-461.
Gould, H. W. (1996). "A Set of Polynomials Associated with the Higher Derivatives of y = xx". Rocky Mountain Journal of Mathematics. 26 (2): 615. doi:10.1216/rmjm/1181072076.
^ It appears in Borwein, Bailey & Girgensohn 2004.
^ Bernoulli 1697.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sophomore%27s_dream&oldid=1053905038"
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Markdown · Liduan Worklog
Liduan Worklog
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Dictionary:Magnetization - SEG Wiki
Magnetic moment per unit volume (occasionally per unit mass), a vector quantity. Also called magnetic polarization or intensity of magnetization. Designated by symbols M or I. A measure of the effect of the medium on the magnetic field B when subject to a magnetizing force H:
{\displaystyle \mathbf {B} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\qquad in\;the\;SI\;System,}
{\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {I} \qquad in\;the\;cgs\;System,}
where μo is the permeability of free space. The proportionality between magnetization and H is the magnetic susceptibility (q.v.), k or k′. See Figure M-1.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Magnetization&oldid=48202"
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Response to Comment by Charles W. McCutchen | J. Biomech Eng. | ASME Digital Collection
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division January 9, 2004; revision received January 16, 2004.
J Biomech Eng. Aug 2004, 126(4): 537 (1 pages)
Ateshian , G. A. (September 27, 2004). "Response to Comment by Charles W. McCutchen ." ASME. J Biomech Eng. August 2004; 126(4): 537. https://doi.org/10.1115/1.1785816
bone, physiological models, lubrication, biorheology, friction, biomechanics, porosity, mechanical contact, orthopaedics, permeability, elastic moduli
Biomechanics, Biorheology, Bone, Elastic moduli, Friction, Lubrication, Orthopedics, Permeability, Physiology, Porosity, Fluids, Stress
We thank Dr. McCutchen for his comment. We are keenly aware of his hypotheses for cartilage lubrication 1,2 and agree with one of the fundamental premises he first proposed, namely that hydrostatic pressurization of the interstitial fluid of cartilage supports most of the contact load transmitted across articular surfaces, thereby reducing the interfacial frictional force exerted on the solid matrix of the opposing surfaces. We have previously formulated a cartilage friction model 3,4, within the frameworks of the biphasic 5 and triphasic 6 theories of cartilage, to quantify this mechanism, and we have recently found excellent agreement between the model predictions and experimental data 7, thus supporting Dr. McCutchen’s original hypothesis.
In our model, the direction of interstitial fluid flow at the contact interface is irrelevant to the mechanism of friction reduction by interstitial fluid load support. The only relevant issue is the magnitude and duration of interstitial fluid pressurization. Our recent experimental studies confirm that this pressurization can last for hundreds of seconds or longer 7,8. From our perspective, it is thus inconsequential whether the interstitial fluid “weeps” out of the tissue or is “boosted” into it. It is only because the direction of flow had been debated historically in the literature that we referred to it in our discussion.
We respectfully disagree with Dr. McCutchen’s statement that “Were [weeping flow] not present cartilage would be, in effect, an impervious bearing…” Cartilage is a highly porous tissue, particularly near the articular surface where the porosity can approach 90% 9,10. As demonstrated in our model 3, it is not necessary for a layer of fluid to separate the articular surfaces in order for its lubrication to be effective. When two cartilage layers come into contact, each with a surface porosity of 90% (or solid fractional content
φs=0.1)
, a fraction of
1−φs2,
or 99%, of the apparent contact area consists of fluid-against-fluid or fluid-against-solid contact. Since the fluid is typically highly pressurized under loading, due to the relatively low modulus and permeability of cartilage, most of the contact load at the interface will be supported by fluid, producing negligible friction. Only 1% of the contact area will consist of solid-against-solid contact, where friction occurs. Trapped lubricant pools would further decrease solid-against-solid contact, and thus frictional forces. Thus, even in the absence of weeping flow, our model predicts that the porous nature of cartilage will maintain effective lubrication.
Sponge-Hydrostatic and Weeping Bearings
Krishnan, R., Kopacz, M., and Ateshian, G. A., “Experimental Verification of the Role of Interstitial Fluid Pressurization in Cartilage Lubrication,” J Orthop Res, In Press.
Cartilage Interstitial Fluid Load Support in Unconfined Compression
Torzilli, P. A., Askari, E., and Jenkins, J. T., 1990, “Water Content and Solute Diffusion Properties in Articular Cartilage,” Biomechanics of Diarthrodial Joints, V. C. Mow et al., eds., Springer-Verlag, New York, pp. 363–390.
3D Nonlinear Finite Element Analysis of Knee Joint Implants
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Coax Impedance: Characteristic Impedance » Electronics Notes - Interesting - 2022
Coax Impedance: Coaxial Cable Characteristic Impedance
The characteristic impedance of a length of coaxial cable the most important parameter in the selection of any length of coax.
To ensure the correct operation of a system using a signal source , e.g. a transmitter, a length of feeder and a load, e.g. an antenna, the feeder impedance must match the source and the load. In this way the maximum transfer of power is achieved between the source and the feeder and then the feeder and the load.
All feeders possess a characteristic impedance. For coaxial cable there are two main standards that have been adopted over the years. These are 75 Ω and 50 Ω
The 50 Ω coax cable is used for professional and commercial applications whereas the 75 Ω coax cable is used almost exclusively for domestic TV and VHF FM applications.
The reason for the choice of these two impedance standards is largely historical but arises from the properties provided by the two impedance levels:
75 ohm coax cable gives the minimum weight for a given loss
50 ohm coax cable gives the minimum loss for a given weight.
Although these two standards are used for the vast majority of coax cable which is produced it is still possible to obtain other impedances for specialist applications. Higher values are often used for computer installations, but other values including 25, 95 and 125 ohms are available. 25 ohm miniature RF cable is extensively used in magnetic core broadband transformers. These values and more are available through specialist coax cable suppliers.
Capacitance of coax cable
A length of coax cable exhibits a capacitance between the inner conductor and outer shield. The capacitance varies with the spacing of the conductors, the dielectric constant, and as a result the impedance of the line.
The lower the impedance, the higher the coax capacitance for a given length because the conductor spacing is decreased. The coax capacitance also increases with increasing dielectric constant, as in the case of an ordinary capacitor.
c=\frac{24.1{\varepsilon }_{r}}{{\mathrm{log}}_{10}\left(\frac{D}{d}\right)}
C = Capacitance in pF / metre
εr = Relative permeability of the dielectric
D = Inner diameter of the outer conductor
d = Diameter of the inner conductor
Inductance of coax cable
The inductance of the line can also be calculated. Again this is proportional to the length of the line.
However the inductance is independent of the dielectric constant for the material between the conductors, and is proportional to the logarithm of the ratio of the diameters of the two conductors.
L = Inductance in µH / metre
Coax impedance calculation
The impedance of the RF coax cable is chiefly governed by the diameters of the inner and outer conductors. On top of this the dielectric constant of the material between the conductors of the RF coax cable has a bearing. The relationship needed to calculate the impedance is given simply by the formula:
{Z}_{0}=\frac{138{\mathrm{log}}_{10}\left(\frac{D}{d}\right)}{\sqrt{{\varepsilon }_{r}}}
Zo = Characteristic impedance in Ω
Note: The units of the inner and outer diameters can be anything provided they are the same, because the equation uses a ratio.
Importance of coax impedance
The coax impedance is one of the major specifications associated with any piece of coax cable. As it will determine the matching within the system and hence the level of standing waves and power transfer, it is a crucial element. It is therefore necessary to ensure that the correct coax impedance is chosen for any system.
Watch the video: Determining Coax Impedance with a TDR (May 2022).
Copyright 2022 \ Coax Impedance: Coaxial Cable Characteristic Impedance...
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Compute reference currents for Maximum Torque Per Ampere (MTPA) and field-weakening operation - Simulink - MathWorks América Latina
Mathematical Model of PMSM
Stator resistance per phase (Ohm)
Stator d-axis inductance (H)
Stator q-axis inductance (H)
Input signal units
Base speed (rpm)
Compute reference currents for Maximum Torque Per Ampere (MTPA) and field-weakening operation
The MTPA Control Reference block computes the d-axis and q-axis reference current values for maximum torque per ampere (MTPA) and field-weakening operations. The computed reference current values results in efficient output for the permanent magnet synchronous motor (PMSM).
The block accepts the reference torque and feedback mechanical speed and outputs the corresponding d- and q-axes reference current values for MTPA and field-weakening operations.
The block computes the reference current values by solving mathematical relationships. The calculations use SI unit system. When working with the Per-Unit (PU) system, the block converts PU input signals to SI units to perform computations, and converts them back to PU values at the output.
These equations describe the computation of reference d-axis and q-axis current values by the block:
These model equations describe dynamics of PMSM in the rotor flux reference frame:
{v}_{d}={i}_{d}{R}_{s}+ \frac{d{\lambda }_{d}}{dt} - {\omega }_{e}{L}_{q}{i}_{q}
{v}_{q}={i}_{q}{R}_{s}+ \frac{d{\lambda }_{q}}{dt} + {\omega }_{e}{L}_{d}{i}_{d}+ {\omega }_{e}{\lambda }_{pm}
{\lambda }_{d}={L}_{d}{i}_{d}+ {\lambda }_{pm}
{\lambda }_{q}={L}_{q}{i}_{q}
{T}_{e}= \frac{3}{2}p\left({\lambda }_{pm}{i}_{q}+\left({L}_{d}- {L}_{q}\right){i}_{d}{i}_{q}\right)
{T}_{e}- {T}_{L}=J\frac{d{\omega }_{m}}{dt} +B{\omega }_{m}
{v}_{d}
{v}_{q}
{i}_{d}
{i}_{q}
{R}_{s}
{\lambda }_{pm}
{\lambda }_{d}
is the d-axis flux linkage (Weber).
{\lambda }_{q}
is the q-axis flux linkage (Weber).
{\omega }_{e}
{\omega }_{m}
{L}_{d}
{L}_{q}
{T}_{e}
is the electromechanical torque produced by the PMSM (Nm).
{T}_{L}
is the load torque (Nm).
p
J
is the inertia coefficient (kg-m2).
B
is the friction coefficient (kg-m2/ sec).
Base speed is the maximum motor speed at the rated voltage and rated load, outside the field-weakening region. These equations describe the computation of the motor base speed.
{v}_{do}=- {\omega }_{e}{L}_{q}{i}_{q}
{v}_{qo}={\omega }_{e}\left({L}_{d}{i}_{d}+ {\lambda }_{pm}\right)
{v}_{max}= \frac{{v}_{dc} }{\sqrt{3}} -{R}_{s}{i}_{max} \ge \sqrt{{v}_{do}^{2}+ {v}_{qo}^{2}}
{i}_{max}^{2}= {i}_{d }^{2}+ {i}_{q}^{2}
{i}_{d}
{i}_{d}
{i}_{q}
{\omega }_{base}= \frac{1}{p}\cdot \frac{\text{ }{v}_{max}}{\sqrt{{\left({L}_{q}{i}_{q}\right)}^{2}+{\left({L}_{d}{i}_{d}+ {\lambda }_{pm}\right)}^{2}}}
{\omega }_{e}
{\omega }_{base}
{i}_{d}
{i}_{q}
{v}_{do}
{i}_{d}
{v}_{qo}
{i}_{q}
{L}_{d}
{L}_{q}
{R}_{s}
{\lambda }_{pm}
{v}_{d}
{v}_{q}
{v}_{max}
{v}_{dc}
{i}_{max}
p
For a surface PMSM, you can achieve maximum torque by using zero d-axis current when the motor is below the base speed. For field-weakening operation, the reference d-axis current is computed by constant-voltage-constant-power control (CVCP) algorithm defined by these equations:
{\omega }_{m}\le {\omega }_{base}
{i}_{d_mtpa}= 0
{i}_{q_mtpa}= \frac{{T}^{ref}}{\frac{3}{2}\cdot p\cdot {\lambda }_{pm}}
{i}_{d_sat}= {i}_{d_mtpa}= 0
{i}_{q_sat}=\text{sat}\left({i}_{q_mtpa}, {i}_{\text{max}}\right)
{\omega }_{m}> {\omega }_{base}
{i}_{d_fw}= \frac{\left({\omega }_{e_base}- {\omega }_{e}\right){\lambda }_{pm}}{{\omega }_{e}{L}_{d}}
{i}_{d_sat}=\text{max}\left({i}_{d_fw}, -{i}_{\text{max}}\right)
{i}_{q_fw}= \frac{{T}^{ref}}{\frac{3}{2}\cdot p\cdot {\lambda }_{pm}}
{i}_{q_lim}= \sqrt{{i}_{max}^{2}- {i}_{d_sat}^{2}}
{i}_{q_sat}=\text{sat}\left({i}_{q_fw}, {i}_{\text{q}_\text{lim}}\right)
The saturation function used to compute
{i}_{q_sat}
is described below:
{i}_{q_fw}< -{i}_{q_lim}
{i}_{q_sat}= -{i}_{q_lim}
{i}_{q_fw}>{i}_{q_lim}
{i}_{q_sat}= {i}_{q_lim}
-{i}_{q_lim}\le {i}_{q_fw}\ge {i}_{q_lim}
{i}_{q_sat}= {i}_{q_fw}
The block outputs the following values:
{I}_{d}^{ref}={i}_{d_sat}
{I}_{q}^{ref}={i}_{q_sat}
{\omega }_{e}
{\omega }_{m}
{\omega }_{base}
{\omega }_{e_base}
is the electrical base speed of the motor (Radians/ sec).
{i}_{d_mtpa}
is the d-axis phase current corresponding to MTPA (Amperes).
{i}_{q_mtpa}
is the q-axis phase current corresponding to MTPA (Amperes).
{T}^{ref}
is the reference torque (Nm).
p
{\lambda }_{pm}
{i}_{d_fw}
is the d-axis field weakening current (Amperes).
{i}_{q_fw}
is the q-axis field weakening current (Amperes).
{L}_{d}
{i}_{max}
{i}_{d_sat}
is the d-axis saturation current (Amperes).
{i}_{q_sat}
is the q-axis saturation current (Amperes).
{I}_{d}^{ref}
is the d-axis current corresponding to the reference torque and reference speed (Amperes).
{I}_{q}^{ref}
is the q-axis current corresponding to the reference torque and reference speed (Amperes).
For an interior PMSM, you can achieve maximum torque by computing the d-axis and q-axis reference currents from the torque equation. For field-weakening operation, the reference d-axis current is computed by voltage and current limited maximum torque control (VCLMT) algorithm.
The reference currents for MTPA and field weakening operations are defined by these equations:
{i}_{m_ref}= \frac{2\cdot {T}^{ref}}{3\cdot p\cdot {\lambda }_{pm}}
{i}_{m}=\mathrm{min}\left({i}_{m_ref}, {i}_{\text{max}}\right)
{i}_{d_mtpa}=\frac{{\lambda }_{pm}}{4\left({L}_{q}-{L}_{d}\right)}-\sqrt{\frac{{\lambda }_{pm}^{2}}{16{\left({L}_{q}-{L}_{d}\right)}^{2}}+\frac{{i}_{m}^{2}}{2}}
{i}_{q_mtpa}=\sqrt{{i}_{m}^{2}-{\left({i}_{d_mtpa}\right)}^{2}}
{v}_{do}=-{\omega }_{e}{L}_{q}{i}_{q}
{v}_{qo}={\omega }_{e}\left({L}_{d}{i}_{d}+ {\lambda }_{pm}\right)
{v}_{do}^{2}+{v}_{qo}^{2}={v}_{\mathrm{max}}^{2}
{\left({L}_{q}{i}_{q}\right)}^{2}+{\left({L}_{d}{i}_{d}+ {\lambda }_{pm}\right)}^{2}\le \frac{{v}_{max}^{2}}{{\omega }_{e}^{2}}
{i}_{q}= \sqrt{{i}_{max}^{2}- {i}_{d}^{2}}
\left({L}_{d}^{2}- {L}_{q}^{2}\right){i}_{d}^{2}+2{\lambda }_{pm}{L}_{d}{i}_{d}+ {\lambda }_{pm}^{2}+ {L}_{q}^{2}{i}_{max}^{2}-\frac{{v}_{max}^{2}}{{\omega }_{e}^{2}}=0
{i}_{d_fw}= \frac{-{\lambda }_{pm}{L}_{d}+ \sqrt{{\left({\lambda }_{pm}{L}_{d}\right)}^{2}- \left({L}_{d}^{2}- {L}_{q}^{2}\right)\left({\lambda }_{pm}^{2}+ {L}_{q}^{2}{i}_{max}^{2}- \frac{{v}_{max}^{2}}{{\omega }_{e}^{2}} \right)}}{\left({L}_{d}^{2}- {L}_{q}^{2}\right)}
{i}_{q_fw}= \sqrt{{i}_{max}^{2}- {i}_{d_fw}^{2}}
{\omega }_{m}\le {\omega }_{base}
{I}_{d}^{ref}= {i}_{d_mtpa}
{I}_{q}^{ref}= {i}_{q_mtpa}
{\omega }_{m}> {\omega }_{base}
{I}_{d}^{ref}=\mathrm{max}\left({i}_{d_fw},-{i}_{\mathrm{max}}\right)
{i}_{q_fw}= \sqrt{{i}_{max}^{2}- {i}_{d_fw}^{2}}
{i}_{q_fw}<{i}_{m}
{I}_{q}^{ref}={i}_{q_fw}
{i}_{q_fw}\ge {i}_{m}
{I}_{q}^{ref}={i}_{m}
For negative reference torque values, the sign of
{i}_{m}
{I}_{q}^{ref}
are updated and equations are modified accordingly.
{i}_{m_ref}
is the estimated maximum current to produce the reference torque (Amperes).
{i}_{m}
is the saturated value of estimated maximum current (Amperes).
{i}_{d_\mathrm{max}}
is the maximum d-axis phase current (peak) (Amperes).
{i}_{q_\mathrm{max}}
is the maximum q-axis phase current (peak) (Amperes).
{T}^{ref}
{I}_{d}^{ref}
is the d-axis current component corresponding to the reference torque and reference speed (Amperes).
{I}_{q}^{ref}
is the q-axis current component corresponding to the reference torque and reference speed (Amperes).
p
{\lambda }_{pm}
{i}_{d_mtpa}
{i}_{q_mtpa}
{L}_{d}
{L}_{q}
{i}_{max}
{v}_{max}
{v}_{do}
{i}_{d}
{v}_{qo}
{i}_{q}
{\omega }_{e}
{i}_{d}
{i}_{q}
{i}_{d_fw}
{i}_{q_fw}
{\omega }_{base}
Idref — Reference d-axis current
Reference d-axis phase current that can efficiently generate the input torque and speed values.
Iqref — Reference q-axis current
Reference q-axis phase current that can efficiently generate the input torque and speed values.
Type of motor — Type of PMSM
Interior PMSM (default) | Surface PMSM
Type of PMSM based on the location of the permanent magnets.
Number of pole pairs — Number of available pole pairs
Number of pole pairs available in the motor.
Stator resistance per phase (Ohm) — Resistance of stator phase winding (ohms)
Resistance of the stator phase winding (ohms).
To enable this parameter, set Type of motor to Interior PMSM.
Stator d-axis inductance (H) — d-axis stator winding inductance
Stator winding inductance (henry) along the d-axis of the rotating dq reference frame.
Stator q-axis inductance (H) — q-axis stator winding inductance
Stator winding inductance (henry) along the q-axis of the rotating dq reference frame.
Permanent magnet flux linkage (Wb) — Magnetic flux linkage of permanent magnets
Magnetic flux linkage between the stator windings and permanent magnets on the rotor (weber).
Maximum phase current limit for the motor (amperes).
DC voltage (V) — DC bus voltage (volts)
DC bus voltage (volts)
Input signal units — Unit of block input values
Unit of the block input values.
Base speed (rpm) — Base speed of motor (rpm)
Speed of the motor at the rated voltage and rated current outside the field weakening region.
Base current (A) — Base current for per-unit conversion (amperes)
Current corresponding to 1 per-unit. We recommend that you use the maximum current detected by an Analog to Digital Converter (ADC) as the base current.
To enable this parameter, set Input signal units to Per-Unit (PU).
Base torque (Nm) — Base torque for per-unit conversion (Nm)
Torque corresponding to 1 per-unit. See Per-Unit System page for more details.
To display this parameter, set Input signal units to Per-Unit (PU).
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Integral - Uncyclopedia, the content-free encyclopedia
“I integrated your MOM over the interval [a,b]!”
~ Oscar Wilde on Integrals
“Come to think of it, your mom is an open interval.”
~ Oscar Wilde on Intervals
“My favorite part about integrals is how you work for half an hour on one of them only to be told that you forgot the plus C so you are wrong and clearly suck”
~ Sir Isaac Fig Newton on Integrals
“Foreplay is simply taking the surface integral over your partner before introducing any flux.”
~ Christian Preacher Ron Jeremy on the mathematics of human behavior
“Integration by parts is just a fancy way of saying gangbang.”
~ Rabbi Ron Jeremy on teaching Oscar Wilde new methods by which to pleasure himself
A sheet of red paper falling. Identified by some as a surface integral.
The Integral was a medieval torture device, likely invented by Isaac Newton or Albert Einstein. Somewhat surprisingly, it is still in use today in many Schools across the globe.
The opposite of the Chainsaw, the Integral takes small pieces that seem to be Nonsense, or just might be dried apples. The Integral follows the concept of the Elizabeth Taylor Series, but then again, who knows?
3 The Other Perspective
6 Toxicology information of the integrals
Oscar Wilde, the greatest town crier to ever have lived, has nothing to say on the subject. If he did, he would probably not say it, as he prefers to huff kittens in his spare time. Also, he believes that integrals are for babies (strangely enough, as his agent strongly recommends that he deny the existence of babies.)
When the Integral was first invented, it was applied to victims that had been seized for heresy. Some of this unfortunate number spent the (short) remainder of their lives, screaming, "Noooo...Calculus." Due to the extreme cruelty displayed, the Integral was replaced by other, kinder forms of torture, such as being forced to drink 32 pints of water. Somewhat later, a revolution in torture (as well as the fact that many of the sissies who went soft on felons had died) rendered all previous methods of Integral Flogging or Lynching obsolete.
Currently, the most deadly integral known to man, marmot, and carrier pidgeon is "dcabin/cabin." While many have tried to identify the solution as "ln(abs(cabin))+C", many regurgitate their entrails and vomit profusely before the words escape.
The Other Perspective[edit]
In the USA, India and 133714nd, the integral has been the subject of a religious cult known as The Holy Order of Calculus Freaks has come to praise the Integral and its counterpart, the derivative. Their holy book is only known as "The Textbook" and is rumoured to contain nothing but equations. The cult members have been accused of being aliens, since no human mind could survive the horror of
{\displaystyle \int _{0}^{1}\sin(\tan(x))\,dx}
. It even has sin in the equation! These allegations are of course totally unfounded (NOT!). However, it has been proven that a linking of the world's most cool supercomputers would be able to solve it in 12,676,923,764,234 years, give or take a few billenia. Fortunately, these calculus freaks are few and far between. Indeed, only one Uncyclopedia user is one. It is Loopquanta137.
It sucks to the core. The supercomputers aforementioned would have to be constructed with 64 core processors, as 64=(2*L*J)^6, whereas L = love, J = Jesus, and since these are two "touchy feely" words, they cancel. Much in the way that two negatives cancel, the two words of hope and joy must cancel or else this just wouldn't be calculus, now would it?
