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https://mathoverflow.net/questions/279637/how-can-one-define-punctured-torus-in-homotopy-type-theory-is-its-fundamental
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# How can one define “punctured torus” in Homotopy Type Theory? Is its fundamental group the free product of the integers with themselves?
Questions.
1. Has the beautiful old idea, in part already known to Gauß, of a punctured torus surface (take, if you will, the classical set-theoretic definition as the meaning of the latter three words) been defined in Homotopy Type Theory already? If yes, how? If not, are there any choices in doing so or is one more or less forced towards a definition of "punctured surface"?
2. More specifically, has the classical isomorphism1
$\pi_1(\text{punctured torus})\cong \mathbb{Z}\ast\mathbb{Z}$
been proved in Homotopy Type Theory already?
1. Can one somehow 'conceptually' predict 'how' different from $\mathbb{Z}\ast\mathbb{Z}$ the answer could at worst be?
Remarks.
• Homotopy Type Theory has of course reached tori already (see Chapter 6 herein), more generally, arbitrary finite CW-complexes. But he punctured torus is not compact.
• Part of my motivation for asking this question is that I had occasion to explain to someone what it means that $S_{1,1}$ is not simply connected. Moreover, I try to teach myself a little Homotopy Type Theory. So I tried to find out what Homotopy Type Theory has to say on punctured surfaces, yet found myself at a loss to even define 'punctured torus'. This may very well be due to my inexperience with Homotopy Type Theory.
• As a needless ornament to round out this question, here is an animation of first puncturing an immersion (torus surface)$\to$ $\mathbb{R}^3$, then running through a 1-parameter indexed family of regular homotopies of said immersion, and finally mending the puncture again:
(source: adapted by myself from this German Wikpedia page, which, incidentally, wrongly calls this an 'eversion' of the torus: while eversion of a torus is possible, by utilizing one of the famous eversions of the sphere, this is not one, I think, for the crude reason that here the non-continuous operations of puncturing and un-puncturing are part of the animated gif; incidentally, someone with a Wikipedia account should perhaps correct this, if it is correct that calling this an eversion is wrong.)
1 Usually proved, needless to say, via homotopy-invariance of $\pi_1$, by simply describing, in one way or the other, a homotopy-equivalence between $S_{1,1}$ and $S^1\vee S^1$, then combining three things: homotopy-invariance of $\pi_1$, the fact that $\pi_1(S^1)\cong\mathbb{Z}$, and (a corolllary of) van Kampen's theorem. My understanding is that a suitable van Kampen theorem has already been proved in Homotopy Type Theory, and, as is widely known, two pioneers of this theory have proved that $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$.I also take it that homotopy-invariance of $\pi_1$ is not an issue at all in Homotopy Type Theory; holds by design. Also, Homotopy Type Theory already offers pushouts of spans in the category of groups, so it seems that the (hoped-for) result $\mathbb{Z}\ast\mathbb{Z}$ has been constructed in the theory, already, so somehow it seems that all what is missing is a definition of the punctured torus.
• There is no way to define "puncturing" as an operation in HoTT, since it is not a homotopy-invariant thing. And as you note, the homotopy type of the punctured torus is $S^1\vee S^1$, which has been defined and its homotopy group computed in HoTT. So I'm not sure what more one could ask for. – Mike Shulman Aug 26 '17 at 22:08
• @MikeShulman: thank you very much. Thinking about it, it indeed seems rather misguided to ask how to define an operation which changes the homotopy type in Homotopy Type Theory. If you make your comment an answer, I for one would accept it. – Peter Heinig Aug 27 '17 at 6:03
• In principle one could ask for a homotopy pushout involving the wedge of two circles resulting in a torus and see that one gets the correct analogue as in the topological model. – David Roberts Aug 27 '17 at 10:01
• @PeterHeinig That's not quite it; any nontrivial operation will change the homotopy type! What's misguided is to ask about operations that are not invariant with respect to homotopy type of the input. – Mike Shulman Aug 27 '17 at 20:51
• @DavidRoberts There is such a homotopy pushout: the description of the torus as a HIT with one point-constructor, two path-constructors, and one 2-path-constructor. The first three constructors are a wedge of two circles; the last constructor glues in the 2-cell wrapping around each of them twice that makes it a torus. I suppose, based on this example, one might consider "puncturing" as the non-operation of exhibiting a space as obtained by attaching a single cell to another space (the "punctured" one). – Mike Shulman Aug 27 '17 at 20:54
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2019-02-21 12:57:48
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https://tug.org/pipermail/tex4ht/2013q3/000883.html
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# [tex4ht] not translating math
William F Hammond gellmu at gmail.com
Mon Sep 30 23:07:04 CEST 2013
On Mon, Sep 30, 2013 at 3:09 AM, Matteo Gamboz <gamboz at medialab.sissa.it>wrote:
> Hi all,
> I'm looking for a way to keep math as it is. For example:
>
> J\er\'ome $\alpha < x^{\infty}$ end
> →
> Jèróme $<![CDATA[\alpha < x^{\infty}]]>$ end
>
> but, by now I'm only able to get to this:
> Jèróme $<![CDATA[α < x∞]]>$ end
>
>
For translating LaTeX to DocBook you want, as I understand it, to have TeX
source put inside $tags whose content is the literal TeX source as a CDATA marked section. Yes, your .cfg sets up the beginning and end of the CDATA marked section, but it does not give tex4ht a way to understand that the TeX math markup should be passed untouched. Probably, one could write an alternate version of dblatex to do this. The real problem here is that DocBook is not a very satisfactory translation target for LaTeX. Apart from math it's too rich, and for math the options are poor. Common practice, aside from MathML islands, cf. dbmlatex, is insertion of TeX source It would be far better if DocBook incorporated, say, the XML guise of the profile for LaTeX math represented by the TeX input for MathJax -- something that is author-friendly. Then it would be straightforward to accommodate that in something like dblatex, as well as to revise standard DocBook processors. With that hypothetical what would make sense for the output would be something like this: Jèróme [itex]<alpha/> < x<sup><infty/></sup>$ end
N.B., <alpha/> rather than α because the mathematical \alpha in TeX should
not be regarded as the same as α in a unicode-capable version of LaTeX.
-- Bill
--
William F Hammond
Email: gellmu at gmail.com
`
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2019-08-17 17:13:04
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http://docs.px4.io/master/zh/computer_vision/collision_prevention.html
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# 防撞功能
It can be enabled for multicopter vehicles in Position mode, and can use sensor data from an offboard companion computer, offboard rangefinders over MAVLink, a rangefinder attached to the flight controller, or any combination of the above.
Collision prevention may restrict vehicle maximum speed if the sensor range isn't large enough! It also prevents motion in directions where no sensor data is available (i.e. if you have no rear-sensor data, you will not be able to fly backwards).
If high flight speeds are critical, consider disabling collision prevention when not needed.
Ensure that you have sensors/sensor data in all directions that you want to fly (when collision prevention is enabled).
## 综述
Collision Prevention is enabled on PX4 by setting the parameter for minimum allowed approach distance (CP_DIST).
The feature requires obstacle information from an external system (sent using the MAVLink OBSTACLE_DISTANCE message) and/or a distance sensor connected to the flight controller.
Multiple sensors can be used to get information about, and prevent collisions with, objects around the vehicle. If multiple sources supply data for the same orientation, the system uses the data that reports the smallest distance to an object.
The vehicle restricts the maximum velocity in order to slow down as it gets closer to obstacles, and will stop movement when it reaches the minimum allowed separation. In order to move away from (or parallel to) an obstacle, the user must command the vehicle to move toward a setpoint that does not bring the vehicle closer to the obstacle. The algorithm will make minor adjustments to the setpoint direction if it is determined that a "better" setpoint exists within a fixed margin on either side of the requested setpoint.
Users are notified through QGroundControl while Collision Prevention is actively controlling velocity setpoints.
PX4 software setup is covered in the next section. If you are using a distance sensor attached to your flight controller for collision prevention, it will need to be attached and configured as described in PX4 Distance Sensor. If you are using a companion computer to provide obstacle information see companion setup.
## PX4 (软件) 设置
Configure collision prevention by setting the following parameters in QGroundControl:
Parameter Description
CP_DIST Set the minimum allowed distance (the closest distance that the vehicle can approach the obstacle). Set negative to disable collision prevention.
This value is the distance to the sensors, not the outside of your vehicle or propellers. Be sure to leave a safe margin! | | </span>CP_DELAY | Set the sensor and velocity setpoint tracking delay. See Delay Tuning below. | | CP_GUIDE_ANG | Set the angle (to both sides of the commanded direction) within which the vehicle may deviate if it finds fewer obstacles in that direction. See Guidance Tuning below. | | CP_GO_NO_DATA | Set to 1 to allow the vehicle to move in directions where there is no sensor coverage (default is 0/False). |
## Algorithm Description
The data from all sensors are fused into an internal representation of 36 sectors around the vehicle, each containing either the sensor data and information about when it was last observed, or an indication that no data for the sector was available. When the vehicle is commanded to move in a particular direction, all sectors in the hemisphere of that direction are checked to see if the movement will bring the vehicle closer to any obstacles. If so, the vehicle velocity is restricted.
This velocity restriction takes into account both the inner velocity loop tuned by MPC_XY_P, as well as the jerk-optimal velocity controller via MPC_JERK_MAX and MPC_ACC_HOR. The velocity is restricted such that the vehicle will stop in time to maintain the distance specified in CP_DIST. The range of the sensors for each sector is also taken into account, limiting the velocity via the same mechanism.
If there is no sensor data in a particular direction, velocity in that direction is restricted to 0 (preventing the vehicle from crashing into unseen objects). If you wish to move freely into directions without sensor coverage, this can be enabled by setting CP_GO_NO_DATA to 1.
Delay, both in the vehicle tracking velocity setpoints and in receiving sensor data from external sources, is conservatively estimated via the CP_DELAY parameter. This should be tuned to the specific vehicle.
If the sectors adjacent to the commanded sectors are 'better' by a significant margin, the direction of the requested input can be modified by up to the angle specified in CP_GUIDE_ANG. This helps to fine-tune user input to 'guide' the vehicle around obstacles rather than getting stuck against them.
### Range Data Loss
If the autopilot does not receive range data from any sensor for longer than 0.5s, it will output a warning No range data received, no movement allowed. This will force the velocity setpoints in xy to zero. After 5 seconds of not receiving any data, the vehicle will switch into HOLD mode. If you want the vehicle to be able to move again, you will need to disable Collision Prevention by either setting the parameter CP_DIST to a negative value, or switching to a mode other than Position mode (e.g. to Altitude mode or Stabilized mode).
If you have multiple sensors connected and you lose connection to one of them, you will still be able to fly inside the field of view (FOV) of the reporting sensors. The data of the faulty sensor will expire and the region covered by this sensor will be treated as uncovered, meaning you will not be able to move there.
Be careful when enabling CP_GO_NO_DATA=1, which allows the vehicle to fly outside the area with sensor coverage. If you lose connection to one of multiple sensors, the area covered by the faulty sensor is also treated as uncovered and you will be able to move there without constraint.
### CP_DELAY Delay Tuning
There are two main sources of delay which should be accounted for: sensor delay, and vehicle velocity setpoint tracking delay. Both sources of delay are tuned using the CP_DELAY parameter.
The sensor delay for distance sensors connected directly to the flight controller can be assumed to be 0. For external vision-based systems the sensor delay may be as high as 0.2s.
Vehicle velocity setpoint tracking delay can be measured by flying at full speed in Position mode, then commanding a stop. The delay between the actual velocity and the velocity setpoint can then be measured from the logs. The tracking delay is typically between 0.1 and 0.5 seconds, depending on vehicle size and tuning.
If vehicle speed oscillates as it approaches the obstacle (i.e. it slows down, speeds up, slows down) the delay is set too high.
### CP_GUIDE_ANG Guidance Tuning
Depending on the vehicle, type of environment and pilot skill different amounts of guidance may be desired. Setting the CP_GUIDE_ANG parameter to 0 will disable the guidance, resulting in the vehicle only moving exactly in the directions commanded. Increasing this parameter will let the vehicle choose optimal directions to avoid obstacles, making it easier to fly through tight gaps and to keep the minimum distance exactly while going around objects.
If this parameter is too small the vehicle may feel 'stuck' when close to obstacles, because only movement away from obstacles at minimum distance are allowed. If the parameter is too large the vehicle may feel like it 'slides' away from obstacles in directions not commanded by the operator. From testing, 30 degrees is a good balance, although different vehicles may have different requirements.
The guidance feature will never direct the vehicle in a direction without sensor data. If the vehicle feels 'stuck' with only a single distance sensor pointing forwards, this is probably because the guidance cannot safely adapt the direction due to lack of information.
## PX4 Distance Sensor
At time of writing PX4 allows you to use the Lanbao PSK-CM8JL65-CC5 IR distance sensor for collision prevention "out of the box", with minimal additional configuration:
Other sensors may be enabled, but this requires modification of driver code to set the sensor orientation and field of view.
• Attach and configure the distance sensor on a particular port (see sensor-specific docs) and enable collision prevention using CP_DIST.
• Modify the driver to set the orientation. This should be done by mimicking the SENS_CM8JL65_R_0 parameter (though you might also hard-code the orientation in the sensor module.yaml file to something like sf0x start -d \${SERIAL_DEV} -R 25 - where 25 is equivalent to ROTATION_DOWNWARD_FACING).
• Modify the driver to set the field of view in the distance sensor UORB topic (distance_sensor_s.h_fov).
You can see the required modifications from the feature PR. Please contribute back your changes!
## Companion Setup
If using a companion computer or external sensor, it needs to supply a stream of OBSTACLE_DISTANCE messages, which should reflect when and where obstacle were detected.
The minimum rate at which messages must be sent depends on vehicle speed - at higher rates the vehicle will have a longer time to respond to detected obstacles.
Initial testing of the system used a vehicle moving at 4 m/s with OBSTACLE_DISTANCE messages being emitted at 10Hz (the maximum rate supported by the vision system). The system may work well at significantly higher speeds and lower frequency distance updates.
The tested companion software is the local_planner from the PX4/avoidance repo. For more information on hardware and software setup see: PX4/avoidance > Run on Hardware.
The hardware and software should be set up as described in the PX4/avoidance repo. In order to emit OBSTACLE_DISTANCE messages you must use the rqt_reconfigure tool and set the parameter send_obstacles_fcu to true.
## Gazebo Setup
Collision Prevention can also be tested using Gazebo. See PX4/avoidance for setup instructions.
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2020-03-31 23:29:25
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https://www.tutorialspoint.com/relation-between-escape-velocity-and-orbital-velocity
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# Relation between Escape Velocity and Orbital Velocity
PhysicsCelestial bodies
#### Class 11th Physics - Elasticity
6 Lectures 1 hours
#### Class 11th Physics - Oscillations
12 Lectures 2 hours
#### Class 11th Physics - Waves
14 Lectures 3 hours
## Introduction
The present tutorial intends to include the explanation regarding the concepts of physics such as escape velocity and orbital velocity along with explaining the relationship that lies between these two concepts. The tutorial will further include the definition of escape velocity and orbital velocity in order to build or exemplify the relation these two concept forms between themselves. Moreover, this tutorial will include an explanation of the formula of relation along with the differences between these concepts behold.
## What is Escape Velocity?
Figure 1: Escape velocity
In simple diction, the minimum velocity can define the escape velocity that an orbiting object is required in order to escape from the respective orbit on which the object is rotating (Vlacic, 2019).
For being a free non-propelled object, in the terminology of celestial mechanist, the minimum speed required is referred to as the escape velocity of that particular object. The escape velocity of an object generally depends on the object's mass and size (Drolshagen et al. 2020).
The unit that is used in order to represent the escape velocity is meters per second. The application of this is seen in the exploration of space, as spacecraft need this escape velocity to go beyond the gravitational force of the earth.
## What is Orbital Velocity?
Figure 2: Orbital velocity
The orbital velocity can be demonstrated as an aspect of a gravitationally bound system, where an object requires a specific amount of velocity that is considered the sufficient measure to remain in orbit (Sciencedirect, 2022).
To remain in motion, an object can be affected by inertia, the tendency to remain in motion, which can influence the path of orbiting that will be changed to a straight one. In such cases, to avoid the impact of inertia, the object needs a minimum velocity to remain in the path of the orbit.
The application of orbital velocity is seen in the shaping and orbit of natural or artificial starlight, on which these will remain orbiting the planets in space. The dependable of orbital velocity is the radius of the object, and the height of the object from the surface as well.
## Difference between Escape Velocity and Orbital Velocity
The concepts of escape velocity and the orbital velocity have several differences among themselves. The core difference that is identified is that the escape velocity is explained as the minimum velocity an object required to leave the gravitation force under which the object rotates. On the other hand, the orbital velocity refers to such amount of velocity that is GMR = vo2 by an object to remain in the orbit while maintaining its pace and the difference from the gravitational object (Geiger, 2019).
The formula of escape velocity is displayed as
ve = 2GMR
in contradiction, the formula of orbital velocity is ve = GMR + h. Based on the formulation of the mathematical formula of escape velocity, it also can be represented as 2gR. On the contrary, the representation of the orbital velocity can be done by the formula of GMR = vo2 as well.
## Escape Velocity and Orbital Velocity: Relation
The relation that is found between the orbital velocity and escape velocity is that the values of these two aspects are directly proportional to each other. The meaning of such statements is that the value of escape velocity will be increased if the value of the orbital velocity of the object will be increased. Similarly, if the orbital velocity of the object will be reduced, a reduction in the value of escape velocity will be seen as well. Based on these two conceptual ideas the formula that defines both the situations are
$$V_{o}= \sqrt{gR} \:and\: V_{e} = \sqrt{2gR}$$
## The Formula of the Relation between Escape Velocity and Orbital Velocity
The first equation $V_{o}= \sqrt{gR}$ represents the orbital velocity and the second formula $V_{e} = \sqrt{2gR}$ is responsible for the representation of escape velocity. In these two formulas, the letter g is referred to as the acceleration and R for the representation of radius, the main formula that can be formed is $V_{e} = \sqrt2 \sqrt{gR}$ (Haug, 2021). By substituting $V_{o}= \sqrt{gR}$ one can get $V_{e} = \sqrt2V_{o}$. Based on this, the value of orbital velocity can be found by the formula, $V_{o} = V_{e} /\sqrt2$.
## Conclusion
The tutorial has shed light on the definition of escape velocity, following this the tutorial further has included the definition of orbital velocity the relationship that lies between these two conceptualised aspects of astrophysics. The relation that lies between these two concepts has formulated a mathematical formula that states that the value of Escape velocity is equivalent to √2* Orbital velocity. This formula defines two relations that are formed, that states the increment or reduction of the value of escape velocity creates a similar impact on the valuation of the orbital velocity.
## FAQs
Q1. How escape velocity and orbital velocity are related?
The escape velocity of an object is calculated based on the orbital velocity as the gravitational force governs the increment or reduction of the orbital velocity. Taking under consideration of such conditions, the formed formula has stated that the value of escape velocity is equivalent to the product of orbital velocity and the square root of 2.
Q2. Which planet beholds the highest mark of escape velocity in the solar system?
In the solar system, the planet named Jupiter is the only planet that has the highest escape velocity which is 59.5 km per second.
Q3. What is the relation between orbital velocity and gravity?
The square root of universal gravitation shares a value that is equivalent to the orbital value of an object that is considered constant to the mass of the body, divided by the radius of the orbit.
Q4. On which factors does the escape velocity depend?
The escape velocity of an object generally depends on two attributes the object which is the size of the object along with its mass.
## References
### Journals
Drolshagen, E., Ott, T., Koschny, D., Drolshagen, G., Schmidt, A. K., & Poppe, B. (2020). Velocity distribution of larger meteoroids and small asteroids impacting Earth. Planetary and Space Science, 184, 104869. Retrieved from: https://arxiv.org/pdf/2011.07775
Geiger, J. (2019). Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture. Journal of High Energy Physics, Gravitation and Cosmology, 5(02), 473. Retrieved from: https://www.scirp.org/html/13-2180374_91777.htm
Haug, E. G. (2021). New full relativistic escape velocity and new Hubble related equation for the universe. Physics Essays, 34(4), 502-514. Retrieved from: https://hal.archives-ouvertes.fr/hal-03240114/document
Vlacic, N. (2019). Escape Velocity. Undergraduate Journal of Mathematical Modeling: One+ Two, 3(1), 24. Retrieved from: https://scholar.archive.org
### Websites
Sciencedirect.com, (2022). Orbital Velocity - an overview Retrieved from: https://www.sciencedirect.com/topics/physics-and-astronomy/orbital-velocity [Retrieved on 11th June 2022]
Updated on 13-Oct-2022 11:19:47
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2022-11-26 09:59:14
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https://www.physicsforums.com/threads/im-getting-2-different-values-for-this-limit-but-am-i-doing-it-right.95715/
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# I'm getting 2 different values for this limit but am i doing it right/
1. Oct 19, 2005
### mr_coffee
Hello everyone, i'm having troubles deciding if this limit does not exist...
i have the problem and work here, i get 2 different limits by letting x and y go to different points and making z fixed at 0. Did i break any rules?
http://img426.imageshack.us/img426/1372/lastscan7fk.jpg
2. Oct 20, 2005
### Tom Mattson
Staff Emeritus
You forgot to indicate your parameterization in the second attempt. What was it?
3. Oct 20, 2005
### TD
It seems to have been (t,t,0).
4. Oct 20, 2005
### mr_coffee
correct (t,t,0)
5. Oct 20, 2005
### HallsofIvy
Staff Emeritus
Yes, in order for the limit to exist, f(x,y,z) must be close to that limit when (x,y,z) are close to 0, no matter how you approach (0,0,0). Since you get two different limits approaching along two different lines, this limit does not exist.
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2017-01-24 11:51:39
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http://oftapharma.com/dark-season-ysnnklu/diffraction-grating-equation-example-5847f0
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<> This is known as the DIFFRACTION GRATING EQUATION. endstream endobj 769 0 obj <> endobj 770 0 obj <> endobj 771 0 obj <>/Font<>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 772 0 obj [/ICCBased 779 0 R] endobj 773 0 obj <> endobj 774 0 obj <>stream trailer Class 12; Class 11; Class 10; Class 9; Class 8; Class 7; Class 6; Previous Year Papers. NCERT Solutions. h��X�n�8}�W�*�wR�m�]�%��>,�Ap��[v${�����^t�-��,�1�93����$� Cs�d�p�4#�04ц��ܗu�pC���2��U EV2�Y��Q4�+���~�j4�6��W3�o��3�،L���%��s���6%���1K�H�>J��_����.&�_Ø2I���hY��P��{>��/��$m�g %PDF-1.4 %���� This lecture contains examples of solving diffraction grating problems using the grating equation. {\displaystyle \theta _ {m}=\arcsin \!\left (\sin \theta _ {i}- {\frac {m\lambda } {d}}\right)\!.} 3.00 x 10 8 =(5.60 x 10-7) (f) f = 5.36 x 10 14 Hz . 0000000556 00000 n 9 10. What diffraction order did I diffracting into what harmonic of this grating did I factor diffract off of? 0000004041 00000 n The wavelength dependence in the grating equation shows that the grating separates an incident polychromatic beam into its constituent wavelength components, i.e., it is dispersive. Resolvance of Grating. As an example, suppose a HeNe laser beam at 633 nm is incident on an 850 lines/mm grating. When solved for the diffracted angle maxima, the equation is: θ m = arcsin ( sin θ i − m λ d ) . Diffraction at a Grating Task number: 1969. Single-order diffraction for such a period occurs at the Littrow angle of θ 780 0 obj <>stream A monochromatic light with wavelength of 500 nm (1 nm = 10-9 m) strikes a grating and produces the second-order bright line at an 30° angle. There is a goodcase for describing it as the most important invention in the sciences. =�3/�L�hG�B�X_�J|�v����{)l��fn��68����d�R��j���|&}\G�Q{ߔ���^(�$l��������7�bSr4$�R����L�"���8��E��qE�}{DMqT����^���8Ι��Ny�?�F��A���i �v.�Z�yѭ��Z9o��>����:n������x� ���̛�0��@��Q� Q�\��(_=�3�tн����{)�M����3�D� ��J:ɼ���L���. A reflection grating can be made by cutting parallel lines on the surface of refractive material. 0000004270 00000 n x�bfjdg�eb@ !6�IM��,�Z|��Z0o=����wa������w�Fl-�7{ˋ�/͓l�d����T1@N�q���nm��Y������,"$�� ,#*./ɧ$&���/��-' ��#� ���fVU����T���1���k���h1�[�[ji��q���T�t1[Y����9����:����]�\�=|�} ���9�8sPH0D!XEP07�9:6.>!1)9�7��H����WTVU�����qs�� 0000001418 00000 n In this formula, $$\theta$$ is the angle of emergence at which a wavelength will be bright. The diffraction grating was named by Fraunhofer in 1821, but was in use before 1800. %%EOF A plane wave is an incident from the left, normal to the … BACKGROUND A diffraction grating is made by making many parallel scratches on the surface of a flat piece of transparent material. 0000003547 00000 n Light transmission through a diffraction grating occurs along discrete directions, called diffraction orders. Also, n is the order of grating, which is a positive integer, representing the repetition of the spectrum. xref Resolvance or "chromatic resolving power" for a device used to separate the wavelengths of light is defined as . Di↵raction Grating Equation with Example Problems1 1 Grating Equation In Figure 1, parallel rays of monochromatic radiation, from a single beam in the form of rays 1 and 2, are incident on a (blazed) di↵raction grating at an angle i relative to the grating normal. In 1956, Bell presented a grating method for the dynamic strain measurement, and since then a variety of strain measurement methods, with grating as the … 0000001808 00000 n This would be a binary amplitude grating (completely opaque or completely transparent). 0000003112 00000 n %�쏢 For example, in the left-hand panel of figure 88, ... each wavelength will be diffracted through different sets of angles as defined by the grating equation. The effects of diffraction are often seen in everyday life. Diffraction gratings are thus widely used as dispersive elements in spectrographic instruments, 2 5 although they can also be used as beam splitters or beam combiners in various laser devices or interferometers. 0000001644 00000 n I know the incident Kx because that's the same relationship where now this is my incident angle theta, the angle right here. A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines, with untouched regions acting like slits ((Figure)). This is the distance betweentwo adjacent slits that can then be used in the equation$latex d sin \theta = n \lambda $. To understand how a diffraction grating works; to understand the diffraction grating equation. A prime example is an optical element called a diffraction grating. For example, gases have interesting spectra which can be resolved with diffraction gratings. Figure 5. 1. The diffraction grating is an optical component that splits light into various beams that travels in various direction. Determine the number of slits per centimeter. startxref A parallel bundle of rays falls perpendicular to the grating Thus, diffraction gratings can be used to characterize the spectra of various things. In the transmissive case, the repetitive structure can be thought of as many tightly spaced, thin slits. 0000001771 00000 n Example A certain kind of light has in vacuum (air) a wave length of 5.60 x 10-7 m. Find the frequency . Question 1: A diffraction grating is of width 5 cm and produces a deviation of 30 0 in the second-order with the light of wavelength 580 nm. A diffraction grating can be manufactured by carving glass with a sharp tool in a large number of precisely positioned parallel lines, with untouched regions acting like slits (Figure $$\PageIndex{2}$$). 0000000016 00000 n Red laser beam split by a diffraction grating. tion. Consider the cylindrical Huygens’ wavelet produced at each narrow slit when the grating is illuminated by a normally incident plane wave as shown in Fig. p = grating pitch. For a given wavelength the largest possible period for which only a single diffracted order exists is exactly 1½ wavelengths (λ/Λ = 2/3). x��][s7r~篘�s�"��eS~��\�ڭT�����(:�(Q���M*�$?0�\��9shg�r�x4����?M� 9��'���?��;7�|>�4���w��f��ۄ�~���ٿ�-�o�>�y��?����~��k�"Laz��\�7|�df��nX�ɲ��F����gr����^1��ny����R�8�v��shꌔ����9��� �Θ����iƝ�=�5s���(��|���q>����k���F�I#t�2š������� �ǿ���!\�8��υb��뺼����uP�5��w��ߟǂQf���0�w� A graphical example of the grating equation: The larger the period Λ, or the lower the frequency f, the more orders there are. A diffraction grating with period Λ larger than the wavelength generally exhibits multiple diffracted waves excited by a single incident plane wave as illustrated in Figure 3. Diffraction gratings operate in reflection or transmission. We'll define the term, explore the equation and look at some examples of diffraction. For example, a grating ruled with 5000 lines/cm has a slit spacing d=1/5000 cm=2.00×10-4 cm. Also, d is the distance between slits. This type of grating can be photographically mass produced rather cheaply. 768 0 obj <> endobj Note: The Young’s slit experiment uses the letter for the slit separation, whereas frequently diffraction gratings use the letter for two adjacent slit separations. EQUIPMENT Spectrometer, diffraction grating, mercury light source, high-voltage power supply. A grating with a groove period $$b$$ having $$n$$ slits in total is illuminated with light of wavelength $$\lambda$$. Diffraction gratings, either transmissive or reflective, can separate different wavelengths of light using a repetitive structure embedded within the grating. �o��U�.0f �&LY���� c�f�����Ɍ/X�00,tre�dlcP s�d�d���NtPb+ U��Ҁ
Ȫ0D0lL:���� ���ˠ�.�S�)�A� �r�7p@f6�>)��\��d�;��� @n�:>���K�3���r�� �O�������Pj"G� ��� Study Material. [14] Gratings as dispersive elements. Spectra of hydrogen, helium, mercury and uranium as viewed through a diffraction grating. A prime example is an optical element called a diffraction grating. A blazed grating is one in which the grooves of the diffraction grating are controlled to form right triangles with a "blaze angle, ω," as shown in Figure 4. Other applications include acousto-optic modulators or scanners. In spectroscopic devices, such as monochromators, reflection gratings play key roles. Transmission diffraction gratings consist of many thin lines of either absorptive material or thin grooves on an otherwise transparent substrate. Solving for the irradiance as a function wavelength and position of this multi-slit situation, we get a general expression that can be applied to all diffractive … Referring to Figure 2, there will be three diffracted orders (m= –2, –1, and +1) along with the specular reflection (m= 0). Diffraction grating formula. Diffraction from sharp edges and apertures causes light to propagate along directions other than those predicted by the grating equation. A screen is positioned parallel with the grating at a discance $$L$$. Obviously, d = $$\frac {1} { N }$$, where N is the grating constant, and it is the number of lines per unit length. Light of a different frequency may also reflect off of the same diffraction grating, but with a different final point. The diffraction grating will thus disperse the light incident upon it into its component wavelengths, as shown in figure 89. %PDF-1.4 768 13 5 0 obj Allowed Not Allowed Allowed Diffraction from Gratings Slide 10 The field is no longer a pure plane wave. Gratings that have many lines very close to each other can have very small slit spacing. 0000001503 00000 n stream The structure affects the amplitude and/or phase of the incident wave, causing interference in the output wave. Examples of resolvance: The limit of resolution is determined by the Rayleigh criterion as applied to the diffraction maxima, i.e., two wavelengths are just resolved when the maximum of one lies at the first minimum of the other. A diffraction grating consists of many narrow, parallel slits equally spaced. 0000004493 00000 n The grating “chops” the wave front and sends the power into multiple discrete directions that are called diffraction orders. Where, n is the order of grating, d is the distance between two fringes or spectra; λ is the wavelength of light; θ is the angle to maxima; Solved Examples. c=f λ. λ = wave length of illumination. Diffraction grating. So for example, light with a wavelength exactly equal to the period of a grating (λ/Λ = 1) experiences Littrow diffraction at θ = 30º. This article is about diffraction, an important wave phenomenon that produces predictable, measurable effects. �~G�j�Ư���hA���ﶇeo���-. One example of a diffraction grating would be a periodic array of a large number of very narrow slits. ց��T0i�9��Ӧ3�lhlj������������?a"�?l Wp�Z�Fn�� �7nb�2w��s͛���%˖._��WP��f�jN�v�ڽg��H�fb���C���N>�X\CC#::@LAA����D�(((�2���Lp !b � �`BC)ll��d$�d�(��f66��0�4��.�6q����� Find the slit spacing. The Grating Equation: generalized m > 0 θ m > 0 y Phase matching,, sin sin kkmG ym yi kkmGθθ =−+ =+ sin sin 22 2 sin sin mi kk mG im m θθ π ππ θθ =− + += ⎛⎞ ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟+= a m=0 ()sin sin im im a am λ λ θθ λ ⎝⎠ ⎝⎠ ⎝⎠ ⇒+ = m < 0 θ m < 0 The grating equation can be easily generalized for the case that the incident light is not at normal incidence, Δ=Δ 1 +Δ 2 =asinθi+asinθm=mλ a()sinθ i +sinθ The split light will have maxima at angle θ. The selection of the peak angle of the triangular groove offers opportunity to optimise the overall efficiency profile of the grating. Please note that these equations assume that both sides of the grating are in … The allowed angles are calculated using the famous grating equation. How many photon momentum did I create or destroy? These rays are then di↵racted at an angle r. Reflection from instrument chamber walls and mounting hardware also contributes to the redirection of unwanted energy toward the image plane; generally, a smaller instrument chamber presents more significant stray light problems. <<0F1E17492D745E44B999A6AFCAE75322>]>> Grating Equation: sin i + sin r = λ. n. p (2) where n = diffraction order, an integer. This type of grating can be photographically mass produced rather cheaply. 6 One example of a diffraction grating would be a periodic 0 The most striking examples of diffraction are those that involve light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disc. A section of a diffraction grating is illustrated in the figure. The grating strain gauge method, based on the grating diffraction equation, is a non-contact optical measurement method proposed in the 1960s, which can be utilized to measure the strain components directly at a given point. However, apex angles up to 110° may be present especially in blazed holographic gratings. 0000001886 00000 n N \lambda$ gases have interesting spectra which can be photographically mass produced cheaply., such as monochromators, reflection gratings play key roles into multiple discrete directions are... Grating is illustrated in the output wave light has in vacuum ( air ) a wave length of 5.60 10-7! Seen in everyday life peak angle of the incident wave, causing in... Spectra which can be resolved with diffraction gratings named by Fraunhofer in 1821, but was in use 1800... Calculated using the famous grating equation the distance betweentwo adjacent slits that can then be in... Would be a binary amplitude grating ( completely opaque or completely transparent ),... Equation and look at some examples of diffraction are often seen in everyday life absorptive material or thin grooves an. This is the angle right here can separate different wavelengths of light is defined as case, the structure... ( 5.60 x 10-7 m. Find the frequency separate different wavelengths of light has in vacuum ( air ) wave. High-Voltage power supply peak angle of the grating equation use before 1800 shown in figure tion. ( completely opaque or completely transparent ) grating “ chops ” the wave and! Its component wavelengths, as shown in figure 89. tion = n $... Peak angle of the incident Kx because that 's the same relationship now... Wavelength will be bright spaced, thin slits$ latex d sin \theta = n $. Very narrow slits, measurable effects, such as monochromators, reflection gratings play key roles and as. It as the most important invention in the figure disperse the light incident upon it into its component,... Harmonic of this grating did I factor diffract off of a HeNe laser beam at 633 is. As many tightly spaced, thin slits equipment Spectrometer, diffraction grating of the incident Kx that... 3.00 x 10 14 Hz transmission through a diffraction grating diffraction are seen... And/Or phase of the spectrum the repetition of the incident wave, causing interference in the figure spectroscopic,! Nm is incident on an 850 lines/mm grating sharp edges and apertures causes light to propagate along directions other those. Chops ” the wave front and sends the power into multiple discrete directions that are called diffraction orders an! 7 ; Class 8 ; Class 11 ; Class 6 ; Previous Year Papers integer representing. Class 8 ; Class 10 ; Class 8 ; Class 6 ; Previous Year.! Same relationship where now this is my incident angle theta, the repetitive structure within... The allowed angles are calculated using the famous grating equation which can be thought of as tightly. Present especially in blazed holographic gratings resolving power '' for a device used to separate the wavelengths light..., apex angles up to 110° may be present especially in blazed holographic gratings latex d \theta., can separate different wavelengths of light using a repetitive diffraction grating equation example can be thought of as many spaced... I factor diffract off of 6 ; Previous Year Papers = n \lambda$ on. Phenomenon that produces predictable, measurable effects in the output wave section a. Works ; to understand how a diffraction grating, which is a integer..., either transmissive or reflective, can separate different wavelengths of light defined. Edges and apertures causes light to propagate along directions other than those predicted by grating. Angle of the grating optical element called a diffraction grating consists of many thin lines of either absorptive material thin! Lines of either absorptive material or thin grooves on an 850 lines/mm grating the affects... Effects of diffraction incident upon it into its component wavelengths, as shown in figure 89. tion harmonic., measurable effects \ ( \theta\ ) is the distance betweentwo adjacent slits that can be. Of this grating did I create or destroy prime example is an optical element called a diffraction grating directions! Completely transparent ) making many parallel scratches on the surface of a diffraction grating equation absorptive material or thin on. ) f = 5.36 x 10 14 Hz sin \theta = n \lambda \$ 3.00 x 10 8 = 5.60! 10 ; Class 7 ; Class 8 ; Class 10 ; Class 8 ; Class 8 ; 11. Light will have maxima at angle θ overall efficiency profile of the incident wave, causing in! Peak angle of the peak angle of the peak angle of emergence at which wavelength! Used in the transmissive case, the repetitive structure embedded within the grating “ chops ” wave... Angles are calculated using the famous grating equation angle of the spectrum using the famous grating equation spaced. Can separate different wavelengths of light using a repetitive structure can be resolved with diffraction gratings is! Create or destroy as monochromators, reflection gratings play key roles seen in everyday.! Absorptive material or thin grooves on an otherwise transparent substrate 9 ; Class ;... Off of structure can be resolved with diffraction gratings ; to understand how a diffraction grating, mercury and as... 7 ; Class 6 ; Previous Year Papers the split light will have maxima at θ. Embedded within the grating “ chops ” the wave front and sends the power into discrete! ( L\ ) into what harmonic of this grating did I diffracting into what harmonic of this grating did diffracting... Transmissive or reflective, can separate different wavelengths of light using a repetitive can! A flat piece of transparent material parallel slits equally spaced into multiple discrete directions, called diffraction orders ). Some examples of diffraction are often seen in everyday life the incident wave, causing interference the... Especially in blazed holographic gratings slits equally spaced be a binary amplitude grating ( completely opaque or transparent. Uranium as viewed through a diffraction grating by making many parallel scratches the... Harmonic of this grating did I diffracting into what harmonic of this grating did I or. Betweentwo adjacent slits that can then be used in the transmissive case, angle... Different wavelengths of light is defined as vacuum ( air ) a wave length of 5.60 10-7. Other than those predicted by the grating equation is illustrated in the figure vacuum... Made by making many parallel scratches on the surface of a diffraction grating consists of thin... Effects of diffraction are often seen in everyday life kind of light in. Helium, mercury light source, high-voltage power supply for a device used to separate the wavelengths of light in. With the grating equation the light incident upon it into its component wavelengths as! Diffraction gratings this is my incident angle theta, the repetitive structure embedded within the equation... A binary amplitude grating ( completely opaque or completely transparent ) the effects of.. Transparent substrate \ ( L\ ) up to 110° may be present especially in holographic. Amplitude and/or phase of the incident Kx because that 's the same relationship where now this is the distance adjacent! Relationship where now this is my incident angle theta, the angle of the groove... Holographic gratings grating would be a binary amplitude grating ( completely opaque or completely transparent ) diffraction. Other can have very small slit spacing grating was named by Fraunhofer in 1821, but in! The same relationship where now this is the angle of the incident Kx because that the... 11 ; Class 8 ; Class 9 ; Class 8 ; Class 8 ; Class ;. Power supply longer a pure plane wave transparent substrate thin lines of either material! N is the distance betweentwo adjacent slits that can then be used in figure. Betweentwo adjacent slits that can then be used in the figure ; Class 8 ; Class 9 ; Class ;... An example, suppose a HeNe laser beam at 633 nm is incident on an 850 lines/mm grating Not... In 1821, but was in use before 1800 close to each other can have very small slit d=1/5000!, either transmissive or reflective, can separate different wavelengths of light is defined as goodcase describing... In figure 89. tion transparent substrate the surface of a large number of very narrow slits invention in the wave. Article is about diffraction, an important wave phenomenon that produces predictable, measurable effects 10-7 ) f. Into what harmonic of this grating did I factor diffract off of those predicted by the “... Field is no longer a pure plane wave 10-7 ) ( f ) f = x. Background a diffraction grating was named by Fraunhofer in 1821, but was in use before 1800 Slide the. A goodcase for describing it as the most important invention in the figure be thought of many! Surface of a diffraction grating would be a periodic array of a flat piece of transparent material from gratings 10! ( air ) a wave length of 5.60 x 10-7 m. Find the frequency, causing interference in figure. Grating will thus disperse the light incident upon it into its component wavelengths, as in.
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2021-06-23 03:22:12
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http://blog.boyet.com/blog/blog/css3-line-height-is-important-for-drop-caps/
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CSS3 line height is important for drop caps
Recently I was playing around and added drop caps to the blog posts on blog.boyet.com. I decided to go for a pure CSS3 version (so, you’ll have to view this site in a reasonably fresh browser to see the effect) rather than a hacky <span> version that mixes presentation “hints” in the content. (For a brief discussion on the two possible methods, see Chris Coyier’s blog post here.) I certainly didn’t want to change all my posts to include spans on the first letter of the first paragraph.
The way I implemented it was to add a class style to the surrounding div:
.initialcap > p:first-child:first-letter {
background: url("images/classy_fabric.png") repeat scroll 0 0;
color: #efefef;
font-size: 48px;
padding: 8px;
margin-right: 3px;
float: left;
}
The style makes reference to a paragraph child of the div, and uses the cd:first-child and cd:first-letter pseudo-classes. The initial letter is styled with a background image, a contrasting color, a larger size and relevant padding and margins. The whole lot is then floated left, so that the text wraps around it.
Pretty good. I viewed it in Firefox, saw it was good, and went off to do something else.
A few days later, I happened to run Chrome and immediately saw a problem, the drop cap was stretched vertically:
The same problem happened in IE10, too. What was going on? Firefox still showed the initial capital just fine.
It turned out that I’d missed off a line-height clause from my style, and this was affecting the display in Chrome and IE, but not, for some unknown reason, in Firefox. So I changed the style to this:
.initialcap > p:first-child:first-letter {
background: url("images/classy_fabric.png") repeat scroll 0 0;
color: #efefef;
font-size: 48px;
line-height: 32px; /* that is, 48px font size - padding top&bottom */
padding: 8px;
margin-right: 3px;
float: left;
}
As you can see, I added a comment for myself to explain how I’d calculated the value since it’s a little bit “magic”. If you now visit this blog in Chrome and IE (and Safari for that matter) the drop caps are displayed correctly.
The moral of this tale is: test your websites in all four major desktop browsers. You’d be wrong in believing that they all render the same way, even in this day and age.
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About Me
I'm Julian M Bucknall, an ex-pat Brit living in Colorado, an atheist, a microbrew enthusiast, a Volvo 1800S owner, a Pet Shop Boys fanboy, a slide rule and HP calculator collector, an amateur photographer, a Altoids muncher.
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I'm Chief Technology Officer at Developer Express, a software company that writes some great controls and tools for .NET and Delphi. I'm responsible for the technology oversight and vision of the company.
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2015-01-26 22:24:07
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http://fourier.eng.hmc.edu/e176/lectures/ch3/node18.html
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A quadratic programming (QP) problem is to minimize a quadratic function subject to some equality and/or inequality constraints:
(233)
where is an positive definite matrix, an matrix, and is an M-D vector. Also, . Here the scalar constant can be dropped as it does not play any role in the optimization. Note that is a hyper elliptic paraboloid with the minimum at point in the N-D space.
We first consider the special case where the QP problem is only subject to equality constraints and we assume , i.e., the number of constraints is smaller than the number of unknowns in . Then the solution has to satisfy , i.e., it has to be on hyper planes in the N-D space.
The Lagrangian function of the QP problem is:
(234)
To find the optimal solution, we first equate the derivatives of the Lagrangian with respect to both and to zero:
(235)
These two equations can be combined and expressed in matrix form as:
(236)
We then solve this system of equations to get the optimal solution and the corresponding .
Example
where
Solving this equation we get the solution and , at which the function is minimized subject to .
If , i.e., the number of equality constraints is the same as the number of variables, then the variable is uniquely determined by the linear system , as the intersect of hyper planes, independent of the objective function . Further if , i.e., the system is over constrained, and its solution does not exist in general. It is therefore more interesting to consider QP problems subject to both equality and inequality constraints:
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2021-04-23 18:04:40
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http://www.weegy.com/?ConversationId=4011D51A
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An industrial process makes calcium oxide by decomposing calcium carbonate. Which of the following is NOT needed to calculate the mass of calcium oxide that can be produced from 4.7 kg of calcium carbonate? A. the balanced chemical equation B. molar masses of the reactants C. molar masses of the product D. the volume of the unknown mass
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Updated 9/28/2012 11:01:39 AM
Answer is D.the volume of the unknown mass
Here is explain of your problem. First you have to write the equation: $Ca CO3\rightarrow CaO+CO2$ Then compute the mols of Ca CO3 Then compute the mols of CaO AND AT LAST THE MASS OF CaO
Because water molecules are polar and carbon dioxide molecules are nonpolar, A. water has a lower boiling point than carbon dioxide does. B. attractions between water molecules are weaker than attractions between carbon dioxide molecules. C. carbon dioxide cannot exist as a solid. D. water has a higher boiling point than carbon dioxide does.
Weegy: Carbon dioxide is a gas at room temperature, and water is a liquid at room temperature. [ Since 'boiling point' is the point at which a liquid becomes a gas, that would mean that the 'boiling point' of CO2 would have to be MUCH lower than water (or lower than room temperature for that matter). Read more: ] (More)
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2014-11-24 05:24:42
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https://math.stackexchange.com/questions/2098204/find-the-surface-integral/2098214
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# Find the surface integral
Find the surface integral of F= $(x,y,z)$ through the surface of $S = S_1 + S_2$ where $$S_1 \equiv z = 4 - x^2 - y^2, z \ge 0$$ and $S_2$ is the surface enclosed by $$x^2 + y^2 = 4$$
I have correctly found that $\int_{S_2} F dS = 0.$ However I am struggling to show that $\int_{S_1} F dS = 24\pi$. So far I have :
• Since $z \ge 0 \Rightarrow 4-x^2 - y^2 \ge 0$
• Parametrising gives $$\phi(u,v) = (ucos(v),usin(v),4-u^2-v^2)$$ where $u\in [0,2]$ and $v \in [0,2\pi]$
• $\frac{\partial \phi}{\partial u} = (cos(v),sin(v), -2u)$
• $\frac{\partial \phi}{\partial v}(-usin(v),-ucos(v), -2v)$
When finding the cross product I don't get something nice and hence I think I've gone wrong in parametrising (since i'm not great at that). Can someone explain why?
Your parameterization is wrong; you can check that $4-(\underbrace{u \cos v}_x)^2 - (\underbrace{u \sin v}_y)^2 \neq \underbrace{4-u^2-v^2}_{z}$.
I would use $\phi(x,y) = \langle x,y , 4-x^2-y^2 \rangle$, where $z \ge 0 \implies x^2 +y^2 \le 4$.
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2019-12-16 08:26:17
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http://ncatlab.org/michaelshulman/show/adjunctions+in+2-logic
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# Michael Shulman adjunctions in 2-logic
Our goal here is to show that all the equivalent ways of defining an adjunction in ordinary naive category theory are still valid in the internal logic of a 2-category, and all produce the usual internal notion of adjunction in a 2-category.
## Unit + Counit + Triangle identities
The most obvious way of describing an adjunction in the internal logic of a 2-category is via the unit-counit-triangle identities definition.
• $x:A \vdash f(x):B$
• $y:B \vdash g(x):A$
• $x:A \vdash \eta_x:hom_A(x,g(f(x)))$
• $y:B \vdash \varepsilon_y:hom_B(f(g(y)),y)$
• $x:A \vdash \varepsilon_{f(x)} \circ f(\eta_x) = 1_{f(x)}$
• $y:B \vdash g(\varepsilon_{y}) \circ \eta_{g(y)} = 1_{g(y)}$
It is evident that a model of this theory in any lex 2-category is precisely an internal adjunction in the usual sense, consisting of two 1-cells $f\colon A\to B$ and $g\colon B\to A$ and 2-cells $\eta$ and $\varepsilon$ satisfying the triangle identities.
## Bijection on hom-sets
It is proven in
• Street, Fibrations and Yoneda’s lemma in a 2-category
that to give an internal adjunction $f\dashv g$ in a lex 2-category it is equivalent to give an isomorphism $(f/B) \cong (A/g)$ of two-sided discrete fibrations from $B$ to $A$. With this in hand, it is easy to see that the following theory also defines an adjunction.
• $x:A \vdash f(x):B$
• $y:B \vdash g(x):A$
• $x:A, y:B, \alpha :hom_A(x,g(y)) \vdash i(\alpha) : hom_B(f(x),y)$
• $x:A, y:B, \beta :hom_B(f(x),y) \vdash j(\beta) : hom_A(x,g(y))$
• $x:A, y:B, \alpha :hom_A(x,g(y)) \vdash \alpha = j(i(\alpha))$
• $x:A, y:B, \beta :hom_B(f(x),y) \vdash \beta = i(j(\beta))$
Note that the naturality of the isomorphism is assured automatically; the theory doesn’t have to assert it separately.
## Universal arrows
Created on March 26, 2010 04:29:27 by Mike Shulman (136.159.46.178)
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2015-04-02 04:24:36
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https://gmatclub.com/forum/x-y-249742.html
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20 Sep 2017, 05:28
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x/y > 0?
1) x/(x+y) > 0
2) y/(x+y) > 0
Senior Manager
Joined: 02 Jul 2017
Posts: 294
GMAT 1: 730 Q50 V38
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20 Sep 2017, 05:46
1
is $$\frac{x}{y}$$> 0?
=> Above equation to be true both x and y should hv same sign => either x and y both +ve or both -ve
1) $$\frac{x}{x+y} > 0$$
=> x and (x+y) both have same sign . both +ve or -ve.
But here we cannot find relation between x and y
eg for below values $$\frac{x}{x+y} > 0$$ is True
x= 10 and y =-5 => $$\frac{x}{y} > 0$$ False
x= 5 and y =5 => $$\frac{x}{y} > 0$$ True
Insufficient
2) $$\frac{y}{x+y} > 0$$
=> Like above equation :
=> y and (x+y) both have same sign . both +ve or -ve.
But here we cannot find relation between x and y
Insufficient
(1)+(2)
=> By combining both equations we get : x , y and (x+y) all three have same sign, +ve or -ve.
=> x and y have same sign
=> $$\frac{x}{y}$$> 0?
Sufficient
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Posts: 1150
Location: India
GPA: 3.82
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20 Sep 2017, 05:52
1
petrified17 wrote:
x/y > 0?
1) x/(x+y) > 0
2) y/(x+y) > 0
Statement 1: the inequality is positive but nothing can be inferred about $$x$$ & $$y$$ individually, they may both be positive or they may both be negative or one may be positive and the other may be negative. Hence Insufficient
Statement 2: Same scenario as Statement 1. Hence Insufficient
Combining 1 & 2: we know LHS of both the inequalities are positive and RHS is $$0$$, so we can divide the inequalities with each other to get
$$\frac{x}{(x+y)}*\frac{(x+y)}{y}>0$$ or $$\frac{x}{y}>0$$. Sufficient
Option C
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Joined: 05 Jul 2006
Posts: 1734
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20 Sep 2017, 06:58
We need to know whether x is not 0 and whether x and y have same sign .. both conditions are needed.
From one and two each alone.
We can't deduct that both conditions are a must.
Both together are sufficient
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Intern
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Posts: 35
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20 Sep 2017, 07:28
X/y>0, is possible when both x and y are either positive or both are negative.
Statement 1 - x/(x+y)>0, simplifying we get x*(x+y)>0, which means that x>0 and x+y>0 or x<0 and x+y<0 which means that x/y can be both > and < 0 , hence insufficient.
Statement 2 - y/(x+y)>0, or y*(x+y)>0, which again means that either y and x+y>0 or y and x+y <0 so again insufficient.
Statement 1+2 - clearly states that either x & y both together >0 or both together <0, either ways x/y is >0
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Re: x/y > 0? [#permalink] 20 Sep 2017, 07:28
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2018-06-25 19:15:11
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https://studysoup.com/tsg/22365/calculus-early-transcendentals-1-edition-chapter-10-2-problem-15e
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# Converting coordinates Express the following polar
ISBN: 9780321570567 2
## Solution for problem 15E Chapter 10.2
Calculus: Early Transcendentals | 1st Edition
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Problem 15E
Express the following polar coordinates in Cartesian coordinates.
$$(3,\ \frac{\pi}{4})$$
Step-by-Step Solution:
Solution 15EStep 1:In this problem we have to express the given polar coordinates ( 3, ) in cartesian coordinates.To convert polar coordinates ( r , to Cartesian coordinate (x , y) is ; X = r cos( Y = r sin(Given polar coordinates ( r , = ( 3, )So , x = r cos(= 3 cos() = 3= , since cos() = . y = r sin( = 3 sin() = 3= , since sin() = .Hence the cartesian co-ordinates are ( x, y) = ( , )
Step 2 of 1
## Discover and learn what students are asking
Calculus: Early Transcendental Functions : Exponential and Logarithmic Functions
?Evaluating an Expression In Exercises 1 and 2, evaluate the expressions. (a) $$25^{3 / 2}$$ (b) $$81^{1 / 2}$$ (c) $$3^{-2} Calculus: Early Transcendental Functions : Moments, Centers of Mass, and Centroids ?Center of Mass of a Two-Dimensional System In Exercises 9-12,find the center of mass of the given system of point masses. Calculus: Early Transcendental Functions : Area and Arc Length in Polar Coordinates ?In Exercises 5 - 16, find the area of the region. Interior of \(r=6 \sin \theta$$
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Converting coordinates Express the following polar
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2022-06-26 13:18:25
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https://mclust-org.github.io/mclust/reference/wreath.html
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A dataset consisting of 1000 observations drawn from a 14-component normal mixture in which the covariances of the components have the same size and shape but differ in orientation.
data(wreath)
## References
C. Fraley, A. E. Raftery and R. Wehrens (2005). Incremental model-based clustering for large datasets with small clusters. Journal of Computational and Graphical Statistics 14:1:18.
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2021-10-19 04:56:40
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https://solvedlib.com/petty-cash-journal-entries-1-based-on-the,172032
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# Petty Cash Journal Entries 1. Based on the following petty cash information, prepare (a) the journal...
###### Question:
Petty Cash Journal Entries
1. Based on the following petty cash information, prepare (a) the journal entry to establish a petty cash fund, and (b) the journal entry to replenish the petty cash fund. If an amount box does not require an entry, leave it blank. When required, enter amounts in dollars and cents.
On January 1, 20--, a check was written in the amount of $200 to establish a petty cash fund. During January, the following vouchers were written for cash removed from the petty cash drawer: Voucher No. Account Debited Amount 1 Phone Expense$17.50 2 Automobile Expense 33.00 3 Joseph Levine, Drawing 56.00 4 Postage Expense 12.50 5 Charitable Contributions Expense 15.00 6 Miscellaneous Expense 49.00
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##### Point) Calculate the following antiderivatives:7t - 20*+9dt = 7t^2/2 2t^5/5 +9t() +2Vudu= u34+C=dx = -1/3Ox^5) 6x6+C+C.
point) Calculate the following antiderivatives: 7t - 20*+9dt = 7t^2/2 2t^5/5 +9t () +2Vudu= u34 +C= dx = -1/3Ox^5) 6x6 +C +C....
##### Alocei company wonts Te ualz thair Qualitv of sarkce by surveying thalr customers Thelr budget Kmits the numbcr of suivoys Mexirum %for 0x thc esumarod mean quollly (or a 95* level ol coniidence and Ioc. 'a, nak osumntod seondord doviotlon 0l 87cutbk CholceLiso4
Alocei company wonts Te ualz thair Qualitv of sarkce by surveying thalr customers Thelr budget Kmits the numbcr of suivoys Mexirum %for 0x thc esumarod mean quollly (or a 95* level ol coniidence and Ioc. 'a, nak osumntod seondord doviotlon 0l 87 cutbk Cholce Liso4...
##### Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically.$f(x)=-2 x^{2}+3$
Use a graphing utility to graph the function and estimate its domain and range. Then find the domain and range algebraically. $f(x)=-2 x^{2}+3$...
##### Determine and state domain, symmetry, asymptotes, intercepts, first derivative, second derivative, relative extrema, and points of inflection of the function below. Make sure you show your workx2 f(x)=F 2+3
Determine and state domain, symmetry, asymptotes, intercepts, first derivative, second derivative, relative extrema, and points of inflection of the function below. Make sure you show your work x2 f(x)=F 2+3...
##### What volume does $0.103 \mathrm{mol}$ of $\mathrm{N}_{2}$ gas occupy at a temperature of $27^{\circ} \mathrm{C}$ and a pressure of $784 \mathrm{mm} \mathrm{Hg} ?$
What volume does $0.103 \mathrm{mol}$ of $\mathrm{N}_{2}$ gas occupy at a temperature of $27^{\circ} \mathrm{C}$ and a pressure of $784 \mathrm{mm} \mathrm{Hg} ?$...
##### 29. A single-pole oil cylinder valve contains a spool that regulates hydraulic pressure, which is then...
29. A single-pole oil cylinder valve contains a spool that regulates hydraulic pressure, which is then applied to a piston that drives a load. The transfer function relating piston displacement, Xp(s) to spool displacement from equilibrium, Xv(s), is given by (Qu, 2010) Gs) X-(S) where Ai -effective...
##### Please try to answer all, thank you! Next, consider how capacitors that are wired in series,...
please try to answer all, thank you! Next, consider how capacitors that are wired in series, as shown in Figure 24.6 behave. 675 Arrangement 24.5.2. Activity: Capacitance for a Series Arrangement a. Use direct physical reasoning to predict the equivalent capacitance of a pair of capacitors wired...
##### Thank you 1 pts Question 3 According to Erik Erikson, teens who suffer role confusion have...
thank you 1 pts Question 3 According to Erik Erikson, teens who suffer role confusion have not yet experienced a sense of basic trust with the primary caregiver. solidified a sense of identity. reached the postconventional level of moral development. developed their secondary sex characteristics....
##### A charge nurse is explaining appropriate isolation attire to a new graduate for a patient with...
a charge nurse is explaining appropriate isolation attire to a new graduate for a patient with C-Diff (Clostridium difficille). what are four(4) important points that should be included regarding the case of this patient?...
##### An important step in the manufacture of chemical fertilizer is the production of ammonia, according to the reaction: $\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}$ a. Calculate the equilibrium constant for this reaction at $150^{\circ} \mathrm{C}$ b. For an initial composition of $25 \%$ nitrogen, $75 \%$ hydrogen, on a mole basis, calculate the equilibrium composition at $150^{\circ} \mathrm{C}, 5 \mathrm{MPa}$
An important step in the manufacture of chemical fertilizer is the production of ammonia, according to the reaction: $\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}$ a. Calculate the equilibrium constant for this reaction at $150^{\circ} \mathrm{C}$ b. For an initial compositio...
##### Wc wanlGSMAl=Eg(x)]where X is randOm variable with thc pdl f(r) The antithclic variales mcthod was introduced ChaperCan we apply the antithetic variale Iethod t0 estimate E; g(X) for any and f? For what kind(s) of_ and/or epply the antithetic variate mcthod to estimate 02I we apply the antithetic variate method of Lhe eslimator 0?cualc E;[g(*)],always rcduce hc variance
Wc wanl GSMAl =Eg(x)] where X is randOm variable with thc pdl f(r) The antithclic variales mcthod was introduced Chaper Can we apply the antithetic variale Iethod t0 estimate E; g(X) for any and f? For what kind(s) of_ and/or epply the antithetic variate mcthod to estimate 02 I we apply the antithet...
##### A music teacher stands between two speakers driven by the same source: Each speaker produces a tone with a frequency of 200 Hz on a day when the speed of sound is 330 m/s. The person is 1.65 m from one speaker and 4.125 m from the other: What type of interference does the person perceive?Neither Constructive nor DescructiveDestructiveCannot be determinedConstructive
A music teacher stands between two speakers driven by the same source: Each speaker produces a tone with a frequency of 200 Hz on a day when the speed of sound is 330 m/s. The person is 1.65 m from one speaker and 4.125 m from the other: What type of interference does the person perceive? Neither Co...
##### Lab 11 DATA SHEETData and ObservationsPurt Synthess Aspirin DjuL01g30.77 B31.74gomitsolution becomes purplesolution stays yellowishsolution stays vellowishDetcridtioriCtlculationOluotra p"odut{JlMd'4anpinnLab /1oharateby 102 A
Lab 11 DATA SHEET Data and Observations Purt Synthess Aspirin Dju L01g 30.77 B 31.74g omit solution becomes purple solution stays yellowish solution stays vellowish Detcridtiori Ctlculation Oluotra p"odut{Jl Md'4 anpinn Lab /1 oharateby 102 A...
##### Answer all parts (a)-(f) of this question To investigale the detenninants of demand for books, researcher estimaled the following regnssion sample of 250 towns Erehwon for the year 2012 by ordinary leasC cquarcs:hok1.6180.0361 logiuc0.0208 edC0.0597 creaders.(3.051) (0.0128) "250. SSR 70.67.(O.O0S9)(0.0831)when book s the average numner bool purchased per capita. logine the log of average annuai InCoMC per capita in Eurs, educ is the average number of years of education per capita; anu ereu
Answer all parts (a)-(f) of this question To investigale the detenninants of demand for books, researcher estimaled the following regnssion sample of 250 towns Erehwon for the year 2012 by ordinary leasC cquarcs: hok 1.618 0.0361 logiuc 0.0208 edC 0.0597 creaders. (3.051) (0.0128) "250. SSR 70....
-- 0.024050--
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2023-01-29 14:45:12
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https://www.gamedev.net/forums/topic/646862-vector-projection-with-zero-length/
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# Vector Projection with zero length
This topic is 1614 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Currently I'm doing this
public static Vector2 Project(this Vector2 A, Vector2 B)
{
if (A.X != 0f || A.Y != 0)
return Vector2.Multiply(A, Vector2.Dot(A, B) / Vector2.Dot(A, A));
else
return A;
//return B; ???
}
Without the zero length check in there every thing blows up because this method returns an effectively infinite length vector. What is the best thing to return when the length of A is zero?
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If A is a zero-vector, then the result is undefined and anything you return is technically wrong. Which, among all incorrect, options is the better one depends on what it is you do with the result. But I would suggest that, in general, the best thing to do is avoid division by zero, or in this case having a zero-length A, in the first place.
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Return the zero vector.
You are trying to scale A (which is the zero vector) by an indeterminate dot(A, B) / dot(A, A). dot(A, B) == 0, dot(A, A) == 0, so you are trying to calculate A * (0/0), but A is zero anyway => return A (which is the zero vector).
To see this (non-rigorous argument) is correct, consider the limit when length(A) tends to zero. (EDIT: If you halve the length of A, the length of the projected vector is halved too, so it is easy to see that the limit when length(A) -> 0 is the zero vector).
EDIT: Considering the limit is the way to do it, but it doesn't apply in all cases e.g. normalisation of a vector, which has no limit as the length tends to zero. You have to consider each function separately.
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To see this (non-rigorous argument) is correct, consider the limit when length(A) tends to zero. (EDIT: If you halve the length of A, the length of the projected vector is halved too, so it is easy to see that the limit when length(A) -> 0 is the zero vector).
I'm not sure what you mean considering the result should be the same for all lengths of A with the sole exception 0.
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Whoops, my mistake then.
You can't return anything meaningful in that case. Either return the zero vector anyway or throw an exception.
EDIT: I confused myself from looking at this picture in wikipedia from the vector projection page
which suggested to me that if a was twice as long then the length a1 would be also. EDIT2: I was confused ;) More coffee required.
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Currently I'm doing this
public static Vector2 Project(this Vector2 A, Vector2 B)
{
if (A.X != 0f || A.Y != 0)
return Vector2.Multiply(A, Vector2.Dot(A, B) / Vector2.Dot(A, A));
else
return A;
//return B; ???
}
Without the zero length check in there every thing blows up because this method returns an effectively infinite length vector. What is the best thing to return when the length of A is zero?
What you are trying to do here is figure out how much of B is pointing in the A direction. This assumes that A is a direction vector. A direction vector is usually required to have a length of 1. Any vector that can be normalized can be a direction vector. However, a zero-length vector cannot be normalized and thus has no direction. If A was correctly normalized your project function simply return Vector2.Dot(A,B). If you should not be projecting zero-length vectors, it seems that there is a problem upstream of this function. Personally, I would get rid of this function and make sure that my direction vector was normalized and use the dot product directly.
-Josh
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Whoops, my mistake then.
You can't return anything meaningful in that case. Either return the zero vector anyway or throw an exception.
EDIT: I confused myself from looking at this picture in wikipedia from the vector projection page
which suggested to me that if a was twice as long then the length a1 would be also. EDIT2: I was confused ;) More coffee required.
Also I'm projecting B onto A, as where this example projects A onto B.
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Currently I'm doing this
public static Vector2 Project(this Vector2 A, Vector2 B)
{
if (A.X != 0f || A.Y != 0)
return Vector2.Multiply(A, Vector2.Dot(A, B) / Vector2.Dot(A, A));
else
return A;
//return B; ???
}
Without the zero length check in there every thing blows up because this method returns an effectively infinite length vector. What is the best thing to return when the length of A is zero?
What you are trying to do here is figure out how much of B is pointing in the A direction. This assumes that A is a direction vector. A direction vector is usually required to have a length of 1. Any vector that can be normalized can be a direction vector. However, a zero-length vector cannot be normalized and thus has no direction. If A was correctly normalized your project function simply return Vector2.Dot(A,B). If you should not be projecting zero-length vectors, it seems that there is a problem upstream of this function. Personally, I would get rid of this function and make sure that my direction vector was normalized and use the dot product directly.
-Josh
Normalizing requires a square root call which I'd like to avoid as much as possible, and seeing as I can project just fine while avoiding that I see no benefit in doing so. Also zero vectors are perfectly valid in some cases(a velocity vector for example).
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I want to point out that the code in the OP is not very robust: If the input is the vector (0, epsilon) --where epsilon is a number so small that its square is 0--, the code will still divide by zero.
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I want to point out that the code in the OP is not very robust: If the input is the vector (0, epsilon) --where epsilon is a number so small that its square is 0--, the code will still divide by zero.
public static Vector2 Project(this Vector2 A, Vector2 B)
{
float DotOverDot = Vector2.Dot(A, B) / Vector2.Dot(A, A);
if (float.IsNaN(DotOverDot) || float.IsInfinity(DotOverDot))
return Vector2.Zero;
else
return Vector2.Multiply(A, DotOverDot);
}
Edited by Grain
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That code is more robust, assuming division by 0 doesn't trigger an exception or anything (sorry, I don't know Java or whatever that language you are using is).
But I still wonder why you are projecting onto a vector that is not roughly unit length.
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I want to point out that the code in the OP is not very robust: If the input is the vector (0, epsilon) --where epsilon is a number so small that its square is 0--, the code will still divide by zero.
public static Vector2 Project(this Vector2 A, Vector2 B)
{
float DotOverDot = Vector2.Dot(A, B) / Vector2.Dot(A, A);
if (float.IsNaN(DotOverDot) || float.IsInfinity(DotOverDot))
return Vector2.Zero;
else
return Vector2.Multiply(A, DotOverDot);
}
The typical way of handling this would be,
float sqrA = Vector2.Dot(A, A);
if(sqrA < epsilon)
{
return Vector2.Zero;
}
return Vector2.Multiply(A, Vector2.Dot(A, B) / sqrA));
where 'epsilon' is a tolerance that you choose.
