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Helmholtz resonance - Wikipedia
(Redirected from Helmholtz resonator)
Helmholtz resonance or wind throb is the phenomenon of air resonance in a cavity, such as when one blows across the top of an empty bottle. The name comes from a device created in the 1850s by Hermann von Helmholtz, the Helmholtz resonator, which he used to identify the various frequencies or musical pitches present in music and other complex sounds.[1]
2 Quantitative explanation
3.4 Music (instruments and amplification)
A selection of Helmholtz resonators from 1870, at Hunterian Museum and Art Gallery in Glasgow.
Helmholtz described in his 1862 book On the Sensations of Tone an apparatus able to pick out specific frequencies from a complex sound. The Helmholtz resonator, as it is now called, consists of a rigid container of a known volume, nearly spherical in shape, with a small neck and hole in one end and a larger hole in the other end to emit the sound.
A set of varied size resonators was sold to be used as discrete acoustic filters for the spectral analysis of complex sounds. There is also an adjustable type, called a universal resonator, which consists of two cylinders, one inside the other, which can slide in or out to change the volume of the cavity over a continuous range. An array of 14 of this type of resonator has been employed in a mechanical Fourier sound analyzer. This resonator can also emit a variable-frequency tone when driven by a stream of air in the "tone variator" invented by William Stern, 1897.[2]
The port (the neck of the chamber) is placed in the ear, allowing the experimenter to hear the sound and to determine its loudness. The resonant mass of air in the chamber is set in motion through the second hole, which is larger and doesn't have a neck.
A gastropod seashell can form a Helmholtz resonator with low Q factor, amplifying many frequencies, resulting in the "sounds of the sea".
The term Helmholtz resonator is now more generally applied to include bottles from which sound is generated by blowing air across the mouth of the bottle. In this case the length and diameter of the bottle neck also contribute to the resonance frequency and its Q factor.
By one definition a Helmholtz resonator augments the amplitude of the vibratory motion of the enclosed air in a chamber by taking energy from sound waves passing in the surrounding air. In the other definition the sound waves are generated by a uniform stream of air flowing across the open top of an enclosed volume of air.
Quantitative explanationEdit
See also: Acoustic resonance § Resonance of a sphere of air (vented)
It can be shown[3] that the resonant angular frequency is given by:
{\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A^{2}}{m}}{\frac {P_{0}}{V_{0}}}}}}
(rad/s),
{\displaystyle \gamma }
(gamma) is the adiabatic index or ratio of specific heats. This value is usually 1.4 for air and diatomic gases.
{\displaystyle A}
is the cross-sectional area of the neck;
{\displaystyle m}
is the mass in the neck;
{\displaystyle P_{0}}
is the static pressure in the cavity;
{\displaystyle V_{0}}
is the static volume of the cavity.
For cylindrical or rectangular necks, we have
{\displaystyle A={\frac {V_{n}}{L_{eq}}}}
{\displaystyle L_{eq}}
is the equivalent length of the neck with end correction, which can be calculated as :
{\displaystyle L_{eq}=L_{n}+0.3D}
{\displaystyle L_{n}}
is the actual length of the neck and
{\displaystyle D}
is the hydraulic diameter of the neck;[4]
{\displaystyle V_{n}}
is the volume of air in the neck,
{\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A}{m}}{\frac {V_{n}}{L_{eq}}}{\frac {P_{0}}{V_{0}}}}}}
From the definition of mass density (
{\displaystyle {\rho }}
{\displaystyle {\frac {V_{n}}{m}}={\frac {1}{\rho }}}
The speed of sound in a gas is given by:
{\displaystyle v={\sqrt {\gamma {\frac {P_{0}}{\rho }}}}}
thus, the resonance frequency is:
{\displaystyle f_{H}={\frac {v}{2\pi }}{\sqrt {\frac {A}{V_{0}L_{eq}}}}}
The length of the neck appears in the denominator because the inertia of the air in the neck is proportional to the length. The volume of the cavity appears in the denominator because the spring constant of the air in the cavity is inversely proportional to its volume.[5] The area of the neck matters for two reasons. Increasing the area of the neck increases the inertia of the air proportionately, but also decreases the velocity at which the air rushes in and out.
Depending on the exact shape of the hole, the relative thickness of the sheet with respect to the size of the hole and the size of the cavity, this formula can have limitations. More sophisticated formulae can still be derived analytically, with similar physical explanations (although some differences matter). See for example the book by F. Mechels.[6] Furthermore, if the mean flow over the resonator is high (typically with a Mach number above 0.3), some corrections must be applied.
Helmholtz resonance sometimes occurs when a slightly open single car window makes a very loud sound, also called side window buffeting or wind throb.[7]
Helmholtz resonance finds application in internal combustion engines (see airbox), subwoofers and acoustics. Intake systems described as 'Helmholtz Systems' have been used in the Chrysler V10 engine built for both the Dodge Viper and the Ram pickup truck, and several of the Buell tube-frame series of motorcycles.
The theory of Helmholtz resonators is used in motorcycle and car exhausts to alter the sound of the exhaust note and for differences in power delivery by adding chambers to the exhaust. Exhaust resonators are also used to reduce potentially loud and obnoxious engine noise where the dimensions are calculated so that the waves reflected by the resonator help cancel out certain frequencies of sound in the exhaust.
Helmholtz resonators are also used to build acoustic liners for reducing the noise of aircraft engines, for example. These acoustic liners are made of two components:
a simple sheet of metal (or other material) perforated with little holes spaced out in a regular or irregular pattern; this is called a resistive sheet;
a series of so-called honeycomb cavities (holes with a honeycomb shape, but in fact only their volume matters).
Such acoustic liners are used in most of today's aircraft engines. The perforated sheet is usually visible from inside or outside the airplane; the honeycomb is just under it. The thickness of the perforated sheet is of importance, as shown above. Sometimes there are two layers of liners; they are then called "2-DOF liners" (DOF meaning Degrees Of Freedom), as opposed to "single DOF liners".
This effect might also be used to reduce skin friction drag on aircraft wings by 20%.[8]
The Roman Theatre according to Vitruvius, from Wikisource:Ten Books on Architecture/Book V
Vitruvius, a 1st-century B.C. Roman architect, described the use of bronze or pottery resonators in classical theater design.[9][10]
Helmholtz resonators are used in architectural acoustics to reduce undesirable low frequency sounds (standing waves, etc.) by building a resonator tuned to the problem frequency, thereby eliminating it.[citation needed]
Music (instruments and amplification)Edit
In stringed instruments as old as the veena or sitar, or as recent as the guitar and violin, the resonance curve of the instrument has the Helmholtz resonance as one of its peaks, along with other peaks coming from resonances of the vibration of the wood. An ocarina[11] is essentially a Helmholtz resonator where the combined area of the opened finger holes determines the note played by the instrument.[12] The West African djembe is the original Helmholtz resonator with a small neck area, giving it a deep bass tone. It has been in use for thousands of years.[citation needed] Conversely, the human mouth is effectively a Helmholtz resonator when it is used in conjunction with a jaw harp,[13] shepherd's whistle,[citation needed] nose whistle, nose flute. The nose blows air through an open nosepiece, into an air duct, and across an edge adjacent to the open mouth, creating the resonator. The volume and shape of the mouth cavity augments the pitch of the tone.[14] In some two-stroke engines, a Helmholtz resonator is used to remove the need for a reed valve. A similar effect is also used in the exhaust system of most two-stroke engines, using a reflected pressure pulse to supercharge the cylinder (see Kadenacy effect.)
Helmholtz resonance is also used in Bass-reflex speaker enclosures, with the compliance of the air mass inside the enclosure and the mass of air in the port forming a helmholtz resonator. By tuning the resonant frequency of the helmholtz resonator to the lower end of the loudspeaker's usable frequency range, the speaker's low-frequency performance is improved.
Helmholtz resonance is one of the principles behind the way piezoelectric buzzers work: a piezoelectric disc acts as the excitation source, but it relies on the acoustic cavity resonance to produce an audible sound.[15]
Acoustic resonance#Resonance of a sphere of air (vented) for more detailed acoustics (physics perspective)
Vessel flute for more detailed acoustics (musical perspective)
Xun (instrument), an instrument that is a Helmholtz resonator with holes
^ Helmholtz, Hermann von (1885), On the sensations of tone as a physiological basis for the theory of music, Second English Edition, translated by Alexander J. Ellis. London: Longmans, Green, and Co., p. 44. Retrieved 2010-10-12.
^ "Helmholtz resonator at Case Western Reserve University". Helmholtz Resonator. Archived from the original on 15 April 2016. Retrieved 16 February 2016.
^ "Derivation of the equation for the resonant frequency of an Helmholtz resonator". lightandmatter.com. Archived from the original on February 28, 2017.
^ "End correction at flue pipe mouth". Johan Liljencrants on organs, pipes, air supply. September 30, 2006. Archived from the original on February 19, 2020. Retrieved October 29, 2018.
^ Greene, Chad A.; Argo IV, Theodore F.; Wilson, Preston S. (2009). A Helmholtz resonator experiment for the Listen Up project. Proceedings of Meetings on Acoustics. ASA. p. 025001. doi:10.1121/1.3112687.
^ Formulas of Acoustics
^ Torchinski, Jason (October 21, 2013). "Why Do Slightly Opened Car Windows Make That Awful Sound?". Jalopnik. Retrieved 2019-11-20.
^ "Wings That Waggle Could Cut Aircraft Emissions By 20%". ScienceDaily. May 22, 2009. Retrieved 2019-11-20.
^ Wikisource:Ten Books on Architecture/Book V, Chapter V: " Sounding Vessels in the Theater". (full text link)
^ Relevant quotes in Vitruvius article @Wikiquote
^ For a survey of prehistoric ocarina-type instruments and a linguistic analysis of the possible origins of the word ocarina, cf. Perono Cacciafoco, Francesco. (2019). A Prehistoric 'Little Goose': A New Etymology for the Word 'Ocarina'. Annals of the University of Craiova: Series Philology, Linguistics, XLI, 1-2: 356-369, Paper.
^ "Ocarina Physics - How Ocarinas Work". ocarinaforest.com. Archived from the original on 2013-03-14. Retrieved 2012-12-31.
^ Nikolsky, Aleksey (2020), Masataka, Nobuo (ed.), ""Talking Jew's Harp" and Its Relation to Vowel Harmony as a Paradigm of Formative Influence of Music on Language", The Origins of Language Revisited, Singapore: Springer Singapore, pp. 217–322, doi:10.1007/978-981-15-4250-3_8, ISBN 978-981-15-4249-7, retrieved 2020-08-24
^ Ukeheidi (2014-09-21). "noseflute.org: Nose Flute Physics - I". noseflute.org. Retrieved 2019-11-20.
^ Audio, PUI. "Design of a Helmholtz Chamber". PUI Audio. Retrieved October 29, 2018.
Wikimedia Commons has media related to Helmholtz resonators.
Oxford Physics Teaching, History Archive, "Exhibit 3 - Helmholtz resonators Archived 2020-12-26 at the Wayback Machine" (archival photograph)
HyperPhysics Acoustic Laboratory Archived 2021-02-11 at the Wayback Machine
HyperPhysics Cavity Resonance Archived 2021-02-10 at the Wayback Machine
Beverage Bottles as Helmholtz Resonators Archived 2016-04-11 at the Wayback MachineScience Project Idea for Students
That Vibrating ‘Wub Wub Wub’ That Comes From Cracking One Car Window? It’s Not Just You!
Helmholtz Resonance Archived 2020-06-08 at the Wayback Machine (web site on music acoustics)
Helmholtz's Sound Synthesiser on '120 years Of Electronic Music' Archived 2014-11-18 at the Wayback Machine
Perono Cacciafoco, Francesco. (2019). A Prehistoric 'Little Goose': A New Etymology for the Word 'Ocarina'. Annals of the University of Craiova: Series Philology, Linguistics, XLI, 1-2: 356-369, Paper
Retrieved from "https://en.wikipedia.org/w/index.php?title=Helmholtz_resonance&oldid=1075982356"
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04.08 Multiple Subplots
Sometimes it is helpful to compare different views of data side by side. To this end, Matplotlib has the concept of subplots: groups of smaller axes that can exist together within a single figure. These subplots might be insets, grids of plots, or other more complicated layouts. In this section we'll explore four routines for creating subplots in Matplotlib.
The most basic method of creating an axes is to use the plt.axes function. As we've seen previously, by default this creates a standard axes object that fills the entire figure. plt.axes also takes an optional argument that is a list of four numbers in the figure coordinate system. These numbers represent [left, bottom, width, height] in the figure coordinate system, which ranges from 0 at the bottom left of the figure to 1 at the top right of the figure.
For example, we might create an inset axes at the top-right corner of another axes by setting the x and y position to 0.65 (that is, starting at 65% of the width and 65% of the height of the figure) and the x and y extents to 0.2 (that is, the size of the axes is 20% of the width and 20% of the height of the figure):
The equivalent of this command within the object-oriented interface is fig.add_axes(). Let's use this to create two vertically stacked axes:
We now have two axes (the top with no tick labels) that are just touching: the bottom of the upper panel (at position 0.5) matches the top of the lower panel (at position 0.1 + 0.4).
Aligned columns or rows of subplots are a common-enough need that Matplotlib has several convenience routines that make them easy to create. The lowest level of these is plt.subplot(), which creates a single subplot within a grid. As you can see, this command takes three integer arguments—the number of rows, the number of columns, and the index of the plot to be created in this scheme, which runs from the upper left to the bottom right:
The command plt.subplots_adjust can be used to adjust the spacing between these plots. The following code uses the equivalent object-oriented command, fig.add_subplot():
We've used the hspace and wspace arguments of plt.subplots_adjust, which specify the spacing along the height and width of the figure, in units of the subplot size (in this case, the space is 40% of the subplot width and height).
The approach just described can become quite tedious when creating a large grid of subplots, especially if you'd like to hide the x- and y-axis labels on the inner plots. For this purpose, plt.subplots() is the easier tool to use (note the s at the end of subplots). Rather than creating a single subplot, this function creates a full grid of subplots in a single line, returning them in a NumPy array. The arguments are the number of rows and number of columns, along with optional keywords sharex and sharey, which allow you to specify the relationships between different axes.
Here we'll create a
2 \times 3
grid of subplots, where all axes in the same row share their y-axis scale, and all axes in the same column share their x-axis scale:
Note that by specifying sharex and sharey, we've automatically removed inner labels on the grid to make the plot cleaner. The resulting grid of axes instances is returned within a NumPy array, allowing for convenient specification of the desired axes using standard array indexing notation:
In comparison to plt.subplot(), plt.subplots() is more consistent with Python's conventional 0-based indexing.
To go beyond a regular grid to subplots that span multiple rows and columns, plt.GridSpec() is the best tool. The plt.GridSpec() object does not create a plot by itself; it is simply a convenient interface that is recognized by the plt.subplot() command. For example, a gridspec for a grid of two rows and three columns with some specified width and height space looks like this:
From this we can specify subplot locations and extents using the familiary Python slicing syntax:
This type of flexible grid alignment has a wide range of uses. I most often use it when creating multi-axes histogram plots like the ones shown here:
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Ayşe Betül Koç, Musa Çakmak, Aydın Kurnaz, "A Matrix Method Based on the Fibonacci Polynomials to the Generalized Pantograph Equations with Functional Arguments", Advances in Mathematical Physics, vol. 2014, Article ID 694580, 5 pages, 2014. https://doi.org/10.1155/2014/694580
Ayşe Betül Koç,1 Musa Çakmak,2 and Aydın Kurnaz1
1Department of Mathematics, Faculty of Science, Selcuk University, 42003 Konya, Turkey
2Yayladağı Vocational School, Mustafa Kemal University, 31550 Hatay, Turkey
A pseudospectral method based on the Fibonacci operational matrix is proposed to solve generalized pantograph equations with linear functional arguments. By using this method, approximate solutions of the problems are easily obtained in form of the truncated Fibonacci series. Some illustrative examples are given to verify the efficiency and effectiveness of the proposed method. Then, the numerical results are compared with other methods.
Many phenomena in applied branches that fail to be modeled by the ordinary differential equations can be described by the delay differential equations. Many researchers have studied different applications of those equations in variety of applied sciences such as biology, physics, economy, and electrodynamics (see [1–4]). Pantograph equations with proportional delays play an important role in this context. The existence and uniqueness of the analytic solutions of the multipantograph equation are investigated in [5]. A numerical approach to multipantograph equations with variable coefficients is also studied in [6]. An extension of the multipantograph equation is known to be the generalized pantograph equation with functional arguments defined as under the mixed conditions where proportional delay-, constant delay-, , , and are real and/or complex coefficients, and the coefficients of th order unknown function- and known are the analytical functions defined in the interval .
In recent years, many researchers have developed different numerical approaches to the generalized pantograph equations as variational iteration method [7], differential transform approach [8], Taylor method [9], collocation method based on Bernoulli matrix [10], and Bessel collocation method [11]. In this study, we investigate a collocation method based on the Fibonacci polynomial operational matrix for the numerical solution of the generalized pantograph equation (1). Even the Fibonacci numbers have been known for a long time; the Fibonacci polynomials are very recently defined to be an important agent in the world of polynomials [12, 13]. Compared to the methods of the orthogonal polynomials, the Fibonacci approach has proved to give more precise and reliable results in the solution of differential equations [14].
This study is organized as follows. In the second part, a short review of the Fibonacci polynomials is presented. A Fibonacci operational matrix for the solution of the pantograph equation is developed in Section 3. Some numerical examples are given in Section 4 to illustrate efficiency and effectiveness of the method.
2. Operational Matrices of the Fibonacci Polynomials
The Fibonacci polynomials are determined by following general formula [12, 13]: with and . Now, we will mention some matrix relations in terms of Fibonacci polynomials.
2.1. Fibonacci Series Expansions
To obtain an expansion form of the analytic solution of the pantograph equation, we use the Fibonacci collocation method as follows.
Suppose that (1) has a continuous function solution that can be expressed in the Fibonacci polynomials Then, a truncated expansion of -Fibonacci polynomials can be written in the vector form where the Fibonacci row vector and the unknown Fibonacci coefficients column vector are given, respectively, by
2.2. Matrix Relations of the Derivatives
The th order derivative of (5) can be written as where , , and is the coefficient vector of the polynomial approximation of th order derivative. Then, there exists a relation between the Fibonacci coefficients as where is operational matrix for the derivative defined by [14] Making use of (7) and (9) yields
3. Solution Procedure for the Pantograph Differential Equations
Let us recall the th order linear pantograph differential equation, The first step in the solution procedure is to define the collocation points in the domain, so that Then, collocating problem (12) at the points in (13) yields The system (14) can, alternatively, be rewritten in the matrix form where Therefore, the th order derivative of the unknown function at the collocation points can be written in the matrix form as or, equivalently, To express the functional terms of (1) as in the form (5), let we put instead of in the relation (18) and then obtain or where are Fibonacci operational matrices corresponding to the coefficients . Therefore, replacing (18) and (20) in (15) gives the fundamental matrix equation for the problem (12) as which corresponds to a system of algebraic equations for the unknown Fibonacci coefficients , . In other words, when we denote the expression in the sum by , for and , we get Thus, the augmented matrix of (22) becomes
On the other hand, in view of (11), the conditions (2) can be taken into account by forming the following matrix equation: where Therefore, the augmented matrix of the specified conditions is Consequently, (23) together with (26) can be written in the new augmented matrix form This form can also be achieved by replacing some rows of the matrix (23) by the rows of (26) or adding those rows to the matrix (23) provided that . Finally, the vector (thereby vector of the coefficients ) is determined by applying some numerical methods designed especially to solve the system of linear equations. On the other hand, when the singular case appears, the least square methods are inevitably available to reach the best possible approximation. Therefore, the approximated solution can be obtained. This would be the Fibonacci series expansion of the solution to the problem (12) with the specified conditions.
3.1. Accuracy of the Results
We can, now, proceed with a short accuracy analysis of the problem in a similar way to [18]. As the truncated Fibonacci series expansion is an approximate solution of (1) with (2), it must satisfy the following equality for , : or When ( is any integer) is prescribed, the truncation limit is increased until the difference at each of the collocation points becomes smaller than the desired value .
In this part, three illustrative examples are given in order to clarify the findings of the previous section. The errors of the proposed method are compared with those of the errors that occurred in the solutions by some other methods in Tables 1–3 for two sample examples. It is noted here that the number of collocation points in the examples is indicated by the capital letter .
Present method Exponential approach [15] Taylor polynomial approach [16]
Comparison of the absolute errors of different approximation techniques to Example 2.
Present method Taylor polynomial approach [16] Taylor method [9]
Present method Taylor matrix method [6] Boubaker matrix method [17]
Example 1 (see [10, 11]). Consider the following linear pantograph type problem equation: with the initial conditions
The exact solution of this problem is known to be . When the solution procedure in Section 3 is applied to the problem, the solution of the linear algebraic system gives the numerical approximation of the solution to the problem. It is noteworthy that the method reaches the exact solution even for .
Example 2 (see [9, 15, 16]). Now, consider the following equation with variable coefficient given by and the condition
The exact solution is also known to be . A comparison of the absolute errors of the proposed approach Taylor method [16] and the exponential approach [15] is given in Table 1 for . Another comparison of the present method with the methods of Taylor polynomials [9, 16] is also given for and in Table 2. These results show that the Fibonacci approach has better accuracy, at least one decimal place, than the other methods.
Example 3 (see [6, 17]). Finally, let us consider the pantograph equation with variable coefficients and the condition which has the exact solution . Computed results are compared with the results of the Taylor [6] and Boubaker [17] matrix methods in Table 3.
This study was supported by Research Projects Center (BAP) of Selcuk University. The authors would like to thank Selcuk University and TUBITAK for their support. Also, the authors would like to thank the editor and referees for their valuable comments and remarks which led to a great improvement of the article. They have denoted here that a minor part of this study was presented orally at the “2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013),” Sarajevo, August, 2013.
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Copyright © 2014 Ayşe Betül Koç et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Global Constraint Catalog: Csoft_cumulative
<< 5.360. soft_alldifferent_var5.362. soft_same_interval_var >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃},\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻},\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\begin{array}{c}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚎𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\hfill \end{array}\right)
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}
\left(2,\mathrm{𝚃𝙰𝚂𝙺𝚂},\left[\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗},\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗},\mathrm{𝚎𝚗𝚍}\right]\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\ge 0
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le \mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚎𝚗𝚍}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\ge 0
\mathrm{𝙻𝙸𝙼𝙸𝚃}\ge 0
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}\ge 0
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}\le \mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}\ge 0
𝒯
n
tasks described by the
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection, where
{\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}}_{j}
{\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}}_{j}
{\mathrm{𝚎𝚗𝚍}}_{j}
{\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}}_{j}
are shortcuts for
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚎𝚗𝚍}
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
. In addition let
\alpha
\beta
respectively denote the earliest possible start over all tasks and the latest possible end over all tasks. The
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint forces the three following conditions:
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right]
\left(1\le j\le n\right)
𝒯
{\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}}_{j}+{\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}}_{j}={\mathrm{𝚎𝚗𝚍}}_{j}
At each point in time, the cumulated height of the set of tasks that overlap that point, does not exceed a given limit
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\forall i\in \left[\alpha ,\beta \right]:{\sum }_{j\in \left[1,n\right]|{\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}}_{j}\le i<{\mathrm{𝚎𝚗𝚍}}_{j}}{\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}}_{j}\le \mathrm{𝙻𝙸𝙼𝙸𝚃}
The surface of the profile resource utilisation, which is greater than
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}
{\sum }_{i\in \left[\alpha ,\beta \right]}max\left(0,\left({\sum }_{j\in \left[1,n\right]|{\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}}_{j}\le i<{\mathrm{𝚎𝚗𝚍}}_{j}}{\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}}_{j}\right)-\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}\right)
=
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}
\left(\begin{array}{c}〈\begin{array}{cccc}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-1\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-4\hfill & \mathrm{𝚎𝚗𝚍}-5\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-1\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-1\hfill & \mathrm{𝚎𝚗𝚍}-2\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-2,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-3\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-3\hfill & \mathrm{𝚎𝚗𝚍}-6\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-2\hfill \end{array}〉,3,2,3\hfill \end{array}\right)
Figure 5.361.1 shows the cumulated profile associated with the example. To each task of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint corresponds a set of rectangles coloured with the same colour: the sum of the lengths of the rectangles corresponds to the duration of the task, while the height of the rectangles (i.e., all the rectangles associated with a task have the same height) corresponds to the resource consumption of the task. The
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
For each task we have that its end is equal to the sum of its origin and its duration.
At each point in time we do not have a cumulated resource consumption strictly greater than the upper limit
\mathrm{𝙻𝙸𝙼𝙸𝚃}=3
enforced by the second argument of the
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
The surface of the cumulated profile located on top of the intermediate level
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}=2
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}=3
Figure 5.361.1. Resource consumption profile associated with the three tasks of the Example slot, where parts on top of the intermediate level 2 are marked by a cross
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚎𝚗𝚍}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)>1
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}>0
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}>0
\mathrm{𝙻𝙸𝙼𝙸𝚃}<
\mathrm{𝚜𝚞𝚖}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}>0
\mathrm{𝙸𝙽𝚃𝙴𝚁𝙼𝙴𝙳𝙸𝙰𝚃𝙴}_\mathrm{𝙻𝙴𝚅𝙴𝙻}<\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚂𝚄𝚁𝙵𝙰𝙲𝙴}_\mathrm{𝙾𝙽}_\mathrm{𝚃𝙾𝙿}>0
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
\mathrm{𝚎𝚗𝚍}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint was initially introduced in CHIP [Cosytec97] as a variant of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint. An extension of this constraint where one can restrict the surface on top of the intermediate level on different time intervals was first proposed in [PetitPoder09] and was generalised in [DeClercq12].
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint type: predefined constraint, soft constraint, scheduling constraint, resource constraint, temporal constraint, relaxation.
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Implied Volatility - Premia
One of the key challenges in enabling new option markets is estimating volatility levels.
NOTE: This is an open area of research. If you would like to help us work on this problem, reach out to us on discord, or [email protected]
Option pricing volatility, often called implied volatility, is a chicken and egg problem:
Implied Volatility (IVOL), aptly named, is the annualized volatility implied by an option's price. One can determine the implied volatility of an option price by plugging the final price into the original Black Scholes equation and solving for
\sigma
, the volatility parameter implied by the price. However, prices in a modern Black Scholes model are often determined by plugging in an implied volatility value to the model in place of
\sigma
. Therein lies the catch 22.
Other than from option prices, implied volatility cannot be directly observed. IVOL per option is a 3-dimensional creature, with its value depending on the the maturity date of the option and the "in-the-moneyness" vs. "out-of-the-moneyness" of the option's strike price.
Because of this three-dimensional nature, the volatility used to price a range of options is often referred to as the IVOL surface, or simply the Volatility Surface.
An example BTC Call option volatility surface from Oct, 2021
We must assume that, for any asset, there exists a theoretical market IVOL surface that is known only to the market participants, and cannot be exactly observed or directly derived from any external set of data. Until this true IVOL surface is known to the pools, there will exist a level of IVOL approximation error (if the
TrueIVOL \neq ProposedIVOL
), which will skew the price of any given trade in favor of either the LP or the option buyer.
However, given the decision nature of option buying, an option buyer is likely to only engage with trades they perceive favorable, so the majority of this inefficiency will fall upon the LPs underwriting the options. Because of that, the objective of our approach aims to achieve 2 things:
Propose the best possible starting point for IVOL
Minimize the cost of adjustment to converge to the true market IVOL
If you'd like to skip the research, you can go straight to how the Volatility Surface Oracle works.
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Global Constraint Catalog: Kdegree_of_diversity_of_a_set_of_solutions
<< 3.7.78. Demand profile3.7.80. Derived collection >>
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚐𝚛𝚎𝚊𝚝𝚎𝚛}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
modelling: degree of diversity of a set of solutions A constraint that allows finding a set of solutions with a certain degree of diversity. As an example, consider the problem of finding 9 diverse solutions for the 10-queens problem. For this purpose we create a 10 by 9 matrix
ℳ
of domain variables taking their values in interval
\left[0,9\right]
. Each row of
ℳ
corresponds to a solution to the 10-queens problem. We assume that the variables of
ℳ
are assigned row by row, and that within a given row, they are assigned from the first to the last column. Moreover values are tried in increasing order. We first post for each row of
ℳ
the 3
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraints related to the 10-queens problem (see Figure 5.12.2 for an illustration of the 3
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraints). With a
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚑𝚊𝚒𝚗}_\mathrm{𝚕𝚎𝚜𝚜}
constraint, we lexicographically order the first two variables of each row of
ℳ
in order to enforce that the first two variables of any pair of solutions are always distinct. We then impose a
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
constraint on the variables of each column of
ℳ
{C}_{i}
denote the corresponding cost variable associated with the
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
constraint of the
i
ℳ
(i.e., the first argument of the
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
constraint). We put a maximum limit (e.g., 3 in our example) on these cost variables. We also impose that the sum of these cost variables should not exceed a given maximum value (e.g., 8 in our example). Finally, in order to balance the diversity over consecutive variables we state that the sum of two consecutive cost variables should not exceed a given threshold (e.g., 2 in our example). As one of the possible results we get the following nine solutions depicted below.
{S}_{1}=〈0,2,5,7,9,4,8,1,3,6〉
{S}_{2}=〈0,3,5,8,2,9,7,1,4,6〉
{S}_{3}=〈1,3,7,2,8,5,9,0,6,4〉
{S}_{4}=〈2,4,8,3,9,6,1,5,7,0〉
{S}_{5}=〈3,6,9,1,4,7,0,2,5,8〉
{S}_{6}=〈5,9,2,6,3,1,8,4,0,7〉
{S}_{7}=〈6,8,1,5,0,2,4,7,9,3〉
{S}_{8}=〈8,1,4,9,7,0,3,6,2,5〉
{S}_{9}=〈9,5,0,4,1,8,6,3,7,2〉
The costs associated with the
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
constraints of columns
1,2,\cdots ,10
are respectively equal to 1, 1, 1, 0, 1, 0, 1, 1, 1, and 1. The different types of constraints between the previous 9 solutions are illustrated by Figure 3.7.23. The nine diverse solutions
{S}_{1},{S}_{2},\cdots ,{S}_{9}
are shown by Figure 3.7.24. Figure 3.7.25 depicts the distribution of all the queens of the nine solutions on a unique chessboard.
Figure 3.7.23. Constraint network associated with the problem of finding 9 diverse solutions for the 10-queens problem where variables are fixed to the solutions corresponding to
{S}_{1}=〈0,2,5,7,9,4,8,1,3,6〉
{S}_{2}=〈0,3,5,8,2,9,7,1,4,6〉
\cdots
{S}_{9}=〈9,5,0,4,1,8,6,3,7,2〉
, and where each type of constraint (hyperedge) is drawn with its own colour
Figure 3.7.24. Nine diverse solutions to the 10-queens problem
Figure 3.7.25. Distribution of the queens on the chessboard of the nine diverse solutions depicted by Figure 3.7.24 to the 10-queens problem: a red queen means two queens from two different solutions that are placed on a same cell, non-red queens of a same colour are queens that belong to a same solution; out of the
10·10
cells of the original chessboard,
9·10-2·8=74
cells are occupied by a single queen, 8 by two queens, and 18 by no queen at all.
Approaches for finding diverse and similar solutions based on the Hamming distances between each pair of solutions are presented by E. Hebrard et al. [HebrardHnichSullivanWalsh05].
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Table 1.5.1 lists formulae that combine both dot and cross products.
Triple Scalar (Box) Product
\left[\mathbf{ABC}\right]=\mathbf{A}·\left(\mathbf{B}×C\right)= |\begin{array}{ccc}{a}_{1}& {a}_{2}& {a}_{3}\\ {b}_{1}& {b}_{2}& {b}_{3}\\ {c}_{1}& {c}_{2}& {c}_{3}\end{array}|
Area of parallelogram with edges A and B
\left|\mathbf{A}×\mathbf{B}\right|
Area of triangle with edges A and B
\frac{1}{2} \left|\mathbf{A}×\mathbf{B}\right|
Volume of parallelepiped with edges A, B, and C
\left|\left[\mathbf{ABC}\right]\right|
Distance from point P to line through points Q and R.
A is the vector from Q to R; B, the vector from Q to P
\frac{∥\mathbf{A}×\mathbf{B}∥}{∥\mathbf{A}∥}
Distance from point P to plane through points Q, R, and S.
A is the vector from Q to R; B, from Q to S; and C from Q to P
\frac{\left|\left[\mathbf{ABC}\right]\right|}{∥\mathbf{A}×\mathbf{B}∥}
\left\{{\mathbf{V}}_{1},{\mathbf{V}}_{2},{\mathbf{V}}_{3}\right\}
\left\{{\mathbf{U}}_{1},{\mathbf{U}}_{2},{\mathbf{U}}_{3}\right\}
are reciprocal sets of vectors if
{\mathbf{U}}_{i}·{\mathbf{V}}_{j}={\mathrm{δ}}_{\mathrm{ij}}=\left\{\begin{array}{cc}1& i=j\\ 0& i≠j\end{array}\right\.
{\mathbf{V}}_{1}=\frac{{\mathbf{U}}_{2}×{\mathbf{U}}_{3}}{\mathrm{λ}}
{\mathbf{V}}_{2}=\frac{{\mathbf{U}}_{3}×{\mathbf{U}}_{1}}{\mathrm{λ}}
{\mathbf{V}}_{3}=\frac{{\mathbf{U}}_{1}×{\mathbf{U}}_{2}}{\mathrm{λ}}
\mathrm{λ}=\left[{\mathbf{U}}_{1}{\mathbf{U}}_{2}{\mathbf{U}}_{3}\right]
\mathbf{τ}
exerted about O by a force F acting at the head of
\mathbf{r}
, a vector of length
r
with tail at O
\mathbf{τ}=\mathbf{r}×\mathbf{F}
Table 1.5.1 Formulas involving dot and cross products
\mathbf{A}=3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{j}
\mathbf{B}=2 \mathbf{i}+5 \mathbf{j}-4 \mathbf{k}
\mathbf{C}=5 \mathbf{i}+7 \mathbf{j}+6\mathbf{ }\mathbf{k}
\left[\mathbf{ABC}\right]
, the Triple Scalar (or Box) Product
\mathbf{A}·\mathbf{B}×\mathbf{C}
\mathbf{A}·\mathbf{B}×\mathbf{C}=\mathbf{A}×\mathbf{B}·\mathbf{C}
for the Triple Scalar Product.
For the vectors A, B, and C of Example 1.5.1, and
\mathbf{D}=4 \mathbf{i}+3 \mathbf{j}-2 \mathbf{k}
\left(\mathbf{A}×\mathbf{B}\right)×\left(\mathbf{C}×\mathbf{D}\right)=\left[\mathbf{ACD}\right]\mathbf{B}-\left[\mathbf{BCD}\right]\mathbf{A}
\left(\mathbf{A}×\mathbf{B}\right)×\left(\mathbf{C}×\mathbf{D}\right)=\left[\mathbf{ABD}\right]\mathbf{C}-\left[\mathbf{ABC}\right]\mathbf{D}
Use the appropriate formula from Table 1.5.1 to calculate the area of the parallelogram whose vertices are the four points P:
\left(4,13\right)
, Q:
\left(12,29\right)
, R:
\left(16,57\right)
, and S:
\left(8,41\right)
Use the appropriate formula from Table 1.5.1 to calculate the area of the triangle whose vertices are the three points P:
\left(1,2,3\right)
\left(-5,3,2\right)
, and R:
\left(7,-5,4\right).
Prove that the point S:
\left(1,-1,4\right)
does not lie in the plane determined by the points P, Q, and R given in Example 1.5.4.
Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P:
\left(1,2,3\right)
from the line through points Q:
\left(5,-3,7\right)
and R:
\left(4,1,-6\right)
\left(2,-3,4\right)
to the plane through the three points Q:
\left(1,2,-3\right)
\left(5,4,7\right)
\left(6,-5,-1\right).
Derive the formula given in Table 1.5.1 for the distance from a point to a plane.
Using the formulas in Table 1.5.1 for reciprocal vectors, obtain
\left\{{\mathbf{V}}_{1},{\mathbf{V}}_{2},{\mathbf{V}}_{3}\right\}
, the set of vectors reciprocal to the vectors
{\mathbf{U}}_{1}=\mathbf{A}
{\mathbf{U}}_{2}=\mathbf{B}
{\mathbf{U}}_{3}=\mathbf{C}
, where A, B, and C are given in Example 1.5.1.
Solve the appropriate set of four equations in four unknowns to find
\left\{{\mathbf{V}}_{1},{\mathbf{V}}_{2}\right\}
, the set of vectors reciprocal to
{\mathbf{U}}_{1}=2 \mathbf{i}-3 \mathbf{j}
{\mathbf{U}}_{2}=3 \mathbf{i}+4 \mathbf{j}
\mathbf{F}=2 \mathbf{i}-3 \mathbf{j}+4 \mathbf{k}
is applied to the head of the position vector
\mathbf{r}=3 \mathbf{i}+2 \mathbf{j}-5 \mathbf{k}
\mathbf{τ}
, the torque vector, and
\mathrm{τ}
, its magnitude.
What is the angle between F and r?
Find a unit vector in the direction of the axis of rotation.
\mathbf{A}≠\mathbf{0}
\mathbf{A}×\mathbf{B}=\mathbf{A}×\mathbf{C}
\mathbf{B}=\mathbf{C}
\mathbf{A}≠\mathbf{0}
\mathbf{A}×\mathbf{B}=\mathbf{A}×\mathbf{C}
\mathbf{A}·\mathbf{B}=\mathbf{A}·\mathbf{C}
together imply that
\mathbf{B}=\mathbf{C}
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Where's George? - Wikipedia
US dollar note tracking website
For the 1935 British comedy, see Where's George? (film).
A screenshot taken from version 4.0 of the Where's George? website, 2007
Money circulation tracker
Where's George? LLC
Where's George? is a website that tracks the natural geographic circulation of American paper money. Its popularity has led to the establishment of a number of other currency tracking websites and sites that track other objects, such as used books. Statistics generated by the website have been used in at least one research paper to study patterns of human travel in the United States.[1]
4 Where's George? and geocaching
5 George Score
The site was established in December 1998 by Hank Eskin, a database consultant in Brookline, Massachusetts.[2][3] Where's George? refers to George Washington, whose portrait appears on the $1 bill. In addition to the $1 bill, $2, $5, $10, $20, $50, and $100 denominations can be tracked. The $1 bill is by far the most popular denomination, accounting for over 70% of bills with "hits" (explained below), followed by $20 bills, and the $5 bill a close third.[4]
As of November 2021, the site says more than 305,500,000 bills, with a total face value of more than $1.64 billion, have been entered into the site's database;[5] the daily influx of bills was noted in November 2020 as about 10,000 new bills a day.[6] To track a bill, users enter their local ZIP code, the serial number of the bill, and series designation of any US currency denomination. Users outside the US also can participate by using an extensive database of unique codes assigned to non-American/Canadian locations. Once a bill is registered, the site reports the time between sightings, the distance between sightings, and any comments from the finders (called "user notes"). The site does not track bills older than series 1963.[7]
Where's George? is supported by advertising, sales of memorabilia, and by users who pay a fee for extra features.[2] The "Friends of Where's George?" (FOG) program allows these users to access the website free of advertisements, access certain features others cannot, and refresh reports on the user's entered bills. The standard FOG costs $8/month, while FOG+ costs $13/month.[8] Eskin states that the "Friends of Where's George?" program will always be optional and payment to use the site will always be at the individual's option.[8]
A "hit" occurs when a registered bill is re-entered into the database after its initial entry. Where's George? does not have specific goals other than tracking currency movements, but many users like to collect interesting patterns of hits, called "bingos".[9] One of the most commonly sought after bingos involves getting at least one hit in all 50 states (called "50 State Bingo"). Another common bingo, called "FRB Bingo", occurs when a user gets hits on bills from all 12 Federal Reserve Banks.[10]
Most bills do not receive any responses, or hits, but many bills receive two or more hits. As of November 2020, the site recorded about 5,000 entries for found bills daily.[6] The approximate hit rate is around 11.4%. Double- and triple-hitters are common, and bills with 4 or 5 hits are not unheard of. Almost daily, a bill receives its 6th hit. As of June 2021[update], the site record is held by a $1 bill with 18 entries.[11]
To increase the chance of having a bill reported, users (called "Georgers") may write or stamp text on the bills encouraging bill finders to visit www.wheresgeorge.com and track the bill's travels.[2] Bills that are entered into the database, but not marked, are known as "naturals", "stealths", or "ghosts". If a bill entry violates the established rules of "natural circulation" (e.g. a user has found the same bill twice, a user has had more than 20 bills wind up with another user, etc.), it is flagged as an "alternate entry". If a user claims a "wild" (A bill found in circulation that has been marked or stamped with wheresgeorge.com as having already been entered on the site.),[further explanation needed] he or she is the submitters' "child".[12]
The site does not encourage the defacement of US currency.[13] In October 1999, when interviewed for The New York Times, Eskin commented on why the Secret Service has not bothered the webmaster over possible defacement of US currency: "They've got better things to do. They want to catch counterfeiters counterfeiting billions of dollars."[3]
In April 2000, the site was investigated by the United States Secret Service, which informed Eskin that the selling of "Where's George?" rubber stamps on the web site is considered "advertising" on United States currency, which is illegal under 18 U.S.C. § 475.[14] The site's administrators immediately ceased selling the rubber stamps; no further action against the site was taken.[2] At least one spokesperson for the US Secret Service has pointed out in print that marking US bills, even if not defacement, can still be illegal if it falls under "advertisement".[15] However, a Secret Service spokesman in Seattle, Washington, told The Seattle Times in 2004: "Quite frankly, we wouldn't spend too much looking into this."[2]
Where's George? and geocaching[edit]
Examples of marked bills, with one bearing the site's address in handwriting, and the others marked by rubber stamping.
The Where's George? site says it "prohibits trading or exchanging bills with friends, family or anyone known to the bill distributor for the purpose of re-entry".[16][17] This rule is to encourage natural circulation of the currency, and to prevent multiple fake hits from happening on any bill. As a result, all bill entry notes containing the word "geocache" or "cache" are tagged as a geocache bill. The site has also dropped a separate listing of "Top 10 Geocache bills" and is cautioning that, if geocache sites are used too often, "all Geocache bills will be removed from this site".[18][further explanation needed]
George Score[edit]
The "George Score" is a method of rating users based on how many bills they have entered, and by how many total hits they have had. The formula is as follows:[2]
{\displaystyle 100\times \left[{\sqrt {\ln({\rm {bills\ entered}})}}+\ln({\rm {hits}}+1)\right]\times [1-({\rm {days\ of\ inactivity}}/100)]}
This logarithmic formula means the more bills a user enters and the more hits the user receives, the less the user's score increases for each entered bill or new hit. Thus, a user's score does not increase as quickly when the user has entered many bills. User ranking can decrease based on inactivity (failure to refresh the "Your Bills" report). Wattsburg Gary, the user with the most bills entered (over 2 million entries per the Wheresgeorge database) has an official George Score of over 1,700 when refreshed, and often holds the #1 rank.[19] While bulk entry is allowed, the site prohibits marking bills and depositing them into financial institutions.[16][20]
Where's George? includes a community of users who interact via forums. The forums divide into several categories, from regional to new-member-help threads. Some members of the site participate in "gatherings," held in cities around the United States. Several gatherings have become annual events, varying in scope and size.[21]
The 2006 documentary by Brian Galbreath, WheresGeorge.com, gives insight into the hobby, hobbyists, and their gatherings. The twenty-seven minute movie features interviews with site creator Hank Eskin, "Georgers" at a St. Louis, Missouri gathering, and narrated information and statistics about the site and culture. The film aired on PBS affiliates WTTW Chicago and WSIU-TV Carbondale, IL.[22]
Although Where's George? does not officially recognize the bills that travel the farthest or fastest, some have approached it as a semi-serious way to track patterns in the flow of American currency.[23]
Money flow displayed through Where's George was used in a 2006 research paper published by theoretical physicist Dirk Brockmann and his coworkers. The paper described statistical laws of human travel in the United States, and developed a mathematical model of the spread of infectious disease resulting from such travel. The article was published in the January 26, 2006, issue of the journal Nature.[24] Researchers found that 57% of the nearly half a million dollar bills studied traveled between 30 and 500 miles (48 and 805 km) over approximately nine months in the United States.[25] A short clip of a Brockmann's presentation on the subject from the IdeaFestival has been posted on YouTube.[26] More recently, "Where's George?" data have been used to attempt to predict the rapidity and pattern of projected spread of the 2009 swine flu pandemic.[27]
Twenty Bucks – 1993 film about the fictional travels of a $20 bill
Where's Willy? – the site's Canadian counterpart
The Money Tracker – the site's Australian counterpart
BookCrossing -- Service to track used book circulation
^ BJS (January 25, 2006). "Web game provides breakthrough in predicting spread of epidemics". Science Blog. Retrieved April 28, 2006.
^ a b c d e f Lacitis, Erik (October 4, 2004). "Where's George? Tracking the travels of paper currency". Local News. The Seattle Times Company. Retrieved July 16, 2008.
^ a b Flaherty, Julie (October 28, 1999). "Making It Easy to Find Where the Money Goes". The New York Times.
^ Eskin, Hank (2008). "Bill Statistics by Denomination". George's Top 10. Where's George? LLC. Retrieved July 19, 2008.
^ Eskin, Hank. "Where's George? ❝Currency Tracking Project❞".
^ a b Stinson, Antonio (November 17, 2020). "Man builds online community after creating website that can track dollar bills". WCBD News 2. Retrieved November 22, 2020.
^ "Where's George? – Currency Tracking Project – FAQs, Rules/User Guidelines, and Privacy Policy". Wheresgeorge.com. Retrieved December 27, 2013.
^ a b Eskin, Hank (2008). "The 'Friends of Where's George?' Program". Tools/Fun. Where's George LLC. Retrieved July 19, 2008.
^ "Main Page – Where's George? Wiki". Archived from the original on June 4, 2011.
^ "Encyclopædia Georgetannica". Slowpoke. 2007. Retrieved August 23, 2007.
^ "Top 20 Bills Report - All Denominations". Where's George?. June 7, 2021. Retrieved June 7, 2021. {{cite web}}: CS1 maint: url-status (link)
^ "The 'Friends of Where's George?' Program". 2014. Retrieved July 26, 2014.
^ "Where's George? ® 2.4 Frequently Asked Questions". Wheresgeorge.com. Retrieved December 6, 2011.
^ "§ 475. Imitating obligations or securities; advertisements". Cornell Law School. 2006. Retrieved September 29, 2006.
^ Moyer, Laura (September 29, 2004). "Following the money". News. The Free Lance-Star Publishing Company. Archived from the original on June 4, 2011.
^ a b User Guidelines/Terms of Service/Rules, no. 4
^ User Guidelines/Terms of Service/Rules, no. 1
^ Eskin, Hank (2008). "Rules for using Where's George? with Geocaching". Where's George? 2.2. Where's George? LLC. Archived from the original on June 5, 2008. Retrieved July 3, 2008.
^ Eskin, Hank. "Where's George? ❝Currency Tracking Project❞". www.wheresgeorge.com.
^ User Guidelines/Terms of Service/Rules, no. 7.
^ Eskin, Hank (2008). "Unofficial Where's George?/Where's Willy? Gatherings". Where's George?/Where's Willy? Discussion. Wheres George? LLC. Retrieved July 3, 2008.
^ Galbreath, Brian (2006). Wheresgeorge.com. Brian Galbreath Productions.
^ "Where's George?: The Trail Of $1 Bills Across The U.S." NPR. March 24, 2013. Retrieved December 27, 2013.
^ Brockmann, D; L. Hufnagel; T. Geisel (January 26, 2006). "The scaling laws of human travel" (PDF). Nature. 439 (7075): 462–465. arXiv:cond-mat/0605511. Bibcode:2006Natur.439..462B. doi:10.1038/nature04292. PMID 16437114. S2CID 4330122. Retrieved April 28, 2006.
^ "Researchers' plan to track disease: follow 'Where's George' cash trail". Health and medicine. St. Petersburg Times. Associated Press. January 26, 2006. Retrieved July 16, 2008.
^ Brockmann, Dirk. "Money Circulation Science" (Flash). IdeaFestival 2007. YouTube.com. Retrieved October 17, 2007.
^ McNeil, Donald G. (May 3, 2009). "Predicting Flu With the Aid of (George) Washington". The New York Times. Retrieved May 5, 2009.
Where's George? web site
Directory of Where's George related pages – a wealth of information regarding Where's George
WG? Virtual Museum – a collection of marked bill images.
WG? Videos on YouTube – Videos relating to Where's George by Brian Galbreath.
The Dollar Project (1986–1988) – an earlier art project involving tracking stamped bills.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Where%27s_George%3F&oldid=1072689384"
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Global Constraint Catalog: Ksmallest_rectangle_area
<< 3.7.234. Smallest square for packing consecutive 3.7.236. Smallest square for packing rectangles w >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
Denotes that a constraint can be used for finding the smallest rectangle area where one can pack a given set of rectangles (or squares). A first example of such packing problem attributed to S. W. Golomb is to find the smallest square that can contain the set of consecutive squares from
1×1
n×n
so that these squares do not overlap each other. A program using the
\mathrm{𝚍𝚒𝚏𝚏𝚗}
constraint was used to construct such a table for
n\in \left\{1,2,\cdots ,25,27,29,30\right\}
in [BeldiceanuBourreauChanRivreau97]. New optimal solutions for this problem were found in [SimonisSullivan08] for
n=26,31,35
. Figure 3.7.66 gives the solution found for
n=35
by H. Simonis and B. O'Sullivan. Algorithms and lower bounds for solving the same problem can also be respectively found in [CapraraLodiMartelloMonaci06] and in [Korf04].
Figure 3.7.66. Smallest square (of size 123) for packing squares of size
1,2,\cdots ,35
In his paper (i.e., [Korf04]), Richard E. Korf also considers the problem of finding the minimum-area rectangle that can contain the set of consecutive squares from
1×1
n×n
and solve it up to
n=25
. In 2008 this value was improved up to
n=27
by H. Simonis and B. O'Sullivan [SimonisSullivan08]. Figure 3.7.67 gives the solution found for
n=27
by H. Simonis and B. O'Sullivan.
Figure 3.7.67. Rectangle with the smallest surface (of size
148×47
) for packing squares of size
1,2,\cdots ,27
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Extremum Seeking Control for Reference Model Tracking of Uncertain Systems - MATLAB & Simulink - MathWorks Benelux
Adaptive Gain Tuning for Uncertain Linear Systems
Extremum Seeking Control Adaptive Gain Tuning
This example shows the design of feedback and feedforward gains for a system, a common controller design technique. Here, you use an extremum seeking controller to track a given reference plant model by adapting feedback and feedforward gains for an uncertain dynamical system.
For this example, consider the following first-order linear system.
\underset{}{\overset{˙}{x}}\left(t\right)={a}_{0}x\left(t\right)+{b}_{0}u\left(t\right)
\mathit{x}\left(\mathit{t}\right)
\mathit{u}\left(\mathit{t}\right)
are the state and control input of the system, respectively. The constants
{\mathit{a}}_{0}
{\mathit{b}}_{0}
The goal of this example is to track the performance of the following reference plant model, which defines the required transient and steady-state behavior.
{\underset{}{\overset{˙}{x}}}_{ref}\left(t\right)={a}^{*}{x}_{ref}\left(t\right)+{b}^{*}r\left(t\right)
{\mathit{x}}_{\mathrm{ref}}\left(\mathit{t}\right)
is the state of the reference plant and
\mathit{r}\left(\mathit{t}\right)
is the reference signal.
The aim of the control signal
u\left(t\right)
is to make the states
x\left(t\right)
of the uncertain system track the reference states
{\mathit{x}}_{\mathrm{ref}}\left(\mathit{t}\right)
u\left(t\right)=Kx\left(t\right)-Kr\left(t\right)
The designed controller contains a feedback term,
\mathit{Kx}\left(\mathit{t}\right)
, and a feedforward term,
-\mathit{Kr}\left(\mathit{t}\right)
Substitute this control signal into the unknown linear system dynamics.
\underset{}{\overset{˙}{x}}\left(t\right)={a}_{0}x\left(t\right)+{b}_{0}\left(Kx\left(t\right)-Kr\left(t\right)\right)
You can rewrite this expression as shown in the following equation.
\underset{}{\overset{˙}{x}}\left(t\right)=\left({a}_{0}+{b}_{0}K\right)x\left(t\right)-{b}_{0}Kr\left(t\right)
In the ideal case, if the coefficients
{\mathit{a}}_{0}
{\mathit{b}}_{0}
of the nominal system dynamics are known, then you can determine the controller gain
\mathit{K}
using pole-placement techniques. Doing so produces the following matching condition.
{a}_{0}+{b}_{0}K={a}^{*},{b}_{0}K={b}^{*}
When you use a single gain value as both the feedforward and feedback gain, this matching condition might not be satisfied for all the possible values of
{\mathit{a}}_{0}
{\mathit{b}}_{0}
. For a more general solution, you can tune two different gain values (multiparameter tuning).
For this example, use the following unknown system and reference dynamics.
\underset{}{\overset{˙}{x}}\left(t\right)=-1x\left(t\right)+u\left(t\right)
{\underset{}{\overset{˙}{x}}}_{ref}\left(t\right)=-3{x}_{ref}\left(t\right)+2r\left(t\right)
In this case, the ideal control gain is
K=-2
To implement an extremum seeking control (ESC) approach to the preceding problem, you define an objective function, which the ESC controller then maximizes to find the controller gain
\mathit{K}
For this example, use the following objective function.
J=-10\int {\left(x\left(t\right)-{x}_{ref}\left(t\right)\right)}^{2}dt
The following figure shows the setup for extremum seeking control.
The cost function is computed from the outputs of the reference system and the actual system.
The extremum seeking controller updates the gain parameter.
The control action is updated using the new gain value.
This control action is applied to the actual system.
The firstOrderRefTracking_Esc Simulink model implements this problem configuration.
mdl = 'firstOrderRefTracking_Esc';
In this model, you use the Extremum Seeking Control block to optimize the gain value.
The System Dynamics and Objective subsystem contains the reference model, the plant (including the actual system and control action), and the objective function computation. These elements are all implemented using MATLAB Function blocks.
open_system([mdl '/System Dynamics and Objective'])
Specify an initial guess for the gain value.
Configure the demodulation and modulation signals by specifying their frequency (omega), phases (phi_1 and phi_2), and their amplitudes (a and b).
omega = 5; % Forcing frequency
a = 1; % Demodulation amplitude
b = 0.1; % Modulation amplitude
phi_1 = 0; % Demodulation phase
For this example, the Extremum Seeking Control block is configured to remove high-frequency noise from the demodulated signal. Set the cutoff frequency for the corresponding low-pass filter.
To check the reference tracking performance, view the state trajectories from the simulation. The actual trajectory converges to the reference trajectory in less than five seconds.
open_system([mdl '/System Dynamics and Objective/State'])
To examine the behavior of the ESC controller, first view the objective function, which reaches its maximum value quickly.
open_system([mdl '/System Dynamics and Objective/Cost'])
By maximizing the objective function, the ESC controller optimizes the control gain value near its ideal value of –2. The fluctuations in the gain value are due to the modulation signal from the Extremum Seeking Control block.
open_system([mdl '/System Dynamics and Objective/Gain K'])
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Two-Dimensional Topology Optimization of a Horn Antenna
Two-Dimensional Topology Optimization of a Horn Antenna ()
Kingswood Rd, West Hartford, CT, USA.
A two-dimensional horn antenna is used as a model for topology optimization. In order to employ the topology optimization, each point in the domain is controlled by a function which is allowed to take values between 0 and 1. Each point’s distinct value then gives it an effective permittivity, either close to that of polyimide or that of air, two materials considered in this study. With these settings, the optimization problem becomes finding the optimal distribution of materials in a given domain, and is solved under constraints of reflection and material usage by the Method of Moving Asymptotes. The final configuration consists of two concentric arcs of air while polyimide takes up the rest of the domain, a result relatively unsensitive to the choice of constraints and initial values. Compared to the unoptimized antenna, a slimmer main lobe is observed and the gain boosts.
Topology Optimization, Horn Antenna, Material
Dan, H. (2020) Two-Dimensional Topology Optimization of a Horn Antenna. Open Journal of Optimization, 9, 39-46. doi: 10.4236/ojop.2020.93004.
Antennas are devices acting as a transition between the free space and the power source [1] [2]. They play an important role in wireless communication, where these devices transmit and receive signals from afar. In their long history, many types of antennas have been developed and constructed over nearly a century [3] - [8]. Among them, horn antennas are built to achieve high gain in frequencies above very high frequency.
Topology optimization is a method that optimizes material layout within a given design space, under a set of constraints. In order to generate the optimal topologies, microstructures, composites of material and void, are employed to form the domain. Therefore, the topology optimization problem in fact turns into a material distribution problem, whose calculation requires considerably less computing costs. Typical algorithm employed is either gradient-based methods such as the optimality criteria algorithm and the method of moving asymptotes (MMA) or non-gradient-based algorithms such as particle swarm or genetic algorithms. Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering [9] [10].
In this work, we adopt topology optimization method for designing a two-dimensional horn antenna.
Optimization results show that a highly directive antenna with non-intuitive materials distribution can be obtained with this powerful method, opening new ways for future antenna design.
2. Background and Model Definition
In electromagnetics, Maxwell’s equations govern. Together with proper boundary conditions, the radiation pattern of antennas can be solved.
The simulation implemented in this paper seeks to find an optimal material distribution for a two-dimensional horn antenna in order to reach a high gain. Leaving the geometry unchanged, the material configuration can be another angle for antenna design and whose optimization might improve the antenna’s performances, particularly its gain as being discussed in this paper.