Multiple Integrals[edit]
After the raging success of the integral, the world's leading nerds began to discover ways to take the concept further. The Double Integral is not a torture device, but serves two purposes. It can either be a set of fancy poles for strip dancers or it can be used to find the volume under surfaces such as your mom as a function of x and y in 3 space. There is a formal definition involving limits, sigmas and flying cows, but no one cares. The Triple Integral is a superfluous piece of crap of which Jesus does not approve. Instead of going under your mom, the Triple Integral will cut your mom up into cubes to find the actual volume of him/her.
Line Integrals[edit]
The path way into a black hole.
Caution: Line integrals are known to rapidly draw lines through vector fields, possibly causing bodily harm. Line integrals are for external use only and should not be taken with alcohol.
Toxicology information of the integrals[edit]
CAS number: x^69-69x+69 ; Classified in the halogen family under Tennessine; Use extreme caution due to the quite extreme oxidation involved in the process of integrating some transcendent functions. It is strongly recommended to ask your doctor if integrals are made for you because a black hole will be most likely to be created in your brain if you have never experienced derivatives before integrating. Also, please use integrals responsibly or you may become a physics teacher and want to integrate your wife from the center of your conductor to the infinity.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Integral&oldid=5994675"
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Analysis and optimization of mutual influence of single channel tunnel construction blasting | JVE Journals
Junfeng Zhang1 , Ming Li2 , Xiaolin Yang3
1Chongqing Branch of China Railway 9st Bureau Group Co. Ltd., Chongqing 401121, China
2, 3State Key Laboratory Cultivation Base for Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, China
In the current urban rail transit project in the vigorous development of the community, due to the needs of the site environment or line selection, it has been a common phenomenon in the actual construction of the tunnel encountered in the parallel operation and tunnel overlap situation, the new tunnel or tunnel group on the existing tunnel or other structure of the interaction problem has become very prominent! Based on the blasting excavation of the single-channel caverns in the first phase of the Chongqing Rail Transit Line 10 project, this paper makes a detailed investigation of the actual engineering, and uses ANSYS/LS-DYNA to simulate to comparative analysis and optimization the tunnel lining.
Keywords: single channel, the tunnel group, blasting construction, ANSYS/LS-DYNA.
Nowadays, the urban rail transit project is in the stage of vigorous development, but it’s quite common to encounter the parallel operation of the tunnel and the overlapping of the tunnel in the tunnel construction project by the restrictions of the site and the line. The problem of mutual interaction between a new tunnel and existing structures has become very prominent. In the parallel small clear distance and large cross section shallow tunnel group, the construction process, the safety construction section spacing, the influence of blasting vibration on the supporting structure, the reasonable timing of support has become an urgent problem to be solved.
The construction of underground caverns has attracted the attention of scholars at home and abroad, and put a lot of effort into the research. The research on the small-scale tunnels has yielded fruitful results [1-6]. In China, there are few studies on the construction technology of small horizontal tunnels with parallel holes of three holes and above, which mainly rely on very few concrete engineering examples.
According to the design summary of the 3-hole tunnel group of Yuexiu Park Station of Guangzhou Metro Line 2, Zhang Hongwei [7] has carried on the preliminary discussion on the construction of 3-hole tunnel scheme. Based on the combination of model test and finite element method, Wang Mingnian [8-9] studied the construction mechanics characteristics of 3-hole parallel tunnel with shallow buried and small spacing in weak surrounding rock. Guangzhou Metro Design and Research Institute [10] summarized the successful design and construction experience of Guangzhou Metro project, and provided technical guidance for similar projects in the future. Yang Hong she [11] also carried out in-depth analysis of the Guangzhou subway project, summed up the subway shallow buried hole construction process and key technology. Sun Zhongcheng [12] also combined with the project summed up the construction of the tunnel hole tunnel technology. In the study of the caverns, the study of blasting vibration and seismic effect is one of the focuses.
In summary, for more than 3 tunnels of single-channel tunnel group construction, reference experience needs to be studied and accumulated. Based on the engineering examples and the industry standard, this paper studies the blasting vibration of the parallel tunnel group in the five tunnels by combining the measured data of the field and the numerical simulation results. In order to obtain reasonable tunneling distance and support time, the mechanism of blasting vibration propagation in tunnel group is explored, which can provide reference for the safety and design of similar construction
2. Propagation effect analysis model of shock wave in group chamber
The main two main models of the shock wave propagation effect model of the shock wave generated by the blasting in the construction of the cavern group are the model of the dampness wave propagation attenuation model (Fig. 1) and the surrounding rock mass response model (Fig. 2). The former mainly studies the propagation vibration of the shock wave along the roadway in the surrounding rock medium during the tunneling process, and explores the construction of the single tunnel. The latter mainly studies the influence law of the vibration of the blasting vibration wave in the adjacent tunnel group, and qualitatively reveals the response characteristics of the surrounding rock of the different distance tunnel to the shock wave in order to arrange the longitudinal space of the construction.
Fig. 1. Attenuation law of blasting seismic wave propagation models
Fig. 2. Shock response characteristics of surrounding rock of tunnel and Chambers model
Yuelai station to Wangjiazhuang station interval and Wangjiazhuang parking lot entry and exit project is Chongqing Rail Transit Line 10 part of the project, the starting point for both Yuelai station, terminal for Wangjiazhuang station. The tunnel passes through the strata mainly sand and mudstone interbed, the surrounding rock level IV-VI level, the groundwater is bedrock fissure water. Wangjiazhuang parking lot entry and exit tunnel dome depth of 8.8-43.5 m, interval line tunnel dome depth of 4.8-40.1 m. The distance between the right and left of the line is 1.5 m, which is a small clear distance tunnel. The distance between the left line and the out-of-section structure is 3-6.5 m, which is a small-distance tunnel. The interval line set two contact channel, a channel construction. Due to the tight construction period, the construction land is limited, a single channel is used as the auxiliary channel construction, the left and right of the five independent channels, forming a unique single channel tunnel construction characteristics. Among them, the construction traffic organization is difficult, the ventilation pressure is big, the interaction crosses the big influence, especially the blast has the influence to the construction of the small clear distance tunnel.
4. Field test scheme design
For the sake of generality, according to the blasting theory model and the existing experimental research results, arranged vibration equipment for testing in the 5, 6, 7, 9 tunnel. In the process of testing, considered the purpose and requirements of the test project, combined with the economic and effectiveness testing. The specific test results are shown in Fig. 3.
Fig. 3. Schematic diagram of the original layout of horizontal tunnel test. Dotted lines represent constraints; digital number for the channel signal;
X
for test instrument
5.1. Longitudinal test results and analysis
Table 1 is the result of the peak value of the maximum vibration velocity of the longitudinal test, as can be seen from the table:
1) According to the measured data in Table 1, in the longitudinal direction of the tunnel, the more close to the working face of the maximum vibration velocity of vibration, with the increase of the distance from the explosion source center, the maximum vibration velocity of the subject is reduced, in accordance with the general law of blasting vibration.
Table 1. Longitudinal test to maximum vibration velocity peak summary table
Maximum peak velocity of each direction (cm/s)
RCK 0+121
ZK 43+265
YK 44+099
2) The vibration velocity of the measuring point has a very significant difference in three different directions of the space. Among them, the vibration velocity of the vertical direction is obviously larger than that of the horizontal direction, this is mainly due to the particle in the direction perpendicular to the surface of the media when the movement of the constraints of small, while the constraint in the direction along the surface is large. Therefore, the vibration velocity in the direction perpendicular to the surface of the medium is the key to the blasting vibration hazard control.
6. Transfer law analysis of numerical simulation and field test result
6.1. Model geometry boundary
According to the engineering geological investigation report of the tunnel and the design dimension of the tunnel, a three-dimensional numerical analysis model is established. The upper surface of the model is chosen according to the actual terrain size, and the left and right boundaries of the model are about 3 times the tunnel diameter. The left and right boundaries of the model are set as
X
direction displacement constraints, the front and rear boundaries are set as
Y
direction displacement constraints, the upper boundary is free, and the lower boundary is set as
Z
direction displacement constraint.
The physical and mechanical parameters of surrounding rock and supporting structure in simulation analysis are determined according to relevant specifications and tunnel design data, at the same time steel arch according to the principle of equal consideration, that is to improve the elastic modulus of shotcrete to replace the role of steel arch. The physical and mechanical parameters of surrounding rock and tunnel support structure are shown in Table 2.
Table 2. Material physical and mechanical parameters
E
\gamma
\rho
(KN/m3)
Steel arches spray concrete
6.3. The establishment and calculation result analysis of group hole model
5 hole and 7 hole dislocation 40 meters, 5 hole and 9 hole dislocation 30 meters, 7 holes and 9 holes dislocation 70 meters of the situation:
Fig. 4. Excavation surface of each tunnel dislocation distance sketch
Under this condition, 5 tunnel face vault
Z
direction vibration velocity 33.12 cm/s, combined vibration velocity is 33.17 cm/s. No. 7 tunnel face vault
Z
direction vibration velocity 1.24 cm/s, combined vibration velocity is1.50 cm/s. No. 9 tunnel face vault
Z
direction vibration velocity 1.16 cm/s, combined vibration velocity is 1.21 cm/s. This can be seen when the dislocation of No. 5 tunnel and No. 7 tunnel face 40 m, No. 5 and No. 9 tunnel face dislocation 30 m. The 5 hole is the explosion source, No. 7 tunnel face vault
Z
direction of vibration speed and combined vibration speed meets the current national standard of newly poured concrete age 3 days allowed by the 3.0 cm/s velocity requirements, and No. 7 tunnel face vault
Z
direction of vibration speed and combined vibration speed meets the current national standard of newly poured concrete age 3 days allowed by the 3.0 cm/s velocity requirements.
6.4. Comparison chart of single hole measured trend chart and simulation under the condition of group tunnel simulation
As can be seen from Table 3, when the No. 5 tunnel and the blasting source have different longitudinal distance, the combined vibration velocity of the vault is different, when the distance of explosion source distance of 12 m, 28 m, 40 m, 48 m, 52 m, 68 m, 80 m, 92 m, 140 m, vault combined vibration velocity were 12.8 cm/s, 9.3 cm/s, 6.9 cm/s, 3.2 cm/s, 2.4 cm/s, 2.5 cm/s, 2.3 cm/s, 1.6 cm/s, 1.8 cm/s. It can be seen that the combination of the No. 5 number of the tunnel vault at the distance of the explosion source 48 m is 3.2 cm/s, more than the current national standard, but when the No. 5 tunnel vault distance explosion source 52 m, the combination of vibration velocity for the 2.4 cm/s to meet the engineering requirements.
From Table 3 can be drawn out of the vault combination speed trend chart 5. From the figure it is obvious that the trend of distance blasting source closer, vibration velocity increasing; blasting source distance farther, vibration velocity is small, there is a sharp drop in the middle process, after falling slowly. Taking into account the factors of blasting damage and other factors, the best distance of the two lining of the No. 5 tunnel should be about the distance from the explosion source 65 m, at this time, it can not only ensure that the impact of blasting on the two lining is in line with the requirements of the specification, but also can effectively make the two secondary lining construction operation, at this time of the second lining of the construction work distance is more reasonable.
Table 3. No. 5 tunnel dome of the measured vibration value
Vibration velocity (cm/s)
explosion source distance (m)
Fig. 5. Vault combination speed trend chart
Fig. 6. The trend chart of the combined vibration velocity of single tunnel vault
Fig. 7. July 9, Field monitoring data and simulation data contrast trend chart
Can be seen from the Figs. 6-9, blasting seismic waves in the transmission process from the blasting source near the blasting seismic wave velocity sharp attenuation, the more distant regions tend to be gentle. In a certain total charge, charge structure, blasting pitch cases, the trend of simulated vault combination velocity is basically the same as the trend of measured vault, there is a certain error in the specific value. This is mainly to simulate the parameter selection and the actual situation there is a certain discrepancy, which is mainly the difference between the charge, simulation of the charge density and the actual blasting construction of the charge is different.
Fig. 8. July 16, Field monitoring data and simulation data contrast trend chart
When blasting, the two lining is generally the new cast concrete, by the current national standard “blasting safety procedures GB6722-2014” article 13.2.2 of the provisions can be safely allowed to allow the value of 3 cm/s. The comparison of the simulation with the actual monitoring as well as the accumulation of the damage considered can be drawn: the secondary lining of the tunnel is about 65 meters away from the blast source and is generally safe, and basically meet the requirements of the specification. If the distance of the second lining is less than 65 meters away from the blast source, the combined vibration velocity of the tunnel dome is generally more than the specification requirements. This is disadvantageous for the second lining, so the second lining of the tunnel group is somewhat lagging behind. Should be a timely lining of the second operation, to meet the requirements of the second lining immediately.
Zheng Yuchao Three Holes Parallel Nearly by Shield Tunneling Effect Research. Southwest Jiaotong University, Chengdu, 2006. [Search CrossRef]
Mi Decai The Stability of Surrounding Rock of Shallow Buried Large Span Cavern Group Engineering Geological Research. Chengdu University of Technology, Chengdu, 2006. [Search CrossRef]
Jing Chunyan, Huang Hongwei, Zhang Zi-xin Dynamic monitoring and numerical simulation of small spacing tunnels construction analysis. Journal of Underground Space and Engineering, Vol. 3, Issue 3, 2007, p. 503-507. [Search CrossRef]
Jiang Quan, Feng Xiating, Zhou Hui Screen deep secondary hydropower station diversion tunnel group allowed minimum distance study. Rock and Soil Mechanics, Vol. 29, Issue 3, 2008, p. 656-661. [Search CrossRef]
Tang Jun-feng Large Hydropower Station Underground Cavities Construction Mechanical Behavior Research. Central South University, Changsha, 2010. [Search CrossRef]
Sun Wenzhao Porous Tunnel Interaction Research. Dalian University of Technology, Dalian, 2012. [Search CrossRef]
Zhang Hong-wei The design of Guangzhou metro station underground running tunnel of Yuexiu park to explore. Journal of Tunnel Construction, Vol. 21, 2001, p. 19-23. [Search CrossRef]
Wang Mingnian, Li Zhi-ye, Guan Baoshu 3 shallow tunnel digging hole small spacing tunnel surface subsidence control technology research. Rock and Soil Mechanics, Vol. 23, Issue 6, 2002, p. 821-824. [Search CrossRef]
Wang Mingnian, Li Zhi-ye, Liu Zhi-cheng 3 hole small spacing parallel shallow buried in weak rock tunnel construction mechanics study. Railway Construction Technology. Vol. 4, 2002, p. 11-14. [Search CrossRef]
Small Spacing Porous Underground Running Tunnel Group of Integrated Technology. Design and Research Institute of Guangzhou Underground Railway, Guangzhou, 2004. [Search CrossRef]
Yang Hongshe The subway construction process and key technology of shallow buried hole. Journal of Urban Rail Transit Research, Vol. 1, 2004, p. 70-72. [Search CrossRef]
Sun Zhongcheng Connecting the small clear distance of hole in the tunnel construction technology. Railway Standard Design, Vol. 12, 2006, p. 66-69. [Search CrossRef]
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Solution for a Semi-Permeable Interface Crack in Elastic Dielectric/Piezoelectric Bimaterials | J. Appl. Mech. | ASME Digital Collection
Solution for a Semi-Permeable Interface Crack in Elastic Dielectric/Piezoelectric Bimaterials
School of Aerospace, MOE Laboratory,
, Xian, Shanxi 710049, P.R.C.
e-mail: yhchen2@mail.xjtu.edu.cn
Li, Q., and Chen, Y. H. (January 11, 2008). "Solution for a Semi-Permeable Interface Crack in Elastic Dielectric/Piezoelectric Bimaterials." ASME. J. Appl. Mech. January 2008; 75(1): 011010. https://doi.org/10.1115/1.2745397
A semi-permeable interface crack in infinite elastic dielectric/piezoelectric bimaterials under combined electric and mechanical loading is studied by using the Stroh complex variable theory. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and on the energy release rate (ERR). Thirty five kinds of such bimaterials are considered, which are constructed by five kinds of elastic dielectrics and seven kinds of piezoelectrics, respectively. Numerical results for the interface crack tip singularities are calculated. We demonstrate that, whatever the dielectric phase is much softer or much harder than the piezoelectric phase, the structure of the singular field near the semi-permeable interface crack tip in such bimaterials always consists of the singularity
r−1∕2
and a pair of oscillatory singularities
r−1∕2±iε
. Calculated values of the oscillatory index
ε
for the 35 kinds of bimaterials are presented in tables, which are always within the range between 0.046 and 0.088. Energy analyses for five kinds of such bimaterials constructed by PZT-4 and the five kinds of elastic dielectrics are studied in more detail under four different cases: (i) the crack is electrically conducting, (ii) the crack gap is filled with air/vacuum, (iii) the crack gap is filled with silicon oil, and (iv) the crack is electrically impermeable. Detailed comparisons on the variable tendencies of the crack tip ERR against the applied electric field are given under some practical electromechanical loading levels. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR. We also conclude that the previous investigations under the impermeable crack model are incorrect since the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack, whereas the previous investigations under the conducting crack model may be accepted in a tolerant way since the results of the ERR show very small discrepancies from those for the semi-permeable crack, especially when the crack gap is filled with silicon oil. In all cases under consideration the curves of the ERR for silicon oil are more likely tending to those for the conducting crack rather than to those for air or vacuum. Finally, we conclude that the variable tendencies of the ERR against the applied electric field have an interesting load-dependent feature when the applied mechanical loading increases. This feature is due to the nonlinear relation between the normal electric displacement component and the applied electromechanical loadings from a quadratic equation.
piezoceramics, interface phenomena, cracks, elasticity, electromechanical effects, permittivity, oils, interface crack, elastic dielectric, piezoelectric, permittivity, the crack tip ERR, oscillatory singularity
Fracture (Materials), Electrical properties, Displacement, Dielectric materials
Monolithic Multilayer Piezoelectric Ceramic Transducer
Proceedings of the IEEE Sixth International Symposium on Applied Ferromagnetics
Modified-Lead-Titanate/Polymer Composites for Hydrophone Applications
On the Modeling and Design of Piezocomposites With Prescribed Properties
Stress Singularities of Interface Cracks in Bonded Piezoelectric Half-Spaces
Fracture Mechanics of Piezoelectric Ceramics
Near-Tip and Intensity Factors for Interfacial Cracks in Dissimilar Anisotropic Piezoelectric Media
Interfacial Debonding of a Circular Inhomogeneity in Piezoelectric Materials
An Arbitrarily-Oriented Plane Crack Terminating an Interface Between Dissimilar Piezoelectric Materials
Anti-Plane Shear Crack in a Piezoelectric Layer Bonded to Dissimilar Half Spaces
Solutions for Transversely Isotropic Piezoelectric Infinite Body, Semi-Infinite Body and Bimaterial Infinite Body Subjected to Uniform Ring Loading and Charges
Interaction Between an Interface Crack and Subinterface Microcracks in Metal/Piezoelectric Bimaterials
Weight Functions for Interface Cracks in Dissimilar Anisotropic Piezoelectric Materials
Conducting Cracks in Dissimilar Piezoelectric Media
Permeable Cracks Between Two Dissimilar Piezoelectric Materials
Singularity Parameters ε and κ for Interface Cracks in Transversely Isotropic Piezoelectric Bimaterials
On the Crack Tip Stress Singularity of Interfacial Cracks in Transversely Isotropic Piezoelectric Bimaterials
Solution of a Semi-Permeable Interface Crack in Dissimilar Piezoelectric Materials
On the Influence of the Electric Permeability on an Interface Crack in a Piezoelectric Bimaterial Compound
Interface Crack Problem in Elastic Dielectric/Piezoelectric Bimaterials
The Energy Release Rate for a Griffith Crack in a Piezoelectric Material
Electromagnetoelasticity
A New Electric Boundary Condition of Electric Fracture Mechanics and its Application
New Development Concerning Piezoelectric Materials With Defects
On a Plane Crack in Piezoelectric Solids
A Fracture Criterion for Conducting Cracks in Homogeneously Poled Piezoelectric PZT-PIC 151 Ceramics
Effect of Electric Fields on Fracture of Piezoelectric Ceramics
Interface Crack in Anisotropic Bimaterials
A Crack Between Dissimilar Media
Stress Intensity Factors and Energy Release Rate for Interfacial Cracks Between Dissimilar Anisotropic Materials
Interfacial Dislocation and its Application to Interface Cracks in Anisotropic Bimaterials
Current Understanding on Fracture Behaviors of Ferroelectric/Piezoelectric Materials
Some Basic Problems of Mathematical Theory of Elasticity
Noordhoof
Solution for a Semi-Permeable Interface Crack Between Two Dissimilar Piezoelectric Material
Nonlinear Behavior and Critical State of a Penny-Shaped Dielectric Crack in a Piezoelectric Solid
M -Integral for Calculating Intensity Factors of Cracked Piezoelectric Materials Using the Exact Boundary Conditions
Three-Dimensional Electroelastic Analysis of a Piezoelectric Material With a Penny-Shaped Dielectric Crack
Dielectric and Piezoelectric Ceramics for High Temperature Applications
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Landsat-derived bathymetry of lakes on the Arctic Coastal Plain of northern...
Simpson, Claire E.; Arp, Christopher D.; Sheng, Yongwei; Carroll, Mark L.; Jones, Benjamin M.; Smith, Laurence C.
The Pleistocene sand sea on the Arctic Coastal Plain (ACP) of northern Alaska is underlain by an ancient sand dune field, a geological feature that affects regional lake characteristics. Many of these lakes, which cover approximately 20 % of the Pleistocene sand sea, are relatively deep (up to 25 m). In addition to the natural importance of ACP sand sea lakes for water storage, energy balance, and ecological habitat, the need for winter water for industrial development and exploration activities makes lakes in this region a valuable resource. However, ACP sand sea lakes have received little prior study. Here, we collect in situ bathymetric data to test 12 model variants for predicting sand sea lake depth based on analysis of Landsat-8 Operational Land Imager (OLI) images. Lake depth gradients were measured at 17 lakes in midsummer 2017 using a Humminbird 798ci HD SI Combo automatic sonar system. The field-measured data points were compared to red–green–blue (RGB) bands of a Landsat-8 OLI image acquired on 8 August 2016 to select and calibrate the most accurate spectral-depth model for each study lake and map bathymetry. Exponential functions using a simple band ratio (with bands selected based on lake turbidity and bed substrate) yielded the most successful model variants. For each lake, the most accurate model explained 81.8 % of the variation in depth, on average. Modeled lake bathymetries were integrated with remotely sensed lake surface area to quantify lake water storage volumes, which ranged from inline-formula
M1inlinescrollmathmlnormal 1.056×{normal 10}^{-normal 3}
63pt14ptsvg-formulamathimgfba03344b9262a44537d5b4a576feaa4 essd-13-1135-2021-ie00001.svg63pt14ptessd-13-1135-2021-ie00001.png to inline-formula
M2inlinescrollmathmlnormal 57.416×{normal 10}^{-normal 3}
69pt14ptsvg-formulamathimg525011b9d532e104cabe88e5e7179f91 essd-13-1135-2021-ie00002.svg69pt14ptessd-13-1135-2021-ie00002.png kminline-formula3. Due to variations in depth maxima, substrate, and turbidity between lakes, a regional model is currently infeasible, rendering necessary the acquisition of additional in situ data with which to develop a regional model solution. Estimating lake water volumes using remote sensing will facilitate better management of expanding development activities and serve as a baseline by which to evaluate future responses to ongoing and rapid climate change in the Arctic. All sonar depth data and modeled lake bathymetry rasters can be freely accessed at https://doi.org/10.18739/A2SN01440https://doi.org/10.18739/A2SN01440 (Simpson and Arp, 2018) and https://doi.org/10.18739/A2HT2GC6Ghttps://doi.org/10.18739/A2HT2GC6G (Simpson, 2019), respectively.