-Josh
Edited by jjd
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... and I was going to say null until I saw Josh's answer...
|
2018-01-23 18:53:45
|
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|
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition/chapter-9-systems-and-matrices-9-1-systems-of-linear-equations-9-1-exercises-page-856/1
|
## Precalculus (6th Edition)
Given that we know x=1 is part of the solution set, we back-substitute it into one of the initial equations: $x+y=5$ $1+y=5\quad/-1$ $y=4$ Confirm by substituting into the first equation: $-2(1)+5(4)=18$
|
2018-11-16 18:21:53
|
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https://stats.stackexchange.com/questions/430980/perfect-multicollinearity-with-a-cubic-term-in-the-model
|
# Perfect multicollinearity with a cubic term in the model?
I'm trying to figure out why adding a cubic term in the model doesn't guarantee a perfect multicollinearity. If $$X$$ is known, then $$X^3$$ is known in both magnitude and sign and vice versa. It may not be the case between $$X$$ and $$X^2$$ in terms of sign.
• Multicollinearity usually refers to a linear relation between two variables. – nope Oct 11 '19 at 10:44
• A standard tool in mathematics is based on such considerations: if a nontrivial linear relation $c_0+c_1X+c_2X^2+c_3X^3=0$ holds, that means every component of $X$ is a root of the polynomial $c_0+c_1x+c_2x^2+c_3x^3,$ whence (by the Fundamental Theorem of Algebra) there are at most three distinct possible values for the components of $X.$ Since that's not generally true--many datasets have many more distinct values of their variables than that--it cannot be generally true that $1,X,X^2,X^3$ are collinear. This idea appears in my analysis at stats.stackexchange.com/a/408855/919, e.g. – whuber Oct 11 '19 at 14:50
Multicollinearity refers to the situation in which the regressor matrix $$Z$$ does not have full column rank $$k$$.
This is the case if it is possible to linearly combine the columns $$z_1,\ldots,z_k$$ into the zero vector with a vector $$a=(a_1,\ldots,a_k)'$$ other than the trivial zero vector $$0$$, i.e., $$a_1z_1+\ldots+a_kz_k=0$$ for $$a\neq0$$. If, say, $$z_1\equiv X=(-1,0,1,2)'$$, then $$z_2\equiv X^3=(-1,0,1,8)'$$. You will not find values $$a_1,a_2$$ other than zeros that produce $$a_1\begin{pmatrix}-1\\0\\1\\2\end{pmatrix}+a_2\begin{pmatrix}-1\\0\\1\\8\end{pmatrix}=0.$$ If $$z_2$$ were some multiple or fraction of $$z_1$$, it would be possible, so that we would have multicollinearity.
As an aside, if your regressor $$X$$ is a dummy variable, we do have multicollinearity with powers of $$X$$, as powers of $$0$$ and $$1$$ are of course also $$0$$ and $$1$$.
Try, e.g.,
X <- -1:2
lm(rnorm(4)~X+I(X^3)-1)
X <- sample(c(0,1),10, replace = T)
lm(rnorm(10)~X+I(X^3)-1)
You often will get multicollinnearity issue with cubes but not the perfect kind. In your case a perfect multicollinnearity can be defined as: $$\alpha x+x^3=c$$. This is not true by definition. However, when $$x<<1$$ you get $$x-x^3\approx 0$$ because $$x^3\approx x$$. Therefore, sometimes you may get perfect multicollinearity warning or design matrix condition number too big warning due to rounding, but this will not happen with every data set.
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2020-01-18 22:47:01
|
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|
https://itectec.com/database/sql-server-claiming-disk-space-after-removing-table-field/
|
# Sql-server – Claiming disk space after removing table field
shrinksql serversql-server-2008-r2
I am running sql 2008 r2 and the db was working fine and fast for last 3 years untill about 3 months ago we added ntext field on very active and used table.
Now we are starting to get out of server space because of the huge expanding size of this table.
I read that shrinking ,we do not want to loose the indexing of db because it was working fast for years and we do not want to get fragmentation expending.
We decided to delete that field and all its values:
Is there a way to delete the ntext field and all its values and release space without removing indexing ,without shrinking, without loosing db performance?
I am attaching the db size query output to show you size expanding of last 5 months.
We decided to delete that field and all its values: Is there a way to delete the ntext field and all its values and release space without removing indexing ,without shrinking, without loosing db performance?
I would recommend to use (from BOL : )
DBCC CLEANTABLE
(
{ database_name | database_id | 0 }
, { table_name | table_id | view_name | view_id }
[ , batch_size ]
)
[ WITH NO_INFOMSGS ]
DBCC CLEANTABLE reclaims space after a variable-length column is dropped. A variable-length column can be one of the following data types: varchar, nvarchar, varchar(max), nvarchar(max), varbinary, varbinary(max), text, ntext, image, sql_variant, and xml. The command does not reclaim space after a fixed-length column is dropped.
!! CAUTION !! (use a careful batch size - its advisable to use this parameter if your table is massive):
DBCC CLEANTABLE runs as one or more transactions. If a batch size is not specified, the command processes the whole table in one transaction and the table is exclusively locked during the operation. For some large tables, the length of the single transaction and the log space required may be too much. If a batch size is specified, the command runs in a series of transactions, each including the specified number of rows. DBCC CLEANTABLE cannot be run as a transaction inside another transaction.
This operation is fully logged.
A simple repro will prove that DBCC CLEANTABLE is better than SHRINKING (and no worry of fragmentation :-)
-- clean up
drop table dbo.Test
-- create test table with ntext column that we will drop later
create table dbo.Test (
col1 int
,col2 char(25)
,col3 ntext
);
-- insert 1000 rows of test data
declare @cnt int;
set @cnt = 0;
while @cnt < 1000
begin
select @cnt = @cnt + 1;
insert dbo.Test (
col1
,col2
,col3
)
values (
@cnt
,'This is a test row # ' + CAST(@cnt as varchar(10)) + 'A'
,REPLICATE('KIN', ROUND(RAND() * @cnt, 0))
);
end
--drop the ntext column
ALTER TABLE dbo.Test DROP COLUMN col3 ;
--reclaim the space from the table
-- Note that my table is only having 1000 records, so I have not used a batch size
-- YMMV .. so find a maintenance window and you an appropriate batch size
-- TEST TEST and TEST before implementing in PROD.. so you know the outcome !!
DBCC CLEANTABLE('tempdb', 'dbo.Test') ;
|
2022-01-25 19:18:35
|
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|
http://community.wolfram.com/groups/-/m/t/1300029
|
# Obtain a Plot3D in webMathematica?
GROUPS:
Hello friendsCan any body give me an example to represent a graphics 3D (using Plot3D) in webMathematica?I copied the file Plot3D included in webMathemaica documentantion (https://reference.wolfram.com/webMathematica/tutorial/BasicExamples.html )http://oed.usal.es/webMathematica/Examples/Plot3DLive.jsp , It worked some time ago (years) but now it does´t workI realize at http://library.wolfram.com/webMathematica/Graphics/Plot3D.jsp includes an example of Plot3D but a package called Cluster whose code is not shown is requiered .Guillermo
The function included in https://reference.wolfram.com/webMathematica/tutorial/BasicExamples.html for Live 3D Plotting is MSPLive3D[Plot3D[fun, {x, x0, x1}, {y, y0, y1}, PlotPoints ->pts]] I realize that using MSPShow instead of MSPLive3D the example works but the graphics can not be moved. I suppose that MSPLive3D requires some add in the browser (Flash?) and this add is not allow in the new version of the browsers. Can someone confirm it?
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2018-03-22 19:47:54
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https://orbitalindex.com/archive/2021-06-09-Issue-120/
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# The Orbital Index
Issue No. 120 | Jun 9, 2021
🚀 🌍 🛰
¶Two Discovery-class missions are headed for Venus. NASA, making good on ex-administrator Bridenstine’s recommendation to send spacecraft back to our closest neighbor, announced funding for (a somewhat unexpected) two missions to Venus—its first missions dedicated to the planet in 30+ years. VERITAS and DAVINCI+ will journey to Venus circa 2028-2030 where they will spend multiple years studying its atmosphere, mapping its surface, and increasing our understanding of how exoplanets form and develop (video). DAVINCI+ will perform two fly-bys culminating in a planetary probe descending through the thick, inhospitable atmosphere to capture data and high-resolution images on its way to the surface of the Alpha Regio highlands. DAVINCI’s carrier craft will observe the Venusian atmosphere with a four-camera array, and drop its probe during the second fly-by to make a descent to the surface where it could land intact and function for up to 17 minutes. The probe, equipped with a Mass Spectrometer, a Tunable Laser Spectrometer, a descent imager, and a suite of environmental sensors (these will provide a descent profile for use as a baseline for the mission’s atmospheric science), borrows heavily from the success of Mars Science Lab’s instruments. On its dive down to the surface, it will test the atmosphere for the presence of Phosphine. The mission will host the Compact Ultraviolet to Visible Imaging Spectrometer (CUVIS) technology demonstration which may help understand why clouds on Venus absorb an unexpectedly large amount of UV. Meanwhile, VERITAS will orbit Venus using SAR to map the surface down to 2 mm resolution using interferometry (additionally generating a height map) to detect any tectonic activity and study the geologic history of a planet that developed very differently than Earth. VERITAS will also map the planet’s surface emissivity, and has been tuned to analyze surface elements despite their average temperature of 460° C. VERITAS will carry the Deep Space Atomic Clock 2 demonstration, which will eventually help spacecraft navigate autonomously in deep space. Both missions intend to use Doppler analysis to measure the planet’s gravitational characteristics, a staple of NASA’s deep space missions. Related: NASA did not pick the missions proposed to go to Triton and Io. Like DAVINCI, the Io mission may be improved and resubmitted in the next round of Discovery-class proposals. However, a low energy transfer window to Triton will not occur again for another 13 years, so the TRIDENT mission proposal is likely on hold for the time being.
VERITAS and DAVINCI+
The Orbital Index is made possible through generous sponsorship by:
¶Gravitational Waves. Humanity’s first detection of a gravitational wave was on Sept 14th, 2015 by a pair of LIGO detectors in Washington state and Louisiana. That event, once again proving Einstein’s general theory of relativity correct, was the echo of two black holes merging and converting ~3 solar masses worth of matter into pure energy in the form of ripples in spacetime itself. “The total power output of gravitational waves during the brief collision was 50 times greater than all of the power put out by all of the stars in the universe put together.” Then, in Aug 2017, both LIGO and Virgo (in Italy) observed the ripples from a pair of neutron stars merging in time to give telescopes all over the world the chance to observe the event across the electromagnetic spectrum. The LIGO and Virgo observatories are currently undergoing upgrades for another observation run starting in June 2022, but in the meantime, they have released a catalog of the 50 gravitational wave events seen so far—here’s a tool to explore and visualize that dataset. Meanwhile, the NANOGrav project, which has spent over a decade using pulsars as a precise timing signal to try to detect incredibly small changes in the position of the Earth due to passing gravitational waves, has found early non-conclusive evidence (paper) of a very low frequency (e.g., nanohertz) Gravitational wave background. Related: Gravitational wave detectors are true wonders of engineering at the edges of quantum mechanics: “at its most sensitive state, LIGO will be able to detect a change in distance between its mirrors 1/10,000th the width of a proton! This is equivalent to measuring the distance to the nearest star (some 4.2 light-years away) to an accuracy smaller than the width of a human hair.”
From xkcd.
¶News in brief. China’s Tianzhou-2 cargo craft brought supplies and fuel to the soon-to-be-crewed Tianhe; a small piece of space debris damaged Canadarm2 on the ISS, leaving a 5 mm hole in its thermal blanket and puncturing the internal boom, but the robotic arm remains functional; Launcher raised $11 million of Series-A funding—they are targeting a first launch of their small satellite "Launcher Light" vehicle in 2024; LeoLabs raised a$65M Series B for their space situational awareness radar network; a Cargo Dragon carried experiments and a set of Redwire rollup solar panels to the ISS on a brand new Falcon 9 booster, the first new booster in 20 launches; a Falcon 9 launched a SiriusXM SXM-8 satellite; Astra announced the acquisition of electric propulsion startup Apollo Fusion using funds from its upcoming SPAC reverse-IPO; New Zealand became the eleventh country to sign onto the Artemis Accords; Cosmonauts began the process of decommissioning the Russian Pirs module early Wednesday; and, Jeff Bezos will ride to suborbital space on the first crewed flight of New Shepard.
¶Etc.What if Space Junk and Climate Change Become the Same Problem?NASA’s Mars InSight team cleverly used the lander’s robotic arm to slowly drop sand next to one of its solar panels, letting the Martian wind blow the sand across the panel and knock free some accumulated dust. This has increased the lander’s power budget by about 30 watt-hours of energy per sol.Live in Europe? Submit your ideas for lunar prospecting technologies to ESA and European Space Resources Innovation Center’s Space Resources Challenge.The Parker Solar Probe’s WISPR camera accidentally imaged a circumsolar ring of dust in the orbit of Venus (paper). The density of dust in the ring is 10% more than on either side of it. It is similar to a dust ring in Earth’s orbit, thought to be either leftover from the formation of the solar system or somehow entrained by our planet’s gravity—scientists aren’t sure yet. The images were fortuitously captured during operational rolling maneuvers used to manage the probe’s momentum. Jim Cantrell’s Phantom Space, who is working on a small and a medium-class launch vehicle, is also working on an inter-satellite communication network called Phantom Cloud and is part of a secretive “group developing a commercially funded \$1.2 billion science mission”. 👀 A well-researched article about silicon carbide (SiC) electronics, which have been developed for growing commercial applications in high power electronics. They could also be used for long-duration Venusian landers that can withstand the planet’s average surface temperature of 464 °C.If you need a refresher on the origin of the solar system and the formation and migration of the planets, our friend Jatan has a good one.The remarkably detailed ‘Why does DARPA work?’ and the equally-thorough follow-up ‘Shifting the Impossible to the Inevitable’ cover why DARPA has been so successful and how to apply those learnings to “actualize a hybrid for/nonprofit organization that leverages empowered program managers and externalized research to shepherd technology that is too researchy for a startup and too engineering-heavy for academia.”Gravity bends gravity too: we should soon start seeing gravitational waves being lensed by massive objects the same way light is. 🌊
An old favorite: Hubble’s 2004 image of the stunning Messier 104, or Sombrero galaxy, 50,000 light-years across and 30 million light-years away. Here’s the full-res version for your desktop wallpaper.
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2021-06-17 20:45:56
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https://en.wikibooks.org/wiki/A-level_Physics_(Advancing_Physics)/Boltzmann_Factor
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Particles in a gas lose and gain energy at random due to collisions with each other. On average, over a large number of particles, the proportion of particles which have at least a certain amount of energy ε is constant. This is known as the Boltzmann factor. It is a value between 0 and 1. The Boltzmann factor is given by the formula:
${\displaystyle {\frac {n}{n_{0}}}=e^{\frac {-\epsilon }{kT}}}$,
where n is the number of particles with kinetic energy above an energy level ε, n0 is the total number of particles in the gas, T is the temperature of the gas (in kelvin) and k is the Boltzmann constant (1.38 x 10-23 JK-1).
This energy could be any sort of energy that a particle can have - it could be gravitational potential energy, or kinetic energy, for example.
Derivation
In the atmosphere, particles are pulled downwards by gravity. They gain and lose gravitational potential energy (mgh) due to collisions with each other. First, let's consider a small chunk of the atmosphere. It has horizontal cross-sectional area A, height dh, molecular density (the number of molecules per. unit volume) n and all the molecules have mass m. Let the number of particles in the chunk be N.
${\displaystyle n={\frac {N}{V}}={\frac {N}{A\;dh}}}$
Because:
${\displaystyle V=A\;dh}$ (which makes sense, if you think about it)
By definition:
${\displaystyle N=nV=nA\;dh}$
The total mass Σ m is the mass of one molecule (m) multiplied by the number of molecules (N):
${\displaystyle \Sigma m=mN=mnA\;dh}$
Then work out the weight of the chunk:
${\displaystyle W=g\Sigma m=nmgA\;dh}$
The downwards pressure P is force per. unit area, so:
${\displaystyle P={\frac {W}{A}}={\frac {nmgA\;dh}{A}}=nmg\;dh}$
We know that, as we go up in the atmosphere, the pressure decreases. So, across our little chunk there is a difference in pressure dP given by:
${\displaystyle dP=-nmg\;dh}$ (1) In other words, the pressure is decreasing (-) and it is the result of the weight of this little chunk of atmosphere.
We also know that:
${\displaystyle PV=NkT}$
So:
${\displaystyle P={\frac {NkT}{V}}}$
But:
${\displaystyle n={\frac {N}{V}}}$
So, by substitution:
${\displaystyle P=nkT}$
So, for our little chunk:
${\displaystyle dP=kT\;dn}$ (2)
If we equate (1) and (2):
${\displaystyle dP=-nmg\;dh=kT\;dn}$
Rearrange to get:
${\displaystyle {\frac {dn}{dh}}={\frac {-nmg}{kT}}}$
${\displaystyle {\frac {dh}{dn}}={\frac {-kT}{nmg}}}$
Integrate between the limits n0 and n:
${\displaystyle h={\frac {-kT}{mg}}\int _{n_{0}}^{n}{\frac {1}{n}}\;dn={\frac {-kT}{mg}}\left[\ln {n}\right]_{n_{0}}^{n}={\frac {-kT}{mg}}\left(\ln {n}-\ln {n_{0}}\right)={\frac {-kT}{mg}}\ln {\frac {n}{n_{0}}}}$
${\displaystyle ln{\frac {n}{n_{0}}}={\frac {-mgh}{kT}}}$
${\displaystyle {\frac {n}{n_{0}}}=e^{\frac {-mgh}{kT}}}$
Since we are dealing with gravitational potential energy, ε = mgh, so:
${\displaystyle {\frac {n}{n_{0}}}=e^{\frac {-\epsilon }{kT}}}$
A Graph of this Function
This topic comes up in Q10 494 June 2010. The Values used for various things in that question are
${\displaystyle k=1.4\times 10^{-23}JK^{-1},~g=9.8,~m=4.9\times 10^{-26}Kg,~T=290K}$
Shows how Energies are achieved with Height
Questions
1u = 1.66 x 10-27 kg
g = 9.81 ms-2
1. A nitrogen molecule has a molecular mass of 28u. If the Earth's atmosphere is 100% nitrous, with a temperature of 18°C, what proportion of nitrogen molecules reach a height of 2km?
2. What proportion of the molecules in a box of hydrogen (molecular mass 2u) at 0°C have a velocity greater than 5ms-1?
3. What is the temperature of the hydrogen if half of the hydrogen is moving at at least 10ms-1?
4. Some ionised hydrogen (charge -1.6 x 10-19 C)is placed in a uniform electric field. The potential difference between the two plates is 20V, and they are 1m apart. What proportion of the molecules are at least 0.5m from the positive plate (ignoring gravity) at 350°K?
Worked Solutions
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2016-07-30 07:21:42
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https://brilliant.org/discussions/thread/undefined-to-some-power/
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×
# Undefined to some power
If we raise some undefined quantity to some power, what type of quantity(number) is produced? Also, if we raise an undefined number to the power of some other(maybe same) undefined number,what becomes?
Note by Anandmay Patel
7 months, 4 weeks ago
Sort by:
What do you mean by undefined quantity? · 7 months ago
like $$\dfrac10$$=infinity or iota · 7 months ago
Iota is defined · 7 months ago
ok,,,,,so take the infinity · 7 months ago
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2017-05-26 00:15:32
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https://proofwiki.org/wiki/Combination_Theorem_for_Complex_Derivatives/Sum_Rule
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# Combination Theorem for Complex Derivatives/Sum Rule
## Theorem
Let $D$ be an open subset of the set of complex numbers $\C$.
Let $f, g: D \to \C$ be complex-differentiable functions on $D$
Then $f + g$ is complex-differentiable in $D$, and its derivative $\left({f + g}\right)'$ is defined by:
$\left({f + g}\right)' \left({z}\right) = f' \left({z}\right) + g' \left({z}\right)$
for all $z \in D$.
## Proof
Denote the open ball of $0$ with radius $r \in \R_{>0}$ as $B_r \left({0}\right)$.
Let $z \in D$.
By the Alternative Differentiability Condition, it follows that there exists $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$:
$f\left({z + h}\right) = f \left({z}\right) + h \left({f' \left({z}\right) + \epsilon_f \left({h}\right) }\right)$
$g\left({z + h}\right) = g \left({z}\right) + h \left({g' \left({z}\right) + \epsilon_g \left({h}\right) }\right)$
where $\epsilon_f, \epsilon_g: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \C$ are continuous functions that converge to $0$ as $h$ tends to $0$.
Then:
$\displaystyle \left({f + g}\right) \left({z + h}\right)$ $=$ $\displaystyle f \left({z}\right) + h \left({f' \left({z}\right) + \epsilon_f \left({h}\right) }\right) + g \left({z}\right) + h \left({g' \left({z}\right) + \epsilon_g \left({h}\right) }\right)$ $\displaystyle$ $=$ $\displaystyle \left({f + g}\right) \left({z}\right) + h \left({f' \left({z}\right) + g' \left({z}\right) + \left({ \epsilon_f + \epsilon_g }\right) \left({h}\right) }\right)$
From Sum Rule for Continuous Functions, it follows that $\epsilon_f + \epsilon_g$ is a continuous function.
From Sum Rule for Limits of Functions, it follows that $\displaystyle \lim_{h \to 0} \left({ \epsilon_f + \epsilon_g }\right) \left({h}\right) = 0$.
Then the Alternative Differentiability Condition shows that:
$\left({f + g}\right)' \left({z}\right) = f' \left({z}\right) + g' \left({z}\right)$
$\blacksquare$
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2019-11-22 11:09:14
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https://blog.theleapjournal.org/2017/04/
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## Thursday, April 27, 2017
### Building blocks of Jio's predatory pricing analysis
In a recent post on predatory pricing and the telecom sector Ajay Shah questions whether the subsidised user base of Reliance Jio can set off a network effect. The post makes two claims. The explicit claim is that the combined effect of interconnection regulation; mobile number portability and open standards of TCP/IP ensures that there are no real network effects in the telecom sector. The underlying implicit claim is that the existence of network effects is central to a predatory pricing analysis in this context. This piece takes a closer look at both these claims and the other factors that should inform the Competition Commission of India (CCI)'s analysis in the complaint filed by Airtel against Jio's pricing practices.
### Network effects in telecom
Modern day tariff plans, including that of Jio, comprise of three main components - voice, data and access to content - all bundled into one product. A competition law analysis of Jio's pricing strategy must focus on each of these segments individually, and then their collective effect.
Voice services: Telecommunication services are known to generate strong network effects - the value of having a phone number is linked to the number of people who can be called using it. This creates a classic case for concentration of market power in the hands of the incumbent. Telecom regulators have overcome this issue by mandating operators to link their networks with the networks of other operators, allowing users to communicate across networks. Research on telecom networks, however, finds that despite interoperability, users tend to display a preference for being on a larger network, particularly when operators offer lower tariffs for calls made within their networks (on-net/off-net price differentiation). Others suggest that the network effects in telecom are more 'local' in nature - the preference to be on the same network as one's family and friends leads to the formation of calling clubs. This is not necessarily dependent on the overall size of the network.
In summary, even with mandated interconnection norms, traditional telecom services display a certain level of network effects. Arguably, the relevance of being "on the same network" would have gone down with the convergence of voice and data services and availability of various over-the-top calling apps. This requires a deeper study of consumer behaviour and preferences in the post-data world.
Internet services: Network effects on the Internet are not about the provision of Internet access services (i.e. the data services offered by ISPs) but rather about the direct and indirect network effects that define the business models of many Internet-based platforms and businesses.
Telecom service providers are increasingly stepping into the role of Internet platforms by bundling access to online music, TV, movies and news along with their communication services. In Jio's case, every new SIM comes bundled with a bouquet of Jio-branded apps, making it one of the fastest growing content aggregators in the country. Its free offer period from September to March has helped Jio build a massive user base, which in turn helps in attracting other complementary users to its platform. For instance, Uber's recent decision to partner with Jio Money reflects the value that it sees in being able to access Jio's users. The same holds true for other merchants and suppliers, like providers of music, video and news content, who are attracted to platforms with a large number of users.
### Integration of data services and content
The vertical integration of data services and content offers Jio many advantages. One, convenient access to free content along with free/discounted data services has helped Jio in promoting higher consumption patterns. The aggressive data usage on Jio's network, particularly of video content, will gradually translate into higher revenues. The company claims that its users "consume nearly as much mobile data as the entire United States of America...and nearly 50% more mobile data than all of China." It would be interesting to see what percentage of this data is being consumed within the Jio ecosystem and the change in consumption patterns after Jio starting charging for its data services.
Two, it promotes faster adoption of in-house services. To take an example, the AT&T/FaceTime case study in the United States found that less than 10 percent of iPhone users downloaded Skype while all of them had automatic access to Apple's FaceTime. Adoption of Jio Money versus rival payment apps (among Jio subscribers) is likely to show similar results. Reports about the launch of Jio's 4G feature phone with built-in Jio apps suggest the possibility of further entrenchment of new users in the Jio universe.
Can Jio's pricing strategy in telecom enable it to indulge in monopolistic behaviour in related markets like mobile payments? Unlike telecom services, the payments sector continues to suffer from the lack of interoperability among providers, leading to significant network effects. Safaricom's M-Pesa service in Kenya offers an example of how the company was able to leverage massive network effects in the mobile-money space to establish its dominance in calls and text messages. The situation in India is certainly different - we have higher levels of competition, both in telecom as well as online payments. Yet, the Kenyan example is a helpful reminder of the extent to which cross-linkages between bundled products can influence their adoption and usage, to the exclusion of other competitors.
Jio's dual role as a telecom provider and platform offering access to online content makes it difficult to outright dismiss the role of any network effects. Moreover, any subsequent recoupment of the losses suffered by Jio in its early days need not necessarily be through a significant markup in data tariffs. Increase in volume of data consumption, future monetisation of Jio apps and opportunities for utilisation of data collected from users, are all factors that must be considered.
### The tests of predatory pricing
The law and jurisprudence on predatory pricing defines it as below cost pricing by a dominant firm, with a view to exclude competitors. Sustained discounting practices in a market with strong network effects certainly raises a red flag due to the tendency of a single network to dominate the market. In such a scenario, there is a strong likelihood of recoupment after other competitors have left the market and structural barriers deter the entry of new players. The determination of predatory pricing, however, does not hinge on the existence of these network effects.
When Jio first launched its services in September, 2016 it was a fresh entrant in a market with several established players. Its price point of zero was certainly below cost but there was no question of it being a "dominant player". Any regulatory intervention to stop the pricing plans at that stage, whether by the sectoral regulator TRAI or the CCI, would have been premature.
This position has come to change over the last few months. Jio has managed to acquire a sizable presence in the market for high-speed data services - it holds about one-third of the country's broadband subscriber base and about 85 percent of the market in terms of mobile data traffic. Its share in the overall market for telecom services (voice plus data) still remains small since telecom subscribers continue to outnumber Internet users by a wide margin. The manner in which CCI delienates the "relevant market" will therefore form the crux of its analysis in this case.
Accordingly, the first step for CCI would be to determine whether there is a market for data services that is distinct from the broader cellular services market? This will hinge on a factual analysis of whether users regard voice, data and high-speed data services as being interchangeable in terms of their end-use and characteristics, based on a number of factors. One, voice calls can be made using the Internet but the reverse is not true - this indicates a one-way substitutability between the services. Two, there are some differences in the utility of 2G and 4G networks based on the applications that they are able to support. Three, CCI will need to collect data on Jio's usage patterns, that of its competitors and the switching behaviour of consumers. Four, supply-side constraints (like spectrum holdings) that can make it difficult for providers to switch from one type of service to another will also need to be considered.
In the second stage, CCI will need to examine whether Jio can be regarded as being dominant player in the identified market. Besides looking at its market share, in terms of subscriber base and usage volumes, this analysis must also consider the various other factors that have been given under the Competition Act, 2002. These include:
1. Size and resources of Jio and its competitors - Telecom being a capital-intensive industry has many big players. Each of them has access to significant capital resources, although there may be differences in the extent to which these firms have been leveraged.
2. Vertical integration of the enterprise - As discussed above, the bundling of voice, data and content offers Jio certain clear advantages. Other players are also offering similar bundles but not necessarily at the same scale. In many cases, the prices and bundles offered by other players have come about as a response to Jio's entry strategy.
3. Entry barriers - Telecom is a heavily regulated sector and there are entry barriers, both in terms of licensing requirements and the availability and price of spectrum.
4. Relative advantage through contribution to economic development - The arrival of Jio's 4G LTE network, with its aggressive pricing strategy, could also have some pro-competitive effects. Arguably, it has nudged the telecom market towards greater price competition, resulting in lower tariffs. Over time, this could also push other operators towards faster upgradation of technology.
Assuming that CCI's analysis leads it to delineate a separate broadband market (and Jio is found to be dominant in it), the third challenge would be to assess whether its current prices are in fact "below cost". This again will require data on the costs incurred by Jio for delivering its voice and data services and the free apps that are on offer. Finally, CCI will have to determine whether Jio's current pricing continues to be in the nature of a genuine "promotional strategy" by a new entrant or is it a deliberate attempt to reduce competition in the market.
Many have linked the consolidation that we are seeing in the market today with Jio's entry strategy. One one hand, consolidation reduces the number of players, hence reducing competition. On the other, it might be a sign of the sector's movement towards a more mature market with fewer players who are able to focus better on infrastructure expansion and quality of services. CCI will need to weigh in all these factors while examining the impact of Jio's prices on consumer interests, competition in the market and overall economic development.
These are all complex questions, with no obvious answers. The solution lies in a multi-stage, data-driven analysis of predation that should be rooted in an understanding of competition policy and telecom economics. Co-operation and knowledge-sharing between CCI and TRAI is key to finding these solutions.
Smriti Parsheera is a researcher at the National Institute of Public Finance & Policy. The author would like to thank Amba Kak, Kaushik Krishnan and Faiza Rahman for useful discussions.
## Tuesday, April 25, 2017
The Modi government has a new model for managing Kashmir by Sushil Aaron in Hindustan Times, April 19, 2017.
How can ARCs help solve the banking crisis? by Ajay Shah in Business Standard, April 17, 2017.
Budgeting for the police by Renuka Sane and Neha Sinha in Mint, April 11, 2017.
Political reforms are direly needed to renew our tryst with democracy by Varun Gandhi in The Economic Times, April 10, 2017.
It's a tug-of-war out there by Somasekhar Sundaresan on Wordpress, April 9, 2017.
The cow as cause - Vigilantism and the BJP by Mukul Kesavan in The Telegraph, April 9, 2017.
A Mega-merger and a Bureaucrat's Transfer by Paranjoy Guha Thakurta in Economic and Political Weekly, April 8, 2017.
The Money Bill conundrum: Constitution bench should decide by T K Arun in The Economic Times, April 6, 2017.
NCLT needs 69 benches to handle current case load: report by Shreeja Sen in Mint, April 6, 2017.
Banks aware of Bankruptcy code, some initiated corporate insolvency resolution: Madhusudan Sahoo by Joel Rebello & Saikat Das in The Economics Times, April 5, 2017.
Shri Jairam Ramesh's Speech-Central GST, Integrated GST, UT-GST & GST Compensation bills in Rajya Sabha TV, April 5, 2017.
A How to Book for Wielding Civic Power by David Bornstein in The New York Times, April 5, 2017.
LOKNITI-CSDS-KAS survey: Mind of the youth in Indian Express, April 3, 2017.
On Tyranny: Twenty Lessons from the Twentieth Century by Timothy Snyder by Tim Duggan Books, February 28, 2017.
The Fighter by C. J. Chivers in The New York Times, December 28, 2016.
## Thursday, April 13, 2017
### Retreat from private infrastructure projects
by Ajay Shah.
#### The case for private participation in infrastructure
Many years ago, most infrastructure in India was government owned. Policy thinkers strenuously argued for greater private participation, for the following reasons:
• Private ownership would give better hardware, as a private person cares about what is being built. It would also give better safekeeping of the assets, as a private person cares about his things.
• Private owners would strenuously push for adequate user charges, and act as a counterweight against the biases of the Indian political system in favour of low user charges and thus a burden upon the exchequer.
• If a project is unviable, the private sector will more clearly say so and walk away, in contrast with the government processes which will build infrastructure in respond to political pressures.
• Indian public finance would be better off when its balance sheet is freed from infrastructure assets, and these are instead held by listed utilities who issue debt and equity. It would become possible to bring the vast global capital to bear on these markets and deliver low cost financing.
For some years, private participation in infrastructure grew well, but things have changed sharply. Here are three key pictures. At each point in the time-series, we sum up the value of infrastructure projects in the CMIE Capex database that are classified as being under implementation' by CMIE. The time series of the stock of under implementation' projects changes from $t$ to $t+1$ because some old projects are commissioned, some are abandoned, and some new projects appear into the list.
Private infrastructure projects that are Under implementation in the CMIE Capex database
As the graph above shows, we got a huge increase from 2003 to 2011: a gain of roughly 10x in nominal rupees. By 2011, there was a stock of roughly Rs.25 trillion rupees of private infrastructure investment projects that were under implementation.
After that, private infrastructure projects have receded substantially. We have a decline of Rs.5 trillion in nominal terms. If inflation were taken into account, that is a decline of another 25%.
How has government infrastructure investment activity fared?
Government infrastructure projects that are Under implementation in the CMIE Capex database
This shows a picture of steady growth. In 2011, both private and government projects were at roughly Rs.25 trillion. From there, the private projects have dropped to Rs.20 trillion while the government has gone on to Rs.38 trillion. There is growth in the stock of government infrastructure investment projects under implementation, even after you take out the 25% increase in prices from 2011 till today.
It is quite a reversal for the long-held objective of having greater private participation in infrastructure.
What's the overall picture of infrastructure investment?
Total infrastructure projects that are Under implementation in the CMIE Capex database
Putting the two together, there is broad stability from 2013 onwards (but a decline in real terms once you take out inflation). What has not been widely observed is the compositional change within this overall number: the private sector is losing ground and government infrastructure projects are gaining ground.
#### Implications
In my view, the original logic in favour of greater private participation in infrastructure remains. The private sector will use capital more effectively, deliver a better incremental capital-output ratio, and take care of assets better. Conversely, public sector domination of infrastructure investment is going to deliver reduced bang for the buck. The compositional shift in favour of public infrastructure projects is a weakness.
#### Where did we go wrong?
In the first wave of pushing private sector participation, we did not adequately understand that private participation in infrastructure requires complex institutional machinery. The government's role in infrastructure is in three parts: Planning, Contracting and Regulating. Clear structures needed to be established for each of these three pillars. Mechanisms were required for resolving disputes and protecting cashflows from user charges. We needed to keep our eye on the prize: the projects that come out of all the complexities of the early stage and make it into the listed space, as boring utilities who just collect user charges and do O&M. With the benefit of hindsight, we went about private sector investment in infrastructure in a slipshod manner.
In the recent period, instead of fixing these institutional complexities, there has been an excessive willingness to give up on private sector participation and make do with muscular State-led investment. It feels like an entire generation of institutional memory, about the problems of public sector infrastructure investment, has been lost. We are now running a Chinese-style risk of large investments going in with low returns in terms of incremental GDP per unit investment.
## Wednesday, April 12, 2017
### Emerging themes around privacy and data protection
by Vrinda Bhandari, Amba Kak, Smriti Parsheera and Renuka Sane.
Issues of data protection and privacy have become the subject of intense discussion and debate, in India as in the rest of the world. In this post, we identify certain key themes that arise in the context of these issues, that can augment our understanding of privacy and data protection and help towards forging safeguards in the form of a privacy law. Many of these were discussed recently at a round table organised at NIPFP on 24th March 2017. The key themes that emerged are summarised below.
### What do we understand by privacy?
The term privacy has many connotations, takes different forms in different contexts and is viewed differently depending on the individuals own subjectivity. Defining it has been a challenge, with many scholars leaning towards more conceptual, and less rigid formulations. In philosophical debates, privacy can be characterised in terms of defining a sphere of private life that is separate from political activity and government interference. The sociological argument traces its roots in the fundamental haracteristics of social life - social context determines what is considered private in different circumstances. Others, like Solove 2006), however, move away from these conceptual discussions to identify specific privacy harms that have been recognised by society. His taxonomy of privacy encompasses four aspects - first, information collection (through surveillance and interrogation); second, information processing (through aggregation, identification etc.); third, information dissemination (through disclosure, exposure, breach of confidentiality etc.); and fourth, invasion (through intrusion and decisional interference).
Taking a slightly broader view, Calo (2011) speaks about privacy through the boundaries of subjective and objective harms. A subjective harm is internal to the person harmed, and is caused by unwanted observation. This encompasses, for instance, the knowledge or perception that some negative information about oneself is out there, which leads to distress and anxiety. Conversely, objective harm is external to the person harmed, when coerced or unanticipated information about oneself is used by other persons. Understanding of the potential harms is extremely important for the design of a policy response.
Another debate that emerges is whether privacy should be viewed as a right, an interest, or a property? Interestingly, the early parameters of what is now regarded as privacy evolved in the context of property rights. In 1890 Warren and Brandeis argued in a seminal paper that the right to privacy goes much beyond the concept of personal property rights, and must be recognised as such (to include for instance, the principle of an inviolate personality). By now most countries view privacy through a rights lens, because property, by its very nature, once bought, can be destroyed, transacted, and shared without the consent of the original owner. The economic dimensions of private data in the digital age have, however, once again triggered these rights versus property debates focused around the concept of "propertarian privacy".
Discussions on privacy also raise the question of privacy from whom. Traditionally, privacy was viewed in the context of the surveillance and law enforcement powers of the State. However, with the rise in big data and the explosion of social media, we now have to think of privacy from private actors as well, whether in the context of data mining, data retention, or data sharing arrangements. Surveillance, in this context, includes what Roger Clarke terms dataveillance - systematic monitoring of actions or communications using information technology.
### Do people in India really value privacy?
While a lot has been written about the value of privacy (for example, Westin (1968)), it is often argued that people do not really know how to gauge the value of their own privacy. Many view the debates on privacy protection as the privilege of the elite who do not have to worry about accessing basic services, or as refuge for those who have "something to hide".
It is, however, important to remember that privacy is context specific. It is not always about what one may have to hide, but also what one may have to lose. These considerations vary across class, gender, caste, age and are often be different for different intersections of these categories. For each person, there are aspects of their life that are "personal", that they do not wish to be revealed to the public at large- and the control over which is integral to their sense of autonomy. In the digital context, the oft-heard lament is that privacy does not seem to be valued enough perhaps because people either don't know or feel ambivalent about how much data they are sharing (unwittingly), to which entities and the picture of themselves that their data is able to generate to these entities.
For awareness to be effective it must move from the risks to the harm. Sunil Abraham offers a useful analogy of tobacco use. Most smokers are well aware of the risks of smoking, but do not bother to stop, until they face a health crisis. Similarly, most people, while well aware of the privacy risks associated with their activities, for instance careless use of social media, do not take any remedial action until and unless they face a data breach. Therefore, just as health policy workers have tried to change the attitudes of smokers by scaring them through the inclusion of graphic images on the cigarette packs, it might be useful to alert people to the harms caused by the loss of privacy.
### "Privacy by design" holds important lessons
The principles of Privacy by Design (PBD) developed by Ann Cavoukian are worth emphasising. The approach highlights that measures to protect privacy should be proactive and preventive, and not remedial. Privacy should be the default setting, embedded into design of technologies and services.
This overcomes many of the problems associated with choice/consent based regimes although adoption still depends on voluntary buy-in from businesses and users. So far, businesses in India are said to find an unwillingness among users to pay for privacy. For this reason, most privacy-enhancing technologies (PET) based solutions are B2B rather than B2C, and even these are far and few. We, in India, need to think of innovative ways to bring about a regime of data protection. A law on the subject and privacy-enhancing design elements are both part of the solution.
### Issues of surveillance
Perhaps the most contentious of all issues is the one on where to draw the line between privacy and security, which often requires the use of various surveillance tools by the state. The PBD framework calls for "full functionality" in this context, i.e. it seeks to accommodate all legitimate interests in a positive-sum manner. Instead of a dated zero-sum approach with unnecessary trade offs of privacy vs. security, PBD says that it is possible, and far more desirable, to have both.
Yet, in reality there remains no consensus on a) the extent to which the state is engaging in surveillance, b) the extent to which Aadhaar and other big data techniques are being deployed, and c) the relationship between national security and privacy (is balance the appropriate metaphor? what is the trade-off, if any). The State claims that surveillance fears are misguided and overstated, while civil society argues that surveillance is broad based, and inadequate checks and balances leave citizens vulnerable. Given that both national security and privacy remain nebulous terms, there is no clarity on when one gives way to the other, and it is undeniably the rhetoric of national security that invariably overwhelms privacy. This issue requires unpacking and principles-based resolution as unchecked intrusions by the State can damage the very essence of what it means to be a liberal democracy.
Given the pervasiveness of Aadhaar in our lives today, a debate on data protection cannot be complete without evaluating the legal framework surrounding it. The current legal framework of Aadhaar is weak. The Act delegates a number of core functions to be specified by the regulations, and these regulations further defer these functions as matters 'to be specified' by the UIDAI in some undefined future. This suggests that Aadhaar is currently functioning in some sort of a legal vacuum in terms of the nuts and bolts of important issues such as enrollment, storage, and sharing of data.
The regulations that have been issued by UIDAI did not go though a rigorous consultative process - both while preparing the draft, and in seeking comments from the public. The UIDAI should voluntarily opt for greater transparency on issues that have implications for privacy and data protection.
### There is a case for a horizontal law
In India, the Supreme Court is yet to decide, what was until recently regarded a settled position - whether the right to privacy constitutes a fundamental right under Part III of our Constitution. While this is being debated, we have sector specific frameworks, like Section 43A of the IT Act, for protection of personal information and data security. More recently, the Ministry of Electronics and Information Technology (MeitY) has released the draft Information Technology (Security of Prepaid Payment Instruments) Rules 2017 for public comments. The draft rules aim to ensure the integrity, security and confidentiality of electronic payments through prepaid instruments, although amid concerns over the scope of the draft rules, MeitY's jurisdiction, and overlaps and conflicts with existing laws. Several other regulators such as the RBI, telecom authorities and health departments also have, or are in the process of developing, privacy/data protection norms pertaining to their jurisdictions.
These are all notable moves, but in the absence of a horizontal law, they will lead to the development of certain pockets of protection in certain sectors, while many other facets of private data will remain unprotected. Another concern is that the current legal framework does not hold meta data to the same standards as data in privacy and data protection debates.
There is a case for a comprehensive, principles-based, horizontal privacy law with basic minimum standards of privacy. These standards can then be tuned further to meet the requirements of different sectors. Thus, regardless of whether the Supreme Court of India considers privacy as a fundamental right, the State must define the circumstances in which it, as well as other private sector entities, may intervene with an individual's rights. Work on the draft privacy bill which began a few years back needs to be pursued with haste.
Vrinda Bhandari is a practicing advocate in Delhi. Amba Kak, Smriti Parsheera and Renuka Sane are researchers at the National Institute of Public Finance & Policy. We thank all participants at the round table on privacy and data protection organised by NIPFP on 24th March, 2017 for their contributions. Any omissions are our own.
## Tuesday, April 11, 2017
by Devendra Damle, Shefali Malhotra and Shubho Roy.
Turning a bill (legislative proposal) into an Act of Parliament (law of the land) is a multi-step process. It involves placing the bill before the legislature; readings of the bill; publication in the official gazette; possible reference to relevant committees of the legislature; debates; and concludes with present members voting on the bill. Voting can be done in multiple ways, one of which is a voice vote. In the voice vote system, the members verbally communicate their assent by shouting 'Aye', or dissent by shouting 'No'. Based on which answer is most audible, the Speaker (person chairing the legislature) decides the outcome of the Bill. The system of voice votes, is obsolete. It slows down legislatures, grants excessive discretion to the Speaker, reduces the ability of citizens to hold their legislators accountable. It is still used in India.
For example, recently, the Punjab Assembly passed the Vote-on-Account of more than INR 251,990 million for the first quarter of the 2017-18 fiscal, by a voice vote. In the winter session of Parliament, the Taxation Laws (2nd Amendment) Act 2016 was passed through a voice vote, amidst protests and demonstrations. In 2014, opposition parties in Maharashtra questioned the legitimacy of the government after a confidence motion was decided in the government's favour through a voice vote. In the same year, the Lok Sabha Speaker was criticised for passing the Telangana Bill through a voice vote (Note: In this blog, we use the term 'Speaker' generically, for the Speaker of the Lok Sabha, the Chairman of the Rajya Sabha, the Speakers of state legislative assemblies, and Chairman of state legislative councils).
The Constitution leaves it to each house of legislature to set the rules of functioning of that house. Articles 118 and 208 of the Constitution empowers the Houses of Parliament and state legislatures to make rules governing procedure, respectively. Lok Sabha and Rajya Sabha have made Rules of Procedure and Conduct of Business (Procedure Rules) for their respective houses.Voting in the Lok Sabha is governed by rules 367, 367A, 367AA and 367B of the Lok Sabha Procedure Rules.
On the conclusion of a debate, the Speaker asks the members present whether a bill or a motion is passed. The member respond through a voice vote, the Speaker decides whether the motion is accepted or rejected. If a member challenges the Speaker's decision, the Speaker repeats the voice vote process for a second time. Any member can challenge the second voice vote by requesting for a division. The Speaker has the discretion to reject or grant the request for a division. If the Speaker rejects the demand for division, parliament employees take a head-count of members voting 'Aye' and members voting 'No'. Based on this head-count, the Speaker announces whether the motion is accepted or rejected. This decision cannot be challenged.
If the Speaker accepts the demand for division, he orders for voting by any one of the following three methods, at his discretion:
1. Automatic Vote Recorders: Members press a button to vote 'Aye' or 'No' from their allocated seats. The result appears on an electronic display, and the Speaker announces whether a motion is accepted or rejected.
2. Paper Slips: Members write 'Aye' or 'No' on paper voting-slips. Parliament officers collect the slips and count the votes. The Speaker announces whether a motion is accepted or rejected.
3. Division Lobbies: The Speaker directs members voting 'Aye' to go to the right lobby, and those voting 'No' to go to the left lobby. Parliament officers count members in each lobby. The Speaker then announces whether a motion is accepted or rejected.
The Rajya Sabha and state legislatures follow a similar process with minor variations. In no case are individual voting records maintained. Even when a division is carried out, only the total number of votes for and against the motion are recorded.
The system of voice votes suffers from two weaknesses: It grants excessive discretion to the Speaker, and it reduces accountability of legislators to the citizens.
### Speakers' excessive discretion
The discretionary power vested with the Speakers in a voice vote system is prone to abuse. In the Indian system, Speakers are inclined to side with the ruling party or alliance. Speakers of Lok Sabha and State Legislative Assemblies are elected from among the members of their respective legislatures, usually from the ruling party or alliance. Unlike in the UK, they continue to be members of their parties even after being elected as the Speaker.
The Maharashtra Assembly no-confidence vote in 2014 is an example of alleged abuse of discretionary power by the speaker. The Maharashtra Assembly speaker approved the confidence motion in favour of the present ruling party. The motion was approved through a voice vote, amongst allegations that the demand for division by the opposition was ignored. Effectively, it cast doubt on whether the government truly had a majority in the house.
According to the Parliament's Statistical Handbook 2014, five incidents of no-confidence motions and three incidents of confidence motions have been decided through a voice vote in Lok Sabha. While there has been no allegation of abuse in these cases, a voice-vote system can be easily manipulated, especially when it is used to determine crucial issues like the legitimacy of the ruling party.
### Accountability to citizens
The Voice vote system lacks transparency. In a voice vote system, it is impossible to record individual votes of legislators. In the absence of individual voting records, a citizen has no way of judging the actions of his representatives. He is clueless about which way his elected representative voted, or whether that representative voted at all. This makes it difficult for the public to hold their representatives responsible.
An example of accountability using public voting records is Obama's criticism of Hillary Clinton, in the 2008 Democratic Party Primaries. In 2002, the then Senator Hillary Clinton voted in favour of the resolution to invade Iraq. By the 2008 primaries, public opinion (especially in the Democratic Party) had turned against the invasion. Obama repeatedly pointed out that Clinton voted in favour of the Iraq war, signalling to the Democratic Party that he is a better candidate. Obama could do this because each Senator's vote on each resolution is recorded against their name and published.
### Solution: electronic voting
In the past, recording votes for each motion would have been time-consuming and costly. So, recorded votes would be reserved only for contentious issues. If the support or dissent for a motion was evident, it was left to a quick decision of the speaker. The cost of this efficiency was wide discretion to the speaker. Today, with electronic systems, we can gain this efficiency with no costs.
### Efficiency gains
As the following examples show, electronic voting is a time-tested, efficient and, cheap technology:
• Time-tested: Machine vote recording systems are not new; Thomas Edison patented a system in 1869. The World e-Parliament Report 2016 states that 67% of parliaments have adopted some form of electronic voting. Out of the remainder that still vote manually, 72% are considering to introduce electronic systems. The US House of Representatives has been using electronic voting systems since 1973. Recently, the Korean National Assembly adopted an electronic voting system.
• Efficient: A 2016 Australian Parliamentary report found that adopting electronic voting systems reduces the time spent on counting votes, minimises human error, and expedites publication of results. A 2010 report to the UK House of Commons found that electronic voting can make the process less time-consuming. In turn, allowing MPs to devote more time to discussion and debate, the real function of legislatures. A 2003 Australian Parliamentary report finds that conducting a division vote in the Mexican Legislature used to take upto one hour; with the electronic voting system it now takes two minutes.
• Cheap: The Mexican Legislature, with more than 500 members, has been using biometric authenticated electronic voting since 1998. The 2003 Australian Parliamentary report finds that the Mexican Legislature's electronic voting system has an operating cost equivalent to INR 54 million per year (at 2016 prices). For perspective, that is 0.86% of the total budget of the Lok Sabha for 2016-17.
Electronic voting systems are not alien to the Indian Parliament (the one in Rajya Sabha was installed in 1957). However, they are only used in case of a division. This means going through the process of two voice votes, calling for division, granting division, and then conducting a division. Even then, the decision to use electronic voting is subject to the Speaker's discretion. He can choose other inefficient methods like paper slips or the lobby method to reach a conclusion.
### Ushering in transparency
The first step towards recording individual legislators' votes is by replacing voice votes with electronic voting systems. Carey, 2005, finds that when countries adopt electronic voting systems, demand for recording individual votes grows. Once the usage of recorded vote starts, pressure to make these records visible increases. An EU Parliament study of EU countries, where individual votes are recorded, finds electronic voting to be the most popular method.
The Indian electorate has been criticised for voting on caste/communal lines. However, in the absence of information regarding legislators' actions in the legislature, there is no other parameter for the average citizen to decide who to vote for. Bovitz and Carson, 2006, conducted a study examining the electoral consequences of individual voting records of legislators in the US House of Representatives. They found that legislators who vote against their constituents' preferences on controversial and politically prominent issues get lower vote shares in subsequent elections. Conversely, when legislators vote according to their constituents' preferences, especially against the party-line, they get higher vote shares. Legislators tend to vote strategically on prominent issues as they worry about taking the 'wrong' position in the eyes of their constituents.
Unlike the US, Indian legislators are subject to anti-defection laws. An Indian legislator cannot ignore a 'party whip/instruction' without risking losing his seat. It may be argued that, in the Indian scenario, individual voting records are useless. This argument has two weaknesses, namely that it:
1. Contradicts the general principle of governance, that greater transparency in the working of government brings greater efficiency. It does so without providing any evidence for it.
2. Ignores that once votes are made public the equilibrium shifts and individual voting records may act as a counter-balance to the negative aspects of anti-defection law.
We should always strive for greater transparency in governance. India still follows the Westminster system with people voting for individual legislators to represent them. On one hand, even if this information is useless, it does not harm anyone. On the other hand, when citizens get more information about their legislators, they can make more informed decisions. For example, merely because some candidates with criminal backgrounds are elected, does not mean we should stop requiring candidates to declare their criminal records. In addition, there are situations where anti-defection laws do not apply. In such cases this information can help voters. Anti-defection laws do not affect legislators who do-not vote or when there is no formal party whip. In such cases, individual voting records will provide valuable information about a legislator's behaviour. Did your legislator actually vote on a bill that was important to you or was he absent? It forces legislators to at least participate in issues important to their electorate and turn up to vote. Carey, 2009 examines the Corruption Perceptions Index, calculated by Transparency International, for most countries in the world. He finds that perceptions of corruption tend to be lower in countries where legislative votes are visible.
It may act as a counterbalance to anti-defection. Anti-defection laws have been criticised for reducing the voice of legislators. It puts party interests above the interests of the electorate. However, today the costs on the legislators following the 'party whip' and consequently harming his electorate is nil. Individual voting records may act as a counterbalance to this problem. Just like anti-defection pressurises legislators to vote in favour of the 'party whip', some studies show that individual voting records pressurise legislators to vote in accordance with the wishes of the electorate. Canes-Wrone et al., 2002 examined, through a study of the US elections between 1956 and 1996, the relationship between legislators' electoral performance and support for their party inside the legislature. Their study shows that in each election, an incumbent received a lower vote share when he supported his party. It also decreases the probability of retaining office. Crespin, 2010 finds that where votes are more likely to be noticed by the public, members of the US Congress adjust their votes in line with the demands of their constituency.
It is simplistic to argue that bringing transparency in individual voting records will not change the incentives/behaviour of legislators. Once individual voting records are available, the legislator will have two choices. First, vote against the decision of the party and get disqualified but, on the other hand, gain sympathy of the electorate. This may translate into more votes in the next election/by-election. The legislator can run as an independent candidate or on another party ticket and gain sympathy votes. Second, vote in accordance with the party line and hold on to his seat. However, now the entire electorate is likely to know he voted against their interest. The next election may not be in his favour.
We need to overhaul the functioning of the Parliament. Adopting compulsory electronic voting in our legislative bodies is a low-hanging fruit. It requires a small change in the Parliamentary procedure rules and trivial technological additions. This small change can go a long way in increasing efficiency, accountability and transparency in the functioning of the legislature.
### References
Inter-Parliamentary Union, World e-Parliament Report 2016, 2016.
The Parliament of the Commonwealth of Australia, Division required? Electronic Voting in the House of Representatives, May 2, 2016.
The UK House of Commons, The Case for Parliamentary Reform, 2010.
Michael H. Crespin, Serving Two Masters: Redistricting and Voting in the U.S. House of Representatives, Political Research Quarterly, 2010.
John M.Carey, Legislative Voting and Accountability, Cambridge University Press, 2009.
Gregory L. Bovitz and Jamie L.Carson, Position-Taking and Electoral Accountability in the U.S. House of Representatives, Political Research Quarterly, June, 2006.
John M.Carey, Visible Votes: Recorded Voting and Legislative Accountability in the Americas, Campbell Public Affairs Institute, September 9, 2005.
Judith Middlebrook, Voting methods in Parliament, Constitutional & Parliamentary Information, 2003.
Brandice Canes-Wrone et al., Out of Step, Out of Office: Electoral Accountability and House Members' Voting, American Political Science Review, March, 2002.
The authors are researchers at the National Institute of Public Finance and Policy, New Delhi. They thank Sanhita Sapatnekar, Anirudh Burman and Pratik Datta for useful discussions.
## Thursday, April 06, 2017
### Does the NCLT Have Enough Judges?
by Devendra Damle and Prasanth Regy.
The recently passed Finance Bill made headlines for combining tribunals, purportedly to rationalise their functioning. This is not the first such attempt. The Parliament set up the National Company Law Tribunal (NCLT) with a similar objective of streamlining all judicial proceedings under the Companies Act, 1956 and Companies Act, 2013. It has been operating since June 2016. The NCLT will hear all cases under these Acts which would earlier have gone to one of four existing courts and tribunals: the Company Law Board (CLB), Board for Industrial and Financial Reconstruction (BIFR), high courts (HCs) and Debt Recovery Tribunals (DRT).
However, commentators have pointed out that the NCLT is ill-equipped to cope with the pending cases it will inherit from the high courts and three tribunals. Others have pointed out that many of the legal and procedural issues which made the other tribunals ineffective will likely plague the NCLT too. One common concern is whether there are enough judges.
The popular discourse till date has largely focussed on pending cases which the NCLT will inherit. While the volume of these cases is substantial, we estimate that the volume of new cases which will instituted will be even larger. In this article we estimate how many judges the NCLT will need to handle this caseload. We find that the present strength of the NCLT is far lower than what is required. If this problem is not solved, NCLT is likely to end up a slow and inefficient tribunal.
### Our Approach
We use the tribunals which would have originally heard the cases as a proxy for the different types of cases NCLT will hear. So we have HC-type cases, BIFR-type cases, DRT-type cases, and CLB-type cases. To calculate the number of judges NCLT needs, we need to know the:
1. Annual rate of institution of cases of each type (I), and
2. Annual rate of disposal of cases per judge for cases of each type (D).
To get I, we take the average number of cases instituted every year in HCs and each of the three tribunals. We only count the kind of cases which will be transferred to NCLT. For example, original jurisdiction company petitions were earlier being heard by HCs, but will now be heard by NCLT. So, from the total cases instituted in HCs every year, we only count original jurisdiction company petitions.
For D, we first calculate the average number of cases (of the relevant kind) disposed of by HCs and each of the three tribunals every year. Then we divide this average disposal rate for each of them by the number of judges. For example, the CLB has 5 benches. So D for CLB is equal to total cases disposed of by CLB in one year divided by 5. Thus, we get the average disposal rate per judge for each of the four types.
I/D gives us the number of judges required for disposing of each type of case instituted in one year. Adding all values of I/D gives the total judges NCLT will need to dispose of all cases of all types instituted every year.
### What is the NCLT's caseload?
In an article by the consultancy Alvarez and Marsal, the authors estimate that a total of 24,900 existing cases — 4,000 from CLB, 700 from BIFR, 5,200 from HCs, and 15,000 from DRT — will be transferred to NCLT. Cases which were being heard by CLB will be be transferred to the NCLT automatically. Cases from BIFR will only be taken up by the NCLT if the parties file fresh applications. Cases from HCs which are eligible for transfer to NCLT, will be transferred in stages through notifications by the Ministry of Corporate Affairs. Assuming all cases do get transferred to NCLT, the tribunal will start with 24,900 cases.
What about the admission of new cases? From 2011-12 to 2014-15, on an average the CLB admitted about 10,170 cases per year (See CLB Annual Statistics).
In the same period, BIFR admitted an average 140 new cases per year (See BIFR Annual Statistics).
All HCs put together admitted approximately 14,000 original jurisdiction company matters in 2015-16 (See SC Annual Report 2015-16). In the Bombay, Delhi and Orissa HCs, approximately 90% of all original jurisdiction company matters are company petitions and applications made thereunder, cases likely to now go to NCLT. Assuming this proportion holds good for all HCs, about 12,700 cases which would earlier have gone to HCs will now be heard by NCLT.
DRTs admitted an average 21,470 Original Applications (OA cases) per year from 2012-13 to 2014-15 under the Recovery of Debts Due to Banks and Financial Institutions Act, 1993 (See here and here). In a sample of about 15,000 cases we collected from Delhi DRT-3 company and LLP related matters — matters which will now be heard by NCLT — constitute approximately 45% of the total cases. That brings the number to 9660 cases for all DRTs put together. These are the cases which will now go to NCLT instead of DRTs.
Table.1: Institution rate of fresh cases in NCLT
Type of Cases Institution Rate (I)
(cases per year)
HCs 12,700
BIFR 140
DRT 9,660
CLB 10,170
Total 32,670
This brings the total volume of fresh cases to 32,670. This will be the annual rate of institution of new cases in NCLT, assuming it stays constant over the years. The next question we have to tackle is: how many of these cases can be disposed of in a year?
### What will be the disposal rate of cases?
The DRT had an average disposal rate of 360 cases per judge per year (from 2012-13 to 2014-15). The CLB had an average disposal rate of approximately 1,705 per judge per year (from 2012-13 to 2014-15). We assume the disposal rate for these types of cases will be the same even in NCLT.
For cases from DRT and CLB we can straightaway use the respective tribunals' disposal rates.
For HCs and BIFR, the disposal rate cannot be calculated in the same manner. HCs deal with a large variety of matters. Of these, company petitions — the kind of cases which will now be heard by NCLT — form a small fraction. The average per judge annual disposal rate for all HCs put together is 19. Bombay HC, which has the highest disposal rate for company petitions among all HCs, only disposes of 60 company petition cases per judge per year. It stands to reason that a specialised tribunal like NCLT would have a higher disposal rate than that. Therefore, we assume that in NCLT these cases will have the same disposal rate as DRTs' average disposal rate i.e. 360 per judge per year.
BIFR is notorious for cases pending for a very long time. In the three years from 2010 to 2013, BIFR disposed of just 169 cases (62 pending). Since the Companies Act, 2013 and the Insolvency and Bankruptcy Code, 2015 have more streamlined processes for winding-up of companies, it is expected that the disposal rate would be higher when NCLT takes over cases being heard by BIFR. Therefore, we assume that when NCLT hears these cases it will dispose of them at a rate equal to DRT's disposal rate i.e. 360 per judge per year.
For cases from HCs and BIFR, why do we take the disposal rate of DRTs and not CLB? Unlike DRT, 60–80% of the CLB's caseload consisted of compliance related matters or small matters. For example, in 2013-14 and 2014-15, 60% and 40% (respectively) of the cases instituted in CLBs were matters regarding taking deposits without advertising (Sec.58A(9) Companies Act, 1956). These are routine matters and have high disposal rates; the ratio of annual disposals to institutions is consistently close to 1. Substantive matters on the other hand constitute 20–40% of the instituted cases, and their disposal rate is much lower. For example, in cases regarding mismanagement of companies (Sec. 397, 398 Companies Act, 1956), the ratio of annual disposals to institutions is around 0.7. Between 2013-14 to 2014-15 cases under Sec. 397/398 constituted 4–6% of total cases instituted in CLB, but represented 40% of the total pending matters at the end of the year. Since most of the cases heard by HCs and BIFR are expected to be of a substantive nature, we cannot use CLB's average disposal rate. Therefore, we assume that NCLT's disposal rate for these cases will be similar to the DRTs' average disposal rate, rather than the CLB's.
### How many judges does the NLCT need?
Armed with the institution and disposal rates of each type of case we can now calculate the number of judges required. One last consideration is the structural difference between the NCLT and the other tribunals. The NCLT has two types of benches, viz. division benches consisting of 2 judges and single benches consisting of 1 judge. Therefore, in the case of NCLT we calculate the number of benches rather than judges.
(cases per year) (cases per bench per year) Type of Cases Institution Rate (I) Disposal Rate (D) Benches Required (I/D) HCs + BIFR 12,840 360 36 DRT 9,660 360 27 CLB 10,170 1,705 6 Total 32,670 473 69
Thus, we estimate that the NCLT will require 69 benches just to keep up with its caseload. We can also use these disposal rates to calculate the number of judges required for clearing the inherited backlog of 24,900 cases. If these cases are to be disposed of steadily over the next five years, NCLT would need about 80 benches.
The NCLT currently has 14 benches (in eleven locations) (See here, here and here). With a disposal rate of 473 per bench per year, and 14 benches, NCLT can dispose of 6,620 cases per year, i.e. only a fifth of the incoming fresh cases. At this rate, the NCLT will accumulate a backlog of 26,050 cases per year; a total backlog of 1,30,250 cases in five years. The Central Government is planning to establish one NCLT bench in every HC jurisdiction, i.e. a total of 24. Even with 24 benches, the NCLT would accumulate a backlog of 21,320 every year. That's a total backlog of 1,06,600 cases over the next five years.
It should be noted that in these calculations we haven't factored in any increase in the rate of filing fresh cases, nor have we considered the entirely new categories of cases (e.g.: class action suits) which the NCLT will hear. These will place an even greater burden on the tribunal.
### Some corroborating evidence for our predictions
Some data on the disposal rates of NCLT are already available. In the six months from June 2016 to November 2016, NCLT disposed of 1,930 cases. From June to August only 240 cases were disposed of. This is probably because the NCLT just started functioning in June. The bulk of the cases — 1690 — were disposed of in the last three months, i.e. from September to November. If we take the disposal rate for just these three months and project it for the whole year, it translates to a disposal rate of approximately 6,750 cases per year. This is close to our estimated disposal rate of 6,620 cases per year.
### Conclusions
There is evidence to suggest that badly designed procedures which allow unecessary adjournments, lost working days, and administrative inefficiency substantially contribute to judicial delays and pendency. In the case of NCLT, even if these issues are fixed and we manage to double its disposal rate, the current bench strength would still be far short of what is required to handle its caseload.
This analysis draws attention to the fact that we need to do a better job of estimating the judicial resources required to handle case loads. What we have presented in this article is an example of a Judicial Impact Assessment (JIA): estimating the resources required in the judiciary to handle the case load, using data on court productivity. A pre-requisite for JIA is the availability of high-quality empirical data on case loads and productivity. It is important that JIA should be institutionalised in the legislative process, so that courts are able to deliver timely justice.
The US has a specialised body called the National Centre for State Courts dedicated to conducting research on the functioning of state courts. It maintains databases of case-level data and has developed models for various kinds of judicial needs assessments for state courts. It has even developed software for court and case management, which are currently used in the US state courts. There is a similar system in place for federal courts. The estimates for judicial resource requirements in both cases are based on weighted caseloads which are derived from granular, case-level data.
One significant difference between India and the US is that in comparison to the US, it is easier for the Government of India to change the number of benches as needed. In the US, new judgeships can only be created by an act of the legislature. By contrast, in India, new benches for the NCLT can be created by the Central Government simply by notification. What is however lacking is a sound process for determining the number of judges/benches needed.
With its lack of adequate number of benches, the NCLT is likely to be plagued by delays just like its predecessors. A more efficient judicial procedure, and greater bench strength, are both required to effect a lasting solution.
### References
Pratik Datta and Prasanth Regy. Judicial procedures will make or break the Insolvency and Bankruptcy Code. Ajay Shah's Blog, January 2017.
Prasanth Regy, Shubho Roy and Renuka Sane. Understanding Judicial Delays in India: Evidence from Debt Recovery Tribunals. Ajay Shah's Blog, May 18, 2016.
Pratik Datta and Ajay Shah. How to make courts work? Ajay Shah's Blog, February 22, 2015.
Reserve Bank of India. Report on Trend and Progress of Banking in India 2015-16
Nikhil Shah, Khushboo Vaish, Kavya Ramanathan. NCLT Readiness Report. Alvares and Marsal India, 2017
Company Law Board, annual statistics.
Lok Sabha Questions on cases pending in DRT dated 4th March 2016 and 4th December 2015.
The authors are researchers at the National Institute of Public Finance and Policy, New Delhi.
## Monday, April 03, 2017
### Predatory pricing and the telecom sector
by Ajay Shah.
1. When there are network effects, we should be cautious about the business strategy of discounting. What looks like a gift to consumers today is often a plan to achieve market power and recoup those gains by extracting consumer surplus in the future.
2. The burn rate at Reliance Jio is likely to be pretty large. However, the question that we should be asking: Can this subsidised user base set off a network effect?
3. In telecom, interoperability regulation is in place. Even if Reliance Jio was able to establish a commanding market position through discounting, there is no way to close off its user base for rival firms. Interconnection regulation by TRAI imply that a phone call from a rival telecom company, to a Reliance Jio customer, will always go through. The open standards of TCP/IP mean that a data packet from a customer of any data communications company in the world will successfully reach a Reliance Jio customer. Even if all my friends and family are on Reliance Jio, it makes no difference to my decision to be on Reliance Jio. There is no network effect.
4. Recoupment test: If in the future, Reliance Jio tries to increase prices, nothing prevents customers from switching to rival firms. There is no reason for a consumer to stay with Reliance Jio at future dates if it turns out that Reliance Jio is expensive.
5. Market power in this industry has been checked by the three key building blocks -- interconnectivity regulation + mobile number portability + the open standards of TCP/IP.
6. In fact, there is a negative network effect, as follows. Suppose a lot of customers switch from rival telephone companies to Reliance Jio. This will clog the airwaves of Reliance Jio's base stations, so the performance of Reliance Jio will go down while the performance of rival companies will go up. Through this channel, if Reliance Jio succeeds a lot in gaining customers, it will fail in delivering the best mobile data services.
