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Obtaining Optimal Solution by Using Very Good Non-Basic Feasible Solution of the Transportation and Linear Programming Problem
Obtaining Optimal Solution by Using Very Good Non-Basic Feasible Solution of the Transportation and Linear Programming Problem ()
Department of Industrial and Management Engineering, Indian Institute of Technology, Kanpur, India.
For the transportation problem, Sharma and Sharma [1] have given a very computationally efficient heuristic (runs in O(c*n2) time) to give very good dual solution to transportation problem. Sharma and Prasad [2] have given an efficient heuristic (complexity O(n3) procedure to give a very good primal solution (that is generally non-basic feasible solution) to transportation problem by using the very good dual solution given by Sharma and Sharma [2]. In this paper we use the solution given by Sharma and Prasad [2] to get a very good Basic Feasible Solution to transportation problem, so that network simplex (worst case complexity (O(n3*(log(n))) can be used to reach the optimal solution to transportation problem. In the second part of this paper, we give a simple heuristic procedure to get a very good BFS to linear programming problem from the solution given by Karmarkar [3] (that generally produces a very good non-basic feasible solution in polynomial time (O(n5.5)). We give a procedure to obtain a good BFS for LP by starting from the solution given by Karmarkar [3]. We note that this procedure (given here) is significantly different from the procedure given in [4].
Optimal BSF Solution, Transportation Problem, Linear Programming Problem
Sharma, R. (2017) Obtaining Optimal Solution by Using Very Good Non-Basic Feasible Solution of the Transportation and Linear Programming Problem. American Journal of Operations Research, 7, 285-288. doi: 10.4236/ajor.2017.75021.
1.1. For Transportation Problem
Sharma and Prasad [2] gave a very good non-basic primal solution to the transportation problem. We give a procedure (named as PNP-1) here to make use of this (this non-basic feasible solution to get a good basic feasible solution). This enables Network Simplex to lead to optimality by making use of advanced start.
1.2. For Linear Program
Similarly, Karmarkar [3] gives a very good interior point solution to Linear Program in polynomial time. We give a procedure (PNP 2) to get very good BFS for Linear program (by using solution given in [3] ). And this enables Simplex Procedure to lead to optimality by making use of this advanced start.
2. Formulation of Transportation Problem
Problem P1.
\mathrm{min}{\sum }_{k}{\sum }_{i}{X}_{ik}\cdot {C}_{ik}
{\sum }_{i}{X}_{ik}={d}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}k=1,\cdots ,K
{\sum }_{k}{X}_{ik}={b}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}i=1,\cdots ,I
{X}_{ik}\ge 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}k
For the balanced transportation problem, method due to Sharma and Prasad [2] gives a very good feasible solution with exactly “p” Xik > 0. It is to be noted that in a BFS exactly K + I − 1, Xik need to be greater than zero.
3. The Proposed New Procedure PNP 1
Step 1: Prepare a list L of “p” Cik such that Xik’s > 0.
Then sort (“p”) Cik’s with Xik’s > 0 in increasing order. Put first K + I − 1 items of L in list L1 and remaining Cik’s with Xik’s > 0 in list L2. It is to be noted that the associated cells in L1 form a basis.
Step 2: Then we take first item from L2. It can be easily seen that cells of L1 + first cell (CE*) taken out of L2 has a unique cycle. Perform the pivot operation to determine the leaving cell (by using Cik’s as relative cost co-efficients. It can be seen that resulting partial solution may increase or decrease.
Step 3: Remove the cell CE* from the list L2. If L2 is not empty, then go to Step 2, else stop.
Step 4: We have good BFS for the balanced transportation problem (P1). Now network simplex for the transportation problem can take over.
4. Formulation of Linear Program
\mathrm{min}CX
AX=b
X\ge 0
Matrices A, X and b are matrices of conformable dimensions (A is of the size m × n; X is of size n × 1; and b is of size m × 1). It is to be noted that a BFS to problem P2 is associated with a matrix B of size m × m. Optimal BFS has at most “m” positive entries in X (that is of size n × 1; such that m < n).
It is to be noted that Karmarkar’s algorithm [3] to solve problem P2 gives a very good interior (feasible) point that has exactly “p” positive entries in X1 such that p > m.
Step 1: It is to be noted that there are exactly p C m bases associated with the solution given to problem P2 by methods of [3] . Get one feasible basis (B1) with exactly “m” columns in X1. Intuitively it is felt that this would be a good BFS.
Step 2: Choose other column (a2) in A that is not included in B1 such that a2 belongs to X1. Set an = a2.
Step 3: Perform pivot with an as entering variable. It can be seen that an will enter the basis only if its relative cost co-efficient is strictly negative. It can be seen that the solution will improve or remain the same.
Step 4: If all columns associated with X1 belonging to (p-m) are considered for entering variable, then stop; we have a good BFS and usual simplex can proceed further to get optimal solution; else choose other column in X1 not considered earlier and call it an. Go to step 3.
We now have the optimal BFS to problem P2. It can be seen that this approach is different than the one given in [4] .
Algorithms (that give very good solutions (non-BFS in general) in competitive times: transportation problem (O(n3) & simplex (for linear program) in O(n5.5)) are available. We give a procedure that makes use for above; and gives very good BFS. Then control is given over to optimizing procedures (like Network Simplex (for transportation runs in O(n3*log(n) time) & ordinary simplex procedure for linear program (exponential time complexity) to get optimal solutions. But since we start from a very good solution, it is hoped that the optimizing procedure will take significantly less time (compared to its worst case complexity given above). An empirical investigation has been undertaken and we will submit results as early as possible.
[1] Sharma, R.R.K. and Sharma, K.D. (2000) A New Dual Based Procedure for the Transportation Problem. European Journal of Operational Research, 122, 611-624.
[2] Sharma, R.R.K. and Prasad, S. (2003) Obtaining a Good Solution to the Uncapacitated Transportation Problem. European Journal of Operational Research, 144, 560-564.
[3] Karmarkar, N. (1984) A New Polynomial Time Algorithm for Linear Programming. Combinatorica, 4, 373-395.
[4] Ye, Y.Y. (1990) Recovering Optimal Basic Variables in Karmarkar's Polynomial Algorithm for Linear Program. Mathematics of Operations Research, 15, 564-572.
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Track and extract RPM profile from vibration signal - MATLAB rpmtrack - MathWorks Nordic
RPM Profile of Vibration Signal
RPM Profile of Revving Engine
Fan Switchoff RPM Profile
Track and extract RPM profile from vibration signal
rpm = rpmtrack(___,Name=Value)
rpm = rpmtrack(x,fs,order,p) returns a time-dependent estimate of the rotational speed rpm from a vibration signal x sampled at a rate fs.
The two-column matrix p contains a set of points that lie on a time-frequency ridge corresponding to a given order. Each row of p specifies one coordinate pair. If you call rpmtrack without specifying both order and p, the function opens an interactive plot that displays the time-frequency map and enables you to select the points.
rpm = rpmtrack(xt,order,p) returns a time-dependent estimate of the rotational speed from a signal stored in the MATLAB® timetable xt.
rpm = rpmtrack(___,Name=Value) specifies additional options for any of the previous syntaxes using name-value arguments. Options include the method used to estimate the time-frequency map and the starting time for the RPM profile.
[rpm,tout] = rpmtrack(___) also returns the time vector at which the RPM profile is computed.
rpmtrack(___) with no output arguments plots the power time-frequency map and the estimated RPM profile on an interactive figure.
Generate a vibration signal with three harmonic components. The signal is sampled at 1 kHz for 16 seconds. The signal's instantaneous frequency resembles the runup and coastdown of an engine. Compute the instantaneous phase by integrating the frequency using the trapezoidal rule.
The harmonic components of the signal correspond to orders 1, 2, and 3. The order-2 sinusoid has twice the amplitude of the others.
Extract and visualize the RPM profile of the signal using a point on the order-2 ridge.
Generate a signal that resembles the vibrations caused by revving a car engine. The signal is sampled at 1 kHz for 30 seconds and contains three harmonic components of orders 1, 2.4, and 3, with amplitudes 5, 4, and 0.5, respectively. Embed the signal in unit-variance white Gaussian noise and store it in a MATLAB® timetable. Multiply the instantaneous frequency by 60 to obtain an RPM profile. Plot the RPM profile.
Derive the RPM profile from the vibration signal. Use four points at 5 second intervals to specify the ridge corresponding to order 2.4. Display a summary of the output timetable.
Plot the reconstructed RPM profile and the points used in the reconstruction.
Use the extracted RPM profile to generate the order-RPM map of the signal.
Reconstruct and plot the time-domain waveforms that compose the signal. Zoom in on a time interval occurring after the transients have decayed.
Estimate the RPM profile of a fan blade as it slows down after switchoff.
An industrial roof fan spinning at 20,000 rpm is turned off. Air resistance (with a negligible contribution from bearing friction) causes the fan rotor to stop in approximately 6 seconds. A high-speed camera measures the x-coordinate of one of the fan blades at a rate of 1 kHz.
Idealize the fan blade as a point mass circling the rotor center at a radius of 50 cm. The blade experiences a drag force proportional to speed, resulting in the following expression for the phase angle
\varphi =2\pi {f}_{0}T\left(1-{e}^{-t/T}\right),
{\mathit{f}}_{0}
\mathit{T}=0.75
Compute and plot the x- and y-coordinates of the blade. Add white Gaussian noise of variance
0.{1}^{2}
Use the rpmtrack function to determine the RPM profile. Type
at the command line to open the interactive figure.
Use the slider to adjust the frequency resolution of the time-frequency map to about 11 Hz. Assume that the signal component corresponds to order 1 and set the end time for ridge extraction to 3.0 seconds. Use the crosshair cursor in the time-frequency map and the Add button to add three points lying on the ridge. Alternatively, double-click the cursor to add the points at the locations you choose. Click Estimate to track and extract the RPM profile.
Verify that the RPM profile decays exponentially. On the Export tab, click Export and select Generate MATLAB Script. The script appears in the Editor.
% Generated by MATLAB 9.12 and Signal Processing Toolbox 8.7
% Generated on 12-Oct-2021 09:36:49
Run the script. Display the RPM profile in a semilogarithmic plot.
Example: cos(pi/4*(0:159))+randn(1,160) specifies a noisy sinusoid sampled at 2π Hz.
order — Ridge order
Ridge order, specified as a positive real scalar.
p — Ridge points
Ridge points, specified as a two-column matrix containing one time-frequency coordinate on each row. The coordinates describe points on the time-frequency map belonging to the order ridge of interest.
Input timetable. xt must contain increasing, finite, and equally spaced row times of type duration. The timetable must contain only one numeric data vector with signal values.
Example: "Method","fsst","PowerPenalty",10 specifies that the time-frequency map is estimated using the synchrosqueezed Fourier transform, allowing up to 10 decibels of power difference between adjacent points on a ridge.
Method — Type of time-frequency map
"stft" (default) | "fsst"
Type of time-frequency map used in the estimation process, specified as either "stft" or "fsst".
"stft" — Use the short-time Fourier transform to compute a power spectrogram time-frequency map. For more details about the short-time Fourier transform, see pspectrum.
"fsst" — Use the synchrosqueezed Fourier transform to compute a time-frequency map. For more details about the synchrosqueezed Fourier transform, see fsst.
Frequency resolution bandwidth used to compute the time-frequency map, specified as a numeric scalar expressed in hertz.
PowerPenalty — Maximum difference in power between adjacent ridge points
Inf (default) | numeric scalar in dB
Maximum difference in power between adjacent ridge points, specified as a numeric scalar expressed in decibels.
Use this parameter to ensure that the ridge-extraction algorithm of rpmtrack finds the correct ridge for the corresponding order. PowerPenalty is useful when the order ridge of interest crosses other ridges or is very close in frequency to other ridges, but has a different power level.
FrequencyPenalty — Penalty in coarse ridge extraction
Penalty in coarse ridge extraction, specified as a nonnegative scalar.
Use this parameter to ensure that the ridge-extraction algorithm of rpmtrack avoids large jumps that could make the ridge estimate move to an incorrect time-frequency location. FrequencyPenalty is useful when you want to differentiate order ridges that cross or are closely spaced in frequency.
StartTime — Start time for RPM profile estimation
input signal start time (default) | scalar in seconds | duration scalar
Start time for RPM profile estimation, specified as a numeric or duration scalar.
EndTime — End time for RPM profile estimation
input signal end time (default) | scalar in seconds | duration scalar
End time for RPM profile estimation, specified as a numeric or duration scalar.
rpm — Rotational speed estimate
Rotational speed estimate, returned as a vector expressed in revolutions per minute.
If the input to rpmtrack is a timetable, then rpm is also a timetable with a single variable labeled rpm. The row times of the timetable are labeled tout and are of type duration.
tout — Time values
Time values at which the RPM profile is estimated, returned as a vector.
rpmtrack uses a two-step (coarse-fine) estimation method:
Compute a time-frequency map of x and extract a time-frequency ridge based on a specified set of points on the ridge, p, the order corresponding to that ridge, and the optional penalty parameters PowerPenalty and FrequencyPenalty. The extracted ridge provides a coarse estimate of the RPM profile.
Compute the order waveform corresponding to the extracted ridge using a Vold-Kalman filter and calculate a new time-frequency map from this waveform. The isolated order ridge from the new time-frequency map provides a fine estimate of the RPM profile.
Any name-value argument character or string input must be a constant at compile time.
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Global Constraint Catalog: Cinterval_and_count
<< 5.196. int_value_precede_chain5.198. interval_and_sum >>
[Cousin93]
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}\left(\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃},\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂},\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\right)
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}\ge 0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\left[\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗},\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right]\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\ge 0
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}>0
First consider the set of tasks of the
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection, where each task has a specific colour that may not be initially fixed. Then consider the intervals of the form
\left[k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻},k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right]
k
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
constraint forces that, for each interval
{I}_{k}
previously defined, the total number of tasks, which both are assigned to
{I}_{k}
and take their colour in
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
, does not exceed the limit
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
\left(\begin{array}{c}2,〈4〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-1\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-4,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-0\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-9,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-10\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-4,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-4\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-4\hfill \end{array}〉,5\hfill \end{array}\right)
Figure 5.197.1 shows the solution associated with the example. The constraint
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
holds since, for each interval, the number of tasks taking colour 4 does not exceed the limit 2.
Figure 5.197.1. The
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
solution to the Example slot with the use of each interval
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}>0
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}<|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
|\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}|>0
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)>1
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}>1
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
that belongs to the
k
-th interval, of size
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
, can be replaced by any other value of the same interval.
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
This constraint was originally proposed for dealing with timetabling problems. In this context the different intervals are interpreted as morning and afternoon periods of different consecutive days. Each colour corresponds to a type of course (i.e., French, mathematics). There is a restriction on the maximum number of courses of a given type each morning as well as each afternoon.
If we want to only consider intervals that correspond to the morning or to the afternoon we could extend the
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
constraint in the following way:
We introduce two extra parameters
\mathrm{𝚁𝙴𝚂𝚃}
\mathrm{𝚀𝚄𝙾𝚃𝙸𝙴𝙽𝚃}
that correspond to non-negative integers such that
\mathrm{𝚁𝙴𝚂𝚃}
\mathrm{𝚀𝚄𝙾𝚃𝙸𝙴𝙽𝚃}
We add the following condition to the arc constraint:
\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\right)\equiv \mathrm{𝚁𝙴𝚂𝚃}\left(\mathrm{mod}\mathrm{𝚀𝚄𝙾𝚃𝙸𝙴𝙽𝚃}\right)
Now, if we want to express a constraint on the morning intervals, we set
\mathrm{𝚁𝙴𝚂𝚃}
\mathrm{𝚀𝚄𝙾𝚃𝙸𝙴𝙽𝚃}
K
denote the index of the last possible interval where the tasks can be assigned:
K=⌊\frac{{max}_{i\in \left[1,|\mathrm{𝚃𝙰𝚂𝙺𝚂}|\right]}\left(\overline{\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}}\right)+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1}{\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}}⌋
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
\left(\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃},\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂},\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\right)
constraint can be expressed in term of a set of reified constraints and of
K
arithmetic constraints (i.e.,
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
constraints).
For each task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
\left(i\in \left[1,|\mathrm{𝚃𝙰𝚂𝙺𝚂}|\right]\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection we create a 0-1 variable
{B}_{i}
that will be set to 1 if and only if task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
takes a colour within the set of colours
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
{B}_{i}⇔\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}=\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}\left[1\right].\mathrm{𝚟𝚊𝚕}\vee
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}=\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}\left[2\right].\mathrm{𝚟𝚊𝚕}\vee
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}=\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}\left[|\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}|\right].\mathrm{𝚟𝚊𝚕}
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
\left(i\in \left[1,|\mathrm{𝚃𝙰𝚂𝙺𝚂}|\right]\right)
and for each interval
\left[k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻},k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right]
\left(k\in \left[0,K\right]\right)
we create a 0-1 variable
{B}_{ik}
that will be set to 1 if and only if, both task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
, and the origin of task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
is assigned within interval
\left[k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻},k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right]
{B}_{ik}⇔{B}_{i}\wedge
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\ge k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\wedge
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1
Finally, for each interval
\left[k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻},k·\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right]
\left(k\in \left[0,K\right]\right)
, we impose the sum
{B}_{1k}+{B}_{2k}+\cdots +{B}_{|\mathrm{𝚃𝙰𝚂𝙺𝚂}|k}
to not exceed the maximum allowed capacity
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
assignment dimension removed:
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
(assignment dimension corresponding to intervals is removed).
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚜𝚞𝚖}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint replaced by
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
characteristic of a constraint: coloured, automaton, automaton with array of counters.
constraint type: timetabling constraint, resource constraint, temporal constraint.
modelling: assignment dimension, interval.
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1},\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}\right)
\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}=\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
\begin{array}{c}\mathrm{𝖲𝖴𝖢𝖢}↦\hfill \\ \left[\begin{array}{c}\mathrm{𝚜𝚘𝚞𝚛𝚌𝚎},\hfill \\ \mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}-\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)\right]\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\left(0,\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}\right)
We use a bipartite graph where each class of vertices corresponds to the different tasks of the
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection. There is an arc between two tasks if their origins belong to the same interval. Finally we enforce an
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint on each set
𝒮
of successors of the different vertices of the final graph. This put a restriction on the maximum number of tasks of
𝒮
for which the colour attribute takes its value in
\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}
Parts (A) and (B) of Figure 5.197.2 respectively show the initial and final graph associated with the Example slot. Each connected component of the final graph corresponds to items that are all assigned to the same interval.
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
constraint. Let
{\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁}}_{i}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
attribute of the
{i}^{th}
item of the
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection. To each pair
\left(\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂},{\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁}}_{i}\right)
{S}_{i}
as well as the following signature constraint:
{\mathrm{𝙲𝙾𝙻𝙾𝚄𝚁}}_{i}\in \mathrm{𝙲𝙾𝙻𝙾𝚄𝚁𝚂}⇔{S}_{i}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚌𝚘𝚞𝚗𝚝}
|
2.4.2. Defining an automaton >>
As we previously said, we focus on those global constraints that can be checked by scanning once through their variables. This is for instance the case of:
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
[VanHentenryckCarillon88],
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
[Beldiceanu01],
\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}
[BourdaisGalinierPesant03],
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
[Maher02],
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
[FrischHnichKiziltanMiguelWalsh02],
\mathrm{𝚊𝚖𝚘𝚗𝚐}
[BeldiceanuContejean94],
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
Since they illustrate key points needed for characterising the set of solutions associated with a global constraint, our discussion will be based on the last five constraints for which we now recall the definition:
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\left(\mathrm{𝚟𝚊𝚛𝚜}\right)
constraint forces the sequence of 0-1 variables
\mathrm{𝚟𝚊𝚛𝚜}
to have at most one group of consecutive 1. For instance, the constraint
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\left(〈0,1,1,0〉\right)
holds since we have only one group of consecutive 1.
The lexicographic ordering constraint
\stackrel{\to }{x}{\le }_{\mathrm{lex}}\stackrel{\to }{y}
(see
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
) over two vectors of variables
\stackrel{\to }{x}=〈{x}_{0},\cdots ,{x}_{n-1}〉
\stackrel{\to }{y}=〈{y}_{0},\cdots ,{y}_{n-1}〉
n=0
{x}_{0}<{y}_{0}
{x}_{0}={y}_{0}
〈{x}_{1},\cdots ,{x}_{n-1}〉{\le }_{\mathrm{lex}}〈{y}_{1},\cdots ,{y}_{n-1}〉
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\left(\mathrm{𝚗𝚟𝚊𝚛},\mathrm{𝚟𝚊𝚛𝚜},\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}\right)
constraint restricts the number of variables of the sequence of variables
\mathrm{𝚟𝚊𝚛𝚜}
that take their values in a given set
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
, to be equal to the variable
\mathrm{𝚗𝚟𝚊𝚛}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\left(3,〈4,5,5,4,1〉,〈1,5,8〉\right)
holds since exactly 3 values of the sequence 45541 are located in the set of values
\left\{1,5,8\right\}
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
\left(\mathrm{𝚗𝚒𝚗𝚏},\mathrm{𝚟𝚊𝚛𝚜}\right)
constraint forces the number of inflexions of the sequence of variables
\mathrm{𝚟𝚊𝚛𝚜}
\mathrm{𝚗𝚒𝚗𝚏}
. An inflexion is described by one of the following patterns: a strict increase followed by a strict decrease or, conversely, a strict decrease followed by a strict increase. For instance,
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
\left(4,〈3,3,1,4,5,5,6,5,5,6,3〉\right)
holds since we can extract from the sequence 33145565563 the four subsequences 314, 565, 6556 and 563, which all follow one of these two patterns.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚟𝚊𝚛𝚜}\right)
constraint forces all pairs of distinct variables of the collection
\mathrm{𝚟𝚊𝚛𝚜}
to take distinct values. For instance
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈6,1,5,9〉\right)
holds since we have four distinct values.
Figure 2.4.1. Five checkers and their corresponding automata
Parts (A1), (B1), (C1), (D1) and (E1) of Figure 2.4.1 depict the five checkers respectively associated with
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
. Within the corresponding automata an initial state is indicated by an arc coming from no state and an accepting state is denoted graphically by a double circle. For each checker we observe the following facts:
Within the checker depicted by part (A1) of Figure 2.4.1, the values of the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
are successively compared against 0 and 1 in order to check that we have at most one group of consecutive 1. This can be translated to the automaton depicted by part (A2) of Figure 2.4.1. The automaton takes as input the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
, and triggers successively a transition for each term of this sequence. Transitions labelled by 0 and 1 are respectively associated with the conditions
\mathrm{𝚟𝚊𝚛𝚜}\left[i\right]=0
\mathrm{𝚟𝚊𝚛𝚜}\left[i\right]=1
.Transitions leading to failure are systematically skipped. This is why no transition labelled with a 1 starts from state
z
Within the checker given by part (B1) of Figure 2.4.1, the components of vectors
\stackrel{\to }{x}
\stackrel{\to }{y}
are scanned in parallel. We first skip all the components that are equal and then perform a final check. This is represented by the automaton depicted by part (B2) of Figure 2.4.1. The automaton takes as input the sequence
〈x\left[0\right],y\left[0\right]〉,\cdots ,〈x\left[n-1\right],y\left[n-1\right]〉
and triggers a transition for each term of this sequence. Unlike the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
constraint, some transitions now correspond to a condition (e.g.,
x\left[i\right]=y\left[i\right]
x\left[i\right]<y\left[i\right]
) between two variables of the
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\left(\mathrm{𝚗𝚟𝚊𝚛},
\mathrm{𝚟𝚊𝚛𝚜},
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}\right)
constraint involves a variable
\mathrm{𝚗𝚟𝚊𝚛}
whose value is computed from a given collection of variables
\mathrm{𝚟𝚊𝚛𝚜}
. The checker depicted by part (C1) of Figure 2.4.1 counts the number of variables of
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
that take their values in
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
. For this purpose it uses a counter
c
, which is possibly tested against the value of
\mathrm{𝚗𝚟𝚊𝚛}
. This convinced us to allow the use of counters in an automaton. Each counter has an initial value, which can be updated while triggering certain transitions. The accepting states of an automaton can force a variable of the constraint to be equal to a given counter. Part (C2) of Figure 2.4.1 describes the automaton corresponding to the code given in part (C1) of the same figure. The automaton uses the counter variable
c
initially set to 0 and takes as input the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
. It triggers a transition for each variable of this sequence and increments
c
when the corresponding variable takes its value in
\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
. The accepting state returns a success when the value of
c
\mathrm{𝚗𝚟𝚊𝚛}
. At this point we want to stress the following fact: it would have been possible to use an automaton that avoids the use of counters. However, this automaton would depend on the effective value of the argument
\mathrm{𝚗𝚟𝚊𝚛}
. In addition, it would require more states than the automaton of part (C2) of Figure 2.4.1. This is typically a problem if we want to have a fixed number of states in order to save memory as well as time.
\mathrm{𝚊𝚖𝚘𝚗𝚐}
constraint, the
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
\left(\mathrm{𝚗𝚒𝚗𝚏},\mathrm{𝚟𝚊𝚛𝚜}\right)
\mathrm{𝚗𝚒𝚗𝚏}
whose value is computed from a given sequence of variables
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],
\cdots ,
\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
. Therefore, the checker depicted in part (D1) of Figure 2.4.1 uses also a counter
c
for counting the number of inflexions, and compares its final value to the
\mathrm{𝚗𝚒𝚗𝚏}
argument. The automaton depicted by part (D2) of Figure 2.4.1 represents this program. It takes as input the sequence of pairs
〈\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\mathrm{𝚟𝚊𝚛𝚜}\left[1\right]〉,
〈\mathrm{𝚟𝚊𝚛𝚜}\left[1\right],\mathrm{𝚟𝚊𝚛𝚜}\left[2\right]〉
,\cdots ,
〈\mathrm{𝚟𝚊𝚛𝚜}\left[n-2\right],\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]〉
and triggers a transition for each pair. Note that a given variable may occur in more than one pair. Each transition compares the respective values of two consecutive variables of
\mathrm{𝚟𝚊𝚛𝚜}\left[0..n-1\right]
and increments the counter
c
when a new inflexion is detected. The accepting state returns a success when the value of
c
\mathrm{𝚗𝚒𝚗𝚏}
The checker associated with
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
is depicted by part (E1) of Figure 2.4.1. It first initialises an array of counters to 0. The entries of the array correspond to the potential values of the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
. In a second phase the checker computes for each potential value its number of occurrences in the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
. This is done by scanning this sequence. Finally in a third phase the checker verifies that no value is used more than once. These three phases are represented by the automaton depicted by part (E2) of Figure 2.4.1. The automaton depicted by part (E2) takes as input the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
. Its initial state initialises an array of counters to 0. Then it triggers successively a transition for each element
\mathrm{𝚟𝚊𝚛𝚜}\left[i\right]
of the input sequence and increments by 1 the entry corresponding to
\mathrm{𝚟𝚊𝚛𝚜}\left[i\right]
. The accepting state checks that all entries of the array of counters are strictly less than 2, which means that no value occurs more than once in the sequence
\mathrm{𝚟𝚊𝚛𝚜}\left[0\right],\cdots ,\mathrm{𝚟𝚊𝚛𝚜}\left[n-1\right]
Synthesising all the observations we got from these examples leads to the following remarks and definitions for a given global constraint
𝒞
For a given state, no transition can be triggered indicates that the constraint
𝒞
Since all transitions starting from a given state are mutually incompatible all automata are deterministic. Let
ℳ
denote the set of mutually incompatible conditions associated with the different transitions of an automaton.
{𝒮}_{0},\cdots ,{𝒮}_{m-1}
denote the sequence of subsets of variables of
𝒞
on which the transitions are successively triggered. All these subsets contain the same number of elements and refer to some variables of
𝒞
. Since these subsets typically depend on the constraint, we leave the computation of
{𝒮}_{0},\cdots ,{𝒮}_{m-1}
outside the automaton. To each subset
{𝒮}_{i}
of this sequence corresponds a variable
{S}_{i}
with an initial domain ranging over
\left[\mathrm{𝑚𝑖𝑛},\mathrm{𝑚𝑖𝑛}+|ℳ|-1\right]
\mathrm{𝑚𝑖𝑛}
is a fixed integer. To each integer of this range corresponds one of the mutually incompatible conditions of
ℳ
. The sequences
{S}_{0},\cdots ,{S}_{m-1}
{𝒮}_{0},\cdots ,{𝒮}_{m-1}
are respectively called the signature and the signature argument of the constraint. The constraint between
{S}_{i}
and the variables of
{𝒮}_{i}
is called the signature constraint and is denoted by
{\Psi }_{𝒞}\left({S}_{i},{𝒮}_{i}\right)
From a pragmatic point the view, the task of writing a constraint checker is naturally done by writing down an imperative program where local variables, arrays, assignment statements and control structures are used. This suggested us to consider deterministic finite automata augmented with local variables and assignment statements on these variables. Regarding control structures, we did not introduce any extra feature since the deterministic choice of which transition to trigger next seemed to be good enough.
Many global constraints involve a variable whose value is computed from a given collection of variables. This convinced us to allow the accepting state of an automaton to optionally return a result. In practice, this result corresponds to the value of a local variable of the automaton in the accepting state.
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Scalability - CC Doc
In the context of parallel programming, scalability refers to the capacity of a program to efficiently use added computing resources, i.e. CPU cores. One might naively imagine that doubling the number of CPU cores devoted to a calculation will halve the duration of the calculation, this is rarely the case. Instead we observe that the gain in performance depends on the nature of the problem, the algorithm or program used to solve it, the underlying hardware (notably memory and network), and the number of CPU cores being used. For this reason when you are planning to use a parallel program on a particular cluster we recommend that you conduct a scalability analysis where the software is tested using a fixed problem while varying the number of CPU cores according to some method (e.g. 2, 4, 8, 16, 32, 64 cores). The run time is obtained for each number of cores, and the resulting curve plotted.
Why is the scalability usually worse than what we might hope for? There are two major reasons:
Firstly, in the parallelization of the code not every operation can be done in parallel and so some percentage of the program's execution remains serial. This percentage represents an ultimate limit for the parallel efficiency of the software. Suppose the serial version of some program needs an hour to do a calculation, and six minutes of that (10%) are spent in operations which cannot be parallelized. Even with an infinite number of cores we could not have the program's duration sink below six minutes. The best we can hope for is that this "serial percentage" shrinks as we increase the size of the problem the software is working on.
Secondly, the parallelization of the program normally requires a certain amount of communication and synchronization among the parallel processes and the cost of this "parallel overhead" will increase with the number of processes working together, typically as a power of the number of cores,
{\displaystyle T_{c}\propto n^{\alpha }}
{\displaystyle \alpha >1}
. If we now suppose that the scientific part of the program's run time is divided equally among the number of cores apart from a residual serial part, so
{\displaystyle T_{s}=A+B/n}
, the total duration of the program
{\displaystyle T=T_{s}+T_{c}=A+B/n+Cn^{\alpha }}
{\displaystyle A}
{\displaystyle B}
{\displaystyle C}
positive real numbers whose value depends on the particular cluster, program and test problem) will ultimately be dominated by this final parallel overhead factor as
{\displaystyle n\to \infty }
{\displaystyle A}
{\displaystyle B}
are much larger than
{\displaystyle C}
, when we plot the curve of the run time versus the number of CPU cores we will obtain something that looks like the accompanying figure.
The most important point to note about this curve is that while for smaller numbers of cores the run time falls, at a certain number of cores a minimum is reached (for
{\displaystyle n\approx 22}
), and after that the program duration starts to increase as we add more processes: too many cooks spoil the broth, according to the proverb. When you are using a parallel program, it's crucial to carry out such a scalability analysis in order to know, for the nature and size of problem you're working on and the cluster you're using, what is the optimal choice of the number of CPU cores: 4, 128, 1024, or some other figure?
It is up to you to choose a test problem for the scalability analysis. You want a problem that is relatively small so that these tests can be carried out quickly, but not so small as to be completely unrepresentative of a production run. A test problem that requires 30 to 60 minutes to finish on one or two cores is probably a good choice. One which runs in under ten minutes is almost certainly too short to be of value. In certain contexts, such as an analysis of the program's behaviour under weak scaling (see below), you also want to have a test problem whose size can be easily increased, ideally in a fairly continuous manner.
There is one class of problems for which the factor
{\displaystyle C}
is for all practical purposes zero, so that there is no parallel overhead to speak of. Such problems are called "embarrassingly parallel". A good example might be running an analysis on 500 different files, in which the analysis of an individual file is independent of any others and simply generates a single number that can be stored in an array. In this case there is no need to synchronize the operations of the various processes analyzing the files nor will any communication among these processes be necessary, so that we can achieve perfect scaling out to any number of processes; the only limitation is the number of files we have.
In the next two sections, we will consider two different forms of scaling, strong and weak. When the term scaling is used without any qualification "strong scaling" is normally what is meant. However, weak scaling may be more important depending on how you intend to use the multiple cores.
Do you wish to perform the same size of simulations as before, but more quickly? Then strong scaling applies.
Or do you wish to simulate larger or more detailed models, and are willing to wait just as long as before, but for better results? Then weak scaling applies.
In this case the problem to be used for the scalability analysis is fixed while the number of CPU cores increases. Ideally, we would expect the scaling to be linear, i.e. the decrease in the program's run time compared to some reference value is the reciprocal of the number of cores added compared to that used for the reference calculation. As a concrete example of doing an analysis of the strong scalability of a program, imagine a parallel program which we have tested on the same cluster using the same input parameters, obtaining the results in the table below:
The efficiency in this table is calculated by dividing the reference run time at two cores by the run time at
{\displaystyle n}
cores, then dividing the result by
{\displaystyle n/2}
and finally multiplying by a hundred to get a percentage. This percentage measures the degree to which the parallel performance scales linearly, i.e. doubling the number of cores halves the run time, which corresponds to an efficiency of 100%.
In the table above, we notice that when going from 2 to 4 cores, we achieve greater than 100% efficiency. This is called "superlinear scaling". It occurs rarely, but when it does it is usually due to the presence of a CPU cache which functions more effectively as each CPU core has less to do.[1]
The test with 128 cores actually took longer than with 64 cores, 238 seconds versus 197 seconds. The 128-core efficiency is therefore terrible, only 18%.
An efficiency of 75% or more is good, so we would advise the user of this software with input like this test case to submit jobs which use 16 CPU cores. The run time does continue to decrease up to 64 cores, but the improvement in run time beyond 16 cores would not be a good use of resources.
The number and range of data points that you obtain for your scalability analysis is up to you. We recommend at least five or six values, although if you find the program runs more slowly with added cores, you should obviously not pursue the analysis beyond that number of cores.
In weak scaling, the problem size is increased in proportion to the increase in the number of CPU cores so that in an ideal situation of linear scaling the program's run time will always remain the same. The definition of "problem size" depends on the nature of the problem: in a molecular simulation it might be the number of atoms, in a fluid dynamics simulation it might be the number of cells or nodes in the mesh. We can create a table of run times as in the preceding section, increasing the problem size by an amount equal to the increase in the number of cores:
12 12,000 3107 99.0
128 128,000 3966 77.6
The formula for the efficiency here is quite simple, just the reference run time divided by the run time at
{\displaystyle n}
cores then multiplied by a hundred to obtain a percentage. Once again, the goal is to achieve an efficiency of at least 75%. As is often the case, efficiency remains high up to larger core counts than with strong scaling.
Weak scaling tends to be especially pertinent for applications that are memory-bound. If the parallel program has been designed to privilege communications between nearest neighbours then the weak scaling is usually good. An application which performs a lot of nonlocal communication (e.g. a fast Fourier transform[2]) may exhibit poor performance in a weak scalability analysis.
↑ Wikipedia, "Super-linear speedup": https://en.wikipedia.org/wiki/Speedup#Super-linear_speedup
↑ Wikipedia, "Fast Fourier transform: https://en.wikipedia.org/wiki/Fast_Fourier_transform
Retrieved from "https://docs.alliancecan.ca/mediawiki/index.php?title=Scalability&oldid=67674"
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Global Constraint Catalog: Cpolyomino
<< 5.323. place_in_pyramid5.325. power >>
\mathrm{𝚙𝚘𝚕𝚢𝚘𝚖𝚒𝚗𝚘}\left(\mathrm{𝙲𝙴𝙻𝙻𝚂}\right)
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\begin{array}{c}\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\hfill \\ \mathrm{𝚛𝚒𝚐𝚑𝚝}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚕𝚎𝚏𝚝}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚞𝚙}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚍𝚘𝚠𝚗}-\mathrm{𝚍𝚟𝚊𝚛}\hfill \end{array}\right)
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙲𝙴𝙻𝙻𝚂}|
|\mathrm{𝙲𝙴𝙻𝙻𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙲𝙴𝙻𝙻𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚛𝚒𝚐𝚑𝚝},\mathrm{𝚕𝚎𝚏𝚝},\mathrm{𝚞𝚙},\mathrm{𝚍𝚘𝚠𝚗}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙲𝙴𝙻𝙻𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚛𝚒𝚐𝚑𝚝}\ge 0
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚛𝚒𝚐𝚑𝚝}\le |\mathrm{𝙲𝙴𝙻𝙻𝚂}|
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚕𝚎𝚏𝚝}\ge 0
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚕𝚎𝚏𝚝}\le |\mathrm{𝙲𝙴𝙻𝙻𝚂}|
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚞𝚙}\ge 0
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚞𝚙}\le |\mathrm{𝙲𝙴𝙻𝙻𝚂}|
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚍𝚘𝚠𝚗}\ge 0
\mathrm{𝙲𝙴𝙻𝙻𝚂}.\mathrm{𝚍𝚘𝚠𝚗}\le |\mathrm{𝙲𝙴𝙻𝙻𝚂}|
Enforce all cells of the collection
\mathrm{𝙲𝙴𝙻𝙻𝚂}
to be connected and to form a single block. Each cell is defined by the following attributes:
\mathrm{𝚒𝚗𝚍𝚎𝚡}
attribute of the cell, which is an integer between 1 and the total number of cells, is unique for each cell.
\mathrm{𝚛𝚒𝚐𝚑𝚝}
attribute that is the index of the cell located immediately to the right of that cell (or 0 if no such cell exists).
\mathrm{𝚕𝚎𝚏𝚝}
attribute that is the index of the cell located immediately to the left of that cell (or 0 if no such cell exists).
\mathrm{𝚞𝚙}
attribute that is the index of the cell located immediately on top of that cell (or 0 if no such cell exists).
\mathrm{𝚍𝚘𝚠𝚗}
attribute that is the index of the cell located immediately above that cell (or 0 if no such cell exists).
This corresponds to a polyomino [Golomb65].
\left(\begin{array}{c}〈\begin{array}{ccccc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚛𝚒𝚐𝚑𝚝}-0\hfill & \mathrm{𝚕𝚎𝚏𝚝}-0\hfill & \mathrm{𝚞𝚙}-2\hfill & \mathrm{𝚍𝚘𝚠𝚗}-0,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚛𝚒𝚐𝚑𝚝}-3\hfill & \mathrm{𝚕𝚎𝚏𝚝}-0\hfill & \mathrm{𝚞𝚙}-0\hfill & \mathrm{𝚍𝚘𝚠𝚗}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚛𝚒𝚐𝚑𝚝}-0\hfill & \mathrm{𝚕𝚎𝚏𝚝}-2\hfill & \mathrm{𝚞𝚙}-4\hfill & \mathrm{𝚍𝚘𝚠𝚗}-0,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚛𝚒𝚐𝚑𝚝}-5\hfill & \mathrm{𝚕𝚎𝚏𝚝}-0\hfill & \mathrm{𝚞𝚙}-0\hfill & \mathrm{𝚍𝚘𝚠𝚗}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚛𝚒𝚐𝚑𝚝}-0\hfill & \mathrm{𝚕𝚎𝚏𝚝}-4\hfill & \mathrm{𝚞𝚙}-0\hfill & \mathrm{𝚍𝚘𝚠𝚗}-0\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚙𝚘𝚕𝚢𝚘𝚖𝚒𝚗𝚘}
constraint holds since all the cells corresponding to the items of the
\mathrm{𝙲𝙴𝙻𝙻𝚂}
collection form one single group of connected cells: the
{i}^{th}
i\in \left[1,4\right]
) cell is connected to the
{\left(i+1\right)}^{th}
cell. Figure 5.324.1 shows the corresponding polyomino.
Figure 5.324.1. Polyomino corresponding to the Example slot where each cell contains the index of the corresponding item within the
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\left(\mathrm{𝚛𝚒𝚐𝚑𝚝},\mathrm{𝚕𝚎𝚏𝚝}\right)
\left(\mathrm{𝚞𝚙}\right)
\left(\mathrm{𝚍𝚘𝚠𝚗}\right)
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\left(\mathrm{𝚛𝚒𝚐𝚑𝚝}\right)
\left(\mathrm{𝚕𝚎𝚏𝚝}\right)
\left(\mathrm{𝚞𝚙},\mathrm{𝚍𝚘𝚠𝚗}\right)
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\left(\mathrm{𝚞𝚙},\mathrm{𝚕𝚎𝚏𝚝},\mathrm{𝚍𝚘𝚠𝚗},\mathrm{𝚛𝚒𝚐𝚑𝚝}\right)
Enumeration of polyominoes.
combinatorial object: pentomino.
final graph structure: strongly connected component.
geometry: geometrical constraint.
puzzles: pentomino.
\mathrm{𝙲𝙴𝙻𝙻𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(\ne \right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1},\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}\right)
\bigvee \left(\begin{array}{c}\bigwedge \left(\begin{array}{c}\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚛𝚒𝚐𝚑𝚝}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\hfill \\ \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚕𝚎𝚏𝚝}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\hfill \end{array}\right),\hfill \\ \bigwedge \left(\begin{array}{c}\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚕𝚎𝚏𝚝}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\hfill \\ \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚛𝚒𝚐𝚑𝚝}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\hfill \end{array}\right),\hfill \\ \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚞𝚙}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\wedge \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚍𝚘𝚠𝚗}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\hfill \\ \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚍𝚘𝚠𝚗}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\wedge \mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{2}.\mathrm{𝚞𝚙}=\mathrm{𝚌𝚎𝚕𝚕𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\hfill \end{array}\right)
•
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
=|\mathrm{𝙲𝙴𝙻𝙻𝚂}|
•
\mathrm{𝐍𝐂𝐂}
=1
The graph constraint models the fact that all the cells are connected. We use the
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}\left(\ne \right)
arc generator in order to only consider connections between two distinct cells. The first graph property
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}=|\mathrm{𝙲𝙴𝙻𝙻𝚂}|
avoid the case isolated cells, while the second graph property
\mathrm{𝐍𝐂𝐂}=1
enforces to have a single group of connected cells.
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
graph property the vertices of the final graph are stressed in bold. Since we also use the
\mathrm{𝐍𝐂𝐂}
graph property we show the unique connected component of the final graph. An arc between two vertices indicates that two cells are directly connected.
\mathrm{𝚙𝚘𝚕𝚢𝚘𝚖𝚒𝚗𝚘}
From the graph property
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}=|\mathrm{𝙲𝙴𝙻𝙻𝚂}|
and from the restriction
|\mathrm{𝙲𝙴𝙻𝙻𝚂}|\ge 1
we have that the final graph is not empty. Therefore it contains at least one connected component. So we can rewrite
\mathrm{𝐍𝐂𝐂}=1
\mathrm{𝐍𝐂𝐂}\le 1
\underline{\overline{\mathrm{𝐍𝐂𝐂}}}
\underline{\mathrm{𝐍𝐂𝐂}}
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Global Constraint Catalog: Ccrossing
<< 5.94. covers_sboxes5.96. cumulative >>
Inspired by [CormenLeisersonRivest90].
\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}\left(\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂},\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}\right)
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚡}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚘𝚢}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚎𝚡}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚎𝚢}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}\ge 0
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}\le \left(|\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}|*|\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}|-|\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}|\right)/2
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂},\left[\mathrm{𝚘𝚡},\mathrm{𝚘𝚢},\mathrm{𝚎𝚡},\mathrm{𝚎𝚢}\right]\right)
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}
is the number of line segments intersections between the line segments defined by the
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
collection. Each line segment is defined by the coordinates
\left(\mathrm{𝚘𝚡},\mathrm{𝚘𝚢}\right)
\left(\mathrm{𝚎𝚡},\mathrm{𝚎𝚢}\right)
of its two extremities.
\left(\begin{array}{c}3,〈\begin{array}{cccc}\mathrm{𝚘𝚡}-1\hfill & \mathrm{𝚘𝚢}-4\hfill & \mathrm{𝚎𝚡}-9\hfill & \mathrm{𝚎𝚢}-2,\hfill \\ \mathrm{𝚘𝚡}-1\hfill & \mathrm{𝚘𝚢}-1\hfill & \mathrm{𝚎𝚡}-3\hfill & \mathrm{𝚎𝚢}-5,\hfill \\ \mathrm{𝚘𝚡}-3\hfill & \mathrm{𝚘𝚢}-2\hfill & \mathrm{𝚎𝚡}-7\hfill & \mathrm{𝚎𝚢}-4,\hfill \\ \mathrm{𝚘𝚡}-9\hfill & \mathrm{𝚘𝚢}-1\hfill & \mathrm{𝚎𝚡}-9\hfill & \mathrm{𝚎𝚢}-4\hfill \end{array}〉\hfill \end{array}\right)
Figure 5.95.1 provides a picture of the example with the corresponding four line segments of the
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
collection. The
\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}
is set to 3, which is actually the number of line segments intersections.
Figure 5.95.1. Illustration of the Example slot: intersection, in red, between the four line segments
{S}_{1}
{S}_{2}
{S}_{3}
{S}_{4}
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}=3
|\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}|>1
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\left(\mathrm{𝚘𝚡},\mathrm{𝚘𝚢}\right)
\left(\mathrm{𝚎𝚡},\mathrm{𝚎𝚢}\right)
\mathrm{𝚘𝚡}
\mathrm{𝚎𝚡}
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝚘𝚢}
\mathrm{𝚎𝚢}
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝚐𝚛𝚊𝚙𝚑}_\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚕𝚊𝚢𝚎𝚛}_\mathrm{𝚎𝚍𝚐𝚎}_\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}
final graph structure: acyclic, no loop.
\mathrm{𝚂𝙴𝙶𝙼𝙴𝙽𝚃𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(<\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚜\mathtt{1},𝚜\mathtt{2}\right)
•\mathrm{𝚖𝚊𝚡}\left(𝚜\mathtt{1}.\mathrm{𝚘𝚡},𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)\ge \mathrm{𝚖𝚒𝚗}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚡},𝚜\mathtt{2}.\mathrm{𝚎𝚡}\right)
•\mathrm{𝚖𝚊𝚡}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚡},𝚜\mathtt{2}.\mathrm{𝚎𝚡}\right)\ge \mathrm{𝚖𝚒𝚗}\left(𝚜\mathtt{1}.\mathrm{𝚘𝚡},𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)
•\mathrm{𝚖𝚊𝚡}\left(𝚜\mathtt{1}.\mathrm{𝚘𝚢},𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)\ge \mathrm{𝚖𝚒𝚗}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢},𝚜\mathtt{2}.\mathrm{𝚎𝚢}\right)
•\mathrm{𝚖𝚊𝚡}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢},𝚜\mathtt{2}.\mathrm{𝚎𝚢}\right)\ge \mathrm{𝚖𝚒𝚗}\left(𝚜\mathtt{1}.\mathrm{𝚘𝚢},𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)
•\bigvee \left(\begin{array}{c}\begin{array}{c}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚡}-𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)*\left(𝚜\mathtt{1}.\mathrm{𝚎𝚢}-𝚜\mathtt{1}.\mathrm{𝚘𝚢}\right)-\hfill \\ \left(𝚜\mathtt{1}.\mathrm{𝚎𝚡}-𝚜\mathtt{1}.\mathrm{𝚘𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢}-𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)\hfill \end{array}=0,\hfill \\ \begin{array}{c}\left(𝚜\mathtt{2}.\mathrm{𝚎𝚡}-𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢}-𝚜\mathtt{1}.\mathrm{𝚘𝚢}\right)-\hfill \\ \left(𝚜\mathtt{2}.\mathrm{𝚘𝚡}-𝚜\mathtt{1}.\mathrm{𝚘𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚎𝚢}-𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)\hfill \end{array}=0,\hfill \\ \begin{array}{c}\mathrm{𝚜𝚒𝚐𝚗}\left(\begin{array}{c}\begin{array}{c}\left(𝚜\mathtt{2}.\mathrm{𝚘𝚡}-𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)*\left(𝚜\mathtt{1}.\mathrm{𝚎𝚢}-𝚜\mathtt{1}.\mathrm{𝚘𝚢}\right)-\hfill \\ \left(𝚜\mathtt{1}.\mathrm{𝚎𝚡}-𝚜\mathtt{1}.\mathrm{𝚘𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢}-𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)\hfill \end{array}\hfill \end{array}\right)\ne \hfill \\ \mathrm{𝚜𝚒𝚐𝚗}\left(\begin{array}{c}\begin{array}{c}\left(𝚜\mathtt{2}.\mathrm{𝚎𝚡}-𝚜\mathtt{1}.\mathrm{𝚎𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚘𝚢}-𝚜\mathtt{1}.\mathrm{𝚘𝚢}\right)-\hfill \\ \left(𝚜\mathtt{2}.\mathrm{𝚘𝚡}-𝚜\mathtt{1}.\mathrm{𝚘𝚡}\right)*\left(𝚜\mathtt{2}.\mathrm{𝚎𝚢}-𝚜\mathtt{1}.\mathrm{𝚎𝚢}\right)\hfill \end{array}\hfill \end{array}\right)\hfill \end{array}\hfill \end{array}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=\mathrm{𝙽𝙲𝚁𝙾𝚂𝚂}
•
\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}
•
\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}
Each line segment is described by the
𝚡
𝚢
coordinates of its two extremities. In the arc generator we use the restriction
<
in order to generate a single arc for each pair of segments. This is required, since otherwise we would count more than once a given line segments intersection.