Simpson, Claire E. / Arp, Christopher D. / Sheng, Yongwei / et al: Landsat-derived bathymetry of lakes on the Arctic Coastal Plain of northern Alaska. 2021. Copernicus Publications.
Rechteinhaber: Claire E. Simpson et al.
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Cv=SD(t)t̄
where SD(t) is the standard deviation and
t̄
is the mean of the measured unit thicknesses. Thickness variation as a function of stratigraphic interval (Fig. 11) records greater variability in the hanging wall to the thrust, particularly higher in the stratigraphy; as can be observed in Figure 10. This trend conforms to the interpretation of the structure (Fig. 9), which records greater complexity of deformation in the hanging wall to the thrust than in the footwall counterpart: multiple minor fold hinges, beds truncated by minor thrusting and accommodation of disharmonic folding by thickening of fine-grained material above and below the sand-rich units h4 and h5 (Fig. 9) are evident in the hanging wall but not in the footwall.
Lo=Lr+Ae/Tud
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Tariff - ecl@ss - Classification - Specification
Tariff (2845 views - Classification & Specification)
A tariff is a tax on imports or exports (an international trade tariff). In other languages and very occasionally in English, tariff or its equivalent may also be used to describe any list of prices (e.g., electrical tariff).
Licensed under Creative Commons Attribution-Share Alike 4.0 (James 4).
For the policies of using tariffs, see Protective tariff. For other uses, see Tariff (disambiguation).
2.1 Calculation of customs duty
2.2 Harmonized System of Nomenclature
2.3 Customs authority
2.5 Duty-free goods
2.6 Duty calculation for companies in real life
5 Within technology strategies
6.2 Trade dynamics
6.3 Trade liberalisation
The small Spanish town of Tarifa is sometimes credited with being the origin of the word "tariff," since it was the first port in history to charge merchants for the use of its docks[1] The name "Tarifa" itself is derived from the name of the Berber warrior, Tarif ibn Malik. However, other sources assume that the origin of tariff is the Italian word tariffa translated as "list of prices, book of rates," which is derived from the Arabic ta'rif meaning "making known" or "to define."[2]
A customs duty or due is the indirect tax levied on the import or export of goods in international trade. In economic sense, a duty is also a kind of consumption tax. A duty levied on goods being imported is referred to as an import duty. Similarly, a duty levied on exports is called an export duty. A tariff, which is actually a list of commodities along with the leviable rate (amount) of customs duty, is popularly referred to as a customs duty.
In the Kingdom of England, customs duties were typically part of the customary revenue of the king, and therefore did not need parliamentary consent to be levied, unlike excise duty, land tax, or other forms of taxes. This is no longer the case.
Calculation of customs duty
Customs duty is calculated on the determination of the assessable value in case of those items for which the duty is levied ad valorem. This is often the transaction value unless a customs officer determines assessable value in accordance with the Harmonized System.
For the purpose of assessment of customs duty, products are given an identification code that has come to be known as the Harmonized System code. This code was developed by the World Customs Organization based in Brussels. A Harmonized System code may be from four to ten digits. For example, 17.03 is the HS code for molasses from the extraction or refining of sugar. However, within 17.03, the number 17.03.90 stands for "Molasses (Excluding Cane Molasses)".
Introduction of Harmonized System code in 1990s has largely replaced the Standard International Trade Classification (SITC), though SITC remains in use for statistical purposes. In drawing up the national tariff, the revenue departments often specifies the rate of customs duty with reference to the HS code of the product. In some countries and customs unions, 6-digit HS codes are locally extended to 8 digits or 10 digits for further tariff discrimination: for example the European Union uses its 8-digit CN (Combined Nomenclature) and 10-digit TARIC codes.
A Customs authority in each country is responsible for collecting taxes on the import into or export of goods out of the country. Normally the Customs authority, operating under national law, is authorized to examine cargo in order to ascertain actual description, specification volume or quantity, so that the assessable value and the rate of duty may be correctly determined and applied.
Main article: Tax avoidance
Evasion of customs duties takes place mainly in two ways. In one, the trader under-declares the value so that the assessable value is lower than actual. In a similar vein, a trader can evade customs duty by understatement of quantity or volume of the product of trade. A trader may also evade duty by misrepresenting traded goods, categorizing goods as items which attract lower customs duties. the Evasion of customs duty may take place with or without the collaboration of customs officials. Evasion of customs duty does not necessarily constitute smuggling.[citation needed]
Many countries allow a traveler to bring goods into the country duty-free. These goods may be bought at ports and airports or sometimes within one country without attracting the usual government taxes and then brought into another country duty-free. Some countries impose allowances which limit the number or value of duty-free items that one person can bring into the country. These restrictions often apply to tobacco, wine, spirits, cosmetics, gifts and souvenirs. Often foreign diplomats and UN officials are entitled to duty-free goods. Duty-free goods are imported and stocked in what is called a bonded warehouse.
Duty calculation for companies in real life
With many methods and regulations, businesses at times struggle to manage the duties. In addition to difficulties in calculations, there are challenges in analyzing duties; and to opt for duty free options like using a bonded warehouse.
Companies use ERP software to calculate duties automatically to, on one hand, avoid error-prone manual work on duty regulations and formulas and on the other hand, manage and analyze the historically paid duties. Moreover, ERP software offers an option for customs warehouse, introduced to save duty and VAT payments. In addition, the duty deferment and suspension is also taken into consideration.
Neoclassical economic theorists tend to view tariffs as distortions to the free market. Typical analyses find that tariffs tend to benefit domestic producers and government at the expense of consumers, and that the net welfare effects of a tariff on the importing country are negative. Normative judgements often follow from these findings, namely that it may be disadvantageous for a country to artificially shield an industry from world markets and that it might be better to allow a collapse to take place. Opposition to all tariff aims to reduce tariffs and to avoid countries discriminating between differing countries when applying tariffs. The diagrams to the right show the costs and benefits of imposing a tariff on a good in the domestic economy.
When incorporating free international trade into the model we use a supply curve denoted as
{\displaystyle P_{tariff}}
(diagram 1) or
{\displaystyle P_{w}}
(diagram 2). This curve represents the assumption that the international supply of the good or service is perfectly elastic and that the world can produce at a near infinite quantity of the good. Before the tariff, there is a quantity demanded of Qc1 (diagram 1) or D (diagram 2). The difference between quantity demanded and quantity supplied (between D and S on diagram 2, respectively) was filled by importing from abroad. This is shown on diagram 1 as Quantity of Imports (without tariff). After the imposition of a tariff, domestic price rises, but foreign export prices fall due to the difference in tax incidence on the consumers (at home) and producers (abroad).
The new price level at Home is Ptariff or Pt, which is higher than the world price. More of the good is now produced at Home – it now makes Qs2 (diagram 1) or S* (diagram 2) of the good. Due to the higher price, only Qc2 or D* of the good is demanded by Home. The difference between the quantity supplied and the quantity demanded is still filled by importing from abroad. However, the imposition of the tariff reduces the quantity of imports from D − S to D* − S* (diagram 2). This is also shown in diagram 1 as Quantity of Imports (with tariff).
Domestic producers enjoy a gain in their surplus. Producer surplus, defined as the difference between what the producers were willing to receive by selling a good and the actual price of the good, expands from the region below Pw to the region below Pt. Therefore, the domestic producers gain an amount shown by the area A.
Domestic consumers face a higher price, reducing their welfare. Consumer surplus is the area between the price line and the demand curve. Therefore, the consumer surplus shrinks from the area above Pw to the area above Pt, i.e. it shrinks by the areas A, B, C and D. This includes the gained producer surplus, the deadweight loss, and the tax revenue.
The government gains from the taxes. It charges an amount Pt − Pt* of tariff for every good imported. Since D* − S* goods are imported, the government gains an area of C and E. However, there is a deadweight loss of the triangles B and D, or in diagram 1, the triangles labeled Societal Loss. Deadweight loss is also called efficiency loss. This cost is incurred because tariffs reduce the incentives for the society to consume and produce.
The net loss to the society due to the tariff would be given by the total costs of the tariff minus its benefits to the society. Therefore, the net welfare loss due to the tariff is equal to:
Consumer Loss − Government Revenue − Producer Gain
or graphically, this gain is given by the areas shown by:
{\displaystyle (A+B+C+D)-(C+E)-A=B+D-E}
That is, tariffs are beneficial to the society if the area given by the rectangle E is greater than the deadweight loss. Rectangle E is called the terms of trade gain.
The model above is completely accurate only in the extreme case where no consumer belongs to the producers group and the cost of the product is a fraction of their wages. If the opposite extreme is taken, assuming that all consumers come from the producers' group, consumers' only purchasing power comes from the wages earned in production, and the product costs their whole wage, the graph looks radically different. Without tariffs, only those producers/consumers able to produce the product at the world price will have the money to purchase it at that price.
See also: Tariffs in United States history, List of tariffs in the United States, and Protectionism in the United States
The tariff has been used as a political tool to establish an independent nation; for example, the United States Tariff Act of 1789, signed specifically on July 4, was called the "Second Declaration of Independence" by newspapers because it was intended to be the economic means to achieve the political goal of a sovereign and independent United States.[3]
The political impact of tariffs is judged depending on the political perspective,; for example the 2002 United States steel tariff imposed a 30% tariff on a variety of imported steel products for a period of three years and American steel producers supported the tariff.[4]
Tariffs can emerge as a political issue prior to an election. In the leadup to the 2007 Australian Federal election, the Australian Labor Party announced it would undertake a review of Australian car tariffs if elected.[5] The Liberal Party made a similar commitment, while independent candidate Nick Xenophon announced his intention to introduce tariff-based legislation as "a matter of urgency".[6]
Unpopular tariffs are known to have ignited social unrest, for example the 1905 meat riots in Chile that developed in protest against tariffs applied to the cattle imports from Argentina.[7][8]
Within technology strategies
This section contains wording that promotes the subject in a subjective manner without imparting real information. Please remove or replace such wording and instead of making proclamations about a subject's importance, use facts and attribution to demonstrate that importance. (September 2015) (Learn how and when to remove this template message)
When tariffs are an integral element of a country's technology strategy, some economist believe that such tariffs can be highly effective in helping to increase and maintain the country's economic health. Other economists might be less enthusiastic, as tariffs may reduce trade and there may be many spillovers and externalities involved with trade and tariffs. The existence of these externalities makes the imposition of tariffs a rather ambiguous strategy. As an integral part of the technology strategy, tariffs are effective in supporting the technology strategy's function of enabling the country to outmaneuver the competition in the acquisition and utilization of technology in order to produce products and provide services that excel at satisfying the customer needs for a competitive advantage in domestic and foreign markets. The notion that government and policy would be effective at finding new and infant technologies, rather than supporting existing politically motivated industry, rather than, say, international technology venture specialists, is however, unproven.
This is related to the infant industry argument.
In contrast, in economic theory tariffs are viewed as a primary element in international trade with the function of the tariff being to influence the flow of trade by lowering or raising the price of targeted goods to create what amounts to an artificial competitive advantage. When tariffs are viewed and used in this fashion, they are addressing the country's and the competitors' respective economic healths in terms of maximizing or minimizing revenue flow rather than in terms of the ability to generate and maintain a competitive advantage which is the source of the revenue. As a result, the impact of the tariffs on the economic health of the country are at best minimal but often are counter-productive.
A program within the US intelligence community, Project Socrates, that was tasked with addressing America's declining economic competitiveness, determined that countries like China and India were using tariffs as an integral element of their respective technology strategies to rapidly build their countries into economic superpowers. However, the US intelligence community tends to have limited inputs into developing US trade policy. It was also determined that the US, in its early years, had also used tariffs as an integral part of what amounted to technology strategies to transform the country into a superpower.[9]
Bound tariff rate
This article uses material from the Wikipedia article "Tariff", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Home : Support : Online Help : Programming : DeepLearning Package : Tensors : Operations on Tensors : abs
compute the absolute value of entries in a Tensor
DeepLearning/Tensor/ceil
compute the ceiling of entries in a Tensor
DeepLearning/Tensor/erf
compute the erf of entries in a Tensor
DeepLearning/Tensor/erfc
compute the erfc of entries in a Tensor
DeepLearning/Tensor/exp
compute the exponential of entries in a Tensor
DeepLearning/Tensor/floor
compute the floor of entries in a Tensor
DeepLearning/Tensor/log
compute the logarithm of entries in a Tensor
DeepLearning/Tensor/lnGAMMA
compute the lnGAMMA of entries in a Tensor
DeepLearning/Tensor/Psi
compute the Psi of entries in a Tensor
DeepLearning/Tensor/round
compute the rounded value of entries in a Tensor
DeepLearning/Tensor/sign
compute the sign of entries in a Tensor
DeepLearning/Tensor/sqrt
compute the square root of entries in a Tensor
DeepLearning/Tensor/Zeta
compute the Hurwitz zeta function on entries in a Tensor
abs(t,opts) ceil(t,opts) erf(t,opts)
erfc(t,opts) exp(t,opts) floor(t,opts)
log(t,opts) lnGAMMA(t,opts) Psi(t,opts)
round(t,opts) sign(t,opts) sqrt(t,opts)
Zeta(t,opts)
The abs(t) command computes the abs of entries in a Tensor.
The ceil(t) command computes the complex ceil of entries in a Tensor.
The erf(t) command computes the erf of entries in a Tensor.
The erfc(t) command computes the erfc of entries in a Tensor.
The exp(t) command computes the exponential of entries in a Tensor.
The floor(t) command computes the floor of entries in a Tensor.
The log(t) command computes the logarithm of entries in a Tensor.
The lnGAMMA(t) command computes the lnGAMMA of entries in a Tensor.
The Psi(t) command computes the Psi of entries in a Tensor.
The round(t) command computes the rounded value of entries in a Tensor.
The sign(t) command computes the sign of entries in a Tensor.
The sqrt(t) command computes the square root of entries in a Tensor.
The Zeta(t) command computes the Hurwitz zeta of entries in a Tensor.
\mathrm{with}\left(\mathrm{DeepLearning}\right):
V≔\mathrm{Matrix}\left([[11.0,18.3],[12.1,20.3]],\mathrm{datatype}=\mathrm{float}[8]\right)
\textcolor[rgb]{0,0,1}{V}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{11.}& \textcolor[rgb]{0,0,1}{18.3000000000000}\\ \textcolor[rgb]{0,0,1}{12.1000000000000}& \textcolor[rgb]{0,0,1}{20.3000000000000}\end{array}]
t≔\mathrm{Tensor}\left(V\right)
\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
\mathrm{abs}\left(t\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
\mathrm{ceil}\left(t\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
\mathrm{erf}\left(t\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
\mathrm{exp}\left(t\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
\mathrm{round}\left(t\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{DeepLearning Tensor}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Name: none}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Shape: undefined}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type: float\left[8\right]}}\end{array}]
The DeepLearning/Tensor/abs, DeepLearning/Tensor/ceil, DeepLearning/Tensor/erf, DeepLearning/Tensor/erfc, DeepLearning/Tensor/exp, DeepLearning/Tensor/floor, DeepLearning/Tensor/log, DeepLearning/Tensor/lnGAMMA, DeepLearning/Tensor/Psi, DeepLearning/Tensor/round, DeepLearning/Tensor/sign, DeepLearning/Tensor/sqrt and DeepLearning/Tensor/Zeta commands were introduced in Maple 2018.
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MainVariable - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : MainVariable
main variable of a nonconstant polynomial
MainVariable(p, R)
The function call MainVariable(p,R) returns the greatest variable of p with respect to the variable ordering of R.
This command is part of the RegularChains package, so it can be used in the form MainVariable(..) only after executing the command with(RegularChains). However, it can always be accessed through the long form of the command by using RegularChains[MainVariable](..).
\mathrm{with}\left(\mathrm{RegularChains}\right):
R≔\mathrm{PolynomialRing}\left([x,y,z]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔\left(y+1\right){x}^{3}+\left(z+4\right)x+3
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}
\mathrm{Initial}\left(p,R\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Rank}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Tail}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
R≔\mathrm{PolynomialRing}\left([z,y,x]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔\mathrm{expand}\left(\left(y+1\right){x}^{3}+\left(z+4\right)x+3\right)
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Initial}\left(p,R\right)
\textcolor[rgb]{0,0,1}{x}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{Rank}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Tail}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
R≔\mathrm{PolynomialRing}\left([z,y,x],3\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{polynomial_ring}}
p≔{\left(x+y\right)}^{3}{z}^{3}+3{z}^{2}+2z+y+4
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
\mathrm{MainVariable}\left(p,R\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{Initial}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{MainDegree}\left(p,R\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Rank}\left(p,R\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Tail}\left(p,R\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
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Theory of everything - Uncyclopedia, the content-free encyclopedia
For those without comedic tastes, the so-called experts at Wikipedia think they have an article about Theory of everything.
The theory of everything is a single theory which tries to have an all-encompassing explanation of reality. Smart people were tipped off to the existence of such a theory way back when they realized that there was such a thing as reality. This has thus been destroyed by a secretive cult known only by their brain-melting machines; Atari.
3 Candidates for the "Theory of Everything"
4 Philosophical theory of everything
4.9 Life extension, transhumanism and singularity
Philosophers challenged the existence of this reality, declaring that it must be proven to be true or until then be only referred to as the Hypothesis of Everything. Stephen Hawking believed this theory to be reachable by assessing the trajectory of gigantic hexahedron-like objects through 42-dimensional space according to Newtonian physics. This theory, however, was debunked a hundred years after Hawking's death by notorious topiarist and dockworker Albert Einstein in his famous essay, God doesn't play craps with the Universe. At the time of the article's writing, God wasn't available for comment, as it occurred during the interlude of God's activity on Earth (Roughly 33 AD-6666 AD, although some recent theorists postulate that the period of Divine inactivity actually spans from
{\displaystyle 10^{10^{100}}}
BC to the Final Destiny of the Universe). The theory can in fact be described in less than one page of advanced mathematical notation, but thanks to limitations in the
{\displaystyle L^{A}T_{E}X}
typesetting language and the general ignorance of Uncyclopedia editors about the subject, will remain undisclosed.
Modern Theory[edit]
The most obvious and simple explanation for reality is, of course, that the theory of everything is the theory of everything! Although we all know what the theory of everything is in theory, we still don't know what it really is, as 'reality' has been extraordinary difficult to observe or duplicate in laboratory conditions. That is the difference between the reality that the theory is trying to explain and the explanation; however, there is no known explanation for the theory itself other than the theory itself. Because of this fact, the theory of everything is also known as the theory on the theory of everything. The theory of everything can be applied to all things, so whatever your thinking right now can be explained by it theoretical application. The actual theory in scientifically reviwable terms has never actually been published, except once, shaved on the back of a kitten found in Wyoming in 1939 which was promptly huffed by Irving Berlin. The Theory's final resting place is theorized to be Berlin's grave in Munich, inside the unfortunate singer/phisiculturist's left nostril.
Candidates for the "Theory of Everything"[edit]
TOE is sometimes also called a supergrand unified theory.
A theory of everything is needed to explain phenonmenon such as the big bang or gravitational singularities in which the current theories of general relativity and quantum mechanics self-destruct. Theoreticial motivations for finding a theory of everything include the Platonic belief that the ultimate nature of the universe is simple and therefore the current models of the universe such as the standard model cannot be complete because they are too complicated.
Popular candidates for a theory of everything at the moment include loop quantum gravity, string theory, and M-theory. Most of these theories attempt to deal with the renormalization problem by setting up some lower bound on the length scales possible. Also early 21st century theories of everything tend to suppose that the universe actually has more dimensions than the easily observed three of space and one of time. The motivation behind this approach began with the Kaluza-Klein theory in which it was noted that adding one dimension to general relativity would produce the electromagnetic Maxwell's equations. This has led to efforts to work with theories with large number of dimensions in the hopes that this would produce equations which are similar to known laws of physics.
Theories of everything must be distinguished from grand unified theories (or GUTs), which attempt to unite all the fundamental forces except gravity. A theory that unites the electromagnetic and weak nuclear forces into a single electroweak force has already been established; GUTs attempt to unify the strong nuclear and electroweak forces.
Attempts to create theories of everything are common among people outside the professional physics community. Unfortunately these theories, like those of professional physics, suffer from the inability to make quantifiable and falsifible predictions. Unlike professional physicists who are generally aware that their pet theory is incomplete, untested, and probably wrong, amateurs who create TOE's tend to be unaware of the need and mechanisms for testing scientific theories and the fact that most proposed theories are just wrong.
Then there's Quantum Murphydynamics, a new theory of everything invented by a brilliant young Uncyclopedia-obsessed particle physicist with no life.
Philosophical theory of everything[edit]
The status of a physical TOE is open to philosophical debate. For instance, if physicalism is true, a physical TOE would coincide with a philosophical theory of everything. Some philosophers — Aristotle, Plato, Hegel, Whitehead — have attempted to construct all-encompassing systems. Others are highly dubious about the very possibility of such an exercise.
Conservation of energy states that everything is energy and energy is conserved. The experience of the observer is energy. Energy can not be created or destroyed. There is as much positive energy as there is negative energy. If they canceled each other there would be nothing. Experience is also energy. There is no free energy or perpetual motion device because energy can not be created or destroyed. Since the observer (you) is energy as well, the observers happiness can not be created or destroyed. Happiness, satisfaction, love, pleasure or what ever you call it, can not be created or destroyed, but only changes form where various forms of pleasure are counteracted with displeasure, pain, suffering, unhappiness, etc. Happiness can not be created, the only time people are happy is when they are creating their own doom and are not aware of it. So the happier you are the more unhappy you will be in the future and vice versa.
Several decades ago computer scientists have realized that programming is greatly simplified by creating of what is called an object. An object is quite simply, everything. Every code, process, algorithm, function etc. is the same thing, a thing, or an object as called in computer lingo. It is thus greatly simplified to code when everything is treated as the same thing. Similarly in life, everything is the same thing. In other words, everything is alive and everything is equal. Everything is an experience. Because everything is an experience, it doesn't matter who's experience it it, everyone is essentially equal, from the greatest minds in the universe to the Tutsi fly. Nobody's experience is any better or worse that anyone else's because everyone's energy or happinesses are conserved. Since energy is conserved, it doesn't matter who or what you are, everything is conserved. Happiness equates unhappiness.
This coincides with theories of quantum entanglement that everything is the same thing, that everything is connected or entangled. Since everything and everyone is equal, we have a lot in common, we're all the same. We are all experiences and all experiences are related to each other. Time is an illusion just like space. There is no time, it's only an experience of how different physical perceptions relate to each other. There is infinite time and space dimensions and time travel is possible and easy once physical mechanisms of hyperspace or multidimensionality are mastered.
There is nothing physical. Scientists have never been able to find a single atom of physical presence. Matter breaks down into atoms, atoms break down into subatomic particles and so on and so forth. In the end, there really is no matter, only information. Belief systems are the same thing, just information. No one belief system is any better or worse that another, because they are all just information, when it all comes down to it. In other words, the Flying Spaghetti Monster belief is no different that the Big Bang or the Grand Unified Field Theory. What matters in not weather it's scientifically proven, but weather you believe in it. If you can imagine it, it's information. It's no different from anything else because nothing but information has ever been found in search of something real, physical or matter. It doesn't mean that by believing in something you can easily make it real. In the end you'll find that it's hardest to believe in what you want the most and the benefits balances out, refer to Conservation of Energy.
Since everything is energy and everything is conserved and is merely information, there's no such thing as someone being better or worse than anyone else. Everyone's experience is infinite. No matter how smart or how dumb someone is their experience is just as rich as someone who has a higher IQ or is perceived to be smarter. This brings us to the deduction that no matter how dumb someone is they are just like us. Since everything is the same and everything is an object, anything you can imagine is just as alive as anything. In other words, a mosquito, rock, Santa Clause, X or anything at all is just as alive as human beings. You'll never know what's it's really like to be a grain of sand, just like you'll never know what it's like to be Whitney Houghston until you actually become it. Meanwhile we'll keep thinking that we are the only ones alive because we can only understand those who are similar to us. In actuality, everything is alive and going through it's own chalanges we just don't realize it because we live in our world.