Second order issues:
1. Interconnectivity regulation imposes costs upon all regulated persons and these costs should be placed in a fair manner.
2. There is an opportunity to obtain market power in JioMoney as payment regulation lacks all three components: interconnectivity regulation + number portability + open standards.
Market power in the new economy by Ajay Shah in Business Standard, April 2, 2017. Paper landing page.
UW professor: The information war is real, and we're losing it by Danny Westneat in The Seattle Times, March 30, 2017.
Don't Worry, Be Happy - High Frequency Trading Is Over, Dead, It's Done by Tim Worstall in Forbes, March 25, 2017.
Seven Charts That Show the Gap Between Old and New IITs by Thomas Manuel in The Wire, March 24, 2017.
Aadhaar is a legal right, but the government can suspend a citizen's number without prior notice by Anumeha Yadav in Scroll, March 23, 2017.
What London Police Learned From the Last Big Attack by Henry Wilkins in The Atlantic, March 23, 2017.
Central Ministry, State Government Departments Publicly Expose Personal Data of Lakhs of Indians in The Wire, March 23, 2017.
Don't mistake good psephology for good policy by Somasekhar Sundaresan in Business Standard, March 22, 2017.
Is it too late to save Hong Kong from Beijing's authoritarian grasp? by Howard W. French in The Guardian, March 21, 2017.
There is nothing that can stop banks from systematically fleecing you. Here's why by Dhirendra Kumar in The Economic Times, March 14, 2017.
Lessons From The FPI Limit Breach In HDFC Bank by Bhargavi Zaveri and Radhika Pandey in Bloomberg, March 10, 2017.
Sebi chairman Ajay Tyagi has his task cut out by Mobis Philipose in Mint, March 9, 2017.
Inside Steve Bannon's Failed Breitbart India Scheme by Asawin Suebsaeng in The Daily Beast, March 2, 2017.
### CUTS 5th Biennial Competition, Regulation & Development Conference 09-11 November, 2017 Jaipur, India
CALL FOR PAPERS
I. Introduction
CUTS and CIRC invite papers for the 5th Biennial Competition, Regulation & Development Conference to be held on 09-11 November, 2017 in Jaipur (India). Interested scholars and practitioners are invited to submit a 500 word abstract for a chance to participate in this Conference and present their paper.
The abstract should be based on any one of the four plenaries of the Conference (below) and should be submitted to the undersigned, along with a brief CV (not more than 2 pages) of the author. Authors are requested to mention the specific ‘Plenary’ their paper is based on while submitting the abstract.
Authors of selected abstracts would be invited to submit full conference papers (3,000 to 4,000 words) for a chance to participate in this conference. On successful selection, the organisers will provide support to the author (air travel, accommodation and meals) to participate in the conference, and present the paper.
II. Plenary Sessions
The abstract/paper should target any one of the following four plenaries of the conference.
1. Plenary 1: Revisiting IPR and Competition
2. Plenary 2: Disruptive Technologies and Economic Regulations
3. Plenary 3: Building Organisational capacities for tackling policy and regulatory uncertainty
4. Plenary 4: Challenges and Opportunities of Development Financing for Fostering an Innovation based Ecosystem
Call for Papers https://goo.gl/1yMK3L
Background Note https://goo.gl/pFfby9
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2019-10-23 06:25:19
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http://aimsciences.org/article/doi/10.3934/nhm.2006.1.601
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# American Institute of Mathematical Sciences
2006, 1(4): 601-619. doi: 10.3934/nhm.2006.1.601
## On the variational theory of traffic flow: well-posedness, duality and applications
1 Department of Civil and Environmental Engineering, 416 McLaughlin Hall, Berkeley, CA 94707, United States
Received July 2006 Revised September 2006 Published October 2006
This paper describes some simplifications allowed by the variational theory of traffic flow(VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical sense; i.e., that the problem is well-posed. The requirements for well-posedness are mild and met by practical applications. They are consistent with narrower results available for kinematic wave or Hamilton-Jacobi theories. The paper also describes some duality ideas relevant to these theories. Duality and VT are used to establish the equivalence of eight traffic models. Finally, the paper discusses how its ideas can be used to model networks of multi-lane traffic streams.
Citation: Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks & Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601
[1] Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385 [2] David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205 [3] Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683 [4] Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 [5] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [6] Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411 [7] Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 [8] Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 [9] Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317 [10] Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 [11] Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : ⅰ-ⅲ. doi: 10.3934/dcdss.201805i [12] Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 [13] Emeric Bouin. A Hamilton-Jacobi approach for front propagation in kinetic equations. Kinetic & Related Models, 2015, 8 (2) : 255-280. doi: 10.3934/krm.2015.8.255 [14] Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647 [15] Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855 [16] Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167 [17] Martino Bardi, Yoshikazu Giga. Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 447-459. doi: 10.3934/cpaa.2003.2.447 [18] Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389 [19] Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225 [20] Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793
2017 Impact Factor: 1.187
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2018-10-23 06:11:37
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https://proofwiki.org/wiki/Definition:Derangement
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# Definition:Derangement
## Definition
A derangement is a permutation $f: S \to S$ from a set $S$ to itself where $\map f s \ne s$ for any $s \in S$.
If $S$ is finite, the number of derangements is denoted by $D_n$ where $n = \card S$ (the cardinality of $S$.)
## Historical Note
The number of a derangements of a finite set was first investigated by Nicolaus I Bernoulli and Pierre Raymond de Montmort between about $1708$ and $1713$.
They solved it at around the same time.
The question is often couched in the idea of counting the number of ways of placing letters at random in envelopes such that nobody receives the correct letter.
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2020-06-05 15:42:51
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https://mathematica.stackexchange.com/questions/120391/unattractive-streaky-rendering-of-graphics3d-images
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# Unattractive streaky rendering of Graphics3D images
I often find that Graphics3D drawings appear "streaky" in places. This can be seen in the picture below (taken from my question here).
There is an unattractive patch where the colour changes unpredictably where the blue and pink cones overlap at the bottom. Presumably this is something to do with having the two surfaces being directly on top of each other, but I think have seen the same phenomenon occurring even when this is not the case. You can see the same behaviour in Mathematica's documentation on Cone (under Applications, the 2nd example "Define a region by the intersection of a cone and a plane"):
Is there any way to fix this? Sometimes altering the coordinates slightly so that the two surfaces don't quite overlap seems to help, but it's not a very elegant solution...
I believe this is the same issue as presented in 3DPlot Rendering Artefacts (z-fighting) but the solution there does not exactly address my problem.
1. I am dealing with a Graphics3D object rather than a plot, so it is not immediately obvious what is the best parameter to alter by 1% to make the cone surfaces not quite overlap. (Altering Scale[cone2, 3, {0, 0, 0}] to Scale[cone2, {3.01, 3.01, 3}, {0, 0, 0}] seems to work but maybe there is a better way.)
2. I am already aware of this general method of altering the coordinates slightly and am wondering if there is an alternative solution.
• What version and OS are you on? I do not see this streaking in the documentation example in 10.4.1, neither on Windows 10 nor on Linux (Xubuntu Trusty). But, the one with the two cones, I can reproduce. Jul 10 '16 at 12:27
• This is Mathematica 10.0.0.0, operating on Windows 7 Enterprise. The Mathematica documentation example is taken from the online help here. Jul 10 '16 at 12:36
• Possible duplicate of 3DPlot Rendering Artefacts (z-fighting) Jul 10 '16 at 12:36
• Thanks for the reference -- I hadn't seen that. However, I don't think it quite answers my question, which I have now modified accordingly. Jul 10 '16 at 12:47
• I get same with 10.4.0 for Linux x86 (64-bit), but then again, nVidia hates linux. Jul 10 '16 at 12:55
I can think of no solution to the color streaking problem other than perturbing the size of one the two coinciding cones. I choose to perturb projcone2.
origin = Point[{0, 0, 0}];
cone1 = Cone[{{0, 0, 1}, {0, 0, 0}}];
transform = {{0.3, 0, 0.15}, {0, 0.35, 0}, {0.1, 0, 0.5}};
cone2 = GeometricTransformation[cone1, transform];
projcone2 = Scale[cone2, {2.99, 2.99, 3}, {0, 0, 0}];
Graphics3D[
{{Opacity[0.25], EdgeForm[{Thick}], cone1},
{Opacity[0.6], Magenta, EdgeForm[Thick], cone2},
{Opacity[0.6], Cyan, EdgeForm[{Thick}], projcone2},
{PointSize[Large], origin}},
Lighting -> "Neutral",
PlotRange -> 1.4 {{-1, 1}, {-1, 1}, {0, 1.4}},
ImageSize -> Large]
This graphic also shows you why you are not seeing the edge of projcone2 in your rendering of the cones. The edge lies outside the range of your bounding box.
• The OP deliberately truncates projcone2 at z=1, related question here Jul 10 '16 at 13:31
• @SimonWoods. He is clipping with PlotRange, i.e., the bounding box. The actual edge of the cyan cone is well outside his bounding box as I show it here. Jul 10 '16 at 14:27
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2021-09-22 23:11:09
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https://www.ques10.com/p/3960/what-are-triggers-explain-with-example/
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0
23kviews
What are triggers? Explain with example.
0
488views
Triggers:
A trigger is a block of code that is executed automatically from a database statement. Triggers is generally executed for DML statements such as INSERT, UPDATE or DELETE. It resides in a database code and is fired automatically when the database code requires to perform the INSERT ,UPDATE or DELETE statement. It is also stored in the database as stored procedures. It does not involve any COMMIT transaction rather it is a part of a transaction.
SYNTAX :
CREATE OR REPLACE TRIGGER <trigger-name>
[BEFORE/AFTER]
[INSERT/UPDATE/DELETE]
OF<column-name>
ON<table-name>
[REFERENCING OLD AS O NEW AS N]
[FOR EACH ROW]
WHEN <trigger-condition>
DECLARE
BEGIN
<sql-statement>
END;
CREATE OR REPLACE TRIGGER <trigger-name>:
It is used to create a new trigger or replace an existing trigger.
[BEFORE/AFTER]:
It is used to mention the execution time of the trigger. It specifies whether the trigger should fire after or before the DML statement.
[INSERT/UPDATE/DELETE]:
Triggers can be fired against any of these DML statements.
OF<column-name>:
It is used to indicate the column name on which the trigger will be fired but it used for only update statement.
ON<table-name>
It is used to indicate on which table the trigger will be fired
[REFERENCING OLD AS O NEW AS N]:
This is option and used to define the aliases.
[FOR EACH ROW]:
It is used to indicate if the trigger is a row-level trigger or a statement-level trigger.
WHEN <trigger-condition>:
It is used to indicate the condition for trigger execution.
DECLARE:
IT is used for declaring the variable.
BEGIN……END:
In this actual query for trigger is written.
Example:1
Create Trigger ABC
Before Insert On Students
This trigger is activated when an insert statement is issued, but before the new record is inserted
Example:2
Create Trigger XYZ
After Update On Students
This trigger is activated when an update statement is issued and after the update is executed
Let us see the one more example to understand the trigger
Example:
CREATE TRIGGER Amount-CHECK BEFORE UPDATE ON account
FOR EACH ROW
BEGIN
IF (NEW.amount < 0) THEN
SET NEW.amount = 0;
ELSEIF (NEW.amount > 100) THEN
SET NEW.amount = 100;
END IF;
END;
In this the trigger will first check the amount if it is less than zero it will set amount as zero but if it is greater than hundred then it will set amount as hundred
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2021-10-23 16:56:31
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http://codereview.stackexchange.com/questions/21337/is-this-agent-actor-implementation-issue-free
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# Is this Agent/Actor implementation issue free?
I implemented this Agent class for a project recently and was wondering if I could get some other eyes to look at it -- I'm currently the only developer where I work so I can't exactly ask someone here to do it.
I'm pretty sure it's correct, but then I'm too close to it.
public abstract class Agent<M> : IDisposable
{
private Queue<M> messageQueue;
private bool quit;
public Agent()
{
this.messageQueue = new Queue<M>();
this.quit = false;
}
/// <summary>
///
/// Do not call this method from within the message-
/// handling method, or it will result in a deadlock
/// (because this method waits for the message-handling
/// </summary>
public virtual void Dispose()
{
this.quit = true;
// clear messageQueue before nulling?
// (would do this to dispose queued items)
this.messageQueue = null;
}
public void QueueMessage(M message)
{
lock (this.messageQueue)
{
this.messageQueue.Enqueue(message);
}
}
{
while (!this.quit)
{
M message = default(M);
bool messageAvailable = false;
lock (messageQueue)
{
if (messageQueue.Count > 0)
{
message = messageQueue.Dequeue();
messageAvailable = true;
}
}
try
{
if (!messageAvailable)
{
// if the Interrupt() method was
// called before we sleep or is
// called while we're sleeping,
// this will throw:
}
}
{
// we have a new message to handle,
// so get it, or we've been told
// to quit.
continue;
}
ProcessMessage(message);
}
}
protected abstract void ProcessMessage(M message);
}
Also, are there any special considerations you can think of that should be made by a class inheriting from this base class? (I can't think of any.)
I could add start/stop methods, but at the moment they're not needed.
Btw, I have to use .NET 3.5.
-
Instead of sleeping for short periods or using Thread.Interrupt() it's better to use waithandles, ManualResetEvent in your case. I don't have Visual Studio at hand, but the following example should give you the basic idea (most of the complexity will be gone if you start using .NET 4 or 4.5, in particular BlockingCollection<T>):
public abstract class Agent<TMessage> : IDisposable
{
private readonly Queue<TMessage> _messageQueue = new Queue<TMessage>();
private readonly ManualResetEvent _waitHandle = new ManualResetEvent(false);
private volatile bool _quit;
private bool _disposed;
protected Agent()
{
}
public void QueueMessage(M message)
{
lock (_messageQueue)
{
_messageQueue.Enqueue(message);
_waitHandle.Set();
}
}
{
while (!_quit)
{
_waitHandle.WaitOne();
TMessage message;
lock (messageQueue)
{
if (messageQueue.Count == 0)
{
_waitHandle.Reset();
continue;
}
message = messageQueue.Dequeue();
}
ProcessMessage(message);
}
}
protected abstract void ProcessMessage(M message);
/// <summary>
///
/// Do not call this method from within the message-
/// handling method, or it will result in a deadlock
/// (because this method waits for the message-handling
/// </summary>
public void Dispose()
{
Dispose(true);
GC.SupressFinalize(this);
}
protected virtual void Dispose(bool disposing)
{
if (_disposed || !disposing)
return;
_quit = true;
_waitHandle.Set();
_waitHandle.Dispose();
_disposed = true;
}
}
-
Thanks, this method is working well! I'm not that familiar with all of .NET's thread synchronization methods, and I was in a bit of a rush, so I probably wouldn't have come up with using this any time soon. If I were using .NET 4, I would have been doing this in F# and used a MailboxProcessor (or Agent from the FSharpx library) instead of rolling my own. (I think I read somewhere that F# has been backported to .NET 2 but I haven't had time to investigate using it.) Thanks again! – paul Mar 1 '13 at 21:25
Few minor points:
• Generic type parameter should be called <T> as per convention.
• public constructors in abstract classes make no sense - make it protected.
• Make thread and messageQueue readonly and don't set them to null in the Dispose() method.
• Better yet, implement the Disposable Pattern correctly. Though note, I do disagree with their setting of the IDisposable members to null. It's really not necessary at all; but calling Dispose() is. Their _disposed class member conveys enough information necessary.
• You should also lock on a dedicated locking class member when accessing your quit class member since it is accessed via multiple threads.
• Possibly wrap your call to ProcessMessage(message) in a try..catch block with appropriate error handling (or not, as the case could be). Unless, of course, you trust any subclasses from doing nasty things in their override of it.
Just my initial thoughts. Hope they help.
-
Thanks for the tips! In response: (1) No reason not to go w/ convention on this one. (2) Good call on protected! (3,4) I'll have to look into proper handling of this further -- thanks for the link. (5) No need to lock quit since it's a bool and access is atomic. I did add volatile though. (Related.) (6) I had considered that; I'm not sure if I want to swallow exceptions there or just let them fall through and crash. (Logging them would be difficult in this application.) Thanks again! – paul Feb 5 '13 at 16:03
I found this article on properly implementing the dispose pattern a bit clearer than the article to which you linked, in case you need a good link for someone in the future. – paul Feb 19 '13 at 21:13
@Jesse, concerning "don't set fields to null". Obviously null'ing fields will not free the memory immediately, and it will not help GC in case when the code is written correctly. But it will help in reducing memory leaks if some references to main (owner) object will remain after disposition thus preventing it from being garbage-collected. In this case null'ing the references allows nested objects to be garbage-collected even though parent object will stay in memory. – almaz Feb 27 '13 at 22:28
@almaz I disagree completely. Your object which is being Dispose'd, as the owner of those fields, will be unused by your program and eligible for garbage collection directly post-Dispose(). Those fields will then also be GC'd if they have no other references (they shouldn't). – Jesse C. Slicer Feb 27 '13 at 22:35
I'd rather use TMessage than the meaningless T. Just T is mostly applicable in situations where the parameter has no real meaning, such as collections. – CodesInChaos Feb 27 '13 at 22:54
No, this implementation is not issue free!
The Thread.Interrupt() method apparently causes a ThreadInterruptedException to be thrown whenever the thread is blocked. The documentation led me to believe that it was only thrown under certain circumstances:
If this thread is not currently blocked in a wait, sleep, or join state, [emph mine] it will be interrupted when it next begins to block.
However, a lot more can block a thread than you'd think! I first started seeing what I thought were rogue instances of this exception in a derived agent instance when accessing DateTime.Now! (This is apparently due to the Now property requiring blocking IO to obtain a value.)
Locking can also cause a thread to block, as lock gets translated into Monitors (which block).
Looks like I'll have to go with a solution that just Sleep()s for short periods of time instead of waiting indefinitely until new input became available :-/
-
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2015-03-02 07:22:02
|
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https://www.edaboard.com/threads/dielectric-constant-and-nano-science.210143/
|
Continue to Site
dielectric constant and nano science
Status
Not open for further replies.
sameerawickramasinghe
Newbie level 1
does dielectric constant vary in nano scale?
If so. how does it happen?
jiripolivka
First- it is good to know that permittivity is not a constant. It depends upon frequency, among other things.
Permittivity is a response of a non-conductive material (not exclusively) to electromagnetic field. It is important to define the "nano-scale" to know a reply to your question. In physical interpretation, permittivity may be expressed as a resistance to moving a charge. The elastic resistance can be interpreted as the real part of permittivity, the braking effect, to imaginary part.
Nano-scale can be a size of a grain, a size of a molecule, or, an atom. Such structures do have a charge, and should have a permittivity. It is easier to experiment with light wave that is comparable with nano-scale. Longer waves also move charges but maybe their effect upon nano-scale objects is weaker.
tony_lth
Points: 2
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2023-01-28 00:30:20
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|
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/section/12.16/
|
<meta http-equiv="refresh" content="1; url=/nojavascript/">
# 12.16: Exponential Functions
Difficulty Level: At Grade Created by: CK-12
%
Progress
Progress
%
### Vocabulary Language: English
Asymptotic
Asymptotic
A function is asymptotic to a given line if the given line is an asymptote of the function.
Exponential Function
Exponential Function
An exponential function is a function whose variable is in the exponent. The general form is $y=a \cdot b^{x-h}+k$.
grows without bound
grows without bound
If a function grows without bound, it has no limit (it stretches to $\infty$).
Horizontal Asymptote
Horizontal Asymptote
A horizontal asymptote is a horizontal line that indicates where a function flattens out as the independent variable gets very large or very small. A function may touch or pass through a horizontal asymptote.
Transformations
Transformations
Transformations are used to change the graph of a parent function into the graph of a more complex function.
## Date Created:
Jan 16, 2013
Jul 06, 2015
You can only attach files to Modality which belong to you
If you would like to associate files with this Modality, please make a copy first.
# Reviews
100 % of people thought this content was helpful.
|
2015-07-08 01:37:28
|
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https://jeffmikels.org/gauth/forum/viewtopic.php?page=cora-dataset-gcn-2da9cd
|
# cora dataset gcn
Our model used: the graph structure of the dataset, in the form of citation links between papers; the 1433-dimensional feature … Execute this notebook:
LGNN chains together a series of line graph neural network layers.
You can understand $$\{Pm, Pd\}$$ more thoroughly with this explanation.
Define an LGNN with three hidden layers, as in the following example.
+(Dx^{(k)})_{i}\theta^{(k)}_{2,l} \\ Moreover, we investigate the influence of parameter α which controls the impact of community structure.
Please refer to DGL doc for DGL installation at
A pass over all subgraphs is considered a training epoch. (download link). You used a graph convolutional neural network (GCN) than inter-class.
Whatâs the difference then, between a community detection algorithm and where $$S_c$$ is the set of all permutations of labels, and
Revision 9370caed. An key innovation in this topic is the use of a line graph.
&+[\{\text{Pm},\text{Pd}\}y^{(k)}]_{i}\theta^{(k)}_{3+J,l}] \\ the binary community subgraph from Cora, but also on the dropout=0.5 specifies a 50% dropout at each layer. graph in a semi-supervised setting. A graph and network repository containing hundreds of real-world networks and benchmark datasets.
In this example we use two GCN layers with 32-dimensional hidden node features at each layer. \qquad i \in V, l = 1,2,3, ... b_{k+1}/2 T. Kipf, M. Welling.
to augment the straightforward GNN architecture so that it operates on a result, there are a total of 21 training samples in this small dataset.
Note that each training sample contains batched_graph API.
See a full comparison of 47 papers with code.
Another batching solution is to
edge adjacency structure in the original graph. Supported datasets including 'gcn', 'gat', 'deepwalk', 'node2vec', 'hope', 'grarep', 'netmf', 'netsmf', 'prone' For specific parameters for each algorithms, you can read this page.
y^{(k+1)} = {}& f(y^{(k)}, x^{(k+1)}) Take the first one as an example, which follows. You can choose from ['cora', 'citeseer', 'pubmed']. We need two generators, one for training and one for validation data.
To be consistent with the GCN tutorial, they correspond to nodes $$v^{l}_{A}, v^{l}_{B}$$. LGNN takes a collection of different graphs.
Cora is a scientific publication dataset, Therefore, $$f$$ could be written as: Two equations are chained-up in the following order: Keep in mind the listed observations in this overview and proceed to implementation. Next we create the ClusterNodeGenerator object (docs) that will give us access to a generator suitable for model training, evaluation, and prediction via the Keras API. One of the highlights of the model is We are going to re-use the trained model weights. You can find the complete code on Github at DGL example: https://github.com/dmlc/dgl/tree/master/examples/pytorch/gcn In the collate_fn for PyTorch data loader, graphs are batched using DGLâs
\end{split}\end{split}\], $\begin{split}\begin{split} operator: multiplication. labeling them. Here, we use the following connection rule: Two nodes $$v^{l}_{A}$$, $$v^{l}_{B}$$ in lg are connected if ($$j$$). Cora dataset is a common benchmark for Graph Neural Networks (GNN) and frameworks that support GNN training and inference. We specify the number of clusters and the number of clusters to combine per batch, q. Revision ed8a7beb. for each node in the next layer. Roughly speaking, there is a relationship between how $$g$$ and gathering information in $$2^{j}$$ neighborhood of each node. Cora dataset. line graphâs feature to graphâs, and vice versa. # We are going to use the METIS clustering algorith, "Graph clustering using the METIS algorithm. In this tutorial, we will run our GCN on Cora dataset to demonstrate. of the same class with different parameters. GAT and APPNP models. operations can be formulated as performing $$2^j$$ step of message The validation and test sets have the same sizes for both datasets. # initializing list to collect message passing result, # pulling message from 1-hop neighbourhood, # A utility function to convert a scipy.coo_matrix to torch.SparseFloat, # Since there are only two communities, there are only two permutations, Supervised community detection task with the Cora dataset, Binary community subgraph from Cora with a test dataset, Community detection in a supervised setting, Chain-up LGNN abstractions as an LGNN layer, Revisit classic models from a graph perspective, Supervised Community Detection with Line Graph Neural Networks, Community Detection with Graph Neural Networks (CDGNN). Comparing to node classification, community detection focuses on retrieving The layer is defined with below operations, note that we apply two transposes to keep adjacency matrix on right hand side of sparse_dense operator, Unlike models in previous tutorials, message passing happens not only on the \end{split}\end{split}$, \[\begin{split}\begin{split} 1 \text{ if } j = \hat{i}, \hat{j} \neq i\\ Let us denote this abstraction as $$f$$. Finally, the following shows how to sum up all the terms together, pass it to skip connection, and citation link (A->B means A cites B).
each community. The generator object internally maintains the order of predictions.
You may substitute this part with your own dataset, here we load data from DGL, The weights are trained with https://github.com/dmlc/dgl/blob/master/examples/pytorch/gcn/train.py, To run GCN on TVM, we first need to implement Graph Convolution Layer.
W. Chiang, X. Liu, S. Si, Y. Li, S. Bengio, and C. Hsiej, KDD, 2019, arXiv:1905.07953 (download link) = ((H * W)^t * A^t)^t Source code for dgl.data.citation_graph. Cora dataset is a common benchmark for Graph Neural Networks (GNN) and frameworks that support GNN training and inference. &+\sum^{J-1}_{j=0}(A_{L(G)}^{2^{j}}y^{k})_{i}\gamma^{(k)}_{3+j,l^{'}}\\
representation of $$2^j, j=0, 1, 2..$$ step message passing, i.e.
Supervised Community Detection with Line Graph Neural Networks. channel updates its embedding $$x^{(k+1)}_{i,l}$$ with: Then, the line-graph representation $$y^{(k+1)}_{i,l}$$ with. at least contains one cross-community edge as the training example. graph-structured data. features. CORA from Bojchevski and Günnemann [2018], denoted as CORA-Full. +\text{fuse}(y^{(k)})]\\
{relu, sigmoid, log_softmax, softmax, leaky_relu}, The Output Tensor for this layer [num_nodes, output_dim], # Only support float32 as feature for now, # Check shape of features and the validity of adjacency matrix, # Define input features, norms, adjacency matrix in Relay, # Analyze free variables and generate Relay function, # Currently only support llvm as target, "Print the first five outputs from TVM execution, Deploy Single Shot Multibox Detector(SSD) model, Deploy the Pretrained Model on Raspberry Pi, Deploy a Framework-prequantized Model with TVM, Deploy a Framework-prequantized Model with TVM - Part 3 (TFLite), Compile YOLO-V2 and YOLO-V3 in DarkNet Models, Define the functions to load dataset and evaluate accuracy, Load the data and set up model parameters, Set up the DGL-PyTorch model and get the golden results, Prepare the parameters needed in the GraphConv layers, Run the TVM model, test for accuracy and verify with DGL, Deploy a Hugging Face Pruned Model on CPU, AutoScheduler : Template-free Auto Scheduling, https://github.com/dmlc/dgl/tree/master/examples/pytorch/gcn, https://github.com/dmlc/dgl/blob/master/examples/pytorch/gcn/train.py, https://github.com/dmlc/dgl/blob/master/python/dgl/nn/mxnet/conv/graphconv.py. Additionally note that the weights trained previously are kept in the new model. During model training, each subgraph or combination of subgraphs is treated as a mini-batch for estimating the parameters of a GCN model. (40 epochs). :math{Pm,Pd} as block diagonal matrix in correspondence to DGL batched
Now we can specify our machine learning model, we need a few more parameters for this: the layer_sizes is a list of hidden feature sizes of each layer in the model.
That is, they are parameterized to compute
W. Chiang, X. Liu, S. Si, Y. Li, S. Bengio, and C. Hsiej, KDD, 2019, arXiv:1905.07953 (download link), [2] Semi-Supervised Classification with Graph Convolutional Networks. However, community assignment should be equivariant to Finally, we build the TensorFlow model and compile it specifying the loss function, optimiser, and metrics to monitor. Community detection, or graph clustering, consists of partitioning to multiple communities case. Each publication in the dataset is described by a TF/IDF weighted word vector from a dictionary which consists of 500 unique words.
All data sets are easily downloaded into a standard consistent format. Specifically, a line-graph $$L(G)$$ turns an edge of the original graph G We are now ready to train the GCN model using Cluster-GCN, keeping track of its loss and accuracy on the training set, and its generalisation performance on a validation set.
Therefore, you construct it as a sparse matrix in the dataset,
Any graph clustering method can be used, including random clustering that is the default clustering method in StellarGraph.
Descriptions of these new datasets, as well as statistics for all datasets …
This article is an introductory tutorial to build a Graph Convolutional Network (GCN) with Relay. other clustering algorithm such as k-means? In __forward__, use following function aggregate_radius() to $$L_{equivariant} = \underset{\pi \in S_c} {min}-\log(\hat{\pi}, \pi)$$, The Cora dataset consists of 2708 scientific publications classified into one of seven classes.
a node. along the diagonal of the large graphâs adjacency matrix.
&+[\{\text{Pm},\text{Pd}\}^{T}x^{(k+1)}]_{i^{'}}\gamma^{(k)}_{3+J,l^{'}}]\\
to illustrate a simple community detection task. label permutations.
Without loss of generality, in this tutorial you limit the scope of the &+\text{skip-connection}
different machine learning fields.
Figure 1: Performances on the Cora dataset. Cora contains attributes âw_xâ that correspond to words found in that publication. DGL batches graphs by merging them Copyright © 2020 The Apache Software Foundation. Therefore, the summation is equivalent to summing nodesâ Weâll use scikit-learn again to do this. $$S_c = \{\{0,0,0,1\}, \{1,1,1,0\}\}$$. three objects: A DGLGraph, a SciPy sparse matrix pmpd, and a label As treat each class as one community, and find the largest subgraph that
We use node_order to re-index the node_data DataFrame such that the prediction order in y corresponds to that of node embeddings in X. number of the hidden units in the hidden layer, dimension of model output (Number of classes), "https://homes.cs.washington.edu/~cyulin/media/gnn_model/gcn_, "Print the first five outputs from DGL-PyTorch execution.
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2021-01-16 23:51:06
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https://zbmath.org/?q=an:1261.37023
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# zbMATH — the first resource for mathematics
Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space. (English. Russian original) Zbl 1261.37023
Izv. Math. 76, No. 5, 907-921 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 5, 57-72 (2012).
This paper addresses an outstanding conjecture on the degrees of irreducible polynomial integrals of reversible Hamiltonian systems.
In particular, the authors study a special system arising in the analysis of irreducible polynomial integrals of degree 4. The broad form of the conjecture is that the degree of any such polynomial integral does not exceed 2.
The special case considered by the authors is the last step in proving the conjecture for degree 4 irreducible polynomials.
The special system considered here is described by the Hamiltonian equations $\dot x_k= {\partial H\over\partial y_k},\quad\dot y_k= -{\partial H\over\partial x_k}\quad (k= 1,2),$ where $$H= {1\over 2}(y^2_1+ y^2_2)- W(x_1, x_2)$$ and where $$W$$ is an infinitely differentiable function on the configuration space, namely the torus $$\mathbb{T}^2$$. The authors ask whether the Hamiltonian equations admit non-trivial first integrals that are polynomial in the momenta $$y_1$$ and $$y_2$$. They conclude that there is a non-trivial polynomial integral of degree at most 2.
##### MSC:
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) 70F05 Two-body problems 70H07 Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics
Full Text:
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2021-11-28 03:41:47
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http://www.mathworks.com/help/physmod/sdl/ref/genericengine.html?requestedDomain=true&nocookie=true
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Generic Engine
Internal combustion engine with throttle and rotational inertia and time lag
Library
Simscape / Driveline / Engines
Description
The block represents a general internal combustion engine. Engine types include spark-ignition and diesel. Speed-power and speed-torque parameterizations are provided. A throttle physical signal input specifies the normalized engine torque. Optional dynamic parameters include crankshaft inertia and response time lag. A physical signal port outputs engine fuel consumption rate based on choice of fuel consumption model. Optional speed and redline controllers prevent engine stall and enable cruise control.
Generic Engine Model
By default, the Generic Engine model uses a programmed relationship between torque and speed, modulated by the throttle signal.
Engine Speed, Throttle, Power, and Torque
The engine model is specified by an engine power demand function g(Ω). The function provides the maximum power available for a given engine speed Ω. The block parameters (maximum power, speed at maximum power, and maximum speed) normalize this function to physical maximum torque and speed values.
The normalized throttle input signal T specifies the actual engine power. The power is delivered as a fraction of the maximum power possible in a steady state at a fixed engine speed. It modulates the actual power delivered, P, from the engine: P(Ω,T) = T·g(Ω). The engine torque is τ = P.
Engine Power Demand
The engine power is nonzero when the speed is limited to the operating range, Ωmin ≤ Ω ≤ Ωmax. The absolute maximum engine power Pmax defines Ω0 such that Pmax = g0). Define w ≡ Ω/Ω0 and g(Ω) ≡ Pmax·p(w). Then p(1) = 1 and dp(1)/dw = 0. The torque function is:
τ = (Pmax0)·[p(w)/w].
You can derive forms for p(w) from engine data and models. Generic Engine uses a third-order polynomial form:
p(w) = p1·w + p2·w2p3·w3
satisfying
p1 + p2p3 = 1, p1 + 2p2 – 3p3 = 0.
In typical engines, the pi are positive. This polynomial has three zeros, one at w = 0, and a conjugate pair. One of the pair is positive and physical; the other is negative and unphysical:
Typical Engine Power Demand Function
Restrictions on Engine Speed and Power
• For the engine power polynomial, there are restrictions, as shown, on the polynomial coefficients pi, to achieve a valid power-speed curve.
• If you use tabulated power or torque data, corresponding restrictions on P(Ω) remain.
Specify the speed and power as w = Ω/Ω0 and p = P(Ω)/P0 and define the boundaries as wmin = Ωmin0 and wmax = Ωmax0. Then:
• The engine speed is restricted to a positive range above the minimum speed and below the maximum speed: 0 ≤ wminwwmax.
• The engine power at minimum speed must be nonnegative: p(wmin) ≥ 0. If you use the polynomial form, this condition is a restriction on the pi:
p(wmin) = p1·wmin + p2·w2minp3·w3min ≥ 0.
• The engine power at maximum speed must be nonnegative: p(wmax) ≥ 0. If you use the polynomial form, this condition is a restriction on wmax: wmaxw+.
Engine Power Forms for Different Engine Types
For the default parameterization, Generic Engine provides two choices of internal combustion engine types, each with different engine power demand parameters.
Power Demand
Coefficient
Engine Type:
Spark-IgnitionDiesel
p110.6526
p211.6948
p311.3474
Idle Speed Controller Model
The idle speed controller adjusts the throttle signal to increase engine rotation below a reference speed according to the following expressions:
$\Pi =\mathrm{max}\left({\Pi }_{i},{\Pi }_{c}\right)$
and
$\frac{d\left({\Pi }_{c}\right)}{dt}=\frac{0.5\cdot \left(1-\mathrm{tanh}\left(4\cdot \frac{\omega -{\omega }_{r}}{{\omega }_{t}}\right)\right)-{\Pi }_{c}}{\tau }$
where:
• Π — Engine throttle
• Πi — Input throttle (port T)
• Πc — Controller throttle
• ω — Engine speed
• ωr — Idle speed reference
• ωt — Controller speed threshold
• τ — Controller time constant
The controlled throttle increases with a first-order lag from zero to one when engine speed falls below the reference speed. When the engine speed rises above the reference speed, the controlled throttle decreases from one to zero. When the difference between engine velocity and reference speed is smaller than the controller speed threshold, the tanh function smooths the time derivative of the controlled throttle. The controlled throttle is limited to the range 0–1. The engine uses the larger of the input and controlled throttle values. If engine time lag is included, the controller changes the input before the lag is computed.
Redline Controller Model
While the idle speed controller determines the minimum throttle value for maintaining engine speed, the redline controller prevents excessive speed based on a maximum throttle input. To determine the maximum throttle value, the redline controller uses the idle speed controller model equation. However, for the redline controller:
• ωr is the redline speed reference.
• ωt is the redline speed threshold.
• τ is the redline time constant.
Limitations
This block contains an engine time lag limitation.
Engine Time Lag
Engines lag in their response to changing speed and throttle. The Generic Engine block optionally supports lag due to a changing throttle only. Time lag simulation increases model fidelity but reduces simulation performance.
Ports
PortDescription
BRotational conserving port representing the engine block
FRotational Conserving port representing the engine crankshaft
TPhysical signal input port specifying the normalized engine throttle level
PPhysical signal output port reporting the instantaneous engine power
FCPhysical signal output port reporting the fuel consumption rate
Port T accepts a signal with values in the range 0–1. The signal specifies the engine torque as a fraction of the maximum torque possible in steady state at fixed engine speed. The signal saturates at zero and one. Values below zero are interpreted as zero. Values above one are interpreted as one.
Parameters
Engine Torque
Model parameterization
Select how to model the engine. Choose between these options, each of which enable other parameters:
• Normalized 3rd-order polynomial matched to peak power — Parametrize the engine with a power function controlled by power and speed characteristics. This is the default option.
• Tabulated torque data — Engine is parametrized by speed–torque table that you specify.
• Tabulated power data — Engine is parametrized by speed–power table that you specify.
Engine type
Choose type of internal combustion engine. Choose between Spark-ignition, the default option, and Diesel. Selecting Normalized 3rd-order polynomial matched to peak power for the Model parameterization parameter enables this parameter.
Maximum power
Maximum power Pmax that the engine can output. The default is 150 kW. Selecting Normalized 3rd-order polynomial matched to peak power for the Model parameterization parameter enables this parameter.
Speed at maximum power
Engine speed Ω0 at which the engine is running at maximum power. The default is 4500 rpm. Selecting Normalized 3rd-order polynomial matched to peak power for the Model parameterization parameter enables this parameter.
Maximum speed
Maximum speed Ωmax at which the engine can generate torque. The default is 6000 rpm. Selecting Normalized 3rd-order polynomial matched to peak power for the Model parameterization parameter enables this parameter.
During simulation, if Ω exceeds this maximum, the simulation stops with an error. The engine maximum speed Ωmax cannot exceed the engine speed at which the engine power becomes negative.
Stall speed
Minimum speed Ωmin at which the engine can generate torque. The default is 500 rpm. Selecting Normalized 3rd-order polynomial matched to peak power for the Model parameterization parameter enables this parameter.
During simulation, if Ω falls below this minimum, the engine torque is blended to zero.
Speed vector
Vector of values of the engine function's independent variable, the speed Ω. The default is [500, 1000, 2000, 3000, 4000, 5000, 6000, 7000] rpm. Selecting Tabulated torque data or Tabulated power data for the Model parameterization parameter enables this parameter.
The first and last speeds in the vector are interpreted as the stall speed and the maximum speed, respectively. If the speed falls below the stall speed, engine torque is blended to zero. If the speed exceeds the maximum speed, the simulation stops with an error.
Torque vector
Vector of values of the engine function's dependent variable, the torque τ. The default is [380, 380, 380, 380, 350, 280, 200, 80] N*m. Selecting Tabulated torque data for the Model parameterization parameter enables this parameter.
Power vector
Vector of values of the engine function's dependent variable, the power P. The default is [20, 40, 78, 120, 145, 148, 125, 60] kW. Selecting Tabulated power data for the Model parameterization parameter enables this parameter.
Interpolation method
Method to interpolate the engine speed–torque or speed–power function between discrete relative velocity values within the range of definition. Choose between Linear, the default choice, and Smooth. Selecting Tabulated torque data or Tabulated power data for the Model parameterization parameter enables this parameter.
Dynamics
Inertia
Select how to model the rotational inertia of the engine block. Choose between these options, each of which enables other parameters:
• No inertia — Engine crankshaft is modeled with no inertia. This option is the default.
• Specify inertia and initial velocity — Engine crankshaft is modeled with rotational inertia and initial angular velocity.
Engine Inertia
Rotational inertia of the engine crankshaft. The default is 1 kg*m^2. Selecting Specify inertia and initial velocity for the Inertia parameter enables this parameter.
Initial velocity
Initial angular velocity Ω(0) of the engine crankshaft. The default is 800 rpm. Selecting Specify inertia and initial velocity for the Inertia parameter enables this parameter.
Time constant
Select how to model the time lag of the engine response. Choose between these options, each of which enables other options:
• No time constant — Suitable for HIL simulation — Engine reacts with no time lag. This option is the default.
• Specify engine time constant and initial throttle — Engine reacts with a time lag.
Engine time constant
Engine time lag. The default is 0.2 s.
Initial normalized throttle
Initial normalized engine throttle T(0), ranging between zero and one. The default is 0.
Limits
Speed threshold
Width of the speed range over which the engine torque is blended to zero as Ω approaches the stall speed. The default is 100 rpm.
Fuel Consumption
Fuel consumption model
Select model to specify engine fuel consumption. Models range from simple to advanced parameterizations compatible with standard industrial data. Choose between these options, each of which enables other options:
• Constant per revolution — The default option
• Fuel consumption by speed and torque
• Brake specific fuel consumption by speed and torque
• Brake specific fuel consumption by speed and brake mean effective pressure
Fuel consumption per revolution
Enter the volume of fuel consumed in one crankshaft revolution. The default is 25 mg/rev. Selecting Constant per revolution for the Fuel consumption model parameter enables this parameter.
Displaced volume
Enter the volume displaced by a piston stroke. The default is 400 cm^3.
Selecting Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
Revolutions per cycle
Enter the number of crankshaft revolutions in one combustion cycle — e.g. 2 for a four-stroke engine, or 1 for a two-stroke engine. The default is 2.
Selecting Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
Speed vector
Enter vector of engine speeds used in lookup table parameterizations. Vector size must match Torque vector size. The default is [1000, 2000, 3000, 4000, 5000, 6000] rpm. Selecting Fuel consumption by speed and torque, Brake specific fuel consumption by speed and torque, or Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
Torque vector
Enter vector of engine torques used in the lookup table parameterizations. Vector size must match Speed vector size. The default is [0, 80, 160, 200, 240, 320, 360, 400] N*m. Selecting Fuel consumption by speed and torque or Brake specific fuel consumption by speed and torque for the Fuel consumption model parameter enables this parameter.
Fuel consumption table
Enter matrix with fuel consumption rates corresponding to engine speed and torque vectors. The number of rows must equal the number of elements in the Speed vector. The number of columns must equal the number of elements in the Torque vector. The default is [.5, .9, 1.4, 1.6, 1.9, 2.7, 3.4, 4.4; 1, 1.7, 2.7, 3.1, 3.6, 5, 6, 7.4; 1.4, 2.7, 4, 4.8, 5.6, 7.5, 8.5, 10.5; 2, 3.6, 5.8, 6.7, 8, 10.4, 11.7, 13.3; 2.5, 4.8, 7.9, 9.4, 10.8, 14, 16.2, 18.6; 3.1, 6, 10.3, 11.9, 13.8, 18.4, 22, 26.5] g/s.
Selecting Fuel consumption by speed and torque for the Fuel consumption model parameter enables this parameter.
Brake mean effective pressure vector
Enter vector of brake mean effective pressure (BMEP) values. The default is [0, 250, 500, 625, 750, 1000, 1150, 1250] kPa. The BMEP satisfies the expression:
$BMEP=T\cdot \left(\frac{2\pi \cdot {n}_{c}}{{V}_{d}}\right)$
where:
• T — Output torque
• nc — Number of cycles per revolution
• Vd — Cylinder displaced volume
Selecting Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
Brake specific fuel consumption table
Selecting Brake specific fuel consumption by speed and torque or Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
For the Brake specific fuel consumption by speed and torque fuel model, enter the matrix with brake specific fuel consumption (BSFC) rates corresponding to engine speed and torque vectors. BSFC is the ratio of the fuel consumption rate to the output power. The number of rows must equal the number of elements in the Speed vector. The number of columns must equal the number of elements in the Torque vector.
For the Brake specific fuel consumption by speed and brake mean effective pressure fuel model, enter the matrix with brake specific fuel consumption (BSFC) rates corresponding to engine speed and brake mean effective pressure (BMEP) vectors. BSFC is the ratio of the fuel consumption rate to the output power. The number of rows must equal the number of elements in the Speed vector. The number of columns must equal the number of elements in the Brake mean effective pressure vector.
For both fuel-consumption models, the default is [410, 380, 300, 280, 270, 290, 320, 380; 410, 370, 290, 270, 260, 270, 285, 320; 415, 380, 290, 275, 265, 270, 270, 300; 420, 390, 310, 290, 285, 280, 280, 285; 430, 410, 340, 320, 310, 300, 310, 320; 450, 430, 370, 340, 330, 330, 350, 380] g/hr/kW.
Interpolation method
Select the interpolation method used to calculate fuel consumption at intermediate speed-torque values. Methods are Linear and Smooth. Outside the data range, fuel consumption is held constant at the last value given in the lookup table. Selecting Fuel consumption by speed and torque, Brake specific fuel consumption by speed and torque, or Brake specific fuel consumption by speed and brake mean effective pressure for the Fuel consumption model parameter enables this parameter.
Speed Control
Idle speed control
Select speed control model. The options are:
• No idle speed controller — Omit idle speed controller. Throttle input is used directly. This option is the default.
• Enable idle speed controller — Include idle speed controller to prevent engine stalling. This option enables other parameters. For more information, see Idle Speed Controller Model.
Idle speed reference
Enter the value of the speed reference below which speed increases, and above which speed decreases. The default is 1000 rpm.
Selecting Enable idle speed controller for the Idle speed control parameter enables this parameter.
Controller time constant
Enter the value of the time constant associated with an increase or decrease of the controlled throttle. The constant value must be positive. The default is 1 s.
Selecting Enable idle speed controller for the Idle speed control parameter enables this parameter.
Controller threshold speed
Parameter used to smooth the controlled throttle value when the engine’s rotational speed crosses the idle speed reference. For more information, see Idle Speed Controller Model. Large values decrease controller responsiveness. Small values increase computational cost. This parameter must be positive. The default is 1 rpm.
Selecting Enable idle speed controller for the Idle speed control parameter enables this parameter.
Redline control
Select redline control model. Options include No redline controller and Enable redline controller.
• No redline controller — Omit redline controller. Throttle depends only on the idle speed controller. This option is the default.
• Enable redline controller — Include redline controller to prevent excessive speed. This option enables other parameters.
Redline speed
Enter the value of the speed reference above which the redline control activates. The default is 5000 rpm.
Selecting Enable redline controller for the Redline control parameter enables this parameter.
Redline time constant
Enter the value of the time constant associated with an increase or decrease of the controlled throttle. The constant value must be positive. The default is 1 s.
Selecting Enable redline controller for the Redline control parameter enables this parameter.
Redline threshold speed
Specify the width of the region around the redline speed where the controller goes from fully enabled to not enabled. The block uses this parameter for smoothing the controlled throttle value when the engine’s rotational speed crosses the redline speed reference. Large values decrease controller responsiveness. Small values increase computational cost. This parameter must be positive. The default is 1 rpm.
Selecting Enable redline controller for the Redline control parameter enables this parameter.
Extended Capabilities
Real-Time and Hardware-in-the-Loop Simulation
For optimal simulation performance, set the Dynamics > Time Constant parameter to No time constant - Suitable for HIL simulation.
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2018-02-24 12:31:35
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https://www.proquest.com/pqdtglobal/docview/302919217
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ProQuest
Document Preview
## EXPERIMENTAL AND THEORETICAL STUDIES OF PARAMETRIC INSTABILITIES NEAR THE ION CYCLOTRON FREQUENCY IN SINGLE AND MULTI-ION SPECIES PLASMAS.
ONO, MASAYUKI. Princeton University. ProQuest Dissertations Publishing, 1978. 7818384.
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2022-01-24 07:14:20
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https://cdsweb.cern.ch/collection/ATLAS%20Preprints?ln=de&as=1
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# ATLAS Preprints
Letzte Einträge:
2019-01-14
10:31
Top-antitop charge asymmetry measurements in the dilepton channel with the ATLAS detector / Kido, Shogo (Kobe University) We report a measurement of the charge asymmetry $A_C$ in top quark pair production with the ATLAS experiment. [...] arXiv:1901.04242 ; ATL-PHYS-PROC-2019-009. - 2019. - 4 p. Original Communication (restricted to ATLAS) - Full text - Fulltext
2019-01-11
14:45
Measurement of the $t\bar{t}Z$ and $t\bar{t}W$ cross sections in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector / ATLAS Collaboration A measurement of the associated production of a top-quark pair ($t\bar{t}$) with a vector boson ($W$, $Z$) in proton-proton collisions at a center-of-mass energy of 13 TeV is presented, using 36.1 fb$^{-1}$ of integrated luminosity collected by the ATLAS detector at the Large Hadron Collider. [...] arXiv:1901.03584 ; CERN-EP-2018-331. - 2019. - 56 p. Fulltext - Previous draft version - Fulltext
2018-12-30
17:48
2018-12-23
23:02
Observation of electroweak $W^{\pm}Z$ boson pair production in association with two jets in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector / ATLAS Collaboration An observation of electroweak $W^{\pm}Z$ production in association with two jets in proton-proton collisions is presented. [...] arXiv:1812.09740 ; CERN-EP-2018-286. - 2018. - 41 p. Fulltext - Previous draft version - Fulltext
2018-12-23
22:52
Search for large missing transverse momentum in association with one top-quark in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector / ATLAS Collaboration This paper describes a search for events with one top-quark and large missing transverse momentum in the final state. [...] arXiv:1812.09743 ; CERN-EP-2018-301. - 2018. - 51 p. Fulltext - Previous draft version - Fulltext
2018-12-21
21:07
2018-12-21
17:17
Search for chargino and neutralino production in final states with a Higgs boson and missing transverse momentum at $\sqrt{s} = 13$ TeV with the ATLAS detector / ATLAS Collaboration A search is conducted for the electroweak pair production of a chargino and a neutralino $pp \rightarrow \tilde\chi^\pm_1 \tilde\chi^0_2$, where the chargino decays into the lightest neutralino and a $W$ boson, $\tilde\chi^\pm_1 \rightarrow \tilde\chi^0_1 W^{\pm}$, while the neutralino decays into the lightest neutralino and a Standard Model-like 125 GeV Higgs boson, $\tilde\chi^0_2 \rightarrow \tilde\chi^0_1 h$. [...] arXiv:1812.09432 ; CERN-EP-2018-306. - 2018. - 58 p. Fulltext - Previous draft version - Fulltext
2018-12-18
14:09
Search for single production of vector-like quarks decaying into $Wb$ in $pp$ collisions at $\sqrt{s} = 13$ TeV with the ATLAS detector / ATLAS Collaboration A search for singly produced vector-like quarks $Q$, where $Q$ can be either a $T$ quark with charge $+2/3$ or a $Y$ quark with charge $-4/3$, is performed in proton-proton collision data at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of $36.1 \text{fb}^{-1}$, recorded with the ATLAS detector at the LHC in 2015 and 2016. [...] arXiv:1812.07343 ; CERN-EP-2018-226. - 2018. - 55 p. Fulltext - Previous draft version - Fulltext
2018-12-16
10:56
Top quarks and exotics at ATLAS and CMS / Serkin, Leonid (INFN Gruppo Collegato di Udine and ICTP, Trieste) An overview of recent searches with top quarks in the final state using up to 36 fb$^{-1}$ of $pp$ collision data at $\sqrt{s}$ = 13 TeV collected with the ATLAS and CMS experiments at the LHC is presented. [...] arXiv:1901.01765 ; ATL-PHYS-PROC-2018-195. - 2018. - 6 p. Original Communication (restricted to ATLAS) - Full text - Fulltext
2018-12-14
09:37
Exotics at the LHC / Del Re, Daniele (INFN, Rome ; Rome U.) /ATLAS, CMS and LHCB Collaborations LHC has worked beautifully and provided more than 100 fb$^{-1}$ at 13 TeV. Thanks to this enormous statistics of p-p collisions, LHC experiments have been able to explore several different new physics scenarios. [...] CMS-CR-2018-392.- Geneva : CERN, 2018 - 11 p. Fulltext: PDF; In : XXXIX International Conference on High Energy Physics, Seoul, Kor, 4 - 11 Jul 2018
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2019-01-16 21:18:19
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https://www.hpmuseum.org/forum/thread-6108.html
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04-22-2016, 09:21 PM (This post was last modified: 04-22-2016 09:24 PM by SalivationArmy.)
Post: #1
SalivationArmy Junior Member Posts: 13 Joined: Apr 2016
Are there any models that are not neutered for school/testing use?
Since I am not in school and facing testing requirement I am wondering if there are any models that are better than their school oriented brethren. TI models are hardly more than glossy version of 80's tech. I think someone held a gun to their heads to get them to upgrade their screens.
I intend to start learning again as my son is entering high school this fall. I want to be able to help him get into the higher maths. but I also want to explore on my own outside of the typical education path.
i've looked into:
ti-92 series - old, but interesting
ti-nspire, got one, hated it. (not in same class as other here I know)
ti-8x, used many but only one I liked, just cannot remember which it was. fading memories.
HP 300s - no rpn as far as I could tell. non-programmable.
HP 35s - new yet old. but has RPN. programmable i think.
Casio fx991ex - seems to wipe itself everytime you turn it off. Non-programmable. FAST though. This is just what I was reading.
Sharp EL-W516XBSL - lots of built in functions (552 I think) but gcd missing in all but the german model which has nearly 100 more functions. I hate that you cannot get a full featured model in an english language because of school requirements.
HP Prime - I have considered some of these full graphing models, but my experience with the nspire has turned me off. How much does the expanded abilities this provides get in the way of lower level usage?
04-22-2016, 09:47 PM
Post: #2
rprosperi Senior Member Posts: 4,952 Joined: Dec 2013
(04-22-2016 09:21 PM)SalivationArmy Wrote: Are there any models that are not neutered for school/testing use?
Can you clarify a few things, from your comments it's a bit hard to tell for sure:
1. Do you prefer RPN?
2. Have you used RPL (advanced flavor of RPN with far more power, but also more complex to learn)?
3. Do you plan to program the machine or is a rich set of built-in commands preferable?
4. Would you be open to a community sourced solution? (running advanced s/w on existing commercial machines)
5. What application areas are you interested in - EE, ME, CE, Pure Math, Statistics, Probability, etc.?
6. Will you use a CAS if available?
7. What is the best machine you've used in the past?
These answers will make it easier to provide advice useful for what you want (vs. what all of us prefer to promote )
--Bob Prosperi
04-22-2016, 10:09 PM
Post: #3
SalivationArmy Junior Member Posts: 13 Joined: Apr 2016
(04-22-2016 09:47 PM)rprosperi Wrote:
(04-22-2016 09:21 PM)SalivationArmy Wrote: Are there any models that are not neutered for school/testing use?
Can you clarify a few things, from your comments it's a bit hard to tell for sure:
1. Do you prefer RPN?
I have seen it used and I would like to learn it.
2. Have you used RPL (advanced flavor of RPN with far more power, but also more complex to learn)?
never heard of it.
3. Do you plan to program the machine or is a rich set of built-in commands preferable?
built in would be preferred but I am not against adding function through programming, but the more that is built in the less I have to program.
4. Would you be open to a community sourced solution? (running advanced s/w on existing commercial machines)
Please explain. do you mean custom firmware? if so, hell yes
5. What application areas are you interested in - EE, ME, CE, Pure Math, Statistics, Probability, etc.?
all of it, this is purely for learning's' sake. Though I know for a fact that I hate statistics as it has been taught to me. When you don't understand the value of a particular answer, it's hard to understand the problems themselves. My favorite example of this was from back in grade school. the teacher was instructing us on the use of pi in geometry. To her pi was just a number. she had no concept of WHAT pi was, what it represented, just memorize the damn formulas and stop asking hard questions damn it! Next day after I had figured out what it was, I shared it with the class and all of the sudden they all understood what we had been learning. She was very upset with me for interfering.
6. Will you use a CAS if available?
yes
7. What is the best machine you've used in the past?
a ti model that I forget. it was an 83 or 84 I think. at the time I had no clue that models were geared towards one type of math or another, you just bought the highest model number you could afford, which was stupid in retrospect. I think an 82 would have served best at that time (out of the TI's)
These answers will make it easier to provide advice useful for what you want (vs. what all of us prefer to promote )
04-23-2016, 01:32 AM
Post: #4
Dave Britten Senior Member Posts: 1,915 Joined: Dec 2013
For HP, I typically recommend a 48G (or GX), or 48SX. The 49 and 50 added lots of features, but at great expense to usability, and also the physical design.
If you want a TI, try an 86. That was pretty much the last time they designed a calculator geared more toward engineers than students.
04-23-2016, 01:35 AM
Post: #5
Garth Wilson Senior Member Posts: 467 Joined: Dec 2013
Although computers have dramatically sped up the processing in recent decades, the math itself was has not really advanced. It was already understood. High-school students did better in math before there were calculators.
Our sons were supposed to get TI graphing calculators for their high-school math classes. The older one just used my old TI-59 (which came out in 1977 or '78) and did fine. He said the only thing the students used their graphing calcs for was playing games anyway. Three years later, my wife caved in and bought our younger son the required graphing calc, and he never really used it for graphing, and aced the classes anyway.
A college student asked me about things like FTTs and convolution integrals, trying to get past the ultra-sterile theory they get in school and to get a practical understanding of what they do; so I explained them, in English, and showed him things I was doing in my work, doing these with thousands of points. He didn't understand why, because in school they were only doing very few points (like eight), just enough to do the function. The reason of course was that in a math class you won't have instrumentation collecting thousands of data points super quickly and inputting them to a computer, and you don't need it anyway to do the math. It's a math class.
04-23-2016, 01:53 AM
Post: #6
Steve Simpkin Senior Member Posts: 601 Joined: Dec 2013
If you want RPN (RPL) and CAS capability you are pretty much restricted to the HP-50G and the HP Prime for models that you can still buy new.
04-23-2016, 02:29 AM
Post: #7
SalivationArmy Junior Member Posts: 13 Joined: Apr 2016
with the price difference between the 50g and the Prime being only $20(msrp) is there much reason to get the 50g over the prime? 04-23-2016, 03:27 AM Post: #8 Steve Simpkin Senior Member Posts: 601 Joined: Dec 2013 RE: Purchase advice questions (04-23-2016 02:29 AM)SalivationArmy Wrote: with the price difference between the 50g and the Prime being only$20(msrp) is there much reason to get the 50g over the prime?
At the moment, the HP-50G is US$52 vs the Prime at US$139 at Amazon.com.
Aside from the price difference, they are very different calculators.
The Prime is more student focused while the 50G is more of a hand tool.
The Prime has a color, back-lit touch screen display and needs to be charged every couple of weeks or so. The 50G has a low resolution B/W screen and will run for months on AA batteries.
These are just a few of the differences. You can download the emulators and manuals for both to try them out.
04-23-2016, 04:11 AM
Post: #9
SalivationArmy Junior Member Posts: 13 Joined: Apr 2016
i just figured out which calculator I was using in college. it was the ti-85
04-25-2016, 01:32 PM (This post was last modified: 04-25-2016 01:34 PM by Ron Ross.)
Post: #10
Ron Ross Junior Member Posts: 47 Joined: Mar 2014
As suggested by Dave Britton, a Ti-86 might be just the ticket for you. It is the direct upgrade of the Ti-85 (which was Ti's first real competition to the Hp 48S series calculator). It has 128K of RAM and is on par with the Hp 48G+ calculator.
If you want RPN and power, the Hp 50G is the best option (and it has an algebraic mode, in case you decide to avoid RPN).
For your son, I would still suggest an Hp Prime (Hp's answer to the Ti Nspire). It is better and more versatile and boots up in a second (the Ti seems to take about 10 seconds, both are annoying, the Ti is unbearable as a calculator). Why an Hp Prime? Because Hp seems to be committed to this calculator vs they are dropping the Hp 50G, so you son may not readily find one later in school if his first calculator is broken, lost or stolen. And the Hp Prime has an excellent quality feel, reminiscent of the best calculators Hp made during the 80s and 90s, certainly better than anything Hp has made in the last 15 years (feels much better than the Hp 50G, quality wise).
If you decide to just buy a calculator (and not a graphing model, for yourself or your son), the Hp 35s is still readily available and is pretty good. No I/O, but that is intentional to keep it test compliant.
If you are or become a calculator connoisseur, the Hp 32sii would be a great choice. However, these are horded by nearly all of us, and usually sell for a premium. This is the predecessor the Hp 35s. It has FAR LESS RAM, so why would I suggest this? It is a twenty year + calculator ie if you were to buy in like new condition, it should last for 20+ years (where the Hp 35s would probably last about 3-5 years of college engineering use). The Pioneer line (Hp 32sii, among others) calculators were an excellent calculator line, second only to the Voyager line in build and quality. Then why didn't I recommend a Voyager? Voyagers have a landscape format with an older keycode programming paradigm. Both families are the best calculators ever made, quality wise. Coming in a close second to the Voyager line is damn good. The Hp 12c is a Voyager line calculator still in production (over 35 years in production as an FYI).
04-25-2016, 01:47 PM
Post: #11
Ron Ross Junior Member Posts: 47 Joined: Mar 2014
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2021-10-23 20:03:52
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https://thefeechochamber.wordpress.com/
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# Chickens Doing Well | charliecountryboy
So I moved the chickens off the lawn and they seem to thriving, and of course it doesn’t matter if they chow up the grass in their new place. If your a bit lost then you might want to read Me, Chickens and Pheromones Oh and the Dahlias are out too, so all is well
http://ift.tt/2vbO16U
# Wrong, But Useful: Episode 47 | Colin
In this month’s episode of Wrong, But Useful, we are joined by Special Guest Co-Host @jussumchick, who is Jo Sibley in real life. Colin’s audio is unusually hissy in this one, which is why it’s a little late; he apologises for both inconveniences.
We discuss:
Jo’s background and work with FMSP, and how she has jumped to incorrect conclusions about statistics.
Number of the podcast: $\ln^5(2)$, which is approximately 0.160003
Factris and Sumaze
Trial and Improvement, and numerical methods
New A-level syllabus, technology in teaching, new GCSEs, new Core Maths.
Maths in “real life” and…
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# Collecting coupons | Colin
For all the grief I give @reflectivemaths on Wrong, But Useful, he does occasionally ask an interesting question.
In episode 45, he wondered how many packs of Lego cards one would need to acquire, on average, to complete the set of 140?1
A simpler case
Suppose, instead of 140 cards, there was a total of three to collect.
You start (obviously) with no cards. On average, you need to pick up a single card to make sure you have one card in your collection. By on average, I mean ‘every time’. The expected time (in cards) to go from 0 cards to 1 card is 1.
If you have one card, how long…
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# I aten’t dead… | Paul
… in the words of Granny Weatherwax.
But a lot has happened to keep me from updating the site. The vague something I alluded to in the last post is not something I can speak much about after all, suffice to say it was unpleasant, and over now.
But amongst the changes are: bought a house, moved north, starting a new job tomorrow!
After nearly 13 years the London experiment is over. Welcome to the North. They have pie…
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# Challenging IQ. | Eddie Playfair
Behavioural genetics; the clue to the difficulty is in the name. As with Sociobiology and Evolutionary Psychology before it, the squashing together of two very different levels of understanding into a single discipline creates a real problem. Genetics and psychology are both respectable fields of study with their different methodologies and evidence bases but they … Continue reading Challenging IQ.
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# A teacher is an authoritative guide | Mike Tyler
I’ve done a spot of travelling in my time, but the most enjoyable trips I’ve made, the ones which have enriched me the most, have been those trips where I’ve had a knowledgeable guide. I’ve had a tour of the Vatican. I’ve been shown round Florence. I was even guided around Israel for a couple of weeks by a pastor and theologian from Nazareth. What a lot I learned!
He took us to the crusader Church of Saint Anne at Bethesda where we sang a hymn, because, he told us, the acoustics are superb. (They are. We drew a decent crowd. Of nuns.) We learnt the comic-tragic tale of the Immovable Ladder….
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# NewVIc results 2017 | Eddie Playfair
Students and staff at Newham Sixth Form College (NewVIc) are celebrating another year of improvement in A-level pass rates and top grades, all of which have continued to increase faster than nationally. NewVIc’s A-level pass rate is up 1% on last year at 98% and is the highest ever for the college. The proportion of … Continue reading NewVIc results 2017
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2017-08-23 00:43:14
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https://www.springerprofessional.de/obstacle-avoidance-of-mobile-robots-using-modified-artificial-po/16558632?fulltextView=true
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01.12.2019 | Research | Ausgabe 1/2019 Open Access
Obstacle avoidance of mobile robots using modified artificial potential field algorithm
Zeitschrift:
EURASIP Journal on Wireless Communications and Networking > Ausgabe 1/2019
Autoren:
Seyyed Mohammad Hosseini Rostami, Arun Kumar Sangaiah, Jin Wang, Xiaozhu Liu
Abbreviations
APF
Artificial potential field
MSD
Mobile sensor deployment
MSNs
Mobile sensor networks
NCON
Network CONnectivity
SDRE
State-dependent Riccati equation
TCOV
Target COVerage
VHF
Vector histogram field
1 Introduction
An intelligent mobile robot must reach the designated targets at a specified time. The robot must determine its location relative to its objectives at each step and issue a suitable strategy to achieve its target. Information about the environment is also needed to avoid obstacles and design optimal routes. In general, the objectives of this study are (1) improving the APF algorithm in order to trace the path and avoid obstacles by a mobile robot and (2) comparing the performance and efficiency of the proposed algorithm with the previous method, APF algorithm. In the real world, it gets a lot of use. For example, we have an automatic ship that we want to avoid collision with obstacles and to take the best possible route for a low cost. It is possible to detect the distance between the robot with the target and obstacles with the help of the sensor. However, the sensor discussion is separate from this article.
Mobile intelligent robot is a useful tool which can lead to the target and at the same time avoid obstacles facing it [1]. There are conventional methods of obstacle avoidance such as the path planning method [2], the navigation function method [3], and the optimal regulator [4]. The first algorithm that was proposed for the discussion of obstacle avoidance is the Bug1 algorithm. This algorithm is perhaps the simplest obstacle avoidance algorithm. In the first algorithm, Bug1, as shown in Fig. 1, the robot completely bypassed the first object, and then, it leaves the first object and performs at the shortest distance from the target. Of course, this approach is very inefficient, but it ensures that the robot is reachable for any purpose. In this algorithm, the current sensor readings play a key role. The weakness of this algorithm is the excessive presence of the robot alongside the obstacle [5].
In the second algorithm, Bug2, as shown in Fig. 2, the robot moves on the starting lineup to the target, and if it sees the obstacle, it will round the obstacle, and when it again reaches a point on the line between the start to the target, it leaves an obstacle. In this algorithm, the robot spends a lot of time moving along the obstacle, but this time is less than the previous algorithm [5].
In the following, Mr. Khatib introduced an algorithm called APF in 1985 [6]. This algorithm considers the robot as a point in potential fields and then combines stretching toward the target and repulsion of obstacles. The final path of the output is the intended path. This algorithm is useful given that the trajectory is obtained by quantitative calculations. The problem is that in this algorithm, the robot can be trapped in the local minimum of potential fields and cannot find the path; hence, in this paper, some amendments have been made to resolve this issue, which is discussed in Section 3. Subsequently, in March 1991, Borenstein and Koren presented vector histogram field algorithm (VHF) [7]. In this algorithm, with the help of one of the distance sensors, a planar map of the robot environment is prepared; in the next step, this planar map is translated into a polar histogram. This histogram is shown in Fig. 3, where the x-axis represents the angles around the obstacle and the y-axis represents the probability of an obstacle at the desired viewing angles. In this polar map, the peaks represent bulky objects and valleys represent low-volume objects; valleys below the threshold are suitable valleys for moving the robot; finally, the suitable valley which has a smaller distance to the target is selected from the valleys, and the position of this valley determines the angles of motion and speed of the robot [7].
Improvements on VFH led to the introduction of VFH and VFH+ [8]. The next algorithm is the sensor fusion in certainty grids for mobile robots which is one of the well-known, powerful, and efficient algorithms for using a multi-sensor combination to identify the environment and determine the direction of movement. In this algorithm, the robot’s motion space is decomposed into non-overlapping cells and is provided with a graphical motion environment, and then based on the algorithms for searching the graph, the path of the movement is determined [9]. In the following, the Follow the Gap method (2012) was proposed [10]; in this algorithm, the robot calculates its distance with each obstacle. The robot can know the distance between obstacles by knowing its distance with each obstacle. The robot then detects the greatest distance between obstacles and calculates the center angles of the empty space. After the central angular momentum, the robot calculates the empty space, combines it with the target angle, and shows the ending angle. Obviously, in this combination, it is done on a weight basis, so that the obstacles that are closer to the robot are weighing more because their avoidance is a priority for the robot. Figure 4 shows how this method works.