In this model, every point in the domain of the horn is assigned a η variable, whose value is to be changed by the optimization. η determines the permittivity, or equivalently what material is used, at one point according to the εramp function. η ranges from 0 to 1; when η decreases from 0.5, the value of εramp reduces rapidly to 1, suggesting a usage of air; when η increases from 0.5, the value of εramp rises sharply to 3.5, implying a usage of polyimide (PI). εramp as a function of η is shown in Equation (1). When the η value at one point equals to 0.5, it is saying that a composite of 50% air and 50% polyimide is used. However, in practice this sort of composite may be difficult to comprehend or implement. A definite usage of material is preferred. To ensure that most points are assigned a η value away from ambiguity, a lower bound of the average of all points’ Weight function’s value is applied to the domain in the optimization. Weight function yields a large value when η at that specific point is very different from 0.5, whereas when η is close to 0.5, it contributes negligibly to the congregation. The graphs of the εramp and Weight functions are shown in Figure 1 and Figure 2, respectively. The reason why only permittivity changed is that both permeability and conductivity of air and polyimide are nearly equal. All parameters used in this work are listed in Table 1.
{\epsilon }_{\text{ramp}}={\epsilon }_{\text{air}}+\frac{{\epsilon }_{\text{polymide}}-{\epsilon }_{\text{air}}}{1+\left[1+{e}^{-3p1\left(\eta -0.5\right)}\right]}
\text{Weight}=1+\frac{1}{1+{e}^{-7p\left(\eta -0.68\right)}}-\frac{1}{1+{e}^{-7p\left(\eta -0.32\right)}}
Figure 1. The graph of function εramp.
Figure 2. The graph of function weight.
Table 1. Parameters used in this study.
\text{Objective}=\frac{2\text{π}{r}_{\text{integration}}{\int }_{A}^{B}{E}_{\text{normalized}}\left(\theta \right)\text{d}\theta }{{L}_{\text{integration}}{\oint }^{\text{}}{E}_{\text{normalized}}\left(\theta \right)\text{d}\theta }
\text{Reflection}={\left(S11\right)}^{2}
The layout of the antenna and its corresponding simulation region is shown in Figure 3. A perfectly matched layer with circular shape surrounds the entire simulation region to eliminate any reflection that could occur at the outer boundary.
The optimization objective expressed in Equation (3) is defined as the ratio of the radiation intensity in the far field at one particular direction to the average radiation intensity of an isotropic antenna: the linear integration of the normalized far field strength over the arc AB over the linear integration of the normalized far field strength over the entire circle, on which the arc AB lies, namely the total power. In the optimization, the Reflection is given an upper bound of 0.45, because excessive reflection from the hardware not only influences the power output, but can also damage the antenna and the transmitting line in practice. A uniform lumped port is used on arc CD to feed the antenna. Perfect electric conductor is applied to CE, DF, EG and FH, as shown in Figure 3. Finally, outside the antenna is defined as the far field, with the outermost layer being the perfectly matched layer to absorb radiation. Figure 4 depicts the mesh of the model. The Optimization module is set to find the maximum of the variable Objective, using the Method of Moving Asymptotes.
The final electric field norm and η distribution given by the topology optimization are shown in Figure 5. Particularly, on the right side, dark color denotes a low η value, indicating at that point, the permittivity is the permittivity of air. On the contrary, light color suggests a high η value, giving the point the permittivity of PI. In the horn, bands of air and PI appear alternately. Notably, dark points scatter in bands of air. The optimized results are quite non-intuitive, indicating the power of topology optimization.
Figures 6 to 8 show three variables as a function of iteration number during
Figure 3. Schematic of the antenna and outer simulation area under study.
Figure 4. Meshing for finite element method simulation. The antenna region is intentionally fine meshed to give better resolution.
Figure 5. Electromagnetic field distribution (left) and η value (right) after topology optimization.
Figure 6. Reflection vs. iteration number.
Figure 7. Weight/Area vs. iteration number.
optimization. As the optimization iterated, reflection drops and approaches 0.45, Weight/Area increases and approaches 0.94, and Objective approaches 105. Even though the reflection was still high in the end, the polar graph demonstrates that the power was indeed concentrated on the desired direction, as shown in Figure 9.
Figure 8. Objective vs. iteration number.
Figure 9. Far field projection of the antenna after topology optimization.
The topology optimization shows that a distribution of strips of air and polyimide occurring alternatingly provides a high gain for the horn antenna in this study. The materials distribution is non-intuitive, opening new ways for antenna performance optimization. This method could be developed to more complicated scenarios, such as phase arrays. Further researches can be conducted to find more combination of materials which may yield even better results.
[1] Balanis, C.A. (2016) Antenna Theory: Analysis and Design. John Wiley & Sons, Hoboken, New Jersey.
[2] Mailloux, R.J. (2017) Phased Array Antenna Handbook. Artech House.
[3] Eggleston, M.S., Messer, K., Zhang, L., Yablonovitch, E. and Wu, M.C. (2015) Optical Antenna Enhanced Spontaneous Emission. Proceedings of the National Academy of Sciences, 112, 1704-1709.
[4] Karim, T., Hirokawa, J., Oogimoto, K., Nagatsuma, T., Seto, H., Inoue, I. and Saito, M. (2016) Corporate-Feed Slotted Waveguide Array Antenna in the 350-GHz Band by Silicon Process. IEEE Transactions on Antennas and Propagation, 65, 217-225.
[5] Lin, Y. and Wang, H. (2016) A Low Phase and Gain Error Passive Phase Shifter in 90 nm CMOS for 60 GHz Phase Array System Application. In 2016 IEEE MTT-S International Microwave Symposium (IMS), San Francisco, California, May 2016, 1-4.
[6] Sheta, A. and Mahmoud, S.F. (2008) A Widely Tunable Compact Patch Antenna. IEEE Antennas and Wireless Propagation Letters, 7, 40-42.
[7] Tong, J., Muthee, M., Chen, S., Yngvesson S.K. and Jun, Y. (2015) Antenna Enhanced Graphene THz Emitter and Detector. Nano Letters, 15, 5295-5301.
[9] Traviss, D.J., Schmidt, M.K., Aizpurua, J. and Muskens, O.L. (2015) Antenna Resonances in Low Aspect Ratio Semiconductor Nanowires. Optics Express, 23, 22771-22787.
[10] Zhu, J., Zhang, W. and Xia, L. (2016) Topology Optimization in Aircraft and Aerospace Structures Design. Archives of Computational Methods in Engineering, 23, 595-622.
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Yuan-Cheng_Fung Knowpia
Yuan-Cheng "Bert" Fung (September 15, 1919 – December 15, 2019) was a Chinese-American bioengineer and writer. He is regarded as a founding figure of bioengineering, tissue engineering, and the "Founder of Modern Biomechanics".[1]
馮元楨
Luna Yu Hsien-Shih
Otto Laporte Award (1977)
Jordan Allen Medal (1991)
Fung was born in Jiangsu Province, China in 1919. He earned a bachelor's degree in 1941 and a master's degree in 1943 from the National Central University (later renamed Nanjing University in mainland China and reinstated in Taiwan), and earned a Ph.D. from the California Institute of Technology in 1948. Fung was Professor Emeritus and Research Engineer at the University of California San Diego. He published prominent texts along with Pin Tong who was then at Hong Kong University of Science & Technology. Fung died at the Jacobs Medical Center in San Diego, California, aged 100, on December 15, 2019.[2][3]
He is the author of numerous books including Foundations of Solid Mechanics, Continuum Mechanics, and a series of books on Biomechanics. He is also one of the principal founders of the Journal of Biomechanics and was a past chair of the ASME International Applied Mechanics Division. In 1972, Fung established the Biomechanics Symposium under the American Society of Mechanical Engineers. This biannual summer meeting, first held at the Georgia Institute of Technology, became the annual Summer Bioengineering Conference. Fung and colleagues were also the first to recognize the importance of residual stress on arterial mechanical behavior.[4]
Fung's LawEdit
Fung's famous exponential strain constitutive equation for preconditioned soft tissues is
{\displaystyle w={\frac {1}{2}}\left[q+c\left(e^{Q}-1\right)\right]}
{\displaystyle q=a_{ijkl}E_{ij}E_{kl}\qquad Q=b_{ijkl}E_{ij}E_{kl}}
{\displaystyle E_{ij}}
{\displaystyle a_{ijkl}}
{\displaystyle b_{ijkl}}
{\displaystyle c}
{\displaystyle w}
is a strain energy function per volume unit, which is the mechanical strain energy for a given temperature. Materials that follow this law are known as Fung-elastic.[6]
Worcester Reed Warner Medal, 1984[7]
Jean-Leonard-Marie Poiseuille Award, 1986[8]
Landis Award, from Microcirculation Society
Alza Award, from BMES
Melville Medal, 1994[9]
United States National Academy of Engineering Founders Award (NAE Founders Award), 1998
Fritz J. and Dolores H. Russ Prize, 2007 ("for the characterization and modeling of human tissue mechanics and function leading to prevention and mitigation of trauma.")[10]
Revelle Medal, from UC San Diego, 2016[11]
Fung was elected to the United States National Academy of Sciences (1993), the National Academy of Engineering (1979), the Institute of Medicine (1991), the Academia Sinica (1968), and was a Foreign Member of the Chinese Academy of Sciences (1994 election).
^ YC “Bert” Fung: The Father of Modern Biomechanics (pdf) Archived 2007-12-02 at the Wayback Machine
^ Robbins, Gary (18 December 2019). "UC San Diego's Y.C. Fung, the lifesaving 'father of biomechanics', dies at 100". San Diego Union Tribune. Retrieved 28 December 2019.
^ Chiang, Yi-ching; Hsu, Phoenix (27 December 2019). "Father of biomechanics Fung Yuan-Cheng dies at 100". Central News Agency. Retrieved 28 December 2019. Republished as: "'Father of biomechanics' has passed away at 100". Taipei Times. 28 December 2019. Retrieved 28 December 2019.
^ Chuong,C.J. and Y.C. Fung (1986). "On Residual Stress in Arteries". Journal of Biomechanical Engineering. 108 (2): 189–192. doi:10.1115/1.3138600. PMID 3079517. S2CID 46231605.
^ Fung, Y.-C. (1993). Biomechanics: Mechanical Properties of Living Tissues. New York: Springer-Verlag. p. 568. ISBN 978-0-387-97947-2.
^ Humphrey, Jay D. (2003). The Royal Society (ed.). "Continuum biomechanics of soft biological tissues". Proceedings of the Royal Society of London A. 459 (2029): 3–43. Bibcode:2003RSPSA.459....3H. CiteSeerX 10.1.1.729.5207. doi:10.1098/rspa.2002.1060. S2CID 108637580.
^ WORCESTER REED WARNER MEDAL RECIPIENTS
^ The International Society of Biorheology: Yuan-Cheng Fung, 1986 Recipient of the Jean-Leonard-Marie Poiseuille Award
^ MELVILLE MEDAL RECIPIENTS
^ Recipient of the Fritz J. and Dolores H. Russ Prize
^ UCSD Chancellor's Announcement of 2016 Revelle Medalists
Classical and Computational Solid Mechanics [1]
Profile at UCSD
Y.C. Fung, Mechanics of Man, Acceptance Speech for the Timoshenko Medal.
YC Fung Young Investigator Award
Molecular & Cellular Biomechanics: In Honor of The 90th Birthday of Professor Yuan Cheng Fung
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a chocolate in the form of a quad with sides 6cm , 10 cm, 5cm, 5 cm is cut - Maths - Quadrilaterals - 10790067 | Meritnation.com
a chocolate in the form of a quad. with sides 6cm , 10 cm, 5cm, 5 cm is cut into 2 pieces along one of its diagonal by a lady ,part I is given to her maid and part II is equally divided among her driver and mala.
a. is this distribution fair? justify?
\mathrm{a}\right) \mathrm{Distribution} \mathrm{is} \mathrm{fair}\phantom{\rule{0ex}{0ex}}\mathrm{b}\right)\mathrm{BD}=\sqrt{{\mathrm{AD}}^{2}-{\mathrm{AB}}^{2}}=\sqrt{100-36}=\sqrt{64}\phantom{\rule{0ex}{0ex}}\mathrm{BD}=8\mathrm{cm}\phantom{\rule{0ex}{0ex}}\mathrm{Ar}\left(\mathrm{ABD}\right)=\frac{1}{2}×6×8=24{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{Ar}\left(\mathrm{BCD}\right)\phantom{\rule{0ex}{0ex}}\mathrm{S}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}=\frac{5+5+8}{2}=9\mathrm{cm}\phantom{\rule{0ex}{0ex}}\mathrm{Area}=\sqrt{9×\left(9-5\right)×\left(9-5\right)×\left(9-8\right)}\phantom{\rule{0ex}{0ex}}=\sqrt{9×4×4×1}\phantom{\rule{0ex}{0ex}}=12{\mathrm{cm}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{c}\right) \mathrm{helpfulness} , \mathrm{caring}
Thanya.s answered this
1.This distribution is unfair according to me bcoz she shouldn't be partial to either the maid , her driver, Mala.
2.Through this act she shows her kind heartedness
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Home : Support : Online Help : Programming : Multithreaded Programming : Add
a parallel implementation of add
Add( f, i = m..n )
Add( f, i = x )
Add( f, i in x )
Add[ tasksize = s ]( ... )
The Add command is a parallel implementation of the add command. For a complete description of the calling sequence of Add see the add help page.
Add is implemented using the Task programming model.
Add attempts to determine how to divide the input into separate tasks to spread the work across all available cores. However in some situations Add may not choose the optimal size. In particular a small number of long running tasks may not be spread evenly over all threads. In this case, you can specify the maximum task size using the tasksize option. Add will divide the input into tasks that compute at most tasksize elements within a single task.
\mathrm{with}\left(\mathrm{Threads}\right):
\mathrm{Add}\left({i}^{2},i=1..5\right)
\textcolor[rgb]{0,0,1}{55}
L≔[\mathrm{seq}\left(i,i=1..5\right)]
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]
\mathrm{Add}\left({i}^{2},i=L\right)
\textcolor[rgb]{0,0,1}{55}
\mathrm{Add}\left(a[i]{x}^{i},i=0..5\right)
{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{0}}
\mathrm{Add}\left(i,i=\mathrm{\infty }..0\right)
\textcolor[rgb]{0,0,1}{0}
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Control theory - CodeDocs
To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the error signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. This is the basis for the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is feedback control, which involves taking measurements using a sensor and making calculated adjustments to keep the measured variable within a set range by means of a "final control element", such as a control valve.[1]
Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell.[2] Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky.[3] Although a major application of mathematical control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and operation research.[4]
Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868, entitled On Governors.[5] A centrifugal governor was already used to regulate the velocity of windmills.[6] Maxwell described and analyzed the phenomenon of self-oscillation, in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, Edward John Routh, abstracted Maxwell's results for the general class of linear systems.[7] Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh–Hurwitz theorem.[8][9]
A notable application of dynamic control was in the area of manned flight. The Wright brothers made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.
By World War II, control theory was becoming an important area of research. Irmgard Flügge-Lotz developed the theory of discontinuous automatic control systems, and applied the bang-bang principle to the development of automatic flight control equipment for aircraft.[10][11] Other areas of application for discontinuous controls included fire-control systems, guidance systems and electronics.
In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable (PV). In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (the set point - SP). This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.[12]
The definition of a closed loop control system according to the British Standard Institution is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."[13]
Likewise; "A Feedback Control System is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."[14]
{\displaystyle Y(s)=P(s)U(s)}
{\displaystyle U(s)=C(s)E(s)}
{\displaystyle E(s)=R(s)-F(s)Y(s).}
{\displaystyle Y(s)=\left({\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\right)R(s)=H(s)R(s).}
{\displaystyle H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}}
{\displaystyle |P(s)C(s)|\gg 1}
{\displaystyle |F(s)|\approx 1}
{\displaystyle u(t)=K_{P}e(t)+K_{I}\int e(\tau ){\text{d}}\tau +K_{D}{\frac {{\text{d}}e(t)}{{\text{d}}t}}.}
{\displaystyle u(s)=K_{P}e(s)+K_{I}{\frac {1}{s}}e(s)+K_{D}se(s)}
{\displaystyle u(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right)e(s)}
{\displaystyle C(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right).}
{\displaystyle P(s)={\frac {A}{1+sT_{P}}}}
{\displaystyle F(s)={\frac {1}{1+sT_{F}}}}
{\displaystyle K_{P}=K\left(1+{\frac {T_{D}}{T_{I}}}\right)}
{\displaystyle K_{I}={\frac {K}{T_{I}}}}
{\displaystyle C(s)=K\left(1+{\frac {1}{sT_{I}}}\right)(1+sT_{D})}
{\displaystyle K={\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}
However, in practice, a pure differentiator is neither physically realizable nor desirable[15] due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.
Nonlinear control theory – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by nonlinear differential equations. The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems. These include limit cycle theory, Poincaré maps, Lyapunov stability theorem, and describing functions. Nonlinear systems are often analyzed using numerical methods on computers, for example by simulating their operation using a simulation language. If only solutions near a stable point are of interest, nonlinear systems can often be linearized by approximating them by a linear system using perturbation theory, and linear techniques can be used.[16]
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.[17][18]
{\displaystyle x}
{\displaystyle \rho }
{\displaystyle \ x[n]=0.5^{n}u[n]}
{\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}}
{\displaystyle z=0.5}
{\displaystyle \ x[n]=1.5^{n}u[n]}
{\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}}
{\displaystyle z=1.5}
{\displaystyle Re[\lambda ]<-{\overline {\lambda }}}
{\displaystyle {\overline {\lambda }}}
{\displaystyle Re[\lambda ]<0}
Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after).
{\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)}
Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the nonlinear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.[19]
Intelligent control uses various AI computing approaches like artificial neural networks, Bayesian probability, fuzzy logic,[20]machine learning, evolutionary computation and genetic algorithms or a combination of these methods, such as neuro-fuzzy algorithms, to control a dynamic system.
Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design.[21] The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. Examples of modern robust control techniques include H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover, Sliding mode control (SMC) developed by Vadim Utkin, and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.[22] Robust methods aim to achieve robust performance and/or stability in the presence of small modeling errors.
Richard Bellman developed dynamic programming since the 1940s.[23]
Rudolf Kalman pioneered the state-space approach to systems and control. Introduced the notions of controllability and observability. Developed the Kalman filter for linear estimation.
Cut-insertion theorem
^ Minorsky, Nicolas (1922). "Directional stability of automatically steered bodies". Journal of the American Society of Naval Engineers. 34 (2): 280–309. doi:10.1111/j.1559-3584.1922.tb04958.x.
^ Maxwell, J.C. (1868). "On Governors". Proceedings of the Royal Society of London. 16: 270–283. doi:. JSTOR 112510.
^ Control Theory: History, Mathematical Achievements and Perspectives | E. Fernandez-Cara1 and E. Zuazua
^ Ang, K.H.; Chong, G.C.Y.; Li, Y. (2005). "PID control system analysis, design, and technology". IEEE Transactions on Control Systems Technology. 13 (4): 559–576. doi:10.1109/TCST.2005.847331. S2CID 921620.
^ trim point
^ Terrell, William (1999). "Some fundamental control theory I: Controllability, observability, and duality —AND— Some fundamental control Theory II: Feedback linearization of single input nonlinear systems". American Mathematical Monthly. 106 (9): 705–719 and 812–828. doi:10.2307/2589614. JSTOR 2589614.
^ Gu Shi; et al. (2015). "Controllability of structural brain networks (Article Number 8414)". Nature Communications. 6 (6): 8414. arXiv:. Bibcode:2015NatCo...6.8414G. doi:10.1038/ncomms9414. PMC . PMID 26423222. Here we use tools from control and network theories to offer a mechanistic explanation for how the brain moves between cognitive states drawn from the network organization of white matter microstructure.
^ Liu, Jie; Wilson Wang; Farid Golnaraghi; Eric Kubica (2010). "A novel fuzzy framework for nonlinear system control". Fuzzy Sets and Systems. 161 (21): 2746–2759. doi:10.1016/j.fss.2010.04.009.
^ Melby, Paul; et., al. (2002). "Robustness of Adaptation in Controlled Self-Adjusting Chaotic Systems". Fluctuation and Noise Letters. 02 (4): L285–L292. doi:10.1142/S0219477502000919.
^ N. A. Sinitsyn. S. Kundu, S. Backhaus (2013). "Safe Protocols for Generating Power Pulses with Heterogeneous Populations of Thermostatically Controlled Loads". Energy Conversion and Management. 67: 297–308. arXiv:. doi:10.1016/j.enconman.2012.11.021. S2CID 32067734.
^ Richard Bellman (1964). "Control Theory" (PDF). Scientific American. Vol. 211 no. 3. pp. 186–200.
Andrei, Neculai (2005). "Modern Control Theory – A historical Perspective" (PDF). Retrieved October 10, 2007. Cite journal requires |journal= (help)
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Home : Support : Online Help : Programming : Logic : Boolean : verify : subset
verify that the first set is a subset of the second
verify(expr1, expr2, `subset`)
verify(expr1, expr2, '`subset`(ver)')
The verify(expr1, expr2, `subset`) and verify(expr1, expr2, '`subset`(ver)') calling sequences return true if for every operand in the first set it can be determined that there is an operand in the second set which satisfies a relationship that is determined either by testing with equality or by using the verification ver.
If true is returned, then it has been determined that each operand of the first set satisfies the relationship with at least one element in the second set. If false is returned, then there is at least one operand in the first set which does not satisfy the relationship (a result of type verify(false)) with each operand of the second set. Otherwise, FAIL is returned.
The relation of a proper subset can be implemented as the verification And(`subset`(ver), Not(set(ver))). This verification is not symmetric.
The name subset is a keyword and therefore it must be enclosed by backquotes in a call to verify.
\mathrm{verify}\left({a,b,c},{a,b,c,d,e},\mathrm{`subset`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
{a,b,c}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{subset}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a,b,c,d,e}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({a,b,c,f},{a,b,c,d,e},\mathrm{`subset`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left({a,b,x\left(x-1\right)},{a,b,{x}^{2}-x},\mathrm{`subset`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
{a,b,x\left(x-1\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{subset}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a,b,{x}^{2}-x}
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The following examples use the expand and float verifications, explained on the help pages verify,expand and verify,float, respectively.
\mathrm{verify}\left({a,b,x\left(x-1\right)},{a,b,{x}^{2}-x},'\mathrm{`subset`}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({a,b,x\left(x-1\right)},{a,b,c,{x}^{2}-x},'\mathrm{`subset`}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({0.10222,0.2333},{0.102221,0.2334},\mathrm{`subset`}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left({0.10222,0.2333},{0.102221,0.2334},'\mathrm{`subset`}\left(\mathrm{float}\left({10}^{6}\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
Consider the lattice of subsets of the set
{a,b,c,d,e,f,g}
. Given the point
{a,b,d}
, select all points which precede this point:
\mathrm{select}\left(\mathrm{verify},\mathrm{combstruct}[\mathrm{allstructs}]\left(\mathrm{Subset}\left({a,b,c,d,e}\right)\right),{a,b,d},\mathrm{And}\left(\mathrm{`subset`},\mathrm{Not}\left(\mathrm{set}\right)\right)\right)
{\textcolor[rgb]{0,0,1}{\varnothing }\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}}}
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Two-Dimensional Semi-Infinite Constraint - MATLAB & Simulink - MathWorks Deutschland
\begin{array}{l}{K}_{1}\left(x,w\right)=\mathrm{sin}\left({w}_{1}{x}_{1}\right)\mathrm{cos}\left({w}_{2}{x}_{2}\right)-\frac{1}{1000}{\left({w}_{1}-50\right)}^{2}-\mathrm{sin}\left({w}_{1}{x}_{3}\right)-{x}_{3}+...\\ \text{ }\mathrm{sin}\left({w}_{2}{x}_{2}\right)\mathrm{cos}\left({w}_{1}{x}_{1}\right)-\frac{1}{1000}{\left({w}_{2}-50\right)}^{2}-\mathrm{sin}\left({w}_{2}{x}_{3}\right)-{x}_{3}\le 1.5,\end{array}
for all values of w1 and w2 over the ranges
starting at the point x = [0.25,0.25,0.25].
Note that the semi-infinite constraint is two-dimensional, that is, a matrix.
First, write a file that computes the objective function.
Second, write a file for the constraints, called mycon.m. Include code to draw the surface plot of the semi-infinite constraint each time mycon is called. This enables you to see how the constraint changes as X is being minimized.
Next, invoke an optimization routine.
After nine iterations, the solution is
and the function value at the solution is
The goal was to minimize the objective f(x) such that the semi-infinite constraint satisfied K1(x,w) ≤ 1.5. Evaluating mycon at the solution x and looking at the maximum element of the matrix K1 shows the constraint is easily satisfied.
This call to mycon produces the following surf plot, which shows the semi-infinite constraint at x.
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Home : Support : Online Help : Mathematics : Finance : Date Arithmetic : Calendar
Calendar(name)
string or name; calendar type
The Calendar command creates a calendar of the specified type and returns a module representing the new calendar. This data structure can be manipulated using the AddHoliday, JoinBusinessDays, JoinHolidays, and RemoveHoliday commands, and can also be used as a parameter to other commands from the Finance package.
The parameter name is the type of the calendar. At present only Western-style calendars are supported. This includes Bratislava, Budapest, Copenhagen, Frankfurt, Helsinki, Milan, Johannesburg, London, Oslo, Prague, Stockholm, Sydney, Tokyo, Toronto, NewYork, Warsaw, Wellington and Zurich.
In addition, two special calendars can be created: Null and Simple. Holidays in the Simple calendar are Saturdays, Sundays, and January 1st. The Null calendar does not have any holidays. Other calendars can be constructed using Null or Simple calendars as a base.
Day of Goodwill, December 26th (possibly moved to Monday)
New Year's Day, January 1st (possibly moved to Monday if it falls on Sunday)
Independence Day, July 4th (moved to Monday if it falls on Sunday or Friday if it falls on Saturday)
Christmas, December 25th (moved to Monday if it falls on Sunday or Friday if it falls on Saturday)
National Independence Day, May 17st
ANZAC Day, April 25th (possibly moved to Monday)
Coming of Age Day, 2nd Monday in January
Marine Day, 3rd Monday in July
Spring Bank Holiday, last Monday of May.
Veterans' Day, November 11th (moved to Monday if it falls on Sunday or Friday if it falls on Saturday)
New Year's Day, January 1st (possibly moved to Monday if it falls on Sunday, or to Friday if it falls on Saturday)
Independence Day, July 4th (moved to Monday if it falls on Sunday or to Friday if it falls on Saturday)
Veterans' Day, November 11th (moved to Monday if it falls on Sunday or to Friday if it falls on Saturday)
Christmas, December 25th (moved to Monday if it falls on Sunday or to Friday if it falls on Saturday)
\mathrm{with}\left(\mathrm{Finance}\right):
C≔\mathrm{Calendar}\left(\mathrm{NewYork}\right):
\mathrm{IsHoliday}\left("December 26, 2006",C\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsBusinessDay}\left("December 26, 2006",C\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{AdvanceDate}\left("December 24, 2006",5,\mathrm{Days},C,\mathrm{output}=\mathrm{formatted}\right)
\textcolor[rgb]{0,0,1}{"January 2, 2007"}
Here is a Beijing calendar for the year 2004.
\mathrm{C2}≔\mathrm{Calendar}\left(\mathrm{Simple}\right):
\mathrm{NewYearDay}≔\mathrm{seq}\left(\mathrm{AdvanceDate}\left("January 1, 2004",i\right),i=0..6\right):
\mathrm{SpringFestival}≔\mathrm{seq}\left(\mathrm{AdvanceDate}\left("January 22, 2004",i\right),i=0..6\right):
\mathrm{LaborDay}≔\mathrm{seq}\left(\mathrm{AdvanceDate}\left("May 1, 2004",i\right),i=0..6\right):
\mathrm{NationalDay}≔\mathrm{seq}\left(\mathrm{AdvanceDate}\left("October 1, 2004",i\right),i=0..6\right):
\mathrm{AddHoliday}\left(\mathrm{C2},[\mathrm{NewYearDay},\mathrm{SpringFestival},\mathrm{LaborDay},\mathrm{NationalDay}]\right):
\mathrm{AdjustDate}\left("January 23, 2004",\mathrm{C2},\mathrm{convention}=\mathrm{Following},\mathrm{output}=\mathrm{formatted}\right)
\textcolor[rgb]{0,0,1}{"January 29, 2004"}
\mathrm{AdjustDate}\left("January 23, 2004",\mathrm{C2},\mathrm{convention}=\mathrm{Preceding},\mathrm{output}=\mathrm{formatted}\right)
\textcolor[rgb]{0,0,1}{"January 21, 2004"}
This calendar can be joined with the New York calendar.
\mathrm{C3}≔\mathrm{JoinBusinessDays}\left(C,\mathrm{C2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{C3}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{module}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end module}}
\mathrm{C4}≔\mathrm{JoinHolidays}\left(C,\mathrm{C2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{C4}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{module}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end module}}
\mathrm{IsHoliday}\left("January 23, 2004",C\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsHoliday}\left("January 23, 2004",\mathrm{C2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsHoliday}\left("January 23, 2004",\mathrm{C3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsHoliday}\left("January 23, 2004",\mathrm{C4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The Finance[Calendar] command was introduced in Maple 15.
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Since May 2022, we implement the research programme
Duality Symmetries as Key to Quantum Spacetime
funded by a SONATA BIS grant awarded by the National Science Centre (the largest grant-making agency in Poland for fundamental research and basic science). With these funds from the NCN, we establish a new research, independent research team at the University of Wrocław. We are always looking for new team members! For details please check the vacancies below.
The Einstein Hilbert action is not renormalisable in more than two dimensions. Thus, it can only represent the leading contribution to an effective action that receives higher-order (curvature/derivative) corrections. These corrections are vital when gravity dominates the other three fundamental forces, i.e. close to the singularity of a black hole or at the beginning of our universe. They eradicate the standard geometric notion of space and time, which must be replaced by the more elaborate concept of quantum spacetime. One of the big puzzles of modern theoretical physics therefore is:
What are the fundamental properties of quantum spacetime?
What is the appropriate mathematical framework to describe it?
We address both systematically.
Interactions with matter and energy dynamically shape spacetime, but currently, progress is only possible for a particular subclass of spacetimes governed by powerful symmetries. They provide the computational control inevitable to approach the questions above but can be very restrictive at the same time and kill all non-trivial phenomena. We manage the balancing act by introducing the quantum version of Poisson-Lie symmetry inspired by the closed string
\sigma
-model and its tight connection to gravity. Thereby, we obtain consistent quantum spacetimes, which are crucial in understanding central quantum gravity phenomena, like the resolution of singularities, holography, and flux vacua in string theory.
We study quantum corrections for the large class of Poisson-Lie symmetric closed string
\sigma
-models. The relevant computations are infamous for their complexity, and despite significant effort, only results for the first subleading order are known. We make progress by employing the working hypothesis: "Poisson-Lie symmetry and T-duality can be defined beyond the classical limit." for which considerable evidence piled up during the last two years. All relevant spacetimes will be embedded in an appropriately extended version of double field theory to make their hidden Poisson-Lie symmetry manifest. Once manifest, this symmetry is so powerful that it facilitates explicit loop calculations up to high orders in perturbation theory. It can appear in combination with the integrability of the σ-model or supersymmetry. We will determine how these symmetries constrain higher derivative corrections of the effective action and ultimately induce the transition from spacetime to quantum spacetime. To achieve these objectives, we pursue three main research tasks:
Develop quantum,
\alpha
'-corrected, Poisson-Lie symmetry and T-duality
Understand implications of 1. for integrable
\sigma
-models and holography
Formulate resulting target space structures in terms of non-commutative geometry
Falk Hassler, group leader
It would be great if you could join our team at the University of Wrocław (UWr)! The UWr is one of the leading universities in Poland, currently teaching over 26,000 students and around 1,300 doctoral students across ten faculties. It is situated on the banks of the Oder River in the city of Wrocław. With more than 130,000 students, Wrocław is one of the most youth-oriented cities in the heart of Europe. Our department offers a specialised master's course in theoretical physics with a large variety of advanced lectures (also available to PhD students) that prepare our students ideally to contribute to cutting edge research.
Please find below all our vacancies and do not hesitate to contact me in case of any questions.
If you are interested in mathematical physic and want to apply it to actual research problems, you might consider writing your master thesis in our group. A solid background in Quantum Field Theory and General Relativity is an advantage, but even more important is that you are motivated and excited about the questions we approach in our group. If this is the case, please reach out to me, and I am sure we will find a project well suited to your interests and abilities.
We have currently one opening for a fully-funded PhD student position (poster). Two more PhD students will be most likely hired next year.
PhD Position in String Theory and Quantum Gravity
Wroclaw U. • Europe
hep-thgr-qcmath-phPHD
Deadline on Jun 5, 2022
The Institute of Theoretical Physics at the University of Wrocław, Poland, invites applications for a 4-year PhD scholarship in String Theory and Quantum Gravity within the project "Duality Symmetries as Key to Quantum Spacetime" funded by the National Science Centre Poland.
Research Tasks: Besides training in advanced theoretical physics and mathematics, you will be immersed in the group's research efforts from the first day. You will contribute to calculations, discussions and publications. Generous travel funds will allow you to attend schools and workshops/conferences to present your results. Topics you will be working on include:
quantum corrections to non-linear σ-models
supergravity and supergroups
master's or equivalent degree in physics or a closely related field
working knowledge of general relativity and quantum field theory
previous exposure to string theory would be an advantage
Terms of Employment: You can only receive this scholarship after being admitted to the Doctoral School of the University of Wrocław. We will help the successful candidate with the required application. For more details, please get in touch.
Starting Date: 1st October 2022 (negotiable)
Salary: The scholarship is at least 2500 PLN (tax-free) per month. It is supplemented by the regular scholarship for PhD students at the University of Wrocław of approximately 2100 PLN per month net. Therefore, you can expect around 4600 PLN net salary per month. PhD students qualify for housing in the university's residential halls, which costs between 850 and 1150 PLN/month (single room).
How to apply: Please send the following application documents via email to the principal investigator Falk Hassler ( ), with the subject "PhD Student":
transcript of grades and diplomas
reference letters (at least two)
include at the end of your email the following clause regarding the personal data protection laws in the EU: "I hereby authorise you to process my personal data included in my job application solely for the purpose of the selection process in accordance with the article 7 of the Regulation of the European Parliament and Council (EU) 2016/679 of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation) [OJ EU L. 2016.119.1 of 4 May 2016]."
The revision of applications starts now and continues until a suitable candidate is found, but no longer than 5 June 2022.
The University of Wrocław is one of the leading universities in Poland, currently teaching over 26,000 students and around 1,300 doctoral students. It is situated on the banks of the Oder River in the city of Wrocław. With more than 130,000 students, Wrocław is one of the most youth-oriented cities in the heart of Europe.
For more details contact Falk Hassler at .
Contact: Hassler, Falk ( )
We plan to fill at least one three-year postdoc position in 2023 (starting date September/October 2023). The official announcements for this position will be posted by the end of this year. We will keep you posted.
If you want to join our group as a postdoc, you might also consider applying for an individual fellowship:
The POLONEZ BIS programme is co-funded by the European Commission and the Polish NCN under the Marie Skłodowska-Curie COFUND grant. Three calls, announced in 2021 and 2022, will recruit 120 experienced researchers worldwide who move to Poland for 24 months to conduct their basic research in public or private institutions of their choice. This grant is a great opportunity to develop your profile as an independent researcher. We would be happy to support you with your application and act as the host. If you are interested, please let me know. The calls this year are:
2nd call closes on 5 June 2022 (4:00 p.m. CEST)
3rd call opens on 15 September 2022 and closes on 15 December 2022
Marie-Curie European Postdoctoral Fellowships also provide ideal working conditions. Moreover, even if your proposal is not funded, you can still obtain a seal of excellence that helps you get funding from other sources. The call usually opens in June and closes in October.
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<< 2.1.6. Reification view2.1.8. Property view >>
Considering a constraint for which not all variables are fixed yet, a question is to count, or estimate, its number of solutions. This is useful for writing heuristics that take into account the tightness of the constraints in order, for example, to select the next variable to assign. Considering a pure functional dependency constraint it is interesting to consider how the number of solutions of that constraint varies depending on the value of the pure functional dependency parameter (e.g., in the context of the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(𝙽,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
constraint, its number of solutions if extremely low when
𝙽=1
, then increase as
𝙽
increases up to a point where it decreases again and ends up with
𝙽=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
like an
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
). This is for instance useful for ranking pure functional dependency constraints in the context of the constraint seeker [BeldiceanuSimonis11].
Counting the number of solutions to an
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint is equivalent to counting the number of maximum matchings in a bipartite graph, which is #P-complete [Valiant79]. Consequently faster approximations for estimating the number of solutions are used in practice [ZanariniPesant10].
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View the Spectrogram Using Spectrum Analyzer - MATLAB & Simulink - MathWorks í•œêµ
Spectrograms are a two-dimensional representation of the power spectrum of a signal as this signal sweeps through time. They give a visual understanding of the frequency content of your signal. Each line of the spectrogram is one periodogram computed using either the filter bank approach or the Welch’s algorithm of averaging modified periodogram.
{N}_{samples}=\frac{\left(1â\frac{{O}_{p}}{100}\right)ÃNENBWÃ{F}_{s}}{RBW}
{P}_{dBW}=10\mathrm{log}10\left(power\text{â}in\text{â}watt/1\text{â}watt\right)
{P}_{dBm}=10\mathrm{log}10\left(power\text{â}in\text{â}watt/1\text{â}milliwatt\right)
\begin{array}{l}{P}_{Watts}={A}^{2}/2\\ {P}_{Watts}=1/2\end{array}
\begin{array}{l}{P}_{dBm}=10\mathrm{log}10\left(power\text{â}in\text{â}watt/1\text{â}milliwatt\right)\\ {P}_{dBm}=10\mathrm{log}10\left(0.5/{10}^{â3}\right)\end{array}
\begin{array}{l}{P}_{whitenoise}={P}_{unitbandwidth}*number\text{â}of\text{â}frequency\text{â}bins,\\ {P}_{whitenoise}=\left({10}^{â4}\right)*\left(\frac{Fs/2}{RBW}\right),\\ {P}_{whitenoise}=\left({10}^{â4}\right)*\left(\frac{22050}{21.53}\right)\end{array}
{P}_{dBFS}=20â
{\mathrm{log}}_{10}\left(\sqrt{{P}_{watts}}/Full_Scale\right)
{P}_{FS}=20â
{\mathrm{log}}_{10}\left(\sqrt{{P}_{watts}}/FS\right)
\begin{array}{l}{P}_{Watts}={A}^{2}/2\\ {P}_{Watts}=1/2\end{array}
{P}_{FS}=20â
{\mathrm{log}}_{10}\left(\sqrt{1/2}/1\right)
{P}_{dBm}=10\mathrm{log}10\left(power\text{â}in\text{â}watt/1\text{â}milliwatt\right)
{V}_{rms}={10}^{{P}_{dBm}/20}\sqrt{{10}^{â3}}
{V}_{rms}={10}^{26.9897/20}\sqrt{0.001}
|
Asymptotic_freedom Knowpia
In particle physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the corresponding length scale decreases.
Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the strong interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing perturbative calculations. At low energies, the interaction becomes strong, leading to the confinement of quarks and gluons within composite hadrons.
The asymptotic freedom of QCD was discovered in 1973 by David Gross and Frank Wilczek,[1] and independently by David Politzer in the same year.[2] For this work all three shared the 2004 Nobel Prize in Physics.[3]
Asymptotic freedom in QCD was discovered in 1973 by David Gross and Frank Wilczek,[1] and independently by David Politzer in the same year.[2] The same phenomenon had previously been observed (in quantum electrodynamics with a charged vector field, by V.S. Vanyashin and M.V. Terent'ev in 1965;[4] and Yang–Mills theory by Iosif Khriplovich in 1969[5] and Gerard 't Hooft in 1972[6][7]), but its physical significance was not realized until the work of Gross, Wilczek and Politzer, which was recognized by the 2004 Nobel Prize in Physics.[3]
The discovery was instrumental in "rehabilitating" quantum field theory.[7] Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become infinitely strong at short distances. This phenomenon is usually called a Landau pole, and it defines the smallest length scale that a theory can describe. This problem was discovered in field theories of interacting scalars and spinors, including quantum electrodynamics (QED), and Lehman positivity led many to suspect that it is unavoidable.[8] Asymptotically free theories become weak at short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale.
The Standard Model is not asymptotically free, with the Landau pole a problem when considering the Higgs boson. Quantum triviality can be used to bound or predict parameters such as the Higgs boson mass. This leads to a predictable Higgs mass in asymptotic safety scenarios. In other scenarios, interactions are weak so that any inconsistency arises at distances shorter than the Planck length.[9]
Screening and antiscreeningEdit
Charge screening in QED
The variation in a physical coupling constant under changes of scale can be understood qualitatively as coming from the action of the field on virtual particles carrying the relevant charge. The Landau pole behavior of QED (related to quantum triviality) is a consequence of screening by virtual charged particle–antiparticle pairs, such as electron–positron pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases.
In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Each gluon carries both a color charge and an anti-color magnetic moment. The net effect of polarization of virtual gluons in the vacuum is not to screen the field but to augment it and change its color. This is sometimes called antiscreening. Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance.
Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors.
Calculating asymptotic freedomEdit
Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's coupling constant under the renormalization group. For sufficiently short distances or large exchanges of momentum (which probe short-distance behavior, roughly because of the inverse relationship between a quantum's momentum and De Broglie wavelength), an asymptotically free theory is amenable to perturbation theory calculations using Feynman diagrams. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in such theories, which is thought to produce confinement.
Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. Essentially, the beta-function describes how the coupling constants vary as one scales the system
{\displaystyle x\rightarrow bx}
. The calculation can be done using rescaling in position space or momentum space (momentum shell integration). In non-abelian gauge theories such as QCD, the existence of asymptotic freedom depends on the gauge group and number of flavors of interacting particles. To lowest nontrivial order, the beta-function in an SU(N) gauge theory with
{\displaystyle n_{f}}
kinds of quark-like particle is
{\displaystyle \beta _{1}(\alpha )={\alpha ^{2} \over \pi }\left(-{11N \over 6}+{n_{f} \over 3}\right)}
{\displaystyle \alpha }
is the theory's equivalent of the fine-structure constant,
{\displaystyle g^{2}/(4\pi )}
in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), one has
{\displaystyle N=3,}
and the requirement that
{\displaystyle \beta _{1}<0}
{\displaystyle n_{f}<{33 \over 2}.}
Thus for SU(3), the color charge gauge group of QCD, the theory is asymptotically free if there are 16 or fewer flavors of quarks.
Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear
{\displaystyle \sigma }
-model in 2 dimensions, which has a structure similar to the SU(N) invariant Yang–Mills theory in 4 dimensions.
Finally, one can find theories that are asymptotically free and reduce to the full Standard Model of electromagnetic, weak and strong forces at low enough energies.[10]
^ a b D.J. Gross; F. Wilczek (1973). "Ultraviolet behavior of non-abelian gauge theories". Physical Review Letters. 30 (26): 1343–1346. Bibcode:1973PhRvL..30.1343G. doi:10.1103/PhysRevLett.30.1343.
^ a b H.D. Politzer (1973). "Reliable perturbative results for strong interactions". Physical Review Letters. 30 (26): 1346–1349. Bibcode:1973PhRvL..30.1346P. doi:10.1103/PhysRevLett.30.1346.
^ a b "The Nobel Prize in Physics 2004". Nobel Web. 2004. Retrieved 2010-10-24.
^ V.S. Vanyashin; M.V. Terent'ev (1965). "The vacuum polarization of a charged vector field" (PDF). Journal of Experimental and Theoretical Physics. 21 (2): 375–380. Bibcode:1965JETP...21..375V. Archived from the original (PDF) on 2016-03-04. Retrieved 2015-05-28.
^ I.B. Khriplovich (1970). "Green's functions in theories with non-Abelian gauge group". Soviet Journal of Nuclear Physics. 10: 235–242.
^ G. 't Hooft (June 1972). "Unpublished talk at the Marseille conference on renormalization of Yang–Mills fields and applications to particle physics". {{cite journal}}: Cite journal requires |journal= (help)
^ a b Gerard 't Hooft, "When was Asymptotic Freedom discovered? or The Rehabilitation of Quantum Field Theory", Nucl. Phys. Proc. Suppl. 74:413–425, 1999, arXiv:hep-th/9808154
^ D.J. Gross (1999). "Twenty Five Years of Asymptotic Freedom". Nuclear Physics B: Proceedings Supplements. 74 (1–3): 426–446. arXiv:hep-th/9809060. Bibcode:1999NuPhS..74..426G. doi:10.1016/S0920-5632(99)00208-X.
^ G. F. Giudice; G. Isidori; A. Salvio; A. Strumia (2015). "Softened Gravity and the Extension of the Standard Model up to Infinite Energy". Journal of High Energy Physics. 2015 (2): 137. arXiv:1412.2769. Bibcode:2015JHEP...02..137G. doi:10.1007/JHEP02(2015)137.
S. Pokorski (1987). Gauge Field Theories. Cambridge University Press. ISBN 0-521-36846-4.
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Global Constraint Catalog: Csort_permutation
<< 5.371. sort5.373. stable_compatibility >>
[Zhou97]
\mathrm{𝚜𝚘𝚛𝚝}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}\left(\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽},\mathrm{𝚃𝙾}\right)
\mathrm{𝚜𝚘𝚛𝚝}
\mathrm{𝚎𝚡𝚝𝚎𝚗𝚍𝚎𝚍}_\mathrm{𝚜𝚘𝚛𝚝𝚎𝚍𝚗𝚎𝚜𝚜}
\mathrm{𝚜𝚘𝚛𝚝𝚎𝚍𝚗𝚎𝚜𝚜}
\mathrm{𝚜𝚘𝚛𝚝𝚎𝚍}
\mathrm{𝚜𝚘𝚛𝚝𝚒𝚗𝚐}
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚃𝙾}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|=|\mathrm{𝙵𝚁𝙾𝙼}|
|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|=|\mathrm{𝚃𝙾}|
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}.\mathrm{𝚟𝚊𝚛}\ge 1
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}.\mathrm{𝚟𝚊𝚛}\le |\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙾},\mathrm{𝚟𝚊𝚛}\right)
The variables of collection
\mathrm{𝙵𝚁𝙾𝙼}
correspond to the variables of collection
\mathrm{𝚃𝙾}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝙵𝚁𝙾𝙼}\left[i\right].\mathrm{𝚟𝚊𝚛}=\mathrm{𝚃𝙾}\left[\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[i\right].\mathrm{𝚟𝚊𝚛}\right].\mathrm{𝚟𝚊𝚛}
). The variables of collection
\mathrm{𝚃𝙾}
are also sorted in increasing order.
\left(〈1,9,1,5,2,1〉,〈1,6,3,5,4,2〉,〈1,1,1,2,5,9〉\right)
\mathrm{𝚜𝚘𝚛𝚝}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
The first item
\mathrm{𝙵𝚁𝙾𝙼}\left[1\right].\mathrm{𝚟𝚊𝚛}=1
of collection
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[1\right].\mathrm{𝚟𝚊𝚛}={1}^{th}
item of collection
\mathrm{𝚃𝙾}
The second item
\mathrm{𝙵𝚁𝙾𝙼}\left[2\right].\mathrm{𝚟𝚊𝚛}=9
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[2\right].\mathrm{𝚟𝚊𝚛}={6}^{th}
\mathrm{𝚃𝙾}
The third item
\mathrm{𝙵𝚁𝙾𝙼}\left[3\right].\mathrm{𝚟𝚊𝚛}=1
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[3\right].\mathrm{𝚟𝚊𝚛}={3}^{th}
\mathrm{𝚃𝙾}
The fourth item
\mathrm{𝙵𝚁𝙾𝙼}\left[4\right].\mathrm{𝚟𝚊𝚛}=5
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[4\right].\mathrm{𝚟𝚊𝚛}={5}^{th}
\mathrm{𝚃𝙾}
The fifth item
\mathrm{𝙵𝚁𝙾𝙼}\left[5\right].\mathrm{𝚟𝚊𝚛}=2
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[5\right].\mathrm{𝚟𝚊𝚛}={4}^{th}
\mathrm{𝚃𝙾}
The sixth item
\mathrm{𝙵𝚁𝙾𝙼}\left[6\right].\mathrm{𝚟𝚊𝚛}=1
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[6\right].\mathrm{𝚟𝚊𝚛}={2}^{th}
\mathrm{𝚃𝙾}
The items of collection
\mathrm{𝚃𝙾}=〈1,1,1,2,5,9〉
are sorted in increasing order.
Figure 5.372.1. Illustration of the correspondence between the items of the
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚃𝙾}
collections according to the permutation defined by the items of the
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
collection of the Example slot (note that the items of the
\mathrm{𝚃𝙾}
collection are sorted in increasing order)
|\mathrm{𝙵𝚁𝙾𝙼}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝙵𝚁𝙾𝙼}.\mathrm{𝚟𝚊𝚛}\right)>1
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\right)
\mathrm{𝚟𝚊𝚛}
attributes of all items of
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚃𝙾}
\mathrm{𝚃𝙾}
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚃𝙾}
This constraint is referenced under the name
\mathrm{𝚜𝚘𝚛𝚝𝚒𝚗𝚐}
in SICStus Prolog.
[Zhou97].
n
denote the number of variables in the collection
\mathrm{𝙵𝚁𝙾𝙼}
\mathrm{𝚜𝚘𝚛𝚝}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
constraint can be reformulated as a conjunction of the form:
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[1\right],\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\left[1\right]\right)
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[2\right],\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\left[2\right]\right)
\cdots
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\left[n\right],\mathrm{𝙵𝚁𝙾𝙼},\mathrm{𝚃𝙾}\left[n\right]\right)
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚃𝙾}\right)
To enhance the previous model, the following necessary condition was proposed by P. Schaus.
\forall i\in \left[1,n\right]:{\sum }_{j=1}^{j=n}\left(\mathrm{𝙵𝚁𝙾𝙼}\left[j\right]<\mathrm{𝚃𝙾}\left[i\right]\right)\le i-1
(i.e., at most
i-1
variables of the collection
\mathrm{𝙵𝚁𝙾𝙼}
are assigned a value strictly less than
\mathrm{𝚃𝙾}\left[i\right]
). Similarly, we have that
\forall i\in \left[1,n\right]:{\sum }_{j=1}^{j=n}\left(\mathrm{𝙵𝚁𝙾𝙼}\left[j\right]>\mathrm{𝚃𝙾}\left[i\right]\right)\ge n-i
n-i
\mathrm{𝙵𝚁𝙾𝙼}
are assigned a value are strictly greater than
\mathrm{𝚃𝙾}\left[i\right]
sorted in Gecode, sorting in SICStus.
\mathrm{𝚘𝚛𝚍𝚎𝚛}
(sort, permutation).
\mathrm{𝚌𝚘𝚛𝚛𝚎𝚜𝚙𝚘𝚗𝚍𝚎𝚗𝚌𝚎}
\mathrm{𝚜𝚘𝚛𝚝}
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
parameter removed).
used in reformulation:
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
characteristic of a constraint: sort, derived collection.
combinatorial object: permutation.
constraint arguments: constraint between three collections of variables.
modelling: functional dependency.
Derived Collection
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙵𝚁𝙾𝙼}_\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚒𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝙵𝚁𝙾𝙼}.\mathrm{𝚟𝚊𝚛},\mathrm{𝚒𝚗𝚍}-\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}.\mathrm{𝚟𝚊𝚛}\right)\right]\hfill \end{array}\right)
\mathrm{𝙵𝚁𝙾𝙼}_\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
\mathrm{𝚃𝙾}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗},\mathrm{𝚝𝚘}\right)
•\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚝𝚘}.\mathrm{𝚟𝚊𝚛}
•\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}.\mathrm{𝚒𝚗𝚍}=\mathrm{𝚝𝚘}.\mathrm{𝚔𝚎𝚢}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|
\mathrm{𝚃𝙾}
\mathrm{𝑃𝐴𝑇𝐻}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚘}\mathtt{1},\mathrm{𝚝𝚘}\mathtt{2}\right)
\mathrm{𝚝𝚘}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\le \mathrm{𝚝𝚘}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚃𝙾}|-1
Parts (A) and (B) of Figure 5.372.2 respectively show the initial and final graph associated with the first graph constraint of the Example slot. In both graphs the source vertices correspond to the items of the derived collection
\mathrm{𝙵𝚁𝙾𝙼}_\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
, while the sink vertices correspond to the items of the
\mathrm{𝚃𝙾}
collection. Since the first graph constraint uses the
\mathrm{𝐍𝐀𝐑𝐂}
graph property, the arcs of its final graph are stressed in bold. The
\mathrm{𝚜𝚘𝚛𝚝}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
The first graph constraint holds since its final graph contains exactly
\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}
Finally the second graph constraint holds also since its corresponding final graph contains exactly
|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}-1|
arcs: all the inequalities constraints between consecutive variables of
\mathrm{𝚃𝙾}
\mathrm{𝚜𝚘𝚛𝚝}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}
Consider the first graph constraint where we use the
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
arc generator. Since all the
\mathrm{𝚔𝚎𝚢}
attributes of the
\mathrm{𝚃𝙾}
collection are distinct, and because of the second condition
\mathrm{𝚏𝚛𝚘𝚖}_\mathrm{𝚙𝚎𝚛𝚖𝚞𝚝𝚊𝚝𝚒𝚘𝚗}.\mathrm{𝚒𝚗𝚍}=\mathrm{𝚝𝚘}.\mathrm{𝚔𝚎𝚢}
of the arc constraint, each vertex of the final graph has at most one successor. Therefore the maximum number of arcs of the final graph is equal to
|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|
. So we can rewrite the graph property
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|
\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝙿𝙴𝚁𝙼𝚄𝚃𝙰𝚃𝙸𝙾𝙽}|
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
Consider now the second graph constraint. Since we use the
\mathrm{𝑃𝐴𝑇𝐻}
arc generator with an arity of 2 on the
\mathrm{𝚃𝙾}
collection, the maximum number of arcs of the corresponding final graph is equal to
|\mathrm{𝚃𝙾}|-1
. Therefore we can rewrite
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝚃𝙾}|-1
\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝚃𝙾}|-1
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
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student(deprecated)/middlebox - Maple Help
Home : Support : Online Help : student(deprecated)/middlebox
middlebox(f(x), x=a..b, <plot options>)
middlebox(f(x), x=a..b, n, 'shading'=<color>, <plot options>)
The function middlebox will generate a plot of rectangular boxes used to approximate a definite integral. The height of each rectangle (box) is determined by the value of the function at the centre of each interval.
The value of the corresponding numerical approximation can be obtained by the Maple procedure middlesum.
The command with(student,middlebox) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{student}\right):
\mathrm{middlebox}\left({x}^{4}\mathrm{ln}\left(x\right),x=2..4,\mathrm{color}=\mathrm{YELLOW}\right)
\mathrm{middlebox}\left(\mathrm{sin}\left(x\right)x+\mathrm{sin}\left(x\right),x=0..2\mathrm{\pi },5,\mathrm{shading}=\mathrm{BLUE}\right)
\mathrm{middlebox}\left(\mathrm{sin}\left(x\right)x+\mathrm{sin}\left(x\right),x=0..2\mathrm{\pi },5,\mathrm{color}=\mathrm{GREEN}\right)
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Answer pls Q There are 24 students in a class They voted to choose a colour for the - Maths - Rational Numbers - 12343953 | Meritnation.com
Q. There are 24 students in a class.
They voted to choose a colour for the class shirts.
\frac{1}{4}
of the class chose blue.
\frac{1}{6}
of the class chose yellow.
The rest of the class chose red.
How many students chose red?
\mathrm{Number} \mathrm{of} \mathrm{students} \mathrm{choose} \mathrm{blue}=\frac{1}{4}×24=6\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{students} \mathrm{choose} \mathrm{yellow}=\frac{1}{6}×24=4\phantom{\rule{0ex}{0ex}}\mathrm{Number} \mathrm{of} \mathrm{students} \mathrm{choose} \mathrm{red}=24-4-6=14\phantom{\rule{0ex}{0ex}}\mathrm{Corect} \mathrm{option} \mathrm{is} \mathrm{D}.
Satyam Sinha answered this
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Unconditional expected shortfall backtest by Acerbi and Szekely - MATLAB unconditional - MathWorks Australia
Run an ES Unconditional Test
Unconditional expected shortfall backtest by Acerbi and Szekely
TestResults = unconditional(ebts)
[TestResults,SimTestStatistic] = unconditional(ebts,Name,Value)
TestResults = unconditional(ebts) runs the unconditional expected shortfall (ES) backtest of Acerbi-Szekely (2014).
[TestResults,SimTestStatistic] = unconditional(ebts,Name,Value) adds an optional name-value pair argument for TestLevel.
Generate the ES unconditional test report.
esbacktestbysim (ebts) object, contains a copy of the given data (the PortfolioData, VarData, ESData, and Distribution properties) and all combinations of portfolio ID, VaR ID, and VaR levels to be tested. For more information on creating an esbacktestbysim object, see esbacktestbysim.
Example: [TestResults,SimTestStatistic] = unconditional(ebts,'TestLevel',0.99)
'Unconditional'— Categorical array with categories 'accept' and 'reject' that indicate the result of the unconditional test
'PValue'— P-value of the unconditional test
'TestStatistic'— Unconditional test statistic
'CriticalValue'— Critical value for the unconditional test
SimTestStatistic — Simulated values of the test statistic
The unconditional test is also known as the second Acerbi-Szekely test.