\mathrm{𝐍𝐀𝐑𝐂}
graph property, the arcs of the final graph are stressed in bold. An arc constraint expresses the fact the two line segments intersect. It is taken from [CormenLeisersonRivest90]. Each arc of the final graph corresponds to a line segments intersection.
\mathrm{𝚌𝚛𝚘𝚜𝚜𝚒𝚗𝚐}
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Global Constraint Catalog: Klimited_discrepancy_search
<< 3.7.134. Lexicographic order3.7.136. Linear programming >>
\mathrm{𝚍𝚒𝚜𝚌𝚛𝚎𝚙𝚊𝚗𝚌𝚢}
A constraint for simulating limited discrepancy search [GinsbergHarvey95]. Limited discrepancy search is useful for problems for which there is a successor ordering heuristics that usually leads directly to a solution. It consists of systematically searching all paths that differ from the heuristic path in at most a very small number of discrepancies. Figure 3.7.37 illustrates the successive search steps (B), (C), (D), (E) and (F) on the search tree depicted by part (A). We successively explore the subtree of (A) corresponding to a discrepancy of 0, 1, 2, 3 and 4. The number on each leave indicates the total number of discrepancies to reach a leave.
Figure 3.7.37. Illustration of limited discrepancy search
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Category of relations — Wikipedia Republished // WIKI 2
Category of Relations Rel.
Rel's opposite Relop.
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.
A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A × B.
The composition of two relations R: A → B and S: B → C is given by
(a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b, c) ∈ S.[1]
Rel has also been called the "category of correspondences of sets".[2]
Aristotle on Qualities, Relations, and Knowledge (Categories, c. 8) - Philosophy Core Concepts
(Abstract Algebra 1) Partitions and Equivalence Relations
The Hawthorne Effect | Better Business Communication through Human Resources and Relations
The category Rel has the category of sets Set as a (wide) subcategory, where the arrow f : X → Y in Set corresponds to the relation F ⊆ X × Y defined by (x, y) ∈ F ⇔ f(x) = y.[3][4]
A morphism in Rel is a relation, and the corresponding morphism in the opposite category to Rel has arrows reversed, so it is the converse relation. Thus Rel contains its opposite and is self-dual.[5]
The involution represented by taking the converse relation provides the dagger to make Rel a dagger category.
The category has two functors into itself given by the hom functor: A binary relation R ⊆ A × B and its transpose RT ⊆ B × A may be composed either as R RT or as RT R. The first composition results in a homogeneous relation on A and the second is on B. Since the images of these hom functors are in Rel itself, in this case hom is an internal hom functor. With its internal hom functor, Rel is a closed category, and furthermore a dagger compact category.
The category Rel can be obtained from the category Set as the Kleisli category for the monad whose functor corresponds to power set, interpreted as a covariant functor.
Perhaps a bit surprising at first sight is the fact that product in Rel is given by the disjoint union[5]: 181 (rather than the cartesian product as it is in Set), and so is the coproduct.
Rel is monoidal closed, with both the monoidal product A ⊗ B and the internal hom A ⇒ B given by cartesian product of sets.
The category Rel was the prototype for the algebraic structure called an allegory by Peter J. Freyd and Andre Scedrov in 1990.[6] Starting with a regular category and a functor F: A → B, they note properties of the induced functor Rel(A,B) → Rel(FA, FB). For instance, it preserves composition, conversion, and intersection. Such properties are then used to provide axioms for an allegory.
David Rydeheard and Rod Burstall consider Rel to have objects that are homogeneous relations. For example, A is a set and R ⊆ A × A is a binary relation on A. The morphisms of this category are functions between sets that preserve a relation: Say S ⊆ B × B is a second relation and f: A → B is a function such that
{\displaystyle xRy\implies f(x)Sf(y),}
then f is a morphism.[7]
The same idea is advanced by Adamek, Herrlich and Strecker, where they designate the objects (A, R) and (B, S), set and relation.[8]
^ Mac Lane, S. (1988). Categories for the Working Mathematician (1st ed.). New York: Springer-Verlag. p. 26. ISBN 0-387-90035-7.
^ Pareigis, Bodo (1970). Categories and Functors. Pure and Applied Mathematics. Vol. 39. Academic Press. p. 6. ISBN 978-0-12-545150-5.
^ This category is called SetRel by Rydeheard and Burstall.
^ George Bergman (1998), An Invitation to General Algebra and Universal Constructions, §7.2 RelSet, Henry Helson Publisher, Berkeley. ISBN 0-9655211-4-1.
^ a b Michael Barr & Charles Wells (1998) Category Theory for Computing Science Archived 2016-03-04 at the Wayback Machine, page 83, from McGill University
^ Peter J. Freyd & Andre Scedrov (1990) Categories, Allegories, pages 79, 196, North Holland ISBN 0-444-70368-3
^ David Rydeheard & Rod Burstall (1988) Computational Category Theory, page 41, Prentice-Hall ISBN 978-0131627369
^ Juri Adamek, Horst Herrlich, and George E. Strecker (2004) [1990] Abstract and Concrete Categories, section 3.3, example 2(d) page 22, from Research group KatMAT at University of Bremen
Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 115. ISBN 978-0-521-44179-7.
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Global Constraint Catalog: Kdynamic_programming
<< 3.7.92. Duplicated variables3.7.94. Empty intersection >>
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}
\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}
A constraint for which a filtering algorithm uses dynamic programming. Note that dynamic programming was also used by M. A. Trick within the context of linear constraints [Trick03].
|
Global Constraint Catalog: Call_min_dist
<< 5.10. all_incomparable5.12. alldifferent >>
[Regin97]
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}\left(\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚍𝚒𝚜𝚝𝚊𝚗𝚌𝚎}
\mathrm{𝚒𝚗𝚝𝚎𝚛}_\mathrm{𝚍𝚒𝚜𝚝𝚊𝚗𝚌𝚎}
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}>0
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|<2\vee \mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}<
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
Enforce for each pair
\left({\mathrm{𝚟𝚊𝚛}}_{i},{\mathrm{𝚟𝚊𝚛}}_{j}\right)
of distinct variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
|{\mathrm{𝚟𝚊𝚛}}_{i}-{\mathrm{𝚟𝚊𝚛}}_{j}|\ge \mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}
\left(2,〈5,1,9,3〉\right)
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
constraint holds since the following expressions
|5-1|
|5-9|
|5-3|
|1-9|
|1-3|
|9-3|
are all greater than or equal to the first argument
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}=2
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
Figure 5.11.1 gives all solutions to the following non ground instance of the
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
{V}_{1}\in \left[0,5\right]
{V}_{2}\in \left[3,9\right]
{V}_{3}\in \left[5,7\right]
{V}_{4}\in \left[2,10\right]
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\left(\mathbf{3},〈{V}_{1},{V}_{2},{V}_{3},{V}_{4}〉\right)
Figure 5.11.1. All solutions corresponding to the non ground example of the
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}>1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}
\ge 1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
Two distinct values of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
can be swapped.
\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
constraint was initially created for handling frequency allocation problems. In [ArtiouchineBaptiste05] it is used for scheduling tasks that all have the same fixed duration in the context of air traffic management in the terminal radar control area of airports.
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
constraint can be modelled as a set of tasks that should not overlap. For each variable
\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection we create a task
t
\mathrm{𝚟𝚊𝚛}
\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}
respectively correspond to the origin and the duration of
t
Some solvers use in a pre-processing phase, while stating constraints of the form
|{X}_{i}-{X}_{j}|\ge {D}_{ij}
{X}_{i}
{X}_{j}
are domain variables and
{D}_{ij}
is a constant), an algorithm for automatically extracting large cliques [BronKerbosch73] from such inequalities in order to state
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
K. Artiouchine and P. Baptiste came up with a cubic time complexity algorithm achieving bound-consistency in [ArtiouchineBaptiste05], [ArtiouchineBaptiste07] based on the adaptation of a feasibility test algorithm from M.R. Garey et al. [GareyJohnsonSimonsTarjan81]. Later on, C.-G. Quimper et al., proposed a quadratic algorithm achieving the same level of consistency in [QuimperOrtizPesant06].
n
Solutions 8 24 120 720 5040 40320 362880
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
0..n
n
Total 8 24 120 720 5040 40320 362880
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
0..n
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
, of same length, replaced by orthotope),
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
, of same length, replaced by
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
\mathrm{𝚖𝚞𝚕𝚝𝚒}_\mathrm{𝚒𝚗𝚝𝚎𝚛}_\mathrm{𝚍𝚒𝚜𝚝𝚊𝚗𝚌𝚎}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
parameter introduced to specify capacity
\ge
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚍𝚒𝚜𝚝𝚊𝚗𝚌𝚎}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
application area: frequency allocation problem, air traffic management.
characteristic of a constraint: sort based reformulation.
constraint type: value constraint, decomposition, scheduling constraint.
final graph structure: acyclic.
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}\left(\mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\left(𝙽,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
𝙽\ge |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(<\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚊𝚋𝚜}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}-\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right)\ge \mathrm{𝙼𝙸𝙽𝙳𝙸𝚂𝚃}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|*\left(|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1\right)/2
•
\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}
•
\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}
We generate a clique with a minimum distance constraint between each pair of distinct vertices and state that the number of arcs of the final graph should be equal to the number of arcs of the initial graph.
Parts (A) and (B) of Figure 5.11.2 respectively show the initial and final graph associated with the Example slot. The
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
constraint holds since all the arcs of the initial graph belong to the final graph: all the minimum distance constraints are satisfied.
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
|
Global Constraint Catalog: Calldifferent_between_sets
<< 5.12. alldifferent5.14. alldifferent_consecutive_values >>
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚗𝚞𝚕𝚕}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚜𝚎𝚌𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚘𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚘𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
Enforce all sets of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(〈\mathrm{𝚟𝚊𝚛}-\left\{3,5\right\},\mathrm{𝚟𝚊𝚛}-\varnothing ,\mathrm{𝚟𝚊𝚛}-\left\{3\right\},\mathrm{𝚟𝚊𝚛}-\left\{3,5,7\right\}〉\right)
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
constraint holds since all the sets
\left\{3,5\right\}
\varnothing
\left\{3\right\}
\left\{3,5,7\right\}
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>2
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
This constraint was available in some configuration library offered by Ilog.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
is proposed by C.-G. Quimper and T. Walsh in [QuimperWalsh05] and a longer version is available in [QuimperWalshReport05] and in [QuimperWalsh06].
\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚜𝚎𝚝}_\mathrm{𝚝𝚘}_\mathrm{𝚋𝚘𝚘𝚕𝚎𝚊𝚗𝚜}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚜𝚎𝚝}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚎𝚚}_\mathrm{𝚜𝚎𝚝}
final graph structure: one_succ.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚎𝚚}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
\le 1
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
We generate a clique with binary set equalities constraints between each pair of vertices (including a vertex and itself) and state that the size of the largest strongly connected component should not exceed 1.
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
holds since all the strongly connected components have at most one vertex.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚜𝚎𝚝𝚜}
|
<< 3.7.120.1. Dual strategy for rectangle placement pr
This is a four-phase search procedure that can be used even when the slack is not equal to zero. We first gradually restrict all the
x
-coordinates and then, in the second phase, all
y
-coordinates without fixing them immediately. Then in the third phase we fix all the
x
-coordinates by trying each value (or by making a binary search). Finally in the last phase we fix all the
y
-coordinates as in the third phase. The intuitions behind this heuristics are:
To restrict the
x
-coordinate of each rectangle
R
in order to just create some compulsory part for
R
x
axis. The hope is that it will trigger the filtering algorithm associated with the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint involved by the non-overlapping constraint, even though the starts of the rectangles on the
x
axis are not yet completely fixed.
Again, as in the previous heuristics, to decrease the combinatorial aspect of the problem by first focussing on all
x
Restricting gradually the
x
-coordinates in phase one is done by partitioning the domain of the
x
R
into intervals whose sizes induce a compulsory part on the
x
axis for rectangle
R
. To achieve this, the size of an interval has to be less than or equal to the size of rectangle
R
x
axis. Picking the best fraction of the size of a rectangle on the
x
axis depends on the problem as well as on the filtering algorithms behind the scene. Within the context of the smallest rectangle area problem [SimonisSullivan08] and of the SICStus implementation of
\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}\mathtt{2}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
H. Simonis and B. O'Sullivan have shown empirically that the best fraction was located within interval
\left[0.2,0.3\right]
. Restricting the
y
-coordinates in phase two can be done in a way similar to restricting the
x
-coordinates in phase one.
|
Analyze computational results using Python, Pandas and LaTeX
Every researcher may use its own set of tools for analyzing the data of their experiments and computing statistics. Writing scripts to automatically compute and include tables in your article can make you save a lot of time. A great tool used by scientists is the R project. The latter is free software and a programming language designed for statistical analysis and graphics. I strongly recommend having a look at this tutorial if you want to learn how to use it. In this post, we will investigate another great programming language: Python. An advantage of Python over R is that it is extremely popular and even more general-purpose. Hence, it offers a huge number of libraries for almost every use case. In particular, we will use a well-known data analysis and manipulation tool: Pandas. This library can perform a lot of stuff, such as loading/saving data in various formats (CSV, Excel...), filtering, and computing numerous statistics. If you are not familiar with Python, there exist plenty of tutorials online, even interactive ones.
Suppose we implemented three algorithms to solve a minimization problem. We run experiments on three instances. Let's name our algorithms alice, bob, and carol, and our instances inst1, inst2, and inst3. For the sake of simplicity, we assume that our algorithms are run only once on each instance. We want to compare the algorithms in a beautiful LaTeX article. In particular, we would like to obtain for each algorithm the average and the maximum relative gaps with respect to the best-known solution, as well as the average computational times. To simplify, we stored the raw results in a single CSV file (data.csv) where each row contains the characteristics of a single run: instance, algorithm, obj (objective value), and time (computational time).
instance,algorithm,obj,time
inst1,alice,7,16
inst1,bob,16,19
inst1,carol,6,14
inst2,bob,3,18
inst2,carol,15,17
Now, the data analysis can start. We create a python script named analyze.py and load the CSV file.
# We need to import Pandas to use it
# Load the contents of "data.csv" in a DataFrame object
# Display the DataFrame object
The last instruction should display the following:
instance algorithm obj time
0 inst1 alice 7 16
1 inst1 bob 16 19
2 inst1 carol 6 14
4 inst2 bob 3 18
5 inst2 carol 15 17
The contents of data.csv are stored in an object df of type DataFrame. This object offers powerful features to transform existing data or create new data. I recall that we want to compute the minimum/average relative gaps and the average computational times.
Computing new data
Best-known solution
First of all, let's compute the best-known solution for each instance, required to compute the relative gaps. To do so, we group rows by instance and take the minimum obj value in each group, as follows:
# Obtain a Series object that contains the best-known solutions
bks = df.groupby("instance")["obj"].transform(min)
Don't panic, let's decompose this instruction. First, df.groupby("instance") creates a GroupBy object. From the latter, we tell Pandas that we need only the obj column, so we use the [...] operator. Finally, the .transform(...) method applies a function (min) on each group. It returns a Series filled with transformed values but the original shape is preserved. As a consequence, we can directly use the bks (best-known solution) variable as a new column for df, using the .assign(...) method.
# Add the "bks" column
df = df.assign(bks=bks)
# Display the updated DataFrame
We can observe that the column has been successfully added:
instance algorithm obj time bks
0 inst1 alice 7 16 6
1 inst1 bob 16 19 6
2 inst1 carol 6 14 6
4 inst2 bob 3 18 2
5 inst2 carol 15 17 2
Legit question: why did we add the bks column? Well, it is not mandatory but doing so makes it easy to compute our relative gap given by the formula:
\frac{\text{obj} - \text{bks}}{\text{obj}}
. Since we are familiar with .assign(...), let's do it in one line:
# Add the "gap" column
df = df.assign(gap=(df["obj"] - df["bks"]) / (df["obj"]))
If obj can have a zero value, you need to adapt the formula (e.g. (df["obj"] - df["bks"]) / (df["obj"] + 1)). In our case, the result is:
instance algorithm obj time bks gap
0 inst1 alice 7 16 6 0.142857
1 inst1 bob 16 19 6 0.625000
2 inst1 carol 6 14 6 0.000000
4 inst2 bob 3 18 2 0.333333
5 inst2 carol 15 17 2 0.866667
We are ready to compute the average/maximum relative gap and the average computational time for each algorithm. First, group the rows by algorithm in a temporary variable df_g.
# Group by algorithm
df_g = df.groupby("algorithm")
Remember that the df_g is an object of type GroupBy. The GroupBy class offers handy methods to compute the most common statistics, in particular, .min(), .max(), and .mean(). We create a new DataFrame summarizing the results:
# Compute the average and the maximum gap, plus the average time
df_summary = pd.DataFrame(
"avg_gap": df_g["gap"].mean().mul(100),
"max_gap": df_g["gap"].max().mul(100),
"avg_time": df_g["time"].mean(),
# Display the summary
print(df_summary)
Let's decompose the previous code a little bit. We create a DataFrame from a dictionary where each element is a Series, thus df_summary will contain three columns. The df_g["gap"].mean() instruction tells that we want to operate on the gap column only, then compute the average of gap in each group. We call .mul(100) to multiply values by 100, since they are percentages. Similarly, it is easy to understand the meaning of the other columns. See the contents of df_summary:
avg_gap max_gap avg_time
alice 4.761905 14.285714 14.000000
bob 47.630719 62.500000 15.666667
carol 46.432749 86.666667 14.666667
Looks pretty, isn't it?
I know two ways of importing the table into a LaTeX article. One way is to call the .to_latex() method. This converts the table into a valid LaTeX code. The other way consists in saving the table to a CSV file (.to_csv()) and letting LaTeX processing it. Both approaches have their pros and their cons, depending on your case. If you need to reuse the same data for several outputs (e.g. a table and a figure), I recommend the second approach. Otherwise, the first approach is fine for common situations.
The LaTeX table we want
Let's start using the .to_latex() method.
# Display the DataFrame in LaTeX format
print(df_summary.to_latex())
# Export the DataFrame to a LaTeX file
df_summary.to_latex("summary.tex")
This outputs the following LaTeX code.
{} & avg\_gap & max\_gap & avg\_time \\
algorithm & & & \\
alice & 4.761905 & 14.285714 & 14.000000 \\
bob & 47.630719 & 62.500000 & 15.666667 \\
carol & 46.432749 & 86.666667 & 14.666667 \\
Well, we require some modifications:
Our algorithms are so great, we want to name them with a capital letter.
Remove the row starting with algorithm.
Rename the columns as avg gap (%), max gap (%), and avg time (s).
All values need two decimals only.
Since the table is pretty small, we could just edit it by hand. But what if we had one hundred algorithms to compare? We can exploit Pandas to automatically post-process our table. First, change the name of the rows and columns. To do so, we use the .rename(...) method.
# Rename rows (indexes) and columns
df_summary.rename(
"alice": "Alice",
"bob": "Bob",
"carol": "Carol"
"avg_gap": "avg gap (%)",
"max_gap": "max gap (%)",
"avg_time": "avg time (s)",
Note that the inplace argument indicates the method should modify directly df_summary. Next, we format the values according to our needs. Through its formatters argument, .to_latex() allows us to apply a function on each column as follows.
# Export the DataFrame to LaTeX
df_summary.to_latex(
"summary.tex",
"avg gap (%)": "{:.2f}".format,
"max gap (%)": "{:.2f}".format,
"avg time (s)": "{:.2f}".format,
Yeah! It provides the result I want. Yet, I think I can improve this Python script. In particular, I don't like to copy-paste the transformed column names, because it means I need to edit them in several places in case I change my mind. Thus, I prefer to define the formatting using the original CSV column names (avg_gap, ...). We can do this elegantly with the magic of dict comprehension:
index = {"alice": "Alice", "bob": "Bob", "carol": "Carol"}
"avg_gap": "{:.2f}".format,
"max_gap": "{:.2f}".format,
"avg_time": "{:.2f}".format,
df_summary.rename(index=index, columns=columns, inplace=True)
formatters={columns[c]: f for c, f in formatters.items()},
index_names=False,
Finally, we set index_names=False to remove the algorithm row. Here we go! The content of summary.tex should be:
{} & avg gap (\%) & max gap (\%) & avg time (s) \\
Alice & 4.76 & 14.29 & 14.00 \\
Bob & 47.63 & 62.50 & 15.67 \\
Carol & 46.43 & 86.67 & 14.67 \\
We can copy-paste this code into our LaTeX article. Personally, I prefer to save it as a separate file and use \input{summary}. Here is my LaTeX template:
\caption{Comparison of algorithms}
From LaTeX
The previous method exports the table to ready-to-use LaTeX using Python. Whereas editing the table from Python is often handier than editing it from LaTeX, I still like the second approach. LaTeX has the ability to import CSV files thanks to packages such as csvsimple and pgfplotstable. One of the great advantages of using a CSV file is that we can use the latter as a single source of truth. Why this is an advantage? For example, we can display the same data simultaneously as a table and as a chart. In the following, we assume that our df_summary table has been unchanged (its columns are still named avg_gap, max_gap, and avg_time). Although we might be able to do it in LaTeX, we choose to rename the algorithm names in the Python script before exporting the table to summary.csv.
# Rename rows (indexes)
df_summary.rename(index=index, inplace=True)
# Export the DataFrame to CSV
df_summary.to_csv("summary.csv")
From now, we use pgfplotstable to load the CSV file. Our LaTeX article follows this template:
\pgfplotstableread[col sep=comma]{summary.csv}{\summarytable}
\pgfplotstabletypeset[<options...>]{\summarytable}
The \pgfplotstableread command imports the CSV file to the \summarytable variable. Note that we need to specify col sep=comma, otherwise it is assumed that values are separated by white spaces. The \pgfplotstabletypeset command outputs a table from the \summarytable variable. All we need to do is to define the options to satisfy our requirements. Since there are plenty of them, we will go through them step by step.
First, we can specify which columns of the CSV file we are using. Although this is optional (we are using all the columns), I recommend doing so in the case we update our CSV file with more columns.
columns={algorithm, avg_gap, max_gap, avg_time},
Next, let's format the column algorithm:
columns/{algorithm}/.style={
string type},
We decided that the column algorithm has no name and its content is aligned to the left (l). We specified that this column contains strings, not numbers (otherwise it will raise an error). Similarly, we format the column avg_gap:
columns/{avg_gap}/.style={
column name={avg gap (\%)},
fixed zerofill},
This time, we want the column to be aligned to the right (r). The precision argument determines the number of decimals to show. Moreover, the numbers should be in fixed notation and filled with zeros. Except for the column name, the options for max_gap and avg_time are identical. To make our table look pretty, we add some \toprule, \midrule, and \bottomrule.
Please find here the complete code of the table:
columns/{max_gap}/.style={
column name={max gap (\%)},
columns/{avg_time}/.style={
column name={avg time (s)},
]{\summarytable}
Previously, I said that we can display a chart using the same source data. This can be done by including the pgfplots package.
The following code creates a simple vertical bar chart embedded in a figure for the average gap.
xlabel={Algorithm},
xticklabels from table={\summarytable}{algorithm},
ylabel={Average gap (\%)},
bar width=40,
enlarge x limits=0.25]
\addplot table [x expr=\coordindex, y={avg_gap}]{\summarytable};
LaTeX figure based on \summarytable
Commands and arguments of pgfplots are detailed in the manual. And voilà! We obtain both a table and a chart from a single source of data.
I described a method for computing statistics and creating LaTeX tables programmatically using Python and Pandas. We discussed two ways to include tables in a LaTeX article. Preparing such scripts can help saving time and avoiding mistakes, compared to writing values manually. Along with R, Pandas is a very mature and powerful library that is not limited to our use case. Check out the manual for more details.
|
FINDOFF
Performs pattern-matching between position lists related by simple offsets
This routine is designed to determine which positions in many unaligned and unlabelled lists match, subject to the condition that the transformations between the lists are well modelled by simple translations. Although the position lists are written in pixel coordinates, the objects can be related by translations in the Current coordinate system of the associated images.
The results from this routine are labelled position lists (one for each input list) which may be used to complete image registration using the REGISTER routine. The estimated offsets are reported, but REGISTER should be used to get accurate values.
findoff inlist error outlist
COMPLETE = _DOUBLE (Read)
A completeness threshold for rejecting matched position list pairs. A completeness factor is estimated by counting the number of objects in the overlap region of two lists, taking the minimum of these two values (this adjusts for incompleteness due to a different object detection threshold) and comparing this with the number of objects actually matched. Ideally a completeness of 1 should be found, the lower this value the lower the quality of the match. [0.5]
ERROR = _DOUBLE (Read)
The error, in pixels, in the X and Y positions. This value is used to determine which positions match within an error box (SLOW) or as a bin size (FAST). An inaccurate value may result in excessive false or null matches. [1.0]
FAILSAFE = _LOGICAL (Read)
If FAST is TRUE then this parameter indicates whether the SLOW algorithm is to be used when FAST fails. [TRUE]
FAST = _LOGICAL (Read)
If TRUE then the FAST matching algorithm is used, otherwise just the SLOW algorithm is used. [TRUE]
If NDFNAMES is FALSE then the actual names of the position lists should be given. These may not use wildcards but may be specified using indirection (other CCDPACK position list processing routines will write the names of their results file into files suitable for use in this manner) the indirection character is "
\text{^}
MAXDISP = _DOUBLE (Read)
This parameter gives the maximum acceptable displacement (in pixels) between the original alignment of the images and the alignment in which the objects are matched. If frames have to be displaced more than this value to obtain a match, the match is rejected. This will be of use when USEWCS is set and the images are already fairly well aligned in their Current coordinate systems. It should be set to the maximum expected inaccuracy in that alignment. If null, arbitrarily large displacements are allowed, although note that a similar restriction is effectively imposed by setting the RESTRICT parameter. [!]
MINMATCH = _INTEGER (Read)
This parameter specifies the minimum number of positions which must be matched for a comparison of two lists to be deemed successful. Small values (especially less than 3) of this parameter can lead to a high probability of false matches, and are only advisable for very sparsely populated lists and/or small values of the MAXDISP parameter (presumably in conjunction with USEWCS). [3]
MINSEP = _DOUBLE (Read)
Positions which are very close may cause false matches by being within the error box of other positions. The value of this parameter controls how close objects may be before they are both rejected (this occurs before pattern-matching). [Dynamic – 5.0
\ast
ERROR]
The name of a file to contain the names of the output position lists. The names written to this file are those generated using the expression given to the OUTLIST parameter. This file may be used in an indirection expression to input all the position lists output from this routine into another routine (say REGISTER), if the associating position lists with images option is not being used. [FINDOFF.LIS]
A list of names specifying the result files. These contain labelled positions which can be used in registration. The names of the lists may use modifications of the input names (image names if available otherwise the names of the position lists). So if you want to call the output lists the same name as the input images except to add a type use:
>
\ast
If no image names are given (NDFNAMES is FALSE) then if you want to change the extension of the files (from ".find" to ".off" in this case) use:
>
\ast
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This parameter controls whether to continue and create an incomplete solution. Such solutions will result when only a subset of the input position lists have been matched.
If the associating position lists with NDFs option has been chosen, an position list will still be written for each input NDF, but for NDFs which were not matched the output list will be empty (will consist only of comment lines).
Incomplete matching would ideally indicate that one, or more, of the input lists are from positions not coincident with the others, in which case it is perfectly legimate to proceed. However, it is equally possible that they have too few positions and have consequently been rejected. [TRUE]
RESTRICT = _LOGICAL (Read)
This parameter determines whether the Current coordinate system is used to restrict the choice of objects to match with each other. If set TRUE, then the only objects which are considered for matching are those which would appear in the overlap of two frames given that they are correctly aligned in their Current coordinate system. If it is set FALSE, then all objects in both frames are considered for matching.
This parameter should therefore be set TRUE if the frames are quite well aligned in their Current coordinate systems (especially in the case that there are many objects and a small overlap), and FALSE if they are not.
This parameter is ignored if USEWCS is FALSE. [FALSE]
USECOMP = _LOGICAL (Read)
This parameter specifies whether the completeness value will be used to weight the number of matches between a pair, when determining the graph connecting all input datasets. Using a completeness weight increases the chance of selecting high quality matches, but may reduce the chance of selecting matches with the highest counts in favour of those with lower counts. [TRUE]
This parameter determines whether Set header information should be used in the object matching. If USESET is true, FINDOFF will try to group position lists according to the Set Name attribute of the image to which they are attached. All lists coming from images which share the same (non-blank) Set Name attribute, and which have a CCD_SET coordinate frame in their WCS component, will be grouped together and treated by the program as a single position list. Thus no attempt is made to match objects between members of the same Set; it is assumed that the relative alignment within a Set is already known and has been fixed.
This parameter specifies whether the coordinates in the position lists should be transformed from Pixel coordinates into the Current coordinate system of the associated image before use. If the Current coordinates are related to pixel coordinates by a translation, the setting of this parameter is usually unimportant (but see also the RESTRICT parameter).
This parameter is ignored if NDFNAMES is false. [TRUE]
findoff inlist=’
\ast
’ error=1 outlist=’
\ast
.off’
In this example all the images in the current directory are accessed and their associated position lists are used. The coordinates used for object matching are those in the position lists transformed into the Current frames of the WCS components of the images. The matched position lists are named
\ast
.off. The method used is to try the FAST algorithm, switching to SLOW if FAST fails. The completeness measure is used when forming the spanning tree. Matches with completenesses less than 0.5 and or with less than three positions, are rejected.
findoff fast nofailsafe
In this example the only the FAST algorithm is used.
findoff usecomp=false
In this example the completeness factor is derived but not used to weight the edges of the spanning tree.
findoff error=8 minsep=100
In this example very fuzzy measurements (or small pixels) are being used. The intrinsic error in the measurements is around 8 pixels and positions within a box 100 pixels of each other are rejected.
findoff inlist=’data
\ast
\ast
.off’ restrict=true
This form would be used if the images ’data
\ast
’ are already approximately aligned in their Current coordinates. Setting the RESTRICT parameter then tells FINDOFF to consider only objects in the region which overlaps in the Current coordinates of each pair of frames. This can save a lot of time if there are many objects and a small overlap, but will result in failure of the program if the images are not translationally aligned reasonably well in the first place.
\ast
\ast
.off’ restrict minmatch=2 maxdisp=20 minsep=30
In this example the images are sparsely populated, and a pair will be considered to match if as few as two matching objects can be found. The images have been initially aligned in their Current coordinate systems to an accuracy of 20 or better. As an additional safeguard, no objects within 30 units (in coordinates of the Current frame) of each other in the same image are used for matching.
“Automated registration”.
The column one value must be an integer and is used to identify positions. In the output position lists from one run of FINDOFF, lines with the same column-1 value in different files represent the same object. In the input position lists column-1 values are ignored. If additional columns are present they must be numeric, and there must be the same number of them in every line. These have no effect on the calculations, but FINDOFF will propagate them to the corresponding lines in the output list.
If NDFNAMEs is TRUE then the names of the input position lists will be gotten from the item "CURRENT_LIST" of the CCDPACK extension of the input NDFs. On exit this item will be updated to contain the name of the appropriate output lists.
The pattern-matching process uses two main algorithms, one which matches all the point pair-offsets between any two input lists, looking for the matches with the most common positions, and one which uses a statistical method based on a histogram of the differences in the offsets (where the peak in the histogram is assumed the most likely difference). In each case an estimate of the positional error must be given as it is used when deciding which positions match (given an offset) or is used as the bin size when forming histograms.
Which algorithm you should use depends on the number of points your position lists contain and the expected size of the overlaps between the datasets. Obviously it is much easier to detect two lists with most of their positions in common. With small overlaps a serious concern is the likelihood of finding a ‘false’ match. False matches must be more likely the larger the datasets and the smaller the overlap.
The first algorithm (referred to as SLOW) is more careful and is capable of selecting out positions when small overlaps in the data are present (although a level of false detections will always be present) but the process is inherently slow (scaling as n
\ast
\ast
3log2(n)). The second algorithm (referred to as FAST) is an n
\ast
n process so is much quicker, but requires much better overlapping.
Because the FAST process takes so little CPU time it is better to try this first (without the SLOW process as a backup), only use the SLOW algorithm when you have small datasets and do not expect large areas (numbers of positions) of overlap.
A third algorithm, referred to as SNGL, is used automatically if one or both of the lists in a pair contains only a single object. In this case object matching is trivial and, of course, may easily be in error. SNGL can only be used if the MINMATCH parameter has been set to 1, which should be done with care. The SNGL algorithm may be useful if there really is only one object, correctly identified, in all the frames. If this is not the case, it should only be used when USEWCS is true and MAXDISP is set to a low value, indicating that the alignment of the images in their Current coordinate systems is already fairly accurate.
The global registration process works by forming a graph with each position list at a node and with connecting edges of weight the number of matched position-pairs. The edge weights may be modified by a completeness factor which attempts to assess the quality of the match (this is based on the ratio of the expected number of matches in the overlap region to the actual number, random matches shouldn’t return good statistics when compared with genuine ones). This still leaves a possibility of false matches disrupting any attempt to register the datasets so a single "spanning tree" is chosen (this is a graph which just visits each node the minimum number of times required to get complete connectivity, no loops allowed) which has the highest possible number of matched positions (rejecting edges with few matched positions/low completenesses where possible). This gives a most likely solution to the offsets between the position lists, rather than the "best" solution which could well include false matches; compare this solution with a median as opposed to a mean. The final registration is then used to identify all the objects which are the same in all datasets (using a relaxation method), resulting in labelled position lists which are output for use by REGISTER.
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Micromechanics-based rock-physics model for inorganic shaleMicromechanical model for shale | Geophysics | GeoScienceWorld
Igor Sevostianov;
, Department of Mechanical and Aerospace Engineering, Las Cruces, New Mexico 88003, USA. E-mail: igor@nmsu.edu.
, Houston, Texas 77082, USA. E-mail: verniklev@gmail.com (corresponding author).
Igor Sevostianov, Lev Vernik; Micromechanics-based rock-physics model for inorganic shale. Geophysics 2021;; 86 (2): MR105–MR116. doi: https://doi.org/10.1190/geo2020-0500.1
The full set of transversely isotropic elastic stiffness constants of inorganic shale (mudrock with total organic carbon less than 1.5%) can be successfully modeled and, therefore, predicted based on the mineral composition, mineral stiffnesses, clay platelet orientation distribution function, and microgeometry of the pore space. A fundamentally novel concept drawing from the Maxwell homogenization scheme allows a zero-porosity mineral matrix of the mudrock to be expressed as a polycrystal of variable composition and clay mineral alignment. Introduction of the brine-saturated pore space allows us to account for realistic 3D pore types and their combinations as well as elastic interactions, opening the way for better integration of rock physics and geomechanics with modern petrographic investigations and better shale velocity/anisotropy prediction as a function of diagenetic porosity reduction. We were able to calibrate the model using a limited subset of high-quality ultrasonic measurements on shale and constrain main pore geometries such as tetrahedra and irregular spheroids, often reported in modern scanning electron microscopy images. The model is then used to constrain the anisotropy tensor elements of illite-dominated clay, impossible to measure directly, and explore the main compositional and microstructural controls on the anisotropic elasticity of inorganic shale, including the most troublesome
C13
stiffness and its derivative — the anisotropy parameter
δ
, which is of paramount importance in quantitative seismic interpretation.
Rock-physics templates for clay-rich source rocks
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Draws position markers on a graphics display
This routine draws a variety of markers (crosses, circles, squares etc.) at positions specified in series of position lists. Before this application can be run an image (or other graphical output such as a contour image) must have been displayed using a suitable routine such as KAPPA’s DISPLAY or CCDPACK’s DRAWNDF.
For a more interactive display of markers on an Xwindows display, you can use the IDICURS program instead.
plotlist inlist [device]
This parameter controls whether or not the display device is cleared before plotting the markers. Setting this TRUE could be useful if plotting in a device overlay. [FALSE]
DEVICE = DEVICE (Write)
The name of the device on which to plot the markers. [Current display device]
\text{^}
MSIZE = _REAL (Read)
The size of the marker which will be drawn as a multiple of the default value. So for instance doubling the value of this parameter will increase the size of the markers by a factor of two. The default marker size is around 1/40 of the lesser of the width or height of the plot. [2.5]
MTYPE = _INTEGER (Read)
The type of marker to plot at the positions given in the input files. PGPLOT Graph Markers are drawn if the value lies in the range 0-31 (a value of 2 gives a cross, 7 a triangle, 24-27 various circles etc. see the PGPLOT manual). If the value of this parameter is less than zero then the identifier values, which are in column one of the input file, will be written over the objects. [2]
If TRUE then the routine will assume that the names of the position lists are stored in the image CCDPACK extensions under the item "CURRENT_LIST".
PALNUM = _INTEGER (Read)
The pen number to use when drawing the markers. The colours associated with these pens are the default PGPLOT pens (see the PGPLOT manual for a complete description). These are:
0 – background colour
1 – foreground colour
and so on up to pen 16 (or up to the number available on the current graphics device). After PLOTLIST has been run these colours can be superseded by using the KAPPA palette facilities PALDEF and PALENTRY, but note that any subsequent runs of PLOTLIST will reinstate the PGPLOT default colours. The KAPPA palette pen numbers correspond to PALNUM values (hence the parameter name). [3]
THICK = _INTEGER (Read)
The thickness of the lines used to draw the markers. This may take any value in the range 1-21. [1]
plotlist inlist=’
\ast
In this example all the images in the current directory are accessed and their associated lists of positions are plotted onto the current display device.
plotlist ndfnames=false inlist=one_list.dat
In this example the position list one_list.dat is opened and its position are plotted on the current display device.
plotlist in=’aligned_
\ast
’ mtype=-1 palnum=4 msize=1 thick=3
In this example the images aligned_
\ast
have their associated position lists accessed and the positions are plotted on the current display device. The pen colour used is blue. The text is drawn at a relative size of 1 (the normal default is 2.5) with a line thickness of 3.
If NDFNAMES is TRUE then the item "CURRENT_LIST" of the .MORE.CCDPACK structure of the input images will be located and assumed to contain the names of the lists whose positions are to be plotted.
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Home : Support : Online Help : Connectivity : Maple T.A. : MapleTA Package : Builtin : binomial
The binomial command computes the rth binomial coefficient of degree n. This corresponds to the number of ways of choosing r objects from a set of n, ignoring order.
For example, binomial(22, 3) returns the coefficient of x^19 in (x+1)^22 or 1540.
\mathrm{MapleTA}:-\mathrm{Builtin}:-\mathrm{binomial}\left(22,3\right)
\textcolor[rgb]{0,0,1}{1540}
The MapleTA[Builtin][binomial] command was introduced in Maple 18.
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Get Property - Maple Help
Home : Support : Online Help : Science and Engineering : Scientific Constants : Functions : Get Property
GetProperty( name )
symbol; property name
The GetProperty( name ) command returns the property if name is a property name in the ScientificConstants package. Otherwise, an error is returned.
The expression sequence returned consists of the property name and an equation of the form
'\mathrm{isotopic}'=
true or false.
If 'isotopic' = false, the property is an element property. If 'isotopic' = true, the property is an isotope property.
\mathrm{with}\left(\mathrm{ScientificConstants}\right):
\mathrm{GetProperty}\left(\mathrm{atomicweight}\right)
\textcolor[rgb]{0,0,1}{\mathrm{atomicweight}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{isotopic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{GetProperty}\left(\mathrm{atomicmass}\right)
\textcolor[rgb]{0,0,1}{\mathrm{atomicmass}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{isotopic}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Global Constraint Catalog: Celement_greatereq
<< 5.139. element5.141. element_lesseq >>
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}\left(\mathrm{𝙸𝚃𝙴𝙼},\mathrm{𝚃𝙰𝙱𝙻𝙴}\right)
\mathrm{𝙸𝚃𝙴𝙼}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚟𝚊𝚕𝚞𝚎}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚃𝙰𝙱𝙻𝙴}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚟𝚊𝚕𝚞𝚎}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙸𝚃𝙴𝙼},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚟𝚊𝚕𝚞𝚎}\right]\right)
\mathrm{𝙸𝚃𝙴𝙼}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙸𝚃𝙴𝙼}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝚃𝙰𝙱𝙻𝙴}|
|\mathrm{𝙸𝚃𝙴𝙼}|=1
|\mathrm{𝚃𝙰𝙱𝙻𝙴}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝙱𝙻𝙴},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚟𝚊𝚕𝚞𝚎}\right]\right)
\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝚃𝙰𝙱𝙻𝙴}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚃𝙰𝙱𝙻𝙴},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙸𝚃𝙴𝙼}\left[1\right].\mathrm{𝚟𝚊𝚕𝚞𝚎}
is greater than or equal to one of the entries (i.e., the
\mathrm{𝚟𝚊𝚕𝚞𝚎}
attribute) of the table
\mathrm{𝚃𝙰𝙱𝙻𝙴}
\left(\begin{array}{c}〈\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\mathrm{𝚟𝚊𝚕𝚞𝚎}-8〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚟𝚊𝚕𝚞𝚎}-6,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚟𝚊𝚕𝚞𝚎}-9,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚟𝚊𝚕𝚞𝚎}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚟𝚊𝚕𝚞𝚎}-9\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝙸𝚃𝙴𝙼}\left[1\right].\mathrm{𝚟𝚊𝚕𝚞𝚎}=8
\mathrm{𝚃𝙰𝙱𝙻𝙴}\left[\mathrm{𝙸𝚃𝙴𝙼}\left[1\right].\mathrm{𝚒𝚗𝚍𝚎𝚡}\right].\mathrm{𝚟𝚊𝚕𝚞𝚎}=\mathrm{𝚃𝙰𝙱𝙻𝙴}\left[1\right].\mathrm{𝚟𝚊𝚕𝚞𝚎}=6
|\mathrm{𝚃𝙰𝙱𝙻𝙴}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚟𝚊𝚕𝚞𝚎}\right)>1
\mathrm{𝚃𝙰𝙱𝙻𝙴}
\mathrm{𝙸𝚃𝙴𝙼}.\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙸𝚃𝙴𝙼}.\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚅𝙰𝙻}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\left(〈\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝙸𝙽𝙳𝙴𝚇}\mathrm{𝚟𝚊𝚕𝚞𝚎}-\mathrm{𝚅𝙰𝙻𝚄𝙴}〉,\mathrm{𝚃𝙰𝙱𝙻𝙴}\right)
\mathrm{𝚎𝚕𝚎𝚖}
\left(〈\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝙸𝙽𝙳𝙴𝚇}\mathrm{𝚟𝚊𝚕𝚞𝚎}-\mathrm{𝚅𝙰𝙻}〉,\mathrm{𝚃𝙰𝙱𝙻𝙴}\right)
constraint and of an inequality constraint
\mathrm{𝚅𝙰𝙻𝚄𝙴}\ge \mathrm{𝚅𝙰𝙻}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}
\mathrm{𝚎𝚕𝚎𝚖}
\mathrm{𝙸𝚃𝙴𝙼}
\mathrm{𝚃𝙰𝙱𝙻𝙴}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚝𝚎𝚖},\mathrm{𝚝𝚊𝚋𝚕𝚎}\right)
•\mathrm{𝚒𝚝𝚎𝚖}.\mathrm{𝚒𝚗𝚍𝚎𝚡}=\mathrm{𝚝𝚊𝚋𝚕𝚎}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•\mathrm{𝚒𝚝𝚎𝚖}.\mathrm{𝚟𝚊𝚕𝚞𝚎}\ge \mathrm{𝚝𝚊𝚋𝚕𝚎}.\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝐍𝐀𝐑𝐂}
=1
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
constraint except that the equality constraint of the second condition of the arc constraint is replaced by a greater than or equal to constraint.
\mathrm{𝐍𝐀𝐑𝐂}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚃𝙰𝙱𝙻𝙴}
are distinct and because of the first arc constraint the final graph cannot have more than one arc. Therefore we can rewrite
\mathrm{𝐍𝐀𝐑𝐂}=1
\mathrm{𝐍𝐀𝐑𝐂}\ge 1
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝙸𝙽𝙳𝙴𝚇}
\mathrm{𝚅𝙰𝙻𝚄𝙴}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚟𝚊𝚕𝚞𝚎}
attributes of the unique item of the
\mathrm{𝙸𝚃𝙴𝙼}
collection. Let
{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i}
{\mathrm{𝚅𝙰𝙻𝚄𝙴}}_{i}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚟𝚊𝚕𝚞𝚎}
{i}^{th}
\mathrm{𝚃𝙰𝙱𝙻𝙴}
collection. To each quadruple
\left(\mathrm{𝙸𝙽𝙳𝙴𝚇},\mathrm{𝚅𝙰𝙻𝚄𝙴},{\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i},{\mathrm{𝚅𝙰𝙻𝚄𝙴}}_{i}\right)
{S}_{i}
\left(\left(\mathrm{𝙸𝙽𝙳𝙴𝚇}={\mathrm{𝙸𝙽𝙳𝙴𝚇}}_{i}\right)\wedge \left(\mathrm{𝚅𝙰𝙻𝚄𝙴}\ge {\mathrm{𝚅𝙰𝙻𝚄𝙴}}_{i}\right)\right)⇔{S}_{i}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
|
Synthesis and Molecular Structure of cis-Tetracarbonyl[N-(diphenylphosphino-kP)-naphthalen-1-yl-P,P-diphenylphosphinous amide-kP]chromium(0)
Harbi Tomah Al-Masri, "Synthesis and Molecular Structure of cis-Tetracarbonyl[N-(diphenylphosphino-kP)-naphthalen-1-yl-P,P-diphenylphosphinous amide-kP]chromium(0)", Journal of Crystallography, vol. 2014, Article ID 495845, 4 pages, 2014. https://doi.org/10.1155/2014/495845
Harbi Tomah Al-Masri1
1Department of Chemistry, Faculty of Science, Taibah University, Madinah 30002, Saudi Arabia
The reaction of N,N-bis(diphenylphosphanyl)naphthylamine C10H7-1-N(PPh2)2 with (C5H10NH)2Cr(CO)4 (1 : 1 molar ratio) in dichloromethane afforded cis-[Cr(CO)4C10H7-1-N(PPh2)2] (1). This complex was crystallized in the monoclinic space group P21/n. The structure was solved by direct methods and refined by full-matrix least squares techniques to an factor of 0.0313 for 6488 observed reflections. The Cr-metal is coordinated by four terminal CO molecules and a P,P′-bidentate N,N-bis(diphenylphosphanyl)naphthylamine ligand in a distorted octahedral array. The N-atom adopts a planar geometry with the two P-atoms and C-atom attached to it. The four-membered metallacycle ring P2CrN is nearly planar.
As a continuation of our work on the synthesis and solid-state structures of phosphorus(III) ligands containing direct P–N bonds and derivatives [1–4], as these have shown a broad spectrum of anticancer, herbicidal, neuroactive, and antimicrobial activities [5–9], thereof, herein we report the synthesis and crystal structure of cis-[Cr(CO)4{C10H7-1-N(PPh2)2}] (1).
All experiments were carried out under purified dry nitrogen using standard Schlenk and vacuum line techniques. Solvents were dried and freshly distilled under nitrogen [10]. The chemicals Cr(CO)6 were used as purchased. C10H7-1-N(PPh2)2 [1] and (C5H10NH)2Cr(CO)4 [11] were prepared according to the literature methods. Infrared spectra were recorded on a Shimadzu FTIR-8400S spectrometer between 4000 and 400 cm−1 using KBr disks. The NMR spectra were recorded at 25°C on a Bruker-Avance-DRX-400 MHz NMR spectrometer operating at 400.17 (1H) and 100.63 (13C) using tetramethylsilane as external standard. Melting point was carried out on a Gallenkamp apparatus with open capillaries.
2.2. Preparation of cis-Tetracarbonyl[N-(diphenylphosphino-kP)-naphthalen-1-yl-P,P-diphenylphosphinous amide-kP]chromium(0) (1)
A solution of N,N-bis(diphenylphosphanyl)naphthylamine (0.26 g, 0.51 mmol) and (C5H10NH)2Cr(CO)4 (0.17 g, 0.51 mmol) in 20 mL of CH2Cl2 was refluxed for 2 h. The orange solution was concentrated to ca. 5 mL under reduced pressure, and n-hexane (5 mL) was added. Cooling this solution to 0°C gave 3 as yellow crystals in 85% yield. Mp 180–183°C. 1H NMR (CDCl3, /ppm): 6.40–7.78 (m, 27 H, C10H7 and 4C6H5). 13C NMR (CDCl3, /ppm): 124.8, 125.2, 125.6, 125.9, 127.1, 127.9, 128.6, 129.6, 130.5, 131.3, 132.6, 134.1, 135.4, 138.8 (C10H6 and 4C6H5), 221.2 (), 228.2 (). 31P NMR (CDCl3, /ppm): 117.41 (s, 2P). IR (selected bands, KBr, cm−1): (s, br), 1920 (s), 2008 (s) ().