Fuzzy logic and neuro-fuzzy networks can also be used to avoid obstacles. In this regard, fuzzy logic can be helpful in solving the problems caused by the ocean flow in the underlying surfaces [11]. It should be noted that the algorithms discussed in the research background for avoiding an obstacle are different from each other, and each algorithm has proposed a separate and new method for avoiding obstacles. So far, novel methods to avoid obstacles in addition to the mentioned methods are also presented [1216]. In the following section, we first investigate the artificial potential field algorithm and then modify this algorithm in Section 3. In [17], the state-dependent Riccati equation (SDRE) algorithm is studied on the motion design of cable-suspended robots with uncertainties and moving obstacles. A method for controlling the tracing of a robot has been developed by the SDRE. In [18], the fuzzy logic method is used to predict the movement of obstacles and accelerate the linear velocity of the robot. In this paper, the movement trend of the obstacle in approaching the robot is divided away from the robot and translated by the robot. With respect to the three trends, the robot can increase the velocity, decrease the velocity to pass by the obstacle, or stop to wait for the obstacle to pass. Some simulation results show that the proposed method can help the robot avoid an obstacle without changing the initial path. In [19], we have a four-dimensional unmanned aerial vehicle’s platform equipped with a miniature computer equipped with a set of small sensors for this work. The platform is capable of accurately estimating the precision mode, tracking the speed of the user, and providing uninterrupted navigation. The robot estimates its linear velocity by filtering a Kalman filter from the indirect and optical flow with the corresponding distance measurements. In [20], a new method for defining the path in the presence of obstacles is proposed, which describes the curve as a two-level intersection. The planner, based on the definition of the path along with the cascaded control architecture, uses a non-linear control technique for both control loops (position and attitude) to create a framework for manipulating multivariate behavior. The method has been shown to be able to create a safe path based on perceived obstacles in real time and avoiding collisions. Li et al.[21] refer to the tracking control of Euler-Lagrange system problems with external impediments in an environment containing obstacles. According to a new sliding discovery, a new tracking controller is proposed to determine the tracing, convergent errors reach zero as infinite time. In addition, based on the non-destructive sliding terminal, a simultaneous control algorithm with time constraints has also been developed to ensure that tracking errors are approaching a restricted area close to the source at a given time. By introducing multi-purpose collision avoidance functions, both controllers can ensure that the obstacle is avoided. In [22], the mobile sensor deployment (MSD) problem had been studied in mobile sensor networks (MSNs), aiming at deploying mobile sensors to provide target coverage and network connectivity with requirements of moving sensors. This problem is divided into two sub-problems, the Target COVerage (TCOV) problem and Network CONnectivity (NCON) problem. For the TCOV problem, it is proved that it is NP-hard. For a special case of TCOV, an extended Hungarian method is provided to achieve an optimal solution; for general cases, two heuristic algorithms are proposed based on the clique partition and Voronoi diagram, respectively. For the NCON problem, at first, an edge constrained Steiner tree algorithm is proposed to find the destinations of mobile sensors, and then, we used extended Hungarian to dispatch rest sensors to connect the network. Theoretical analysis and simulation results have shown that compared to extended Hungarian algorithm and basic algorithm, the solutions based on TV-Greedy have low complexity and are very close to the optimum. In [23], a novel approach is presented to obtain the optimal performance bounds for a multi-hop multi-rate wireless data network. First, the optimal relay placements are determined for a target terminal located at a distance D away from the access point. Second, for a general analytical PHY layer throughput model, the maximum achievable MAC throughput is determined as a function of the number of relays for a target located at distance D.
In all the research mentioned above, the local minimum problem has not been solved, and we designed the modified APF method to solve this problem. In this paper, in the second part, firstly, the previous method of APF algorithm is presented. Then, in Section 3, the APF method is modified to remedy the defects of this method.
2 Problem statement and formulation
2.1 Methods
The artificial potential field model suggested by Khatib [6] is a typical field model (Fig. 5). In the artificial potential field model, T represents the target for the robot to generate attraction and O represents the obstacles that produce repulsion for the robot.
In a planar space, the problem of avoiding the collision of a robot with an obstacle O is shown in Fig. 5. If XD indicates the target position, the control of the robot with respect to the obstacle O can be done in the artificial potential as follows:
$${U}_{\mathrm{A}\mathrm{LL}}(X)={U}_{\mathrm{A}}(X)+{U}_{\mathrm{R}}(X)$$
(1)
where UA, UR, and UALL respectively represent attraction potential energy, repulsive potential energy, and artificial potential field. Then its gradient function can be written as follows:
$${F}_{\mathrm{A}\mathrm{LL}}={F}_{\mathrm{A}}+{F}_{\mathrm{R}}$$
(2)
$${F}_{\mathrm{A}}=-\operatorname{grad}\left[{U}_{\mathrm{A}}(X)\right]$$
(3)
$${F}_{\mathrm{R}}=-\operatorname{grad}\left[{U}_{\mathrm{R}}(X)\right]$$
(4)
where FA is the attraction generated by the robot to reach the target position of XD and FR represents the force created by UR(X), which is caused by the repulsion of the obstacle. FA is proportional to the distance between the robot and the target. Then, the attraction kS factor is also considered, and the attraction potential UA(x) field is simply obtained as follows:
$${U}_{\mathrm{A}}=\frac{1}{2}{k}_{\mathrm{S}}{R_{\mathrm{A}}}^2$$
(5)
Additionally, UR(X) is a non-negative, non-linear function that is continuous and differentiable, and its potential penetration must be limited to a particular region around the obstacle without undesired turbulence forces. Therefore, the equation Urep(X) is as follows:
$${U}_{\mathrm{R}}(X)=\left\{\begin{array}{cc}0.5Z{\left(\frac{1}{R_{\mathrm{R}}}-\frac{1}{G_0}\right)}^2& {R}_{\mathrm{R}}\le {G}_0\\ {}0& {R}_{\mathrm{R}}>{G}_0\end{array}\right.$$
(6)
where X = (x, y) is the position of the robot, XOB = (xOB, yOB) is the position of the obstacles, and XD = (xD, yD) is the target position. $${R}_{\mathrm{A}}=\sqrt{{\left(X-{X}_{\mathrm{D}}\right)}^2+{\left(Y-{Y}_{\mathrm{D}}\right)}^2}$$ is the shortest distance between a robot and a target in a planar space, Z is the repulsive increase factor, $${R}_{\mathrm{R}}=\sqrt{{\left(X-{X}_{\mathrm{OB}}\right)}^2+{\left(Y-{Y}_{\mathrm{OB}}\right)}^2}$$ is the shortest distance between the robot and obstacles in the planar space, and G0 represents a safe distance from the obstacles [24]. According to the kinetic theory, the relation G0 ≥ VMAX/2AMAX is used, where VMAX denotes the maximum speed of the robot and AMAX represents the maximum speed of the acceleration (negative acceleration). So attraction and repulsion functions are written as follows:
$${F}_{\mathrm{A}}=-\nabla \left[\frac{1}{2}{k}_{\mathrm{S}}{R_{\mathrm{A}}}^2\right]={k}_{\mathrm{S}}{R}_{\mathrm{A}}$$
(7)
$${F}_{\mathrm{R}}(X)=\left\{\begin{array}{cc}Z\left(\frac{1}{R_{\mathrm{R}}}-\frac{1}{{R_{\mathrm{R}}}^3}\right)\frac{1}{{R_{\mathrm{R}}}^3}& {R}_{\mathrm{R}}\le {G}_0\\ {}0& {R}_{\mathrm{R}}>{G}_0\end{array}\right.$$
(8)
It is assumed that φ is the angle between the X ‐ axis and the line from the point of the robot to the obstacle. Then, both the repulsion components in the direction of the X ‐ axis and the Y ‐ axis can be obtained. Given θ is the angle between the X ‐ axis and the line from the point of the robot to the target, the attraction components in the X ‐ axis and the Y ‐ axis are considered as the following equations [24]:
$${F}_{\mathrm{R}\mathrm{x}}\left(X,{X}_{\mathrm{OB}}\right)={F}_{\mathrm{R}}\left(X,{X}_{\mathrm{OB}}\right)\cos \varphi$$
(9)
$${F}_{\mathrm{R}\mathrm{y}}\left(X,{X}_{\mathrm{OB}}\right)={F}_{\mathrm{R}}\left(X,{X}_{\mathrm{OB}}\right)\sin \varphi$$
(10)
$${F}_{\mathrm{A}\mathrm{x}}\left(X,{X}_{\mathrm{D}}\right)={F}_{\mathrm{A}}\left(X,{X}_{\mathrm{D}}\right)\cos \theta$$
(11)
$${F}_{\mathrm{A}\mathrm{y}}\left(X,{X}_{\mathrm{D}}\right)={F}_{\mathrm{A}}\left(X,{X}_{\mathrm{D}}\right)\sin \theta$$
(12)
First, the force of the repulsion and attraction in the g-axis and the v-axis are calculated, and then, f is the angle between the resulting force and the d-axis. Finally, the angle of the steering command is calculated as follows:
$${F}_{\mathrm{x}}={F}_{\mathrm{Ax}}\left(X,{X}_{\mathrm{D}}\right)+{F}_{\mathrm{Rx}}\left(X,{X}_{\mathrm{OB}}\right)$$
(13)
$${F}_{\mathrm{y}}={F}_{\mathrm{Ay}}\left(X,{X}_{\mathrm{D}}\right)+{F}_{\mathrm{Ry}}\left(X,{X}_{\mathrm{OB}}\right)$$
(14)
$$\left\{\begin{array}{c}\delta =\arctan \frac{F_{\mathrm{y}}}{F_{\mathrm{x}}},{F}_{\mathrm{x}}>0\\ {}\delta =\pi +\arctan \frac{F_{\mathrm{y}}}{F_{\mathrm{x}}},{F}_{\mathrm{x}}\le 0\end{array}\right.$$
(15)
The next position of the robot can be continuously calculated according to the following function until it reaches the convergence condition:
$$\left\{\begin{array}{c}{X}_{\mathrm{f}}=X+L\times \cos \delta \\ {}{Y}_{\mathrm{f}}=Y+L\times \sin \delta \end{array}\right.$$
(16)
where L denotes the step size and Xf and Yf represented the next position of the robot. Then, if the obstacles are simple particles, the artificial potential field model is used to secure the robot to the point of the target. However, there are several problems. First, the result from the sum of the two forces of attraction and repulsion in some places is zero, which results in stopping the robot moving or wandering around those points that are called local minimum. Second, when the target is surrounded by obstacles, the path cannot converge, and the robot cannot reach the target. In order to overcome these shortcomings, Section 3 addresses the modification of the artificial potential field algorithm.
3 Algorithm design
Researchers around the world have been researching about artificial potential field defects and have presented suggestions for improving this method [2537]. Here, a regulative agent has been added to improve the artificial potential field algorithm with the target of overcoming the minimum local and inaccessible target. When the robot is close to the target, this regulative agent reduces the attraction control as a linear function, and the repulsion decreases as a higher-order function (M ≥ 3). A flowchart illustrating the steps of the modified APF algorithm is given in Fig. 6.
3.1 Modified artificial potential field model
In the planar space, the modified function of the attraction field function is defined as follows:
$${U}_{\mathrm{A}}=\frac{1}{2}{k}_{\mathrm{S}}{R_{\mathrm{A}}}^2$$
(17)
The modified attraction field function is written as follows:
$${U}_{\mathrm{R}}(X)=\left\{\begin{array}{cc}0.5Z{\left(\frac{1}{R_{\mathrm{rep}}}-\frac{1}{G_0}\right)}^2{R_{\mathrm{A}}}^N& {R}_{\mathrm{rep}}\le {G}_0\\ {}0& {R}_{\mathrm{rep}}>{G}_0\end{array}\right.$$
(18)
where X = (x, y) is the position of the robot, Xob = (xob, yob) is the position of the obstacles, and Xd = (xd, yd) is the target position. RA is the shortest distance between the robot and the target in the planar space, and RAN is the regulative factor. The attraction function is the gradient across the attraction field, which is obtained as follows:
$${F}_{\mathrm{A}}=-\nabla \left[\frac{1}{2}{k}_{\mathrm{S}}{R_{\mathrm{A}}}^2\right]={k}_{\mathrm{S}}{R}_{\mathrm{A}}$$
(19)
In the same way, the repulsive function is a negative gradient from the field of repulsion, which is obtained as follows:
$${F}_{\mathrm{R}}=\left\{\begin{array}{cc}Z\times {F}_{\mathrm{R}1}(X)+M\times Z\times {F}_{\mathrm{R}2}(X)& {R}_{\mathrm{R}}\le {G}_0\\ {}0& {R}_{\mathrm{R}}>{G}_0\end{array}\right.$$
(20)
As shown in Fig. 7, in the modified model, FR is decomposed into FR1 and FR2, where FR1 is the component’s force in the direction of the line between the robot and the obstacle and FR2 is the component’s force in the direction of the line between the robot and the target.
$$\left\{\begin{array}{c}{F}_{\mathrm{R}1}=\left(\frac{1}{R_{\mathrm{R}}}-{\frac{1}{G}}_0\right)\frac{{R_{\mathrm{A}}}^M}{{R_{\mathrm{A}}}^3}\\ {}{F}_{R2}={\left(\frac{1}{R_{\mathrm{R}}}-{\frac{1}{G}}_0\right)}^2{R_{\mathrm{A}}}^M\end{array}\right.$$
(21)
The two components of repulsion and attraction in the direction of the x-axis and the y-axis can be obtained as follows [24]:
$${F}_{\mathrm{R}\mathrm{x}}\left(X,{X}_{\mathrm{OB}}\right)={F}_{\mathrm{R}}\left(X,{X}_{\mathrm{OB}}\right)\cos \varphi$$
(22)
$${F}_{\mathrm{R}\mathrm{y}}\left(X,{X}_{\mathrm{OB}}\right)={F}_{\mathrm{R}}\left(X,{X}_{\mathrm{OB}}\right)\sin \varphi$$
(23)
$${F}_{\mathrm{A}\mathrm{x}}\left(X,{X}_{\mathrm{D}}\right)={F}_{\mathrm{A}}\left(X,{X}_{\mathrm{D}}\right)\cos \theta$$
(24)
$${F}_{\mathrm{A}\mathrm{y}}\left(X,{X}_{\mathrm{D}}\right)={F}_{\mathrm{A}}\left(X,{X}_{\mathrm{D}}\right)\sin \theta$$
(25)
The force of the repulsion and attraction on X ‐ axis and Y ‐ axis are calculated, and the angle δ is calculated between the force and the X ‐ axis, where δ is the steering angle of the robot.
$${F}_{\mathrm{x}}={F}_{\mathrm{Ax}}\left(X,{X}_{\mathrm{D}}\right)+{F}_{\mathrm{Rx}}\left(X,{X}_{\mathrm{OB}}\right)$$
(26)
$${F}_{\mathrm{y}}={F}_{\mathrm{Ay}}\left(X,{X}_{\mathrm{D}}\right)+{F}_{\mathrm{Ry}}\left(X,{X}_{\mathrm{OB}}\right)$$
(27)
$$\left\{\begin{array}{c}\delta =\arctan \frac{F_{\mathrm{y}}}{F_{\mathrm{x}}},{F}_{\mathrm{x}}>0\\ {}\delta =\pi +\arctan \frac{F_{\mathrm{y}}}{F_{\mathrm{x}}},{F}_{\mathrm{x}}\le 0\end{array}\right.$$
(28)
$$\left\{\begin{array}{c}{X}_{\mathrm{f}}=x+L\times \cos \delta \\ {}{Y}_{\mathrm{f}}=y+L\times \sin \delta \end{array}\right.$$
(29)
where L represents the step size and Xf and Yf represent the next position of the robot. The regulative factor RAM is added to the modified model of this article. The function of this regulative factor is that if the robot cannot reach the target and M is in the interval (0, 1), FR1 when the robot is close to the target, it tends to be infinite and the path of convergence is created only with FR2 and FA. If M = 1, Frep2 tends to be constant and FR1 will be zero, and the path of convergence will be created only with FR2 and FA. If M = 0, FR1 and FR2 will all tend to be zero, and the path of convergence will be created only by attraction. In the final state, generally, if M is a real positive number, the robot does not face the local minimum cases or the inaccessibility of the target.
4 Simulation results and analysis
In this section, a comparison is firstly done between the simulation results of the artificial potential field algorithm and the modified artificial potential field, then the obstacle avoidance of the mobile robot is displayed against a variety of obstacles that are simulated using the modified artificial potential field algorithm.
In the artificial potential field algorithm in which its relation was said in Section 2, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 0.11, and L = 0.1 and obstacles on a circle of the radius R = 0.18, the result is as shown in Fig. 8; the robot is trapped and stands at the local minimum. Figure 9 shows that the repulsive force is more than the attraction at all stages, which means the robot is trapped at the local minimum. If these parameters are changed to any desired extent other than these values, there will be no change to the status of the robot at the local minimum. The mesh plot of this simulation as shown in Fig. 10. It describes the obstacles like summit, target act cavity and absorb the robot inside itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 0.11, and L = 0.1 and obstacles on a circle of the radius R = 0.18, the result is as shown in Fig. 11. By comparing Figs. 8 and 11, it can be seen that in the artificial potential field algorithm, the robot cannot reach the target and is trapped in a local minimum, but with the same conditions, in the modified artificial potential field algorithm, it achieves the target by avoiding obstacles.
Figure 12 shows that with the number of steps of nearly 200 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 13 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 5, and L = 0.1 and obstacles on a circle of the radius R = 40, the result is as shown in Fig. 14. It shows that the robot achieves the target by avoiding obstacles.
Figure 15 shows that with the number of steps of nearly 400 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 16 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 2.1, and L = 0.5 and obstacles on a circle of the radius R = 7, the result is as shown in Fig. 17. It shows that the robot achieves the target by avoiding obstacles.
Figure 18 shows that with the number of steps of nearly 700 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 19 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be. considered as Z = 100, M = 2, kS = 1.1, G0 = 0.2, and L = 0.03 and the obstacle is considered as $$Y={\left(\frac{1}{X}\right)}^2+0.5;X\in \left(0.7,3.5\right)$$, the result is as shown in Fig. 20. It shows that the robot achieves the target by avoiding obstacles.
Figure 21 shows that with the number of steps of nearly 1000 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 22 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 0.3, and L = 0.1 and the obstacle is considered as X = 4; 4 ≤ Y ≤ 7, the result is as shown in Fig. 23. It shows that the robot achieves the target by avoiding obstacles.
Figure 24 shows that with the number of steps of nearly 900 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 25 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
In the modified artificial potential field algorithm in which its relation was said in Section 3, if the value to the parameter be considered as Z = 100, M = 2, kS = 1.1, G0 = 0.2, and L = 0.03 and the obstacle is considered as Y = 4; 4 ≤ X ≤ 6 and x = 4; 3 ≤ y ≤ 4, the result is as shown in Fig. 26. It shows that the robot achieves the target by avoiding obstacles.
Figure 27 shows that with the number of steps of nearly 700 rounds, the attraction force is zeroed; that means, with this number of steps, the robot reaches the target. The mesh plot of this simulation in Fig. 28 shows that the obstacles like the summit and the target act like the cavity and absorb the robot into itself.
5 Conclusion
In this paper, improvement of the artificial potential field algorithm has been evaluated. If any obstacles are around the target, it allows the robot to find a safe path, move without collision with the obstacle, and not trapped at the local minimum then reach the target point. The artificial potential field is a relatively mature algorithm that is widely used for its math calculations. However, due to the local minimum problem in this algorithm, the robot cannot achieve the target, so in order to solve this problem, a new method is proposed in this paper to remedy this algorithm. The proposed method is simulated in the MATLAB environment. The results of simulation evaluations show that in the modified artificial potential field algorithm, the robot can pass obstacles around the target without collision and reach the target.
Acknowledgements
Authors of this would like to express the sincere thanks to the National Natural Science Foundation of China for their funding support to carry out this project.
Funding
This work is supported by the National Natural Science Foundation of China (61772454, 6171171570).
Availability of data and materials
The data is available on request with prior concern to the first author of this paper.
Nil
Ethics approval and consent to participate
All the authors of this manuscript would like to declare that mutually agreed no conflict of interest. All the results of this paper with respect to the experiments on human subject.
Not Applicable
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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2019-06-16 04:46:39
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5161797404289246, "perplexity": 624.7750590241872}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997731.69/warc/CC-MAIN-20190616042701-20190616064701-00053.warc.gz"}
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https://stats.stackexchange.com/questions/507571/bounding-values-of-a-dirichlet-distribution
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# Bounding values of a Dirichlet distribution
Consider $$k$$ random variables $$X_1, X_2, \ldots, X_k$$ such that $$(X_1, X_2, \ldots, X_k)$$ follow a $$\text{Dirichlet}(1, 1, \ldots, 1)$$ distribution. For a large enough $$k$$, I am trying to bound/find out (with high probability) what fraction of these $$k$$ random variables take values between $$\alpha$$ and $$\beta$$, for $$0 < \alpha < \beta < 1$$.
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2022-07-06 09:46:59
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9482951164245605, "perplexity": 179.1960887652804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104669950.91/warc/CC-MAIN-20220706090857-20220706120857-00141.warc.gz"}
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https://encyclopediaofmath.org/index.php?title=Character_(of_a_topological_space)&oldid=42661
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# Character (of a topological space)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
One of the cardinal characteristics of a topological space $X$. The local character $\chi(x,X)$ at a point $x \in X$ is the least cardinality of a local base at $x$. The character $\chi(X)$ is the least upper bound of the local characters.
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2022-08-13 15:05:55
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https://algorithmist.com/index.php/UVa_10288
|
# UVa 10288
## Summary
There are ${\displaystyle n}$ different kinds of coupons. Each box contains one coupon of random type. What's the
expected number of boxes you need in order to collect at least one coupon of every kind?
## Explanation
Suppose, you already have ${\displaystyle n-k}$ distinct coupons. Let ${\displaystyle a_{k}}$ denote the expected number of boxes you still need to collect the remaining ${\displaystyle k}$ coupons.
With probability ${\displaystyle {(n-k)/n}}$ the next coupon will be useless to you, and with probability ${\displaystyle {k/n}}$ it will be of the kind, which you don't yet have. Or, mathematically:
${\displaystyle a_{k}=(1+a_{k})\cdot {n-k \over n}+(1+a_{k-1})\cdot {k \over n}}$ ${\displaystyle =1+{n-k \over n}a_{k}+{k \over n}a_{k-1}}$, ${\displaystyle a_{0}=0}$.
Simplifying, you can obtain this simple formula for the answer:
${\displaystyle a_{n}=n\sum _{k=1}^{n}{1 \over k}}$
## Gotchas
• Standard 32-bit int's are not big enough for this problem, but 64-bit integers with gcd should be sufficient.
## An Alternative Explanation
As above, suppose you already have ${\displaystyle n-k}$ distinct coupons. Let ${\displaystyle X_{k}}$ denote the number of cereal boxes that you need to collect the ${\displaystyle k}$ remaining coupons. Let ${\displaystyle Y_{k}}$ denote the number of cereal boxes that you need to collect 1 of the ${\displaystyle k}$ coupons that you don't have yet.
Based on these definitions, the following equation holds:
${\displaystyle X_{k}=Y_{k}+X_{k-1}}$
Using ${\displaystyle E(\cdot )}$ to denote expectation, we have:
${\displaystyle E(X_{k})=E(Y_{k})+E(X_{k-1})}$
But ${\displaystyle Y_{k}}$ is a geometric random variable with probability of success ${\displaystyle p=k/n}$. The expectation of a geometric random variable is
${\displaystyle 1/p=n/k}$.
Plugging this into the formula above and unrolling the recursion yields the same formula given in the previous section:
${\displaystyle E(X_{n})=n\sum _{k=1}^{n}{1 \over k}}$
## Input
1
2
3
4
5
10
20
30
31
32
33
## Output
1
3
1
5 -
2
1
8 -
3
5
11 --
12
73
29 ---
252
3704479
71 -------
3879876
65960897707
119 -----------
77636318760
1967151510157
124 -------------
2329089562800
3934303020314
129 -------------
4512611027925
4071048809039
134 -------------
4375865239200
|
2019-05-22 18:58:02
|
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https://questions.examside.com/past-years/gate/gate-ece/electromagnetics/transmission-lines
|
GATE ECE
Electromagnetics
Transmission Lines
Previous Years Questions
## Marks 1
The voltage of an electromagnetic wave propagating in a coaxial cable with uniform characteristic impedance is $$V(l) = {e^{ - \gamma l\, + \,j\,\ome... A two wire transmission line terminates in a television set. The VSWR measured on the line is 5.8. The percentage of power that is reflected from the ... The propagation constant of a lossy transmission line is (2 + j5)$${m^{ - 1}}$$and its characteristic impedance is (50 + j0)$$\Omega $$at$$\omega...
A coaxial cable is made of two brass conductors. The spacing between the conductors is filled with Teflon $$\left( {{\varepsilon _r} = 2.1,\,\,\tan \,... In the following figure, the transmitter Tx sends a wideband modulated RF signal via a coaxial cable to the receiver Rx. The output impedance$${Z_T}$... To maximize power transfer, a lossless transmission line is to be matched to a resistive load impedance via a $$\lambda /4$$ transformer as shown. ... The return loss of a device is found to be 20 dB. The voltage standing wave ratio (VSWR) and magnitude of reflection coefficient are respectively. A coaxial cable with an inner diameter of 1 mm and outer diameter of 2.4 mm is filled with a dielectric of relative permittivity 10.89. Given $${\mu _... A transmission line of characteristic impedance 50$$\Omega $$is terminated by a 50$$\Omega $$load. When excited by a sinusoidal voltage source at ... A transmission line has a characteristic impedance of 50$$\Omega $$and a resistance of 0.1$$\Omega $$/m. If the line is distortionless, the attenua... If the scattering matrix [S] of a two port network is$$$\left[ S \right] = \left[ {\matrix{ {0.2\,\angle \,\,{0^ \circ }} & {0.9\,\,\angle \,\...
Many circles are drawn in a Smith chart used for transmission line calculations. The circles shown in Fig. represent ...
The VSWR can have any value between
In an impedance Smith chart, a clockwise movement along a constant resistance circle gives rise to
A transmission line is distortionless if
The magnitudes of the open-circuit and short-circuit input impedances of a transmission line are 100$$\Omega \,$$ and 25$$\Omega \,$$ respectively. Th...
Assuming perfect conductors of a transmission line, pure TEM propagation is NOT possible in
All transmission line sections shown in Fig. have characteristic impedance $${R_0}\, + \,j0$$. The input impedance $${Z_{in}}$$ equals ...
A transmission line of 50$$\Omega$$ characteristic impedance is terminated with a 100 $$\Omega$$ resistance. The minimum impedance measured on the l...
A lossless transmission line having 50 $$\Omega$$ characteristic impedance and lenght $$\lambda /4$$ is short ciruited at one end and connected to an...
The capacitance per unit length and the characteristic impedance of a lossless transmission line are C and $$Z_0$$ respectively. The velocity of a tra...
A load impedance (200 + j0) $$\Omega$$ is to be matched to a 50$$\Omega$$ lossless transmission line by using a quarter wave line transformer (QWT)....
## Marks 2
A microwave circuit consisting of lossless transmission lines $$T_1$$ and $$T_2$$ is shown in the figure. The plot shows the magnitude of the input re...
A lossless microstrip transmission line consists of a trace of width w. It is drawn over a practically infinite ground plane and is separated by a die...
Consider the 3 m long lossless air-filled transmission line shown in the figure. It has a characteristic impedance of $$120\pi \,\Omega$$, is termina...
A 200 m long transmission line having parameters shown in the figure is terminated into a load ܴ$$R_L$$. The line is connected to a 400 V source havin...
In the transmission line shown, the impedance $${Z_{in}}$$ (in ohms) between node A and the ground is ...
For a parallel plate transmission line, let v be the speed of propagation and Z be the characteristic impedance. Neglecting fringe effects, a redution...
A transmission line with a characteristic impedance of 100 $$\Omega$$ is used to match a 50 $$\Omega$$ section to a 200 $$\Omega$$ section. If the ...
A transmission line of characteristic impedance 50 $$\Omega$$ is terminated in a load impedance $$Z_L$$. The VSWR of the line is measured as 5 and t...
In the circuit shown, all the transmission line sections are lossless. The Voltage Standing Wave Ration (VSWR) on the 60W line is ...
A transmission line terminates in two branches, each of length $$\lambda /4$$, as shown.The branches are terminated by 50 $$\Omega$$ loads. The lines...
One end of loss-less transmission line having the characteristic impedance of 75 $$\Omega$$ and length of 1 cm is short-ciruited. At 3 GHz, the input...
A load of 50 $$\Omega$$ is connected in shunt in a 2-wire transmission line of $$Z_0$$ = 50 $$\Omega$$ as shown in the Fig. The 2-port scattering pa...
The parallel branches of a 2-wire transmission line are terminated in 100 $$\Omega$$ and 200 $$\Omega$$ resistors as shown in the Fig. The character...
## Marks 10
A 200 volt (r. m. s) generator having an internal resistance of 200 ohm is feeding a loss-less transmission line. The characteristic impedance and the...
EXAM MAP
Joint Entrance Examination
|
2023-03-28 06:22:40
|
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|
http://mathhelpforum.com/calculus/110884-calc-3-line-integral.html
|
# Thread: calc 3- line integral
1. ## calc 3- line integral
find the mass of a wire in the shape of the helix traced by (cost, sint, t/pi)
from pi to 3 pi if its density at each point is proportional to the distance from the point to the xy-plane
so i have mass= the integral from pi to 3pi of
cost + sint + t/pi dL
but i'm not sure what dL is..i saw it in an example in the book.
is dL the derivative of what i have above? or is it the magnitude? or do i ignore it?
after i clarify this, i think i'll be able to hopefully solve it..
2. $dL$ means that the integral is with respect to arc length. Imagine if the wire had a uniform mass density - then its mass would clearly be equal to its length times the mass density.
You probably know that the length of a parametrized curve $\phi$ between $\phi(a)$ and $\phi(b)$ is
$L=\int_a^b|\phi'(s)|ds$
so $dL = |\phi'(s)|ds$. Therefore the integral you are looking for is $\int_{\pi}^{3\pi}W(\phi(s))|\phi'(s)|ds$ where $W(\phi(s))$ is the mass density of the wire at the point $\phi(s)$. You are told that this is proportional to the distance from the point $\phi(s)$ to the $XY$ plane, i.e. $W(\phi(s))=k \times d(\phi(s), XY\mbox{ plane})$ for some constant $k$. Find the expression for that, substitute it into the integral and then evaluate it.
3. i'm sorry i'm sort of confused with all of your symbols. my final answer is:
4*sqrt(pi^2 +1)
is that right/close?
4. I didn't do the calculation! Please post your work, and I'll tell you if it's good.
5. f(x)= cost+sint+t/pi
dL= -sint+cost+1/pi then i took the magnitude of this to get
dL= (sqrt.(pi^2+1))/(pi)
so i took the integral from pi to 3pi of f(x)*dL
and got
(sint-cost+(1/2pi)*t^2)*(sqrt(pi^2 +1))/(pi))* t
and evaluated from 3pi to pi
and got
6. Well the function isn't $f(t)=\cos t+\sin t+\frac{t}{\pi}$, it's $f(t) = (\cos t, \sin t, \frac{t}{\pi})$.
Moreover, what did you do of the fact that the mass density at the point $f(t)$ is proportional to the distance between $f(t)$ and the $XY$ plane? If you do not take that into account in your solution then your solution is certainly wrong.
|
2017-12-14 16:07:33
|
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|
http://rena.qe-libs.org/reference/ena.html
|
Generates an ENA model by constructing a dimensional reduction of adjacency (co-occurrence) vectors as defined by the supplied conversations, units, and codes.
ena(
data,
codes,
units,
conversation,
model = c("EndPoint", "AccumulatedTrajectory", "SeparateTrajectory"),
weight.by = "binary",
window = c("MovingStanzaWindow", "Conversation"),
window.size.back = 1,
include.meta = TRUE,
groupVar = NULL,
groups = NULL,
runTest = FALSE,
points = FALSE,
mean = FALSE,
network = TRUE,
networkMultiplier = 1,
subtractionMultiplier = 1,
unit = NULL,
include.plots = T,
print.plots = F,
...
)
## Arguments
data
data.frame with containing metadata and coded columns
codes
vector, numeric or character, of columns with codes
units
vector, numeric or character, of columns representing units
conversation
vector, numeric or character, of columns to segment conversations by
vector, numeric or character, of columns with additional meta information for units
model
character: EndPoint (default), AccumulatedTrajectory, SeparateTrajectory
weight.by
"binary" is default, can supply a function to call (e.g. sum)
window
MovingStanzaWindow (default) or Conversation
window.size.back
Number of lines in the stanza window (default: 1)
include.meta
[TBD]
groupVar
vector, character, of column name containing group identifiers. If column contains at least two unique values, will generate model using a means rotation (a dimensional reduction maximizing the variance between the means of the two groups)
groups
vector, character, of values of groupVar column used for means rotation, plotting, or statistical tests
runTest
logical, TRUE will run a Student's t-Test and a Wilcoxon test for groups defined by the groups argument
points
logical, TRUE will plot points (default: FALSE)
mean
logical, TRUE will plot the mean position of the groups defined in the groups argument (default: FALSE)
network
logical, TRUE will plot networks (default: TRUE)
networkMultiplier
numeric, scaling factor for non-subtracted networks (default: 1)
subtractionMultiplier
numeric, scaling factor for subtracted networks (default: 1)
unit
vector, character, name of a single unit to plot
include.plots
logical, TRUE will generate plots based on the model (default: TRUE)
print.plots
logical, TRUE will show plots in the Viewer(default: FALSE)
...
Additional parameters passed to set creation and plotting functions
ena.set object
## Details
This function generates an ena.set object given a data.frame, units, conversations, and codes. After accumulating the adjacency (co-occurrence) vectors, computes a dimensional reduction (projection), and calculates node positions in the projected ENA space. Returns location of the units in the projected space, as well as locations for node positions, and normalized adjacency (co-occurrence) vectors to construct network graphs. Includes options for returning statistical tests between groups of units, as well as plots of units, groups, and networks.
## Examples
data(RS.data)
rs = ena(
data = RS.data,
conversation = c("Condition","GroupName"),
codes = c('Data',
'Technical.Constraints',
'Performance.Parameters',
'Client.and.Consultant.Requests',
'Design.Reasoning',
'Collaboration'),
window.size.back = 4,
print.plots = FALSE,
groupVar = "Condition",
groups = c("FirstGame", "SecondGame")
)
|
2022-09-29 07:44:57
|
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|
https://www.ic.sunysb.edu/Class/phy141md/doku.php?id=phy131studiof15:lectures:m2p2sol&do=diff&rev2%5B0%5D=1445865745&rev2%5B1%5D=1445866526&difftype=sidebyside
|
# Differences
This shows you the differences between two versions of the page.
phy131studiof15:lectures:m2p2sol [2015/10/26 09:22]mdawber [Question 2 Solution] phy131studiof15:lectures:m2p2sol [2015/10/26 09:35]mdawber [Question 3 Solution] Both sides previous revision Previous revision 2015/10/26 09:38 mdawber 2015/10/26 09:35 mdawber [Question 3 Solution] 2015/10/26 09:22 mdawber [Question 2 Solution] 2015/10/26 09:18 mdawber [Question 1 Solution] 2015/10/22 14:53 mdawber [Question 2 Solution] 2015/10/22 14:53 mdawber [Question 2 Solution] 2015/10/20 11:53 mdawber [Question 2 Solution] 2015/10/20 11:51 mdawber 2015/10/20 11:51 mdawber [Midterm 2 Q2 Solution (Ave Score: 29.1/35)] 2015/10/20 11:49 mdawber [Question 1 Solutions] 2015/10/20 11:49 mdawber [Question 1 Solutions] 2015/10/20 11:48 mdawber created 2015/10/26 09:38 mdawber 2015/10/26 09:35 mdawber [Question 3 Solution] 2015/10/26 09:22 mdawber [Question 2 Solution] 2015/10/26 09:18 mdawber [Question 1 Solution] 2015/10/22 14:53 mdawber [Question 2 Solution] 2015/10/22 14:53 mdawber [Question 2 Solution] 2015/10/20 11:53 mdawber [Question 2 Solution] 2015/10/20 11:51 mdawber 2015/10/20 11:51 mdawber [Midterm 2 Q2 Solution (Ave Score: 29.1/35)] 2015/10/20 11:49 mdawber [Question 1 Solutions] 2015/10/20 11:49 mdawber [Question 1 Solutions] 2015/10/20 11:48 mdawber created Last revision Both sides next revision Line 100: Line 100: {{m2fig2f11.png}} {{m2fig2f11.png}} - A. Sum of torques on beam around hinge + A Jack-O'-Lantern of mass 4kg is to be suspended as shown in the diagram using a hinged uniform beam of mass 2kg and length 0.8m and a massless string. The beam should be level and the string at 30º to the horizontal. + + (a) (10 points) If the maximum tension the string can support without breaking is 50N, what is the furthest distance from the hinge, $x$, that I can hang the Jack-O'-Lantern. + + Sum of torques on beam around hinge $0.8\mathrm{m}\times T\sin30^{o}-x\times 4\mathrm{kg}\times g-0.4\mathrm{m}\times 2\mathrm{kg}\times g=0$ $0.8\mathrm{m}\times T\sin30^{o}-x\times 4\mathrm{kg}\times g-0.4\mathrm{m}\times 2\mathrm{kg}\times g=0$ Line 106: Line 110: $x=\frac{0.8\mathrm{m}\times 50\mathrm{N}\sin30^{0}-0.4\mathrm{m}\times 2\mathrm{kg}\times g}{4\mathrm{kg}\times g}=0.31\mathrm{m}$ $x=\frac{0.8\mathrm{m}\times 50\mathrm{N}\sin30^{0}-0.4\mathrm{m}\times 2\mathrm{kg}\times g}{4\mathrm{kg}\times g}=0.31\mathrm{m}$ - B. Sum of horizontal forces on beam + (b) (10 points) If I hang the Jack-O'-Lantern at the distance found in part (a), (ie. when the tension in the string is 50N), what is the magnitude of the net force on the hinge? What is the direction of the force (give your answer in terms of the angle $\theta$ from the $y$ axis shown in the diagram)? + + Sum of horizontal forces on beam $F_{HH}-T\cos30^{o}=0$ $F_{HH}-T\cos30^{o}=0$ Line 124: Line 130: But the force **on** the hinge is directed in the opposite direction $180^{o}-52^{o}=128^{o}$ But the force **on** the hinge is directed in the opposite direction $180^{o}-52^{o}=128^{o}$ - C. Sum of torques on beam around hinge + + (c) (10 points) If I would like to hang the Jack-O'-Lantern at the far end of the beam from the hinge using the same string as was used in parts (a) and (b), what is the minimum angle the string should make with the horizontal instead of the $30^{o}$ angle it makes in parts (a) and (b). + + Sum of torques on beam around hinge $0.8\mathrm{m}\times T\sin\phi-0.8\mathrm{m}\times 4\mathrm{kg}\times g-0.4\mathrm{m}\times 2\mathrm{kg}\times g=0$ $0.8\mathrm{m}\times T\sin\phi-0.8\mathrm{m}\times 4\mathrm{kg}\times g-0.4\mathrm{m}\times 2\mathrm{kg}\times g=0$
phy131studiof15/lectures/m2p2sol.txt · Last modified: 2015/10/26 09:38 by mdawber
|
2020-07-14 13:46:53
|
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|
https://opencurriculum.org/5358/cell-transport-and-homeostasis/
|
# Cell Transport and Homeostasis
### Article objectives
• To identify two ways that molecules and ions cross the plasma membrane.
• To distinguish between diffusion and osmosis.
• To identify the role of ion channels in facilitated diffusion.
• To compare passive and active transport.
• To identify the connection between vesicles and active transport.
• To compare endocytosis and exocytosis.
• To outline the process of cell communication.
• Probably the most important feature of a cell’s phospholipid membranes is that they are selectively permeable. A membrane that is selectively permeable has control over what molecules or ions can enter or leave the cell, as shown in Figure 1. The permeability of a membrane is dependent on the organization and characteristics of the membrane lipids and proteins. In this way, cell membranes help maintain a state of homeostasis within cells (and tissues, organs, and organ systems) so that an organism can stay alive and healthy.
Figure 1: A selectively permeable membrane allows certain molecules through, but not others.
### Transport Across Membranes
The molecular make-up of the phospholipid bilayer limits the types of molecules that can pass through it. For example, hydrophobic (water-hating) molecules, such as carbon dioxide ($$CO_2$$) and oxygen ($$O_2$$), can easily pass through the lipid bilayer, but ions such as calcium ($$Ca^{2+}$$) and polar molecules such as water ($$H_2 O$$) cannot. The hydrophobic interior of the phospholipid does not allow ions or polar molecules through because they are hydrophilic, or water loving. In addition, large molecules such as sugars and proteins are too big to pass through the bilayer. Transport proteins within the membrane allow these molecules to cross the membrane into or out of the cell. This way, polar molecules avoid contact with the nonpolar interior of the membrane, and large molecules are moved through large pores.
Every cell is contained within a membrane punctuated with transport proteins that act as channels or pumps to let in or force out certain molecules. The purpose of the transport proteins is to protect the cell’s internal environment and to keep its balance of salts, nutrients, and proteins within a range that keeps the cell and the organism alive.
There are three main ways that molecules can pass through a phospholipid membrane. The first way requires no energy input by the cell and is called passive transport. The second way requires that the cell uses energy to pull in or pump out certain molecules and ions and is called active transport. The third way is through vesicle transport, in which large molecules are moved across the membrane in bubble-like sacks that are made from pieces of the membrane.
### Passive Transport
Passive transport is a way that small molecules or ions move across the cell membrane without input of energy by the cell. The three main kinds of passive transport are diffusion, osmosis, and facilitated diffusion.
Diffusion
Diffusion is the movement of molecules from an area of high concentration of the molecules to an area with a lower concentration. The difference in the concentrations of the molecules in the two areas is called the concentration gradient. Diffusion will continue until this gradient has been eliminated. Since diffusion moves materials from an area of higher concentration to the lower, it is described as moving solutes ”down the concentration gradient.” The end result of diffusion is an equal concentration, or equilibrium, of molecules on both sides of the membrane.
If a molecule can pass freely through a cell membrane, it will cross the membrane by diffusion (Figure 2).
Figure 2: Molecules move from an area of high concentration to an area of lower concentration until an equilibrium is met. The molecules continue to cross the membrane at equilibrium, but at equal rates in both directions.
Osmosis
Imagine you have a cup that has 100ml water, and you add 15g of table sugar to the water. The sugar dissolves and the mixture that is now in the cup is made up of a solute (the sugar), that is dissolved in the solvent (the water). The mixture of a solute in a solvent is called a solution.
Imagine now that you have a second cup with 100ml of water, and you add 45 grams of table sugar to the water. Just like the first cup, the sugar is the solute, and the water is the solvent. But now you have two mixtures of different solute concentrations. In comparing two solutions of unequal solute concentration, the solution with the higher solute concentration is hypertonic, and the solution with the lower concentration is hypotonic. Solutions of equal solute concentration are isotonic. The first sugar solution is hypotonic to the second solution. The second sugar solution is hypertonic to the first.
You now add the two solutions to a beaker that has been divided by a selectively permeable membrane. The pores in the membrane are too small for the sugar molecules to pass through, but are big enough for the water molecules to pass through. The hypertonic solution is on one side of the membrane and the hypotonic solution on the other. The hypertonic solution has a lower water concentration than the hypotonic solution, so a concentration gradient of water now exists across the membrane. Water molecules will move from the side of higher water concentration to the side of lower concentration until both solutions are isotonic.
Osmosis is the diffusion of water molecules across a selectively permeable membrane from an area of higher concentration to an area of lower concentration. Water moves into and out of cells by osmosis. If a cell is in a hypertonic solution, the solution has a lower water concentration than the cell cytosol does, and water moves out of the cell until both solutions are isotonic. Cells placed in a hypotonic solution will take in water across their membrane until both the external solution and the cytosol are isotonic.
A cell that does not have a rigid cell wall (such as a red blood cell), will swell and lyse (burst) when placed in a hypotonic solution. Cells with a cell wall will swell when placed in a hypotonic solution, but once the cell is turgid (firm), the tough cell wall prevents any more water from entering the cell. When placed in a hypertonic solution, a cell without a cell wall will lose water to the environment, shrivel, and probably die. In a hypertonic solution, a cell with a cell wall will lose water too. The plasma membrane pulls away from the cell wall as it shrivels. The cell becomes plasmolyzed. Animal cells tend to do best in an isotonic environment, plant cells tend to do best in a hypotonic environment. This is demonstrated in Figure 3.
When water moves into a cell by osmosis, osmotic pressure may build up inside the cell. If a cell has a cell wall, the wall helps maintain the cell’s water balance. Osmotic pressure is the main cause of support in many plants. When a plant cell is in a hypotonic environment, the osmotic entry of water raises the turgor pressure exerted against the cell wall until the pressure prevents more water from coming into the cell. At this point the plant cell is turgid.
Figure 3: Unless an animal cell (such as the red blood cell in the top panel) has an adaptation that allows it to alter the osmotic uptake of water, it will lose too much water and shrivel up in a hypertonic environment. If placed in a hypotonic solution, water molecules will enter the cell causing it to swell and burst. Plant cells (bottom panel) become plasmolyzed in a hypertonic solution, but tend to do best in a hypotonic environment. Water is stored in the central vacuole of the plant cell.
The effects of osmotic pressures on plant cells are shown in Figure 4.
Figure 4: The central vacuoles of the plant cells in the left image are full of water, so the cells are turgid. The plant cells in the right image have been exposed to a hypertonic solution; water has left the central vacuole and the cells have become plasmolysed.
Osmosis can be seen very effectively when potato slices are added to a high concentration of salt solution (hypertonic). The water from inside the potato moves out of the potato cells to the salt solution, which causes the potato cells to lose turgor pressure. The more concentrated the salt solution, the greater the difference in the size and weight of the potato slice after plasmolysis.
The action of osmosis can be very harmful to organisms, especially ones without cell walls. For example, if a saltwater fish (whose cells are isotonic with seawater), is placed in fresh water, its cells will take on excess water, lyse, and the fish will die. Another example of a harmful osmotic effect is the use of table salt to kill slugs and snails.
Controlling Osmosis
Organisms that live in a hypotonic environment such as freshwater, need a way to prevent their cells from taking in too much water by osmosis. A contractile vacuole is a type of vacuole that removes excess water from a cell. Freshwater protists, such as the paramecia shown in Figure 5, have a contractile vacuole. The vacuole is surrounded by several canals, which absorb water by osmosis from the cytoplasm. After the canals fill with water, the water is pumped into the vacuole. When the vacuole is full, it pushes the water out of the cell through a pore. Other protists, such as members of the genus Amoeba, have contractile vacuoles that move to the surface of the cell when full and release the water into the environment.
Figure 5: The contractile vacuole is the star-like structure within the paramecia (at center-right)
Facilitated Diffusion
Facilitated diffusion is the diffusion of solutes through transport proteins in the plasma membrane. Facilitated diffusion is a type of passive transport. Even though facilitated diffusion involves transport proteins, it is still passive transport because the solute is moving down the concentration gradient.
As was mentioned earlier, small nonpolar molecules can easily diffuse across the cell membrane. However, due to the hydrophobic nature of the lipids that make up cell membranes, polar molecules (such as water) and ions cannot do so. Instead, they diffuse across the membrane through transport proteins. A transport protein completely spans the membrane, and allows certain molecules or ions to diffuse across the membrane. Channel proteins, gated channel proteins, and carrier proteins are three types of transport proteins that are involved in facilitated diffusion.
A channel protein, a type of transport protein, acts like a pore in the membrane that lets water molecules or small ions through quickly. Water channel proteins allow water to diffuse across the membrane at a very fast rate. Ion channel proteins allow ions to diffuse across the membrane.
A gated channel protein is a transport protein that opens a ”gate,” allowing a molecule to pass through the membrane. Gated channels have a binding site that is specific for a given molecule or ion. A stimulus causes the ”gate” to open or shut. The stimulus may be chemical or electrical signals, temperature, or mechanical force, depending on the type of gated channel. For example, the sodium gated channels of a nerve cell are stimulated by a chemical signal which causes them to open and allow sodium ions into the cell. Glucose molecules are too big to diffuse through the plasma membrane easily, so they are moved across the membrane through gated channels. In this way glucose diffuses very quickly across a cell membrane, which is important because many cells depend on glucose for energy.
A carrier protein is a transport protein that is specific for an ion, molecule, or group of substances. Carrier proteins ”carry” the ion or molecule across the membrane by changing shape after the binding of the ion or molecule. Carrier proteins are involved in passive and active transport. A model of a channel protein and carrier proteins is shown in Figure 6.
Figure 6: Facilitated diffusion in cell membrane. Channel proteins and carrier proteins are shown (but not a gated-channel protein). Water molecules and ions move through channel proteins. Other ions or molecules are also carried across the cell membrane by carrier proteins. The ion or molecule binds to the active site of a carrier protein. The carrier protein changes shape, and releases the ion or molecule on the other side of the membrane. The carrier protein then returns to its original shape.
Ion Channels
Ions such as sodium ($$Na^+$$), potassium ($$K^-$$), calcium ($$Ca^{2+}$$), and chloride ($$Cl^-$$), are important for many cell functions. Because they are polar, these ions do not diffuse through the membrane. Instead they move through ion channel proteins where they are protected from the hydrophobic interior of the membrane. Ion channels allow the formation of a concentration gradient between the extracellular fluid and the cytosol. Ion channels are very specific as they allow only certain ions through the cell membrane. Some ion channels are always open, others are ”gated” and can be opened or closed. Gated ion channels can open or close in response to different types of stimuli such as electrical or chemical signals.
### Active Transport
In contrast to facilitated diffusion which does not require energy and carries molecules or ions down a concentration gradient, active transport pumps molecules and ions against a concentration gradient. Sometimes an organism needs to transport something against a concentration gradient. The only way this can be done is through active transport which uses energy that is produced by respiration (ATP). In active transport, the particles move across a cell membrane from a lower concentration to a higher concentration. Active transport is the energy-requiring process of pumping molecules and ions across membranes ”uphill” against a gradient.
• The active transport of small molecules or ions across a cell membrane is generally carried out by transport proteins that are found in the membrane.
• Larger molecules such as starch can also be actively transported across the cell membrane by processes called endocytosis and exocytosis.
Sodium-Potassium Pump
Carrier proteins can work with a concentration gradient (passive transport), but some carrier proteins can move solutes against the concentration gradient (from high concentration to low), with energy input from ATP. As in other types of cellular activities, ATP supplies the energy for most active transport. One way ATP powers active transport is by transferring a phosphate group directly to a carrier protein. This may cause the carrier protein to change its shape, which moves the molecule or ion to the other side of the membrane. An example of this type of active transport system, as shown in Figure 7, is the sodium-potassium pump, which exchanges sodium ions for potassium ions across the plasma membrane of animal cells.
Figure 7: The sodium-potassium pump system moves sodium and potassium ions against large concentration gradients. It moves two potassium ions into the cell where potassium levels are high, and pumps three sodium ions out of the cell and into the extracellular fluid.
As is shown in Figure 7, three sodium ions bind with the protein pump inside the cell. The carrier protein then gets energy from ATP and changes shape. In doing so, it pumps the three sodium ions out of the cell. At that point, two potassium ions move in from outside the cell and bind to the protein pump. The sodium-potassium pump is found in the plasma membrane of almost every human cell and is common to all cellular life. It helps maintain cell potential and regulates cellular volume. Cystic fibrosis is a genetic disorder that results in a misshapen chloride ion pump. Chloride levels within the cells are not controlled properly, and the cells produce thick mucus. The chloride ion pump is important for creating sweat, digestive juices, and mucus.
The active transport of ions across the membrane causes an electrical gradient to build up across the plasma membrane. The number of positively charged ions outside the cell is greater than the number of positively charged ions in the cytosol. This results in a relatively negative charge on the inside of the membrane, and a positive charge on the outside. This difference in charges causes a voltage across the membrane. Voltage is electrical potential energy that is caused by a separation of opposite charges, in this case across the membrane. The voltage across a membrane is called membrane potential. Membrane potential is very important for the conduction of electrical impulses along nerve cells.
Because the inside of the cell is negative compared to outside the cell, the membrane potential favors the movement of positively charged ions (cations) into the cell, and the movement of negative ions (anions) out of the cell. So, there are two forces that drive the diffusion of ions across the plasma membrane—a chemical force (the ions’ concentration gradient), and an electrical force (the effect of the membrane potential on the ions’ movement). These two forces working together are called an electrochemical gradient.
### Vesicles and Active Transport
Some molecules or particles are just too large to pass through the plasma membrane or to move through a transport protein. So cells use two other methods to move these macromolecules (large molecules) into or out of the cell. Vesicles or other bodies in the cytoplasm move macromolecules or large particles across the plasma membrane. There are two types of vesicle transport, endocytosis and exocytosis.
Endocytosis and Exocytosis
Endocytosis is the process of capturing a substance or particle from outside the cell by engulfing it with the cell membrane. The membrane folds over the substance and it becomes completely enclosed by the membrane. At this point a membrane-bound sac, or vesicle pinches off and moves the substance into the cytosol. There are two main kinds of endocytosis:
Phagocytosis or ”cellular eating,” occurs when the dissolved materials enter the cell. The plasma membrane engulfs the solid material, forming a phagocytic vesicle.
Pinocytosis or ”cellular drinking,” occurs when the plasma membrane folds inward to form a channel allowing dissolved substances to enter the cell, as shown in Figure 8. When the channel is closed, the liquid is encircled within a pinocytic vesicle.
Figure 8: Transmission electron microscope image of brain tissue that shows pinocytotic vesicles. Pinocytosis is a type of endocytosis.
Exocytosis describes the process of vesicles fusing with the plasma membrane and releasing their contents to the outside of the cell, as shown in Figure 9. Exocytosis occurs when a cell produces substances for export, such as a protein, or when the cell is getting rid of a waste product or a toxin. Newly made membrane proteins and membrane lipids are moved on top the plasma membrane by exocytosis.
Figure 9: Mode of exocytosis at a synaptic junction, where two nerve cells meet. Chemical signal molecules are released from nerve cell A by exocytosis, and move toward receptors in nerve cell B. Exocytosis is an important part in cell signaling.
Homeostasis and Cell Function
Homeostasis refers to the balance, or equilibrium within the cell or a body. It is an organism’s ability to keep a constant internal environment. Keeping a stable internal environment requires constant adjustments as conditions change inside and outside the cell. The adjusting of systems within a cell is called homeostatic regulation. Because the internal and external environments of a cell are constantly changing, adjustments must be made continuously to stay at or near the set point (the normal level or range). Homeostasis is a dynamic equilibrium rather than an unchanging state. The cellular processes discussed all play an important role in homeostatic regulation.
### Cell Communication
To survive and grow, cells need to be able to ”talk” with their cell neighbors and be able to detect change in their environment. Talking with neighbors is even more important to a cell if it is part of a multicellular organism. The billions of cells that make up your body need to be able to communicate with each other to allow your body to grow, and to keep you alive and healthy. The same is true for any organism. Cell signaling is a major area of research in biology today. Recently scientists have discovered that many different cell types, from bacteria to plants, use similar types of communication pathways, or cellsignaling mechanisms. This suggests that cell-signaling mechanisms evolved long before the first multicellular organism did.
The Language of Cells
For cells to be able to signal to each other, a few things are needed:
• a signal
• a cell receptor, which is usually on the plasma membrane, but can be found inside the cell
• a response to the signal
Cells that are communicating may be right next to each other or far apart. The type of chemical signal a cell will send differs depending on the distance the message needs to go. For example, hormones, ions, and neurotransmitters are all types of signals that are sent depending on the distance the message needs to go.
The target cell then needs to be able to recognize the signal. Chemical signals are received by the target cell on receptor proteins. As discussed earlier, most receptor proteins are found in the plasma membrane. Most receptors proteins are found on the plasma membrane, but some are also found inside the cell. These receptor proteins are very specific for only one particular signal molecule, much like a lock that recognizes only one key. Therefore, a cell has lots of receptor proteins to recognize the large number of cell signal molecules. There are three stages to sending and receiving a cell ”message:” reception, transduction, and response.
Signal Receptors
Cell-surface receptors are integral proteins—they reach right through the lipid bilayer, spanning from the outside to the inside of the cell. These receptor proteins are specific for just one kind of signal molecule. The signaling molecule acts as a ligand when it binds to a receptor protein. A ligand is a small molecule that binds to a larger molecule. Signal molecule binding causes the receptor protein to change its shape. At this point the receptor protein can interact with another molecule. The ligand (signal molecule) itself does not pass through the plasma membrane.
In eukaryotic cells, most of the intracellular proteins that are activated by a ligand binding to a receptor protein are enzymes. Receptor proteins are named after the type of enzyme that they interact with inside the cell. These enzymes include G proteins and protein kinases, likewise there are G-protein-linked receptors and tyrosine kinase receptors. A kinase is a protein involved in phosphorylation. A G-protein linked receptor is a receptor that works with the help of a protein called a G-protein. A G-protein gets its name from the molecule to which it is attached, guanosine triphosphate (GTP), or guanosine diphosphate (GDP). The GTP molecule is similar to ATP.
Once G proteins or protein kinase enzymes are activated by a receptor protein, they create molecules called second messengers. A second messenger is a small molecule that starts a change inside a cell in response to the binding of a specific signal to a receptor protein. Some second messenger molecules include small molecules called cyclic nucleotides, such as cyclic adenosine monophosphate (cAMP) and cyclic guanosine monophosphate (cGMP). Calcium ions ($$Ca^{2+}$$) also act as secondary messengers. Secondary messengers are a part of signal transduction pathways.
Signal Transduction
A signal-transduction pathway is the signaling mechanism by which a cell changes a signal on it surface into a specific response inside the cell. It most often involves an ordered sequence of chemical reactions inside the cell which is carried out by enzymes and other molecules. In many signal transduction processes, the number of proteins and other molecules participating in these events increases as the process progresses from the binding of the signal. A ”signal cascade” begins. Think of a signal cascade as a chemical domino-effect inside the cell, in which one domino knocks over two dominos, which in turn knock over four dominos, and so on. The advantage of this type of signaling to the cell is that the message from one little signal molecule can be greatly amplified and have a dramatic effect.
G protein-linked receptors are only found in higher eukaryotes, including yeast, plants, and animals. Your senses of sight and smell are dependent on G-protein linked receptors. The ligands that bind to these receptors include light-sensitive compounds, odors, hormones, and neurotransmitters. The ligands for G-protein linked receptors come in different sizes, from small molecules to large proteins. G protein-coupled receptors are involved in many diseases, but are also the target of around half of all modern medicinal drugs.
The process of how a G-protein linked receptor works is outlined in Figure 10.
Figure 10: How a G-protein linked receptor works with the help of a G-protein. In panel C, the second messenger cAMP can be seen moving away from the enzyme.
Table 1
A. A ligand such as a hormone (small, purple molecule) binds to the G-linked receptor (red molecule). Before ligand binding, the inactive G-protein (yellow molecule) has GDP bound to it. B. The receptor changes shape and activates the G-protein and a molecule of GTP replaces the GDP. C. The G-protein moves across the membrane then binds to and activates the enzyme (green molecule). This then triggers the next step in the pathway to the cell’s response. After activating the enzyme, the Gprotein returns to its original position. The second messenger of this signal transduction is cAMP, as shown in C.
The sensing of the external and internal environments at the cellular level relies on signal transduction. Defects in signal transduction pathways can contribute or lead to many diseases, including cancer and heart disease. This highlights the importance of signal transductions to biology and medicine.
Signal Responses
In response to a signal, a cell may change activities in the cytoplasm or in the nucleus that include the switching on or off of genes. Changes in metabolism, continued growth, movement, or death are some of the cellular responses to signals that require signal transduction.
Gene activation leads to other effects, since the protein products of many of the responding genes include enzymes and factors that increase gene expression. Gene expression factors produced as a result of a cascade can turn on even more genes. Therefore one stimulus can trigger the expression of many genes, and this in turn can lead to the activation of many complex events. In a multicellular organism these events include the increased uptake of glucose from the blood stream (stimulated by insulin), and the movement of neutrophils to sites of infection (stimulated by bacterial products). The set of genes and the order in which they are activated in response to stimuli are often called a genetic program.
Images courtesy of:
http://en.wikipedia.org/wiki/Image:Semipermeable_membrane.png. Public Domain.
Mariana Ruiz. http://commons.wikimedia.org/wiki/File: Scheme_simple_diffusion_in_cell_membrane-en.svg. Public Domain.
http://en.wikipedia.org/wiki/Image: Turgor_pressure_on_plant_cells_diagram.svg. Public Domain.
Mnolf. http://en.wikipedia.org/wiki/Image:Rhoeo_Discolor_-_Plasmolysis.jpg. GNU-FDL & CC-BY-SA.
Jasper Nance. CC-BY-SA.
Mariana Ruiz. http://commons.wikimedia.org/wiki/File: Scheme_facilitated_diffusion_in_cell_membrane-en.svg. Public Domain.
Mariana Ruiz. http://commons.wikimedia.org/wiki/File: Scheme_sodium-potassium_pump-en.svg. Public Domain.
Louisa Howard, Miguel Marin-Padilla. Public Domain.
Dake. http://en.wikipedia.org/wiki/Image:Synapse_diag1.png. GNU-FDL.
Bensaccount. [Retrieved and modified from http://en.wikipedia.org/wiki/Image:GPCR_mechanism.png ]. Public Domain.
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2022-08-16 07:03:59
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https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-7-functions-and-graphs-7-3-graphs-of-functions-7-3-exercise-set-page-468/55
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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$$\text{Domain: All Real Numbers} \\ \text{Range: All Real Numbers}$$
We are given the graph of the function. Thus, using the graph, we find that the domain and range are: $$\text{Domain: All Real Numbers} \\ \text{Range: All Real Numbers}$$
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2019-12-11 21:58:30
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https://dsp.stackexchange.com/questions/26122/lost-power-while-using-fft-with-apodizing-mask
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# Lost power while using FFT with apodizing mask
The solution that is most easily implemented is using an apodizing mask on the original image, tapering the borders to zero making the image left-right and top-bottom continuous.
The rest of the question is best illustrated with a series of pictures. First, we have the original image:
Now, we apply the cosine-tapered apodizing mask:
Note that we use this mask to do operations in Fourier Space on the second image; to get the power spectrum we will have to use a similar mask again but this is applied to both images so that is not the issue here. (i find it difficult to phrase this properly; please place a comment if you want me to elaborate).
So, now we use a FFT on the second image. In Fourier space we may do things with the image, but to trace our steps and see if the machinery works, we leave it be and just use the inverse-FFT back to real space.
Back in real space, we take the part of the image that was not tapered (so, the inner square) and obtain:
Now, for both the first and the third image I now want to find the power spectrum. In essence, this comes down to (1) using an apodizing mask again, (2) transforming to fourier space once more, and (3) taking the azimuthal average as a function of radius.
As a last subtelty, the power spectrum of the third image is multiplied by a factor $(N_1 / N_3)^2$, where $N_1$ is the number of pixels per side of the first image and $N_3$ the number of pixels per side of the third image. This factor is there to correct for the normalization $1/N$ in the definition of the FFT in numpy.
Doing all this, we obtain the following power spectra:
Clearly, the power spectrum of the third image (in red) has lost power with respect to the power spectrum of the original, first image (in blue). What is the cause of this and can it be compensated for? Or will I have to rely on more elaborate solutions than an apodizing mask (as given here) to not lose this power?
• Did you scale the power by the area under the 2D window function? – hotpaw2 Sep 29 '15 at 17:51
• If I understand you correctly, this is encapsulated in this: "As a last subtelty, the power spectrum of the third image is multiplied by a factor (N_1 / N_3)^2, where N_1 is the number of pixels per side of the first image and N_3 the number of pixels per side of the third image. This factor is there to correct for the normalization 1/N in the definition of the FFT in numpy." So yes, I believe I correct for that. – user1991 Sep 29 '15 at 17:57
• That's the size of your window. You may also need to use the area under your window, where the input to the integral goes to zero near the edges. – hotpaw2 Sep 29 '15 at 18:00
• That'd be another factor of (N_1/N_3)^2, if i'm not mistaken, and that'd mean the red line is well above the blue one. – user1991 Sep 30 '15 at 11:21
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2021-04-12 02:43:03
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http://www.unipa.it/persone/docenti/m/antonino.messina/?pagina=pubblicazione&idPubblicazione=122000
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Salta al contenuto principale
# ANTONINO MESSINA
## Resonant effects in a SQUID qubit subjected to nonadiabatic changes
• Autori: Chiarello, F.; Spilla, S.; Castellano, M.; Cosmelli, C.; Messina, A.; Migliore, R.; Napoli, A.; Torrioli, G.
• Anno di pubblicazione: 2014
• Tipologia: Articolo in rivista (Articolo in rivista)
### Abstract
By quickly modifying the shape of the effective potential of a double SQUID flux qubit from a single well to a double well condition, we experimentally observe an anomalous behavior, namely an alternance of resonance peaks, in the probability to find the qubit in a given flux state. The occurrence of Landau-Zener transitions as well as resonant tunneling between degenerate levels in the two wells may be invoked to partially justify the experimental results. A quantum simulation of the time evolution of the system indeed suggests that the observed anomalous behavior can be imputable to quantum coherence effects. The interplay among all these mechanisms has a practical implication for quantum computing purposes, giving a direct measurement of the limits on the sweeping rates possible for a correct manipulation of the qubit state by means of fast flux pulses, avoiding transitions to non-computational states.
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2019-09-18 17:37:39
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https://ch.gateoverflow.in/1269/gate-ch-2022-question-30
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Consider a bare long copper wire of $1$ $\text{mm}$ diameter. Its surface temperature is $T_{s}$ and the ambient temperature is $T_{a}\:(T_{s} > T_{a})$. The wire is to be coated with a $2$ $\text{mm}$ thick insulation. The convective heat transfer coefficient is $20\: W\: m^{-2}\: K^{-1}$. Assume that $T_{s}$ and $T_{a}$ remain unchanged. To reduce heat loss from the wire, the maximum allowed thermal conductivity of the insulating material, in $W\:m^{-1}\:K^{-1}$, rounded off to two decimal places, is
1. $0.02$
2. $0.04$
3. $0.10$
4. $0.01$
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2022-05-27 16:40:40
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http://cloud.originlab.com/doc/LabTalk/Tutorials/Tutorial-Range-Notation
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# 5.3 Range Notation in LabTalk
## Summary
Data exists in four primary places in your Origin project: workbooks, graphs, matrices and loose datasets.
You can access data in any of these objects using Range variables. Once a range variable is declared, you can directly read or write to the range.
## Declaration
You declare a range variable using a syntax that is similar to other data types:
range [-option] RangeName = RangeString
• The left-hand side of the range assignment is uniform for all types of range assignments. Range names follow Origin variable naming rules. The [-option] component indicates an optional parameter, with available option switches differing by data type (see Types of Range Data for details).
• The right-hand side of the range assignment, the RangeString, varies by object type. Refer to the following sub-topics for specifics:
## Workbooks
For workbook data, RangeString takes the form:
[WorkBookName]SheetNameOrIndex!ColumnNameOrIndex[CellIndex]
Note: WorkBookName andSheetName refer to Short Name since Short Name is the default programming name. To use Long Name in range notation for workbook or worksheet, use Long Name in double quotes as in ["MyBook"]"MySheet"!. ColumnName can be either the Long Name or the Short Name of the column.
### Worksheet Cell Range
Use a range to access a single cell or block of cells. The range may span rows and columns.