The whole question of extraterrestrials and aliens is redundant because not only do other life forms exist, they are everywhere. The only question is which life forms do we choose to relate to and accept.
A specific question is if there are other humanoid life forms elsewhere in the universe. The answer is probably yes, but if they have superior technology and better lifestyles, they would not have any reason in coming here. Just like we have no reason of going to underdeveloped countries unless there's a benefit in it for us. The answer is probably that if you have yet not seen aliens, thank you lucky starts that they've left you alone because if you're not careful, you might lust get what you wished for. All these government conspiracies and cover ups really are for your own protection designed to preserve our ways of life by those for whom it is already too late.
Point zero[edit]
So where does this leave us? Well apparently none of it really does matter. The universe is perfectly fair and at the same time, you just can't get a break no matter what you do. So should we all now run for the hills and wait for the sky to fall, or summon the aliens and await the horrendous consequence that will come? Well apparently one again, according to this theory, it really does not matter because aliens are really just like us. One bean of hope however remains, maybe I'm wrong about all this. Maybe the universe is not balanced and there is a way to obtain free energy after all, despite that fact that no one has done it and it is the most impossible thing to do imaginable. Obtaining this free energy, this free lunch, something for nothing, free happiness without consequence is what is called point zero.
The impossible dream[edit]
Obtaining free energy will be the hardest technological, sociological or spiritual undertaking that has never been done by man, aliens, god or any other life forms anywhere ever since the beginning of time because it would have only had to have been done once. By all logic, rationale and sanity, it is impossible no matter what you do. Obtaining this point zero is the only thing left imaginable to do once we realize that nothing really matters and that we are all the same and we all just want to be happy. By definition, it is not easy and it is the most impossible thing ever conceivable or more accurately inconceivable. It is the most impossible, hardest most ridiculous thing imaginable (or more accurately, unimaginable) that anyone could ever try to do and by definition guaranteed not to work, and it is the only choice we have.
Life extension, transhumanism and singularity[edit]
Technology factors in when we accept that nothing matters and we might as well try to stay alive to keep our memories and try as hard as possible to evolve in hopes of finding free energy, which is impossible. Life extension is important as a precautionary measure and a form of insurance. It is done just in case this is all wrong and the truth is completely different from what we could have ever imagined. It is basically a way of saying that life isn't that great, and might not be desirable, but for all we know, death isn't that great either, and not possible anyway, so might as well just stay alive in hopes of finding the ever elusive happiness in the future. Once immortality has been achieved, transhumanism and artificial intelligence that is the last invention that will ever have to be made will come in handy if we are to achieve infinity, god and beyond to do that which has never been done and by all definition never will be. At the same time we have no other choice, plus there's always hope that this whole theory is wrong and there really is a pot of gold at the end of the rainbow in the distant future against all odds.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Theory_of_everything&oldid=6127645"
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Motion in Two Dimensions | Boundless Physics | Course Hero
An object moving with constant velocity must have a constant speed in a constant direction.
Examine the terms for constant velocity and how they apply to acceleration
Constant velocity means that the object in motion is moving in a straight line at a constant speed.
This line can be represented algebraically as:
\text{x}=\text{x}_0 + \text{vt}
\text{x}_0
represents the position of the object at
\text{t}=0
, and the slope of the line indicates the object's speed.
The velocity can be positive or negative, and is indicated by the sign of our slope. This tells us in which direction the object moves.
constant velocity: Motion that does not change in speed nor direction.
Motion with constant velocity is one of the simplest forms of motion. This type of motion occurs when an an object is moving (or sliding) in the presence of little or negligible friction, similar to that of a hockey puck sliding across the ice. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion to a straight path.
Newton's second law (
\text{F}=\text{ma}
) suggests that when a force is applied to an object, the object would experience acceleration. If the acceleration is 0, the object shouldn't have any external forces applied on it. Mathematically, this can be shown as the following:
\text{a} = \frac{\text{dv}}{\text{dt}} = 0~\Rightarrow~ \text{v} = \text{const}
If an object is moving at constant velocity, the graph of distance vs. time (
\text{x}
\text{t}
) shows the same change in position over each interval of time. Therefore the motion of an object at constant velocity is represented by a straight line:
\text{x}=\text{x}_0+\text{vt}
\text{x}_0
is the displacement when
\text{t}=0
(or at the y-axis intercept).
Motion with Constant Velocity: When an object is moving with constant velocity, it does not change direction nor speed and therefore is represented as a straight line when graphed as distance over time.
You can also obtain an object's velocity if you know its trace over time. Given a graph as in, we can calculate the velocity from the change in distance over the change in time. In graphical terms, the velocity can be interpreted as the slope of the line. The velocity can be positive or negative, and is indicated by the sign of our slope. This tells us in which direction the object moves.
Analyzing two-dimensional projectile motion is done by breaking it into two motions: along the horizontal and vertical axes.
Analyze a two-dimensional projectile motion along horizontal and vertical axes
Constant acceleration in motion in two dimensions generally follows a projectile pattern.
Projectile motion is the motion of an object thrown or projected into the air, subject to only the (vertical) acceleration due to gravity.
We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes.
Projectile motion is the motion of an object thrown, or projected, into the air, subject only to the force of gravity. The object is called a projectile, and its path is called its trajectory. The motion of falling objects is a simple one-dimensional type of projectile motion in which there is no horizontal movement. In two-dimensional projectile motion, such as that of a football or other thrown object, there is both a vertical and a horizontal component to the motion.
Projectile Motion: Throwing a rock or kicking a ball generally produces a projectile pattern of motion that has both a vertical and a horizontal component.
The most important fact to remember is that motion along perpendicular axes are independent and thus can be analyzed separately. The key to analyzing two-dimensional projectile motion is to break it into two motions, one along the horizontal axis and the other along the vertical. To describe motion we must deal with velocity and acceleration, as well as with displacement.
We will assume all forces except for gravity (such as air resistance and friction, for example) are negligible. The components of acceleration are then very simple:
\text{a}_\text{y} = -\text{g} = -9.81 \frac{\text{m}}{\text{s}^2}
(we assume that the motion occurs at small enough heights near the surface of the earth so that the acceleration due to gravity is constant). Because the acceleration due to gravity is along the vertical direction only,
\text{a}_\text{x} = 0
. Thus, the kinematic equations describing the motion along the
\text{x}
\text{y}
directions respectively, can be used:
\text{x} = \text{x}_0 + \text{v}_\text{x} \text{t}
\text{v}_\text{y}=\text{v}_{0\text{y}}+\text{a}_\text{y} \text{t}
\text{y}=\text{y}_0+\text{v}_{0\text{y}} \text{t}+\frac{1}{2}\text{a}_\text{y} \text{t}^2
\text{v}_\text{y}^2=\text{v}_{0\text{y}}^2+2\text{a}_\text{y}(\text{y}-\text{y}_0)
We analyze two-dimensional projectile motion by breaking it into two independent one-dimensional motions along the vertical and horizontal axes. The horizontal motion is simple, because
\text{a}_\text{x} = 0
\text{v}_\text{x}
is thus constant. The velocity in the vertical direction begins to decrease as an object rises; at its highest point, the vertical velocity is zero. As an object falls towards the Earth again, the vertical velocity increases again in magnitude but points in the opposite direction to the initial vertical velocity. The
\text{x}
\text{y}
motions can be recombined to give the total velocity at any given point on the trajectory.
Velocity. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Velocity. License: CC BY-SA: Attribution-ShareAlike
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//physics/definition/constant-velocity. License: CC BY-SA: Attribution-ShareAlike
Acceleration. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Acceleration. License: CC BY-SA: Attribution-ShareAlike
kinematic. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/kinematic. License: CC BY-SA: Attribution-ShareAlike
Practice-3-Report-Physics-Laboratory.pdf
FISICA 3 1255 • Autonomous University of Nuevo León
Physics Honors Paper (1).docx
SCIENCE 101 • International Studies Preparatory Academy
ajdhd.pdf
FIN 30600 • University of Notre Dame
Lily_Gao-projectile
ENGLISH 214 • Christian Academy of Knoxville
projectile motion.pdf
PHYS 201 • Claflin University
Topic 2.1 Motion - Lab Report (Buggy Lab).docx
PH MECHANICS • Guest Community Ed. Center
Lab-Motion in Two Dimensions ( Projectile Motion).docx
PHYSICS 1401 • Tarleton State University
admin_physics-3-physics-for-scientists-and-engineers-serway-calc-414-ch-04-motion-in-two-dimensions-
PHYS MISC • Columbus State University
Olivia Gaillard - Motion in Two Dimensions PPT.pptx.pdf
PHYSICS Physics • Paul Vi Catholic High School
Unit IV - Motion in Two Dimensions Animated (1).ppt
PHYSICS 1 • American University in Cairo
2.1 Motion in two dimensions - Scale diagrams.filled.pdf
PHYSICS 101L • Kettering University
2.2 Motion in two dimensions - Algebra.filled.pdf
4. Motion in Two Dimensions.pptx
Chapter 4 - Motion in two dimensions (3).pptx
Motion in Two Dimensions.docx
HW3-Motion in Two Dimensions.pdf
PHYSICS 1401 • Dallas County Community College
Lab 3 Motion in Two Dimensions
PHYSICS 161 • Temple University
PHYSICS 1061 • Bucks County Community College
Physics-2A-Chapter-3-Notes-Motion-in-Two-Dimensions.pdf
PHYSICS PHYSICS 2A • Riverside City College
May 6 - U1 L6 - Motion in Two Dimensions.pdf
PHYSICS SPH3U • York University
Motion in Two Dimensions-Ave
PHYSICS 1BB3 • McMaster University
2 - Motion in Two Dimensions.pdf
Motion_In_Two_Dimensions_Solution
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PHYSICS 1 • University of the Philippines Los Baños
Motion in one and two dimensions.pdf
PHYSICS 123 • Universitas Padjadjaran
Motion in Two Dimensions-May21-Ave
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Quiz_3__Motion_in_Two-Dimensions.pdf
SCIENCE PHYSICS • Piscataway Twp High
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Correspondence to: #E-mail: sanghu@pusan.ac.kr, TEL: +82-51-510-1011
Triboelectric sensor, Dielectric polarization characteristic, High sensitivity, Flexible device
마찰전기 센서, 유전분극 특성, 고감도, 유연 장치
\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{ }\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{ }\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{ }\left(\text{S}\right)=4\pi ×{r}^{3}×n
Pv\mathrm{ }\left(%\right)=\left[\left(4/3\pi ×{r}^{2}\right)×n\right]×\left[1/\left(\text{a}×\text{b}×\text{h}\right)\right]
Pang, C., Lee, G.-Y., Kim, T.-I., Kim, S. M., Kim, H. N., Ahn, S.-H., Suh, K.-Y., (2012), A flexible and highly sensitive strain-gauge sensor using reversible interlocking of nanofibres, Nature Materials, 11(9), 795-801. [https://doi.org/10.1038/nmat3380]
Zhu, G., Chen, J., Liu, Y., Bai, P., Zhou, Y. S., Jing, Q., Pan, C., Wang, Z. L., (2013), Linear-grating triboelectric generator based on sliding electrification, Nano Letters, 13(5), 2282-2289. [https://doi.org/10.1021/nl4008985]
Cao, L.-M., Li, Z.-X., Guo, C., Li, P.-P., Meng, X.-Q., Wang, T.-M., (2019), Design and test of the MEMS coupled piezoelectric–electromagnetic energy harvester, International Journal of Precision Engineering and Manufacturing, 20(4), 673-686. [https://doi.org/10.1007/s12541-019-00051-x]
Khoo, Z. X., Teoh, J. E. M., Liu, Y., Chua, C. K., Yang, S., An, J., Leong, K. F., Yeong, W. Y., (2015), 3D printing of smart materials: A review on recent progresses in 4D printing, Virtual and Physical Prototyping, 10(3), 103-122. [https://doi.org/10.1080/17452759.2015.1097054]
Su, M., Brugger, J., Kim, B., (2020), Simply structured wearable triboelectric nanogenerator based on a hybrid composition of carbon nanotubes and polymer layer, International Journal of Precision Engineering and Manufacturing-Green Technology, 7(3), 683-698. [https://doi.org/10.1007/s40684-020-00212-8]
Kim, J., Lee, M., Shim, H. J., Ghaffari, R., Cho, H. R., Son, D., Jung, Y. H., Soh, M., Choi, C., Jung, S., (2014), Stretchable silicon nanoribbon electronics for skin prosthesis, Nature Communications, 5(1), 1-11. [https://doi.org/10.1038/ncomms6747]
Liu, H., Zhang, H., Han, W., Lin, H., Li, R., Zhu, J., Huang, W., (2021), 3D printed flexible strain sensors: From printing to devices and signals, Advanced Materials, 33(8), 2004782. [https://doi.org/10.1002/adma.202004782]
Leigh, S. J., Bradley, R. J., Purssell, C. P., Billson, D. R., Hutchins, D. A., (2012), A simple, low-cost conductive composite material for 3D printing of electronic sensors, PloS One, 7(11), e49365. [https://doi.org/10.1371/journal.pone.0049365]
Tang, Z., Jia, S., Wang, F., Bian, C., Chen, Y., Wang, Y., Li, B., (2018), Highly stretchable core–sheath fibers via wet-spinning for wearable strain sensors, ACS Applied Materials & Interfaces, 10(7), 6624-6635. [https://doi.org/10.1021/acsami.7b18677]
Yeo, J. C., Lim, C. T., (2016), Emerging flexible and wearable physical sensing platforms for healthcare and biomedical applications, Microsystems & Nanoengineering, 2(1), 16043. [https://doi.org/10.1038/micronano.2016.43]
Li, T., Xu, Y., Willander, M., Xing, F., Cao, X., Wang, N., Wang, Z. L., (2016), Lightweight triboelectric nanogenerator for energy harvesting and sensing tiny mechanical motion, Advanced Functional Materials, 26(24), 4370-4376. [https://doi.org/10.1002/adfm.201600279]
Fan, F.-R., Lin, L., Zhu, G., Wu, W., Zhang, R., Wang, Z. L., (2012), Transparent triboelectric nanogenerators and selfpowered pressure sensors based on micropatterned plastic films, Nano Letters, 12(6), 3109-3114. [https://doi.org/10.1021/nl300988z]
Jang, D., Kim, Y., Kim, T. Y., Koh, K., Jeong, U., Cho, J., (2016), Force-assembled triboelectric nanogenerator with high-humidity-resistant electricity generation using hierarchical surface morphology, Nano Energy, 20, 283-293. [https://doi.org/10.1016/j.nanoen.2015.12.021]
Davies, D. K., (1969), Charge generation on dielectric surfaces, Journal of Physics D: Applied Physics, 2(11), 1533. [https://doi.org/10.1088/0022-3727/2/11/307]
Khanafer, K., Duprey, A., Schlicht, M., Berguer, R., (2009), Effects of strain rate, mixing ratio, and stress-strain definition on the mechanical behavior of the polydimethylsiloxane (PDMS) material as related to its biological applications, Biomedical Microdevices, 11(2), 503-508. [https://doi.org/10.1007/s10544-008-9256-6]
So, S.-R., Park, S.-H., Park, S.-H., (2018), Relationship between mechanical properties and porosity of porous polymer sheet fabricated using water-soluble particles, Journal of the Korean Society of Manufacturing Process Engineers, 17(6), 16-23. [https://doi.org/10.14775/ksmpe.2018.17.6.016]
Kim, G., Kim, J., Javaid, M. U., Cho, H., Kim, S. Y., Park, J., (2021), Development and characterization of double-contact triboelectric nanogenerator with improved energy harvesting performance, Journal of the Korean Society for Precision Engineering, 38(4), 287-294. [https://doi.org/10.7736/JKSPE.021.005]
He is a Professor in the School of Mechanical Engineering at Pusan National University. He earned his M.S. and Ph.D. in the Mechanical Engineering at Korea Advanced Institute of Science and Technology (KAIST) in 1996 and 2006, respectively. His research fields are the engineering for additive manufacturing including design, processing, and post-processing of mechanical parts.
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Maasakkers, Joannes D.; Jacob, Daniel J.; Sulprizio, Melissa P.; Scarpelli, Tia R.; Nesser, Hannah; Sheng, Jianxiong; Zhang, Yuzhong; Lu, Xiao; Bloom, A. Anthony; Bowman, Kevin W.; Worden, John R.; Parker, Robert J.
We use 2010–2015 Greenhouse Gases Observing Satellite (GOSAT) observations of atmospheric methane columns over North America in a high-resolution inversion of methane emissions, including contributions from different sectors and their trends over the period. The inversion involves an analytical solution to the Bayesian optimization problem for a Gaussian mixture model (GMM) of the emission field with up to inline-formula
M1inlinescrollmathmlnormal 0.5{}^{\circ }×normal 0.625{}^{\circ }
64pt11ptsvg-formulamathimg9fd588606ec2b110e8d31a527531ec46 acp-21-4339-2021-ie00001.svg64pt11ptacp-21-4339-2021-ie00001.png resolution in concentrated source regions. The analytical solution provides a closed-form characterization of the information content from the inversion and facilitates the construction of a large ensemble of solutions exploring the effect of different uncertainties and assumptions in the inverse analysis. Prior estimates for the inversion include a gridded version of the Environmental Protection Agency (EPA) Inventory of US Greenhouse Gas Emissions and Sinks (GHGI) and the WetCHARTs model ensemble for wetlands. Our best estimate for mean 2010–2015 US anthropogenic emissions is 30.6 (range: 29.4–31.3) Tg ainline-formula−1, slightly higher than the gridded EPA inventory (28.7 (26.4–36.2) Tg ainline-formula−1). The main discrepancy is for the oil and gas production sectors, where we find higher emissions than the GHGI by 35 % and 22 %, respectively. The most recent version of the EPA GHGI revises downward its estimate of emissions from oil production, and we find that these are lower than our estimate by a factor of 2. Our best estimate of US wetland emissions is 10.2 (5.6–11.1) Tg ainline-formula−1, on the low end of the prior WetCHARTs inventory uncertainty range (14.2 (3.3–32.4) Tg ainline-formula−1), which calls for better understanding of these emissions. We find an increasing trend in US anthropogenic emissions over 2010–2015 of 0.4 % ainline-formula−1, lower than previous GOSAT-based estimates but opposite to the decrease reported by the EPA GHGI. Most of this increase appears driven by unconventional oil and gas production in the eastern US. We also find that oil and gas production emissions in Mexico are higher than in the nationally reported inventory, though there is evidence for a 2010–2015 decrease in emissions from offshore oil production.
Maasakkers, Joannes D. / Jacob, Daniel J. / Sulprizio, Melissa P. / et al: 2010–2015 North American methane emissions, sectoral contributions, and trends: a high-resolution inversion of GOSAT observations of atmospheric methane. 2021. Copernicus Publications.
Rechteinhaber: Joannes D. Maasakkers et al.
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Element (mathematics) - Wikipedia
Any one of the distinct objects that make up a set in set theory
For elements in category theory, see Element (category theory).
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.
3 Cardinality of sets
5 Formal relation
{\displaystyle A=\{1,2,3,4\}}
means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example
{\displaystyle \{1,2\}}
, are subsets of A.
Sets can themselves be elements. For example, consider the set
{\displaystyle B=\{1,2,\{3,4\}\}}
. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set
{\displaystyle \{3,4\}}
The elements of a set can be anything. For example,
{\displaystyle C=\{\mathrm {\color {red}red} ,\mathrm {\color {green}green} ,\mathrm {\color {blue}blue} \}}
is the set whose elements are the colors red, green and blue.
The relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing
{\displaystyle x\in A}
means that "x is an element of A".[1] Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, although some authors use them to mean instead "x is a subset of A".[2] Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.[3]
For the relation ∈ , the converse relation ∈T may be written
{\displaystyle A\ni x,}
meaning "A contains or includes x".
The negation of set membership is denoted by the symbol "∉". Writing
{\displaystyle x\notin A}
means that "x is not an element of A".
The symbol ∈ was first used by Giuseppe Peano, in his 1889 work Arithmetices principia, nova methodo exposita.[4] Here he wrote on page X:
Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b; …
The symbol ∈ means is. So a ∈ b is read as a is a b; …
The symbol itself is a stylized lowercase Greek letter epsilon ("ϵ"), the first letter of the word ἐστί, which means "is".[4]
ELEMENT OF NOT AN ELEMENT OF CONTAINS AS MEMBER DOES NOT CONTAIN AS MEMBER
Unicode 8712 U+2208 8713 U+2209 8715 U+220B 8716 U+220C
UTF-8 226 136 136 E2 88 88 226 136 137 E2 88 89 226 136 139 E2 88 8B 226 136 140 E2 88 8C
Numeric character reference ∈ ∈ ∉ ∉ ∋ ∋ ∌ ∌
Named character reference ∈, ∈, ∈, ∈ ∉, ∉, ∉ ∋, ∋, ∋, ∋ ∌, ∌, ∌
LaTeX \in \notin \ni \not\ni or \notni
Wolfram Mathematica \[Element] \[NotElement] \[ReverseElement] \[NotReverseElement]
Cardinality of sets[edit]
The number of elements in a particular set is a property known as cardinality; informally, this is the size of a set.[5] In the above examples, the cardinality of the set A is 4, while the cardinality of set B and set C are both 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers {1, 2, 3, 4, ...}.
Using the sets defined above, namely A = {1, 2, 3, 4 }, B = {1, 2, {3, 4}} and C = {red, green, blue}, the following statements are true:
yellow ∉ C
Formal relation[edit]
As a relation, set membership must have a domain and a range. Conventionally the domain is called the universe denoted U. The range is the set of subsets of U called the power set of U and denoted P(U). Thus the relation
{\displaystyle \in }
is a subset of U x P(U). The converse relation
{\displaystyle \ni }
is a subset of P(U) x U.
^ Weisstein, Eric W. "Element". mathworld.wolfram.com. Retrieved 2020-08-10.
^ Eric Schechter (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8. p. 12
^ George Boolos (February 4, 1992). 24.243 Classical Set Theory (lecture) (Speech). Massachusetts Institute of Technology.
^ a b Kennedy, H. C. (July 1973). "What Russell learned from Peano". Notre Dame Journal of Formal Logic. Duke University Press. 14 (3): 367–372. doi:10.1305/ndjfl/1093891001. MR 0319684.
^ "Sets - Elements | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-10.
Halmos, Paul R. (1974) [1960], Naive Set Theory, Undergraduate Texts in Mathematics (Hardcover ed.), NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not fully axiomatized, not that it is silly or easy (Halmos's treatment is neither).
Jech, Thomas (2002), "Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University
Suppes, Patrick (1972) [1960], Axiomatic Set Theory, NY: Dover Publications, Inc., ISBN 0-486-61630-4 - Both the notion of set (a collection of members), membership or element-hood, the axiom of extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".