The unconditional test is based on the unconditional relationship
E{S}_{t}=-{E}_{t}\left[\frac{{X}_{t}{I}_{t}}{{p}_{VaR}}\right]
The unconditional test statistic is defined as:
{Z}_{uncond}=\frac{1}{N{p}_{VaR}}\sum _{t=1}^{N}\frac{{X}_{t}{I}_{t}}{E{S}_{t}}+1
Under the assumption that the distributional assumptions are correct, the expected value of the test statistic Zuncond is 0.
This is expressed as
E\left[{Z}_{uncond}\right]=0
Negative values of the test statistic indicate risk underestimation. The unconditional test is a one-sided test that rejects when there is evidence that the model underestimates risk (for technical details on the null and alternative hypotheses, see Acerbi-Szekely, 2014). The unconditional test rejects the model when the p-value is less than 1 minus the test confidence level.
For more information on the steps to simulate the test statistics and the details for the computation of thep-values and critical values, see simulate.
The unconditional test statistic takes a value of 1 when there are no VaR failures in the data or in a simulated scenario.
1 is also the maximum possible value for the test statistic. When the expected number of failures NpVaR is small, the distribution of the unconditional test statistic has a discrete probability jump at Zuncond = 1, and the probability that Zuncond ≤ 1 is 1. The p-value is set to 1 in these cases, and the test result is to 'accept', because there is no evidence of risk underestimation. Scenarios with no failures are more likely as the expected number of failures NpVaR gets smaller.
summary | runtests | conditional | quantile | minBiasRelative | minBiasAbsolute | simulate | esbacktestbysim | esbacktestbyde
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4. Case Study: Interface Design - Think Julia [Book]
Think Julia by Ben Lauwens, Allen B. Downey
Get full access to Think Julia and 60K+ other titles, with free 10-day trial of O'Reilly.
Chapter 4. Case Study: Interface Design
It introduces turtle graphics, a way to create programmatic drawings. Turtle graphics are not included in the standard library, so to use them you’ll have to add the ThinkJulia module to your Julia setup.
The examples in this chapter can be executed in a graphical notebook on JuliaBox, which combines code, formatted text, math, and multimedia in a single document (see Appendix B).
A module is a file that contains a collection of related functions. Julia provides some modules in its standard library. Additional functionality can be added from a growing collection of packages.
Packages can be installed in the REPL by entering the Pkg REPL mode using the key ] and using the add command:
Before we can use the functions in a module, we have to import it with a using statement:
julia> 🐢 = Turtle()
Luxor.Turtle(0.0, 0.0, true, 0.0, (0.0, 0.0, 0.0))
The ThinkJulia module provides a function called Turtle that creates a Luxor.Turtle object, which we assign to a variable named 🐢(\:turtle: TAB).
Once you create a turtle, you can call a function to move it around. For example, to move the turtle forward:
forward(🐢, 100)
The @svg keyword runs a macro that draws an SVG picture (Figure 4-1). Macros are an important but advanced feature of Julia.
Figure 4-1. Moving the turtle forward
The arguments of forward are the turtle and a distance in pixels, so the actual size of the line that’s drawn depends on your display.
Each turtle is holding a pen, which is either down or up; if the pen is down (the default), the turtle leaves a trail when it moves. Figure 4-1 shows the trail left behind by the turtle. To move the turtle without drawing a line, first call the function penup. To start drawing again, call pendown.
Another function you can call with a turtle as an argument is turn for turning. The second argument for turn is an angle in degrees.
To draw a right angle, modify the macro call:
🐢 = Turtle()
turn(🐢, -90)
Now modify the macro to draw a square. Don’t go on until you’ve got it working!
We can do the same thing more concisely with a for statement:
The syntax of a for statement is similar to a function definition. It has a header and a body that ends with the keyword end. The body can contain any number of statements.
The following is a series of exercises using turtles. They are meant to be fun, but they have a point, too. While you are working on them, think about what the point is.
The following sections contain solutions to the exercises, so don’t look until you have finished (or at least tried them).
Write a function call that passes 🐢 as an argument to square, and then run the macro again.
Add another parameter, named len, to square. Modify the body so the length of the sides is len, and then modify the function call to provide a second argument. Run the macro again. Test with a range of values for len.
Make a copy of square and change the name to polygon. Add another parameter named n and modify the body so it draws an
-sided regular polygon.
The exterior angles of an
-sided regular polygon are
\frac{360}{n}
Figure out the circumference of the circle and make sure that len * n == circumference.
Make a more general version of circle called arc that takes an additional parameter angle, which determines what fraction of a circle to draw. angle is in units of degrees, so when angle = 360, arc should draw a complete circle.
function square(t)
forward(t, 100)
turn(t, -90)
square(🐢)
The innermost statements, forward and turn, are indented twice to show that they are inside the for loop, which is inside the function definition.
Inside the function, t refers to the same turtle 🐢, so turn(t, -90) has the same effect as turn(🐢, -90). In that case, why not call the parameter 🐢? The idea is that t can be any turtle, not just 🐢 so you could create a second turtle and pass it as an argument to square:
🐫 = Turtle()
square(🐫)
Wrapping a piece of code up in a function is called encapsulation. One of the benefits of encapsulation is that it attaches a name to the code, which serves as a kind of documentation. Another advantage is that if you reuse the code, it is more concise to call a function twice than to copy and paste the body!
The next step is to add a len parameter to square. Here is a solution:
function square(t, len)
square(🐢, 100)
Adding a parameter to a function is called generalization because it makes the function more general. In the previous version, the square is always the same size; in this version it can be any size.
polygon(🐢, 7, 70)
function circle(t, r)
circumference = 2 * π * r
len = circumference / n
polygon(t, n, len)
The first line computes the circumference of a circle with radius r using the formula 2πr. n is the number of line segments in our approximation of a circle, so len is the length of each segment. Thus, polygon draws a 50-sided polygon that approximates a circle with radius r.
The interface of a function is a summary of how it is used: What are the parameters? What does the function do? And what is the return value? An interface is “clean” if it allows the caller to do what he wants without dealing with unnecessary details.
Rather than cluttering up the interface, it is better to choose an appropriate value of n depending on circumference:
n = trunc(circumference / 3) + 3
Now the number of segments is an integer near circumference/3, so the length of each segment is approximately 3, which is small enough that the circles look good but big enough to be efficient, and acceptable for any size circle.
Adding 3 to n guarantees that the polygon has at least three sides.
When I wrote circle, I was able to reuse polygon because a many-sided polygon is a good approximation of a circle. But arc is not as cooperative; we can’t use polygon or circle to draw an arc.
function arc(t, r, angle)
arc_len = 2 * π * r * angle / 360
n = trunc(arc_len / 3) + 1
step_len = arc_len / n
forward(t, step_len)
turn(t, -step_angle)
The second half of this function looks like polygon, but we can’t reuse polygon without changing the interface. We could generalize polygon to take an angle as a third argument, but then polygon would no longer be an appropriate name! Instead, let’s call the more general function polyline:
function polyline(t, n, len, angle)
polyline(t, n, len, angle)
polyline(t, n, step_len, step_angle)
This process—rearranging a program to improve interfaces and facilitate code reuse—is called refactoring. In this case, we noticed that there was similar code in arc and polygon, so we “factored it out” into polyline.
Once you get the program working, identify a coherent piece of it, encapsulate the piece in a function, and give it a name.
Repeat steps 1–3 until you have a set of working functions. Copy and paste working code to avoid retyping (and redebugging).
A docstring is a string before a function that explains the interface (“doc” is short for “documentation”). Here is an example:
Draws n line segments with the given length and
Documentation can be accessed in the REPL or in a notebook by typing ? followed by the name of a function or macro, and pressing Enter:
help?> polyline
Draws n line segments with the given length and angle (in degrees) between them. t is a turtle.
Docstrings are often triple-quoted strings, also known as “multiline” strings because the triple quotes allow the string to span more than one line.
A docstring contains the essential information someone would need to use the function. It explains concisely what the function does (without getting into the details of how it does it). It explains what effect each parameter has on the behavior of the function and what type each parameter should be (if it is not obvious).
For example, polyline requires four arguments: t has to be a turtle; n has to be an integer; len should be a positive number; and angle has to be a number, which is understood to be in degrees.
An external library with additional functionality.
Enter the code in this chapter in a notebook.
Draw a stack diagram that shows the state of the program while executing circle(🐢, radius). You can do the arithmetic by hand or add print statements to the code.
The version of arc in “Refactoring” is not very accurate because the linear approximation of the circle is always outside the true circle. As a result, the turtle ends up a few pixels away from the correct destination. The solution shown here illustrates a way to reduce the effect of this error. Read the code and see if it makes sense to you. If you draw a diagram, you might see how it works.
Draws an arc with the given radius and angle:
arc_len = 2 * π * r * abs(angle) / 360
turn(t, -step_angle/2)
turn(t, step_angle/2)
Write an appropriately general set of functions that can draw flowers as in Figure 4-2.
Figure 4-2. Turtle flowers
Write an appropriately general set of functions that can draw shapes as in Figure 4-3.
Figure 4-3. Turtle pies
You should write one function for each letter, with names draw_a, draw_b, etc., and put your functions in a file named letters.jl.
Read about spirals at https://en.wikipedia.org/wiki/Spiral; then write a program that draws an Archimedean spiral as in Figure 4-4.
Figure 4-4. Archimedean spiral
Get Think Julia now with the O’Reilly learning platform.
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Obtain the arc-length function for the curve
x={p}^{2}-p/2,y=4/3 {p}^{3/2}
p∈\left[0,\infty \right).
s=s\left(p\right)
p=p\left(s\right)
and reparametrize the curve with the arc length
s
as the parameter.
∥\frac{d\mathbf{R}}{\mathrm{ds}}∥=1
\mathbf{R}\left(t\right)=x\left(t\right) \mathbf{i}+y\left(t\right) \mathbf{j}
\stackrel{.}{\mathbf{R}}=\stackrel{.}{x} \mathbf{i}+\stackrel{.}{y} \mathbf{j}
t
\sqrt{{\stackrel{.}{x}}^{2}+{\stackrel{.}{y}}^{2}}
∥\stackrel{.}{\mathbf{R}}∥
Figure 2.2.6(a) Graph of the given plane curve
∥\frac{d\mathbf{R}}{\mathrm{dp}}∥
=\sqrt{{\left(\frac{d}{\mathrm{dp}}\left({p}^{2}-p/2\right)\right)}^{2}+{\left(\frac{d}{\mathrm{dp}}\left(4/3 {p}^{3/2}\right)\right)}^{2}}
=\sqrt{{\left(2 p-1/2\right)}^{2}+{\left(2\sqrt{p}\right)}^{2}}
=\sqrt{4 {p}^{2}+2 p+1/4}
=\sqrt{{\left(4 p+1\right)}^{2}/4}
=2 p+1/2
The arc-length function is then
s\left(p\right)={∫}_{0}^{p}\left(2 u+1/2\right) ⅆu={p}^{2}+p/2
, where the variable of integration is chosen as
u
because the upper limit of integration is itself
p
p=p\left(s\right)
s\left(p\right)={p}^{2}+p/2
p
by the quadratic formula.
p=\left(-1 ±\sqrt{1+16 s}\right)/4
, but the restriction
p∈\left[0,\infty \right)
p\left(s\right)=\left(-1+\sqrt{1+16 s}\right)/4
\mathbf{R}\left(s\right)=\mathbf{R}\left(p\left(s\right)\right)
\mathbf{R}\left(s\right)=
\mathbf{R}\left(p\left(s\right)\right)
\left[\begin{array}{c}{\left(-\frac{1}{4}+\frac{1}{4}\sqrt{1+16s}\right)}^{2}+\frac{1}{8}-\frac{1}{8}\sqrt{1+16s}\\ \frac{4}{3}{\left(-\frac{1}{4}+\frac{1}{4}\sqrt{1+16s}\right)}^{3/2}\end{array}\right]
=\left[\begin{array}{c}\frac{1}{4}-\frac{1}{4}\sqrt{1+16s}+s\\ \frac{1}{6}{\left(-1+\sqrt{1+16s}\right)}^{3/2}\end{array}\right]
The relevant calculations are as follows.
\frac{d}{\mathrm{ds}}\mathbf{R}\left(s\right)=\left[\begin{array}{c}-\frac{2}{\sqrt{1+16s}}+1\\ \frac{2\sqrt{-1+\sqrt{1+16s}}}{\sqrt{1+16s}}\end{array}\right]
{∥\frac{d}{\mathrm{ds}}\mathbf{R}\left(s\right)∥}^{2}
={\left(-\frac{2}{\sqrt{1+16s}}+1\right)}^{2}+{\left(\frac{2\sqrt{-1+\sqrt{1+16s}}}{\sqrt{1+16s}}\right)}^{2}
=\frac{4}{1+16 s}+1-\frac{4}{\sqrt{1+16 s}}+\frac{4\left(-1+\sqrt{1+16 s}\right)}{1+16 s}
=\frac{4}{1+16 s}+1-\frac{4}{\sqrt{1+16 s}}-\frac{4}{1+16 s}+\frac{4}{\sqrt{1+16 s}}
=1
Define the curve as the position vector R
〈{p}^{2}-p/2,4/3 {p}^{3/2}〉
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{R}
Using vertical strokes for norm bars and the Calculus palette for the differentiation operator, write the norm of
{R}^{}\prime
Context Panel: Constructions≻Definite Integral≻p ≻ Set range from 0 to t
∥\frac{ⅆ}{ⅆ p} \mathbf{R}∥
\stackrel{\text{assuming positive}}{\to }
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{p}
\stackrel{\text{integrate w.r.t. p}}{\to }
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{t}}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{p}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{p}
=
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}
Interpret the result
s\left(t\right)={t}^{2}+t/2
s\left(p\right)={p}^{2}+p/2
s=\dots \phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
Solve≻Obtain Solutions for≻
p
s={p}^{2}+p/2
\stackrel{\text{solutions for p}}{\to }
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}{\textcolor[rgb]{0,0,1}{4}}
Control-drag the solution with the positive radical.
P
-\frac{1}{4}+\frac{1}{4}\sqrt{1+16s}
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{P}
Re-parametrize R by making the substitution
p=p\left(s\right)=P
Write R, the name of the position vector
\mathbf{R}\left(p\right)
, and press the Enter key.
p→P
Context Panel: Assign to a Name≻Rs
\mathbf{R}
\left[\begin{array}{c}{p}^{2}-\frac{1}{2}\textcolor[rgb]{0,0,1}{}p\\ \frac{4}{3}\textcolor[rgb]{0,0,1}{}{p}^{3/2}\end{array}\right]
\stackrel{\text{evaluate at point}}{\to }
\left[\begin{array}{c}{\left(-\frac{1}{4}+\frac{1}{4}\textcolor[rgb]{0,0,1}{}\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}\right)}^{2}+\frac{1}{8}-\frac{1}{8}\textcolor[rgb]{0,0,1}{}\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}\\ \frac{4}{3}\textcolor[rgb]{0,0,1}{}{\left(-\frac{1}{4}+\frac{1}{4}\textcolor[rgb]{0,0,1}{}\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}\right)}^{3/2}\end{array}\right]
\stackrel{\text{simplify}}{=}
\left[\begin{array}{c}\frac{1}{4}-\frac{1}{4}\textcolor[rgb]{0,0,1}{}\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}+s\\ \frac{1}{6}\textcolor[rgb]{0,0,1}{}{\left(-1+\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}\right)}^{3/2}\end{array}\right]
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{\mathrm{Rs}}
Apply to
\mathbf{R}\left(s\right)
∥\frac{ⅆ}{ⅆ s} \mathbf{Rs}∥
\sqrt{{\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}\right)}{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}
\stackrel{\text{simplify}}{=}
\textcolor[rgb]{0,0,1}{1}
\mathrm{with}\left(\mathrm{Student}:-\mathrm{MultivariateCalculus}\right):
\mathbf{R}≔〈{p}^{2}-p/2,4/3 {p}^{3/2}〉:
Apply the int, Norm, and diff commands, imposing the positivity assumption on
t
\mathrm{int}\left(\mathrm{Norm}\left(\mathrm{diff}\left(\mathbf{R},p\right)\right),p=0..t\right) assuming t>0
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}
The result of these calculations is
s\left(t\right)={t}^{2}+t/2
s\left(p\right)={p}^{2}+p/2
Use the solve command to obtain
p=p\left(s\right)
s=s\left(p\right)
Assign the sequence of solutions to the name
P
P≔\mathrm{solve}\left(s={p}^{2}+p/2,p\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{s}}}{\textcolor[rgb]{0,0,1}{4}}
Apply the eval command to obtain
\mathbf{R}\left(p\left(s\right)\right)=\mathbf{R}\left(s\right)
In addition, apply the simplify command.
\mathrm{Rs}≔\mathrm{simplify}\left(\mathrm{eval}\left(R,p={P}_{1}\right)\right)
\left[\begin{array}{c}\frac{1}{4}-\frac{1}{4}\textcolor[rgb]{0,0,1}{}\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}+s\\ \frac{1}{6}\textcolor[rgb]{0,0,1}{}{\left(-1+\sqrt{1+16\textcolor[rgb]{0,0,1}{}s}\right)}^{3/2}\end{array}\right]
Apply simplify to the Norm command applied to
\frac{d}{\mathrm{ds}}\mathbf{R}\left(s\right)
, obtained in turn by an application of the diff command.
\mathrm{simplify}\left(\mathrm{Norm}\left(\mathrm{diff}\left(\mathrm{Rs},s\right)\right)\right)
\textcolor[rgb]{0,0,1}{1}
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PDA_RNPOI
Returns pseudo-random numbers from a Poisson distribution
This is a simple random-number generator providing deviates in the from a Poisson distribution, with a period of 2
\ast
\ast
RESULT = PDA_RNPOI( MEAN )
The mean value of the Poisson distribution.
PDA_RNPOI = INTEGER
The pseudo-random deviate. A value of -1 is returned if the supplied mean is not positive.
Ahrens, J.H., & Dieter, U. 1973, "Computer Generation of Poisson Deviates from modified Normal distributions", ACM Trans. Math. Software, 8(2), pp.163–179.
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This is the mandatory Quantum Field Theory course of the Master in Theoretical Physics at the University of Wrocław. It is tailored towards master and PhD students who are familiar with
There are many good books on the subject, four that I like are
[PeskSchr] Peskin, Schroeder: An introduction to quantum field theory
[Ryder] Ryder: Quantum Field Theory
Weinberg: The Quantum Theory of Fields, Volume 1 & 2
Zee: Quantum Field Theory in a Nutshell.
We will follow mostly the first one in the lectures. There will be 2 hours of lectures and 2 hours of seminar each week. Exercises will be posted here a week before the seminar they are discussed in. Please keep in mind that active participation in the seminars is important to pass the course. For more information, please refer to the syllabus (in Polish) or contact me directly.
Important: Students will be assigned to exercise problems by the system described below at the Thursday, 9:00 pm, before the tutorial. Please indicate your preferences by then.
Additional material for the individual lectures, including the exercises which we discuss in the tutorials, is given below:
Reminder of spin 0 and 1/2 fields, notes, exercise
Suppl. reading: [PeskSchr] sections one and two
Please note that there are no exercise assignments for the first seminar. Instead, we continue with the lecture.
Reminder of spin 1 fields and abelian gauge symmetries, notes, exercise
Suppl. reading: [Ryder] sections 3.3 and 4.4
Non-abelian gauge symmetries and Lie groups, notes, exercise
Suppl. reading: [PeskSchr] section 15 except for 15.3 or alternatively [Ryder] sections 3.5 and 3.6
Path integral in quantum mechanics and for the scalar field, notes, exercise
Suppl. reading: [PeskSchr] section 9.1
Generating functional, interactions and Feynman rules, notes, exercise
Path integral for fermions, Grassmann numbers, chiral anomaly (in the exercise), notes, exercise
Path integral for spin-1 bosons, ghost fields, notes, exercise
One loop effects in QED: field-strength renormalisation and self-energy, notes, exercise
Dimensional regularisation and superficial degree of divergence, notes, exercise
Suppl. reading: Peskin&Schroeder sections 7.5 and 10.1
One-loop renoramlised
\phi^4
theory, notes, exercise
Suppl. reading: [PeskSchr] section 10.2
Renormalisation scale,
\beta
-function, RG flow
Suppl. reading: [PresSchr] sections 12.1-12.3
Renormalised perturbation theory of QED,
g − 2
Suppl. reading: [PresSchr] sections 10.3 and 6.3, Status of the Fermilab Muon g − 2 Experiment
String theory in a nutshell or the one-loop
\beta
-functions of a 2d
\sigma
Suppl. reading: David Tong's lecture notes on String Theory section 7.1, original article Phys.Rev.Lett. 45 (1980) 1057
BRST symmetry, physical Hilbert space
Suppl. reading: [PeskSchr] section 16.2-16.4
Non-Abelian gauge theory, Feynman rules, QCD at one loop
Lie algebras describe infinitesimal symmetries of physical systems. Therefore, they and their representation theory are extensively used in physics, most notably in quantum mechanics and particle physics. This course introduces semi-simple Lie algebras and the associated Lie groups for physicists. We discuss the essential tools, like the root and weight system, to efficiently work with them and their representations. As on explicit application of the mathematical framework, we discuss Grand Unified Theories (GUT). Moreover, we show how modern computer algebra tools like LieART can significantly help in all explicit computations throughout the course.
A simple example is the visualisation of the root system of
E_6
projected on the Coxeter plane, which you can see here. If you want to understand how it is created and connected to particle physics, you should take this course. Basic knowledge of core concepts in linear algebra, like vector spaces, eigenvalues and eigenvectors, is assumed. Some good books about the topic are:
Fuchs and Schweigert: Symmetries, Lie Algebras and Representations
Gilmore: Lie Groups, Lie Algebras, and Some of Their Applications
Fulton and Harris: Representation Theory
Georgi: Lie Algebras In Particle Physics: from Isospin To Unified Theories
The article Phys. Rep. 79 (1981) 1 by Slansky and the manual of the LieART package are good references, too.
Note: At multiple occations, we will use Mathematica and it might be a good idea to set it up on your computer. Following the instructions below you should be eventually able to run the notebook, which generates the projection of the
E_6
root system above.
Introduction and motivation, notes, notebook
Some mathematical preliminaries, notes
Classical matrix groups, notes
Cartan subalgebra, notes
Root system, notes, notebook
Simple root and Cartan matrix, notes
Classification and Dynkin diagrams, notes
Irreducible representations, notes
\mathfrak{su}
N
) representations and Young tableaux, notes
Highest weight representations, notes
Characters and self-conjugate modules
Particle theory and the standard model
The Georgi–Glashow model
This course introduces bosonic string theory in the elite master course "Theoretical and Mathematical Physics" at LMU Munich. This semester, it is taught by Dieter Lüst. In coordinating with Prof. Lüst, I prepare the excercises, grade them and discuss all solutions with the students in the tutorials. Below is a list of all exercises.
Light-cone coordinates and compact dimensions, exercise
Point particle action and reparameterization, exercise
The 2-sphere, exercise
Polyakov action and conformal transformations, exercise
Classical relativistic string, exercise
Virasoro algebra, exercise
Quantization of the relativistic string, exercise
Path integrals, ghosts and Grassmann numbers, exercise
Conformal field theory, exercise
Vertex operators and the complex plane, exercise
String compactification on the circle, exercise
String compactification on the torus, exercise
D-branes, exercise
An introduction to classical mechanics for bachelor students at the LMU Munich, taught in German by Dieter Lüst. Due to a large number of participating students, there are multiple tutorials for this course, coordinated by James Gray. My responsiblity is to help James in preparing excercises and the exam in German and to give one of the tutorials. A list of all twelve excercises follows.
Vektoren und Kinematik, exercise
Die Newton’schen Axiome, exercise
Erhaltungsgrößen und die Newton’schen Axiome, exercise
Drehimpulserhaltung und Streuung, exercise
Flugbahn und Streuung, exercise
Orbits und der Runge-Lenz Vektor, exercise
Rotierende Koordinatensysteme, exercise
Starre Körper, exercise
Starre Körper und Lagrange-Gleichung erster Art, exercise
Lagrange-Formalismus, exercise
Lagrange-Formalismus II, exercise
Lagrange-Formalismus III, exercise
For computations we use Mathematica. It is a very powerful tool with unfortunately a quite high price for a license. Students can get a discount on licenses. If your budget is not sufficient for a student license, you can use the Wolfram Engine for Developers. After creating a Wolfram ID, it can be downloaded for free. The Wolfram Engine implements the Wolfram Language, which Mathematica is based on. But, it lacks the graphical notebook interface. Fortunately, some excellent free software called Jupyter Notebook fills the gap. Both can be connected with the help of Wolfram Language for Jupyter project on GitHub. Getting everything running might require a little bit of tinkering. But in the end, you get a very powerful computer algebra system for free. Finally, you should install LieART by following the "Manual Installation" instructions.
Assignment of problems
Solving the exercise problems for a course is very important. It helps to practice the concepts and ideas introduced during the lecture, and you will also be graded for the solutions you present during the tutorials. But at the same time, one of the most annoying questions for the students and the lecturer is: "Who would like to present his solutions to the next problem?".
Therefore, we use the following system to assign students to problems based on their preferences:
You need to log in with your USOS account (every student at the University of Wrocław should have one). To do so, click on the small closed door in the top left corner of this window and enter your credentials in the window which pops up. The first time you do this, you will be asked to give this website minimal access to your USOS profile.
After a successful login, you can go to your course above and find the new link "manage" after each lecture with an exercise. If you do not see this link, check if you are logged in (the small closed door you clicked in the last step should be slightly open now). Second, verify that the course you are looking at is indeed your course. When the problem still persists, please get in touch with me.
If you click "manage", you will find a list of all the problems that we discuss during the exercises. If problems have not yet been assigned to students, you can indicate your preferences by sorting this list. Entries on the top have the highest priority, while those at the bottom have the lowest. You sort them by drag&drop with either the mouse or your finger if you work with a touchscreen. Once this is done, do not forget to click the "Save" button at the bottom.
You can always come back later and revisit your choice or change it until the assignments are fixed. Once this happens, you will get an email and see the students' names to present the various problems. The backup candidate (the second name) should be ready to take over if required.
Make sure you log out at the end of your session by clicking on the now slightly opened door you arleady used to login.
The assignments made by this system are binding. For every problem you present you can get up to three points. These points are added up and used at the end of the course to calculate your grade. If you cannot present an assigned problem, you will get zero points and put your backup on the spot. Therefore: Please prepare properly and in case of any emergencies, let us know timely. Backup candidates can earn extra points (up to 1.5) by presenting a problem. But they can also lose the same amount if they are not prepared.
Problems are assigned completely automatically after the criteria: Everybody in the course should present the same number of problems. The indicated preferences are taken into account. If several students have the same preferences, the one who submitted them the earliest wins. You do not have to submit any preferences at all. In this case, the system assumes that you do not care which problem you have to present.
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Global Constraint Catalog: KPartridge
<< 3.7.181. Path3.7.183. Pattern sequencing >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
Denotes that a constraint can be used for solving the Partridge problem: the Partridge problem consists of tiling a square of size
\frac{n·\left(n+1\right)}{2}
\frac{n·\left(n+1\right)}{2}
squares of respective size
1 square of size 1,
2 squares of size 2,
\cdots
n
squares of size
n
It was initially proposed by R. Wainwright and is based on the identity
1·{1}^{2}+2·{2}^{2}+\cdots +n·{n}^{2}={\left(\frac{n·\left(n+1\right)}{2}\right)}^{2}
. The problem is described in http://mathpuzzle.com/partridge.html. Part (A) of Figure 3.7.49 gives a solution for
n=12
found with
\mathrm{𝚐𝚎𝚘𝚜𝚝}
[AgrenCarlssonBeldiceanuSbihiTruchetZampelli09a], while Part (B) provides a solution for
n=13
found by S. Hougardy [Hougardy12].
Figure 3.7.49. (A) a solution to the Partridge problem for
n=12
, and (B) a solution for
n=13
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Expansion of Symbolic Polynomials
Several enhancements were made to zip.
Direct hardware-float callbacks are recognized for certain routines. These can bypass much of the overhead of ordinary interpreted function evaluation resulting in speed similar to that of compiled code. The following now takes 0.084 seconds in Maple 12 compared to 3.900 seconds on the same machine in Maple 11.
A := Matrix(1..N,1..N,datatype=float[8]):
B := Matrix(1..N,1..N,datatype=float[8]):
time(zip(`+`,A,B));
\textcolor[rgb]{0,0,1}{0.027}
Special case detection for some built-in non-hardware float operations. For example, the following now takes 0.104s compared to 3.532s before.
A := Matrix(1..N,1..N):
B := Matrix(1..N,1..N):
\textcolor[rgb]{0,0,1}{0.025}
Automatic procedure inlining is applied when possible. Most arrow-operator functions bypass normal function evaluation overhead resulting in better speed. The following now takes 1.104s compared to 4.248s before.
time(zip((x,y)->x+y+1,A,B));
\textcolor[rgb]{0,0,1}{0.115}
With the introduction of round bracket array indexing, certain block-copy operations can be done much more efficiently as no intermediate object needs to be created. For example the second timing below using round brackets for indexing into A and B is much faster than the first timing, which uses square brackets. The result of a square bracket index assignment is always the assigned value so an intermediate object needs to be computed. The result of a round bracket index assignment is always the whole array, so, if the right side is a large block subselection, the intermediate copy can be bypassed in most cases resulting in memory savings.
A := LinearAlgebra:-RandomMatrix(N,N):
B := Matrix(N,N):
B[1001..4000,1001..4000]:=A[1..3000,1..3000]:
time()-t;
\textcolor[rgb]{0,0,1}{0.407}
B(1001..4000,1001..4000):=A(1..3000,1..3000):
\textcolor[rgb]{0,0,1}{0.031}
Here is an example of a small dimension 1 system for which the solve command in Maple 11 was unable to find a solution. A solution could be found using FGb by calling the Groebner[Solve] command. Now in Maple 12, solve uses FGb and finds the 22 solutions in about 30 seconds.
symmetric5:={d^5-e^5,c^5-d^5,b^5-c^5,a^5-b^5,a^4*b+b^4*c+c^4*d+d^4*e+a*e^4}:
time(solve(symmetric5,{a,b,c,d,e}));
\textcolor[rgb]{0,0,1}{19.337}
In Maple 10, before the FGb library was used by the Groebner package, solving this system would have taken about twice as long on the same machine.
Expansion of large polynomials via the expand command has been improved. The following is a subset of a larger example. The original example used to take over 1400s to compute and now finishes in under 200s on the same machine. The example below has about a 3 times speed-up compared to Maple 11.
(-12*c^2*x^2*f+18*c^2*x^2*b+18*c*x^2*g*f+18*x^2*b*a^2+18*c^2*x^2*a-12*x^2*b^2*f-12*c^2*x^2*g+18*x^2*b^2*a+18*c*x^2*b^2-4+6*g-12*x*a*g-6*x^3*b*a^2-27*x^2*g*f-12*x*a*f+63*x*g*f-12*c*x*f+12*x*a*g^2-6*c*x^2*a*f-6*c*x^2*b*f-6*c*x^2*b*g+6*c*x*g*f-6*c*x^2*a*g-6*x^2*b*a*f+6*x*b*g*f-6*x^2*b*a*g+6*x*a*g*f-18*x*b*a*f-18*x*b*a*g-18*c*x*a*f-18*c*x*a*g-18*c*x*b*f+54*c*b*a*x-18*c*x*b*g+4*x^3*b^3+4*x^3*a^3+4*c^3*x^3+6*c^2*x^2+6*x^2*b^2+18*x^2*f-54*c*b*a*x^2+9*x+6*f+9*x*b*a+24*c*b*a*x^3+18*x^2*b*g*f+18*x^2*a*g*f+18*c*x^2*a^2-12*x^2*b^2*g-12*x^2*a^2*f-12*x^2*a^2*g)^6:
time(expand(ee));
\textcolor[rgb]{0,0,1}{0.192}
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CChart - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : ProcessControl Package : CChart
CChart(X, n, options, plotoptions)
(optional) equation(s) of the form option=value where option is one of color, confidencelevel, controllimits, or ubar; specify options for generating the C chart
The CChart command generates the control chart for the number of nonconformities (C chart) for the specified observations. The chart also contains the upper control limit (UCL), the lower control limit (LCL), and the average fraction of nonconforming items (represented by the center line) of the underlying quality characteristic. Unless explicitly given, the average fraction of nonconforming items and the control limits are computed based on the data.
color=list -- This option specifies colors of the various components of the C chart. The value of this option must be a list containing the color of the control limits, center line, data to be plotted, and the specification limits.
\mathrm{with}\left(\mathrm{ProcessControl}\right):
\mathrm{infolevel}[\mathrm{ProcessControl}]≔1:
A≔[12,8,6,9,10,12,11,16,10,6,20,15,9,8,6,8,10,7,5,8,5,8,10,6,9]
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]
\mathrm{CChart}\left(A,100\right)
Estimated Control Limits: [.18176485369759, 18.5382351463024]
\mathrm{CControlLimits}\left(A,100\right)
[\textcolor[rgb]{0,0,1}{0.181764854164603}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{18.5382351458354}]
l≔\mathrm{CControlLimits}\left(A,100,\mathrm{confidencelevel}=0.90\right)
\textcolor[rgb]{0,0,1}{l}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{4.32771555575638}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{14.3922844442436}]
\mathrm{CChart}\left(A,100,\mathrm{controllimits}=l\right)
ProcessControl[CControlLimits]
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Do objects gain protons to become positive? | Brilliant Math & Science Wiki
Sravanth C., Gian Ralph, Jimin Khim, and
Positively charged objects necessarily gained protons.
Why some people say it's true: We know that protons are positively charged particles, and therefore a positively charged object must have gained electrons.
Why some people say it's false: Protons are largely stuck in the nucleus and it is difficult for them to leave.
\color{#D61F06}{\textbf{false}}
The answer to our question lies in the basics of the atomic structure. Atoms are made up of neutrons, protons, and electrons. Neutrons and protons are called nucleons because they make up the tightly bound nucleus of atoms. Electrons exist in clouds around the nucleus called orbitals. For our purposes, we can think of an atom using the cartoon model below.
Simple model of an atom. The electrons can jump shells by releasing/gaining energy.
If an atom develops a positive charge, how could it have happened? It couldn't possibly have taken on additional protons because protons and neutrons are bound by the strong nuclear force, which keeps the nucleus stable.
Instead, we look to the loosely bound electrons. Electrons are not as tightly bound as nucleons and can be kicked out of the atom given sufficient energy. The ionization energy is the threshold energy required to kick out the most loosely bound electron in an atom, e.g. in the case of hydrogen, the energy required to make the
\ce{H -> H^+}
transition.
For an atom in the ground state, the number of protons is equal to the number of electrons so that the net charge is zero. But when an atom loses an electron, the number of protons becomes one greater than the number of electrons, so that the net charge is
+1\, q_e.
Thus the net charge of the ion will be positive.
_\square
Note: It is possible for atoms to gain protons under certain conditions, like in a particle accelerator.
Atoms, Molecules, Elements, and Compounds
Cite as: Do objects gain protons to become positive?. Brilliant.org. Retrieved from https://brilliant.org/wiki/do-objects-gain-protons-to-become-positive/
|
Range and angle calculation - MATLAB rangeangle - MathWorks Benelux
The rangeangle function returns the path distance and path angles in either the global or local coordinate systems. By default, the rangeangle function determines the angle a signal path makes with respect to global coordinates. If you add the refaxes argument, you can compute the angles with respect to local coordinates. As an illustration, this figure shows a 5-by-5 uniform rectangular array (URA) rotated from the global coordinates (xyz) using refaxes. The x' axis of the local coordinate system (x'y'z') is aligned with the main axis of the array and moves as the array moves. The path length is independent of orientation. The global coordinate system defines the azimuth and elevations angles (Φ,θ) and the local coordinate system defines the azimuth and elevations angles (Φ',θ').
The free-space signal propagation model states that a signal propagating from one point to another in a homogeneous, isotropic medium travels in a straight line, called the line-of-sight or direct path. The straight line is defined by the geometric vector from the radiation source to the destination.
The figure illustrates two propagation paths. From the source position, ss, and the receiver position, sr, you can compute the arrival angles of both paths, θ′los and θ′rp. The arrival angles are the elevation and azimuth angles of the arriving radiation with respect to a local coordinate system. In this case, the local coordinate system coincides with the global coordinate system. You can also compute the transmitting angles, θlos and θrp. In the global coordinates, the angle of reflection at the boundary is the same as the angles θrp and θ′rp. The reflection angle is important to know when you use angle-dependent reflection-loss data. You can determine the reflection angle by using the rangeangle (Phased Array System Toolbox) function and setting the reference axes to the global coordinate system. The total path length for the line-of-sight path is shown in the figure by Rlos which is equal to the geometric distance between source and receiver. The total path length for the reflected path is Rrp= R1 + R2. The quantity L is the ground range between source and receiver.
\begin{array}{l}\stackrel{\to }{R}={\stackrel{\to }{x}}_{s}-{\stackrel{\to }{x}}_{r}\\ {R}_{los}=|\stackrel{\to }{R}|=\sqrt{{\left({z}_{r}-{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{1}=\frac{{z}_{r}}{{z}_{r}+{z}_{z}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{2}=\frac{{z}_{s}}{{z}_{s}+{z}_{r}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{rp}={R}_{1}+{R}_{2}=\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ \mathrm{tan}{\theta }_{los}=\frac{\left({z}_{s}-{z}_{r}\right)}{L}\\ \mathrm{tan}{\theta }_{rp}=-\frac{\left({z}_{s}+{z}_{r}\right)}{L}\\ {{\theta }^{\prime }}_{los}=-{\theta }_{los}\\ {{\theta }^{\prime }}_{rp}={\theta }_{rp}\end{array}
|
Relations | Brilliant Math & Science Wiki
Agnishom Chattopadhyay, 展豪 張, Geoff Pilling, and
Relations are a structure on a set that pairs any two objects that satisfy certain properties. Examples of familiar relations in this context are 7 is greater than 5, Alice is married to Bob, and 3
\clubsuit
\clubsuit
. For each of these statements, the elements of a set are related by a statement.
A function is a special kind of relation and derives its meaning from the language of relations.
Symmetry, Reflexivity, and Transitivity
A (binary) relation
\Re
X
Y
X \times Y.
One way to think about this definition is to think of it as that the ordered pairs correspond to the edges in a graph which links the related things.
This graph could be pictured as a relation between the set
\left \{ A, B, C \right \}
\left \{ X, Y, Z \right \}
Every member of a set is related to every member of the other set. So, in roster form, our relation is
\Re = \left \{ (A, X), (A, Y), (A, Z), (B, X), (B, Y), (B, Z), (C, X), (C, Y), (C, Z) \right \}.
Notice that something like
(Z,A)
is not a part of the relation since we defined the relation to be from
\left \{ A, B, C \right \}
\left \{ X, Y, Z \right \}.
Also, there is no problem with an edge connecting itself, since it is always possible that an object is related to itself. Consider the following:
This definition is also very useful when describing entities in the Cartesian plane, i.e. relations on
\mathbb{R}.
Note: When we have a definition from a set to the same set, we say that the relation is on that set.
The circle above is an illustration of the relation
\Re = \left \{ (x,y) \in \mathbb{R}: x^2 + y^2 = 4^2 \right \}.
Finally, we add a logician/computer scientist's definition of a relation.
\Re
is a predicate with (at least) two arguments of arbitrary types
X
Y.
_\square
While the set-theoretic definition and the logician's definition differ at an abstract level, they practically mean the same.
The same relation which describes the circle could also be interpreted as the following Haskell predicate:
r :: (Num a, Eq a) => a -> a -> Bool
r x y = x^2 + y^2 == 4^2
The usage of
\sim
is more popular to denote a relation. Very often, we will say things like
x \sim y
x
y
, or according to the set-theoretic definition,
\left ( x, y \right )
\sim.
So far, we have only talked about relations between two objects. Such relations are called binary relations. As you can see, it is not difficult to extend the notion of a relation to an
-ary relation, where there are n objects involved. However, we will mostly restrict the discussion of relations to binary relations, unless otherwise specified.
An ideal gas follows the ternary relation
P V = nRT .
This could be pictured as the following subset of the
(P,V,T)
\left \{ \left ( P, V, T \right ) : P V = nRT \right \}.
Symmetry, reflexivity, and transitivity are some interesting properties that are possessed by relations defined on elements of the same set.
\sim
A
is said to be symmetric, if
\forall a,b \in A \quad a\sim b \implies b \sim a.
"is married to" is a symmetric relation. Alice is married to Bob implies Bob is married to Alice.
"is older than" isn't symmetric. If Alice is older than Bob, Bob isn't older than Alice.
In fact, this relation is anti-symmetric, meaning that
A \sim B \implies B \not \sim A .
\sim
A
\forall a \in A \quad a \sim a.
Equality is a reflexive relation. Everything is equal to itself.
Coprimeness is not reflexive. 1 is coprime to itself. But other integers are not.
\sim
A
\forall a,b,c \in A \quad \big( (a \sim b ) \wedge (b \sim c ) \big) \implies a \sim c .
Divisiblity is transitive. If
a \mid b
b \mid c
a \mid c
If friendship was transitive, you wouldn't have the concept of a mutual friend on Facebook.
Alice and Bob are friends. Also, Bob and Carol are friends. But Alice and Carol might not necessarily be friends.
Can you think of relations which are symmetric but not transitive, transitive but not symmetric, symmetric but not reflexive, reflexive and transitive but not symmetric, and so on?
\Re \subseteq S \times T
be a binary relation.
The inverse relation
\Re^{-1}
\Re^{-1} \subseteq T \times S: \{(t,s):(s,t) \in \Re \}.
\Re
be the relation "smaller than" on the real number set, i.e.
\Re = \{(s,t):s<t,\;\;s,t\in \mathbb R\}.
\Re^{-1}
is the relation "greater than".
s<t \Leftrightarrow t>s
>
is also called the dual ordering of
<
\Re
\Re^{-1}
\Re
This notation is consistent with the inverse function notation
f^{-1}
Equivalence relations are those relations which are reflexive, symmetric, and transitive at the same time.
Parallelness is an equivalence relation.
Symmetry: If
a \parallel b
b \parallel a.
Reflexivity: All lines are parallel to itself.
a \parallel b
b \parallel c
a \parallel c.
Other equivalence relations include the following:
We define the concept of an equivalence class as follows:
The equivalence class of an element
a \in A
\sim ,
[a] ,
[a] = \left \{ x \in A \mid x \sim a \right \}.
The two triangles on the left are congruent and in the same equivalence class. The other two triangles form two other distinct equivalence classes.
This is what makes the equivalence classes a useful idea:
Any equivalence
\sim
X
X
into equivalence classes, and conversely, corresponding to any partition of
X
, there is an equivalence relation
\sim
X.
By partition, we mean breaking up the set into disjoint subsets whose union is the set itself.
Consider the equivalence classes
[i]
formed by the relation
\sim
X.
Since by reflexivity
i \in [i]\ \forall i
, the union of all the equivalence classes are
S.
They are disjoint because if
x \in [a]
x \in [b],
x \sim a
by definition, and by symmetry,
a \sim x
x \sim b
by definition. But then
a \sim b
by transitivity, so
a\in [b]
y \in [a]
[b]
again by transitivity. This means that
[a]\subseteq [b]
. Swapping
and
b
, proceeding in the same way, we conclude
[a]=[b]
_\square
Conversely, given any partitioning scheme, we could define
\sim
a \sim b
and
belong to the same set.
Reflexivity: All elements belong to the set in which they belong. So,
a \sim a.
a \sim b \implies b \sim a
since what this means is that
and
b
are essentially in the same set.
(a \sim b) \wedge (b \sim c) \implies ( a \sim c )
, since once again
a, b, c
are all in the same set.
_\square
How many distinct equivalence relations are there on a set of 6 elements?
See order theory
Orderings are different from equivalences in that they are (mostly) antisymmetric.
A (non-strict) partial order is a relation
\preceq
S
which satisfies the following
\forall a, b, c \in S
a \preceq a
Antisymmetry:
(a \preceq b) \wedge (b \preceq a) \implies a = b
(a \preceq b) \wedge (b \preceq c) \implies (a \preceq c).
Here, we say non-strict, since we allow reflexivity.
A set equipped with a partial ordering is called a partially-ordered set or a poset.
Divisibility of positive integers forms posets. This is a Hasse diagram, a representation for posets. If you choose any walk on this graph, each number is divisible by the number before it.
Notice that not all pairs of elements are related, though. This is why we say the ordering is partial.
A different example would be the vertices of a directed acyclic graph ordered by reachability.
To illustrate,
7 \preceq 11 \preceq 10
11 \preceq 9.
We define the notion of totality of a relation as follows:
\sim
X
is total if
\forall a, b \in X
a \sim b
b \sim a.
A total order is simply a partial order which also satisfies totality.
The set of words in a dictionary equipped with a lexicographic ordering
\mathbb{N}
\leq
People standing in a queue, equipped with their position in the queue
How many total orderings of a set consisting of
n
elements are there?
\clubsuit
\clubsuit
See Function Terminology
A special case of relations are functions.
from
X
Y
f : X \to Y,
is a relation between
X
Y
such that for every element
x \in X
(called pre-image), there exists exactly one
y \in Y
(called image) which is related to
x
X
Y
are called the domain and co-domain of
f,
When we say
f(x) = y
, we mean
(x,y) \in f
. A metaphorical description of a function could be a black-box or a machine that takes in an input and returns a corresponding output.
A function mapping objects to their colors
A depiction of a function on a Cartesian plane
Not all relations are functions.
For example, the relation
\Re = \left \{ (x,y) \in \mathbb{R}: x^2 + y^2 = 4^2 \right \}
is not a function since there are multiple values of
y
possible for each
x
The definition of a function requires us to have exactly one image for each pre-image.
This is why the inverse of a function is not necessarily a function. For example, the inverse of
x \mapsto x^2
When a student is first introduced to the notion of a function, he tends to believe that a function is an algorithm being run upon the image. However, this is not so. There is no constraint of computability imposed on the notion of a function.
They are just arbitrary mappings. It is perfectly possible that a function is non-computable. Popular examples include a function which could tell if a program halts (halting problem) or the minimum length of a program that outputs a given string (Kolmogorov complexity).
Cite as: Relations. Brilliant.org. Retrieved from https://brilliant.org/wiki/relations/
|
167 (one hundred [and] sixty-seven) is the natural number following 166 and preceding 168.
(one hundred sixty-seventh)
167 is a Chen prime, a Gaussian prime, a safe prime,[1] and an Eisenstein prime with no imaginary part and a real part of the form
{\displaystyle 3n-1}
167 is the smallest prime which can not be expressed as a sum of seven or fewer cubes. It is also the smallest number which requires six terms when expressed using the greedy algorithm as a sum of squares, 167 = 144 + 16 + 4 + 1 + 1 + 1,[2] although by Lagrange's four-square theorem its non-greedy expression as a sum of squares can be shorter, e.g. 167 = 121 + 36 + 9 + 1.
167 is a full reptend prime in base 10, since the decimal expansion of 1/167 repeats the following 166 digits: 0.00598802395209580838323353293413173652694610778443113772455089820359281437125748502994 0119760479041916167664670658682634730538922155688622754491017964071856287425149700...
167 is a highly cototient number, as it is the smallest number k with exactly 15 solutions to the equation x - φ(x) = k. It is also a strictly non-palindromic number.
167 is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits, i. e., 1×6×7 = 3×(1+6+7)
167 is the smallest positive integer d such that the imaginary quadratic field Q(√–d) has class number = 11.[3]
167 Urda is a main belt asteroid
167P/CINEOS is a periodic comet in the Solar System
IC 167 is interacting galaxies
Marine Light Attack Helicopter Squadron 167 is a United States Marine Corps helicopter squadron
Martin Model 167 was a U.S.-designed light bomber during World War II
USCGC Acushnet (WMEC-167) was a U.S. Navy Diver-class rescue and salvage ship during World War II
USS Acree (DE-167) was a U.S. Navy Cannon-class destroyer escort during World War II
USS Caledonia (AK-167) was a U.S. Navy Alamosa-class cargo ship during World War II
USS Cowell (DD-167) was a U.S. Navy Wickes-class destroyer during World War I
USS Freestone (APA-167) was a U.S. Navy Haskell-class attack transport during World War II
USS Leyden (IX-167) was a transport ship during World War II
USS Narwhal (SS-167) was a U.S. Navy Narwhal-class submarine during World War II
Martina Navratilova has 167 tennis titles, an all-time record for men or women
SMRT Bus Service 167 in Singapore
167th Street is an elevated local station in the Bronx on the IRT Jerome Avenue Line, 4 train, of the New York City Subway.
167th Street is an underground local station in the Bronx on the IND Concourse Line, B and D trains, of the New York City Subway.
The Universal Disk Format (or ECMA-167) format of a file system for optical media storage
C167 family is a 16-bit microcontroller architecture from Infineon
Pips are dots on the face of a die, denoting its value. The pip count at the start of a backgammon game is 167[importance?]
M167 (disambiguation)
^ "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
^ Sloane, N. J. A. (ed.). "Sequence A006892 (Representation as a sum of squares requires n squares with greedy algorithm)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ "Tables of imaginary quadratic fields with small class number". numbertheory.org.
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Global Constraint Catalog: Ktwo_dimensional_orthogonal_packing
<< 3.7.263. Tuple3.7.265. Unary constraint >>
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
A constraint that can be used to model the two-dimensional orthogonal packing problem. Given a set of rectangles pack them into a rectangular placement space. Borders of the rectangles should be parallel to the borders of the placement space and rectangles should not overlap. Some variants of strip packing allow to rotate rectangles from 90 degrees. Benchmarks can be obtained from a generator described in the following paper [ClautiauxCarlierMoukrim06].
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Member journals (current)
Save the Date : open access day on 28 June
For its 5th anniversary, the Mersenne centre is organising a day on new open access publishing models on 28 June 2022 at the Institut Henri Poincaré (IHP) in Paris.
Registration to the day is free, upon registration
Welcome Cynthia & Romain
Centre Mersenne welcomes Cynthia, software developer, and Romain, translator, to work on a semi-automatic translation project for scientific articles with the Académie des Sciences.
Mersenne centre hires a translator
The Mersenne centre hires a translator.
Mission: to carry out the translation of a corpus of scientific articles from English into French (and, to a lesser extent, from French into English) using a customised CAT (Computer Assisted Translation) system, and to participate in the adaptation of this system by making recommendations and evaluating various development proposals. More
The Centre Mersenne
An open access publishing platform for scientific publications.
The Centre Mersenne for Open Scientific Publishing aims at supporting and fostering open access scientific publishing. It offers tools and services for scholars and editorial teams of open access journals formatted with LaTeX.
The Comptes Rendus. are open access and peer-reviewed electronic scientific journals to enable researchers to publish original studies in French and in English. more...
Publishes original articles of a high level in all fields of mathematics. more...
Publishes original research articles in mathematics. more...
On the monoidal invariance of the cohomological dimension of Hopf algebras
Issue 360 (2022) no. G5 p. 561-582
Orthogonalization of Positive Operator Valued Measures
On the Thom–Sebastiani Property of Quasi-Homogeneous Isolated Hypersurface Singularities
Epure, Raul
Nonlinear Helmholtz equations with sign-changing diffusion coefficient
Mandel, Rainer; Moitier, Zoïs; Verfürth, Barbara
The subword complexity of polynomial subsequences of the Thue–Morse sequence
On the Largest intersecting set in
{\mathrm{GL}}_{2}\left(q\right)
and some of its subgroups
Ahanjideh, Milad
|
Global Constraint Catalog: Ccycle_card_on_path
<< 5.103. cycle5.105. cycle_or_accessibility >>
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}\left(\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴},\mathrm{𝙽𝙾𝙳𝙴𝚂},\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃},\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃},\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝚍𝚟𝚊𝚛}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}\ge 1
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}\le |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙽𝙾𝙳𝙴𝚂},\left[\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚜𝚞𝚌𝚌},\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\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}|
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}\ge 0
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}\le \mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}\ge \mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}
\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}\ge 0
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
Consider a digraph
G
described by the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
is the number of circuits for covering
G
in such a way that each vertex belongs to a single circuit. In addition the following constraint must also hold: on each set of
\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}
consecutive distinct vertices of each final circuit, the number of vertices for which the attribute colour takes his value in the collection of values
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
should be located within the range
\left[\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃},\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}\right]
\left(\begin{array}{c}2,〈\begin{array}{ccc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-7\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-8\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-9\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-5\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-2,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-6\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-7\hfill & \mathrm{𝚜𝚞𝚌𝚌}-5\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-8\hfill & \mathrm{𝚜𝚞𝚌𝚌}-6\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-9\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1\hfill \end{array}〉,1,2,3,\hfill \\ 〈1〉\hfill \end{array}\right)
The constraint
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}
holds since the vertices of the
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection correspond to a set of disjoint circuits and since, for each set of 3 (i.e.,
\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}=3
) consecutive vertices, colour 1 (i.e., the value provided by the
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
collection) occurs at least once (i.e.,
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}=1
) and at most twice (i.e.,
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}=2
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>2
\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}<|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}<\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}>0
\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}>1
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}>0\vee \mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}<\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
that belongs to
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
(resp. does not belong to
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
) can be replaced by any other value in
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
(resp. not in
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃}
\ge 0
\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
Assume that the vertices of
G
are partitioned into the following two categories:
Clients to visit.
Depots where one can reload a vehicle.
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}
constraint we can express a constraint like: after visiting three consecutive clients we should visit a depot. This is typically not possible with the
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
constraint since we do not know in advance the set of variables involved in the
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
This constraint is a special case of the
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}
parameter of the
\mathrm{𝚌𝚢𝚌𝚕𝚎}
constraint of CHIP [Bourreau99].
\mathrm{𝚌𝚢𝚌𝚕𝚎}
(graph partitioning constraint).
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
characteristic of a constraint: coloured.
combinatorial object: sequence.
constraint type: graph constraint, graph partitioning constraint, sliding sequence constraint.
final graph structure: connected component, one_succ.
\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{𝐍𝐓𝐑𝐄𝐄}
=0
•
\mathrm{𝐍𝐂𝐂}
=\mathrm{𝙽𝙲𝚈𝙲𝙻𝙴}
\mathrm{𝙾𝙽𝙴}_\mathrm{𝚂𝚄𝙲𝙲}
\begin{array}{c}\mathrm{𝖯𝖠𝖳𝖧}_\mathrm{𝖫𝖤𝖭𝖦𝖳𝖧}\left(\mathrm{𝙿𝙰𝚃𝙷}_\mathrm{𝙻𝙴𝙽}\right)↦\hfill \\ \left[\begin{array}{c}\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{𝙽𝙾𝙳𝙴𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)\right]\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\left(\mathrm{𝙰𝚃𝙻𝙴𝙰𝚂𝚃},\mathrm{𝙰𝚃𝙼𝙾𝚂𝚃},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝐍𝐂𝐂}
graph property, we show the two connected components of the final graph. The constraint
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}
holds since all the vertices belong to a circuit (i.e.,
\mathrm{𝐍𝐓𝐑𝐄𝐄}
=
0) and since for each set of three consecutive vertices, colour 1 occurs at least once and at most twice (i.e., the
\mathrm{𝚊𝚖𝚘𝚗𝚐}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
constraint holds).
\mathrm{𝚌𝚢𝚌𝚕𝚎}_\mathrm{𝚌𝚊𝚛𝚍}_\mathrm{𝚘𝚗}_\mathrm{𝚙𝚊𝚝𝚑}
|
Global Constraint Catalog: Ccoloured_cumulative
<< 5.72. colored_matrix5.74. coloured_cumulatives >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
\mathrm{𝚌𝚘𝚕𝚘𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\begin{array}{c}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚎𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛},\hfill \\ \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\hfill \end{array}\right)
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎}_\mathrm{𝚊𝚝}_\mathrm{𝚕𝚎𝚊𝚜𝚝}
\left(2,\mathrm{𝚃𝙰𝚂𝙺𝚂},\left[\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗},\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗},\mathrm{𝚎𝚗𝚍}\right]\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\ge 0
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le \mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚎𝚗𝚍}
\mathrm{𝙻𝙸𝙼𝙸𝚃}\ge 0
𝒯
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint forces that, at each point in time, the number of distinct colours of the set of tasks that overlap that point, does not exceed a given limit. A task overlaps a point
if and only if (1) its origin is less than or equal to
i
i
. For each task of
𝒯
it also imposes the constraint
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}+\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}=\mathrm{𝚎𝚗𝚍}
\left(\begin{array}{c}〈\begin{array}{cccc}\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-1\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-2\hfill & \mathrm{𝚎𝚗𝚍}-3\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-1,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-2\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-9\hfill & \mathrm{𝚎𝚗𝚍}-11\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-2,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-3\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-10\hfill & \mathrm{𝚎𝚗𝚍}-13\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-3,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-6\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-6\hfill & \mathrm{𝚎𝚗𝚍}-12\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-2,\hfill \\ \mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-7\hfill & \mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-2\hfill & \mathrm{𝚎𝚗𝚍}-9\hfill & \mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}-3\hfill \end{array}〉,2\hfill \end{array}\right)
Figure 5.73.1. The coloured cumulative solution to the Example slot with at most two distinct colours in parallel
Figure 5.73.1 shows the solution associated with the example. Each rectangle of the figure corresponds to a task of the
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint. Tasks that have their colour attribute set to 1, 2 and 3 are respectively coloured in yellow, blue and pink. The
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint holds since at each point in time we do not have more than
\mathrm{𝙻𝙸𝙼𝙸𝚃}=2
distinct colours.
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚎𝚗𝚍}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)>1
\mathrm{𝙻𝙸𝙼𝙸𝚃}<
\mathrm{𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
\mathrm{𝚎𝚗𝚍}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
Useful for scheduling problems where a machine can only proceed in parallel a maximum number of tasks of distinct type. This condition cannot be modelled by the classical
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint. Also useful for coloured bin packing problems (i.e.,
\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}=1
) where each item has a colour and no bin contains items with more than
\mathrm{𝙻𝙸𝙼𝙸𝚃}
distinct colours [DawandeKalagnanamSethuraman98], [GarganiRefalo07], [HeinzSchlechteStephanWinkler12].