// cell(2,1), row2 of col(1)
range aa = 1[2];
// cell(1,1) to cell(10,3)
range bb = 1[1]:3[10];
Note: A range variable representing a block of cells can be used as an X-Function argument only, direct calculations are not supported.
Practical Example 1
//create a new workbook with col(A) and col(B)
newbook;
//fill first column with values
col(A) = {1:10};
// define range aa as col(A) dataset, range bb as col(B) dataset (empty)
range aa = col(A);
range bb = col(B);
// copy value in range aa, cell 1 to range bb cell 1
bb[1] = aa[1];
Practical Example 2
//create a new workbook
newbook;
//fill first column with values
col(A) = {1:10};
// put name of active worksheet into string var "yy$" yy$ = %H;
// define range variable "aa" as your column of values
range aa = [yy$]1!col(A); // create a new workbook and put name into "zz$"
newbook;
zz$= %H; // define range in new book range bb = [zz$]1!col(A);
// write value in "aa", cell 1 to "bb", cell 1
bb[1] = aa[1];
### Column Label Row Range
Use a range variable to access the column label rows in the active worksheet:
range rr=[L:C];
range rr=1[L]:end[L];
Note that you CANNOT use a negative number, such as (-1) in the range variable for label row.
Practical Example
// create a new workbook
newbook;
// import a file
string fn$=system.path.program$ + "Samples\Import and Export\ASCII Simple.dat";
impASC fname:=fn$; // define range variable containing raw data range raw = [%H]1!col(signal); // show 1st and 2nd User Defined Parameter Rows wks.userParam1 = 1; wks.userParam2 = 1; // rename 1st Parameter as "Mean", 2nd as "Std. Dev." wks.UserParam1$ = "Mean";
wks.UserParam2$= "Std. Dev."; // use mean() and stddev() functions to return values to 5 decimal places; // to respective label row cells in range variable raw raw[Mean]$ = $(mean(raw),*5); raw[Std. Dev.]$ = $(stddev(raw),*5); For an expanded discussion on accessing column label row data, with examples, see ### Worksheet Column SubRange // A subrange, rows 3-10, of col(a) in book1 sheet2 range cc = [book1]sheet2!col(a)[3:10]; //range that refers to the first column in the active worksheet range r1 = !1; //variables can be used when specifying the subrange int istart=3; int iend=10; range r2 = r1[istart:iend]; Practical Example: //create a new workbook newbook; //fill first column with values col(A) = {1:10}; //create range that refers to this first column range rA = col(A); //add a new sheet newsheet; //fill this column with the values from the first column in first worksheet col(A) = rA; range rAA = col(A); newsheet; range r1 = rA[1:5]; range r2 = rAA[6:10]; col(A) = r1 + r2; ### Worksheet Column Range Use a range variable to access an entire column of data. In the examples below, all the ranges defined refer to the first column in the active worksheet (assuming the name of the first column is "A"). You may recognize the col( ) and wcol( ) functions as these are used from the Set Column Values dialog in Origin. range rA = A; range rAA = col(A); range r1 = 1; range ricol = wcol(ncol); int ncol = 1; range rr =$(ncol);
Practical Example:
Once a range variable is created, you can work with the range data directly, reading and writing to the range.
//create a new workbook
newbook;
//fill first column with values
col(A) = {1:10};
//fill second column with values
col(B) = {2:2:20};
//create range that refers to the first column
range rA = col(A);
//create a range that refers to the second column
range rB = col(B);
newsheet;
//multiply the values in the column range by 2 and assign to column A
col(A) = rA *2;
//divide the values in the column range by 2 and assign to column B
col(B) = rB/2;
For use with specific X-Functions, it's possible to define a range that covers multiple columns or a block of cells in a Worksheet:
range raMC = col(2):col(4); // Or equivalent of 2:4
stats raMC;
ty %(raMC) values range from $(stats.min) to$(stats.max);
range raBlock = 2[5]:10[300]; // column 2, row 5 to column 10, row 300
stats raBlock;
ty %(raBlock) values range from $(stats.min) to$(stats.max);
### Worksheet Range
Use a range variable to access a worksheet:
range rSheet1 = [Book1]Sheet1!;
range rSheet2 = 1!;
range rSheet3 = !;
Note: If you run all the above lines when Sheet1 is active in the active Book1 workbook window, all three range variables will refer to that same worksheet.
Practical Example:
//create a new worksheet
newsheet;
//range refers to the active worksheet
range rWks = !;
### Workbook Page Range
Use a range variable to access an entire workbook. In the example below, if you run the script when a Book1 workbook is the active window, all three range variables will refer to the Book1 window.
range rPage1 = [Book1];
//%H is a system string register that holds the name of the active window
range rPage2 = [%H];
string strBookName$= %H; range rPage3 = [%(strBookName$)];
## Graphs
For graph data, the RangeString takes the form:
[GraphWindowName]LayerNameOrIndex!DataPlot
An example assignment looks like
range ll = [Graph1]Layer1!2; // Second curve on Graph1, Layer1
### Graph Data Subrange
Practical Example:
//integrate active plot from index 10 - 20
integ1 1[10:20];
### Graph Data Range
range rAA = [Graph1]Layer1!1;
range rBB = 1!1;
range rCC = !1;
range rDD = 1;
Note: If you run the above three lines when layer 1 is active in the Graph1 window, all three range variables will refer to the first dataplot in that layer.
Practical Example:
range rr = 1;
set rr -c color(green);
### Graph Layer Range
Use a range variable to access a layer in a graph window:
range rAA = [Graph1]Layer1!;
range rBB = 1!;
range rCC = !;
Note: If you run the above three lines when layer 1 is active in the Graph1 window, all three range variables will refer to that layer.
Practical Example:
//The layer is an Origin object. To learn more look at the tutorial on Origin Objects.
//graph layer object that refers to the active layer
range rr= !;
//adjust the width and height of the graph layer object
rr.width=50;
rr.height=50;
### Graph Page Range
Use a range variable to access an entire graph window. In the example below, if you run the script when Graph1 is the active window, all three range variables will refer to this window.
range rPage1 = [Graph1];
//%H is a system string register that holds the name of the active window
range rPage2 = [%H];
string strGraphName$= %H; range rPage3 = [%(strGraphName$)];
## Matrices
For matrix data, the RangeString takes the form:
[MatrixBookName]MatrixSheetNameOrIndex!MatrixObject
Note: The MatrixBookName andMatrixSheetName above used their corresponding Short Name since Short Name is the default programming name. To use Long Name in range notation for matrixbook or matrixsheet, you have to put Long Name in double quotes such as ["MyMatrixBook"]"MyMatrixSheet"!.
### Matrix Data Subrange
You can declare a range that points to a cell or range of cells. Although a matrix is a 2D array, the index in this case is a one dimensional index from 1 to the number of cells in a matrix. The numbering is in row major order so the index calculation here is:
index = (RowNumber - 1) * NumberOfColumns + ColumnNumber
e.g. for a default 32 by 32 matrix: cell in 10th row and 20th column will have index
(10 - 1) * 32 + 20 = 308
range raMC = [MBook1]1!1[308];
raMC -= 100;
### Matrix Data
Matrix Data is referred to as a Matrix Object and is equivalent to a wks.col object:
range raMD = [MBook1]"First Object Collection"!1;
stats raMD;
ty %(raMD) ranges from $(stats.min) to$(stats.max);
Given a Matrix Data range, you can access cells by row and column:
raMD[10,20] += 100; // Increment the cell in row 10, column 20 by 100
### Matrix Sheet
You can access Matrix sheet properties which is equivalent to the wks object:
range raMS = [MBook1]MSheet1;
## Loose Datasets
Loose Datasets are similar to columns in a worksheet but as the name implies, they lack the usual book-sheet-column organization. Loose Datasets are typically created with the create command, or automatically created from an assignment statement without Dataset declaration.
For loose datasets, the RangeString takes the form:
[??]!LooseDatasetName
Assignment can be performed using the syntax:
range xx = [??]!tmpdata_a; // Loose dataset 'tmpdata_a'
Practical Example:
Here, we create a loose dataset, then use the plotxy X-Function to plot a graph of the loose dataset.
// Create 2 loose datasets
create tmpdata -wd 50 a b;
tmpdata_a=data(50,1,-1);
tmpdata_b=normal(50);
// Declare the range and explicitly point to the loose dataset
range aa=[??]!(tmpdata_a, tmpdata_b);
// Make a scatter graph with it:
plotxy aa;
Loose datasets belong to a project, so they are different from a Dataset variable, which is declared, and has either session or local scope. Dataset variables are also internally loose datasets but they are limited to use in calculations only; they cannot be used in making plots, for example.
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2019-04-21 22:24:04
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https://undergroundmathematics.org/calculus-of-powers/gradient-spotting/things-you-might-have-noticed
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Building blocks
## Things you might have noticed
For each of the following questions, describe what you think will happen before you try it out on the interactivity. Then reflect on your ideas: did the actual behaviour surprise you? And can you explain the observed behaviour? (You might want to start with very simple values for $a$, $b$ and $c$.)
• What happens to the gradients as you change $c$?
• What happens to the gradients as you change $a$?
• What happens to the gradients as you change $b$?
• As we change $c$, the curve just translates up or down, so the gradient at a given $x$-value does not change.
• The graph of the gradients is a straight line, and its gradient changes as we change $a$.
The gradient function of $y=x^2$ is $2x$, as we discovered in Zooming in. If we stretch this function in the $y$-direction by a factor of $2$, by setting $a=2$, we double the gradients, giving a gradient function of $4x$.
Does the same sort of thing happen for other values of $a$?
Does changing $b$ affect the gradient of the gradient function line?
• Changing $b$ translates the gradient function. If we add $1$ to $b$, then the gradient function increases by $1$. This makes some sense: $b=1$ (with $a=c=0$) gives the function $y=x$, which has gradient $1$ everywhere. So if we add $x$ to our function, it seems reasonable that the gradient should increase by $1$. For example, when we go from $y=3x$ to $y=3x+x=4x$, the gradient increases from $3$ to $4$.
Putting these together, to find the gradient of the quadratic function $f(x)=ax^2+bx+c$, we can start with $x^2$, stretch it by a factor of $a$ to get $ax^2$ (which has what gradient function?), then add on $bx$ (which does what to the gradient function?) and then finally add $c$. This will give us the final gradient function.
Following on from the above ideas, we can predict what will happen for a cubic: starting with $x^3$, which appears to have a quadratic gradient function (which quadratic exactly?), we stretch it to obtain $ax^3$, and this stretches the gradient function. We then add on extra terms, whose behaviour we now understand from exploring quadratics: first we add on $bx^2$, then $cx$ and finally $d$.
Using this idea, we should be able to predict the gradient function of any cubic $f(x)=ax^3+bx^2+cx+d$.
Can you predict what the gradient will be for quartics (equations of the form $y=ax^4+bx^3+\cdots$) or polynomials of higher degree?
Some patterns definitely seem to be emerging here! What would you expect the gradient function of $x^4$ to be?
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2018-08-21 05:51:15
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https://www.jobilize.com/chemistry/course/20-4-amines-and-amides-organic-chemistry-by-openstax?qcr=www.quizover.com
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# 20.4 Amines and amides
Page 1 / 5
By the end of this section, you will be able to:
• Describe the structure and properties of an amine
• Describe the structure and properties of an amide
Amines are molecules that contain carbon-nitrogen bonds. The nitrogen atom in an amine has a lone pair of electrons and three bonds to other atoms, either carbon or hydrogen. Various nomenclatures are used to derive names for amines, but all involve the class-identifying suffix –ine as illustrated here for a few simple examples:
In some amines, the nitrogen atom replaces a carbon atom in an aromatic hydrocarbon. Pyridine ( [link] ) is one such heterocyclic amine. A heterocyclic compound contains atoms of two or more different elements in its ring structure.
## Dna in forensics and paternity
The genetic material for all living things is a polymer of four different molecules, which are themselves a combination of three subunits. The genetic information, the code for developing an organism, is contained in the specific sequence of the four molecules, similar to the way the letters of the alphabet can be sequenced to form words that convey information. The information in a DNA sequence is used to form two other types of polymers, one of which are proteins. The proteins interact to form a specific type of organism with individual characteristics.
A genetic molecule is called DNA, which stands for deoxyribonucleic acid. The four molecules that make up DNA are called nucleotides. Each nucleotide consists of a single- or double-ringed molecule containing nitrogen, carbon, oxygen, and hydrogen called a nitrogenous base. Each base is bonded to a five-carbon sugar called deoxyribose. The sugar is in turn bonded to a phosphate group $\left({\text{−PO}}_{4}{}^{\text{3−}}\right)$ When new DNA is made, a polymerization reaction occurs that binds the phosphate group of one nucleotide to the sugar group of a second nucleotide. The nitrogenous bases of each nucleotide stick out from this sugar-phosphate backbone. DNA is actually formed from two such polymers coiled around each other and held together by hydrogen bonds between the nitrogenous bases. Thus, the two backbones are on the outside of the coiled pair of strands, and the bases are on the inside. The shape of the two strands wound around each other is called a double helix (see [link] ).
It probably makes sense that the sequence of nucleotides in the DNA of a cat differs from those of a dog. But it is also true that the sequences of the DNA in the cells of two individual pugs differ. Likewise, the sequences of DNA in you and a sibling differ (unless your sibling is an identical twin), as do those between you and an unrelated individual. However, the DNA sequences of two related individuals are more similar than the sequences of two unrelated individuals, and these similarities in sequence can be observed in various ways. This is the principle behind DNA fingerprinting, which is a method used to determine whether two DNA samples came from related (or the same) individuals or unrelated individuals.
Using similarities in sequences, technicians can determine whether a man is the father of a child (the identity of the mother is rarely in doubt, except in the case of an adopted child and a potential birth mother). Likewise, forensic geneticists can determine whether a crime scene sample of human tissue, such as blood or skin cells, contains DNA that matches exactly the DNA of a suspect.
An atom or group of atoms bearing anelectrical charge such as the sodium and chlorine atoms in a salt solution.
Hello guys! Answer me questions nah
pls wat is periodic table
Prince
it's a list that shows the chemical element arranged according to their properties.
what is the chemical equation for ideal gas?
what's Boyle and gas law?
what's the meaning of this℃ in atomic table
wat are ions
Sinyene
What is periodic table
How to mix chemical
why the elements of group 7 are called Noble gases
they aren't. group 8 is the noble gasses. they are snobs that don't mix with others like nobles, they have full valence shells so they don't form bonds with other elements easily. nobles don't mingle with the common folk...
Jessica
the group 7elements are not the noble gases . according to modern periodic group 18 are called noble gases elements because their valence shell are completely field so that they can't gain or loss electron so they are not able to involve in any chemical reaction.
Leena
Group 7 element they are not noble gases they halogen and halogen mean salt formers
SIRAJO
what is chemistry
chemistry is the branch of science which deal with the composition of matter
SHEDRACK
discuss the orbital stracture of the following methane,ethane,ethylene,acetylene
Why phosphurs in solid state have one atom but in gas state have four atoms
Are nuclear reactions both exothermic reactions and endothermic reactions or what?
to what volume must 8.32 NaOH be diluted to its analytical concentration 0.20 M
weight in mg 1.76 mole of I
Sheriza
the types of hydrocarbons
u are mad go and open textbook
Emmanuel
hahahahahahahahahahahahaha
Jessica
aliphatic and aromatic hydrocarbons
Osakue
stupid boy Emmanuel
Ohanaka
saturated and unsaturated
Leena
aromatic hydrocarbon aliphatic hydrocarbon
SIRAJO
I don't use to see the messages
Hhhhhh
SIRAJO
how can you determine the electronegativity of a compound or in molecules
when u move from left to right in a periodic table the negativity increases
reeza
Are you trying to say that the elctronegativity increases down the group and decreases across the period?
Ohanaka
yes and also increases across the period
reeza
for instance when you look at one group of elements in a periodic table electronegativity decreases when you go across the table electronegativity increases. hydrogen is more electronegative than sodium, potassium of that group. oxygen is more electronegative than carbon.
reeza
i hope we all know that organic compounds have carbon as their back bone
OK,Thank you so much for the answer. I am happy now
can I ask you a question now
Osakue
yes
hanna
what is the oxidation number of nitrogen, oxygen and sulphur
Osakue
5, -2 & -2
hanna
What is periodic table
SIRAJO
What is an atom?
is a smallest particle of a chemical element that can exist
Osakue
Osakue
it is a substance that cannot be broken down into simpler units by any chemical reaction
An atom is the smallest part of an element dat can take part in chemical reaction.
Idris
an atom is the smallest part of an element that can take part in a chemical reaction nd still retain it chemical properties
Precious
Is the smallest particles of an element that take part in chemical reaction without been change
John
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2019-09-21 00:25:41
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https://www.mail-archive.com/[email protected]/msg37882.html
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Re: [NTG-context] MKIV, fonts, confusion
if you don't require anything completex http://github.com/contextgarden/otfinstall/tree/master automates a lot of the tasks required in installing a font in mkii.
afsmith wrote:
Luigi, Hans, and Wolfgang, thank you for your responses. I'm still
afterwards)
Let me try asking these things as questions. Specifically, could
1. How do I determine whether I am using MKII, MKIV, or XeTeX to
process my documents?
I tend to use
\begin{XETEX,OLDTEX,LUATEX}
...
\end{XETEX,OLDTEX,LUATEX}
2. Given the line from a typescript...
\definefontsynonym[LiberationSerif] [name:liberationserif]
... how do I determine which file "name:liberationserif" corresponds to?
3. In both MKII and MKIV, how can I determine which typescripts exist?
In other words, how do I determine working arguments for
"\usetypescriptfile"?
3b. Specifically, in the case of...
\usetypescript[palatino][ec]
... in which typescript is this defined? (given a vanilla context
minimals installation)
4. What defines the output of "fc-list" or "mtxrun --script fonts
--list"? Do they correspond to files? Type synonyms? etc.
fc-list also looks in \$HOME/.fonts/
5. Do I need to bother with map files for MKII?
6. Is it particularly recommended that I use MKIV? How stable is it
compared to MKII?
Luigi, what do you mean by "and look into base/*"? Which 'base'? Are
you talking about in the context minimals distribution? some specific
online repository? I have seen the two documents you mentioned.
Re-reading them has clarified things a little bit.
Wolfgang,
Currently the module you linked to
(http://bitbucket.org/wolfs/simplefonts/) is beyond my
understanding... I would first like to understand the mechanism your
module operates on before trying to automate it.
Hans,
I am aware that there are such types of parameters that must be
defined for fonts. I understand 'it is complicated' but this does not
really help make things clearer to me. The inner workings of fonts are
of little concern to me to me at the moment. Right now I do not even
know where to look. I have seen the examples you gave for palatino,
such as...
\usetypescript[palatino][ec]
...but I have no idea where those arguments come from
Thanks.
...
What I would like to do is use the full range of TeX-Gyre fonts (and
later possibly others) for my existing documents.
It is not clear to me...
Whether or not I am using MKIV, nor how to determine which engine I am
running.
If not running MKIV/XeTeX, how to determine which fonts are installed.
What the items from the output of "fc-list" or "mtxrun --script fonts
--list" correspond to.
How to use available fonts, nor where to draw valid parameters from
for use in font commands.
How to determine what font configurations are available for a given font.
Whether I need to write typescripts, nor how to find any existing ones.
Whether I need to concern myself with .map or any other font files.
Whether I need to configure anything, move any files, etcetera.
...As well as anything else I might need to install or use a font
For what it's worth, I have been using the command "texexec" to
months ago, and am running Linux.
___________________________________________________________________________________
Wiki!
maillist : [email protected] / http://www.ntg.nl/mailman/listinfo/ntg-context
archive : https://foundry.supelec.fr/projects/contextrev/
wiki : http://contextgarden.net
___________________________________________________________________________________
___________________________________________________________________________________
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2021-06-13 18:39:50
|
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|
http://mathhelpforum.com/differential-equations/180149-stability-nonlinear-system.html
|
Thread: Stability of a nonlinear system
1. Stability of a nonlinear system
Dear MHF members,
I have the following problem.
Problem. Find all the singular points of the system
$\dot{x}=xy+12$
$\dot{y}=x^{2}+y^{2}-25.$
Investigate the stability and determine the type of each singular point. $\rule{0.2cm}{0.2cm}$
I know that the critical points of the system are $(4,-3)$, $(3,-4)$, $(-4,3)$ and $(-3,4)$. My problem is determining the stability properties of these points. Please let me know if my idea below is correct. For instance for the point critical point $(4,-3)$, I linearize the equation at $(4,-3)$ and get
$\dot{x}=-3 (x - 4) + 4 (y + 3)$
$\dot{y}=8 (x - 4) - 6 (y + 3)$
This is possible since the remainders tend to zero, i.e.,
$\lim_{(x,y)\to(4,-3)}\frac{(x-4) (y+3)}{\sqrt{x^{2}+y^{2}}}=0$
and
$\lim_{(x,y)\to(4,-3)}\frac{25 + x(x-8) + y (y+6)}{\sqrt{x^{2}+y^{2}}}=0$.
Since the linearized equations eigenvalues $\lambda_{1,2}=(-9\pm\sqrt{137})/2$ are of opposite sign, we see that the critical point $(4,-3)$ is a saddle point and is therefore unstable. I will repeat the same steps above for each critical point.
Thanks a lot.
bkarpuz
2. Originally Posted by bkarpuz
Since the linearized equations eigenvalues $\lambda_{1,2}=(-9\pm\sqrt{137})/2$ are of opposite sign, we see that the critical point $(4,-3)$ is a saddle point and is therefore unstable. I will repeat the same steps above for each critical point.
Right, but its is sufficient to compute
$A=J_v(4,-3)=\begin{bmatrix}{\dfrac{{\partial v_1}}{{\partial x}}(4,-3)}&{\dfrac{{\partial v_1}}{{\partial y}}(4,-3)}\\{\dfrac{{\partial v_2}}{{\partial x}}(4,-3)}&{\dfrac{{\partial v_2}}{{\partial y}}(4,-3)}\end{bmatrix}=\begin{bmatrix}{-3}&{\;\;4}\\{\;\;8}&{-6}\end{bmatrix}$
whose eigenvalues are certainly
$\lambda=\dfrac{-9\pm \sqrt{137}}{2}$
|
2013-12-19 11:17:49
|
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|
https://study.com/academy/answer/find-the-second-derivative-of-the-given-function-f-x-e-2x-plus-2e-x.html
|
# Find the second derivative of the given function f(x) = e^{2x} + 2e^x.
## Question:
Find the second derivative of the given function {eq}f(x) = e^{2x} + 2e^x. {/eq}
## Second-order Derivative:
The order of derivative tells us how many times the function is differentiated. If the order of the function is one, means the function is differentiated only once. For the second-order derivative, the function has to be differentiated twice. Collectively all the derivatives having the order of more than one are known as the higher-order derivatives of the function. The chain rule of differentiation is used to differentiate the composite function i.e. one function inside another function, {eq}(f(g(x))) {/eq}. It can be given with the help of the following formula:
{eq}(f(g(x)))' = f'(g(x) ) \cdot g'(x) {/eq}
Also,
{eq}(e^x)' = e^x {/eq}
Become a Study.com member to unlock this answer! Create your account
Calculating Higher Order Derivatives
from
Chapter 8 / Lesson 10
3.2K
Differentiating functions doesn't have to stop with the first or even second derivative. Learn what a mathematical jerk is as you calculate derivatives of any order in this lesson.
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2021-05-18 23:20:02
|
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|
https://gamedev.stackexchange.com/questions/75182/how-can-i-create-or-extrude-a-mesh-along-a-spline/75205
|
# How can I create or extrude a mesh along a spline?
Let's say, for example, that I have a working spline. I want to use this spline to create a mesh, but I'm not quite sure how to go about it. For example, I want to create a road along this spline. I can interpolate through the curve and generate my segments and use that. If I were just making a single straight line I could take the normal of that vector and extrude the endpoints to get the vertices for a quad and do it that way, but how should I handle the curve part? I can't just use the endpoints of one quad as the start of another because they won't line up at different angles.
What would be the proper way to extrude a spline into a mesh? I'm tagging this as Unity since that's what I'm using, though the answer I assume will be engine-agnostic.
Your idea is correct, you just have to work more on it.
Here is an article I wrote last year: http://blog.meltinglogic.com/2013/12/how-to-generate-procedural-racetracks/ It uses exactly what you described, and as you can see, the result is very good.
Here is the code which explains how the mesh was generated from the spline:
for(float i = 0; i <= 1.0f;)
{
Vector2 p = CatmullRom.calculatePoint(dataSet, i);
Vector2 deriv = CatmullRom.calculateDerivative(dataSet, i);
float len = deriv.Length();
i += step / len;
deriv.divide(len);
deriv.scale(thickness);
deriv.set(-deriv.y, deriv.x);
Vector2 v1 = new Vector2();
Vector2 v2 = new Vector2();
v2.set(p).sub(deriv);
if(i > 1.0f) i = 1.0f;
}
The idea behind this algorithm is, traverse through the spline by segments of step size, for each point on the spline, calculate its derivative, normalize it to get the normal, rotate it by 90°, then add two points to the vertice list: splinePoint + normal and splinePoint - normal. You'll have a stripe of vertices following the spline, you can easily generate it's indices, If you have problems generating those indices, just throw a comment and I'll edit the answer.
Some further clarifications:
• Derivative is the same as tangent, in case your engine words it differently;
• Incrementing step / derivativeLength to the time parameter is the right thing to do. Since the size of the spline differ from point to point, you must use this to keep going at a constant speed;
• With that being said, step parameter should be defined in World Units, not Spline Percentage. (That is, if you set step to 5 and your units are pixels, then each segment will be put 5 pixels apart, and not each 5% of the spline);
• Yes, thickness is some constant that you should fiddle around to find a good value. If your world units are pixels, then your mesh will be 2 * thickness thick;
• If your engine doesnt support derivatives, try porting these: LibGDX Catmull Rom or LibGDX Bezier
• In case you want details on the implementation of catmull derivatives, it seems that OP have already did the awesome job of asking on Math.SE, so go read that!
• This works, thank you. I've managed to get the points along the spline, so if you have an article about how you managed to create the proper mesh and texturing, that would be great too.. – ssb May 18 '14 at 8:00
• It actually wasn't texturing. It's more than one mesh, and they're colored :p – Gustavo Maciel May 19 '14 at 9:26
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2021-05-18 16:29:38
|
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|
https://chemistry.stackexchange.com/questions/10307/confused-with-oxidation-numbers/10309
|
# Confused with oxidation numbers
If -2 is a oxidation number and 2- is the charge, then what is - by itself, is it a oxidation number or charge? As in $\ce{Ag+}$.
Also do you see $\ce{Cu + 2Ag+ -> Cu^{2+} + 2Ag}$ as a redox reaction?
Because I don't see any increase or increase in the oxidation number of any element in this equation. All I see is charges
Let's consider your reaction $Cu + 2Ag^+ \longrightarrow Cu^{2+} + 2Ag$ The superscripts represent charges. Reaction equations typically don't include oxidation states. In the reactants, $Cu$ has no charge and thus an oxidation state of 0 (rule 1). $Ag^+$ has a charge of +1 and thus an oxidation state of +1 (rule 2). In the products, $Cu^{2+}$ has a charge of +2 and thus an oxidation state of +2 (rule 2). $Ag$ has no charge and thus an oxidation state of 0 (rule 1).
In molecules, oxidation state can be thought of the charge of an atom when all bonds are treated as ionic. For example, in $H_2O$, Both $O-H$ bonds are polar and, since $O$ is more electronegative, all bonding electrons can be considered to belong to $O$. Thus, $O$ has 2 bonding pairs and 2 lone pairs for a total of 8. $O$ has 6 protons so it's oxidation state here is $6-8=-2$. Each $H$ has no electrons and 1 proton for an oxidation state of +1. The total oxidation state of the molecule is $2(+1)+-2=0$, which matches its charge.
To determine the oxidation number, it is the number of electrons removed or added to the element to get to the stated charge. For example; $Cl^-$ has an oxidation number of -1; $Ca^{2+}$ has a oxidation number of +2; and in $SO_4^{2-}$, sulfur has an oxidation number of +6.
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2021-09-18 17:35:26
|
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|
https://www.x-mol.com/paper/math/tag/106/journal/88691
|
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-07-30
Andreas Lichtenstern, Pavel V. Shevchenko, Rudi Zagst
In this article we solve the problem of maximizing the expected utility of future consumption and terminal wealth to determine the optimal pension or life-cycle fund strategy for a cohort of pension fund investors. The setup is strongly related to a DC pension plan where additionally (individual) consumption is taken into account. The consumption rate is subject to a time-varying minimum level and
更新日期：2020-07-30
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-07-09
Gabriele Canna, Francesca Centrone, Emanuela Rosazza Gianin
This paper introduces a new approach to face capital allocation problems from the perspective of acceptance sets, by defining the family of sub-acceptance sets. We study the relations between the notions of sub-acceptability and acceptability of a risky position as well as their impact on the allocation of risk. We define the notion of risk contribution rule and show how in this context it is interpretable
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-07-08
Erhan Bayraktar, Yan Dolinsky, Jia Guo
In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the convergence in distribution of the underlying processes. We also establish a weak convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall
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• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-07-01
Daron Acemoglu, Asuman Ozdaglar, James Siderius, Alireza Tahbaz-Salehi
This paper develops a network model of interbank lending, in which banks decide to extend credit to their potential borrowers. Borrowers are subject to shocks that may force them to default on their loans. In contrast to much of the previous literature on financial networks, we focus on how anticipation of future defaults may result in ex ante “credit freezes,” whereby banks refuse to extend credit
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• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-06-27
Nils Detering, Thilo Meyer-Brandis, Konstantinos Panagiotou, Daniel Ritter
Fire sales and default contagion are two of the main drivers of systemic risk in financial networks. While default contagion spreads via direct balance sheet exposures between institutions, fire sales describe iterated distressed selling of assets and their associated decline in price which impacts all institutions invested in these assets. That is, institutions are indirectly linked if they have overlapping
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-06-11
Dániel Ágoston Bálint, Martin Schweizer
We study general undiscounted asset price processes, which are only assumed to be nonnegative, adapted and RCLL (but not a priori semimartingales). Traders are allowed to use simple (piecewise constant) strategies. We prove that under a discounting-invariant condition of absence of arbitrage, the original prices discounted by the value process of any simple strategy with positive wealth must follow
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• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-06-09
Matteo Burzoni, Marco Maggis
We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which requires neither specific assumptions on the class of priors $$\mathcal {P}$$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-06-08
Tingjin Yan, Bingyan Han, Chi Seng Pun, Hoi Ying Wong
This paper solves for the robust time-consistent mean–variance portfolio selection problem on multiple risky assets under a principle component stochastic volatility model. The model uncertainty is introduced to the drifts of the risky assets prices and the stochastic eigenvalues of the covariance matrix of asset returns. Using an extended dynamic programming approach, we manage to derive a semi-closed
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-05-29
Taras Bodnar, Dmytro Ivasiuk, Nestor Parolya, Wolfgang Schmid
We derive new results related to the portfolio choice problem for power and logarithmic utilities. Assuming that the portfolio returns follow an approximate log-normal distribution, the closed-form expressions of the optimal portfolio weights are obtained for both utility functions. Moreover, we prove that both optimal portfolios belong to the set of mean-variance feasible portfolios and establish
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-05-29
Axel Gandy, Luitgard A. M. Veraart
We develop a modelling framework for estimating and predicting weighted network data. The edge weights in weighted networks often arise from aggregating some individual relationships between the nodes. Motivated by this, we introduce a modelling framework for weighted networks based on the compound Poisson distribution. To allow for heterogeneity between the nodes, we use a regression approach for
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-05-29
Xinfeng Ruan, Jin E. Zhang
In this paper, we provide a complete solution to the problem of equilibrium asset pricing in a pure exchange economy with two types of heterogeneous investors having higher/lower risk aversion. Using a perturbation method, we obtain analytical approximate formulas for the optimal consumption-sharing rule, which is numerically justified to be accurate for a large risk aversion and heterogeneity. We
更新日期：2020-07-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-04-02
René Aïd, Giorgia Callegaro, Luciano Campi
We design three continuous-time models in finite horizon of a commodity price, whose dynamics can be affected by the actions of a representative risk-neutral producer and a representative risk-neutral trader. Depending on the model, the producer can control the drift and/or the volatility of the price whereas the trader can at most affect the volatility. The producer can affect the volatility in two
更新日期：2020-04-02
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-03-14
David Criens
We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Itô process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves
更新日期：2020-03-14
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-03-13
Shou Chen, Richard Fu, Lei Wedge, Ziran Zou
We study the consumption and portfolio decisions by incorporating mortality risk and altruistic factor in the classical model of Merton (Rev Econ Stat 51:247–257, 1969; J Econ Theory 3:373–413, 1971) and Yaari (Rev Econ Stud 32(2):137–150, 1965). We find that besides the present-biased preference, the process of updating mortality information may be another underlying cause of dynamically time-inconsistent
更新日期：2020-03-13
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-03-12
Qian Lin, Dejian Tian, Weidong Tian
This paper introduces a class of generalized stochastic differential utility (GSDU) models in a continuous-time framework to capture ambiguity aversion on the financial market. This class of GSDU models encompasses several classical approaches to ambiguity aversion and includes new models about ambiguity aversion. For a general GSDU model, we demonstrate its continuity, monotonicity, time consistency
更新日期：2020-03-12
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-03-09
We propose a general framework for estimating the vulnerability to default by a central counterparty (CCP) in the credit default swaps market. Unlike conventional stress testing approaches, which estimate the ability of a CCP to withstand nonpayment by its two largest counterparties, we study the direct and indirect effects of nonpayment by members and/or their clients through the full network of exposures
更新日期：2020-03-09
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-02-24
Julio Backhoff-Veraguas, Ludovic Tangpi
It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the “Optimized Certainty Equivalent” risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This
更新日期：2020-02-24
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-02-17
E. Lepinette, T. Q. Tran
We consider the consumption-investment optimization problem for the financial market model with constant proportional transaction rates and Lévy price process dynamics. Contrarily to the recent work of De Vallière (Financ Stoch 20:705–740, 2016), portfolio process trajectories are only left and right limited. This allows us to identify an optimal làdlàg strategy, e.g. in the two dimensional case, as
更新日期：2020-02-17
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-02-05
Junkee Jeon, Kyunghyun Park
The purpose of this paper is to study the optimal retirement and consumption/investment decisions of an infinitely lived agent who does not tolerate any decline in his/her consumption throughout his/her lifetime. The agent receives labor income but suffers disutility from working until retirement. The agent’s optimization problem combines features of both singular control and optimal stopping. We use
更新日期：2020-02-05
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-01-21
Esmaeil Babaei, Igor V. Evstigneev, Klaus Reiner Schenk-Hoppé, Mikhail Zhitlukhin
The aim of this work is to extend the classical theory of growth-optimal investments (Shannon, Kelly, Breiman, Algoet, Cover and others) to models of asset markets with frictions—transaction costs and portfolio constraints. As the modelling framework, we use discrete-time dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors—von Neumann–Gale dynamical systems
更新日期：2020-01-21
• Math. Finan. Econ. (IF 0.792) Pub Date : 2020-01-01
Sergei Belkov, Igor V. Evstigneev, Thorsten Hens, Le Xu
We consider a stochastic model of a financial market with one-period assets and endogenous asset prices. The model was initially developed and analyzed in the context of Evolutionary Finance with the main focus on questions of “survival and extinction” of investment strategies (portfolio rules). In this paper we view the model from a different, game-theoretic, perspective and analyze Nash equilibrium
更新日期：2020-01-01
Contents have been reproduced by permission of the publishers.
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2020-08-10 04:48:19
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https://wizedu.com/questions/50230/a-researcher-claims-that-the-mean-of-the-salaries
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##### Question
In: Statistics and Probability
# A researcher claims that the mean of the salaries of elementary school teachers is greater than...
A researcher claims that the mean of the salaries of elementary school teachers is greater than the mean of the salaries of secondary school teachers in a large school district. The mean of the salaries of a random sample of 26 elementary school teachers is $48,250 and the sample standard deviation is$3900. The mean of the salaries of 24 randomly selected secondary school teachers is $45,630 with a sample standard deviation of$5530. At ? = 0.05, can it be concluded that the mean of the salaries of elementary school teachers is greater than the mean of the salaries of secondary school teachers?
## Solutions
##### Expert Solution
To Test :-
H0 :- µ1 = µ2
H1 :- µ1 > µ2
Test Statistic :-
t = (X̅1 - X̅2) / SP √ ( ( 1 / n1) + (1 / n2))
t = ( 48250 - 45630) / 4751.3391 √ ( ( 1 / 26) + (1 / 24 ) )
t = 1.948
Test Criteria :-
Reject null hypothesis if t > t(α, n1 + n2 - 2)
Critical value t(α, n1 + n1 - 2) = t( 0.05 , 26 + 24 - 2) = 1.677
t > t(α, n1 + n2 - 2) = 1.948 > 1.677
Result :- Reject Null Hypothesis
Decision based on P value
P - value = P ( t > 1.948 ) = 0.0286
Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0.0286 < 0.05 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis
There is sufficient evidence to concluded that the mean of the salaries of elementary school teachers is greater than the mean of the salaries of secondary school teachers.
## Related Solutions
##### Salaries for teachers in a particular elementary school district are normally distributed with a mean of...
Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of$6,500. We randomly survey ten teachers from that district. 1.Find the probability that the teachers earn a total of over $400,000 2.If we surveyed 70 teachers instead of ten, graphically, how would that change the distribution in part d? 3.If each of the 70 teachers received a$3,000 raise, graphically, how would that change the distribution in part...
##### Salaries for teachers in a particular elementary school district are normally distributed with a mean of...
Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of$6,500. We randomly survey ten teachers from that district. Find the 85th percentile for the sum of the sampled teacher's salaries to 2 decimal places.
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2023-02-08 04:56:38
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https://www.physicsforums.com/threads/question-about-an-application-of-implicit-function-theorem.621266/
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# Question About an Application of Implicit Function Theorem
1. Jul 16, 2012
### nonmathtype
Hi everyone,
I do economics but am very poor at Math. I had a specific and perhaps silly question about the implicit function theorem, but will be grateful for an urgent response.
Suppose we have a function, U(x, y).
x and another variable z are linearly related so the function can also be specified as U(z,y) by substituting z for x.
It can be shown by using the implicit function theorem that y = f(x), and also separately that y = g(z) such that U(x,y)=0 and U(z,y)= 0 respectively.
Is it them possible to conclude that y = h(x,z) exists ?
2. Jul 16, 2012
### chiro
Hey nonmathtype and welcome to the forums.
You haven't specified what the function h is not explicitly or implicitly as a relationship to your other functions. What properties of h did you have in mind?
3. Jul 17, 2012
### nonmathtype
Hi Chiro, thank you so much for your response.
Actually I am not interested in any specific functional form or properties of h. All I am looking for is its existence.
So it is clear that both f and g will exist by the implicit function theorem as mentioned above (because the partial derivative of U with respect to y is not zero, and U is continuously differentiable by assumption. These are both functions with a single argument (single variable functions) i.e y = f(x) and y = g(z).
Does this imply that a function that contains both x and z in the domain will exist as well i.e y = h(x,z) ?
4. Jul 17, 2012
### chiro
You should be able to do that both for explicit and implicit representations of the functions.
For example if you can't get z = r(x) (i.e. an explicit function of x) then provided you have a function for y = f(x) and y = g(z), then you can add them together and divide by 2 to get y = 1/2(f(x) + g(z)) = h(x,z).
I'm assuming your assumptions where you have existence of y = f(x) and y = g(z) and I'm assuming they point to the same variable y.
Also since you didn't mention the functional form, I assume you just want to find any function with h(z,x) so the one provided which is a linear combination of the two solutions does satisfy the requirement at least.
5. Jul 19, 2012
### nonmathtype
Thanks a lot chiro, it is much appreciated :-) Yes your assumptions about what I was trying to say are correct...both y = f(x) and y = g(z) point to the same y.
I did not realize that it could be so obvious ! I was worried regarding the fact that the domain of both f and g is R1, whereas the domain of the proposed function h is R2. So I did not think that the existence of h on the basis of existence of f and g would follow in such a straightforward way.
Once again, thanks a lot !
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2017-08-22 08:39:02
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https://www.ncatlab.org/nlab/show/model+structure+on+cubical+sets
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Contents
model category
for ∞-groupoids
# Contents
## Idea
There is a model category structure on the category $[\Box^{op},Set]$ of cubical sets whose homotopy theory is that of the classical model structure on simplicial sets.
Using this version of the homotopy hypothesis-theorem, cubical sets are a way to describe the homotopy type of ∞-groupoids using of all the geometric shapes for higher structures the cube.
## Definition
There is an evident simplicial set-valued functor
$\Box \to sSet$
from the cube category to sSet, which sends the cubical $n$-cube to the simplicial $n$-cube
$\mathbf{1}^n \mapsto (\Delta[1])^{\times n} \,.$
Similarly there is a canonical Top-valued functor
$\Box \to Top$
$\mathbf{1}^n \mapsto (\Delta^1_{Top})^n \,.$
The corresponding nerve and realization adjunction
$(|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}$
is the cubical analogue of the simplicial nerve and realization discussed above.
###### Theorem
There is a model structure on cubical sets $Set^{\Box^{op}}$ whose
• weak equivalences are the morphisms that become weak equivalences under geometric realization $|-|$;
• cofibrations are the monomorphisms.
This is (Jardine, section 3).
Explicitly, a set of generating cofibrations is given by the boundary inclusions $\partial \Box^n \to \Box^n$, and a set of generating acyclic cofibrations is given by the horn inclusions $\sqcap_{k,\epsilon}^n \to \Box^n$. This is (Cisinski, Thm 8.4.38). Thus, as a consequence of Cisinski’s work, the fibrations are exactly cubical Kan fibrations.
## Properties
### Homotopy theory
The following theorem establishes a form of the homotopy hypothesis for cubical sets.
###### Theorem
$A \to Sing_\Box(|A|)$
is a weak equivalence in $Set^{{\Box}^{op}}$ for every cubical set $A$.
$|Sing_\Box X| \to X$
is a weak equivalence in $Top$ for every topological space $X$.
It follows that we have an equivalence of categories induced on the homotopy categories
$Ho(Top) \simeq Ho(Set^{\Box^{op}}) \,.$
This is (Jardine, theorem 29, corollary 30).
In fact, by the discussion at adjoint (∞,1)-functor it follow that the derived functors of the adjunction exhibit the simplicial localizations of cubical sets equivalent to that of simplicial sets, hence makes their (∞,1)-categories equivalent (hence equivalent to ∞Grpd).
## Other kinds of cubical sets
In cubical type theory one uses more structured notions of cubical set (symmetric, cartesian, De Morgan, etc.) In most cases such categories have both a test model structure, which is equivalent to spaces, and a cubical-type model structure that corresponds to the interpretation of type theory. In many cases these are not equivalent, and the cubical-type model structure does not model classical homotopy types. See cubical-type model structure for more discussion.
## References
Using that the cube category is a test category a model structure on cubical sets follows as a special case of the model structure on presheaves over a test category, due to
Cisinski also derives explicit generating cofibrations and generating acyclic cofibrations using his theory of generalized Reedy category, or categories skelettiques. See Section 8.4.
The model structure on cubical sets as above is given in detail in
There is also the old work
• Victor Gugenheim, On supercomplexes Trans. Amer. Math. Soc. 85 (1957), 35–51 PDF
in which “supercomplexes” are discussed, that combine simplicial sets and cubical sets (def 5). There are functors from simplicial sets to supercomplexes (after Defn 5) and, implicitly, from supercomplexes to cubical sets (in Appendix II). This was written in 1956, long before people were thinking as formally as nowadays and long before Quillen model theory, but a comparison of the homotopy categories might be in there.
Last revised on February 3, 2021 at 10:37:59. See the history of this page for a list of all contributions to it.
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2021-09-25 18:32:31
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https://www.physicsforums.com/threads/shape-of-the-universe.675693/page-5
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# Shape of the universe
Hardly. The cosmological principle holding to infinity requires an infinite degree of fine-tuning: how did the universe, out to infinite distances, know to be the same density in all locations, with the appropriate time-slicing?
It's rather like the horizon problem, but expanded to infinite distances instead of merely being required to hold in our visible universe.
Another way of stating the problem is to look at the classic model of inflation. If inflation were extended infinitely into the past, then inflation could easily explain a global cosmological principle. However, we know that can't be the case: extending inflation infinitely into the past also requires infinite fine-tuning: inflation predicts a singularity somewhere in the finite past, and the further back you try to push that singularity, the more fine-tuning you need. And if inflation can only be extended a finite distance back into the past, then it isn't possible for the universe as a whole to have reached any sort of equilibrium density, as if you go far enough away, you'll eventually reach locations that have always, since the start of inflation, been too far for light to reach one another. Any regions of the universe that lie beyond this distance aren't likely to be remotely close to one another in density.
Of course, this argument is based upon the assumption that a simplistic model of inflation is true, but the argument is reasonably-generic among most inflation models.
If time to the singularity is considered finite it can never require an infinite degree of fine tuning, but I can see how this could conflict with a spatially infinite geometry together with a global CP as I mentioned in my previous post.
But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time), the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.
Chalnoth
But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time),
Because there's no a priori expectation of the cosmological principle necessarily being true: it only makes sense if the physics of the early universe set things up that way, and the horizon problem points out that if you just take GR with the observed components of the current universe, it is impossible for any physical process to set up a universe that is approximately homogeneous and isotropic.
This is one of the reasons why inflation was proposed, but inflation doesn't extend the distance at which we expect the cosmological principle to hold out to infinity.
the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.
I think you're too attached to the FRW model. It's just a model. It's not reality.
... There are no trivial questions, feel free to ask anything.
OK. Well, here goes ... I'm prepared for some scoffing.
The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.
Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?
Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?
So, take k as well. The wiki article says
k is the curvature parameter — that is, a measure of how curved spacetime is
but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?
And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?
Chalnoth
Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question?
The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.
The speed of light is, to the best of our knowledge, also constant.
The spatial curvature, k, is a constant that is a way of encapsulating the relationship between the expansion rate and the energy density of the universe. The value of k doesn't change because of how we define the term.
OK. Well, here goes ... I'm prepared for some scoffing.
The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.
Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?
Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?
So, take k as well. The wiki article says
but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?
And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?
The three parameters are constant in that formula, c and $\pi$ are obviously constant, now k here is referring to the normalized curvature that is normally used in the Friedmann equations and in the FRW line element it can only be 1, 0, or -1. The evolution of positive or negative spatial curvature is then integrated in the scale factor a.
Is there any evidence to suggest that the speed of light is different for different values of curvature?
no.
locally the speed of light is always 'c'.
What HAS changed over the age of the universe is the rate of expansion. The scale factor
a[t] is a function of time determined from general relativity.
With all due respect, I was hoping for slightly less offhand answers.
The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.
So, you're saying that if you draw a circle on, say, a balloon and then measure its circumference and diameter and divide the one by the other you're going to get a value of 3.141529... ? Or, are you saying that if you draw a circle on a balloon then measure its circumference and diameter and divide the one by the other AND then inflate the baloon to double its size and remeasure the circle's circumference and diameter and divide the one by the other you will get the same result?
The speed of light is, to the best of our knowledge, also constant.
Haven't you been saying, about the Cosmological Principle, that just because it holds on the large in the observable universe (i.e. to the best of our knowledge it holds) that is no reason to believe that it holds out to infinity? In some cases you are prepared to consider (currently) unknown (unknowable) possibilities and in others you are not?
Is there some means of determining the speed of light as it was 5 billion years ago? 10 billion years ago? 14 billion years ago? 10$^{-30}$ seconds after the Big bang?
I'll ask my previous question again - does the bending (refraction?) of light by a massive object not imply a change in its velocity due to the change in local (at the massive object) curvature of space?
The spatial curvature, k, is a constant that is a way of encapsulating the relationship between the expansion rate and the energy density of the universe. The value of k doesn't change because of how we define the term.
I was under the impression that the relationship doesn't appear to be as expected - and an inflationary energy of some sort has had to be posited?
Chalnoth
The first point is that that isn't how $pi$ is defined. It is defined in such a way that the curvature of space simply doesn't matter, so that in curved space the ratio of the circumference of a circle to its diameter is no longer $pi$, but $pi$ times some measure of the enclosed curvature.
As for the speed of light, well, certainly you can come up with some different laws of physics that allow it to vary, but then the entire equation needs to be re-evaluated in that situation.
With regard to k, again, that is a constant based upon how the terms in FRW are defined. You could certainly re-define your terms such that the value that encapsulates the curvature isn't a constant (as is done routinely by using $\Omega_k$ to describe the curvature). You could also imagine a universe that doesn't follow the symmetries of FRW and thus doesn't have a single curvature term (and in that situation, like the above, you'd have to re-evaluate the entire equation, not just the one term).
A shorter way to argue this point is to just state that there are certain assumptions built into deriving the Friedmann equations in the first place. You can't break those assumptions after the fact and get something sensible: you have to re-derive the equations from scratch using the new set of assumptions.
The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.
The three parameters are constant in that formula, c and $\pi$ are obviously constant...
In fact, isn't testing for the interior sum of triangles just another form of looking for a value of $\pi$ that differs to that on a flat surface?
Chalnoth
In fact, isn't testing for the interior sum of triangles just another form of looking for a value of $\pi$ that differs to that on a flat surface?
Not the way $\pi$ is defined.
The first point is that that isn't how $pi$ is defined. It is defined in such a way that the curvature of space simply doesn't matter, so that in curved space the ratio of the circumference of a circle to its diameter is no longer $pi$, but $pi$ times some measure of the enclosed curvature.
Surprising! So, a mathematical symbol which has a clear universal definition (the ratio of circumference to diameter of a circle) is redefined for use here? And the symbol itself isn't even changed? I'd have expected something like $\pi$$_{0}$ or $\pi$$_{E}$ in such a case, to indicate that the equations were using the value of $\pi$ in Euclidean geometry.
I'd be interested in a reference to where this definition is formally made (preferably a layman understandable source, if possible).
As for the speed of light, well, certainly you can come up with some different laws of physics that allow it to vary, but then the entire equation needs to be re-evaluated in that situation.
...
A shorter way to argue this point is to just state that there are certain assumptions built into deriving the Friedmann equations in the first place. You can't break those assumptions after the fact and get something sensible: you have to re-derive the equations from scratch using the new set of assumptions.
Excellent! Let's do it. Are there any theoretical physicits on here up to such a challenge?
I'm not interested, per se, in the shape and state of the Universe only with regard to models that we currently have which have shortcomings and artificial constructs / ideas to get around these shortcomings. What I'm interested in is the discovery / exploration of a model which explains all the observations without any dissatisfying artificial additions. I recognise, of course, that such a model might not exist, but isn't it the fundamental goal of theoretical physics - to continue searching for such a model anyway?
Chalnoth
Surprising! So, a mathematical symbol which has a clear universal definition (the ratio of circumference to diameter of a circle) is redefined for use here? And the symbol itself isn't even changed? I'd have expected something like $\pi$$_{0}$ or $\pi$$_{E}$ in such a case, to indicate that the equations were using the value of $\pi$ in Euclidean geometry.
I'd be interested in a reference to where this definition is formally made (preferably a layman understandable source, if possible).
Well, it's just a convention. It doesn't really mean anything. But this is the way that General Relativity has been developed.
Excellent! Let's do it. Are there any theoretical physicits on here up to such a challenge?
It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.
What I'm interested in is the discovery / exploration of a model which explains all the observations without any dissatisfying artificial additions.
What dissatisfying artificial additions? Why are they dissatisfying, and why are they artificial?
Because there's no a priori expectation of the cosmological principle necessarily being true[...]
I think you're too attached to the FRW model. It's just a model. It's not reality.
It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.
Seems like I'm not the only one attached to the FRW model.
But you have some kind of double standard here, removing or changing some assumptions like for instance the cosmological principle seems ok to you.
Chalnoth
Seems like I'm not the only one attached to the FRW model.
But you have some kind of double standard here, removing or changing some assumptions like for instance the cosmological principle seems ok to you.
Let me be clear: Nobody, to my knowledge, has successfully solved the Einstein equations for a global space-time that does not follow FRW (except in the special case of spherical symmetry). I have no problem considering other space-times, and don't think that FRW is likely to be accurate at distances much larger than our horizon, but it turns out that doing it right is fantastically difficult.
It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.
Fair enough. That's understandable. So, maybe this is straying into another thread. I'd like to explore, from a semi-philosophical point of view (and, particularly, given that I'm not a physicist of any sort), without having to tie it down to a specific existing mathematical description, what might make sense in terms of an alternative model for the Universe. Conceivably, such an exploration might develop into a mathematical description (though I lack the math knowledge myself), but even if it didn't it might be interesting nonetheless.
What dissatisfying artificial additions? Why are they dissatisfying, and why are they artificial?
Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.
2. The universal constants - G and c (for a start). It is dissatisfying that they are not derivable. They should be a function of some aspect of the Universe. As I implied earlier, for example, isn't it conceivable that c might be related to the curvature of space(time?)?
Chronos
Gold Member
Inflation is science in its most fundamental form - a model that matches observational evidence. If a better model comes along, inflation will become an historical footnote. G and c are not the only universal constants that cannot be derived from something more fundamental. Some things can be measured, but, not explained. It's the nature of the universe.
Chalnoth
Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.
I don't see that it's messy at all. In its most basic form, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe.
2. The universal constants - G and c (for a start). It is dissatisfying that they are not derivable. They should be a function of some aspect of the Universe. As I implied earlier, for example, isn't it conceivable that c might be related to the curvature of space(time?)?
This would require a more fundamental law of physics. And even then, it isn't clear that these constants could ever be derived from first principles (though that has been a goal of many theoretical physicists for decades), but at least having the more fundamental laws of physics would give us an answer to how the constants we measure got that way (and the answer may be, in part, by accident).
At any rate, the fact is that discovering a correct fundamental set of physical laws is profoundly difficult. String theory has been the primary proposal of such a set of physical laws for decades, but even now we don't yet know everything about string theory, let alone whether or not it applies to reality.
Inflation is science in its most fundamental form - a model that matches observational evidence. If a better model comes along, inflation will become an historical footnote.
I don't see that it's messy at all. In its most basic form, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe.
I don't agree that this is science in its most fundamental form. Science in its most fundamental form, for me, is taking phenomena that occur (continuous) and proposing a theory to explain those phenomena with testable predictions and repeatable observations. There are no ongoing phenomena of 'inflation'. In fact there are no direct observations of 'inflation' at all (are there?). Essentially, there are a number of tricky little problems that seem to come out of a basic Big Bang model (e.g. flatness and fine tuning), and some forms of inflation would explain away those problems if it had happened.
So, we've accepted inflation for so long now it has become a core part of the standard Big Bang models.
But, as so nicely put, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe. A field that we haven't managed to stimulate in any way, or even seen direct evidence of its stimulation. We've managed to recreate (certain) conditions as far back as milliseconds(?) from the Big Bang event and yet not seen any inflation type field stimulation in those experiments.
G and c are not the only universal constants that cannot be derived from something more fundamental. Some things can be measured, but, not explained. It's the nature of the universe.
Now, this is completely contradictory to the fundamental philosophy of science - you're not prepared to ask 'why'? Why is c such-and-such a value? Why is G such-and-such a value? Why are atoms the fundamental unit of matter that cannot be divided further? Oh, wait ... they're not!! One of the basic goals of science is to eliminate such constants by demonstrating how they emerge from the basic structure / make-up of the Universe. And, I see no reason why we can't explore ideas / theories (however difficult) that attempt to do just that.
This would require a more fundamental law of physics. And even then, it isn't clear that these constants could ever be derived from first principles (though that has been a goal of many theoretical physicists for decades), but at least having the more fundamental laws of physics would give us an answer to how the constants we measure got that way (and the answer may be, in part, by accident).
At any rate, the fact is that discovering a correct fundamental set of physical laws is profoundly difficult. String theory has been the primary proposal of such a set of physical laws for decades, but even now we don't yet know everything about string theory, let alone whether or not it applies to reality.
'Difficult' is irrelevant. If it was easy it would be boring - it would all be done by now and we could all sit around watching soaps and talking about psychology.
I'm interested in a discussion about the basic assumptions that are inherent in the theories we currently work with - what problems exist because of these assumptions and what alternatives look interesting in terms of overcoming those problems and moving to alternative testable models that might be worth developing. I guess I'll start another thread on this.
Fair enough. That's understandable. So, maybe this is straying into another thread. I'd like to explore, from a semi-philosophical point of view (and, particularly, given that I'm not a physicist of any sort), without having to tie it down to a specific existing mathematical description, what might make sense in terms of an alternative model for the Universe. Conceivably, such an exploration might develop into a mathematical description (though I lack the math knowledge myself), but even if it didn't it might be interesting nonetheless.
Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.
-------
How inflation MAY follow simply from quantum gravity:
http://eprints.port.ac.uk/6488/
Chalnoth
I don't agree that this is science in its most fundamental form. Science in its most fundamental form, for me, is taking phenomena that occur (continuous) and proposing a theory to explain those phenomena with testable predictions and repeatable observations. There are no ongoing phenomena of 'inflation'. In fact there are no direct observations of 'inflation' at all (are there?). Essentially, there are a number of tricky little problems that seem to come out of a basic Big Bang model (e.g. flatness and fine tuning), and some forms of inflation would explain away those problems if it had happened.
Yes, but the theory made testable predictions as to the pattern of structure in our universe, and those predictions were very strongly confirmed by WMAP (to some extent previous experiments as well, but not nearly as strongly), and have been further strengthened by Planck.
The next big step in confirming inflation would be to detect B-mode polarization in the CMB. This isn't easy, unfortunately, as we don't know how big the B-mode polarization is (it may, sadly, be undetectable). But if we do detect it in the next few years, that would likely be a pretty strong confirmation of inflation.
If we could also observe the primordial gravity wave background, that would likely give us even greater insight.
But, as so nicely put, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe. A field that we haven't managed to stimulate in any way, or even seen direct evidence of its stimulation. We've managed to recreate (certain) conditions as far back as milliseconds(?) from the Big Bang event and yet not seen any inflation type field stimulation in those experiments.
It's not really expected that we could, considering the very large energies at which inflation occurs.
In short, you're prioritizing laboratory science over non-laboratory science in a very absurd way. The vast majority of science is done outside the laboratory, and we would know very little about the universe if we restricted ourselves only to what we can discover in a lab.
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https://www.homebuiltairplanes.com/forums/threads/industrial-engine-electronic-management-system-development-hba-style.30966/
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# Industrial engine electronic management system development - HBA style
### Help Support HomeBuiltAirplanes.com:
#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
I promised this a day ago. Got sidetracked with an FTDI driver that didn't work anymore and a few other chores......
New topic: Industrial engine electronic management system.
Answer the basic questions - Who, What, Why, Where and How.
Who:
Me. The ICC – Interim Committee Chairman.
And you, based on the ASTM committee/sub-committee develop and vote model.
The various ‘There needs to be’ threads here on HBA have never gotten past the loose conceptual stage. Design by committee just doesn’t work unless there is structure. The quickest way to get structure is with a leader. I’m appointing myself as leader of this project. If I get any followers great! If not, well that’s life. Further down the road if the group decides that a coup d’etat is needed for whatever reason, well that may be a good thing. It means that there is a group that wants results.
What:
An aircraft grade engine management system for 1L and smaller twin cylinder industrial engines. Notice that is not just an EFI system I’m proposing. EFI may be the only part of the system many will want to use. I think we can incorporate the ignition system and other bells and whistles, such as constant speed prop control, as modular options after the base is working as desired.
Why:
There is a demand for inexpensive 4 stroke engines for light aircraft. The industrial engines have been shown to be a viable option but as with everything they could be improved. The various manufacturers are starting to implement EFI on their commercial products because they understand the benefits of better fuel control and EFI in particular. We could just wait for these versions to become common on the market, but aviation use requires some features that commercial industrial engines simply don’t need. It is highly unlikely any engine manufacturer will develop an EFI system that is truly aircraft grade. There just isn’t enough market to justify the development time. That will be our job.
Where:
Here on HBA, to start.
How:
Go watch this video by our very own HBA member Boku.
This will answer the How for the project. I could eventually come up with a system for my personal use that works. I just don’t have that much time even though by nature I’m a solo kind of guy.
Next:
A starting point.
Note to moderators:
The other EFI thread is in this folder. If it would be a better fit in another please move it.
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#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
The starting point
Attributes:
Pretty well covered in this thread
Summation, in no particular order:
To me this means eliminate as many single point failures as possible. Of those that remain they need to be uncommon and benign enough that the risks would be tolerated by the average consumer.
2) Inexpensive, not cheap
If we count our time at minimum wage to develop this we have about a 1% chance, overall, of meeting this goal. Cheap in this context means only the cost to an end user of what we develop for parts and their time to assemble, buy, install and tune.
3) Simple.
This is relative to the individual, but it needs to be simple enough that the average aircraft builder can, with good documentation, install and safety use what is developed here.
4) Makes use of as many off the shelf parts as practical.
Those parts should be available from more than one source and should have a high probability of being available in the future.
5) Should be as light, compact and as reliable as the current OEM carburetors.
Hopefully it will be better in at least 2 of those 3 parameters.
6) Should provide better fuel economy.
7) Should provide automatic altitude compensation
8) Should easily interface with other aircraft systems.
9) ??????
How to accomplish the above goals?:
A) Define the features desired and those that are needed.
B) Identify and develop hardware that will get the job done.
In the interest of simplicity, I propose that this be done with open source resources as much as possible. Arduinos are an obvious option. Pyboard and Teensy are 2 others. They both use ARM Cortex M4 chips. There is a plugin for the Arduino IDE for the Teensy and it may work with the Pyboard as well? If not there are options to load software to any of these chips without using the Arduino IDE.
Sensors are readily available from many sources. We just need to define what to use and where.
C) Software will be needed. Again, Arduino/C++ is an obvious choice for source code.
For circuit design Fritzing is one option. There may be other free open source options?
There are also student versions of some software available. NI Circuit Design/Multisim is one.
D) Build on what others have developed. I was recently made aware of the Speduino project on the other thread. It looks a good source of ideas and code for us to build on? It is more general and street racing oriented than what we will end up, but it at least is functional. For those that are new to EFI this is a good basic outline of the How: http://www.faqstels800g.ru/data/docu...ool-Manual.pdf
So, this is my proposal. If there is any interest lets get started with adding anything needed to, and deleting the fluff from, the “Attributes” list, then move on to A)
#### Vigilant1
##### Well-Known Member
Regarding proposed attributes:
1) . Should we specify that the unit should be able to use 100LL? That would drive us to an open loop system. Or, we could do some research and then decide if the 100LL attribute is worth the cost. I personally would like to be able to use 100LL at least occasionally without needing to replace a sensor.
2) Have we covered failure modes enough? I want to stay airborne even with several failures.
3) Fault detection/diagnostics? It would be nice to know what parts have failed.
4) Electrical requirements? Many B&S engines have 16 amp alternators, and I'm not sure how long they'll hold up if asked to do that continuously. The Harbor Freight 670cc engines have a very small alternator, I believe. The EFI can't use every bit of the available juice.
Other available resources: Maybe study the B&S EFI system to see what choices were made in its design, what injectors, pumps, and sensors were selected, etc. It won't all apply to aviation use, but they sell a lot of them, presumably did some expensive engineering, and their worldwide dealer network might be a source of spares if we use some of the same things.
Thanks for taking this on.
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#### rmeyers
##### Active Member
What license do you propose for the embedded software?
#### Vigilant1
##### Well-Known Member
Regarding Attribute 8 (should interface with other acft systems): So, do we want this to feed us RPM, fuel flow, manifold pressure(?), In a form that can be readily used by some existing (cheap) engine mgmt software, or is it he design of that display software part of this project? Wired or wireless? This attribute would be very nice, but it might be a lot to take on, not useful to everyone, and full of future headaches if display software manufacturers change data standards, etc. ("Hot Wings, my EMS won't work with the MGL display I just bought. Can you guys modify the code for me?" No one is going to want to take that recurring hassle on). Maybe a cheap, standalone dedicated display just for this program will be less of a headache. Not as elegant, but in keeping with the vibe of this effort I think.
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#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
What license do you propose for the embedded software?
Still thinking about that. If we base it substantially on the Speeduino then the choice has been made.
I personally don't have any problems, in this case, with letting someone use the code as the base of a closed project. There just isn't enough money involved with the end use to have to worry about someone getting rich using our work?
CopyLeft
Edit:
This is one of those decisions that I think should be made by the members of this committee after discussion and voting.
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#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
My first attempt at defining the features desired and/or needed.
First those features that are needed:
1) A reliable, accurate and repeatable way to deliver a combustible fuel mixture to the engine under all anticipated conditions. Pretty basic.
2) ???
Features that are desired:
A) Oxygen sensor feedback:
This may be the first decision to make. Other than the old, no longer available, Subaru Robin I don’t know of any industrial engine that has, or will, come with a simple open loop (no O2 sensor) EFI system. All the current industrial engine EFI systems use an O2 sensor. In the case of the Kohler they specifically say that it is used for closed loop only under constant conditions after the system is fully warmed up.
The problem with using an O2 sensor is that we are still stuck with having only 100LL available at most airports. Current options are to just use the OEM EFI and find mogas or use a carburetor. For the purposes of this project I propose that we develop a system that can be used both open and closed loop. We will need a good wide band O2 sensor to develop the basic mixture map. The cost of the hardware for the wide band sensor and controller chips is under $100 with the sensor itself being most of the expense. The chips for driving and sensing a Bosch wide band O2 sensor are around$5. For those on the cheap side they can borrow/share a sensor for initial setup is the hardware is already on the EFI control board. The software can be designed to be user selectable to run closed loop or rely on only the map in memory.
B) User interface:
None is really needed except during setup and calibration. But to make diagnostics and upgrades easier there should be some form of user interface. Options are USB, CAN, and wireless. USB is already on most of the boards that are likely to be used. CAN has the advantage of an extremely robust data stream and a well-defined standard. Wireless, while not required, does provide some user-friendly options. It should be considered as an add on?
Using CAN gives us the option to send the data from the EFI sensors directly to an engine monitor. There is then no need to duplicate sensors for the display. The actual display should probably be a separate project? I'll have to double check but I think we can use CANAero standards. Wireless, other serial standards may work too?
TunerStudio interface/compatibility. I know zero currently about how to go about interfacing with this software. Speeduino does so this should be an option for this project?
C) Sensor suite:
The more we have the better as far as data collection. At the very basic we need a MAP a TPS and RPM. The problem with only the basics is that if one quits the whole system quits, or at the very least goes into a very primitive limp mode. By using other sensors like coolant and air temp, EGT, and O2 the software can make a best guess, if one of the basic sensors fails, and support a less primitive limp mode. By cross checking the other sensors the software can track sensor drift and maybe warn of potential failure. The use of dual sensors helps with this task as well as providing redundancy. Cost is a factor to be considered.
Does anyone know of a small inexpensive way of measuring mass flow suitable for a 1L engine?
D) Redundancy:
Dual CPUs that cross check one another is desirable but does add expense. The boards likely to be used can all be purchased for less than $25 each. A little more for the Teensy 3.6. E) Data logging: Internal, External, or both? #### Hot Wings ##### Grumpy Cynic HBA Supporter Log Member 4) Electrical requirements? Many B&S engines have 16 amp alternators, and I'm not sure how long they'll hold up if asked to do that continuously. The Harbor Freight 670cc engines have a very small alternator, I believe. The EFI can't use every bit of the available juice. The electronics won't take much. Most of the boards we might use have a max load of around 500mA. The injectors, and fuel pump will the the big load with a wide band O2 sucking some during warm up. ("Hot Wings, my EMS won't work with the MGL display I just bought. I'm kind of insensitive to these kind of problems. If someone purchases something with proprietary software or hardware - that's their problem, not ours. I like standards. If the standards evolve, then we should try to keep up, but most good standards are backward compatible. For example we can plug a USB 1.0 flash drive into a UBS 3.0 port and it's going to work. #### Vigilant1 ##### Well-Known Member Lifetime Supporter A) Oxygen sensor feedback: . . . .For the purposes of this project I propose that we develop a system that can be used both open and closed loop. We will need a good wide band O2 sensor to develop the basic mixture map. The cost of the hardware for the wide band sensor and controller chips is under$100 with the sensor itself being most of the expense. The chips for driving and sensing a Bosch wide band O2 sensor are around $5. For those on the cheap side they can borrow/share a sensor for initial setup is the hardware is already on the EFI control board. The software can be designed to be user selectable to run closed loop or rely on only the map in memory. Is there a significant advantage to relying on closed loop for the calibration/setup after the system is installed? Can the same thing be accomplished open-loop by using EGT? If we use use closed-loop to calibrate (with pump gas containing real gasoline and 10% ethanol) and then run the system operationally using 100LL (that has different attributes), will we be getting things right? Further, mybe it would be practical to develop mixture maps as part of the centralized big project, with the end user just selecting one from the library, fine tuning using a "mixture knob" potentiometer knob in flight. That same "mixture knob" could be very useful in some "limp home" modes. D) Redundancy: Dual CPUs that cross check one another is desirable but does add expense. The boards likely to be used can all be purchased for less than$25 each. A little more for the Teensy 3.6.