Retrieved from "https://en.wikipedia.org/w/index.php?title=Element_(mathematics)&oldid=1080721917"
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Construct a trapezium ABCD in which AB is parallel to DC having AB=9 cm, CD=5 cm, angle BAD=70 degree , - Maths - Practical Geometry - 10040141 | Meritnation.com
Construct a trapezium ABCD in which AB is parallel to DC having AB=9 cm, CD=5 cm, angle BAD=70 degree , angle DBA=50 degree
\angle \mathrm{BAD}+\angle \mathrm{ADC}=180° \left(\mathrm{Adjacents} \mathrm{angles} \mathrm{are} \mathrm{supplementary} \mathrm{in} \mathrm{trapezium}\right)\phantom{\rule{0ex}{0ex}}70°+\angle \mathrm{ADC}=180°\phantom{\rule{0ex}{0ex}}\angle \mathrm{ADC}=110°\phantom{\rule{0ex}{0ex}}\mathrm{steps} \mathrm{of} \mathrm{construction} :\phantom{\rule{0ex}{0ex}}\left(1\right) \mathrm{Draw} \mathrm{a} \mathrm{line} \mathrm{segment} \mathrm{AB}=9 \mathrm{cm} .\phantom{\rule{0ex}{0ex}}\left(2\right) \mathrm{Draw} \mathrm{a} \mathrm{line} \mathrm{AP} \mathrm{at} \mathrm{an} \mathrm{angle} 70° \mathrm{at} \mathrm{point} \mathrm{P} \mathrm{and} \mathrm{Draw} \mathrm{a} \mathrm{line} \mathrm{BQ} \mathrm{at} \mathrm{an} \mathrm{angle} 50° \mathrm{at} \mathrm{point} \mathrm{Q}.\phantom{\rule{0ex}{0ex}}\left(3\right) \mathrm{Line} \mathrm{AP} \mathrm{and} \mathrm{BQ} \mathrm{intersects} \mathrm{at} \mathrm{point} \mathrm{D}.\phantom{\rule{0ex}{0ex}}\left(4\right) \mathrm{At} \mathrm{point} \mathrm{draw} \mathrm{a} \mathrm{line} \mathrm{DR} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} 110° .\phantom{\rule{0ex}{0ex}}\left(5\right) \mathrm{Taking} \mathrm{D} \mathrm{as} \mathrm{centre} , \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{of} 5 \mathrm{cm} , \mathrm{which} \mathrm{cuts} \mathrm{line} \mathrm{DR} \mathrm{at} \mathrm{point} \mathrm{C} .\phantom{\rule{0ex}{0ex}}\left(6\right) \mathrm{Join} \mathrm{BC} .\phantom{\rule{0ex}{0ex}}\mathrm{Thus} \mathrm{ABCD} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{trapezium} .
Plz expert send me the right answer don't send me the similar link as question is different to draw
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Continuous Lyapunov equation solution - MATLAB lyap - MathWorks Benelux
AX+X{A}^{T}+Q=0
AX+XB+C=0
AX{E}^{T}+EX{A}^{T}+Q=0
{\alpha }_{1},{\alpha }_{2},...,{\alpha }_{n}
{\beta }_{1},{\beta }_{2},...,{\beta }_{n}
{\alpha }_{i}+{\beta }_{j}\ne 0\text{ }for\text{\hspace{0.17em}}all\text{\hspace{0.17em}}pairs\text{\hspace{0.17em}}\left(i,j\right)
AX+X{A}^{T}+Q=0
A=\left[\begin{array}{cc}1& 2\\ -3& -4\end{array}\right]\text{ }\text{ }Q=\left[\begin{array}{cc}3& 1\\ 1& 1\end{array}\right]
AX+XB+C=0
A=5\text{ }\text{ }B=\left[\begin{array}{cc}4& 3\\ 4& 3\end{array}\right]\text{ }\text{ }C=\left[\begin{array}{cc}2& 1\end{array}\right]
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Characterizing the origins of dissolved organic carbon in coastal seawater...
Han, Heejun; Hwang, Jeomshik; Kim, Guebuem
In order to determine the origins of dissolved organic matter (DOM) occurring in the seawater of Sihwa Lake, we measured the stable carbon isotope ratios of dissolved organic carbon (DOC-inline-formulaδ13C) and the optical properties (absorbance and fluorescence) of DOM in two different seasons (March 2017 and September 2018). Sihwa Lake is enclosed by a dike along the western coast of South Korea, and the water is exchanged with the Yellow Sea twice a day through the sluice gates. The DOC concentrations were generally higher in lower-salinity waters in both periods, and excess of DOC was also observed in 2017 in high-salinity waters. Here, the excess DOC represents any DOC concentrations higher than those in the incoming open-ocean seawater. The excess DOC occurring in the lower-salinity waters originated mainly from marine sediments of tidal flats, based on the DOC-inline-formulaδ13C values (inline-formula
M3inlinescrollmathml-normal 20.7±normal 1.2
58pt10ptsvg-formulamathimg1d72abcf6d66ff54473613b71b21263c bg-18-1793-2021-ie00001.svg58pt10ptbg-18-1793-2021-ie00001.png ‰) and good correlations among the DOC, humic-like fluorescent DOM (inline-formulaFDOMH), and inline-formula
M5inlinescrollmathmlchem{\mathrm{normal NH}}_{normal 4}^{+}
24pt15ptsvg-formulamathimg68d940fa21d9c6691de36bd82f3e56d8 bg-18-1793-2021-ie00002.svg24pt15ptbg-18-1793-2021-ie00002.png concentrations. However, the origins of the excess DOC observed in 2017 appear to be from two different sources: one mainly from marine sources such as biological production based on the DOC-inline-formulaδ13C values (inline-formula−19.1 ‰ to inline-formula−20.5 ‰) and the other mainly from terrestrial sources by land–seawater interactions based on its depleted DOC-inline-formulaδ13C values (inline-formula−21.5 ‰ to inline-formula−27.8 ‰). This terrestrial DOM source observed in 2017 was likely associated with DOM on the reclaimed land, which experienced extended exposure to light and bacterial degradation as indicated by the higher spectral slope ratio (inline-formulaSR) of light absorbance and no concurrent increases in the inline-formulaFDOMH and inline-formula
M14inlinescrollmathmlchem{\mathrm{normal NH}}_{normal 4}^{+}
24pt15ptsvg-formulamathimgcb88f58b3b25b473f7c5a29ace587a7f bg-18-1793-2021-ie00003.svg24pt15ptbg-18-1793-2021-ie00003.png concentrations. Our study demonstrates that the combination of these biogeochemical tools can be a powerful tracer of DOM sources and characteristics in coastal environments.
Han, Heejun / Hwang, Jeomshik / Kim, Guebuem: Characterizing the origins of dissolved organic carbon in coastal seawater using stable carbon isotope and light absorption characteristics. 2021. Copernicus Publications.
Rechteinhaber: Heejun Han et al.
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Making a movie rating predictor based on just the length and release date of movies is pretty limited. There are so many more interesting pieces of data about movies that we could use! So let’s add another dimension.
Let’s say this third dimension is the movie’s budget. We now have to find the distance between these two points in three dimensions.
What if we’re not happy with just three dimensions? Unfortunately, it becomes pretty difficult to visualize points in dimensions higher than 3. But that doesn’t mean we can’t find the distance between them.
The generalized distance formula between points A and B is as follows:
\sqrt{(A_1-B_1)^2+(A_2-B_2)^2+ \dots+(A_n-B_n)^2}
Here, A1-B1 is the difference between the first feature of each point. An-Bn is the difference between the last feature of each point.
Using this formula, we can find the K-Nearest Neighbors of a point in N-dimensional space! We now can use as much information about our movies as we want.
We will eventually use these distances to find the nearest neighbors to an unlabeled point.
Modify your distance function to work with any number of dimensions. Use a for loop to iterate through the dimensions of each movie.
Return the total distance between the two movies.
We’ve added a third dimension to each of our movies.
Print the new distance between Star Wars and Raiders of the Lost Ark.
Print the new distance between Star Wars and Mean Girls.
Which movie is Star Wars closer to now?
4. Distance Between Points - 3D
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The wave equation - SEG Wiki
The wave equation is based on two fundamental laws. Hooke’s law says that stress is proportional to strain, and Newton’s law says that force equals mass times acceleration. From the wave equation, we can predict the existence of compressional waves and shear waves and their properties. These properties include Snell’s law of reflection and refraction, the partition of energy at an interface into compressional and shear components, the generation of surface waves and their characteristics, the diffraction of waves, the attenuation of waves as they travel in the earth, and many other facts of wave propagation. All migration procedures invoke the wave equation (Loewenthal et al., 1976[1]). A simplified model based on raypaths obeying Snell’s law at interfaces and following least-time paths has become a standard model used in geophysics.
What is a plane wave? A plane wave is perhaps the simplest example of a 3D wave. It exists at a given instant and can be visualized as a propagating plane surface of constant phase, so that the plane surface remains perpendicular to a given direction of propagation. A plane wave is essentially one-dimensional because spatial variation occurs only along the direction of propagation. We have quite practical reasons for studying this sort of disturbance, one of which is that an actual observed spherical wave can be decomposed into its constituent plane waves. This procedure is called plane-wave decomposition, and it is useful because plane waves have much simpler properties than spherical waves do. The special significance of this approach is that any 3D wave can be expressed as a combination of plane waves, each having a distinct amplitude and direction of propagation. Of all the 3D waves, only the plane wave (whether it is sinusoidal or not) moves through space with an unchanging profile.
When the 3D wave equation is written in terms of Cartesian coordinates (x,y,z), the position variables x, y, and z appear symmetrically — a fact to be kept in mind. Cartesian coordinates are suited particularly for describing plane waves. However, as various physical situations arise, we often can take better advantage of existing symmetries by using other coordinate representations.
What is a ray? There are several ways to think about waves. One way is to think of wave motion in terms of rays. The concept of a ray is very useful. Picture the wave as traveling in very narrow pencils, or beams, called rays. This ray model gives an understanding of various wave phenomena, especially reflection and refraction. A ray is a line drawn in space corresponding to the direction of flow of radiant energy. As such, it is a mathematical device rather than a physical entity. In practice, we sometimes can produce very narrow beams, or pencils, and we might imagine a ray to be the unattainable limit on the narrowness of such a beam. Within homogeneous isotropic materials, a ray is a straight line because, by symmetry, there is no preferred direction that would cause the ray to form a curved path. Rays are bent at the interface between two media with different velocities, and rays can be curved in a medium in which velocity is a function of position.
What is a wavefront? The dual of the ray is the wavefront (Figure 3). Consider the wavefront representing the front surface of the wave traveling away from an explosive source. We can visualize this wavefront physically and we can measure it within narrow limits. For example, when we look at the waves emanating from a point at which we have dropped a stone into a still pond, we see the wave motion traveling outward as wavefronts, and the raypaths are only mental constructions. Likewise, in seismic interpretation, the rays are visualized only as mathematical abstractions, whereas wavefronts can be seen on the seismic sections as reflection events. For that reason, we turn our attention to the geometric study of wavefronts.
Figure 3. A ray and a wavefront.
Points at which a single ray intersects a set of wavefronts are called corresponding points. Evidently, the separation in time between any two corresponding points on any two sequential wavefronts is identical. In other words, if wavefront S transforms into
{\displaystyle S^{'}}
{\displaystyle t^{'}}
, the distance between corresponding points on any and all rays will be traversed in that same time
{\displaystyle t^{'}}
. This obviously will be true even if the wavefronts pass from one homogeneous isotropic medium into another. This simply means that each point on S can be imagined as following the path of a ray to arrive at
{\displaystyle S^{'}}
in the same time
{\displaystyle t^{'}}
. Within a homogeneous isotropic medium, the velocity of propagation is identical in all directions. In such a medium, the spatial separation between two wavefronts, measured along a ray, must be the same everywhere.
A wavefront can be defined as the surface (in three dimensions) or curve (in two dimensions) over which the phase of a traveling-wave disturbance is the same. In an isotropic medium — one whose properties are the same in all directions — rays form trajectories that are orthogonal to the wavefronts. That is to say, rays are lines normal to the wavefronts at every point of intersection. Evidently, in such a medium, a ray follows the direction of propagation. However, that is not true in anisotropic materials.
A wavefront also can refer to the leading edge of a waveform. A wavefront chart (Figure 4) is a plot of horizontal distance x versus depth z on which wavefronts have been drawn emanating from a source. The wavefronts show the position of a traveling seismic disturbance at successive times. The shapes of the wavefronts depend on the velocity distribution of the rocks. On such charts, raypaths corresponding to different move-outs also can be drawn. These raypaths are perpendicular to the wavefronts for isotropic media. Sometimes, wavefront charts are drawn that are specific for a particular location where velocity varies laterally; in such cases, the wavefront charts are asymmetric. For the case in which velocity varies only with depth, the wavefront chart is symmetric with respect to the vertical raypath.
Figure 4. Wavefronts and raypaths.
An idealized point source is one for which the radiation emanating from it streams out radially and uniformly in all directions. The source is said to be isotropic, and the resulting wavefronts are again concentric spheres that increase in diameter as they expand into the surrounding space. One solution of the wave equation gives a spherical wave progressing radially outward from the origin at a constant velocity and having an arbitrary functional form. Another solution is given by the case in which the wave converges toward the origin.
Notice that the amplitude of any spherical wavefront is a function of its distance from the center because the radial term serves as an extrinsic attenuation factor. This extrinsic attenuation factor is a direct consequence of energy conservation, a phenomenon known as geometric spreading. That is, unlike a plane wave, a spherical wave decreases in amplitude as it expands and travels away from its source, thereby changing its profile. As a spherical wavefront propagates outward, its radius increases. Far enough away from the source, a small area of the wavefront therefore will resemble closely a portion of a plane wave (Figure 5).
Figure 5. Spherical spreading. Energy flowing across the upper area later flows across the lower area. Explain why the two figures are different. It is necessary to correct the amplitude for spherical spreading. The correction is obtained by multiplying the amplitude by the distance traveled. The correction can be obtained approximately by multiplying the amplitude by the traveltime.
Much of current seismic processing methodology is based on the plane-wave approximation to a spherical wave in particular as well as to other types of curved wavefronts in general. The temporal frequency, or the number of cycles per unit time, is the Fourier dual for the time variable. The spatial frequency, or wavenumber, is the Fourier dual for the space variable. The wavenumber gives the number of cycles per unit distance.
When irregular discontinuities exist in a nonuniform medium, they give rise to complex reflection, refraction, and diffraction phenomena. Huygens’ principle (or construction) is particularly useful for treating problems of this type. Huygens’ principle states that every point on a primary wavefront serves as the source of spherical secondary wavelets, and the primary wavefront at some later time is the envelope of these wavelets. Moreover, the wavelets advance with a speed and frequency equal to those of the primary wave at each point in space. If the medium is homogeneous, the wavelets can be constructed with finite radii, whereas if the medium is inhomogeneous, the wavelets must have infinitesimal radii.
Huygens’ construction allows a backward-traveling wave moving toward the source — something that is not observed. This difficulty was taken care of theoretically by Fresnel and Kirchhoff, who showed that only the forward-moving wave can exist in the real world. Thus, we merely can use the forward waves when applying Huygens’ construction.
Seismic analysis almost always is carried out through a process of abstraction in which the data are reduced to the essentials that admit of mathematical treatment. The accuracy required in the results influences the extent to which such reductions are made, and that accuracy is limited by the reliability of the data. Today, analytic methods are implemented with the digital computer, and their relative values are judged by their speed, reliability, and clarity.
Two important processes in seismic data processing are stacking and migration (Berkhout and de Jong, 1981[2]). Much can be learned about them by geometric lines of thought. In dealing with stacking and migration from a geometric point of view, one can take either the raypath concept or the wavefront concept as the point of departure. In geometric seismology (i.e., the theory of seismic waves with wavelengths that are small compared with the undulations of the subsurface interfaces), a ray can be visualized as a narrow beam along which energy is transported. This concept is especially valid in treating a long series of waves with wavelengths on the order of a fraction of the dimension of any object encountered.
However, when the dimension of some obstacle is of the same order of magnitude as the wavelength, then the phenomenon of diffraction occurs. In the case of an impulsive source (e.g., a dynamite explosion), the seismic energy is transported in the form of a short compressional wavelet whose breadth is of the same order of magnitude as the geologic bodies whose shape we wish to determine. Thus, within the usual distance ranges, velocities, and frequencies encountered in seismic work, we do not have pure raypaths in the form of infinitely narrow beams, so we must take into account diffraction effects.
Absorption loss and transmission loss Wave velocity
↑ Loewenthal, D., L. Lu, R. Roberson, and J. Sherwood, 1976, The wave equation applied to migration: Geophysical Prospecting, 24, 380–399.
↑ Berkhout, A. J., and B. A. de Jong, 1981, Recursive migration in three dimensions: Geophysical Prospecting, 29, 758–781.
Retrieved from "https://wiki.seg.org/index.php?title=The_wave_equation&oldid=165935"
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CliqueCover - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : CliqueCover
find a minimal vertex clique cover for a graph
return the size of a minimal vertex clique cover for a graph
CliqueCover(G)
CliqueCover(G, k)
CliqueCoverNumber(G)
(optional) non-negative integer (the number of cliques)
The CliqueCover(G) command returns a minimum vertex clique cover for the graph G.
The CliqueCover(G,k) command attempts to produce a clique cover for G of no more than k cliques. If such a partition is possible, a list of cliques is returned. If not possible, an error message is displayed.
The CliqueCoverNumber(G) command returns the size of a minimum vertex clique cover for G. Note this equivalent to computing the chromatic number for the graph complement of G.
The problem of finding a clique cover of size k for a graph is NP-complete, meaning that no polynomial-time algorithm is presently known. The exhaustive search will take an exponential time on some graphs.
A clique cover or vertex clique cover of a graph G is a partition of the vertices of G into cliques. That is, it is a set of mutually disjoint subsets of the vertices of G, each of which is a clique. Each clique cover is a coloring of the graph complement of G.
A minimum clique cover of a graph G is a clique cover of minimum size for the graph G.
The clique cover number of a graph G is the cardinality of a minimum clique cover of G. It is equal to the chromatic number of the graph complement of G.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{SpecialGraphs}\right):
P≔\mathrm{PetersenGraph}\left(\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 10 vertices and 15 edge\left(s\right)}}
\mathrm{CliqueCover}\left(P\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]]
\mathrm{C5}≔\mathrm{CycleGraph}\left(5\right)
\textcolor[rgb]{0,0,1}{\mathrm{C5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: an undirected unweighted graph with 5 vertices and 5 edge\left(s\right)}}
\mathrm{CliqueCoverNumber}\left(\mathrm{C5}\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{CliqueCover}\left(\mathrm{C5}\right)
[[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}]]
The GraphTheory[CliqueCover] and GraphTheory[CliqueCoverNumber] commands were introduced in Maple 2016.
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Numbers - archetype
Three types of numbers are available in Archetype: integers, naturals and rationals
Naturals are positive integers. They are defined as follows:
var n1 : nat = 345;
var n2 : nat = 9999999999999999999999999999999999999999999999999999999;
var n3 : nat = 10_000_000;
Note that it is possible to structure large numbers by packs of 3 digits using the underscore character.
3 arithmetics operations + * % and the 6 comparison operators = <> < > <= >= are available. Note however that the difference operator returns an integer value (see integers section below).
var a : nat = 5;
var b : nat = 7;
var c : int = a - b; /* -2 typed as intger */
var d : int = b - a; /* 2 typed as integer */
var e : nat = sub_nat(a,b); /* fails with "NegResult" */
var f : nat = sub_nat(b,a); /* f = 2 */
Note that the div operator is the euclidean division and returns a nat value.
The abs function is used to convert a integer value to a natural value.
var a = 5i; /* a is typed 'int'*/
var b = abs(5); /* b is typed 'nat' */
Integers are defined as follows:
var n1 : int = 3_458i;
var n2 : int = -5i;
var n3 : int = 99999999999999999999999999999999999999999999999999999999i;
var n4 : int = 10_000_000i;
Note that the literal syntax of positive integers uses the i suffix.
Integer values are big integers, meaning there is no real constraint on the value and it can be negative.
The 5 arithmetics operations + * - div % and the 6 comparison operators = <> < > <= >= are available.
Tez is the type to specify an amount in Tezos cryptocurrency.
var t1 := 1tz; // one tezis
var t2 := 1mtz: // one 0.001 tezis
var t3 := 1utz; // one mutez
It is more constrained than integers since you can only add and subtract tez values. All comparison operators are available.
A rational is the quotient or fraction of two integers. You can define a rational with :
a quotient of two integers
a floating point notation
a percentage notation
The following presents an example for each:
var r1 := 6/4;
var r2 := 1.5;
var r3 := 150%;
r1 r2 and r3 represent the same rational,
3/2
. These values are transcoded to the pair of integers (3,2). A fractions is simplified during transcoding process.
Archetype provides the 4 arithmetic operations + - * / and the minus sign on rationals. They all return a rational. For example:
var r1 := 8 / 6; // transcoded to (4,3)
var r2 := 1.8; // transcoded to (9,5)
var rpl := r1 + r2; // will execute to (6,15)
var rmi := r1 - r2; // will execute to (-7,15)
var rmu := r1 * r2; // will execute to (4,5)
var rdi := r1 / r2: // will execute to (20,27)
var rms := -r1 // will execute to (-4,3)
It is also possible to mix integers and rationals:
var r := 5 / 3;
var rtn := r * n; // will execute to (20/3)
var rpn := r + n; // will execute to (17/3)
It is also possible to mix rationals and tezies values, in that order. However, the result is a value in tezies.
var r := 80%;
var a := 56tz;
var res := r * a;
In the example above, the res value is 44800000utz . The process is the following:
convert a to utez (smallest unit)
compute the rational (here 4 * 56000000 / 5 = 224000000 / 5)
execute the euclidean division (44800000)
Note that the term a * r is not accepted as res value (line 4 above): rationals come first.
Rationals are comparable with = <> < <= > >= operators.
var r1 := 3/12;
var r2 := 0.25;
if r1 = r2 and r3 > r2 then transfer 1tz to coder;
It is possible to compare rationals and integers. It is not possible to compare rationals and tez values.
Rational are converted to integers with the floor and ceil operators with the expected behaviour.
Conversion to other types
There is no explicit cast (conversion operator) from a rational to a tez value, a duration or a date. You may just multiply by 1tez for example.
var a := 2.5;
transfer (a * 1tz) from source to dest;
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(Redirected from P-wave)
This article is on the type of seismic wave. For the term used in electrocardiography, see P wave (electrocardiography). P-wave can also refer to a type of electronic wave function in atomic physics; see atomic orbital.
A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any affected location or at a seismograph. P waves may be transmitted through gases, liquids, or solids.
Plane P wave
Representation of the propagation of a P wave on a 2D grid (empirical shape)[clarification needed]
2 Seismic waves in the Earth
2.1 P-wave shadow zone
2.2 As an earthquake warning
The name P wave can stand for either pressure wave (as it is formed from alternating compressions and rarefactions) or primary wave (as it has high velocity and is therefore the first wave to be recorded by a seismograph).[1] The name S wave represents another seismic wave propagation mode, standing for secondary or shear wave.
Seismic waves in the EarthEdit
See also: Core-mantle boundary, Mohorovičić discontinuity, Low-velocity zone, and Lehmann discontinuity
Velocity of seismic waves in the Earth versus depth.[2] The negligible S-wave velocity in the outer core occurs because it is liquid, while in the solid inner core the S-wave velocity is non-zero.
Primary and secondary waves are body waves that travel within the Earth. The motion and behavior of both P and S waves in the Earth are monitored to probe the interior structure of the Earth. Discontinuities in velocity as a function of depth are indicative of changes in phase or composition. Differences in arrival times of waves originating in a seismic event like an earthquake as a result of waves taking different paths allow mapping of the Earth's inner structure.[3][4]
P-wave shadow zoneEdit
P-wave shadow zone (from USGS)
Almost all the information available on the structure of the Earth's deep interior is derived from observations of the travel times, reflections, refractions and phase transitions of seismic body waves, or normal modes. P waves travel through the fluid layers of the Earth's interior, and yet they are refracted slightly when they pass through the transition between the semisolid mantle and the liquid outer core. As a result, there is a P-wave "shadow zone" between 103° and 142°[5] from the earthquake's focus, where the initial P waves are not registered on seismometers. In contrast, S waves do not travel through liquids.