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
For each pair of tasks
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right],\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right]
\left(i,j\in \left[1,|\mathrm{𝚃𝙰𝚂𝙺𝚂}|\right]\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection we create a variable
{C}_{ij}
which is set to the colour of task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right]
if task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right]
overlaps the origin attribute of task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
, and to the colour of task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
i=j
{C}_{ij}=\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
i\ne j
{C}_{ij}=\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\vee {C}_{ij}=\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}
\left(\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le \mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\wedge
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚎𝚗𝚍}>\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)\wedge \left({C}_{ij}=\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)\right)\vee
\left(\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}>\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\vee
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[j\right].\mathrm{𝚎𝚗𝚍}\le \mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\right)\wedge \left({C}_{ij}=\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right].\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛}\right)\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
\left(i\in \left[1,|\mathrm{𝚃𝙰𝚂𝙺𝚂}|\right]\right)
we create a variable
{N}_{i}
which gives the number of distinct colours associated with the tasks that overlap the origin of task
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
\mathrm{𝚃𝙰𝚂𝙺𝚂}\left[i\right]
overlaps its own origin) and we impose
{N}_{i}
to not exceed the maximum number of distinct colours
\mathrm{𝙻𝙸𝙼𝙸𝚃}
allowed at each instant:
{N}_{i}\ge 1\wedge {N}_{i}\le \mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left({N}_{i},〈{C}_{i1},{C}_{i2},\cdots ,{C}_{i|\mathrm{𝚃𝙰𝚂𝙺𝚂}|}〉\right)
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚝𝚛𝚊𝚌𝚔}
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚝𝚊𝚜𝚔𝚜}
(a colour is assigned to each collection of tasks of constraint
\mathrm{𝚍𝚒𝚜𝚓𝚘𝚒𝚗𝚝}_\mathrm{𝚝𝚊𝚜𝚔𝚜}
and a limit of one single colour is enforced).
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
modelling: number of distinct values, zero-duration task.
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\right)
\mathrm{𝚝𝚊𝚜𝚔𝚜}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}+\mathrm{𝚝𝚊𝚜𝚔𝚜}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}=\mathrm{𝚝𝚊𝚜𝚔𝚜}.\mathrm{𝚎𝚗𝚍}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1},\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}\right)
•\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}>0
•\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}\le \mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
•\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{1}.\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}<\mathrm{𝚝𝚊𝚜𝚔𝚜}\mathtt{2}.\mathrm{𝚎𝚗𝚍}
•
\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}
•
\mathrm{𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴}
•
\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}
\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{𝚗𝚟𝚊𝚕𝚞𝚎𝚜}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},\le ,\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
, except that we use another constraint for computing the resource consumption at each time point.
Parts (A) and (B) of Figure 5.73.2 respectively show the initial and final graph associated with the second graph constraint of the Example slot. On the one hand, each source vertex of the final graph can be interpreted as a time point. On the other hand the successors of a source vertex correspond to those tasks that overlap that time point. The
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint holds since for each successor set
𝒮
of the final graph the number of distinct colours of the tasks in
𝒮
does not exceed the
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝐍𝐀𝐑𝐂}
=
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}
\ge
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
|
There are three new visualizations in Statistics:
VennDiagrams are a method of data visualization showing the relationships between multiple sets of data by depicting these sets as regions inside closed curves.
FurryAnimals := {"Bat", "Cat", "Caterpillar", "Dog", "Gerbil"}:
Pets := {"Cat", "Dog", "Gerbil", "Goldfish", "Lizard", "Parrot", "Snake"}:
FlyingAnimals := {"Bat", "Butterfly", "Eagle", "Parrot", "Vulture", "Wasp"}:
VennDiagram(FurryAnimals, Pets, FlyingAnimals, legend=["Furry Animals", "Pets", "Flying Animals"]);
ViolinPlots are a visualization of the distribution of data consisting of a rotated kernel density plot and markers for the quartiles and the mean.
C := [seq(Sample(Normal(ln(i), 3), 60), i = 1 .. 20)]:
F := [seq(Sample(Normal(sin(i*Pi), 3), 120), i = 1 .. 20)]:
ViolinPlot( C[1..3], F[1..3], size = [800,400], color = "LightBlue" .. "red", scale = area('pairwise', [1,1,2], [2,3,1]));
Weibull plots are used to verify whether a particular data set follows the Weibull distribution and provides additional information about the estimated shape and scale parameters.
X := RandomVariable(Weibull(1, 0.6)):
A := Sample(X, 100):
WeibullPlot(A, scale = 1, shape = 0.6, color = [blue,magenta]);
More updates to Statistics Visualizations
Several existing visualizations have also been updated or have new optional arguments.
BarCharts and ColumnGraphs
The BarChart / ColumnGraph routines have been updated to support the colorscheme option:
BarChart( < 1, 3, 5, 2, 4, 6 >, colorscheme = [ "valuesplit", [ 1..2 = "DarkGrey", 3..4 = "WhiteSmoke", 5..6 = "Crimson" ] ] );
BarCharts and ColumnGraphs also now support individual color specification for each bar or column.
ColumnGraph(< 6,3,9; 3,5,9>, color = Matrix( [ [ "Red", "OrangeRed", "Orange"], ["Blue", "Navy", "Purple" ] ] ) );
DensityPlots
The DensityPlot command has a new discont option for the detection of discontinuities.
f:=x -> piecewise(-1 < x and x < 1,3/2 *x^2,0);
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{↦}{\begin{array}{cc}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
F:=Distribution(PDF=f):
Z:=RandomVariable(F):
DensityPlot(Z,range=-2..2,thickness=2,color=red,discont=true);
There are several new commands for working with DataFrames and DataSeries: Remove and SubsDatatype. Several existing commands and packages have also been updated to support DataFrames and DataSeries, including sort, Describe, and the CurveFitting package.
The Remove command makes it easier to remove one or more columns from a DataFrame. For example, many commands in the Statistics package assume that the supplied data is strictly numeric. In the case that there are one or more columns that contain non-numeric data, this command makes it possible to "remove" those columns and produce a result.
IrisData := Import( "datasets/iris.csv", base=datadir );
\textcolor[rgb]{0,0,1}{\mathrm{IrisData}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Sepal Length}}& \textcolor[rgb]{0,0,1}{\mathrm{Sepal Width}}& \textcolor[rgb]{0,0,1}{\mathrm{Petal Length}}& \textcolor[rgb]{0,0,1}{\mathrm{Petal Width}}& \textcolor[rgb]{0,0,1}{\mathrm{Species}}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5.1}& \textcolor[rgb]{0,0,1}{3.5}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4.9}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4.7}& \textcolor[rgb]{0,0,1}{3.2}& \textcolor[rgb]{0,0,1}{1.3}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{4.6}& \textcolor[rgb]{0,0,1}{3.1}& \textcolor[rgb]{0,0,1}{1.5}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{3.6}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5.4}& \textcolor[rgb]{0,0,1}{3.9}& \textcolor[rgb]{0,0,1}{1.7}& \textcolor[rgb]{0,0,1}{0.4}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{4.6}& \textcolor[rgb]{0,0,1}{3.4}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.3}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{3.4}& \textcolor[rgb]{0,0,1}{1.5}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{"setosa"}\\ \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}\end{array}]
Note that the fifth column, "Species", contains strings. Attempting to plot this DataFrame as is would result in an error. However, if the "Species" column is removed, a result is returned.
BoxPlot( :-Remove( IrisData, Species ) );
Note that the Remove command does not act in-place. In order to permanently remove a column, re-assignment is necessary.
SubsDataType
The SubsDatatype command changes the specified datatype of a DataSeries. It will also attempt to coerce any data in the DataSeries into the given datatype.
ds := DataSeries( < 1, 2, 3 >, datatype = integer );
\textcolor[rgb]{0,0,1}{\mathrm{ds}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\end{array}]
Datatype( ds );
\textcolor[rgb]{0,0,1}{\mathrm{integer}}
ds := SubsDatatype( ds, float );
\textcolor[rgb]{0,0,1}{\mathrm{ds}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{1.}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2.}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3.}\end{array}]
{\textcolor[rgb]{0,0,1}{\mathrm{float}}}_{\textcolor[rgb]{0,0,1}{8}}
SubsDatatype( IrisData, Species, name, conversion = ( (x) -> convert(x,name) ) );
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Sepal Length}}& \textcolor[rgb]{0,0,1}{\mathrm{Sepal Width}}& \textcolor[rgb]{0,0,1}{\mathrm{Petal Length}}& \textcolor[rgb]{0,0,1}{\mathrm{Petal Width}}& \textcolor[rgb]{0,0,1}{\mathrm{Species}}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5.1}& \textcolor[rgb]{0,0,1}{3.5}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4.9}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4.7}& \textcolor[rgb]{0,0,1}{3.2}& \textcolor[rgb]{0,0,1}{1.3}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{4.6}& \textcolor[rgb]{0,0,1}{3.1}& \textcolor[rgb]{0,0,1}{1.5}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{3.6}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{5.4}& \textcolor[rgb]{0,0,1}{3.9}& \textcolor[rgb]{0,0,1}{1.7}& \textcolor[rgb]{0,0,1}{0.4}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{4.6}& \textcolor[rgb]{0,0,1}{3.4}& \textcolor[rgb]{0,0,1}{1.4}& \textcolor[rgb]{0,0,1}{0.3}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{3.4}& \textcolor[rgb]{0,0,1}{1.5}& \textcolor[rgb]{0,0,1}{0.2}& \textcolor[rgb]{0,0,1}{\mathrm{setosa}}\\ \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}& \textcolor[rgb]{0,0,1}{\mathrm{...}}\end{array}]
More Updates for DataFrames and DataSeries
There have also been more updates to expand the number of commands that support DataFrame and DataSeries objects. The CurveFitting package now supports each and is also available from the right-click context menu.
The sort command can also be used with DataFrames.
berries := DataFrame( < < 220, 288, 136 > | < 11.94, 18.1, 7.68 > |
< Russia, China, USA > | < "Rubus", "Vitis", "Fragaria" > >,
columns = [ Energy, Carbohydrates, `Top Producer`, Genus ],
rows = [ Raspberry, Grape, Strawberry ] );
\textcolor[rgb]{0,0,1}{\mathrm{berries}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220}& \textcolor[rgb]{0,0,1}{11.94}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288}& \textcolor[rgb]{0,0,1}{18.1}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136}& \textcolor[rgb]{0,0,1}{7.68}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\end{array}]
Here the DataFrame is sorted in order of ascending energy level:
sort( berries, Energy );
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136}& \textcolor[rgb]{0,0,1}{7.68}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220}& \textcolor[rgb]{0,0,1}{11.94}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288}& \textcolor[rgb]{0,0,1}{18.1}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\end{array}]
sort( berries, Genus );
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{}& \textcolor[rgb]{0,0,1}{\mathrm{Energy}}& \textcolor[rgb]{0,0,1}{\mathrm{Carbohydrates}}& \textcolor[rgb]{0,0,1}{\mathrm{Top Producer}}& \textcolor[rgb]{0,0,1}{\mathrm{Genus}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Strawberry}}& \textcolor[rgb]{0,0,1}{136}& \textcolor[rgb]{0,0,1}{7.68}& \textcolor[rgb]{0,0,1}{\mathrm{USA}}& \textcolor[rgb]{0,0,1}{"Fragaria"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Raspberry}}& \textcolor[rgb]{0,0,1}{220}& \textcolor[rgb]{0,0,1}{11.94}& \textcolor[rgb]{0,0,1}{\mathrm{Russia}}& \textcolor[rgb]{0,0,1}{"Rubus"}\\ \textcolor[rgb]{0,0,1}{\mathrm{Grape}}& \textcolor[rgb]{0,0,1}{288}& \textcolor[rgb]{0,0,1}{18.1}& \textcolor[rgb]{0,0,1}{\mathrm{China}}& \textcolor[rgb]{0,0,1}{"Vitis"}\end{array}]
The Describe command returns a printed description for procedures, modules and objects. In Maple 2017, the Describe command has been extended to provide a description for DataFrames and DataSeries that includes the number of observations ( rows ), the number of variables ( columns ), as well as the type of each column (if specified). In addition, for numeric columns, the minimum and maximum values are displayed, and for truefalse, string or name columns, the distinct levels are given.
Describe( berries );
berries :: DataFrame: 3 observations for 4 variables
Energy: Type: anything Min: 136.00 Max: 288.00
Carbohydrates: Type: anything Min: 7.68 Max: 18.10
Top Producer: Type: anything Tally: [Russia = 1, China = 1, USA = 1]
Genus: Type: anything Tally: ["Vitis" = 1, "Fragaria" = 1, "Rubus" = 1]
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Global Constraint Catalog: Kscheduling_with_machine_choice_calendars_and_preemption
<< 3.7.224. Scheduling constraint3.7.226. Shared table >>
\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
modelling: scheduling with machine choice, calendars and preemption A set of constraints that can be used for modelling a scheduling problem where:
We have tasks that have both to be assigned to machine and time.
Each task has a fixed duration.
Machines can run at most one task at a given instant.
Each machine has its own fixed unavailability periods (i.e., a calendar of unavailability periods).
An unavailability period that allows (respectively forbids) a task to be interrupted and resumed just after is called crossable (respectively non-crossable). A task that can be (respectively cannot be) interrupted by a crossable unavailability period is called resumable (respectively non-resumable).
We have a precedence constraint between specific pairs of tasks. Each precedence forces that a given task ends before the start of another given task.
This model illustrates the use of two time coordinates systems:
The first coordinate system, so called the virtual coordinate system, does not consider at all the crossable unavailability periods associated with the different machines. Since resumable tasks can be preempted by machine crossable unavailability, all resource scheduling constraints (i.e.,
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
) are expressed within this first coordinate system. This stands from the fact that resource scheduling constraints like
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚐𝚎𝚘𝚜𝚝}
do not support preemption.
The second coordinate system, so called the real coordinate system, considers all timepoints whether they correspond or not to crossable unavailability periods. All temporal constraints (i.e., precedence constraints represented by
\mathrm{𝚕𝚎𝚚}
constraints in this model) are expressed with respect to this second coordinate system.
Consequently, each task has a start and an end that are expressed within the virtual coordinate system as well as within the real coordinate system.
Each task, whether it is resumable or not, is passed to the resource scheduling constraints as well as to the precedence constraints. In addition, we represent each non-crossable unavailability period as a fixed task that is also passed to the resource scheduling constraints.
\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}
constraint ensures the link between variables (i.e., the start and the end of the tasks no matter whether they are resumable or not) expressed in these two coordinate systems with respect to the crossable unavailability periods.
We now provide the corresponding detailed model. Given:
A set of machines
ℳ=\left\{{m}_{1},{m}_{2},\cdots ,{m}_{p}\right\}
, where each machine has a list of fixed unavailability periods. An unavailability
{u}_{i}
is defined by the following attributes:
The crossable flag
{c}_{i}
tells whether unavailability
{u}_{i}
is crossable (
{c}_{i}=1
{c}_{i}=0
The machine
{r}_{i}
indicates the machine (i.e., a value in
\left[1,p\right]
) to which unavailability
{u}_{i}
corresponds (i.e., since different machines may have different unavailability periods).
The start
{s}_{i}
of the unavailability
{u}_{i}
which indicates the first unavailable timepoint of the unavailability.
{e}_{i}
{u}_{i}
which gives the last unavailable timepoint of the unavailability.
A set of tasks
𝒯=\left\{{t}_{1},{t}_{2},\cdots ,{t}_{n}\right\}
, where each task
{t}_{i}
i\in \left[1,n\right]
) has the following attributes which are all domain variables except the resumable flag and the virtual duration:
The resumable flag
{r}_{i}
tells whether task
{t}_{i}
is resumable (
{r}_{i}=1
{r}_{i}=0
{m}_{i}
\left[1,p\right]
) to which task
{t}_{i}
will be assigned.
The virtual start
{\mathrm{𝑣𝑠}}_{i}
gives the start of task
{t}_{i}
in the virtual coordinate system.
The virtual duration
{\mathrm{𝑣𝑑}}_{i}
corresponds to the duration of task
{t}_{i}
without counting the eventual unavailability periods crossed by task
{t}_{i}
The virtual end
{\mathrm{𝑣𝑒}}_{i}
provides the end of task
{t}_{i}
in the virtual coordinate system. We have that
{\mathrm{𝑣𝑠}}_{i}+{\mathrm{𝑣𝑑}}_{i}={\mathrm{𝑣𝑒}}_{i}
The real start
{\mathrm{𝑟𝑠}}_{i}
{t}_{i}
in the real coordinate system.
The real duration
{\mathrm{𝑟𝑑}}_{i}
{t}_{i}
including the eventual unavailability periods crossed by task
{t}_{i}
. When task
{t}_{i}
is non-resumable (i.e.,
{r}_{i}=0
) its real duration is equal to its virtual duration (i.e.,
{\mathrm{𝑟𝑑}}_{i}={\mathrm{𝑣𝑑}}_{i}
The real end
{\mathrm{𝑟𝑒}}_{i}
indicates the end of task
{t}_{i}
in the real coordinate system. We have that
{\mathrm{𝑟𝑠}}_{i}+{\mathrm{𝑟𝑑}}_{i}={\mathrm{𝑟𝑒}}_{i}
The link between the virtual starts (respectively virtual ends) and the real starts (respectively real ends) of the different tasks of
𝒯
is ensured by a
\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}
\left(\mathrm{𝙸𝙽𝚂𝚃𝙰𝙽𝚃𝚂},\mathrm{𝙼𝙰𝙲𝙷𝙸𝙽𝙴𝚂}\right)
constraint. More precisely, for each task
{t}_{i}
i\in \left[1,n\right]
), no matter whether it is resumable or not, we create the following items for the collection
\mathrm{𝙸𝙽𝚂𝚃𝙰𝙽𝚃𝚂}
\begin{array}{c}〈\begin{array}{cccc}\mathrm{𝚖𝚊𝚌𝚑𝚒𝚗𝚎}-{m}_{i}\hfill & \mathrm{𝚟𝚒𝚛𝚝𝚞𝚊𝚕}-{\mathrm{𝑣𝑠}}_{i}\hfill & \mathrm{𝚒𝚛𝚎𝚊𝚕}-{\mathrm{𝑟𝑠}}_{i}\hfill & \mathrm{𝚏𝚕𝚊𝚐𝚎𝚗𝚍}-0\hfill \end{array}〉,\hfill \\ 〈\begin{array}{cccc}\mathrm{𝚖𝚊𝚌𝚑𝚒𝚗𝚎}-{m}_{i}\hfill & \mathrm{𝚟𝚒𝚛𝚝𝚞𝚊𝚕}-{\mathrm{𝑣𝑒}}_{i}\hfill & \mathrm{𝚒𝚛𝚎𝚊𝚕}-{\mathrm{𝑟𝑒}}_{i}\hfill & \mathrm{𝚏𝚕𝚊𝚐𝚎𝚗𝚍}-1\hfill \end{array}〉.\hfill \end{array}
The first item links the virtual and the real start of task
{t}_{i}
, while the second item relates the virtual and real ends. For each machine
{m}_{i}
i\in \left[1,p\right]
) and its corresponding list of crossable unavailability periods, denoted
\mathrm{𝑐𝑟𝑜𝑠𝑠𝑎𝑏𝑙𝑒}_{\mathrm{𝑢𝑛𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦}}_{i}
, we create the following item of the collection
\mathrm{𝙼𝙰𝙲𝙷𝙸𝙽𝙴𝚂}
\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚍}-i\hfill & \mathrm{𝚌𝚊𝚕}-\mathrm{𝑐𝑟𝑜𝑠𝑠𝑎𝑏𝑙𝑒}_{\mathrm{𝑢𝑛𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑖𝑙𝑖𝑡𝑦}}_{i}\hfill \end{array}〉.\hfill \end{array}
To express the resource constraint, i.e., the fact that two tasks assigned to the same machine should not overlap in time, we use a
\mathrm{𝚐𝚎𝚘𝚜𝚝}
\left(2,\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂},\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}\right)
constraint. For each task
{t}_{i}
i\in \left[1,n\right]
) we create one item for the
\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}
collection as well as one item for the
\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}
\begin{array}{c}〈\begin{array}{ccc}\mathrm{𝚘𝚒𝚍}-i\hfill & \mathrm{𝚜𝚒𝚍}-i\hfill & 𝚡-〈{m}_{i},{\mathrm{𝑣𝑠}}_{i}〉\hfill \end{array}〉,\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚜𝚒𝚍}-i\hfill & 𝚝-〈0,0〉\hfill & 𝚕-〈1,{\mathrm{𝑣𝑑}}_{i}〉\hfill \end{array}〉.\hfill \end{array}
The first item corresponds to an object with
i
as unique identifier, with a rectangular shape identifier
i
{m}_{i},{\mathrm{𝑣𝑠}}_{i}
as the coordinates of its lower left corner. The second item corresponds to a rectangular shape with
i
as unique identifier,
〈0,0〉
as shift offset with respect to its lower left corner, and
〈1,{\mathrm{𝑣𝑑}}_{i}〉
as the sizes of the rectangular shape.
Similarly, to express that each task does not overlap a non-crossable unavailability period, we create for each non-crossable unavailability period
i
one item for the
\mathrm{𝙾𝙱𝙹𝙴𝙲𝚃𝚂}
\mathrm{𝚂𝙱𝙾𝚇𝙴𝚂}
\begin{array}{c}〈\begin{array}{ccc}\mathrm{𝚘𝚒𝚍}-n+i\hfill & \mathrm{𝚜𝚒𝚍}-n+i\hfill & 𝚡-〈{r}_{i},{s}_{i}〉\hfill \end{array}〉,\hfill \\ 〈\begin{array}{ccc}\mathrm{𝚜𝚒𝚍}-n+i\hfill & 𝚝-〈0,0〉\hfill & 𝚕-〈1,{e}_{i}-{s}_{i}+1〉\hfill \end{array}〉.\hfill \end{array}
Finally, a precedence constraint between two distinct tasks
{t}_{i}
{t}_{j}
i,j\in \left[1,n\right]
) is modelled by an inequality constraint between the real end of task
{t}_{i}
and the real start of task
{t}_{j}
{\mathrm{𝑟𝑒}}_{i}\le {\mathrm{𝑟𝑠}}_{j}
. Figure 3.7.60 provides a toy example of such problem with:
Four machines, numbered from 1 to 4, where:
{m}_{1}
has two crossable unavailability periods respectively corresponding to intervals
\left[2,2\right]
\left[6,7\right]
{m}_{2}
\left[2,2\right]
\left[6,7\right]
, as well as one non-crossable unavailability period corresponding to interval
\left[3,3\right]
{m}_{3}
has a single non-crossable unavailability corresponding to interval
\left[6,8\right]
{m}_{4}
has a single crossable unavailability period corresponding to interval
\left[3,4\right]
Five tasks, numbered from 1 to 5, where:
{t}_{1}
is a non-resumable task that has a virtual duration of 3.
{t}_{2}
is a resumable task that has a virtual duration of 2.
{t}_{3}
{t}_{4}
{t}_{5}
Finally, (1) all five tasks should not overlap, (2) task
{t}_{3}
should precedes task
{t}_{2}
and (3) task
{t}_{1}
{t}_{5}
Figure 3.7.60. Illustration of the scheduling problem with crossable and non-crossable unavailability periods as well as with resumable and non-resumable tasks: part (A) gives the real time coordinate system where all precedence constraints are stated, while part (B) provides the virtual time coordinate system – from which all crossable unavailability periods are removed – where the non-overlapping constraint is stated
A survey on machine scheduling problems with unavailability constraints both in the deterministic and stochastic cases can be found in [SaidyTaghaviFard08]. Unavailability can have multiple causes such as:
In the context of production scheduling, machine unavailability corresponds to accepted orders that were already scheduled for a given date. This can typically corresponds to unavailability periods at the beginning of the planning horizon. Preemptive maintenance can also be another cause of machine unavailability.
In the context of timetabling, unavailability periods may come from work regulation which enforces not to work in a continuous way more than a given limit. Unavailability periods may also come from scheduled meetings during the working day.
In the context of distributed computing where cpu time is donated for performing huge tasks, machines are typically partially available [DiedrichJansenSchwarzTrystram09].
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Second-order (biquadratic) IIR digital filtering - MATLAB sosfilt - MathWorks España
Second-Order Section Filtering
y = sosfilt(sos,x,dim)
y = sosfilt(sos,x) applies the second-order section digital filter sos to the input signal x.
If x is a matrix, then the function operates along the first dimension and returns the filtered data for each column.
If x is a multidimensional array, then the function operates along the first array dimension with size greater than 1.
y = sosfilt(sos,x,dim) operates along the dimension dim.
Design a seventh-order Butterworth highpass filter to attenuate the components of the signal below Fs/4. Use a normalized cutoff frequency of 0.48π rad/sample. Express the filter coefficients in terms of second-order sections.
[zhi,phi,khi] = butter(7,0.48,'high');
soshi = zp2sos(zhi,phi,khi);
freqz(soshi)
outhi = sosfilt(soshi,y);
title('Highpass-Filtered Signal')
Design a lowpass filter with the same specifications. Filter the signal and compare the result to the original. Use the same y-axis scale for both plots. The result is mostly noise.
[zlo,plo,klo] = butter(7,0.48);
soslo = zp2sos(zlo,plo,klo);
outlo = sosfilt(soslo,y);
title('Lowpass-Filtered Signal')
sos — Second-order section digital filter
Second-order section digital filter, specified as an L-by-6 matrix where L is the number of second-order sections. The matrix
\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1& {a}_{1L}& {a}_{2L}\end{array}\right]
represents the second-order section digital filter
H\left(z\right)=\prod _{k=1}^{L}{H}_{k}\left(z\right)=\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}.
Example: [b,a] = butter(3,1/32); sos = tf2sos(b,a) specifies a third-order Butterworth filter with a normalized 3 dB frequency of π/32 rad/sample.
Example: x = [2 1].*sin(2*pi*(0:127)'./[16 64]) specifies a two-channel sinusoid.
Dimension to operate along, specified as a positive integer scalar. By default, the function operates along the first array dimension of x with size greater than 1.
Filtered signal, returned as a vector, matrix, or N-D array. y has the same size as x.
[1] Bank, Balázs. "Converting Infinite Impulse Response Filters to Parallel Form". IEEE Signal Processing Magazine. Vol. 35, Number 3, May 2018, pp. 124-130.
[2] Orfanidis, Sophocles J. Introduction to Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1996.
Input filter sos must be stable. Use isstable to check for filter stability.
All second-order subsections of the input filter must be IIR.
The gpuArray version of sosfilt uses a parallel algorithm [1] which is different from the MATLAB® version. The algorithms give different results for complex-valued input with NaN or Inf values:
In the MATLAB version, the NaNs and Infs propagate only in the real part.
In the gpuArray version, the NaNs and Infs propagate in both the real part and the imaginary part.
filter | medfilt1 | sgolayfilt
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UWB Localization Using IEEE 802.15.4z - MATLAB & Simulink - MathWorks Benelux
One-Way Ranging / Time-Difference of Arrival (OWR/TDOA)
This example shows how to estimate the location of a single device as per the IEEE® 802.15.4z™ standard [ 2 ], using 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 ] is a MAC and PHY specification designed for 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 3 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. This technique is demonstrated in the UWB Ranging Using IEEE 802.15.4z example.
Double-Sided Two-Way Ranging (DS-TWR) - Both devices estimate the distance between the 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 example demonstrates the OWR/TDOA technique for uplink transmissions, by using MAC and PHY frames 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. For more information on generating IEEE 802.15.4 MAC frames, see the IEEE 802.15.4 - MAC Frame Generation and Decoding example.
One-way ranging (OWR) involves frame transmission either in the uplink or in the downlink direction. In the uplink case, the device to be localized periodically broadcasts short messages referred to as blinks. The IEEE 802.15.4z amendment [ 2 ] does not stipulate a specific frame format for the blinks, however it states that blinks should be as short as possible. These blink messages are received by a set of infrastructure nodes that are synchronized either through a wired backbone or via an UWB wireless communications link. In the downlink case, the synchronized nodes periodically transmit broadcast messages with a known time offset.
The time-difference of arrival (TDOA) between the periodic messages places the device in one hyperbolic surface for each pair of synchronized nodes [ 3 ]. The intersection of all hyperbolic surfaces (for every pair of synchronized nodes) gives the location estimate for the device.
This example demonstrates the uplink OWR case.
Confirm installation of the Communications Toolbox™ Library for ZigBee® and UWB add-on.
% Check if the 'Communications Toolbox Library for ZigBee and UWB' support package is installed:
commSupportPackageCheck('ZIGBEE');
Set up a network with 3 synchronized nodes and 1 device, in a 100x100 plane:
deviceLoc = [50 50]; % place device at the center
nodeLoc = [40 41;
TDOA = nan(numNodes);
helperShowLocations(deviceLoc,nodeLoc);
Calculate the actual distance and time of flight (TOF) between nodes and the device.
actualDistances = sqrt(sum((nodeLoc - deviceLoc).^2, 2));
c = physconst('LightSpeed'); % speed of light (m/s)
actualTOF = actualDistances/c;
SNR = 30; % in dB
Configure Blinks
Use a short (IEEE 802.15.4 MAC) data frame as a blink.
numBlinks = 1;
% MAC layer:
payload = '00';
cfg = lrwpan.MACFrameConfig( ...
FrameType='Data', ...
SourceAddressing='Short address', ...
SourcePANIdentifier='AB12', ...
SourceAddress='CD77');
blinkMAC = lrwpan.MACFrameGenerator(cfg,payload);
% PHY layer:
% Ensure the Ranging field is enabled.
% Also set the proper PSDU length.
blinkPHYConfig = lrwpanHRPConfig( ...
Mode='HPRF', ...
STSPacketConfiguration=1, ...
PSDULength=length(blinkMAC), ...
Ranging=true);
blinkPHY = lrwpanWaveformGenerator( ...
blinkMAC, ...
blinkPHYConfig);
% Cache preamble, to use in preamble detection.
% Get the 1st instance out of the Nsync=PreambleDuration repetitions.
indices = lrwpanHRPFieldIndices(blinkPHYConfig); % length (start/end) of each field
blinkPreamble = blinkPHY( ...
1:indices.SYNC(end)/blinkPHYConfig.PreambleDuration); % 1 of the Nsync repetitions
In the simulation loop, a blink propagates to each node with a propagation delay that is determined by their distinct distance. Next, each pair of nodes calculates the difference of their blink arrival times. As a result, the position of the device is estimated within a hyperbolic surface for each pair of nodes. The intersection of all surfaces gives the position estimate for the device. Here, a plot of 2D curves shows the intersection point to indicate the position estimate for the device.
vfd = dsp.VariableFractionalDelay;
arrivalTime = zeros(1,numNodes);
plotStr = {'r--','b--','g--'};
[x, y] = deal(cell(1, 3));
for idx = 1:numBlinks
for node = 1:numNodes
% Transmission and reception of blink
% Each node receives a specifically delayed version of the blink
tof = actualTOF(node);
samplesToDelay = tof * blinkPHYConfig.SampleRate;
reset(vfd);
vfd.MaximumDelay = ceil(1.1*samplesToDelay);
delayedBlink = vfd( ...
[blinkPHY; zeros(ceil(samplesToDelay), 1)], ...
samplesToDelay);
% Add white Gaussian noise
receivedBlink = awgn(delayedBlink,SNR);
% Node receiver detection of preamble
preamPos = helperFindFirstHRPPreamble( ...
receivedBlink,blinkPreamble,blinkPHYConfig);
% Transmit each blink at t=0 of each period. The blink arrives at different
% instances at each node, due to their dissimilar distance to the device.
arrivalTime(node) = ( ...
preamPos - indices.SYNC(end) / ...
blinkPHYConfig.PreambleDuration)/blinkPHYConfig.SampleRate;
% Localization: Estimate position at synchronized backbone for each pair
pairCnt = 1;
for node1 = 1:numNodes
for node2 = (node1+1):numNodes
% Calculate Time Difference of Arrival (TDOA)
TDOA(node1, node2) = arrivalTime(node1)-arrivalTime(node2);
% Get hyperbolic surface for the TDOA between node1 and node2
[x{pairCnt}, y{pairCnt}] = helperGetHyperbolicSurface( ...
nodeLoc(node1,:), ...
TDOA(node1,node2));
plot(x{pairCnt},y{pairCnt},plotStr{pairCnt});
pairCnt = pairCnt + 1;
% Find intersection points between hyperbolic surfaces
[xC,yC] = helperFindHyperbolicIntersection(x,y);
% Estimate location as the center of intersection triangle
xO = mean(xC, 2);
yO = mean(yC, 2);
plot(xO, yO, 'ro')
plot(xC',yC','rx')
leg = legend('Device', 'Synchronized nodes','A-B', 'A-C', 'B-C', 'Estimation', 'Intersections', 'location', 'northwest');
Zoom in to estimation area:
zoomInToEstimationArea(deviceLoc, xC, yC, xO, yO, leg);
The IEEE 802.15.4z localization algorithm allows for multiple intersection points between 2 hyperbolic surfaces, thus either one or two possible localization answers exist. Calculate the localization error for each answer:
for idx = 1:numel(xO)
locError = sqrt(sum([xO(idx) yO(idx)]-deviceLoc).^2);
fprintf('Localization error #%d = %0.3f m.\n', idx, locError);
Localization error #1 = 0.012 m.
For localization methods that rely on estimating the time of arrival, errors in the distance estimate are primarily caused when the arrival time is not an integer multiple of the sample time. The largest distance error for such localization methods occurs when the arrival time lasts half a sample time more than an integer multiple of sample time. The smallest distance error occurs when the arrival time 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. The maximum distance estimation error is
0.5×c/\left(499.2×10\right)
, which is approximately 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 distance error for narrowband communication, estimates for the localization error lie between 0 and 10 cm for 160 MHz and between 0 and 75 cm for 20 MHz. For more information regarding positioning with IEEE 802.11az, see the 802.11az Positioning Using Super-Resolution Time of Arrival Estimation (WLAN Toolbox) example.
This example uses these objects and functions from the Communications Toolbox™ Library for ZigBee® and UWB add-on.
lrwpan.MACFrameConfig: Create configuration for 802.15.4 MAC frames
lrwpan.MACFrameGenerator: Generate 802.15.4 MAC frames
lrwpanHRPConfig: HRP waveform configuration
lrwpanWaveformGenerator: Create an IEEE 802.15.4a/z HRP UWB waveform
These utilities are undocumented and their API or functionality may change in the future.
function zoomInToEstimationArea(deviceLoc, xC, yC, xO, yO, leg)
% Zoom 2D plane into region around device location
allX = [deviceLoc(1); xO(:); xC(:)];
allY = [deviceLoc(2); yO(:); yC(:)];
minX = min(allX);
maxX = max(allX);
minY = min(allY);
maxY = max(allY);
axis([minX-0.1*(maxX-minX), maxX+0.1*(maxX-minX), minY-0.1*(maxY-minY), maxY+0.1*(maxY-minY)])
leg.Location = 'NorthEast';
1 - "IEEE Standard for Low-Rate Wireless Networks," in IEEE Std 802.15.4-2020 (Revision of IEEE Std 802.15.4-2015), pp.1-800, 23 July 2020, doi: 10.1109/IEEESTD.2020.9144691.
2 - "IEEE Standard for Low-Rate Wireless Networks--Amendment 1: Enhanced Ultra Wideband (UWB) Physical Layers (PHYs) and Associated Ranging Techniques," in IEEE Std 802.15.4z-2020 (Amendment to IEEE Std 802.15.4-2020), pp.1-174, 25 Aug. 2020, doi: 10.1109/IEEESTD.2020.9179124.
3 - Wong, S.; Zargani, R. Jassemi; Brookes, D. & Kim, B. "Passive target localization using a geometric approach to the time-difference-of-arrival method", Defence Research and Development Canada Scientific Report, DRDC-RDDC-2017-R079, June 2017, pp. 1-77
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Configurational entropy: Level 3-4 Challenges Practice Problems Online | Brilliant
Configurational entropy: Level 3-4 Challenges
A token is placed at one of 9 positions in a
3 \times 3
grid according to a probability distribution
P.
After a token is placed into one of the positions of the grid, it is then moved uniformly at random to one of the horizontally, vertically, or diagonally adjacent positions. For each position on the board, the probability that the token is in that position after being moved is also given by the distribution
P.
If two tokens are placed into the grid according to the distribution
P
and then moved, the probability that the set of occupied positions is the same before and after the tokens are moved can be expressed as
\frac{a}{b}
and
b
a + b?
A weight hangs at a steady height from a rubber band that is attached to the ceiling. Now, you turn on an air conditioner, significantly lowering the temperature of the room. What happens to the position of the weight?
It stays the same It rises It lowers
If you could read the title just fine, it is because the English language (as well as all natural languages) are redundant. This isn't to say that there are multiple words that mean the same thing (although there are), but that if you compare the number of questions you would have to ask to uniquely identify a word I'm thinking of, and the number of possible words, the second number is much bigger than the first.
For example, by the time you read the first four letters of a word starting with calc, you can be pretty sure it's going to end up being calculus, calcium, calculate, or calculation. You don't need all the extra letters to distinguish the remaining possibilities. Concretely, if we take a list of all the words of a given length that exist, and sort them into alphabetical order, we only need
\log_2 N
questions to identify any given word in the list (where
N
is the number of words in the list). On the other hand, if we were making full use of the language, we could manage
\displaystyle 26^L
unique words of length
L
Using the English language dictionary built in to UNIX operating systems, and filtering for words of length 5, I find 10230 unique words. Taking words of length 5 as a proxy for the entire English language, how short, on average, could we make five letter words before someone with perfect reasoning couldn't read them anymore?
One thousand dust particles are trapped on a surface. The two states that a particle can occupy are
absorbed on the surface in a zero energy state.
excited with energy
E_1
Suppose the particles are in thermal equilibrium at temperature
T
, and 20 % of the particles are in the excited state. Find the value of
\frac{1}{k_BT}\sum_i E_i
The particles are identical.
k_B
Proteins are molecules responsible for catalyzing chemical reactions, regulating gene expression, sensing changes in the extracellular environment, giving superstructure to the genome, and many other important tasks. They take the form of long chains that fold into compact structures by seeking out shapes that minimize the energy of self-interaction.
Consider a simple protein that has two states, completely folded, and completely unfolded, which have a free energy gap of
\Delta = 3.74
kJ/mole.
You have an ensemble of several million copies of this protein dissolved in buffer. At temperature
T = 37^{\circ}
C, what fraction of the proteins are folded?
Note: You may wish to read about partition functions.
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Global Constraint Catalog: Csame_and_global_cardinality
<< 5.334. same5.336. same_and_global_cardinality_low_up >>
Conjoin
\mathrm{𝚜𝚊𝚖𝚎}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚜𝚐𝚌𝚌}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚐𝚌𝚌}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚌𝚌}
\mathrm{𝚜𝚠𝚌}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚠𝚒𝚝𝚑}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚒𝚎𝚜}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-\mathrm{𝚍𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
collection according to a permutation. In addition, each value
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
i\in \left[1,|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right]
) should be taken by exactly
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
collection. Finally, each variable of
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
should be assigned a value of
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
i\in \left[1,|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right]
\left(\begin{array}{c}〈1,9,1,5,2,1〉,\hfill \\ 〈9,1,1,1,2,5〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚟𝚊𝚕}-1\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-3,\hfill \\ \mathrm{𝚟𝚊𝚕}-2\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-7\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-0,\hfill \\ \mathrm{𝚟𝚊𝚕}-9\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
The values 1, 9, 1, 5, 2, 1 assigned to
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
correspond to a permutation of the values 9, 1, 1, 1, 2, 5 assigned to
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
The values 1, 2, 5, 7 and 6 are respectively used 3, 1, 1, 0 and 1 times.
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\right)>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}\right)>1
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right)>1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}\right)
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
The filtering algorithm presented in [BeldiceanuKatrielThiel05b] can be reused for pruning the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
collection. This algorithm does not restrict the
\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚜𝚊𝚖𝚎}
𝚔_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(two overlapping
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
plus restriction on values).
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚏𝚒𝚡𝚎𝚍}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
combinatorial object: permutation, multiset.
constraint arguments: constraint between two collections of variables.
filtering: flow.
modelling: equality between multisets.
problems: demand profile.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}
•\text{for}\text{all}\text{connected}\text{components:}
\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}
=
\mathrm{𝐍𝐒𝐈𝐍𝐊}
•
\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}
=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}|
•
\mathrm{𝐍𝐒𝐈𝐍𝐊}
=|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{2}|
For all items of
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\mathtt{1}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
Parts (A) and (B) of Figure 5.335.1 respectively show the initial and final graph associated with the first graph constraint of the Example slot. Since we use the
\mathrm{𝐍𝐒𝐎𝐔𝐑𝐂𝐄}
\mathrm{𝐍𝐒𝐈𝐍𝐊}
graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since there is a constraint on each connected component of the final graph we also show the different connected components. Each of them corresponds to an equivalence class according to the arc constraint.
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚊𝚗𝚍}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
|
G
is Abelian (or commutative) if every pair of elements of
G
commute with each other. That is, for all
and
b
G
a·b=b·a
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔\mathrm{SmallGroup}\left(32,1\right):
\mathrm{IsAbelian}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsAbelian}\left(\mathrm{SmallGroup}\left(32,5\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
G≔〈a|{a}^{6}=1〉
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}⟨\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{∣}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{}⟩
\mathrm{IsAbelian}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsCommutative}\left(〈〈a,b,c〉|〈a·b=b·a,a·c=c·a,b·c=c·b〉〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsAbelian}\left(〈〈a,b〉|〈{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1〉〉\right)
|
Global Constraint Catalog: Kreified_constraint
<< 3.7.208. Reified automaton constraint3.7.210. Relation >>
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}_\mathrm{𝚛𝚎𝚒𝚏𝚒𝚎𝚍}
(reified version of
\mathrm{𝚒𝚗}_\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
The reified version
\mathrm{𝐶𝑅}
of a given constraint
C
\mathrm{𝐶𝑅}
has as arguments all arguments of
C
plus one extra 0-1 variable. This 0-1 variable is set to 1 when constraint
C
holds, and 0 otherwise. Note that constraint
\mathrm{𝐶𝑅}
inherits from all restrictions of constraint
C
(i.e., incorrect parameters for constraint
C
are also incorrect for constraint
\mathrm{𝐶𝑅}
). Within the context of linear programming the extra 0-1 variable is often called an indicator variable.
It was shown in [BeldiceanuCarlssonFlenerPearson12] how to reify a global constraint by reformulating it as a conjunction of pure functional dependency constraints together with a constraint that can be easily reified (e.g., an automaton with or without counter, or a Boolean combination of linear arithmetic equalities and inequalities and 0-1 variables).
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2.3.2. Ingredients used for describing global c >>
Within the graph-based representation, a global constraint is represented as a digraph where each vertex corresponds to a variable and each arc to a binary arc constraint between the variables associated with the extremities of the corresponding arc. The main difference with classical constraint networks [DechterPearl87], stems from the fact that we do not force any more all arc constraints to hold. We rather consider this graph from which we discard all the arc constraints that do not hold as well as all isolated vertices (i.e, vertices not involved any more in any arc) and impose one or several graph properties on this remaining graph. These properties can for instance be a restriction on the number of connected components, on the size of the smallest connected component or on the size of the largest connected component.
Figure 2.3.1. Illustration of the link between graph-properties and global constraints
EXAMPLE: We give an example of interpretation of such graph properties in terms of global constraints. For this purpose we consider the sequence
s
1311288236883
from which we construct the following graph
G
To each value associated with a position in
s
G
There is an arc from a vertex
{v}_{1}
{v}_{2}
if these vertices correspond to the same value.
Figure 2.3.1 depicts graph
G
G
is symmetric, we omit the directions of the arcs. We have the following correspondence between graph properties and constraints on the sequence
s
The number of connected components of
G
corresponds to the number of distinct values of
s
The size of the smallest connected component of
G
is the smallest number of occurrences of the same value in
s
The size of the largest connected component of
G
is the largest number of occurrences of the same value in
s
As a result, in this context, putting a restriction on the number of connected components of
G
can been seen as a global constraint on the number of distinct values of a sequence of variables. Similar global constraints can be associated with the two other graph properties.
We now explain how to generate the initial graph associated with a global constraint. A global constraint has one or more arguments, which usually correspond to an integer value, to one variable or to a collection of variables. Therefore we have to describe the process that allows for generating the vertices and the arcs of the initial graph from the arguments of a global constraint under consideration. For this purpose we will take a concrete example.
Consider the constraint
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝙽𝚅𝙰𝙻}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
respectively correspond to a domain variable and to a collection of domain variables
〈\mathrm{𝚟𝚊𝚛}-{V}_{1},\mathrm{𝚟𝚊𝚛}-{V}_{2},\cdots ,\mathrm{𝚟𝚊𝚛}-{V}_{m}〉
\mathrm{𝚟𝚊𝚛}
corresponds to the name of the attribute used in the collection of variables. This constraint holds if
\mathrm{𝙽𝚅𝙰𝙻}
is equal to the number of distinct values assigned to the variables
{V}_{1},{V}_{2},\cdots ,{V}_{m}
. We first show how to generate the initial graph associated with the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint. We then describe the arc constraint associated with each arc of this graph. Finally, we give the graph property we impose on the final graph.
To each variable of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
corresponds a vertex of the initial graph. We generate an arc between each pair of vertices. To each arc, we associate an equality constraint between the variables corresponding to the extremities of that arc. We impose that
\mathrm{𝙽𝚅𝙰𝙻}
, the variable corresponding to the first argument of
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
, be equal to the number of strongly connected components of the final graph. This final graph consists of the initial graph from which we discard all arcs such that the corresponding equality constraint does not hold.
Part (A) of Figure 2.3.2 shows the graph initially generated for the constraint
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝚅𝙰𝙻},〈\mathrm{𝚟𝚊𝚛}-{V}_{1},\mathrm{𝚟𝚊𝚛}-{V}_{2},\mathrm{𝚟𝚊𝚛}-{V}_{3},\mathrm{𝚟𝚊𝚛}-{V}_{4}〉\right)
\mathrm{𝙽𝚅𝙰𝙻}
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
are domain variables. Part (B) presents the final graph associated with the ground instance
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(3,〈\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-1,\mathrm{𝚟𝚊𝚛}-8〉\right)
. For each vertex of the initial and final graph we respectively indicate the corresponding variable and the value assigned to that variable. We have removed from the final graph all the arcs associated with equalities that do not hold. The constraint
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(3,〈\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-1,\mathrm{𝚟𝚊𝚛}-8〉\right)
holds since the final graph contains three strongly connected components, which in the context of the definition of the
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
constraint, can be reinterpreted as the fact that
\mathrm{𝙽𝚅𝙰𝙻}
is the number of distinct values assigned to variables
{V}_{1},{V}_{2},{V}_{3},{V}_{4}
Figure 2.3.2. (A) Initial and (B) final graph associated with the constraint
\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(3,〈\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-5,\mathrm{𝚟𝚊𝚛}-1,\mathrm{𝚟𝚊𝚛}-8〉\right)
Now that we have illustrated the basic ideas for describing a global constraint in terms of graph properties, we go into more details.
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Global Constraint Catalog: KHungarian_method_for_the_assignment_problem
<< 3.7.120. Heuristics for two-dimensional rectangle3.7.122. Hybrid-consistency >>
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}_\mathrm{𝚠𝚎𝚒𝚐𝚑𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
A constraint that can use the Hungarian method for the assignment problem [Kuhn55] in order to evaluate the minimum or maximum value of one of its argument. Given
persons,
n
tasks and a corresponding
n
n
cost matrix, the assignment problem is the search for an assignment of persons to tasks so that the sum of the costs is maximised.
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Halfband decimator - MATLAB - MathWorks Deutschland
Specific to dsp.FIRHalfbandDecimator
Filter Input into Lowpass and Highpass Subbands Using FIR Halfband Decimator
The dsp.FIRHalfbandDecimator System object™ performs an efficient polyphase decimation of the input signal by a factor of two. You can use dsp.FIRHalfbandDecimator to implement the analysis portion of a two-band filter bank to filter a signal into lowpass and highpass subbands. dsp.FIRHalfbandDecimator uses an FIR equiripple design or a Kaiser window design to construct the halfband filters and a polyphase implementation to filter the input.
Create the dsp.FIRHalfbandDecimator object and set its properties.
firhalfbanddecim = dsp.FIRHalfbandDecimator
firhalfbanddecim = dsp.FIRHalfbandDecimator(Name=Value)
firhalfbanddecim = dsp.FIRHalfbandDecimator returns a halfband decimator, firhalfbanddecim, with the default settings. Under the default settings, the System object filters and downsamples the input data with a halfband frequency of 11025 Hz, a transition width of 4.1 kHz, and a stopband attenuation of 80 dB. The design method is set to "Auto".
firhalfbanddecim = dsp.FIRHalfbandDecimator(Name=Value) returns a halfband decimator, with additional properties specified by one or more Name=Value pair arguments.
Example: firhalfbanddecim = dsp.FIRHalfbandDecimator(Specification="Filter order and stopband attenuation") creates an FIR halfband decimator object with filter order set to 52 and stopband attenuation set to 80 dB.
The filter is designed using the optimal equiripple filter design method or the kaiser-window-based design method.
firhalfband('minorder',0.407,1e-4) (default) | row vector
FIR halfband filter coefficients, specified as a row vector. The coefficients must comply with the FIR halfband impulse response format. For details on this format, see Halfband Filters and FIR Halfband Filter Design. If half the order of the filter, (length(Numerator) - 1)/2, is even, every other coefficient starting from the first coefficient must be a zero except for the center coefficient which must be a 0.5. If half the order of the filter is odd, the sequence of alternating zeros with a 0.5 at the center starts at the second coefficient.
Input sample rate in Hz, specified as a positive real scalar. The input sample rate defaults to 44100 Hz. If you specify transition width as one of your filter design parameters, the transition width cannot exceed 1/2 the input sample rate.
ylow = firhalfbanddecim(x)
[ylow,yhigh] = firhalfbanddecim(x)
ylow = firhalfbanddecim(x) filters the input signal x using the FIR halfband filter, firhalfbanddecim, and downsamples the output by a factor of 2.
[ylow,yhigh] = firhalfbanddecim(x) computes the ylow and yhigh, of the analysis filter bank, firhalfbanddecim for input x. A Ki-by-N input matrix is treated as N independent channels. The System object generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively. ylow and yhigh are of the same size (Ko-by-N) and data type. Ko = Ki/2, where 2 is the decimation factor.
Data input, specified as a column vector or a matrix. If the input signal is a matrix, each column of the matrix is treated as an independent channel. The number of rows in the input signal must be a multiple of 2.
This object supports variable-size input signal.
hfirhalfbanddecim = dsp.FIRHalfbandDecimator(...
TransitionWidth=2000,...
SampleRate=44.1e3);
Filter a two-channel input into low and highpass subbands.
[ylow,yhigh] = hfirhalfbanddecim(x);
h\left(n\right)=\frac{1}{2\pi }{\int }_{-\pi /2}^{\pi /2}{e}^{j\omega n}d\omega =\frac{\mathrm{sin}\left(\frac{\pi }{2}n\right)}{\pi n}.
g\left(n\right)=\frac{1}{2\pi }{\int }_{-\pi }^{-\pi /2}{e}^{j\omega n}d\omega +\frac{1}{2\pi }{\int }_{\pi /2}^{\pi }{e}^{j\omega n}d\omega .
g\left(n\right)=\frac{\mathrm{sin}\left(\pi n\right)}{\pi n}-\frac{\mathrm{sin}\left(\frac{\pi }{2}n\right)}{\pi n}.
{\ell }^{\infty }
w\left(n\right)=\frac{{I}_{0}\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^{2}}\right)}{{I}_{0}\left(\beta \right)},\text{ }\text{ }0\le n\le N,
\beta =\left\{\begin{array}{ll}0.1102\left(\alpha -8.7\right),\hfill & \alpha >50\hfill \\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.07886\left(\alpha -21\right),\hfill & 50\ge \alpha \ge 21\hfill \\ 0,\hfill & \alpha <21\hfill \end{array}
n=\frac{\alpha -7.95}{2.285\left(\Delta \omega \right)}
{H}_{0}\left(z\right)=\sum _{n}h\left(2n\right){z}^{-n}.
{H}_{1}\left(z\right)=\sum _{n}h\left(2n+1\right){z}^{-n}.
H\left(z\right)={H}_{0}\left({z}^{2}\right)+{z}^{-1}{H}_{1}\left({z}^{2}\right).
dsp.FIRHalfbandInterpolator | dsp.IIRHalfbandDecimator | dsp.DyadicAnalysisFilterBank | dsp.Channelizer
FIR Halfband Decimator | FIR Halfband Interpolator | IIR Halfband Decimator | Dyadic Analysis Filter Bank
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Mechanically Assisted Low-Temperature Pyrolysis of Hydrocarbons
1Sintos Systems Svenska Filial, Norsborg, Sweden
2Ukrnaphtohimsynthes, Lisichansk, Ukraine
3Observatory of Belarussian State University, Minsk, Belarus
4School of Industrial Technology and Management, KTH, Stockholm, Sweden
5Department of Material Science and Engineering, KTH, Stockholm, Sweden
6Department of Chemistry, KTH, Stockholm, Sweden
The focus of the study is experimental setups and conditions leading to pyrolysis (cracking) of such gaseous hydrocarbons as methane, mixed propane and butane, at the temperatures of the heater below 200˚C. The process was mechanically assisted by putting the substances being decomposed into a dynamic interaction with the fractal interfaces of cracks in titanium dioxide films, as well as in tin and bismuth alloy. During a trial, the alloy was periodically heated and cooled so that it changed its phase state, and fractal interfaces were created between its surface and the gases. The interaction of the gases with fractal surfaces of the alloy being produced by mechanical fracturing made it possible to obtain gas cracking even at the lower temperatures of the heater (150˚C). It should be noted that at this temperature, the heater couldn’t melt the alloy in the heated volume with the gas.
\frac{Q}{m}=-2{D}^{2}\nabla \left[\frac{\Delta \sqrt{P}}{\sqrt{P}}\right]
E=Dw+\frac{1}{2}{a}^{2}{w}^{2}
Alevanau, A.Y., Vyhoniailo, O.I., Kuznechik, O.P., Jönsson, P., Ersson, M. and Kantarelis, E. (2018) Mechanically Assisted Low-Temperature Pyrolysis of Hydrocarbons. Energy and Power Engineering, 10, 133-153. https://doi.org/10.4236/epe.2018.104010
1. Chen, G., Sjostrom, K. and Bjornbom, E. (1992) Pyrolysis/Gasification of Wood in a Pressurized Fluidized Bed Reactor. Industrial & Engineering Chemistry Research, 31, 2164-2168. https://doi.org/10.1021/ie00012a021
2. Gutfraind, R. and Sheintuch, S. (1992) Scaling Approach to Study Diffusion and Reaction Processes on Fractal Catalysts. Chemical Engineering Science, 47, 4425-4433. https://doi.org/10.1016/0009-2509(92)85120-Z
3. Kantarelis, E., Yang, W. and Blasiak, W. (2013) Effects of Silica-Supported Nickel and Vanadium on Liquid Products of Catalytic Steam Pyrolysis of Biomass. Energy & Fuels, 28, 591-599. https://doi.org/10.1021/ef401939g
4. Shostak, T.A. and Vyhoniailo, O.I. (2007) Process for the Preparation of Synthesis Gas from Fossil Coal, EPA. http://v3.espacenet.com/origdoc?DB=EPODOC&IDX=UA79575&F=0&QPN=UA79575
5. Imre, A., Pajkossy, T. and Nyikos, L. (1992) Electrochemical Determination of the Fractal Dimension of Fractured Surfaces. Acta Metallurgica et Materialia, 40, 1819-1826. https://doi.org/10.1016/0956-7151(92)90168-E
6. De Castro, W.B. (2005) Undercooling of Eutetic Sn-57wt% Bi Alloy. Materials Science Forum, 480-481, 201-206. https://doi.org/10.4028/http://www.scientific.net/MSF.480-481.201
7. Weimer, A.W., Dahl, J., Tamburini, J., Lewandowski, A., Pitts, R., Bingham, C. and Glatzmaier, G.C. (2000) Thermal Dissociation of Methane Using a Solar Coupled Aerosol Flow Reactor. Proceedings of the 2000 DOE Hydrogen Program Review NREL/CP-570-28890, 1-23.
8. Mudry, S., Shtablavyi, I. and Shevernoga, I. (2013) Structural Disordering in Sn-Pb(Bi) Eutectic Melts Induced by Heating. Polish Journal of Chemical Technology, 15, 61-64. https://doi.org/10.2478/pjct-2013-0045
9. Sanyal, D., Ramachandrarao, P. and Gupta, O.P. (2005) A Fractal Description of Transport Phenomena in Dendritic Porous Net-work. Chemical Engineering Science, 61, 307-315. https://doi.org/10.1016/j.ces.2005.06.005
10. Nottale, L. (2008) Scale Relativity and Fractal Space-Time: Theory and Applications. Proceedings of 1st International Conference on the Evolution and Development of the Universe, Paris, Vol. 15, 101-152.