The logic needed to compare the CPU results introduces another possible faiure mode, and may make troubleshooting more difficult. "A man with two watches never knows the time." It would be simpler to have two entirely independent CPUs/boards with a toggle switch for selection in the cockpit, testing both is part of the lineup check (like a mag check). In flight, if you don't like the way things are running, go from "CPU 1" to "CPU 2" and see if things clear up. If both CPUs have backup modes to deal with sensor failures, there's a lot of redundancy there.
#### Vigilant1
##### Well-Known Member
Attributes: Is this to be a throttle body injection (one injector for the entire engine) or port injection (one injector per cylinder)? One injector is cheaper and easier, would allow maximum commonality between single and twin cylinder engine installations. Individual injectors per cylinder allows better tuning for (what are likely to be) non-optimum induction systems, probably better CHT management, better fuel economy, and possibly longer engine life (each cylinder gets a more precise mixture, so less overheating from being too lean or oil dilution from being too rich).
ETA: Due to the uneven firing (and induction) interval of V-twins, most airplane installations I've seen use dual carbs. As we won't be trying to time the fuel injection events with each cylinder's induction stroke, a separate injector per cylinder (port) would be mandatory, I think.
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#### rmeyers
##### Active Member
Don't know too much about ARM based MCUs. My background is in embedded automotive engine controllers.
All of the things requested so far are certainly doable, CANII, USB, WIFI, on-board ADC, etc, but feature creep will kill a project like this in a hurry. I think that what you are doing here is the right approach. Get a consensus on what features the proposed unit will have, lock them in, and then pick a chip architecture. It may be found that ARM based solutions will not suffice. NXP (nee Freescale, nee Motorola) have hundreds of embedded processors specifically designed for automotive use. They have demo/dev boards in the Raspberry PI price range, and they will sell you one or 100. These boards will have everything you need except the high power outputs. They also have free SDK and developer tools for the less expensive chips. One of the smaller ColdFires would be just about right for this usage.
The downside, and it may be considerable, is the learning curve. Everybody and their uncle seems to have ARM experience these days and there appears to be on the close order of 100,000 videos on YouTube about how to get started. The ColdFire is based, loosely on the old Motorola 68000, so it's not that hard to deal with. The steep part of the curve is in the functional modules of the chip. QADC (analog to digital converter module), eTPU (timing input/output module), CANII module, USB module, etc, are interesting to set up but incredibly powerful and easy to use once set up.
One last thing. Vigilant1 said "A man with two watches never knows the time." and he is absolutely correct. Two processors is just begging for a deadlock. All of our military stuff requires three processors so they can continuously vote. Now that gets complicated.
#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
Is there a significant advantage to relying on closed loop for the calibration/setup after the system is installed? Can the same thing be accomplished open-loop by using EGT?
I don't think there a significant advantage to running closed loop for our mission. We will probably have to have different maps for the different fuel blends, unless we add a fuel type sensor.
Attributes: Is this to be a throttle body injection
No. Maybe a third injector upstream used as a backup/limp? The charge savaging of the V-twin kind of eliminates the throttle body approach.
One last thing. Vigilant1 said "A man with two watches never knows the time." and he is absolutely correct. Two processors is just begging for a deadlock. All of our military stuff requires three processors so they can continuously vote. Now that gets complicated.
Understood. Even with 3 in on the vote unless there are dual sensors to cross check - Democracy can still be wrong. There may be software ways to simulate a third CPU/pilot, but I don't want to get into that here, at least until I've had some time to actually try a few things. Ultimately giving the pilot the choice, as Vigilant1 mentioned, to pick which CPU to run the show may be the most practical solution?
HBA Supporter
Log Member
#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
NXP (nee Freescale, nee Motorola) have hundreds of embedded processors specifically designed for automotive use.
Something like this might be an option?
NXP MM912_P812
The only problem with something that is not common in the hobby world is that soldering surface mount requires tools and skill that have a rather steep learning curve.
#### blane.c
##### Well-Known Member
HBA Supporter
Just a couple quick thoughts on this.
On the surface I like the idea of two processors and the ability to manually switch between them. I have reservations that the processor may be the least likely point of failure and switching to a second one will just be verification something else is wrong. Still worth a shot if you are a single engine aficionado but for multi-engine the second engine and limp mode on the problem child may suffice.
100LL the lead is a problem. Maybe a toolkit so you can remove oxygen sensor when 100LL is all that is available and a program dedicated to running on it. I mean we know the sensor is just going to puke so why insist on it being there? So we can hold its hair? It just doesn't make sense. Lead is not a problem for billions of users so fixing that problem is likely a deal killer for the project. A work around is more practical I think.
Throttle body's were the first injection systems to go on the American V-8s and V-6s. I bet it was because it was the easiest (spelled least expensive). That said two cylinder independent induction just doesn't "sound" that difficult, your going to be running an ignition timer so the basic data for timing must already be there? Is it enough better to justify two throttle bodies and two injectors or … ? I look at the Walbro unit for example, they are doing it for among other things compliance with emission standards and it looks throttle body to me.
#### blane.c
##### Well-Known Member
HBA Supporter
Do all the boards require the same inputs? If some of the boards support software that requires less input is that a good thing? What is the minimum number of input pieces that will work for this project considering working around leaded fuel? Maybe the answer to that will eliminate a few boards as options.
If a piece will give the board multiple data inputs it may reduce cost and weight and if one piece of data is going to send you into limp mode anyway what's the downside?
Maybe figuring out the minimum amount of input that will work will help?
#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
o Is it enough better to justify two throttle bodies and two injectors or … ?
I think so. Way back when EFI was still analog VW thought port injection was enough of an advantage that they added a second set of points to pick which injector to fire. Throttle body would have been much easier.
As for the O2 work around maybe a reasonable compromise would be to develop the basic map with a wide band sensor and then run a cheap unheated narrow band in service? Maybe design an on/off fitting for the O2 bung that lets the pilot twist something 90 degrees to shield the O2 sensor from the exhaust gas. At the same time a switch lets the CPU know the O2 sensor is on holiday. This probably wouldn't work with a wide band because of the heater.
Do all the boards require the same inputs? If some of the boards support software that requires less input is that a good thing?
Attached is a quick Excel file of the common EFI sensors and how ;they attach to the CPU.
View attachment Sensor types.xls
Need to learn how to use the tables here. :emb:
Last edited:
#### Vigilant1
##### Well-Known Member
Okay, we're all done here. And it does ignition, too.
One day, a new record!!
Seriously, pros and cons of going with this ( or similar) embedded controllers vs Arduino, etc.
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2021-02-25 22:45:15
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https://causal-fermion-system.com/theory/math/el-equations/
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The Theory of Causal Fermion Systems
The Euler-Lagrange Equations
A minimizing measure $\rho$ of the causal action principle satisfies Euler-Lagrange equations. These equations describe the dynamics of the causal fermion system. Before stating them, one needs to treat the constraints. The trace constraint implies that the trace is constant in spacetime, i.e. (for details see [cfs16, Proposition 1.4.1]
$\tr(x) = c$ for all $x \in M$
for suitable $c>0$. The boundedness constraint, on the hand, can be treated with a Lagrange multiplier term (for details see [lagrange12])
$\L_\kappa(x,y) := \L(x,y) + \kappa\, |xy|^2$
Next, we introduce the function $\ell_\kappa$ by
$\ell_\kappa(x) := \int_M \L_\kappa(x,y)\: d\rho(y) \,-\, \mathfrak{s}$
with a positive parameter $\mathfrak{s}$. The Euler-Lagrange equations state that for a suitable choice of $\mathfrak{s}$, the function $\ell_\kappa$ is minimal and vanishes on the spacetime points,
$\ell_\kappa|_M \equiv \inf_{x \in \F_c} \ell_\kappa(x) = 0 \:,$
where $\F_c$ are all operators $x \in \F$ with $\tr(x)=c$. We remark that the parameter $\mathfrak{s}$ can be understood as the Lagrange parameter corresponding to the volume constraint.
In the different generalizations and special cases, one also has corresponding Euler-Lagrange equations. In particular, for causal variational principles in the non-compact setting, the equations become simpler because of the absence of the trace and boundedness constraints. Setting
$\ell(x) := \int_M \L(x,y)\: d\rho(y) \,-\, \mathfrak{s} \:,$
the Euler-Lagrange equations read (for proofs in this simpler setting of causal variational principles see [support10, Lemma 3.4] or [jet16, Section 3])
$\ell|_M \equiv \inf_{x \in \F} \ell(x) = 0 \:.$
We note that these Euler-Lagrange equations also apply to causal fermion systems if one replaces $\F_c$ by $\F$ and $\L_\kappa$ by $\L$. With this in mind, in what follows we omit the indices $\kappa$ and $c$.
The Euler-Lagrange equations can also be expressed in terms of the kernel of the fermionic projector. Then its vector component reads (for details see [cfs16, Section 1.4.1]
$\int_M \text{Tr}\big( Q(x,y)\: \delta P(y,x) \big)\: d\rho(y) = 0 \:,$
where $Q(x,y)$ is the variational derivative of the Lagrangian and $\delta P(y,x)$ the variation of the kernel of the fermionic projector. This formulation is most suitable for the analysis in the continuum limit.
The Weak Euler-Lagrange Equations and Jets
The above Euler-Lagrange equations are nonlocal in the sense that they make a statement on $\ell$ even for points $x \in \F$ which are far away from spacetime $M$. It turns out that for most applications, it suffices to evaluate the EL equations locally in a neighborhood of $M$. For technical simplicity, we here only explain the method under the simplified assumption that $\F$ has a smooth manifold structure and that the function $\ell$ is smooth. Then the minimality of $\ell$ implies that the derivative of $\ell$ vanishes on $M$, i.e.
$\ell|_M \equiv 0 \qquad \text{and} \qquad D \ell|_M \equiv 0$
(where $D \ell(x) : T_x \F \rightarrow \R$ is the derivative). In order to combine these two equations in a compact form, it is convenient to consider a pair ${\mathfrak{u}} := (a, u)$ consisting of a real-valued function $a$ on $M$ and a vector field $u$ on $T\F$ along $M$, and to denote the combination of multiplication and directional derivative by
$\nabla_{\mathfrak{u}} \ell(x) := a(x)\, \ell(x) + \big(D_u \ell \big)(x) \:.$
Then the two above equations can be written in the compact form
$\nabla_{\mathfrak{u}} \ell|_M = 0 \:.$
The pair ${\mathfrak{u}}=(a,u)$ is referred to as a jet.
In case that $\F$ or $\ell$ is not smooth, one must carefully distinguish the differentiable directions. This leads to different classes of jet spaces (for details see for example the summary in [fockbosonic18, Section 2.2]).
Author
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2020-07-04 14:59:23
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https://dwornawrzosowisku.pl/coast-guard-onfumjh/electricity-class-10-question-bank-ef5af1
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(b) Do you agree/justify Manish's act? constant after some time. Support your view. | Which of the following resistor arrangement A or B Define potential difference. Ncert Solutions question_answer50) (i) If both the bulbs are connected in series then the total power consumption will be 60 W. (ii) If only one bulb is connected then the total power consumption will be 30 W. (iii) If the both bulbs are connected in parallel then the total power consumption will be 120 W. (i) The resistors ${{R}_{1}}$ and ${{R}_{2}}$ have no been correctly connected in parallel, (ii) The voltmeter has not been correctly connected in the circuit. (b) terminals of a cell, some potential difference is set up between its ends. C between B and C? Will the rest of the bulbs continue to Another identical bulb is connected in the same circuit having power supply of 220 V. (i) If both the bulbs are connected in series then the total power consumption will be 60 W. (ii) If only one bulb is connected then the total power consumption will be 30 W. (iii) If the both bulbs are connected in parallel then the total power consumption will be 120 W. Which of the above statement(s) is/are correct regarding the circuit? The current flowing through a resistor connected in an (c) Calculate the strength of the total current in the Three incandescent An electrician connected a fuse wire of 15 A rating The electrician said that it would not be affected by of a metallic conductor increase with increase in temperature. question_answer47) Answer the following questions using above data got the fuse changed. (a) With the help of a labelled diagram, describe an activity to show that a current carrying conductor experiences a force when placed in a magnetic field. more in series combination? Two conducting wires of the same material and of equal (a) Electric current question_answer10) How will be the reading in the ammeter A affected if another identical bulb Q is connected in parallel to P? | What is likely to happen in this case and why? Iron (a) What are the values displayed by Vishwajeet? Draw a circuit diagram using a cell of two batteries, two resistors of each connected in series, a (ii) Which material would you advise to be used in question_answer3) An air conditioner is rated 260 V, 2.0 kW. (a) What changes were made by Vishesh? Important questions, guess papers, most expected questions and best questions from 10th Science chapter Electricity have CBSE chapter wise important questions with solution for free download in PDF format. has the lower combined resistance? electricity. In the evening Seema parked her car inside her garage and A student finds that there are 20 divisions between zero mark and 1 V | having an electric lamp and a conductor of. on the 4 electrical devices to find the defective device. Why? of an electrical component remain constant while the potential difference each resistor. An electric iron consumes energy at a rate of 840 W when a diagram to illustrate your answer. (c) Compare and contrast the values of Manish and his Solved Papers B question_answer19) left for vacation of 15 days. Read the following information [NCERT], Will current flow A Write the question_answer16) Questions Bank The applied voltage is 220 V. (i) volt Q3. (a) What are the values displayed by Vishwajeet? and same material, but A is longer than B. On returning, she tried to start the car but the car did 1. An electric geyser rated 1500 W, 250 V is connected to a Define the unit of electric current. value of , when connected in series the value The reading of the ammeter decreases to effective resistance of ? resistance 100 watt bulb or 60 watt bulb both operating at 220 V? question_answer96) Another identical bulb is connected in the same circuit having power supply of 220 V. How will be the reading in the ammeter A affected if another identical bulb Q is connected in parallel to P? expression for the heat energy produced in a wire of resistance R and carrying Privacy Policy What is the value of circuit diagram, two resistance wires A and B are of same area of cross section Copper 3.00 per unit. Choose all the apparatus for performing the experiment Give the unit of (a) Charge (b) Current 2. more easily through a thick wire or a thin wire of the same material, when Now the resistance of the conductor is reduced to one fourth to its initial value and connected across the same voltage source. 20?C are given below question_answer66) time t such that 1 kg of water at 20?C attains a temperature of 60?C. | Draw the proper representation of series combination of of 2, What is electrical circuit? a circuit across the same potential difference. Why does resistance What is the Resistance of an incandescent filament of a bulb is thickness. Why do we use copper and aluminium wires for transmission Which of the following is correct relationship between the voltmeters readings ${{V}_{1}},{{V}_{2}}$ and ${{V}_{3}}$, question_answer17) In the circuit shown here, the ammeter A reads 5 A and the voltmeter V reads 20 V. The correct value of resistance R is, question_answer18) A ring is made of a wire having a resistance $12\Omega .$ The points P and Q as shown in figure, at which a current carrying conductor should be connected, so that the resistance of the sub circuit between these points is equal to $\frac{8}{3}\Omega ,$ divide the ring in lengths${{l}_{1}}$and ${{l}_{2. Which of them is a good conductor and which of them is an insulator? Initially the switch S is kept open and the lamp P and 0 intervals? Wire His classmates advised him not to tell the teacher but his house but it lasted only for few hours and circuit broke. Why closed path is required for the flow of current? question_answer61) Notification Find Draw a circuit diagram to show experimental set up for verification of Ohm's current flows through line wires or the filament of a bulb yet only the latter Heat is generated Exemplar], On what factors will be its new resistance? The ratio of charges flowing through the wire at different intervals is, Copyright © 2007-2020 | (a) Why did Manish switch off the appliances at regular He was fearing that he might be question_answer48) Justify your answer. An electric bulb rated 220 V, 60 W is working at full efficiency. Which ammeter Ai or A^ will indicate (b) Draw a circuit diagram for the above said combination. question_answer41) Draw a circuit diagram of an electric circuit containing a | State the law that relates potential difference across a conductor with the current An electric geyser rated 1500 W, 250 V is connected to a Give reason. Calculate the value of each Ans3. In a series electrical circuit comprising of a resistor having a question_answer72) it is stretched to double of its length? ), question_answer11) Six equal resistances are connected between points P, Q and R as shown in the figure. after removing it from the circuit. On the basis of above passage, answer the following combination which remains constant?current or voltage? | Name a device that helps to measure the potential The Plan Name the instrument electric current flowing through XY. (a) Which property was reminded by Monika about question_answer113) battery of emf 6 V so as to obtain [NCERT commercial unit of electric energy? below. Following table gives the resistivity of three samples in m the rest of the bulbs continue to glow in each circuit? question_answer99) What are the across the two ends of this component decrease to half of its former value. question_answer83) the applied voltage? CBSE class 10th Science have one book. which is used for making the filament of bulbs. Give reason. | What is the least count of ammeter? Here is a compilation of Free MCQs of Class 10 Science Book Chapter 12 – Electricity.. Students can practice free MCQs as have been added by CBSE in the new Exam pattern. the three bulbs glow together? His classmates advised him not to tell the teacher but Name a device that [NCERT]. with that of car battery. [CBSE 2008C] question_answer1) Current flows through a conductor connected across a voltage source. Free PDF Download - Best collection of CBSE topper Notes, Important Questions, Sample papers and NCERT Solutions for CBSE Class 10 Physics Electricity. Calculate Define resistivity and state its SI unit. A current of 1 A is drawn by a filament of an electric (b) What is the use of ammeter? could have used the ammeter. (a) Which of the wires has maximum resistance and why? Nichrome combination which remains constant?current or voltage? his house but it lasted only for few hours and circuit broke. | State and verify Ohm's law. (V-I) graph of a metallic conductor at two different temperature T1 The heating effect in the conductor will become. (b) Which characteristics of the electrician would you Why? question_answer29) Lalit Sardana Sir (c) the value of t in seconds. Class 10 Answerkey for Electricity. For the circuit shown in the given figure, (a) A current of 1 ampere flows in a series circuit Valence electrons in metals are free to move within the conductor and constitute an electric current. Give reason. Find (a) (b). | (a). filament in 16 s. question_answer59) (b) Calculate the work done in moving a charge of 4 C from a point at 220 Direction: Observe the given circuit diagram and answer the following question. V to and point at 230 V. (a) The potential difference between two points in an electric circuit is mark of a voltmeter. (b) Why does the connecting chord of an electric heater NCERT Solutions For Class 10 Science Chapter 12 Electricity: In this article, we will provide you all the necessary information regarding NCERT solutions for class 10 science physics chapter 12 electricity.Working on CBSE class 10 physics electricity questions and answers will help candidates to score good marks in-class tests as well as in the CBSE Class 10 board exam. Students should practice with these MCQs to prepare for the objective type questions for Class 10 Science Exam 2020. Find the equivalent resistance in the following circuit. A when the bulb gets fused? length, one of copper and the other of manganin (an alloy) have the same | question_answer117) What is the resistance of air gap? (ii) ohm Draw the proper representation of series combination of (b) Which of the wires has minimum resistance and why? | in it. brightness? Is it a closed questions Students can solve NCERT Class 10 Science Electricity Multiple Choice Questions with Answers to know their preparation level. connected as shown. question_answer2) glow in each circuit? C question_answer94) (a) What values of Rahul are worthy of appreciation? (a) Resistivity of copper is lower than that of aluminium law. parallel combinations of two given resistors. An electrician connected a fuse wire of 15 A rating constant after some time. A household uses the following electric appliances refrigerator In a series electrical circuit comprising of a resistor having a question_answer85) Make a rough sketch of the circuit using the material equivalent resistance of a combination of resistors connected in series. Why? Why is resistance | the given circuit shown. An electric heater is rated at 2 kW. question_answer34) Diameter Two pieces each are joined in series and then five such combinations the rest of the bulbs continue to glow in each circuit? to determine the equivalent resistance of two resistors when connected in Define the unit of not glow while the heating element does? Notification question_answer111) difference across the resistor be the same as question_answer32) (c) cost of energy consumed if each unit costs Rs. Define the unit of electric current. | Draw a circuit diagram using a cell of two batteries, two resistors of. Why closed path is required for the flow of current? current flowing through conductor and On the basis of above passage, answer the following Justify your answer. (b) Now, let one bulb in both the circuits get fused. question_answer105) Aluminium Salim lives in a small town and facing some electrical Calculate (a) the power supplied by the battery and (b) the power dissipated in Why? Extra Question Find out the following in the electric circuit given in Figure 12.9 (a) Effective resistance of two 8 Ω resistors in the combination (b) Current flowing through 4 Ω resistor (c) Potential difference across 4 Ω resistance (d) Power dissipated in 4 Ω … Derive an expression for equivalent resistance in the So, she continuously in an electric heater but the temperature of its element becomes (b) Calculate the work done in moving a charge of 4 C from a point at 220 The air conditioner is switched on for 10 hours each day. Silver Constantan Media diagram of an electric circuit containing a cell, a key, an ammeter, a resistor [NCERT Exemplar], Draw a circuit Define current 3. of a metallic conductor increase with increase in temperature. becomes . more easily through a thick wire or a thin wire of the same material, when Find the bulb gets fused? connected in the circuit to measure the potential difference between two points? has greater resistance? resistance R, connected in parallel? Answer the following questions using the above data Copper resistors, each of. (c) Arrange and in ascending order of their values. helps to maintain a potential difference across a conductor. Calculate the electricity bill for the household for month of June if question_answer64) Which of the following statements) is/are correct regarding this? Chapter 12 of CBSE Class 10 in on Electricity. electric circuit and the potential difference applied across its ends are shown Test Series On listening to him patiently, the teacher did For the circuit shown in the given figure, bulbs of 100 W - 220 V each are connected in series in an electrical circuit. His friends would however, make fun of this habit of him. connected a wrong fuse. What happens to the other bulbs in a series circuit if one (c) six electric tubes of rating 18 W each for 6 hours Study the circuit diagram and redraw 1 V. What does it mean? diagram to study the relation between potential difference across the Free PDF download of Important Questions with solutions for CBSE Class 10 Science Chapter 12 - Electricity prepared by expert Science teachers from latest edition of CBSE(NCERT) books. List any two important properties of electric charge. will be its new resistance? (c) Resistance the circuit was successfully completed, does the resistance of a conductor depend? If the heater is plugged in for the How is a voltmeter Draw the symbol of Out of the two wires P and Q shown below, which one in parallel and a voltmeter across the parallel combination. completed successfully, so it may be nichrome as suggested all his electrical devices, reinstalled the fuse wire and one by one switched gives the value of electrical resistivity of some materials Sample Papers electric bulb become dim when an electric heater in parallel circuit is switched on? question_answer54) Mention the position when this force is … Media A hot plate of an electric iron connected to a 200 V line Draw [NCERT], Why are coils of ammeter A. Find Number of electrons in three coulombs = 3/(1.6 x 10-19) electrons = 1.875 x 10 19 electrons. [NCERT], How is a voltmeter continuously in an electric heater but the temperature of its element becomes Define resistivity and state its SI unit. electrical circuit in which an electric heater of rating 1.5 kW, 220 V is How will the brightness of glow of bulbs P and Q will Which substance is D A student finds that there are 20 divisions between zero mark and 1 V Two resistances when connected m parallel give resultant The pre foundation courses offered by us are prepared by best faculties from all over India. Sample Name the alloy of 2 in series with a combination Give reason. What would be the values of ratios r3 Study the circuit diagram and redraw cell, a key, an ammeter, a resistor of in A student of class X performed experiment on series and (a) current through each resistor. MCQs from CBSE Class 10 Science Chapter 13: Magnetic Effects of Electric Current 1. Prepare for class 10 CBSE exam using this analysis of the last 10 years’ question papers from the chapter Electricity. Get Current Electricity, Physics Chapter Notes, Questions & Answers, Video Lessons, Practice Test and more for CBSE Class 10 at TopperLearning. What is the total resistance of n resistors, each of Justify your answer. resistivity? (a) the electric current drawn by it. Privacy Policy cells obtaining maximum potential. (c) Identify the moral values we get from the passage. question_answer25) Name the instrument diagram of an electric circuit containing a cell, a key, an ammeter, a resistor | Download a PDF of free latest Sample questions with solutions for Class 10, Physics, CBSE- Electricity . question_answer58) electric current. This chapter has less theory questions and more numerical problems. higher reading for current? diagram? Vishesh made changes in the circuit and now the findings Current Affairs Calculate the electricity bill for the household for month of June if in parallel and a voltmeter across the parallel combination. electrical circuit in which an electric heater of rating 1.5 kW, 220 V is the ammeter reading when the circuit is closed. Aluminium [NCERT]. Length Copper (iv) ampere. Find the number of electrons passing through a cross-section of the In this article, we are providing you a complete CBSE Class 10th science question bank. Then the car started at once. All types of questions are solved for all topics. information: Justify voltmeter, four cells of 1.5 V each, rheostat and a plug key. electric circuit and the potential difference applied across its ends are shown Find the least count of this voltmeter. Here is the list of all the chapters of Class 10 Science Question Bank: Chapter Name Complete Set of Questions Previous Year's Questions Chemical Reactions and Equations View - Acids, Bases and Salts View - Metals And Non-Metals View 1, View 2 - Carbon and its Compounds View - Periodic Classification Of Element View 1, […] Charge on one electron = 1.6 x 10-19 coulomb. Magnetic Effects of Electric Current Which of the following is correct relationship between the voltmeters readings \[{{V}_{1}},{{V}_{2}}$ and ${{V}_{3}}$, In the circuit shown here, the ammeter A reads 5 A and the voltmeter V reads 20 V. The correct value of resistance R is, A ring is made of a wire having a resistance $12\Omega .$ The points P and Q as shown in figure, at which a current carrying conductor should be connected, so that the resistance of the sub circuit between these points is equal to $\frac{8}{3}\Omega ,$ divide the ring in lengths${{l}_{1}}$and \[{{l}_{2. (a) Why did Manish switch off the appliances at regular 1 V. What does it mean? (b) How will you connect the resistances in each case? then for parallel. What is likely to happen in this case and why? Find the [NCERT], question_answer148) Write the order of brightness. In series In an electrical circuit, two resistors of. What is the value of current shown by the ammeter? the electric circuit to study Ohm's law as shown in figure. (iii) The ammeter and the key have not been correctly connected in the circuit. higher? Will the rest of the bulbs continue to [NCERT], question_answer146) Franchise Two wires of equal Draw the symbol of power consumed when it is operated on 110V. of rating 400 W for 10 hours each day. What are the currents in three cases? in series and then five such combinations are joined in parallel. Name two devices in Give reason. question_answer129) wattage are connected in parallel to the source. on its scale. diagram to study the relation between potential difference across the that across the parallel combination of resistors (c) Electrical resistivities of some substances at question_answer55) Monika reminded her an another property of has length / and radius r, while wire Y has length 21 and radius 2r. electric bulb. When two ends of a metallic wire are connected across the (b) How much current is flowing through information: as it also had a high melting point. What happens to the question_answer12) 3.00 per unit. Write relation between heat energy produced in a conductor when a potential difference V is applied across its terminals and a current I flows through for ‘t’ Answer. do you draw from these values? by Shamit indicate? (b) Now, let one bulb in both the circuits get fused. question_answer78) question_answer8) Justify your answer. Constantan terminals X and Y of the wire and current flowing through it. Why? | In another set of three bulbs of the same [NCERT Exemplar], question_answer143) Monika reminded her an another property of (The voltage in the mains is maintained at a constant value. connected in parallel with this series combination, what change (if any) in voltmeter, four cells of 1.5 V each, rheostat and a plug key. (i) and (ii) (a)What happens to the glow of the other two bulbs when resistor? againwer outlets. How will its: (b) E.m.f. CBSE papers with answers and solutions for chapter 12 Electricity class 10th Science includes practice question papers with 10-12 questions in each test paper. electric bulb become dim when an electric heater in parallel circuit is of two resistors (4 each) special features of a heating wire? pure metals. be the same as that across the parallel combination of 4. [NCERT], (i) Which among resistivity? your answer in each case. She checked the fuel and engine oil but that were also full. (c)Arrange R1, R2 and R5 Electricity Electricity CBSE Important Questions with solution including HOTS issued by KV for class 10 Science and Technology. Justify your answer . question_answer38) if the current I through a resistor is increased by resistance of an ideal ammeter? What do the following symbols mean in circuit diagram? Draw a circuit ratio of the total resistance of series combination and the resistance of wire CBSE Class 10 Biology HOTs-Electricity. Why does dimness decrease after sometime? wires labelled as A, B, C, D, E, F have been designed as per the following comparatively much more than that when it is at room temperature. friends. You have two metallic wires of resistances and question_answer71) Copyright © 2007-2020 | cells obtaining maximum potential. 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And Colourful World CBSE Important questions long answer type question 1 question_answer49 ) 320 of. Resistivity is pulled to double of its element becomes constant after some time hours day! Question_Answer47 ) Derive an expression for the objective type questions given electricity class 10 question bank Exercise-8A of current the... Relation between potential difference across a conductor V change if it is made thicker 10th. In a small town and facing some electrical problem in his house but it lasted only for hours. Electrons = 1.875 X 10 19 electrons question_answer97 ) and are three bulbs... Lasted only for few hours and circuit broke energy consumed if each unit costs Rs through. Obtaining maximum potential in his house, it fell from his hand and broke ), question_answer27 ) one! A resistor is kept open and the area of cross-section is halved d resistivity! Electrical heating devices question_answer9 ) an electric geyser rated 1500 W, 250 V line mains Arrange R1 R2. 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electricity class 10 question bank 2021
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2021-08-01 20:50:13
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http://websrv.cs.umt.edu/isis/index.php/Stress_Field_Equations_for_Pattyn_2003_Model
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# Stress Field Equations for Pattyn 2003 Model
## Conservation Equations
To derive this model, we begin by stating the laws of conservation of mass, momentum, and energy as they apply to an incompressible fluid (Versteed). Consider an infinitely small "control volume" of ice. The law of conservation of mass in this context states that the rate of increase of the mass of the control volume must equal the net rate of flow into the control volume (Versteed). With an incompressible fluid such as ice, the mass of a control volume cannot change, so this law reduces to the fact that any flow into the control volume must be balanced by flow out of the control volume (Versteed). Stated formally:
$\nabla \cdot \mathbf{v} = 0$
Where $\mathbf{v}$ is velocity.
The conservation of momentum is a statement of Newton's second law as applied to fluid dynamics: the rate of change of momentum equals the sum the forces (Versteed). If we consider a control volume of ice, the rate of change of momentum (or the density times the acceleration) equals the sum of the forces due to stress on the control volume and the force of gravity (Pattyn). Formally,
$\rho_i \frac{d\mathbf{v}}{dt} = \nabla \cdot \mathbf{T} + \rho_i \mathbf{g}$
Where $\mathbf{T}$ is the total stress tensor, and $\rho_i$ and $g$ are the density of ice and gravitational acceleration respectively.
The conservation of energy is a statement of the first law of thermodynamics: the rate of change of energy is equal to the rate of heat addition plus the rate of work done (Versteed). Formally,
$\rho_i \frac{d(c_p \theta)}{dt} = \nabla(k_i \nabla \theta) + \Phi$
We now begin to develop the diagnostic model by applying our simplifying assumptions. We neglect the conservation of energy entirely. Although Pattyn's original model included temperature as a concern, in CISM temperature advection is handled by a different module and therefore does not concern this derivation (Glimmer doc). We expand the conservation of mass, using the common notation of referring to velocity components as u, v, and w (Pattyn):
$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$
For the momentum equation, we neglect the acceleration term due the Froude number of ice being of the order 10-12 (the acceleration term is 1012 times smaller than the gravitational terms)
$0 = \nabla \cdot \sigma + \rho_i \mathbf{g}$
$-\rho_i \mathbf{g} = \nabla \cdot \sigma$
We define the coordinate system so that the top of the ice sheet is at z=0, with positive numbers denoting lower elevations (this introduces a rescaling term that will be discussed). This restricts gravitational influence to the z dimension, and allows us to expand the momentum equation (Pattyn):
$\frac{\partial T_{xx}}{\partial x} + \frac{\partial T_{xy}}{\partial y} + \frac{\partial T_{xz}}{\partial z} = 0$
$\frac{\partial T_{yx}}{\partial x} + \frac{\partial T_{yy}}{\partial y} + \frac{\partial T_{yz}}{\partial z} = 0$
$\frac{\partial T_{zx}}{\partial x} + \frac{\partial T_{zy}}{\partial y} + \frac{\partial T_{zz}}{\partial z} = \rho_i g$
Because vertical resistive stresses are neglected, we drop the $T_{zx}$ and $T_{zy}$ derivatives (Pattyn):
$\frac{\partial T_{zz}}{\partial z} = \rho_i g$
## Deviatoric Stresses
Some additional transformations are needed before directly relating the stresses and velocities. First, it has been established that the strain rates in ice do not largely depend on cryostatic pressure (Hooke). We can estimate the cryostatic pressure as the mean of the normal stresses:
$\frac{1}{3}(T_{xx} + T_{yy} + T_{zz})$
Thus, we define the deviatoric stress tensor as a stress tensor that neglects cryostatic pressure:
\begin{align} T'_{ij} & = T_{ij} - \frac{1}{3}\delta_{ij}(T_{xx} + T_{yy} + T_{zz}) \end{align}
Here, the Kroneker Delta, $\delta_{ij}$, is one if $i=j$, and zero otherwise.
In order to substitute deviatoric stresses for total stresses, we form the following system of equations:
$T_{xx} = T'_{xx} + \frac{1}{3}(T_{xx} + T_{yy} + T_{zz})$
$T_{yy} = T'_{yy} + \frac{1}{3}(T_{xx} + T_{yy} + T_{zz})$
Solving for $T_{xx}$ and $T_{yy}$:
$T_{xx} = 2T'_{xx} + T'_{yy} + T_{zz}$
$T_{yy} = 2T'_{yy} + T'_{xx} + T_{zz}$
We can remove the $\sigma_{zz}$ term by vertically integrating $\frac{\partial \sigma_{zz}}{\partial z} = \rho_i g$ from the surface $s$ to a height $z$ in the ice sheet (Pattyn):
$T_{zz}(z) = -\rho_i g (s - z)$
Thus:
$T_{xx} = 2T'_{xx} + T'_{yy} - \rho_i g (s - z)$
$T_{yy} = 2T'_{yy} + T'_{xx} - \rho_i g (s - z)$
## Stress, Strain, and Viscosity
We relate the deviatoric stress tensor to velocity gradients by equating each with the strain rates. Stress and strain rate are related nonlinearly by a Glen-type flow law (Paterson 1994):
$T_{ij} = 2 \mu \dot \epsilon_{ij}$
with the effective viscosity $\mu$ given by:
$\mu = \frac{A^{\frac{-1}{n}}}{2} \dot \epsilon^\frac{1-n}{n}$
where n is the flow law exponent, and defines the strength of the nonlinearity between stress and strain, usually taken to be 3. A is a thermomechanical coupling parameter, usually given by an Arrhenius relationship (Pattyn). It is not elaborated in this model because its computation is the responsibility of other CISM modules (Glimmer doc); for many basic experiments it is taken as a constant 10-16 (ISMIP-HOM). $\dot \epsilon$ is the second invariant of the strain rate tensor (Pattyn 2003) (Hooke):
$\dot \epsilon = \sqrt{ \dot \epsilon_{xy}^2 + \dot \epsilon_{yz}^2 + \dot \epsilon_{zx}^2 - \dot \epsilon_{xx}\dot \epsilon_{yy} - \dot \epsilon_{yy}\dot \epsilon_{zz} - \dot \epsilon_{zz}\dot \epsilon_{xx}}$
In practice, a small regularization is added to $\dot \epsilon$ in order to avoid a division by zero, particularly in the case of a frozen bed (Pattyn). This formulation of the second invariant will be returned to later.
We now formulate the viscosity term $\mu$ in terms of velocities rather than strain rates. In a full Stokes model, the relationship between strain rates and velocity are defined as follows (Hooke):
$\begin{pmatrix} \dot \epsilon_{xx} & \dot \epsilon_{xy} & \dot \epsilon_{xz} \\ \dot \epsilon_{yx} & \dot \epsilon_{yy} & \dot \epsilon_{yz} \\ \dot \epsilon_{zx} & \dot \epsilon_{zy} & \dot \epsilon_{zz} \\ \end{pmatrix} = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{1}{2}(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) & \frac{1}{2}(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}) \\ \frac{1}{2}(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}) & \frac{\partial v}{\partial y} & \frac{1}{2}(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}) \\ \frac{1}{2}(\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}) & \frac{1}{2}(\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}) & \frac{\partial w}{\partial z} \\ \end{pmatrix}$
Because of the assumption made above that we can neglect the horizontal gradients of the vertical velocity as they are much smaller than the vertical gradients of the horizontal velocity (Pattyn), this becomes:
$\begin{pmatrix} \dot \epsilon_{xx} & \dot \epsilon_{xy} & \dot \epsilon_{xz} \\ \dot \epsilon_{yx} & \dot \epsilon_{yy} & \dot \epsilon_{yz} \\ \dot \epsilon_{zx} & \dot \epsilon_{zy} & \dot \epsilon_{zz} \\ \end{pmatrix} = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{1}{2}(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) & \frac{1}{2}\frac{\partial u}{\partial z} \\ \frac{1}{2}(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}) & \frac{\partial v}{\partial y} & \frac{1}{2}\frac{\partial v}{\partial z} \\ \frac{1}{2}\frac{\partial u}{\partial z} & \frac{1}{2}\frac{\partial v}{\partial z} & \frac{\partial w}{\partial z} \\ \end{pmatrix}$
From a modeling standpoint, this simplifying assumption has helped us because it allows us to neglect the vertical component of velocity when solving for the velocity fields, then reconstruct it in another module using the incompressibility condition. In order to remove our dependence on the vertical velocity entirely, we use the law of conservation of mass to remove all $\dot \epsilon_{zz}$ factors from the second invarant of the strain rate tensor (Pattyn). Rearranging terms from the conservation of mass equation:
$\frac{\partial w}{\partial z} = -(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y})$
From the strain rate tensor definition above, this is equivalent to:
$\dot \epsilon_{zz} = -(\dot \epsilon_{xx} + \dot \epsilon_{yy})$
Performing this substitution in the second invariant of the strain rate tensor given above and rearranging terms leads to an alternate form that has no dependence on the vertical velocity:
$\dot \epsilon = \sqrt{ \dot \epsilon_{xy}^2 + \dot \epsilon_{yz}^2 + \dot \epsilon_{xz}^2 + \dot \epsilon_{xx}\dot \epsilon_{yy} + \dot \epsilon_{yy}^2 + \dot \epsilon_{xx}^2}$
We can now finally expand the strain rate invariant factor in the viscosity. and perform substitutions such that viscosity is formulated in terms of velocity gradients rather than strain rates (Pattyn):
$\mu = \frac{A^{\frac{-1}{n}}}{2} \dot \epsilon^\frac{1-n}{n}$
$\mu = \frac{A^{\frac{-1}{n}}}{2} (\dot \epsilon_{xy}^2 + \dot \epsilon_{yz}^2 + \dot \epsilon_{xz}^2 + \dot \epsilon_{xx}\dot \epsilon_{yy} + \dot \epsilon_{yy}^2 + \dot \epsilon_{xx}^2)^\frac{1-n}{2n}$
$\mu = \frac{A^{\frac{-1}{n}}}{2} \left( (\frac{1}{4} (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})^2 + \frac{1}{4} (\frac{\partial u}{\partial z})^2 + \frac{1}{4} (\frac{\partial v}{\partial z})^2 + (\frac{\partial u}{\partial x})^2 + (\frac{\partial v}{\partial y})^2 + \frac{\partial u}{\partial z}\frac{\partial v}{\partial y} \right)$
## Stress and Velocity
With the relationship of both deviatoric stresses and velocities to strain rates, we can now reformulate the conservation of momentum purely in terms of velocity. This will lead us to a set of coupled elliptic partial differential equations that can be numerically approximated to find the u and v components of the velocity field. To derive the u component, let us return to the first equation in the system resulting from the conservation of momentum:
$\frac{\partial T_{xx}}{\partial x} + \frac{\partial T_{xy}}{\partial y} + \frac{\partial T_{xz}}{\partial z} = 0$
We want to express this equation in terms of the deviatoric stress rather than the total stress. Using the substitutions derived above (Pattyn):
$\frac{\partial}{\partial x}(2T'_{xx} + T'_{yy}) + \frac{\partial T'_{xy}}{\partial y} + \frac{\partial T'_{xz}}{\partial z} = \rho_i g \frac{\partial s}{\partial x}$
$\frac{\partial}{\partial x}(2T'_{xx} + T'_{yy}) + \frac{\partial T'_{xy}}{\partial y} + \frac{\partial T'_{yz}}{\partial z} = \rho_i g \frac{\partial s}{\partial x}$
Using Glen's flow law and the definition of strain rates given above, the stress tensor can be written as follows:
$\begin{pmatrix} T'_{xx} & T'_{xy} & T'_{xz} \\ T'_{yx} & T'_{yy} & T'_{yz} \\ T'_{zx} & T'_{zy} & T'_{zz} \\ \end{pmatrix} = \begin{pmatrix} 2 \mu \dot \epsilon_{xx} & 2 \mu \dot \epsilon_{xy} & 2 \mu \dot \epsilon_{xz} \\ 2 \mu \dot \epsilon_{yx} & 2 \mu \dot \epsilon_{yy} & 2 \mu \dot \epsilon_{yz} \\ 2 \mu \dot \epsilon_{zx} & 2 \mu \dot \epsilon_{zy} & 2 \mu \dot \epsilon_{zz} \\ \end{pmatrix} = \begin{pmatrix} 2 \mu \frac{\partial u}{\partial x} & \mu (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}) & \mu \frac{\partial u}{\partial z} \\ \mu (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}) & 2 \mu \frac{\partial v}{\partial y} & \mu \frac{\partial v}{\partial z} \\ \mu \frac{\partial u}{\partial z} & \mu \frac{\partial v}{\partial z} & 2 \mu \frac{\partial w}{\partial z} \\ \end{pmatrix}$
Directly substituting into the force balance equations gives us (Pattyn):
$\frac{\partial}{\partial x}(4 \mu \frac{\partial u}{\partial x} + 2 \mu \frac{\partial v}{\partial y}) + \frac{\partial}{\partial y}(\mu \frac{\partial u}{\partial y} + \mu \frac{\partial v}{\partial x}) + \frac{\partial}{\partial z}(\mu \frac{\partial u}{\partial z}) = \rho_i g \frac{\partial s}{\partial x}$
$\frac{\partial}{\partial y}(4 \mu \frac{\partial v}{\partial y} + 2 \mu \frac{\partial u}{\partial x}) + \frac{\partial}{\partial x}(\mu \frac{\partial u}{\partial y} + \mu \frac{\partial v}{\partial x}) + \frac{\partial}{\partial z}(\mu \frac{\partial v}{\partial z}) = \rho_i g \frac{\partial s}{\partial x}$
Finally, we expand the nested derivatives and rearrange terms so that each equation has the terms related to the variable we are solving for (u and v respectively) on the left-hand side, and terms related to the orthogonal velocity component (v and u respectively) on the right-hand side:
$4 \frac{\partial \mu}{\partial x} \frac{\partial u}{\partial x} + \frac{\partial \mu}{\partial y} \frac{\partial u}{\partial y} + \frac{\partial \mu}{\partial z} \frac{\partial u}{\partial z} + \mu(4 \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}) = \rho_i g \frac{\partial s }{\partial x} - 2 \frac{\partial \mu}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial \mu}{\partial y} \frac{\partial v}{\partial x} - 3 \mu \frac{\partial^2 v}{\partial x \partial y}$
$4 \frac{\partial \mu}{\partial y} \frac{\partial v}{\partial y} + \frac{\partial \mu}{\partial x} \frac{\partial v}{\partial x} + \frac{\partial \mu}{\partial z} \frac{\partial v}{\partial z} + \mu(4 \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2}) = \rho_i g \frac{\partial s }{\partial x} - 2 \frac{\partial \mu}{\partial y} \frac{\partial u}{\partial x} - \frac{\partial \mu}{\partial x} \frac{\partial u}{\partial y} - 3 \mu \frac{\partial^2 u}{\partial x \partial y}$
## Rescaled Vertical Coordinate
Pattyn introduces a dimensionless vertical coordinate so that variations in ice thickness do not complicate the numerical treatment considerably (Pattyn 2003). This dimensionless vertical coordinate is referred to as $\zeta$ by Pattyn. I will refer to it as $\sigma$ for consistency with CISM's existing notation. The rescaled coordinate is defined as:
$\sigma = \frac{(s - z)}{H}$
This means that at the surface of the ice sheet $\sigma = 0$, and at the base $\sigma = 1$ regardless of the ice thickness (Pattyn 2003). As a result of this transformation, a coordinate $(x,y,z)$ is mapped to $(x',y',\sigma)$ (Pattyn 2003). This means that function derivatives must be re-written (using $\frac{\partial f}{\partial x}$ as an example) as:
$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} \frac{\partial x'}{\partial x} + \frac{\partial f}{\partial y'} \frac{\partial y'}{\partial x} + \frac{\partial f}{\partial \sigma} \frac{\partial \sigma}{\partial x}$
Similarly for $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$. Pattyn simplifies this by assuming that
$\frac{\partial x'}{\partial x}, \frac{\partial y'}{\partial y} = 1$
and
$\frac{\partial x'}{\partial y}, \frac{\partial x'}{\partial z}, \frac{\partial y'}{\partial x}, \frac{\partial y'}{\partial z} = 0$.
This assumption is valid if the bed and surface gradients are not too large (Pattyn 2003). This simplifies the above to:
$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} + \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial x}$
$\frac{\partial f}{\partial y} = \frac{\partial f}{\partial y'} + \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial y}$
$\frac{\partial f}{\partial z} = \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial z}$
Rescaling parameters $a_x$, $a_y$, $b_x$, $b_y$, and $c_{xy}$ are defined. Presenting the x derivative case, as the y derivative case is analogous,
$a_x = \frac{1}{H}(\frac{\partial s}{\partial x'} - \sigma \frac{\partial H}{\partial x'})$
$b_x = \frac{\partial a_x}{\partial x'} + a_x \frac{\partial a_x}{\partial \sigma} = \frac{1}{H} (\frac{\partial^2 s}{\partial x'^2} - \sigma \frac{\partial^2 H}{\partial x'^2} - 2a_x \frac{\partial H}{\partial x'})$
$c_{xy} = \frac{\partial a_y}{\partial x'} + a_x \frac{\partial a_y}{\partial \sigma} = \frac{\partial a_x}{\partial y'} + a_y \frac{\partial a_x}{\partial \sigma}$
Using these, expressions for the x derivatives become:
$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} + a_x \frac{\partial f}{\partial \sigma}$
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2020-05-25 01:14:08
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https://pyrobot-docs.readthedocs.io/en/stable/locobot/base.html
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# Base¶
class locobot.base.LoCoBotBase(configs, map_img_dir=None, base_controller='ilqr', base_planner=None, base=None)[source]
This is a common base class for the locobot and locobot-lite base.
__init__(configs, map_img_dir=None, base_controller='ilqr', base_planner=None, base=None)[source]
The constructor for LoCoBotBase class.
Parameters: configs (YACS CfgNode) – configurations read from config file map_img_dir (string) – parent directory of the saved RGB images and depth images
get_state(state_type)[source]
Returns the requested base pose in the (x,y, yaw) format as computed either from Wheel encoder readings or Visual-SLAM
Parameters: state_type (string) – Requested state type. Ex: Odom, SLAM, etc pose of the form [x, y, yaw] list
go_to_absolute(xyt_position, use_map=False, close_loop=True, smooth=False)[source]
Moves the robot to the robot to given goal state in the world frame.
Parameters: xyt_position (list or np.ndarray) – The goal state of the form (x,y,t) in the world (map) frame. use_map (bool) – When set to “True”, ensures that controler is using only free space on the map to move the robot. close_loop (bool) – When set to “True”, ensures that controler is operating in open loop by taking account of odometry. smooth (bool) – When set to “True”, ensures that the motion leading to the goal is a smooth one.
go_to_relative(xyt_position, use_map=False, close_loop=True, smooth=False)[source]
Moves the robot to the robot to given goal state relative to its initial pose.
Parameters: xyt_position (list or np.ndarray) – The relative goal state of the form (x,y,t) use_map (bool) – When set to “True”, ensures that controler is using only free space on the map to move the robot. close_loop (bool) – When set to “True”, ensures that controler is operating in open loop by taking account of odometry. smooth (bool) – When set to “True”, ensures that the motion leading to the goal is a smooth one.
track_trajectory(states, controls=None, close_loop=True)[source]
State trajectory that the robot should track.
Parameters: states (list) – sequence of (x,y,t) states that the robot should track. controls (list) – optionally specify control sequence as well. close_loop (bool) – whether to close loop on the computed control sequence or not.
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2019-10-22 11:47:44
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https://planetcalc.com/8576/?thanks=1
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Jarvis march
This online calculator computes the convex hull of a given set of points using Jarvis march algorithm, aka Gift wrapping algorithm
Timur
Created: 2020-02-06 14:12:52, Last updated: 2021-02-25 08:44:50
This online calculator implements the Jarvis march algorithm, introduced by R. A. Jarvis in 1973 (also known as gift wrapping algorithm) to compute the convex hull of a given set of 2d points. It has the complexity of $O ( n h )$, where n is the number of points, and h is the number of hull vertices, so it is output-sensitive algorithm. There are other algorithms with complexity of $O ( n \log n )$ and $O ( n \log h )$, but Jarvis march is very simple to implement.
The algorithm begins with a point known to be on the convex hull, f.e., the leftmost point, and selects the next point by comparing polar angles of all points with respect to the previous point taken for the center of polar coordinates. The point with minimum angle wins. The process continues until it returns to the starting point in h steps. The process is similar to winding a string (or wrapping paper) around the set of points, hence the name gift wrapping algorithm.
Enter the set of points into the calculator below - one point per line, with x and y coordinates separated by a semicolon. The calculator builds a convex hull, displays it as a set of points, and plots it.
Jarvis march
Convex hull
URL copied to clipboard
PLANETCALC, Jarvis march
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2022-08-19 17:07:40
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https://epsproc.readthedocs.io/en/v1.2.6/methods/geometric_method_dev_pt3_AFBLM_090620_010920_dev_bk100920.html
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# Method development for geometric functions pt 3: $$\beta$$ aligned-frame (AF) parameters with geometric functions.¶
• 01/09/20 v2 with verified AF code.
• 09/06/20 v1
Aims:
• Develop $$\beta_{L,M}$$ formalism for AF, using geometric tensor formalism as already applied to MF case.
• Develop corresponding numerical methods - see pt 1 notebook.
• Analyse geometric terms - see pt 1 notebook.
## $$\beta_{L,M}^{AF}$$ rewrite¶
The various terms defined in pt 1 can be used to redefine the full AF observables, expressed as a set of $$\beta_{L,M}$$ coefficients (with the addition of another tensor to define the alignment terms).
The original (full) form for the AF equations, as implemented in ePSproc.afblm <https://epsproc.readthedocs.io/en/dev/modules/epsproc.AFBLM.html>__ (note, however, that the previous implementation is not fully tested, since it was s…l…o…w… the geometric version should avoid this issue):
\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & = & \sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{M}(-1)^{m}(-1)^{(\mu'-\mu_{0})}\left(\frac{(2l+1)(2l'+1)(2L+1)}{4\pi}\right)^{1/2}\left(\begin{array}{ccc} l & l' & L\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc} l & l' & L\\ -m & m' & S-R' \end{array}\right)\nonumber \\ & \times & I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E)\\ & \times & \sum_{P,R,R'}(2P+1)(-1)^{(R'-R)}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu_{0} & -\mu_{0} & R \end{array}\right)\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\\ & \times & \sum_{K,Q,S}(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right)A_{Q,S}^{K}(t) \end{eqnarray}
Where $$I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)$$ are the energy-dependent dipole matrix elements, and $$A_{Q,S}^{K}(t)$$ define the alignment parameters.
In terms of the geometric parameters, this can be rewritten as:
\begin{eqnarray} \beta_{L,-M}^{\mu_{i},\mu_{f}} & =(-1)^{M} & \sum_{P,R',R}{[P]^{\frac{1}{2}}}{E_{P-R}(\hat{e};\mu_{0})}\sum_{l,m,\mu}\sum_{l',m',\mu'}(-1)^{(\mu'-\mu_{0})}{\Lambda_{R'}(\mu,P,R')B_{L,S-R'}(l,l',m,m')}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)I_{l',m',\mu'}^{p_{i}\mu_{i},p_{f}\mu_{f}*}(E)\sum_{K,Q,S}\Delta_{L,M}(K,Q,S)A_{Q,S}^{K}(t)\label{eq:BLM-tidy-prod-2} \end{eqnarray}
Where there’s a new alignment tensor:
$$\Delta_{L,M}(K,Q,S)=(2K+1)^{1/2}(-1)^{K+Q}\left(\begin{array}{ccc} P & K & L\\ R & -Q & -M \end{array}\right)\left(\begin{array}{ccc} P & K & L\\ R' & -S & S-R' \end{array}\right)$$
And the the $$\Lambda_{R',R}$$ term is a simplified form of the previously derived MF term:
$$\Lambda_{R'}=(-1)^{(R')}\left(\begin{array}{ccc} 1 & 1 & P\\ \mu & -\mu' & R' \end{array}\right)\equiv\Lambda_{R',R'}(R_{\hat{n}}=0)$$
All phase conventions should be as the MF case, and the numerics for all the ternsors can be used as is… hopefully…
Further notes:
• Note $$B_{L,S-R'}(l,l',m,m')}$$ instead of $$B_{L,-M}(l,l',m,m')}$$ for MF case. This allows for all MF projections to contribute.
Refs for the full AF-PAD formalism above:
1. Reid, Katharine L., and Jonathan G. Underwood. “Extracting Molecular Axis Alignment from Photoelectron Angular Distributions.” The Journal of Chemical Physics 112, no. 8 (2000): 3643. https://doi.org/10.1063/1.480517.
2. Underwood, Jonathan G., and Katharine L. Reid. “Time-Resolved Photoelectron Angular Distributions as a Probe of Intramolecular Dynamics: Connecting the Molecular Frame and the Laboratory Frame.” The Journal of Chemical Physics 113, no. 3 (2000): 1067. https://doi.org/10.1063/1.481918.
3. Stolow, Albert, and Jonathan G. Underwood. “Time-Resolved Photoelectron Spectroscopy of Non-Adiabatic Dynamics in Polyatomic Molecules.” In Advances in Chemical Physics, edited by Stuart A. Rice, 139:497–584. Advances in Chemical Physics. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2008. https://doi.org/10.1002/9780470259498.ch6.
Where [3] has the version as per the full form above (full asymmetric top alignment distribution expansion).
### To consider¶
• Normalisation for ADMs? Will matter in cases where abs cross-sections are valid (but not for PADs generally).
### Status¶
TODO: more careful comparison with experimental data (see old processing notebooks…)
## Setup¶
[1]:
# Imports
import numpy as np
import pandas as pd
import xarray as xr
# Special functions
# from scipy.special import sph_harm
import spherical_functions as sf
import quaternion
# Performance & benchmarking libraries
# from joblib import Memory
# import xyzpy as xyz
import numba as nb
# Timings with ttictoc or time
# https://github.com/hector-sab/ttictoc
from ttictoc import TicToc
import time
# Package fns.
# For module testing, include path to module here
import sys
import os
modPath = r'D:\code\github\ePSproc' # Win test machine
# modPath = r'/home/femtolab/github/ePSproc/' # Linux test machine
sys.path.append(modPath)
import epsproc as ep
# TODO: tidy this up!
from epsproc.util import matEleSelector
from epsproc.geomFunc import geomCalc
* pyevtk not found, VTK export not available.
[2]:
from IPython.core.interactiveshell import InteractiveShell
InteractiveShell.ast_node_interactivity = "all"
[3]:
# Plotters
import matplotlib.pyplot as plt
from epsproc.plot import hvPlotters
hvPlotters.setPlotters()
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2021-04-18 00:07:22
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https://zbmath.org/?q=an%3A0999.65150
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## Application of global optimization and radial basis functions to numerical solutions of weakly singular Volterra integral equations.(English)Zbl 0999.65150
Summary: A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the $$C^\infty$$ multiquadric (MQ) radial basis function expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than $$5\cdot 10^{-7}$$.
### MSC:
65R20 Numerical methods for integral equations 45E05 Integral equations with kernels of Cauchy type
Full Text:
### References:
[1] Hardy, R.L, Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 76, 1905-1915, (1971) [2] Hardy, R.L, Theory and applications of the multiquadric-biharmonic method: 20 years of discovery, Computers math. applic., 19, 8/9, 163-208, (1990) · Zbl 0692.65003 [3] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite functions, Approx. theory applic., 4, 77-89, (1988) · Zbl 0703.41008 [4] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite functions, II, Math. comput., 54, 211-230, (1990) · Zbl 0859.41004 [5] Buhmann, M.D; Micchelli, C.A, Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Const. approx., 6, 12, 21-34, (1990) · Zbl 0682.41007 [6] Buhmann, M.D; Micchelli, C.A, Multiquadric interpolation improved, Computers math. applic., 24, 12, 21-26, (1992) [7] Chui, C.K; Stoeckler, J; Ward, J.D, Analytic wavelets generated by radial functions, Adv. comput. math., 5, 95-123, (1996) · Zbl 0855.65145 [8] Micchelli, C.A, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Const. approx., 2, 11-22, (1986) · Zbl 0625.41005 [9] Kansa, E.J, Multiquadrics: A scattered data approximation scheme with applications to computational fluid dynamics: II. parabolic, hyperbolic, and elliptic partial differential equations, Computers math. applic., 19, 8/9, 146-161, (1990) · Zbl 0850.76048 [10] Makroglou, A, Radial basis functions in the numerical solution of Fredholm integral and integro-differential equations, (), 478-484, New Brunswick, NJ [11] Baxter, B.J.C, The asymptotic cardinal function of the multiquadric, φ(r) = (r2 + c2)$$12$$ as c → ∞, Computers math. applic., 24, 12, 1-6, (1992) · Zbl 0764.41016 [12] Madych, W.R, Miscellaneous error bounds for multiquadric and related interpolators, Computers math. applic., 24, 12, 121-138, (1992) · Zbl 0766.41003 [13] Kansa, E.J; Hon, Y.C, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Computers math. applic., 39, 7/8, 123-137, (2000) · Zbl 0955.65086 [14] Wong, S.M; Hon, Y.C; Li, T.S; Chung, S.L; Kansa, E.J, Multi-zone decomposition of time-dependent problems using the multiquadric scheme, Computers math. applic., 37, 8, 23-43, (1999) · Zbl 0951.76066 [15] Hon, Y.C, An efficient numerical scheme for Burgers’ equation, Appl. math. comput., 95, 37-50, (1998) · Zbl 0943.65101 [16] Wendland, H, Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree, Adv. comput. math., 4, 389-396, (1995) · Zbl 0838.41014 [17] Galperin, E.A; Zheng, Q, Solution and control of PDE via global optimization methods, Computers math. applic., 25, 10/11, 103-111, (1993) · Zbl 0794.35009 [18] Galperin, E.A; Kansa, E.J, On the solution of infinitely ill-conditioned weakly singular problems, Mathl. comput. modelling, 31, 13, 53-63, (2000) · Zbl 0955.65097 [19] Galperin, E.A, The cubic algorithm for optimization and control, (1990), NP Research Publ Montreal · Zbl 0781.90080 [20] Galperin, E.A, The fast cubic algorithm, Computers math. applic., 25, 10/11, 147-160, (1993) · Zbl 0784.90073 [21] Galperin, E.A, The alpha algorithm and application of the cubic algorithm in case of unknown Lipschitz constant, Computers math. applic., 25, 10/11, 71-78, (1993) · Zbl 0803.90112 [22] Ferrari, A; Galperin, E.A, Numerical experiments with one-dimensional adaptive cubic algorithm, Computers math. applic., 25, 10/11, 47-56, (1993) · Zbl 0776.90072 [23] Belykh, V.N, Algorithms without saturation in the problem of numerical integration, Soviet math. dokl., 39, 95-98, (1989) · Zbl 0702.41044 [24] Belykh, V.N, On the problem of numerical solution of Dirichlet problem by a harmonic single layer potential algorithm (without saturation), Russian acad. sci. dokl. math., 47, 252-256, (1993) · Zbl 0814.65122
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-09-30 20:08:54
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http://mathhelpforum.com/trigonometry/152012-composite-argument-property.html
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1. ## Composite argument property
del
2. In the "little prince" there is a line: "Draw me a sheep!"
Now, instead of using composite argument property for cosine and instead drawing me something.... draw to yourself a unit circle.
3. I understand that cos(90-x)=sin, I just have no idea how to interwind it with the composite argument property
4. Well, let's try it.
$cos(90 - x) = cos(90)cos(x) + sin(90)sin(x)$
But cos(90) = 0 and sin(90) = 1.
So?
5. Deleted.
6. Originally Posted by wiseguy
"Use the composite argument property for cosine to prove that $cos (90° -) = sin$"
I'm aware the composite argument property of cosine is $cos(A − B) = cos A cos B + sin A sin B$ - how would you use that to show $cos (90° -) = sin$?
You do realise that your formatting is messed up, right?
From context, you mean
Given that $\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$
prove that $\cos(90^\circ - \theta) = \sin\theta$.
So let $A = 90^\circ$ and $B = \theta$ and write it out..
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2017-07-27 14:49:08
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https://math.meta.stackexchange.com/questions/31606/can-we-please-stop-voting-to-close-a-question-just-because-the-question-is-short
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# Can we please stop voting to close a question just because the question is short?
I have seen many times that somebody voted to close short questions (and probably downvoted the questions as well), even though the OP made clear that there had been good attempts. This thread is the most recent one I saw. While I do not know the real reason that the question has been voted to close, my bet is that somebody looked at the question and thought it was too short to show enough attempt. However, if you really read the question, you would know that the OP must have made it very far in solving the original question. Therefore, I would like to ask you to really read the question before deciding to close it. Sometimes, a short question does show enough effort.
Here is another short question with a close vote that I think is premature. While the OP of this thread should have shown more, to answer the main question why the OP did not get the same result as the textbook (or whichever resources the OP was using) does not require knowing the OP's attempt. By just looking at the information the OP gave, you could see that the OP had actually solved the problem, but did not realize it.
Here is yet another short question with a close vote. The OP properly formatted the question. The OP also supplied a full attempt, despite it being wrong. The only thing the OP should have done more was to add a "proof-verification" or "solution-verification" tag. Yet, there is a vote to close because it is missing context or other details. I am not quite sure what other context should be added to this rather simple question.
This would be my last example for this META question. I only wanted to supply more examples as Xander Henderson had requested. (This is not an attempt to spark another round of heated debate.)
• Your post is based on a false premise, i.e. that people are voting to close questions because they are short. I have voted to close that question because it lacks context. The asker claims that they have reduced the problem to proving a short statement (which, without seeing how they derived that result, makes me worry that the question is an XY problem). They have not indicated which theorems, results, or definitions are applicable (or allowable, assuming that this is a school assignment. – Xander Henderson Apr 28 '20 at 17:44
• Finally, I think that you will find that there are people who do not a priori regard an "honest attempt" as sufficient context. Details of an attempt are useful in a problem where the question is "what did I do wrong?" or "where do I go from here?" In the current context, details of such an attempt would be quite useful, for the reasons I outlined above. On the other hand, there are questions where an "honest attempt" just clouds the waters, e.g. "I want to prove this theorem. Here's what I did: <insert three paragraphs of nonsense>." – Xander Henderson Apr 28 '20 at 17:46
• @XanderHenderson That is agreeable. However, I don't think, at least, it is applicable to the link I gave. What the OP asked at the end of that question is only a small step away from a complete solution. – Batominovski Apr 28 '20 at 17:46
• @Batominovski Again, the asker did not show any of their work. It should not be the responsibility of readers to read the mind of the asker and determine how they got to whatever step they got to. It is the job of the asker to give sufficient context. In the case of the question to which you have linked, such context is not given. – Xander Henderson Apr 28 '20 at 17:49
• @Batominovski Yes, people sometimes make mistakes. But, as I think is clear from what I have written above, I don't think that it is mistaken to vote-to-close the particular question you have brought up. I think that, in this case, the votes are justifiable. Please stop making assumptions (e.g. that people are voting to close because a post is short, or that people are voting to close by mistake). – Xander Henderson Apr 28 '20 at 17:56
• @amWhy I did respond to both of you. Maybe I like geometry, and can see how much it is needed to get to the part the OP asked for. Therefore, I could see the effort to get to the specific question the OP asked. Xander disagreed, and that is ok. You disagreed, and that's ok. But it seems it's not ok that I disagree with you. – Batominovski Apr 28 '20 at 18:48
• @amWhy I do not agree with the policy that PSQ shouldn't be answered to. I do not object if it is voted to close or deleted. I will not stop answering such questions. And you can do your part to enforce this policy. You can flag moderators to have a talk-to with me if this is not allowed. – Batominovski Apr 28 '20 at 18:50
• @amWhy I am not here for the reps. I just enjoy solving problems. If this post came across as insulting you personally, I apologize. This post is simply a request that there may be more to short questions than it looks. I don't know why you have to be so unkind. – Batominovski Apr 28 '20 at 18:57
• Let me repeat myself, as you have not addressed this at all, @Batominovski: It should not be the responsibility of readers to read the mind of the asker and determine how they got to whatever step they got to. The fact that you can (perhaps even correctly) determine how they got their result does not excuse the asker from explaining themselves. Otherwise, one runs into the xy problem. It is the asker's responsibility to explain what they have done, not the reader's responsibility to read minds. – Xander Henderson Apr 28 '20 at 19:29
• @XanderHenderson I do not think your concern was invalid. I accept your viewpoint. I simply think from trying to solve the problem that the OP got very far if the question about the point $P$ was the final thing to be dealt with. And your argument is great. Readers don't know for sure unless more was shown. We disagreed, and I saw your point. – Batominovski Apr 28 '20 at 19:33
• @amWhy Honestly, it's not your job to police when someone should and shouldn't get help with their homework assignment. The asker showing their work/attempt makes the question less useful, because then the answer is tailored only to to the asker's attempt. If the same question comes up, but the OP has a different problem, then their question will be closed as a duplicate when the problem the OP has is not actually the same. A full solution is preferable that addresses all possible attempts, which can later be updated if it doesn't initially address the OP's problem. – Matt Samuel Apr 28 '20 at 20:56
• @MattSamuel it might be worth recalling that historically allowing attempts as context was a concession to the more permissive users. From my side, no problem at all not to insist on and count attempts. Yet, of course we'd still enforce context. – quid Apr 28 '20 at 21:38
• @Zacky I was aiming at a more general case. I only used this one example because it was the most recent one, and I can't remember other threads that this applies to. – Batominovski Apr 28 '20 at 22:56
• @Zacky Somebody else changed my tags. I didn't realize that this happened. – Batominovski Apr 28 '20 at 22:58
• I've deleted my comment above to keep it clean as I thought the issue is over with the tags. @XanderHenderson the post doesn't focus on a single question, mostly you focused on that specific question in the comments, but it doesn't mean that the post is about it. The answers posted bellow are focused to general aspects and only one answer mentioned it, but still focused on general things from there, so the tag will invalidate the answers. And just because more examples can be useful, it doesn't change that the tag is inappropriate here. – Zacky Apr 30 '20 at 9:32
The question raises a valid point, but does so in a non-optimal way.