Crystallographic data are given in Table 1. Single-crystal X-ray diffraction data were collected using an Oxford Diffraction Supernova dual-source diffractometer equipped with a 135 mm Atlas CCD area detector. Crystals were selected under Paratone-N oil, mounted on micromount loops, and quench-cooled using an Oxford Cryosystems open flow N2 cooling device [12]. Data were collected at 150 K using mirror monochromated radiation ( = 1.5418 Å) and processed using the CrysAlisPro package, including unit cell parameter refinement and interframe scaling (which was carried out using SCALE3 ABSPACK within CrysAlisPro) [13]. Equivalent reflections were merged and diffraction patterns were processed with the CrysAlisPro. The structure was subsequently solved using direct methods and refined on F2 using the SHELXL 97-2 package [14–16]. All nonhydrogen atoms were refined with anisotropic displacement parameters. All H-atoms bonded to carbon atoms were placed in geometrically optimized positions and refined with an isotropic displacement parameter relative to the attached atoms. Crystallographic data (excluding structure factors) for the structure in this paper has been deposited with the Cambridge Crystallographic Data Centre, CCDC, 12 Union Road, Cambridge, UK. Copies of the data can be obtained free of charge on quoting the depository number CCDC-973175 for 1 (Fax: +44-1223-336-033; E-Mail: deposit@ccdc.cam.ac.uk, http://www.ccdc.cam.ac.uk/).
Formula C38H27CrNO4P2 4
675.55 [Mg m−3] 1.342
Temp [K] 150(2) (000) 1392
Crystal system Monoclinic Abs coeff [mm−1] 4.053
Space group P21/n No. of rflns coll. 24513
[Å] 12.209(5) No. of indep rflns 6488
[Å] 20.003(5) 0.0223
[Å] 13.954(5) No. of params 415
α [°] 90 1 0.0313
β [°] 101.085(5) w 2 (all data) 0.0869
γ [°] 90 0.200
[Å3] 3344(2) −0.323
Complex 1 was previously prepared by the reaction of Cr(CO)6 with C10H7-1-N(PPh2)2 in refluxing toluene for 36 hours [1]. Here, an alternative synthetic methodology was devised using (C5H10NH)2Cr(CO)4 instead of Cr(CO)6 in refluxing CH2Cl2 to avoid prolonged refluxing time and further purification of the desired product. This methodology gave 1 in high yield and without the need for further purification. The NMR data are in agreement with those reported previously in the literature [1].
Yellow colored crystals of 1 were obtained as described in the Experimental Section. 1 crystallizes in the monoclinic space group P21/n. Selected interatomic distances and angles are collected in Table 2. The molecular structure is depicted in Figure 1.
Cr1−C11 1.883(2) C14−Cr1−C11 175.52(8)
Cr1−C12 1.859(2) P1−Cr1−C12 100.26(7)
Cr1−C14 1.881(2) P1−Cr1−C14 88.05(6)
Cr1−P1 2.347(1) P2−Cr1−C11 93.51(6)
Cr1−P2 2.347(1) P2−Cr1−C12 168.99(6)
P1−N1 1.719(1) P2−Cr1−C13 97.99(7)
P1−C15 1.829(2) P1−N1−P2 100.70(7)
P1−C21 1.818(2) P1−Cr1−P2 68.74(3)
P2−C27 1.819(2) P1−Cr1−C11 92.79(6)
P2−C33 1.830(2) N1−P1−Cr1 94.26(5)
N1−C1 1.443(2) N1−P2−Cr1 94.19(5)
C11−O1 1.136(2) N1−P1−C15 105.49(7)
C12−Cr1−C11 86.43(9) C15−P1−C21 100.97(8)
C12−Cr1−C14 89.09(8) angles at N1 359.91(9)
C13−Cr1−C11 90.05(9) P2−Cr1−P1−N1 −9.66(5)
C13−Cr1−C14 90.15(9) C3−C2−C1−N1 −177.61(2)
Selected bond lengths (Å) and bond angles (deg.) for 1.
An ORTEP diagram of complex 1 showing an atom numbering scheme. Thermal ellipsoids are drawn at the 30% probability level and arbitrary spheres for the hydrogen atoms.
The crystal structure of 1 shows a distorted octahedral environment around the Cr-metal surrounded by four terminal CO ligands and two phosphorus centers (Figure 1).
The ability of the N,N-bis(diphenylphosphanyl)naphthylamine ligand to act as bidentate P,P′-chelating ligand to the Cr-metal results in the formation of a four-membered metallacycle, that is, P–Cr–P–N, that is approximately planar with a torsion angle P–Cr–P–N of −9.66(5)°, with a smaller P–Cr–P bite angle [68.74(3)°] and larger P–N–P bond angle [100.70(7)°] (Table 2). The nitrogen atom is displaced out of the plane (Cr1, P1, P2) by 0.2873(14) .
To the best of our knowledge, there are other three structurally characterized monocyclic four-membered ring complexes of bidentate P,P′-chelating bis(phosphino)amine ligand, namely, cis-[Cr(CO)4{((o-MeOC6H4)2P)2NCH3}] (2) [17], cis-[Cr(CO)4{(Ph2P)2Pr}] (3) [18], and cis-[Cr(CO)4{Ph2P)2NH}] (4) [19].
A comparison of the structural data of the P–Cr–P and P–N–P bond angles in 1 (Table 2) with those of the four-membered rings in similar cis-chelated tetracarbonylchromium(0) 2–4 shows that the P–Cr–P bite angle in 1 is larger than those in 2 [67.54(2)°], 3 [67.82(4)°], and 4 [68.58(2)°]. The P–N–P bond angle in 1 is smaller than those in 2 [101.24(7)°] and 4 [103.24(9)°] and larger than those in 3 [99.86(11)°].
The P–N–P [100.70(7)°] bond angle is significantly smaller than those in the free diphosphinoamine ligands [19, 20] due to the formation of a strained four-membered chelate ring.
The C-napthyl skeleton in 1 is almost planar and virtually perpendicular to the P–Cr–P–N plan. A planar environment would be expected for the three-coordinate nitrogen atoms in 1 and the sum of bond angles is indeed close to 360° (Table 2). The P–Cr–C trans angles of 1 [166.57(7) and 168.99(6)°] differ significantly from 180°.
The average P–N bond distance in 1 [av. 1.721 Å] is slightly longer than those in 2 [av. 1.699 Å], 3 [av. 1.713 Å], and 4 [av. 1.692 Å] and significantly shorter than the sum of Pauling covalent radii (1.77 Å) as expected due to P–N -bonding. Consistent with this, the nitrogen atom is planar as evidenced by the sum of angles about nitrogen being 359.91(9)° for 1. Also, the P–N bond distances in 1 are slightly shorter than those in the free diphosphinoamine ligands [19, 20] which clearly indicate an enhancement of -bonding in the P–N unit.
The two Cr–P bond distances [2.347(1) and 2.347(1) Å] in 1 are equal. The average Cr–P bond distance in 1 [av. 2.347 Å] is slightly shorter than those in cis-chelated tetracarbonylchromium(0) 2 [av. 2.364 Å], 3 [av. 2.350 Å], and 4 [av. 2.354 Å]. The atoms P1, P2, Cr1, C12, and C13 are essentially coplanar with a maximum deviation from the mean plane of 0.0246(7) for C13.
The Cr–C bond distances are 1.859(2)–1.883(2) Å for 1. The two Cr–C bonds mutually trans are longer (weaker) than those trans to Cr–P bonds (Table 2). This result reflects the difference in the strength of the metal-to-ligand -bonding [21]. The aromatic rings in 1 as expected have usual bond lengths and angles.
In conclusion, we have shown the synthesis and molecular structure of cis-chelated tetracarbonylchromium(0) complex 1. The Cr-atom has a distorted octahedral arrangement with four CO ligands and two P-centers. The two Cr–C bonds mutually trans are longer (weaker) than those trans to Cr–P bonds due to the various strengths of the metal-to-ligand -bonding.
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Copyright © 2014 Harbi Tomah Al-Masri. 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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American Option Pricing - Premia
A few notes on American vs European option pricing.
There exist several models that often do a better job of pricing American options than vanilla Black-Scholes. The Binomial Option Pricing Model (BOPM) is a good example. However, due to the computation costs on Ethereum mainnet, a BOPM implementation is not feasible.
There exist several variants of the Black Scholes models adjusted for American options pricing. Most of them, however, simply add an adjustment coefficient that takes the unobservable "probability of exercise" as an input value. In other words, the mapping from vanilla
BS
to American-adjusted-
BS
is usually done in the form of a coefficient that may or may not be stable across time and cannot be computed on-chain efficiently. In any case, given the nature of this relationship, there is no reason that the C-value should converge to a level that disregards this European-American adjustment.
|
Global Constraint Catalog: Cstrict_lex2
<< 5.377. stretch_path_partition5.379. strictly_decreasing >>
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝}_\mathrm{𝚕𝚎𝚡}\mathtt{2}\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇},\mathrm{𝚟𝚎𝚌}\right)
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}
\left(\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇},\mathrm{𝚟𝚎𝚌}\right)
Given a matrix of domain variables, enforces that both adjacent rows, and adjacent columns are lexicographically ordered (adjacent rows and adjacent columns cannot be equal).
\left(〈\mathrm{𝚟𝚎𝚌}-〈2,2,3〉,\mathrm{𝚟𝚎𝚌}-〈2,3,1〉〉\right)
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝}_\mathrm{𝚕𝚎𝚡}\mathtt{2}
〈2,2,3〉
is lexicographically strictly less than the second row
〈2,3,1〉
〈2,2〉
is lexicographically strictly less than the second column
〈2,3〉
〈2,3〉
is lexicographically strictly less than the third column
〈3,1〉
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|>1
|\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}|>1
\mathrm{𝚟𝚊𝚛}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}.\mathrm{𝚟𝚎𝚌}
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝}_\mathrm{𝚕𝚎𝚡}\mathtt{2}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝙼𝙰𝚃𝚁𝙸𝚇}
strict_lex2 in MiniZinc.
\mathrm{𝚊𝚕𝚕𝚙𝚎𝚛𝚖}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}\mathtt{2}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
|
Determining Kinetic Energy Lost in Inelastic Collisions | Brilliant Math & Science Wiki
Adam Strandberg, Tim O'Brien, Christopher Williams, and
A perfectly inelastic collision is one in which two objects colliding stick together, becoming a single object. For instance, two balls of sticky putty thrown at each other would likely result in perfectly inelastic collision: the two balls stick together and become a single object after the collision.
5 \sqrt{2}
2 \sqrt{5}
A
B
m
15
A
B
20
-10
g= 10
^{2}
General Solution for Perfectly Inelastic Collisions
Consider two particles of mass
m_{1}
m_{2}
moving at velocities
\vec{v}_{1}
\vec{v}_{2}
, respectively. Before they collide, they have a combined energy of
E_{\text{init}} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{1} v_{2}^{2}
and a combined momentum of
\vec{p}_{\text{init}} = m_{1} \vec{v}_{1} + m_{2} \vec{v}_{2}
Since the collision is perfectly inelastic, after the collision there is a single combined object of mass
m_{1} + m_{2}
. Since momentum is conserved, this object has momentum equal to the total intitial momentum
\vec{p} = (m_{1} + m_{2}) \vec{v}_{f}
. The velocity of the combined object
\vec{v}_f
\begin{aligned} (m_{1} + m_{2}) \vec{v}_{f} &= m_{1} \vec{v}_{1} + m_{2} \vec{v}_{2}\\ \vec{v}_{f} &= \frac{m_1}{m_1 + m_2} \vec{v}_1 + \frac{m_{2}}{m_1 + m_2} \vec{v}_2. \end{aligned}
The energy depends on the squared magnitude of
\vec{v}_f
, which is the dot product of
\vec{v}_f
with itself. If the angle between
\vec{v}_1
\vec{v}_2
\theta
, then this equals
\| \vec{v}_f \|^2 = \frac{m_1^2}{(m_1 + m_2)^2} v_1^2 + \frac{m_2^2}{(m_1 + m_2)^2} v_2^2 + \frac{m_1 m_2}{(m_1 + m_2)^2} v_1 v_2 \text{cos} \theta.
The final energy
E_f
\begin{aligned} E_f &= \frac{1}{2} (m_1 + m_2) \| \vec{v}_f \|^2\\ &= \frac{1}{2} \left[\frac{m_1^2}{(m_1 + m_2)} v_1^2 + \frac{m_2^2}{(m_1 + m_2)} v_2^2 + 2 \frac{m_1 m_2}{(m_1 + m_2)} v_1 v_2 \text{cos} \theta\right]. \end{aligned}
This equation is the general solution for perfectly inelastic collisions. It's somewhat ugly, but exploring how it works in particular simplified cases can help build intuition for what it says.
m_{1} = m_{2}
\vec{v}_{1} = -\vec{v}_{2}
m_1 = m_2
) and identical velocities (
\vec{v}_1 = \vec{v}_2
), what is the final energy?
What is the energy difference
\Delta E = E_ f - E_ i
m_2
is much much smaller than
m_1
? Physicists express this with symbols as
m_2 \ll m_1
m_1 + m_2 \approx m_1
. This simplifies the equation to
\begin{aligned} E_f &= \frac{1}{2} \left[\frac{m_1^2}{m_1} v_1^2 + \frac{m_2}{m_1} v_2^{2} + 2 \frac{m_1 m_2}{m_1} v_1 v_2 \text{cos} \theta\right] \\ &= \frac{1}{2} \left(m_1 v_1^2 + 2 m_2 v_1 v_2 \text{cos} \theta + \frac{m_2}{m_1} m_1 v_2^2\right). \end{aligned}
m_2 \ll m_1
\frac{m_2}{m_1} \ll 1
, the last term is small if in addition
v_2
is smaller than or not much larger than
v_1
. These combined assumptions allow
E_f
to be further simplified to
\begin{aligned} E_f &= \frac{1}{2} \left(m_1 v_1^2 + 2 m_2 v_1 v_2 \text{cos} \theta\right)\\\\ \Delta E &= E_f - E_i \\ &= m_2 v_1 v_2 \text{cos} \theta - \frac{1}{2} m_2 v_2^2. \end{aligned}
This equation gives a nice interpretation for this limiting case. The second term "eliminates" the energy of the original particle, while the first term "creates" a particle of mass
m_2
with velocity projected in the direction of the more massive
m_1
, because it's stuck to
m_1
. The energy of the mass
m_1
is left unchanged.
Take special care that this simplification required that the velocity of the smaller particle was not too high. If it were, then the larger particle would have its energy changed as well, and
Cite as: Determining Kinetic Energy Lost in Inelastic Collisions. Brilliant.org. Retrieved from https://brilliant.org/wiki/determining-kinetic-energy-lost-in-inelastic/
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✅ Ansys ICEM CFD - Import Points NACA Airfoil 4412 - CFD.NINJA
The NACA airfoils are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics (NACA). The shape of the NACA airfoils is described using a series of digits following the word “NACA”. The parameters in the numerical code can be entered into equations to precisely generate the cross-section of the airfoil and calculate its properties.
Equation for a symmetrical 4-digit NACA airfoil[edit]
The formula for the shape of a NACA 00xx foil, with “x” being replaced by the percentage of thickness to chord, is[3]
{\displaystyle y_{t}=5t\left[0.2969{\sqrt {x}}-0.1260x-0.3516x^{2}+0.2843x^{3}-0.1015x^{4}\right],}
{\displaystyle y_{t}}
{\displaystyle r=1.1019{\frac {t^{2}}{c}}.}
{\displaystyle (x_{U},y_{U})}
{\displaystyle (x_{L},y_{L})}
{\displaystyle x_{U}=x_{L}=x,\quad y_{U}=+y_{t},\quad y_{L}=-y_{t}.}
Equation for a cambered 4-digit NACA airfoil[edit]
The simplest asymmetric foils are the NACA 4-digit series foils, which use the same formula as that used to generate the 00xx symmetric foils, but with the line of mean camber bent. The formula used to calculate the mean camber line is[3]
{\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&0\leq x\leq pc,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&pc\leq x\leq c,\end{cases}}}
{\displaystyle (x_{U},y_{U})}
{\displaystyle (x_{L},y_{L})}
, of respectively the upper and lower airfoil surface, become[7]
{\displaystyle {\begin{aligned}x_{U}&=x-y_{t}\,\sin \theta ,&y_{U}&=y_{c}+y_{t}\,\cos \theta ,\\x_{L}&=x+y_{t}\,\sin \theta ,&y_{L}&=y_{c}-y_{t}\,\cos \theta ,\end{aligned}}}
{\displaystyle \theta =\arctan {\frac {dy_{c}}{dx}},}
{\displaystyle {\frac {dy_{c}}{dx}}={\begin{cases}{\dfrac {2m}{p^{2}}}\left(p-{\dfrac {x}{c}}\right),&0\leq x\leq pc,\\{\dfrac {2m}{(1-p)^{2}}}\left(p-{\dfrac {x}{c}}\right),&pc\leq x\leq c.\end{cases}}}
In this first tutorial, you will learn how to import points (NACA Airfoil 4412) from Excel to Ansys ICEM CFD.
Download NACA Airfoil
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PDA_CHE2D
Evaluates a 2-dimensional Chebyshev polynomial
This routine evaluates a two-dimensional Chebyshev polynomial for one or more arguments. It uses Clenshaw’s recurrence relationship twice.
CALL PDA_CHE2D( NPTS, XMIN, XMAX, X, YMIN, YMAX, Y, XDEG, YDEG, NCOEF, CC, NW, WORK, EVAL, IFAIL )
XMIN = DOUBLE PRECISION (Given)
The lower endpoint of the range of the fit along the first dimension. The Chebyshev series representation is in terms of a normalised variable, evaluated as (2x - (XMAX
+
XMIN) ) / (XMAX - XMIN), where x is the original variable. XMIN must be less than XMAX.
XMAX = DOUBLE PRECISION (Given)
The upper endpoint of the range of the fit along the second dimension. See XMIN.
X( NPTS ) = DOUBLE PRECISION (Given)
The co-ordinates along the first dimension for which the Chebyshev polynomial is to be evaluated.
YMIN = DOUBLE PRECISION (Given)
The lower endpoint of the range of the fit along the first dimension. The Chebyshev series representation is in terms of a normalised variable, evaluated as (2y - (YMAX
+
YMIN) ) / (YMAX - YMIN), where y is the original variable. YMIN must be less than YMAX.
YMAX = DOUBLE PRECISION (Given)
The upper endpoint of the range of the fit along the second dimension. See YMIN.
Y = DOUBLE PRECISION (Given)
The co-ordinate along the second dimension for which the Chebyshev polynomial is to be evaluated.
XDEG = INTEGER (Given)
The degree of the polynomial along the first dimension.
YDEG = INTEGER (Given)
The degree of the polynomial along the second dimension.
MCOEF = INTEGER (Given)
The number of coefficients. This must be at least the product of (XDEG
+
\ast
(YDEG
+
CC( MCOEF ) = DOUBLE PRECISION (Given)
The Chebyshev coefficients. These should be the order such that CCij is in CC( i
\ast
+
+
+
1 ) for i=0,XDEG; j=0,YDEG. In other words the opposite order to Fortran standard.
NW = INTEGER (Given)
The number of elements in the work array. It must be at least XDEG
+
WORK( NW ) = DOUBLE PRECISION (Returned)
EVAL( NPTS ) = DOUBLE PRECISION (Returned)
The evaluated polynomial for the supplied arguments. Should an element of argument X lie beyond the range [XMIN,XMAX], IFAIL=7 is returned.
The status. A value of 0 indicates that the routine completed successfully. Positive values indicate the following errors:
IFAIL = 1 XMAX less than or equal to XMIN IFAIL = 2 YMAX less than or equal to YMIN IFAIL = 3 NCOEF less than 1. IFAIL = 4 XDEG or YDEG less than 1. IFAIL = 5 Number of coefficients is too great, namely (XDEG
+
\ast
+
1) is greater than NCOEF. IFAIL = 6 Y lies outside the range YMIN to YMAX. IFAIL = 7 An element of X lies outside the range XMIN to XMAX.
A single precision version of this function is available, named PDA_CHE2R.
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Subtyping - CodeDocs
{\displaystyle P_{T}}
{\displaystyle P_{s}}
{\displaystyle T=\{v\in D\mid \ P_{T}(v)\}}
{\displaystyle S=\{v\in D\mid \ P_{T}(v){\text{ and }}P_{s}(v)\}}
{\displaystyle \mathbf {T} =P_{T}}
{\displaystyle P_{s}}
{\displaystyle \mathbf {S} =\mathbf {T} \land P_{s}=P_{T}\land P_{s}}
{\displaystyle {\mathit {Felinae=\{cat\in Felidae\mid \ ofSubfamily(cat,felinaeSubfamilyName)\}}}}
{\displaystyle {\mathit {Felis=\{cat\in Felinae\mid \ ofGenus(cat,felisGenusName)\}}}}
{\displaystyle s\in S}
as an operand (input argument or term) will therefore be able to operate over that value as one of type T, because
{\displaystyle s\in T}
Subtyping schemes
Width and depth subtyping
{\displaystyle T_{1}\leq :S_{1}\quad S_{2}\leq :T_{2} \over S_{1}\rightarrow S_{2}\leq :T_{1}\rightarrow T_{2}}
The argument type of S1 → S2 is said to be contravariant because the subtyping relation is reversed for it, whereas the return type is covariant. Informally, this reversal occurs because the refined type is "more liberal" in the types it accepts and "more conservative" in the type it returns. This is what exactly works in Scala: a n-ary function is internally a class that inherits the
{\displaystyle {\mathtt {Function_{N}(-A_{1},-A_{2},\dots ,-A_{n},+B)}}}
{\displaystyle {\mathtt {A_{1},A_{2},\dots A_{n}}}}
The subtyping of mutable references is similar to the treatment of function arguments and return values. Write-only references (or sinks) are contravariant, like function arguments; read-only references (or sources) are covariant, like return values. Mutable references which act as both sources and sinks are invariant.
Relationship with inheritance
The third case is a consequence of function subtyping input contravariance. Assume a super class of type T having a method m returning an object of the same type (i.e. the type of m is T → T, also note that the first argument of m is this/self) and a derived class type S from T. By inheritance, the type of m in S is S → S. In order for S to be a subtype of T the type of m in S must be a subtype of the type of m in T, in other words: S → S ≤: T → T. By bottom-up application of the function subtyping rule, this means: S ≤: T and T ≤: S, which is only possible if S and T are the same. Since inheritance is an irreflexive relation, S can't be a subtype of T.
In coercive subtyping systems, subtypes are defined by implicit type conversion functions from subtype to supertype. For each subtyping relationship (S <: T), a coercion function coerce: S → T is provided, and any object s of type S is regarded as the object coerceS → T(s) of type T. A coercion function may be defined by composition: if S <: T and T <: U then s may be regarded as an object of type u under the compound coercion (coerceT → U ∘ coerceS → T). The type coercion from a type to itself coerceT → T is the identity function idT
Coercion functions for records and disjoint union subtypes may be defined componentwise; in the case of width-extended records, type coercion simply discards any components which are not defined in the supertype. The type coercion for function types may be given by f'(s) = coerceS2 → T2(f(coerceT1 → S1(t))), reflecting the contravariance of function arguments and covariance of return values.
Cook, William R.; Hill, Walter; Canning, Peter S. (1990). Inheritance is not subtyping. Proc. 17th ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages (POPL). pp. 125–135. CiteSeerX . doi:10.1145/96709.96721. ISBN 0-89791-343-4.
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Global Constraint Catalog: Csoft_alldifferent_var
<< 5.359. soft_alldifferent_ctr5.361. soft_cumulative >>
[PetitReginBessiere01]
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}\left(𝙲,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
𝙲
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
𝙲\ge 0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
𝙲
is greater than or equal to the minimum number of variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
for which the value needs to be changed in order that all variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
take a distinct value.
\left(3,〈5,1,9,1,5,5〉\right)
\left(1,〈5,1,9,6,5,3〉\right)
\left(0,〈8,1,9,6,5,3〉\right)
Within the collection
〈5,1,9,1,5,5〉
of the first example, 3 and 2 items are respectively fixed to values 5 and 1. Therefore one must change the values of at least
\left(3-1\right)+\left(2-1\right)=3
items to get back to 6 distinct values. Consequently, the corresponding
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
constraint holds since its first argument
𝙲
is greater than or equal to 3.
𝙲>0
2*𝙲\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
𝙲
can be increased.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
All occurrences of two distinct values of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
can be swapped; all occurrences of a value of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
can be renamed to any unused value.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
A soft
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint.
Since it focus on the soft aspect of the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint, the original article [PetitReginBessiere01], which introduce this constraint, describes how to evaluate the minimum value of
𝙲
and how to prune according to the maximum value of
𝙲
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
constraint is called
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
in [HebrardMarxSullivanRazgon09].
A first filtering algorithm presented in [PetitReginBessiere01] achieves arc-consistency. A second filtering algorithm also achieving arc-consistency is described in [Cymer12], [CymerPhD13].
By introducing a variable
M
that gives the number of distinct values used by variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
\left(𝙲,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
constraint can be expressed as a conjunction of the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(M,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
constraint and of the linear constraint
𝙲\ge |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-M
Length (
n
) 2 3 4 5 6 7 8
Solutions 24 212 2470 35682 614600 12286024 279472266
Number of solutions for
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
: domains
0..n
n
Total 24 212 2470 35682 614600 12286024 279472266
2 9 64 620 7320 97440 1404480 21530880
3 - 64 625 7770 116340 1992480 37406880
4 - - 625 7776 117642 2093616 42550704
5 - - - 7776 117649 2097144 43037568
6 - - - - 117649 2097152 43046712
7 - - - - - 2097152 43046721
8 - - - - - - 43046721
Solution count for
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
0..n
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝𝚎𝚍}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚊𝚕}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}
(soft constraint).
hard version:
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
implied by:
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
characteristic of a constraint: all different, disequality.
constraint type: soft constraint, value constraint, relaxation, variable-based violation measure.
filtering: bipartite matching.
final graph structure: strongly connected component, equivalence.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐒𝐂𝐂}
\ge |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-𝙲
We generate a clique with binary equalities constraints between each pairs of vertices (this include an arc between a vertex and itself) and we state that
𝙲
is equal to the difference between the total number of variables and the number of strongly connected components.
Parts (A) and (B) of Figure 5.360.1 respectively show the initial and final graph associated with the first example of the Example slot. Since we use the
\mathrm{𝐍𝐒𝐂𝐂}
graph property we show the different strongly connected components of the final graph. Each strongly connected component of the final graph includes all variables that take the same value. Since we have 6 variables and 3 strongly connected components the cost variable
𝙲
6-3
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
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✅ Ansys CFX - NACA 4412 (Structured Mesh) - CFD.NINJA
Ansys CFX – NACA 4412 (Structured Mesh)
by cfd.ninja | Mar 19, 2020 | Ansys CFX
The NACA four-digit wing sections define the profile by:[1]
First digit describing maximum camber as percentage of the chord.
Second digit describing the distance of maximum camber from the airfoil leading edge in tenths of the chord.
Last two digits describing maximum thickness of the airfoil as percent of the chord.[2]
For example, the NACA 2412 airfoil has a maximum camber of 2% located 40% (0.4 chords) from the leading edge with a maximum thickness of 12% of the chord.
The NACA 0015 airfoil is symmetrical, the 00 indicating that it has no camber. The 15 indicates that the airfoil has a 15% thickness to chord length ratio: it is 15% as thick as it is long.
Plot of a NACA 0015 foil generated from formula
The formula for the shape of a NACA 00xx foil, with “x” being replaced by the percentage of thickness to chord, is
{\displaystyle y_{t}=5t\left[0.2969{\sqrt {x}}-0.1260x-0.3516x^{2}+0.2843x^{3}-0.1015x^{4}\right],}
x is the position along the chord from 0 to 1.00 (0 to 100%),
{\displaystyle y_{t}}
is the half thickness at a given value of x (centerline to surface),
t is the maximum thickness as a fraction of the chord (so t gives the last two digits in the NACA 4-digit denomination divided by 100).
Note that in this equation, at x/c = 1 (the trailing edge of the airfoil), the thickness is not quite zero. If a zero-thickness trailing edge is required, for example for computational work, one of the coefficients should be modified such that they sum to zero. Modifying the last coefficient (i.e. to −0.1036) will result in the smallest change to the overall shape of the airfoil. The leading edge approximates a cylinder with a radius of
{\displaystyle r=1.1019{\frac {t^{2}}{c}}.}
Now the coordinates
{\displaystyle (x_{U},y_{U})}
of the upper airfoil surface and
{\displaystyle (x_{L},y_{L})}
of the lower airfoil surface are
{\displaystyle x_{U}=x_{L}=x,\quad y_{U}=+y_{t},\quad y_{L}=-y_{t}.}
Symmetrical 4-digit series airfoils by default have maximum thickness at 30% of the chord from the leading edge.
Equation for a cambered 4-digit NACA airfoil
Plot of a NACA 2412 foil. The camber line is shown in red, and the thickness – or the symmetrical airfoil 0012 – is shown in purple.
The simplest asymmetric foils are the NACA 4-digit series foils, which use the same formula as that used to generate the 00xx symmetric foils, but with the line of mean camber bent. The formula used to calculate the mean camber line is
{\displaystyle y_{c}={\begin{cases}{\dfrac {m}{p^{2}}}\left(2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&0\leq x\leq pc,\\{\dfrac {m}{(1-p)^{2}}}\left((1-2p)+2p\left({\dfrac {x}{c}}\right)-\left({\dfrac {x}{c}}\right)^{2}\right),&pc\leq x\leq c,\end{cases}}}
m is the maximum camber (100 m is the first of the four digits),
p is the location of maximum camber (10 p is the second digit in the NACA xxxx description).
For this cambered airfoil, because the thickness needs to be applied perpendicular to the camber line, the coordinates
{\displaystyle (x_{U},y_{U})}
{\displaystyle (x_{L},y_{L})}
, of respectively the upper and lower airfoil surface, become
{\displaystyle {\begin{aligned}x_{U}&=x-y_{t}\,\sin \theta ,&y_{U}&=y_{c}+y_{t}\,\cos \theta ,\\x_{L}&=x+y_{t}\,\sin \theta ,&y_{L}&=y_{c}-y_{t}\,\cos \theta ,\end{aligned}}}
{\displaystyle \theta =\arctan {\frac {dy_{c}}{dx}},}
{\displaystyle {\frac {dy_{c}}{dx}}={\begin{cases}{\dfrac {2m}{p^{2}}}\left(p-{\dfrac {x}{c}}\right),&0\leq x\leq pc,\\{\dfrac {2m}{(1-p)^{2}}}\left(p-{\dfrac {x}{c}}\right),&pc\leq x\leq c.\end{cases}}}
In this tutorial you will learn to simulate a NACA Airfoil (4412) using ANSYS CFX. First, we will import the points of the NACA profile and then we will generate the geometry using DesignModeler and SpaceClaim, the mesh using an unstructured mesh in Ansys Meshing. You can download the file in the following link.
Ansys CFX Tutorial | Flow through Porous Media
by cfd.ninja | Dec 1, 2020 | Ansys CFX, Ansys for Beginners
In this tutorial you will learn how to simulate a Flow through Porous Media using Ansys CFX.
OpenFOAM vs ANSYS CFX
by cfd.ninja | Mar 19, 2020 | Ansys CFX, OpenFOAM
We share the same tutorial using ANSYS Fluent.
BladeGen + Turbogrid + Ansys CFX – Centrifugal Pump
Ansys CFX – Compressible Flow
Compressibility effects are encountered in gas flows at high velocity and/or in which there are large pressure variations. When the flow velocity approaches or exceeds the speed of sound of the gas or when the pressure change in the system ( $\Delta p /p$) is large, the variation of the gas density with pressure has a significant impact on the flow velocity, pressure, and temperature.
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Global Constraint Catalog: Kbipartite_matching_in_convex_bipartite_graphs
<< 3.7.33. Bipartite matching3.7.35. Boolean channel >>
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚜𝚝}
Denotes that, for a given constraint, a bipartite matching algorithm using Glover's rule for constructing a maximum matching of a convex bipartite graph can be used. Given a convex bipartite graph
G=\left(U,V,E\right)
U=\left\{{u}_{1},{u}_{2},\cdots ,{u}_{n}\right\}
V=\left\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\right\}
, Glover [Glover67] showed how to efficiently compute a maximum matching in such a graph:
First start with the empty matching.
Second for each vertex
{v}_{j}
V
\left(j=1,2,\cdots ,m\right)
{v}_{j}
has still a free neighbour in
U
, then add to the current matching the edge
\left({u}_{i},{v}_{j}\right)
{u}_{i}
is free and
{\alpha }_{i}=max\left\{j:\left({x}_{i},{y}_{j}\right)\in E,{y}_{j}\in V\right\}
is as small as possible.
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Operation on the subsets of a set
This article is about closures in general. For the specific use in topology, see Closure (topology). For the use in computer science, see closure (computer science).
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the positive integers are closed under addition, but not under subtraction: 1 − 2 is not a positive integer even though both 1 and 2 are positive integers.
Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.
The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest subset that is closed under these operations. It is often called the span (for example linear span) or the generated set.
1.1 In algebraic structures
2 In topology
3 Binary relations
5 Closure operator
5.1 Closure operator vs. closed sets
Let consider a set S equipped with one or several methods for producing elements of S from other elements of S. Operations and (partial) multivariate function are examples of such methods. If S is a topological space, the limit of a sequence of elements of S is an example, where there are an infinity of input elements and the result is not always defined. If S is a field the roots in S of a polynomial with coefficients in S is another example where the result may be not unique.
A subset X of S is said closed under these methods, if, when all input elements are in X, then all possible results are also in X. Sometimes, one say also that X has the closure property.
The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set. It follows that for every subset Y of S, there is a smallest closed subset X of S such that
{\displaystyle Y\subseteq X}
(it is the intersection of all closed subsets that contain Y). Depending on the context, X is called the closure of Y or the set generated or spanned by Y.
The concept of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in
{\displaystyle \mathbb {C} ^{n},}
a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the Zariski closure of a set V of points is the smallest algebraic set that contains V.
In algebraic structures[edit]
An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers
{\displaystyle \exists ;}
in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas. See Algebraic structure for details.
In this context, given an algebraic structure S, a substructure of S is a subset that is closed under all operations of S, including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as S. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.
Given a subset X of an algebraic structure S, the closure of X is the smallest substructure of S that is closed under all operations of S. In the context of algebraic structures, this closure is generally called the substructure generated or spanned by X, and one says that X is a generating set of the substructure.
For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element. Here, the auxiliary operations are the nullary operation that results in the identity element and the unary operation of inversion. A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non empty. So, a nonempty subset of a group that is closed under multiplication and inversion is a group that is called a subgroup. The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group.
In linear algebra, the closure of a nonempty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset. It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology. For example, in a commutative ring, the closure of a single element under ideal operations is called a principal ideal.
In topology[edit]
In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.
Binary relations[edit]
A binary relation on a set A can be defined as a subset R of
{\displaystyle A\times A,}
the set of the ordered pairs of elements of A. The notation
{\displaystyle xRy}
is commonly used for
{\displaystyle (x,y)\in R.}
Many properties or operations on relations can be used to define closures. Some of the most common ones follows.
A relation R on the set A is reflexive if
{\displaystyle (x,x)\in R}
{\displaystyle x\in A.}
As every intersection of reflexive relations is reflexive, this defines a closure. The reflexive closure of a relation R is thus
{\displaystyle R\cup \{(x,x)\mid x\in A\}.}
Symmetry is the unary operation on
{\displaystyle A\times A}
{\displaystyle (x,y)}
{\displaystyle (y,x).}
A relation is symmetric if it is closed under this operation, and the symmetric closure of a relation R is its closure under this relation.
Transitivity is defined by the partial binary operation on
{\displaystyle A\times A}
{\displaystyle (x,y)}
{\displaystyle (y,z)}
{\displaystyle (x,z).}
A relation is transitive if it is closed under this operation, and the transitive closure of a relation is its closure under this operation.
A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it. Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation that contains it.
In matroid theory, the closure of X is the largest superset of X that has the same rank as X.
The transitive closure of a set.[1]
The algebraic closure of a field.[2]
The integral closure of an integral domain in a field that contains it.
The radical of an ideal in a commutative ring.
In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.[3]
In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set.
In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.
Main article: Closure operator
In the preceding sections, closures are considered for subsets of a given set. The subsets of a set form a partially ordered set (poset) for inclusion. Closure operators allow generalizing the concept of closure to any partially ordered set.
Given a poset S whose partial order is denoted with ≤, a closure operator on S is a function
{\displaystyle C:S\to S}
that is increasing (
{\displaystyle x\leq C(x)}
{\displaystyle x\in S}
), idempotent (
{\displaystyle C(C(x))=C(x)}
), and monotonic (
{\displaystyle x\leq y\implies C(x)\leq C(y)}
Equivalently, a function from S to S is a closure operator if
{\displaystyle x\leq C(y)\iff C(x)\leq C(y)}
{\displaystyle x,y\in S.}
An element of S if closed if it is its own closure, that is, if
{\displaystyle x=C(x).}
By idempotency, an element is closed if and only if it is the closure of some element of S.
An example of a closure operator that does not operates on subsets is given by the ceiling function, which maps every real number x to the smallest integer that is not smaller than x.
Closure operator vs. closed sets[edit]
A closure on the subsets of a given set may be defined either by a closure operator or by a set of closed sets that is stable under intersection and includes the given set. These two definitions are equivalent.
Indeed, the defining properties of a closure operator C implies that an intersection of closed sets is closed: if
{\textstyle X=\bigcap X_{i}}
is an intersection of closed sets, then
{\displaystyle C(X)}
must contain X and be contained in every
{\displaystyle X_{i}.}
{\displaystyle C(X)=X}
by definition of the intersection.
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that
{\displaystyle C(X)}
is the intersection of the closed sets containing X.
This equivalence remains true for partially ordered sets with the greatest-lower-bound property, if one replace "closet sets" by "closed elements" and "intersection" by "greatest lower bound".
^ Weisstein, Eric W. "Transitive Closure". mathworld.wolfram.com. Retrieved 2020-07-25.
^ Weisstein, Eric W. "Algebraic Closure". mathworld.wolfram.com. Retrieved 2020-07-25.
^ Bernstein, Dennis S. (2005). Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory. Princeton University Press. p. 25. ISBN 978-0-691-11802-4. ...convex hull of S, denoted by coS, is the smallest convex set containing S.
^ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. Vol. 25. Am. Math. Soc. p. 111. ISBN 9780821889534.
Weisstein, Eric W. "Algebraic Closure". MathWorld.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Closure_(mathematics)&oldid=1086627626"
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Hedging with Constrained Portfolios - MATLAB & Simulink
Example: Fully Hedged Portfolio
Example: Minimize Portfolio Sensitivities
Example: Under-Determined System
Example: Portfolio Constraints with hedgeslf
Both hedging functions cast the optimization as a constrained linear least-squares problem. (See the function lsqlin for details.) In particular, lsqlin attempts to minimize the constrained linear least squares problem
\begin{array}{ccc}\underset{x}{\mathrm{min}}\frac{1}{2}{‖Cx-d‖}_{2}^{2}& \text{such that}& A\cdot x\le b\\ & & Aeq\cdot x=beq\\ & & lb\le x\le ub\end{array}
where C, A, and Aeq are matrices, and d, b, beq, lb, and ub are vectors. For Financial Instruments Toolbox™ software, x is a vector of asset holdings (contracts).
Depending on the constraint and the number of assets in the portfolio, a solution to a particular problem may or may not exist. Furthermore, if a solution is found, it may not be unique. For a unique solution to exist, the least squares problem must be sufficiently and appropriately constrained.
Recall that hedgeopt allows you to allocate an optimal hedge by one of two goals:
As an example, reproduce the results for the fully hedged portfolio example.
FixedInd = [1 4 5 7 8];
[Sens,Cost,Quantity] = hedgeopt(Sensitivities, Price,...
This example finds a unique solution at a cost of just over $23,000. The matrix C (formed internally by hedgeopt and passed to lsqlin) is the asset Price vector expressed as a row vector.
C = Price' == [98.72 97.53 0.05 98.72 100.55 6.28 0.05 3.69]
The vector d is the current portfolio value Value0 = 23674.62. The example maintains, as closely as possible, a constant portfolio value subject to the specified constraints.
In the absence of any additional constraints, the least squares objective involves a single equation with eight unknowns. This is an under-determined system of equations. Because such systems generally have an infinite number of solutions, you need to specify additional constraints to achieve a solution with practical significance.
The additional constraints can come from two sources:
User-specified equality constraints
Target sensitivity equality constraints imposed by hedgeopt
The example in Fully Hedged Portfolio specifies five equality constraints associated with holding assets 1, 4, 5, 7, and 8 fixed. This reduces the number of unknowns from eight to three, which is still an under-determined system. However, when combined with the first goal of hedgeopt, the equality constraints associated with the target sensitivities in TargetSens produce an additional system of three equations with three unknowns. This additional system guarantees that the weighted average of the delta, gamma, and vega of assets 2, 3, and 6, together with the remaining assets held fixed, satisfy the overall portfolio target sensitivity needs in TargetSens.
Combining the least-squares objective equation with the three portfolio sensitivity equations provides an overall system of four equations with three unknown asset holdings. This is no longer an under-determined system, and the solution is as shown.
If the assets held fixed are reduced, for example, FixedInd = [1 4 5 7], hedgeopt returns a no cost, fully hedged portfolio (Sens = [0 0 0] and Cost = 0).
If you further reduce FixedInd (for example, [1 4 5], [1 4], or even []), hedgeopt always returns a no cost, fully hedged portfolio. In these cases, insufficient constraints result in an under-determined system. Although hedgeopt identifies no cost, fully hedged portfolios, there is nothing unique about them. These portfolios have little practical significance.
Constraints must be sufficient and appropriately defined. Additional constraints having no effect on the optimization are called dependent constraints. As a simple example, assume that parameter Z is constrained such that
Z\le 1
. Furthermore, assume that you somehow add another constraint that effectively restricts
Z\le 0
Z\le 1
now has no effect on the optimization.
To illustrate using hedgeopt to minimize portfolio sensitivities for a given maximum target cost, specify a target cost of $20,000 and determine the new portfolio sensitivities, holdings, and cost of the rebalanced portfolio.
MaxCost = 20000;
Holdings, [1 4 5 7 8], [], MaxCost);
This example corresponds to the $20,000 point along the cost axis in the figures Rebalancing Cost Profile, Funds Available for Rebalancing, and Rebalancing Cost.
When minimizing sensitivities, the maximum target cost is treated as an inequality constraint; in this case, MaxCost is the most you are willing to spend to hedge a portfolio. The least-squares objective matrix C is the matrix transpose of the input asset sensitivities
C = Sensitivities'
a 3-by-8 matrix in this example, and d is a 3-by-1 column vector of zeros,
[0 0 0]'.
Without any additional constraints, the least-squares objective results in an under-determined system of three equations with eight unknowns. By holding assets 1, 4, 5, 7, and 8 fixed, you reduce the number of unknowns from eight to three. Now, with a system of three equations with three unknowns, hedgeopt finds the solution shown.
Reducing the number of assets held fixed creates an under-determined system with meaningless solutions. For example, see what happens with only four assets constrained.
FixedInd = [1 4 5 7];
You have spent $20,000 (all the funds available for rebalancing) to achieve a fully hedged portfolio.
With an increase in available funds to $50,000, you still spend all available funds to get another fully hedged portfolio.
MaxCost = 50000;
Holdings, FixedInd, [],MaxCost);
-0.00 0.00 0.00
All solutions to an under-determined system are meaningless. You buy and sell various assets to obtain zero sensitivities, spending all available funds every time. If you reduce the number of fixed assets any further, this problem is insufficiently constrained, and you find no solution (the outputs are all NaN).
Note also that no solution exists whenever constraints are inconsistent. Inconsistent constraints create an infeasible solution space; the outputs are all NaN.
The other hedging function, hedgeslf, attempts to minimize portfolio sensitivities such that the rebalanced portfolio maintains a constant value (the rebalanced portfolio is hedged against market moves and is closest to being self-financing). If a self-financing hedge is not found, hedgeslf tries to rebalance a portfolio to minimize sensitivities.
From a least-squares systems approach, hedgeslf first attempts to minimize cost in the same way that hedgeopt does. If it cannot solve this problem (a no cost, self-financing hedge is not possible), hedgeslf proceeds to minimize sensitivities like hedgeopt. Thus, the discussion of constraints for hedgeopt is directly applicable to hedgeslf as well.
To illustrate this hedging facility using equity exotic options, consider the portfolio CRRInstSet obtained from the example MAT-file deriv.mat. The portfolio consists of eight option instruments: two stock options, one barrier, one compound, two lookback, and two Asian.
The hedging functions require inputs that include the current portfolio holdings (allocations) and a matrix of instrument sensitivities. To create these inputs, start by loading the example portfolio into memory
Next, compute the prices and sensitivities of the instruments in this portfolio.
Extract the current portfolio holdings (the quantity held or the number of contracts).
Holdings = instget(CRRInstSet, 'FieldName', 'Quantity');
Each row of the Sensitivities matrix is associated with a different instrument in the portfolio and each column with a different sensitivity measure.
8.29 10.00 0.59 0.04 53.45
2.50 5.00 -0.31 0.03 67.00
12.13 1.00 0.69 0.03 67.00
3.32 3.00 -0.12 -0.01 -98.08
7.60 7.00 -0.40 -45926.32 88.18
11.78 9.00 -0.42 -112143.15 119.19
4.18 4.00 0.60 45926.32 49.21
3.42 6.00 0.82 112143.15 41.71
The first column contains the dollar unit price of each instrument, the second contains the holdings of each instrument, and the third, fourth, and fifth columns contain the delta, gamma, and vega dollar sensitivities, respectively.
Suppose that you want to obtain a delta, gamma, and vega neutral portfolio using hedgeslf.
[Sens, Value1, Quantity]= hedgeslf(Sensitivities, Price, ...
hedgeslf returns the portfolio dollar sensitivities (Sens), the value of the rebalanced portfolio (Value1) and the new allocation for each instrument (Quantity).
If Value0 and Value1 represent the portfolio value before and after rebalancing, respectively, you can verify the cost by comparing the portfolio values.
Value0= Holdings' * Price
In this example, the portfolio is fully hedged (simultaneous delta, gamma, and vega neutrality) and self-financing (the values of the portfolio before and after balancing (Value0 and Value1) are the same.
Suppose now that you want to place some upper and lower bounds on the individual instruments in your portfolio. By using function portcons, you can specify these constraints, along with various general linear inequality constraints.
As an example, assume that, in addition to holding instrument 1 fixed as before, you want to bound the position of all instruments to within +/- 20 contracts (for each instrument, you cannot short or long more than 20 contracts). Applying these constraints disallows the current position in the fourth instrument (long 26.13). All other instruments are currently within the upper/lower bounds.
LowerBounds = [-20 -20 -20 -20 -20 -20 -20 -20];
UpperBounds = [20 20 20 20 20 20 20 20];
To impose these constraints, call hedgeslf with ConSet as the last input.
[Sens, Cost, Quantity1] = hedgeslf(Sensitivities, Price, ...
Holdings, 1, ConSet)
Quantity1 =
Observe that hedgeslf enforces the upper bound on the fourth instrument, and the portfolio continues to be fully hedged and self-financing.
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Find the coordinates of circumcenter of a tri ABC where A(1,2) , B(3,-4) , C(5,-6) No link pls - Maths - Coordinate Geometry - 12762439 | Meritnation.com
Find the coordinates of circumcenter of a tri.ABC where A(1,2) , B(3,-4) , C(5,-6).
No link pls...
Let the coordinates of the circumcentre of the triangle be (x, y).
Circumcentre of a triangle is equidistant from each of the vertices.
Distance between (1,2) and (x, y) = Distance between (3,-4) and (x, y)
\sqrt{{\left(\mathrm{x}-1\right)}^{2}+{\left(\mathrm{y}-2\right)}^{2}}=\sqrt{{\left(\mathrm{x}-3\right)}^{2}+{\left(\mathrm{y}+4\right)}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{square} \phantom{\rule{0ex}{0ex}}{\left(\mathrm{x}-1\right)}^{2}+{\left(\mathrm{y}-2\right)}^{2}={\left(\mathrm{x}-3\right)}^{2}+{\left(\mathrm{y}+4\right)}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}-2\mathrm{x}+1+{\mathrm{y}}^{2}+4-4\mathrm{y}= {\mathrm{x}}^{2}+9-6\mathrm{x}+{\mathrm{y}}^{2}+16+8\mathrm{y} \phantom{\rule{0ex}{0ex}}4\mathrm{x}-12\mathrm{y}-20 = 0 \phantom{\rule{0ex}{0ex}}\mathrm{x}-3\mathrm{y} = 5 ....\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{also} \mathrm{distance} \mathrm{between} \left(3,-4\right) \mathrm{and} \left(\mathrm{x},\mathrm{y}\right)= \mathrm{distance} \mathrm{between} \left(5,-6\right) \mathrm{and} \left(\mathrm{x},\mathrm{y}\right)\phantom{\rule{0ex}{0ex}}\sqrt{{\left(\mathrm{x}-3\right)}^{2}+{\left(\mathrm{y}+4\right)}^{2}}=\sqrt{{\left(\mathrm{x}-5\right)}^{2}+{\left(\mathrm{y}+6\right)}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{square} \phantom{\rule{0ex}{0ex}}{\left(\mathrm{x}-3\right)}^{2}+{\left(\mathrm{y}+4\right)}^{2}={\left(\mathrm{x}-5\right)}^{2}+{\left(\mathrm{y}+6\right)}^{2}\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}-6\mathrm{x}+9+{\mathrm{y}}^{2}+16+8\mathrm{y}= {\mathrm{x}}^{2}+25-10\mathrm{x}+{\mathrm{y}}^{2}+36+12\mathrm{y} \phantom{\rule{0ex}{0ex}}4\mathrm{x}-4\mathrm{y}=36 \phantom{\rule{0ex}{0ex}}\mathrm{x}-\mathrm{y} = 9 ....\left(2\right)\phantom{\rule{0ex}{0ex}}\mathrm{subtract} \mathrm{equation} 1 \mathrm{and} 2 \phantom{\rule{0ex}{0ex}}2\mathrm{y} = 4 \phantom{\rule{0ex}{0ex}}\mathrm{y} = 2 \mathrm{so} \mathrm{putting} \mathrm{this} \mathrm{in} \mathrm{any} \mathrm{equation} \phantom{\rule{0ex}{0ex}}\mathrm{x} = 9+2 = 11\phantom{\rule{0ex}{0ex}}
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Initial condition response of state-space model - MATLAB initial - MathWorks Australia
\begin{array}{cc}\stackrel{˙}{x}=Ax,& x\left(0\right)={x}_{0}\\ y=Cx& \end{array}
\begin{array}{rcl}\left[\begin{array}{l}{\underset{}{\overset{˙}{x}}}_{1}\\ {\underset{}{\overset{˙}{x}}}_{2}\end{array}\right]& =& \left[\begin{array}{cc}-0.5572& -0.7814\\ 0.7814& 0\end{array}\right]\left[\begin{array}{l}{x}_{1}\\ {x}_{2}\end{array}\right]\\ y& =& \left[\begin{array}{cc}1.9691& 6.4493\end{array}\right]\left[\begin{array}{l}{x}_{1}\\ {x}_{2}\end{array}\right].\end{array}
x\left(0\right)=\left[\begin{array}{l}1\\ 0\end{array}\right].