As an earthquake warningEdit
Advance earthquake warning is possible by detecting the nondestructive primary waves that travel more quickly through the Earth's crust than do the destructive secondary and Rayleigh waves.
The amount of warning depends on the delay between the arrival of the P wave and other destructive waves, generally on the order of seconds up to about 60 to 90 seconds for deep, distant, large quakes such as the 2011 Tohoku earthquake. The effectiveness of a warning depends on accurate detection of the P waves and rejection of ground vibrations caused by local activity (such as trucks or construction). Earthquake early warning systems can be automated to allow for immediate safety actions, such as issuing alerts, stopping elevators at the nearest floors, and switching off utilities.
In isotropic and homogeneous solids, a P wave travels in a straight line longitudinal; thus, the particles in the solid vibrate along the axis of propagation (the direction of motion) of the wave energy. The velocity of P waves in that kind of medium is given by
{\displaystyle v_{\mathrm {p} }\;=\;{\sqrt {\frac {\,K+{\tfrac {4}{3}}\mu \;}{\rho }}}\;=\;{\sqrt {\frac {\,\lambda +2\mu \;}{\rho }}}}
where K is the bulk modulus (the modulus of incompressibility), μ is the shear modulus (modulus of rigidity, sometimes denoted as G and also called the second Lamé parameter), ρ is the density of the material through which the wave propagates, and λ is the first Lamé parameter.
In typical situations in the interior of the Earth, the density ρ usually varies much less than K or μ, so the velocity is mostly "controlled" by these two parameters.
The elastic moduli P-wave modulus,
{\displaystyle M}
, is defined so that
{\displaystyle \,M=K+{\tfrac {4}{3}}\mu \,}
{\displaystyle v_{\mathrm {p} }={\sqrt {\frac {\,M\;}{\rho }}}}
Typical values for P wave velocity in earthquakes are in the range 5 to 8 km/s. The precise speed varies according to the region of the Earth's interior, from less than 6 km/s in the Earth's crust to 13.5 km/s in the lower mantle, and 11 km/s through the inner core.[6]
Velocity in Common Rock Types[7]
Unconsolidated Sandstone 4,600–5,200 15,000–17,000
Consolidated Sandstone 5,800 19,000
Shale 1,800–4,900 6,000–16,000
Limestone 5,800–6,400 19,000–21,000
Geologist Francis Birch discovered a relationship between the velocity of P waves and the density of the material the waves are traveling in:
{\displaystyle v_{\mathrm {p} }=a(\,{\bar {M}}\,)+b\,\rho }
which later became known as Birch's law. (The symbol a() is an empirically tabulated function, and b is a constant.)
^ Milsom, J. (2003). Field Geophysics. The geological field guide series. Vol. 25. John Wiley and Sons. p. 232. ISBN 978-0-470-84347-5. Retrieved 2010-02-25.
^ GR Helffrich & BJ Wood (2002). "The Earth's Mantle" (PDF). Nature. 412 (2 August): 501–7. doi:10.1038/35087500. PMID 11484043. S2CID 4304379.
^ Justin L Rubinstein, DR Shelly & WL Ellsworth (2009). "Non-volcanic tremor: A window into the roots of fault zones". In S. Cloetingh, Jorg Negendank (ed.). New Frontiers in Integrated Solid Earth Sciences. Springer. p. 287 ff. ISBN 978-90-481-2736-8. The analysis of seismic waves provides a direct high-resolution means for studying the internal structure of the Earth...
^ CMR Fowler (2005). "§4.1 Waves through the Earth". The solid earth: an introduction to global geophysics (2nd ed.). Cambridge University Press. p. 100. ISBN 978-0-521-58409-8. Seismology is the study of the passage of elastic waves through the Earth. It is arguably the most powerful method available for studying the structure of the interior of the Earth, especially the crust and mantle.
^ Lowrie, William. The Fundamentals of Geophysics. Cambridge University Press, 1997, p. 149.
^ Dziewonski, Adam M.; Anderson, Don L. (1981). "Preliminary reference Earth model". Physics of the Earth and Planetary Interiors. 25 (4): 297–356. Bibcode:1981PEPI...25..297D. doi:10.1016/0031-9201(81)90046-7.
^ "Acoustic Logging". Geophysics. U.S. Environmental Protection Agency. 2011-12-12. Retrieved 2015-02-03.
"Photo Glossary of Earthquakes". U.S. Geological Survey". Archived from the original on February 27, 2009. Retrieved March 8, 2009.
Animation of a P wave
P-wave velocity calculator
Purdue's catalog of animated illustrations of seismic waves
Animations illustrating simple wave propagation concepts by Jeffrey S. Barker
Bayesian Networks for Earthquake Magnitude Classification in a Early Warning System
Retrieved from "https://en.wikipedia.org/w/index.php?title=P_wave&oldid=1087543012"
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How to Calculate Opportunity Cost: 10 Steps (with Pictures)
1 Calculating Opportunity Cost
2 Evaluating Business Decisions
3 Assessing Personal Decisions
Opportunity cost is a formula to help you calculate the difference of you make one choice over another. It gives you feedback you can use to compare what is lost with what is gained, based on your decision. It's often used to give you an advantage when you're trying to understand the returns of an investment, and you may be given a table or graph to pull your data from. Our guide will help you understand what opportunity cost is and how to calculate it!
Calculating Opportunity Cost Download Article
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Identify your different options. When faced with a choice between two options, calculate the potential returns of both options. Since you can only choose one option, you forfeit the potential returns from the other option. That loss is your opportunity cost.[1] X Research source
For example, suppose your company has $100,000 in extra funds, and you have to decide between investing in securities or purchasing new capital equipment.
If you decide to invest in the securities, you may see a return on that investment. But, you forfeit any profit you might have earned from purchasing new equipment.
On the other hand, if you decide to purchase new equipment, you may see a return on that investment in the form of increased sales. But, you forfeit any profit you might have earned from investing in the securities.
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Calculate the potential returns on each option. Research each option and estimate the financial return on each. In the above example, suppose the expected return on the investment in the stock market is 12 percent. Your potential return is $12,000. The new equipment, on the other hand, might result in a 10 percent increase in your profit margin. Your potential return for that investment would be $10,000.
Choose the best option. Sometimes the best option is not the most lucrative, especially in the short term. Decide which option is best for you based on long term goals, not just on the potential return. The company in the above example may choose to invest the funds in new equipment instead of the stock market. Although the stock market investment has the higher potential return in the short term, the new equipment will allow them to increase efficiency and lower opportunity costs. This will have a long term impact on their profit margin.
Calculate the opportunity cost. The opportunity cost is the difference between the most lucrative option and the chosen option. In the above example, the most lucrative option is investing in the securities, which has a potential return of $12,000. The option the company chose, however, was to invest in new equipment, for a return of $10,000.
The opportunity cost = most lucrative option – chosen option.
{\displaystyle \$12,000-\$10,000=\$2,000}
The opportunity cost of choosing to purchase new equipment is $2,000.
Evaluating Business Decisions Download Article
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Establish the capital structure of your business. Capital structure is how a company funds its operations and growth. It is a mix of the company’s debt and equity. Debt can be in the form of bonds issued or loans from financial institutions. Equity can be in the form of stock or retained earnings.[2] X Research source
Companies must evaluate opportunity cost when choosing between debt and equity.
If a company chooses to borrow money to fund an expansion, then the money used to repay the principal and interest on the loan is not available to be invested into stocks.
The company must evaluate the opportunity cost to see if the expansion made possible with the debt will generate enough revenue in the long term to justify passing on the stock investments.[3] X Research source
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Evaluate non-financial resources. Opportunity cost is often calculated to evaluate financial decisions. However, companies can use opportunity cost to govern their use of other resources, such as man hours, time or mechanical output. Opportunity cost can be defined with any resource that is limited in the company.[4] X Research source
Companies must make decisions about how to allocate these resources to different projects. The time spent on one project is taken away from something else.
Suppose, for example, a furniture company with 450 available man hours per week uses 10 man hours per chair to produce 45 chairs per week. They decide to produce 10 sofas per week that take 15 man hours per sofa. This will use 150 man hours and produce 10 sofas.
They will have 300 hours left to produce chairs, which will yield 30 chairs. The opportunity cost of the 10 sofas, therefore, is 15 chairs
{\displaystyle (45-30=15)}
Determine what your time is worth if you are an entrepreneur. If you are an entrepreneur, you will spend all of your time at your new business. However, this is time that you could have spent working at a different job. This is your opportunity cost. If you have high earning potential in a different line of work, you must decide whether or not it is worthwhile for you to open your new business.[5] X Research source
For example, suppose you are a chef earning $23 per hour and you decide to leave your job to open your own restaurant. Before you ever earn a penny from the new business, you will spend time buying food, hiring staff, renting the building and opening the restaurant. You will eventually earn revenues, but the opportunity cost will be how much you would have earned working at your old job during all of that time.
Assessing Personal Decisions Download Article
Decide whether or not to hire a housekeeper. Identify which household chores are using up your time. Decide whether the time spent on these chores takes away from time spent doing something else that you consider more valuable. Chores such as laundry and cleaning may interfere with work if you work from home a lot. Also, time spent on housework may hinder your ability to partake in other more enjoyable activities, such as being with your children or pursuing a hobby.[6] X Research source
Calculate the financial opportunity cost. Suppose you work from home and earn $25 per hour. If you hired a housekeeper, you would have pay $20 per hour. The opportunity cost of doing the housework yourself is $5 per hour
{\displaystyle (\$25-\$20=\$5)}
Calculate the opportunity cost in time. Suppose you spend 5 hours each Saturday on laundry, food shopping and cleaning. If a housekeeper came once per week to clean and help with laundry, you would only have to spend 3 hours on Saturday finishing the laundry and food shopping. The opportunity cost of doing the housework yourself is 2 hours.
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Determine the true cost of going to college. Suppose you are going to pay $4,000 per year to attend an in-state college. The government will subsidy an additional $8,000 for your tuition. But, you must also factor in the opportunity cost of not working while you’re in college. Suppose you could earn $20,000 per year at a job instead of going to college. This means that the true cost of a year of college is the tuition plus the opportunity cost of not working.[7] X Research source
The total tuition is the amount you pay ($4,000) plus the government subsidy ($8,000), which equals a total of $12,000.
The opportunity cost of not working is $20,000.
Therefore, the total cost of a year of college is
{\displaystyle \$12,000+\$20,000=\$32,000}
Other opportunity costs associated with going to college include the value of four years’ real-world work experience, the value of time spent on studying instead of other activities, or the value of what you could have purchased with the money you spent on tuition or the interest that money could have earned if you had invested it.[8] X Research source
However, consider the other side of the coin. Median weekly earnings is $400 higher for a person with a college degree than for a person with only a high school diploma. If you decide not to go to college, the opportunity cost is the value of your future increased earnings.[9] X Trustworthy Source US Bureau of Labor Statistics U.S. government agency that collects and reports labor-related information Go to source
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Recognize opportunity costs in daily choices. Whenever you make a choice, you are foregoing something else. The opportunity cost is the value of the option you do not choose. That value can refer to something personal, financial or environmental.[10] X Research source
If you choose to buy a new car instead of a used car, the opportunity cost is the money you could have saved on the used car and how you could have used that money differently.
Suppose you decide to spend your tax return on a family vacation instead of saving or investing the money. The opportunity cost is the value of the savings account interest or the potential return on an investment.
Remember that the value does not necessarily just refer to money or tangible assets. Consider how a choice will impact your intangible assets, such as happiness, health, and your free time.
Categories: Accounting | Business Skills
To calculate opportunity cost, identify your different options and their potential returns. Do this by calculating how much interest they will earn or how much money they will save. Then, subtract the potential gain of the chosen option from the potential gain of the most lucrative option. For example, if option A could earn you $100, and option B could earn you $80, then option B has an opportunity cost of $20 because $100 minus $80 is $20. For more information from our reviewer on calculating opportunity cost, including how to evaluate non-financial resources, read on!
"I want to start my own cloth business, so I wanted this knowledge first. "
"This has helped me to know how to calculate opportunity cost. "
Kimana Nicholas
"I have grasped something small within a short period of time."
Fardeen Hans
"It was a good explanation."
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Dictionary:Magnetic susceptibility - SEG Wiki
A measure of the degree to which a substance may be magnetized; the ratio k or k′ of the magnetization M or I to the magnetizing force H that is responsible for it:
{\displaystyle kH=M\;\qquad in\;the\;SI\;System,}
{\displaystyle k'H=I\;\qquad in\;the\;cgs\;System,}
The susceptibility is dimensionless but of different magnitude in the two systems:
{\displaystyle k=4\pi k'}
The susceptibility is related to the magnetic permeability μ
{\displaystyle k=\mu -1\;,}
{\displaystyle k'={\frac {\mu -1}{4\pi }}\;.}
Susceptibility in cgs units is sometimes measured in units of 10–6 (‘‘micro-cgs’’). Rock susceptibility usually ranges from 0 to 0.01 cgs units (0 to 10 000 micro-cgs) and it is often proportional to the fraction of magnetite present. See Figure M-1.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Magnetic_susceptibility&oldid=50054"
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Voltage_divider Knowpia
Figure 1: A simple voltage divider
Resistor voltage dividers are commonly used to create reference voltages, or to reduce the magnitude of a voltage so it can be measured, and may also be used as signal attenuators at low frequencies. For direct current and relatively low frequencies, a voltage divider may be sufficiently accurate if made only of resistors; where frequency response over a wide range is required (such as in an oscilloscope probe), a voltage divider may have capacitive elements added to compensate load capacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.
A voltage divider referenced to ground is created by connecting two electrical impedances in series, as shown in Figure 1. The input voltage is applied across the series impedances Z1 and Z2 and the output is the voltage across Z2. Z1 and Z2 may be composed of any combination of elements such as resistors, inductors and capacitors.
If the current in the output wire is zero then the relationship between the input voltage, Vin, and the output voltage, Vout, is:
{\displaystyle V_{\mathrm {out} }={\frac {Z_{2}}{Z_{1}+Z_{2}}}\cdot V_{\mathrm {in} }}
Proof (using Ohm's Law):
{\displaystyle V_{\mathrm {in} }=I\cdot (Z_{1}+Z_{2})}
{\displaystyle V_{\mathrm {out} }=I\cdot Z_{2}}
{\displaystyle I={\frac {V_{\mathrm {in} }}{Z_{1}+Z_{2}}}}
{\displaystyle V_{\mathrm {out} }=V_{\mathrm {in} }\cdot {\frac {Z_{2}}{Z_{1}+Z_{2}}}}
The transfer function (also known as the divider's voltage ratio) of this circuit is:
{\displaystyle H={\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {Z_{2}}{Z_{1}+Z_{2}}}}
In general this transfer function is a complex, rational function of frequency.
Resistive dividerEdit
Figure 2: Simple resistive voltage divider
A resistive divider is the case where both impedances, Z1 and Z2, are purely resistive (Figure 2).
Substituting Z1 = R1 and Z2 = R2 into the previous expression gives:
{\displaystyle V_{\mathrm {out} }={\frac {R_{2}}{R_{1}+R_{2}}}\cdot V_{\mathrm {in} }}
{\displaystyle V_{\mathrm {out} }={\frac {1}{2}}\cdot V_{\mathrm {in} }}
If Vout = 6V and Vin = 9V (both commonly used voltages), then:
{\displaystyle {\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {R_{2}}{R_{1}+R_{2}}}={\frac {6}{9}}={\frac {2}{3}}}
and by solving using algebra, R2 must be twice the value of R1.
To solve for R1:
{\displaystyle R_{1}={\frac {R_{2}\cdot V_{\mathrm {in} }}{V_{\mathrm {out} }}}-R_{2}=R_{2}\cdot \left({{\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}-1}\right)}
{\displaystyle R_{2}=R_{1}\cdot {\frac {1}{\left({{\frac {V_{\mathrm {in} }}{V_{\mathrm {out} }}}-1}\right)}}}
Any ratio Vout/Vin greater than 1 is not possible. That is, using resistors alone it is not possible to either invert the voltage or increase Vout above Vin.
Low-pass RC filterEdit
Figure 3: Resistor/capacitor voltage divider
Consider a divider consisting of a resistor and capacitor as shown in Figure 3.
Comparing with the general case, we see Z1 = R and Z2 is the impedance of the capacitor, given by
{\displaystyle Z_{2}=-\mathrm {j} X_{\mathrm {C} }={\frac {1}{\mathrm {j} \omega C}}\ ,}
where XC is the reactance of the capacitor, C is the capacitance of the capacitor, j is the imaginary unit, and ω (omega) is the radian frequency of the input voltage.
This divider will then have the voltage ratio:
{\displaystyle {\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}={\frac {Z_{\mathrm {2} }}{Z_{\mathrm {1} }+Z_{\mathrm {2} }}}={\frac {\frac {1}{\mathrm {j} \omega C}}{{\frac {1}{\mathrm {j} \omega C}}+R}}={\frac {1}{1+\mathrm {j} \omega RC}}\ .}
The product τ (tau) = RC is called the time constant of the circuit.
The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic (first-order) lowpass filter. The ratio contains an imaginary number, and actually contains both the amplitude and phase shift information of the filter. To extract just the amplitude ratio, calculate the magnitude of the ratio, that is:
{\displaystyle \left|{\frac {V_{\mathrm {out} }}{V_{\mathrm {in} }}}\right|={\frac {1}{\sqrt {1+(\omega RC)^{2}}}}\ .}
Inductive dividerEdit
Inductive dividers split AC input according to inductance:
{\displaystyle V_{\mathrm {out} }={\frac {L_{2}}{L_{1}+L_{2}}}\cdot V_{\mathrm {in} }}
(with components in the same positions as Figure 2.)
The above equation is for non-interacting inductors; mutual inductance (as in an autotransformer) will alter the results.
Inductive dividers split DC input according to the resistance of the elements as for the resistive divider above.
Capacitive dividerEdit
Capacitive dividers do not pass DC input.
For an AC input a simple capacitive equation is:
{\displaystyle V_{\mathrm {out} }={\frac {Xc_{2}}{Xc_{1}+Xc_{2}}}\cdot V_{\mathrm {in} }={\frac {1/C_{2}}{1/C_{1}+1/C_{2}}}\cdot V_{\mathrm {in} }={\frac {C_{1}}{C_{1}+C_{2}}}\cdot V_{\mathrm {in} }}
Any leakage current in the capactive elements requires use of the generalized expression with two impedances. By selection of parallel R and C elements in the proper proportions, the same division ratio can be maintained over a useful range of frequencies. This is the principle applied in compensated oscilloscope probes to increase measurement bandwidth.
Loading effectEdit
The output voltage of a voltage divider will vary according to the electric current it is supplying to its external electrical load. The effective source impedance coming from a divider of Z1 and Z2, as above, will be Z1 in parallel with Z2 (sometimes written Z1 // Z2), that is: (Z1 Z2) / (Z1 + Z2)=HZ1.
To obtain a sufficiently stable output voltage, the output current must either be stable (and so be made part of the calculation of the potential divider values) or limited to an appropriately small percentage of the divider's input current. Load sensitivity can be decreased by reducing the impedance of both halves of the divider, though this increases the divider's quiescent input current and results in higher power consumption (and wasted heat) in the divider. Voltage regulators are often used in lieu of passive voltage dividers when it is necessary to accommodate high or fluctuating load currents.
Voltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and for measurement of voltages. A Wheatstone bridge and a multimeter both include voltage dividers. A potentiometer is used as a variable voltage divider in the volume control of many radios.
Sensor measurementEdit
Voltage dividers can be used to allow a microcontroller to measure the resistance of a sensor.[1] The sensor is wired in series with a known resistance to form a voltage divider and a known voltage is applied across the divider. The microcontroller's analog-to-digital converter is connected to the center tap of the divider so that it can measure the tap voltage and, by using the measured voltage and the known resistance and voltage, compute the sensor resistance. This technique is commonly used to measure the resistance of temperature sensors such as thermistors and RTDs.
Another example that is commonly used involves a potentiometer (variable resistor) as one of the resistive elements. When the shaft of the potentiometer is rotated the resistance it produces either increases or decreases, the change in resistance corresponds to the angular change of the shaft. If coupled with a stable voltage reference, the output voltage can be fed into an analog-to-digital converter and a display can show the angle. Such circuits are commonly used in reading control knobs.
High voltage measurementEdit
High voltage (HV) resistor divider probe. The HV to be measured (VIN) is applied to the corona ball probe tip and ground is connected to the other end of the divider via the black cable. The divider output (VOUT) appears on the connector adjacent to the cable.
A voltage divider can be used to scale down a very high voltage so that it can be measured by a volt meter. The high voltage is applied across the divider, and the divider output—which outputs a lower voltage that is within the meter's input range—is measured by the meter. High voltage resistor divider probes designed specifically for this purpose can be used to measure voltages up to 100 kV. Special high-voltage resistors are used in such probes as they must be able to tolerate high input voltages and, to produce accurate results, must have matched temperature coefficients and very low voltage coefficients. Capacitive divider probes are typically used for voltages above 100 kV, as the heat caused by power losses in resistor divider probes at such high voltages could be excessive.
Logic level shiftingEdit
A voltage divider can be used as a crude logic level shifter to interface two circuits that use different operating voltages. For example, some logic circuits operate at 5V whereas others operate at 3.3V. Directly interfacing a 5V logic output to a 3.3V input may cause permanent damage to the 3.3V circuit. In this case, a voltage divider with an output ratio of 3.3/5 might be used to reduce the 5V signal to 3.3V, to allow the circuits to interoperate without damaging the 3.3V circuit. For this to be feasible, the 5V source impedance and 3.3V input impedance must be negligible, or they must be constant and the divider resistor values must account for their impedances. If the input impedance is capacitive, a purely resistive divider will limit the data rate. This can be roughly overcome by adding a capacitor in series with the top resistor, to make both legs of the divider capacitive as well as resistive.
^ "A very quick and dirty introduction to Sensors, Microcontrollers, and Electronics" (PDF). Retrieved 2 November 2015.
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When the electron is excited from K level to M level we get
\gamma
- rays
B. cathode rays
D. absorption spectra
Sitting Arrangement MCQ Photo-Chemistry MCQ Line Graph MCQ Operating System MCQ Engineering Drawing MCQ Urologic problems in pregnancy MCQ Rearrangement Set 5 MCQ Power Plant Engineering MCQ Cause & Effect MCQ Problems on Ages MCQ Java Programming MCQ PageMaker MCQ
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SetCopyrightHolder - Maple Help
Home : Support : Online Help : Programming : eBookTools Package : SetCopyrightHolder
set copyright holder of the book
SetCopyrightHolder(book, copyrightHolder)
The SetCopyrightHolder command sets the copyright holder of the book.
On Windows only the following characters are allowed in the copyright holder: A-Z, a-z, 0-9, space " ", comma ",", period ".", underscore "_" and dash "-".
In order to specify non English characters for copyright holder use the eBook Publisher Assistant.
\mathrm{with}\left(\mathrm{eBookTools}\right):
\mathrm{book}≔\mathrm{NewBook}\left("ProgrammingGuide","Maple Programming Guide","",""\right):
\mathrm{SetCopyrightHolder}\left(\mathrm{book},"Maplesoft, a division of Waterloo Maple Inc."\right)
\textcolor[rgb]{0,0,1}{"Maplesoft, a division of Waterloo Maple Inc."}
The eBookTools[SetCopyrightHolder] command was introduced in Maple 16.
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David Eugene Smith - Wikiquote
David Eugene Smith (January 21, 1860 – July 29, 1944) was an American mathematician, educator, and editor.