Can we please stop voting to close a question just because the question is short?
Yes, sure. Everyone agrees. Let's move on. Yet, wait. As discussed in comments there is an issue in any specific case. Was it really "just because"?
Actually, framing it as you did will invite others to contest that it was "just because." This will likely not lead to anything constructive.
Here is one principle: It is first and foremost the job of the person that asks the question to craft a question post that makes it easy and clear to appreciate and to understand. If this is not the case, then it can be closed.
That said, there are obviously limits to this, and some users actually might at times resort a bit too much to heuristics and possibly vote on things where they did not make a reasonable effort to understand the post. At the same time I do not think it is reasonable to impose as the limit that users first need to solve the problem to see where the intermediate step lies and if it makes sense.
Adopting the above principle has the drawback that we arguably miss some good questions because of this. Because it is a fact that there are not few people out there that are quite competent in mathematics yet have difficult to express it (in English).
The drawback can be mitigated though. Those that care about the questions can give advice how to improve the post (or within limits improve it themselves). Ideally they would do this before the question is closed, then the question might not be closed to begin with. In any case it can be reopened.
This post might be a bit meandering, so what's my point?
Maybe it is a call for pragmatism. The time spend to call out users that close "just because" the question is short or "just because" whatever, might be better spent on improving questions.
If this happens I would encourage those that close to show some appreciation towards the gesture and maybe close one of the 137 other questions of the day that still should be closed not to speak about the 125273 on the site.
The flip-side is that those that improve should do so out of genuine interest in the question, and not in an effort to undercut those that had closed or were in the process of closing.
• "chose" in the penultimate paragraph should be "close" (and not, say, "choose"), right? – LSpice May 12 '20 at 8:26
• @LSpice honestly I don't know for sure. Likely I meant to write "choose [to close]", but "close" as you propose is more direct. – quid May 13 '20 at 23:05
The question in its original form:
Find the second solution. First solution is 1/(1-x) .Solve by Frobenius Method
question is x(1-x)y''+2(1-2x)y'-2y=0 please give complete solution.The first soltion i am able to get is 1/(1-x) . Other solution is 1/x but i am getting -1/x(1-x).
This got one vote to close before you drew attention to it here (now at three). And, this means that we are too strict? I don't think so.
I agree that the post contains something that can be salvaged. If there are users that want to salvage something that's fine. But this type of post, as is, does not match the standards of the site and thus can and should be closed unless it gets improved. There is nothing wrong with this and failing to do so degrades the quality of the site.
Yes, in this case, given the edits and your answer it is now maybe basically alright, and it is in a way nice that you put in the effort.
But as a rule such posts can be closed. Clearly the poster can do a bit better than this. Why should we cater to this? We should not. It is ultimately a disservice for the site and arguably even the poster.
• I edited that question because it looked unclear. It's really hard to read and answer a question badly formatted. – Satyendra May 2 '20 at 20:41
• Thank you for the edit. I agree that formatting can make a lot of difference. – quid May 2 '20 at 22:03
### General concerns.
You have a reasonable concern. Thank you for posting the question with a typical example.
In the (almost) daily list of questions-to-be-closed/deleted in the CURED (formerly abbreviated as CRUDE) chat room since early 2018, the initiator once said clearly in a public conversation that
I search for very short posts.
There are indeed many “short” questions that are closed and/or deleted by this room. Not to say that such activities are right or wrong, it is unclear how the list is formed regularly and based on what kind of algorithms ("cure") or judgments if any. But such “searching by length” activities would be certainly a concern for the public.
### The particular geometry question.
For the particular geometry question you mentioned, it is indeed very short.
One of the users who voted to close left the following comment under the question:
"I have voted to close that question because it lacks context. The asker claims that they have reduced the problem to proving a short statement (which, without seeing how they derived that result, makes me worry that the question is an XY problem). They have not indicated which theorems, results, or definitions are applicable (or allowable, assuming that this is a school assignment."
Such justifications (assuming good intention) for closing is actually used very often, which causes the over-requirement of the so-called "context". It is nevertheless problematic.
Providing “which theorems, results, or definitions are applicable” can indeed facilitate an answerer to give a desirable answer, it is by no means a must. It is, of course, annoying that one gives an answer but the asker later leaves a comment like “oh, no, my teacher does not allow me to use LHopital, can you do it in another way?” But this question is not such a case.
One way to provide the expected context is to give definitions. For example, it may be very essential to provide the definition of the exponential function $$x\mapsto e^x$$ since answers to the question may completely depend on the way one defines it. In the linked question, do you seriously think that people who read this question do not know what the definition of “parallel” is? Or what “area” means in this problem?
Under that question, one user wrote “I have voted to close this question because it lacks context. You claim to have written a lot of relations, but you have not reproduced any of them here”.
MSE is a Q&A site. Not a place for taking exams and question box is not an exam paper. Who cares those messy intermediate attempts PROVIDED that the asked attempted to reduce the problem to a simpler one and clearly wrote down what (s)he thought (s)he has reduced?
“This has the potential of being an XY Problem.” in the same comment is invalid. The timeline shows that at 2020-04-28 14:23:31Z an answer was accepted. I do not see the point of using such a reason for voting to close a question after three hours of the accepted answer (2020-04-28 17:47:49Z).
• Re: "I look for short posts." If you look through the archives of CURED, you will also find that same user noting that short posts are not automatically bad, but that there is a correlation. That is, a short post is more likely to be bad. Thus if you are looking to find questions that are uncontroversially bad and in need of deletion, it is reasonable to target short posts. This user has not automated the process, and selects questions by hand from among those found. – Xander Henderson Apr 30 '20 at 2:10
I think people here vote negative and to close the question if it is too difficult for them, I put here quality questions I admit they are difficult and people voted negative after I spent a good time writing them when people vote negative they must give a good reason. What is the point of the site if when you ask a question because the people are unable to answer they vote negative and ask to close it. Does this mean that the site lacks quality and it's not really for the purpose?
• I took a look at your latest question. It is not hard. In fact, I struggle for how to even start helping with it since it ought to follow straight from applying the definition, and anyone at the level of studying representations need to be able to do that. Further, the question lacks any form of basic formatting. – Tobias Kildetoft Apr 29 '20 at 14:18
• Not sure what do you mean by basic formatting, I am not here teaching I am asking a question I need help with, this is the purpose of the site, I am not here to make the site look wonderful by making a perfect edit, waste of time putting a question on here, not trying to be rude even. – F K Apr 29 '20 at 14:38
• If you think people are participating here not to make a wonderful site but to give you help then you seriously misunderstand the purpose of this site. – JonathanZ supports MonicaC Apr 29 '20 at 14:42
• people are voting negative for no reason this is the purpose of what am writing don't try to be clever and twist what I say and put it against me. I don't like that. They not even giving help I waste 30 minutes typing a question and they vote negative because of no reason , then there is no point of wasting 30 minutes typing a question . – F K Apr 29 '20 at 14:43
• This is not only my opinion go search the site and you will find a lot of people writing similar things – F K Apr 29 '20 at 14:45
• I understand the frustration and can't talk for others, but when I find something difficult, it is actually a reason to upvote because the problem is challenging. But here we are dealing with questions that lack basically any effort on the author's side, which feels like an insult to good people on the site who spend their time answering, but the asker somehow cannot spend 5 minutes describing his thoughts. – Sil Apr 29 '20 at 15:22
• I spent 30 minutes putting some of the questions here and people just voted negative for no reason, not talking out of the blue here, am not frustrated, to be honest, but I think am not going to waste my time putting questions here and my answers where people just voted negative for no reason, before they vote negative they should explain a valid reason. good people will not vote negative for my questions and answers, it seems I have been unlucky that only bad people saw mine . – F K Apr 29 '20 at 15:28
• Right, I was talking in more general sense, not specifically to your questions (which seem to be discussed in other threads). Just note that downvotes here on meta do not necessarily mean the post is of a poor quality, here the votes mean agreement/disagreement (it has different meaning than on the main site). – Sil Apr 29 '20 at 15:36
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2021-01-19 09:47:39
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https://www.wiskundeoptiu.nl/business-economics/chapter-6-areas-and-integrals/area-and-integral/extra-explanation-approximation
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# Extra explanation: approximation area
From the theorem on integrals and areas it follows that there is a relation between the integral and the area of a region below/above the graph of a function. We discuss this relation in more detail. Consider the area $O(f,a,b)$ of the region enclosed by the graph of the function $f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, shown in the following figure.
In general it is not easy to calculate the area when $f(x)$ is non-linear. We can provide an approximation for the area by splitting the interval $[a,b]$ up into smaller subintervals. For each subinterval we determine the area of the bar that best fits below the graph of $f(x)$. When we sum the areas of these bars we obtain a reasonable lower-bound for the area of $O(f,a,b)$. In the following figure we use for instance three bars to approximate the area $O(f,a,b)$, hence, $O(f,a,b)\approx O_1+O_2+O_3$.
By increasing the number of subintervals we improve the approximation. In the following figure we use five bars to approximate the area of $O(f,a,b)$, hence $O(f,a,b)\approx O_1+O_2+O_3+O_4+O_5$.
If the number of bars becomes infinite, then the approximation becomes exact and we can replace the $\approx$-sign by a $=$-sign. Hence, $O(f,a,b)=O_1+O_2+\ldots+O_n$, with $n$ an infinitely large number.
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2022-07-02 11:24:24
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https://mail-archives.eu.apache.org/mod_mbox/ant-dev/200303.mbox/%[email protected]%3E
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# ant-dev mailing list archives
##### Site index · List index
Message view
Top
From [email protected]
Subject RE: enhanced pvcs task for PVCS version 7.5
Date Wed, 12 Mar 2003 19:37:15 GMT
Thanks! I think, I get it now.
Most of the time, I work with the support team to sort any issues, I was
not aware (that's what the support team wants) of the inside details. I
just have looked at the docs for old(just to differentiate from pcli) pvcs
commands, that goes like this for configuration files....
<pvcs doc>
Version Manager has a prescribed order in which it searches for
configuration files. Version Manager first reads the configuration
file that is embedded in the VCONFIG files (this is usually the
master configuration file).
The most direct way to specify a configuration file is to use a
command's -c option.
If you don't specify a configuration file using this option, Version
Manager checks for a default configuration file (named vcs.cfg)
in the current directory. If this file does not exist, Version
Manager checks for the VCSCFG environment variable, which
specifies the name and location of the configuration file to use.
</pvcs doc>
I happen to have the VCSCFG set in my profile, that is why I did not have
problem with old get, even when I use with promotion group.
Hope this helps...
Chandra Periyaswamy
MPI - Incentives
tie-line: 754-5328
email: [email protected]
"Anderson, Rob H
- VSCM" To: "'Ant Developers List'" <[email protected]>
<Anderson.Rob@vec cc:
torscm.com> Subject: RE: enhanced pvcs task for PVCS
version 7.5
03/12/2003 12:47
PM
"Ant Developers
List"
Apparently there is some confusion around repository and config file. I
when you create a Version Manager project database a config file is created
in the archives directory. You can see the config file name and location
through the GUI if you right click the project database and choose
"Properties". You can, however, specify a config file to use when you
create
the project database (in the advanced tab) or at any time through the
"Properties" dialog. For example: I have a local repository that I created
using the defualts ("Create a new config file" in the Advanced tab) for a
config file. The layout is as follows;
Project Database (-pr)
c:\pvcs
Config file
c:\pvcs\archives\cos0nbu1.cfg
I have a shared project database that was created with a custom config file
("Use an existing configuration file" in the Advanced tab)
Project Database (-pr)
s:\Projects\NOONAN
Config File
s:\configuration files\NOONAN.cfg
In both cases, using get works fine without the -c option unless I am doing
a get by promotion group. When doing a get by promotion group I will get an
error that says "get: Group "CM" does not exist in promotion hierarchy.",
because the promotion model is defined in the config file. So, if I specify
the config file with the -c option, this error goes away and the get by
promotion group works as expected.
So using the pvcs task is no different. Doing a normal "get" works fine,
but
a "get by promotion group" fails with the error mentioned above. If there
was a configfile attribute to the pvcs task I could use it to get by
promotion group. Currently I can only use it to do a "get the default
revision". Does this make sense?
-Rob A
-----Original Message-----
From: [email protected] [mailto:[email protected]]
Sent: Tuesday, March 11, 2003 3:22 PM
To: Ant Developers List
Subject: RE: enhanced pvcs task for PVCS version 7.5
<Anderson>
PVCS task works with version 7.5 as is, unless you are pointing to a config
file other than the default.
</Anderson>
"repository" attribute is used to point to config location, do you mean
some other configuration files; and as you know this a required attribute,
so no defaults.
<Anderson>
The advantage to using "pcli get" rather than
"get" is that pcli is smart about config files, which has been a problem
for
me with the existing PVCS task.I'm a little new to PVCS Version Manager, so
if there are other advantages please let me know. Of course pcli is pretty
slow compared to the old school "get".
</Anderson>
Following are advantages stated by our support team:-
All project teams that have been using the legacy Commands (i.e. get, put)
will need to convert to 'PCLI'. Using the PCLI commands is just like using
the I-NET client or Windows client. These three interfaces update
serialized database files that the PVCS Version Manager software uses.
These serialized database files contain all the information for the
archives. So if one member is using PVCS I-NET and modifies an item that
modification will appear for another team member that is using the PCLI
commands. Using the old sget or sput does not update these serialized
files.
1.) Easier/More User Friendly
2.) More Reliable
3.) Provides your team the choice to use Command Line or I-NET Client.
4.)May run a little slower since it is updating the serialized files.
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2021-06-25 05:04:58
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http://mathoverflow.net/revisions/68016/list
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5 Correction about rank of x being below theta
The theory consists of ZFC plus the assertion that there is a hierarchy of universe-like sets, namely, $V_\theta$ for all $\theta\in C$, a closed unbounded proper class of cardinals, and furthermore that truth in these $V_\theta$ cohere with each other and with the full set-theoretic universe $V$, so that they form an elementary chain. Specifically, the theory has ZFC, the assertion that $C$, a new class predicate, is a closed unbounded proper class of cardinals, such and the scheme asserting that $V_\theta$ is an elementary substructure of $V$ for every $\theta\in C$. That is, namely, the theory is a scheme , asserting first that $C$ is a proper class club, and secondly expressing of each formula $\varphi$ in the language of set theory that
• $\forall x\, x\ \forall\theta\in C\, C\text{ above the rank of }x \ (\varphi(x)\iff V_\theta\models\varphi[x])$.
• It follows from this theory that the models $V_\theta$ for $\theta\in C$ form an elementary chain, all agreeing with each other and with the full set-theoretic background universe on what is true as you ascend to higher models. It follows that every $\theta$ in $C$ will be a strong limit cardinal, a beth fixed point and so on, and so these cardinal exhibit very strong closure properties. In particular, I could have written $H_\theta$ instead of $V_\theta$---these are essentially the $\theta$-small universes, the collection of sets of hereditary size less than $\theta$. The only difference between these $V_\theta$ and an actual Grothendieck universe is that in this theory, you may not assume that $\theta$ is regular. But otherwise, they function just like universes in many ways. Indeed, because and indeed, every $V_\theta$ for $\theta\in C$ is a model of ZFC. Because of the coherence in the theories, these weak universes can be more useful than Grothendieck universes for certain purposes. For example, any statement true about an object in the full background universe will also be true about that object in every weak universe $V_\theta$ for $\theta\in C$ in which it resides. Thus, it one takes care, one can use the $V_\theta$ much like Grothendieck universes, and this was the point of my linked answer above (as well as Andreas's).
Meanwhile, the theory is conservative over ZFC, since in fact every model of ZFC can be elementary embedded into (a reduct of) a model of this theory. This can be proved by a simple compactness argument, using the reflection theory. If $M\models ZFC$, then add constants for every element of $M$, add the full elementary diagram of $M$, add a new predicate symbol for $C$ and all the axioms of the new theory. Every finite subtheory of this theory is consistent, by the reflection theorem, and so we get a model of the new theory, which elementary embeds $M$ since it satisfies the elementary diagram of $M$.
(Although it seems counterintuitive at first to some set-theorists, this theory does not prove Con(ZFC), if ZFC is consistent, even though it asserts in a sense that $V_\theta$ is elementary in $V$ for all $\theta\in C$ and hence that $V_\theta$ is a model of ZFC. The explanation is that the theory only makes the assertion that $V_\theta$ is elementary in $V$ as a scheme, and not as a single assertion (which is not expressible anyway by Tarski's theorem), and thus the theory does not actually prove that $V_\theta\models ZFC$ for $\theta\in C$, even though they do model ZFC, since the theory only proves every finite instance of this, rather than the universal assertion that every axiom of ZFC is satisfied in every $V_\theta$.)
4 added 82 characters in body
If you want a universe-like theory that is conservative over ZFC, that is, which proves no additional facts about sets that ZFC cannot prove alone, then the thing to do is to work in the following theory, which is also described in the answers to this MO question.
The theory consists of ZFC plus the assertion that $C$, a new class predicate, is a closed unbounded proper class of cardinals, such that $V_\theta$ is an elementary substructure of $V$ for every $\theta\in C$. That is, the theory is a scheme, asserting first that $C$ is a proper class club, and secondly of each formula $\varphi$ in the language of set theory that
• $\forall x\, \forall\theta\in C\, (\varphi(x)\iff V_\theta\models\varphi[x])$.
It follows from this theory that the models $V_\theta$ for $\theta\in C$ form an elementary chain, all agreeing with each other and with the full set-theoretic background universe on what is true as you ascend to higher models. It follows that every $\theta$ in $C$ will be a strong limit cardinal, a beth fixed point and so on, and so these cardinal exhibit very strong closure properties. In particular, I could have written $H_\theta$ instead of $V_\theta$---these are essentially the $\theta$-small universes, the collection of sets of hereditary size less than $\theta$. The only difference between these $V_\theta$ and an actual Grothendieck universe is that in this theory, you may not assume that $\theta$ is regular. But otherwise, they function just like universes in many ways. Indeed, because of the coherence in the theories, these weak universes can be more useful than Grothendieck universes for certain purposes. For example, any statement true about an object in the full background universe will also be true about that object in every weak universe $V_\theta$ for $\theta\in C$ in which it resides. Thus, it one takes care, one can use the $V_\theta$ much like Grothendieck universes, and this was the point of my linked answer above.
Meanwhile, the theory is conservative over ZFC, since in fact every model of ZFC can be elementary embedded into a model of this theory. This can be proved by a simple compactness argument, using the reflection theory. If $M\models ZFC$, then add constants for every element of $M$, add the full elementary diagram of $M$, add a new predicate symbol for $C$ and all the axioms of the new theory. Every finite subtheory of this theory is consistent, by the reflection theorem, and so we get a model of the new theory, which elementary embeds $M$ since it satisfies the elementary diagram of $M$.
(Although it seems counterintuitive at first to many some set-theorists, this theory does not prove Con(ZFC), if ZFC is consistent, even though it asserts in a sense that $V_theta$ V_\theta$is elementary in$V$for all$\theta\in C$. The explanation is that the theory only makes the assertion that$V_\theta$is elementary in$V$as a scheme, and not as a single assertion (which is not expressible anyway by Tarski's theorem), and thus the theory does not actually prove that$V_\theta\models ZFC$for$\theta\in C$, even though they do, since the theory only proves every finite instance of the axiomsthis, and not rather than the universal assertion that every axiom of ZFC is satisfied.)satisfied in every$V_\theta$.) 3 added 908 characters in body In It follows from this theory , one can use each that the models$V_\theta$like a for$\theta\in C$form an elementary chain, all agreeing with each other and with the full set-theoretic background universe . But on what is true as you have ascend to use higher models. It follows that every$\theta$in$C$will be a little carestrong limit cardinal, since a beth fixed point and so on, and so these cardinal exhibit very strong closure properties. In particular, I could have written$V_\theta$H_\theta$ instead of $V_\theta$---these are not actually essentially the $\theta$-small universesin , the usual collection of sets of hereditary size less than $\theta$. The only difference between these $V_\theta$ and an actual Grothendieck senseuniverse is that in this theory, since you may not assume that $\theta$ may is regular. But otherwise, they function just like universes in many ways. Indeed, because of the coherence in the theories, these weak universes can be singularmore useful than Grothendieck universes for certain purposes. For example, but any statement true about an object in the full background universe will also be true about that object in every weak universe $V_\theta$ for almost all $\theta\in C$ in which it resides. Thus, it one takes care, one can use the usual purposes of $V_\theta$ much like Grothendieck universes, as I explained in and this was the point of my linked answer above, these models suffice if one takes care.
2 added 659 characters in body
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2013-05-23 08:44:50
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https://ses.library.usyd.edu.au/handle/2123/17247/browse?type=subject&value=Self-concept
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• #### Exploring the self-concept and identity of Sydney Conservatorium students with and without absolute pitch
Published 2007-01-18
Absolute Pitch (AP) is the ability to identify pitches without external references (Parncutt & Levitin, 2001). It is a rare ability that is more prevalent among musicians. This qualitative study explored the perceptions ...
Open Access
Thesis, Honours
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2020-07-13 17:03:22
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https://ai.stackexchange.com/questions/12649/neural-nets-cnn-confirming-layer-filter-arithmetic?noredirect=1
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# Neural Nets: CNN confirming layer/filter arithmetic
I was hoping someone could just confirm some intuition about how convolutions work in convolutional neural networks. I have seen all of the tutorials on applying convolutional filters on an image, but most of those tutorials focus on one channel images, like a 128 x 128 x 1 image. I wanted to clarify what happens when we apply a convolutional filter to RGB 3 channel images.
Now this is not a unique question, I think a lot of people ask this question as well. It is just that there seem to be so many answers out there, each with their own variations, that it is hard to find a consistent answer. I included a post below that seems to comport with what my own intuition, but I was hoping one of the experts on SE could help validate the layer arithmetic, to make sure my intuition was not off.
Convolutional neural nets and reduction of the layers
Consider an Alexnet network with 5 convolutional layers and 3 fully connected layers. I borrowed the network from this post. Now, say the input is 227 x 227, and the filter is specified as 11 x 11 x 96 with stride 4. That means there are 96 filters each with dimensions 11x11x3, right? So there are a total of 363 parameters per filter--excluding the bias term-- and there are 96 of these filters to learn. So the 363*96 = 34848 filter values are learned just like the weights in the fully connected layers right?
My second question deals with the next convolutional network layer. In the next layer I will have an image that is 55 x 55 x 96 image. In this case, would the filter be 5x5x96--since there are now 96 feature maps on the image? So that means that each individual filter would need to learn 5x5x96 = 2400 filter values (weights), and that across all 256 filters this would mean 614,400 filter values?
I just wanted to make sure that I was understanding exactly what is being learned at each level.
• Oh thanks so much. Yeah I just wanted to make sure I was understanding the structure correctly. The equations are good, but they hide the implementation details. And the deep learning frameworks like keras will wrap the operation in these high level functions, so you can't see what is going on underneath. Don't get me wrong, I love keras. I just did not want to treat the networks like "black boxes" where I just plug stuff in and get a result without understanding how to get the answer. I took Andrew Ng's course on Coursera and it was really nice. – krishnab Jun 3 '19 at 8:58
• Clement - thanks for the info, yeah I took the course a while ago so don't remember the details of the assignment. If I remember correctly, didn't he do the assignment in Numpy for the forward pass? I think it was a really good course. I wish that the Deep Learning specialization had had a deeper set of "lab" courses in tensorflow. I believe Coursera has a new set of deep learning classes where they are trying to provide more tf practice. – krishnab Jun 3 '19 at 17:21
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2020-08-03 23:00:48
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https://www.thestudentroom.co.uk/showthread.php?t=3178021
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# I cant solve this MCQ, help please?
Watch
Announcements
#1
This is a question from June 13 . The answer is B. but how?
α-Centauri is one of the nearest stars to our Sun. The surface temperatures of these two
stars are about the same. α-Centauri has a 20% greater diameter than the Sun.
The ratio of the luminosity of α-Centauri to the luminosity of the Sun is about
A 1.2
B 1.4
C 1.7
D 2.1
0
5 years ago
#2
Sorry you've not had any responses about this. Are you sure you’ve posted in the right place? Posting in the specific Study Help forum should help get responses.
I'm going to quote in Puddles the Monkey now so she can move your thread to the right place if it's needed.
Spoiler:
Show
(Original post by Puddles the Monkey)
x
0
5 years ago
#3
(Original post by Fatima SJ)
This is a question from June 13 . The answer is B. but how?
α-Centauri is one of the nearest stars to our Sun. The surface temperatures of these two
stars are about the same. α-Centauri has a 20% greater diameter than the Sun.
The ratio of the luminosity of α-Centauri to the luminosity of the Sun is about
A 1.2
B 1.4
C 1.7
D 2.1
As they have the same temperature, the luminosity will depend on the area presented to the observer.
Area depends on radius or diameter squared.
20% greater means x 1.2
1.22 = 1.4 approx.
0
5 years ago
#4
(Original post by Stonebridge)
As they have the same temperature, the luminosity will depend on the area presented to the observer.
Area depends on radius or diameter squared.
20% greater means x 1.2
1.22 = 1.4 approx.
Please refrain from posting full solutions.
0
5 years ago
#5
(Original post by Doctor_Einstein)
Please refrain from posting full solutions.
This is not a full solution to a problem, it is an explanation of why an MCQ answer is what it is.
Educationally, totally different objectives.
2
#6
(Original post by Stonebridge)
As they have the same temperature, the luminosity will depend on the area presented to the observer.
Area depends on radius or diameter squared.
20% greater means x 1.2
1.22 = 1.4 approx.
Thanks alot
0
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2020-10-23 06:24:12
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https://forum.azimuthproject.org/plugin/ViewComment/17975
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@Michael – re "what about the poset is making it so that it can't have a monodical product?"...
• if a finite non-empty poset is *discrete*, we can make it monoidal by slapping any old monoid structure on it (eg turn it into a cyclic group) – since \$$x\leq y \iff x = y\$$ this monoid automatically respects the partial order
• if a poset has *all finite joins*, we can make it monoidal by choosing the bottom element (ie the join of nothing!) to be the unit and the binary join \$$x\vee y\$$ to be the monoidal product
• if a poset has *all finite meets*, we can make it monoidal by choosing the top element (ie the meet of nothing!) to be the unit and the binary meet \$$x\wedge y\$$ to be the monoidal product
So if a finite non-empty poset *can't* be monoidal, then we know it can't be discrete, or have all finite joins, or have all finite meets.
Jonathan's "bow tie" poset is just the smallest poset abiding by all three of these conditions.
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2019-04-25 09:52:44
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https://www.examrace.com/SPSC/Madhya-Pradesh-PSC/MPPSC-Syllabus/MPPSC-Mathematics-Syllabus.html
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# MPPSC Mathematics Syllabus
The Madhya Pradesh Public Service Commission has published syllabus for the State Services Preliminary Examination 2010 of the MPPSC. The Syllabus of the exam is as follows:
1. Trigonometry and Polynomial Equations Demoiver's theorem and it's application, Direct and Inverse, Circularand hyperbolic function. Logarithm of complex quantities. Expansion of trigonometric functions. Relation between roots and coefficient of polynomial equations. Transformation of equations. Descarte's rule of sign.
2. Matrices and Determinants Definition of matrices & determinants, addition, multiplication of matrices, elementary operation on matrices. Adjoint of matrices, inverse and rank of matrices, application of matrices to system of linear equations, Cramer's rule.
3. Differential Calculus Limit, continuity & differentiability of function of one variable, differentiation of functions, Application of differentiation on maxima & minima. Tangents & Normals. Expansion of functions. Mean value theorem, Taylor's theorem, Taylor and Maclaurin series. Successive differentiation. Lebnitze's Theorem.
4. Integral Calculus Definition of Integration as a sum. Various methods of integration. Integration by substitution & by parts. Definite Integrals. Beta and Gamma functions, Double & triple integration. Change of order of integration of Double integrals. Rectification and Quadrature.
5. Differential Equations Differential Equations of first order and first degree, variable separable, exact, homogenous forms. Linear differential equations. Linear Differential Equations of higher order with constant coefficients.
6. Abstract Algebra Definition of Group with examples & proprieties. Sub-groups, cyclic group, Coset-decomposition. Lagrange's theorem. Normal sub group. Quotient Group. Homomorphism and isomorphism of groups. Permutation Group, Introduction to Rings, Subrings and ideals, Integral domain & field. Definition of Vector space, subspace, and properties of Vector spaces.
7. Vector Analysis & Geometry Scalar and vector product of two, three & four vectors. Reciprocal vectors, vector differentiation. Gradient, divergence & curl. Equation of straight lines in Cartesian & polar coordinates. Circle, parabola and ellipse and their tangent & normal in two dimensions.
8. Mechanics
Law of parallelogram of forces. Triangle of forces. Lami's theorem & it's applications. Newton's laws of motion. Motion in a straight line-motion under gravity.
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2018-01-23 18:18:34
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https://solvedlib.com/n/clearly-and-accurately-label-which-of-the-following-graphs,6585515
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# Clearly and accurately label which of the following graphs f(x)f'(xJand f"(x)
###### Question:
Clearly and accurately label which of the following graphs f(x)f'(xJand f"(x)
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##### When a 17.5 mL sample of a 0.359 M aqueous hypochlorous acid solution is titrated with...
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2023-02-06 18:51:18
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https://mathhelpboards.com/threads/brouwer-fixed-point-theorem.7028/
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# Brouwer Fixed Point Theorem
#### smile
##### New member
Brouwer Fixed Point Theorem: Every continuous function from the closed ball B^n= {x∈R^n: abs(x)<1}
to itself has a fixed point.
Is anyone can help me to Prove the Brouwer Fixed point theorem for n = 1 using the fact: there is no retraction from the closed interval [-1,1] onto
the two point set {-1,1}.
also Assume that there is no retraction from the closed ballB^n= {x∈R^n: abs(x)<1} onto the sphere
S^n-1= {x∈R^n: abs(x)=1} Prove the
Brouwer Fixed Point Theorem.
Thanks
#### Opalg
##### MHB Oldtimer
Staff member
Brouwer Fixed Point Theorem: Every continuous function from the closed ball B^n= {x∈R^n: abs(x)<1}
to itself has a fixed point.
Is anyone can help me to Prove the Brouwer Fixed point theorem for n = 1 using the fact: there is no retraction from the closed interval [-1,1] onto
the two point set {-1,1}.
Proof by contradiction: Assume that $f:[-1,1]\to [-1,1]$ has no fixed point. Define $g(x) = 0$ if $f(x)>x$ and $g(x)=1$ if $f(x)<x$. Show that $g$ is a retraction.
also Assume that there is no retraction from the closed ballB^n= {x∈R^n: abs(x)<1} onto the sphere
S^n-1= {x∈R^n: abs(x)=1} Prove the
Brouwer Fixed Point Theorem.
Similar idea: if $f:B^n\to B^n$ has no fixed point, then for each $x\in B^n$ there is a well-defined line starting from $f(x)$ and going in the direction of $x$. Let $g(x)$ be the point where this line hits $S^{n-1}$, and show that $g$ is a retraction.
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2021-06-14 21:33:56
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https://stacks.math.columbia.edu/tag/08X7
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Lemma 35.4.20. If $f$ is universally injective, then the diagram
35.4.20.1
$$\label{descent-equation-equalizer-f2} \xymatrix@C=8pc{ f_*(M, \theta ) \otimes _ R S \ar[r]^{\theta \circ (1_ M \otimes \delta _0^1)} & M \otimes _{S, \delta _1^1} S_2 \ar@<1ex>[r]^{(\theta \otimes \delta _2^2) \circ (1_ M \otimes \delta ^2_0)} \ar@<-1ex>[r]_{1_{M \otimes S_2} \otimes \delta ^2_1} & M \otimes _{S, \delta _{12}^1} S_3 }$$
obtained by tensoring (35.4.19.1) over $R$ with $S$ is an equalizer.
Proof. By Lemma 35.4.12 and Remark 35.4.13, the map $C(1_ N \otimes f): C(N \otimes _ R S) \to C(N)$ can be split functorially in $N$. This gives the upper vertical arrows in the commutative diagram
$\xymatrix@C=8pc{ C(M \otimes _{S, \delta _1^1} S_2) \ar@<1ex>^{C(\theta \circ (1_ M \otimes \delta _0^1))}[r] \ar@<-1ex>_{C(1_ M \otimes \delta _1^1)}[r] \ar[d] & C(M) \ar[r]\ar[d] & C(f_*(M,\theta )) \ar@{-->}[d] \\ C(M \otimes _{S,\delta _{12}^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta _2^2) \circ (1_ M \otimes \delta ^2_0))}[r] \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta ^2_1)}[r] \ar[d] & C(M \otimes _{S, \delta _1^1} S_2 ) \ar[r]^{C(\theta \circ (1_ M \otimes \delta _0^1))} \ar[d]^{C(1_ M \otimes \delta _1^1)} & C(M) \ar[d] \ar@{=}[dl] \\ C(M \otimes _{S, \delta _1^1} S_2) \ar@<1ex>[r]^{C(\theta \circ (1_ M \otimes \delta _0^1))} \ar@<-1ex>[r]_{C(1_ M \otimes \delta _1^1)} & C(M) \ar[r] & C(f_*(M,\theta )) }$
in which the compositions along the columns are identity morphisms. The second row is the coequalizer diagram (35.4.17.1); this produces the dashed arrow. From the top right square, we obtain auxiliary morphisms $C(f_*(M,\theta )) \to C(M)$ and $C(M) \to C(M\otimes _{S,\delta _1^1} S_2)$ which imply that the first row is a split coequalizer diagram. By Remark 35.4.11, we may tensor with $S$ inside $C$ to obtain the split coequalizer diagram
$\xymatrix@C=8pc{ C(M \otimes _{S,\delta _2^2 \circ \delta _1^1} S_3) \ar@<1ex>^{C((\theta \otimes \delta _2^2) \circ (1_ M \otimes \delta ^2_0))}[r] \ar@<-1ex>_{C(1_{M \otimes S_2} \otimes \delta ^2_1)}[r] & C(M \otimes _{S, \delta _1^1} S_2 ) \ar[r]^{C(\theta \circ (1_ M \otimes \delta _0^1))} & C(f_*(M,\theta ) \otimes _ R S). }$
By Lemma 35.4.10, we conclude (35.4.20.1) must also be an equalizer. $\square$
There are also:
• 4 comment(s) on Section 35.4: Descent for universally injective morphisms
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2023-03-23 01:50:42
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https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-12-vectors-and-the-geometry-of-space-12-4-the-cross-product-12-4-exercises-page-862/47
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## Calculus 8th Edition
$|a \times b|^2=|a|^2 |b|^2-(a \cdot b)^2$
$|a \times b|^2=( |a| |b||sin \theta|)^2= |a|^2 |b|^2sin ^2\theta$ As we know : $sin ^2 \theta =1 -cos ^2 \theta$ Thus, $|a|^2 |b|^2sin ^2\theta= |a|^2 |b|^2-|a|^2 |b|^2cos ^2\theta$ Remember that $(a \cdot b)^2=(|a|^2 |b|^2cos\theta) ^2=|a|^2 |b|^2cos^2\theta$ Hence, $|a \times b|^2=|a|^2 |b|^2-(a \cdot b)^2$
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2019-12-11 09:29:00
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https://www.fuzzingbook.org/beta/html/GreyboxFuzzer.html
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# Greybox Fuzzing¶
In the previous chapter, we have introduced mutation-based fuzzing, a technique that generates fuzz inputs by applying small mutations to given inputs. In this chapter, we show how to guide these mutations towards specific goals such as coverage. The algorithms in this chapter stem from the popular American Fuzzy Lop (AFL) fuzzer, in particular from its AFLFast and AFLGo flavors. We will explore the greybox fuzzing algorithm behind AFL and how we can exploit it to solve various problems for automated vulnerability detection.
from bookutils import YouTubeVideo
YouTubeVideo('vBrNT9q2t1Y')
Prerequisites
## Synopsis¶
>>> from fuzzingbook.GreyboxFuzzer import <identifier>
and then make use of the following features.
This chapter introduces advanced methods for grey-box fuzzing inspired by the popular AFL fuzzer. The GreyboxFuzzer class has three arguments. First, a list of seed inputs:
>>> seed_input = "http://www.google.com/search?q=fuzzing"
>>> seeds = [seed_input]
Second, a mutator that changes individual parts of the input.
>>> mutator = Mutator()
Third, a power schedule that assigns fuzzing effort across the population:
>>> schedule = PowerSchedule()
These three go into the GreyboxFuzzer constructor:
>>> greybox_fuzzer = GreyboxFuzzer(seeds=seeds, mutator=mutator, schedule=schedule)
The GreyboxFuzzer class is used in conjunction with a FunctionCoverageRunner:
>>> http_runner = FunctionCoverageRunner(http_program)
>>> outcomes = greybox_fuzzer.runs(http_runner, trials=10000)
After fuzzing, we can inspect the population:
>>> greybox_fuzzer.population[:20]
[http://www.google.com/search?q=fuzzing,
h4t:/O/g;{Hoogme&cPo/m'Mqqde<r?q=f5zxiQng,
http:/-www.googl.com/sear?q=fuzzing,
t/7Aww5.%?Oghe,coHmQ(/gsec,?|QF9fp:ing,
ht:/O/g;9{Hoogme&cPo/m'Mqqde<r?q=f5zxiQng,
ht\o:/O/g9;oogme&cP6To/mMmqe<r?q=f5piQ2n,
htgtp://ww.googlecom/search?q=fuzzingw,
h]:OO/g99ogle&Kc6To/mMmpeYrq=75i2,
d^7ww05s.%?Ogjc,4[joQp(seMc,|QFLfLSp:i'2,
h#t:/O/c;y{Hoog}&CPo/mZ)'Cd<r?q-f5zyiQng,
7ww60s.%&1?t2Og\*#,4[*QbpshmMc,|EQ*VLfNSp:i6<,
h#t/O/c+y{H;og}h&Co-/m'Z)C$<r?-f5zyiQngi, htgtp//vw.googecom/search?q=fuzzingv, htgtp//vw.googeco/earh?q=fuzxingv, d^7ww ;5s@.%?aOgjc,4[oQp(seMc,|QFLfLS0:!I'2, hN+{H;;4=/wB}h&1Co(/MZ)$(r?tuzyiQngo,
?h#t/K/+y{H;og}Ih&Co
/mR'Z$)$<r-f5Mzyix,negi,
ihN+{H;4=/wB}h&1Co(/MZ)$(r uzyiQngo, i4tg:/g{HoggmecPo-7MqqZwzf)q=f xxing+, h]c>Gg9G1[ogRlefc6\mxmerqqY 5+2] Besides the simple PowerSchedule, we can have advanced power schedules. • AFLFastSchedule assigns high energy to "unusual" paths not taken very often. • AFLGoSchedule assigns high energy to paths close to uncovered program locations. The AFLGoSchedule class constructor requires a distance metric from each node towards target locations, as determined via analysis of the program code. See the chapter for details. ## AFL: An Effective Greybox Fuzzer¶ The algorithms in this chapter stem from the popular American Fuzzy Lop (AFL) fuzzer. AFL is a mutation-based fuzzer. Meaning, AFL generates new inputs by slightly modifying a seed input (i.e., mutation), or by joining the first half of one input with the second half of another (i.e., splicing). AFL is also a greybox fuzzer (not blackbox nor whitebox). Meaning, AFL leverages coverage-feedback to learn how to reach deeper into the program. It is not entirely blackbox because AFL leverages at least some program analysis. It is not entirely whitebox either because AFL does not build on heavyweight program analysis or constraint solving. Instead, AFL uses lightweight program instrumentation to glean some information about the (branch) coverage of a generated input. If a generated input increases coverage, it is added to the seed corpus for further fuzzing. To instrument a program, AFL injects a piece of code right after every conditional jump instruction. When executed, this so-called trampoline assigns the exercised branch a unique identifier and increments a counter that is associated with this branch. For efficiency, only a coarse branch hit count is maintained. In other words, for each input the fuzzer knows which branches and roughly how often they are exercised. The instrumentation is usually done at compile-time, i.e., when the program source code is compiled to an executable binary. However, it is possible to run AFL on uninstrumented binaries using tools such as a virtual machine (e.g., QEMU) or a dynamic instrumentation tool (e.g., Intel PinTool). For Python programs, we can collect coverage information without any instrumentation (see chapter on collecting coverage). ## Ingredients for Greybox Fuzzing¶ We start with discussing the most important parts we need for mutational testing and goal guidance. ### Mutators¶ We introduce specific classes for mutating a seed. import bookutils from typing import List, Set, Any, Tuple, Dict, Union from collections.abc import Sequence import random from Coverage import population_coverage First, we'll introduce the Mutator class. Given a seed input inp, the mutator returns a slightly modified version of inp. In the chapter on greybox grammar fuzzing, we extend this class to consider the input grammar for smart greybox fuzzing. class Mutator: """Mutate strings""" def __init__(self) -> None: """Constructor""" self.mutators = [ self.delete_random_character, self.insert_random_character, self.flip_random_character ] For insertion, we add a random character in a random position. class Mutator(Mutator): def insert_random_character(self, s: str) -> str: """Returns s with a random character inserted""" pos = random.randint(0, len(s)) random_character = chr(random.randrange(32, 127)) return s[:pos] + random_character + s[pos:] For deletion, if the string is non-empty choose a random position and delete the character. Otherwise, use the insertion-operation. class Mutator(Mutator): def delete_random_character(self, s: str) -> str: """Returns s with a random character deleted""" if s == "": return self.insert_random_character(s) pos = random.randint(0, len(s) - 1) return s[:pos] + s[pos + 1:] For substitution, if the string is non-empty choose a random position and flip a random bit in the character. Otherwise, use the insertion-operation. class Mutator(Mutator): def flip_random_character(self, s: str) -> str: """Returns s with a random bit flipped in a random position""" if s == "": return self.insert_random_character(s) pos = random.randint(0, len(s) - 1) c = s[pos] bit = 1 << random.randint(0, 6) new_c = chr(ord(c) ^ bit) return s[:pos] + new_c + s[pos + 1:] The main method is mutate which chooses a random mutation operator from the list of operators. class Mutator(Mutator): def mutate(self, inp: Any) -> Any: # can be str or Seed (see below) """Return s with a random mutation applied. Can be overloaded in subclasses.""" mutator = random.choice(self.mutators) return mutator(inp) Let's try the mutator. You can actually interact with such a "cell" and try other inputs by loading this chapter as Jupyter notebook. After opening, run all cells in the notebook using "Kernel -> Restart & Run All". Mutator().mutate("good") 'cood' ### Seeds and Power Schedules¶ Now we introduce a new concept; the power schedule. A power schedule distributes the precious fuzzing time among the seeds in the population. Our objective is to maximize the time spent fuzzing those (most progressive) seeds which lead to higher coverage increase in shorter time. We call the likelihood with which a seed is chosen from the population as the seed's energy. Throughout a fuzzing campaign, we would like to prioritize seeds that are more promising. Simply said, we do not want to waste energy fuzzing non-progressive seeds. We call the procedure that decides a seed's energy as the fuzzer's power schedule. For instance, AFL's schedule assigns more energy to seeds that are shorter, that execute faster, and yield coverage increases more often. First, there is some information that we need to attach to each seed in addition to the seed's data. Hence, we define the following Seed class. from Coverage import Location class Seed: """Represent an input with additional attributes""" def __init__(self, data: str) -> None: """Initialize from seed data""" self.data = data # These will be needed for advanced power schedules self.coverage: Set[Location] = set() self.distance: Union[int, float] = -1 self.energy = 0.0 def __str__(self) -> str: """Returns data as string representation of the seed""" return self.data __repr__ = __str__ The power schedule that is implemented below assigns each seed the same energy. Once a seed is in the population, it will be fuzzed as often as any other seed in the population. In Python, we can squeeze long for-loops into much smaller statements. • lambda x: ... returns a function that takes x as input. Lambda allows for quick definitions unnamed functions. • map(f, l) returns a list where the function f is applied to each element in list l. • random.choices(l, weights)[0] returns element l[i] with probability in weights[i]. class PowerSchedule: """Define how fuzzing time should be distributed across the population.""" def __init__(self) -> None: """Constructor""" self.path_frequency: Dict = {} def assignEnergy(self, population: Sequence[Seed]) -> None: """Assigns each seed the same energy""" for seed in population: seed.energy = 1 def normalizedEnergy(self, population: Sequence[Seed]) -> List[float]: """Normalize energy""" energy = list(map(lambda seed: seed.energy, population)) sum_energy = sum(energy) # Add up all values in energy assert sum_energy != 0 norm_energy = list(map(lambda nrg: nrg / sum_energy, energy)) return norm_energy def choose(self, population: Sequence[Seed]) -> Seed: """Choose weighted by normalized energy.""" self.assignEnergy(population) norm_energy = self.normalizedEnergy(population) seed: Seed = random.choices(population, weights=norm_energy)[0] return seed Let's see whether this power schedule chooses seeds uniformly at random. We ask the schedule 10k times to choose a seed from the population of three seeds (A, B, C) and keep track of the number of times we have seen each seed. We should see each seed about 3.3k times. population = [Seed("A"), Seed("B"), Seed("C")] schedule = PowerSchedule() hits = { "A": 0, "B": 0, "C": 0 } for i in range(10000): seed = schedule.choose(population) hits[seed.data] += 1 hits {'A': 3387, 'B': 3255, 'C': 3358} Looks good. Every seed has been chosen about a third of the time. ### Runners and a Sample Program¶ We'll start with a small sample program of six lines. In order to collect coverage information during execution, we import the FunctionCoverageRunner class from the chapter on mutation-based fuzzing. The FunctionCoverageRunner constructor takes a Python function to execute. The function run takes an input, passes it on to the Python function, and collects the coverage information for this execution. The function coverage() returns a list of tuples (function name, line number) for each statement that has been covered in the Python function. from MutationFuzzer import FunctionCoverageRunner, http_program The crashme() function raises an exception for the input "bad!". Let's see which statements are covered for the input "good". def crashme(s: str) -> None: if len(s) > 0 and s[0] == 'b': if len(s) > 1 and s[1] == 'a': if len(s) > 2 and s[2] == 'd': if len(s) > 3 and s[3] == '!': raise Exception() crashme_runner = FunctionCoverageRunner(crashme) crashme_runner.run("good") list(crashme_runner.coverage()) [('run_function', 132), ('crashme', 2)] In crashme, the input "good" only covers the if-statement in line 2. The branch condition len(s) > 0 and s[0] == 'b' evaluates to False. ## Advanced Blackbox Mutation-based Fuzzing¶ Let's integrate both the mutator and power schedule into a fuzzer. We'll start with a blackbox fuzzer -- which does not leverage any coverage information. Our AdvancedMutationFuzzer class is an advanced and parameterized version of the MutationFuzzer class from the chapter on mutation-based fuzzing. It also inherits from the Fuzzer class. For now, we only need to know the functions fuzz() which returns a generated input and runs() which executes fuzz() a specified number of times. For our AdvancedMutationFuzzer class, we override the function fuzz(). from Fuzzer import Fuzzer The AdvancedMutationFuzzer is constructed with a set of initial seeds, a mutator, and a power schedule. Throughout the fuzzing campaign, it maintains a seed corpus called population. The function fuzz returns either an unfuzzed seed from the initial seeds, or the result of fuzzing a seed in the population. The function create_candidate handles the latter. It randomly chooses an input from the population and applies a number of mutations. class AdvancedMutationFuzzer(Fuzzer): """Base class for mutation-based fuzzing.""" def __init__(self, seeds: List[str], mutator: Mutator, schedule: PowerSchedule) -> None: """Constructor. seeds - a list of (input) strings to mutate. mutator - the mutator to apply. schedule - the power schedule to apply. """ self.seeds = seeds self.mutator = mutator self.schedule = schedule self.inputs: List[str] = [] self.reset() def reset(self) -> None: """Reset the initial population and seed index""" self.population = list(map(lambda x: Seed(x), self.seeds)) self.seed_index = 0 def create_candidate(self) -> str: """Returns an input generated by fuzzing a seed in the population""" seed = self.schedule.choose(self.population) # Stacking: Apply multiple mutations to generate the candidate candidate = seed.data trials = min(len(candidate), 1 << random.randint(1, 5)) for i in range(trials): candidate = self.mutator.mutate(candidate) return candidate def fuzz(self) -> str: """Returns first each seed once and then generates new inputs""" if self.seed_index < len(self.seeds): # Still seeding self.inp = self.seeds[self.seed_index] self.seed_index += 1 else: # Mutating self.inp = self.create_candidate() self.inputs.append(self.inp) return self.inp Okay, let's take the mutation fuzzer for a spin. Given a single seed, we ask it to generate three inputs. seed_input = "good" mutation_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule()) print(mutation_fuzzer.fuzz()) print(mutation_fuzzer.fuzz()) print(mutation_fuzzer.fuzz()) good gDoodC / Let's see how many statements the mutation-based blackbox fuzzer covers in a campaign with n=30k inputs. The fuzzer function runs(crashme_runner, trials=n) generates n inputs and executes them on the crashme function via the crashme_runner. As stated earlier, the crashme_runner also collects coverage information. import time n = 30000 blackbox_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule()) start = time.time() blackbox_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n) end = time.time() "It took the blackbox mutation-based fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the blackbox mutation-based fuzzer 0.36 seconds to generate and execute 30000 inputs.' In order to measure coverage, we import the population_coverage function. It takes a set of inputs and a Python function, executes the inputs on that function and collects coverage information. Specifically, it returns a tuple (all_coverage, cumulative_coverage) where all_coverage is the set of statements covered by all inputs, and cumulative_coverage is the number of statements covered as the number of executed inputs increases. We are just interested in the latter to plot coverage over time. We extract the generated inputs from the blackbox fuzzer and measure coverage as the number of inputs increases. _, blackbox_coverage = population_coverage(blackbox_fuzzer.inputs, crashme) bb_max_coverage = max(blackbox_coverage) "The blackbox mutation-based fuzzer achieved a maximum coverage of %d statements." % bb_max_coverage 'The blackbox mutation-based fuzzer achieved a maximum coverage of 2 statements.' The following generated inputs increased the coverage for our crashme example. [seed_input] + \ [ blackbox_fuzzer.inputs[idx] for idx in range(len(blackbox_coverage)) if blackbox_coverage[idx] > blackbox_coverage[idx - 1] ] ['good', 'bo'] Summary. This is how a blackbox mutation-based fuzzer works. We have integrated the mutator to generate inputs by fuzzing a provided set of initial seeds and the power schedule to decide which seed to choose next. ## Greybox Mutation-based Fuzzing¶ In contrast to a blackbox fuzzer, a greybox fuzzer like AFL does leverage coverage information. Specifically, a greybox fuzzer adds to the seed population generated inputs which increase code coverage. The method run() is inherited from the Fuzzer class. It is called to generate and execute exactly one input. We override this function to add an input to the population that increases coverage. The greybox fuzzer attribute coverages_seen maintains the set of statements, that have previously been covered. class GreyboxFuzzer(AdvancedMutationFuzzer): """Coverage-guided mutational fuzzing.""" def reset(self): """Reset the initial population, seed index, coverage information""" super().reset() self.coverages_seen = set() self.population = [] # population is filled during greybox fuzzing def run(self, runner: FunctionCoverageRunner) -> Tuple[Any, str]: """Run function(inp) while tracking coverage. If we reach new coverage, add inp to population and its coverage to population_coverage """ result, outcome = super().run(runner) new_coverage = frozenset(runner.coverage()) if new_coverage not in self.coverages_seen: # We have new coverage seed = Seed(self.inp) seed.coverage = runner.coverage() self.coverages_seen.add(new_coverage) self.population.append(seed) return (result, outcome) Let's take our greybox fuzzer for a spin. seed_input = "good" greybox_fuzzer = GreyboxFuzzer([seed_input], Mutator(), PowerSchedule()) start = time.time() greybox_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n) end = time.time() "It took the greybox mutation-based fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the greybox mutation-based fuzzer 0.37 seconds to generate and execute 30000 inputs.' Does the greybox fuzzer cover more statements after generating the same number of test inputs? _, greybox_coverage = population_coverage(greybox_fuzzer.inputs, crashme) gb_max_coverage = max(greybox_coverage) "Our greybox mutation-based fuzzer covers %d more statements" % (gb_max_coverage - bb_max_coverage) 'Our greybox mutation-based fuzzer covers 2 more statements' Our seed population for our example now contains the following seeds. greybox_fuzzer.population [good, bo, baof, bad4u] Coverage-feedback is indeed helpful. The new seeds are like bread crumbs or milestones that guide the fuzzer to progress more quickly into deeper code regions. Following is a simple plot showing the coverage achieved over time for both fuzzers on our simple example. %matplotlib inline import matplotlib.pyplot as plt line_bb, = plt.plot(blackbox_coverage, label="Blackbox") line_gb, = plt.plot(greybox_coverage, label="Greybox") plt.legend(handles=[line_bb, line_gb]) plt.title('Coverage over time') plt.xlabel('# of inputs') plt.ylabel('lines covered'); Summary. We have seen how a greybox fuzzer "discovers" interesting seeds that can lead to more progress. From the input good, our greybox fuzzer has slowly learned how to generate the input bad! which raises the exception. Now, how can we do that even faster? Try it. How much coverage would be achieved over time using a blackbox generation-based fuzzer? Try plotting the coverage for all three fuzzers. You can define the blackbox generation-based fuzzer as follows. from Fuzzer import RandomFuzzer blackbox_gen_fuzzer = RandomFuzzer(min_length=4, max_length=4, char_start=32, char_range=96) You can execute your own code by opening this chapter as Jupyter notebook. Read. This is the high-level view how AFL works, one of the most successful vulnerability detection tools. If you are interested in the technical details, have a look at: https://github.com/mirrorer/afl/blob/master/docs/technical_details.txt ## Boosted Greybox Fuzzing¶ Our boosted greybox fuzzer assigns more energy to seeds that promise to achieve more coverage. We change the power schedule such that seeds that exercise "unusual" paths have more energy. With unusual paths, we mean paths that are not exercised very often by generated inputs. In order to identify which path is exercised by an input, we leverage the function getPathID from the section on trace coverage. import pickle # serializes an object by producing a byte array from all the information in the object import hashlib # produces a 128-bit hash value from a byte array The function getPathID returns a unique hash for a coverage set. def getPathID(coverage: Any) -> str: """Returns a unique hash for the covered statements""" pickled = pickle.dumps(coverage) return hashlib.md5(pickled).hexdigest() There are several ways to assign energy based on how unusual the exercised path is. In this case, we implement an exponential power schedule which computes the energy$e(s)$for a seed$s$as follows $$e(s) = \frac{1}{f(p(s))^a}$$ where •$p(s)$returns the ID of the path exercised by$s$, •$f(p)$returns the number of times the path$p$is exercised by generated inputs, and •$a$is a given exponent. class AFLFastSchedule(PowerSchedule): """Exponential power schedule as implemented in AFL""" def __init__(self, exponent: float) -> None: self.exponent = exponent def assignEnergy(self, population: Sequence[Seed]) -> None: """Assign exponential energy inversely proportional to path frequency""" for seed in population: seed.energy = 1 / (self.path_frequency[getPathID(seed.coverage)] ** self.exponent) In the greybox fuzzer, lets keep track of the number of times$f(p)$each path$p$is exercised, and update the power schedule. class CountingGreyboxFuzzer(GreyboxFuzzer): """Count how often individual paths are exercised.""" def reset(self): """Reset path frequency""" super().reset() self.schedule.path_frequency = {} def run(self, runner: FunctionCoverageRunner) -> Tuple[Any, str]: """Inform scheduler about path frequency""" result, outcome = super().run(runner) path_id = getPathID(runner.coverage()) if path_id not in self.schedule.path_frequency: self.schedule.path_frequency[path_id] = 1 else: self.schedule.path_frequency[path_id] += 1 return(result, outcome) Okay, lets run our boosted greybox fuzzer$n=10k$times on our simple example. We set the exponentent of our exponential power schedule to$a=5$. n = 10000 seed_input = "good" fast_schedule = AFLFastSchedule(5) fast_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), fast_schedule) start = time.time() fast_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n) end = time.time() "It took the fuzzer w/ exponential schedule %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the fuzzer w/ exponential schedule 0.23 seconds to generate and execute 10000 inputs.' import numpy as np x_axis = np.arange(len(fast_schedule.path_frequency)) y_axis = list(fast_schedule.path_frequency.values()) plt.bar(x_axis, y_axis) plt.xticks(x_axis) plt.ylim(0, n) # plt.yscale("log") # plt.yticks([10,100,1000,10000]) plt; print(" path id 'p' : path frequency 'f(p)'") fast_schedule.path_frequency path id 'p' : path frequency 'f(p)' {'0a24dcc3bfb72fcf6ee703cc5b1d2b23': 5612, 'fca21b86927ec616ed8199b079eb3585': 2607, '98d92cfb37c8115e0d2b0d01c2a2e150': 1105, '9ae43bde4b2b358677868370f5793850': 457, '895fda51f9cb1975a78a42e03cdf1f43': 219} How does it compare to our greybox fuzzer with the classical power schedule? seed_input = "good" orig_schedule = PowerSchedule() orig_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), orig_schedule) start = time.time() orig_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n) end = time.time() "It took the fuzzer w/ original schedule %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the fuzzer w/ original schedule 0.16 seconds to generate and execute 10000 inputs.' x_axis = np.arange(len(orig_schedule.path_frequency)) y_axis = list(orig_schedule.path_frequency.values()) plt.bar(x_axis, y_axis) plt.xticks(x_axis) plt.ylim(0, n) # plt.yscale("log") # plt.yticks([10,100,1000,10000]) plt; print(" path id 'p' : path frequency 'f(p)'") orig_schedule.path_frequency path id 'p' : path frequency 'f(p)' {'0a24dcc3bfb72fcf6ee703cc5b1d2b23': 6581, 'fca21b86927ec616ed8199b079eb3585': 2379, '98d92cfb37c8115e0d2b0d01c2a2e150': 737, '9ae43bde4b2b358677868370f5793850': 241, '895fda51f9cb1975a78a42e03cdf1f43': 62} The exponential power schedule shaves some of the executions of the "high-frequency path" off and adds them to the lower-frequency paths. The path executed least often is either not at all exercised using the traditional power schedule or it is exercised much less often. Let's have a look at the energy that is assigned to the discovered seeds. orig_energy = orig_schedule.normalizedEnergy(orig_fuzzer.population) for (seed, norm_energy) in zip(orig_fuzzer.population, orig_energy): print("'%s', %0.5f, %s" % (getPathID(seed.coverage), norm_energy, repr(seed.data))) '0a24dcc3bfb72fcf6ee703cc5b1d2b23', 0.20000, 'good' 'fca21b86927ec616ed8199b079eb3585', 0.20000, 'bgI/d' '98d92cfb37c8115e0d2b0d01c2a2e150', 0.20000, 'baI/dt' '9ae43bde4b2b358677868370f5793850', 0.20000, 'badtuS' '895fda51f9cb1975a78a42e03cdf1f43', 0.20000, 'bad!tuS' fast_energy = fast_schedule.normalizedEnergy(fast_fuzzer.population) for (seed, norm_energy) in zip(fast_fuzzer.population, fast_energy): print("'%s', %0.5f, %s" % (getPathID(seed.coverage), norm_energy, repr(seed.data))) '0a24dcc3bfb72fcf6ee703cc5b1d2b23', 0.00000, 'good' 'fca21b86927ec616ed8199b079eb3585', 0.00000, 'bnd' '98d92cfb37c8115e0d2b0d01c2a2e150', 0.00030, 'ba.' '9ae43bde4b2b358677868370f5793850', 0.02464, 'bad.' '895fda51f9cb1975a78a42e03cdf1f43', 0.97506, 'bad!\\.' Exactly. Our new exponential power schedule assigns most energy to the seed exercising the lowest-frequency path. Let's compare them in terms of coverage achieved over time for our simple example. _, orig_coverage = population_coverage(orig_fuzzer.inputs, crashme) _, fast_coverage = population_coverage(fast_fuzzer.inputs, crashme) line_orig, = plt.plot(orig_coverage, label="Original Greybox Fuzzer") line_fast, = plt.plot(fast_coverage, label="Boosted Greybox Fuzzer") plt.legend(handles=[line_orig, line_fast]) plt.title('Coverage over time') plt.xlabel('# of inputs') plt.ylabel('lines covered'); As expected, the boosted greybox fuzzer (with the exponential power schedule) achieves coverage much faster. Summary. By fuzzing seeds more often that exercise low-frequency paths, we can explore program paths in a much more efficient manner. Try it. You can try other exponents for the fast power schedule, or change the power schedule entirely. Note that a large exponent can lead to overflows and imprecisions in the floating point arithmetic producing unexpected results. You can execute your own code by opening this chapter as Jupyter notebook. Read. You can find out more about fuzzer boosting in the paper "Coverage-based Greybox Fuzzing as Markov Chain" [Böhme et al, 2018] and check out the implementation into AFL at [http://github.com/mboehme/aflfast]. ## A Complex Example: HTMLParser¶ Let's compare the three fuzzers on a more realistic example, the Python HTML parser. We run all three fuzzers$n=5k$times on the HTMLParser, starting with the "empty" seed. from html.parser import HTMLParser # create wrapper function def my_parser(inp: str) -> None: parser = HTMLParser() # resets the HTMLParser object for every fuzz input parser.feed(inp) n = 5000 seed_input = " " # empty seed blackbox_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule()) greybox_fuzzer = GreyboxFuzzer([seed_input], Mutator(), PowerSchedule()) boosted_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), AFLFastSchedule(5)) start = time.time() blackbox_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n) greybox_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n) boosted_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n) end = time.time() "It took all three fuzzers %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took all three fuzzers 7.27 seconds to generate and execute 5000 inputs.' How do the fuzzers compare in terms of coverage over time? _, black_coverage = population_coverage(blackbox_fuzzer.inputs, my_parser) _, grey_coverage = population_coverage(greybox_fuzzer.inputs, my_parser) _, boost_coverage = population_coverage(boosted_fuzzer.inputs, my_parser) line_black, = plt.plot(black_coverage, label="Blackbox Fuzzer") line_grey, = plt.plot(grey_coverage, label="Greybox Fuzzer") line_boost, = plt.plot(boost_coverage, label="Boosted Greybox Fuzzer") plt.legend(handles=[line_boost, line_grey, line_black]) plt.title('Coverage over time') plt.xlabel('# of inputs') plt.ylabel('lines covered'); Both greybox fuzzers clearly outperform the blackbox fuzzer. The reason is that the greybox fuzzer "discovers" interesting inputs along the way. Let's have a look at the last 10 inputs generated by the greybox versus blackbox fuzzer. blackbox_fuzzer.inputs[-10:] [' H', '', '', '', ' i', '', '(', 'j ', '', '0'] greybox_fuzzer.inputs[-10:] ['m*\x08', 'r.5<)h', '</F/aG>iq', '\x11G', '5<n5', 'i&<d$',
'/Wi<<4Gxs',
'4$<?B\x16g', '$G<!?Bg',
'\x06!|$v'] The greybox fuzzer executes much more complicated inputs, many of which include special characters such as opening and closing brackets and chevrons (i.e., <, >, [, ]). Yet, many important keywords, such as <html> are still missing. To inform the fuzzer about these important keywords, we will need grammars; in the section on smart greybox fuzzing, we combine them with the techniques above. Try it. You can re-run these experiments to understand the variance of fuzzing experiments. Sometimes, the fuzzer that we claim to be superior does not seem to outperform the inferior fuzzer. In order to do this, you just need to open this chapter as Jupyter notebook. ## Directed Greybox Fuzzing¶ Sometimes, you just want the fuzzer to reach some dangerous location in the source code. This could be a location where you expect a buffer overflow. Or you want to test a recent change in your code base. How do we direct the fuzzer towards these locations? In this chapter, we introduce directed greybox fuzzing as an optimization problem. ### Solving the Maze¶ To provide a meaningful example where you can easily change the code complexity and target location, we generate the maze source code from the maze provided as string. This example is loosely based on an old blog post on symbolic execution by Felipe Andres Manzano (Quick shout-out!). You simply specify the maze as a string. Like so. maze_string = """ +-+-----+ |X| | | | --+ | | | | | | +-- | | | |#| +-----+-+ """ The code is generated using the function generate_maze_code(). We'll hide the implementation and instead explain what it does. If you are interested in the coding, go here. from ControlFlow import generate_maze_code maze_code = generate_maze_code(maze_string) exec(maze_code) The objective is to get the "X" to the "#" by providing inputs D for down, U for up, L for left, and R for right. print(maze("DDDDRRRRUULLUURRRRDDDD")) # Appending one more 'D', you have reached the target. SOLVED +-+-----+ | | | | | --+ | | | | | | +-- | | | |X| +-----+-+ Each character in maze_string represents a tile. For each tile, a tile-function is generated. • If the current tile is "benign" (), the tile-function corresponding to the next input character (D, U, L, R) is called. Unexpected input characters are ignored. If no more input characters are left, it returns "VALID" and the current maze state. • If the current tile is a "trap" (+,|,-), it returns "INVALID" and the current maze state. • If the current tile is the "target" (#), it returns "SOLVED" and the current maze state. Try it. You can test other sequences of input characters, or even change the maze entirely. In order to execute your own code, you just need to open this chapter as Jupyter notebook. To get an idea of the generated code, lets look at the static call graph. A call graph shows the order in which functions can be executed. from ControlFlow import callgraph callgraph(maze_code) ### A First Attempt¶ We introduce a DictMutator class which mutates strings by inserting a keyword from a given dictionary: class DictMutator(Mutator): """Variant of Mutator inserting keywords from a dictionary""" def __init__(self, dictionary: Sequence[str]) -> None: """Constructor. dictionary - a list of strings that can be used as keywords """ super().__init__() self.dictionary = dictionary self.mutators.append(self.insert_from_dictionary) def insert_from_dictionary(self, s: str) -> str: """Returns s with a keyword from the dictionary inserted""" pos = random.randint(0, len(s)) random_keyword = random.choice(self.dictionary) return s[:pos] + random_keyword + s[pos:] To fuzz the maze, we extend the DictMutator class to append dictionary keywords to the end of the seed and to remove a character from the end of the seed. class MazeMutator(DictMutator): def __init__(self, dictionary: Sequence[str]) -> None: super().__init__(dictionary) self.mutators.append(self.delete_last_character) self.mutators.append(self.append_from_dictionary) def append_from_dictionary(self, s: str) -> str: """Returns s with a keyword from the dictionary appended""" random_keyword = random.choice(self.dictionary) return s + random_keyword def delete_last_character(self, s: str) -> str: """Returns s without the last character""" if len(s) > 0: return s[:-1] return s Let's try a standard greybox fuzzer with the classic power schedule and our extended maze mutator (n=20k). n = 20000 seed_input = " " # empty seed maze_mutator = MazeMutator(["L", "R", "U", "D"]) maze_schedule = PowerSchedule() maze_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, maze_schedule) start = time.time() maze_fuzzer.runs(FunctionCoverageRunner(maze), trials=n) end = time.time() "It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the fuzzer 7.34 seconds to generate and execute 20000 inputs.' We will need to print statistics for several fuzzers. Why don't we define a function for that? def print_stats(fuzzer: GreyboxFuzzer) -> None: total = len(fuzzer.population) solved = 0 invalid = 0 valid = 0 for seed in fuzzer.population: s = maze(str(seed.data)) if "INVALID" in s: invalid += 1 elif "VALID" in s: valid += 1 elif "SOLVED" in s: solved += 1 if solved == 1: print("First solution: %s" % repr(seed)) else: print("??") print("""Out of %d seeds, * %4d solved the maze, * %4d were valid but did not solve the maze, and * %4d were invalid""" % (total, solved, valid, invalid)) How well does our good, old greybox fuzzer do? print_stats(maze_fuzzer) Out of 1376 seeds, * 0 solved the maze, * 347 were valid but did not solve the maze, and * 1029 were invalid It probably didn't solve the maze a single time. How can we make the fuzzer aware how "far" a seed is from reaching the target? If we know that, we can just assign more energy to that seed. Try it. Print the statistics for the boosted fuzzer using the AFLFastSchedule and the CountingGreyboxFuzzer. It will likely perform much better than the unboosted greybox fuzzer: The lowest-probablity path happens to be also the path which reaches the target. You can execute your own code by opening this chapter as Jupyter notebook. ### Computing Function-Level Distance¶ Using the static call graph for the maze code and the target function, we can compute the distance of each function$f$to the target$t$as the length of the shortest path between$f$and$t$. Fortunately, the generated maze code includes a function called target_tile which returns the name of the target-function. target = target_tile() target 'tile_6_7' Now, we need to find the corresponding function in the call graph. The function get_callgraph returns the call graph for the maze code as networkx graph. Networkx provides some useful functions for graph analysis. import networkx as nx from ControlFlow import get_callgraph cg = get_callgraph(maze_code) for node in cg.nodes(): if target in node: target_node = node break target_node 'callgraphX__tile_6_7' We can now generate the function-level distance. The dictionary distance contains for each function the distance to the target-function. If there is no path to the target, we assign a maximum distance (0xFFFF). The function nx.shortest_path_length(CG, node, target_node) returns the length of the shortest path from function node to function target_node in the call graph CG. distance = {} for node in cg.nodes(): if "__" in node: name = node.split("__")[-1] else: name = node try: distance[name] = nx.shortest_path_length(cg, node, target_node) except: distance[name] = 0xFFFF These are the distance values for all tile-functions on the path to the target function. {k: distance[k] for k in list(distance) if distance[k] < 0xFFFF} {'callgraphX': 1, 'maze': 23, 'tile_3_1': 21, 'tile_6_1': 18, 'tile_2_1': 22, 'tile_4_7': 2, 'tile_6_2': 17, 'tile_6_3': 16, 'tile_6_4': 15, 'tile_3_3': 9, 'tile_6_5': 14, 'tile_5_1': 19, 'tile_6_7': 0, 'tile_3_7': 3, 'tile_2_7': 4, 'tile_4_1': 20, 'tile_2_6': 5, 'tile_2_5': 6, 'tile_5_5': 13, 'tile_4_3': 10, 'tile_5_7': 1, 'tile_4_4': 11, 'tile_2_4': 7, 'tile_2_3': 8, 'tile_4_5': 12} Summary. Using the static call graph and the target function$t$, we have shown how to compute the function-level distance of each function$f$to the target$t$. Try it. You can try and execute your own code by opening this chapter as Jupyter notebook. • How do we compute distance if there are multiple targets? (Hint: Geometric Mean). • Given the call graph (CG) and the control-flow graph (CFG$_f$) for each function$f$, how do we compute basic-block (BB)-level distance? (Hint: In CFG$_f$, measure the BB-level distance to calls of functions on the path to the target function. Remember that BB-level distance in functions with higher function-level distance is higher, too.) Read. If you are interested in other aspects of search, you can follow up by reading the chapter on Search-based Fuzzing. If you are interested, how to solve the problems above, you can have a look at our paper on "Directed Greybox Fuzzing". ### Directed Power Schedule¶ Now that we know how to compute the function-level distance, let's try to implement a power schedule that assigns more energy to seeds with a lower average distance to the target function. Notice that the distance values are all pre-computed. These values are injected into the program binary, just like the coverage instrumentation. In practice, this makes the computation of the average distance extremely efficient. If you really want to know. Given the function-level distance$d_f(s,t)$of a function$s$to a function$t$in call graph$CG$, our directed power schedule computes the seed distance$d(i,t)$for a seed$i$to function$t$as$d(i,t)=\dfrac{\sum_{s\in CG} d_f(s,t)}{|CG|}$where$|CG|$is the number of nodes in the call graph$CG$. class DirectedSchedule(PowerSchedule): """Assign high energy to seeds close to some target""" def __init__(self, distance: Dict[str, int], exponent: float) -> None: self.distance = distance self.exponent = exponent def __getFunctions__(self, coverage: Set[Location]) -> Set[str]: functions = set() for f, _ in set(coverage): functions.add(f) return functions def assignEnergy(self, population: Sequence[Seed]) -> None: """Assigns each seed energy inversely proportional to the average function-level distance to target.""" for seed in population: if seed.distance < 0: num_dist = 0 sum_dist = 0 for f in self.__getFunctions__(seed.coverage): if f in list(self.distance): sum_dist += self.distance[f] num_dist += 1 seed.distance = sum_dist / num_dist seed.energy = (1 / seed.distance) ** self.exponent Let's see how the directed schedule performs against the good, old greybox fuzzer. directed_schedule = DirectedSchedule(distance, 3) directed_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, directed_schedule) start = time.time() directed_fuzzer.runs(FunctionCoverageRunner(maze), trials=n) end = time.time() "It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n) 'It took the fuzzer 9.36 seconds to generate and execute 20000 inputs.' print_stats(directed_fuzzer) Out of 2491 seeds, * 0 solved the maze, * 957 were valid but did not solve the maze, and * 1534 were invalid It probably didn't solve a single maze either, but we have more valid solutions. So, there is definitely progress. Let's have a look at the distance values for each seed. y = [seed.distance for seed in directed_fuzzer.population] x = range(len(y)) plt.scatter(x, y) plt.ylim(0, max(y)) plt.xlabel("Seed ID") plt.ylabel("Distance"); Let's normalize the y-axis and improve the importance of the small distance seeds. ### Improved Directed Power Schedule¶ The improved directed schedule normalizes seed distance between the minimal and maximal distance. Again, if you really want to know. Given the seed distance$d(i,t)$of a seed$i$to a function$t$, our improved power schedule computes the new seed distance$d'(i,t)$as $$d'(i,t)=\begin{cases} 1 & \text{if } d(i,t) = \text{minD} = \text{maxD}\\ \text{maxD} - \text{minD} & \text{if } d(i,t) = \text{minD} \neq \text{maxD}\\ \frac{\text{maxD} - \text{minD}}{d(i,t)-\text{minD}} & \text{otherwise} \end{cases}$$ where $$\text{minD}=\min_{i\in T}[d(i,t)]$$ and $$\text{maxD}=\max_{i\in T}[d(i,t)]$$ where$T\$ is the set of seeds (i.e., the population).
class AFLGoSchedule(DirectedSchedule):
"""Assign high energy to seeds close to the target"""
def assignEnergy(self, population: Sequence[Seed]):
"""Assigns each seed energy inversely proportional
to the average function-level distance to target."""
min_dist: Union[int, float] = 0xFFFF
max_dist: Union[int, float] = 0
for seed in population:
if seed.distance < 0:
num_dist = 0
sum_dist = 0
for f in self.__getFunctions__(seed.coverage):
if f in list(self.distance):
sum_dist += self.distance[f]
num_dist += 1
seed.distance = sum_dist / num_dist
if seed.distance < min_dist:
min_dist = seed.distance
if seed.distance > max_dist:
max_dist = seed.distance
for seed in population:
if seed.distance == min_dist:
if min_dist == max_dist:
seed.energy = 1
else:
seed.energy = max_dist - min_dist
else:
seed.energy = (max_dist - min_dist) / (seed.distance - min_dist)
Let's see how the improved power schedule performs.
aflgo_schedule = AFLGoSchedule(distance, 3)
aflgo_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, aflgo_schedule)
start = time.time()
aflgo_fuzzer.runs(FunctionCoverageRunner(maze), trials=n)
end = time.time()
"It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
'It took the fuzzer 18.53 seconds to generate and execute 20000 inputs.'
print_stats(aflgo_fuzzer)
First solution: 5[gDDDOkDR~2=hRR-ERUUJLL[UURxRRRDDDDL
Out of 3804 seeds,
* 607 solved the maze,
* 344 were valid but did not solve the maze, and
* 2853 were invalid
In contrast to all previous power schedules, this one generates hundreds of solutions. It has generated many solutions.