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Global Constraint Catalog: sec3.7.226
<< 3.7.225. Scheduling with machine choice, calendar3.7.227. Schur number >>
\mathrm{𝚌𝚊𝚜𝚎}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚜}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚜}_\mathrm{𝚜𝚙𝚊𝚛𝚜𝚎}
A constraint for which the same table is shared by several
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
constraints. Within the context of the
\mathrm{𝚌𝚊𝚜𝚎}
constraint, the same directed acyclic graph can be shared by several tuples of variables. This happen for instance when the
\mathrm{𝚌𝚊𝚜𝚎}
constraint is used for encoding all the transitions of an automaton [BeldiceanuCarlssonPetit04].
Within the context of planning, the idea of reusing the same constraint for encoding the transitions of an automatonEven though the original work was not presented in the context of automata, it can be partly reinterpreted as the encoding of an automaton. was proposed under the name slice encoding by C. Pralet and G. Verfaillie in [PraletVerfaillie09]. The motivation behind was to avoid to completely unfold the behaviour of the automaton (i.e., the successive triggered transitions) over the full planning horizon. From an implementation point of view, this encoding requires the possibility to reset the domains of the variables to some initial state.
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Home : Support : Online Help : Connectivity : MTM Package : int
int(M, v)
int(M, a, b)
int(M, v, a, b)
The int(M) function computes the element-wise integral of M. The result, R, is formed as R[i,j] = int(M[i,j], v, a, b).
F = int(f) is the indefinite integral of the scalar f. If f is a constant, the variable of integration is x.
int(f,v) is the indefinite integral of the scalar f with respect to v.
int(f,a,b) is the definite integral of the scalar f on the interval [a,b].
int(f,v,a,b) is the definite integral of the scalar f on the interval [a,b] with respect to v.
\mathrm{with}\left(\mathrm{MTM}\right):
M≔\mathrm{Matrix}\left(2,3,'\mathrm{fill}'=\mathrm{exp}\left(x\right)+3{x}^{2}+5\right):
\mathrm{int}\left(M\right)
[\begin{array}{ccc}{\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}& {\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}& {\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\\ {\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}& {\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}& {\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}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\end{array}]
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The ranking is constructed from independent sources, here we make the assumption that the data is trustworthy. The score is summarized below. To study the data points, simply enter the table Masters in Finance Rankings , the code is fully reproducible, see the →Ranking Methodology drop-down, we also include the scrapers used to collect the data and an easily accessible notebook.
\textbf{Overall} = Reputation Score \times 1/3 + Remuneration Score \times 1/3 + Quality Score \times 1/3
= Company offers role and applications are open → quant prep for more details.
= Company offers role but applications are not open yet
SWE = Software Engineering or Quantitative Development
QR = Quantitative Research
QT = Quantitative Trading
quant_graduate
TransMarketGroup
The market making arm of Citadel. The general perception is that the culture tends to be better at Citadel Securities vs Citadel LLC.
Applications are open for the QR role!
Similarly to Aquatic, Headlands and Radix, a small but prestigious firm that is willing to pay more than most competitors. Fairly young firm. Uses Python and C++ for QR.
Aquatic also has a Research Engineer position here.
XTX Market
Branched out of GSA capital.
Also hires people out of the US. Very focused on research with a lot of people working on cutting-edge Machine Learning.
Used to only recruit grad students but recently started recruiting undergrads as well.
Where Jeff Bezos worked before founding Amazon. Tends to be extremely school-selective, mostly recruiting from ivy leagues and similar.
The legendary OG quant fund (medallion). Good luck with this one. You can email your resume and they do interview normal SWEs sometimes but it's very unlikely you'll get an interview for any other role unless you're extra-extraordinary.
Fairly school-selective recruiting. Very engineering-focused with siloed teams.
Mako Trading
Headlands Tech
Founded by ex-Citadel people. Max Dama works here. Similarly to Ansatz, Aquatic, Headlands and Radix, a small firm that is willing to pay more than most competitors. Known to have a large focus on C++, both among QRs and SWEs.
Perception is that DRWers tend to have a good WLB but teams are usually siloed. One of the first traditional firms that started going into crypto.
The hedge fund side of Ken Griffin's Citadel. The quants are mostly on the Global Quantitative Strategies team (GQS).
Applications for QR and QT are open!
Primary focus is on ETFs. Pay tends be lower than IMC and Optiver but the office is NYC instead of Chicago.
Only offers internships in Europe.
Founded by an ex-Jane-Street guy. Very school-selective, mostly recruiting out of MIT.
Has a huge poker culture. Fairly chill culture but pay tends to be on the lower side.
Founded by ex-DE-Shaw people. Collaborative and chill culture. The org is mostly composed of QRs and SWEs. More of a quant hedge fund, with a a smaller market making arm and a venture capital team.
Founded by ex-Citadel people. Radix calls SWEs Quantitative Technologists. They don't publicize internships but they do select a handful of interns every year. Just email your resume. Rumor is that they offer the highest internship salaries. Culture is likely most similar to Renaissance.
Founded by ex-optiver traders. Meritocratic culture where people are promoted quickly. On the flip-side, stress levels and pressure tend to be on the higher side. Also try to have competing offers otherwise they might low-ball you.
Tends to have the highest pay out of the three big dutch firms (IMC, Optiver, Flow Traders) due their marble bonus system. Larger focus on traders. Traders generally tend to earn considerably more than SWEs at the Dutch firms, especially over time. QR roles are open for their Delta One team, mainly grad students.
Founded by ex-SIG people. Tends to be more school-agnostic than other firms. Also offers a lot of first-round interviews. General perception of JS is that it has a very comfortable and quirky culture. Specializes in market making.
Founded by Pete Muller, collaborative but secretive culture (similar to Rentech). Internships are SWE + SWE/QR combo roles. New grad roles are solely PhD for QR.
Founded by ex-Jump people.
Cliff Asness's firm.
Austin, Chicago, NYC
The Algo Dev role at HRT is essentially Quantitative Research. HRT's culture seems to be pretty similar to JS and pay is similar too. HRT has a bigger focus on ML with their dedicated HRT AI Labs. HRT, JS and Two Sigma are also known to recruit QRs out of undergrad.
The quant arm of Steve Cohen's Point72 hedge fund.
Specializes in Machine Learning and recruits college students for SWE roles but primarily recruits PhD students for research roles.
Pretty traditional portfolio manager culture.
Day Ralio's firm. Very unique culture.
The internships are geared towards women. The new grad roles are open to everyone.
Word is that IMC has the chillest culture out of the three big dutch firms (IMC, Optiver, Flow Traders). Pay tends to be in between Flow Traders and Optiver. Tends to be more focused on the quantiative side than Optiver and Flow. Has one of the highest intern salaries.
Founded by ex-optiver traders.
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Global Constraint Catalog: Kpallet_loading
<< 3.7.178. Packing almost squares3.7.180. Partition >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
A constraint that can be used for modelling the pallet loading problem. The pallet loading problem consists of packing a maximum number of identical rectangular boxes onto a rectangular pallet in such a way that boxes are placed with their edges parallel to the edges of the pallet. The problem often arises in distribution, when many boxes must be shipped and an increase of the number of boxes on a pallet saves costs. Even though the complexity of the problem is not yet known [Nelissen94], many solutions have been developed over the past years:
Exact algorithms based on tree search procedures extend a partial solution by positioning a new box according to different heuristics. One of the most used heuristics is the so called G4 heuristics [ScheithauerTerno96] which recursively divides the placement space into four huge rectangles. Beside the use of an appropriate heuristics, the key point is the use of upper bounds on the maximum number of boxes that can be packed. Some bounds like the Barnes [Barnes79] and the Keber [Keber85] bounds consider the geometric structure of the problem. Some other bounds are obtained by solving a linear programming problem [Isermann91].
Approximate algorithms are based on constructive methods (i.e., methods that either divide the pallet into blocks or methods that divide the pallet in a recursive way) or metaheuristics based on genetic algorithms or tabu search [AlvarezValdesParrenoTamarit05].
Both in the context of exact and approximates algorithms, the problem is usually first normalised in order to reduce the set of possible solutions [Dowsland84], [Dowsland85].
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Convert camera intrinsic parameters from MATLAB to OpenCV - MATLAB cameraIntrinsicsToOpenCV - MathWorks Deutschland
cameraIntrinsicsToOpenCV
Calibrate Camera in MATLAB and Convert Intrinsic Parameters to OpenCV
distortionCoefficients
Convert camera intrinsic parameters from MATLAB to OpenCV
[intrinsicMatrix,distortionCoefficients] = cameraIntrinsicsToOpenCV(intrinsics)
[intrinsicMatrix,distortionCoefficients] = cameraIntrinsicsToOpenCV(intrinsics) converts a MATLAB® cameraIntrinsics or cameraParameters object, specified by intrinsics, to OpenCV camera intrinsic parameters.
The OpenCV spatial coordinate system specifies the upper-left pixel center at (0,0), whereas the MATLAB spatial coordinate system specifies the pixel center at (1,1). The cameraIntrinsicsToOpenCV function compensates for this difference by subtracting 1 from both the x and y-values for the converted principal point.
OpenCV camera intrinsic parameters do not include the skew of a pinhole camera model. Therefore, only the intrinsics that were estimated without the skew can be exported to OpenCV.
Detect the checkerboard calibration pattern in the images.
Convert the intrinsic parameters to OpenCV format
[intrinsicMatrix,distortionCoefficients] = cameraIntrinsicsToOpenCV(params);
Camera intrinsic parameters, specified as a cameraIntrinsics or a cameraParameters object.
intrinsicMatrix — Camera intrinsic matrix
Camera intrinsics matrix formatted for OpenCV, returned as a 3-by-3 matrix of the form:
\left[\begin{array}{ccc}fx& 0& cx\\ 0& fy& cy\\ 0& 0& 1\end{array}\right]
where fx and fy are the focal lengths in the x and y-directions, and (cx,cy) is the principal point in OpenCV.
distortionCoefficients — Camera radial and tangential distortion coefficients
Camera radial and tangential distortion coefficients, returned as a five-element vector in the form [k1 k2 p1 p2 k3]. The values of k1, k2, and k3 describe the radial distortion and p1 and p2 describe the tangential distortion, specified in OpenCV.
undistortImage | stereoAnaglyph | stereoParametersToOpenCV | cameraIntrinsicsFromOpenCV | stereoParametersFromOpenCV | cameraIntrinsicsFromOpenCV
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Global Constraint Catalog: Cconnected
<< 5.85. connect_points5.87. consecutive_groups_of_ones >>
\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
G
\mathrm{𝙽𝙾𝙳𝙴𝚂}
G
i
j
j
i
) and connected (i.e., there is a path between any pair of vertices of
G
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,2,3\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,3\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,2,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3,5,6\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{4\right\}\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection depicts a symmetric graph involving a single connected component.
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>1
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}
constraint is sketched in [Dooms06]. Beside the pruning associated with the fact that the final graph is symmetric, it is based on the fact that all bridges and cut vertices on a path between two vertices that should for sure belong to the final graph should also belong to the final graph.
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}
(symmetric).
\mathrm{𝚜𝚝𝚛𝚘𝚗𝚐𝚕𝚢}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
final graph structure: connected component, symmetric.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)
\mathrm{𝐍𝐂𝐂}
=1
\mathrm{𝚂𝚈𝙼𝙼𝙴𝚃𝚁𝙸𝙲}
\mathrm{𝚜𝚞𝚌𝚌}
\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}
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The “native” data format that CCDPACK uses is the Starlink NDF (the N-dimensional data format (SUN/33), which is based on HDS, the Hierarchical Data System (SUN/92)). This is portable between the operating systems that CCDPACK runs on (Digital UNIX, Solaris and Linux at this time) so it can be copied, accessed via NFS and ftp’d (using binary transfer) between these systems.
If you have your data stored on tapes in FITS format then you can use the KAPPA application FITSIN to read them into NDFs. You can also get a complete list of all the FITS headers of all the files on a tape (or list of files) using the KAPPA application FITSHEAD.
If you already have (or want to keep) your data in another astronomical format such as IRAF, disk FITS or old FIGARO then CCDPACK can also process these, but with an additional processing overhead. To use data in these formats you should initialise the CONVERT package (SUN/55):
% convert
and the necessary facilities will be set up. Having done this, the simplest way to proceed is to use the full name of the unconverted files (i.e. including an extension such as ‘.fits’) when giving filenames to CCDPACK; it will transparently convert them to NDF format as required without any further effort from you.
An alternative is to convert the files explicitly to NDF format before using CCDPACK commands on them. Depending on how you are using the commands, this may save processing time by preventing CCDPACK from doing the same conversion more than once. A full description of how to do this is given in SUN/55, but normally it just consists of running a command called ‘
<
>
2NDF’. For instance converting all the FITS files in the current directory to NDF files with the same name can be done like this:
% fits2ndf in=’*.fits’ out=’*’
One useful tip for FITS2NDF is to set the CONTAINER parameter to true if you are using Multi-Extension FITS files (MEFs), i.e. type instead
% fits2ndf in=’*.fits’ out=’*’ container=true
This will convert an MEF into a single HDS container file containing each of the HDUs in the MEF — the upshot of this is that you can pass a single file name to CCDPACK programs and it will process all the images contained therein. It may also make Set processing (see section 9) easier.
One point that you should take note of is that not all formats are as flexible as NDF and cannot therefore store all the information that can be generated by CCDPACK. In particular IRAF data files cannot store additional data arrays such as variance and quality, so for instance you cannot gain any useful information about how errors propagate in your data. Also the extension information stored by CCDPACK has to be converted into native headers, this makes it less obvious what header information CCDPACK is using.
If you are unfortunate enough to have data in a format not supported by the CONVERT package then you will need to consult SSN/20 about how to proceed. You should bear in mind that the requirements of CCDPACK are that your format supports the storage of an image and some associated header information (this is essential for registration).
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O. Adebimpe1*, L. M. Erinle-Ibrahim2, A. F. Adebisi3
1 Department of Physical Sciences, Landmark University, Omu-Aran, Nigeria.
2 Department of Mathematics, Tai Solarin University of Education, Ijebu-Ode, Nigeria.
3 Department of Mathematical and Physical Science, Osun State University, Osogbo, Nigeria.
Abstract: A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium exists when the basic reproduction number R0, is less or greater than unity respectively. The global stability of the disease-free and endemic equilibrium is proved using Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion respectively.
Keywords: SIQS Epidemic Model, Saturated Incidence Rate, Basic Reproduction Number, Lyapunov Function, Poincare-Bendixson, Dulac Criterion
The isolation and treatment of symptomatic individuals coupled with the quarantining of individuals that have a high risk of having been infected, constitute two commonly used epidemic control measures. Mass quarantine can inflict significant social, psychological and economic costs without resulting in the detection of many infected individuals. Day et al. [1] , Hethcote et al. [2] considered SIQS and SIQR epidemic models with three forms of incidence, which include the bilinear, standard and quarantined-adjusted incidences.
Feng and Thieme [3] considered SEIQR models with arbitrarily distributed periods of infection, including quarantine and a general incidence assumed that all infected individuals go through the quarantine stage and investigated the model dynamics. Settapat and Wirawah [4] discussed the SIQ epidemic model with constant immigration. Yang et al. [5] also studied an SIQ epidemic model with isolation and nonlinear incidence rate. El-Marouf and Alihaby [6] studied the equilibrium points and their local stability for SIQ and SIQR epidemic models with three forms of incidence rates. They also studied the global stability of the equilibrium by constructing the new forms of Lyapunov functions.
Gbadamosi and Adebimpe investigated an SIQ epidemic model with nonlinear incidence rate. They introduced the concept that describes the present and past states of the disease.
We extended the work of Gbadamosi and Adebimpe to include the rates at which individuals recover and return to susceptible compartment from compartments I and Q respectively and we apply Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion to prove the global stability of disease-free and endemic equilibria respectively.
The model that governs a system of differential equation is presented as follows:
\begin{array}{l}\frac{\text{d}S}{\text{d}t}=\left(1-p\right)A-\frac{\beta SI}{1+mI}-dS+\gamma I+\epsilon Q\\ \frac{\text{d}I}{\text{d}t}=\frac{\beta SI}{1+mI}+pA-\left(\gamma +\delta +d+\alpha \right)I\\ \frac{\text{d}Q}{\text{d}t}=\delta I-\left(\epsilon +d+\alpha \right)Q\end{array}
S\left(0\right)={S}_{0}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}I\left(0\right)={I}_{0}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Q\left(0\right)={Q}_{0}\ge 0
The parameters with their descriptions are presented in Table 1.
The addition of the system (1), gives
\frac{\text{d}N}{\text{d}t}=A-dN-\alpha I-\alpha Q\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}\text{\hspace{0.17em}}N=S+I+Q
\begin{array}{l}0\le \underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}N\left(t\right)\le {N}_{0}\\ \text{with}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}N\left(t\right)={N}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{only}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}I\left(t\right)=0\end{array}
From the first equation of the system (1), it follows
0\le \underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}S\left(t\right)\le {S}_{0}
And the second equation gives
Table 1. Descriptions of parameters.
0\le \underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}I\left(t\right)\le {I}_{0}
So, from the above, if
N>{N}_{0}
\frac{\text{d}N}{\text{d}t}<0
\Omega =\left\{\left(S,I,Q\right)\in {R}_{+}^{3}:S+I+Q\le {N}_{0},\text{\hspace{0.17em}}S\le {S}_{0},\text{\hspace{0.17em}}I\le {I}_{0}\right\}
The system (1) has always the disease-free equilibrium at
{E}_{0}=\left({S}_{0},{I}_{0},{Q}_{0}\right)=\left(\frac{\left(1-p\right)A}{d},0,0\right)
Endemic Equilibrium:
{E}_{*}=\left({S}_{*},{I}_{*},{Q}_{*}\right)
In this section, we discussed the local stability of the disease-free equilibrium and endemic equilibrium for the system (1).
We state and prove the following results:
Theorem 1: At
{E}_{0}
, the disease-free equilibrium of the system (1) is locally asymptotically stable when
{R}_{0}<1
Proof: The Jacobian matrix at the point
{E}_{0}
through linearization is given by
{J}_{0}=\left(\begin{array}{ccc}-d& -\left(\beta {S}_{0}-\gamma \right)& \epsilon \\ 0& \beta {S}_{0}-\left(\gamma +\delta +d+\alpha \right)& 0\\ 0& \delta & -\left(\epsilon +d+\alpha \right)\end{array}\right)
By finding the eigenvalues, we have the following
\lambda s
{\lambda }_{1}=-d,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}=\beta {S}_{0}-\left(\gamma +\delta +d+\alpha \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{3\text{\hspace{0.17em}}\text{\hspace{0.17em}}}=-\left(\epsilon +d+\alpha \right)
{\lambda }_{2}
to be negative
\beta {S}_{0}<\left(\gamma +\delta +d+\alpha \right)
\frac{\beta {S}_{0}}{\gamma +\delta +d+\alpha }<1
{R}_{0}=\frac{\beta \left(1-p\right)A}{d\left(\gamma +\delta +d+\alpha \right)}
{R}_{0}=\frac{\beta \left(1-p\right)A}{d\left(\gamma +\delta +d+\alpha \right)}<1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }_{2}<0
{\lambda }_{1}<0,{\lambda }_{3}<0
{\lambda }_{2}<0
{R}_{0}<1
, the disease-free equilibrium is locally asymptotically stable.
Theorem 3.1: The system (1) is locally asymptotically stable at
{E}_{*}
{R}_{0}>1
, otherwise unstable.
Proof: At the endemic equilibrium
{E}_{*}
, the Jacobian matrix of the system (1) is given by:
{J}_{*}=\left(\begin{array}{ccc}-\left(\beta {I}_{*}+d\right)& -\beta {S}_{0}+\gamma & \epsilon \\ \beta {I}_{*}& \beta {S}_{*}-\left(\gamma +\delta +d+\alpha \right)& 0\\ 0& \delta & -\left(\epsilon +d+\alpha \right)\end{array}\right)
The characteristic equation of the Jacobian matrix
{J}_{*}
{\lambda }^{3\text{\hspace{0.17em}}}+{a}_{1}{\lambda }^{2}+{a}_{2}\lambda +{a}_{3}
{a}_{1}=3\alpha +2\delta +\gamma +3d+\beta {I}_{*}+\epsilon -2\beta {S}_{*}
\begin{array}{c}{a}_{2}=2\beta {I}_{*}\alpha +\beta {I}_{*}d+\beta {I}_{*}\delta +3{d}^{2}+4\alpha d+{\alpha }^{2}+\gamma \alpha +\delta \alpha +2\delta d+2\gamma d\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3d\epsilon +\gamma \delta +\beta {I}_{*}\epsilon +\beta {I}_{*}d-\beta {S}_{*}d-2\beta {S}_{*}\epsilon \end{array}
\begin{array}{c}{a}_{3}=\beta {I}_{*}d\epsilon +\beta {I}_{*}\alpha \epsilon +\gamma d\epsilon +\delta d\epsilon +{d}^{2}\epsilon +\alpha d\epsilon +\beta {I}_{*}\delta d+\beta {I}_{*}\alpha d+\gamma {d}^{2}+{d}^{3}+2\alpha {d}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\beta {I}_{*}\alpha \delta +\beta {I}_{*}d\alpha +\beta {I}_{*}{\alpha }^{2}+\gamma \alpha d+\delta \alpha d+{\alpha }^{2}d-\beta {S}_{*}\alpha d\end{array}
{a}_{1}>0,{a}_{2}>0,\text{\hspace{0.17em}}{a}_{3}>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}{a}_{2}-{a}_{3}>0
{a}_{1}{a}_{2}>{a}_{3}
, by Routh Hurwitz criterion, all the eigenvalues of the system (1) has negative real part. Therefore, the endemic equilibrium of the system (1) at
{E}_{*}
is locally asymptotically stable.
In this section, we study the global stability of the disease-free equilibrium and endemic equilibrium by Lyapunov function and Poincare-Bendixson theorem respectively.
Theorem 3: (Dulac’s Criterion)
Consider the following general nonlinear autonomous system of de
x\left(t\right)=f\left(x\right),x\in E
f={C}^{1}\left(E\right)
where E is a simple connected region in R2. If the exists a function it
H\in {C}^{1}\left(E\right)
\nabla \cdot \left(Hf\right)
is not identically zero and does not change sign in E, the system (*) has no close orbit lying entirely in E. if A is an annular region contained in E on which
\nabla \cdot \left(Hf\right)
does not change sign, then there is at most one limit cycle of the system (*) in A.
Theorem 4: (The Poincare-Bendixson Theorem):Suppose that
f\in {C}^{1}\left(E\right)
where E is an open subset of Rn and that the system (*) has a rejecting
\Gamma
contained in a compact subset f of E. assume that the system (*) has only one unique equilibrium point x0 in f, then one of the following possibilities holds.
w\left(\Gamma \right)
is the equilibrium point x0
w\left(\Gamma \right)
is a periodic orbit
w\left(\Gamma \right)
is a graphic
Theorem 5: The disease-free equilibrium of the model (1) is globally asymptotically stable if
{R}_{0}<1
Proof: To prove this result, we construct the following Lyapunov function
L={u}_{1}\left(S-{S}_{0}\right)+{u}_{2}\left(I-{I}_{0}\right)+{u}_{3}Q
{u}_{1},\text{\hspace{0.17em}}{u}_{2}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{u}_{3}
are positive constants to be determined later. Differentiating equation (3) with respect to t, we obtain
\begin{array}{c}{L}^{\prime }={u}_{1}\left[\left(1-p\right)A-\frac{\beta SI}{1+mI}-dS+\gamma I+\epsilon Q\right]+{u}_{2}\left[\frac{\beta SI}{1+mI}+pA-\left(\gamma +\delta +d+\alpha \right)I\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{u}_{3}\left[\delta I-\left(\epsilon +d+\alpha \right)Q\right]\end{array}
After rearrangements, we get
\begin{array}{c}{L}^{\prime }=\frac{\beta SI}{1+mI}\left({u}_{2}-{u}_{1}\right)+pA\left({u}_{2}-{u}_{1}\right)+\gamma I\left({u}_{2}-{u}_{1}\right)+\epsilon Q\left({u}_{3}-{u}_{1}\right)+\delta I\left({u}_{3}-{u}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{u}_{1}A-{u}_{1}dS-{u}_{2}dI-{u}_{3}dQ-{u}_{2\text{\hspace{0.17em}}}\alpha I-{u}_{3}\alpha Q\end{array}
Let us choose the constants
{u}_{1}={u}_{2}={u}_{3}=1
. Finally, we obtain
\begin{array}{l}{L}^{\prime }=A-d\left(S+I+Q\right)-\alpha \left(I+Q\right)\\ {L}^{\prime }=-\left(dN-A\right)-\alpha \left(N-S\right)<0\end{array}
Thus, the disease-free equilibrium of the system (1) is globally asymptotically stable if
{R}_{0}<1
In the next theorem, we present the global stability of the endemic equilibrium of the system (1) at
{E}_{*}
Theorem 6: The endemic equilibrium
{E}_{*}
of the system (1) is globally asymptotically stable if
{R}_{0}>1
Proof: In order to prove the result, we use Dulac plus Poincare Bendixson theorem as follow
H\left(S,I,Q\right)=\frac{1}{SIQ}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}I>0,\text{\hspace{0.17em}}Q>0.
\begin{array}{l}\nabla \cdot \left(HF\right)=\frac{\partial }{\partial S}\left(H\cdot {F}_{1}\right)+\frac{\partial }{\partial I}\left(H\cdot {F}_{2}\right)+\frac{\partial }{\partial Q}\left(H\cdot {F}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\partial }{\partial S}\left[\frac{1}{SIQ}\left(\left(1-p\right)A-\frac{\beta SI}{1+mI}-dS+\gamma I+\epsilon Q\right)\right]+\frac{\partial }{\partial I}\left[\frac{\beta SI}{1+mI}+pA-\left(\gamma +\delta +d+\alpha \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}I\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial Q}\left[\delta I-\left(\epsilon +d+\alpha \right)Q\right]\\ \nabla \cdot \left(HF\right)=-\frac{\left(1-p\right)A}{{S}^{2}IQ}-\frac{\gamma }{{S}^{2}Q}-\frac{\epsilon }{{S}^{2}I}-\frac{pA}{S{I}^{2}Q}-\frac{\delta }{S{Q}^{2}}<0\end{array}
Hence, by Dulac’s criterion, there is no closed orbit in the first quadrant. Therefore, the endemic equilibrium is globally asymptotically stable.
The mathematical and stability analysis of SIQS epidemic model with saturated incidence rate and temporary immunity has been presented. We investigated the local stability of the disease-free equilibrium and endemic equilibrium using the basic reproduction number,
{R}_{0}
. We observed that, when
{R}_{0}<1
, the disease-free equilibrium is stable at
{E}_{0}
locally and endemic equilibrium is unstable which means there is tendency for the disease to die out in the long run. We proved the global stability of the disease free equilibrium and endemic equilibrium of the model using Lyapunov function and Dulac’s criterion plus Poincare-Bendixson theorem respectively.
Cite this paper: Adebimpe, O. , Erinle-Ibrahim, L. and Adebisi, A. (2016) Stability Analysis of SIQS Epidemic Model with Saturated Incidence Rate. Applied Mathematics, 7, 1082-1086. doi: 10.4236/am.2016.710096.
[1] Day, T., Park, A., Madras, N., Gumel, A. and Wu, J. (2006) When Is Quarantine a Useful Control Strategy for Emerging Infectious Diseases? American Journal of Epidemiology, 163, 479-485.
[2] Hethcote, H., Ma, Z. and Liao, S. (2002) Effects of Quarantine in Six Endemic Models for Infectious Diseases. Mathematical Biosciences, 180, 141-160.
[3] Feng, Z. and Thieme, H.R. (2000) Endemic Models with Arbitrarily Distributed Periods of Infection, I. General Theory. SIAM Journal on Applied Mathematics, 61, 803-833.
[4] Settapat, C. and Wirawan, C. (2007) Global Stability of an SIQ Epidemic Model. Kasetsart Journal: Natural Science, 41, 225-228.
[5] Yang, X., Li, F. and Cheng, Y. (2012) Global Stability Analysis on the Dynamics of an SIQ Model with Nonlinear Incidence Rate. In: Advances in Future Computer and Control Systems, 2, Volume 160 of the series Advances in Intelligent and Soft Computing, 561-565.
[6] El-marouf, S.A.A. and Alihaby, S.M. (2011) Global Analysis of an Epidemic Model with Nonlinear Incidence Rate. Journal of Mathematics and Statistics, 7, 319-325.
http://dx.doi.org/10.3844/jmssp.2011.319.325
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Geometry for Elementary School/Angles - Wikibooks, open books for an open world
Geometry for Elementary School/Angles
This is the latest reviewed version, checked on 23 September 2018.
Lines Angles Plane shapes
The corresponding material in Euclid's elements can be found on page 26 of Book I in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.
In this section, we will talk about angles.
2 Angles at a point
3 Adjacent angles on a straight line
4 Vertically opposite angles
Angles[edit | edit source]
An angle (∠) is made up of a vertex (a point), two arms (rays), and an arc. They are arranged so that the endpoint of the arms are the same as the vertex, and the arc runs from one arm to another. The size of an angle depends on how big the arms are opened, and they are measured in degrees. You can measure them by putting your protractor on the vertex and looking at the degrees your second arm has reached.
An angle that is less than 90° is known as an acute angle. A 90° angle is known as a right angle. Those between 90° and 180° are obtuse angles. Exactly 180° angles are called straight angles. Those between 180° and 360° are reflex angles, while angles at 360° are round angles
An angle is usually named by the points it contains. The format is as follows:
"∠" + a point on one arm + vertex + a point on the other arm
However, sometimes there are no angles on that vertex, and we can omit the point on the arms. In fact, when we are lazy, we can even use a lowercase letter to represent a certain angle. Note that in this case, ∠ must be omitted. Although the lowercase letter represents the value of the angle, all of these names can be used as unknowns in equations.
Adjacent angles (adj. ∠s) are angles where:
Their opposite arms coincide (overlap);
Their arcs do not coincide (overlap);
Their vertices coincide (overlap).
Sometimes, two angles may add up to 90° or 180°. They are called complementary angles and supplementary angles respectively. As many angles have such properties, these will be quite handy in the future.
Angles at a point[edit | edit source]
Sometimes, two or more angles share a common vertex, and their sizes add up to 360o. They are called angles at a point (∠s at a pt.). This can be very useful when we write proofs or find out angles.
For example, imagine that O is a point in the figure. The three points, A, B, and C are around the point O, and a ray shoots out of O to A, B and C respectively. Given that ∠AOB = 120° and ∠BOC = 150°,
{\displaystyle {\begin{aligned}\angle AOB+\angle BOC+\angle COA&=360^{\circ }(\angle s{\text{ at a pt.}})\\120^{\circ }+150^{\circ }+\angle COA&=360^{\circ }\\\angle COA&=360^{\circ }-120^{\circ }-150^{\circ }\\&=90^{\circ }\end{aligned}}}
Adjacent angles on a straight line[edit | edit source]
When the sizes of adjacent angles add up to 180°, they are adjacent angles on a straight line. They are used when finding out the value of one of the angles. (Or more, for that matter, when you have angles that are equal or related.) The abbreviation, adj. ∠s on st. line, can be used as a reference that the angles add up to 180°.
Look at the image on the right as an example. Here, b and a are supplementary. The sum of b and a is equal to c. b and a are adjacent angles on a straight line. If we know the value of b, we can find out the value of a easily. Note that a, b, and c are angles at a point.
Vertically opposite angles[edit | edit source]
Vertically opposite angles are very simple. If two straight lines run into each other, the opposite angles produced must be vertically opposite angles (vert. opp. ∠s). They must be equal to each other. Note that you cannot assume that something is a straight line just by observation, so be sure that it's mentioned in the question before you do anything. Vertically opposite angles is a very common reference and will come in handy in many situations, so before you are stuck on a problem, see if you can find some vertically opposite angles first.
Look at the figure on the right. As indicated in this figure, D is equal to C and A is equal to B. This is because they are vertically opposite angles. Note that here, D and A, A and C, C and B, and D and B are all pairs of adjacent angles on a straight line. Also, the four angles are angles at a point.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Geometry_for_Elementary_School/Angles&oldid=3470333"
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Global Constraint Catalog: Cincreasing_nvalue
<< 5.186. increasing_global_cardinality5.188. increasing_nvalue_chain >>
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝚅𝙰𝙻}\ge \mathrm{𝚖𝚒𝚗}\left(1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)
\mathrm{𝙽𝚅𝙰𝙻}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
are increasing. In addition,
\mathrm{𝙽𝚅𝙰𝙻}
is the number of distinct values taken by the variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(2,〈6,6,8,8,8〉\right)
\left(1,〈6,6,6,6,6〉\right)
\left(5,〈0,2,3,6,7〉\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint (see Figure 5.187.1 for a graphical representation) holds since:
〈6,6,8,8,8〉
\mathrm{𝙽𝚅𝙰𝙻}=2
is set to the number of distinct values occurring within the collection
〈6,6,8,8,8〉
Figure 5.187.1. Illustration of the first example of the Example slot: five variables
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
{V}_{5}
respectively fixed to values 6, 6, 8, 8 and 8, and the corresponding number of distinct values
\mathrm{𝙽𝚅𝙰𝙻}=2
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
A complete filtering algorithm in a linear time complexity over the sum of the domain sizes is described in [BeldiceanuHermenierLorcaPetit10].
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint can be expressed in term of a conjunction of a
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
constraints (i.e., a chain of non strict inequality constraints on adjacent variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
). But as shown by the following example,
{V}_{1}\in \left[1,2\right]
{V}_{2}\in \left[1,2\right]
{V}_{1}\le {V}_{2}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
2,〈{V}_{1},{V}_{2}〉
), this hinders propagation (i.e., the unique solution
{V}_{1}=1
{V}_{2}=2
is not directly obtained after stating all the previous constraints).
A better reformulation achieving arc-consistency uses the
\mathrm{𝚜𝚎𝚚}_\mathrm{𝚋𝚒𝚗}
𝙽
𝚇
𝙲
𝙱
\mathrm{𝚜𝚎𝚚}_\mathrm{𝚋𝚒𝚗}
\left(𝙽,𝚇,𝙲,𝙱\right)
𝙽
𝙲
𝚇
𝙱
𝚇
𝙲
𝙲
𝙲
𝚇
𝙲
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚜𝚎𝚚}_\mathrm{𝚋𝚒𝚗}
\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},=,\le \right)
n
Solutions 6 20 70 252 924 3432 12870
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
n
Total 6 20 70 252 924 3432 12870
3 - 4 30 120 350 840 1764
4 - - 5 60 350 1400 4410
5 - - - 6 105 840 4410
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
increasingNValue in Choco.
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
(remove
\mathrm{𝙽𝚅𝙰𝙻}
parameter from
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚒𝚜𝚒𝚋𝚕𝚎}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚜𝚝𝚊𝚛𝚝}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛}
\le
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
characteristic of a constraint: automaton, automaton without counters, reified automaton constraint.
constraint type: counting constraint, value partitioning constraint, order constraint.
modelling: number of distinct equivalence classes, number of distinct values, functional dependency.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐒𝐂𝐂}
=\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝙴𝚀𝚄𝙸𝚅𝙰𝙻𝙴𝙽𝙲𝙴}
\mathrm{𝐍𝐒𝐂𝐂}
graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a value that is assigned to some variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection. The 2 following values 6 and 8 are used by the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
A first systematic approach for creating an automaton that only recognises the solutions to the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint could be to:
First, create an automaton that recognises the solutions to the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
Second, create an automaton that recognises the solutions to the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
Third, make the product of the two previous automata and minimise the resulting automaton.
However this approach is not going to scale well in practice since the automaton associated with the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint has a too big size. Therefore we propose an approach where we directly construct in a single step the automaton that only recognises the solutions to the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint. Note that we do not have any formal proof that the resulting automaton is always minimum.
Without loss of generality, assume that the collection of variables
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
contains at least one variable (i.e.,
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\ge 1
l
m
n
\mathrm{𝑚𝑖𝑛}
\mathrm{𝑚𝑎𝑥}
respectively denote the minimum and maximum possible value of variable
\mathrm{𝙽𝚅𝙰𝙻}
, the number of variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
, the smallest value that can be assigned to the variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
, and the largest value that can be assigned to the variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
s=\mathrm{𝑚𝑎𝑥}-\mathrm{𝑚𝑖𝑛}+1
denote the total number of potential values. Clearly, the maximum number of distinct values that can be assigned to the variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
cannot exceed the quantity
d=min\left(m,n,s\right)
\frac{s·\left(s+1\right)}{2}-\frac{\left(s-d\right)·\left(s-d+1\right)}{2}+1
states of the automaton that only accepts solutions to the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint can be defined in the following way:
{s}_{00}
\frac{s·\left(s+1\right)}{2}-\frac{\left(s-d\right)·\left(s-d+1\right)}{2}
states labelled by
{s}_{ij}
\left(1\le i\le d,i\le j\le s\right)
. The first index
i
of a state
{s}_{ij}
corresponds to the number of distinct values already encountered, while the second index
j
denotes the the current value (i.e., more precisely the index of the current value, where the minimum value has index 1).
Terminal states depend on the possible values of variable
\mathrm{𝙽𝚅𝙰𝙻}
and correspond to those states
{s}_{ij}
is a possible value for variable
\mathrm{𝙽𝚅𝙰𝙻}
. Note that we assume no further restriction on the domain of
\mathrm{𝙽𝚅𝙰𝙻}
(otherwise the set of accepting states needs to be reduced in order to reflect the current set of possible values of
\mathrm{𝙽𝚅𝙰𝙻}
). Three classes of transitions are respectively defined in the following way:
\mathrm{𝑚𝑖𝑛}+j-1
, from the initial state
{s}_{00}
{s}_{1j}
\left(1\le j\le s\right)
There is a loop, labelled by
\mathrm{𝑚𝑖𝑛}+j-1
for every state
{s}_{ij}
\left(1\le i\le d,i\le j\le s\right)
\forall i\in \left[1,d-1\right],\forall j\in \left[i,s\right],\forall k\in \left[j+1,s\right]
\mathrm{𝑚𝑖𝑛}+k-1
{s}_{ij}
{s}_{i+1k}
We respectively have
s
transitions of class 1,
\frac{s·\left(s+1\right)}{2}-\frac{\left(s-d\right)·\left(s-d+1\right)}{2}
transitions of class 2, and
\frac{\left(s-1\right)·s·\left(s+1\right)}{6}-\frac{\left(s-d\right)·\left(s-d+1\right)·\left(s-d+2\right)}{6}
transitions of class 3.
Note that all states
{s}_{ij}
i+s-j<l
can be discarded since they do not allow to reach the minimum number of distinct values required
l
Part (A) of Figure 5.187.3 depicts the automaton associated with the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint of the first example of the Example slot. For this purpose, we assume that variable
\mathrm{𝙽𝚅𝙰𝙻}
is fixed to value 2 and that variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
take their values within interval
\left[6,8\right]
. Part (B) of Figure 5.187.3 represents the simplified automaton where all states that do not allow to reach an accepting state were removed. The corresponding
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
constraint holds since the corresponding sequence of visited states,
{s}_{00}
{s}_{11}
{s}_{11}
{s}_{23}
{s}_{23}
{s}_{23}
, ends up in an accepting state (i.e., accepting states are denoted graphically by a double circle).
Figure 5.187.3. Automaton – Part (A) – and simplified automaton – Part (B) – of the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(2,〈6,6,8,8,8〉\right)
constraint of the first example of the Example slot: the path corresponding to the second argument
〈\mathbf{6},\mathbf{6},\mathbf{8},\mathbf{8},\mathbf{8}〉
is depicted by thick orange arcs, where the self-loop on state
{s}_{23}
is applied twice
Figure 5.187.4 depicts a second deterministic automaton with one counter associated with the
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚟𝚊𝚕𝚞𝚎}
constraint for a non-empty collection of variables
|
FINDCENT
Centroids image features
This routine determines the centroids of image features located in the data components of a list of images. It is useful for locating accurate values for the positions of stars given hand selected positions. It can also be used for centroiding any sufficiently peaked image features.
The initial positions associated with each image are given in formatted files whose names are determined either using the CCDPACK image extension item CURRENT_LIST (which is maintained by list processing CCDPACK applications) or from an explicit list of names.
findcent in outlist
AUTOSCALE = _LOGICAL (Read)
Whether to "automatically" adjust the centroid location parameters to reflect the fact that picking good initial positions is less likely when dealing with very large images (these tend to be displayed using one display pixel to represent many image pixels).
If TRUE then the values of the parameters ISIZE, TOLER and MAXSHIFT are scaled by an amount that maps the largest dimension of each input image to an image of size 1024 square (so an image of size 2048 square will have these parameters increased by a factor of two). [FALSE]
The names of the images whose data components contain image features which are to be centroided. The image names should be separated by commas and may include wildcards.
If NDFNAMES is FALSE then this parameter will be used to access the names of the lists which contain the initial positions. The format of the data in the files is described in the notes section.
The names of the input lists may use modifications of the input image names, so for instance if the position lists are stored in files with the same name as the input images but with a file type of ".dat" instead of ".sdf" then use:
INLIST
>
\ast
.dat
If the input list names are a modification of the image names say with a trailing type of "_initial.positions". Then a response of:
>
\ast
_initial.positions
will access the correct files. Names may also use substitution elements, say the image names are
\ast
_data and the position lists are
\ast
_pos.dat, then a response like:
>
\ast |
|
pos.dat
|
may be used. If a naming scheme has not been used then an explicit list of names should be returned (wildcards cannot be used to specify list names). These names should be given in the same order as the input image names and may use indirection elements as well as names separated by commas. A listing of the input image name order (after any wildcard expansions etc. have been made) is shown to make sure that the order is correct.
The size of a box side (in pixels) centered on current position which will be used to form the marginal profiles used to estimate the centroid. [9]
The maximum number of iterations which may be used in estimating the centroid. Only used if the tolerance criterion is not met in this number of iterations. [3]
MAXSHIFT = _DOUBLE (Read)
The maximum shift (in pixels) allowed from an initial position. [5.5]
Only used if NDFNAMES is FALSE. If this is the case then this specifies the name of a file to contain a listing of the names of the output lists. This file may then be used to pass the names onto another CCDPACK application using indirection. [FINDCENT.LIS]
If TRUE then the routine will assume that the names of the input position lists are stored in the CCDPACK extension item "CURRENT_LIST" of the input images. The names will be present in the extension if the positions were located using a CCDPACK application (such as IDICURS). Using this facility allows the transparent propagation of position lists through processing chains.
If TRUE then the image features have increasing values otherwise they are negative. [TRUE]
A list of names specifying the centroid result files. The names of the lists may use modifications of the input image names. So if you want to call the output lists the same name as the input images except to add a type use:
>
\ast
.cent
Or alternatively you can use an explicit list of names. These may use indirection elements as well as names separated by commas. [
\ast
.cent]
The required tolerance in the positional accuracy of the centroid. On each iteration the box of data from which the centroid is estimated is updated. If the new centroid does not differ from the previous value by more than this amount (in X and Y) then iteration stops. Failure to meet this level of accuracy does not result in the centroid being rejected, the centroiding process just stops after the permitted number of iterations (MAXITER). [0.05]
findcent in=’
\ast
\ast
In this example all the images in the current directory are processed. It is assumed that the images are associated with position lists of inaccurate positions (via the item CURRENT_LIST in the image CCDPACK extensions). These position lists are accessed and centroided with the appropriate images. On exit the new lists are named
\ast
.cent and are associated with the images (instead of the original "input" lists).
findcent ndfnames=false in=’"image1,image2,image3"’
inlist=’"image1.pos,image2.pos,image3.pos"’ outlist=’
\ast
.acc’ namelist=new_position_lists
In this example the position list names are not previously associated with the images and must have their names given explicitly (and in the same order as the image names). The output lists are called the same names as the input images except with the extension .acc. The names of the output lists are written into the file new_position_lists which can be used to pass these names onto another application using indirection (in which invoke the next application with ndfnames=false inlist=
\text{^}
new_position_lists).
CCDPACK format – the first three columns are interpreted as the following.
EXTERNAL format – positions are specified using just an X and a Y entry and no other entries.
Data following the third column is copied without modification into the results files
If NDFNAMES is TRUE then the item "CURRENT_LIST" of the .MORE.CCDPACK structure of the input images will be located and assumed to contain the names of the lists whose positions are to be centroided. On exit this item will be updated to reference the name of the centroided list of positions.
This routine correctly processes the DATA and QUALITY components of an NDF data structure. Bad pixels and all non-complex numeric data types can be handled.
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Materials | Free Full-Text | Impact of the Loading Conditions and the Building Directions on the Mechanical Behavior of Biomedical β-Titanium Alloy Produced In Situ by Laser-Based Powder Bed Fusion
Influence of Adaptive Gap Control Mechanism and Tool Electrodes on Machining Titanium (Ti-6Al-4V) Alloy in EDM Process
In Situ Observation of Liquid Solder Alloys and Solid Substrate Reactions Using High-Voltage Transmission Electron Microscopy
Ben Boubaker, H.
Laheurte, P.
Biriaie, S.
Lohmuller, P.
Housseme Ben Boubaker
Gael Le Coz
Seyyed-Saeid Biriaie
Abdelhadi Moufki
Universite de Lorraine, CNRS, LEM3, Arts et Metiers ParisTech, 57070 Metz, France
Universite de Lorraine, CNRS, LEM3, IMT, GIP InSIC, 88100 Saint Die des Vosges, France
Academic Editor: Filippo Berto
In order to simulate micromachining of Ti-Nb medical devices produced in situ by selective laser melting, it is necessary to use constitutive models that allow one to reproduce accurately the material behavior under extreme loading conditions. The identification of these models is often performed using experimental tension or compression data. In this work, compression tests are conducted to investigate the impact of the loading conditions and the laser-based powder bed fusion (LB-PBF) building directions on the mechanical behavior of
\beta
-Ti42Nb alloy. Compression tests are performed under two strain rates (1 s
{}^{-1}
and 10 s
{}^{-1}
) and four temperatures (298 K, 673 K, 873 K and 1073 K). Two LB-PBF building directions are used for manufacturing the compression specimens. Therefore, different metallographic analyses (i.e., optical microscopy (OM), scanning electron microscopy (SEM), energy-dispersive X-ray (EDX), electron backscatter diffraction (EBSD) and X-ray diffraction) have been carried out on the deformed specimens to gain insight into the impact of the loading conditions on microstucture alterations. According to the results, whatever the loading conditions are, specimens manufactured with a building direction of 45
{}^{\circ }
exhibit higher flow stress than those produced with a building direction of 90
{}^{\circ }
, highlighting the anisotropy of the as-LB-PBFed alloy. Additionally, the deformed alloy exhibits at room temperature a yielding strength of 1180 ± 40 MPa and a micro-hardness of 310 ± 7 HV
{}_{0.1}
. Experimental observations demonstrated two strain localization modes: a highly deformed region corresponding to the localization of the plastic deformation in the central region of specimens and perpendicular to the compression direction and an adiabatic shear band oriented with an angle of ±45 with respect to same direction. View Full-Text
Keywords: Ti-Nb alloy; additive manufacturing; selective laser melting; characterization; thermomechanical behavior Ti-Nb alloy; additive manufacturing; selective laser melting; characterization; thermomechanical behavior
Ben Boubaker, H.; Laheurte, P.; Le Coz, G.; Biriaie, S.-S.; Didier, P.; Lohmuller, P.; Moufki, A. Impact of the Loading Conditions and the Building Directions on the Mechanical Behavior of Biomedical β-Titanium Alloy Produced In Situ by Laser-Based Powder Bed Fusion. Materials 2022, 15, 509. https://doi.org/10.3390/ma15020509
Ben Boubaker H, Laheurte P, Le Coz G, Biriaie S-S, Didier P, Lohmuller P, Moufki A. Impact of the Loading Conditions and the Building Directions on the Mechanical Behavior of Biomedical β-Titanium Alloy Produced In Situ by Laser-Based Powder Bed Fusion. Materials. 2022; 15(2):509. https://doi.org/10.3390/ma15020509
Ben Boubaker, Housseme, Pascal Laheurte, Gael Le Coz, Seyyed-Saeid Biriaie, Paul Didier, Paul Lohmuller, and Abdelhadi Moufki. 2022. "Impact of the Loading Conditions and the Building Directions on the Mechanical Behavior of Biomedical β-Titanium Alloy Produced In Situ by Laser-Based Powder Bed Fusion" Materials 15, no. 2: 509. https://doi.org/10.3390/ma15020509
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Estimate state-space model by reduction of regularized ARX model - MATLAB ssregest - MathWorks India
\begin{array}{l}\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+Ke\left(t\right)\\ y\left(t\right)=Cx\left(t\right)+Du\left(t\right)+e\left(t\right)\end{array}
\left({\lambda }_{1},\sigma ±j\omega ,{\lambda }_{2}\right)
\left[\begin{array}{cccc}{\lambda }_{1}& 0& 0& 0\\ 0& \sigma & \omega & 0\\ 0& -\omega & \sigma & 0\\ 0& 0& 0& {\lambda }_{2}\end{array}\right]
P\left(s\right)={s}^{n}+{\alpha }_{1}{s}^{n-1}+\dots +{\alpha }_{n-1}s+{\alpha }_{n}
A=\left[\begin{array}{ccc}\begin{array}{l}0\\ 1\\ 0\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{ccc}\begin{array}{cc}\begin{array}{l}0\\ 0\\ 0\\ 1\\ ⋮\\ 0\end{array}& \begin{array}{l}\dots \\ \dots \\ \dots \\ \dots \\ \ddots \\ \dots \end{array}\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\\ ⋮\\ 1\end{array}& \begin{array}{l}-{\alpha }_{n}\\ -{\alpha }_{n-1}\\ -{\alpha }_{n-2}\\ -{\alpha }_{n-3}\\ \text{ }⋮\\ -{\alpha }_{1}\end{array}\end{array}\end{array}\right]
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Global Constraint Catalog: Cdisjunctive
<< 5.125. disjoint_tasks5.127. disjunctive_or_same_end >>
[Carlier82]
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
\mathrm{𝚘𝚗𝚎}_\mathrm{𝚖𝚊𝚌𝚑𝚒𝚗𝚎}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\left[\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗},\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\right]\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\ge 0
All the tasks of the collection
\mathrm{𝚃𝙰𝚂𝙺𝚂}
that have a duration strictly greater than 0 should not overlap.