11. Notalle, L. (2009) Generalized Quantum Potentials. Journal of Physics A: Mathematical and Theoretical, 42, 275-306. https://doi.org/10.1088/1751-8113/42/27/275306
12. Nottale, L. and Lehner, T. (2012) Numerical Simulation of a Macroscopic Quantum-Like Experiment: Oscillating Wave Packet. International Journal of Modern Physics C, 23, 1-27. https://doi.org/10.1142/S0129183112500350
13. Moon, K.W., Boettinger, W.J., Kattner, U.R., Handwerker, C.A. and Lee, D.J. (2001) The Effect of Pb Contamination on the Solidification Behavior of Sn-Bi Solders. Journal of Electronic Materials, 30, 45-52. https://doi.org/10.1007/s11664-001-0213-x
14. Alevanau, A., Yang, W., Blasiak, W., Kuznechik, O.P. and Vyhoniailo, O.I. (2013) Prospective Side Effects of the Heat and Mass Transfers in Micro-Porous Structure of Char during Intermediate and Final Stages of the High-Temperature Pyrolysis. Nonlinear Phenomena in Complex Systems, 16, 287-301.
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Glossary of elementary quantum mechanics - Wikipedia
Glossary of elementary quantum mechanics
{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
Different authors may have different definitions for the same term.
The discussions are restricted to Schrödinger picture and non-relativistic quantum mechanics.
{\displaystyle |x\rangle }
- position eigenstate
{\displaystyle |\alpha \rangle ,|\beta \rangle ,|\gamma \rangle ...}
- wave function of the state of the system
{\displaystyle \Psi }
- total wave function of a system
{\displaystyle \psi }
- wave function of a system (maybe a particle)
{\displaystyle \psi _{\alpha }(x,t)}
- wave function of a particle in position representation, equal to
{\displaystyle \langle x|\alpha \rangle }
1.1 Kinematical postulates
1.2.1 Dynamics related to single particle in a potential / other spatial properties
1.3 Measurement postulates
1.4 Indistinguishable particles
1.5 Quantum statistical mechanics
2 Nonlocality
3 Rotation: spin/angular momentum
5 Historical Terms / semi-classical treatment
6 Uncategorized terms
Kinematical postulates[edit]
a complete set of wave functions
A basis of the Hilbert space of wave functions with respect to a system.
The Hermitian conjugate of a ket is called a bra.
{\displaystyle \langle \alpha |=(|\alpha \rangle )^{\dagger }}
. See "bra–ket notation".
The bra–ket notation is a way to represent the states and operators of a system by angle brackets and vertical bars, for example,
{\displaystyle |\alpha \rangle }
{\displaystyle |\alpha \rangle \langle \beta |}
Physically, the density matrix is a way to represent pure states and mixed states. The density matrix of pure state whose ket is
{\displaystyle |\alpha \rangle }
{\displaystyle |\alpha \rangle \langle \alpha |}
Mathematically, a density matrix has to satisfy the following conditions:
{\displaystyle \operatorname {Tr} (\rho )=1}
{\displaystyle \rho ^{\dagger }=\rho }
Synonymous to "density matrix".
Synonymous to "bra–ket notation".
Given a system, the possible pure state can be represented as a vector in a Hilbert space. Each ray (vectors differ by phase and magnitude only) in the corresponding Hilbert space represent a state.[nb 1]
A wave function expressed in the form
{\displaystyle |a\rangle }
is called a ket. See "bra–ket notation".
A mixed state is a statistical ensemble of pure state.
Pure state:
{\displaystyle \operatorname {Tr} (\rho ^{2})=1}
Mixed state:
{\displaystyle \operatorname {Tr} (\rho ^{2})<1}
Normalizable wave function
{\displaystyle |\alpha '\rangle }
is said to be normalizable if
{\displaystyle \langle \alpha '|\alpha '\rangle <\infty }
. A normalizable wave function can be made to be normalized by
{\displaystyle |a'\rangle \to \alpha ={\frac {|\alpha '\rangle }{\sqrt {\langle \alpha '|\alpha '\rangle }}}}
{\displaystyle |a\rangle }
is said to be normalized if
{\displaystyle \langle a|a\rangle =1}
A state which can be represented as a wave function / ket in Hilbert space / solution of Schrödinger equation is called pure state. See "mixed state".
a way of representing a state by several numbers, which corresponds to a complete set of commuting observables.
A common example of quantum numbers is the possible state of an electron in a central potential:
{\displaystyle (n,l,m,s)}
, which corresponds to the eigenstate of observables
{\displaystyle H}
{\displaystyle r}
{\displaystyle L}
(magnitude of angular momentum),
{\displaystyle L_{z}}
(angular momentum in
{\displaystyle z}
-direction), and
{\displaystyle S_{z}}
Spin wave function
Part of a wave function of particle(s). See "total wave function of a particle".
Synonymous to "spin wave function".
Spatial wave function
A state is a complete description of the observable properties of a physical system.
Sometimes the word is used as a synonym of "wave function" or "pure state".
synonymous to "wave function".
A large number of copies of a system.
A sufficiently isolated part in the universe for investigation.
Tensor product of Hilbert space
When we are considering the total system as a composite system of two subsystems A and B, the wave functions of the composite system are in a Hilbert space
{\displaystyle H_{A}\otimes H_{B}}
, if the Hilbert space of the wave functions for A and B are
{\displaystyle H_{A}}
{\displaystyle H_{B}}
Total wave function of a particle
For single-particle system, the total wave function
{\displaystyle \Psi }
of a particle can be expressed as a product of spatial wave function and the spinor. The total wave functions are in the tensor product space of the Hilbert space of the spatial part (which is spanned by the position eigenstates) and the Hilbert space for the spin.
The word "wave function" could mean one of following:
A vector in Hilbert space which can represent a state; synonymous to "ket" or "state vector".
The state vector in a specific basis. It can be seen as a covariant vector in this case.
The state vector in position representation, e.g.
{\displaystyle \psi _{\alpha }(x_{0})=\langle x_{0}|\alpha \rangle }
{\displaystyle |x_{0}\rangle }
is the position eigenstate.
See "degenerate energy level".
If the energy of different state (wave functions which are not scalar multiple of each other) is the same, the energy level is called degenerate.
There is no degeneracy in 1D system.
The energy spectrum refers to the possible energy of a system.
For bound system (bound states), the energy spectrum is discrete; for unbound system (scattering states), the energy spectrum is continuous.
related mathematical topics: Sturm–Liouville equation
{\displaystyle {\hat {H}}}
The operator represents the total energy of the system.
{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\alpha \rangle ={\hat {H}}|\alpha \rangle }
(1) is sometimes called "Time-Dependent Schrödinger equation" (TDSE).
Time-Independent Schrödinger Equation (TISE)
A modification of the Time-Dependent Schrödinger equation as an eigenvalue problem. The solutions are energy eigenstate of the system.
{\displaystyle E\alpha \rangle ={\hat {H}}|\alpha \rangle }
Dynamics related to single particle in a potential / other spatial properties[edit]
In this situation, the SE is given by the form
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi _{\alpha }(\mathbf {r} ,\,t)={\hat {H}}\Psi =\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right)\Psi _{\alpha }(\mathbf {r} ,\,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi _{\alpha }(\mathbf {r} ,\,t)+V(\mathbf {r} )\Psi _{\alpha }(\mathbf {r} ,\,t)}
It can be derived from (1) by considering
{\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }
{\displaystyle {\hat {H}}:=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\hat {V}}}
A state is called bound state if its position probability density at infinite tends to zero for all the time. Roughly speaking, we can expect to find the particle(s) in a finite size region with certain probability. More precisely,
{\displaystyle |\psi (\mathbf {r} ,t)|^{2}\to 0}
{\displaystyle |\mathbf {r} |\to +\infty }
{\displaystyle t>0}
There is a criterion in terms of energy:
{\displaystyle E}
be the expectation energy of the state. It is a bound state iff
{\displaystyle E<\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}
Position representation and momentum representation
Position representation of a wave function:
{\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }
momentum representation of a wave function:
{\displaystyle {\tilde {\Psi }}_{\alpha }(p,t):=\langle p|\alpha \rangle }
{\displaystyle |x\rangle }
is the position eigenstate and
{\displaystyle |p\rangle }
the momentum eigenstate respectively.
The two representations are linked by Fourier transform.
A probability amplitude is of the form
{\displaystyle \langle \alpha |\psi \rangle }
Having the metaphor of probability density as mass density, then probability current
{\displaystyle J}
is the current:
{\displaystyle J(x,t)={\frac {i\hbar }{2m}}(\psi {\frac {\partial \psi ^{*}}{\partial x}}-{\frac {\partial \psi }{\partial x}}\psi )}
The probability current and probability density together satisfy the continuity equation:
{\displaystyle {\frac {\partial }{\partial t}}|\psi (x,t)|^{2}+\nabla \cdot \mathbf {J(x,t)} =0}
Given the wave function of a particle,
{\displaystyle |\psi (x,t)|^{2}}
is the probability density at position
{\displaystyle x}
{\displaystyle t}
{\displaystyle |\psi (x_{0},t)|^{2}\,dx}
means the probability of finding the particle near
{\displaystyle x_{0}}
The wave function of scattering state can be understood as a propagating wave. See also "bound state".
{\displaystyle E}
be the expectation energy of the state. It is a scattering state iff
{\displaystyle E>\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}
Square-integrable is a necessary condition for a function being the position/momentum representation of a wave function of a bound state of the system.
Given the position representation
{\displaystyle \Psi (x,t)}
of a state vector of a wave function, square-integrable means:
1D case:
{\displaystyle \int _{-\infty }^{+\infty }|\Psi (x,t)|^{2}\,dx<+\infty }
{\displaystyle \int _{V}|\Psi (\mathbf {r} ,t)|^{2}\,dV<+\infty }
A stationary state of a bound system is an eigenstate of Hamiltonian operator. Classically, it corresponds to standing wave. It is equivalent to the following things:[nb 2]
an eigenstate of the Hamiltonian operator
an eigenfunction of Time-Independent Schrödinger Equation
a state of definite energy
a state which "every expectation value is constant in time"
a state whose probability density (
{\displaystyle |\psi (x,t)|^{2}}
) does not change with respect to time, i.e.
{\displaystyle {\frac {d}{dt}}|\Psi (x,t)|^{2}=0}
Measurement postulates[edit]
Main article: Measurement in quantum mechanics
The probability of the state
{\displaystyle |\alpha \rangle }
collapse to an eigenstate
{\displaystyle |k\rangle }
of an observable is given by
{\displaystyle |\langle k|\alpha \rangle |^{2}}
"Collapse" means the sudden process which the state of the system will "suddenly" change to an eigenstate of the observable during measurement.
An eigenstate of an operator
{\displaystyle A}
is a vector satisfied the eigenvalue equation:
{\displaystyle A|\alpha \rangle =c|\alpha \rangle }
{\displaystyle c}
Usually, in bra–ket notation, the eigenstate will be represented by its corresponding eigenvalue if the corresponding observable is understood.
{\displaystyle <M>}
of the observable M with respect to a state
{\displaystyle |\alpha }
is the average outcome of measuring
{\displaystyle M}
with respect to an ensemble of state
{\displaystyle |\alpha }
{\displaystyle <M>}
{\displaystyle <M>=\langle \alpha |M|\alpha \rangle }
If the state is given by a density matrix
{\displaystyle \rho }
{\displaystyle <M>=\operatorname {Tr} (M\rho )}
An operator satisfying
{\displaystyle A=A^{\dagger }}
{\displaystyle \langle \alpha |A|\alpha \rangle =\langle \alpha |A^{\dagger }|\alpha \rangle }
for all allowable wave function
{\displaystyle |\alpha \rangle }
Mathematically, it is represented by a Hermitian operator.
Indistinguishable particles[edit]
Intrinsically identical particles
If the intrinsic properties (properties that can be measured but are independent of the quantum state, e.g. charge, total spin, mass) of two particles are the same, they are said to be (intrinsically) identical.
If a system shows measurable differences when one of its particles is replaced by another particle, these two particles are called distinguishable.
Bosons are particles with integer spin (s = 0, 1, 2, ... ). They can either be elementary (like photons) or composite (such as mesons, nuclei or even atoms). There are five known elementary bosons: the four force carrying gauge bosons γ (photon), g (gluon), Z (Z boson) and W (W boson), as well as the Higgs boson.
Fermions are particles with half-integer spin (s = 1/2, 3/2, 5/2, ... ). Like bosons, they can be elementary or composite particles. There are two types of elementary fermions: quarks and leptons, which are the main constituents of ordinary matter.
Anti-symmetrization of wave functions
Symmetrization of wave functions
Quantum statistical mechanics[edit]
Bose–Einstein distribution
Bose–Einstein condensation state (BEC state)
Nonlocality[edit]
Rotation: spin/angular momentum[edit]
singlet state and triplet state
Approximation methods[edit]
Historical Terms / semi-classical treatment[edit]
A theorem connecting the classical mechanics and result derived from Schrödinger equation.
{\displaystyle x\to {\hat {x}},\,p\to i\hbar {\frac {\partial }{\partial x}}}
Uncategorized terms[edit]
List of mathematical topics in quantum theory
List of quantum-mechanical potentials
^ Exception: superselection rules
^ Some textbooks (e.g. Cohen Tannoudji, Liboff) define "stationary state" as "an eigenstate of a Hamiltonian" without specific to bound states.
Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN 0-306-44790-8.
Claude Cohen-Tannoudji; Bernard Diu; Frank Laloë (2006). Quantum Mechanics. Wiley-Interscience. ISBN 978-0-471-56952-7.
Graduate textook
Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0-201-53929-2.
Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel, eds. (2009). Compendium of Quantum Physics - Concepts, Experiments, History and Philosophy. Springer. ISBN 978-3-540-70622-9.
d'Espagnat, Bernard (2003). Veiled Reality: An Analysis of Quantum Mechanical Concepts (1st ed.). US: Westview Press.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glossary_of_elementary_quantum_mechanics&oldid=1034881609"
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Sierpiński carpet - Wikipedia
Plane fractal built from squares
"Sierpinski snowflake" redirects here. For other uses, see Sierpinski curve.
6 steps of a Sierpinski carpet.
The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust.
The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing recursively can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the Sierpiński triangle. In three dimensions, a similar construction based on cubes is known as the Menger sponge.
3 Brownian motion on the Sierpiński carpet
4 Wallis sieve
The construction of the Sierpiński carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position, using the infinitesimal number representation of
{\displaystyle 0.1111\dots =0.2}
The process of recursively removing squares is an example of a finite subdivision rule.
Variant of the Peano curve with the middle line erased creates a Sierpiński carpet
The area of the carpet is zero (in standard Lebesgue measure).
Proof: Denote as ai the area of iteration i. Then ai + 1 = 8/9ai. So ai = (8/9)i, which tends to 0 as i goes to infinity.
The interior of the carpet is empty.
Proof: Suppose by contradiction that there is a point P in the interior of the carpet. Then there is a square centered at P which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of 1/3k for some k. But, this square must have been holed in iteration k, so it cannot be contained in the carpet – a contradiction.
The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.[2]
Sierpiński demonstrated that his carpet is a universal plane curve.[3] That is: the Sierpinski carpet is a compact subset of the plane with Lebesgue covering dimension 1, and every subset of the plane with these properties is homeomorphic to some subset of the Sierpiński carpet.
This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve. However, in 1958 Gordon Whyburn[4] uniquely characterized the Sierpiński carpet as follows: any curve that is locally connected and has no 'local cut-points' is homeomorphic to the Sierpinski carpet. Here a local cut-point is a point p for which some connected neighborhood U of p has the property that U − {p} is not connected. So, for example, any point of the circle is a local cut point.
In the same paper Whyburn gave another characterization of the Sierpiński carpet. Recall that a continuum is a nonempty connected compact metric space. Suppose X is a continuum embedded in the plane. Suppose its complement in the plane has countably many connected components C1, C2, C3, ... and suppose:
the diameter of Ci goes to zero as i → ∞;
the boundary of Ci and the boundary of Cj are disjoint if i ≠ j;
the boundary of Ci is a simple closed curve for each i;
the union of the boundaries of the sets Ci is dense in X.
Then X is homeomorphic to the Sierpiński carpet.
Brownian motion on the Sierpiński carpet[edit]
The topic of Brownian motion on the Sierpiński carpet has attracted interest in recent years.[5] Martin Barlow and Richard Bass have shown that a random walk on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to √n after n steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to β√n for some β > 2. They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years.
Wallis sieve[edit]
Third iteration of the Wallis sieve
A variation of the Sierpiński carpet, called the Wallis sieve, starts in the same way, by subdividing the unit square into nine smaller squares and removing the middle of them. At the next level of subdivision, it subdivides each of the squares into 25 smaller squares and removes the middle one, and it continues at the ith step by subdividing each square into (2i + 1)2 (the odd squares[6]) smaller squares and removing the middle one. By the Wallis product, the area of the resulting set is π/4, unlike the standard Sierpiński carpet which has zero limiting area. Although the Wallis sieve has positive Lebesgue measure, no subset that is a Cartesian product of two sets of real numbers has this property, so its Jordan measure is zero.[7][8]
Mobile phone and Wi-Fi fractal antennas have been produced in the form of few iterations of the Sierpiński carpet. Due to their self-similarity and scale invariance, they easily accommodate multiple frequencies. They are also easy to fabricate and smaller than conventional antennas of similar performance, thus being optimal for pocket-sized mobile phones.
^ Semmes, Stephen (2001). Some Novel Types of Fractal Geometry. Oxford Mathematical Monographs. Oxford University Press. p. 31. ISBN 0-19-850806-9. Zbl 0970.28001.
^ Sierpiński, Wacław (1916). "Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée". C. R. Acad. Sci. Paris (in French). 162: 629–632. ISSN 0001-4036. JFM 46.0295.02.
^ Whyburn, Gordon (1958). "Topological chcracterization of the Sierpinski curve". Fund. Math. 45: 320–324. doi:10.4064/fm-45-1-320-324.
^ Barlow, Martin; Bass, Richard, Brownian motion and harmonic analysis on Sierpiński carpets (PDF)
^ Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. JSTOR 2324662. MR 1247533.
^ Weisstein, Eric W. "Wallis Sieve". MathWorld.
Wikimedia Commons has media related to Sierpinski carpet.
Sierpiński Cookies
Sierpinski Carpet solved by means of modular arithmetics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sierpiński_carpet&oldid=1063418722"
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Pfam: Family: NO_synthase (PF02898)
1326 structures 1024 species 0 interactions 2340 sequences 52 architectures
Family: NO_synthase (PF02898)
Summary: Nitric oxide synthase, oxygenase domain
Wikipedia: Nitric oxide synthase
This is the Wikipedia entry entitled "Nitric oxide synthase". More...
Nitric oxide synthase Edit Wikipedia article
The Nitric Oxide Synthases (NOS) are a group of enzymes responsible for the synthesis of [Nitric Oxide] from the terminal guanidino nitrogen atom of the semi-essential exclusive precursor amino acid L-Arginine in the presence of O2 and the cofactors Nicotinamide adenine dinucleotide phosphate (NADPH) and Flavin adenine dinucleotide (FAD) and tetrahydrobiopterin(BH4) and calmodulin. L-citrulline is a byproduct.
The different forms of NO synthase have been classified as
a. Neuronal NOS (nNOS or NOS1) which produces NO in neuronal tissue in both the central and peripheral nervous system. Neuronal NOS also performs a role in cell communication.
b. Inducible NOS (iNOS or NOS2) which can be found in the immune system but is also found in the cardiovascular system. It uses the oxidative stress of NO (a free radical) to be used by macrophages in immune defence against pathogens.
c. Endothelial NOS (eNOS or NOS3) generates NO in blood vessels and is involved with regulating vascular function. A constitutive Ca
{\displaystyle 2+}
dependent NOS provides a basal release of NO.
All three isoforms (each of which is presumed to function as a homodimer during activation) share a carboxyl-terminal reductase domain homologous to the cytochrome p450 reductases. They also share an amino-terminal oxygenase domain containing a heme prosthetic group, which are linked in the middle of the protein by a calmodullin-binding domain. Binding of calmodullin appears to act as a "molecular switch" to enable electron flow from flavin prosthetic groups in the reductase domain to heme. This facilitates the conversion of O2 and L-arginine to NO and L-citrulline. The reductase domain of each NOS isoform also contains an H4B prosthetic group, which is required for the efficient generation of NO. Unlike other enzymes where H4B is used as a source of reducting equivalents and is recycled by dihyrobiopterin reductase H4B, H4B appears to be necessary to maintain a stable conformation for electron transport possible by promoting homodimerization.
The originally identified nitric oxide synthase was the NOS isoform identified in neuronal tissue known as nNOS or NOS1 followed by the endothelial NOS called eNOS or NOS3. They were originally classified as "constituitvely expressed" and "Ca2+ sensitive" but it is now known that they are present in may different cell types and that expression is regulated under specific physiological conditions.
In NOS1 (neuronal) and NOS3 (endothelial), physiological concentrations of Ca2+ in cells regulate the binding of calmodullin to the "latch domains" thereby initiating electron transfer from the flavins to the heme moieties. In contrast, calmodullin remains tightly bound to the inducible and Ca2+ insensitive isoform termed iNOS or NOS2 even at a low intracellular Ca2+ activity, acting essentially as a subunit of this isoform.
It is interesting that NO may itself regulate NOS expression and activity and has been shown to play an important negative feedback regulatory role on endothelial NO synthase, and therefore vascular endothelial cell function. Both NOS1 and NOS2 have been shown to form ferrous-nitrosyl complexes in their heme prosthetic groups that may act partially to self inactivate these enzymes under certain conditions. The rate-limiting step for the production of Nitric Oxide may well be the availability of L-arginine in some cell types. This is may particularly be important after the induction of NOS2
Nitric Oxide Release and Deactivation
Nitric Oxide generally exists as a lipophilic inorganic gas and is usually able to diffuse from producer to target cell. On reaching the vascular smooth muscle cells nitric oxide activates the soluble cyclic guanylate cyclase, which results in the formation of soluble cGMP. Nitric Oxide has an extremely short half-life, which has been estimated to be less than 4 seconds in biological solutions due to the rapid reaction with oxygen-derived free radicals, in particular the superoxide anion and oxyhaemoglobin. Nitric Oxide is rapidly oxidised by oxygenated haemoglobin to nitrite and then nitrate before being excreted in the urine. In addition to the direct effects of nitric oxide there is also evidence that it may well exert its effects through the formation of S-nitroso-thiols and metal-nitrosyl complexes, which can act as circulating reservoirs of nitric oxide.
The increase in soluble cGMP is matched with a decrease in intracellular calcium, which results in relaxation of the vascular smooth muscle. Nitric oxide can diffuse across the endothelial cell membrane to enter the adjacent vascular smooth muscle cells or alternatively pass into the lumen where it prevents platelet adhesion and aggregation by raising the level of cGMP in platelets. Nitric oxide also interacts with enzymes of the respiratory chain including aconitase and complex I and II and by this way can later tissue mitochondrial respiration.
Under resting conditions, it has been shown by studies on forearm blood flow that there is a continuous basal release of nitric oxide from the vascular endothelium. Infusing the NOS inhibitor L-NMMA into forearm vessels demonstrated a 50% fall in basal blood flow and attenuated the dilator response to infused acetylcholine but did not attenuate the vasodilatation due to glyceryl trinitrate. Basal release of NO is primarily due to shear stress or "viscous drag" which is determined by the vasoconstriction and flow rate and viscosity of the blood. This acts to balance the neurogenic and myogenic mediated vasoconstriction. The release of NO can be increased rapidly following the activation of specific receptors on endothelial cells resulting in an increased intracellular concentration of free calcium.
The other important physiological stimulants of NO release are factors released from aggregating platelets, such as serotonin and adenosine nucleotides. This strong release of NO upon exposure to platelet products and thrombin is of crucial importance in preventing unwarranted intravascular coagulation assuming the endothelium is intact.
The most important role of nitric oxide as we have discussed is in the control of vascular tone. Aside from the response to acetylcholine, the release of NO is stimulated by shear stress, serotonin and ADP as mentioned above. Other stimulants include thrombin and histamine. Only a few vasodilators work independently of the endothelium, most notably the nitrovasodilators such as nitroprusside and nitroglycerine, and other mediators such as prostacyclin and adenosine
Nitric oxide synthase, oxygenase domain Provide feedback
Crane BR, Arvai AS, Ghosh DK, Wu C, Getzoff ED, Stuehr DJ, Tainer JA; , Science 1998;279:2121-2126.: Structure of nitric oxide synthase oxygenase dimer with pterin and substrate. PUBMED:9516116 EPMC:9516116
SCOP: 1nos
This entry represents the N-terminal of the nitric oxide synthases.
Nitric oxide synthase ( EC ) (NOS) enzymes produce nitric oxide (NO) by catalysing a five-electron oxidation of a guanidino nitrogen of L-arginine (L-Arg). Oxidation of L-Arg to L-citrulline occurs via two successive monooxygenation reactions producing N(omega)-hydroxy-L-arginine as an intermediate. 2 mol of O(2) and 1.5 mol of NADPH are consumed per mole of NO formed [ PUBMED:8782597 ].
Arginine-derived NO synthesis has been identified in mammals, fish, birds, invertebrates, plants, and bacteria [ PUBMED:8782597 ]. Best studied are mammals, where three distinct genes encode NOS isozymes: neuronal (nNOS or NOS-1), cytokine-inducible (iNOS or NOS-2) and endothelial (eNOS or NOS-3) [ PUBMED:7510950 ]. iNOS and nNOS are soluble and found predominantly in the cytosol, while eNOS is membrane associated. The enzymes exist as homodimers, each monomer consisting of two major domains: an N-terminal oxygenase domain, which belongs to the class of haem-thiolate proteins, and a C-terminal reductase domain, which is homologous to NADPH:P450 reductase ( EC ). The interdomain linker between the oxygenase and reductase domains contains a calmodulin (CaM)-binding sequence. NOSs are the only enzymes known to simultaneously require five bound cofactors animal NOS isozymes are catalytically self-sufficient. The electron flow in the NO synthase reaction is: NADPH --> FAD --> FMN --> haem --> O(2).
eNOS localisation to endothelial membranes is mediated by cotranslational N-terminal myristoylation and post-translational palmitoylation [ PUBMED:9199168 ]. The subcellular localisation of nNOS in skeletal muscle is mediated by anchoring of nNOS to dystrophin. nNOS contains an additional N-terminal domain, the PDZ domain [ PUBMED:7535955 ]. Some bacteria, like Bacillus halodurans, Bacillus subtilis or Deinococcus radiodurans, contain homologues of NOS oxygenase domain.
Molecular function nitric-oxide synthase activity (GO:0004517)
Biological process nitric oxide biosynthetic process (GO:0006809)
For those sequences which have a structure in the Protein DataBank, we use the mapping between UniProt, PDB and Pfam coordinate systems from the PDBe group, to allow us to map Pfam domains onto UniProt sequences and three-dimensional protein structures. The table below shows the structures on which the NO_synthase domain has been found. There are 1326 instances of this domain found in the PDB. Note that there may be multiple copies of the domain in a single PDB structure, since many structures contain multiple copies of the same protein sequence.
A2BIN8 View 3D Structure Click here
B1B557 View 3D Structure Click here
F1QML0 View 3D Structure Click here
K0EMM7 View 3D Structure Click here
Q8T8C0 View 3D Structure Click here
Q9RR97 View 3D Structure Click here
Q9TUX8 View 3D Structure Click here
Q9Z0J4 View 3D Structure Click here
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Global Constraint Catalog: Kcompulsory_part
<< 3.7.48. Coloured3.7.50. Conditional constraint >>
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚠𝚘}_𝚍
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}
A constraint for which the filtering algorithm may use the notion of compulsory part. The notion of compulsory part was introduced by A. Lahrichi within the context of cumulative scheduling problems [Lahrichi79], [Lahrichi82], [Lahrichi82a] as well as within the context of rectangles placement problems [LahrichiGondran84]. Within these two contexts, the compulsory part respectively corresponds to the intersection of all feasible instances of a task or to the intersection of all feasible instances of a rectangle.
Figure 3.7.13. Illustration of the notion of compulsory part
Figure 3.7.13 illustrates the notion of compulsory part in the context of scheduling and placement problems. The first, second and third rows respectively corresponds to the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
[AggounBeldiceanu93], the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚛𝚊𝚙𝚎𝚣𝚎}
[Poder02], [PoderBeldiceanuSanlaville04] and the
\mathrm{𝚍𝚒𝚏𝚏𝚗}
[BeldiceanuGuoThiel01] constraints. The first, second and third columns respectively correspond to the shape of the object for which we compute the compulsory part, to the extreme positions of the object and to the corresponding compulsory part. When both, the shape of an object is convex and the domain of its origin is also convex, we do not need to consider all feasible instances of the object to compute its compulsory part. We only need to position the object to the extreme positions of its domain and to compute the intersection to get its compulsory part [BeldiceanuGuoThiel01].
This is the case of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint where a task is positioned to its earliest and latest starts
{s}_{\mathrm{𝑚𝑖𝑛}}
{s}_{\mathrm{𝑚𝑎𝑥}}
(see the first row and second column of Figure 3.7.13).
This is also the case of the
\mathrm{𝚍𝚒𝚏𝚏𝚗}
constraint where an orthotope is positioned to its
2·n
extreme positions, where
n
is the number of dimensions of the placement space (see the third row and second column of Figure 3.7.13, where the origin of the rectangle is fixed to the extreme positions
\left({s}_{{x}_{\mathrm{𝑚𝑖𝑛}}},{s}_{{y}_{\mathrm{𝑚𝑖𝑛}}}\right)
\left({s}_{{x}_{\mathrm{𝑚𝑎𝑥}}},{s}_{{y}_{\mathrm{𝑚𝑖𝑛}}}\right)
\left({s}_{{x}_{\mathrm{𝑚𝑖𝑛}}},{s}_{{y}_{\mathrm{𝑚𝑎𝑥}}}\right)
\left({s}_{{x}_{\mathrm{𝑚𝑎𝑥}}},{s}_{{y}_{\mathrm{𝑚𝑎𝑥}}}\right)
But this is not the case of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚝𝚛𝚊𝚙𝚎𝚣𝚎}
constraint with a task that has a valley, i.e. a task for which a resource consumption decrease is followed by a resource consumption increase. In addition of computing the intersection between the two extreme positions
{\mathrm{I}}^{\mathrm{min}}
{\mathrm{I}}^{\mathrm{max}}
of a task, we must also consider the valleys to further reduce
{\mathrm{I}}^{\mathrm{min}}\cap {\mathrm{I}}^{\mathrm{max}}
as explained now [PoderBeldiceanuSanlaville04]. The end of a valley is the lowest rightmost point of a valley. We must remove from
{\mathrm{I}}^{\mathrm{min}}\cap {\mathrm{I}}^{\mathrm{max}}
all parts that are located both (1) between the earliest start and latest end of the valley end, (2) and on top of the valley. Figure 3.7.14 illustrates this point for the task used in the second row and first column of Figure 3.7.13.
Figure 3.7.14. Illustrating the computation of the compulsory part of a task with a valley: (A) the task shape and its valley end in red, (B) in cyan the intersection between the task positioned at its earliest start (in dashed) and its latest start (in dotted); in pink the part located (1) between the earliest and latest positions of the valley end, and (2) on top of the valley, (C) the compulsory part of the task, i.e.,
{\mathrm{I}}^{\mathrm{min}}\cap {\mathrm{I}}^{\mathrm{max}}
from which we remove the pink part on top of the valley.
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Sebaran normal - Wikipédia Sunda, énsiklopédi bébas
Normal distribution (distribusi normal) mangrupa hal anu penting dina probability distribution di loba widang. Biasa ogé disebut Gaussian distribution, hususna dina widang fisika jeung rékayasa. Dina kaayaan sabenerna kumpulan distribusi mibanda bentuk anu sarupa, bédana ngan dina paraméter location jeung scale: mean jeung simpangan baku. Standard normal distribution nyaéta distribusi normal anu mibanda nilai mean sarua jeung nol sarta nilai standar deviasi sarua jeung hiji. Sabab bentuk grafik dénsitas probabilitas mangrupa bell, sering disebut bell curve.
Probability density function of Gaussian distribution (bell curve).
2 Spesifikasi sebaran normal
2.1 Fungsi probabiliti densiti
2.2 Fungsi Sebaran Kumulatif
2.3.2 Fungsi karakteristik
3.1 Standardizing normal random variables
3.2 Generating normal random variables
4.1 Photon counts
4.3 Physical characteristics of biological specimens
4.4 Financial variables
4.6 Test scores
6 Tumbu kaluar jeung rujukan
Distribusi normal mimiti dikenalkeun ku de Moivre dina artikel taun 1733 (dicitak ulang edisi kaduana dina The Doctrine of Chances, 1738) dina kontek "pendekatan" sebaran binomial keur n anu loba. Hasil de Moivre diteruskeun ku Laplace dina bukuna Analytical Theory of Probabilities (1812), mangsa kiwari disebut Theorem of de Moivre-Laplace.
Laplace ngagunakeun distribusi normal keur analysis of errors dina percobaanna. Method of least squares nu kacida pentingna dikenalkeun ku Legendre dina taun 1805. Gauss, ogé ngakukeun yén manéhna geus maké métodeu anu sarua ti mimiti taun 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors.
Istilah "bell curve" ngacu ka Jouffret nu ngagunakeun watesan "bell surface" dina taun 1872 keur bivariate normal dina komponen bébas (independent). Istilah "sebaran normal" "ditemukan" sacara sewang-sewangan ku Charles S. Peirce, Francis Galton jeung Wilhelm Lexis kira-kira taun 1875 [Stigler]. This terminology is unfortunate, since it reflects and encourages the fallacy that "everything is Gaussian". (See the discussion of "occurrence" below).
Yen sebaran disebut sebaran normal atawa Gaussian, ngagantikeun sebaran de Moivrean, is just an instance of Stigler's law of eponymy: "No scientific discovery is named after its original discoverer".
Spesifikasi sebaran normalÉdit
Aya sababaraha jalan keur nangtukeun random variable. Anu paling ngagambarkeun nyaéta probability density function (plot at the top), which represents how likely éach value of the random variable is. The cumulative density function is a conceptually cléaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the moments, the cumulants, the characteristic function, the moment-generating function, and the cumulant-generating function. Some of these are very useful for théoretical work, but not intuitive. See probability distribution for a discussion.
All of the cumulants of the normal distribution are zero, except the first two.
Fungsi probabiliti densitiÉdit
Fungsi dénsitas probabilitas dina sebaran normal nu mana méan μ jeung simpangan baku σ (sarua jeung, varian σ2) mangrupa conto fungsi Gauss,
{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}}
(Tempo ogé fungsi eksponensial jeung pi.) Lamun variabel acak X mibanda distribusi ieu, bisa dituliskeun X ~ N(μ, σ2). Lamun μ = 0 jeung σ = 1, distribusi disebut distribusi standar normal, rumusna
{\displaystyle f(x)={1 \over {\sqrt {2\pi }}}\,e^{-{x^{2}/2}}}
Gambar diluhur nunjukkeun grafik probability density function tina sebaran normal nu mana μ = 0 jeung sababaraha nila σ.
For all normal distributions, the density function is symmetric about its méan value. About 68% of the aréa under the curve is within one standard deviation of the méan, 95.5% within two standard deviations, and 99.7% within three standard deviations. The inflection points of the curve occur at one standard deviation away from the méan.
Fungsi Sebaran KumulatifÉdit
Fungsi sebaran kumulatif (saterusna disebut cdf) hartina probabilitas di mana nilai variabel X leuwih leutik tinimbang x, jeung digambarkeun dina watesan fungsi densiti nyaéta
{\displaystyle \Pr(X\leq x)=\int _{-\infty }^{x}{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-(u-\mu )^{2}/(2\sigma ^{2})}\,du}
Standar normal cdf, sacara konvensional dilambangkeun ku
{\displaystyle \Phi }
, mangrupa nilai cdf umum di-evaluasi ku
{\displaystyle \mu =0}
{\displaystyle \sigma =1}
{\displaystyle \Phi (z)=\int _{-\infty }^{z}{1 \over {\sqrt {2\pi }}}\,e^{-{x^{2}/2}}\,dx}
The standard normal cdf can be expressed in terms of a special function called the error function, as
{\displaystyle \Phi (z)={\frac {1}{2}}\left(1+\operatorname {erf} \,{\frac {z}{\sqrt {2}}}\right)}
The following graph shows the cumulative distribution function for values of z from -4 to +4:
On this graph, we see the probability that a standard normal variable has a value less than 0.25 is approximately equal to 0.60.
Generating functionsÉdit
Moment generating functionÉdit
Fungsi karakteristikÉdit
Fungsi karakteristik dihartikeun salaku nilai ekspektasi
{\displaystyle e^{itX}}
. Keur sebaran normal, ieu bisa ditembongkeun dina fungsi karakteristik nyaéta
{\displaystyle \phi _{X}(t)=E\left[e^{itX}\right]=\int _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\,e^{itx}\,dx=e^{i\mu t-\sigma ^{2}t^{2}/2}}
saperti nu katempo ku kuadrat eksponen nu lengkep.
Lamun X ~ N(μ, σ2) sarta a sarta b mangrupa wilangan riil, mangka aX + b ~ N(aμ + b, (aσ)2).
If X1 ~ N(μ1, σ12) and X2 ~ N(μ2, σ22), and X1 and X2 are independent, then X1 + X2 ~ N(μ1 + μ2, σ12 + σ22).
If X1, ..., Xn are independent standard normal variables, then X12 + ... + Xn2 has a sebaran chi-kuadrat with n degrees of freedom.
Standardizing normal random variablesÉdit
As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.
If X is a normal random variable with méan μ and variance σ2, then
{\displaystyle Z={\frac {X-\mu }{\sigma }}}
is a standard normal random variable: Z~N(0,1). An important consequence is that the cdf of a general normal distribution is therefore
{\displaystyle \Pr(X<x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left(1+{\mbox{erf}}\,\left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right)}
Conversely, if Z is a standard normal random variable, then
{\displaystyle X=\sigma Z+\mu \,}
is a normal random variable with méan μ and variance σ2.
The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.
Generating normal random variablesÉdit
For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods; the most basic is to invert the standard normal cdf. More efficient methods are also known. One such method is the Box-Muller transform. The Box-Muller transform takes two uniformly distributed values as input and maps them to two normally distributed values. This requires generating values from a uniform distribution, for which many methods are known. See also random number generators.
The Box-Muller transform is a consequence of Property 3 and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which is éasy to generate).
The central limit theoremÉdit
The normal distribution has the very important property that under certain conditions, the distribution of a sum of a large number of independent variables is approximately normal. This is the so-called central limit theorem.
The practical importance of the central limit théorem is that the normal distribution can be used as an approximation to some other distributions.
Sebaran binomial mibanda paraméter n sarta p ngadeukeutan kana normal keur n nu badag sarta p teu deukeut ka 1 atawa 0. Pendekatan sebaran normal mibanda méan μ = np sarta simpangan baku σ = (n p (1 - p))1/2.
A Poisson distribution with paraméter λ is approximately normal for large λ. The approximating normal distribution has méan μ = λ and standard deviation σ = √λ.
Approximately normal distributions occur in many situations, as a result of the central limit theorem. When there is réason to suspect the presence of a large number of small effects acting additively, it is réasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption.
Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.
Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the théory of errors (see below).
To summarize, here's a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.
In counting problems (so the central limit théorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as
Binomial random variables, associated to yes/no questions;
Poisson random variables, associates to rare events;
In physiological méasurements of biological specimens:
The logarithm of méasures of size of living tissue (length, height, skin aréa, weight);
Other physiological méasures may be normally distributed, but there is no réason to expect that a priori;
Méasurement errors are assumed to be normally distributed, and any deviation from normality must be explained;
The logarithm of interest rates, exchange rates, and inflation; these variables behave like compound interest, not like simple interest, and so are multiplicative;
Stock-market indices are supposed to be multiplicative too, but some reséarchers claim that they are log-Lévy variables instéad of lognormal;
Other financial variables may be normally distributed, but there is no réason to expect that a priori;
The intensity of laser light is normally distributed;
Thermal light has a Bose-Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit théorem.
Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.
Photon countsÉdit
Light intensity from a single source varies with time, and is usually assumed to be normally distributed. However, quantum mechanics interprets méasurements of light intensity as photon counting. Ordinary light sources which produce light by thermal emission, should follow a Poisson distribution or Bose-Einstein distribution on very short time scales. On longer time scales (longer than the coherence time), the addition of independent variables yields an approximately normal distribution. The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution.
Measurement errorsÉdit
Repéated méasurements of the same quantity are expected to yield results which are clustered around a particular value. If all major sources of errors have been taken into account, it is assumed that the remaining error must be the result of a large number of very small additive effects, and hence normal. Deviations from normality are interpreted as indications of systematic errors which have not been taken into account. Note that this is the central assumption of the mathematical theory of errors.
Physical characteristics of biological specimensÉdit
The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore méasures of body size should at most follow a lognormal rather than normal distribution. Despite common claims of normality, the sizes of plants and animals is approximately lognormal. The evidence and an explanation based on modéls of growth was first published in the classic book
Huxley, Julian: Problems of Relative Growth (1932)
Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further maké the joint distribution of sizes deviate from lognormality.
The assumption that linéar size of biological specimens is normal léads to a non-normal distribution of weight (since weight/volume is roughly the 3rd power of length, and gaussian distributions are only preserved by linéar transformations), and conversely assuming that weight is normal léads to non-normal lengths. This is a problem, because there is no a priori réason why one of length, or body mass, and not the other, should be normally distributed. Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed.
blood pressure of adult humans is supposed to be normally distributed, but only after separating males and females into different populations (éach of which is normally distributed)
The length of inert appendages such as hair, nails, teet, claws and shells is expected to be normally distributed if méasured in the direction of growth. This is because the growth of inert appendages depends on the size of the root, and not on the length of the appendage, and so proceeds by additive increments. Hence, we have an example of a sum of very many small lognormal increments approaching a normal distribution. Another plausible example is the width of tree trunks, where a new thin ring if produced every yéar whose width is affected by a large number of factors.
Financial variablesÉdit
Because of the exponential nature of interest and inflation, financial indicators such as interest rates, stock values, or commodity prices maké good examples of multiplicative behaviour. As such, they should not be expected to be normal, but lognormal.
Benoît Mandelbrot, the popularizer of fractals, has claimed that even the assumption of lognormality is flawed.
LifetimeÉdit
Other examples of variables that are not normally distributed include the lifetimes of humans or mechanical devices. Examples of distributions used in this connection are the sebaran eksponensial (memoryless) and the Weibull distribution. In general, there is no réason that waiting times should be normal, since they are not directly related to any kind of additive influence.
Test scoresÉdit
The IQ score of an individual for example can be seen as the result of many small additive influences: many genes and many environmental factors all play a role.
IQ scores and other ability scores are approximately normally distributed. For most IQ tests, the méan is 100 and the standard deviation is 15.
Criticisms: test scores are discrete variable associated with the number of correct/incorrect answers, and as such they are related to the binomial. Moreover (see this USENET post), raw IQ test scores are customarily 'massaged' to force the distribution of IQ scores to be normal. Finally, there is no widely accepted model of intelligence, and the link to IQ scores let alone a relationship between influences on intelligence and additive variations of IQ, is subject to debate.
Tumbu kaluar jeung rujukanÉdit
A. Kropinski's normal distribution tutorial
S. M.Stigler: Statistics on the Table, Harvard University Press 1999, chapter 22. History of the term "normal distribution".
Earliest Known uses of some of the Words of Mathematics. See: [1] for "normal", [2] for "Gaussian", and [3] for "error".
Earliest Uses of Symbols in Probability and Statistics. See Symbols associated with the Normal Distribution.
Dicomot ti "https://su.wikipedia.org/w/index.php?title=Sebaran_normal&oldid=499082"
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Section Exercises | College Algebra | Course Hero
1. Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction
2. Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)
3. Can you explain how to verify a partial fraction decomposition graphically?
4. You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.
5. Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had
\frac{7x+13}{3{x}^{2}+8x+15}=\frac{A}{x+1}+\frac{B}{3x+5}
, we eventually simplify to
7x+13=A\left(3x+5\right)+B\left(x+1\right)
. Explain how you could intelligently choose an
x
-value that will eliminate either
A
B
A
B
For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.
\frac{5x+16}{{x}^{2}+10x+24}
\frac{3x - 79}{{x}^{2}-5x - 24}
\frac{-x - 24}{{x}^{2}-2x - 24}
\frac{10x+47}{{x}^{2}+7x+10}
\frac{x}{6{x}^{2}+25x+25}
\frac{32x - 11}{20{x}^{2}-13x+2}
\frac{x+1}{{x}^{2}+7x+10}
\frac{5x}{{x}^{2}-9}
\frac{10x}{{x}^{2}-25}
\frac{6x}{{x}^{2}-4}
\frac{2x - 3}{{x}^{2}-6x+5}
\frac{4x - 1}{{x}^{2}-x - 6}
\frac{4x+3}{{x}^{2}+8x+15}
\frac{3x - 1}{{x}^{2}-5x+6}
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
\frac{-5x - 19}{{\left(x+4\right)}^{2}}
\frac{x}{{\left(x - 2\right)}^{2}}
\frac{7x+14}{{\left(x+3\right)}^{2}}
\frac{-24x - 27}{{\left(4x+5\right)}^{2}}
\frac{-24x - 27}{{\left(6x - 7\right)}^{2}}
\frac{5-x}{{\left(x - 7\right)}^{2}}
\frac{5x+14}{2{x}^{2}+12x+18}
\frac{5{x}^{2}+20x+8}{2x{\left(x+1\right)}^{2}}
\frac{4{x}^{2}+55x+25}{5x{\left(3x+5\right)}^{2}}
\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\left(3x+2\right)}^{2}}
\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\left({x}^{2}+12x+36\right)}
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.
\frac{4{x}^{2}+6x+11}{\left(x+2\right)\left({x}^{2}+x+3\right)}
\frac{4{x}^{2}+9x+23}{\left(x - 1\right)\left({x}^{2}+6x+11\right)}
\frac{-2{x}^{2}+10x+4}{\left(x - 1\right)\left({x}^{2}+3x+8\right)}
\frac{{x}^{2}+3x+1}{\left(x+1\right)\left({x}^{2}+5x - 2\right)}
\frac{4{x}^{2}+17x - 1}{\left(x+3\right)\left({x}^{2}+6x+1\right)}
\frac{4{x}^{2}}{\left(x+5\right)\left({x}^{2}+7x - 5\right)}
\frac{4{x}^{2}+5x+3}{{x}^{3}-1}
\frac{-5{x}^{2}+18x - 4}{{x}^{3}+8}
\frac{3{x}^{2}-7x+33}{{x}^{3}+27}
\frac{{x}^{2}+2x+40}{{x}^{3}-125}
\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}
\frac{-50{x}^{2}+5x - 3}{125{x}^{3}-1}
\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}
For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.
\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\left({x}^{2}+4\right)}^{2}}
\frac{{x}^{3}+6{x}^{2}+5x+9}{{\left({x}^{2}+1\right)}^{2}}
\frac{{x}^{3}-{x}^{2}+x - 1}{{\left({x}^{2}-3\right)}^{2}}
\frac{{x}^{2}+5x+5}{{\left(x+2\right)}^{2}}
\frac{{x}^{3}+2{x}^{2}+4x}{{\left({x}^{2}+2x+9\right)}^{2}}
\frac{{x}^{2}+25}{{\left({x}^{2}+3x+25\right)}^{2}}
\frac{2{x}^{3}+11x+7x+70}{{\left(2{x}^{2}+x+14\right)}^{2}}
\frac{5x+2}{x{\left({x}^{2}+4\right)}^{2}}
\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\left({x}^{2}+6\right)}^{2}}
\frac{2x - 9}{{\left({x}^{2}-x\right)}^{2}}
\frac{5{x}^{3}-2x+1}{{\left({x}^{2}+2x\right)}^{2}}
For the following exercises, find the partial fraction expansion.
\frac{{x}^{2}+4}{{\left(x+1\right)}^{3}}
\frac{{x}^{3}-4{x}^{2}+5x+4}{{\left(x - 2\right)}^{3}}
For the following exercises, perform the operation and then find the partial fraction decomposition.
\frac{7}{x+8}+\frac{5}{x - 2}-\frac{x - 1}{{x}^{2}-6x - 16}
\frac{1}{x - 4}-\frac{3}{x+6}-\frac{2x+7}{{x}^{2}+2x - 24}
\frac{2x}{{x}^{2}-16}-\frac{1 - 2x}{{x}^{2}+6x+8}-\frac{x - 5}{{x}^{2}-4x}
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Starts the automated CCD data reduction GUI
This command starts the CCDPACK reduction GUI.
The GUI is specifically designed to help the inexperienced or occasional reducer of CCD data (although others will also find it of use). These aims are meet by providing an easy to use, X based, graphical interface that features contextual help and that limits options to those of immediate relevance. It concentrates on data organization and the definition of any CCD characteristics rather than on the nature and control of the core CCDPACK reduction programs.
The reduction of the actual data is separate to the GUI and uses the automated scheduling facilities of CCDPACK.
The interface can be configured by controlling the values of various CCDxxxxx global variables. These can be set in either a global configuration file called ".ccdpack" which should be placed in the $HOME directory, or by loading as part of a state from a local ".ccdpack" file. The names and functions of the more significant configurations follows.
CCDbrowser, the name of the WWW browser used to show hypertext help. This may only be Mosaic or netscape (or whatever the names of these browsers are on your system) and should be the full path names if they are not located on your PATH.
This option can also be set using the environment variable HTX_BROWSER.
The default is [Mm]osiac followed by [Nn]etscape.
CCDstarhtml, the top directories that contains the Starlink HTML documents (in particular sun139 and ccdpack hypertext help). This defaults to
$CCDPACK_DIR/../../docs:$CCDPACK_DIR/../../help.
CCDprefs, this is an array of values that define widget preferences such as the colour scheme and the reliefs etc. The more interesting elements are:
(priority), this defines the priority of the preferences. If you want to override colours and fonts etc. from your .Xdefaults then set this value to widgetDefault. The normal value is userDefault as I think it looks nice the way it is.
(font_size), this is set to 12 or 14. Normally this is set to 14 if your display has more than 800 pixels in both dimensions.
(scheme_colour), this controls the scheme of colours used by the interface. XREDUCE has its own scheme but you override this by setting this to a new colour for the background, the other colours will be derived from this. For finer control see the palette.tcl script in the Tcl distribution.
(click_for_focus), this controls how the focus moves between the various widgets. If you set this to 0 (false), then the focus follows the cursor position.
CCDdetectorcache, the directory that contains the known detector setups and import tables. Defaults to CCDPACK_DIR. If the variable CCDPACK_CONFIG is set this directory is also used.
An example configuration file follows:
\sim
/.ccdpack
set CCDbrowser netscape
set CCDprefs(priority) widgetDefault
set CCDprefs(scheme_colour) bisque
set CCDprefs(click_for_focus) 0
set CCDdetectorcache /home/user/ccdsetups
This sets the default browser to netscape, allows your .Xdefaults to override any internal preferences, makes the focus follow the mouse and defines a local directory that contains setups and import tables.
“Using the CCDPACK data reduction GUI”, REDUCE.
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Global Constraint Catalog: KCostas_arrays
<< 3.7.66. Core3.7.68. Cost filtering constraint >>
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
A constraint that allows for expressing the Costas arrays problem. A Costas array is a permutation
{p}_{1},{p}_{2},\cdots ,{p}_{n}
n
1,2,\cdots ,n
\forall \delta \in \left[1,n-2\right],\forall i\in \left[1,n-\delta -1\right],\forall j\in \left[i+1,n-\delta \right]:{p}_{i}-{p}_{i+\delta }\ne {p}_{j}-{p}_{j+\delta }
. A. Vellino compares in [Vellino90] three approaches respectively using Prolog, Pascal and CHIP for solving the Costas arrays problem. In fact the weaker formulation
\forall \delta \in \left[1,⌊\frac{n-1}{2}⌋\right],\forall i\in \left[1,n-\delta -1\right],\forall j\in \left[i+1,n-\delta \right]:{p}_{i}-{p}_{i+\delta }\ne {p}_{j}-{p}_{j+\delta }
was shown to be equivalent to the original one in [Chang87].
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Natural language processing - CodeDocs
1.1 Symbolic NLP (1950s - early 1990s)
1.2 Statistical NLP (1990s - 2010s)
Natural language processing has its roots in the 1950s. Already in 1950, Alan Turing published an article titled "Computing Machinery and Intelligence" which proposed what is now called the Turing test as a criterion of intelligence, a task that involves the automated interpretation and generation of natural language, but at the time not articulated as a problem separate from artificial intelligence.
The premise of symbolic NLP is well-summarized by John Searle's Chinese room experiment: Given a collection of rules (e.g., a Chinese phrasebook, with questions and matching answers), the computer emulates natural language understanding (or other NLP tasks) by applying those rules to the data it is confronted with.
1970s: During the 1970s, many programmers began to write "conceptual ontologies", which structured real-world information into computer-understandable data. Examples are MARGIE (Schank, 1975), SAM (Cullingford, 1978), PAM (Wilensky, 1978), TaleSpin (Meehan, 1976), QUALM (Lehnert, 1977), Politics (Carbonell, 1979), and Plot Units (Lehnert 1981). During this time, the first many chatterbots were written (e.g., PARRY).
In the 2010s, representation learning and deep neural network-style machine learning methods became widespread in natural language processing, due in part to a flurry of results showing that such techniques[7][8] can achieve state-of-the-art results in many natural language tasks, for example in language modeling,[9] parsing,[10][11] and many others. This is increasingly important in medicine and healthcare, where NLP is being used to analyze notes and text in electronic health records that would otherwise be inaccessible for study when seeking to improve care.[12]
Despite the popularity of machine learning in NLP research, symbolic methods are still (2020) commonly used
Many different classes of machine-learning algorithms have been applied to natural-language-processing tasks. These algorithms take as input a large set of "features" that are generated from the input data. Increasingly, however, research has focused on statistical models, which make soft, probabilistic decisions based on attaching real-valued weights to each input feature. Such models have the advantage that they can express the relative certainty of many different possible answers rather than only one, producing more reliable results when such a model is included as a component of a larger system.
A major drawback of statistical methods is that they require elaborate feature engineering. Since 2015,[17] the field has thus largely abandoned statistical methods and shifted to neural networks for machine learning. Popular techniques include the use of word embeddings to capture semantic properties of words, and an increase in end-to-end learning of a higher-level task (e.g., question answering) instead of relying on a pipeline of separate intermediate tasks (e.g., part-of-speech tagging and dependency parsing). In some areas, this shift has entailed substantial changes in how NLP systems are designed, such that deep neural network-based approaches may be viewed as a new paradigm distinct from statistical natural language processing. For instance, the term neural machine translation (NMT) emphasizes the fact that deep learning-based approaches to machine translation directly learn sequence-to-sequence transformations, obviating the need for intermediate steps such as word alignment and language modeling that was used in statistical machine translation (SMT). Latest works tend to use non-technical structure of a given task to build proper neural network.[18]
Given a piece of text (typically a sentence), produce a formal representation of its semantics, either as a graph (e.g., in AMR parsing) or in accordance with a logical formalism (e.g., in DRT parsing). This challenge typically includes aspects of several more elementary NLP tasks from semantics (e.g., semantic role labelling, word sense disambiguation) and can be extended to include full-fledged discourse analysis (e.g., discourse analysis, coreference; see Natural language understanding below).
The goal of argument mining is the automatic extraction and identification of argumentative structures from natural language text with the aid of computer programs. [24] Such argumentative structures include the premise, conclusions, the argument scheme and the relationship between the main and subsidiary argument, or the main and counter-argument within discourse. [25][26]
Grammatical error detection and correction involves a great band-width of problems on all levels of linguistic analysis (phonology/orthography, morphology, syntax, semantics, pragmatics). Grammatical error correction is impactful since it affects hundreds of millions of people that use or acquire English as a second language. It has thus been subject to a number of shared tasks since 2011.[30][31][32] As far as orthography, morphology, syntax and certain aspects of semantics are concerned, and due to the development of powerful neural language models such as GPT-2, this can now (2019) be considered a largely solved problem and is being marketed in various commercial applications.[33]
Cognition and NLP
Most more higher-level NLP applications involve aspects that emulate intelligent behaviour and apparent comprehension of natural language. More broadly speaking, the technical operationalization of increasingly advanced aspects of cognitive behaviour represents one of the developmental trajectories of NLP (see trends among CoNLL shared tasks above).
Cognition refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses."[36]Cognitive science is the interdisciplinary, scientific study of the mind and its processes.[37]Cognitive linguistics is an interdisciplinary branch of linguistics, combining knowledge and research from both psychology and linguistics.[38] Especially during the age of symbolic NLP, the area of computational linguistics maintained strong ties with cognitive studies.
{\displaystyle {RMM(token_{N})}={PMM(token_{N})}\times {\frac {1}{2d}}\left(\sum _{i=-d}^{d}{((PMM(token_{N-1})}\times {PF(token_{N},token_{N-1}))_{i}}\right)}
^ Kongthon, Alisa; Sangkeettrakarn, Chatchawal; Kongyoung, Sarawoot; Haruechaiyasak, Choochart (October 27–30, 2009). Implementing an online help desk system based on conversational agent. MEDES '09: The International Conference on Management of Emergent Digital EcoSystems. France: ACM. doi:10.1145/1643823.1643908.
^ Goldberg, Yoav (2016). "A Primer on Neural Network Models for Natural Language Processing". Journal of Artificial Intelligence Research. 57: 345–420. arXiv:. doi:10.1613/jair.4992. S2CID 8273530.
^ Jozefowicz, Rafal; Vinyals, Oriol; Schuster, Mike; Shazeer, Noam; Wu, Yonghui (2016). Exploring the Limits of Language Modeling. arXiv:. Bibcode:2016arXiv160202410J.
^ Choe, Do Kook; Charniak, Eugene. "Parsing as Language Modeling". Emnlp 2016.
^ Vinyals, Oriol; et al. (2014). "Grammar as a Foreign Language" (PDF). Nips2015. arXiv:. Bibcode:2014arXiv1412.7449V.