1.1 History of Mathematics (1923) Vol.1
2 Quotes about David Eugene Smith
The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Furthermore it takes a century or more to weigh men and their discoveries, thus making the judgment of contemporaries often quite worthless.
David Eugene Smith, History of Modern Mathematics, 1896; 1904
1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.
David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics (1906)
History of Mathematics (1923) Vol.1[edit]
David Eugene Smith. History of Mathematics (1923) Vol.1
The fact that arithmetic and geometry took such a notable step forward... was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B.C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales.
The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.
More than any of his predecessors Plato appreciated the scientific possibilities of geometry. .. By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is... clear. ...That Plato should hold the view... is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.
Grégoire de Saint-Vincent... was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son... He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series.
{\displaystyle ds=\!dx{\sqrt {1+({\frac {dy}{dx}})^{2}}}}
Among his [John Wallis'] interesting discoveries was the relation
{\displaystyle {\frac {4}{\pi }}={\frac {3}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {5}{6}}\cdot {\frac {7}{6}}\cdot {\frac {7}{8}}\cdots }
p. 441 Footnote: see his Opera Mathematica, I
David Eugene Smith, History of Mathematics (1925), Vol.2
The Arabs contributed nothing new to the theory, but al-Khowârizmî (c. 825) states the usual rules, and the same is true of his successors.
When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation
{\displaystyle ax^{2}+bx+c=0}
, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the the science was known about 1800 B.C., and probably still earlier.<
Ch. 6: Algebra, p. 378
The first writer on algebra whose works have come down to us is Ahmes. He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical.
Ch. 6: Algebra
There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; Brahmagupta, whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic... Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)... extends the work through quadratic equations.
It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. ...Sun-tzï solved such problems by analysis and was content with a single result...
The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should...
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819).
With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra...
Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations.
The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence.
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work... to appear in print was the General Trattato of Tartaglia...
p. 384; Ch. 6: Algebra
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra.
The first noteworthy attempt to write an algebra in England was made by Robert Recorde, whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde.
The first Italian textbook to bear the title of algebra was Bombelli's work of 1572.
By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675).
So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century.
p. 386, Ch. 6: Algebra,-->
Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due.
Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:
{\displaystyle x^{6}-15x^{4}+85x^{3}-225x^{2}+274x=120}
He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns.
p. 430; footnote
In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation
{\displaystyle x^{2}+ax+b=0}
he placed
{\displaystyle u+z}
{\displaystyle x}
. He then had
{\displaystyle u^{2}+(2z+a)u+(z^{2}+az+b)=0.}
He now let
{\displaystyle 2z+a=0,}
{\displaystyle z=-{\frac {1}{2}}a,}
and this gave
{\displaystyle u^{2}-{\frac {1}{4}}(a^{2}-4b)=0.}
{\displaystyle u=\pm {\frac {1}{2}}{\sqrt {a^{2}-4b}}.}
{\displaystyle x=u+z=-{\frac {1}{2}}a\pm {\sqrt {a^{2}-4b}}.}
[Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics
{\displaystyle x^{3}+ax^{2}=c}
{\displaystyle x^{3}+bx=c}
could now be solved. The reduction of the general cubic
{\displaystyle x^{3}+ax^{2}+bx=c}
to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types
{\displaystyle x^{3}=ax^{2}+c}
{\displaystyle x^{3}+ax^{2}=c}
{\displaystyle x=y+{\frac {1}{3}}a}
{\displaystyle x=y-{\frac {1}{3}}a}
respectively, and transformed the type
{\displaystyle x^{3}+c=ax^{2}}
{\displaystyle x={\sqrt[{3}]{c^{2}/y}},}
thus freeing the equations of the term
{\displaystyle x^{2}}
. This completed the general solution, and he applied the method to the complete cubic in his later problems.
Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type
{\displaystyle x^{3}+bx=c}
; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2
He states that the root of
{\displaystyle x^{3}+6x=20}
{\displaystyle x={\sqrt[{3}]{{\sqrt {108}}+10}}-{\sqrt[{3}]{{\sqrt {108}}-10}}.}
He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam.
Although Cardan reduced his particular equations to forms lacking a term in
{\displaystyle x^{2}}
, it was Vieta who began with the general form
{\displaystyle x^{3}+px^{2}+qx+r=0}
and made the substitution
{\displaystyle x=y-{\frac {1}{3}}p,}
thus reducing the equation to the form
{\displaystyle y^{3}+3by=2c.}
He then made the substitution
{\displaystyle z^{3}+yz=b,}
{\displaystyle y={\frac {b-z^{2}}{z}},}
which led to the form
{\displaystyle z^{6}+2cz^{2}=b^{2},}
a sextic which he solved as a quadratic.
The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. ...This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri [of Deventer].
The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.
Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type
{\displaystyle x^{4}+2gx^{2}+bx=c,}
wrote it as
{\displaystyle x^{4}+2gx^{2}=c-bx,}
{\displaystyle gx^{2}+{\frac {1}{4}}y^{2}+yx^{2}+gy}
to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent.
Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745).
It is difficult to say who it is who first recognized the advantage of always equating to zero in the study of the general equation. It may very likely have been Napier, for he wrote his De Arte Logistica before 1594, and in this there is evidence that he understood the advantage of this procedure. Bürgi also recognized the value of making the second member zero, Harriot may have done the same, and the influence of Descartes was such that the usage became fairly general.
Quotes about David Eugene Smith[edit]
Johannes Tropfke... described the history of those individual parts of mathematics that he believed were most important for mathematics as taught in secondary schools. He intended his history to inform teachers about the origin of special problems, terms, and methods in school mathematics. ...Tropfke's approach to the history of mathematics at this time was new and even now is not yet out of date. The only comparable work is the second volume of D.E Smith's History of Mathematics... which gives far less detailed information.
Menso Folkerts, Cristoph J. Scriba, Hans Wussing, "Germany", Writing the History of Mathematics - Its Historical Development (2002).
Retrieved from "https://en.wikiquote.org/w/index.php?title=David_Eugene_Smith&oldid=2931410"
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What Is the Front-End Debt-to-Income Ratio (DTI)?
How Lenders Use Front-End Debt-to-Income Ratio
Front-End Debt-to-Income FAQs
The front-end debt-to-income ratio (DTI) is a variation of the DTI that calculates how much of a person's gross income is going toward housing costs. If a homeowner has a mortgage, the front-end DTI is typically calculated as housing expenses (such as mortgage payments, mortgage insurance, etc.) divided by gross income. Back-end DTI, sometimes called the back-end ratio, calculates the percentage of gross income going toward additional debt types such as credit cards and car loans. You may also hear these ratios referred to as "Housing 1" and "Housing 2," or "Basic" and "Broad," respectively.
The front-end debt-to-income ratio (DTI), or the housing ratio, calculates how much of a person's gross income is spent on housing costs.
The front-end DTI is typically calculated as housing expenses (such as mortgage payments, mortgage insurance, etc.) divided by gross income.
A back-end DTI calculates the percentage of gross income spent on other debt types, such as credit cards or car loans.
Lenders usually prefer a front-end DTI of no more than 28%.
Back-end DTI, also called the back-end ratio, considers housing expenses as part of the calculation.
Front-End Debt-to-Income Ratio (DTI) Formula and Calculation
The DTI is also known as the mortgage-to-income ratio or the housing ratio. It may be contrasted with the back-end ratio. There's a specific formula for calculating front-end debt-to-income ratio.
Calculating front-end debt-to-income ratio (DTI)
\text{Front-End DTI}=\left(\frac{\text{Housing Expenses}}{\text{Gross Monthly Income}}\right)*100
Front-End DTI=(Gross Monthly IncomeHousing Expenses)∗100
To calculate the front-end DTI, add up your expected housing expenses and divide it by how much you earn each month before taxes (your gross monthly income). Multiply the result by 100, and that is your front-end DTI ratio. For instance, if all your housing-related expenses total $1,000 and your monthly income is $3,000, your DTI is 33%.
What is a desirable front-end debt-to-income ratio (DTI)?
To qualify for a mortgage, the borrower often must have a front-end debt-to-income ratio of less than an indicated level. Paying bills on time, having a stable income, and having a good credit score won't necessarily qualify you for a mortgage loan. In the mortgage lending world, how far you are from financial ruin is measured by your DTI. Simply put, this is a comparison of your housing expenses and your monthly debt obligations versus how much you earn.
Higher ratios tend to increase the likelihood of default on a mortgage. For example, in 2009, many homeowners had front-end DTIs that were significantly higher than average, and consequently, mortgage defaults began to rise. In 2009, the government introduced loan modification programs in an attempt to get front-end DTIs below 31%.
Lenders usually prefer a front-end DTI of no more than 28%. In reality, depending on your credit score, savings, and down payment, lenders may accept higher ratios, although it depends on the type of mortgage loan. However, the back-end DTI is actually considered more important by many financial professionals for mortgage loan applications.
The maximum acceptable DTI for qualified mortgages is 43%.
The main difference between front-end debt-to-income ratio and debt-to-income ratio is how the two are calculated. With the front-end DTI, calculations are based solely on your housing expenses. The back-end DTI, however, takes into account other financial obligations, including:
Monthly payments on installment debts
Monthly payments on revolving debts, such as credit cards or lines of credit
Monthly alimony and child support payments
Monthly payments for rental properties you own
Back-end debt-to-income ratio is more comprehensive in that it takes into all of your debt payments beyond housing. A good back-end DTI ratio is typically no more than 33% to 36%.
Back-end debt-to-income ratio can be used to qualify borrowers for other loans beyond mortgages including personal loans, auto loans, and private student loans.
Lenders use both front-end and back-end debt-to-income ratios to determine your ability to repay a home mortgage loan. A higher DTI can signal to lenders that you might be stretched thin financially, while a lower DTI suggests that you have more disposable income each month that isn't going to debt repayment.
Debt-to-income ratio is just one part of the puzzle, however. Lenders can also look at your income, assets, and employment history to gauge your ability to repay a mortgage loan. Debt-to-income ratios can play a part in decision-making for purchase loans as well as mortgage refinancing.
Paying off credit cards, student loans, or other debts can improve your back-end debt-to-income ratio and potentially increase the amount of home you're able to afford.
When preparing for a mortgage application, the most obvious of strategies for lowering the front-end DTI is to pay off debt. However, most people don’t have the money to do so when they are in the process of getting a mortgage—most of their savings are going toward the down payment and closing costs. If you think you can afford the mortgage, but your DTI is over the limit, a cosigner might help. Keep in mind, however, that if you're unable to meet your mortgage obligations, your credit score as well as your cosigner's could suffer.
What Is Front-End Debt-to-Income Ratio?
Front-end debt-to-income ratio is a measure of how much of monthly income goes toward housing costs. That includes mortgage payments, property taxes, homeowners insurance premiums, and homeowners association fees, if applicable.
What Is a Good Debt-to-Income Ratio to Buy a Home?
Generally, lenders look for a debt-to-income ratio of between 28% and 36% when qualifying a borrower for a mortgage. Qualified mortgage loans, however, may allow a DTI of up to 43%.
How Can I Improve My Debt-to-Income Ratio for a Mortgage?
Some of the best ways to improve debt-to-income ratio include paying down revolving or installment debts, reducing housing costs, and increasing income. A lower DTI can increase the amount of home you may be able to afford when qualifying to mortgage a property.
Prospective borrowers should do everything they can to keep their debt-to-income ratios low. This shows potential creditors that the prospective borrower has a good relationship with debt, and has a monetary cushion between their income and debt in order to absorb unforeseen expenses, which greatly lessens the likelihood of default.
Fannie Mae. "Debt to Income Ratios."
Federal Deposit Insurance Corporation. "How Much Home Can I Afford?" Page 1.
Veterans United. "Debt-to-Income (DTI) Ratio Guidelines for VA Loans."
U.S. Department of Housing and Urban Development. "Making Home Affordable," Pages 1-3.
The front-end ratio is a ratio that indicates what portion of an individual's income is allocated to mortgage payments.
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Ok - Uncyclopedia, the content-free encyclopedia
Okay as a sideways person.
Hand Gesture Signifying that everything is okay {positive} moja (Is) Thumb points upwards, rest of hand in fist shape.
Rosco Peterson's water bottle from that famous baseball game
Rosco Peterson's teddy bear
1 The Obvious Meaning
2 Greek Language Origins
3 In the Medical Field
4 Uses in the United States
5 Common questions in which "OK" is used as a response
The Obvious Meaning[edit]
Obviously, OK is a sideways person. At first it was said to be female, but that was before the species realized they have brains and can do things other than getting pregnant (like, for example giving birth, which did not always require sitting horizontally), so they started a riot. Quick enough, OK became a male. Then Michael Jackson came to be, moments in which everyone was confused and decided to blame its origins on Greek mythology.
Your mom still thinks it's cake on a weird looking table. Nigga Please.
Greek Language Origins[edit]
"O.K." is the abbreviation (spelled correctly) of the Greek expression, Ola Kala (Ολα Καλά, ΟΚ) It is a standard expression in Greece that simply means: "My anus is red". An alternative etymology suggests that Okay is pig latin for K.O. (Knock-out).
Originally teachers used this as a mark on kids school papers in red ink as a taunt, saying that the paper was horrible. The meaning was lost when the children came up with a brilliant idea to stop this abuse; they being saying OK(as we know it)with and around their parents so that their parents became familiar with this expression as a positive meaning, and when they brought home their school papers, the parents thought that "OK" was good. Some teachers still use it to mark good school papers today. The brilliant children completly switched the meaning so that now it actually DOES mean the paper is good!
ok was originally written by a dislexic as K.O. after beating an enemy over the ass so many times his anus went red and he lossed con... the ability to stand up.
The word was introduced to the Western World during the Christmas period of 1689, when the majority of Christmas trees were oak saplings. A travelling pine tree salesman approached a Canadian passerby, and offered to sell him a tree, in a pitch similar to the following excerpt.
Salesman: "Would you like to buy a tree, sir?" Canadian: "Oak, eh."
The Canadian was given a pine tree to use that Christmas, as were many other Canadian families. Thousands of oak loving children cried that Christmas.
In the Medical Field[edit]
The abbreviation "OK" was informally used to communicate some type of anus rash (light to severe) was present, mainly used between and among doctors. Also, for a doctor, hearing the Ola Kala was a quick way to take stock of a situation. Doctors had to mentally prepare themselves before entering a room with a patient that had "OK". It was common for a patient with "OK" to sit for great lengths of time waiting to be treated, as every doctor tried as hard as possible to avoid having to be the one to treat those patients.
Uses in the United States[edit]
"OK," sometimes spelled as "okay," is the most recognized word in the United States of America today, and rightly so. It was first used in the United States in the mid 1960's when the coach of the Birmingham Firehoses, a professional baseball team, told a player named Rosco Peterson, "Go get them Niggers!" to which Rosco replied, "OK". To this day the word has been widely used in the United States. It is often heard in gay bars (this usually is accompanied by a flick of the wrist) and even known to be the first words spoken when a prostitute asks a virgin if he wants to conduct business.
Today, the word is also used in the U.S. when a question is asked to a person that does not speak the language in which the question was asked;
Englishman - "What type of tea would you like?"...Spaniard - "OK".
Asian - "Small or Large Boba?"...American - "OK".
African American - "You strapped?"...Canadian - "OK".
But some say that O.K. would be the abbreviation of Donkey Kong's long lost brother, Onkey Kong.
Common questions in which "OK" is used as a response[edit]
Alight, I get it. You win.
Often, when a individual is lost in the moment of a ridiculous and also lude thought, s/he will respond paradoxically to any question with the same initial reply (OK). This is a physiological response orchastrated by a severely imbalanced brain chemistry as a result of massive head trauma sustained during the fifth grade, as well as a strong sense of shame from the act of masterbation. A similar condition with the exact same symptoms as described above is caused by a cell phone somehow being accidentally glued or hammered into the side of the face. The only known cure for this unfortunate condition is a swift kick, punch, slap, or similar impulsive event, to the opposite side of the head. Also, a pail of hot or cold water can be applied to the area. If you've done it right, the battery will short circuit, inducing a sizeable shock to the brain. This kills enough brain cells in precisely the correct region to render the patient brain dead, incapable of any linguistic vocalizations in the first place.
The following are common situations in which a sufferer will exhibit the response "OK":
"Paper or plastic, sir?"
"And what kind of drink with that?"
"I'm gonna bash you, gay!"
"Can I have the rest of your french fries?"
"I'm an American, so I smell like a poof."
"According to my calculations, if you
{\displaystyle 67/8x93+rfha+67}
Should make all of us want to make a lot of maths!"
"What the hell are you doing on me?!?! GET THE HELL OFF!"
"Stop raping me!"
"This isn't making much sense."
"Eat something,"
"AAGGHHHHHH FUCK YOU!!!"
"GOD DAMN STOP TEABAGGING ME!!!"
"Hello, I am a gay. You fancy a bum?"
"I'm considering whether you need a good punch in the weiner."
Retrieved from "https://uncyclopedia.com/w/index.php?title=Ok&oldid=5979519"
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Bernoulli Distribution Vs Binomial
Bernoulli Distribution vs Binomial Distribution
A Bernoulli random variable has two possible outcomes:
0 or 1
. A binomial distribution is the sum of independent and identically distributed Bernoulli random variables.
The Bernoulli Distribution posist the success or failure of a single Bernoulli trial whereas the Binomial Distribution represents the number of successes and failures in n independent Bernoulli trials for some given value of n.
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Relativism - Uncyclopedia, the content-free encyclopedia
“All is relative, except the absolute”
~ Oscar Wilde on relativism
Relativist research articles
According to relativism, Life, the Universe and Everything is relative. Relativism is the exact opposite of objectivism, another theory which says that everything is objective. However, the astute reader might have noticed that both of these theories are based on personal opinions. The only logical conclusion is that relativism is in fact, objectivism, which makes this article useless. Some people believe that relativism has nothing to do with philosophy, and is, in reality an astronomic theory.
1 The Origin of Relativism
2.1 Explanation of the Theory of Relativity (
{\displaystyle E=mc^{2}\,\!}
The Origin of Relativism[edit]
It is well known that relativism is old, but there are varying opinions on exactly how old it is. In fact there are also varying opinions on exactly how many varying opinions there are. Some think that it was created by the sophists of ancient Greece, while there are others who believe it is even older than that, and others yet who are of the opinion that it was invented in more recent times.
Theory of Relativity[edit]
Supporters of the Young Relativism, or Relativity theory, state that it was formulated in 1915 by philosopher Albert Einstein, with his famous equation:
{\displaystyle E=mc^{2}\,\!}
However, due to the relative nature of relativity, this has often been misinterpreted as a physical formula, in which E indicates energy, m indicates mass and c indicates the speed of light, which is why Einstein was mistaken for a scientist. In reality the letters in the formula could stand for something else, depending on the interpretation.
Explanation of the Theory of Relativity (
{\displaystyle E=mc^{2}\,\!}
)[edit]
Obviously, each letter in the equation stands for something:
c = Comparative/Confrontable (Experts haven't been able to agree on what c stands for due to differing opinions).
{\displaystyle E=mc^{2}\,\!}
means that Everything is mostly comparative/confrontable, squared. Why the squared? Because to confront or compare something, two differing views are necessary, and therefore c is multiplied by itsself. c cannot be added to itsself because the two different views are opposite, which means that we would have:
{\displaystyle E=m(c-c)\,\!}
which would become
{\displaystyle E=m0\,\!}
{\displaystyle E=0\,\!}
, although this interpretation of the formula, Everything = Nothing has been accepted by some people. Neither can c be multiplied by 2, because, we would still end up with
{\displaystyle E=0\,\!}
m is in the formula because, since everything is relative, this means that some things may not be relative, it all depends on opinions.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Relativism&oldid=4269989"
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How to Calculate the Sum of Interior Angles: 8 Steps
How to Calculate the Sum of Interior Angles
1 Using the Formula
2 Drawing Triangles
A polygon is any closed figure with sides made from straight lines. At each vertex of a polygon, there is both an interior and exterior angle, corresponding to the angles on the inside and outside of the closed figure. Understanding the relationships that govern these angles is useful in various geometrical problems. In particular, it is helpful to know how to calculate the sum of interior angles in a polygon. This can be done using a simple formula, or by dividing the polygon into triangles.
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Set up the formula for finding the sum of the interior angles. The formula is
{\displaystyle sum=(n-2)\times 180}
{\displaystyle sum}
is the sum of the interior angles of the polygon, and
{\displaystyle n}
equals the number of sides in the polygon.[1] X Expert Source David Jia
Academic Tutor Expert Interview. 23 February 2021 [2] X Research source
The value 180 comes from how many degrees are in a triangle. The other part of the formula,
{\displaystyle n-2}
is a way to determine how many triangles the polygon can be divided into. So, essentially the formula is calculating the degrees inside the triangles that make up the polygon.[3] X Research source
This method will work whether you are working with a regular or irregular polygon. Regular and irregular polygons with the same number of sides will always have the same sum of interior angles, the difference only being that in a regular polygon, all interior angles have the same measurement.[4] X Research source In an irregular polygon, some of the angles will be smaller, some of the angles will be larger, but they will still add up to the same number of degrees that are in the regular shape.
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Count the number of sides in your polygon. Remember that a polygon must have at least three straight sides.
For example, if you want to know the sum of the interior angles of a hexagon, you would count 6 sides.
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Plug the value of
<b class="whb">{\displaystyle n}</b>
into the formula. Remember,
{\displaystyle n}
is the number of sides in your polygon.
For example, if you are working with a hexagon,
{\displaystyle n=6}
, since a hexagon has 6 sides. So, your formula should look like this:
{\displaystyle sum=(6-2)\times 180}
<b class="whb">{\displaystyle n}</b>
. To do this, subtract 2 from the number of sides, and multiply the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon.
For example, to find out the sum of the interior angles of a hexagon, you would calculate:
{\displaystyle sum=(6-2)\times 180}
{\displaystyle sum=(4)\times 180}
{\displaystyle sum=(4)\times 180=720}
So, the sum of the interior angles of a hexagon is 720 degrees.
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Draw the polygon whose angles you need to sum. The polygon can have any number of sides and can be regular or irregular.
For example, you might want to find the sum of the interior angles of a hexagon, so you would draw a six-sided shape.
Choose one vertex. Label this vertex A.
A vertex is a point where two sides of a polygon meet.
Draw a straight line from Point A to each other vertex in the polygon. The lines should not cross. You should create a number of triangles.
You do not have to draw lines to the adjacent vertices, since they are already connected by a side.
For example, for a hexagon you should draw three lines, dividing the shape into 4 triangles.
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Multiply the number of triangles you created by 180. Since there are 180 degrees in a triangle, by multiplying the number of triangles in your polygon by 180, you can find the sum of the interior angles of your polygon.
For example, since you divided your hexagon into 4 triangles, you would calculate
{\displaystyle 4\times 180=720}
to find a total of 720 degrees in the interior of your polygon.
How do I find a single interior angle?
Work out what all the interior adds up to, then divide by however many sides the shape has.
How do I calculate the number of sides of a polygon if the sum of the interior angles is 1080?
Divide that sum by 180°, then add 2. In this example, 1080° / 180° = 6. 6 + 2 = 8. The polygon has 8 sides.
If two equilateral triangles are placed together to form a rhombus, how do I calculate the value of each interior angle of this rhombus, and how do I find the sum?
In the rhombus you describe, the two smaller interior angles would each be 60°, and the two larger interior angles would each be 120°. You wouldn't have to calculate the angles. Simple inspection of the rhombus and the two triangles would show what the angles are, given that equilateral triangles have three 60° angles. The sum is 60° + 60° + 120° + 120°.
Does a regular polygon's interior angles add up to 160?