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-1\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-3,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-2\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-0,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-7\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-2,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-4\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-1\hfill \end{array}〉\hfill \end{array}\right)
Figure 5.126.1 shows the tasks with non-zero duration of the example. Since these tasks do not overlap the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
Figure 5.126.1. Tasks with non-zero duration of the Example slot
Figure 5.126.2 gives all solutions to the following non ground instance of the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint:
{O}_{1}\in \left[2,5\right]
{D}_{1}\in \left[2,4\right]
{O}_{2}\in \left[2,4\right]
{D}_{2}\in \left[1,6\right]
{O}_{3}\in \left[3,6\right]
{D}_{3}\in \left[4,4\right]
{O}_{4}\in \left[2,7\right]
{D}_{4}\in \left[1,3\right]
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
\left(〈{O}_{1}{D}_{1},{O}_{2}{D}_{2},{O}_{3}{D}_{3},{O}_{4}{D}_{4}〉\right)
Figure 5.126.2. All solutions corresponding to the non ground example of the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint of the All solutions slot
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|>2
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\ge 1
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}
can be decreased to any value
\ge 0
One and the same constant can be added to the
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
attribute of all items of
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint occurs in many resource scheduling problems in order to model a resource that can not be shared. This means that tasks using this resource can not overlap in time. Quite often
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraints are used together with precedence constraints. A precedence constraint between two tasks models the fact that the processing of a task has to be postponed until an other task is completed. Such mix of disjunctive and precedence constraints occurs for instance in job-shop problems.
Some systems like Ilog CP Optimizer also imposes that zero duration tasks do not overlap non-zero duration tasks.
A soft version of this constraint, under the hypothesis that all durations are fixed, was presented by P. Baptiste et al. in [BaptisteLePapePeridy98]. In this context the goal was to perform as many tasks as possible within their respective due-dates.
When all tasks have the same (fixed) duration the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint can be reformulated as an
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
constraint for which a filtering algorithm achieving bound-consistency is available [ArtiouchineBaptiste05].
Within the context of linear programming [Hooker07book] provides several relaxations of the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
Some solvers use in a pre-processing phase, while stating precedence and cumulative constraints, an algorithm for automatically extracting large cliques [BronKerbosch73] from a set of tasks that should not pairwise overlap (i.e., two tasks
{t}_{i}
{t}_{j}
can not overlap either, because
{t}_{i}
ends before the start of
{t}_{j}
, either because the sum of resource consumption of
{t}_{i}
{t}_{j}
exceeds the capacity of a cumulative resource that both tasks use) in order to state
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
We have four main families of methods for handling the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
Methods based on the compulsory part [Lahrichi82] of the tasks (also called time-tabling methods). These methods determine the time slots which for sure are occupied by a given task, an propagate back this information to the attributes of each task (i.e., the origin and the duration). Because of their simplicity, these methods have been originally used for handling the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint. Even if they propagate less than the other methods they can in practice handle a large number of tasks. To our best knowledge no efficient incremental algorithm devoted to this problem was published up to now (i.e., September 2006).
Methods based on constructive disjunction. The idea is to try out each alternative of a disjunction (e.g., given two tasks
{t}_{1}
{t}_{2}
that should not overlap, we successively assume that
{t}_{1}
finishes before
{t}_{2}
{t}_{2}
{t}_{1}
) and to remove values that were pruned in both alternatives.
Methods based on edge-finding. Given a set of tasks
𝒯
, edge-finding determines that some task must, can, or cannot execute first or last in
𝒯
. Efficient edge-finding algorithms for handling the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint were originally described in [CarlierPinson88], [CarlierPinson90] and more recently in [Vilim04], [PeridyRivreau05].
Methods that, for any task
t
, consider the maximal number of tasks that can end up before the start of task
t
as well as the maximal number of tasks that can start after the end of task
t
[Wolf05].
All these methods are usually used for adjusting the minimum and maximum values of the variables of the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint. However some systems use these methods for pruning the full domain of the variables. Finally, Jackson priority rule [Jackson55] provides a necessary condition [CarlierPinson90] for the
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint. Given a set of tasks
𝒯
, it consists to progressively schedule all tasks of
𝒯
It assigns to the first possible time point (i.e., the earliest start of all tasks of
𝒯
) the available task with minimal latest end. In this context, available means a task for which the earliest start is less than or equal to the considered time point.
It continues by considering the next time point until all the tasks are completely scheduled.
disjunctive in Choco, unary in Gecode.
\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}
\mathrm{𝚍𝚒𝚜𝚓}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚎𝚗𝚍}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚝𝚊𝚛𝚝}
(scheduling constraint).
generalisation:
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚝𝚊𝚜𝚔}
\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}
\mathrm{𝚛𝚎𝚜𝚘𝚞𝚛𝚌𝚎}
\mathrm{𝚕𝚒𝚖𝚒𝚝}
are not necessarly all equal to 1),
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚝𝚊𝚜𝚔}
\mathrm{𝚑𝚎𝚒𝚐𝚝𝚑}
1 replaced by orthotope).
\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎𝚗𝚌𝚎}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚎𝚗𝚍}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}_\mathrm{𝚘𝚛}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚝𝚊𝚛𝚝}
specialisation:
\mathrm{𝚊𝚕𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚍𝚒𝚜𝚝}
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
\mathrm{𝚕𝚒𝚗𝚎}\mathrm{𝚜𝚎𝚐𝚖𝚎𝚗𝚝}
, of same length),
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚝𝚊𝚜𝚔}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
characteristic of a constraint: core, sort based reformulation.
complexity: sequencing with release times and deadlines.
constraint type: scheduling constraint, resource constraint, decomposition.
filtering: compulsory part, constructive disjunction, Phi-tree.
modelling: disjunction, sequence dependent set-up, zero-duration task.
modelling exercises: sequence dependent set-up.
problems: maximum clique.
Cond. implications
•
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\right)>0
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)
•
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\right)>0
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚜𝚝}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝚃𝙰𝚂𝙺𝚂}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(<\right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1},\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}\right)
\bigvee \left(\begin{array}{c}\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}=0,\hfill \\ \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}=0,\hfill \\ \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}+\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\le \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗},\hfill \\ \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}+\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\le \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\hfill \end{array}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚃𝙰𝚂𝙺𝚂}|*\left(|\mathrm{𝚃𝙰𝚂𝙺𝚂}|-1\right)/2
We generate a clique with a non-overlapping constraint between each pair of distinct tasks and state that the number of arcs of the final graph should be equal to the number of arcs of the initial graph.
Parts (A) and (B) of Figure 5.126.3 respectively show the initial and final graph associated with the Example slot. The
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
constraint holds since all the arcs of the initial graph belong to the final graph: all the non-overlapping constraints holds.
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
: checking whether a ground instance holds or not
|
Surface integral - Simple English Wikipedia, the free encyclopedia
In mathematics, a surface integral is a definite integral taken over a surface (which may be a curve set in space). Just as a line integral allows one to integrate over an arbitrary curve (of one dimension), a surface integral can be thought of as a double integral integrating over a two-dimensional surface. Given a surface, one may integrate over its scalar fields (that is, functions which return numbers as values), or its vector fields (that is, functions which return vectors as values).
Surface integrals have applications in physics, particularly with the classical theory of electromagnetism.
An illustration of a single surface element. These elements are made infinitesimally small, by the limiting process, so as to approximate the surface.
1 Surface integrals of scalar fields
2 Surface integrals of vector fields
3 Theorems involving surface integrals
4.1 Changing parametrization
4.2 Parameterizations work on parts of the surface
4.3 Inconsistent surface normals
Surface integrals of scalar fields[change | change source]
Consider a surface S on which a scalar field f is defined. If one thinks of S as made of some material, and for each x in S the number f(x) is the density of material at x, then the surface integral of f over S is the mass per unit thickness of S. (This is only true if the surface is an infinitesimally thin shell.)
One approach to calculating the surface integral is then to split the surface in many very small pieces, assume that the density is approximately constant on each piece, find the mass per unit thickness of each piece (by multiplying the density of the piece by its area), and then sum up the resulting numbers to find the total mass per unit thickness of S.
To find an explicit formula for the surface integral, mathematicians parametrize S by considering on S a system of curvilinear coordinates, like the latitude and longitude on a sphere. Let such a parametrization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by[1][2]
{\displaystyle \int _{S}f\,dS=\iint _{T}f(\mathbf {x} (s,t))\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|ds\,dt}
where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t).
For example, to find the surface area of some general functional shape, say
{\displaystyle z=f\,(x,y)}
{\displaystyle A=\int _{S}\,dS=\iint _{T}\left\|{\partial \mathbf {r} \over \partial x}\times {\partial \mathbf {r} \over \partial y}\right\|dx\,dy}
{\displaystyle \mathbf {r} =(x,y,z)=(x,y,f(x,y))}
{\displaystyle {\partial \mathbf {r} \over \partial x}=(1,0,f_{x}(x,y))}
{\displaystyle {\partial \mathbf {r} \over \partial y}=(0,1,f_{y}(x,y))}
{\displaystyle {\begin{aligned}A&{}=\iint _{T}\left\|\left(1,0,{\partial f \over \partial x}\right)\times \left(0,1,{\partial f \over \partial y}\right)\right\|dx\,dy\\&{}=\iint _{T}\left\|\left(-{\partial f \over \partial x},-{\partial f \over \partial y},1\right)\right\|dx\,dy\\&{}=\iint _{T}{\sqrt {\left({\partial f \over \partial x}\right)^{2}+\left({\partial f \over \partial y}\right)^{2}+1}}\,\,dx\,dy\end{aligned}}}
which is the formula used for the surface area of a general functional shape. One can recognize the vector in the second line above as the normal vector to the surface.
Note that because of the presence of the cross product, the above formulas only work for surfaces embedded in three dimensional space.
Surface integrals of vector fields[change | change source]
A vector field on a surface.
Consider a vector field v on S, that is, for each x in S, v(x) is a vector.
The surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector. For example, this applies to the electric field at some fixed point due to an electrically charged surface, or the gravity at some fixed point due to a sheet of material. It can also calculate the magnetic flux through a surface.
Alternatively, mathematicians can integrate the normal component of the vector field; the result is a scalar. An example is a fluid flowing through S, such that v(x) determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in a unit amount of time.
This illustration implies that if the vector field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S, and neither in nor out. This also implies that if v does not just flow along S, that is, if v has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field as above. This gives the formula
{\displaystyle \int _{S}{\mathbf {v} }\cdot \,d{\mathbf {S} }=\int _{S}({\mathbf {v} }\cdot {\mathbf {n} })\,dS=\iint _{T}{\mathbf {v} }(\mathbf {x} (s,t))\cdot \left({\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right)ds\,dt.}
The cross product on the right-hand side of this expression is a surface normal determined by the parametrization.
This formula defines the integral on the left (note the dot and the vector notation for the surface element).
Theorems involving surface integrals[change | change source]
Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem.
Advanced issues[change | change source]
Changing parametrization[change | change source]
The discussion above defined the surface integral by using a parametrization of the surface S. A given surface might have several parametrizations. For example, when the locations of the North Pole and South Pole are moved on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple: the value of the surface integral will be the same no matter what parametrization one uses.
Integrals of vector fields are more complicated, because the surface normal is involved. Mathematicians have proved that given two parametrizations of the same surface, whose surface normals point in the same direction, both parametrizations give the same value for the surface integral. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. It follows that given a surface, we do not need to stick to any unique parametrization; but, when integrating vector fields, we do need to decide in advance which direction the normal will point to, and then choose any parametrization consistent with that direction.
Parameterizations work on parts of the surface[change | change source]
Another issue is that sometimes surfaces do not have parametrizations which cover the whole surface; this is true for example for the surface of a cylinder (of finite height). The obvious solution is then to split that surface in several pieces, calculate the surface integral on each piece, and then add them all up. This is indeed how things work, but when integrating vector fields, one needs to again be careful in choosing the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. For the cylinder, this means that if we decide that for the side region the normal will point out of the body, then for the top and bottom circular parts, the normal must point out of the body too.
Inconsistent surface normals[change | change source]
Last, there are surfaces which do not have a surface normal at each point with consistent results (for example, the Möbius strip). If such a surface is split into pieces, on each piece a parametrization and corresponding surface normal is chosen, and the pieces are put back together, the normal vectors coming from different pieces cannot be reconciled. This means that at some junction between two pieces will have normal vectors pointing in opposite directions. Such a surface is called non-orientable. Vector fields cannot be integrated on non-orientable surfaces.
Volume and surface area elements in a spherical coordinate system
Volume and surface area elements in a cylindrical coordinate system
Holstein–Herring method
↑ "Surface integrals (article)". Khan Academy. Retrieved 2020-09-19.
Surface Integral -- from MathWorld
Surface Integral -- Theory and exercises
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Surface_integral&oldid=8023291"
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On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables
Tolev, Doychin
Nous considérons une version affaiblie de la conjecture sur la représentation des entiers comme somme de quatre carrés de nombres premiers.
We consider an approximation to the popular conjecture about representations of integers as sums of four squares of prime numbers.
author = {Tolev, Doychin},
title = {On the exceptional set of {Lagrange{\textquoteright}s} equation with three prime and one almost{\textendash}prime variables},
AU - Tolev, Doychin
TI - On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables
Tolev, Doychin. On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 925-948. doi : 10.5802/jtnb.528. http://www.numdam.org/articles/10.5802/jtnb.528/
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[2] J. Brüdern, E. Fouvry, Lagrange’s Four Squares Theorem with almost prime variables. J. Reine Angew. Math. 454 (1994), 59–96. | MR 1288679 | Zbl 0809.11060
[3] G. Greaves, On the representation of a number in the form
{x}^{2}+{y}^{2}+{p}^{2}+{q}^{2}
p
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are odd primes. Acta Arith. 29 (1976), 257–274. | MR 404182 | Zbl 0283.10030
[4] H. Halberstam, H.-E. Richert, Sieve methods. Academic Press, 1974. | MR 424730 | Zbl 0298.10026
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[6] G. Harman, A. V. Kumchev, On sums of squares of primes. Math. Proc. Cambridge Philos. Soc., to appear. | MR 2197572 | Zbl 05012459
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[8] D.R. Heath-Brown, D.I.Tolev, Lagrange’s four squares theorem with one prime and three almost–prime variables. J. Reine Angew. Math. 558 (2003), 159–224. | MR 1979185 | Zbl 1022.11050
[9] L.K. Hua, Some results in the additive prime number theory. Quart. J. Math. Oxford 9 (1938), 68–80. | Zbl 0018.29404
[10] L.K. Hua, Introduction to number theory. Springer, 1982. | MR 665428 | Zbl 0483.10001
[11] L.K. Hua, Additive theory of prime numbers. American Mathematical Society, Providence, 1965. | MR 194404 | Zbl 0192.39304
[12] H. Iwaniec, Rosser’s sieve. Acta Arith. 36 (1980), 171–202. | MR 581917 | Zbl 0435.10029
[13] H. Iwaniec, A new form of the error term in the linear sieve. Acta Arith. 37 (1980), 307–320. | MR 598883 | Zbl 0444.10038
[14] H.D. Kloosterman, On the representation of numbers in the form
a{x}^{2}+b{y}^{2}+c{z}^{2}+d{t}^{2}
. Acta Math. 49 (1926), 407–464.
[15] J. Liu, On Lagrange’s theorem with prime variables. Quart. J. Math. Oxford, 54 (2003), 453–462. | MR 2031178 | Zbl 1080.11071
[16] J. Liu, M.-C. Liu, The exceptional set in the four prime squares problem. Illinois J. Math. 44 (2000), 272–293. | MR 1775322 | Zbl 0942.11044
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\Large \begin{array} {c c c c } & & \color{#69047E}{X}& \color{#69047E}{X} \\ & & \color{#D61F06}{Y} & \color{#D61F06}{Y} \\ + & & \color{#3D99F6}{Z} & \color{#3D99F6}{Z} \\ \hline & \color{#69047E}{X} & \color{#D61F06}{Y} & \color{#3D99F6}{Z} \\ \end{array}
If each letter represents a distinct digit, what is the value of the three-digit number
\overline{XYZ}?
\begin{array}{ccccc} & & & & A&B\\ \times & & & & A &A \\ \hline & & B & A & A &B \end{array}
Solve the above cryptogram. What is the first two-digit number in the product above,
\overline{AB}
Note: A number cannot start with 0, so A and B are non-zero.
by Akshat Sharda
\begin{array} { l l l l l } & & & & & 9 & 9 & 9 \\ \times & & & & & A & B & C \\ \hline & & D & E & F & 1 & 3 & 2 \\ \end{array}
A,B,C,D,E
F
are (not necessarily distinct) single digits. What is the value of
A+B+C+D+E+F?
by dharmendra kumar rai
\begin{array} { l l l l l } & S & E & N & D \\ + & M & O & R & E \\ \hline M & O & N & E & Y \\ \end{array}
In this cryptogram, each letter represents a distinct single digit positive integer except
O
which is equal to 0. Find the value of
\overline{MONEY}.
by Moinul Islam Tanvir
\large{\begin{array}{cccccc} & & & A & B & C&D\\ \times & & & & & &D\\ \hline & & & D& C & B&A\\ \end{array}}
A,B,C
D
are distinct single digit non-negative integers satisfying the cryptogram above, find
A+B+C+D
by saptarshi sen
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Plot simulated time response of dynamic system to arbitrary inputs; simulated response data - MATLAB lsim
\begin{array}{cc}sys\left(s\right)=\frac{{\omega }^{2}}{{s}^{2}+2s+{\omega }^{2}},& \omega =62.83\end{array}.
sys\left({z}^{-1}\right)=\frac{{a}_{0}+{a}_{1}{z}^{-1}+\dots +{a}_{n}{z}^{-n}}{1+{b}_{1}{z}^{-1}+\dots +{b}_{n}{z}^{-n}},
y\left[k\right]={a}_{0}u\left[k\right]+\dots +{a}_{n}u\left[k-n\right]-{b}_{1}y\left[k-1\right]-\dots -{b}_{n}\left[k-n\right].
\begin{array}{c}x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right],\\ y\left[n\right]=Cx\left[n\right]+Du\left[n\right].\end{array}
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Global Constraint Catalog: Cpath_from_to
<< 5.315. path5.317. pattern >>
[AlthausBockmayrElfKasperJungerMehlhorn02]
\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}\left(\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚃𝙾}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝙵𝚁𝙾𝙼}\ge 1
\mathrm{𝙵𝚁𝙾𝙼}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚃𝙾}\ge 1
\mathrm{𝚃𝙾}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
Select some arcs of a digraph
G
so that there is still a path between two given vertices of
G
\left(\begin{array}{c}4,3,〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{5\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{5\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3\right\}\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
constraint holds since within the digraph
G
corresponding to the item of the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection there is a path from vertex
\mathrm{𝙵𝚁𝙾𝙼}=4
\mathrm{𝚃𝙾}=3
: this path starts from vertex 4, enters vertex 5, and ends up in vertex 3.
\mathrm{𝙵𝚁𝙾𝙼}\ne \mathrm{𝚃𝙾}
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
(path),
\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚜𝚎𝚝}_\mathrm{𝚝𝚘}_\mathrm{𝚋𝚘𝚘𝚕𝚎𝚊𝚗𝚜}
(constraint involving set variables),
\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝚝𝚎𝚖𝚙𝚘𝚛𝚊𝚕}_\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
filtering: linear programming.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\right)=1
Within the context of the Example slot, part (A) of Figure 5.316.1 shows the initial graph from which we choose to start. It is derived from the set associated with each vertex. Each set describes the potential values of the
\mathrm{𝚜𝚞𝚌𝚌}
attribute of a given vertex. Part (B) of Figure 5.316.1 gives the final graph associated with the Example slot. Since we use the
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
graph property we show on the final graph the following information:
The vertices that respectively correspond to the start and the end of the required path are stressed in bold.
\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
constraint holds since there is a path from vertex 4 to vertex 3 (4 and 3 refer to the
\mathrm{𝚒𝚗𝚍𝚎𝚡}
attribute of a vertex).
\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
Since the maximum value returned by the graph property
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
is equal to 1 we can rewrite
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\right)=1
\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}
\left(\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\right)\ge 1
. Therefore we simplify
\underline{\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}}
\overline{\mathrm{𝐏𝐀𝐓𝐇}_\mathrm{𝐅𝐑𝐎𝐌}_\mathrm{𝐓𝐎}}
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Global Constraint Catalog: Klogic
<< 3.7.137. Line segments intersection3.7.139. Logigraphe >>
\mathrm{𝚌𝚘𝚗𝚝𝚊𝚒𝚗𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚌𝚘𝚟𝚎𝚛𝚎𝚍𝚋𝚢}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚌𝚘𝚟𝚎𝚛𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
\mathrm{𝚐𝚎𝚘𝚜𝚝}_\mathrm{𝚝𝚒𝚖𝚎}
\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚖𝚎𝚎𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚗𝚘𝚗}_\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚘𝚗}_\mathrm{𝚝𝚘𝚙}_\mathrm{𝚘𝚏}_\mathrm{𝚘𝚛𝚝𝚑}
\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚙𝚕𝚊𝚌𝚎}_\mathrm{𝚒𝚗}_\mathrm{𝚙𝚢𝚛𝚊𝚖𝚒𝚍}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚒𝚗}_\mathrm{𝚌𝚘𝚗𝚝𝚊𝚌𝚝}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚌𝚘𝚕𝚞𝚖𝚗}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚍𝚘}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚒𝚗𝚌𝚕𝚞𝚍𝚎}
A constraint which can be defined with first order logic formula encoded in the dedicated language introduced in [CarlssonBeldiceanuMartin08].
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<< 2.1.5. Cost violation view2.1.7. Counting view >>
Suppose we want to associate a 0-1 domain variable
b
to a constraint
𝒞
and maintain the equivalence
b\equiv 𝒞
. This is called the reification of
𝒞
. For most global constraints this can be achieved by reformulating the global constraint as a conjunction of pure functional dependency constraints together with constraints that can be easily reified, e.g. linear constraints involving at most two variables [BeldiceanuCarlssonFlenerPearson13].
We can reify the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈{x}_{1},{x}_{2},\cdots ,{x}_{n}〉\right)
constraint by using the idea of sorting its variables (i.e., the pure functional dependency part) and by stating that within the sorted list of variables adjacent variables are in strictly increasing order. This leads to the following expression
\mathrm{𝚜𝚘𝚛𝚝}
\left(〈{x}_{1},{x}_{2},\cdots ,{x}_{n}〉,〈{y}_{1},{y}_{2},\cdots ,{y}_{n}〉\right)\wedge \left({y}_{1}<{y}_{2}\wedge {y}_{2}<{y}_{3}\wedge \cdots \wedge {y}_{n-1}<{y}_{n}\right)\equiv b
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Percentages | Brilliant Math & Science Wiki
Ashish Menon, Manjunath Sreedaran, Yash Dev Lamba, and
A percent is a number that represents the fractional part out of 100 (per cent literally means per one hundred). Thus 94% means
\frac{94}{100}
. Likewise we can represent a denominator of 100 with decimals by moving the decimal point by 2 places. So
94 \times \frac{1}{100} = 0.94
94\% = 0.94 = \frac{94}{100}
. The main aim of this wiki is to discuss about percentages.
Percentages - Word Problems
50
25?
\begin{aligned} 25 & =\dfrac{25}{50} × 100 \%\\\\ & = \dfrac{1}{2} × 100 \%\\\\ & = 50\%.\ _\square \end{aligned}
75
100?
\begin{aligned} 100 & =\dfrac{100}{75} × 100 \%\\\\ & = \dfrac{4}{3} × 100 \%\\\\ & = 133.33\%.\ _\square \end{aligned}
What number is
35\%
200?
\begin{aligned} \text{35\% of }200 & = \dfrac {35}{100} × 200\\\\ & = 70.\ _\square \end{aligned}
150\%
50?
\begin{aligned} \text{150\% of }50 & = \dfrac {150}{100} × 50\\\\ & = 75.\ _\square \end{aligned}
\Large {25\%}^{{50\%}}= \, ?
The simplest way to perform arithmetic with percentages is to convert them to their decimal equivalent by moving the decimal two places, then adding the decimals.
If John has
37 \%
of the apples and Sally has
42 \%
, what percentage of the apples do they have if they combine their apple piles?
Converting the percentages to decimal and adding, we see that
0.37 + 0.42 = 0.79 = 79 \%
79 \%
_\square
John and Mac were friends. In the election of the class monitor John secured
30\%
of the votes and Mac secured
45\%
votes. Being friends, they decided to combine their votes and mind the class together. If the total votes polled were
240
, what is the total number of votes they jointly secured?
\begin{aligned} \text{Total votes secured by John and Mac} & = \text {(30+45)\% of 240}\\\\ & = \text{75\% of 240}\\\\ & = \dfrac{75}{100} × 240\\\\ & = 180.\ _\square \end{aligned}
\LARGE\underbrace{2016\%+2016\%+\ldots+2016\%}_{2016\text{ times}}=(2016\%)^2
John has
10\%
20\%
30\%
1000000
bananas. How many bananas does John have?
\begin{aligned} \text{Number of bananas John has} & = \dfrac{10}{100} × \dfrac{20}{100} × \dfrac{30}{100} ×1000000\\\\ & = {\dfrac{10}{\cancel {100}} × \dfrac{20}{\cancel {100}} × \dfrac{30}{\cancel {100}} × \cancel{1000000}}\\\\ & = 6000.\ _\square \end{aligned}
Given two values of some variable
x
taken at different points of time, percentage change
\%\Delta
measures the proportion of the difference between the two values to the original reading of
x
. If the value of
\Delta
is positive, we call it a percentage increase; if negative, then decrease.
The general formula for percentage change is
\%\Delta = \frac {x_1 - x_0} {x_0} \times 100\% .
The price of a commodity changed from
\$20
\$30
. Find the percentage change in price of the commodity.
\begin{aligned} \text {Change in the price of the commodity} & = \$30 - \$20\\ & = \$10\\\\ \text {Original price of the commodity} & = \$20\\\\ \text{\% change in the price of the commodity} & = \dfrac{10}{20} × 100 \%\\ & = 50\%.\ _\square \end{aligned}
Imagine a city with 10000 people at the beginning of a year. Given that the population at the end of that year is 10500, compute the percentage change of the population.
Let us assign symbols to the population readings:
\begin{aligned} x_0 &= 10000\\ x_1 &= 10500. \end{aligned}
We apply the formula for pecentage change:
\begin{aligned} \%\Delta &= \frac {x_1 - x_0} {x_0} \times 100\%\\ &= \frac {10500 - 10000} {10000} \times 100\%\\ &= \frac {500} {10000} \times 100\%\\ &= 0.05 \times 100\%\\ &= 5\%. \end{aligned}
We say that the percentage change of the population from the beginning to the end of the year is an increase of
5\%
. This means that population grew 5% in proportion to the initial count.
_\square
The price of a commodity is
\$10
. If its price increases by
10
%, then what is the new price?
\begin{aligned} \text{10\% of 10} & = \dfrac {10×10}{100}\\ & = \dfrac {100}{100}\\ & = 1\\\\ \Rightarrow \text{(New price of the commodity in dollars)} & = 10+1\\ & = 11. \ _\square \end{aligned}
Let's take our knowledge of percentages to a whole new level by practicing word problems on them. Some examples are given below.
2384
students of the school,
75\%
attempted the examination, of which
25\%
failed. How many students passed the examination?
\begin{aligned} \text{No. of students who attempted the examination} & = \dfrac {75}{100} × 2384\\ & = 1788\\\\ \text {No. of students who failed the examination} & = \text {25\% of 1788}\\\\ \text {No. of students who passed the examination} & = \text {(100 - 25)\% of 1788}\\ & = \text {75\% of 1788}\\ & = \dfrac{75}{100} × 1788\\ & =1341.\ _\square \end{aligned}
Of all the students enrolled at BRCM Public School,
9\%
are in the band. The band has
180
members. How many students are enrolled at the school?
We know that the
180
band members are equal to
9\%
of the total population of the school. We need to find out the total number of students
(100\%)
that go to BRCM.
We can first divide
180
9
20
students equal
1\%
100
to see how many students equal
100\%
\begin{aligned} \text{100 \% of students } &=100 \times (\text{1 \% of students})\\ &=100 \times (\text{20 students})\\ &=\text{2,000 students}. \end{aligned}
2,000
students at the school.
_\square
In Idaho, there are 23 Democratic delegates, to be proportionally distributed to Bernie Sanders and Hillary Clinton based on the popular vote. Bernie Sanders gets 78.8% of the vote. Hillary Clinton gets 21.2% of the vote. How many delegates should be allotted to each candidate?
For Bernie,
0 .788 \times 23 = 18.124
, round down to 18 delegates because you can't have a decimal of a delegate.
For Hillary,
23- 18 = 5
_\square
Option B Both offer the same price Not enough information Option A
You're given 2 gift vouchers for shopping, as shown above. The option A grants you a free second item after purchasing the product in the normal price. On the other hand, the option B gives you the discount of 40% for the first item and allows you to buy the second one at 60% cost of the first one.
Which option will save you more money?
Cite as: Percentages. Brilliant.org. Retrieved from https://brilliant.org/wiki/percentages/
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Global Constraint Catalog: Karc-consistency
<< 3.7.11. Apartition3.7.13. Arithmetic constraint >>
\mathrm{𝚊𝚋𝚜}_\mathrm{𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚜𝚝}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚍𝚒𝚏𝚏}_\mathtt{0}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚖𝚘𝚍𝚞𝚕𝚘}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}
\mathrm{𝚊𝚗𝚍}
\mathrm{𝚊𝚛𝚒𝚝𝚑}
\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚘𝚛}
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚌𝚕𝚊𝚞𝚜𝚎}_\mathrm{𝚊𝚗𝚍}
\mathrm{𝚌𝚕𝚊𝚞𝚜𝚎}_\mathrm{𝚘𝚛}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚐𝚛𝚘𝚞𝚙𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚘𝚗𝚎𝚜}
\mathrm{𝚌𝚘𝚞𝚗𝚝}
\mathrm{𝚌𝚘𝚞𝚗𝚝𝚜}
\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚍𝚒𝚜𝚌𝚛𝚎𝚙𝚊𝚗𝚌𝚢}
\mathrm{𝚍𝚒𝚟𝚒𝚜𝚒𝚋𝚕𝚎}
\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}_\mathrm{𝚌𝚘𝚗𝚜𝚝𝚛𝚊𝚒𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖}
\mathrm{𝚎𝚕𝚎𝚖}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚗}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚖𝚊𝚝𝚛𝚒𝚡}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚜𝚙𝚊𝚛𝚜𝚎}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚜}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚜}_\mathrm{𝚜𝚙𝚊𝚛𝚜𝚎}
\mathrm{𝚎𝚚}
\mathrm{𝚎𝚚}_\mathrm{𝚌𝚜𝚝}
\mathrm{𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚝}
\mathrm{𝚎𝚡𝚊𝚌𝚝𝚕𝚢}
\mathrm{𝚐𝚎𝚚}
\mathrm{𝚐𝚎𝚚}_\mathrm{𝚌𝚜𝚝}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚐𝚝}
\mathrm{𝚒𝚖𝚙𝚕𝚢}
\mathrm{𝚒𝚗}
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚛𝚎𝚒𝚏𝚒𝚎𝚍}
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕𝚜}
\mathrm{𝚒𝚗}_\mathrm{𝚛𝚎𝚕𝚊𝚝𝚒𝚘𝚗}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}
\mathrm{𝚒𝚗𝚟𝚎𝚛𝚜𝚎}_\mathrm{𝚘𝚏𝚏𝚜𝚎𝚝}
\mathrm{𝚕𝚎𝚚}
\mathrm{𝚕𝚎𝚚}_\mathrm{𝚌𝚜𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚕𝚝}
\mathrm{𝚖𝚊𝚡𝚒𝚖𝚞𝚖}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\mathrm{𝚗𝚊𝚗𝚍}
\mathrm{𝚗𝚎𝚚}
\mathrm{𝚗𝚎𝚚}_\mathrm{𝚌𝚜𝚝}
\mathrm{𝚗𝚘𝚛}
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚒𝚗}
\mathrm{𝚘𝚙𝚙𝚘𝚜𝚒𝚝𝚎}_\mathrm{𝚜𝚒𝚐𝚗}
\mathrm{𝚘𝚛}
\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}
\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎𝚗𝚌𝚎}
\mathrm{𝚜𝚊𝚖𝚎}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚐𝚗}
\mathrm{𝚜𝚒𝚐𝚗}_\mathrm{𝚘𝚏}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚝𝚊𝚐𝚎}_\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚝𝚛𝚎𝚎}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚒𝚗}_\mathrm{𝚌𝚘𝚗𝚝𝚊𝚌𝚝}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚍𝚘}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}
\mathrm{𝚞𝚜𝚎𝚍}_\mathrm{𝚋𝚢}
\mathrm{𝚟𝚎𝚌}_\mathrm{𝚎𝚚}_\mathrm{𝚝𝚞𝚙𝚕𝚎}
\mathrm{𝚡𝚘𝚛}
Denotes that, for a given constraint involving only domain variables, there is a filtering algorithm that ensures arc-consistency. A constraint ctr defined on the distinct domain variables
{V}_{1},\cdots ,{V}_{n}
is arc-consistent if and only if for every pair
\left(V,v\right)
V
is a domain variable of ctr and
v\in \mathrm{𝑑𝑜𝑚}\left(V\right)
, there exists at least one solution to ctr in which
V
is assigned the value
v
. As quoted by C. Bessière in [Bessiere06], “a different name has often been used for arc-consistency on non-binary constraints”, like domain consistency, generalised arc-consistency or hyper arc-consistency.
There is also a weaker form of arc-consistency that also try to remove values from the middle of the domain of a variable
V
(i.e., unlike bound-consistency which focus on reducing the minimum and maximum value of a variable), called range consistency in [Bessiere06], that is defined in the following way. A constraint ctr defined on the distinct domain variables
{V}_{1},\cdots ,{V}_{n}
is range-consistent if and only if, for every pair
\left(V,v\right)
V
v\in \mathrm{𝑑𝑜𝑚}\left(V\right)
, there exists at least a solution to ctr in which, (1)
V
v
, and (2) each variable
U\in \left\{{V}_{1},\cdots ,{V}_{n}\right\}
distinct from
V
is assigned a value located in its range
\left[\underline{U},\overline{U}\right]
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1Faculty of Geological Engineering, China University of Geosciences, Wuhan, China.
2Institute of Geology, Earthquake Engineering and Seismology, National Academy of Sciences of Tajikistan, Dushanbe, Tajikistan.
DOI: 10.4236/gep.2021.912011 PDF HTML XML 101 Downloads 1,076 Views Citations
In the valley of the Surkhob River, manifestations of modern geological processes, characterized by various forms of manifestation, are widely developed. Studying these processes is of utmost importance, primarily landslides, which are directly related to the loss of stability of rocks on the slopes. Landslide processes in the Surkhob River valley, regardless of their type, cause significant economic damage to the population and the economy, as well as negatively impacting human living conditions. The primary goal of this project is to map landslide susceptibility using a geographic information system and quantitative and semi-quantitative methods. Landslide susceptibility assessment of this research was conducted using slope (degree), aspect of the slope, curvature, stream power index, topographic wetness index, precipitation and altitude. Except for precipitation, which was collected from the world climate site, most of the causal elements were derived from DEM from the SRTM (Shuttle Radar Topography Mission) database, with cell sizes of 30 m. 416 landslides were discovered from satellite pictures of Google Earth Pro and then validated in the field to analyses the link between causative factors and landslide inventory. To measure the weights of each causal element, the frequency ratio (FR) and the analytical hierarchy process (AHP) were used. The quality of the landslide susceptibility map was determined using the Receiver Operating Characteristics curve (ROC), and the AUC value was determined to be 0.877. It is possible to use the landslide susceptibility map as an engineering and geological basis for establishing a national economic development plan for the territory of the Surkhob River valley.
Mukhammadzoda, S. , Shohnavaz, F. , Ilhomjon, O. and Zhang, G. (2021) Application of Frequency Ratio Method for Landslide Susceptibility Mapping in the Surkhob Valley, Tajikistan. Journal of Geoscience and Environment Protection, 9, 168-189. doi: 10.4236/gep.2021.912011.
TWI=\mathrm{ln}\left(\frac{a}{\mathrm{tan}b}\right)
F{R}_{i}=\frac{\text{Ncell}\left({S}_{i}\right)/\text{Ncell}\left({N}_{i}\right)}{\sum \text{Ncell}\left({S}_{i}\right)/\sum \text{Ncell}\left({N}_{i}\right)}
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RandomCrop - Maple Help
Home : Support : Online Help : Programming : DeepLearning Package : Tensors : Operations on Tensors : RandomCrop
randomly shuffle a Tensor to specified shape
DeepLearning/Tensor/RandomShuffle
randomly shuffle a Tensor along its first dimension
RandomCrop(t,shape,opts)
RandomShuffle(t,opts)
seed=integer[8]
The RandomCrop(t,shape,opts) command constructs a Tensor with shape shape by randomly slicing data from t with uniform probability.
The RandomShuffle(t,opts) command randomly shuffles a Tensor along its first dimension.
Crop a random 3x2 slice from a 4x3 Tensor.
\mathrm{with}\left(\mathrm{DeepLearning}\right):
C≔\mathrm{Constant}\left(〈〈3.5,7.9,-0.5,1〉|〈-3.8,-4.2,9.5,3〉|〈4.,-7.2,-8.2,2〉〉\right)
\textcolor[rgb]{0,0,1}{C}\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{t1}≔\mathrm{RandomCrop}\left(C,[3,2]\right)
\textcolor[rgb]{0,0,1}{\mathrm{t1}}\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{Shape}\left(\mathrm{t1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{undefined}}
Shuffle a Tensor along its first dimension.
\mathrm{t2}≔\mathrm{RandomShuffle}\left(C\right)
\textcolor[rgb]{0,0,1}{\mathrm{t2}}\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}]
The DeepLearning/Tensor/RandomCrop and DeepLearning/Tensor/RandomShuffle commands were introduced in Maple 2018.
DeepLearning,RandomTensor
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Basic principles of CCD data reduction
4 Basic principles of CCD data reduction
The primary aim of CCD data reduction is to remove any effects that are due to the nature of the detector and telescope – the ‘instrumental signature’. This is so that measurements (of intensity and possibly error) can be made that do not require any knowledge about how the data was taken. CCD data requires several corrections to attain this state. The most fundermental of these corrections is the ‘bias level’ subtraction.
The bias level is an electronic offset added to the signal from the CCD that makes sure that the Analogue-to-Digital Converter (ADC) always receives a positive value. The ADC is responsible for converting the signal representing the amount of charge accumulated in a CCD pixel to a digital value. A pixel is one of the discrete elements of the CCD where electrons accumulate during exposure to a light source (you see these as a picture element on your image display). The bias level has an intrinsic noise (induced by the signal amplification) known as the ‘readout noise’ (this one of the features which limits the usefulness of CCDs at very low light levels).
Usually the bias level is removed by the subtraction of specifically taken ‘bias-frames’ (
0
second exposure readouts) or by using estimates derived from bias data that is added in regions around the real data. These regions are known as the bias strips or over/under-scan regions (see Figure §1).
After bias subtraction the data values are now directly related to the number of photons detected in each CCD pixel. The relation between the units of the data and the number of photons is a scale factor known as the gain. In this document the gain factor is referred to as the ADC factor. The units of CCD data before being multiplied by the ADC factor are known as ADUs (Analogue-to-Digital Units). CCD data when calibrated in electrons has a Poissonian noise distribution (if you exclude the readout noise).
Other corrections which are occasionally made to CCD data are dark count subtraction and pre-flash subtraction. These are only usual in older CCD data (but for IR array data the dark current correction is essential). Dark correction is the subtraction of the electron count which accumulates in each pixel due to thermal noise. Modern CCDs usually have dark counts of less than a few ADUs per pixel per hour, so this correction can generally be ignored. Pre-flashing of CCDs has been used to stop loss of signal in CCDs with poor across-chip charge transfer characteristics, the reasoning being that if signal is entered in a pixel before the main exposure, then subsequent losses are less likely to affect the data — note however that this also means that a higher signal to noise level is required for detection.
The final stage in the correction of CCD data for instrument signature is ‘flatfielding’. The sensitivity of CCDs varies from point to point (i.e. the recorded signal per unit of incident flux – photons – is not uniform), so if the data is to be relatively flux calibrated (so that comparison from point to point can be made) this sensitivity variation must be removed. To achieve this correction exposures of a photometrically flat source must be taken, these are known as flatfields. The basic idea of flatfield correction is then to divide the data by a ‘sensitivity map’ created from the calibrations, although in real life noise considerations, together with others (see appendix E), mean that particular care needs to be taken at this stage. After all these corrections have been made your data is usually1 ready for analysis.
Other processes which are frequently undertaken before analysis are registration, alignment, normalisation and combination. Registration is the process of determining the transformations which map the same positions on different datasets. This is essential if measurements, say with different filters, are to be made. In this case registration may be informal and just consists of identifying the same objects on different datasets. However, very accurate measures are often also required; certainly this is the case when data combination is to be performed. ‘Data combination’ is just when aligned datasets are combined by a process of taking the mean or some other estimator at each pixel, this is also frequently referred to as ‘mosaicing’. Aligning datasets means achieving pixel-to-pixel correspondence (in real data it is unlikely that this state is true, even if it was intended). Alignment uses the registering transforms to ‘resample’ the data onto a new pixel grid. If the exposure times, atmospheric transparency or sky brightness have varied, then data must be ‘normalised’ before combination. Normalisation is the determination of the zero points and scale factors which correct for these changes.
1Usually because another correction may also be necessary – the removal of fringing see appendix E.
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Global Constraint Catalog: Cglobal_cardinality_low_up
<< 5.163. global_cardinality5.165. global_cardinality_low_up_no_loop >>
Used for defining
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚐𝚌𝚌}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚐𝚌𝚌}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚘𝚖𝚒𝚗}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚘𝚖𝚊𝚡}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚘𝚖𝚒𝚗},\mathrm{𝚘𝚖𝚊𝚡}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}\le \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
\left(1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚒𝚗}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚊𝚡}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(\begin{array}{c}〈3,3,8,6〉,\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚟𝚊𝚕}-3\hfill & \mathrm{𝚘𝚖𝚒𝚗}-2\hfill & \mathrm{𝚘𝚖𝚊𝚡}-3,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚘𝚖𝚒𝚗}-0\hfill & \mathrm{𝚘𝚖𝚊𝚡}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚘𝚖𝚒𝚗}-1\hfill & \mathrm{𝚘𝚖𝚊𝚡}-2\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint holds since values 3, 5 and 6 are respectively used 2 (
2\le 2\le 3
0\le 0\le 1
1\le 1\le 2
〈3,3,8,6〉
and since no constraint was specified for value 8.
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>1
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}>0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}<|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\mathrm{𝚒𝚗}_\mathrm{𝚊𝚝𝚝𝚛}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}
\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}
\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
Within the context of linear programming [Hooker07book] provides relaxations of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
In MiniZinc (http://www.minizinc.org/) there is also a
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚌𝚕𝚘𝚜𝚎𝚍}
constraint where all variables must be assigned a value from the
\mathrm{𝚟𝚊𝚕}
attribute.
A filtering algorithm achieving arc-consistency for the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint is given in [Regin96]. This algorithm is based on a flow model of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint where there is a one-to-one correspondence between feasible flows in the flow model and solutions of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint. The leftmost part of Figure 3.7.29 illustrates this flow model.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint is entailed if and only if for each value
v
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
) the following two conditions hold:
The number of variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection assigned value
v
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚒𝚗}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection that can potentially be assigned value
v
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚊𝚡}
A reformulation of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
, involving linear constraints, preserving bound-consistency was introduced in [BessiereKatsirelosNarodytskaQuimperWalsh09IJCAI]. For each potential interval
\left[l,u\right]
of consecutive values this model uses
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
0-1 variables
{B}_{1,l,u},{B}_{2,l,u},\cdots ,{B}_{|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|,l,u}
for modelling the fact that each variable of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
is assigned a value within interval
\left[l,u\right]
\forall i\in \left[1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right]:{B}_{i,l,u}⇔l\le \mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}\wedge \mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}\le u
), as well as one domain variable
{C}_{l,u}
for counting how many values of
\left[l,u\right]
are assigned to variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
{C}_{l,u}={B}_{1,l,u}+{B}_{2,l,u}+\cdots +{B}_{|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|,l,u}
). The lower and upper bounds of variable
{C}_{l,u}
are respectively initially set with respect to the minimum and maximum number of possible occurrences of the values of interval
\left[l,u\right]
. Finally, assuming that
s
is the smallest value that can be assigned to the variables of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
, the constraint
{C}_{s,u}={C}_{s,k}+{C}_{k+1,u}
is stated for each
k\in \left[s,u-1\right]
globalCardinality in Choco, global_cardinality_low_up in MiniZinc.
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚏𝚒𝚡𝚎𝚍}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint where the
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
are increasing),
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
(restrictions are done on nested sets of values, all starting from first value).
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}_\mathrm{𝚗𝚘}_\mathrm{𝚕𝚘𝚘𝚙}
(assignment of a
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
to its position is ignored).
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚜𝚎𝚝}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
defines the set of variables that are actually considered).
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(each value should occur at most once).
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
(one
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint for each sliding sequence of
\mathrm{𝚂𝙴𝚀}
consecutive
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
filtering: flow, arc-consistency, bound-consistency, DFS-bottleneck, entailment.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
•
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
\ge \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}
•
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
\le \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
|
Global Constraint Catalog: Krcc8
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\mathrm{𝚌𝚘𝚗𝚝𝚊𝚒𝚗𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚌𝚘𝚟𝚎𝚛𝚎𝚍𝚋𝚢}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚌𝚘𝚟𝚎𝚛𝚜}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚒𝚗𝚜𝚒𝚍𝚎}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚖𝚎𝚎𝚝}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}_\mathrm{𝚜𝚋𝚘𝚡𝚎𝚜}
Region Connection Calculus (i.e., RCC-8) [RandellCuiCohn92] provides eight topological relations (i.e., disjoint, meet, overlap, equal, covers, coveredby, contains, inside) between two fixed objects such that any two fixed objects are in one and exactly one of these topological relations. Figure 3.7.54 illustrates the meaning of each topological relation.
Figure 3.7.54. The eight topological relations of RCC-8 (non-overlapping parts of rectangles A and B are coloured in pink, while overlapping parts are coloured in red)
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Python - Working with predictive analytics · A Blogger who is a Data Science, Web Developer, Game Developer and an enthusiast for anything Tech related.
Posted by Vignesh S on Sun, 8th Sep, 2019
Predictive analytics roadmap
Commonly used Regression Models
We will look at how the companies predict sales, real estate agents predict housing prices or insurance companies predict the healthcare costs.
Let’s look at the Cross Industry Standard process for Data Mining or in short CRISP-DM. The below is the logical steps in order to follow CRISP-DM.
First we need to define the goal. What is our business and what are we trying to predict. Example, How many number of products will we sell?
Now once you have defined your concrete problem statement, you need to have a through understanding of the data you are working with. In this stage you have to make sure you have all the required data to function. If not then acquiring the data or analyzing any alternative must be done in this stage.
Now once you have collected all the necessary data that will be needed to build the model you will need to do clean up of the data and prepare the data ready for the model to run. This includes accounting for missing values, removing outliers, converting to categorical data to one hot encoding, feature scaling, extracting necessary columns etc..
Once you have got the data prepared, you need to split the data into training and test set. And once that is done you can feed the data into the algorithms and then use the test data to test your trained model.
Once you have your model trained you have to evaluate your model using the test data. There are many evaluation metrics. And many times it might not be a perfectly trained model so you may end up going back and forth between Modeling and the Evaluation stages.