^ Turchin, Alexander; Florez Builes, Luisa F. (2021-03-19). "Using Natural Language Processing to Measure and Improve Quality of Diabetes Care: A Systematic Review". Journal of Diabetes Science and Technology. 15 (3): 553–560. doi:10.1177/19322968211000831. ISSN 1932-2968. PMID 33736486.
^ Annamoradnejad, I. and Zoghi, G. (2020). Colbert: Using bert sentence embedding for humor detection. arXiv preprint arXiv:2004.12765.
^ Yi, Chucai; Tian, Yingli (2012), "Assistive Text Reading from Complex Background for Blind Persons", Camera-Based Document Analysis and Recognition, Springer Berlin Heidelberg, pp. 15–28, CiteSeerX , doi:10.1007/978-3-642-29364-1_2, ISBN 9783642293634
^ Kishorjit, N.; Vidya, Raj RK.; Nirmal, Y.; Sivaji, B. (2012). "Manipuri Morpheme Identification" (PDF). Proceedings of the 3rd Workshop on South and Southeast Asian Natural Language Processing (SANLP). COLING 2012, Mumbai, December 2012: 95–108. CS1 maint: location (link)
^ Lippi, Marco; Torroni, Paolo (2016-04-20). "Argumentation Mining: State of the Art and Emerging Trends". ACM Transactions on Internet Technology. 16 (2): 1–25. doi:10.1145/2850417. ISSN 1533-5399. S2CID 9561587.
^ Writer, Beta (2019). Lithium-Ion Batteries. doi:10.1007/978-3-030-16800-1. ISBN 978-3-030-16799-8.
^ "Document Understanding AI on Google Cloud (Cloud Next '19) - YouTube". www.youtube.com. Retrieved 2021-01-11.
^ "About Us | Grammarly". www.grammarly.com. Retrieved 2021-01-11.
^ Socher, Richard; Karpathy, Andrej; Le, Quoc V.; Manning, Christopher D.; Ng, Andrew Y. (2014). "Grounded Compositional Semantics for Finding and Describing Images with Sentences". Transactions of the Association for Computational Linguistics. 2: 207–218. doi:. S2CID 2317858.
Bates, M (1995). "Models of natural language understanding". Proceedings of the National Academy of Sciences of the United States of America. 92 (22): 9977–9982. Bibcode:1995PNAS...92.9977B. doi:10.1073/pnas.92.22.9977. PMC . PMID 7479812.
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I’m all about making math visual. It’s so important to me that a student is able to see why something works and use a method that makes sense to them and is connected to the underlying mathematics, instead of just relying on a rote procedure or formula.
I was recently teaching my students how to solve for missing side lengths using right triangle trigonometry. We were thinking conceptually about what the trig ratio represents, what it tells us about the comparison of sides, and how we interpret ratios given as decimals. But I found that even after students successfully set up a proportion, they struggled to solve for the missing side. I knew I didn’t want to revert to cross-multiplying because I wanted students to think conceptually about what it meant that one number was 0.45 of another. While I used some guiding questions to try to help students, we ended up still going back to how to isolate x in the equation by using inverse operations. There is nothing particularly wrong with this method, but it didn’t feel very related to the rich work we had been doing with ratios. I knew there had to be a better way.
I realized that even when students understood trigonometry conceptually, they might still struggle with proportional reasoning, which would allow them to solve for a missing side. I had been doing some learning about proportional relationships from some resources for teaching the middle school grades, and I knew there had to be a way to apply that work to high school classes. Well, I found it.
The double number line.
The double number line is a visual tool that can help students solve all kinds of proportional relationship problems. And it's not just any tool, it's like the Swiss Army Knife of tools with a million different uses. We’ll start by looking at a classic proportional relationship problem students might encounter in middle school as a way of learning about the tool and its versatility. We’ll then look at how I think it can be leveraged for high school math.
Example: Suppose that the instructions on a jar of instant coffee say to add 4 tablespoons of coffee granules to 5 cups of water. How many cups of water are needed for 10 tablespoons of coffee?
Note that this problem could be represented as the proportion
\frac{4}{5}=\frac{10}{x}
, what most would consider the
\text{\textquotedblleft}
\text{\textquotedblright}
version of proportion solving because the variable is in the denominator. Isolating x is tricky for students, so most teachers teach cross multiplication.
4x=50
x=12.5
. This is the correct answer of course, but the value of 50 does not seem at all related to the problem. Let’s see how using the double number line would provide students with opportunities for strategic reasoning and sense making. Here's how we might represent the problem.
Now let's look at four possible methods students could use to solve the problem using the same visual tool.
We could consider this the method of iterating (repeating) the composed ratio of 4 tablespoons to 5 cups. That means 8 tablespoons would be 10 cups and 12 tablespoons would be 15 cups. Even though 10 is not a multiple of 4, it is half-way between 8 and 12, so the number of cups of water needed is half-way between 10 and 15, so 12.5.
This method could be called partitioning the composed ratio, and in this case the partition creates a unit ratio. If 4 tablespoons require 5 cups of water, then 1 tablespoon requires 1.25 cups of water, so 10 tablespoons requires 12.5 cups of water.
Partitioning does not necessarily mean finding the unit rate. We could just as easily have said that 2 tablespoons of water requires 2.5 cups of water. Since 10 tablespoons of coffee is 5 times the amount of coffee, we will need 2.5(5)=12.5 cups of water.
This next method I would call the multiplicative factor method. How many times bigger is 10 tablespoons than 4 tablespoons? Note here that we are reasoning multiplicatively within a single variable. 10 tablespoons of coffee is 2.5 times bigger than 4 tablespoons of coffee. Thus, the number of cups we need for 10 tablespoons must be 2.5 times bigger than the number of cups we need for 4 tablespoons; 5(2.5)=12.5.
We can also reason multiplicatively between two variables. Now we are asking the question: How many times greater is the number of cups of water than the tablespoons of coffee?
Since the number of cups of water is always 1.25 times greater than the number of tablespoons of coffee, 10 tablespoons of coffee will require 10(1.25)=12.5 cups of water.
The goal here is not to teach students
\text{\textquotedblleft}
the 4 methods for solving a proportion using the double number line
\text{\textquotedblright}
as that would proceduralize the process again. Instead, the goal is introduce a tool (the visual representation of a double number line) that will help students think through any proportional reasoning problem in a way that makes sense to them. It is very likely that students' solutions paths will align with one of the methods described above.
Benefits of using the double number line
It doesn’t matter whether the variable is in the numerator or denominator of a proportion.
Students can easily assess the reasonableness of an answer and detect values that don’t make sense, based on the visual.
A single tool offers a wide variety of solution methods, depending on what makes sense to the student and the values given in the problem.
Using the double number line is intuitive and builds on foundational ideas of multiplication and division; students aren’t relying on a set of procedures or algorithms they don’t understand.
So at this point you might be thinking: This is all great, but proportional relationships are not really a high school standard. Besides just to review proportions, how could I use this in the classes I teach?
Great question. It took me a long time to figure out that so much of what we teach in high school actually is just proportional relationships.
Consider how we might use the double number line to reason about the following problem.
∆FIG\sim∆TEA
, find the length of segment FI.
A student might see that a length of 10 on
\Delta TEA
would correspond with a length of 6 on
\Delta FIG
by halving. It follows that a length of 15 on
\Delta TEA
\Delta FIG
, since 15 is half-way between 10 and 20. Students may also split the 20 into fourths and then find three of those fourths. Additionally, since 15 is three fourths of 20, we need three fourths of 12 to get 9. Or, 20 is 1 ⅔ times 12, so 15 divided by 1 ⅔ gives 9.
And finally, how might a flexibility with the double number line support students in solving proportional relationships when the ratio is not given as a fraction or as a comparison of two concrete things, but as a complicated decimal? Is there hope for trigonometry?
This is the most abstract use of the double number line as we are no longer dealing with two concrete quantities. Note that we have no labels on this double number line because the upper part of the number line represents the ratio itself, which is unit-less.
Possible solution methods
How many groups of 0.237 are in 1? Divide 1 by 0.237. That’s how many groups we need of 12.
How many times bigger is 12 than 0.237? Divide 12 by 0.237. That’s how many times bigger the hypotenuse has to be than 1.
How many times bigger is the whole (1), than the part (0.237)? Divide 1 by 0.237. This is how many times bigger the hypotenuse is than the adjacent side, since the adjacent side represents 0.237 parts of the 1 whole hypotenuse.
I love how this tool allows for flexible problem solving and sense making in a wide variety of contexts. Give it a try with your students and let us know how it goes!
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Global Constraint Catalog: Cgolomb
<< 5.168. global_contiguity5.170. graph_crossing >>
Inspired by [Golomb72].
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\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{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
Given a strictly increasing sequence
{X}_{1},{X}_{2},\cdots ,{X}_{n}
, enforce all differences
{X}_{i}-{X}_{j}
between two variables
{X}_{i}
{X}_{j}
\left(i>j\right)
to be distinct.
\left(〈0,1,4,6〉\right)
Figure 5.169.1 gives a graphical interpretation of the solution given in the example in term of a graph: each vertex corresponds to a value of
〈0,1,4,6〉
, while each arc depicts a difference between two values. The
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}
constraint holds since one can note that these differences 1, 4, 6, 3, 5, 2 are all-distinct.
Figure 5.169.1. Graphical representation of the solution
\mathbf{0}
\mathbf{1}
\mathbf{4}
\mathbf{6}
(differences are displayed in light red and are pairwise distinct).
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>2
\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
This constraint refers to the Golomb ruler problem. We quote the definition from [Shearer96]: “A Golomb ruler is a set of integers (marks)
{a}_{1}<\cdots <{a}_{k}
such that all the differences
{a}_{i}-{a}_{j}
\left(i>j\right)
Different constraints models for the Golomb ruler problem were presented in [SmithStergiouWalsh99].
At a first glance, one could think that, because it looks so similar to the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint, we could have a perfect polynomial filtering algorithm. However this is not true since one retrieves the same variable in different vertices of the graph. This leads to the fact that one has incompatible arcs in the bipartite graph (the two classes of vertices correspond to the pair of variables and to the fact that the difference between two pairs of variables takes a specific value). However one can still reuse a similar filtering algorithm as for the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint, but this will not lead to perfect pruning.
n
) 2 3 4 5 6 7 8 9 10 11
Solutions 3 2 2 4 8 10 2 2 2 4
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}
0..k
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
(all different).
\mathrm{𝚜𝚝𝚛𝚒𝚌𝚝𝚕𝚢}_\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}
characteristic of a constraint: disequality, difference, all different, derived collection.
puzzles: Golomb ruler.
•
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐}_\mathrm{𝚗𝚟𝚊𝚕𝚞𝚎}
\left(\mathrm{𝙽𝚅𝙰𝙻},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝙽𝚅𝙰𝙻}=
\mathrm{𝚗𝚟𝚊𝚕}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)
•
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚜𝚘𝚏𝚝}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚌𝚝𝚛}
\left(𝙲,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙿𝙰𝙸𝚁𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚡-\mathrm{𝚍𝚟𝚊𝚛},𝚢-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(𝚡-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛},𝚢-\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\right]\hfill \end{array}\right)
\mathrm{𝙿𝙰𝙸𝚁𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{1},\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{2}\right)
\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{1}.𝚢-\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{1}.𝚡=\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{2}.𝚢-\mathrm{𝚙𝚊𝚒𝚛𝚜}\mathtt{2}.𝚡
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
\le 1
When applied on the collection of items
〈\mathrm{𝚅𝙰𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁}\mathtt{2},\mathrm{𝚅𝙰𝚁}\mathtt{3},\mathrm{𝚅𝙰𝚁}\mathtt{4}〉
, the generator of derived collection generates the following collection of items:
〈\mathrm{𝚅𝙰𝚁}\mathtt{2}\mathrm{𝚅𝙰𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁}\mathtt{3}\mathrm{𝚅𝙰𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁}\mathtt{3}\mathrm{𝚅𝙰𝚁}\mathtt{2},\mathrm{𝚅𝙰𝚁}\mathtt{4}\mathrm{𝚅𝙰𝚁}\mathtt{1},\mathrm{𝚅𝙰𝚁}\mathtt{4}\mathrm{𝚅𝙰𝚁}\mathtt{2},\mathrm{𝚅𝙰𝚁}\mathtt{4}\mathrm{𝚅𝙰𝚁}\mathtt{3}〉
. Note that we use a binary arc constraint between two vertices and that this binary constraint involves four variables.
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐒𝐂𝐂}
graph property we show one of the largest strongly connected components of the final graph. The constraint holds since all the strongly connected components have at most one vertex: the differences 1, 2, 3, 4, 5, 6 that one can construct from the values 0, 1, 4, 6 assigned to the variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection are all-distinct.
\mathrm{𝚐𝚘𝚕𝚘𝚖𝚋}
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Filter and downsample input signals - Simulink - MathWorks Nordic
Filter and downsample input signals
The FIR Decimation block resamples vector or matrix inputs along the first dimension. The FIR decimator (as shown in the schematic) conceptually consists of an anti-aliasing FIR filter followed by a downsampler. To design an FIR anti-aliasing filter, use the designMultirateFIR function.
The FIR filter filters the data in each channel of the input using a direct-form FIR filter. The downsampler that follows downsamples each channel of filtered data by taking every M-th sample and discarding the M – 1 samples that follow. M is the value of the decimation factor that you specify. The resulting discrete-time signal has a sample rate that is 1/M times the original sample rate.
When the Rate options parameter is set to Enforce single-rate processing, the number of rows in the input must be a multiple of the Decimation factor parameter.
H\left(z\right)={b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{N}{z}^{-N}
You can generate the FIR filter coefficient vector, b = [b0, b1, …, bN], using one of the DSP System Toolbox™ filter design functions such as designMultirateFIR, firnyquist, firhalfband, firgr, or firceqrip.
To act as an effective anti-aliasing filter, the coefficients usually correspond to a lowpass filter with a normalized cutoff frequency no greater than 1/M, where M is the decimation factor. To design such a filter, use the designMultirateFIR function.
Coefficient values obtained through Num are tunable, that is, they can change during simulation, while their properties must remain constant.
Out — Decimator output
Output of the FIR Decimator block, returned as a vector or a matrix.
Enforce single-rate processing — The block maintains the input sample rate and decimates the signal by decreasing the output frame size by a factor of M.
Allow multirate processing — The block decimates the signal such that the output sample rate is M times slower than the input sample rate.
Filter object –– Specify the filter using a dsp.FIRDecimator System object™.
Auto –– When you select Auto, the block designs an FIR decimator using the decimation factor that you specify in Decimation factor. The designMultirateFIR function designs the filter and returns the coefficients used by the block.
Specify the lowpass FIR filter coefficients, in descending powers of z, as a vector. By default, designMultirateFIR(1,2) computes the filter coefficients.
H\left(z\right)={b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{N}{z}^{-N}
You can generate the FIR filter coefficient vector, b = [b0, b1, …, bN], using one of the DSP System Toolbox filter design functions such as designMultirateFIR, firnyquist, firhalfband, firgr, or firceqrip.
Specify the integer factor M. The block decreases the sample rate of the input sequence by this factor.
Filter structure — FIR filter structure
Direct form (default) | Direct form transposed
Specify the FIR filter structure as either Direct form or Direct form transposed.
Specify the name of the multirate filter object that you want the block to implement. You must specify the filter as a dsp.FIRDecimator System object.
Rate options — Method by which block decimates input
Specify the method by which the block should decimate the input. You can select one of the following options:
Enforce single-rate processing — When you select this option, the block maintains the input sample rate and decimates the signal by decreasing the output frame size by a factor of M. To select this option, you must set the Input processing parameter to Columns as channels (frame based).
When you set the Rate options parameter to Enforce single-rate processing, you can use the FIR Decimation block inside triggered subsystems.
Allow multirate processing — When you select this option, the block decimates the signal such that the output sample rate is M times slower than the input sample rate.
When you set the FIR Decimation block to the frame-based processing mode, the block can exhibit one-frame latency. In the case of one-frame latency, this parameter specifies the output of the block until the first filtered input sample is available. Specify this parameter as a scalar value to be applied to all signal channels, or as a matrix containing one value for each channel.
Cases of one-frame latency can occur when the input frame size is greater than one, and you set the Input processing and Rate options parameters of the FIR Decimation block as follows:
Input processing set to Columns as channels (frame based)
Rate options set to Allow multirate processing
For more information on latency in the FIR Decimation block, see Latency.
Specify the minimum value of the filter coefficients. The default value is [] (unspecified). Simulink® software uses this value to perform automatic scaling of fixed-point data types.
Use FIR Decimation block in single-rate processing mode.
Use FIR Decimation block in multirate frame-based processing mode.
Polyphase implementation of the FIR decimator.
y\left(n\right)=\sum _{l=0}^{N}h\left(l\right)x\left(nM-l\right)
When you set the Rate options parameter to Enforce single-rate processing, the input and output of the block have the same sample rate. To decimate the output while maintaining the input sample rate, the block resamples the data in each column of the input such that the frame size of the output (Ko) is 1/M times that of the input (Ko = Ki/M),
In this mode, the input frame size, Ki, must be a multiple of the Decimation factor, M.
For an example of single-rate FIR decimation, see FIR Decimation Using Single-Rate Processing.
When you set the Rate options parameter to Allow multirate processing, the input and output of the FIR Decimation block are of the same size, but the sample rate of the output is M times slower than that of the input. In this mode, the block treats a Ki-by-N matrix input as N independent channels. The block decimates each column of the input over time by keeping the frame size constant (Ki=Ko), and making the output frame period (Tfo) M times longer than the input frame period (Tfo = M*Tfi).
See FIR Decimation Using Multirate Frame-Based Processing for an example that uses the FIR Decimation block in this mode.
When you set the Input processing parameter to Elements as channels (sample based), the block treats a P-by-Q matrix input as P*Q independent channels, and decimates each channel over time. The output sample period (Tso) is M times longer than the input sample period (Tso = M*Tsi), and the input and output sizes are identical.
When you use the FIR Decimation block in the sample-based processing mode, the block always has zero-tasking latency. Zero-tasking latency means that the block propagates the first filtered input sample (received at time t= 0) as the first output sample. That first output sample is then followed by filtered input samples M+1, 2M+1, and so on.
When you use the FIR Decimation block in the frame-based processing mode with a frame size greater than one, the block may exhibit one-frame latency. Cases of one-frame latency can occur when the input frame size is greater than one and you set the Input processing and Rate options parameters of the FIR Decimation block as follows:
In cases of one-frame latency, you can define the value of the first Ki output rows by setting the Output buffer initial conditions parameter. The default value of the Output buffer initial conditions parameter is 0. However, you can enter a matrix containing one value for each channel of the input, or a scalar value to be applied to all channels. The first filtered input sample (first filtered row of the input matrix) appears in the output as sample Ki+ 1. That sample is then followed by filtered input samples M+ 1, 2M+ 1, and so on.
H\left(z\right)={b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{N}{z}^{-N}
H\left(z\right)=\begin{array}{c}\left({b}_{0}+{b}_{M}{z}^{-M}+{b}_{2M}{z}^{-2M}+..+{b}_{N-M+1}{z}^{-\left(N-M+1\right)}\right)+\\ {z}^{-1}\left({b}_{1}+{b}_{M+1}{z}^{-M}+{b}_{2M+1}{z}^{-2M}+..+{b}_{N-M+2}{z}^{-\left(N-M+1\right)}\right)+\\ \begin{array}{c}⋮\\ {z}^{-\left(M-1\right)}\left({b}_{M-1}+{b}_{2M-1}{z}^{-M}+{b}_{3M-1}{z}^{-2M}+..+{b}_{N}{z}^{-\left(N-M+1\right)}\right)\end{array}\end{array}
H\left(z\right)={E}_{0}\left({z}^{M}\right)+{z}^{-1}{E}_{1}\left({z}^{M}\right)+...+{z}^{-\left(M-1\right)}{E}_{M-1}\left({z}^{M}\right)
The FIR Decimation block supports SIMD code generation using Intel AVX2 technology under these conditions:
For a FIR decimation filter with hardware-friendly control signals and simulation of HDL latency in Simulink, or for complex data with complex coefficients, use the FIR Decimation (DSP HDL Toolbox) block instead of this block.
HDL Coder supports Coefficient source options Dialog parameters, Filter object, or Auto. Programmable coefficients are not supported.
HDL Coder supports the use of vector inputs to FIR Decimation blocks, where each element of the vector represents a sample in time. You can use an input vector of up to 512 samples. The frame-based implementation supports fixed-point input and output data types, and uses full-precision internal data types. The output is a column vector of reduced size, corresponding to your decimation factor. You can use real input signals with real coefficients, complex input signals with real coefficients, or real input signals with complex coefficients.
Connect a column vector signal to the FIR Decimation block input port.
Specify Input processing as Columns as channels (frame based).
Set Rate options to Enforce single-rate processing.
Right-click the block and open HDL Code > HDL Block Properties. Set the Architecture to Frame Based. The block implements a parallel HDL architecture. See Frame-Based Architecture (HDL Coder).
To use block-level optimizations to reduce hardware resources, set Architecture to Fully Serial or Partly Serial. See HDL Filter Architectures (HDL Coder).
When you specify SerialPartition for a FIR Decimator block, set Filter structure to Direct form. The Direct form transposed structure is not supported with serial architectures. Accumulator reuse is not supported for FIR Decimation filters.
When you select the Distributed Arithmetic (DA) architecture and use the DALUTPartition and DARadix distributed arithmetic properties, set Filter structure to Direct form. The Direct form transposed structure is not supported with distributed arithmetic.
To improve clock speed, use AddPipelineRegisters to use a pipelined adder tree rather than the default linear adder. This option is supported for Direct form architecture. You can also specify the number of pipeline stages before and after the multipliers. See HDL Filter Architectures (HDL Coder).
Programmable coefficients are not supported.
The following diagram shows the data types used within the FIR Decimation block for fixed-point signals.
This diagram shows that data is stored in the input buffer with the same data type and scaling as the input. The block stores filtered data and any initial conditions in the output buffer using the output data type and scaling that you set in the block dialog box.
When the block input is fixed point, all internal data types are signed fixed-point values.
dsp.FIRDecimator | dsp.CICCompensationDecimator | dsp.FIRHalfbandDecimator
FIR Interpolation | FIR Rate Conversion | FIR Halfband Interpolator | FIR Halfband Decimator | IIR Halfband Interpolator | IIR Halfband Decimator | CIC Compensation Interpolator | CIC Compensation Decimator | Downsample | CIC Decimation | Digital Up-Converter | Digital Down-Converter
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Computer approximation for real numbers
{\displaystyle {\text{significand}}\times {\text{base}}^{\text{exponent}},}
{\displaystyle 1.2345=\underbrace {12345} _{\text{significand}}\times \underbrace {10} _{\text{base}}\!\!\!\!\!\!^{\overbrace {-4} ^{\text{exponent}}}.}
1.2 Alternatives to floating-point numbers
2 Range of floating-point numbers
3 IEEE 754: floating point in modern computers
3.2.1 Signed zero
3.2.2 Subnormal numbers
3.2.4 NaNs
3.2.5 IEEE 754 design rationale
4 Other notable floating-point formats
5 Representable numbers, conversion and rounding
5.2 Binary-to-decimal conversion with minimal number of digits
5.3 Decimal-to-binary conversion
6 Floating-point operations
6.3 Literal syntax
7 Dealing with exceptional cases
8.2 Machine precision and backward error analysis
8.3 Minimizing the effect of accuracy problems
8.4 "Fast math" optimization
Floating-point numbers[edit]
{\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},}
{\displaystyle p=24}
{\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.}
{\displaystyle 11001001\ 00001111\ 1101101{\underline {1}}.}
{\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.5707964\times 2\\\approx {}&3.1415928\end{aligned}}}
Alternatives to floating-point numbers[edit]
{\displaystyle \pi }
{\displaystyle {\sqrt {3}}}
{\displaystyle \sin(3\pi )}
{\displaystyle ^{1}/_{\infty }=0}
{\displaystyle 0\times \infty }
{\displaystyle \pm \infty }
Range of floating-point numbers[edit]
{\displaystyle 2\left(B-1\right)\left(B^{P-1}\right)\left(U-L+1\right)}
{\displaystyle B^{L}}
{\displaystyle \left(1-B^{-P}\right)\left(B^{U+1}\right)}
IEEE 754: floating point in modern computers[edit]
Main article: IEEE 754
Internal representation[edit]
Signed zero[edit]
Main article: Signed zero
Main article: Subnormal numbers
Infinities[edit]
Further information on the concept of infinite: Infinity
Main article: NaN
IEEE 754 design rationale[edit]
Other notable floating-point formats[edit]
The Hopper architecture GPUs provide two FP8 formats: one with the same numerical range as half-precision (E5M2) and one with higher precision, but less range (E4M3).[31]
Bfloat16, TensorFloat-32 and the two FP8 formats specifications, compared with IEEE 754 half-precision and single-precision standard formats
FP8 (E4M3) 1 4 3 8
Representable numbers, conversion and rounding [edit]
Rounding modes[edit]
Binary-to-decimal conversion with minimal number of digits[edit]
Decimal-to-binary conversion[edit]
Floating-point operations[edit]
Multiplication and division[edit]
Literal syntax[edit]
Dealing with exceptional cases [edit]
{\displaystyle R_{tot}}
{\displaystyle R_{\text{tot}}=1/(1/R_{1}+1/R_{2}+\cdots +1/R_{n})}
{\displaystyle R_{1}}
{\displaystyle 1/R_{1}}
{\displaystyle R_{tot}}
Accuracy problems[edit]
{\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}
Machine precision and backward error analysis[edit]
{\displaystyle \mathrm {E} _{\text{mach}}=B^{1-P},\,}
{\displaystyle \mathrm {E} _{\text{mach}}={\tfrac {1}{2}}B^{1-P},}
{\displaystyle \left|{\frac {\operatorname {fl} (x)-x}{x}}\right|\leq \mathrm {E} _{\text{mach}}.}
{\displaystyle x}
{\displaystyle y}
{\displaystyle {\begin{aligned}\operatorname {fl} (x\cdot y)&=\operatorname {fl} {\big (}fl(x_{1}\cdot y_{1})+\operatorname {fl} (x_{2}\cdot y_{2}){\big )},{\text{ where }}\operatorname {fl} (){\text{ indicates correctly rounded floating-point arithmetic}}\\&=\operatorname {fl} {\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )},{\text{ where }}\delta _{n}\leq \mathrm {E} _{\text{mach}},{\text{ from above}}\\&={\big (}(x_{1}\cdot y_{1})(1+\delta _{1})+(x_{2}\cdot y_{2})(1+\delta _{2}){\big )}(1+\delta _{3})\\&=(x_{1}\cdot y_{1})(1+\delta _{1})(1+\delta _{3})+(x_{2}\cdot y_{2})(1+\delta _{2})(1+\delta _{3}),\end{aligned}}}
{\displaystyle \operatorname {fl} (x\cdot y)={\hat {x}}\cdot {\hat {y}},}
{\displaystyle {\begin{aligned}{\hat {x}}_{1}&=x_{1}(1+\delta _{1});\quad {\hat {x}}_{2}=x_{2}(1+\delta _{2});\\{\hat {y}}_{1}&=y_{1}(1+\delta _{3});\quad {\hat {y}}_{2}=y_{2}(1+\delta _{3}),\\\end{aligned}}}
{\displaystyle \delta _{n}\leq \mathrm {E} _{\text{mach}}}
Minimizing the effect of accuracy problems[edit]
{\displaystyle (x+y)(x-y)=x^{2}-y^{2}\,}
{\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }=1\,}
{\textstyle t_{0}=1/{\sqrt {3}}}
{\textstyle t_{i+1}=({\sqrt {t_{i}^{2}+1}}-1)/{t_{i}}}
{\textstyle t_{i+1}={t_{i}}/({\sqrt {t_{i}^{2}+1}}+1)}
{\displaystyle \pi \sim 6\times 2^{i}\times t_{i}}
{\displaystyle i\rightarrow \infty }
"Fast math" optimization[edit]
^ "NVIDIA Hopper Architecture In-Depth".
Retrieved from "https://en.wikipedia.org/w/index.php?title=Floating-point_arithmetic&oldid=1089077532"
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Global Constraint Catalog: Csymmetric_alldifferent_loop
<< 5.396. symmetric_alldifferent_except_05.398. symmetric_cardinality >>
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚜𝚢𝚖𝚖}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚜𝚢𝚖𝚖}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚜𝚢𝚖𝚖}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\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}|
\mathrm{𝚜𝚞𝚌𝚌}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[i\right].\mathrm{𝚜𝚞𝚌𝚌}
j
\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[j\right].\mathrm{𝚜𝚞𝚌𝚌}
i
i
j
are not necessarily distinct. This can be interpreted as a graph-covering problem where one has to cover a digraph
G
with circuits of length two or one in such a way that each vertex of
G
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-1,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-4,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-3,\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-2\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
We have two loops respectively corresponding to
\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[1\right].\mathrm{𝚜𝚞𝚌𝚌}=1
\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[3\right].\mathrm{𝚜𝚞𝚌𝚌}=3
We have one circuit of length 2 corresponding to
\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[2\right].\mathrm{𝚜𝚞𝚌𝚌}=4⇔\mathrm{𝙽𝙾𝙳𝙴𝚂}\left[4\right].\mathrm{𝚜𝚞𝚌𝚌}=2
Figure 5.397.1 provides a second example involving a
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
Figure 5.397.1. (A) Magic square Duerer where cells that belong to a same cycle are coloured identically by a colour different from grey; each cell has an index in its upper left corner (in red) and a value (in blue). (B) Corresponding graph where there is an arc from node
i
j
if and only if the value of cell
is equal to the index of cell
j
. (C) Collection of nodes passed to the
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
constraint: the four self-loops of the graph correspond to the four grey cells of the magic square such that the value of the cell (in blue) is equal to the index of the cell (in red).
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
{S}_{1}\in \left[2,5\right]
{S}_{2}\in \left[1,3\right]
{S}_{3}\in \left[1,4\right]
{S}_{4}\in \left[2,4\right]
{S}_{5}\in \left[1,5\right]
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\left(〈1{S}_{1},2{S}_{2},3{S}_{3},4{S}_{4},5{S}_{5}〉\right)
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
constraint of the All solutions slot; the
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚜𝚞𝚌𝚌}
attribute and self loops are coloured in red.
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|\ge 4
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
constraint is described in [Cymer13], [CymerPhD13]. The algorithm is based on the following ideas:
First, one can map solutions of the
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\left(g,f\right)
g\left(x\right)=0
f\left(x\right)=1
x
which have a self-loop, and
g\left(x\right)=f\left(x\right)=1
\left(g,f\right)
ℳ
of a graph is a subset of edges such that every vertex
ℳ
g\left(x\right)
f\left(x\right)
Second, Gallai-Edmonds decomposition [Gallai63], [Edmonds65] allows to find out all edges that do not belong any perfect
\left(g,f\right)
n
Solutions 2 4 10 26 76 232 764 2620 9496
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
0..n
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚝𝚠𝚒𝚗}
\mathrm{𝚕𝚎𝚡}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}\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{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}
\mathrm{𝚜𝚢𝚖𝚖𝚎𝚝𝚛𝚒𝚌}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}_\mathrm{𝚕𝚘𝚘𝚙}
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}=\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡}
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝐍𝐀𝐑𝐂}\ge |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}
\overline{\mathrm{𝐍𝐀𝐑𝐂}}
|
Global Constraint Catalog: Ccumulative_convex
<< 5.96. cumulative5.98. cumulative_product >>
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
\mathrm{𝙿𝙾𝙸𝙽𝚃𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-\mathrm{𝙿𝙾𝙸𝙽𝚃𝚂},\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙿𝙾𝙸𝙽𝚃𝚂},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝙿𝙾𝙸𝙽𝚃𝚂}|>0
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂},\left[\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜},\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right]\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\ge 0
\mathrm{𝙻𝙸𝙼𝙸𝚃}\ge 0
Cumulative scheduling constraint or scheduling under resource constraints. Consider a set
𝒯
of tasks described by the
\mathrm{𝚃𝙰𝚂𝙺𝚂}
collection where each task is defined by:
A set of distinct points depicting the time interval where the task is actually running: the smallest and largest coordinates of these points respectively give the first and last instant of that time interval.
A height that depicts the resource consumption used by the task from its first instant to its last instant.
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint enforces that, at each point in time, the cumulated height of the set of tasks that overlap that point, does not exceed a given limit. A task overlaps a point
if and only if (1) its origin is less than or equal to
i
, and (2) its end is strictly greater than
i
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈2,1,5〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1,\hfill \\ \mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈4,5,7〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-2,\hfill \\ \mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈14,13,9,11,10〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-2\hfill \end{array}〉,3\hfill \end{array}\right)
Figure 5.97.1 shows the cumulated profile associated with the example. To each set of points defining a task corresponds a rectangle. The height of each rectangle represents the resource consumption of the associated task. The
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint holds since at each point in time we do not have a cumulated resource consumption strictly greater than the upper limit 3 enforced by the last argument of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
Figure 5.97.1. Points defining the three tasks of the Example slot and corresponding resource consumption profile (note that the vertical position of a task does not really matter but is only used for displaying the contribution of a task to the resource consumption profile)
|\mathrm{𝚃𝙰𝚂𝙺𝚂}|>1
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}>0
\mathrm{𝙻𝙸𝙼𝙸𝚃}<
\mathrm{𝚜𝚞𝚖}
\left(\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}
\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
\ge 0
\mathrm{𝙻𝙸𝙼𝙸𝚃}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
A natural use of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint corresponds to problems where a task is defined as the convex hull of a set of distinct points
{P}_{1},\cdots ,{P}_{n}
that are not initially fixed. Note that, by explicitly introducing a start
S
and an end
E
variables, and by using a
\mathrm{𝚖𝚒𝚗𝚒𝚖𝚞𝚖}
\left(S,〈\mathrm{𝚟𝚊𝚛}-{P}_{1},\cdots ,\mathrm{𝚟𝚊𝚛}-{P}_{n}〉\right)
\mathrm{𝚖𝚊𝚡𝚒𝚖𝚞𝚖}
\left(E,〈\mathrm{𝚟𝚊𝚛}-{P}_{1},\cdots ,\mathrm{𝚟𝚊𝚛}-{P}_{n}〉\right)
constraints, one could replace the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint by a
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
constraint. However this hinders propagation.
As a concrete example of use of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint we present a constraint model for a well-known pattern-sequencing problem [FinkVoss99] (also known to be equivalent to the graph path-width [LinharesYanasse02] problem) that is based on a single
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint. The pattern sequencing problem can be described as follows: Given a 0-1 matrix in which each column
j
\left(1\le j\le p\right)
i
\left(1\le i\le c\right)
{c}_{ij}
i
j
is open at point
k
i
k
k
Figure 5.97.2. An input matrix for the pattern sequencing problem (A1), its corresponding cumulated matrix (A2), a view in term of tasks (A3) and the corresponding cumulative profile (A4); a second matrix (B1) where column 4 of (A1) is put at the rightmost position.
Before giving the constraint model, let us first provide an instance of the pattern-sequencing problem. Consider the matrix
{ℳ}_{1}
depicted by part (A1) of Fig. 5.97.2. Part (A2) gives its corresponding cumulated matrix
{ℳ}_{2}
obtained by setting to 1 each 0 of
{ℳ}_{1}
that is both preceded and followed by a 1. Part (A3) depicts the corresponding solution in term of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint: to each row of the matrix
{ℳ}_{1}
corresponds a task defined as the convex hull of the different 1 located on that row. Finally part (A4) gives the cumulated profile associated with part (A3), namely the number of 1 in each column of
{ℳ}_{2}
. The cost 3 of this solution is equal to the maximum number of 1 in the columns of the cumulated matrix
{ℳ}_{2}
. As shown by parts (B1-B4), we can get a lower cost of 2 by pushing the fourth column to the rightmost position.
The idea of the model is to associate to each row (i.e., customer)
i
of the cumulated matrix a stack task that starts at the first 1 on row
i
and ends at the last 1 of row
i
(i.e., the task corresponds to the convex hull of the different 1 located on row
i
). Then the cost of a solution is simply the maximum height on the corresponding cumulated profile.
For each column
j
of the 0-1 matrix initially given there is a variable
{V}_{j}
ranging from 1 to the number of columns
p
{V}_{j}
gives the position of column
j
in a solution. We put all the stack tasks in a
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint, telling that each stack task uses one unit of the resource during all it execution. Since we want to have the same model for different limits on the maximum number of open stacks, and since all variables
{V}_{1},{V}_{2},\cdots ,{V}_{p}
have to be distinct, we have an extra dummy task characterised as the convex hull of
{V}_{1},{V}_{2},\cdots ,{V}_{p}
. This extra dummy task has a height
H
that has to be maximised. For the matrix depicted by (A1) of Fig. 5.97.2 we pass to the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint the following collection of tasks:
〈\begin{array}{cc}\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈{P}_{1},{P}_{2},{P}_{3},{P}_{4},{P}_{6},{P}_{7},{P}_{9}〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1,\hfill \\ \mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈{P}_{2},{P}_{5}〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1,\hfill \\ \mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈{P}_{4},{P}_{7},{P}_{8}〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-1,\hfill \\ \mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}-〈{P}_{1},{P}_{2},{P}_{3},{P}_{4},{P}_{5},{P}_{6},{P}_{7},{P}_{8},{P}_{9}〉\hfill & \mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-0\hfill \end{array}〉
A first natural way to handle the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint is to accumulate the compulsory part [Lahrichi82] of the different tasks in a profile and to prune according to this profile. We give the main ideas for computing the compulsory part of a task and for pruning a task according to the profile of compulsory parts.
Compulsory part of a task Given a task
T
characterised as the convex hull of a set of distinct points
{P}_{1},{P}_{2},\cdots ,{P}_{k}
the compulsory part of
T
corresponds to the, possibly empty, interval
\left[{s}_{T},{e}_{T}\right]
{s}_{T}
is the largest value
v
such that, when all variables
{P}_{1},{P}_{2},\cdots ,{P}_{k}
are greater than or equal to
v
, all variables
{P}_{1},{P}_{2},\cdots ,{P}_{k}
can still take distinct values.
{e}_{T}
is the smallest value
v
{P}_{1},{P}_{2},\cdots ,{P}_{k}
are less than or equal to
v
{P}_{1},{P}_{2},\cdots ,{P}_{k}
Pruning according to the profile of compulsory parts Given two instants
i
j
\left(i<j\right)
and a task
T
{P}_{1},{P}_{2},\cdots ,{P}_{k}
T
cannot overlap
i
j
since this would lead exceeding
\mathrm{𝙻𝙸𝙼𝙸𝚃}
, the second argument of the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint. Furthermore assume that, when all variables
{P}_{1},{P}_{2},\cdots ,{P}_{k}
are both greater than
i
j
{P}_{1},{P}_{2},\cdots ,{P}_{k}
cannot take distinct values. Then all values of
\left[i+1,j-1\right]
can be removed from variables
{P}_{1},{P}_{2},\cdots ,{P}_{k}
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
(resource constraint).
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚖𝚊𝚡}
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
characteristic of a constraint: convex.
constraint type: scheduling constraint, resource constraint, temporal constraint.
filtering: compulsory part.
problems: pattern sequencing.
\mathrm{𝚌𝚘𝚕}\left(\begin{array}{c}\mathrm{𝙸𝙽𝚂𝚃𝙰𝙽𝚃𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚜𝚝𝚊𝚗𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\right),\hfill \\ \mathrm{𝚒𝚝𝚎𝚖}\left(\mathrm{𝚒𝚗𝚜𝚝𝚊𝚗𝚝}-\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}.\mathrm{𝚟𝚊𝚛}\right)\right]\hfill \end{array}\right)
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}\right)
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\left(\mathrm{𝚝𝚊𝚜𝚔𝚜}.\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}\right)
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝚃𝙰𝚂𝙺𝚂}|
\mathrm{𝙸𝙽𝚂𝚃𝙰𝙽𝚃𝚂}
\mathrm{𝚃𝙰𝚂𝙺𝚂}
\mathrm{𝑃𝑅𝑂𝐷𝑈𝐶𝑇}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚜𝚝𝚊𝚗𝚝𝚜},\mathrm{𝚝𝚊𝚜𝚔𝚜}\right)
\mathrm{𝚋𝚎𝚝𝚠𝚎𝚎𝚗}_\mathrm{𝚖𝚒𝚗}_\mathrm{𝚖𝚊𝚡}
\left(\mathrm{𝚒𝚗𝚜𝚝𝚊𝚗𝚝𝚜}.\mathrm{𝚒𝚗𝚜𝚝𝚊𝚗𝚝},\mathrm{𝚝𝚊𝚜𝚔𝚜}.\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}\right)
•
\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}
•
\mathrm{𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴}
•
\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}
\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{𝚑𝚎𝚒𝚐𝚑𝚝}\right)\right]\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}
\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚝𝚛}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜},\le ,\mathrm{𝙻𝙸𝙼𝙸𝚃}\right)
The first graph constraint forces for each task that the set of points defining its time interval are all distinct. The second graph constraint makes sure for each time point
, that the cumulated heights of the tasks that overlap
t
does not exceed the limit of the resource.
Parts (A) and (B) of Figure 5.97.3 respectively show the initial and final graph associated with the second graph constraint of the Example slot. On the one hand, each source vertex of the final graph can be interpreted as a time point corresponding to a point used in the definitions of the different tasks. On the other hand, the successors of a source vertex correspond to those tasks that overlap a given time point. The
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
constraint holds since, for each successor set
𝒮
of the final graph, the sum of the heights of the tasks in
𝒮
does not exceed the limit
\mathrm{𝙻𝙸𝙼𝙸𝚃}=3
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}
|
Global Constraint Catalog: KSAT
<< 3.7.215. Run of a permutation3.7.217. Scalar product >>
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
A constraint for which a reference provides a reformulation in SAT. Encoding for the
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
\mathrm{𝚊𝚖𝚘𝚗𝚐}
constraints were respectively provided in [GentNightingale04] and in [Bacchus07]. Based on Fekete et al. model of the multi-dimensional orthogonal packing problem [FeketeSchepersVeen07], an encoding for the
\mathrm{𝚍𝚒𝚏𝚏𝚗}
constraint when all the sizes of all the orthotopes are fixed was described in [GrandcolasPinto10].
|
Numba JIT benchmark and example - lw1.at
This is an extended answer to a test question that made me test out the results afterwards.
Numba is an JIT compiler for a subset of Python and Numpy that allows boosting the execution of some phython functions.
To try out how much performance improvement is possible, I compared a simple mathematical function with the same function, but prepended with the @njit decorator, that instructs numpy to compile the function the first time it has been executed.
Afterwards I run the function with
n=10^3
n=10^8
, record the timing and plot it with Matplotlib.
As one can see above with
n=100
the compiled function (orange) is 100 times slower than a normal execution (blue) because the compilation on the first function call takes about
0.1s
. But if we ignore the time of the first compilation (because the function is already compiled), then the JIT-compiled function (green) is always 100 times faster than a normal python function.
def testfunction(v):
if v[i, 0] * v[i, 0] + v[i, 1] * v[i, 1] + v[i, 2] * v[i, 2] < 1:
return n / v.shape[0]
def testfunction_jit(v):
mit_jit = []
mit_jit_compiled = []
testfunction_jit(numpy.random.rand(3, 3)) # precompile
n = int(10 ** (i / 2))
x = numpy.random.rand(n, 3)
result = testfunction(x)
normal.append(end - start)
print(result * 6)
testfunction_jit.recompile()
result = testfunction_jit(x)
mit_jit.append(end - start)
mit_jit_compiled.append(end - start)
my_dpi = 80
plt.figure(figsize=(1000 / my_dpi, 500 / my_dpi), dpi=my_dpi)
plt.loglog(ns, normal, label="normal")
plt.loglog(ns, mit_jit, label="jit recompiled every time")
plt.loglog(ns, mit_jit_compiled, label="jit already compiled")
plt.savefig('my_fig.png', dpi=my_dpi)
|
Resamples and mosaics using the drizzling algorithm
This routine transforms a set of images from their pixel into their Current coordinate system. The resulting images are combined together onto a single output grid, which can therefore form a mosaic of the input images. Normalisation of the images can optionally be carried out so that in overlapping regions the scaling and zero point values of the images are consistent with each other.
The algorithm used for combining the images on the output grid is Variable-Pixel Linear Reconstruction, or so-called ‘drizzling’. The user is allowed to shrink the input pixels to a smaller size (drops) so that each pixel of the input image only affects pixels in the output image under the corresponding drop.
drizzle in out
CORRECT = LITERAL (Read)
Name of the sequential file containing the SCALE and ZERO point corrections for the list of input images given by the IN parameter [!]
If GENVAR is set to TRUE and some of the input images supplied contain statistical error (variance) information, then variance information will also be calculated for the output image. [TRUE]
A list of the names of the input images which are to be combined into a mosaic. The image names should be separated by commas and may include wildcards. The input images are accessed only for reading.
LISTIN = _LOGICAL (Read)
If a TRUE value is given for this parameter (the default), then the names of all the images supplied as input will be listed (and will be recorded in the logfile if this is enabled). Otherwise, this listing will be omitted. [TRUE]
MAPVAR = _LOGICAL (Read)
The value of this parameter specifies whether statistical error (variance) information contained in the input images should be used to weight the input image pixels as they are drizzled on to the output image (see the discussion of the drizzling algorithm). If MAPVAR is set to .TRUE. then the ratio of the inverse variance of the input pixel and the the mean inverse variance of the reference frame (or first input image if no reference frame is provided) will be used to weight each pixel as it drizzled onto the output image.
If weighting of the input pixels by the mean inverse variance of the entire input image (rather than the pixels own variance) is required MAPVAR should be set to .FALSE. and USEVAR should be set to .TRUE. (this is the default condition). [FALSE]
MULTI = _DOUBLE (Read)
The linear scaling between the size of the input and output pixels, i.e. for a MULTI of 2.0 then each side of the input pixel is twice that of the sub-sampling output pixel. For large values of MULTI, PIXFRAC must also be larger (e.g. for a MULTI of 4.0 a PIXFRAC of 0.7 is unacceptably small for simgle image drizzling, however for a MULTI of 3.0 a PIXFRAC of 0.7 produces acceptable ouput images). [1.5]
Name of the image to contain the output mosaic.
PIXFRAC = _DOUBLE (Read)
The linear "drop" size, this being the ratio of the linear size of the drizzled drop to that of the input pixel. Interlacing is equivalent to setting PIXFRAC=0.0, while shift-and-add is equivalent to setting PIXFRAC=1.0. For low values of PIXFRAC the MULTI parameter must also be set correspondingly low. [0.9]
If a TRUE value is given for this parameter (the default), then the data type of the output mosaic image will be derived from that of the input image with the highest precision, so that the input data type will be "preserved" in the output image. Alternatively, if a FALSE value is given, then the output image will be given an appropriate floating point data type.
When using integer input data, the former option is useful for minimising the storage space required for large mosaics, while the latter typically permits a wider output dynamic range when necessary. A wide dynamic range is particularly important if a large range of scale factor corrections are being applied (as when combining images with a wide range of exposure times).
If a global value has been set up for this parameter using CCDSETUP, then that value will be used. [TRUE]
REF = NDF (Read)
If the input images being drizzled onto the output image are being weighted by the inverse of their mean variance (see the USEVAR parameter) then by default the first image in the input list (IN) will be used as a reference image. However, if an image is given via the REF parameter (so as to over-ride its default null value), then the weighting will instead be relative to the "reference image" supplied via this parameter.
If scale-factor, zero-point corrections (see the SCALE and ZERO parameters respectively) have not been specified via a sequential file listing (see the CORRECT parameter) then if an image is given via the REF parameter the program will attempt to normalise the input images to the "reference image" supplied.
This provides a means of retaining the calibration of a set of data, even when corrections are being applied, by nominating a reference image which is to remain unchanged. It also allows the output mosaic to be normalised to any externally-calibrated image with which it overlaps, and hence allows a calibration to be transferred from one set of data to another.
If the image supplied via the REF parameter is one of those supplied as input via the IN parameter, then this serves to identify which of the input images should be used as a reference, to which the others will be adjusted. In this case, the scale-factor, zero-point corrections and/or weightings applied to the nominated input image will be set to one, zero and one respectively, and the corrections for the others will be adjusted accordingly.
Alternatively, if the reference image does not appear as one of the input images, then it will be included as an additional set of data in the inter-comparisons made between overlapping images and will be used to normalise the corrections obtained (so that the output mosaic is normalised to it). However, it will not itself contribute to the output mosaic in this case. [!]
SCALE = _LOGICAL (Read)
This parameter specifies whether DRIZZLE should attempt to adjust the input data values by applying scale-factor (i.e. multiplicative) corrections before combining them into a mosaic. This would be appropriate, for instance, if a series of images had been obtained with differing exposure times; to combine them without correction would yield a mosaic with discontinuities at the image edges where the data values differ.
If SCALE is set to TRUE, then DRIZZLE will ask the user for a sequential file containing the corrections for each image (see the CORRECT parameter). If none is supplied the program will attempt to find its own corrections.
DRIZZLE will inter-compare the images supplied as input and will estimate the relative scale-factor between selected pairs of input data arrays where they overlap. From this information, a global set of multiplicative corrections will be derived which make the input data as mutually consistent as possible. These corrections will be applied to the input data before drizzling them onto the output frame.
Calculation of scale-factor corrections may also be combined with the use of zero-point corrections (see the ZERO parameter). By default, no scale-factor corrections are applied. [FALSE]
Title for the output mosaic image. [Output from DRIZZLE]
USEVAR = _LOGICAL (Read)
The value of this parameter specifies whether statistical error (variance) information contained in the input images should be used to weight the input image pixels as they are drizzled on to the output image (see the discussion of the drizzling algorithm). If USEVAR is set to TRUE then the ratio of the mean inverse variance of the input image and the mean inverse variance of the reference frame (or first input image if no reference frame is provided) will be used as a weighting for the image.
If weighting of the input image by the inverse variance map (rather than the mean) then the MAPVAR parameter whould be used. [TRUE]
This parameter specifies whether DRIZZLE should attempt to adjust the input data values by applying zero-point (i.e. additive) corrections before combining them into a mosaic. This would be appropriate, for instance, if a series of images had been obtained with differing background (sky) values; to combine them without correction would yield a mosaic with discontinuities at the image edges where the data values differ.
If ZERO is set to TRUE, then DRIZZLE will ask the user for a sequential file containing the corrections for each image (see the CORRECT parameter). If none is supplied the program will attempt to calculate its own corrections.
DRIZZLE will inter-compare the images supplied as input and will estimate the relative zero-point difference between selected pairs of input data arrays where they overlap. From this information, a global set of additive corrections will be derived which make the input data as mutually consistent as possible. These corrections will be applied to the input data before drizzling them onto the output frame.
Calculation of zero-point corrections may also be combined with the use of scale-factor corrections (see the SCALE parameter). By default, no zero-point corrections are applied. [FALSE]
\ast
out pixfrac=0.7
Drizzles a set of images matching the wild-card "
\ast
" into a mosaic called "out". The drop size of the input pixel is set to 0.7, i.e. it is scaled to 70% of its orginal size before being drizzled onto the output grid.
drizzle in=img
\ast
out=combined scale=true zero=true ref=! multi=4.0
Drizzles a set of images matching the wild-card "img
\ast
" into a mosaic called "combined". Both scaling and zero-point corrections are enabled (the program will request a correction file), however no reference image has been supplied (the program will use the first image supplied in the input list). The multiplicative scaling factor between input and output images is set to 4, i.e. the input pixel is 4 times larger than the output pixel and contains 16 output pixels.
“Combination by drizzling”.
The file containing scale and zero-point corrections (see the CORRECT parameter) must contain one line per frame having the following information
INDEX SCALE ZERO
INDEX = the index number of the frame, this must be the same as its order number in the input list (see the IN parameter)
SCALE = the multiplicative scaling factor for the image
ZERO = the zero-point correction for the image
Comment lines may be added, by must be prefixed with a "#" character.
The format of the file containing scale and zero-point corrections must be correct or the A-task will abort operations.
Taken from Fruchter et al., "A package for the reduction of dithered undersampled images", in Casertano et al. (eds), HST Calibration Workshop, STSCI, 1997, pp. 518–528:
The drizzle algorithm is conceptually straightforward. Pixels in the original input images are mapped into pixels in the subsampled output image, taking into account shifts and rotations between the images and the optical distortion of the camera. However, in order to avoid convolving the image with the larger pixel ‘footprint’ of the camera, we allow the user to shrink the pixel before it is averaged into the output image.
The new shrunken pixels, or ‘drops’, rain down upon the subsampled output. In the case of the Hubble Deep Field (HDF), the drops used had linear dimensions one-half that of the input pixel – slightly larger than the dimensions of the output subsampled pixels. The value of an input pixel is averaged into the output pixel with a weight proportional to the area of overlap between the ‘drop’ and the output pixel. Note that, if the drop size if sufficently small, not all output pixels have data added to them from each input image. One must therefore choose a drop size that is small enough to avoid degrading the image, but large enough so that after all images are ‘dripped’ the coverage is fairly uniform.
The drop pize if controlled by a user-adjustable parameter called PIXFRAC, which is simply the ratio of the linear size of the drop to the input pixel (before any adjustment due to geometric distortion of the camera). Thus interlacing is equivalent to setting PIXFRAC=0.0, while shift-and-add is equivalent to PIXFRAC=1.0.
When a drop with value
{i}_{xy}
and a user-defined weight
{w}_{xy}
is added to an image with pixel value
{I}_{xy}
, weight
{W}_{xy}
, and fractional pixel overlap
0<{a}_{xy}<1
, the resulting value the image
{I}_{xy}^{\prime }
{W}_{xy}^{\prime }
\begin{array}{rcll}{W}_{xy}^{\prime }& =& {a}_{xy}{w}_{xy}+{W}_{xy}& \text{}\\ {I}_{xy}^{\prime }& =& \frac{{a}_{xy}{i}_{xy}{w}_{xy}+{I}_{xy}{W}_{xy}}{{W}_{xy}^{\prime }}& \text{}\end{array}
This algorithm has a number of advantages over standard linear reconstruction methods presently used. Since the area of the pixels scales with the Jacobian of the geometric distortion, drizzle preserves both surface and absolute photometry. Therefore flux can be measured using an aperture whose size is independent of position on the chip. As the method anticipates that a given output pixel may receive no information from a given input pixel, missing data (due for instance to cosmic rays or detector defects) do not cause a substantial problem, so long as there are enough dithered images to fill in the gaps caused by these zero-weight input pixels. Finally the linear weighting scheme is statistically optimum when inverse variance maps are used as weights.
All non-complex numeric data types are supported.
Bad pixels are supported.
The algorithm is restricted to handling 2D images only
|
Global Constraint Catalog: Cset_value_precede
<< 5.343. sequence_folding5.345. shift >>
[YatChiuLawJimmyLee04]
\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}\left(𝚂,𝚃,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)
𝚂
\mathrm{𝚒𝚗𝚝}
𝚃
\mathrm{𝚒𝚗𝚝}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚜𝚟𝚊𝚛}\right)
𝚂\ne 𝚃
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
If there exists a set variable
{v}_{1}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
𝚂
{v}_{1}
𝚃
does, then there also exists a set variable
{v}_{2}
{v}_{1}
𝚂
{v}_{2}
𝚃
does not.
\left(2,1,〈\mathrm{𝚟𝚊𝚛}-\left\{0,2\right\},\mathrm{𝚟𝚊𝚛}-\left\{0,1\right\},\mathrm{𝚟𝚊𝚛}-\varnothing ,\mathrm{𝚟𝚊𝚛}-\left\{1\right\}〉\right)
\left(0,1,〈\mathrm{𝚟𝚊𝚛}-\left\{0,2\right\},\mathrm{𝚟𝚊𝚛}-\left\{0,1\right\},\mathrm{𝚟𝚊𝚛}-\varnothing ,\mathrm{𝚟𝚊𝚛}-\left\{1\right\}〉\right)
\left(0,2,〈\mathrm{𝚟𝚊𝚛}-\left\{0,2\right\},\mathrm{𝚟𝚊𝚛}-\left\{0,1\right\},\mathrm{𝚟𝚊𝚛}-\varnothing ,\mathrm{𝚟𝚊𝚛}-\left\{1\right\}〉\right)
\left(0,4,〈\mathrm{𝚟𝚊𝚛}-\left\{0,2\right\},\mathrm{𝚟𝚊𝚛}-\left\{0,1\right\},\mathrm{𝚟𝚊𝚛}-\varnothing ,\mathrm{𝚟𝚊𝚛}-\left\{1\right\}〉\right)
The following examples are taken from [Law05]:
\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\left(2,1,〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉\right)
constraint holds since the first occurrence of value 2 precedes the first occurrence of value 1 (i.e., the set
\left\{0,2\right\}
occurs before the set
\left\{0,1\right\}
\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\left(0,1,〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉\right)
\left\{0,2\right\}
\left\{0,1\right\}
\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\left(0,2,〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉\right)
constraint holds since “there is no set in
〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉
that contains 2 but not 0”.
\mathrm{𝚜𝚎𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\left(0,4,〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉\right)
constraint holds since no set in
〈\left\{0,2\right\},\left\{0,1\right\},\left\{\right\},\left\{1\right\}〉
contains value 4.
𝚂<𝚃
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
A filtering algorithm for maintaining value precedence on a sequence of set variables is presented in [YatChiuLawJimmyLee04]. Its complexity is linear to the number of variables of the collection
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
precede in Gecode.