Not necessarily. A triangle's sum is 180, a quadrilateral's sum is 360, and a pentagon's sum is 540. These are all polygons. Use the formula 180(n-2) where "n" is the number of the sides of the polygon in question to find your sum.
How do I find the sum of the interior angles of an irregular polygon?
The formula for finding the sum of the interior angles of a polygon is the same, whether the polygon is regular or irregular. So you would use the formula (n-2) x 180, where n is the number of sides in the polygon.
How do I find the missing angle of an irregular polygon?
First calculate the sum of all the interior angles of the polygon by using the formula (n - 2)(180°), where n is the number of sides. Then add together all of the known angles, and subtract that sum from the sum you calculated first. That will give you the missing angle.
Why is the sum of an interior angle 180?
The sum of the angles in a triangle is 180°. The sum of the angles in a square (or other quadrilateral) is 360 °. Since two congruent triangles will combine to form a square or other quadrilateral, the sum of the angles in one of the triangles is half of 360°, or 180°.
If the exterior angle is 72, what is the interior angle?
To find the interior angle, subtract the exterior angle from 180°.
How can we work out 1 or more interior angles of an irregular polygon without a protractor?
Without a protractor you would have to know all of the linear dimensions of the polygon, divide the figure into various right triangles, and then use trigonometric functions to find the interior angles. In other words, it would be better to use a protractor on an irregular polygon.
How do I find the number of triangles in a polygon?
By counting them manually. Drawing a diagram is usually the easiest way to visualize this as described in Method 2 above.
Check your work on a piece of paper using a protractor to sum the interior angles manually. When doing this, be careful while drawing the polygon's sides as they should be linear.
To calculate the sum of interior angles, start by counting the number of sides in your polygon. Next, plug this number into the formula for the "n" value. Then, solve for "n" by subtracting 2 from the number of sides and multiplying the difference by 180. This will give you, in degrees, the sum of the interior angles in your polygon! To learn how to calculate the sum of interior angles by drawing triangles, read on!
Русский:вычислить сумму внутренних углов
"When you explained slowly how the formula works, I completely understood. I have a big math test tomorrow, and I appreciate this! Thank you!"..." more
Holly Flavian
"This has really helped me in remembering what the teacher taught. Thanks a lot."
Afia Subah
"The formula is super helpful, thank you."
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Bayes'_theorem Knowpia
Statement of theoremEdit
{\displaystyle P(A\mid B)={\frac {P(B\mid A)P(A)}{P(B)}}}
{\displaystyle A}
{\displaystyle B}
{\displaystyle P(B)\neq 0}
{\displaystyle P(A\mid B)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle P(B\mid A)}
{\displaystyle B}
{\displaystyle A}
{\displaystyle A}
{\displaystyle B}
{\displaystyle P(B\mid A)=L(A\mid B)}
{\displaystyle P(A)}
{\displaystyle P(B)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle A}
{\displaystyle B}
{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}},{\text{ if }}P(B)\neq 0,}
{\displaystyle P(A\cap B)}
{\displaystyle P(B\mid A)={\frac {P(A\cap B)}{P(A)}},{\text{ if }}P(A)\neq 0,}
{\displaystyle P(A\cap B)}
{\displaystyle P(A\mid B)}
{\displaystyle P(A\mid B)={\frac {P(B\mid A)P(A)}{P(B)}},{\text{ if }}P(B)\neq 0.}
For continuous random variablesEdit
{\displaystyle f_{X\mid Y=y}(x)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}}
{\displaystyle f_{Y\mid X=x}(y)={\frac {f_{X,Y}(x,y)}{f_{X}(x)}}}
{\displaystyle f_{X\mid Y=y}(x)={\frac {f_{Y\mid X=x}(y)f_{X}(x)}{f_{Y}(y)}}.}
Drug testingEdit
{\displaystyle P({\text{User}}\mid {\text{Positive}})}
Suppose, a particular test for whether someone has been using cannabis is 90% sensitive, meaning the true positive rate (TPR)=0.90. Therefore, it leads to 90% true positive results (correct identification of drug use) for cannabis users.
The test is also 80% specific, meaning true negative rate (TNR)=0.80. Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or false positive rate (FPR)=0.20, for non-users.
{\displaystyle P({\text{User}}\mid {\text{Positive}})}
{\displaystyle {\begin{aligned}P({\text{User}}\mid {\text{Positive}})&={\frac {P({\text{Positive}}\mid {\text{User}})P({\text{User}})}{P({\text{Positive}})}}\\&={\frac {P({\text{Positive}}\mid {\text{User}})P({\text{User}})}{P({\text{Positive}}\mid {\text{User}})P({\text{User}})+P({\text{Positive}}\mid {\text{Non-user}})P({\text{Non-user}})}}\\[8pt]&={\frac {0.90\times 0.05}{0.90\times 0.05+0.20\times 0.95}}={\frac {0.045}{0.045+0.19}}\approx 19\%\end{aligned}}}
{\displaystyle P({\text{Positive}})=P({\text{Positive}}\mid {\text{User}})P({\text{User}})+P({\text{Positive}}\mid {\text{Non-user}})P({\text{Non-user}})}
Sensitivity or specificityEdit
Cancer rateEdit
{\displaystyle {\begin{aligned}P({\text{Cancer}}|{\text{Symptoms}})&={\frac {P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})}{P({\text{Symptoms}})}}\\&={\frac {P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})}{P({\text{Symptoms}}|{\text{Cancer}})P({\text{Cancer}})+P({\text{Symptoms}}|{\text{Non-Cancer}})P({\text{Non-Cancer}})}}\\[8pt]&={\frac {1\times 0.00001}{1\times 0.00001+(10/99999)\times 0.99999}}={\frac {1}{11}}\approx 9.1\%\end{aligned}}}
Defective item rateEdit
{\displaystyle P(X_{A})=0.2,\quad P(X_{B})=0.3,\quad P(X_{C})=0.5.}
{\displaystyle P(Y|X_{A})=0.05,\quad P(Y|X_{B})=0.03,\quad P(Y|X_{C})=0.01.}
{\displaystyle P(Y)=\sum _{i}P(Y|X_{i})P(X_{i})=(0.05)(0.2)+(0.03)(0.3)+(0.01)(0.5)=0.024.}
{\displaystyle P(X_{C}|Y)={\frac {P(Y|X_{C})P(X_{C})}{P(Y)}}={\frac {0.01\cdot 0.50}{0.024}}={\frac {5}{24}}}
Bayesian interpretationEdit
Frequentist interpretationEdit
{\displaystyle {\overline {P}}}
{\displaystyle {\begin{aligned}P({\text{Rare}}\mid {\text{Pattern}})&={\frac {P({\text{Pattern}}\mid {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}})}}\\[8pt]&={\frac {P({\text{Pattern}}\mid {\text{Rare}})P({\text{Rare}})}{P({\text{Pattern}}\mid {\text{Rare}})P({\text{Rare}})+P({\text{Pattern}}\mid {\text{Common}})P({\text{Common}})}}\\[8pt]&={\frac {0.98\times 0.001}{0.98\times 0.001+0.05\times 0.999}}\\[8pt]&\approx 1.9\%\end{aligned}}}
Simple formEdit
{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B)}}\cdot }
{\displaystyle P(A|B)\propto P(A)\cdot P(B|A)}
{\displaystyle P(A|B)=c\cdot P(A)\cdot P(B|A){\text{ and }}P(\neg A|B)=c\cdot P(\neg A)\cdot P(B|\neg A).}
{\displaystyle 1=c\cdot (P(B|A)\cdot P(A)+P(B|\neg A)\cdot P(\neg A)),}
{\displaystyle c={\frac {1}{P(B|A)\cdot P(A)+P(B|\neg A)\cdot P(\neg A)}}={\frac {1}{P(B)}}.}
Alternative formEdit
{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}.}
{\displaystyle P(A)}
{\displaystyle P(\neg A)}
{\displaystyle P(\neg A)=1-P(A)}
{\displaystyle P(B|A)}
{\displaystyle P(B|\neg A)}
{\displaystyle P(A|B)}
Extended formEdit
{\displaystyle P(B)={\sum _{j}P(B|A_{j})P(A_{j})},}
{\displaystyle \Rightarrow P(A_{i}|B)={\frac {P(B|A_{i})P(A_{i})}{\sum \limits _{j}P(B|A_{j})P(A_{j})}}\cdot }
{\displaystyle P(A|B)={\frac {P(B|A)P(A)}{P(B|A)P(A)+P(B|\neg A)P(\neg A)}}\cdot }
Random variablesEdit
{\displaystyle P(X{=}x|Y{=}y)={\frac {P(Y{=}y|X{=}x)P(X{=}x)}{P(Y{=}y)}}}
{\displaystyle f_{X|Y{=}y}(x)={\frac {P(Y{=}y|X{=}x)f_{X}(x)}{P(Y{=}y)}}}
{\displaystyle f}
{\displaystyle P(X{=}x|Y{=}y)={\frac {f_{Y|X{=}x}(y)P(X{=}x)}{f_{Y}(y)}}.}
{\displaystyle f_{X|Y{=}y}(x)={\frac {f_{Y|X{=}x}(y)f_{X}(x)}{f_{Y}(y)}}.}
{\displaystyle f_{Y}(y)=\int _{-\infty }^{\infty }f_{Y|X=\xi }(y)f_{X}(\xi )\,d\xi .}
Bayes' rule in odds formEdit
{\displaystyle O(A_{1}:A_{2}\mid B)=O(A_{1}:A_{2})\cdot \Lambda (A_{1}:A_{2}\mid B)}
{\displaystyle \Lambda (A_{1}:A_{2}\mid B)={\frac {P(B\mid A_{1})}{P(B\mid A_{2})}}}
{\displaystyle O(A_{1}:A_{2})={\frac {P(A_{1})}{P(A_{2})}},}
{\displaystyle O(A_{1}:A_{2}\mid B)={\frac {P(A_{1}\mid B)}{P(A_{2}\mid B)}},}
{\displaystyle A_{1}=A}
{\displaystyle A_{2}=\neg A}
{\displaystyle O(A)=O(A:\neg A)=P(A)/(1-P(A))}
{\displaystyle A}
{\displaystyle A}
{\displaystyle O(A\mid B)=O(A)\cdot \Lambda (A\mid B),}
{\displaystyle A}
{\displaystyle A}
{\displaystyle A}
{\displaystyle B}
{\displaystyle \Lambda _{+}=P({\text{True Positive}})/P({\text{False Positive}})=90\%/(100\%-91\%)=10}
The example above can also be understood with more solid numbers: Assume the patient taking the test is from a group of 1000 people, where 91 of them actually have the disease (prevalence of 9.1%). If all these 1000 people take the medical test, 82 of those with the disease will get a true positive result (sensitivity of 90.1%), 9 of those with the disease will get a false negative result (false negative rate of 9.9%), 827 of those without the disease will get a true negative result (specificity of 91.0%), and 82 of those without the disease will get a false positive result (false positive rate of 9.0%). Before taking any test, the patient's odds for having the disease is 91:909. After receiving a positive result, the patient's odds for having the disease is
{\displaystyle {\frac {91}{909}}\times {\frac {90.1\%}{9.0\%}}={\frac {91\times 90.1\%}{909\times 9.0\%}}=1:1}
{\displaystyle P(\neg B\mid A)=1-P(B\mid A)}
{\displaystyle P(\neg B\mid \neg A)}
{\displaystyle P(A\mid B)}
{\displaystyle P(\neg B\mid \neg A)=1-\left(1-P(A\mid B)\right){\frac {P(B)}{P(\neg A)}}}
{\displaystyle P(\neg A)=1-P(A)\neq 0}
{\displaystyle P(A\mid B)=1\implies P(\neg B\mid \neg A)=1}
{\displaystyle B}
{\displaystyle A}
{\displaystyle \neg A}
{\displaystyle \neg B}
{\displaystyle P(B)\neq 0}
{\displaystyle P(A\mid B)}
{\displaystyle B\implies A}
{\displaystyle B\implies A}
{\displaystyle P(A\mid B)=1}
{\displaystyle (B\implies A)\iff (\neg A\implies \neg B)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle a}
{\displaystyle A}
{\displaystyle P(A\mid B)=P(B\mid A){\frac {a(A)}{P(B\mid A)\,a(A)+P(B\mid \neg A)\,a(\neg A)}}}
{\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})=(\omega _{B\mid A}^{S},\omega _{B\mid \lnot A}^{S}){\widetilde {\phi }}a_{A},}
{\displaystyle {\widetilde {\phi }}}
{\displaystyle (\omega _{B\mid A}^{S},\omega _{B\mid \lnot A}^{S})}
{\displaystyle S}
{\displaystyle a_{A}}
{\displaystyle A}
{\displaystyle (\omega _{A{\tilde {|}}B}^{S},\omega _{A{\tilde {|}}\lnot B}^{S})}
{\displaystyle \omega _{A\mid B}^{S}}
{\displaystyle P(A\mid B)}
{\displaystyle S}
{\displaystyle (A\mid B)}
{\displaystyle \omega _{A}^{S}}
{\displaystyle A}
{\displaystyle S}
{\displaystyle P(\omega _{A}^{S})}
{\displaystyle P(\omega _{A{\tilde {|}}B}^{S})={\frac {P(\omega _{B\mid A}^{S})a(A)}{P(\omega _{B\mid A}^{S})a(A)+P(\omega _{B\mid \lnot A}^{S})a(\lnot A)}}.}
Conditioned versionEdit
{\displaystyle C}
{\displaystyle P(A\mid B\cap C)={\frac {P(B\mid A\cap C)\,P(A\mid C)}{P(B\mid C)}}}
{\displaystyle P(A\cap B\cap C)=P(A\mid B\cap C)\,P(B\mid C)\,P(C)}
{\displaystyle P(A\cap B\cap C)=P(B\cap A\cap C)=P(B\mid A\cap C)\,P(A\mid C)\,P(C)}
{\displaystyle P(A\mid B\cap C)}
Bayes' rule with 3 eventsEdit
{\displaystyle P(A\mid B,C)={\frac {P(B\mid A,C)\;P(A\mid C)}{P(B\mid C)}}}
{\displaystyle {\begin{aligned}P(A\mid B,C)&={\frac {P(A,B,C)}{P(B,C)}}\\[1ex]&={\frac {P(B\mid A,C)\,P(A,C)}{P(B,C)}}\\[1ex]&={\frac {P(B\mid A,C)\,P(A\mid C)\,P(C)}{P(B,C)}}\\[1ex]&={\frac {P(B\mid A,C)\,P(A\mid C)P(C)}{P(B\mid C)P(C)}}\\[1ex]&={\frac {P(B\mid A,C)\;P(A\mid C)}{P(B\mid C)}}\end{aligned}}}
Sir Harold Jeffreys put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry".[19]
Use in geneticsEdit
Using pedigree to calculate probabilitiesEdit
Example of a Bayesian analysis table for a female individual's risk for a disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subject's siblings and parents, she is equally likely to be a carrier as to be a non-carrier (this likelihood is denoted by the Prior Hypothesis). However, the probability that the subject's four sons would all be unaffected is 1/16 (½·½·½·½) if she is a carrier, about 1 if she is a non-carrier (this is the Conditional Probability). The Joint Probability reconciles these two predictions by multiplying them together. The last line (the Posterior Probability) is calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities.[26]
Using genetic test resultsEdit
After carrying out the same analysis on the patient's male partner (with a negative test result), the chances of their child being affected is equal to the product of the parents' respective posterior probabilities for being carriers times the chances that two carriers will produce an affected offspring (¼).
Genetic testing done in parallel with other risk factor identification.Edit
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How to Measure Phase Noise with a Spectrum Analyzer - Collections - 2022
How to Measure Phase Noise with a Spectrum Analyzer
Today's spectrum analyzers provide a very effective means of testing phase noise easily and accurately, and can be much easier and more accurate than using approaches using other forms of electronics test instruments.
These electronics test instruments are often designed with routines incorporated into the software to make the testing even easier. When compared with other methods using different forms of test equipment, the spectrum analyzer not only provides a more convenient method of gaining the phase noise measurements, but it is normally more accurate.
Phase noise is growing in importance as a parameter on many RF devices because not only can poor phase noise performance result in increased data errors, but it can also create interference to users on other channels.
Accordingly phase noise measurements are needed for a host of different types of electronics equipment during the design stages. It can be applicable to items from mobile phones, to nodes / units used for the Internet of Things, IoT, short range wireless, radio communications equipment and a large number of other items.
As a result of the variety of items that may need phase noise measurements to be made, a convenient way of achieving this is needed, and the spectrum analyzer is an ideal test instrument to meed this need.
What is phase noise
Phase noise results from the short term phase fluctuations that exist on any signal. This is known as phase jitter and is measured directly in radians.
The phase jitter manifests itself on a signal as sidebands that spread out either side of the main signal. This is known as single sideband phase noise, and when looked at in this manner it is easier to visualise and also measure.
Phase noise is important for a number of reasons:
Degrades performance of data transmissions: Most data transmissions like those used for cellular communications, Wi-Fi, and many other applications use forms of modulation that use phase as part or all of the modulation technique. Any phase noise will reduce the margin between the different states and will impact signal margins and resulting bit error rates. This means it is important to have a good phase noise performance for any local oscillators.
Adjacent channel interference: The phase noise spreads out either side of the main signal and can fall into nearby channels causing interference to other users. As a result, spurious emissions, including phase noise must be kept below certain limits to ensure interference is not a problem.
The phase noise is measured as the noise power in a given bandwidth. The standard is a 1Hz bandwidth. Although the measurement may be made in a wider bandwidth, it can be easily converted to the value for a 1Hz bandwidth.
In addition to this, the value of the noise is related to the carrier level. A given number of decibels down on the carrier. The standard abbreviation indicating this is dBc.
Finally the offset from the carrier must be stated because the noise level varies as the offset from the carrier is changed.
This a typical specification is quoted in terms of decibels down on the carrier in a 1Hz bandwidth at a given frequency offset, i.e. dBc / Hz at xx kHz offset.
Note on the Phase Noise:
Phase noise consists of small random perturbations in the phase of the signal, i.e. phase jitter. These perturbations are effectively phase modulation and as a result, noise sidebands are generated. These spread out either side of the main signal and can be plotted on a spectrum analyzer as single sideband phase noise.
Read more about Phase Noise.
Pre-requisites for measuring phase noise
The main requirement for any phase noise measurement using a spectrum analyser is that it must have a low level of drift compared to the sweep rate. If the level of oscillator drift is too high, then it would invalidate the measurement results.
This means that this technique is ideal for measuring the phase noise levels of frequency synthesizers as they are locked to a stable reference and drift levels are very low.
Many free running oscillators are not sufficiently stable to use this technique. Often they would need to be locked to a reference in some way, and this would alter the phase noise characteristics of at least part of the spectrum.
In addition to this the phase noise performance of the spectrum analyzer must be better than that of the item under test, otherwise the test will measure the phase noise characteristic of the spectrum analyzer.
Whilst it is not essential, it helps if the spectrum analyser has a built in routine for phase noise measurement. Many modern test instruments have these routines built in and t can be a great help.
Although there are many ways of measuring phase noise, the most straightforward is to use a spectrum analyzer.
Essentially the analyzer is connected to the output of the unit under test via any suitable attenuator needed to reduce the power into the analyzer (if the output power from the unit under test is high).
In some instances it may be necessary to lock the oscillator standards of the analyzer and unit under test together. In this way there will be no signal drift which could be an issue for close in measurements.
The analyzer is then set to measure the signal level out from the carrier - often this may from the carrier out to a frequency of 1 MHz or possibly more. Ideally to a point where the noise has reached the noise floor.
The bandwidth of analyzer must be set so that a good balance is achieved between the resolution of the scan and the time taken for the scan to be taken. The noise level can then be converted to that found in a 1Hz bandwidth.
Analyzer filter & detector characteristics
The filter and detector characteristics of the spectrum analyzer have an impact on the phase noise measurement results.
One of the key issues is the bandwidth of the filter used within the spectrum analyser. Analysers do not possess 1 Hz filters, and even if they did measurements with a 1 Hz bandwidth filter would take far too long to make. Accordingly, wider filters are used and the noise level is adjusted to the levels that would be found if a 1 Hz bandwidth filter had been used.
It is possible to use a simple formula to make an adjustment for the filter bandwidth:
{L}_{1Hz}={L}_{\mathrm{filter}}-10{\mathrm{log}}_{10}\left(\frac{\mathrm{BW}}{1}\right)
L1Hz = level in 1 Hz bandwidth, i.e. normalised to 1 Hz, typically in dBm
Lfilt = level in the filter bandwidth, typically in dBm
BW = bandwidth of the measurement filter in Hz
As the filter shape is not a completely rectangular shape and has a finite roll-off, this has an effect on the transformation to give the noise in a 1Hz bandwidth. Typically a known factor for the filter in use needs to be incorporated to ensure a correct transformation.
The type of detector also has an impact. If a sampling detector is used instead of an RMS detector and the trace is averaged over a narrow bandwidth or several measurements, then it is found that the noise will be under-weighted.
Adjustments for these and any other factors are normally accommodated within the spectrum analyser, and often a special phase noise measurement set-up is incorporated within the software capabilities.
Phase noise measurement precautions
There are a few important precautions to remember when measuring phase noise with a spectrum analyzer
Ensure no external noise can be picked up: The spectrum analyzer measures the single sideband phase noise and therefore any amplitude noise that is present will add to this, degrading the result. Ensure that no external noise can be picked up by the analyzer:
Use screened leads: Use screened leads for all signal connections
Keep away from noise sources: Ensure that the test system including the unit under test is located away from any sources of interference. As the signal levels being measured will be very low for some frequencies even a small amount of pick-up can cause erroneous results
Screened room? If an RF screened room is available, it may be possible to use this to perform the test, ensuring no interference is picked up.
Run unit under test from correct power source: A power supply can considerably alter the noise performance of the RF circuitry. Ensure that the power supply for the equipment is used, or at least one with a performance level the same. Switching power supplies often generate more noise than analogue linear ones so this should be remembered.
Ensure analyser performance suitable: There are two main issues; namely the spectrum analyzer phase noise itself, and the dynamic range performance:
Spectrum analyzer phase noise performance : For signals that have very low levels of phase noise, it is possible that the unit under test may approach the performance of the analyzer. In this case the phase noise of the oscillator within the analyzer will add to that of the signal under test and this will distort the result. To prevent this, ensure that the phase noise performance of the analyzer is at least 10dB better than that of the unit under test.
Spectrum analyzer dynamic range: The dynamic range performance of the spectrum analyser must also be sufficient. The analyser must be able to accommodate the carrier level as well as the very low noise levels that exist further out from the carrier. It is easy to check whether the thermal noise is an issue. The trace of the phase noise of the signal source can be taken and stored. Using exactly the same settings, but with no signal, the measurement can be repeated. If at the offset of interest there is a clear difference between the two, then the measurement will not be unduly affected by the analyser thermal noise.
Using these precautions and any others that may be appropriate ensure that it is possible to obtain some very good results when using a spectrum analyzer to measure phase noise.
Spectrum analyzers are ideal test instruments for making phase noise measurements. With many modern high performance analyzers already incorporating routines for undertaking these tests in any RF design or test scenario, the measurements are not only easy to make but also reliable.
In view of the rigours of making phase noise measurements, it is mainly the top end test instruments that can make these measurements and have the routines built in. Nevertheless, it is possible, with care to use other lower end spectrum analyzers to make estimates of the phase noise performance of circuits, modes and systems, provided that there is an understanding of the limitations of the test.
Watch the video: IMS2014 Importance of Phase Noise and Ways to Measure It. Keysight Technologies (May 2022).
Copyright 2022 \ How to Measure Phase Noise with a Spectrum Analyzer...
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