Once we get the right trained model we then find a way to successfully deploy the model and start using the model in real world scenario.
Note: Many times there is also a human factor involved in analyzing the model accuracy. Things like subject matter knowledge etc. So the evaluation metrics alone cant be everything.
The data can be of major 2 types which can also be further broken down,
Depending upon the operations we can perform on the data we can categorize them. For example, if we need to predict how many bulldozers we need to remove trees in an area for a landscaping business and things like what paint color we need and other equipments to decorate the area. Then we will be requiring data such as Color of the house, Door numbers of the house, Temperature of the house, Area of the house.
The Color of the house can be blue, red or green which is a Categorical: Nominal data as it cannot be added, multiplied, compared(< | >), or ordered.
The Door numbers of a house can be #1201, #1202, #1203 which comes under Categorical: Ordinal which cannot be added/subtracted or divided/multiplied because it wont make any sense, But it can be ordered in ascending or descending order.
The Temperature of the house can be 23Deg, 24Deg, -3Deg which can be added/subtracted, compared, or ordered. But cannot be multiplied/divided because they are not on the same scale meaning all positive numbers and hence it comes under Numerical: Interval. For example 0Deg is a valid data point and division by 0 does not make any sense.
The Area of a house can be 800sq.ft, 1200sq.ft, 2400sq.ft which can be divided and multiplied as well because, there cannot be a 0sq.ft house or -200sq.ft house. Hence it comes under Numerical: Ratio.
Below is a summarization of various data types with the various operations that can be performed on them.
== or !=
Nominal Yes No No No
Ordinal Yes Yes No No
Interval Yes Yes Yes No
Ratio Yes Yes Yes Yes
Prediction models need numbers hence we need to convert Categorical Data into numbers.
data = pd.read_csv('./insurance.csv') # Load the insurance dataset
print(data.head(15)) # Print the top 15 rows of data from the loaded dataset
# Check how many missing values (NaN) are in each column
count_nan = data.isnull().sum()
# Print the column names which contains NaN
print(count_nan[count_nan > 0])
# Filling in the missing values using the mean of the column
data['bmi'].fillna(data['bmi'].mean(), inplace=True) # The inplace true will change inplace so that the value in the data frame itself changes.
# Check if there are any NaN still present
To convert Categorical Data into Numbers we can do 2 things,
Label Encoding - Two distinct values (Binomial)
One Hot Encoding - Three or more distinct values
We will be using nd arrays to do the LabelEncoding but it can also be done using pandas Series.
from sklearn.preprocessing import LabelEndoder
# create nd array for LabelEncoding
sex = data.iloc[:, 1:2].values
smoker = data.iloc[:, 4:5].values
# perform LabelEncoding for sex
sex[:, 0] = le.fit_transform(sex[:, 0])
sex = pd.DataFrame(sex)
sex.columns = ['sex']
le_sex_mapping = dict(zip(le.classes_, le,.transform(le.classes_)))
print("sklearn label encoder results for sex")
print(le_sex_mapping)
print(sex[:10])
# perform LabelEncoding for smoker
smoker[:, 0] = le.fit_transform(smoker[:, 0])
smoker = pd.DataFrame(smoker)
smoker.columns = ['smoker']
le_smoker_mapping = dict(zip(le.classes_, le,.transform(le.classes_)))
print("sklearn label encoder results for smoker")
print(le_smoker_mapping)
print(smoker[:10])
We will be using nd arrays to do the OneHotEncoding but it can also be done using pandas Series.
from sklearn.preprocessing import OneHotEncoding
# create nd array for OneHotEncoding
region = data.iloc[:, 5:6].values
# perform OneHotEncoding for region
ohe = OneHotEncoding()
region = ohe.fit_transform(region).toarray()
region = pd.DataFrame(region)
region.columns = ['northeast', 'northwest', 'southeast', 'southwest']
print("sklearn one hot encoding results for region")
print(region[:10])
Once we have built our individual DataFrames as we want we might need to combine them into a single clubbed DataFrame and then split the DataFrame into train and test split. The train split will be used in training the model and the test split will be used to evaluate our trained model. We would also need to define the independent(features) and dependent(output) variable in the DataFrame.
# Take the numerical data from the original data
X_num = data[['age', 'bmi', 'children']]
# Take the encoded data and add it to the numerical data
X_final = pd.concat([X_num, sex, smoker, region], axis=1)
# Take the charges column from the original data and assign it as the y_final
y_final = data[['charges']].copy()
# Do the train test split
X_train, X_test, y_train, y_test = train_test_split(X_final, y_final, test_size=0.33, random_state=0)
The values in each column might be in different ranges which means the model will see each columns as being higher or lower than the other columns, which wont be the case. So we need to bring the numerical values in all the columns in the same scale. This process is called FeatureScaling and there are 2 types of FeatureScaling, they are Normalization and Standardization.
By doing FeatureScaling we primarily achieve 2 things.
We make the features with smaller values play a larger role in the model.
We dilute the biases coming from the outliers.
This method is affected more against the outliers.
z=\frac{x - \text{min}(x)}{[\text{max}(x) - \text{min}(x)]}
n_scaler = MinMaxScaler()
X_train = n_scaler.fit_transform(X_train.astype(np.float))
X_test = n_scaler.transform(X_test.astype(np.float))
This method is affected less against the outliers.
z=\frac{x - \mu}{\sigma}
\mu = \text{Mean}
\sigma = \text{Standard Deviation}
s_scaler = StandardScaler()
X_train = s_scaler.fit_transform(X_train.astype(np.float))
X_test = s_scaler.transform(X_test.astype(np.float))
There are 3 kinds of Modeling.
Regression predicts a numerical value. Example: House sales website predicts the price of the house. Use regression if the predicted output in numerical.
Classification predicts a categorical variable. Example: Bank categorize if an expense is a fraud or not. Use classification if the predicted output is Category.
Discover the inherent groupings in the data. Example: Analyze Customer behavior by grouping customers by their purchasing behavior without prior knowledge.
Discover rules that describe portions of the data. Example: Analyze Customer behavior as in Customers who buys product A also buys product B.
The most commonly used regression models are,
y_train_pred = lr.predict(X_train)
# Printing score
print("lr train score: %.3f, lr test score: %.3f" % (lr.score(X_train, y_train), lr.score(X_test, y_test)))
Polynomial Regression affects over-fitting or under-fitting.
Support Vectors was first used for classification problems as SVC(Support Vector Classification). Later it was also applied for Regression problems as SVR to predict numerical data.
In SVR the plane which separates the 2 classes is called the hyperplane. The data points that are sitting the closest to the hyperplane are called the support vectors. The hyperplane that goes through those closest data points are called margins. If we do not allow any of the points to be in the margin area its called Hard margin, if we allow then its called Soft margin.
Kernal functions help to separate the 2 classes by projecting the data into a higher dimension. So that it’s easier to separate them using a hyperplane rather than simple 2D line.
SVR is very similar to SVC.
Output is a continuous number rather than a category.
Goal is to minimize the error and obtain a minimum margin interval which contains the maximum number of data points.
Commonly used Kernal functions,
svr = SVR(kernal="linear", C=300)
X_train, y_train, X_test, y_test = train_test_split(X_final, y_final, test_size=0.33, random_state=0)
# standard scaler (fit transform on train, fit only on test)
X_train = sc.fit_transform(X_train.astype(np.float))
X_test = sc.transform(X_test.astype(np.float))
svr = svr.fit(X_train, y_train.values.ravel())
y_train_pred = svr.predict(X_train)
y_test_pred = svr.predict(X_test)
print('svr train score %.3f, svr test score: %.3f' % (
svr.score(X_train, y_train),
svr.score(X_test, y_test)))
Decision tree model creates a tree based structured decision model. But it tends to overfit. Decision tree does not require data scaling.
dt = DecisionTreeReggressor(random_state = 0)
X_final, y_final, test_size=0.33, random_state=0)
dt = dt.fit(X_train, y_train.values.ravel())
y_train_pred = dt.predict(X_train)
y_test_pred = dt.predict(X_test)
print("dt train score: %.3f, dt test score: %.3f" % (dt.score(X_train, y_train), dt.score(X_test, y_test)))
Random forest regression is a type of ensemble learning, which means multiple types learning methods are used simultaneously. It is much like the previous decision tree model but instead of using a single tree we use multiple trees and take the opinion of all the trees into consideration. Random forest does not require data scaling.
The main difference between a Regression tree and Classification tree is that the regression tree outputs number and the predicted value is calculated from the mean (average value) and classification tree outputs category and the predicted value is calculated from the mode (most occurring value).
Bagging or bootstrap aggregating, subdivides the data into smaller components. So we take the large dataset and then divide it into smaller chunks of data, then we apply machine learning models to each one of them and then finally do the aggregation.
# the n_estimators is the number of trees and the criterion is feature selection method
# n_jobs mentions the number of processors to run in parallel for both training and prediction. -1 means run on all available processors.
forest = RandomForestRegressor(n_estimators = 100, criterion = "mse", random_state = 1, n_jobs = -1)
forest.fit(X_train, y_train.values.ravel())
y_train_pred = forest.predict(X_train)
y_test_pred = forest.predict(X_test)
print("forest train score: %.3f, forest test score: %.3f" % (forest.score(X_train, y_train), forest.score(X_test, y_test)))
Hyperparameter optimization is the process of deriving the ideal set of parameters for prediction algorithm with optimum performance.
Hyperparameter optimization methods
def print_best_params(gd_model):
param_dict = gd_model.best_estimator_.get_params()
model_str = str(gd_model.estimator).split('(')[0]
print("\n*** {} Best Parameters ***".format(model_str))
for k in param_dict:
print("{} {}".format(k, param_dict[k]))
# SVR parameter grid
param_grid_svr = dict(kernel=['linear', 'poly'],
degree=[2],
C=[600, 700, 800, 900],
epsilon=[0.0001, 0.00001, 0.000001])
svr = GridSearchCV(SVR(), param_grid=param_grid_svr, cv=5, verbose=3)
print('\n\n svr train score %.3f, svr test score: %.3f' % (
print_best_params(svr)
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Global Constraint Catalog: Cbipartite
<< 5.55. binary_tree5.57. calendar >>
\mathrm{𝚋𝚒𝚙𝚊𝚛𝚝𝚒𝚝𝚎}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
G
\mathrm{𝙽𝙾𝙳𝙴𝚂}
G
so that the corresponding graph is symmetric (i.e., if there is an arc from
i
j
j
i
) and bipartite (i.e., there is no cycle involving an odd number of vertices).
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,4,5\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3,6\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3,6\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{4,5\right\}\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚋𝚒𝚙𝚊𝚛𝚝𝚒𝚝𝚎}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection depicts a symmetric graph with no cycle involving an odd number of vertices. The corresponding graph is depicted by Figure 5.56.1.
Figure 5.56.1. Two ways of looking at the bipartite graph given in the Example slot
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙾𝙳𝙴𝚂}
The sketch of a filtering algorithm for the
\mathrm{𝚋𝚒𝚙𝚊𝚛𝚝𝚒𝚝𝚎}
constraint is given in [Dooms06]. Beside enforcing the fact that the graph is symmetric, it checks that the subset of mandatory vertices and arcs is bipartite and removes all potential arcs that would make the previous graph non-bipartite.
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
filtering: DFS-bottleneck.
final graph structure: bipartite, symmetric.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)
•
\mathrm{𝚂𝚈𝙼𝙼𝙴𝚃𝚁𝙸𝙲}
•
\mathrm{𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴}
Part (A) of Figure 5.56.2 shows the initial graph from which we start. It is derived from the set associated with each vertex. Each set describes the potential values of the
\mathrm{𝚜𝚞𝚌𝚌}
attribute of a given vertex. Part (B) of Figure 5.56.2 gives the final graph associated with the Example slot.
\mathrm{𝚋𝚒𝚙𝚊𝚛𝚝𝚒𝚝𝚎}
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Generate or plot ARMA model impulse responses - MATLAB armairf - MathWorks France
{y}_{t}=0.3{y}_{t-1}-0.1{y}_{t-2}+{\epsilon }_{t}+0.05{\epsilon }_{t-1}.
Plot the orthogonalized IRF of
{y}_{t}
Alternatively, create an ARMA model that represents
{y}_{t}
. Specify 1 for the variance of the innovations, and no model constant.
\left(1-0.3L+0.1{L}^{2}\right){y}_{t}=\left(1+0.05L\right){\epsilon }_{t}
\begin{array}{l}\left\{\left[\begin{array}{ccc}1& 0.2& -0.1\\ 0.03& 1& -0.15\\ 0.9& -0.25& 1\end{array}\right]-\left[\begin{array}{ccc}-0.5& 0.2& 0.1\\ 0.3& 0.1& -0.1\\ -0.4& 0.2& 0.05\end{array}\right]{L}^{4}-\left[\begin{array}{ccc}-0.05& 0.02& 0.01\\ 0.1& 0.01& 0.001\\ -0.04& 0.02& 0.005\end{array}\right]{L}^{8}\right\}{y}_{t}=\\ \left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]+\left[\begin{array}{ccc}-0.02& 0.03& 0.3\\ 0.003& 0.001& 0.01\\ 0.3& 0.01& 0.01\end{array}\right]{L}^{4}\right\}{\epsilon }_{t}\end{array}
{y}_{t}={\left[{y}_{1t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{y}_{2t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{y}_{3t}\right]}^{\prime }
{\epsilon }_{t}={\left[{\epsilon }_{1t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon }_{2t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon }_{3t}\right]}^{\prime }
Create a cell vector containing the VAR matrix coefficients. Because this model is a structural model in lag operator notation, start with the coefficient of
{y}_{t}
and enter the rest in order by lag. Construct a vector that indicates the degree of the lag term for the corresponding coefficients (the structural-coefficient lag is 0).
Create a cell vector containing the VMA matrix coefficients. Because this model is in lag operator notation, start with the coefficient of
{\epsilon }_{t}
and enter the rest in order by lag. Construct a vector that indicates the degree of the lag term for the corresponding coefficients.
{y}_{t}=0.3{y}_{t-1}-0.1{y}_{t-2}+{\epsilon }_{t}+0.05{\epsilon }_{t-1}
{y}_{t}
y is a 5-by-1 vector of impulse responses. y(1) is the impulse response for time
t=0
, y(2) is the impulse response for time
t=1
, and so on. The IRF fades after period
t=4
{y}_{t}
{y}_{t}=\left[\begin{array}{cc}1& -0.2\\ -0.1& 0.3\end{array}\right]{y}_{t-1}-\left[\begin{array}{cc}0.75& -0.1\\ -0.05& 0.15\end{array}\right]{y}_{t-2}+\left[\begin{array}{cc}0.55& -0.02\\ -0.01& 0.03\end{array}\right]{y}_{t-3}+{\epsilon }_{t}.
{y}_{t}=\left[{y}_{1,t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{y}_{2,t}{\right]}^{\prime }
{\epsilon }_{t}=\left[{\epsilon }_{1,t}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon }_{2,t}{\right]}^{\prime }
, and, for all t,
{\epsilon }_{t}
is Gaussian with mean zero and covariance matrix
\Sigma =\left[\begin{array}{cc}0.5& -0.1\\ -0.1& 0.25\end{array}\right].
Compute the entire generalized IRF of
{y}_{t}
. Because no MA terms exist, specify an empty array ([]) for the second input argument.
For example, consider
{y}_{t}=0.5{y}_{t-1}-0.8{y}_{t-2}+{\epsilon }_{t}-0.6{\epsilon }_{t-1}+0.08{\epsilon }_{t-2}
. The model is in difference-equation form. To compute the impulse responses, enter the following in the command line.
The ARMA model written in lag-operator notation is
\left(1-0.5L+0.8{L}^{2}\right){y}_{t}=\left(1-0.6L+0.08{L}^{2}\right){\epsilon }_{t}.
The AR coefficients of the lagged responses are negated compared to the corresponding coefficients in difference-equation format. To obtain the same result using lag operator notation, enter the following in the command line.
{\Phi }_{0}{y}_{t}=c+{\Phi }_{1}{y}_{t-1}+...+{\Phi }_{p}{y}_{t-p}+{\Theta }_{0}{\epsilon }_{t}+{\Theta }_{1}{\epsilon }_{t-1}+...+{\Theta }_{q}{\epsilon }_{t-q},
\Phi \left(L\right){y}_{t}=\Theta \left(L\right){\epsilon }_{t}.
Φ(L) is the lag operator polynomial of the autoregressive coefficients, in other words,
\Phi \left(L\right)={\Phi }_{0}-{\Phi }_{1}L-{\Phi }_{2}{L}^{2}-...-{\Phi }_{p}{L}^{p}.
Θ(L) is the lag operator polynomial of the moving average coefficients, in other words,
\Theta \left(L\right)={\Theta }_{0}+{\Theta }_{1}L+{\Theta }_{2}{L}^{2}+...+{\Theta }_{q}{L}^{q}.
{y}_{t}={\Phi }^{-1}\left(L\right)\Theta \left(L\right){\epsilon }_{t}=\Omega \left(L\right){\epsilon }_{t}.
{\psi }_{j}\left(m\right)={C}_{m}{e}_{j}.
{C}_{m}={\Omega }_{m}P,
where P is the lower triangular factor in the Cholesky factorization of Σ.
{C}_{m}={\sigma }_{j}^{-1}{\Omega }_{m}\Sigma ,
\Phi \left(L\right){y}_{t}=c+\Theta \left(L\right){\epsilon }_{t},
\Phi \left(L\right)={\Phi }_{0}-{\Phi }_{1}L-{\Phi }_{2}{L}^{2}-...-{\Phi }_{p}{L}^{p}
, which is the autoregressive, lag operator polynomial.
L is the back-shift operator, in other words,
{L}^{j}{y}_{t}={y}_{t-j}
\Theta \left(L\right)={\Theta }_{0}+{\Theta }_{1}L+{\Theta }_{2}{L}^{2}+...+{\Theta }_{q}{L}^{q}
, which is the moving average, lag operator polynomial.
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How many controllers should be plotted for problem 3c? - Murray Wiki
How many controllers should be plotted for problem 3c?
Q How many controllers should be plotted for problem 3c?
A You want to contrast the effect on the step response from varying
{\displaystyle \rho }
for two fixed
{\displaystyle q_{1}}
values, so constrasting requires you pick at least two values of
{\displaystyle \rho }
for these plots. In addition, contrast the effect on step response from varying
{\displaystyle q_{1}}
for at least two fixed values of
{\displaystyle \rho }
If you use subplot to display the x and
{\displaystyle \theta }
step response in a single figure, you can do the above contrast within four figures.
Retrieved from "https://murray.cds.caltech.edu/index.php?title=How_many_controllers_should_be_plotted_for_problem_3c%3F&oldid=5737"
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Sangaku - Wikipedia
For the ancient Japanese performing art of Sangaku (散楽), see Sarugaku.
A Sangaku dedicated to Konnoh Hachimangu (Shibuya, Tokyo) in 1859.
Sangaku or San Gaku (算額; lit. translation: calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.
A Sangaku dedicated at Emmanji Temple in Nara
Select examples[edit]
The smallest distinct integer solution to the Sangaku puzzle in which three circles touch each other and share a tangent line.
rleft
The six primitive triplets of integer radii up to 1000
A typical problem, which is presented on an 1824 tablet in Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is:
{\displaystyle {\frac {1}{\sqrt {r_{\text{middle}}}}}={\frac {1}{\sqrt {r_{\text{left}}}}}+{\frac {1}{\sqrt {r_{\text{right}}}}}.}
(See also Ford circle.)
Soddy's hexlet, thought previously to have been discovered in the west in 1937, had been discovered on a Sangaku dating from 1822.
One Sangaku problem from Sawa Masayoshi and other from Jihei Morikawa were solved only recently.[1][2]
Japanese theorem for concyclic polygons
Japanese theorem for concyclic quadrilaterals
^ Holly, Jan E.; Krumm, David (2020-07-25). "Morikawa's Unsolved Problem". arXiv:2008.00922 [math.HO].
^ Kinoshita, Hiroshi (2018). "An Unsolved Problem in the Yamaguchi's Travell Diary" (PDF). Sangaku Journal of Mathematics. 2: 43–53.
Fukagawa, Hidetoshi, and Dan Pedoe. (1989). Japanese temple geometry problems = Sangaku. Winnipeg: Charles Babbage. ISBN 9780919611214; OCLC 474564475
__________ and Dan Pedoe. (1991) How to resolve Japanese temple geometry problems? (日本の幾何ー何題解けますか?, Nihon no kika nan dai tokemasu ka) Tōkyō : Mori Kitashuppan. ISBN 9784627015302; OCLC 47500620
__________ and Tony Rothman. (2008). Sacred Mathematics: Japanese Temple Geometry. Princeton: Princeton University Press. ISBN 069112745X; OCLC 181142099
Huvent, Géry. (2008). Sangaku. Le mystère des énigmes géométriques japonaises. Paris: Dunod. ISBN 9782100520305; OCLC 470626755
Rehmeyer, Julie, "Sacred Geometry", Science News, March 21, 2008.
Rothman, Tony; Fugakawa, Hidetoshi (May 1998). "Japanese Temple Geometry". Scientific American. pp. 84–91.
Wikimedia Commons has media related to Sangaku.
Japanese Temple Geometry Problem
Sangaku Journal of Mathematics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sangaku&oldid=1068917854"
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Nitrogen and Phosphorus - Course Hero
General Chemistry/Metals, Metalloids, and Nonmetals/Nitrogen and Phosphorus
Nitrogen and phosphorus are elements from group 15 of the periodic table. Nitrogen is an essential element for living things on Earth because it is crucial for both plant and animal biological function, and phosphorus is very important in industry.
In nature, pure nitrogen is in the form of a diatomic molecule (N2). Nitrogen is abundant in the atmosphere, forming about 78% of it. The nitrogen-nitrogen bond in an N2 molecule is an extremely strong triple bond. This bond cannot be easily broken. This causes nitrogen to be an inert molecule in most situations. Nitrogen has five valence electrons. Nitrogen shows a variety of oxidation states in compounds, ranging from +5 to –3. The most common oxidation states are +5, 0, and –3. Nitrogen is highly electronegative but is less electronegative than oxygen and fluorine. Nitrogen commonly takes on positive oxidation states when in a compound with oxygen or fluorine and negative oxidation states when in a compound with other elements.
The low reactivity of N2 gas makes it important in many industries. N2 is obtained from the atmosphere by separating N2 from air.
Living organisms need a variety of nitrogen compounds. Plants are an important part of the nitrogen cycle, as animals can obtain nitrogen by eating plants. Plants cannot obtain nitrogen from air, due to the high strength of the nitrogen-nitrogen bond. In nature, plants rely on bacteria to break nitrogen down. The availability of non-N2 nitrogen in nature is a major factor limiting plant growth. The process of breaking down N2 into more usable nitrogen compounds is called nitrogen fixation. Nitrogen fixation is used to convert nitrogen in the air to molecules such as ammonia that can be used by plants.
Nitrogen-based artificial fertilizers can be used to overcome this limitation. Nitrogen-based artificial fertilizers use the Haber-Bosch process which breaks down the nitrogen-nitrogen bonds in N2 and converts nitrogen into ammonia (NH3). This process relies on the following reaction:
{\rm N}_2(g)+3{\rm H}_2(g)\rightleftharpoons2{\rm{NH}}_3(g)
This is an equilibrium reaction that heavily favors N2 in normal conditions. Under high pressure and specific catalysts, the equilibrium still favors N2 but becomes more balanced, and some NH3 can be recovered. The ammonia can then be used for various nitrogen compounds, including artificial fertilizers. The Haber-Bosch process also is the biggest consumer of commercial hydrogen gas.
The Haber-Bosch process produces ammonia (NH3) form nitrogen (N2), which is typically nonreactive due to its strong triple bonds, and hydrogen (H2) gases. This process uses a catalyst, high temperature, and high pressure to make ammonia, which was very difficult to make on an industrial scale before the process was developed.
Like nitrogen, phosphorus has five valence electrons and can take oxidation states between +5 and –3. Phosphorus is less electronegative than nitrogen. Phosphorus often takes positive oxidation states in compounds.
Pure phosphorus has multiple allotropes. White phosphorus is molecular, with molecules consisting of four phosphorus atoms tetrahedral in shape. White phosphorus is unstable and very reactive; it spontaneously combusts when it comes in contact with air. Red phosphorus is an amorphous solid and is much less reactive. Black phosphorus is a crystal and is more stable than either white or red phosphorus.
Phosphorus is extracted from minerals that have phosphates. The most common phosphate found in nature is calcium phosphate (Ca3(PO4)2). Phosphorus halides such as PCl3 and PCl5 are the most important commercial phosphorus compounds, used in lubrication oils, paints, pesticides, and flame retardants. These halides are obtained by reacting phosphorus with diatomic molecules of elements of group 17.
White and black phosphorus are allotropes of phosphorus. Phosphate ion is a commercially important form of phosphate.
<Oxygen>Halogens and Noble Gases
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Snub_triheptagonal_tiling Knowpia
Snub triheptagonal tiling
{\displaystyle s{\begin{Bmatrix}7\\3\end{Bmatrix}}}
In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.
Dual tilingEdit
The dual tiling is called an order-7-3 floret pentagonal tiling, and is related to the floret pentagonal tiling.
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-7.
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SPM Add Maths – user's Blog!
Home › Archive for SPM Add Maths
Short Questions (Question 7 & 8)
Posted on May 31, 2020 by Myhometuition
Posted in Circular Measure
9.7 Second-Order Differentiation, Turning Points, Maximum and Minimum Points (Examples)
\begin{array}{l}y=3x\left(4-x\right)\\ y=12x-3{x}^{2}\\ \frac{dy}{dx}=12-6x\\ \text{When }y\text{ is maximum, }\frac{dy}{dx}=0\\ 0=12-6x\\ x=2\end{array}
\begin{array}{l}y=12x-3{x}^{2}\\ \text{When }x=2,\\ y=12\left(2\right)-3{\left(2\right)}^{2}\\ y=12\end{array}
\begin{array}{l}y=2{x}^{3}+3{x}^{2}-12x+7\\ \frac{dy}{dx}=6{x}^{2}+6x-12\\ \text{At turning point, }\frac{dy}{dx}=0\end{array}
\begin{array}{l}\frac{{d}^{2}y}{d{x}^{2}}=12x+6\\ \text{When }x=1,\\ \frac{{d}^{2}y}{d{x}^{2}}=12\left(1\right)+6=18>0\text{ (positive)}\end{array}
\begin{array}{l}\text{When }x=-2,\\ \frac{{d}^{2}y}{d{x}^{2}}=12\left(-2\right)+6=-18<0\text{ (negative)}\end{array}
SPM Practice Question 2
The third term and the sixth term of a geometric progression are 24 and
7\frac{1}{9}
(a) the first term and the common ratio,
(b) the sum of the first five terms,
(c) the sum of the first n terms with n is very big approaching rn ≈ 0.
\begin{array}{l}\text{Given }{T}_{3}=24\\ \text{ }a{r}^{2}=24\text{ }...........\left(1\right)\\ \text{Given }{T}_{6}=7\frac{1}{9}\\ \text{ }a{r}^{5}=\frac{64}{9}\text{ }...........\left(2\right)\\ \frac{\left(2\right)}{\left(1\right)}:\frac{a{r}^{5}}{a{r}^{2}}=\frac{\frac{64}{9}}{24}\\ \text{ }{r}^{3}=\frac{8}{27}\\ \text{ }r=\frac{2}{3}\end{array}
\begin{array}{l}\text{Substitute }r=\frac{2}{3}\text{ into }\left(1\right)\\ \text{ }a{\left(\frac{2}{3}\right)}^{2}=24\\ \text{}a\left(\frac{4}{9}\right)=24\\ \text{ }a=24×\frac{9}{4}\\ \text{ }=54\\ \therefore \text{ the first term 54 and the common ratio is }\frac{2}{3}.\end{array}
\begin{array}{l}{S}_{5}=\frac{54\left[1-{\left(\frac{2}{3}\right)}^{5}\right]}{1-\frac{2}{3}}\\ \text{ }=54×\frac{211}{243}×\frac{3}{1}\\ \text{ }=140\frac{2}{3}\\ \\ \therefore \text{ sum of the first five term is }140\frac{2}{3}.\end{array}
\begin{array}{l}\text{When }-1<r<1\text{ and }n\text{ becomes }\\ \text{very big approaching }{r}^{n}\approx 0,\\ \therefore \text{ }{S}_{n}=\frac{a}{1-r}\\ \text{ }=\frac{54}{\text{ }1\text{ }-\text{ }\frac{2}{3}\text{ }}\\ \text{ }=162\end{array}
Therefore, sum of the first n terms with n is very big approaching rn ≈ 0 is 162.
Posted in Progression
The masses of mangoes in a stall have a normal distribution with a mean of 200 g and a standard deviation of 30 g.
(a) Find the mass, in g, of a mango whose z-score is 0.5.
(b) If a mango is chosen at random, find the probability that the mango has a mass of at least 194 g.
σ = 30 g
Let X be the mass of a mango.
\begin{array}{l}\frac{X-200}{30}=0.5\\ X=0.5\left(30\right)+200\\ X=215g\end{array}
\begin{array}{l}P\left(X\ge 194\right)\\ =P\left(Z\ge \frac{194-200}{30}\right)\\ =P\left(Z\ge -0.2\right)\\ =1-P\left(Z>0.2\right)\\ =1-0.4207\\ =0.5793\end{array}
Diagram below shows a standard normal distribution graph.
The probability represented by the area of the shaded region is 0.3238.
(b) X is a continuous random variable which is normally distributed with a mean of 80 and variance of 9.
Find the value of X when the z-score is k.
P(Z > k) = 0.5 – 0.3238
µ = 80,
σ2 = 9, σ = 3
\begin{array}{l}\frac{X-80}{3}=0.93\\ X=3\left(0.93\right)+80\\ X=82.79\end{array}
Posted in Probability Distribution
\begin{array}{l}\frac{1}{16}+\frac{1}{4}+h+\frac{1}{4}+\frac{1}{16}=1\\ h=1-\frac{5}{8}\\ h=\frac{3}{8}\end{array}
P\left(X\ge 3\right)=\frac{1}{4}+\frac{1}{16}=\frac{5}{16}
\begin{array}{l}\text{Standard deviation}=\sqrt{npq}\\ \text{}=\sqrt{10×\frac{1}{4}×\frac{3}{4}}\\ \text{}=1.875\end{array}
\begin{array}{l}P\left(X=r\right)=C{}_{r}{\left(\frac{1}{4}\right)}^{r}{\left(\frac{3}{4}\right)}^{10-r}\\ P\left(X\ge 1\right)\\ =1-P\left(X<1\right)\\ =1-P\left(X=0\right)\\ =1-C{}_{0}{\left(\frac{1}{4}\right)}^{0}{\left(\frac{3}{4}\right)}^{10}\\ =0.9437\end{array}
SPM Practice 2 (Question 11 & 12)
Question 11 (3 marks):
Diagram 6 shows the graph of a straight line
\frac{{x}^{2}}{y}\text{ against }\frac{1}{x}.
Based on Diagram 6, express y in terms of x.
\begin{array}{l}m=\frac{4-\left(-5\right)}{6-0}=\frac{3}{2}\\ c=-5\\ Y=\frac{{x}^{2}}{y}\text{, }X=\frac{1}{x}\\ \\ Y=mX+c\\ \frac{{x}^{2}}{y}=\frac{3}{2}\left(\frac{1}{x}\right)+\left(-5\right)\\ \frac{{x}^{2}}{y}=\frac{3}{2x}-5\\ \frac{{x}^{2}}{y}=\frac{3-10x}{2x}\\ \frac{y}{{x}^{2}}=\frac{2x}{3-10x}\\ y=\frac{2{x}^{3}}{3-10x}\end{array}
The variables x and y are related by the equation
y=x+\frac{r}{{x}^{2}}
, where r is a constant. Diagram 8 shows a straight line graph obtained by plotting
\left(y-x\right)\text{ against }\frac{1}{{x}^{2}}.
Express h in terms of p and r.
\begin{array}{l}y=x+\frac{r}{{x}^{2}}\\ y-x=r\left(\frac{1}{{x}^{2}}\right)+0\\ Y=mX+c\\ m=r,\text{ }c=0\\ \\ m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\ r=\frac{5p-0}{\frac{h}{2}-0}\\ \frac{hr}{2}=5p\\ hr=10p\\ h=\frac{10p}{r}\end{array}
Posted in Linear Law
SPM Practice 2 (Linear Law) – Question 1
Use a graph to answer this question.
Table 1 shows the values of two variables, x and y, obtained from an experiment. A straight line will be obtained when a graph of
\frac{{y}^{2}}{x}\text{ against }\frac{1}{x}
is plotted.
(a) Based on Table 1, construct a table for the values of
\frac{1}{x}\text{ and }\frac{{y}^{2}}{x}.
\begin{array}{l}\left(b\right)\text{ Plot }\frac{{y}^{2}}{x}\text{ against }\frac{1}{x}\text{, using a scale of 2 cm to 0}\text{.1 unit on the }\frac{1}{x}\text{-axis}\\ \text{ and 2cm to 2 units on the }\frac{{y}^{2}}{x}\text{-axis}\text{.}\\ \text{ Hence, draw the line of best fit}\text{.}\end{array}
(c) Using the graph in 1(b)
(i) find the value of y when x = 2.7,
(ii) express y in terms of x.
\begin{array}{l}\text{When }x=2.7,\text{ }\frac{1}{x}=0.37\\ \text{From graph,}\\ \frac{{y}^{2}}{x}=5.2\\ \frac{{y}^{2}}{2.7}=5.2\\ y=3.75\end{array}
\begin{array}{l}\text{Form graph, }y\text{-intercept, }c\text{ = –4}\\ \text{gradient, }m=\frac{16-\left(-4\right)}{0.8-0}=25\\ Y=mX+c\\ \frac{{y}^{2}}{x}=25\left(\frac{1}{x}\right)-4\\ y=\sqrt{25-4x}\end{array}
The table below shows the values of two variables, x and y, obtained from an experiment. The variables x and y are related by the equation , where k and p are constants.
(a) Based on the table above, construct a table for the values of and . Plot against , using a scale of 2 cm to 0.1 unit on the - axis and 2 cm to 0.2 unit on the - axis. Hence, draw the line of best fit.
(b) Use the graph from (b) to find the value of
(i) a,
(ii) b.
Step 1 : Construct a table consisting X and Y.
Step 2 : Plot a graph of Y against X, using the scale given and draw a line of best fit
Steps to draw line of best fit - Click here
Step 3 : Calculate the gradient, m, and the Y-intercept, c, from the graph
Step 4 : Rewrite the original equation given and reduce it to linear form
Step 5 : Compare with the values of m and c obtained, find the values of the unknown required
The following table shows the corresponding values of two variables, x and y, that are related by the equation , where p and k are constants.
(a) Plot against . Hence, draw the line of best fit
(b) Use your graph in (a) to find the values of p and k.
For steps to draw line of best fit - Click here
The table below shows the corresponding values of two variables, x and y, that are related by the equation , where p and q are constants.
One of the values of y is incorrectly recorded.
(a) Using scale of 2 cm to 5 units on the both axis, plot the graph of xy against . Hence, draw the line of best fit
(b) Use your graph in (a) to answer the following questions:
(i) State the values of y which is incorrectly recorded and determine its actual value.
(ii) Find the value of p and of q.
(b) (i) State the values of y which is incorrectly recorded and determine its actual value.
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Probability Problem on Poisson Distribution: Statistics - Jaber Al-arbash | Brilliant
When a computer disk manufacturer tests a disk, it writes to the disk and then tests it using a certifier. The certifier counts the number of missing pulses or errors. The number of errors in a test area on a disk has a Poisson distribution with
\lambda = 0.2
What percentage of test areas have two or fewer errors?
by Jaber Al-arbash
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Compute electrical position, magnetic flux, and electrical torque of rotor - Simulink - MathWorks Nordic
Discrete step size (s)
Position datatype
Flux datatype
Torque datatype
Compute electrical position, magnetic flux, and electrical torque of rotor
Motor Control Blockset / Sensorless Estimators
The Flux Observer block computes the electrical position, magnetic flux, and electrical torque of a PMSM or an induction motor by using the per unit voltage and current values along the α- and β-axes in the stationary αβ reference frame.
These equations describe how the block computes the electrical position, magnetic flux, and electrical torque for a PMSM.
{\psi }_{\alpha }= {\int }^{\text{}}\left({V}_{\alpha }-{I}_{\alpha }R\right)\text{dt}- \left({L}_{s}\cdot {I}_{\alpha }\right)
{\psi }_{\beta }= {\int }^{\text{}}\left({V}_{\beta }-{I}_{\beta }R\right)\text{dt}- \left({L}_{s}\cdot {I}_{\beta }\right)
\psi = \sqrt{{\psi }_{\alpha }^{2}+{\psi }_{\beta }^{2}}
{T}_{\text{e}}=\frac{3}{2}P\left({\psi }_{\alpha }{I}_{\beta }-{\psi }_{\beta }{I}_{\alpha }\right)
{\theta }_{\text{e}}= {\mathrm{tan}}^{-1}\frac{{\psi }_{\beta }}{{\psi }_{\alpha }}
These equations describe how the block computes the rotor electrical position, rotor magnetic flux, and electrical torque for an induction motor.
{\psi }_{\alpha }=\frac{{L}_{r}}{{L}_{m}} \left({\int }^{\text{}}\left({V}_{\alpha }-{I}_{\alpha }R\right)\text{dt}- \sigma {L}_{s}{I}_{\alpha }\right)
{\psi }_{\beta }=\frac{{L}_{r}}{{L}_{m}} \left({\int }^{\text{}}\left({V}_{\beta }-{I}_{\beta }R\right)\text{dt}- \sigma {L}_{s}{I}_{\beta }\right)
\sigma =1-\frac{{L}_{m}^{2}}{{L}_{r}\cdot {L}_{s}}
\psi = \sqrt{{\psi }_{\alpha }^{2}+{\psi }_{\beta }^{2}}
{T}_{\text{e}}=\frac{3}{2}\cdot P\cdot \frac{{L}_{m}}{{L}_{r}}\left({\psi }_{\alpha }{I}_{\beta }-{\psi }_{\beta }{I}_{\alpha }\right)
{\theta }_{\text{e}}= {\mathrm{tan}}^{-1}\frac{{\psi }_{\beta }}{{\psi }_{\alpha }}
{V}_{\alpha }
{V}_{\beta }
are the α-axis and β-axis voltages (Volts).
{I}_{\alpha }
{I}_{\beta }
are the α-axis and β-axis current (Amperes).
R
is the stator resistance of the motor (Ohms).
{L}_{s}
is the stator inductance of the motor (Henry).
{L}_{r}
is the rotor inductance of the motor (Henry).
{L}_{m}
is the magnetizing inductance of the motor (Henry).
\sigma
P
\psi
is the rotor magnetic flux (Weber).
{\psi }_{\alpha }
{\psi }_{\beta }
are the rotor magnetic fluxes along the α- and β-axes (Weber).
{T}_{e}
is the electrical torque of the rotor (Nm).
{\theta }_{e}
is the electrical position of the rotor (Radians).
Vα — α-axis voltage
Voltage component along the α-axis in the stationary αβ reference frame.
Vβ — β-axis voltage
Voltage component along the β-axis in the stationary αβ reference frame.
Iα — α-axis current
Current along the α-axis in the stationary αβ reference frame.
Iβ — β-axis current
Current along the β-axis in the stationary αβ reference frame.
Rst — Reset block
The pulse (true value) that resets the block algorithm.
θe — Electrical position of motor
The electrical position of the rotor as estimated by the block.
To enable this port, set Block output to Position.
Ψ — Rotor flux of motor
The magnetic flux of the rotor as estimated by the block.
To enable this port, set Block output to Flux.
Te — Electrical torque of motor
The electrical torque of the rotor as estimated by the block.
To enable this port, set Block output to Torque.
Motor selection — Type of motor
PMSM (default) | ACIM
Select the type of motor that the block supports.
Input units — Unit of voltage and current inputs
SI unit (default) | Per-unit
Select the unit of the α and β-axes voltage and current input values.
Block output — Select outputs that block should compute
Position (default) | Flux | Torque
Select one or more quantities that the block should compute and display in the block output.
You must select at least one value. The block displays an error message if you click Ok or Apply without selecting any value.
Pole pairs — Number of pole pairs available in motor
To enable this parameter, set Block output to Torque.
Stator resistance (ohm) — Stator winding resistance
Stator phase winding resistance of the motor in ohms.
Stator d-axis inductance (H) — Stator winding inductance along d-axis
Stator winding inductance of the motor along d-axis in Henry.
To enable this parameter, set Motor selection to PMSM.
Stator leakage inductance (H) — Leakage inductance of stator winding
Leakage inductance of the induction motor stator winding in Henry.
To enable this parameter, set Motor selection to ACIM.
Leakage inductance of the induction motor rotor winding in Henry.
Magnetizing inductance of the induction motor in Henry.
Cutoff frequency (Hz) — Cutoff frequency of internal high-pass filter
Cutoff frequency of the internal high-pass filter (that filters noise) in Hertz.
The Flux Observer block uses an internal first order IIR high-pass filter. You should set the Cutoff frequency (Hz) for this filter to a value that is lower than the lowest frequency corresponding to the minimum speed of the motor. For example, you can enter a value that is one-tenth of the lowest electrical frequency of the stator voltages and the currents. However, you can adjust this value to determine a more accurate cutoff frequency that generates the desired block output.
Discrete step size (s) — Sample time after which block executes again
The fixed time interval in seconds between two consecutive instances of block execution.
Position unit — Unit of electrical position output
Radians (default) | Degrees | Per-unit
Unit of the electrical position output.
To enable this parameter, set Block output to Position.
Position datatype — Data type of electrical position output
single (default) | double | fixed point
Data type of the electrical position output.
Flux unit — Unit of magnetic flux output
Weber (default) | Per-unit
Unit of the magnetic flux output.
To enable this parameter, set Block output to Flux.
Flux datatype — Data type of magnetic flux output
Data type of the magnetic flux output.
Torque unit — Unit of electrical torque output
Nm (default) | Per-unit
Unit of the electrical torque output.
Torque datatype — Data type of electrical torque output
Data type of the electrical torque output.
Uses sensorless position estimation to implement the field-oriented control (FOC) technique to control the speed of a three-phase AC induction motor (ACIM). For details about FOC, see Field-Oriented Control (FOC).
[1] O. Sandre-Hernandez, J. J. Rangel-Magdaleno and R. Morales-Caporal, "Simulink-HDL cosimulation of direct torque control of a PM synchronous machine based FPGA," 2014 11th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE), Campeche, 2014, pp. 1-6. (doi: 10.1109/ICEEE.2014.6978298)
[2] Y. Inoue, S. Morimoto and M. Sanada, "Control method suitable for direct torque control based motor drive system satisfying voltage and current limitations," The 2010 International Power Electronics Conference - ECCE ASIA -, Sapporo, 2010, pp. 3000-3006. (doi: 10.1109/IPEC.2010.5543698)
Sliding Mode Observer | Clarke Transform | Inverse Park Transform | Speed Measurement
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Complete the squares in
3 {x}^{2}-5 {y}^{2}+7 x+4 y-9
3 {x}^{2}-5 {y}^{2}+7 x+4 y-9
=3\left({x}^{2}+\frac{7}{3}x\right)-5\left({y}^{2}-\frac{4}{5}y\right)-9
=3\left({x}^{2}+2\frac{7}{2\cdot 3}x+{\left(\frac{7}{6}\right)}^{2}-{\left(\frac{7}{6}\right)}^{2}\right)-5\left({y}^{2}-\frac{4}{5}y\right)-9
=3\left({\left(x+\frac{7}{6}\right)}^{2}-{\left(\frac{7}{6}\right)}^{2}\right)-5\left({y}^{2}-\frac{4}{5}y\right)-9
=3{\left(x+\frac{7}{6}\right)}^{2}-3{\left(\frac{7}{6}\right)}^{2}-5\left({y}^{2}-\frac{4}{5}y\right)-9
=3{\left(x+\frac{7}{6}\right)}^{2}-5\left({y}^{2}-\frac{4}{5}y\right)-9-3{\left(\frac{7}{6}\right)}^{2}
=3{\left(x+\frac{7}{6}\right)}^{2}-5\left({y}^{2}-2\frac{4}{2\cdot 5}y+{\left(\frac{4}{10}\right)}^{2}-{\left(\frac{4}{10}\right)}^{2}\right)-\frac{157}{12}
=3{\left(x+\frac{7}{6}\right)}^{2}-5\left({\left(y-\frac{2}{5}\right)}^{2}-{\left(\frac{2}{5}\right)}^{2}\right)-\frac{157}{12}
=3{\left(x+\frac{7}{6}\right)}^{2}-5{\left(y-\frac{2}{5}\right)}^{2}+5{\left(\frac{2}{5}\right)}^{2}-\frac{157}{12}
=3{\left(x+\frac{7}{6}\right)}^{2}-5{\left(y-\frac{2}{5}\right)}^{2}+\frac{4}{5}-\frac{157}{12}
=3{\left(x+\frac{7}{6}\right)}^{2}-5{\left(y-\frac{2}{5}\right)}^{2}-\frac{737}{60}
Although the Context Panel contains the Complete Square option, it operates on one variable at a time. Hence, it has to be invoked twice. The underlying command, however, admits syntax wherein the square can be completed in more than one variable with but a single call to the function. This more powerful version can be accessed if the Student Precalculus package is first loaded.
Tools≻Load Package: Student Precalculus
Loading Student:-Precalculus
Complete the square in both variables
Context Panel: Student Precalculus≻Complete the Square≻All Variables
3 {x}^{2}-5 {y}^{2}+7 x+4 y-9
\stackrel{\text{complete square}}{\to }
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}{\left(\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{7}}{\textcolor[rgb]{0,0,1}{6}}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{737}}{\textcolor[rgb]{0,0,1}{60}}
Assign the polynomial to q.
q≔3 {x}^{2}-5 {y}^{2}+7 x+4 y-9
Apply the CompleteSquare command from the Student Precalculus package.
\mathrm{Student}\left[\mathrm{Precalculus}\right]\left[\mathrm{CompleteSquare}\right]\left(q,\left[x,y\right]\right)
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A Lower Bound on the Sinc Function and Its Application
Yue Hu, Cristinel Mortici, "A Lower Bound on the Sinc Function and Its Application", The Scientific World Journal, vol. 2014, Article ID 571218, 4 pages, 2014. https://doi.org/10.1155/2014/571218
Yue Hu 1 and Cristinel Mortici 2,3
1School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China
2Valahia University of Târgovişte, 18 Unirii Boulevard, 130082 Târgovişte, Romania
3Academy of Romanian Scientists, Splaiul Independenţei 54, 050094 Bucharest, Romania
A lower bound on the sinc function is given. Application for the sequence which related to Carleman inequality is given as well.
The sinc function is defined to be
This function plays a key role in many areas of mathematics and its applications [1–6].
The following result that provides a lower bound for the sinc is well known as Jordan inequality [7]: where equality holds if and only if .
This inequality has been further refined by many authors in the past few years [8–35].
In [36], it was proposed that
We noticed that the lower bound in (3) is the fractional function. Similar result has been reported as follows [1]:
To the best of the authors’ knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function.
Theorem 1. For any , one has
In [37], Yang proved that for any positive integer , the following Carleman type inequality holds: whenever , , with , where
From a mathematical point of view, the sequence has very interesting properties. Yang [38] and Gyllenberg and Ping [39] have proved that, for any positive integer ,
In [40], the authors proved that where
As an application of Theorem 1, it is the second aim of the present paper to give a better upper bound on the sequence .
Theorem 2. For any positive integer , one has
The proof is not based on (3). We first prove the following result.
Lemma 3. For any , one has
Proof. Set , . Then inequality (13) is equivalent to To prove (14) by (4), it is enough to prove that namely, Next we prove (16). Let We need only to prove that . Elementary calculations reveal that Noting that, for , we have Thus, from (19) and (18), we get This completes the proof. Now we prove Theorem 1.
Proof. By using the power series expansions of and , we find that where Set , . Consider the function defined by From (21), we get and . Lemma 3 implies where Elementary calculations reveal that for , Hence, for , we have Therefore, If we set then we have The intermediate value theorem implies that there must be at least one root with such that . Using Maple, we find that on the open interval the equation has a unique real root .
Hence, from (28) we get By (21), (24), and (31), Theorem 1 follows.
First, we need an auxiliary result.
Proof. By letting , , the requested inequality can be equivalently written as so it suffices to show that the function is negative on . Theorem 1 implies Hence, The required inequality follows. Now we prove Theorem 2.
Proof. Let We first consider the case .
Taking the natural log gives Taking the second derivative of both sides of (38), we have By Lemma 4, it follows that Thus, and therefore for , we have For the case , since , , and is concave up, it follows that Using (10) from (42) and (43), we have This proves Theorem 2.
The authors are very grateful to the anonymous referees and the editor for their insightful comments and suggestions. The work of the second author was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI project no. PN-II-ID-PCE-2011-3-0087.
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Copyright © 2014 Yue Hu and Cristinel Mortici. 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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Cayley's formula - Wikipedia
Number of spanning trees of a complete graph
The complete list of all trees on 2,3,4 labeled vertices:
{\displaystyle 2^{2-2}=1}
tree with 2 vertices,
{\displaystyle 3^{3-2}=3}
trees with 3 vertices and
{\displaystyle 4^{4-2}=16}
trees with 4 vertices.