\mathrm{𝚒𝚗𝚝}_\mathrm{𝚟𝚊𝚕𝚞𝚎}_\mathrm{𝚙𝚛𝚎𝚌𝚎𝚍𝚎}
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}
\mathrm{𝚜𝚎𝚝}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
\mathrm{𝚜𝚎𝚚𝚞𝚎𝚗𝚌𝚎}
\mathrm{𝚍𝚘𝚖𝚊𝚒𝚗}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
symmetry: symmetry, indistinguishable values, value precedence.
|
Global Constraint Catalog: Ccond_lex_cost
<< 5.79. compare_and_count5.81. cond_lex_greater >>
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴},\mathrm{𝙲𝙾𝚂𝚃}\right)
\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚝𝚞𝚙𝚕𝚎}-\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂}\right)
\mathrm{𝙲𝙾𝚂𝚃}
\mathrm{𝚍𝚟𝚊𝚛}
|\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂}|\ge 1
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|=|\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴},\mathrm{𝚝𝚞𝚙𝚕𝚎}\right)
\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}
\left(\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴},\mathrm{𝚝𝚞𝚙𝚕𝚎}\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴},\left[\right]\right)
\mathrm{𝚒𝚗}_\mathrm{𝚛𝚎𝚕𝚊𝚝𝚒𝚘𝚗}
\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}\right)
\mathrm{𝙲𝙾𝚂𝚃}\ge 1
\mathrm{𝙲𝙾𝚂𝚃}\le |\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}|
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
is assigned to the
{\mathrm{𝙲𝙾𝚂𝚃}}^{th}
item of the collection
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}
\left(\begin{array}{c}〈0,1〉,\hfill \\ 〈\begin{array}{c}\mathrm{𝚝𝚞𝚙𝚕𝚎}-〈1,0〉,\hfill \\ \mathrm{𝚝𝚞𝚙𝚕𝚎}-〈0,1〉,\hfill \\ \mathrm{𝚝𝚞𝚙𝚕𝚎}-〈0,0〉,\hfill \\ \mathrm{𝚝𝚞𝚙𝚕𝚎}-〈1,1〉\hfill \end{array}〉,2\hfill \end{array}\right)
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
is assigned to the second item of the collection
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}
|\mathrm{𝚃𝚄𝙿𝙻𝙴}_\mathrm{𝙾𝙵}_\mathrm{𝚅𝙰𝙻𝚂}|>1
|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|>1
|\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}|>1
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚝𝚞𝚙𝚕𝚎}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚝𝚞𝚙𝚕𝚎}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}.\mathrm{𝚝𝚞𝚙𝚕𝚎}
We consider an example taken from [WallaceWilson06] were a customer has to decide among vacations. There are two seasons when he can travel (
\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}
\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}
) and two locations
\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜}
\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒}
. Furthermore assume that location is more important than season and the preferred period of the year depends on the selected location. The travel preferences of a customer are explicitly defined by stating the preferences ordering among the possible tuples of values
〈\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜},\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}〉
〈\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜},\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}〉
〈\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒},\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}〉
〈\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒},\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}〉
. For instance we may state within the preference table
\mathrm{𝙿𝚁𝙴𝙵𝙴𝚁𝙴𝙽𝙲𝙴}_\mathrm{𝚃𝙰𝙱𝙻𝙴}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
constraint the preference ordering
〈\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜},\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}〉\succ 〈\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒},\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}〉\succ 〈\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒},\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}〉\succ 〈\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜},\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}〉
, which denotes the fact that our customer prefers
\mathrm{𝙽𝚊𝚙𝚕𝚎𝚜}
\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}
\mathrm{𝙷𝚎𝚕𝚜𝚒𝚗𝚔𝚒}
\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}
, and a vacation in
\mathrm{𝚜𝚙𝚛𝚒𝚗𝚐}
is preferred over
\mathrm{𝚜𝚞𝚖𝚖𝚎𝚛}
. Finally a solution minimising the cost variable
\mathrm{𝙲𝙾𝚂𝚃}
will match the preferences stated by our customer.
attached to cost variant:
\mathrm{𝚒𝚗}_\mathrm{𝚛𝚎𝚕𝚊𝚝𝚒𝚘𝚗}
\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}
(preferences).
\mathrm{𝚎𝚕𝚎𝚖𝚎𝚗𝚝}
\mathrm{𝚝𝚞𝚙𝚕𝚎}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}
replaced by single
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
characteristic of a constraint: vector, automaton, automaton without counters, reified automaton constraint.
filtering: arc-consistency, cost filtering constraint.
Figure 5.80.1 depicts the automaton associated with
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}
{\mathrm{𝚅𝙰𝚁}}_{k}
\mathrm{𝚟𝚊𝚛}
{k}^{th}
\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}
collection. Figure 5.80.2 depicts the reformulation of the
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
Figure 5.80.2. Hypergraph of the reformulation corresponding to the automaton of the
\mathrm{𝚌𝚘𝚗𝚍}_\mathrm{𝚕𝚎𝚡}_\mathrm{𝚌𝚘𝚜𝚝}
|
Utility - Eden Network
Symbol: EDEN
Address: 0x1559fa1b8f28238fd5d76d9f434ad86fd20d1559
EDEN Token is the primary coordination unit of the network
Functionally, EDEN is the unit used to conduct the continuous auction for claim to priority blockspace, and provides users access to private relayer.
Economically, it serves two important purposes:
EDEN tokens are the unit used to conduct the continuous auction for claim to priority blockspace, and provides users access to private relayer.
EDEN tokens are burned on a daily basis under the Harberger tax imposed on slot tenants to ensure that there is a frequent rotation of top addresses, creating a constant demand for EDEN tokens
The total supply of EDEN is set at a maximum of 250,000,000 tokens. The inflation parameter is given by the following curve
\max \left\{ \left( 5.595 \times 10^6 \right) - \left( 1.5 \times 10^6 \right) \times \left( \log M^{1.797} \right), 0 \right\}
Monthly inflation is distributed as follows:
Block Producers: 60%
Eden Treasury: 10%
|
On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation
Wenbin Zhang, Jiangbo Zhou, Lixin Tian, Sunil Kumar, "On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation", Journal of Mathematics, vol. 2013, Article ID 578094, 4 pages, 2013. https://doi.org/10.1155/2013/578094
Wenbin Zhang,1 Jiangbo Zhou ,2 Lixin Tian,2 and Sunil Kumar3
1Taizhou Institute of Science and Technology, NUST, Taizhou, Jiangsu 225300, China
2Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China
3Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India
Academic Editor: Ji Gao
It is shown that if is a sufficiently smooth solution to a two-dimensional nonlinear wave equation such that there exists with supp , for , then .
In this paper, we consider the following two-dimensional nonlinear wave equation: where , , , are arbitrary positive constants. Equation (1) was recently derived by Gottwald [1] for large scale motion from the barotropic quasigeostrophic equation as a two-dimensional model for Rossby waves. He [2] showed that (1) has traveling wave solutions via the homotopy perturbation method. Using a subequation method, the traveling wave solutions are also studied by Fu et al. [3]. Aslan [4] constructed solitary wave solutions and periodic wave solutions to (1) by the Exp-function method.
For and in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma. Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ([6–11]). Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically. Recently, Bustamante et al. [13] showed that sufficiently smooth solutions of the ZK equation that have compact support for two different times are identically zero.
The purpose of this paper is to investigate the support of solutions to (1). To solve the problem, we mainly use the ideas of [12–15]. The main result is as follows.
Theorem 1. Assume that and , if is a solution of (1) such that then, .
Lemma 2 (see [13]). Assume that and . (i) If , then (ii) if , then where , , and .
Lemma 3. Assume that , if is a solution of (1) such that ; then, is bounded in .
Proof. Assume that is a decreasing function with if and if . Let for and . It is easy to check that and Multiplying (1) by and integrating by parts in , we obtain Applying Gronwall Lemma and the Monotone Convergence Theorem, we have This proves that is bounded in .
Applying Lemma 2 with and , we have that is bounded in . Here, we used the fact that . This completes the proof of the lemma.
Lemma 4. Assume that , , and is a solution of (1). (i) If , then is bounded in ; (ii) if , then is bounded in .
Proof. Letting and a solution to (1), we have Multiplying (8) by and integrating by parts in , we obtain Note that and It follows from (9) that Since and is bounded in , applying Lemma 2 with and , we have that is also bounded in .
Similarly, we can prove that is bounded in . Let ; then, is a solution to (1) and satisfies , and therefore is bounded in . This proves (i).
Now, we prove (ii). Let ; then, is also a solution of (1) and satisfies the hypothesis of (i). This proves (ii) and completes the proof of the lemma.
Remark 5. In particular, if the conditions for and given in (i) and (ii), respectively, are satisfied, then is bounded in .
Lemma 6 (see [13]). Let , is a function such that is bounded in , and . Then, for all and all , the functions and are absolutely continuous in with derivatives and a.e. , respectively.
Lemma 7. Assume that , , and , if is a function such that is bounded in and . Then,
Proof. Let and ; then, Taking the spatial Fourier transform in (14) and applying Lemma 6, we have where According to (15), when , we have and when , we choose to write Therefore, we have Appling Plancherel formula, we have inequality (12).
Similarly, letting , we can also have (13). This completes the proof of the lemma.
Lemma 8 (see [12, 13]). Assume that , and , if is a solution to (1) such that then, .
Proof. The proof is similar to that of Theorem 1.1 in [12], and we omit the details.
Assume that for where is a nondecreasing function such that for and for . Let ; then, . According to Lemma 7, we obtain that where and Note that the derivatives of are supported in the interval ; then, where is dependent on and .
Combining (21) with (23), we obtain Applying Lemma 4 with , we have that then Since , taking such that , we have Note that for ; we have Letting , we obtain and this proves that in .
Next, we will prove that in . Let ; then, where .
In fact, Let ; it is easy to check that also satisfies the hypotheses of this theorem, and then we find that in Thus, there exists such that for all . Applying Lemma 8, we complete the proof of Theorem 1.
This work was supported by the National Natural Science Foundation of China (no. 11171135), the Natural Science Foundation of Jiangsu (no. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), as well as the Taizhou Social Development Project (no. 2011213).
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\left(2+1\right)
-dimensional Zakharov-Kuznetsov equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1421–1429, 2010. View at: Publisher Site | Google Scholar | MathSciNet
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Copyright © 2013 Wenbin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Price Impact by Size - Premia
Premia pools disincentivize highly disruptive trades through size price impact.
After every transaction, the pool price level updates from
C_t \rarr C_{t+1}
, depending on the size and the direction of the trade. This results in either an increased or decreased price for the next buyer/underwriter. There are no obvious reasons to disincentivize larger blocks of provided liquidity (on the LP side), however, whale-buying behavior needs to be accounted for.
Suppose a whale is waiting on the sidelines for the C-level to fall below their perceived market equilibrium, just to scoop up 50% of the pool's liquidity. Such a trade would cause a significant pool price level disruption; this disruption needs to be accounted for in the price charged to the whale. If the starting value is
C_t
, and the ending value (post whale trade) becomes
C_{t+1}
C_{t+1} > C_{t}
, what price impact penalty should be imposed on the whale?
In discrete form, the whale would end up paying
BS(V_i)* \frac{(C_t+C_{t+1})}{2}
, however the differential form is slightly more accurate:
x_t=\frac{(S_{t+1}-S_t)}{max(S_{t+1};S_t)}
or more intuitively - the normalized step size, relative to the free capital in the pool.
Putting it all together, using
C^*_t=C_t \text{ adjusted for slippage}
\alpha=1.0
as a potential future trade-specific steepness modifier, we get a final pricing function:
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-x_t})
This ensures large traders have no advantage over smaller traders.
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Poisson's equation - Wikipedia
(Redirected from Poisson equation)
Expression frequently encountered in mathematical physics, generalization of Laplace's equation.
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.[1][2]
1 Statement of the equation
2 Newtonian gravity
3.1 Potential of a Gaussian charge density
4 Surface reconstruction
Statement of the equation[edit]
{\displaystyle \Delta \varphi =f}
{\displaystyle \Delta }
is the Laplace operator, and
{\displaystyle f}
{\displaystyle \varphi }
are real or complex-valued functions on a manifold. Usually,
{\displaystyle f}
{\displaystyle \varphi }
is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as
{\displaystyle \nabla ^{2}\varphi =f.}
In three-dimensional Cartesian coordinates, it takes the form
{\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\varphi (x,y,z)=f(x,y,z).}
{\displaystyle f=0}
identically we obtain Laplace's equation.
Poisson's equation may be solved using a Green's function:
{\displaystyle \varphi (\mathbf {r} )=-\iiint {\frac {f(\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\!r',}
where the integral is over all of space. A general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution, such as the relaxation method, an iterative algorithm.
Newtonian gravity[edit]
Main articles: Gravitational field and Gauss's law for gravity
In the case of a gravitational field g due to an attracting massive object of density ρ, Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity,
{\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho ~.}
Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential Φ,
{\displaystyle \mathbf {g} =-\nabla \phi ~.}
Substituting into Gauss's law
{\displaystyle \nabla \cdot (-\nabla \phi )=-4\pi G\rho }
yields Poisson's equation for gravity,
{\displaystyle \nabla ^{2}\phi =4\pi G\rho .}
If the mass density is zero, Poisson's equation reduces to Laplace's equation. The corresponding Green's function can be used to calculate the potential at distance r from a central point mass m (i.e., the fundamental solution). In three dimensions the potential is
{\displaystyle \phi (r)={\dfrac {-Gm}{r}}.}
which is equivalent to Newton's law of universal gravitation.
One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potential φ for a given charge distribution
{\displaystyle \rho _{f}}
The mathematical details behind Poisson's equation in electrostatics are as follows (SI units are used rather than Gaussian units, which are also frequently used in electromagnetism).
Starting with Gauss's law for electricity (also one of Maxwell's equations) in differential form, one has
{\displaystyle \mathbf {\nabla } \cdot \mathbf {D} =\rho _{f}}
{\displaystyle \mathbf {\nabla } \cdot }
is the divergence operator, D = electric displacement field, and ρf = free charge volume density (describing charges brought from outside).
Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive equation,
{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }
where ε is the permittivity of the medium and E is the electric field.
Substituting this into Gauss's law and assuming ε is spatially constant in the region of interest yields
{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} ={\frac {\rho }{\varepsilon }}~.}
{\displaystyle \rho }
is a total volume charge density. In electrostatics, we assume that there is no magnetic field (the argument that follows also holds in the presence of a constant magnetic field). Then, we have that
{\displaystyle \nabla \times \mathbf {E} =0,}
where ∇× is the curl operator. This equation means that we can write the electric field as the gradient of a scalar function φ (called the electric potential), since the curl of any gradient is zero. Thus we can write,
{\displaystyle \mathbf {E} =-\nabla \varphi ,}
where the minus sign is introduced so that φ is identified as the potential energy per unit charge.
The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field,
{\displaystyle \nabla \cdot \mathbf {E} =\nabla \cdot (-\nabla \varphi )=-{\nabla }^{2}\varphi ={\frac {\rho }{\varepsilon }},}
directly produces Poisson's equation for electrostatics, which is
{\displaystyle \nabla ^{2}\varphi =-{\frac {\rho }{\varepsilon }}.}
Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then Laplace's equation results. If the charge density follows a Boltzmann distribution, then the Poisson-Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions.
Using Green's Function, the potential at distance r from a central point charge Q (i.e., the Fundamental Solution) is:
{\displaystyle \varphi (r)={\frac {Q}{4\pi \varepsilon r}}.}
which is Coulomb's law of electrostatics. (For historic reasons, and unlike gravity's model above, the
{\displaystyle 4\pi }
factor appears here and not in Gauss's law.)
The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing φ is no longer sufficient to calculate E, since E also depends on the magnetic vector potential A, which must be independently computed. See Maxwell's equation in potential formulation for more on φ and A in Maxwell's equations and how Poisson's equation is obtained in this case.
Potential of a Gaussian charge density[edit]
If there is a static spherically symmetric Gaussian charge density
{\displaystyle \rho _{f}(r)={\frac {Q}{\sigma ^{3}{\sqrt {2\pi }}^{3}}}\,e^{-r^{2}/(2\sigma ^{2})},}
where Q is the total charge, then the solution φ(r) of Poisson's equation,
{\displaystyle {\nabla }^{2}\varphi =-{\rho _{f} \over \varepsilon },}
{\displaystyle \varphi (r)={\frac {1}{4\pi \varepsilon }}{\frac {Q}{r}}\,\operatorname {erf} \left({\frac {r}{{\sqrt {2}}\sigma }}\right)}
where erf(x) is the error function.
This solution can be checked explicitly by evaluating ∇2φ.
Note that, for r much greater than σ, the erf function approaches unity and the potential φ(r) approaches the point charge potential
{\displaystyle \varphi \approx {\frac {1}{4\pi \varepsilon }}{\frac {Q}{r}},}
as one would expect. Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice for r > 3σ the relative error is smaller than one part in a thousand.
Surface reconstruction[edit]
Surface reconstruction is an inverse problem. The goal is to digitally reconstruct a smooth surface based on a large number of points pi (a point cloud) where each point also carries an estimate of the local surface normal ni.[3] Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction.[4]
The goal of this technique is to reconstruct an implicit function f whose value is zero at the points pi and whose gradient at the points pi equals the normal vectors ni. The set of (pi, ni) is thus modeled as a continuous vector field V. The implicit function f is found by integrating the vector field V. Since not every vector field is the gradient of a function, the problem may or may not have a solution: the necessary and sufficient condition for a smooth vector field V to be the gradient of a function f is that the curl of V must be identically zero. In case this condition is difficult to impose, it is still possible to perform a least-squares fit to minimize the difference between V and the gradient of f.
In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field V. The basic approach is to bound the data with a finite difference grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform [trilinear interpolation] on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ni onto the nodes of the particular staggered grid cell containing pi. Kazhdan and coauthors give a more accurate method of discretization using an adaptive finite difference grid, i.e. the cells of the grid are smaller (the grid is more finely divided) where there are more data points.[4] They suggest implementing this technique with an adaptive octree.
For the incompressible Navier–Stokes equations, given by:
{\displaystyle {\begin{aligned}{\partial {\bf {v}} \over {\partial t}}+{\bf {v}}\cdot \nabla {\bf {v}}&=-{1 \over {\rho }}\nabla p+\nu \Delta {\bf {v}}+{\bf {g}}\\\nabla \cdot {\bf {v}}&=0\end{aligned}}}
The equation for the pressure field
{\displaystyle p}
is an example of a nonlinear Poisson equation:
{\displaystyle {\begin{aligned}\Delta p&=-\rho \nabla \cdot ({\bf {v}}\cdot \nabla {\bf {v}})\\&=-\rho \,\mathrm {Tr} {\big (}(\nabla {\bf {v}})(\nabla {\bf {v}}){\big )}.\end{aligned}}}
Notice that the above trace is not sign-definite.
^ Jackson, Julia A.; Mehl, James P.; Neuendorf, Klaus K. E., eds. (2005), Glossary of Geology, American Geological Institute, Springer, p. 503, ISBN 9780922152766
^ Poisson (1823). "Mémoire sur la théorie du magnétisme en mouvement" [Memoir on the theory of magnetism in motion]. Mémoires de l'Académie Royale des Sciences de l'Institut de France (in French). 6: 441–570. From p. 463: "Donc, d'après ce qui précède, nous aurons enfin:
{\displaystyle {\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0,=-2k\pi ,=-4k\pi ,}
selon que le point M sera situé en dehors, à la surface ou en dedans du volume que l'on considère." (Thus, according to what preceded, we will finally have:
{\displaystyle {\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0,=-2k\pi ,=-4k\pi ,}
depending on whether the point M is located outside, on the surface of, or inside the volume that one is considering.) V is defined (p. 462) as:
{\displaystyle V=\iiint {\frac {k'}{\rho }}\,dx'\,dy'\,dz'}
where, in the case of electrostatics, the integral is performed over the volume of the charged body, the coordinates of points that are inside or on the volume of the charged body are denoted by
{\displaystyle (x',y',z')}
{\displaystyle k'}
is a given function of
{\displaystyle (x',y,'z')}
and in electrostatics,
{\displaystyle k'}
would be a measure of charge density, and
{\displaystyle \rho }
is defined as the length of a radius extending from the point M to a point that lies inside or on the charged body. The coordinates of the point M are denoted by
{\displaystyle (x,y,z)}
{\displaystyle k}
{\displaystyle k'}
(the charge density) at M.
^ Calakli, Fatih; Taubin, Gabriel (2011). "Smooth Signed Distance Surface Reconstruction" (PDF). Pacific Graphics. 30 (7).
^ a b Kazhdan, Michael; Bolitho, Matthew; Hoppe, Hugues (2006). "Poisson surface reconstruction". Proceedings of the fourth Eurographics symposium on Geometry processing (SGP '06). Eurographics Association, Aire-la-Ville, Switzerland. pp. 61–70. ISBN 3-905673-36-3.
Evans, Lawrence C. (1998). Partial Differential Equations. Providence (RI): American Mathematical Society. ISBN 0-8218-0772-2.
Mathews, Jon; Walker, Robert L. (1970). Mathematical Methods of Physics (2nd ed.). New York: W. A. Benjamin. ISBN 0-8053-7002-1.
Polyanin, Andrei D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton (FL): Chapman & Hall/CRC Press. ISBN 1-58488-299-9.
"Poisson equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Poisson Equation at EqWorld: The World of Mathematical Equations
Retrieved from "https://en.wikipedia.org/w/index.php?title=Poisson%27s_equation&oldid=1081030026"
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Conditional Probability | Brilliant Math & Science Wiki
Andy Hayes and Andrew Wang contributed
A conditional probability is a probability that a certain event will occur given some knowledge about the outcome or some other event.
P(A\mid B)
is a conditional probability. It is read as "The probability of
A
B
A
B
are in a uniform sample space, then:
P(A\mid B)=\dfrac{|A\cap B|}{|B|}
A
B
are not in a uniform sample space, then (more generally):
P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}
The concept of conditional probability is closely tied to the concepts of independent and dependent events. Probability problems that provide knowledge about the outcome can often lead to surprising results. A good example of this is the Monty Hall Problem. Often, one's initial intuition is incorrect on these kind of problems. However, an understanding of conditional probability can help one obtain the correct result.
Uniform Conditional Probability
A fair 12-sided die is rolled. What is the probability that the roll is a 3 given that the roll is odd?
A
be the event that a 3 is rolled. Let
B
be the event that an odd number is rolled. Using the definitions above, this problem can be restated as
P(A\mid B)
The entire sample space of the 12-sided die is as follows:
S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
Because the die is fair, the sample space will be uniform.
The event, B, is the following subset of S:
B=\{1,3,5,7,9,11\}
A\cap B
is the event that the roll is 3 and is odd. A is a subset of B and
A\cap B=\{3\}
P(A\mid B)=\dfrac{|A\cap B|}{|B|}=\boxed{\dfrac{1}{6}}
\dfrac{3}{4}
\dfrac{2}{5}
\dfrac{1}{2}
\dfrac{3}{5}
A deck of 10 cards numbered 1 through 10 is shuffled, and a card is drawn from the deck. Let
A
be the event that an odd number is drawn and
B
be the event that a prime number is drawn. What is
P(B\mid A)
A player is dealt 5 cards from a shuffled standard playing card deck. What is the probability that the player will obtain a 3-of-a-kind given that two of the cards are the same rank?
In poker, a 3-of-a-kind is a hand in which 3 cards are of the same rank, and the two other cards are different ranks.
A
be the event that a 3-of-a-kind is obtained.
B
be the event that two of the cards are the same rank.
This problem can be restated as finding
P(A\mid B)
A\cap B=A
, because two of the cards will always be the same rank when the player obtains a 3-of-a-kind.
Using the above formulas, this gives
P(A\mid B)=\dfrac{|A\cap B|}{|B|}=\dfrac{|A|}{|B|}
|A|
, it is important to keep in mind that in a 3-of-a-kind hand of 5 cards, the other 2 cards are different ranks.
|A|=\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}\binom{4}{1}=54912
|B|
can be calculated with the complement of the event that all cards are different.
|B|=\binom{52}{5}-\binom{13}{5}\binom{4}{1}^5=1281072
P(A\mid B)=\dfrac{54912}{1281072}\approx 0.042864
Cite as: Conditional Probability. Brilliant.org. Retrieved from https://brilliant.org/wiki/conditional-probability/
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Global Constraint Catalog: KSLAM_problem
<< 3.7.227. Schur number3.7.229. Sliding cyclic(1) constraint network(1) >>
\mathrm{𝚗𝚟𝚎𝚌𝚝𝚘𝚛}
Denotes that a constraint was used in the context of the simultaneous localisation and map building (SLAM) problem. Given a mobile autonomous robot that, for some reason do not has a direct way to perform self-location (i.e., for instance do not has a GPS), the problem is to dynamically build a map and locate its trajectory on that map from a set of partial snapshots of its environment. Within the context of constraint programming this problem is described in [Jaulin06], [ChabertJaulinLorca09].
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Experimental Study on the Helical Flow in a Concentric Annulus With Rotating Inner Cylinder | J. Fluids Eng. | ASME Digital Collection
Special Section On The Fluid Mechanics And Rheology Of Nonlinear Materials At The Macro, Micro And Nano Scale
Nam-Sub Woo,
Nam-Sub Woo
, 300 Chunchun-dong, Jangan-gu, Suwon 440-746, South Korea
e-mail: nswoo@skku.edu
Young-Ju Kim,
e-mail: kyjp7272@kigam.re.kr
Young-Kyu Hwang
e-mail: ykhwang@skku.edu
J. Fluids Eng. Jan 2006, 128(1): 113-117 (5 pages)
Woo, N., Kim, Y., and Hwang, Y. (September 22, 2005). "Experimental Study on the Helical Flow in a Concentric Annulus With Rotating Inner Cylinder." ASME. J. Fluids Eng. January 2006; 128(1): 113–117. https://doi.org/10.1115/1.2136923
This experimental study concerns the characteristics of vortex flow in a concentric annulus with a diameter ratio of 0.52, whose outer cylinder is stationary and inner one is rotating. Pressure losses and skin friction coefficients have been measured for fully developed laminar flows of water and of 0.4% aqueous solution of sodium carboxymethyl cellulose, respectively, when the inner cylinder rotates at the speed of
0-600rpm
. The results of the present study show the effect of the bulk flow Reynolds number Re and Rossby number Ro on the skin friction coefficients. They also point to the existence of a flow instability mechanism. The effect of rotation on the skin friction coefficient depends significantly on the flow regime. In all flow regimes, the skin friction coefficient is increased by the inner cylinder rotation. The change in skin friction coefficient, which corresponds to a variation of the rotational speed, is large for the laminar flow regime, whereas it becomes smaller as Re increases for transitional flow regime and, then, it gradually approaches to zero for turbulent flow regime. Consequently, the critical bulk flow Reynolds number
Rec
decreases as the rotational speed increases. The rotation of the inner cylinder promotes the onset of transition due to the excitation of Taylor vortices.
confined flow, vortices, friction, laminar flow, flow instability, laminar to turbulent transitions, turbulence, vortex flow, concentric annulus, rotating cylinder, skin friction coefficient
Annulus, Ceramic matrix composites, Cylinders, Flow (Dynamics), Fluids, Laminar flow, Pressure, Reynolds number, Rotation, Skin friction (Fluid dynamics), Water, Vortices, Turbulence, Flow instability
Stability of a Viscous Fluid Contained Between Two Rotating Cylinders
The Stability of a Viscous Fluid Between Rotating Cylinders With a Bulk Flow
Frictional Moment and Pressure Drop of the Flow Through Co-Axial Cylinders With an Outer Rotating Cylinder
Concentric Annular Flow With Centerbody Rotation of a Newtonian and a Shear-Thinning Liquid
Stratabit
Slimhole Drilling Hydraulics
,” SPE Paper No. 24596, pp. 521–541.
Fully Developed Laminar Flow of Purely Viscous Non-Newtonian Liquids Through Annuli, Including the Effects of Eccentricity and Inner-Cylinder Rotation
Laminar Flow Forced Convection
Spatio-Temporal Character of Non-Wavy and Wavy Taylor-Couette Flow
On Preferred Perturbations Selected by Centrifugal Instability
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f=x {y}^{2}{z}^{3}
∇f
f=x {y}^{2}{z}^{3}
∇f=\left[\begin{array}{c}{y}^{2}{z}^{3}\\ 2xy{z}^{3}\\ 3x{y}^{2}{z}^{2}\end{array}\right]
and its curl is the vector
∇×\left(∇f\right)=|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ {∂}_{x}& {∂}_{y}& {∂}_{z}\\ {y}^{2}{z}^{3}& 2 x y {z}^{3}& 3 x {y}^{2}{z}^{2}\end{array}|
\left[\begin{array}{c}{∂}_{y}(3 x {y}^{2}{z}^{2})-{∂}_{z}(2 x y {z}^{3})\\ {∂}_{z}({y}^{2}{z}^{3})-{∂}_{x}(3 x {y}^{2}{z}^{2})\\ {∂}_{x}(2 x y {z}^{3})-{∂}_{y}({y}^{2}{z}^{3})\end{array}\right]
\left[\begin{array}{c}6 x y {z}^{2}-6 x y {z}^{2}\\ 3 {y}^{2}{z}^{2}-3 {y}^{2}{z}^{2}\\ 2 y {z}^{3}-2 y {z}^{3}\end{array}\right]
\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]
Obtain the curl of the gradient of
∇×\left(∇\left(x {y}^{2}{z}^{3}\right)\right)
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{BasisFormat}\left(\mathrm{false}\right):
Compute the curl of the gradient of
Apply the Gradient and Curl commands from the Student VectorCalculus package.
\mathrm{Curl}\left(\mathrm{Gradient}\left(x {y}^{2}{z}^{3}\right)\right)
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
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Global Constraint Catalog: Catmost1
<< 5.39. atmost5.41. atmost_nvalue >>
[SadlerGervet01]
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}\left(\mathrm{𝚂𝙴𝚃𝚂}\right)
\mathrm{𝚙𝚊𝚒𝚛}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(𝚜-\mathrm{𝚜𝚟𝚊𝚛},𝚌-\mathrm{𝚒𝚗𝚝}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚂𝙴𝚃𝚂},\left[𝚜,𝚌\right]\right)
\mathrm{𝚂𝙴𝚃𝚂}.𝚌\ge 1
Given a collection of set variables
{s}_{1},{s}_{2},\cdots ,{s}_{n}
and their respective cardinality
{c}_{1},{c}_{2},\cdots ,{c}_{n}
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
constraint forces the following two conditions:
\forall i\in \left[1,n\right]:|{s}_{i}|={c}_{i}
\forall i,j\in \left[1,n\right]\left(i<j\right):|{s}_{i}\bigcap {s}_{j}|\le 1
\left(\begin{array}{c}〈\begin{array}{cc}𝚜-\left\{5,8\right\}\hfill & 𝚌-2,\hfill \\ 𝚜-\left\{5\right\}\hfill & 𝚌-1,\hfill \\ 𝚜-\left\{5,6,7\right\}\hfill & 𝚌-3,\hfill \\ 𝚜-\left\{1,4\right\}\hfill & 𝚌-2\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
|\left\{5,8\right\}|=2
|\left\{5\right\}|=1
|\left\{5,6,7\right\}|=3
|\left\{1,4\right\}|=2
|\left\{5,8\right\}\bigcap \left\{5\right\}|\le 1
|\left\{5,8\right\}\bigcap \left\{5,6,7\right\}|\le 1
|\left\{5,8\right\}\bigcap \left\{1,4\right\}|\le 1
|\left\{5\right\}\bigcap \left\{5,6,7\right\}|\le 1
|\left\{5\right\}\bigcap \left\{1,4\right\}|\le 1
|\left\{5,6,7\right\}\bigcap \left\{1,4\right\}|\le 1
|\mathrm{𝚂𝙴𝚃𝚂}|>1
\mathrm{𝚂𝙴𝚃𝚂}
\mathrm{𝚂𝙴𝚃𝚂}.𝚜
\mathrm{𝚂𝙴𝚃𝚂}.𝚜
\mathrm{𝚂𝙴𝚃𝚂}
When we have only two set variables the
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
constraint was called
\mathrm{𝚙𝚊𝚒𝚛}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
in [HoeveSabharwal07].
C. Bessière et al. have shown in [BessiereHebrardHnichWalsh04] that it is NP-hard to enforce bound consistency for the
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
constraint. Consequently, following the first filtering algorithm from A. Sadler and C. Gervet [SadlerGervet01], W.-J. van Hoeve and A. Sabharwal have proposed an algorithm that enforces bound-consistency when the
\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}\mathtt{1}
constraint involves only two sets variables [HoeveSabharwal07].
at_most1 in MiniZinc.
constraint arguments: constraint involving set variables.
constraint type: predefined constraint.
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Global Constraint Catalog: Ctour
<< 5.400. temporal_path5.402. track >>
\mathrm{𝚝𝚘𝚞𝚛}\left(\mathrm{𝙽𝙾𝙳𝙴𝚂}\right)
\mathrm{𝚊𝚝𝚘𝚞𝚛}
\mathrm{𝚌𝚢𝚌𝚕𝚎}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|\ge 3
\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)
Enforce to cover an undirected graph
G
\mathrm{𝙽𝙾𝙳𝙴𝚂}
collection with a Hamiltonian cycle.
\left(\begin{array}{c}〈\begin{array}{cc}\mathrm{𝚒𝚗𝚍𝚎𝚡}-1\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-2\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,3\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-3\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{2,4\right\},\hfill \\ \mathrm{𝚒𝚗𝚍𝚎𝚡}-4\hfill & \mathrm{𝚜𝚞𝚌𝚌}-\left\{1,3\right\}\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚝𝚘𝚞𝚛}
\mathrm{𝙽𝙾𝙳𝙴𝚂}
argument depicts the following Hamiltonian cycle visiting successively the vertices 1, 2, 3 and 4.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
When the number of vertices is odd (i.e.,
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
is odd) a necessary condition is that the graph is not bipartite. Other necessary conditions for filtering the
\mathrm{𝚝𝚘𝚞𝚛}
constraint are given in [Cymer13], [CymerPhD13].
\mathrm{𝚌𝚒𝚛𝚌𝚞𝚒𝚝}
(graph partitioning constraint,Hamiltonian),
\mathrm{𝚌𝚢𝚌𝚕𝚎}
(graph constraint),
\mathrm{𝚕𝚒𝚗𝚔}_\mathrm{𝚜𝚎𝚝}_\mathrm{𝚝𝚘}_\mathrm{𝚋𝚘𝚘𝚕𝚎𝚊𝚗𝚜}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
filtering: DFS-bottleneck, linear programming.
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(\ne \right)↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}\right)
\begin{array}{c}\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚜𝚞𝚌𝚌}\right)⇔\hfill \\ \mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}\left(\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{1}.\mathrm{𝚒𝚗𝚍𝚎𝚡},\mathrm{𝚗𝚘𝚍𝚎𝚜}\mathtt{2}.\mathrm{𝚜𝚞𝚌𝚌}\right)\hfill \end{array}
\mathrm{𝐍𝐀𝐑𝐂}
=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|*|\mathrm{𝙽𝙾𝙳𝙴𝚂}|-|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝙽𝙾𝙳𝙴𝚂}
\mathrm{𝐶𝐿𝐼𝑄𝑈𝐸}
\left(\ne \right)↦\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{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}
=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
•
\mathrm{𝐌𝐈𝐍}_\mathrm{𝐈𝐃}
=2
•
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐈𝐃}
=2
•
\mathrm{𝐌𝐈𝐍}_\mathrm{𝐎𝐃}
=2
•
\mathrm{𝐌𝐀𝐗}_\mathrm{𝐎𝐃}
=2
The first graph property enforces the subsequent condition: If we have an arc from the
{i}^{th}
vertex to the
{j}^{th}
vertex then we have also an arc from the
{j}^{th}
{i}^{th}
vertex. The second graph property enforces the following constraints:
We have one strongly connected component containing
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
Each vertex has exactly two predecessors and two successors.
Part (A) of Figure 5.401.1 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 5.401.1 gives the final graph associated with the Example slot. The
\mathrm{𝚝𝚘𝚞𝚛}
constraint holds since the final graph corresponds to a Hamiltonian cycle.
\mathrm{𝚝𝚘𝚞𝚛}
Since the maximum number of vertices of the final graph is equal to
|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
, we can rewrite the graph property
\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}=|\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}\ge |\mathrm{𝙽𝙾𝙳𝙴𝚂}|
\underline{\overline{\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}}}
\overline{\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐒𝐂𝐂}}
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Cryptograms: Level 1 Challenges Practice Problems Online | Brilliant
\Large \begin{array} {c c c } & \color{#3D99F6}A & \color{#D61F06}B & \color{#EC7300}C \\ \times& & & 8 \\ \hline \color{#3D99F6}A & \color{#EC7300}C & \color{#EC7300}C & \color{#EC7300}C \\ \end{array}
In the cryptogram above,
A
B
C
are distinct digits. Find the value of
A+B+C+8.
A, B, C,
D
are distinct digits, find the maximum possible value of the following sum:
\begin{array} {ccc} & A & B \\ + & C & D \\ \hline \end{array}
\large{\begin{array}{lllllllll}&&&&&&&1&1&4&X&9\\\times&&&&&&&1&X&8&2&1\\\hline &\ &&1&2&3&4&5&6&7&8&9 \\\hline\end{array}}
Above shows an incomplete long multiplication for which X represents a single digit integer. What is the value of X?
by Syed Hissaan
\begin{array} { l l l } & & & X & Y & Z \\ & & & X & Y & Z \\ &+ & & X & Y & Z \\ \hline & & & Z & Z & Z \\ \end{array}
X, Y,
Z
X \times Y \times Z ?
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CCDALIGN
Aligns images graphically by interactive object selection
This program aids the registration of images which may not be related by simple offsets (see FINDOFF and PAIRNDF if they are). It also has the capability of dealing with groups of images which are almost registered (frames which have not been moved on the sky) saving effort in re-identification of image features.
The basic method used is to supply a list of images and an optional reference image. The first image or the reference image is initially displayed and you are invited to mark the positions of centroidable image features on it using a graphical interface. This window then remains on the screen for reference while you identify the same features on each of the other images in the same way.
After centroiding you are then given the option to stop. If you decide to, then you will have labelled position lists to use in the other CCDPACK routines (the labelled positions will be called IMAGE_NAME.acc). If you choose the option to continue then a full registration of the images will be attempted. This may only be performed for ’linear’ transformations.
After choosing a transformation type the procedure will then go on to calculate a transformation set between all the images; this is used (with the extended reference set from REGISTER) to approximate the position of all possible image features, which are then located by centroiding and a final registration of all images is performed. The resultant images then have associated lists of labelled positions, and attached coordinate systems which may be used to transform other position lists or when resampling the data.
If the EXTRAS parameter is true you may also enter, for each of the original images, a group of images which is almost registered with it (within the capabilities of centroiding, i.e. a few pixels). In this way similar registration processes can be performed on many almost-aligned images without additional work from the user.
The graphical interface used for marking features on the image should be fairly self-explanatory. The image can be scrolled using the scrollbars, the window can be resized, and there are controls for zooming the image in or out, changing the style of display and altering the percentile cutoff limits. The displayed index numbers of any identified features on each image must match those on the reference image (though it is not necessary to identify all of the features from the reference image on each one), and there is also a control for selecting the number of the next point to mark. Points are added by clicking mouse button 1 (usually the left one) and may be removed by clicking mouse button 3 (usually the right one). It is possible to edit the points marked on the reference image while you are marking points on the other images. When you have selected all the points you wish to on a given image, click the ’Done’ button and you will be presented with the next one.
ccdalign in
CONTINUE = _LOGICAL (Read)
If TRUE then this command will proceed to also work out the registrations of your images. Note that this is only possible if you are intending to use linear transformations (this is the usual case). [FALSE]
EXTRAS = _LOGICAL (Read)
If this parameter is true, then for each image (or Set of images, if USESET is true) from the IN list you will be prompted to enter a group of corresponding names which represent more files of the same type pointing at (almost) the same sky position as the one in the IN list. CCDALIGN will then centroid the marked objects in all the images in the same group so that multiple similar registrations can be done at the same time. [FALSE]
The type of fit which should be used when determining the transformation between the input positions lists. This may take the values
6 – self defined function
A list of the images to be displayed in the GUI for interactive marking of features. The names should be separated by commas and may include wildcards.
If the logging system has been initialised using CCDSETUP then the value specified there will be used. Otherwise, the default is ’CCDPACK.LOG’. [CCDPACK.LOG]
If the logging system has been initialised using CCDSETUP then the value specified there will be used. Otherwise, the default is ’BOTH’. [BOTH]
A string indicating how markers are initially to be plotted in the aligner widget. It consists of a comma-separated list of "attribute=value" type strings. The available attributes are:
This parameter only gives the initial marker type; it can be changed interactively while the program is running. If specifying this value on the command line, it is not necessary to give values for all the attributes; missing ones will be given sensible defaults. [""]
MAXCANV = INTEGER (Read and Write)
MORE = LITERAL (Read)
If EXTRAS is true, this parameter is used to get a list of images corresponding to each one which is named by the IN parameter. These lists are always got interactively; MORE values cannot be given on the command line. For any given response the null value (!) may be supplied, indicating that there are no similarly aligned images. If the original image is included again in the supplied MORE value, it will be ignored, since it already forms part of the group being considered. [!]
The initial low and high percentiles of the data range to use when displaying the images; any pixels with a value lower than the first element will have the same colour, and any with a value higher than the second will have the same colour. Must be in the range 0
<
<
<
= 100. This can be changed from within the GUI. [2,98]
REFNDF = LITERAL (Read)
The name of an additional reference image (or Set); this is the first image displayed and the one which will be visible while you are marking points on all the others. If the null value (!) is supplied then no additional reference image will be used, and the first one in the IN list will be the first displayed. [!]
This parameter determines whether Set header information will be used. If USESET is true, then CCDALIGN will try to group images according to their Set Name attribute before displaying them, rather than treating them one by one. All images in the IN list which share the same (non-blank) Set Name attribute, and which have a CCD_SET attached coordinate system, will be shown together as a single image in the viewer for object marking, plotted in their CCD_SET coordinates.
If USESET is false, then regardless of Set headers, each individual image will be displayed for marking separately. If the input images have no Set headers, or if they have no CCD_SET coordinates in their WCS components, the value of this parameter will make no difference.
If a global value for this parameter has been set using CCDSETUP than that value will be used. [FALSE]
WINX = INTEGER (Read and Write)
WINY = INTEGER (Read and Write)
ZOOM = DOUBLE (Read and Write)
\ast
continue=no
This will display all the images in the current directory and invite you to mark corresponding image features on each one in turn. When you have done this, the centroids will be calculated and you will be left with a position list with the extension ‘.acc’ associated with each one.
ccdalign "x1008,x1009,x1010" refndf=xmos extras=yes continue
Here the EXTRAS parameter is true, so for each of the named images you will be prompted for a list of other images which were taken pointing in the same direction. The file ‘xmos’ is being used as the reference image, so that will be presented first for marking features. When you have marked features on all four images, the program will go on to match them all up and produce a global registration, attaching a new coordinate system in which they are all registered to each file.
“General linear transformations”, IDICURS, PAIRNDF.
All parameters retain their current value as default. The ’current’ value is the value assigned on the last run of the application. If the application has not been run then the ’intrinsic’ defaults, as shown in the parameter help, apply.
Some of the parameters (MAXCANV, PERCENTILES, WINX, WINY, ZOOM, MARKSTYLE) give initial values for quantities which can be modified while the program is running. Although these may be specified on the command line, it is normally easier to start the program up and modify them using the graphical user interface. If the program exits normally, their values at the end of the run will be used as defaults next time the program starts up.
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PAIRNDF
Aligns images graphically by drag and drop
This routine accepts a list of images which may be aligned using simple offsets in their Current coordinate frames. By making use of a graphical user interface you can indicate how pairs of images are aligned with respect to each other, and mark image features to allow accurate alignments. Once enough pairings have been specified to register all frames completely a global merger of all the positions for each image takes place. This results in the output of one list of uniquely labelled positions for each image. These position lists can then be used in a routine such as REGISTER to produce the actual transformation between the images.
If images have been grouped into Sets for alignment purposes by using MAKESET, and the USESET parameter is true, then the program will treat each Set of images as a single image to be aligned.
The graphical interface consists of two parts: a chooser which allows you to nominate pairs of images to be aligned, and an aligner which allows you to move the pair around the screen until they are registered, and to mark points in the overlapping region where the same centroidable features exist on both images.
Operation is as follows. You must first use the chooser window to select a pair of images which have a region in common (if you only have two images this step may be skipped). Use the tabs at either side of the screen to pick the image to appear on that side. You can use the "Show FITS" button to select one or more FITS headers to be displayed alongside each image if this will make it easier to identify which is which. You can use the "Display cutoff" menu to select the percentiles controlling the brightness of each pixel; alignment is easier if the same features are of a similar brightness in different images. The images are displayed resampled into their Current coordinates, so that their orientation will be the same as in the aligner. You can only align them using this program if a simple offset (translation) maps one onto another in these coordinates (or very nearly does so). If that is not the case, you will have to set their Current coordinate system to a different value (see WCSEDIT) or align them using a different method. The whole of each image will be displayed in the chooser window, select a pair with an overlapping region which you wish to align, and click the "Use this pair" button. The aligner window will then appear, displaying the two images which you have selected. The chooser window can normally be resized in the normal way to make the images bigger or smaller. However there is currently a bug which causes this to crash in some window managers which use continuous resizing. In this case you must use the PREVX and PREVY parameters to change the image size.
In the aligner window you can drag either of these images around the display region by holding down mouse button 1 (usually the left one) as you move the mouse; the easiest way to align the pair is to "pick up" one image by an identifiable feature and "drop" it on the same feature in the other image. Where the images overlap their pixels will be averaged. If they are not correctly positioned, you can move them again. Once you are happy that they are aligned about right, then click in the overlap region to mark features which appear in both images. During this part you mark points by clicking with mouse button 1 (usually the left one) and you can remove them by clicking with button 3 (usually the right one).
When you add a point by clicking it will be centroided on both images, and two markers plotted, one for each centroided position. If a centroidable object near that point cannot be identified on both images the program will not allow you to mark a point there. However, note that the centroiding algorithm is capable of locating spurious objects from noise, so the fact that a point can be marked does not prove that a real feature exists on both images. By looking at the two markers it should be possible to see whether a real feature has been located. Though the two markers do not need to be exactly concentric (REGISTER can take care of that later), the offset between them should be similar to that of other marked objects nearby in the overlap region. If you do not think the same object has been identified in both images, you should remove this point (with mouse button 3).
The aligner window can be resized, the magnification changed using the "Zoom" control, the display region scrolled using the scrollbars, and the shape and colour of the point markers selected. When you have aligned the images and marked shared features, or if you decide that the pair cannot be satisfactorily registered, click the "Done" button.
You will then be returned to the chooser window to select another pair and repeat the process. After the first time however, you will only be allowed to select a pair of images to align if at least one of them has already been aligned. Those which have already been done are marked with a ‘
+
’ sign on their selection tabs.
Once you have made enough pairings to register the whole set, the graphical windows will disappear and the program will complete the global matching up of positions without any further user interaction.
pairndf in outlist percentiles
CHOOSER = _LOGICAL (Read)
If only two images are presented in the IN list, then this parameter determines whether they should be previewed in the chooser widget before they are presented for alignment. With only two, the chooser is not normally necessary since there is only one possible pair to select for alignment, but if you want to equalise the image brightnesses using the “Display cutoff” button or preview FITS headers you may wish to respond true to this. [FALSE]
A list of image names whose data are to be transformed. The image names should be separated by commas and may include wildcards.
MARKSTYLE1 = LITERAL (Read and Write)
A string indicating how markers are initially to be plotted in the aligner widget to represent points on the left hand image. It consists of a comma-separated list of "attribute=value" type strings. The available attributes are:
colour – Colour of the marker in Xwindows format.
size – Approximate height of the marker in pixels.
thickness – Approximate thickness of lines in pixels.
shape – One of Plus, Cross, Circle, Square, Diamond.
This parameter only gives the initial marker type; it can be changed interactively while the program is running. If specifying this value on the command line, it is not necessary to give values for all the attributes; missing ones will be given sensible defaults. ["shape=plus"]
A string indicating how markers are initially to be plotted in the aligner widget to represent points on the right hand image. It consists of a comma-separated list of "attribute=value" type strings. The available attributes are:
This parameter only gives the initial marker type; it can be changed interactively while the program is running. If specifying this value on the command line, it is not necessary to give values for all the attributes; missing ones will be given sensible defaults. ["shape=circle"]
MAXCANV = _INTEGER (Read and Write)
A value in pixels for the maximum initial X or Y dimension of the region in which the image is displayed. Note this is the scrolled region, and may be much bigger than the sizes given by WINX and WINY, which limit the size of the window on the X display. It can be overridden during operation by zooming in and out using the GUI controls, but it is intended to limit the size for the case when ZOOM is large (perhaps because the last image was quite small) and a large image is going to be displayed, which otherwise might lead to the program attempting to display an enormous viewing region. If set to zero, then no limit is in effect. [1280]
OVERRIDE = _LOGICAL (Read)
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 paired.
In this case, any images for which matching was not achieved will have their associated position lists removed from their .MORE.CCDPACK extensions. Thus after running PAIRNDF with OVERRIDE set to TRUE, any position list associated with an image is guaranteed to be one which has been matched, and not just one left over from the previously associated unmatched list. [TRUE]
OUTLIST = LITERAL (Read)
An expression which is either a list of names or expands to a list of names for the output position lists.
These may be specified as list of comma separated names, using indirection if required, OR, as a single modification element (of the input names). The simplest modification element is the asterisk "
\ast
" which means call each of the output lists the same name as the corresponding input images (but without the ".sdf" extension). So, IN
>
\ast
>
\ast
signifies that all the images in the current directory should be used and the output lists should have the same names.
Other types of modification can also occur, such as, OUTLIST
>
\ast
_objs.dat which means call the position lists the same as the input images but put "_objs.dat" after the names. Replacement of a specified string with another in the output file names can also be used, OUTLIST
>
\ast
|
_debias
|
_images.dat
|
this replaces the string "_debias" with "_images.dat" in any of the output names.
If wildcarded names for the input images are used then it is recommended that wildcards are also used for the position list names as the correspondence between these may be confusing. [
\ast
.DAT]
The default low and high percentiles of the data range to use when displaying the images; any pixels with a value lower than the first element will have the same colour, and any with a value higher than the second will have the same colour. This parameter gives the default value - the percentile settings can be set for each image individually from within the GUI to accomodate the situation where images have different brightnesses. Must be in the range 0
<
<
<
= 100. [2,98]
PREVX = _INTEGER (Read and Write)
The initial width in pixels of the preview display for each image; two images will be displayed side by side at any one time at this size in the chooser window. This can be effectively changed by resizing the entire chooser window in the normal way using the window manager while the program is running. [350]
PREVY = _INTEGER (Read and Write)
The initial height in pixels of the preview display for each image; two images will be displayed side by side at any one time at this size in the chooser window. This can be effectively changed by resizing the entire chooser window in the normal way using the window manager while the program is running. [350]
TOLER = _DOUBLE (Read)
The tolerance for deduplicating centroided points (in pixels). If two centroided objects on the same image are within this distance of each other they will be identified as the same object. For a bright elliptical object, centroiding arising from any nearby point will normally arrive at the same position, so this can be set to a small value (less than 1), but if the objects being identified cover many pixels and are close to the background noise level it may be advantageous to set it to a larger value so that centroids near to each other are identified as referring to the same object. [0.5]
This parameter determines whether Set header information should be used or not. If USESET is true, PAIRNDF will try to group images according to their Set Name attribute. All images which share the same (non-blank) Set Name attribute, and which have a CCD_SET attached coordinate system, will be grouped together and treated as a single image for alignment. In the graphical part of the program you will view and position this group of images as a single item.
If the input images have no Set headers, or if they have no Set alignment coordinate system (one with a Domain of CCD_SET) the setting of USESET will make no difference.
WINX = _INTEGER (Read and Write)
The initial width in pixels of the aligner window, which contains a space for dragging around a pair of images and associated controls. If the region required for the images is larger than the area allocated for display, it can be scrolled around within the window. The window can be resized in the normal way using the window manager while the program is running. [800]
WINY = _INTEGER (Read and Write)
The initial height in pixels of the aligner window, which contains space for dragging around a pair of images and associated controls. If the region required for the images is larger than the area allocated for display, it can be scrolled around within the window. The window can be resized in the normal way using the window manager while the program is running. [400]
ZOOM = _DOUBLE (Read and Write)
A factor giving the initial level to zoom in to the images displayed in the aligner window, that is the number of screen pixels to use for one image pixel. It will be rounded to one of the values ... 3, 2, 1, 1/2, 1/3 .... The zoom can be changed interactively from within the program. The initial value may be limited by MAXCANV. [1]
\ast
\ast
.dat [1,99]
This example shows the positional nature of the parameters. All the images in the current directory are presented for alignment. Their output position lists have the same name as the images except that they have a file extension of .dat. The default image display cutoff is between the 1st and 99th percentile, which shows bright detail well.
pairndf in="data1,data2" outlist="d1-pos,d2-pos" zoom=2 maxcanv=0 markstyle1="shape=circle,size=8,thickness=1,colour=HotPink"
Only the two images data1 and data2 will be aligned, and the corresponding sets of positions will be written to the files d1-pos and d2-pos. The images will initially be displayed for alignment at a magnification of two screen pixels to each data pixel, even if that results in a very large display area. During alignment, marked points on the left hand image will be shown as little pink circles.
“Semi-automated registration”, CCDALIGN, IDICURS.
On exit the CURRENT_LIST items in the CCDPACK extensions (.MORE.CCDPACK) of the input NDFs are set to the names of the appropriate output lists. These items will be used by other CCDPACK position list processing routines to automatically access the lists.
Output position list format.
CCDPACK format - Position lists in CCDPACK are formatted files whose first three columns are interpreted as the following.
The column one value must be an integer and is used to identify positions which may have different locations but are to be considered as the same point. Comments may be included in the file using the characters # and !. Columns may be separated by the use of commas or spaces.
In all cases, the coordinates in position lists are pixel coordinates.
Certain parameters (LOGTO, LOGFILE and USESET) have global values. These global values will always take precedence, except when an assignment is made on the command line. Global values may be set and reset using the CCDSETUP and CCDCLEAR commands.
Some of the parameters (MARKSTYLE1, MARKSTYLE2, MAXCANV, PERCENTILES, PREVX, PREVY, WINX, WINY) give initial values for quantities which can be modified while the program is running. Although these may be specified on the command line, it is normally easier to start the program up and modify them using the graphical user interface. If the program exits normally, their values at the end of the run will be used as defaults next time the program starts up.
Supports Bad pixel values and all non-complex data types.
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Apple SPG
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Synthesis of Incremental Regions of Attraction
Formal methods in control design for nonlinear systems are often focused on the stabilization of an equilibrium. Modern robotics applications, however, require notions of stability to be defined when tracking reference trajectories. The idea behind an incremental region of attraction is that as long as an attractive region can be defined around any trajectory within some operating regime then all initial conditions of the system that start within the region converge to the trajectory asymptotically or exponentially. The synthesis of the region involves the verification of an incremental Lyapunov function using a branch-and-cut approach as shown below.
Once a suitable incremental region of attraction is found, the attractive region around any trajectory can be efficiently computed. For example, the figure below shows the attractive region around a swing-up trajectory of a torque-limited inverted pendulum. The blue lines indicate simulations started randomly within the attractive region. Some of the randomized simulations plotted above are animated in the following GIF. This is currently unpublished work, please watch this space for more information.
Contraction Theory-Based
\mathcal{L}_1
L1-Adaptive Control
Contraction theory is a tool to analyze incremental stability properties for nonlinear systems and constructively design tracking controllers that stabilize the system around any trajectory. In (Zhao et al. (2021)), we provide synthesis procedures to compute robust control contraction metrics that minimize the
\mathcal{L}_\infty
L∞-gain from the disturbances to the tracking error. Even better performance can be acheived by augmenting the contraction theory-based setup with the
\mathcal{L}_1
L1-adaptive control architeture to compensate for the disturbances within the control channel (Lakshmanan et al. (2020)). We provide certificates in the form of tubes whose width is adjustable based on the adaptive controller parameters. Furthermore, they can be incorporated into motion planners to provide paths that are safe with respect to the disturbances, as shown below. The figure illustrates the path of 10 unicycle systems with randomized initializations that remain within the tube and avoid collisions with the gray obstacles. While the tubes are tunable based on tracking requirements, their adjustment relies on a trade-off with the inherent robustness of the system to model inaccuracies. In (Lakshmanan et al. (2021)), we show that the performance of the contraction theory-based
\mathcal{L}_1
L1-adaptive control architecture can be improved by learning any unmodeled dynamics in the system without sacrificing robustness. The GIF below illustrates the improvement in the peformance a vehicle traversing a race track in the form of tighter tubes. In the video below, we show the performance of an
\mathcal{L}_1
L1-adaptive control augmentation to the tracking control of a quadrotor by injecting disturbances in several scenarios (Wu et al. (2021)).
A. Lakshmanan[1], A. Gahlawat[1], N. Hovakimyan. "Safe Feedback Motion Planning: A Contraction Theory and
\mathcal{L}_1
L1-Adaptive Control Based Approach." Proceedings of the 59th IEEE Conference on Decision and Control (CDC), pp. 1578-1583, 2020.
A. Lakshmanan[1], A. Gahlawat[1], L. Song, A. Patterson, Z. Wu, N. Hovakimyan, E. Theodorou. "Contraction
\mathcal{L}_1
L1-Adaptive Control using Gaussian Processes." Proceedings of the 3rd Conference on Learning for Dynamics and Control, pp. 1027-1040, 2021.
P. Zhao, A. Lakshmanan, K. Ackerman, A. Gahlawat, M. Pavone, & N. Hovakimyan. "Tube-certified trajectory tracking for nonlinear systems with robust control contraction metrics." Under review at ICRA 2022.
Z. Wu, S. Cheng, K. Ackerman, A. Gahlawat, A. Lakshmanan, P. Zhao, & N. Hovakimyan."
\mathcal{L}_1
L1 Adaptive Augmentation for Geometric Tracking Control of Quadrotors." Under review at ICRA 2022.
[1] Equal contribution
Fast Collision Detection for Trajectories
Collision checking is a computational bottleneck in motion planning problems. While there exist several fast methods for exact collision checking between convex objects, collision checking for trajectories are generally more ad-hoc — they are typically checked pointwise up to some resolution. In (Lakshmanan et al. (2019)), we provide fast methods to check for collisions for absolutely continuous curves. In the video below, we describe the prodcedure at a high-level, and show an example of how such methods may be employed in sampling-based planning. The collision detection is extensible to situations when only a probabilistic model of the motion of an obstacle is available as a high-probability result (Patterson et al. (2020)). The following figure shows two paths (green and blue), one of which has a higher probability collision than another with an obstacle. The motion of the obstacle is predicted using a gaussian process model based on past data and probabilistic intention.
A. Lakshmanan, A. Patterson, V. Cichella, N. Hovakimyan. "Proximity Queries for Absolutely Continuous Parametric Curves." Proceedings of Robotics: Science and Systems XV, 2019.
A. Patterson, A. Lakshmanan, N. Hovakimyan. "Intent-Aware Probabilistic Trajectory Estimation for Collision Prediction with Uncertainty Quantification." Proceedings of the 58th IEEE Conference on Decision and Control (CDC), pp. 3827-3832, 2019.
© Arun Lakshmanan. Website built with Franklin.jl and the Julia programming language.
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extended Euclidean algorithm for polynomials
gcdex(A, B, C, x, 's', 't')
polynomials in the variable x
For the first calling sequence (when the number of parameters is less than six), gcdex applies the extended Euclidean algorithm to compute unique polynomials s, t, and g in x such that
sA+tB=g
where g is the monic GCD (Greatest Common Divisor) of A and B. The results computed satisfy
\mathrm{degree}\left(s\right)<\mathrm{degree}\left(\frac{B}{g}\right)
\mathrm{degree}\left(t\right)<\mathrm{degree}\left(\frac{A}{g}\right)
. The GCD g is returned as the function value.
If arguments s and t are specified, they are assigned the cofactors.
In the second calling sequence, gcdex solves the polynomial Diophantine equation
sA+tB=C
for polynomials s and t in x. Let g be the GCD of A and B. The input polynomial C must be divisible by g; otherwise, an error message is displayed. The polynomial s computed satisfies
\mathrm{degree}\left(s\right)<\mathrm{degree}\left(\frac{B}{g}\right)
\mathrm{degree}\left(\frac{C}{g}\right)<\mathrm{degree}\left(\frac{A}{g}\right)+\mathrm{degree}\left(\frac{B}{g}\right)
then the polynomial t will satisfy
\mathrm{degree}\left(t\right)<\mathrm{degree}\left(\frac{A}{g}\right)
. The NULL value is returned as the function value.
In this case, s and t are not optional.
Note that if the input polynomials are multivariate then, in general, s and t will be rational functions in variables other than x.
\mathrm{gcdex}\left({x}^{3}-1,{x}^{2}-1,x,'s','t'\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
s,t
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}
\mathrm{gcdex}\left({x}^{2}+a,{x}^{2}-1,{x}^{2}-a,x,'s','t'\right)
s,t
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
\mathrm{gcdex}\left(1,x,1-2x+4{x}^{2},x,'s','t'\right)
s,t
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}
\mathrm{gcdex}\left({x}^{2}-1,{x}^{3}-1,x,x,'s','t'\right)
Error, (in `gcdex/diophant`) the Diophantine equation has no solution
\mathrm{gcd}\left({x}^{2}-1,{x}^{3}-1\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
The gcdex command was updated in Maple 2018.
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<< 2.2.1. Basic data types2.2.3. Restrictions >>
We provide the following compound data types:
\mathrm{𝚕𝚒𝚜𝚝}\left(T\right)
corresponds to a list of elements of type
T
T
is a basic or a compound data type.
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left({A}_{1},{A}_{2},\cdots ,{A}_{n}\right)
corresponds to a collection of ordered items, where each item consists of
n>0
{A}_{1},{A}_{2},\cdots ,{A}_{n}
. Each attribute is an expression of the form
𝚊-T
𝚊
is the name of the attribute and
T
the type of the attribute (a basic or a compound data type). All names of the attributes of a given collection should be distinct and different from the keyword
\mathrm{𝚔𝚎𝚢}
, which corresponds to an implicitThis attribute is not explicitly defined. attribute. Its value is the position of an item within the collection. The first item of a collection is associated with position 1.
The following notations are used for instantiated arguments:
A list of elements
{e}_{1},{e}_{2},\cdots ,{e}_{n}
\left[{e}_{1},{e}_{2},\cdots ,{e}_{n}\right]
A finite set of integers
{i}_{1},{i}_{2},\cdots ,{i}_{n}
\left\{{i}_{1},{i}_{2},\cdots ,{i}_{n}\right\}
A multiset of integers
{i}_{1},{i}_{2},\cdots ,{i}_{n}
\left\{\left\{{i}_{1},{i}_{2},\cdots ,{i}_{n}\right\}\right\}
n
items, each item having
m
attributes, is denoted by
〈{𝚊}_{1}-{v}_{11}\cdots {𝚊}_{m}-{v}_{1m},{𝚊}_{1}-{v}_{21}\cdots {𝚊}_{m}-{v}_{2m},\cdots ,{𝚊}_{1}-{v}_{n1}\cdots {𝚊}_{m}-{v}_{nm}〉
. Each item is separated from the previous item by a comma. When the items of the collection involve a single attribute
{𝚊}_{1}
〈{v}_{11},{v}_{21},\cdots ,{v}_{n1}〉
can possibly be used as a shortcut for
〈{𝚊}_{1}-{v}_{11},{𝚊}_{1}-{v}_{21},\cdots ,{𝚊}_{1}-{v}_{n1}〉
{i}^{th}
item of a collection
𝚌
𝚌\left[i\right]
The value of the attribute
𝚊
{i}^{th}
𝚌
𝚌\left[i\right].𝚊
. Note that, within an arithmetic expression, we can use the shortcut
𝚌\left[i\right]
when the collection
𝚌
involves a single attribute.
The number of items of a collection
𝚌
|𝚌|
EXAMPLE: Let us illustrate with four examples, the types one can create. These examples concern the creation of a collection of variables, a collection of tasks, a graph variable [Dooms06] and a collection of orthotopes.An orthotope corresponds to the generalisation of a segment, a rectangle and a box to the
n
In the first example we define
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
so that it corresponds to a collection of variables.
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
constraint. The declaration
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}:\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
defines a collection of items, each of which having one attribute
\mathrm{𝚟𝚊𝚛}
that is a domain variable.
In the second example we define
\mathrm{𝚃𝙰𝚂𝙺𝚂}
so that it corresponds to a collection of tasks, each task being defined by its origin, its duration, its end and its resource consumption. Such a collection is for instance used in the
\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}
\mathrm{𝚃𝙰𝚂𝙺𝚂}:\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚎𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}-\mathrm{𝚍𝚟𝚊𝚛}\right)
defines a collection of items, each of which having the four attributes
\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}
\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}
\mathrm{𝚎𝚗𝚍}
\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}
which all are domain variables.
In the third example we define a graph as a collection of nodes
\mathrm{𝙽𝙾𝙳𝙴𝚂}
, each node being defined by its index (i.e., identifier) and its successors. Such a collection is for instance used in the
\mathrm{𝚍𝚊𝚐}
\mathrm{𝙽𝙾𝙳𝙴𝚂}:\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚒𝚗𝚍𝚎𝚡}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚜𝚞𝚌𝚌}-\mathrm{𝚜𝚟𝚊𝚛}\right)
defines a collection of items, each of which having the two attributes
\mathrm{𝚒𝚗𝚍𝚎𝚡}
\mathrm{𝚜𝚞𝚌𝚌}
which respectively are integers and set variables.
In the last example we define
\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}
so that is corresponds to a collection of orthotopes. Each orthotope is described by an attribute
\mathrm{𝚘𝚛𝚝𝚑}
. Unlike the previous examples, the type of this attribute does not correspond any more to a basic data type but rather to a collection of
n
items, where
n
is the number of dimensions of the orthotope.1 for a segment, 2 for a rectangle, 3 for a box, ... . This collection, named
\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}
, defines for a given dimension the origin, the size and the end of the object in this dimension. This leads to the two declarations:
\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚒}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚜𝚒𝚣}-\mathrm{𝚍𝚟𝚊𝚛},\mathrm{𝚎𝚗𝚍}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}-\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚘𝚛𝚝𝚑}-\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴}\right)
\mathrm{𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴𝚂}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
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PDA_RNGAM
Returns pseudo-random numbers from a gamma distribution
This is a simple random-number generator providing deviates in the from a gamma distribution, with a period of 2
\ast
\ast
26, and to 6 or 7 digits accuracy. It is based upon Ahrens, Dieter & Grube’s TOMS599 routines. A value of zero is returned if the argument of the gamma function is not positive.
RESULT = PDA_RNGAM( A )
A = REAL (Given)
The argument (mean) of the gamma function.
PDA_RNGAM = REAL
The pseudo-random deviate. A value of zero is returned if the argument of the gamma function is not positive.
Ahrens, J.H., & Dieter, U. 1982, "Generating gamma variates by a modified rejection technique", Comm. ACM 25(1), pp.47–54. (For A
>
= 1.0, algorithm GD) Ahrens, J.H., & Dieter, U. 1974, "Computer Methods for sampling gamma, Poisson and binomial distributions", Computing. 12), pp.223–246. (For 0.0
<
<
1.0, adapted algorithm GS)
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Solenoid_valve Knowpia
A solenoid valve is an electromechanically operated valve.
Solenoid valves are the most frequently used control elements in fluidics. Their tasks are to shut off, release, dose, distribute or mix fluids. They are found in many application areas. Solenoids offer fast and safe switching, high-reliability, long service life, good medium compatibility of the materials used, low control power and compact design.
There are many valve design variations. Ordinary valves can have many ports and fluid paths. A 2-way valve, for example, has 2 ports; if the valve is open, then the two ports are connected and fluid may flow between the ports; if the valve is closed, then ports are isolated. If the valve is open when the solenoid is not energized, then the valve is termed normally open (N.O.). Similarly, if the valve is closed when the solenoid is not energized, then the valve is termed normally closed (N.C.).[1] There are also 3-way and more complicated designs.[2] A 3-way valve has 3 ports; it connects one port to either of the two other ports (typically a supply port and an exhaust port).
Solenoid valves are also characterized by how they operate. A small solenoid can generate a limited force. An approximate relationship between the required solenoid force Fs, the fluid pressure P, and the orifice area A for a direct acting solenoid valve is:[3]
{\displaystyle F_{s}=P*A=P\pi d^{2}/4}
If the force required is low enough, the solenoid is able to directly actuate the main valve. These are simply called Direct-Acting solenoid valves. When electricity is supplied, electrical energy is converted to mechanical energy, physically moving a barrier to either obstruct flow (if it is N.O.) or allow flow (if it is N.C.). A spring is often used to return the valve to its resting position once power is shut off. Direct-acting valves are useful for their simplicity, although they do require a large amount of power relative to other types of solenoid valves.[4]
If fluid pressures are high and orifice diameter is large, a solenoid may not generate enough force on its own to actuate the valve. To solve this, a Pilot-Operated solenoid valve design can be used.[1] Such a design uses the pressurized fluid itself to apply the forces required to actuate the valve, with the solenoid as a "pilot" directing the fluid (see subsection below). These valves are used in dishwashers, irrigation systems, and other applications where large pressures and/or volumes are desired. Pilot-operated solenoids tend to consume less energy than direct-action, although they will not work at all without sufficient fluid pressure and are more susceptible to getting clogged if the fluid has solid impurities.[4]
A direct-acting solenoid valve typically operates in 5 to 10 milliseconds. Pilot-operated valves are slightly slower; depending on their size, typical values range from 15 to 150 milliseconds.[2]
Power consumption and supply requirements of the solenoid vary with application, being primarily determined by fluid pressure and orifice diameter. For example, a popular 3⁄4-inch 150 psi sprinkler valve, intended for 24 VAC (50–60 Hz) residential systems, has a momentary inrush of 7.2 VA, and a holding power requirement of 4.6 VA.[5] Comparatively, an industrial 1⁄2-inch 10,000 psi valve, intended for 12, 24, or 120 VAC systems in high-pressure fluid and cryogenic applications, has an inrush of 300 VA and a holding power of 22 VA.[6] Neither valve lists a minimum pressure required to remain closed in the unpowered state.
E- Electro Mechanical Solenoid
Pilot-operatedEdit
While there are multiple design variants, the following is a detailed breakdown of a typical pilot-operated solenoid valve. They may use metal seals or rubber seals, and may also have electrical interfaces to allow for easy control.
The diagram to the right shows the design of a basic valve, controlling the flow of water in this example. The top half shows the valve in its closed state. An inlet stream of pressurized water enters at A. B is an elastic diaphragm and above it is a spring pushing it down. The diaphragm has a pinhole through its center which allows a very small amount of water to flow through. This water fills cavity C so that pressure is roughly equal on both sides of the diaphragm. However, the pressurized water in cavity C acts across a much greater area of the diaphragm than the water in inlet A. From the equation
{\displaystyle F=P*A}
, the force from cavity C pushing downward is greater than the force from inlet A pushing upward, and the diaphragm remains closed.
Diaphragm B will stay closed as long as small drain passage D remains blocked by a pin, which is controlled by solenoid E. In a normally closed valve, supplying an electric current to the solenoid will raise the pin via magnetic force, and the water in cavity C drains out through passage D faster than the pinhole can refill it. Less water in cavity C means the pressure on that side of the diaphragm drops, proportionately dropping the force too. With the downward force of cavity C now less than the upward force of inlet A, the diaphragm is pushed upward, thus opening the valve. Water now flows freely from A to F. When the solenoid is deactivated and passage D is closed, water once again accumulates in cavity C, closing the diaphragm once the downward force exerted is great enough.
This process is the opposite for a normally open pilot-operated valve. In that case, the pin is naturally held open by a spring, passage D is open, and cavity C is never able to fill up enough, pushing open diaphragm B and allowing unobstructed flow. Supplying an electric current to the solenoid pushes the pin into a closed position, blocking passage D, allowing water to accumulate in cavity C, and ultimately closing diaphragm B.
In this way, a pilot-operated solenoid valve can be conceptualized as two valves working together: a direct-acting solenoid valve which functions as the "brain" to direct the "muscle" of a much more powerful main valve which gets actuated pneumatically or hydraulically. This is why pilot-operated valves will not work without a sufficient pressure differential between input and output, the "muscle" needs to be strong enough to push back against the diaphragm and open it. Should the pressure at the output rise above that of the input, the valve would open regardless of the state of the solenoid and pilot valve.
Bonnet–diaphragm–body seal
To simplify the sealing issues, the plugnut, core, springs, shading ring, and other components are often exposed to the fluid, so they must be compatible as well. The requirements present some special problems. The core tube needs to be non-magnetic to pass the solenoid's field through to the plugnut and the core. The plugnut and core need a material with good magnetic properties such as iron, but iron is prone to corrosion. Stainless steels can be used because they come in both magnetic and non-magnetic varieties.[13] For example, a solenoid valve might use 304 stainless steel for the body, 305 stainless steel for the core tube, 302 stainless steel for the springs, and 430 F stainless steel (a magnetic stainless steel[14]) for the core and plugnut.[1]
Solenoid valves can be used for a wide array of industrial applications, including general on-off control, calibration and test stands, pilot plant control loops, process control systems, and various original equipment manufacturer applications.[15]
History and commercial developmentEdit
^ a b c "Archived copy" (PDF). Archived from the original (PDF) on 29 October 2013. Retrieved 18 February 2013. {{cite web}}: CS1 maint: archived copy as title (link)
^ a b "Archived copy" (PDF). Archived from the original (PDF) on 25 February 2015. Retrieved 25 February 2013. {{cite web}}: CS1 maint: archived copy as title (link)
^ "The relation ignores the dynamic head" (PDF). Asconumatics.eu. p. V030-1. Retrieved 17 July 2018.
^ a b "Direct Acting vs. Pilot Operated Solenoid Valve | ATO.com". www.ato.com. Retrieved 13 July 2021.
^ "Orbit 3/4 150 PSI Sprinkler" (PDF). homedepot. Home Depot. Retrieved 9 December 2015.
^ "Omega High Pressure Solenoid Valve SVH-111/SVH-112 Series" (PDF). omega. Omega. Retrieved 9 December 2015.
^ "Microelettrovalvole - Asco Numatics Sirai". Sirai.com. Retrieved 17 July 2018.
^ "Elettrovalvole a separazione totale (DRY) - Asco Numatics Sirai". Sirai.com. Retrieved 17 July 2018.
^ Skinner Valve 1997, p. 128, stating "The tube is made of non-magnetic material to make certain that the flux is directed through the plunger rather than around it."
^ Skinner Valve (1997), Two-Way, Three-Way and Four-Way Solenoid Valves (PDF), Parker Hannifin, Catalog CFL00897 [permanent dead link], p. 128
^ "States, "Internal parts in contact with fluids are of non-magnetic 300 and magnetic 400 series stainless steel."" (PDF). Controlandpower.com. p. 450f. Retrieved 17 July 2018.
^ "Crucible Steel 430F Stainless Steel". Matweb.com. Retrieved 17 July 2018.
^ "General Purpose Solenoid Valves - Valcor Engineering". Valcor.com. Retrieved 17 July 2018.
^ Trauthwein, Greg (February 2006). "Propelling W&O Supply to New Heights". Maritime Reporter.
^ "A History of ASCO". Valveproducts.net. Retrieved 11 June 2013.
Solenoid Valve Types- circuit functions and operation types of solenoid valves explained with illustrations
pp. 39–40. Solenoid valve illustration; breakaway pin / kick-off
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PQRS is a square and ABC=90 degrees shown in the figure If AP=BQ=CR , then prove that - Maths - Quadrilaterals - 9033909 | Meritnation.com
PQRS is a square and <ABC=90 degrees shown in the figure. If AP=BQ=CR , then prove that <BAC=45 degrees
\mathrm{Here} \mathrm{AP}=\mathrm{BQ}=\mathrm{RC}....\left(1\right)\phantom{\rule{0ex}{0ex}}\mathrm{Also} \mathrm{PQ}=\mathrm{QR}\phantom{\rule{0ex}{0ex}}\mathrm{So} \mathrm{PB}+\mathrm{BQ}=\mathrm{QC}+\mathrm{CR}\phantom{\rule{0ex}{0ex}}\mathrm{PB}=\mathrm{QC} \left[\mathrm{using} \mathrm{equation} 1\right]\phantom{\rule{0ex}{0ex}}\mathrm{Now} \mathrm{in} \mathrm{triangle} \mathrm{APB} \mathrm{and} \mathrm{BQC}, \phantom{\rule{0ex}{0ex}}\mathrm{AP} = \mathrm{BQ} \phantom{\rule{0ex}{0ex}}\mathrm{PB}=\mathrm{QC} \phantom{\rule{0ex}{0ex}}\angle \mathrm{APB}=\angle \mathrm{BQC}=90 \mathrm{degree}\phantom{\rule{0ex}{0ex}}\mathrm{So} ∆\mathrm{APB} \cong ∆ \mathrm{BQC} \left[\mathrm{SAS} \mathrm{congruency}\right]\phantom{\rule{0ex}{0ex}}\mathrm{SO} \mathrm{AB} = \mathrm{BC}\phantom{\rule{0ex}{0ex}}\angle \mathrm{BAC}=\angle \mathrm{BCA}\phantom{\rule{0ex}{0ex}}\mathrm{So} \mathrm{in} \mathrm{triangle} \mathrm{ABC}, \phantom{\rule{0ex}{0ex}}90+\angle \mathrm{BAC}+\angle \mathrm{BCA}=180\phantom{\rule{0ex}{0ex}}\angle \mathrm{BAC}=\angle \mathrm{BCA}=45 \mathrm{degree}\phantom{\rule{0ex}{0ex}}
Ruby answered this
I have an exam tmr pls could someone answer this
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Presents a list of images to CCDPACK
This routine enters reduction information into the CCDPACK extensions of a list of images. This information is required if an automated reduction schedule is to be produced using SCHEDULE. Before using this routine you should set up the CCDPACK global parameters, describing the CCD characteristics, using the CCDSETUP application.
If the input images have not already been categorised then this routine performs this task for the "frame types" BIAS, TARGET, DARK, FLASH, FLAT, MASTER_BIAS, MASTER_FLAT, MASTER_DARK and MASTER_FLASH (these are input as different groups of images).
Missing exposure times for DARK and FLASH counts can be entered as can filter types.
This routine can also be used to check that a list of images have the minimum amount of information in their CCDPACK extensions to allow an automated scheduling.
present modify=? simple=? in=? bias=? target=? dark=? flash=? flat=? ftype=? filter=? darktime=? flashtime=?
The Analogue-to-Digital conversion factor. CCD readout values are usually given in Analogue-to-Digital Units (ADUs). The ADC factor is the value which converts ADUs back to the number of electrons which were present in each pixel in the CCD after the integration had finished. This value is required to allow proper estimates of the inherent noise associated with each readout value. CCDPACK makes these estimates and stores them in the variance component of the final images. Not supplying a value for this parameter (if prompted) may be a valid response if variances are not to be generated.
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if ADC=value is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
ADDDARK = _LOGICAL (Read)
Whether or not to prompt for a dark exposure time for the input images which require one. [Dynamic default, TRUE if dark count frames are given, FALSE otherwise]
ADDFLASH = _LOGICAL (Read)
Whether or not to prompt for a pre-flash exposure time for the input images which require one. [Dynamic default, TRUE if pre-flash frames are given, FALSE otherwise]
A list of the names of the images which contain the raw bias data. These are the images which are to used to produce a "master" bias image. On exit these images will have their FTYPE extension item set to the value "BIAS". [!]
BIASVALUE = _DOUBLE (Read)
If no raw bias frames exist and the data does not have any bias strips, then the only way to remove the bias contribution is to subtract a constant. If your data has already had its bias contribution subtracted and you want to process it using CCDPACK (so that you can generate variances for instance) then set this value to zero. This parameter defaults to ! and is not prompted for so the only way that a value can be supplied is on the command-line or by using the PROMPT keyword. [!]
The bounds of the detector bias strips (if any exist). The bounds (if given) should be in pixel indices and be given in pairs up to a limit of 2. The sense of the bounds is along the readout direction. For example, 2,16,400,416 means that the bias strips are located between pixels 2 to 16 and 400 to 416 inclusive along the readout direction. The bias strips are used to either offset the master bias image or as an estimate of the bias which is to be interpolated across the image in some way (see DEBIAS). Not supplying values for this parameter may be a valid response if the bias frame is to be directly subtracted from the data without offsetting or if a single constant is to be used as the bias value for the whole image.
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if BOUNDS=[value,...] is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
DARK = LITERAL (Read)
A list of the names of the images which contain the raw dark count data. These are the images which are to used to produce a "master" dark counts image. On exit these images will have their FTYPE extension item set to the value "DARK". [!]
DARKTIME = _DOUBLE (Read)
The time for which the data in the current image collected dark count electrons. The dark count is basically charge which accumulates in the detector pixels due to thermal noise. The effect of dark current is to produce an additive quantity to the electron count in each pixel. Most modern devices only produce a few ADU (or less) counts per pixel per hour and so this effect can generally be ignored. This, however, is not the case for Infra-Red detectors.
The value given does not need to be a number of seconds or minutes and can be ratio of some kind, as long as it is consistently used for all images (so if all your images have the same darktime then the value 1 could be used). Images which have no dark count should be given a DARKTIME of 0. This parameter is only used if ADDDARK is TRUE. [!]
The deferred charge value. Often known as the "fat" or "skinny" zero (just for confusion). This is actually the charge which is not transferred from a CCD pixel when the device is read out. Usually this is zero or negligible and is only included for completeness and for processing very old data.
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if DEFERRED=value is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
The readout direction of the detector. This may take the values X or Y. A value of X indicates that the readout direction is along the first (horizontal) direction, an Y indicates that the readout direction is along the direction perpendicular to the X axis.
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if DIRECTION=value is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
The extent of the useful detector area in pixel indices. The extent is defined as a range in X values and a range in Y values (XMIN, XMAX, YMIN, YMAX). These define a section of an image (see SUN/33). Any parts of the detector surface area outside of this region will not be present in the final output. This is useful for excluding bias strips, badly vignetted parts etc.
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if EXTENT=[XMIN, XMAX, YMIN, YMAX] is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
FILTER = LITERAL (Read)
The filter name associated with the current image. The filter name is stored in the extension item FILTER and is used when determining which flatfields should be used for which data. images with a frame type which is independent of the filter will not use this parameter. The filter type is a case sensitive string. [Current value]
FLASH = LITERAL (Read)
A list of the names of the images which contain the raw pre-flash correction data. These are the images which are to used to produce a "master" pre-flash correction image. On exit these images will have their FTYPE extension item set to the value "FLASH". [!]
FLASHTIME = _DOUBLE (Read)
The time for which the data in the current image was exposed to pre-flash.
The value given does not need to be a number of seconds or minutes and can be ratio of some kind, as long as it is consistently used for all images (so if all your images have the same darktime then the value 1 could be used). Images which have no pre-flash should be given a FLASHTIME of 0. This parameter is only used if ADDFLASH is TRUE. [!]
A list of the names of the images which contain the raw flatfield data. These are the images which are to used to produce "master" flatfields (one for each filter type). On exit these images will have their FTYPE extension item set to the value "FLAT". [!]
FTYPE = LITERAL (Read)
The "frame" type of the current image. Each image is processed in turn and if SIMPLE is TRUE and a frame type extension item does not exist then this parameter will be used to prompt for a value. A prompt will also be made if SIMPLE is TRUE and MODIFY is TRUE regardless of whether the item already exists or not. If SIMPLE is FALSE then this parameter will not be used. [Current value]
A list of the names of the images which contain the raw CCD data. Images entered using this parameter must already have the correct "frame type" information (extension item FTYPE) entered into their CCDPACK extensions. This parameter is only used if SIMPLE is TRUE.
The name of a master bias frame. If this has been created by CCDPACK then there is no need to present it. This parameter is designed for the import of frames created by other packages. [!]
The name of a master dark counts frame. If this has been created by CCDPACK then there is no need to present it (unless for some reason it has been assigned the wrong frame type). This parameter is designed for the import of frames created by other packages.
The name of a master pre-flash frame. If this has been created by CCDPACK then there is no need to present it (unless for some reason it has been assigned the wrong frame type). This parameter is designed for the import of frames created by other packages. [!]
The names of a set of master flatfield frames (one for each filter type used). If these have been created by CCDPACK then there is no need to present them (unless for some reason they have been assigned the wrong frame type or filter). This parameter is designed for the import of frames created by other packages (such as those that specifically process spectral data). [!]
MASTERS = _LOGICAL (Read)
If this parameter is TRUE then prompts will be made for all the master calibration types (MASTERBIAS, MASTERDARK, MASTERFLAT and MASTERFLASH). [FALSE]
MODIFY = _LOGICAL (Read)
If the input images already contain information in their CCDPACK extensions, then this parameter controls whether this information will be overwritten (if a new value exists) or not. [TRUE]
MULTIENTRY = _LOGICAL (Read)
Whether or not the names of the input images, their frame types, filters and related exposure factors are all given in response to the IN parameter (SIMPLE must be TRUE). If this option is selected then the parameters FTYPE, FILTER, DARKTIME and FLASHTIME will be set up with these values as defaults. If MODIFY is TRUE then you will be given an opportunity to modify them, otherwise these values will be entered into the image CCDPACK extensions.
The input record format is five fields separated by commas. These are:
1 Image name
3 Filter name
4 Dark exposure time
5 Flash exposure time
The latter three fields can be specified as "!" in which case they are not set (they may not be relevant). Multiple records can be entered and can be read in from a text file. So for instance if the file "XREDUCE.NDFS" had the following as its contents:
DATA1,target,!,!,!
FF1,flat,!,!,!
BIAS1,bias,!,!,!
Then it would be invoked using parameters
SIMPLE MULTIENTRY IN=
\text{^}
XREDUCE.NDFS
This parameter is intended as an aid when using this program non-interactively (i.e. from scripts) and shouldn’t normally be used, hence its default is FALSE and this can only be overridden by assignment on the command line or in response to a forced prompt. [FALSE]
The name of a file to contain a listing of the name of the input images. This is intended to be of use when using these same names with other applications (such as SCHEDULE). [!]
ONEDARKTIME = _LOGICAL (Read)
If the input data have the same dark count exposure time then this parameter may be set to inhibit repeated prompting for an exposure for every frame. This parameter is of particular use when running from scripts. [FALSE]
ONEFILTER = _LOGICAL (Read)
If the input data have only one filter type then this parameter may be set to inhibit repeated prompting for a filter name for every frame (that is filter dependent). This parameter is of particular use when running from scripts. [FALSE]
ONEFLASHTIME = _LOGICAL (Read)
If the input data have the same pre-flash exposure time then this parameter may be set to inhibit repeated prompting for an exposure for every frame. This parameter is of particular use when running from scripts. [FALSE]
The readout noise of the detector (in ADUs). Usually the readout noise of a detector is estimated by the observatory at which the data was taken and this is the value which should be supplied. Not supplying a value for this parameter may be a valid response if variances are not to be generated.
This parameter normally accesses the value of the related CCDPACK global association (which is the readout noise value). This behaviour can only be superceded if RNOISE=value is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
The saturation value of the detector pixels (in ADUs).
This parameter normally accesses the value of the related CCDPACK global association. This behaviour can only be superceded if SATURATION=value is used on the command-line or if a prompt is forced (using the PROMPT keyword). The value of this parameter will be entered into the extension of the input images only if MODIFY is TRUE or the related extension item does not exist. [!]
SIMPLE = _LOGICAL (Read)
Whether or not the input images already contain "frame type" (extension item FTYPE) information in their CCDPACK extensions or not. Usually images to be presented to CCDPACK do not contain this information, unless it has been imported from FITS information using IMPORT, or the images have already been presented and this pass is to modify existing extension items. [FALSE]
TARGET = LITERAL (Read)
A list of the names of the images which contain the "target" data. These are the images which contain the images or spectra etc. On exit these images will have their FTYPE extension item set to the value "TARGET". [!]
ZEROED = _LOGICAL (Read)
If a master bias frame is given, then this parameter indicates whether or not it has a mean value of zero. If SIMPLE and MULTIENTRY are TRUE then this value (TRUE or FALSE) can be entered as the fourth field to the IN parameter. [FALSE]
present simple in=’
\ast
’ modify
In this example PRESENT processes all the images in the current directory. The images should already have a valid frame type (such as TARGET, FLAT etc.). The any existing global variables describing the detector are accessed and written into the image extension overwriting any values which already exist.
present simple=false bias=’bias
\ast
’ target=’data
\ast
’ dark=! flash=! flat=’ff
\ast
In this example the input images are organised into their respective frame types using the specially designed input parameters. On exit the output images will have the correct frame types entered into their CCDPACK extensions (provided MODIFY is TRUE).
present modify=false simple=true in=’
\ast
In this example all the images in the current directory are accessed. If any required extension or global associated items are missing then they will be entered into the image extension. If all extension items are present then a listing of their values will be made.
present masters simple=false masterflat=2dspectraff
In this example a master flatfield is imported to be used in an automated reduction of spectral data.
“Setting reduction information”.
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Variational mode decomposition - MATLAB vmd - MathWorks Italia
\mathit{x}\left(\mathit{t}\right)=6{\mathit{t}}^{2\text{\hspace{0.17em}}}+\mathrm{cos}\left(4\pi \mathit{t}+10\pi {\mathit{t}}^{2}\right)+\left\{\begin{array}{ll}\mathrm{cos}\left(60\pi \mathit{t}\right),& \mathit{t}\le 0.5,\\ \mathrm{cos}\left(100\pi \mathit{t}-10\pi \right),& \mathit{t}>0.5.\end{array}
The signals labeled in this example are from the MIT-BIH Arrhythmia Database [3] (Signal Processing Toolbox). The signal in the database was sampled at 360 Hz.
x\left(t\right)=\sum _{k=1}^{K}{u}_{k}\left(t\right).
{u}_{k}\left(t\right)={A}_{k}\left(t\right)\mathrm{cos}\left({\varphi }_{k}\left(t\right)\right),
\begin{array}{ccccccc}L\left({u}_{k}\left(t\right),{f}_{k},\lambda \left(t\right)\right)& =& \alpha \sum _{k=1}^{K}{‖\frac{d}{dt}\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast {u}_{k}\left(t\right)\right]{e}^{-j2\pi {f}_{k}t}‖}_{2}^{2}& +& {‖x\left(t\right)-\sum _{k=1}^{K}{u}_{k}\left(t\right)\text{\hspace{0.17em}}‖}_{2}^{2}& +& 〈\lambda \left(t\right),x\left(t\right)-\sum _{k=1}^{K}{u}_{k}\left(t\right)〉\\ & & \left(i\right)& & \left(ii\right)& & \left(iii\right)\end{array},
〈p\left(t\right),q\left(t\right)〉={\int }_{-\infty }^{\infty }{p}^{\ast }\left(t\right)\text{\hspace{0.17em}}q\left(t\right)\text{\hspace{0.17em}}dt
{‖p\left(t\right)‖}_{2}^{2}=〈p\left(t\right),p\left(t\right)〉
x\left(t\right)={\sum }_{k=1}^{K}{u}_{k}\left(t\right)
The algorithm solves the optimization problem using the alternating direction method of multipliers described in [1] (Signal Processing Toolbox).
The Lagrange multiplier introduced in Optimization (Signal Processing Toolbox) has the Fourier transform Ʌ(f). The length of the Lagrange multiplier vector is the length of the extended signal.
\sum _{k}{‖{u}_{k}^{n+1}\left(t\right)-{u}_{k}^{n}\left(t\right)‖}_{2}^{2}/{‖{u}_{k}^{n}\left(t\right)‖}_{2}^{2}<{\epsilon }_{\text{r}}
\sum _{k}{‖{u}_{k}^{n+1}\left(t\right)-{u}_{k}^{n}\left(t\right)‖}_{2}^{2}<{\epsilon }_{\text{a}}
{U}_{k}^{n+1}\left(f\right)=\frac{X\left(f\right)-\sum _{i<k}{U}_{k}^{n+1}\left(f\right)-\sum _{i>k}{U}_{k}^{n}\left(f\right)+\frac{{\Lambda }^{n}}{2}\left(f\right)}{1+2\alpha {\left\{2\pi \left(f-{f}_{k}^{n}\right)\right\}}^{2}},
{U}_{k}^{n+1}\left(f\right)
{f}_{k}^{n+1}
{f}_{k}^{n+1}=\frac{{\int }_{0}^{\infty }{|{U}_{k}^{n+1}\left(f\right)|}^{2}\text{\hspace{0.17em}}f\text{\hspace{0.17em}}df}{{\int }_{0}^{\infty }{|{U}_{k}^{n+1}\left(f\right)|}^{2}\text{\hspace{0.17em}}df}\approx \frac{\sum f{|{U}_{k}^{n+1}\left(f\right)|}^{2}\text{\hspace{0.17em}}}{\sum {|{U}_{k}^{n+1}\left(f\right)|}^{2}\text{\hspace{0.17em}}}.
{\Lambda }^{n+1}\left(f\right)={\Lambda }^{n}\left(f\right)+\tau \left(X\left(f\right)-\sum _{k}{U}_{k}^{n+1}\left(f\right)\right),
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Determine coefficients of Nth-order forward linear predictors - Simulink - MathWorks í•œêµ
Autocorrelation LPC
Output prediction error power (P)
Inherit prediction order from input dimensions
Prediction order (N)
Determine coefficients of Nth-order forward linear predictors
DSP System Toolbox / Estimation / Linear Prediction
The Autocorrelation LPC block determines the coefficients of an N-step forward linear predictor for the time-series in each length-M input channel, u, by minimizing the prediction error in the least squares sense. A linear predictor is an FIR filter that predicts the next value in a sequence from the present and past inputs. This technique has applications in filter design, speech coding, spectral analysis, and system identification.
The Autocorrelation LPC block can output the prediction error for each channel as polynomial coefficients, reflection coefficients, or both. The block can also output the prediction error power for each channel.
Input 1 — Input array
unoriented vector | column vector | matrix
Specify the input u as an unoriented vector, column vector, or a matrix. Row vectors are not valid inputs. The block treats all M-by-N matrix inputs as N channels of length M.
A — Polynomial coefficients
Polynomial coefficients generated when you set the Output(s) parameter to A or A and K. For each input channel, port A outputs an (N+1)-by-1 column vector a = [1 a2a3 ... aN+1]T, containing the coefficients of an Nth-order moving average (MA) linear process that predicts the next value, ûM+1, in the input time-series.
{\stackrel{^}{u}}_{M+1}=â\left({a}_{2}{u}_{M}\right)â\left({a}_{3}{u}_{Mâ1}\right)â...â\left({a}_{N+1}{u}_{MâN+1}\right)
To enable port A, set Output(s) to A or A and K.
Reflection coefficients generated when Output(s) is set to K or A and K. For each input channel, port K outputs a length-N column vector whose elements are the prediction error reflection coefficients.
To enable port K, set Output(s) to A or A and K.
P — Prediction error power
Prediction error power output at port P as a vector whose length is the number of input channels.
To enable port P, select the Output prediction error power (P) parameter.
Output(s) — Type of prediction coefficients
A (default) | A and K | K
Specify the type of prediction coefficients output by the block. The block can output polynomial coefficients (A), reflection coefficients (K), or both (A and K).
When you set Output(s) to A and K, the block enables port A and K and each port outputs its respective set of prediction coefficients for each channel.
Output prediction error power (P) — Output prediction error power
Select this parameter to enable the output port P, which outputs the output prediction error power.
Inherit prediction order from input dimensions — Inherit prediction order from input dimensions
Select this parameter to inherit the prediction order N from the input dimensions.
Prediction order (N) — Prediction order
Specify the prediction order N. Note that N must be a scalar with a value less than the length of the input channels or the block produces an error.
This parameter appears only when you do not select the Inherit prediction order from input dimensions parameter.
Estimate Data Series Using Forward Linear Predictor
Use the Autocorrelation LPC block to estimate the future values of signal.
The Autocorrelation LPC block computes the least squares solution to
\underset{iâ{\mathrm{â}}^{n}}{\mathrm{min}}âU\stackrel{Ë}{a}âbâ
ââ
â
indicates the 2-norm and
U=\left[\begin{array}{cccc}{u}_{1}& 0& \cdots & 0\\ {u}_{2}& {u}_{1}& â±& ⋮\\ ⋮& {u}_{2}& â±& 0\\ ⋮& ⋮& â±& {u}_{1}\\ ⋮& ⋮& ⋮& {u}_{2}\\ ⋮& ⋮& ⋮& ⋮\\ {u}_{M}& ⋮& ⋮& ⋮\\ 0& â±& ⋮& ⋮\\ ⋮& â±& â±& ⋮\\ 0& \cdots & 0& {u}_{M}\end{array}\right],\text{â}\text{â}\text{â}\text{â}\stackrel{Ë}{a}=\left[\begin{array}{c}{a}_{2}\\ ⋮\\ {a}_{n+1}\end{array}\right],\text{â}\text{â}b=\left[\begin{array}{c}{u}_{2}\\ {u}_{3}\\ ⋮\\ {u}_{M}\\ 0\\ ⋮\\ 0\end{array}\right]
Solving the least squares problem via the normal equations
{U}^{â}U\stackrel{Ë}{a}={U}^{â}b
leads to the system of equations
\left[\begin{array}{cccc}{r}_{1}& {r}_{2}^{â}& \cdots & {r}_{n}^{â}\\ {r}_{2}& {r}_{1}& â±& ⋮\\ ⋮& â±& â±& {r}_{2}^{â}\\ {r}_{n}& \cdots & {r}_{2}& {r}_{1}\end{array}\right]\text{â}\left[\begin{array}{c}{a}_{2}\\ {a}_{3}\\ ⋮\\ {a}_{n+1}\end{array}\right]\text{â}=\left[\begin{array}{c}â{r}_{2}\\ â{r}_{3}\\ ⋮\\ â{r}_{n+1}\end{array}\right]
where r = [r1r2r3 ... rn+1]T is an autocorrelation estimate for u computed using the Autocorrelation block, and * indicates the complex conjugate transpose. The normal equations are solved in O(n2) operations by the Levinson-Durbin block.
Note that the solution to the LPC problem is very closely related to the Yule-Walker AR method of spectral estimation. In that context, the normal equations above are referred to as the Yule-Walker AR equations.
[1] Haykin, S. Adaptive Filter Theory. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1996.
[2] Ljung, L. System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice Hall, 1987. Pgs. 278-280.
Autocorrelation | Levinson-Durbin | Yule-Walker Method
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Hi, I am Falk Hassler, an assistant professor at the Institute of Theoretical Physics at the University of Wrocław.
Before coming here in 2021, my life as a postdoc started in 2015, when I finished my PhD at the Ludwig Maximilian University of Munich. Since then, it has not been short of adventures. I had the opportunity to work in many exciting places with great people from all over the world: New York City, Chapel Hill (North Carolina), Philadelphia, Ovideo (Spain) and College Station (Texas) have been home to my family (wife Antje and our daughter Amy who was born in Philly) and me. After growing up in a small town in the northeast of Germany, I would have never dreamed that physics would lead me to all these incredible places one day. You can find more details in my CV.
Imagine we take a coffee mug and zoom in with a very powerful microscope. Eventually, we will discover that it is made out of atoms. These atoms have protons and neutrons in their core which consists of quarks held together by gluons. We don't have machines yet to zoom in much further. But one thing is certain: something dramatic has to happen at the incredibly small scale of
10^{-35}
meters. At this point, the two fundamental ingredients of physics, general relativity and quantum field theory, start to contradict each other.
My research takes us exactly to this point. Although we do not have any experimental data at this scale yet, the last 50 years have produced some incredible ideas of what we might find. All of them are based on the fundamental mechanisms in physics that we already have confirmed experimentally. The most studied idea is that extended objects, strings, should ultimately substitute point particles.
Strings are so fundamental that not only particles are made of them but also the interactions between them and even spacetime itself.
Hence, we face a crucial change of paradigms. Point particles have a natural notion of distance. Take as a simple a free particle on a ring. Its energy spectrum is indirectly proportional to the radius. Thus, we could easily distinguish between large and small rings. Distance between points is also the defining concept in Riemannian geometry that underpins general relativity.
Things become more subtle if we look at strings because, in addition to the centre of mass motion of point particles, they can wind around the circle. Hence, their spectrum is characterised by two quantum numbers. Remarkably, it is the same on two circles, one larger and the other one smaller than the length of the string. Just the role of momentum and winding gets flipped. This effect is called T-duality, and it obfuscates the clear notion of distance neeed to define geometry. Therefore strings require ultimately to work with a generalisation of geometry.
My work has revealed how this adapted version of geometry can capture T-dualities far beyond the simple example we have just discussed. In contrast to a circle or a torus, the spacetimes I am interested in are curved.
Strings in curved backgrounds automatically induce higher curvature corrections that modify the Einstein-Hilbert action of point particles. These corrections are essential to understanding how a quantum theory of gravity might resolve singularities at the centre of black holes or the Big Bang. Thus, my current efforts focus on how T-duality allows to explicitly compute these corrections. Moreover, my work gives a new handle on integrable string models, which are an indispensable tool in the long-standing quest of proving the AdS/CFT correspondence, perhaps the successful spin-offs of string theory.
Using dualities in string theory, I explore quantum field theory and quantum gravity at strong coupling, very high energies and small distances. Important ingredients in my work are double/exceptional field theory, an effective target space description of string/M-theory, which makes T-/U-dualities manifest, and (super)conformal field theories, (S)CFTs, in two and more dimensions. On the formal side, I look into the underlying principles of generalised geometry, non-commutative or even non-associative geometry and, especially, how they naturally arise from strings and higher dimensional membranes probing spacetime. String field theory and worldsheet renormalisation group flow, which allow extracting new mathematical structures from the
\sigma
-model that underlies the string's dynamics, are powerful tools I rely on. Although this is still a perturbative approach, it can point out underlying symmetry principles that allow accessing the non-perturbative regime.
A prominent example is supersymmetry. It allows studying certain protected sectors of a theory (like BPS solutions) at strong coupling. These by the symmetry distinguished sectors are also indispensable for approaching higher-dimensional SCFTs, which usually do not have a weak coupling limit. Besides having all these fundamental aspects in mind, I am interested in applications. They range from flux compactifications over consistent truncations to simple toy models for inflation in cosmology. In the last years, I revealed an elementary link between Poisson-Lie T-duality and generalised geometry. It brings together two thriving research communities and paves the way for important discoveries. Among them is a new approach to one of the biggest questions in contemporary physics: What is the fundamental structure of space and time? I am exploring this and related questions together with my research team.
falkhassler
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Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : HypergeometricTerm Subpackage : PolynomialSolution
return the polynomial solution of linear difference equation depending on a hypergeometric term
PolynomialSolution(eq, var, term)
The PolynomialSolution(eq, var, term) command returns the polynomial solution of the linear difference equation eq. If such a solution does not exist, the function returns NULL.
t=n!
, or specified as a list consisting of the name of term variable and the consecutive term ratio, for example,
[t,n+1]
If the third parameter is omitted, then the input equation can contain a hypergeometric term directly (not a name). In this case, the procedure extracts the term from the equation, transforms the equation to the form with a name representing a hypergeometric term, and then solves the transformed equation.
The term "polynomial solution" means a solution
y\left(x\right)
Q\left(x\right)[t,{t}^{-1}]
, that is, in the form
y={y}_{d}{t}^{d}+...+{y}_{g}{t}^{g}
d\le g
{y}_{d},...,{y}_{g}
Q\left(x\right)
The solution is the function, corresponding to var. The solution involves arbitrary constants of the form, for example, _c1 and _c2.
\mathrm{with}\left(\mathrm{LREtools}[\mathrm{HypergeometricTerm}]\right):
\mathrm{eq}≔y\left(n+2\right)-\left(t+n\right)y\left(n+1\right)+n\left(t-1\right)y\left(n\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{n}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)
\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right),t=n!\right)
\frac{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_C}}}_{\textcolor[rgb]{0,0,1}{1}}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{eq}≔y\left(n+2\right)-\left(n!+n\right)y\left(n+1\right)+n\left(n!-1\right)y\left(n\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{n}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)
\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(n\right)\right)
\frac{\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_C}}}_{\textcolor[rgb]{0,0,1}{1}}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{eq}≔\left(t+{n}^{2}\right)z\left(n+1\right)-\left(2nt+2t+{n}^{2}+2n+1\right)z\left(n\right)
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{-}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\right)
\mathrm{PolynomialSolution}\left(\mathrm{eq},z\left(n\right),t={2}^{n}n!\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_C}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_C}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}]
\mathrm{eq}≔45y\left(x\right)-9y\left(x\right)x-18y\left(x+3\right)+9y\left(x+3\right)x
\textcolor[rgb]{0,0,1}{\mathrm{eq}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{45}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{18}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}
\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(x\right),[t,\frac{9\cdot 1}{10-7x-8{x}^{2}}]\right)
\frac{{\textcolor[rgb]{0,0,1}{\mathrm{_C}}}_{\textcolor[rgb]{0,0,1}{1}}}{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{9}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}}]
LREtools[HypergeometricTerm][SubstituteTerm]
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Global Constraint Catalog: Copen_global_cardinality
<< 5.297. open_atmost5.299. open_global_cardinality_low_up >>
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}\left(𝚂,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\right)
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚌𝚌}
\mathrm{𝚘𝚐𝚌𝚌}
𝚂
\mathrm{𝚜𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚕}-\mathrm{𝚒𝚗𝚝},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-\mathrm{𝚍𝚟𝚊𝚛}\right)
𝚂\ge 1
𝚂\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)
\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\left[\mathrm{𝚟𝚊𝚕},\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right]\right)
\mathrm{𝚍𝚒𝚜𝚝𝚒𝚗𝚌𝚝}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂},\mathrm{𝚟𝚊𝚕}\right)
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\ge 0
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\le |\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚕}
\left(1\le i\le |\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|\right)
should be taken by exactly
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}\left[i\right].\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
𝚂
\left(\begin{array}{c}\left\{2,3,4\right\},\hfill \\ 〈3,3,8,6〉,\hfill \\ 〈\begin{array}{cc}\mathrm{𝚟𝚊𝚕}-3\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1,\hfill \\ \mathrm{𝚟𝚊𝚕}-5\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-0,\hfill \\ \mathrm{𝚟𝚊𝚕}-6\hfill & \mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}-1\hfill \end{array}〉\hfill \end{array}\right)
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
Values 3, 5 and 6 respectively occur 1, 0 and 1 times within the collection
〈3,3,8,6〉
〈3,3,8,6〉
S=\left\{2,3,4\right\}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1
|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|>1
\mathrm{𝚛𝚊𝚗𝚐𝚎}
\left(\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}\right)>1
|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>|\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}|
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚍𝚒𝚏𝚏𝚗}
In their article [HoeveRegin06], W.-J. van Hoeve and J.-C. Régin consider the case where we have no counter variables for the values, but rather some lower and upper bounds (i.e., in fact the
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
(assignment,counting constraint),
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚖𝚘𝚗𝚐}
(open constraint,counting constraint),
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚝𝚕𝚎𝚊𝚜𝚝}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}
(open constraint,value constraint).
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚊𝚕𝚕𝚍𝚒𝚏𝚏𝚎𝚛𝚎𝚗𝚝}
𝚂
. should occur at most once),
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}_\mathrm{𝚕𝚘𝚠}_\mathrm{𝚞𝚙}
\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}
\mathrm{𝚏𝚒𝚡𝚎𝚍}
\mathrm{𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕}
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝑆𝐸𝐿𝐹}
↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\right)
•\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚟𝚊𝚛}=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚟𝚊𝚕}
•
\mathrm{𝚒𝚗}_\mathrm{𝚜𝚎𝚝}
\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}.\mathrm{𝚔𝚎𝚢},𝚂\right)
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
=\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}.\mathrm{𝚗𝚘𝚌𝚌𝚞𝚛𝚛𝚎𝚗𝚌𝚎}
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
𝚂
\mathrm{𝚅𝙰𝙻𝚄𝙴𝚂}
collection of the Example slot. Part (B) of Figure 5.298.1 shows the two corresponding final graphs respectively associated with values 3 and 6 that are both assigned to those variables of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
collection for which the index belongs to
𝚂
(since value 5 is not assigned to any variable of the
\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}
\mathrm{𝐍𝐕𝐄𝐑𝐓𝐄𝐗}
\mathrm{𝚘𝚙𝚎𝚗}_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}
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\large{\begin{array}{ccccccc} && & & \color{#3D99F6}X& \color{#3D99F6}X & \color{#3D99F6}X&\color{#3D99F6}X\\ && & & \color{#20A900}Y& \color{#20A900}Y & \color{#20A900}Y&\color{#20A900}Y\\ +&& & & \color{#D61F06}Z& \color{#D61F06}Z & \color{#D61F06}Z&\color{#D61F06}Z\\ \hline & & & \color{#20A900}Y& \color{#3D99F6}X& \color{#3D99F6}X & \color{#3D99F6}X&\color{#D61F06}Z\\ \hline \end{array}}
\color{#3D99F6}X
\color{#20A900}Y
\color{#D61F06}Z
are distinct digits in the sum above, then find
\color{#D61F06}Z
\begin{array} { l l l l l l } & & A & B & C & D & E \\ \times & & & & & & 4 \\ \hline & & E & D & C & B & A \\ \end{array}
\overline{ABCDE}
by Mohd Sasa
\large{\begin{array}{ccccc} & F & O& R & T&Y\\ &&& T& E&N\\ +&&&T&E&N\\ \hline & S& I& X & T&Y \ \end{array}}
Solve the above cryptarithm given than each letter represent distinct single non-negative integers.
Enter your answer as
\overline{FORTY} + \overline{TEN} + \overline{SIXTY}
\frac { \begin{matrix} & A & B & C & D & E \\ \times & & & & 1 & 2 \end{matrix} }{ C\quad D\quad E\quad 0\quad A\quad B }
\overline { ABCDE }
\large{\begin{array}{ccccccc} && 7& 4& 2& 5 & 8&6\\ +&&8 & 2 & 9& 4 & 3&0\\ \hline & 1& 2& 1 & 2& 0 & 1&6\\ \hline \end{array}}
Obviously, the above addition is incorrect.
However, the display can be corrected simply by changing one digit
d
, wherever it occurs, to another digit
e
d+e
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Tunable two-degree-of-freedom PID controller - MATLAB tunablePID2 - MathWorks India
Kp,Ki,Kd,Tf,b,c
Name changed from ltiblock.pid2
Tunable two-degree-of-freedom PID controller
blk = tunablePID2(name,type)
blk = tunablePID2(name,type,Ts)
blk = tunablePID2(name,sys)
Model object for creating tunable two-degree-of-freedom PID controllers. tunablePID2 lets you parametrize a tunable SISO two-degree-of-freedom PID controller. You can use this parametrized controller for parameter studies or for automatic tuning with tuning commands such as systune, looptune, or the Robust Control Toolbox™ command hinfstruct.
tunablePID2 is part of the family of parametric Control Design Blocks. Other parametric Control Design Blocks include tunableGain, tunableSS, and tunableTF.
blk = tunablePID2(name,type) creates the two-degree-of-freedom continuous-time PID controller described by the equation:
u={K}_{p}\left(br-y\right)+\frac{{K}_{i}}{s}\left(r-y\right)+\frac{{K}_{d}s}{1+{T}_{f}s}\left(cr-y\right).
r is the setpoint command, y is the measured response to that setpoint, and u is the control signal, as shown in the following illustration.
The tunable parameters of the block are:
Scalar gains Kp, Ki, and Kd
Filter time constant Tf
Scalar weights b and c
The type argument sets the controller type by fixing some of these values to zero (see Input Arguments).
blk = tunablePID2(name,type,Ts) creates a discrete-time PID controller with sample time Ts. The equation describing this controller is:
u={K}_{p}\left(br-y\right)+{K}_{i}IF\left(z\right)\left(r-y\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}\left(cr-y\right).
IF(z) and DF(z) are the discrete integrator formulas for the integral and derivative terms, respectively. The values of the IFormula and DFormula properties set the discrete integrator formulas (see Properties).
blk = tunablePID2(name,sys) uses the dynamic system model, sys, to set the sample time, Ts, and the initial values of all the tunable parameters. The model sys must be compatible with the equation of a two-degree-of-freedom PID controller.
PID controller Name, specified as a character vector such as 'C' or '2DOFPID1'. (See Properties.)
Dynamic system model representing a two-degree-of-freedom PID controller.
Parametrization of the PID gains Kp, Ki, Kd, the filter time constant, Tf, and the scalar gains, b and c.
The following fields of blk.Kp, blk.Ki, blk.Kd, blk.Tf, blk.b, and blk.c are used when you tune blk using a tuning command such as systune:
Value Current value of the parameter. blk.b.Value, and blk.c.Value are always nonnegative.
Logical value determining whether the parameter is fixed or tunable. For example:
Maximum Maximum value of the parameter. This property places an upper bound on the tuned value of the parameter. For example, setting blk.c.Maximum = 1 ensures that c does not exceed unity.
blk.Kp, blk.Ki, blk.Kd, blk.Tf, blk.b, and blk.c are param.Continuous objects. For more information about the properties of these param.Continuous objects, see the param.Continuous (Simulink Design Optimization) object reference page.
\frac{{T}_{s}}{z-1}
\frac{{T}_{s}z}{z-1}
\frac{{T}_{s}}{2}\frac{z+1}{z-1}
Tunable Two-Degree-of-Freedom Controller with a Fixed Parameter
Create a tunable two-degree-of-freedom PD controller. Then, initialize the parameter values, and fix the filter time constant.
blk = tunablePID2('pdblock','PD');
blk.b.Value = 1;
blk.c.Value = 0.5;
blk.Tf.Value = 0.01;
blk.Tf.Free = false;
Parametric continuous-time 2-DOF PID controller "pdblock" with equation:
where r,y are the controller inputs and Kp, Kd, b, c are tunable gains.
Type "showBlockValue(blk)" to see the current value and "get(blk)" to see all
Create a tunable two-degree-of-freedom PI controller. Use a two-input, one-output tf model to initialize the parameters and other properties.
sys = [(b*Kp + Ki/s), (-Kp - Ki/s)];
blk = tunablePID2('PI2dof',sys)
Parametric continuous-time 2-DOF PID controller "PI2dof" with equation:
u = Kp (b*r-y) + Ki --- (r-y)
where r,y are the controller inputs and Kp, Ki, b are tunable gains.
blk takes initial parameter values from sys.
If sys is a discrete-time system, blk takes the value of properties, such as Ts and IFormula, from sys.
Controller with Named Inputs and Output
Create a tunable PID controller, and assign names to the inputs and output.
blk = tunablePID2('pidblock','pid');
blk.InputName = {'reference','measurement'};
blk.OutputName = {'control'};
blk.InputName is a cell array containing two names, because a two-degree-of-freedom PID controller has two inputs.
You can modify the PID structure by fixing or freeing any of the parameters. For example, blk.Tf.Free = false fixes Tf to its current value.
To convert a tunablePID2 parametric model to a numeric (nontunable) model object, use model commands such as tf or ss. You can also use getValue to obtain the current value of a tunable model.
R2016a: Name changed from ltiblock.pid2
Prior to R2016a, tunablePID2 was called ltiblock.pid2.
tunablePID | tunableGain | tunableTF | tunableSS | systune | looptune | genss | hinfstruct (Robust Control Toolbox)
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