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer
{\displaystyle n}
, the number of trees o{\displaystyle n}
labeled vertices is
{\displaystyle n^{n-2}}
Many proofs of Cayley's tree formula are known.[1] One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests. A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see Double counting (proof technique) § Counting trees.
Cayley's formula immediately gives the number of labelled rooted forests on n vertices, namely (n + 1)n − 1. Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label n + 1 and connecting it to all roots of the trees in the forest.
There is a close connection with rooted forests and parking functions, since the number of parking functions on n cars is also (n + 1)n − 1. A bijection between rooted forests and parking functions was given by M. P. Schützenberger in 1968.[4]
The following generalizes Cayley's formula to labelled forests: Let Tn,k be the number of labelled forests on n vertices with k connected components, such that vertices 1, 2, ..., k all belong to different connected components. Then Tn,k = k nn − k − 1.[5]
^ Aigner, Martin; Ziegler, Günter M. (1998). Proofs from THE BOOK. Springer-Verlag. pp. 141–146.
^ Borchardt, C. W. (1860). "Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung". Math. Abh. der Akademie der Wissenschaften zu Berlin: 1–20.
^ Cayley, A. (1889). "A theorem on trees". Quart. J. Pure Appl. Math. 23: 376–378.
^ Schützenberger, M. P. (1968). "On an enumeration problem". Journal of Combinatorial Theory. 4: 219–221. MR 0218257.
^ Takács, Lajos (March 1990). "On Cayley's formula for counting forests". Journal of Combinatorial Theory, Series A. 53 (2): 321–323. doi:10.1016/0097-3165(90)90064-4.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cayley%27s_formula&oldid=1084206819"
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Moore-Penrose pseudoinverse - MATLAB pinv - MathWorks India
Solve System of Linear Equations Using Pseudoinverse
pinv returns NaN for nonfinite inputs
B = pinv(A)
B = pinv(A,tol)
B = pinv(A) returns the Moore-Penrose Pseudoinverse of matrix A.
B = pinv(A,tol) specifies a value for the tolerance. pinv treats singular values of A that are smaller than the tolerance as zero.
Compare solutions to a system of linear equations obtained by backslash (\) and pinv.
If a rectangular coefficient matrix A is of low rank, then the least-squares problem of minimizing norm(A*x-b) has infinitely many solutions. Two solutions are returned by x1 = A\b and x2 = pinv(A)*b. The distinguishing properties of these solutions are that x1 has only rank(A) nonzero components, and norm(x2) is smaller than for any other solution.
Create an 8-by-6 matrix that has rank(A) = 3.
64 2 3 61 60 6
8 58 59 5 4 62
Create a vector for the right-hand side of the system of equations.
b = 260*ones(8,1)
The number chosen for the right-hand side, 260, is the value of the 8-by-8 magic sum for A. If A were still an 8-by-8 matrix, then one solution for x would be a vector of 1s. With only six columns, a solution exists since the equations are still consistent, but the solution is not all 1s. Since the matrix is of low rank, there are infinitely many solutions.
Solve for two of the solutions using backslash and pinv.
Both of these solutions are exact, in the sense that norm(A*x1-b) and norm(A*x2-b) are on the order of roundoff error. The solution x1 is special because it has only three nonzero elements. The solution x2 is special because norm(x2) is smaller than it is for any other solution, including norm(x1).
Singular value tolerance, specified as a scalar. pinv treats singular values that are smaller than tol as zeros during the computation of the pseudoinverse.
The default tolerance is max(size(A))*eps(norm(A)).
Example: pinv(A,1e-4)
The Moore-Penrose pseudoinverse is a matrix that can act as a partial replacement for the matrix inverse in cases where it does not exist. This matrix is frequently used to solve a system of linear equations when the system does not have a unique solution or has many solutions.
For any matrix A, the pseudoinverse B exists, is unique, and has the same dimensions as A'. If A is square and not singular, then pinv(A) is simply an expensive way to compute inv(A). However, if A is not square, or is square and singular, then inv(A) does not exist. In these cases, pinv(A) has some (but not all) of the properties of inv(A):
\begin{array}{l}1.\text{\hspace{0.17em}}\text{\hspace{0.17em}}ABA=A\\ 2.\text{\hspace{0.17em}}\text{\hspace{0.17em}}BAB=B\\ 3.\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(AB\right)}^{*}=AB\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{AB}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Hermitian}\right)\\ 4.\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(BA\right)}^{*}=BA\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{BA}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Hermitian}\right)\end{array}
The pseudoinverse computation is based on svd(A). The calculation treats singular values less than tol as zero.
You can replace most uses of pinv applied to a vector b, as in pinv(A)*b, with lsqminnorm(A,b) to get the minimum-norm least-squares solution of a system of linear equations. lsqminnorm is generally more efficient than pinv, and it also supports sparse matrices.
pinv uses the singular value decomposition to form the pseudoinverse of A. Singular values along the diagonal of S that are smaller than tol are treated as zeros, and the representation of A becomes:
\begin{array}{l}A=US{V}^{*}=\left[{U}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{U}_{2}\right]\left[\begin{array}{cc}{S}_{1}& 0\\ 0& 0\end{array}\right]{\left[{V}_{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{V}_{2}\right]}^{*}\\ A={U}_{1}{S}_{1}{V}_{1}^{*}\text{\hspace{0.17em}}.\end{array}
The pseudoinverse of A is then equal to:
B={V}_{1}{S}_{1}^{-1}{U}_{1}^{*}\text{\hspace{0.17em}}.
R2021b: pinv returns NaN for nonfinite inputs
pinv returns NaN values when the input contains nonfinite values (Inf or NaN). Previously, pinv threw an error when the input contained nonfinite values.
inv | qr | rank | svd | lsqminnorm | decomposition
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\overline{x}\pm \text{SD}
\overline{x}\pm \text{SD}
Management of student or team behaviors by discrete gestures or by planned measures with students (/ 5)
Monitoring students during work on various tasks (peripheral vision) to ensure that they remain focused on the work to be performed (/ 5)
Student participation in the establishment of classroom operating standards (/ 5)
Maintaining a climate conducive to learning by fostering cooperation between students rather than competition (/ 5)
Sharing with students and with each other responsibilities for the proper functioning of the class (/ 5)
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RF signal attenuation due to rainfall - MATLAB rainpl - MathWorks Benelux
rainpl
Signal Attenuation Due to Rainfall
Signal Attenuation Due to Rainfall as Function of Frequency
Signal Attenuation Due to Rainfall as Function of Elevation Angle
Signal Attenuation Due to Rainfall as Function of Polarization
RF signal attenuation due to rainfall
L = rainpl(range,freq,rainrate)
L = rainpl(range,freq,rainrate,elev)
L = rainpl(range,freq,rainrate,elev,tau)
L = rainpl(range,freq,rainrate,elev,tau,pct)
L = rainpl(range,freq,rainrate) returns the signal attenuation, L, due to rainfall. In this syntax, attenuation is a function of signal path length, range, signal frequency, freq, and rain rate, rainrate. The path elevation angle and polarization tilt angles are assumed to zero.
The rainpl function applies the International Telecommunication Union (ITU) rainfall attenuation model to calculate path loss of signals propagating in a region of rainfall [1]. The function applies when the signal path is contained entirely in a uniform rainfall environment. Rain rate does not vary along the signal path. The attenuation model applies only for frequencies at 1–1000 GHz.
L = rainpl(range,freq,rainrate,elev) also specifies the elevation angle, elev, of the propagation path.
L = rainpl(range,freq,rainrate,elev,tau) also specifies the polarization tilt angle, tau, of the signal.
L = rainpl(range,freq,rainrate,elev,tau,pct) also specifies the specified percentage of time, pct. pct is a scalar in the range of 0.001–1, inclusive. The attenuation, L, is computed from a power law using the long-term statistical 0.01% rain rate (in mm/h).
Compute the signal attenuation due to rainfall for a 20 GHz signal over a distance of 10 km in light and heavy rain.
Propagate the signal in a light rainfall of 1 mm/hr.
L = rainpl(10000,20.0e9,rr)
Propagate the signal in a heavy rainfall of 10 mm/hr.
Plot the signal attenuation due to a 20 mm/hr statistical rainfall for signals in the frequency range from 1 to 1000 GHz. The path distance is 10 km.
L = rainpl(10000,freq,rr);
Compute the signal attenuation due to heavy rain as a function of elevation angle. Elevation angles vary from 0 to 90 degrees. Assume a path distance of 100 km and a signal frequency of 100 GHz.
Set the rain rate to 10 mm/hr.
Set the elevation angles, frequency, range.
freq = 100.0e9;
rng = 100000.0*ones(size(elev));
L = rainpl(rng,freq,rr,elev);
Compute the signal attenuation due to heavy rainfall as a function of the polarization tilt angle. Assume a path distance of 100 km, a signal frequency of 100 GHz, and a path elevation angle of 0 degrees. Set the rainfall rate to 10 mm/hour. Plot the signal attenuation versus polarization tilt angle.
rng = 100e3*ones(size(tau));
L = rainpl(rng,freq,rr,elev,tau);
nonnegative real-valued scalar | nonnegative real-valued M-by-1 column vector | nonnegative real-valued 1-by-M row vector
Signal path length, specified as a nonnegative real-valued scalar, or as a M-by-1 or 1-by-M vector. Units are in meters.
positive real-valued scalar | nonnegative real-valued N-by-1 column vector | nonnegative real-valued 1-by-N row vector
Signal frequency, specified as a positive real-valued scalar, or as a nonnegative N-by-1 or 1-by-N vector. Frequencies must lie in the range 1–1000 GHz.
Example: [1400.0e6,2.0e9]
rainrate — Long-term statistical rain rate
Long-term statistical rain rate, specified as a nonnegative real-valued scalar. The long-term statistical rain rate is the rain rate that is exceeded 0.01% of the time. You can adjust the percent of time using the pct argument. Units are in mm/hr.
0.0 (default) | real-valued scalar | real-valued M-by-1 column vector | real-valued 1-by-M row vector
Signal path elevation angle, specified as a real-valued scalar, or as an M-by-1 or 1-by- M vector. Units are in degrees between –90° and 90°. If elev is a scalar, all propagation paths have the same elevation angle. If elev is a vector, its length must match the dimension of range and each element in elev corresponds to a propagation range in range.
tau — Tilt angle of polarization ellipse
Tilt angle of the signal polarization ellipse, specified as a real-valued scalar, or as an M-by-1 or 1-by- M vector. Units are in degrees between –90° and 90°. If tau is a scalar, all signals have the same tilt angle. If tau is a vector, its length must match the dimension of range. In that case, each element in tau corresponds to a propagation path in range.
The tilt angle is defined as the angle between the semi-major axis of the polarization ellipse and the x-axis. Because the ellipse is symmetrical, a tilt angle of 100° corresponds to the same polarization state as a tilt angle of -80°. Thus, the tilt angle need only be specified between ±90°.
pct — Exceedance percentage of rainfall
0.01 (default) | positive scalar between 0.001 and 1
Exceedance percentage of rainfall, specified as a positive scalar between 0.001 and 1. The long-term statistical rain rate is the rain rate that is exceeded pct of the time. Units are dimensionless.
{\gamma }_{R}=k{R}^{\alpha },
r=\frac{1}{0.477{d}^{0.633}{R}_{0.01}^{0.073\alpha }{f}^{0.123}-10.579\left(1-\mathrm{exp}\left(-0.024d\right)\right)}
[1] Radiocommunication Sector of International Telecommunication Union. Recommendation ITU-R P.838-3: Specific attenuation model for rain for use in prediction methods. 2005.
[3] Recommendation ITU-R P.837-7: Characteristics of precipitation for propagation modelling
fspl | gaspl | fogpl | cranerainpl
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UWB Ranging Using IEEE 802.15.4z - MATLAB & Simulink - MathWorks Benelux
Single-Sided Two-Way Ranging (SS-TWR)
This example shows how to estimate distance between two devices as per the IEEE® 802.15.4z™ standard [ 2 ] by using features in the Communications Toolbox™ Library for ZigBee® and UWB add-on.
The IEEE 802.15.4z amendment [ 2 ] of the IEEE® 802.15.4 standard [ 1 ] specifies the MAC and PHY layers, and associated ranging and localization using ultra wideband (UWB) communication. The very short pulse durations of UWB allow a finer granularity in the time domain and therefore more accurate estimates in the spatial domain.
The key ranging and localization functionality of the 802.15.4z amendment includes three MAC-level techniques:
Single-Sided Two-Way Ranging (SS-TWR) - One device estimates the distance between two devices by using frame transmission in both directions of a wireless 802.15.4z link.
One-Way Ranging / Time-Difference of Arrival (OWR/TDOA) - Network-assisted localization whereby one device communicates with a set of synchronized nodes to estimate the position of the device. This technique is demonstrated in the UWB Localization Using IEEE 802.15.4z example.
This example demonstrates the SS-TWR technique by using PHY frames that are compatible with the IEEE 802.15.4 standard [ 1 ] and the IEEE 802.15.4z amendment [ 2 ]. For more information on generating PHY-level IEEE 802.15.4z waveforms, see the HRP UWB IEEE 802.15.4a/z Waveform Generation example.
Two-way ranging involves frame transmission in both directions of a wireless 802.15.4z link. Single-sided ranging means that only one of the two devices estimates the distance between them.
Each frame is timed at its ranging marker (RMARKER), which is the time of the first symbol following the start-of-frame delimiter (SFD). For more information on the fields in the transmitted frame, see the HRP UWB IEEE 802.15.4a/z Waveform Generation example. The ranging responder device, transmits the response frame after a certain reply time (Treply). The ranging initiator device computes the round-trip time (Tround) as the time-distance between the RMARKERs of the transmitted and the response frames. Treply is communicated from the ranging responder device to the ranging initiator device, so that the latter estimates the propagation time (Tprop) as Tprop = (Tround - Treply)/2.
The IEEE 802.15.4z amendment [ 2 ] specifies multiple possibilities for sharing Treply:
Communication of Treply from the responder to the initiator is deferred, and performed with another message following the response frame.
Embed Treply in the response frame.
Set Treply to a fixed value known between the initiator and the responder.
This example considers the fixed reply time scenario between the two devices.
IEEE 802.15.4 [ 1 ] specifies that the exchanged frames must be a Data frame and its acknowledgement. The IEEE 802.15.4z amendment [ 2 ] relaxes this specification and allows the ranging measurement to be performed over any pair of transmitted and response frames. However, for the fixed reply time scenario, the 802.15.4z amendment specifies exchange of scrambled timestamp sequence packet configuration option three (SP3) frames. SP3 frames contain a scrambled timestamp sequence (STS) and no PHY header (PHR) or payload.
This example focuses on the basic ranging exchange without demonstrating the preceding set-up and following finish-up activities associated with the ranging procedure.
% Ensure that the ZigBee/UWB add-on is installed:
Determine the actual distance and Tprop, and initialize visualizations. Configure a timescope object to plot the initiator and responder signals.
c = physconst('LightSpeed'); % Speed of light (m/s)
actualDistance = 5; % In meters
actualTprop = actualDistance/c; % In seconds
SNR = 30; % Signal-to-Noise ratio
symbolrate = 499.2e6; % Symbol rate for HRP PHY
sps = 10; % Samples per symbol
ts = timescope( ...
SampleRate=sps*symbolrate, ...
ChannelNames={'Initiator','Responder'}, ...
LayoutDimensions=[2 1], ...
Name='SS-TWR');
ts.YLimits = [-0.25 0.25];
Transmission from Initiator
Generate the waveform containing SP3 PHY frames (with no MAC frame/PSDU) to be transmitted between the devices. Register the transmitted frame on the timeline of the initiator.
sp3Config = lrwpanHRPConfig( ...
PSDULength=0, ...
sp3Wave = lrwpanWaveformGenerator([],sp3Config);
[transmitFrame,responseFrame] = deal(sp3Wave);
% start initiator time at the start of transmission
initiatorView = transmitFrame;
Filter the transmission frame through an AWGN channel and add propagation delay. Then, update timeline for both link endpoints.
samplesToDelay = actualTprop*sp3Config.SampleRate;
receivedTransmitted = lclDelayWithNoise( ...
transmitFrame,samplesToDelay,SNR);
initiatorView = [initiatorView; zeros(ceil(samplesToDelay),1)];
responderView = receivedTransmitted;
Reception at Responder
At the responder side, detect the preamble of the 802.15.4z PHY frame, and then process the transmitted frame. Preamble detection consists of determining the first instance of the preamble out of Nsync = PreambleDuration. Plot the initiator and responder views on a timescope.
ind = lrwpanHRPFieldIndices(sp3Config); % length (start/end) of each field
sp3Preamble = sp3Wave(1:ind.SYNC(end)/sp3Config.PreambleDuration);
receivedTransmitted,sp3Preamble,sp3Config);
ts(initiatorView,responderView);
Transmission from Responder
Set the Treply time to the length of three SP3 frames to specify when to transmit the response frame. Set the first and last RMARKER sample indices on the responder side to be the beginning of first post-SFD symbol and Treply samples later. After Treply samples, transmit the response frame from the responder device.
Treply = 3*length(sp3Wave); % in samples
% Find RMARKERs at responder side
frameStart = 1+preamPos-ind.SYNC(end)/sp3Config.PreambleDuration;
sfdEnd = frameStart + ind.SYNC(end) + diff(ind.SFD);
RMARKER_R1 = sfdEnd+1;
RMARKER_R2 = RMARKER_R1 + Treply;
% Transmit after Treply. Find how long the responder needs to remain idle.
idleResponderTime = Treply - diff(ind.STS)-1 - diff(ind.SHR)-1;
responderView = [responderView; zeros(idleResponderTime,1); responseFrame; zeros(ceil(samplesToDelay),1)];
initiatorView = [initiatorView; zeros(idleResponderTime, 1)];
receivedResponse = lclDelayWithNoise( ...
responseFrame,samplesToDelay,SNR);
initiatorView = [initiatorView; receivedResponse];
Reception at Initiator
Back at the initiator side, detect the preamble of the 802.15.4z PHY frame, and then process the transmitted frame.
txFrameEnd = ind.STS(end);
initiatorView(txFrameEnd+1:end),sp3Preamble,sp3Config);
Estimate the propagation delay and the distance between two devices. Set the first and last RMARKER sample indices on the initiator side to be the start of transmission (which is known at t=0) and the beginning of first post-SFD symbol. Use the RMARKERs, Tround, and Tprop to estimate the distance between initiator and responder.
RMARKER_I1 = 1+ind.SFD(end);
sfdEnd = txFrameEnd + frameStart + ind.SYNC(end) + diff(ind.SFD);
RMARKER_I2 = sfdEnd+1;
Tround = RMARKER_I2 - RMARKER_I1; % In samples
Tprop = (Tround-Treply)/(2*sp3Config.SampleRate); % In seconds
estimatedDistance = c*Tprop; % In meters
This timescope illustrates the frame exchange as in Fig. 6-47a in [ 2 ] with X-axis limit zoomed in to see the propagation delay between the transmitted and response frames.
ts([initiatorView; zeros(ceil(samplesToDelay),1)],responderView);
The estimated distance is a few centimeters different than the actual distance.
fprintf(['Actual distance = %d m.' ...
'\nEstimated Distance = %0.2f m' ...
'\nError = %0.3f m (%0.2f%%)\n'], ...
actualDistance,estimatedDistance, ...
estimatedDistance-actualDistance, ...
100*(estimatedDistance-actualDistance)/actualDistance)
Actual distance = 5 m.
Estimated Distance = 5.01 m
Error = 0.015 m (0.29%)
For ranging methods that rely on estimating the time of flight (TOF), errors in the distance estimate are primarily caused when the propagation time (Tprop) is not an integer multiple of the sample time. The largest distance error for such ranging methods occurs when Tprop lasts half a sample time more than an integer multiple of sample time. The smallest distance error occurs when Tprop is an integer multiple of sample time. For the higher pulse repetition frequency (HRPF) mode of the high rate pulse repetition frequency (HRP) PHY used in this example, the symbol rate is 499.2 MHz and the number of samples per symbol is 10, which results in a maximum error in Tprop estimation of
0.5×c/\left(499.2×10\right)
. So, the default ranging error lies between 0 and 3 cm.
In general, the larger channel bandwidth in UWB corresponds to shorter symbol duration and smaller ranging error as compared to narrowband communication. For the narrowband communication as specified in IEEE 802.11az, the channel bandwidth ranges from 20 MHz to 160 MHz. Considering the maximum Tprop error for narrowband communication, estimates for the ranging error lie between 0 and 10 cm for 160 MHz and between 0 and 75 cm for 20 MHz. For more information regarding ranging with IEEE 802.11az, see the 802.11az Positioning Using Super-Resolution Time of Arrival Estimation (WLAN Toolbox) example.
function received = lclDelayWithNoise(transmitted, samplesToDelay, SNR)
% lclDelayWithNoise Operations of wireless channel (propagation delay, AWGN)
% zero pad @ end, to get entire frame out of VFD
delayedTransmitted = vfd( ...
[transmitted; zeros(ceil(samplesToDelay), 1)],samplesToDelay);
% add white gaussian noise:
received = awgn(delayedTransmitted,SNR);
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Global Constraint Catalog: Cninterval
<< 5.276. next_greater_element5.278. no_peak >>
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\right)
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙽𝚅𝙰𝙻}\ge \mathrm{𝚖𝚒𝚗}\left(1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)
\mathrm{𝙽𝚅𝙰𝙻}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}>0
Consider the intervals of the form
\left[\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}·k,\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}·k+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right]
k
\mathrm{𝙽𝚅𝙰𝙻}
is the number of intervals for which at least one value is assigned to at least one variable of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\left(2,〈3,1,9,1,9〉,4\right)
In the example, the third argument
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}=4
defines the following family of intervals
\left[4·k,4·k+3\right]
k
is an integer. Values 3, 1, 9, 1 and 9 are respectively located within intervals
\left[0,3\right]
\left[0,3\right]
\left[8,11\right]
\left[0,3\right]
\left[8,11\right]
. Since we only use the two intervals
\left[0,3\right]
\left[8,11\right]
the first argument of the
\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
constraint is set to value 2.
\mathrm{𝙽𝚅𝙰𝙻}>1
\mathrm{𝙽𝚅𝙰𝙻}<|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}>1
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}<
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)
\left(
\mathrm{𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)+\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}-1\right)/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}<\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
k
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}=1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>0
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
constraint is useful for counting the number of actually used periods, no matter how many time each period is used. A period can for example stand for a hour or for a day.
\mathrm{𝚗𝚌𝚕𝚊𝚜𝚜}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\in \mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}\mathrm{mod}\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚗𝚙𝚊𝚒𝚛}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚙𝚊𝚒𝚛}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}/\mathrm{𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
modelling: number of distinct equivalence classes, interval, functional dependency.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\begin{array}{c}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}=\hfill \\ \mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}/\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙸𝙽𝚃𝙴𝚁𝚅𝙰𝙻}\hfill \end{array}
\mathrm{𝐍𝐒𝐂𝐂}
=\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝐍𝐒𝐂𝐂}
graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to those values of an interval that are assigned to some variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection. The values 1, 3 and the value 9, which respectively correspond to intervals
\left[0,3\right]
\left[8,11\right]
, are assigned to the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
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In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form
{\displaystyle P(x)}
{\displaystyle P}
is a relation symbol and
{\displaystyle x}
Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments.
The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains).[1][2] Adding a single binary relation symbol to monadic logic, however, results in an undecidable logic.
Relationship with term logic
The need to go beyond monadic logic was not appreciated until the work on the logic of relations, by Augustus De Morgan and Charles Sanders Peirce in the nineteenth century, and by Frege in his 1879 Begriffsschrifft. Prior to the work of these three men, term logic (syllogistic logic) was widely considered adequate for formal deductive reasoning.
Inferences in term logic can all be represented in the monadic predicate calculus. For example the argument
No mammal is a bird.
Thus, no dog is a bird.
can be notated in the language of monadic predicate calculus as
{\displaystyle [(\forall x\,D(x)\Rightarrow M(x))\land \neg (\exists y\,M(y)\land B(y))]\Rightarrow \neg (\exists z\,D(z)\land B(z))}
{\displaystyle D}
{\displaystyle M}
{\displaystyle B}
denote the predicates of being, respectively, a dog, a mammal, and a bird.
Conversely, monadic predicate calculus is not significantly more expressive than term logic. Each formula in the monadic predicate calculus is equivalent to a formula in which quantifiers appear only in closed subformulas of the form
{\displaystyle \forall x\,P_{1}(x)\lor \cdots \lor P_{n}(x)\lor \neg P'_{1}(x)\lor \cdots \lor \neg P'_{m}(x)}
{\displaystyle \exists x\,\neg P_{1}(x)\land \cdots \land \neg P_{n}(x)\land P'_{1}(x)\land \cdots \land P'_{m}(x),}
These formulas slightly generalize the basic judgements considered in term logic. For example, this form allows statements such as "Every mammal is either a herbivore or a carnivore (or both)",
{\displaystyle (\forall x\,\neg M(x)\lor H(x)\lor C(x))}
. Reasoning about such statements can, however, still be handled within the framework of term logic, although not by the 19 classical Aristotelian syllogisms alone.
Taking propositional logic as given, every formula in the monadic predicate calculus expresses something that can likewise be formulated in term logic. On the other hand, a modern view of the problem of multiple generality in traditional logic concludes that quantifiers cannot nest usefully if there are no polyadic predicates to relate the bound variables.
The formal system described above is sometimes called the pure monadic predicate calculus, where "pure" signifies the absence of function letters. Allowing monadic function letters changes the logic only superficially[citation needed], whereas admitting even a single binary function letter results in an undecidable logic.
Monadic second-order logic allows predicates of higher arity in formulas, but restricts second-order quantification to unary predicates, i.e. the only second-order variables allowed are subset variables.
^ Heinrich Behmann, Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem, in Mathematische Annalen (1922)
^ Löwenheim, L. (1915) "Über Möglichkeiten im Relativkalkül," Mathematische Annalen 76: 447-470. Translated as "On possibilities in the calculus of relatives" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press: 228-51.
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A-4: Algebraic Expressions and Operations
Obtain the sum of the coefficients of
{x}^{3}
{x}^{4}
in the sum of
{\left(2 x-a\right)}^{4} {\left(x+a\right)}^{2}
{\left(3 x-1\right)}^{5}
Enter the sum (ensuring a space between the factors) and press the Enter key.
Context Panel: Expand
Context Panel: Collect≻
x
{\left(2 x-a\right)}^{4} {\left(x+a\right)}^{2}+{\left(3 x-1\right)}^{5}
{\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{4}}{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{5}}
\stackrel{\text{expand}}{=}
{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{24}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{243}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{405}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{270}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{90}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
\stackrel{\text{collect w.r.t. x}}{=}
\textcolor[rgb]{0,0,1}{16}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{243}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{24}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{405}\right){\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{8}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{270}\right){\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{9}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{90}\right){\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}\right)\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
Sum of the coefficients of
{x}^{3}
{x}^{4}
by control-drag (or copy/paste). Press the Enter key.
\left(270+8{a}^{3}\right)+\left(-24{a}^{2}-405\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{135}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{24}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}
Assign the sum to
{q}_{1}
{q}_{1}≔{\left(2 x-a\right)}^{4}\cdot {\left(x+a\right)}^{2}+{\left(3 x-1\right)}^{5}
Apply the expand command.
{q}_{2}≔\mathrm{expand}\left({q}_{1}\right)
Apply the collect command.
{q}_{3}≔\mathrm{collect}\left({q}_{2},x\right)
Use the coeff command.
\mathrm{coeff}\left({q}_{3},x,3\right)+\mathrm{coeff}\left({q}_{3},x,4\right)
|
Global Constraint Catalog: Cdom_reachability
<< 5.133. divisible_or5.135. domain >>
[QuesadaVanRoyDevilleCollet06]
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}\left(\begin{array}{c}\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴},\hfill \\ \mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\hfill \\ \mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\hfill \\ \mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}\hfill \end{array}\right)
\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚒𝚗𝚝}\right)
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴}\ge 1
\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴}\le |\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
|\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|=|\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
|\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|=|\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
be three directed graphs respectively called the flow graph, the dominance graph and the transitive closure graph which all have the same vertices. In addition let
\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴}
denote a vertex of the flow graph called the source node (not necessarily a vertex with no incoming arcs). The
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint holds if and only if the flow graph (and its source node) verifies:
The dominance relation expressed by the dominance graph (i.e., if there is an arc
\left(i,j\right)
in the dominance graph then, within the flow graph, all the paths from the source node to
j
i
; note that when there is no path from the source node to
j
then any node dominates
j
The transitive relation expressed by the transitive closure graph (i.e., if there is an arc
\left(i,j\right)
in the transitive closure graph then there is also a path from
i
j
in the flow graph).
\left(\begin{array}{c}1,〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing \hfill \end{array}〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing ,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\varnothing \hfill \end{array}〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,2,3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,3,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{3\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{4\right\}\hfill \end{array}〉\hfill \end{array}\right)
The flow graph, the dominance graph and the transitive closure graph corresponding to the second, third and fourth arguments of the
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint are respectively depicted by parts (A), (B) and (C) of Figure 5.134.1. The
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
holds since the following conditions hold.
The dominance relation expressed by the dominance graph is verified:
\left(1,2\right)
belongs to the dominance graph all the paths from 1 to 2 in the flow graph pass through 1.
\left(1,3\right)
\left(1,4\right)
\left(2,3\right)
\left(2,4\right)
The graph depicted by the fourth argument of the
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint (i.e.,
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
) is the transitive closure of the graph depicted by the second argument (i.e.,
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
Figure 5.134.1. (A) Flow graph, (B) dominance graph and (C) transitive closure graph of the Example slot (taken from [Quesada06])
|\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}|>2
\mathrm{𝙵𝙻𝙾𝚆}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝙳𝙾𝙼𝙸𝙽𝙰𝚃𝙾𝚁}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚃𝚁𝙰𝙽𝚂𝙸𝚃𝙸𝚅𝙴}_\mathrm{𝙲𝙻𝙾𝚂𝚄𝚁𝙴}_\mathrm{𝙶𝚁𝙰𝙿𝙷}
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint was introduced in order to solve reachability problems (e.g., disjoint paths, simple path with mandatory nodes).
Within the name
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
\mathrm{𝚍𝚘𝚖}
stands for domination. In the context of path problems
\mathrm{𝚂𝙾𝚄𝚁𝙲𝙴}
refers to the start of the path we want to build.
It was shown in [Quesada06] that, finding out wether a
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint has a solution or not is NP-hard. This was achieved by reduction to disjoint paths problem [GareyJohnson79].
The first implementation [QuesadaVanRoyDeville05] of the
\mathrm{𝚍𝚘𝚖}_\mathrm{𝚛𝚎𝚊𝚌𝚑𝚊𝚋𝚒𝚕𝚒𝚝𝚢}
constraint was done in Mozart [Mozart06]. Later on, a second implemention [Quesada06] was done in Gecode [Gecode06]. Both implementations consist of the following two parts:
Algorithms [Roditty03] for maintaining the lower bound of the transitive closure graph.
Algorithms for maintaining the upper bound of the transitive closure graph, while respecting the dominance constraints [Georgiadis05].
\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
(path).
combinatorial object: path.
constraint type: predefined constraint, graph constraint.
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Global Constraint Catalog: Csliding_distribution
<< 5.349. sliding_card_skip05.351. sliding_sum >>
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}\left(\mathrm{𝚂𝙴𝚀},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚂𝙴𝚀}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚘𝚖𝚒𝚗}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚘𝚖𝚊𝚡}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚂𝙴𝚀}>0
\mathrm{𝚂𝙴𝚀}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚘𝚖𝚒𝚗},\mathrm{𝚘𝚖𝚊𝚡}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}\le \mathrm{𝚂𝙴𝚀}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}\le \mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}
For each sequence of
\mathrm{𝚂𝙴𝚀}
consecutive variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection, each value
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
\left(1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right)
should be taken by at least
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚒𝚗}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚘𝚖𝚊𝚡}
\left(\begin{array}{c}4,〈0,5,0,6,5,0,0〉,\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚟𝚊𝚕}-0\hfill & \mathrm{𝚘𝚖𝚒𝚗}-1\hfill & \mathrm{𝚘𝚖𝚊𝚡}-2,\hfill \\ \mathrm{𝚟𝚊𝚕}-1\hfill & \mathrm{𝚘𝚖𝚒𝚗}-0\hfill & \mathrm{𝚘𝚖𝚊𝚡}-4,\hfill \\ \mathrm{𝚟𝚊𝚕}-4\hfill & \mathrm{𝚘𝚖𝚒𝚗}-0\hfill & \mathrm{𝚘𝚖𝚊𝚡}-4,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚘𝚖𝚒𝚗}-1\hfill & \mathrm{𝚘𝚖𝚊𝚡}-2,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚘𝚖𝚒𝚗}-0\hfill & \mathrm{𝚘𝚖𝚊𝚡}-2\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
On the first sequence of 4 consecutive values
0506
values 0, 1, 4, 5 and 6 are respectively used 2, 0, 0, 1 and 1 times.
On the second sequence of 4 consecutive values
5065
On the third sequence of 4 consecutive values
0650
On the fourth sequence of 4 consecutive values
6500
\mathrm{𝚂𝙴𝚀}>1
\mathrm{𝚂𝙴𝚀}<|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚒𝚗}
\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚘𝚖𝚊𝚡}
can be increased to any value
\le \mathrm{𝚂𝙴𝚀}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚂𝙴𝚀}=1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚜𝚞𝚖}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}
(sliding sequence constraint).
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚜𝚎𝚚}
(individual values replaced by single set of values).
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint type: decomposition, sliding sequence constraint, system of constraints.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑃𝐴𝑇𝐻}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}
\mathrm{𝚂𝙴𝚀}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\left(\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-\mathrm{𝚂𝙴𝚀}+1
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
constraint is a constraint where the arc constraints do not have an arity of 2.
Parts (A) and (B) of Figure 5.350.1 respectively show the initial and final graph associated with the Example slot. Since all arc constraints hold (i.e., because of the graph property
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-\mathrm{𝚂𝙴𝚀}+1
) the final graph corresponds to the initial graph.
Figure 5.350.1. (A) Initial and (B) final graph of the
\mathrm{𝚜𝚕𝚒𝚍𝚒𝚗𝚐}_\mathrm{𝚍𝚒𝚜𝚝𝚛𝚒𝚋𝚞𝚝𝚒𝚘𝚗}
\left(\mathbf{4},〈0,\mathbf{5},\mathbf{0},\mathbf{6},\mathbf{5},\mathbf{0},\mathbf{0}〉,〈012,104,404,512,602〉\right)
constraint of the Example slot where each ellipse represents an hyperedge involving
\mathrm{𝚂𝙴𝚀}=\mathbf{4}
vertices (to each ellipse corresponds a
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint)
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Global Constraint Catalog: Kpentomino
<< 3.7.183. Pattern sequencing3.7.185. Periodic >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
\mathrm{𝚙𝚘𝚕𝚢𝚘𝚖𝚒𝚗𝚘}
\mathrm{𝚛𝚎𝚐𝚞𝚕𝚊𝚛}
A constraint (i.e.,
\mathrm{𝚙𝚘𝚕𝚢𝚘𝚖𝚒𝚗𝚘}
) that can be used to model a pentomino. A pentomino is an arrangement of five unit squares that are joined along their edges.
Also denotes a constraint (i.e.,
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
\mathrm{𝚛𝚎𝚐𝚞𝚕𝚊𝚛}
) that can be used for solving tiling problems involving pentominoes. For instance, the
\mathrm{𝚐𝚎𝚘𝚜𝚝}
\mathrm{𝚛𝚎𝚐𝚞𝚕𝚊𝚛}
constraints where respectively used in [BeldiceanuCarlssonPoderSadekTruchet07] and in [LagerkvistPesant08] to solve such tiling problems.
Figure 3.7.50 presents a tiling of a rectangle with distinct pentominoes.
Figure 3.7.50. Tiling a rectangle with pentominoes
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Global Constraint Catalog: Kopen_constraint
<< 3.7.172. Open automaton constraint3.7.174. Order constraint >>
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚖𝚘𝚗𝚐}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚖𝚊𝚡𝚒𝚖𝚞𝚖}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\mathrm{𝚜𝚒𝚣𝚎}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚜𝚝𝚊𝚛𝚝𝚒𝚗𝚐}_\mathrm{𝚜𝚎𝚚}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
A constraint from which all its variables are not completely known when the constraint is posted [HoeveRegin06]. In many situations, such as configuration, planning, or scheduling of process dependant activities, the variables of a constraint are not completely known initially when the constraint is posted. Instead, they are revealed during the search process [Bartak03], [FaltingsMachoGonzalez02], [FaltingsMachoGonzalez05]. In practice, an additional argument of the constraint (a set variable or a set of 0-1 variables) provides the initial set of potential variables (the lower bound in the context of a set variable). In Bartak's model [Bartak03], an open constraint admits a sequence of domain variables
{V}_{1}{V}_{2}\cdots {V}_{m}
\left(m\ge 1\right)
as well as an additional variable
C
which gives the index of the last variable that effectively belongs to the constraint (i.e., variables
{V}_{C+1},{V}_{C+2},\cdots ,{V}_{m}
are discarded). This is for instance the case for the
\mathrm{𝚜𝚒𝚣𝚎}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚜𝚝𝚊𝚛𝚝𝚒𝚗𝚐}_\mathrm{𝚜𝚎𝚚}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
Within the context of open constraints, the notion of contractibility was introduced in [Maher09c] in order to characterise a global constraint for which any pruning rule that removes a value from one of its variable (or which enforces any type of condition) can be reused in the context of the corresponding open global constraint (i.e., the pruning rule still makes valid deductions in the context of the open case). Intuitively, many global constraints which impose a kind of at most condition are contractible, while this is typically not the case for global constraints which enforce a kind of at least condition.
See also the keywords open automaton constraint, contractible, and extensible.
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Global Constraint Catalog: Catmost_nvalue
<< 5.40. atmost15.42. atmost_nvector >>
[BessiereHebrardHnichKiziltanWalsh05]
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙽𝚅𝙰𝙻}\ge \mathrm{𝚖𝚒𝚗}\left(1,|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
The number of distinct values taken by the variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝙽𝚅𝙰𝙻}
\left(4,〈3,1,3,1,6〉\right)
\left(3,〈3,1,3,1,6〉\right)
\left(1,〈3,3,3,3,3〉\right)
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint holds since the collection
〈3,1,3,1,6〉
involves at most 4 distinct values (i.e., in fact 3 distinct values).
\mathrm{𝙽𝚅𝙰𝙻}>1
\mathrm{𝙽𝚅𝙰𝙻}<|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
This constraint was introduced together with the
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint by C. Bessière et al. in an article [BessiereHebrardHnichKiziltanWalsh05] providing filtering algorithms for the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
It was shown in [BessiereHebrardHnichWalshO4] that, finding out whether a
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
[Beldiceanu01] provides an algorithm that achieves bound consistency. [BeldiceanuCarlssonThiel02] provides two filtering algorithms, while [BessiereHebrardHnichKiziltanWalsh05] provides a greedy algorithm and a graph invariant for evaluating the minimum number of distinct values. [BessiereHebrardHnichKiziltanWalsh05] also gives a linear relaxation for approximating the minimum number of distinct values.
n
Solutions 12 108 1280 18750 326592 6588344 150994944
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
n
Total 12 108 1280 18750 326592 6588344 150994944
3 - 64 505 3456 20209 104672 496017
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
0..n
atMostNValue in Choco.
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\le
\mathrm{𝙽𝚅𝙰𝙻}
=
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚊𝚡}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚟𝚊𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚟𝚊𝚛}
modelling: number of distinct equivalence classes, number of distinct values.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐒𝐂𝐂}
\le \mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝙴𝚀𝚄𝙸𝚅𝙰𝙻𝙴𝙽𝙲𝙴}
Parts (A) and (B) of Figure 5.41.1 respectively show the initial and final graph associated with the first example of the Example slot. Since we use the
\mathrm{𝐍𝐒𝐂𝐂}
graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a specific value that is assigned to some variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection. The 3 following values 1, 3 and 6 are used by the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
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Global Constraint Catalog: Cglobal_contiguity
<< 5.167. global_cardinality_with_costs5.169. golomb >>
[Maher02]
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\ge 0
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\le 1
Enforce all variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection to be assigned value 0 or 1. In addition, all variables assigned to value 1 appear contiguously.
\left(〈0,1,1,0〉\right)
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
constraint holds since the sequence
0110
contains no more than one group of contiguous 1.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
{V}_{1}\in \left[0,1\right]
{V}_{2}\in \left[0,1\right]
{V}_{3}=1
{V}_{4}\in \left[0,1\right]
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\left(〈{V}_{1},{V}_{2},{V}_{3},{V}_{4}〉\right)
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>2
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\left(2,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},1\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
can be reversed.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
The article [Maher02] introducing this constraint refers to hardware configuration problems.
A filtering algorithm for this constraint is described in [Maher02].
n
) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Solutions 4 7 11 16 22 29 37 46 56 67 79 92 106 121 137 154 172 191 211 232 254 277 301
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
0..1
\mathrm{𝚐𝚛𝚘𝚞𝚙}
\mathrm{𝚒𝚗𝚏𝚕𝚎𝚡𝚒𝚘𝚗}
(sequence).
\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝚖𝚞𝚕𝚝𝚒}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚗𝚘}_\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}
\mathrm{𝚛𝚘𝚘𝚝𝚜}
characteristic of a constraint: convex, automaton, automaton without counters, automaton with same input symbol, reified automaton constraint.
final graph structure: connected component.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>2
\mathrm{𝚜𝚘𝚖𝚎}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑃𝐴𝑇𝐻}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝐿𝑂𝑂𝑃}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=1
\mathrm{𝐍𝐂𝐂}
\le 1
Each connected component of the final graph corresponds to one set of contiguous variables that all take value 1.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
constraint holds since the final graph does not contain more than one connected component. This connected component corresponds to 2 contiguous variables that are both assigned to 1.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
{\mathrm{𝚅𝙰𝚁}}_{i}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
corresponds a signature variable that is equal to
{\mathrm{𝚅𝙰𝚁}}_{i}
. There is no signature constraint.
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
Figure 5.168.4. Hypergraph of the reformulation corresponding to the automaton of the
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
|
Global Constraint Catalog: Kautomaton_without_counters
<< 3.7.24. Automaton with same input symbol3.7.26. Autoref >>
\mathrm{𝚊𝚗𝚍}
\mathrm{𝚊𝚛𝚒𝚝𝚑}
\mathrm{𝚊𝚛𝚒𝚝𝚑}_\mathrm{𝚘𝚛}
\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚖𝚊𝚡}
\mathrm{𝚌𝚕𝚊𝚞𝚜𝚎}_\mathrm{𝚊𝚗𝚍}
\mathrm{𝚌𝚕𝚊𝚞𝚜𝚎}_\mathrm{𝚘𝚛}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
\mathrm{𝚌𝚘𝚗𝚜𝚎𝚌𝚞𝚝𝚒𝚟𝚎}_\mathrm{𝚐𝚛𝚘𝚞𝚙𝚜}_\mathrm{𝚘𝚏}_\mathrm{𝚘𝚗𝚎𝚜}
\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}_\mathrm{𝚌𝚘𝚗𝚜𝚝𝚛𝚊𝚒𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖}
\mathrm{𝚎𝚕𝚎𝚖}_\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚝𝚘}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚖𝚊𝚝𝚛𝚒𝚡}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}_\mathrm{𝚜𝚙𝚊𝚛𝚜𝚎}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝𝚗}
\mathrm{𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚝}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢}
\mathrm{𝚒𝚖𝚙𝚕𝚢}
\mathrm{𝚒𝚗}
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}_\mathrm{𝚌𝚑𝚊𝚒𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛𝚎𝚚}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
\mathrm{𝚖𝚊𝚡𝚒𝚖𝚞𝚖}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚎𝚡𝚌𝚎𝚙𝚝}_\mathtt{0}
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}_\mathrm{𝚝𝚑𝚊𝚗}
\mathrm{𝚗𝚊𝚗𝚍}
\mathrm{𝚗𝚎𝚡𝚝}_\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚗𝚘}_\mathrm{𝚙𝚎𝚊𝚔}
\mathrm{𝚗𝚘}_\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}
\mathrm{𝚗𝚘𝚛}
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚊𝚕𝚕}_\mathrm{𝚎𝚚𝚞𝚊𝚕}
\mathrm{𝚗𝚘𝚝}_\mathrm{𝚒𝚗}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚖𝚊𝚡𝚒𝚖𝚞𝚖}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\mathrm{𝚘𝚛}
\mathrm{𝚙𝚊𝚝𝚝𝚎𝚛𝚗}
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}_\mathrm{𝚏𝚘𝚕𝚍𝚒𝚗𝚐}
\mathrm{𝚜𝚝𝚊𝚐𝚎}_\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}
\mathrm{𝚜𝚝𝚛𝚎𝚝𝚌𝚑}_\mathrm{𝚙𝚊𝚝𝚑}_\mathrm{𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗}
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚊𝚛𝚎}_\mathrm{𝚒𝚗}_\mathrm{𝚌𝚘𝚗𝚝𝚊𝚌𝚝}
\mathrm{𝚝𝚠𝚘}_\mathrm{𝚘𝚛𝚝𝚑}_\mathrm{𝚍𝚘}_\mathrm{𝚗𝚘𝚝}_\mathrm{𝚘𝚟𝚎𝚛𝚕𝚊𝚙}
\mathrm{𝚡𝚘𝚛}
A constraint for which the catalogue provides a deterministic automaton without counters and without array of counters. Note that the filtering algorithm [Pesant04] and the reformulation [BeldiceanuCarlssonPetit04] that were initially done in the context of deterministic automata can also be used for non-deterministic automata. All these constraints are also annotated with the keyword reified automaton constraint.
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Mean predictive measure of association for surrogate splits in regression tree - MATLAB - MathWorks Nordic
Mean predictive measure of association for surrogate splits in regression tree
A regression tree constructed with fitrtree, or a compact regression tree constructed with compact.
Load the carsmall data set. Specify Displacement, Horsepower, and Weight as predictor variables.
Grow a regression tree using MPG as the response. Specify to use surrogate splits for missing values.
{\lambda }_{jk}=\frac{\text{min}\left({P}_{L},{P}_{R}\right)-\left(1-{P}_{{L}_{j}{L}_{k}}-{P}_{{R}_{j}{R}_{k}}\right)}{\text{min}\left({P}_{L},{P}_{R}\right)}.
{P}_{{L}_{j}{L}_{k}}
{P}_{{R}_{j}{R}_{k}}
prune | RegressionTree | fitrtree
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Of all the courses we've developed, Algebra 1 is the one we've spent the most time planning, debating, and brainstorming. Why? Because it sets the stage for all of high school math. Both in terms of content and how a student views themselves as a math learner. For many people, Algebra 1 was the course that caused them to develop a distaste for math. Something that used to be concrete became abstract and elusive. We want to change that! We designed the first three units of our Algebra 1 curriculum to give students opportunities for sense making and time to develop reasoning skills.
We start our year in Algebra 1 by identifying, describing, representing, and generalizing patterns. The goal is to give students multiple opportunities to describe relationships between variables and express how change is occurring. This process is made concrete through the use of various visual pattern tasks as students first
\text{\textquotedblleft}
\text{\textquotedblright}
the change and then describe the change with words, colors, tables, or algebraic expressions. Students will also work on noticing structure that allows them to generalize a rule for a pattern. This involves communicating about ideas at a level where the focus is no longer on a particular instance but rather on patterns and relationships between particular instances that allow for the development of a generalized case. Students will begin to use generalizing language such as always, every time, the pattern is, the rule is, any number, for all numbers, etc. An emphasis is placed on the Standards for Mathematical Practice, specifically Practices 7 and 8, where students are asked to look for and make use of structure and use repeated reasoning.
There are two types of sequences we look at in particular: arithmetic sequences and geometric sequences. This is to provide the foundation for our upcoming units on linear relationships (Unit 2) and exponential functions (Unit 8). While we do not expect students to write explicit formulas for any given arithmetic or geometric sequence, we want to emphasize patterns of growth that rely on repeated addition and repeated multiplication. We introduce the vocabulary words of term, common difference, and common ratio.
Big ideas: Identifying, describing, and representing patterns in multiple representations; problem-solving
Our second unit is all about linear relationships. While students have some experience with lines and linear equations from middle school, our goal is to deepen students’ understanding of linear growth and specifically describing situations that model linear growth. Students will interpret and create graphical, verbal, numerical, and algebraic representations of linear relationships and continually make connections between those representations.
We begin the unit with a look at proportional relationships, primarily to introduce tools and visual models that will help develop students’ reasoning skills that do not rely on memorizing rote procedures. In the second lesson, we graph proportional relationships as a way to review the coordinate plane and basic graphing skills. We want students to understand that ordered pairs do not just denote distinct data points but that they also represent solutions to an equation whose graph can be described as a collection of points.
Throughout the rest of the unit we work to build strong conceptual understanding around the rate of change and the values of a linear relationship, and link these to graphical features such as the slope of a line and the x- and y-intercepts. Most of all, we want students to be flexible with their reasoning, being able to use any features of the linear graph or equation to determine any other feature.
Big idea: Creating and connecting algebraic, tabular, graphical, and verbal representations of linear relationships.
This unit builds upon Unit 2 with a focus on solving linear equations and inequalities–one of the most important skills students need to master in an Algebra 1 course. Instead of dividing these lessons by
\text{\textquotedblleft}
\text{\textquotedblright}
of equation (one-step, two-step, multi-step), we want to introduce students to a variety of strategies that can be used to solve equations, built upon strong conceptual understanding of what an equation is and what a solution to an equation represents. We will use visual models like double number lines and bar models as well as mental models like working backwards, balancing, or isolating the impact in order to build fluency and sense making around solving equations. Finally, we specifically choose contexts that encourage students to use intuition and logic to solve for the unknown quantity. The goal is not to teach students a procedure but to help students collect a variety of flexible strategies that can be applied to many different problems. Our goal in providing many different strategies is not just that each student will be able to choose one strategy that works for them, but that students would begin to make decisions about which strategy works best in any given scenario, based on what information is presented in the problem.
In the second half of the unit we turn to representing situations with inequalities and solving inequalities. In general, our approach to solving inequalities will be to find the boundary point (i.e. solving the related equation) and then reasoning about the direction of the inequality. This can be done by testing values on either side of the solution to the related equation, using the context of the problem to make sense of the relationship between the quantities, or analyzing the structure of the inequality more abstractly using relational reasoning and visual models like a number line.
Big ideas: Understanding equivalence, relational reasoning
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n
n
n-1
n
n^2
\sqrt{n}
\log n
\sqrt[4]{n}
\begin{aligned} 2^{2^0} +1&= & 3 \\ 2^{2^1} +1&= & 5 \\ 2^{2^2} +1&= & 17 \\ 2^{2^3} +1&= & 257 \\ 2^{2^4} +1&= & 65537 \\ \end{aligned}
Fact: The numbers above:
3,5,17,257,65537
2^{2^{5}} + 1
is a prime."
There is insufficient information True False This question is flawed
Find the first 10-digit prime number that is found in consecutive digits of
e
Background: The tech giant Google is known to have setup such billboards in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read
"{first 10-digit prime found in consecutive digits of e}.com".
Solving this problem and visiting the website led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a resume. Unfortunately, that website has been taken down.
Fibonacci sequence is defined as
F_0=0 , F_1=1
n \geq 2
F_n=F_{n-1}+F_{n-2}
So the Fibonacci sequence is
0,1,1,2,3,5,8,13,...
Find the sum of all the terms in the Fibonacci sequence which are less than
\textbf{1 billion}
and are
\textbf{prime numbers}
Details and assumptions:-
\bullet
A prime number is the number which has only
2
positive integer divisors, which are
1
and the number itself. No other positive integer divides the number.
\bullet
1 is not a prime number, 2 is the only even prime number.
This problem is a part of the set Crazy Fibonacci
How many 3-digit numbers satisfy:
\overline { abc } =\quad { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }
by Đức Huy Là Ta
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April 2020 – user's Blog!
10.3 SPM Practice (Long Questions)
The diagram above shows a solid consisting of a right prism and a half-cylinder which are joined at the plane HICB. The base ABCDEF is on a horizontal plane. The rectangle LKJG is an inclined plane. The vertical plane JDEK is the uniform cross-section of the prism. AB = CD = 2 cm. BC = 4 cm. CM = 12 cm.
Draw to full scale
(a) The plan of the solid
(b) The elevation of the solid on a vertical plane parallel to ABCD as viewed from X.
(c) The elevation of the solid on a vertical plane parallel to DE as viewed from Y.
Posted in Plans and Elevations, SPM Add Maths
The diagram above shows a solid consisting of a right prism and a half-cylinder which are joined at the plane EFGH. EF is the diameter of the semi-circle and is equal to 3 cm.
The base ABCD is on a horizontal plane and AB = 6 cm, BC = 4 cm. The vertical plane ABFE is the uniform cross-section of the prism.
Draw to full scale, the plan of the solid.
Diagram above shows a solid right prism with rectangular base ABCD on a horizontal table. The vertical plane ABEHIL is the uniform cross-section of the prism. Rectangle LIJK, IHGJ and HEFG are inclined planes. AL, DK, BE and CF are vertical edges.
Given BC = 4 cm, AB = 6cm. EB = FC = LA = KD = 4cm, The vertical height of I and J from the rectangular base ABCD = 3cm, while the vertical height of H and G from the rectangular base ABCD = 5cm.
Draw to full scale, the elevation of the solid on a vertical plane parallel to BC as viewed from X.
10.2 Plans and Elevations
1. Plan is the image formed when solid is viewed from the top. Its orthogonal projection lies on the horizontal plane.
2. Elevation is the image formed when a solid is viewed from the front or from the side. Its orthogonal projection lies on the vertical plane.
3. In drawing plans and elevations of solids,
(a) Visible edges should be drawn using solid lines(──),
(b) Hidden edges whose views are blocked should be drawn using dashed lines (- - -).
The diagram above shows a right prism attach to a cuboid at one of its plane. ABCG and CDEF, IJNH and KLMJ are horizontal planes, ABIH, AGNH and GCJN are vertical planes while IBCJ is an inclined plane. HA = 3 cm, BC = 3 cm, CD = 1.5 cm, HI = NJ = JK = 3 cm. Draw to full scale
(a) The elevation of the solid on a vertical plane parallel to AB as viewed from X.
(b) A solid semi-cylinder is joined to the solid in A as show in the diagram above. JPON is a vertical plane and JP = 1.5 cm. Draw to full scale
i. The plan of the solid
ii. The elevation of the solid on a vertical plane parallel to BCD as viewed from Y.
9.4.3 Shortest Distance between Two Points
1. The shortest distance between two points on the surface of the earth is the distance measured along a great circle.
Shortest distance between points D and M
= ( θ × 60 ) nautical miles
In the above diagram, calculate
(a) The distance from P to Q, measured along the parallel of latitude 48o S,
(b) The distance from P to Q, measured along the route PSQ, where S is the South Pole.
State the shorter distance.
Distance from Pto Q, measured along the parallel of latitude 48o S
= 180 × 60 × cos 48o← (angle PMQ = 180o)
= 7266.61 n.m.
Distance from Pto Q, measured along the route PSQ, where S is the South Pole
= 84 × 60 ← (angle POQ = 180o – 48o – 48o = 84o)
= 5040 n.m.
The distance from P to Q, measured along the route PSQ in (b), where S is the South Pole, is shorter than the distance measured along the parallel of latitude in (a).
The shortest distance in the above example is
the distance along the arc of a great circle,
which passes through the South (or North) Pole.
Posted in Earth as a Sphere, SPM Add Maths
9.3 Location of a Place
The location of a place is written as an ordered pair of latitude and longitude (latitude, longitude).
Location of point P is (35o N, 27o E).
9.6 SPM Practice (Long Questions)
Diagram below shows four points P, Q, R and M, on the surface of the earth. P lies on longitude of 70oW. QR is the diameter of the parallel of latitude of 40o N. M lies 5700 nautical miles due south of P.
(a) Find the position of R.
(b) Calculate the shortest distance, in nautical miles, from Q to R, measured along the surface of the earth.
(c) Find the latitude of M.
(d) An aeroplane took off from R and flew due west to P along the parallel of latitude with an average speed of 660 knots.
Calculate the time, in hours, taken for the flight.
Latitude of R = latitude of Q = 40o N
Longitude of Q = (70o – 25o) W = 45o W
Longitude of R = (180o – 45o) E = 135o E
Therefore, position of R = (40o N, 135oE).
Shortest distance from Q to R
= (180 – 40 – 40) x 60
= 6000 nautical miles
\begin{array}{l}\angle POM=\frac{5700}{60}\\ \text{}={95}^{o}\\ \therefore \text{Latitude of}M=\left({95}^{o}-{40}^{o}\right)S\\ \text{}={55}^{o}S\end{array}
\begin{array}{l}\text{Time taken =}\frac{\text{distance from}R\text{to}P}{\text{average speed}}\\ \text{}=\frac{\left(180-25\right)×60×\mathrm{cos}{40}^{o}}{660}\\ \text{}=\frac{155×60×\mathrm{cos}{40}^{o}}{660}\\ \text{}=10.79\text{hours}\end{array}
P(25o S, 40o E), Q(θo N, 40o E), R(25o S, 10o W) and K are four points on the surface of the earth. PK is the diameter of the earth.
(a) State the location of point K.
(b) Q is 2220 nautical miles from P, measured along the same meridian.
Calculate the value of θ.
(c) Calculate the distance, in nautical mile, from P due west to R, measured along the common parallel of latitude.
(d) An aeroplane took off from Q and flew due south to P. Then, it flew due west to R. The average speed of the aeroplane was 600 knots.
Calculate the total time, in hours, taken for the whole flight.
As PK is the diameter of the earth, therefore latitude of K = 25o N
Longitude of K= (180o – 40o) W = 140o W
Therefore, location of K = (25o N, 140oW).
Let the centre of the earth be O.
\begin{array}{l}\angle POQ=\frac{2220}{60}\\ \text{}={37}^{o}\\ {\theta }^{o}={37}^{o}-{25}^{o}={12}^{o}\\ \therefore \text{The value of}\theta \text{is 12}\text{.}\end{array}
Distance from P to R
= (40 + 10) × 60 × cos 25o
= 50 × 60 × cos 25o
= distance from Q to P + distance from P to R
= 2220 + 2718.92
= 4938.92 nautical miles
\begin{array}{l}\text{Time taken =}\frac{\text{total distance from}Q\text{to}R}{\text{average speed}}\\ \text{}=\frac{4938.92}{600}\\ \text{}=8.23\text{hours}\end{array}
9.5 SPM Practice (Short Questions)
In diagram below, N is the North Pole and S is the South Pole. The location of point P is (40o S, 70o W) and POQ is the diameter of the earth.
Find the longitude of Q.
Since PQ is a diameter of the earth and the longitude of P is θo W, the longitude of Q is (180o – θo) E.
Longitude of P = 70o W
Longitude of Q = (180o – 70o) E
= 110oE
In diagram below, N is the North Pole and S is the South Pole and NOS is the axis of the earth.
Find the position of point Q.
Latitude of Q = (90o – 42o) N
= 48o N
Longitude of Q = (65o – 30o) E
= 35o E
Therefore, position of Q = (48o N, 35oE).
1. A great circle is a circle with the centre of the Earth as its centre.
2. A meridian is half of a great circle from the North pole to the South pole.
3. The longitude of the Greenwich Meridian is 0o.
4. The longitude of a meridian is determined by:
(a) The angle between the meridian plane and the Greenwich Meridian.
(b) The position of the meridian to the east or west of the Greenwich Meridian.
Longitude of P is 55o W.
Longitude of Q is 30o E.
5. All points on the same meridian have the same longitude.
Difference between Two Longitudes
1. If both the meridians are on the east (or west) of the Greenwich Meridian, subtract the angles of the longitudes.
2. If both the meridians are on the opposite sides of the Greenwich Meridian, add the angles of the longitudes.
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ME Vol.9 No.3 , March 2018
Abstract: Bus has become an integral traffic mode for urban residents along with the development of public transport system; and as a kind of public service, urban public transport is the basic infrastructure closely related to people, and is also required to constantly improve its service levels to better serve people. Therefore, how to evaluate its service level has become one of the important projects that need to be studied. Based on establishing the mathematical model of fuzzy comprehensive evaluation and illustrated by the case of Xi’an, this paper verifies the model rationality, assesses the bus development level in Xi’an, and further puts forward countermeasures for its bus services according to the results.
Keywords: Public Transport, Passenger Satisfaction, Fuzzy Comprehensive Evaluation
Urban public transport is not only the key to the modernization construction, but also an indispensable part of urban infrastructure construction, as well as an important link to build a harmonious society and improve the living standard of residents. As a kind of public service, public transport is also required to conduct marketing for most consumers under the background of continuously improved transport facilities. More importantly, the quality of bus service not only reflects the travel condition, social style, urban and spiritual civilization construction level, but also becomes the main indicator to measure the urban public transport marketing competitiveness.
The service level of urban public transport has been greatly improved in recent years, but the traditional increase in the number of buses and control of passenger fares still cannot solve the problems of low safety, poor comfort and environmental health of the bus. The improved people’s living standard and the reduced costs of cars have made more and more people choose private cars, thus exacerbating the urban traffic congestion, air pollution, noise and safety. As the service level of urban public transport directly affects its sustainable development goals, the research on the passenger satisfaction evaluation can effectively find problems in the process of public transport services from the perspective of “customers”, in order to improve the bus passenger satisfaction and public transport service level.
2. Selection of Evaluation Index of Passenger Satisfaction in Public Transport
There is no complete, unified and effective evaluation index system of public transport passenger satisfaction in China, making different evaluation indicators in various regions. Therefore, based on the foreign evaluation indexes and the actual situation in China, the paper adopts the expert investigation to screen more suitable six major indicators (see Table 1), including safety [1] , convenience [1] , punctuality, rapidity, comfort [1] and economy.
In the process of index selection, this article follows the scientific principle, objectivity principle, operability principle, measurability principle and Importance guarantee principle.
1) Scientific principle: The choice of indicators and the determination of index weights, data selection, calculation and synthesis must be based on accepted scientific theories.
Table 1. Passenger satisfaction evaluation index.
2) Objectivity principle: Ensure the objective and fairness of the evaluation index system, ensure the accuracy of the data sources and the scientificity of the assessment methods.
3) Operability principle: The indicators should try to select daily statistical indicators or easy available indicators so as to provide an intuitive and easy understanding of the stage of public transport and help improve the service level.
4) Measurability principle: Customer Satisfaction Evaluation results must be quantifiable, so the selected indicators must be able to carry out statistical analysis.
5) Importance guarantee principle [2] : Need to grasp the needs of customers accurately, the selected indicators must be considered important by customers.
3. Establishment of the Fuzzy Comprehensive Evaluation Model
Fuzzy comprehensive evaluation method is to regard the fuzzy object and fuzzy concept as the certain fuzzy set, then establish a fuzzy membership function, and conduct the quantitative analysis on fuzzy object through the relevant operations [3] .
1) Determine the factor set of fuzzy comprehensive evaluation object
The set of factors that affects the scores of evaluation objects is called the factor set, which is usually expressed by the letter U,
U=\left\{{u}_{1},{u}_{2},\cdots ,{u}_{m}\right\}
2) Determine the weight set of each influencing factor
The influencing degree of each factor on the value of evaluation object is different, so different factors
{u}_{i}\left(i=1,2,\cdots \right)
should be given the corresponding weight coefficient
{\alpha }_{i}\left(i=1,2,\cdots m\right)
, thus forming the weight set
A=\left\{{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{m}\right\}
And the weight coefficient should be normalized before synthesis, which means
\underset{i=1}{\overset{m}{\sum }}{\alpha }_{i}=1,{\alpha }_{i}\ge 0\left(i=1,2,\cdots ,m\right)
As the weight set is a fuzzy set, in order to clearly represent the correspondence between weight coefficient and various factors, it can be expressed as
A=\frac{{\alpha }_{1}}{{u}_{1}}+\frac{{\alpha }_{2}}{{u}_{2}}+\cdots +\frac{{\alpha }_{m}}{{u}_{m}}
3) Determine the evaluation set of the evaluation object
The evaluation set is a collection of various evaluation results that reviewers may make on the evaluation objects, and usually expressed as capital letter V, then
V=\left\{{V}_{1},{V}_{2},\cdots ,{V}_{n}\right\}
, and each level corresponds to a fuzzy subset.
The evaluation set is desirable as V = {very satisfied, more satisfied, general, not very satisfied, very dissatisfied} in the passenger satisfaction evaluation in public transport.
4) Establish the fuzzy membership matrix
The single-factor fuzzy evaluation is to evaluate separately from a factor, and determine the elements membership of evaluation objects in the evaluation set.
Generally, the i-th element ui should be evaluated in the factor set, and if the membership degree of the j-th element vj is given as γij, the result can be expressed as a fuzzy set:
{R}_{i}=\frac{{\gamma }_{i1}}{{v}_{1}}+\frac{{\gamma }_{i2}}{{v}_{2}}+\cdots +\frac{{\gamma }_{in}}{{v}_{n}}\left(i=1,2,\cdots ,m\right)
Ri is a single-factor evaluation set, and the fuzzy matrix R formed by its membership is the single-factor evaluation matrix,
R=\left[\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& \cdots & {\gamma }_{1n}\\ {\gamma }_{21}& {\gamma }_{22}& \cdots & {\gamma }_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {\gamma }_{m1}& {\gamma }_{m2}& \cdots & {\gamma }_{mn}\end{array}\right]
5) Synthetic fuzzy comprehensive evaluation result vector
The evaluation result vector can be obtained by synthesizing the membership matrix of each subject with the appropriate weight set.
B=A*R=\left({\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{m}\right)\left[\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& \cdots & {\gamma }_{1n}\\ {\gamma }_{21}& {\gamma }_{22}& \cdots & {\gamma }_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {\gamma }_{m1}& {\gamma }_{m2}& \cdots & {\gamma }_{mn}\end{array}\right]
The matrix B can obtain the results according to the fuzzy evaluation method after the normalized process.
4. Evaluation of Passenger Satisfaction in Public Transport-Illustrated by the Case of Xi’an
With the accelerating process of urbanization, economic development has stimulated the development of urban transport and brought tremendous pressure on urban transport. As a result, traffic congestion has become increasingly serious and the contradiction between supply and demand on roads has become increasingly acute. As a political, economic and cultural center of Shanxi Province, Xi’an has such characteristics as dense population, frequent activities, centralized facilities and tight land. It is reflected in the fact that Xi’an has less people and less traffic and fewer cars, these basic contradictions are prominent [4] .
In the process of public transportation development, there is a single public transport system structure, an unreasonable layout of the line network, a low service rate of the site, and a poor service level of the system, which does not fully reflect the idea of “bus priority” and can no longer fully satisfy people’s travel needs. Quality requirements, the downward trend in the attraction of the travel crowd, resulting in the structure of the entire city traffic to the bad direction, reducing the utilization of urban roads.
In order to verify the rationality of the mathematical model, at the same time, put forward reasonable suggestions for the development of public transport in Xi’an. In this paper, Xi’an city as an example, Xi’an city bus passenger satisfaction survey were conducted, a total of 570 questionnaires were distributed, a total of 500 valid questionnaires were obtained, the statistical results in Table 2.
Then makes the quantitative evaluation of its overall level of the according to the fuzzy evaluation method.
1) Factor set
The factor set of the fuzzy comprehensive evaluation can be obtained according to the evaluation index determined above.
U = {safety (u1), convenience (u2), punctuality (u3), rapidity (u4), economy (u5), comfort (u6)}
2) Weight set
Based on the results of the calculation, we can give the evaluation weight set of the six components of bus passenger satisfaction: A = {α1, α2, α3, α4, α5, α6} = {0.03, 0.25, 0.04, 0.15, 0.44, 0.09}
3) Evaluation set
The evaluation set V = {V1, V2, V3, V4, V5} = {very satisfied, more satisfied, general, not satisfied, very dissatisfied}
The corresponding numerical set of the evaluation set is N = {N1, N2, N3, N4, N5} = {95, 85, 75, 65, 55}
4) Single-factor fuzzy evaluation
It can be obtained from the data,
\begin{array}{l}{R}_{1}=\frac{0.4}{{V}_{1}}+\frac{0.3}{{V}_{2}}+\frac{0.21}{{V}_{3}}+\frac{0.05}{{V}_{4}}+\frac{0.04}{{V}_{5}}\\ {R}_{2}=\frac{0.24}{{V}_{1}}+\frac{0.17}{{V}_{2}}+\frac{0.27}{{V}_{3}}+\frac{0.21}{{V}_{4}}+\frac{0.11}{{V}_{5}}\\ {R}_{3}=\frac{0.17}{{V}_{1}}+\frac{0.3}{{V}_{2}}+\frac{0.31}{{V}_{3}}+\frac{0.14}{{V}_{4}}+\frac{0.08}{{V}_{5}}\\ {R}_{4}=\frac{0.04}{{V}_{1}}+\frac{0.12}{{V}_{2}}+\frac{0.32}{{V}_{3}}+\frac{0.42}{{V}_{4}}+\frac{0.10}{{V}_{5}}\\ {R}_{5}=\frac{0.05}{{V}_{1}}+\frac{0.25}{{V}_{2}}+\frac{0.48}{{V}_{3}}+\frac{0.17}{{V}_{4}}+\frac{0.05}{{V}_{5}}\\ {R}_{6}=\frac{0.09}{{V}_{1}}+\frac{0.31}{{V}_{2}}+\frac{0.39}{{V}_{3}}+\frac{0.17}{{V}_{4}}+\frac{0.04}{{V}_{5}}\end{array}
Table 2. The percentage Xi’an citizen’s satisfaction with public transit evaluation results.
The single-factor evaluation matrix is:
R=\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\ 0.24& 0.17& 0.27& 0.21& 0.11\\ 0.17& 0.3& 0.31& 0.14& 0.08\\ 0.04& 0.12& 0.32& 0.42& 0.10\\ 0.05& 0.25& 0.48& 0.17& 0.05\\ 0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]
And then the complex operation of fuzzy matrix is
\begin{array}{c}B=A*R=\left(0.03,0.15,0.04,0.15,0.44,0.09\right)\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\ 0.24& 0.17& 0.27& 0.21& 0.11\\ 0.17& 0.3& 0.31& 0.14& 0.08\\ 0.04& 0.12& 0.32& 0.42& 0.10\\ 0.05& 0.25& 0.48& 0.17& 0.05\\ 0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]\\ =\left(0.19,0.2,0.35,0.19,0.07\right)\end{array}
We can calculate specific scores in safety, convenience, punctuality, rapidity, economy and comfort of public transport in Xi’an according to the obtained weight and the corresponding numerical set of the evaluation set.
The score of safety
{B}_{1}*N=\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=84.7
The score of convenience
{B}_{2}*N=\left[\begin{array}{ccccc}0.24& 0.17& 0.27& 0.21& 0.11\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.2
The score of punctuality
{B}_{3}*N=\left[\begin{array}{ccccc}0.17& 0.3& 0.31& 0.14& 0.08\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=78.4
The score of rapidity
{B}_{4}*N=\left[\begin{array}{ccccc}0.04& 0.12& 0.32& 0.42& 0.10\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=70.8
The score of economy
{B}_{5}*N=\left[\begin{array}{ccccc}0.05& 0.25& 0.48& 0.17& 0.05\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=75.8
The score of comfort
{B}_{6}*N=\left[\begin{array}{ccccc}0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.4
The comprehensive score
B*N=\left[\begin{array}{ccccc}0.19& 0.2& 0.35& 0.19& 0.07\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.5
The above calculation results show that the comprehensive score of bus passenger satisfaction in Xi’an is 77.5 and in the intermediate level, so many public transport services still need to be adjusted. And the scores of safety, convenience, punctuality, economy and comfort index are 84.7, 77.2, 78.4, 75.8 and 77.4, which are basically between the general and the satisfaction, belong to the intermediate level, and still require to be further improved. While the score of rapidity index is 70.8, and between not satisfied and the general, which is very urgent to be improved. Only in this way, we can improve the bus passenger satisfaction level in Xi’an, and make the overall level show a relatively stable upward trend.
As a scientific, accurate and intuitive research method, the quantitative research of fuzzy comprehensive evaluation is widely applied under the background of the continuous developed society and science.
Based on the establishment of the fuzzy comprehensive evaluation model, this paper mainly measures the bus passenger satisfaction in service level of Xi’an, verifies the reason and applicability of the model, and further puts forward suggestions on public transport services according to the results.
1) To strengthen the investment of government in public transport infrastructure, and take measures to ensure its preferential development.
2) To improve the hardware facilities of bus services, strengthen the construction of junction and interchange station.
3) To conduct humanistic bus services, and improve the soft power of public transport.
4) To establish and improve the bus service operation mechanism and enhance the social benefits.
At the same time, there are still some deficiencies in this study:
1) The evaluation lacks consideration of regional environmental issues. This paper did not consider the influence of weather, seasons and urban culture when doing evaluation studies. In particular, Xi’an City, as a world-famous cultural city, has a very limited plan for bus routes. At the same time, due to different weather and seasons, people’s physiological experience is different [5] , which may also have an impact on ride satisfaction.
2) Bus collinear problem evaluation does not consider the problem that how many buses the passengers can choose to the same place. In the case of collinearity, ride satisfaction will affect passengers’ decision-making on line selection. This is also a direction of future research.
Cite this paper: Yin, D. (2018) Research on Fuzzy Comprehensive Evaluation of Passenger Satisfaction in Urban Public Transport. Modern Economy, 9, 528-535. doi: 10.4236/me.2018.93034.
[1] Wen, C.-H., Lan, L. and Chen, C.-H. (2005) Passengers Perception on Service Quality and Their Choice for Intercity Bus Services. The Transportation Research Board 84th Annual Meeting, Washington DC, 9-13 January 2005, 14-15.
[2] Xing, K. (2004) Research on the Evaluation of Urban Public Traffic Service Level. Jilin University, Changchun, 1-2.
[3] Yang, L.B. and Gao, Y. (1995) Fuzzy Mathematics Principle and Application. South China University of Technology Press, Guangzhou.
[4] Fu, L.L., Liang, Y.H. and Wang, Y. (2007) Analysis of Current Traffic Situations and Problems in Xi’an City. Technology of Highway and Transport, No. 6, 117.
[5] Huang, H.B. (2014) The Research on Changsha City Bus Passenger Satisfaction Evaluation by AHP-Fuzzy Comprehensive Evaluation. Central South University of Forestry and Technology, Changsha, 54.
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Global Constraint Catalog: Ccircuit
<< 5.65. change_vectors5.67. circuit_cluster >>
[Lauriere78]
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚊𝚝𝚘𝚞𝚛}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌}\right]\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚒𝚗𝚍𝚎𝚡}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝚒𝚗𝚍𝚎𝚡}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\ge 1
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚜𝚞𝚌𝚌}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
Enforce to cover a digraph
G
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection with one circuit visiting once all vertices of
G
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
constraint holds since its
\mathrm{𝙽𝙾𝙳𝙴𝚂}
argument depicts the following Hamiltonian circuit visiting successively the vertices 1, 2, 3, 4 and 1.
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
{S}_{1}\in \left[3,4\right]
{S}_{2}\in \left[1,2\right]
{S}_{3}\in \left[1,4\right]
{S}_{4}\in \left[2,4\right]
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\left(〈1{S}_{1},2{S}_{2},3{S}_{3},4{S}_{4}〉\right)
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚜𝚞𝚌𝚌}
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙾𝙳𝙴𝚂}
In the original
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
constraint of CHIP the
\mathrm{𝚒𝚗𝚍𝚎𝚡}
attribute was not explicitly present. It was implicitly defined as the position of a variable in a list.
Within the context of linear programming [AlthausBockmayrElfKasperJungerMehlhorn02] this constraint was introduced under the name
\mathrm{𝚊𝚝𝚘𝚞𝚛}
. In the same context [Hooker07book] provides continuous relaxations of the
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
Within the KOALOG constraint system this constraint is called
\mathrm{𝚌𝚢𝚌𝚕𝚎}
Since all
\mathrm{𝚜𝚞𝚌𝚌}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection have to take distinct values one can reuse the algorithms associated with the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint. A second necessary condition is to have no more than one strongly connected component. Pruning for enforcing this condition can be done by forcing all strong bridges to belong to the final solution, since otherwise the strongly connected component would be broken apart. A third necessary condition is that, if the graph is bipartite then the number of vertices of each class should be identical. Consequently if the number of vertices is odd (i.e.,
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
is odd) the graph should not be bipartite. Further necessary conditions (useful when the graph is sparse) combining the fact that we have a perfect matching and a single strongly connected component can be found in [ShufetBerliner94]. These conditions forget about the orientation of the arcs of the graph and characterise new required elementary chains. A typical pattern involving four vertices is depicted by Figure 5.66.2 where we assume that:
There is an elementary chain between
c
and
(depicted by a dashed edge),
b
has exactly 3 neighbours.
In this context the edge between
and
b
is mandatory in any covering (i.e., the arc from
a
b
or the arc from
b
a
) since otherwise a small circuit involving
b
c
and
would be created.
Figure 5.66.2. Reasoning about elementary chains and degrees: if we have an elementary chain between
c
and
b
has 3 neighbours then the edge
\left(a,b\right)
is mandatory.
When the graph is planar [HopcroftTarjan74][Deo76] one can also use as a necessary condition discovered by Grinberg [Grinberg68] for pruning.
Finally, another approach based an the notion of 1-toughness [Chvatal73] was proposed in [KayaHooker06] and evaluated for small graphs (i.e., graphs with up to 15 vertices).
n
{s}_{1},{s}_{2},\cdots ,{s}_{n}
respectively denotes the number of vertices (i.e.,
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
) and the successor variables associated with vertices
1,2,\cdots ,n
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
constraint can be reformulated as a conjunction of one
\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}
constraint, two
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraints, and
n
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
First, we state an
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈{s}_{1},
{s}_{2},\cdots ,{s}_{n}〉\right)
constraint for enforcing distinct values to be assigned to the successor variables.
Second, the key idea is, starting from vertex 1, to successively extract the vertices
{t}_{1},{t}_{2},...,{t}_{n-1}
of the circuit until we come back on vertex 1, where
{t}_{i}
i\in \left[2,n-1\right]
) denotes the successor of
{t}_{i-1}
{t}_{1}
the successor of vertex 1. Since we have one single circuit all the
{t}_{1},{t}_{2},...,{t}_{n-1}
should be different from 1. Consequently we state a
\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}
\left(〈{t}_{1},{t}_{2},...,{t}_{{n}_{1}}〉,2,n\right)
constraint for declaring their initial domains. To express the link between consecutive
{t}_{i}
we also state a conjunction of
n
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
constraints of the form:
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left(1,〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,{t}_{1}\right)
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left({t}_{1},〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,{t}_{2}\right)
\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left({t}_{n-1},〈{s}_{1},{s}_{2},\cdots ,{s}_{n}〉,1\right)
Finally we add a redundant constraint for stating that all
{t}_{i}
i\in \left[1,n-1\right]
) are distinct, i.e.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(〈{t}_{1},{t}_{2},\cdots ,{t}_{n-1}〉\right)
n
Solutions 1 2 6 24 120 720 5040 40320 362880
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
0..n
circuit in Gecode, circuit in JaCoP, circuit in SICStus.
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(permutation),
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}_\mathrm{𝚌𝚕𝚞𝚜𝚝𝚎𝚛}
(graph constraint, one_succ),
\mathrm{𝚙𝚊𝚝𝚑}
(graph partitioning constraint, one_succ),
\mathrm{𝚙𝚛𝚘𝚙𝚎𝚛}_\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
(permutation, one_succ),
\mathrm{𝚝𝚘𝚞𝚛}
(graph partitioning constraint, Hamiltonian).
\mathrm{𝚌𝚢𝚌𝚕𝚎}
(introduce a variable for the number of circuits).
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚙𝚛𝚘𝚙𝚎𝚛}_\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
\mathrm{𝚝𝚠𝚒𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚜𝚝𝚛𝚘𝚗𝚐𝚕𝚢}_\mathrm{𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍}
constraint type: graph constraint, graph partitioning constraint.
filtering: linear programming, planarity test, strong bridge, DFS-bottleneck.
final graph structure: circuit, one_succ.
problems: Hamiltonian.
•
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}=1
•
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>1
\mathrm{𝚍𝚎𝚛𝚊𝚗𝚐𝚎𝚖𝚎𝚗𝚝}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
•
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>1
𝚔_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚅𝙰𝚁𝚂}:\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
•
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
•
\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}
=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
•
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐈𝐃}
\le 1
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
The first graph property enforces to have a single strongly connected component containing
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
vertices. The second graph property imposes to only have circuits. Since each vertex of the final graph has only one successor we do not need to use set variables for representing the successors of a vertex.
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
constraint holds since the final graph consists of one circuit mentioning once every vertex of the initial graph.
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
|
Global Constraint Catalog: Kproducer-consumer
<< 3.7.195. Preferences3.7.197. Product >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
A constraint that can be used for modelling problems where a first set of tasks produces a non-renewable resource, while a second set of tasks consumes this resource so that a limit on the minimum or the maximum stock at each instant is imposed.
Parts (A) and (B) of Figure 3.7.52 describes the simplest variant of the producer-consumer problem [SimonisCornelissens95] where no negative stock is allowed. Given an initial stock, a first set of tasks (i.e., producers) add instantaneously their respective productions to the stock (when they are finished), and a second set of tasks (i.e., consumers) take instantaneously from the stock (when they start) the amount of non-renewable resource they need. The problem is to schedule these tasks (i.e., fix the end of the producers and fix the start of the consumers) and to fix for each task the quantity it produces or consumes, so that no negative stock occurs. Part (A) of Figure 3.7.52 describes an instance of such problem where we respectively have 2 producers and 3 consumers. Part (B) depicts the corresponding cumulative view of the problem. At each timepoint the difference between the top line and the top of the cumulated profile gives the amount of available stock at that timepoint.
Figure 3.7.52. Producer-consumer models (A,C) and corresponding
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
views (B,D) enforcing that, at any point in time, we do not have any negative stock, i.e. at any point in time we do not consume more that we have produced so far; (E)
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint associated with (B)
A fundamental problem with the previous variant of the producer-consumer problem is that it does not allow to handle the fact that a resource is produced or used gradually. Parts (C) and (D) of Figure 3.7.52 describes a second variant where this is in fact possible. This is achieved by replacing the rectangle associated with a producer by a task with a decreasing height. At a given instant the cumulated quantity produced by a producer is the difference between the height of that task at its starting time and the height of that task at the considered instant. Conversely a consumer is modelled by a task with an increasing height. At a particular timepoint the cumulated quantity used by a consumer task is the difference between the height of that task at its end and the height of that task at the considered instant. Part (C) of Figure 3.7.52 describes an instance of such problem where, again, we respectively have 2 producers and 3 consumers. Part (D) depicts the corresponding cumulative view of the problem. As before, at each timepoint the difference between the top line and the top of the cumulated profile gives the amount of available stock at that timepoint.
|
Performance of a Turbine Rim Seal Subject to Rotationally-Driven and Pressure-Driven Ingestion | J. Eng. Gas Turbines Power | ASME Digital Collection
Department of Engineering Science, Oxford Thermofluids Institute, University of Oxford
Oxford OX2 0ES,
e-mail: anna.brurevert@rolls-royce.com
Thermo-Fluid Systems UTC, University of Surrey
Guildford GU2 7XH,
Sebastiaan Bottenheim
P. O. Box 3, Bristol BS34 7QE,
e-mail: Sebastiaan.Bottenheim@Rolls-Royce.com
Bru Revert, A., Beard, P. F., Chew, J. W., and Bottenheim, S. (May 28, 2021). "Performance of a Turbine Rim Seal Subject to Rotationally-Driven and Pressure-Driven Ingestion." ASME. J. Eng. Gas Turbines Power. August 2021; 143(8): 081025. https://doi.org/10.1115/1.4049858
This experimental study considered the performance of a chute rim seal downstream of turbine nozzle guide vanes (but without rotor blades). The experimental setup reproduced rotationally-driven ingestion without vanes and conditions of pressure-driven ingestion with vanes. The maximum rotor speed was 9000 rpm corresponding to a rotational Reynolds number of 3.3 × 106 with a flow coefficient of 0.45. Measurements of mean pressures in the annulus and the disk rim cavity as well as values of sealing effectiveness deduced from gas concentration data are presented. At high values of flow coefficient (low rotational speeds), the circumferential pressure variation generated by the vanes drove relatively high levels of ingestion into the disk rim cavity. For a given purge flow rate, increasing the disk rotational speed led to a reduction in ingestion, shown by higher values of sealing effectiveness, despite the presence of upstream vanes. At
Uax/(Ωb)=0.45
, the sealing effectiveness approached that associated with purely rotationally-driven ingestion. A map of sealing effectiveness against non-dimensional purge flow summarizes the results and illustrates the combined rotational and pressure-driven effects on the ingestion mechanism. The results imply that flow coefficient is a relevant parameter in rim sealing and that rotational effects are important in many applications, especially turbines with low flow coefficient.
Annulus, Cavities, Disks, Flow (Dynamics), Pressure, Rotors, Sealing (Process), Turbines
Proc. Inst. Mech. Eng. Part C J
An Investigation of Turbine Disc Cooling (Experimental Investigation and Observation of Hot Gas Flow Into a Wheel Space)
13th CIMAC Conference
, Vienna, Austria, May 7–10, Paper No. GT-30.
Aerodynamic Aspects of the Sealing of Gas-Turbine Rotor-Stator Systems: Part 2: The Performance of Simple Seals in a Quasi-Axisymmetric External Flow
Paper No. 92-GT-160.10.1115/1992-GT-160
O. J. P.
Turbine Rim Seal Ingestion
, Sussex, UK.
A Transient Flow Facility for the Study of the Thermofluid-Dynamics of a Full Stage Turbine Under Engine Representative Flow Conditions
,” 18th ISROMAC Conference, ISROMAC, Honololu, HI, Apr. 19–23, Paper No.
.https://www.researchgate.net/publication/339912323_Sealing_Performance_of_a_Turbine_Rim_Chute_Seal_Under_Rotationally-Induced_Ingestion
Experimental Measurements of Ingestion Through Turbine Rim Seals. Part II: Rotationally Induced Ingress
|
The Pricing Model - Premia
A liquidity sensitive, Black and Scholes based pricing model.
All models are wrong - but the good ones are useful.
This can also be said about the application of Black-Scholes to the crypto markets. The traditional model requires the following inputs: 1. Strike price 2. Spot price 3. Option maturity 4. Implied volatility 5. Risk free rate of return
The final Premia pool pricing model also depends on the following inputs: 6. Position size 7. Pool capital supply and demand For simplicity, let's just consider all of the inputs required as a 5 dimensional input vector
V_i
. Even though the model cannot be trusted to produce perfect equilibrium pricing, it still contains fundamental embedded relationships about how the change in each input factor affects the output price, due to the specific risk-adjusted metrics of the option.
Suppose there exists a theoretical market pricing curve (at which the crypto option market clears) for all input vectors
V_i
. There is no reason to assume, that the difference (depicted as
X
in the diagram below) between the Black-Scholes
(BS)
model output and the actual market price would be inconsistent across different values of
V_i
. In fact, the deviations from the classical market microstructure assumptions do not appear to be a reason why the fundamental dynamics of option pricing should break down. In other words, there is no reason to assume that the shape of the equilibrium pricing curve is materially different from that of the Black-Scholes dynamic-hedge based model.
Intuitively, suppose we have two different options. They both have different strike prices, different maturities, and different implied volatilities. Suppose that the unobservable real market price of one of these options is 110 DAI, while the
BS
model suggested price is 100 DAI. This implies that
BS
underprices this option by 10%. There is no reason why the other option, with a different maturity and strike price, should be underpriced by the
BS
model by any other amount than 10%.
In other words, we can assume that there exists a linear relationship between the actual market pricing curve and the
BS
-suggested curve, and that this relationship is consistent (practically speaking) across all levels of
V_i
. So in order to find a market clearing pricing curve, we have to uncover this linear relationship. But how can we achieve that?
We let the market demonstrate the relationship between
\bold{\textit{BS}}
and the actual market price curve
The answer to understanding this relationship lies in allowing the market forces to quickly converge towards it. The pricing mechanism used by Premia consists of 3 parts: 1) Original Black-Scholes Model 2) Current pool price level, adjusted for impact of option size 3) Discrete liquidity adjustment coefficient to update the price level
P_{t}(V_i;C_t)=BS(V_i)*C^*_t \\ s.t.\hspace{0.25cm}C^*_t=C_t*\int^0_{x_t}e^{-x} \alpha_x*(\frac{1}{0-\frac{(S_{t+1}-S_t)}{max(S_{t+1};S_t)}}) \\ C_{t+1}=C_t*e^{-\alpha_x\frac{(S_{t+1}-S_t)}{max(S_{t+1};S_t)}}
P_t
- Price quoted by the pool for an option at time t
BS(V_i)
- Black-Scholes model output for the selected option
V_i
- Vector of
BS
model inputs (spot price, strike price, maturity, implied volatility, risk free rate)
C_t
- Pool price level (liquidity adjustment coefficient) at time t (current period)
C^*_t
- Pool price level adjusted for price impact of option size [see Price Impact by Size]
S_t
- Pool size at time t (current period)
S_{t+1}
- Pool size after purchase (at time t+1)
\alpha_x
- Trade-specific steepness parameter, currently defaulted to 1 for all trades (no effect)
One implication of this model is the resulting price can be less than the intrinsic (exercise) value of the option when the C-level is exceedingly low. As such, we have implemented a Minimum Return model for the safety of LPs, such that the final price of an option offered by a pool will always be at least as high as the intrinsic value of the option, plus a minimum annualized return (even if the standard model produces a lower price).
|
Global Constraint Catalog: KPhi_tree
<< 3.7.187. Permutation channel3.7.189. Phylogeny >>
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
A constraint for which one of its filtering algorithms uses a balanced binary tree in order to efficiently evaluate the maximum or minimum value of a formula over all possible subsets of tasks
\Omega
of a given set of tasks
\Phi
\Phi
-trees were introduced by P. Vilím, first in the context of unary resources in [Vilim04] and in [Vilim07], and later on in the context of cumulative resources [Vilim09a], [Vilim09b]. Without loss of generality, let us sketch the main idea behind a
\Phi
-tree in the context of a cumulative resource of capacity
C
. For this purpose we follow the description given in [Vilim09a]. Given a set of tasks
\Phi
where each task has an earliest possible start, a latest possible end, a duration and a resource consumption, assume we need to evaluate the earliest completion time over all tasks of
\Phi
under the hypothesis that we should not exceed the maximum resource capacity
C
. Let us first introduce some notations:
\Omega
denotes any non-empty subset of tasks of
\Phi
{\mathrm{𝑒𝑠𝑡}}_{\Omega }
is the minimum over the earliest starts of the tasks in
\Omega
{e}_{\Omega }
is the sum of the surfaces (i.e., the product of the duration by the resource consumption) of the tasks in
\Omega
Figure 3.7.51. Example of
\Phi
-tree associated with four tasks of respective duration and resource consumption
3×4
1×3
5×5
2×4
and of respective earliest start 1, 3, 8, 9 under the assumption that the maximum capacity of the cumulative resource is equal to 5
A common estimation of the earliest completion time over all tasks of
\Phi
{max}_{\Omega \subseteq \Phi }\left\{{\mathrm{𝑒𝑠𝑡}}_{\Omega }+⌈\frac{{e}_{\Omega }}{C}⌉\right\}
⌈\frac{{max}_{\Omega \subseteq \Phi }\left\{C{\mathrm{𝑒𝑠𝑡}}_{\Omega }+{e}_{\Omega }\right\}}{C}⌉
. The numerator of the last fraction is called the energy envelope of the set of tasks
\Phi
and the purpose of a
\Phi
-tree is to evaluate this quantity efficiently. For a node
n
ℒ\left(n\right)
denote the set of leaves of the sub-tree rooted at
n
. The leaves of the
\Phi
-tree correspond to the tasks of
\Phi
sorted from left to right by increasing earliest start. Each node
n
\Phi
-tree records both, the sum of the surfaces of the tasks in
ℒ\left(n\right)
, as well as the energy envelope of the tasks in
ℒ\left(n\right)
. The sum of the surfaces associated with a non-leave node
n
of the tree corresponds to the sum of the surfaces of the children of
n
, while the energy envelope of
n
is equal to the maximum between on the one hand, the energy envelop of its right child and on the other hand the sum of the energy envelop of its left child and the recorded sum of surfaces of its right child (see [Vilim09a] for a justification of these recursive formulae). Figure 3.7.51 illustrates the construction of a
\Phi
-tree associated with four given tasks.
|
15th IAPR International Conference, DGCI 2009, Montréal, Canada, September 30 - October 2, 2009. Proceedings
Srečko Brlek, Christophe Reutenauer, Xavier Provençal
Discrete and Combinatorial Topology
Lecture Notes in Computer Science > Discrete Geometry for Computer Imagery
Discrete and Combinatorial Tools for Image Segmentation and Analysis
Discrete Shape Representation, Recognition and Analysis
Models for Discrete Geometry
Vanishing Point Detection with an Intersection Point Neighborhood
Frank Schmitt, Lutz Priese
A new technique to automatically detect the vanishing points in digital images is presented. The proposed method borrows several ideas from various papers on vanishing point detection and segmentation in sparse images and recombines them with a new intersection point neighborhood on
{\mathbb Z}^2
Ellipse Detection with Elemental Subsets
We propose a simple method for fitting ellipses to data sets. The method first computes the fitting cost of small samples, called elemental subsets. We then prove that the global fitting cost can be easily derived from the fitting cost of the samples. Since fitting costs are computed from small samples, the technique can be incorporated in many ellipse detection and recognition algorithms, and in...
Multi-Label Simple Points Definition for 3D Images Digital Deformable Model
Alexandre Dupas, Guillaume Damiand, Jacques-Olivier Lachaud
The main contribution of this paper is the definition of multi-label simple points that ensures that the partition topology remains invariant during a deformable partition process. The definition is based on simple intervoxel properties and is easy to implement. A deformation process is carried out with a greedy energy minimization algorithm. A discrete area estimator is used to approach at best standard...
Marching Triangle Polygonization for Efficient Surface Reconstruction from Its Distance Transform
In this paper we propose a new polygonization method based on the classic Marching Triangle algorithm. It is an improved and efficient version of the basic algorithm which produces a complete mesh without any cracks. Our method is useful in the surface reconstruction process of scanned objects. It works over the scalar field distance transform of the object to produce the resulting triangle mesh....
Pixel Approximation Errors in Common Watershed Algorithms
Hans Meine, Peer Stelldinger, Ullrich Köthe
The exact, subpixel watershed algorithm delivers very accurate watershed boundaries based on a spline interpolation, but is slow and only works in 2D. On the other hand, there are very fast pixel watershed algorithms, but they produce errors not only in certain exotic cases, but also in real-world images and even in the most simple scenarios. In this work, we examine closely the source of these errors...
Digital Deformable Model Simulating Active Contours
François Vieilleville, Jacques-Olivier Lachaud
Deformable models are continuous energy-minimizing techniques that have been successfully applied to image segmentation and tracking since twenty years. This paper defines a novel purely digital deformable model (DDM), whose internal energy is based on the minimum length polygon (MLP). We prove that our combinatorial regularization term has “convex” properties: any local descent on the energy leads...
Topology-Preserving Thinning in 2-D Pseudomanifolds
Nicolas Passat, Michel Couprie, Loïc Mazo, Gilles Bertrand
Lecture Notes in Computer Science > Discrete Geometry for Computer Imagery > Discrete and Combinatorial Topology > 217-228
Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on ℤ2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. ℤ n , n ≥ 3), it was proved in the 80’s that the exclusive use...
Discrete Versions of Stokes’ Theorem Based on Families of Weights on Hypercubes
Gilbert Labelle, Annie Lacasse
This paper generalizes to higher dimensions some algorithms that we developed in [1,2,3] using a discrete version of Green’s theorem. More precisely, we present discrete versions of Stokes’ theorem and Poincaré lemma based on families of weights on hypercubes. Our approach is similar to that of Mansfield and Hydon [4] where they start with a concept of difference forms to develop their discrete version...
In this paper, a structural property of the set of lozenge tilings of a 2n-gon is highlighted. We introduce a simple combinatorial value called Hamming-distance, which is a lower bound for the the number of flips – a local transformation on tilings – necessary to link two tilings. We prove that the flip-distance between two tilings is equal to the Hamming-distance for n ≤ 4. We also show, by providing...
Jordan Curve Theorems with Respect to Certain Pretopologies on $\mathbb Z^2$
We discuss four quotient pretopologies of a certain basic topology on
\mathbb Z^2
. Three of them are even topologies and include the well-known Khalimsky and Marcus-Wyse topologies. Some known Jordan curves in the basic topology are used to prove Jordan curve theorems that identify Jordan curves among simple closed ones in each of the four quotient pretopologies.
Decomposing Cavities in Digital Volumes into Products of Cycles
Ainhoa Berciano, Helena Molina-Abril, Ana Pacheco, Paweł Pilarczyk, more
The homology of binary 3–dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators corresponding to these features are represented by nontrivial 0–cycles, 1–cycles and 2–cycles, respectively. In the framework of cubical representation of digital volumes with the topology that corresponds...
Thinning Algorithms as Multivalued
{\mathcal{N}}
-Retractions
Carmen Escribano, Antonio Giraldo, María Asunción Sastre
In a recent paper we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued maps, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In particular, we characterized the deletion of simple...
DIGITAL CURVE (2)
SIMPLE POINT (2)
3D SCANNED OBJECTS (1)
ALEXANDER WHITNEY DIAGONAL (1)
ALGEBRAIC GRADIENT VECTOR FIELD (1)
ANISOTROPIC LATTICE (1)
ARITHMETIC DISCRETE PLANES (1)
ARITHMETIZATION (1)
AUTOMATIC MESH GENERATION (1)
CELL COMPLEX (1)
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CHAMFER DISTANCE (1)
CHAMFER NORM (1)
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CUBICAL COMPLEXES (1)
CUBICAL HOMOLOGY (1)
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DEFORMABLE MODEL (1)
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GEOMETRICAL MODELING (1)
HOMOLOGY COMPUTATION (1)
INTERSECTION POINTS (1)
K-COLOR PROBLEM (1)
LATTICE PATHS (1)
MARCHING TRIANGLE (1)
MEASURABLE LINES (1)
MULTI-LABEL IMAGE (1)
MULTIGRID CONVERGENCE (1)
MULTISCALE GEOMETRY (1)
POINCARÉ LEMMA (1)
POLYGONIZATION ALGORITHM (1)
POLYNOMIAL TIME ALGORITHM (1)
POLYOMINO (1)
PSEUDOMANIFOLDS (1)
SCALAR FIELD DISTANCE TRANSFORM (1)
SELF-INTERSECTION (1)
SIMPLE POINTS (1)
SIMPLE SETS (1)
THINNING ALGORITHM (1)
TOPOLOGICAL SKELETON (1)
TOPOLOGY PRESERVATION (1)
TRIANGLE MESH SURFACE (1)
WORD COMBINATORICS (1)
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