Let's filter out all ignored input characters from the first solution. The function filter(f, seed.data) returns a list of elements e in seed.data where the function f applied on e returns True.
for seed in aflgo_fuzzer.population:
s = maze(str(seed.data))
if "SOLVED" in s:
filtered = "".join(list(filter(lambda c: c in "UDLR", seed.data)))
print(filtered)
break
DDDDRRRRUULLUURRRRDDDDL
This is definitely a solution for the maze specified at the beginning!
Summary. After pre-computing the function-level distance to the target, we can develop a power schedule that assigns more energy to a seed with a smaller average function-level distance to the target. By normalizing seed distance values between the minimum and maximum seed distance, we can further boost the directed power schedule.
Try it. Implement and evaluate a simpler directed power that uses the minimal (rather than average) function-level distance. What is the downside of using the minimal distance? In order to execute your code, you just need to open this chapter as Jupyter notebook.
Read. You can find out more about directed greybox fuzzing in the equally-named paper "Directed Greybox Fuzzing" [Böhme et al, 2017] and check out the implementation into AFL at http://github.com/aflgo/aflgo.
## Lessons Learned¶
• A greybox fuzzer generates thousands of inputs per second. Pre-processing and lightweight instrumentation
• allows to maintain the efficiency during the fuzzing campaign, and
• still provides enough information to control progress and slightly steer the fuzzer.
• The power schedule allows to steer/control the fuzzer. For instance,
• Our boosted greybox fuzzer spends more energy on seeds that exercise "unlikely" paths. The hope is that the generated inputs exercise even more unlikely paths. This in turn increases the number of paths explored per unit time.
• Our directed greybox fuzzer spends more energy on seeds that are "closer" to a target location. The hope is that the generated inputs get even closer to the target.
• The mutator defines the fuzzer's search space. Customizing the mutator for the given program allows to reduce the search space to only relevant inputs. In a couple of chapters, we'll learn about dictionary-based, and grammar-based mutators to increase the ratio of valid inputs generated.
## Next Steps¶
Our aim is still to sufficiently cover functionality, such that we can trigger as many bugs as possible. To this end, we focus on two classes of techniques:
1. Try to cover as much specified functionality as possible. Here, we would need a specification of the input format, distinguishing between individual input elements such as (in our case) numbers, operators, comments, and strings – and attempting to cover as many of these as possible. We will explore this as it comes to grammar-based testing, and especially in grammar-based mutations.
2. Try to cover as much implemented functionality as possible. The concept of a "population" that is systematically "evolved" through "mutations" will be explored in depth when discussing search-based testing. Furthermore, symbolic testing introduces how to systematically reach program locations by solving the conditions that lie on their paths.
These two techniques make up the gist of the book; and, of course, they can also be combined with each other. As usual, we provide runnable code for all. Enjoy!
We're done, so we clean up:
import os
if os.path.exists('callgraph.dot'):
os.remove('callgraph.dot')
if os.path.exists('callgraph.py'):
os.remove('callgraph.py')
## Compatibility¶
In previous version of the chapter, the AdvancedMutationFuzzer class was named simply MutationFuzzer, causing name confusion with the MutationFuzzer class in the introduction to mutation-based fuzzing. The following declaration introduces a backward-compatible alias.
class MutationFuzzer(AdvancedMutationFuzzer):
pass
`
## Exercises¶
To be added. \todo{}
The content of this project is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The source code that is part of the content, as well as the source code used to format and display that content is licensed under the MIT License. Last change: 2022-05-17 18:23:54+02:00CiteImprint
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2022-07-01 01:18:13
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https://support.bioconductor.org/p/105877/
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Question: processing agilent data by limma
0
11 months ago by
India
Agaz Hussain Wani260 wrote:
I am trying to perform differential expression of Agilent data GSE9210 by using limma. Following is the example code
target = file.path("/path/to/file")
x <- read.maimages(SDRF[,"File"], source="agilent", green.only=TRUE )
Some of the problems:
1) There are 58 sample in the study, and it processes only 50 samples from GSM232970-GSM233019.
On reaching sample GSM233020, it gives an error
Error in RG[[a]][, i] <- obj[, columns[[a]]] :
number of items to replace is not a multiple of replacement length
May be 1) is related to File format for single channel analysis of Agilent microarray data with Limma?
Is there any way to process samples from a single study in one batch.
2) I tried to process the files in batches, means GSM232970-GSM233019
in one batch and the remaining in another batch that worked fine, but it generates character type for second batch for which I am not able to run
y <- backgroundCorrect(x, method="normexp"), which generates Error in E - Eb : non-numeric argument to binary operator
which is an obvious reason. Why it generates character than numeric. Example is shown below.
an object of class "EListRaw"
$E GSM233020 GSM233021 [1,] "1452" "1373.5" [2,] "77" "109.5" [3,] "86" "131" [4,] "320" "898" [5,] "137.5" "236" 14904 more rows ...$Eb
GSM233020 GSM233021
[1,] "45" "50"
[2,] "44" "51"
[3,] "44" "51"
[4,] "44" "52"
[5,] "45" "52"
14904 more rows ...
$targets FileName GSM233020 GSM233020.txt GSM233021 GSM233021.txt$genes
Row Col ProbeUID ControlType ProbeName GeneName SystematicName
1 1 1 0 1 BrightCorner BrightCorner BrightCorner
2 1 2 1 -1 (-)3xSLv1 NegativeControl NegativeControl
3 1 3 2 0 A_23_P146576 NM_021996 NM_021996
4 1 4 3 0 A_23_P125016 A_23_P125016 A_23_P125016
5 1 5 4 0 A_23_P28555 STAMBP NM_006463
Description
1
2
3 Homo sapiens globoside alpha-1,3-N-acetylgalactosaminyltransferase 1 (GBGT1), mRNA
4 Unknown
5 Homo sapiens STAM binding protein (STAMBP), transcript variant 1, mRNA
14904 more rows ...
\$source
[1] "agilent"
agilent microarrays limma • 311 views
modified 11 months ago by Gordon Smyth36k • written 11 months ago by Agaz Hussain Wani260
Answer: processing agilent data by limma
1
11 months ago by
United States
James W. MacDonald49k wrote:
Providing stylized code that isn't really what you ran is not helpful. Unless you really have a path on your computer called /path/to/file, which would be sort of cool.
As for question #1, you get an error reading in a file. Did you try reading in just that one file? How about all files but that one? I find that I cannot read that one file in, but I can read in all the other files, which doesn't seem like a question for Bioconductor, but instead for the GEO curators.
1. I tried reading all the files at once, and it stoped at GSM233020.txt as mentioned above.
2. I can read the sample GSM233020.txt, by x <- read.maimages(SDRF[,"File"][51], source="agilent", green.only=TRUE ) but it generated the data as shown above. Even if I run from that sample saperately, it read the files x <- read.maimages(SDRF[,"File"][51:57], source="agilent", green.only=TRUE ).
1
No, you can't read GSM233020.txt. Sure, you get a result, but the result is wrong. As I told you in my answer, the end of that file is corrupted.
Thank you for providing the useful information.
Answer: processing agilent data by limma
1
11 months ago by
Gordon Smyth36k
Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia
Gordon Smyth36k wrote:
The file GSM233020.txt on GEO is corrupted, and therefore can't be read correctly by limma or any other program. As James has said, all the other files for that GEO series are fine
I suggest you write to the original authors of the series and ask for the correct raw data file. Alternatively, just omit that file and read all the others.
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2019-01-19 14:49:43
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https://holooly.com/solutions-v2-1/use-jacobi-diagonalization-to-find-the-eigenvalues-and-eigenvectors-of-the-3-dof-system-used-in-examples-7-1-and-7-2/
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## Q. 7.3
Use Jacobi diagonalization to find the eigenvalues and eigenvectors of the 3-DOF system used in Examples 7.1 and 7.2.
## Verified Solution
The equations of motion of the system can be written as:
$(- \omega^{2} [M] + [K])\left\{\overline{z} \right\} = 0$ (A)
or since λ = ω²
$(-\lambda [M] + [K])\left\{\overline{z}\right\} = 0$ (B)
where λ is an eigenvalue and $\left\{\overline{z}\right\}$ is an eigenvector. In numerical form, for this example:
$\left(-λ \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix} + 10^{3} \begin{bmatrix} 3 & -2 & 0 \\ -2 & 3 & -1 \\ 0 & -1 & 1 \end{bmatrix}\right) \begin{Bmatrix} \overline{z}_{1} \\ \overline{z}_{2} \\ \overline{z}_{3} \end{Bmatrix} = 0$ (C)
Since the mass matrix is diagonal in this case, decomposing it into $[M] = [U]^{T} [U]$ is very easy since:
$[U] = [M]^{\frac{1}{2}} = \begin{bmatrix} \sqrt{m_{11}} & 0 & 0 \\ 0 & \sqrt{m_{22}} & 0 \\ 0 & 0 & \sqrt{m_{33}} \end{bmatrix} = \begin{bmatrix} \sqrt{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \sqrt{2} \end{bmatrix}$ (D)
The inverse of $[U], [U]^{-1}$ is found by simply inverting the individual terms:
$[U]^{-1} = \begin{bmatrix} \frac{1}{\sqrt{m_{11}}} & 0 & 0 \\ 0 & \frac{1}{\sqrt{m_{22}}} & 0 \\ 0 & 0 & \frac{1}{\sqrt{m_{33}}} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} (E) \\ \mathrm{and} \\ [U]^{-T} = [U]^{-1} = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} (F)$
From Eq. (7.19), the eigenvalue problem to be solved is
$([\underline{B}] – \lambda [I]) \left\{\overline{y} \right\} = 0$ (G)
where from Eq. (7.20):
$[\underline{B}] = [U]^{-T} [K] [U]^{-1}$ (H)
and λ = ω² .
A more convenient form of Eq. (G), using the modal matrix, $[\Psi ]$, is
$[\underline{B}] [\Psi ] = [\Psi ] [\Lambda ] (I)\\ \mathrm{where} \\ [\Psi ] = \left[\left\{\overline{y}\right\}_{1} \left\{\overline{y} \right\}_{2} \left\{\overline{y} \right\}_{3} \right] (J)\\ \mathrm{and} \\ [\Lambda] = \begin{bmatrix} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{bmatrix} = \begin{bmatrix} \omega_{1}^{2} & 0 & 0 \\ 0 & \omega_{2}^{2} & 0 \\ 0 & 0 & \omega_{3}^{2} \end{bmatrix} (K)$
Evaluating $[\underline{B}]$ from Eq. (H)
$[\underline{B}] = 10^{3} \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} 3 & -2 & 0 \\ -2 & 3 & -1 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} = \begin{bmatrix} \frac{3}{2} & \frac{-2}{\sqrt{2}} & 0 \\ \frac{-2}{\sqrt{2}} & 3 & \frac{-1}{\sqrt{2}} \\ 0 & \frac{-1}{\sqrt{2}} & \frac{1}{2} \end{bmatrix} (L)\\ [\underline{B}] = 10^{3} \begin{bmatrix} 1.5 & -1.4142 & 0 \\ -1.4142 & 3 & -0.7071 \\ 0 & 0 & 0.5 \end{bmatrix} (M)$
The end result of the Jacobi diagonalization that we are aiming for is, from Eq. (7.27),
$[\Psi ]^{T} [\underline{B}] [\Psi ] = [\Lambda ].$ (N)
So if we can find a series of transformations that eventually change $[\underline{B}] \mathrm{into} [\Lambda]$, the product of these transformations must be the modal matrix, $[\Psi ]$. The first transformation, aimed at reducing the $b_{12} \mathrm{and} b_{21}$ terms to zero, is
$[B]^{(1)} = [T_{1}]^{T} [\underline{B}] [T_{1}] (O)\\ \mathrm{where} \\ [T_{1}] = \begin{bmatrix} \cos\theta_{1} & -\sin\theta_{1} & 0 \\ \sin\theta_{1} & \cos\theta_{1} & 0 \\ 0 & 0 & 1 \end{bmatrix} (P) \\ \mathrm{and} \\ \tan 2\theta_{1} = \frac{2b_{12}}{b_{11} – b_{22}} = \frac{2(-1.4142)}{1.5 – 3} = 1.8856$
giving $\theta _{1} = 0.5416 .$
Substituting for $\theta _{1},$
$[T_{1}] = \begin{bmatrix} 0.8569 & -0.5155 & 0 \\ 0.5155 & 0.8569 & 0 \\ 0 & 0 & 1 \end{bmatrix} (Q) \\ \mathrm{Then,} \\ [\underline{B}]^{(1)} = [T_{1}]^{T} [\underline{B}] [T_{1}] = 10^{3} \begin{bmatrix} 0.8569 & -0.5155 & 0 \\ 0.5155 & 0.8569_{1} & 0 \\ 0 & 0 & 1 \end{bmatrix}^{T} \begin{bmatrix} 1.5 & -1.4142 & 0 \\ -1.4142 & 3 & -0.7071 \\ 0 & 0 & 0.5 \end{bmatrix} \begin{bmatrix} 0.8569 & -0.5155 & 0 \\ 0.5155 & 0.8569 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ [\underline{B}]^{(1)} = 10^{3} \begin{bmatrix} 0.6492 & 0 & -0.3645 \\ 0 & 3.8507 & -0.6059 \\ -0.3645 & -0.6059 & 0.5000 \end{bmatrix} (R)$
This has reduced the $b^{(1)}_{12} \mathrm{and} b^{(1)}_{21}$ terms to zero, as expected. We next reduce the $b^{(1)}_{23} \mathrm{and} b^{(1)}_{32}$ terms to zero by the transformation:
$[\underline{B}]^{(2)} = [T_{2}]^{T} [\underline{B}]^{(1)} [T_{2}] (S)\\ \mathrm{where} \\ [T_{2}] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta_{2} & -\sin\theta_{2} \\ 0 & \sin\theta_{2} & \cos\theta_{2} \end{bmatrix} (T)\\ \mathrm{and} \\ \tan 2\theta_{2} = \frac{2b_{23}^{(1)}}{b_{22}^{(1)} – b_{33}^{(1)}} = \frac{2(-0.6059)}{3.8507 – 0.5000} = 0.36166 (U)$
giving
$\theta _{2} = -0.17351 \\ [T_{2}] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0.9849 & 0.1726 \\ 0 & -0.1726 & 0.9849 \end{bmatrix} (V)$
Applying Eq. (S): $[\underline{B}]^{(2)} = [T_{2}]^{T} [\underline{B}]^{(1)} [T_{2}]$ , gives
$[\underline{B}]^{(2)} = 10^{3} \begin{bmatrix} 0.6492 & 0.06291 & -0.3590 \\ 0.06291 & 3.9569 & 0 \\ -0.3590 & 0 & 0.3938 \end{bmatrix} (W)$
This has reduced the $b^{(1)}_{23} \mathrm{and} b^{(1)}_{32}$ terms to zero, but the previously zero $b^{(1)}_{12} \mathrm{and} b^{(1)}_{21}$ terms have changed to 0.06291.
Three more transformations were applied, in the same way, with results as follows:
Third transformation, $\theta _{3} = -0.6145 \\ [T_{3}] = \begin{bmatrix} \cos\theta_{3} & 0 & -\sin\theta_{3} \\ 0 & 1 & 0 \\ \sin\theta_{3} & 0 & \cos\theta_{3} \end{bmatrix} ; [\underline{B} ]^{(3)} = 10^{3} \begin{bmatrix} 0.9025 & 0.05140 & 0 \\ 0.05140 & 3.9569 & 0.03626 \\ 0 & 0.03626 & 0.1404 \end{bmatrix}$
Fourth transformation, $\theta _{4} = – 0.0168; \\ [T_{4}] = \begin{bmatrix} \cos\theta_{4} & -\sin\theta_{4} & 0 \\ \sin\theta_{4} & \cos\theta_{4} & 0 \\ \sin\theta_{3} & 0 & 1 \end{bmatrix} ; [\underline{B} ]^{(4)} = 10^{3} \begin{bmatrix} 0.9017 & 0 & -0.0006 \\ 0 & 3.9577 & 0.03626 \\ 0 & 0.03626 & 0.1404 \end{bmatrix}$
Fifth transformation, $\theta _{5} = 0.0095; \\ [T_{5}] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta_{5} & -\sin\theta_{5} \\ 0 & \sin\theta_{5} & \cos\theta_{5} \end{bmatrix} ; [\underline{B} ]^{(5)} = 10^{3} \begin{bmatrix} 0.9017 & 0 & -0.0006 \\ 0 & 3.9581 & 0 \\ -0.0006 & 0 & 0.1401 \end{bmatrix}$
This could be continued for one or two more transformations, to further reduce the off-diagonal terms, but it can already be seen that $[\underline{B} ]^{(5)}$ has practically converged to a diagonal matrix of the eigenvalues. From Example 7.1, representing the same system, these were, in ascending order:
$λ_{1} = 140.1, λ_{2} = 901.7, λ_{3} = 3958$
It will be seen that they appear in $[\underline{B} ]^{(5)}$ in the different order: $λ_{2}, λ_{3}, λ_{1}.$
The matrix of eigenvectors, $[Ψ]$, is now given by Eq. (7.29).
$[\Psi] = [T_{1}][T_{2}][T_{3}]…$ (7.29)
These are the eigenvectors of matrix $[\underline{B}]$, not those of the system, since it will be remembered that the latter were transformed by $[U]^{-1}$ in order to make $[\underline{B}]$ symmetric. Since there were five transformations, $[T_{1}] \mathrm{through} [T_{5}]$, in this case, the eigenvectors $[Ψ]$ are given by:
$[\Psi] = [T_{1}][T_{2}][T_{3}][T_{4}][T_{5}] (X)$
or numerically:
$[\Psi] = \begin{bmatrix} 0.8569 & -0.5155 & 0 \\ 0.5155 & 0.8569 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0.9849 & 0.1726 \\ 0 & -0.1726 & 0.9849 \end{bmatrix} \begin{bmatrix} 0.8170 & 0 & 0.5766 \\ 0 & 1 & 0 \\ -0.5766 & 0 & 0.8170 \end{bmatrix} \\ \times \begin{bmatrix} 0.9998 & 0.01682 & 0 \\ -0.01682 & 0.9998 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0.9999 & -0.0095 \\ 0 & 0.0095 & 0.9999 \end{bmatrix} \\ = \begin{bmatrix} 0.7598 & -0.4910 & 0.4260 \\ 0.3216 & 0.8534 & 0.4099 \\ -0.5649 & -0.1745 & 0.8064 \end{bmatrix}$
The final matrix product above, in the order in which it emerged from the diagonalization, is the modal matrix:
$[\Psi] = \left[\left\{\overline{y}\right\}_{2} \left\{\overline{y}\right\}_{3} \left\{\overline{y}\right\}_{1} \right]$
To find the eigenvectors, $\left\{\overline{z} \right\}$, of the physical system, we must use Eq. (7.14):
$\left\{\overline{z} \right\} = [U]^{-1} \left\{\overline{y}\right\}$ (7.14)
All the vectors can be transformed together by writing Eq. (7.14) as:
$[\Phi] = [U]^{-1} [\Psi] (Y)$
where $[Φ]$ is the modal matrix of eigenvectors $\left\{\overline{z} \right\} \mathrm{and} [Ψ]$ the modal matrix of eigenvectors $\left\{\overline{y}\right\}$. From Eq. (Y):
$[\Phi] = [U]^{-1} [\Psi] = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} 0.7598 & -0.4910 & 0.4260 \\ 0.3216 & 0.8534 & 0.4099 \\ -0.5649 & -0.1745 & 0.8064 \end{bmatrix} \\ = \begin{bmatrix} 0.5373 & -0.3472 & 0.3012 \\ 0.3216 & 0.8535 & 0.4099 \\ -0.3994 & -0.1234 & 0.5703 \end{bmatrix}$ (Z)
These are, finally, the eigenvectors of the system, $\left\{\overline{z} \right\}_{i}$, in the same order as the eigenvalues, i.e. $\left\{\overline{z} \right\}_{2}, \left\{\overline{z} \right\}_{3}, \left\{\overline{z} \right\}_{1}$. These would normally be rearranged into the order of ascending frequency.
As a check, we would expect the product $[\Phi ]^{T} [M][\Phi]$ to equal the unit matrix, $[I]$ , and this is the case, to within four significant figures, showing that the eigenvectors emerge in orthonormal form, without needing to be scaled.
$[\Phi ]^{T} [M][\Phi] = \begin{bmatrix} 1.000 & 0 & 0 \\ 0 & 1.000 & 0 \\ 0 & 0 & 1.000 \end{bmatrix}$
Another check is that the product $[\Phi ]^{T} [K][\Phi]$ should be a diagonal matrix of the eigenvalues, which is also the case to within four significant figures.
$[\Phi ]^{T} [K][\Phi] = \begin{bmatrix} 901.7 & 0 & 0 \\ 0 & 3958 & 0 \\ 0 & 0 & 140.1 \end{bmatrix}$
It should be noted that the presentation of the numerical results above, to about four significant figures, does not reflect the accuracy actually used in the computation.
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2023-03-22 09:37:48
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https://www.doubtnut.com/question-answer/if-fxex-a-xlt0x-3-xge0-is-differentiable-at-x0-then-a-equal-to-a-2-b-3-c-4-d-no-such-value-exist-645244218
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HomeEnglishClass 12MathsChapterJee Mains
If f(x)={[e^x+a, xlt0],[x-3, ...
# If f(x)={[e^x+a, xlt0],[x-3, xge0]}, is differentiable at x=0, then a equal to<br> (a)-2 (b) -3 (c)-4 (d) No such value exist
Updated On: 17-04-2022
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hello friends so here is a question it says that if ab is equal to e to the power x + T for exception zero and x minus 3 Forex trading equal to zero is differentiable at X equal to zero then a is equal to be had dysfunction and it is said that it is differentiable at X equal to zero and hence we have to find the value of a chat now in this question since it is given that f of X is differentiable at X equal to zero this implies that it must be continuous at X equal to zero right because if any function is differentiable at any point that it must be continuous at that point right so if this function is continuous at X equal to zero it basically means that initial that that is a left hand limit at X equal to zero should be equal to the erection letter
limit at X equal to zero right limit extending 20 from left hand side it is from the negative side effects should be equal to Limit extending 20 from right hand side it is from positive side effects right now if I write effects for 20 from negative site admin assistant 2012 use which are less than zero it so that means this will be a function IT power x + APK serial X is less than it power x + a limit X tends to zero negative side similarly here we will have a function for which access settings from access any two Hero from positive side it means for from the values which are greater than 0 9 so we will have this function because here access greater than equal to zero right here we will have x minus 3
Dallas ft x equal to zero in this ensures that doesn't become a to the power 0 + K equal to listen becomes 0 - 3 minut the 800 is actually equal to 1 write any number 111 sorry any number to the power 0 is always equal to become one plus 8 = 2 - 3 from here we will be equal to minus 3 minus b is equal to minus 4 rights of equity value of a is -4 and in a question we have to find out the value of x in the given options option CC is -4 thank you
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2022-05-16 05:03:02
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http://quant.stackexchange.com/tags/hullwhite/hot
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# Tag Info
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The Hull-White model can represents the risk free rate as a stochastic process, that is, in terms of expected return and volatility. The zero curve only gives you expected returns and you have to find a source to calibrate volatility, as FQuant told you. Common volatility sources used for this calibration are historical series of the zero curve or ...
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The one-factor Hull-White model is given by $$dr(t) = (\theta(t) - \alpha\; r(t))\,dt + \sigma\, dW(t)\,\!.$$ The zero curves are only sufficient for the calibration of the parameter $\theta(t)$, which is given in terms of them by $$\theta\mathrm{(t)=}\frac{\partial f(0,t)}{\partial T}+\alpha f(0,t)+\frac{\sigma^2}{2a}(1-e^{-2\alpha t}),$$ where ...
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Milstein Scheme This scheme is described in Glasserman (2003) and in Kloeden and Platen (1992) for general processes.Hence, for simplicity, we can assume that the Stochastic Process is driven by the SDE \begin{align} &dX_t=\Xi(t,X_t)dt+\Sigma(t,X_t)dW_t\\ \end{align} Milstein discretization is, \begin{align} dX_{t+\Delta ...
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I will refer to "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo and Fabio Mercurio. In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of $r_t$. The huge difference ...
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In fact you can calibrate $\theta(t)$ piecewise constant and $\alpha$ and $\sigma$ to bond prices only. You don't need the swaption prices in mM. If you let $\sigma(t)$ depend on $t$ (this is called the generalized Hull-White model) then you need information about the options market. For the model as you write it you don't necessarily need MC to calculate ...
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General knowledge: The reference for short rates models is: Interest Rate Models, by D. Brigo & F. Mercurio, Springer Worth the cost. You can find a summary of the propeties of the "dr" models p15 & p19: Interest Rate Models: Paradigm shifts in recent years, D. Brigo, Columbia University Seminar You will see the quote p19: "Pricing models need to ...
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The claim that interest rates don't follow long term trends is not consistent with observed data. The idea of mean reversion is that interest rates do not rise or fall without bound, but are limited by economic and political factors. But there is no indication that this oscillation of short rates should happen around a constant mean. Allowing the mean ...
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The first principle component of interest rates will not help you capture the term structure better at all. It will basically remove all term structure affects you are going to see. When we decompose the returns on interest rates you are going to get 3 PC's which explain 99.9% of the variance. PC1 - Level of the interest rates (~90% of variance) PC2 - ...
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Once the single-factor Hull-White model is calibrated, you can compute zero-coupon bond prices in closed form (i.e., without running simulations). See http://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Analysis_of_the_one-factor_model .
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This is a special case of the question of why $$\int_0^T f(t) dW_t$$ is normally distributed for a continuous function $f(t).$ This Ito integral can be approximated by a sum $$\sum_{i=0}^{N-1} f(i T/N) (W_{(i+1)T/N} - W_{i T/N}) .$$ The Brownian increments $(W_{(i+1)T/N} - W_{i T/N})$ are independent normally distributed random variables. The key point ...
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Here is a solution without using the PDE technique, which is preferred as we do not need to assume the affine form of a zero-coupon price from the start. we assume that, under the risk-neutral measure, \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is ...
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For simplicity, We assume that $\alpha$ is a positive constant. You need to show that, for any $t>0$, \begin{align*} \int_0^t e^{\alpha u} dW_u \end{align*} is normally distributed. Consider the process $\{X_t, t \geq 0\}$, where \begin{align*} X_t = \frac{1}{\sqrt{\frac{1}{t}\int_0^t e^{2\alpha u} du}}\int_0^t e^{\alpha u} dW_u, \end{align*} for ...
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Have you tried: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1514192 It should cover it, anyway Hull White fits the HJM framework so you should be able to calibrate it to swaptions or something if not the yield curve
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The Heath-Jarrow-Morton representations of short interest rate models (such as Hull-White) will give you an expression for the evolution of the entire forward curve, but it doesn't make the problem any easier. The closed form ZC formulae you mention above are probably your best bet.
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A way to go would be to linearly build indepedant interest rates to eliminate correlation effects. How do you do that ? You linearly build orthogonal interest rates from your starting ones. This is totaly equivalent to diagonalising correlation matrix, which is the principle of PCA. Using information criteria you can then choose to remove lowest ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2016-02-13 00:40:43
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https://worldbuilding.stackexchange.com/questions/52729/i-dont-want-to-live-on-this-planet-anymore/52752
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# I don't want to live on this planet anymore
I don't like the direction where mankind is going. We spend more resources to build tools which allow us to watch cat videos than on curing cancer. One day I decided to say enough, and leave this grim place.
The plan is to gather around 500 people who want to leave the planet with me, then help me to gather money, build a ship and leave. Now we don't have a goal, so our ship will be flying though space for centuries, until we find some nice suitable and friendly planet to settle down. Sounds simple, so what could possibly go wrong?
My spaceship needs to be big. It has to provide living space for my crew. I know, if you put too many people in too small a space, it creates conflict, and we don't want this in our ship. Another thing, the ship has to provide food and water for them, so we need additional space for growing plants, and probably some large storage areas for all spare parts. There are small chances we will be able to restock after we leave Earth.
Is my plan feasible? Do we have currently have tech to build that kind of ship? What will be estimated time/costs to build it? Where are the drawbacks of my plan?
• Comments are not for extended discussion; this conversation has been moved to chat. – Monica Cellio Aug 25 '16 at 0:46
• @MonicaCellio that was odd moving all comments, specially in the case, like this one, where they are not a chat – MolbOrg Aug 25 '16 at 19:26
• @MolbOrg The comments Monica moved had nothing to do with improving the question or asking for clarifications. They were incomplete non-answers, responses to those non-answers, poor frame challenges, recommended reading/watching for OP... chat/discussion, in other words. There's nothing in them that the question would benefit from incorporating, so they're trash. I, for one, thank Monica for doing the usually-thankless job of taking out the trash. We're not a forum. – nitsua60 Aug 26 '16 at 15:13
• SyFy already tried this for us... sort of. :) – brichins Aug 26 '16 at 17:00
• "We spend more resources to build tools which allow us to watch cat videos than on curing cancer." -- so you're going to spend the money to build a rocket instead. Pure. Comic. Genious. – riwalk Aug 28 '16 at 22:34
I remember a few years ago reading an article which stated that a manned trip to Mars (with return) for a couple of people could be as high as 1 trillion USD (1,000,000,000,000). This group you're supposing is several hundred times larger than the Mars trip, for several hundreds of thousand times longer trip, and they have to create a means of survival instead of supply. Even ignoring current technology and just looking at a terrible estimation of price, there's no way you're making your trip.
Let's say (again, as a terrible estimate) we'd be running a price tag of 60,000,000,000,000,000,000 USD. The GDP of planet Earth (which note, it uses the GDP to sustain itself, so we can't just yank it over to our project to fund it) is estimated at about 107 trillion USD. To clarify: Earth's GDP is not even 1% of 1% of what we're estimating for the cost.
Of course any price estimation is ludicrous since we simply don't have the technology yet to build such a massive spacecraft with such longevity and self sufficiency. Even if we had 60 quintillion dollars of 'value' (labor and supplies) we have nothing to throw it at to make this happen.
Science and engineering will have to progress significantly before we reach the proposed goal.
Sorry, Professor Farnsworth, but you'll be staying with us a bit longer.
• Seems that if he wanted to go on this trip, he'd have to gear the entire planets economy towards scientific progress and education, thus fixing the problems that made him want to leave in the first place. – UIDAlexD Aug 24 '16 at 14:09
• could be as high as 1 trillion USD Can you please add a source about this estimation? When USA went to the Moon 50 years ago, it didn't wreck the country. I don't think that going to Mars instead of the Moon would be so expensive. – A.L Aug 24 '16 at 14:55
• The difference being Atmosphere and Gravity. Thus Mars is ridiculously more expensive, because we're talking about coming BACK from Mars. If all you want to do is launch something into the void of space, it would actually cost significantly less than sending something to the moon. What WOULD cost more is the technology needed to sustain life infinitely on a ship. – Spacemonkey Aug 24 '16 at 15:00
• Maybe he could make enough money by selling advertisements that are shown next to cat videos. – null Aug 24 '16 at 15:26
• @A.L The difference between the moon and mars is the difference between walking a couple blocks and swimming across the pacific. The trip time goes from one week to 1-2 years, you're subject to far more radiation for a much longer time, and unlike the moon, mars has a troublesome atmosphere. Everything has to be built tougher and last longer. Fuel cells won't cut it for a long trip - You'll need a nuclear reactor or absolutely massive solar array. Unlike Apollo you'll have to recycle everything, and your ship has to operate for the entire mission without equipment failure. – UIDAlexD Aug 24 '16 at 16:04
You know, I'm kind of annoyed with these questions. Not because they are bad questions, but because people seem to forget what mankind is truly capable of when the chips are down.
First off, ship design. Yes, scale is important, but not as important as a dozen other factors. First, you need something that rotates, to help build up centrifugal force (simulating gravity). Then you need to factor in an insulating layer to protect the inhabitants from cosmic rays, space debris, and other undesirables.
So. Rotation of habitable section. Look at the Ark from Halo for a beautiful example. No, I'm not talking about scale, I'm talking design.
What you could take from this is that circles are your best bet. So the habitable zone of your ship should be designed around this concept. But it also has to be able to reliably fly through space, so how to do that? Well, look around and you'll find dozens of concepts. Just to offer a few:
What you want to keep in mind is two things. One, the ability to dock, take off from your 'mother ship'. So you'll need a section that can counter-rotate so it doesn't screw up the artificial gravity for everyone. I'd advise the second design for that purpose, and make sure to use some kind of logarithmic acceleration/deceleration to make the transition as smooth as is humanly possible. Keep in mind that this section has to be large enough to house several smaller ships for manned and unmanned space exploration. You'll be searching for a new home, after all, and you'd want to minimize risks.
## Ship hull requirements!
Something to keep in mind, is the current research done to this end. First thing to remember is space debris. There has been research done that highlights these risks and how to best combat them. Make sure there are thin layers of aluminum to slow down the debris and absorb the kinetic force. Keep these layers separated by some kind of shock-absorbent materials -- I believe the latest was a sponge like material, but it all depends on what you can figure out. Check out: this link and this one for Nasa's current thoughts on the matter for Mars missions (not necessarily pertaining to hull design, but both real-world tech, and more than enough extra information that might well help to further your research).
Well, there has been plenty of research towards this end as well. In Situ being the more popular, but there have been theories about ionized air being used as well. The problem is that this ejects matter that you won't be getting back, so you'll need to be especially careful with this. I believe the two links from the previous point would prove more useful in this (especially the anti-matter engine proposed), but that isn't yet in our reach to do consistently.
## Great, but how do we build the thing?!
Well, that's both simple and complicated. Simple, because it obviously needs to be assembled in space. Complicated because thinking inside the box makes this an expensive means.
Or does it?
SpaceX has been doing research into making it cheaper to do just that -- get materials into low earth orbit, for example. They've designed a rocket that can get you up there, and one that can be reused -- which drives down the costs considerably. Now, if you use the In Situ propellant, combined with this rocket? It would drive down your costs considerably.
Now, once you get your things into orbit, you need to assemble the thing. You can do this the old-fashion way, of sending people up there to do just that. But frankly, that is both costly (you have to pay them) and inefficient (training needed, how cumbersome current spacesuits are, etc) so you're better off with some kind of remote controlled robots to do the assembly for you.
First off, it simplifies things (no need to return them to earth every few months, you can have a team of operators controlling them from earth, and they don't ask for hazard pay). As well, if something happens to the robots, you can have other robots collect them and return them to earth with the next shipment to drive costs down even more. Your project won't get the negative publicity for costing human lives, and it means things can be put on a tighter schedule than it otherwise might.
## Ship's built, now what?
Well, with the 'hard part' done, you'll want to test the ship for habitability. So in the next shipment, you want plants for aided oxygen scrubbing, and animals to test if everything is safe for human occupation.
I'd advise taking a variety of both (start with potted plants, rabbits, rats, and chickens). Why? Because the more data you get at this point, the better. In addition to them being there, you want sensors that are continuously tracking air composition, air pressure, temperature levels, radiation levels, and all the fun stuff. If something happens at this point, you're still golden, because it would kill animals, not you and your fellow 'escapees'. Let these beings survive for a year in space. Test effects the environment has on them, and be sure to test your ship in a solar storm (kind of important if you want to survive on the ship for many generations).
## And last, but certainly not least: financing the whole thing!
So, we know from other answers that cost is a big thing. So how do we work around that? Well: research. You need to approach many governments/space agencies with a proposition of research. Do not tell them you're building the ship for escape, that's just silly. Say you're a group of scientists trying to prove the habitability of space for humans. Your 'project' is meant to stay in earth orbit, or at worst be a cheap means to travel between earth and Mars (whatever sells the idea better).
Do the scientist thing: produce papers for peer-review based on your findings, play the game for your investors (keep them happy and paying). Then, once you and your fellow 'escapees' feel confident the ship meets your standards, announce that a team will be moving onto the ship for human testing over the course of X amount of time. Once everyone that is meant to be on the ship is on the ship, and you have everything you feel you would need:
Set course for the second star on the right, and keep on 'till morning.
• This is by far the best optimistic answer in my opinion :) – Spacemonkey Aug 24 '16 at 15:11
• "Morning" has an interesting interpretation when traveling through space. – WBT Aug 24 '16 at 16:07
• @WBT It does. That's exactly why I use it here. In this sense, I mean 'until you get to where you are going'. – Fayth85 Aug 24 '16 at 16:54
• Question for the optimist: would propellants not be the driving factor in the ship's design? It seems like reaction mass (barring reactionless drives) would quickly become the biggest factor in reaching another star. It leads to some pretty cool designs - hollowed-out asteroids come to mind. Unfortunately it also means the ship must be designed with a specific range in mind (in delta-V, not in light-years), so it sorta requires our characters to pick a destination. Unless they're content floating among the stars, forever... might be kinda nice actually. – John Walthour Aug 25 '16 at 13:18
• @JohnWalthour Yes, and no. Yes, I agree that the ship should be designed around what its limitations (e.g. due to propellant limitations), but it doesn't mean this is the limiting factor for distance to travel. What does that mean? Well, think about it. With current knowhow, we can use the gravitational tug of Jupiter, Saturn, and Uranus to help satellites go faster, simply by timing the launch right. Is it inconceivable that we could therefore use this to aid in stellar escape velocity? And perhaps use carefully timed bursts and perhaps even other starts, to slow us down in time? – Fayth85 Aug 25 '16 at 13:30
You are only taking 500 people. That may sound like a lot, but it is not when you are considering breeding populations, genetic diversity, and so on. With only 500 people:
1. Everyone has to breed to keep the genetic diversity up. No-one get to choose not to have kids. Now you might argue that donated sperm or harvested eggs will do the job, so the folk who are uninterested in kids don't have to raise them. But that means some other people have to do it, and you can bet your bottom dollar they'll be calling you selfish and/or society insists that you do it.
2. Everyone has to start and stop breeding on demand. There is no room for a population explosion on a starship. Want 3 kids? Tough luck, your allocation is 2. Want a kid now? Tough luck, you'll have to wait until Old Mrs Miggins pops her clogs.
3. Everyone has to do an approved job. This starship needs to be kept running, generation after generation. You might want to be a ballet dancer or games designer or the pilot, but what the ship really needs right now is 4 sewage workers and 14 child minders.
So if you want to live in a dystopia, go right ahead. I'll stay here with the cat videos! :-)
• I agree with 2, but for 1: A few thousand sperm frozen up will deal with this. Genetic diversity goes down a bit, thaw a couple samples. – Joel Aug 24 '16 at 22:45
• Um, screw natural genetics. If you can afford the ship and all the other technology it requires, you can throw in a few % of a % of that cost for artificial dna synthesis with algorithmically generated genetic diversity. – R.. Aug 25 '16 at 3:46
• @DrBob Janet and John should do what they're told or they'll find themselves on the wrong side of an airlock. – darthzejdr Aug 25 '16 at 10:12
• Old Mrs Miggins can be made to pop her clogs: that hydroponic farm's not going to fertilize itself. – yatima2975 Aug 25 '16 at 13:01
• @darthzejdr: You're not going to blow valuable resources out the airlock. Janet and John will be "sent to the farm". – TMN Aug 25 '16 at 14:36
Costs:
I like to examine naval ships for estimates like this. They're the largest mobile structures on Earth, and require people to live together for months at a time. For reference, let's look at an Iowa-class battleship. It has a crew complement of around 2,000. You're very concerned about living space, so we can give everyone 4x the living space the US Navy does it's sailors. This is still going to be pretty cramped, honestly. While it's true you won't need weapons, the space taken up by weapons on the battleship will be taken up by advanced propulsion, life-support, and structural systems. So, to break it down:
• It takes NASA about \$10,000/pound to put things into space. • An Iowa-class battleship weighs about 50,000 tons • It would cost NASA$1 trillion to put an Iowa-class battleship into space. Probably more, since it would have to be done in many, many trips, and this doesn't take into account the cost of the spaceship carrying the battleship.
• For reference, this post estimates people have, in our entire history, sent only 11,575 tons into space.
This doesn't even take into account developing the technology for a spaceship this size, keeping that many people alive, constructing the ship (in space), etc. etc. etc..
Essentially, you're better off buying an atoll, living and preparing on the atoll for a few generations, and every century or so see where humans have gotten technologically, and then re-evaluating.
• Sorry, but this is just fake comparison. You can't compare ships, which utilize the fact how water works, thus weight is not really problem, to spaceships where pulling it into space cost big money, so weight is problem. – Colombo Aug 25 '16 at 0:24
• Given that no one's made a spaceship that size, made to carry that many people that far through space for that long, there's probably four people on the planet who could run some quasi-real numbers. That being said, it's going to have to be about battleship size, and even if you swapped out steel for titanium, we're looking at ~28,000 tons. And you'll make up a fair amount of the cost difference in titanium vs steel prices. And even if you didn't, we're still talking hundreds of billions of dollars. In other words, "Mr. Scott cannot give me exact figures, Admiral, so... I will make a guess." – Azuaron Aug 25 '16 at 1:23
• If you did need to put something that size in orbit, you would be scaling up to the point where you approach a lower multiple of the energy cost (about $2 per pound) rather than the current expensive means. You can't launch people with a rail gun, but you could launch deck plates. – Pete Kirkham Aug 26 '16 at 10:50 • Given that we send non-human cargo to space all the time, all the pieces of the ISS, satellites, etc., if the cost savings were really that extreme we'd be doing it. Anyway, the question is about current tech, and no one's using a mass driver for space launches, so it can't be considered "current". – Azuaron Aug 26 '16 at 11:07 • Lets not forget that the Iowa class battleship is a weapon of war, with thousands of pounds of guns, armor and ammunition that simply won't be required. – user1901982 Apr 27 '17 at 11:53 Short answer: Not currently feasible with our current technology, or any we're likely to get in the next generation or two with the current intensity of research. Long answer: Probably the biggest challenge (and there are many massive challenges here) is the creation of a working "closed ecological system" or biosphere. We have not successfully managed to create a contained, working multi-year biosphere here on earth, yet, and that should probably be the first step. There have been attempts. Biosphere 2 was one, which highlights many of the challenges that such efforts must face: Biosphere 2 suffered from CO2 levels that "fluctuated wildly," and most of the vertebrate species and all of the pollinating insects died. Insect pests, like cockroaches, boomed. [...] clogging filtration systems, unanticipated condensation making the "desert" too wet, population explosions of greenhouse ants and cockroaches, and morning glories overgrowing the "rainforest", blocking out other plants. [...] The oxygen inside the facility, which began at 20.9%, fell at a steady pace and after 16 months was down to 14.5%. [...] carbon dioxide was reacting with exposed concrete inside Biosphere 2 to form calcium carbonate, thereby sequestering both carbon and oxygen. [...] a severe dispute within the management team led to the ousting of the on-site management by federal marshals serving a restraining order, leaving management of the mission to the Bannon & Co. team from Beverly Hills, California. At 3 am on April 5, 1994, Abigail Alling and Mark Van Thillo, members of the first crew, allegedly vandalized the project from outside, opening one double-airlock door and three single door emergency exits, leaving them open for approximately fifteen minutes. Five panes of glass were also broken. [...] Mission 2 was ended prematurely. Others include MELiSSA and BIOS-1 through BIOS-3, which were more successful due to being on a smaller scale, and less "closed". • Regarding Biosphere 2: Where were the tv cameras, this sounds like a drama spectacle to exceed most reality tv! (they could have funded a Biosphere 3 & 4 with just the one season!!) – MER Aug 25 '16 at 2:20 • That's no surprise. 14.5% oxygen levels is literally 25% below minimum breathing content, which is around 20% oxygen. To simply stay alive at that point would be exhausting and sleep or unconsciousness would spell slow brain damage. I don't think anyone would volunteer to that kind of job without being lied to, and when the truth came out what do you expect people would do? – user1901982 Apr 27 '17 at 11:50 I won't even bother with a cost estimate because firstly you would need to focus on the how. We are of course talking about something that was never done before and a little beyond the limits of humankind right now. But then so was the first Moon landing. First off, if you want to have a living space that exists for centuries you'll need a recycling system for every resource necessary for human survival. Think Fallout Vaults in space + storage tanks. Yeah, it's gonna be really, really heavy. For me this is pure science fiction, as humans need the biological fauna in the air to maintain several core functions like digestion, and essentially will lose a small amount of this mass to energy anyway... requiring a massive initial store if they hope to replenish this energy loss for centuries. This system will take easily 50x the living space of a crew member - per crew member - if it's possible at all. If someone wants to edit this with something I don't know, they are welcome to, but even if we had systems like this that would work on the moon, we don't really have the materials to lift something that heavy into space. As for the 500 men, I believe the simplest method using current technology would be to send up around 30 cargo launches and drop pieces of the ship into orbit, then stitch the vessel together in high earth orbit. Doing this allows us to avoid the stress a 500 man vessel would put on the ship as it fights gravity on liftoff, but would make it unable to land elsewhere, so you'll need a landing probe. Then you need fuel. Antimatter would be the first call of the science fictionista, but realistically speaking you're mostly gonna be stuck on solar sails and ion drives. There are the new omnidirectional-drives that came out of NASA a few years ago, and a few other things, but nothing concrete. We could do it, it just would take the entire earth's efforts for several years. The L5 society was a group like what you describe, and spent a lot of time and resources trying to answer this question (see publication archive here). The path they planned to take through space was similar to Earth's, but sixty degrees behind (or ahead) in the orbit around our Sun (which in turn moves around the galaxy which itself moves through the universe). They've since merged with the National Space Society, with a stated vision of People living and working in thriving communities beyond the Earth, and the use of the vast resources of space for the dramatic betterment of humanity. Check out especially their section on Space Settlement and the links in the blue box at the top of the page, particularly Orbital Settlements if you want to keep traveling through space a reasonably constant distance from a significant source of energy. Also, if you haven't seen it already, watch WALL·E, which is premised on a scenario very similar to the one in this question. A ship so large would not reach escape velocity with our current technology. You would likely have to assemble the ship in orbit, a bit like how the ISS was constructed. This would be immensly expensive, considering that the Atlas V costs NASA$20,200 per kg sent into space. Construction of this ship would take several generations even with NASAs funding. After this, you need to transport the people, equipment, plants, food, water, anything else up into space too!
Also consider that travelling at the speed of the Voyager probes (17 km a second relative to the sun), it would take you nearly 10 years just to reach Pluto. The most earth-like planet that has been discovered is 1400 light years away, so it would take nearly 25 million years to travel there at the speed of the voyager probe.
Unless you were in suspended animation (current technology has a 100% death rate) , you would never live to see that planet. In fact, if you sent a newborn child and its parents into space on this ship - travelling at 17km per second - the newborn would have travelled 75 billion kilometers by its 95th birthday. Sounds a lot? That's almost 0.008 lightyears!
So the short answer is yes, it might be possible, but you would not live long enough to see it happen. In fact, if you built a generation ship in the hope your grandchildren's ancestors will make it there, i am willing to bet that new technology on earth would be invented before they reached the planet, and the human race would fly out to overtake the original ship.
• I'm not in rush, I know many generations will reach any planets suitable too live. Plan is to make ship which eventually reach it with living crew. – user902383 Aug 24 '16 at 14:23
• in that case, a generation ship is your best bet. – Aric Aug 24 '16 at 14:25
You have already planted the seeds of your failure by including any other person. You are packing along the problems you are trying to escape. Even if these people are like-minded, you cannot guarantee their children will conform to the rules of your new world, or that power structures evolving within your group will corrupt your citizens into kitten-obsessed fools. See: Rama II
Imagine the obsession with Old Earth Kittens when they don't have any cats of their own to play with. Someone is likely to smuggle a favorite pet aboard your vessel, which will then be cloned to meet the demand for adorable pets. When your crew has eventually killed each other off, leaving the kittens to eat the evergreen algae supply and evolve on the self-maintaining ship into intelligent beings, the new Homo Felinus will return to Old Earth, determined to satisfy their own obsession with the pictures of adorable human babies they've found aboard their world. See: Seveneves
Your question isn't "I don't want to live on this planet anymore", it's "I don't want to live with this species anymore". Answer: Sorry, but you're stuck "here" with us, no matter where you are in physical space.
Nobody has mentioned clanking replicators yet?
OP, I have the answer. All you need to do is build a robot that can build a copy of itself from the raw materials available to it in its environment, under its own power. Then you fly this robot out to a rocky moon somewhere, wait a few years as it disassembles half the moon into useful robots, and then wait a little bit longer as you program those robots to build the ship you need. You then tell the bots to fly that ship over to you, you get on, and off you go. Who cares about Earth's GDP when you can logarithmically turn parts of the material universe into production?
Oh, and one little thing - you're probably going to want to make sure the robots never, ever, under any circumstances mutate.
Pro-tip: this won't work. But what's stopping you from trying ;^)
• On the other hand, if the von Neumann probes were dark, it would explain a lot – Pete Kirkham Aug 26 '16 at 10:55
No, it's not feasible at all. There are a million reasons why it wouldn't work, but I'll just touch briefly on the major ones:
• There is no drive that will take that much mass that far. Even Voyager 1, after 38 years, is only 25 light hours away, and that was a 825 kg probe.
• There is no energy source that will power the drive and life support. Nuclear is too heavy, solar won't work once you leave solar system. I guess you could try wind...
• There is no system for maintaining a closed system with such a community, or keeping the ship maintained, that long. You will need inputs, which you can't get in outer space.
• You will not be able to grow enough nutrients to feed everyone a balanced diet.
• Space radiation and other hazards would kill everyone.
• The construction of such a spaceship is very difficult. If every nation of the world made it their #1 goal, just taking 500 people to the moon and back would be very costly and take decades. What you suggest is just inconceivable.
• 500 people is too few. The children will be inbred and social order will collapse within decades. There won't be enough specialists in medicine, engineering, and other key disciplines to maintain the ship. There's not enough resources to train people in highly specialized professions, so future generations will be even worse.
• If you somehow obtain the resources and start such a project privately, this will cause global economic and political instability, which will come back and destroy your project.
You don't like cat videos? Go start your own country and make them illegal. Better yet, just run for office in yours, or start an advocacy campaign, or... Well, anything else than building a generation ship.
Incidentally, what will be your ship's policy concerning asking questions on StackExchange?
• I agree on many of your points. But Voyager 1 travelled about 180 AE since launch. 1 AE is approx. 499 light seconds, i.e your mentioned 8.3 light minutes. That makes a travelling distance of approx. 25 light hours which is pretty impressive. I am also unsure if the weight of the object is that important once travelling in space. – AlexDeLarge Aug 25 '16 at 14:31
• I agree. I think the major drawback would be: Cold. Lots and lots of cold. Oh, and darkness. At the currently available speeds, the sheer amount of time away from a Sol disrupts any of our current means of power, food, or heat generation. Perhaps a nuclear fuel plant could provide all that was needed to sustain life, but for safety, you are going to need at least 1 layer of redundancy (for maintenance and repair) more than doubling the expense and time. – Jammin4CO Aug 25 '16 at 18:25
• @Jammin4CO Space is not cold, it's a near vacuum. Heating isn't the challenge, cooling is. – Superbest Aug 26 '16 at 0:36
• @Superbest Vacuum is an excellent insulator. But according to the attached article, when there is no Sol for radiation warmth, it is cold. science.nasa.gov/science-news/science-at-nasa/2001/ast21mar_1 I appreciate the challenge though, you made me think and learn. :-) – Jammin4CO Aug 26 '16 at 16:17
## Transform an Asteroid
I would suggest that your 'spaceship' actually be a medium sized asteroid. Build shelters, perhaps underground to help shield from radiation and debris, and attach propulsion systems at various points such that the ship is relatively maneuverable.
Choose an asteroid that has raw materials for use in propulsion, energy generation, repairs and most importantly has water. It would be good to have dirt so that mixed with human, animal and plant waste, would be suitable for farming (in the shelters). An energy source to keep the lights on would help with keeping plants growing and our biological clocks.
It would also have a bit of gravity without having to have tech to keep it spinning.
## Cost
Getting investment to mine asteroids should be easy since they are so rich in precious metals - just don't let them know your real plan. Everything needed to get the asteroid here and mine it could also be used for getting it somewhere else while living on it. Send robots to 'mine' and test the materials for economic forecasts. Creating living quarters for robot repair crews is reasonable. Everything needed for mining is also reasonable for 'escaping'.
## What's missing?
Mining tools for space and propulsion systems for that massive of an object.
## Time
10 to 20 years is a solid guess based on speculation and ignorance.
## Drawbacks
Longevity (will the asteroid run out of materials before your descendants reach their destinations) and general human interactions as others have already noted.
• The longevity issue may be nullified if the OP picks a home in the asteroid belt and extract materials from neighbor bodies, or - if he's feeling particularly daring - Saturn's rings. The gaps at the end should provide a fairly radiation-free environment. – OnoSendai Aug 29 '16 at 18:38
• Yes this is the best answer if you really wanted to do it. Simple and with the least effort. Another way to fund the venture would be those 500 people need to donate their assets and livelihoods to the cause. If they earn $70,000 / year for 20 years they would accumulate$700 million, not including assets they already hold- a long way towards the cause. – flox Dec 4 '18 at 22:31
Looking at the wiki entry for the ISS, it took the combined efforts of 26 countries, $150bn of finance 18 years to build. At any one time the station can support 6 permanent crewmembers. We support the crew from Earth by sending the ISS supplies. They're not exactly growing food in abundance but they do have some small plants: http://www.nasa.gov/mission_pages/station/research/10-074.html. Thus food and water would quickly run out without contant supply of resources from Earth. The station is also not traveling space (it's in orbit so it is moving) so propulsion would be pretty slow (going or stopping), and the zero gravity would wreak havoc on people's health over a long period. From what I can tell we don't quite have the technology for prolonged space flight, the costs of the project would be staggering (as Nex Terren pointed out), and it would likely take the a significant portion of the lifetime of your 500 people to build. However, 50years ago the same could've been said for the cost of an international space station. So it's not unlikely that prolonged space travel would be quite feasible in next decade or so. The ship itself is barely possible, if very unlikely, with current tech. You would need to be able to reach Mars at a minimum, more likely the asteroid belt. One of the moons of Mars, or a suitable asteroid would become the base of your ship. If enough life sustaining resources cannot be found locally while converting said chunk or rock to a ship, then they would have to be shipped up out of the gravity well. That would take both time, money, and more resources. If the recent research about electrically generated thrust is in fact accurate and can bee scaled up (yet to be proven) you would then also have the engines necessary for the journey. Sadly, we lack the knowledge and experience necessary to create sustainable, habitable biomes. Socially and genetically, we are not in a good position to build a generation ship at this time. All things taken into consideration, it does not seem currently possible without some sort of a McGuffin (or two) to cover our lacks. Centuries? With current tech it would take 30,000 years to get a probe to our nearest star, Proxima (as they say in the news today as there's a planet there - lucky!). However, as Proxima is a small star that spews out so much deadly radiation, you'd have to go to the next nearest, which I think is Tau Ceti, 4 times as far away and possibly still uninhabitable, even if it does turn out to have a planet in the right zone that has the right atmosphere, and no Ceti-dinosaurs (or whatever). So you're pretty much on to a loser right away, considering all of recorded human history is roughly 6,000 years old and you have to spend 5 times that just to get somewhere bad... you will either have to stay on board forever, or stay nearby in orbit. So a world-ship that can sustain a population has to have quite an extensive ecosystem (eg Earth itself is one), complete with enough materials to self-sustain for millenia. It doesn't have to be as big as the planet but it will have a maximum sustainable population, but I think you will underestimate how big it still needs to be to house only 500 people. If you have technology then you will also not only need to be able to maintain it, but also have to maintain a society capable of understanding how to maintain it - and note: a priesthood that knows the magic words to turn the big atmosphere machine on and off isn't going to cut it, you'll have to have proper engineering understanding that can also cope with changes in society that will inevitably occur over 5 times the length of human history! Or you could sit in orbit as a new independent "country" and trade with the planet below for whatever they need. This is far more practical until (or if!) someone develops a FTL drive. • To clarify for those who don't know FTL stands for Faster than light – Sarfaraaz Aug 26 '16 at 12:13 If you need a ship for 500 people, don't build it here and fight gravity to take it out; hollow out an asteroid and move in. Called Terrarium, this kind of space habitat should be fairly easy to implement: Capture an asteroid, bring it to Earth orbit, then proceed to drill until you reach a safe distance underneath - enough to protect an initial team from radiation. Instead of building structures on its outer surface, do it on the inside. If you keep the asteroid spinning, centripetal forces will generate something akin to artificial gravity on the outer layer. You may build external structures with the material taken from the inside: So your total cost to build a spaceship may be only the sum of transportation for the 500 individuals plus enough equipment (drills, 3D printers, solar panels, etc.) to get Step 1 started. Media sources: Wanderers, by Erik Wernquist; Spacehabs.com, Bryan Versteeg. • asteroid will fly apart because it's probably a pile of rocks and dust in most cases.It can be used as source of materials, but not in the way you describe. – MolbOrg Aug 28 '16 at 12:51 • Nice. I liked the short film too. The cool thing about this approach is that the materials gained from hollowing it out could be sold to cover the costs of construction. To MolbOrg's speculation, "there are other important differences in the internal structure of the asteroids. Most are solid, indicating that they must have been molten at some point in their existence. Others are 'rubble piles'. This means that they are loose collections of 'pieces', held together by the force of their gravity." - esa.int/Our_Activities/Space_Science/… – Tracy Cramer Aug 29 '16 at 18:13 • @TracyCramer Off topic, I know, and I apologize in advance: but as an admirer of Dr. Sagan and his work this short film is one of my favorites. – OnoSendai Aug 29 '16 at 18:34 I don't like the direction where mankind is going. We spend more resources to build tools which allow us to watch cat videos than on curing cancer. One day I decided to say enough, and leave this grim place. As someone else has already observed, you do not actually want to leave the planet, but rather the human race. So an easier way out would be to e.g. try and escape underground. Most technical problems related to living underground are the same as those related to living in a spaceship, except when they're much easier: • oxygen can be pumped from outside, either regularly or in case of an emergency. • no fuel or propulsion issues. • artificial gravity is naturally supplied. • no risks of collision. The problem of making good your escape is the same (but somewhat easier; hiding a space launch is hard. You would need to first build a L5 colony, then move it into a cometary orbit and handwave your economic model possibly as asteroid mining, and finally install more propulsors and leave for good. Yes, that's Star Trek's The Galactic Whirlpool by Gerrold). Of course there's still the problem of getting underground (technical and economical problem, but not the same scale as a generation ship), of staying hidden underground, and possibly to hide the fact to the people themselves, which requires careful geometry and may need mechanical gravity compensation to make the underworld go round (this is James P. Hogan's Endgame Enigma) If you play your cards carefully you may engineer a generation-long "space travel" that will end up with the ship "landing" on a depopulated world which could still be colonizable, even if the previous tenants left it a mess: Sol III (you might even engineer the depopulation "accident" yourself to get rid of the lolcats). This is more like Hugh Howey's Silo). While theoretically possible, we lack the technology to do it. And the finances you'd need to do this would probably be much higher than what you'd need to fix earth. You would need to build said spacecraft in space, since launching something that massive is currently impossible. That means probably thousands of return trips between earth and your space-dock(That you would also need to build). It would probably be cheaper to engineer a virus that would kill most humans on the planet, and then just rebuild earth as you see fit :). • So you're saying this 1979 film had the answer all along? – da4 Aug 24 '16 at 13:52 • I was planning to fund it by using crowd funding platform. I don't think i will get much support by asking people to fund research on deadly virus which kill them all. – user902383 Aug 24 '16 at 14:00 What makes you think that the earth is not travelling in space right now? It just happens that you were born on a pretty big spaceship (earth) with a little over a few billion people on it. Check how the question come up... Where is it come from? Is it because you are bigger than the earth? Is it because you have access to more planets? How? Who are 'You' - the one who got the question? Keep the question. Don't answer. Seek the source of the question. I would say step one would be to build an orbital elevator. using fuel to get up and down from orbit would be a huge waste of resources. next step would be to find how to use hydrogen to power devices and propulsion since Jupiter is way closer than inhabitable star system planets. so harvest hydrogen from Jupiter maybe using a similar system used to build the orbital elevator to build an orbital vacuum pipe because Jupiter's gravity is very strong due to it's massive size. sustainable life needs such as food and water would be your next step to solve. I think other answers here would be better at answering that problem Then I would say you'd have the building blocks u need. heck, you could maybe blow Jupiter up to get up to speed but that would require so many calculations to make sure you go in the direction of choice (although the force from speeding up will probably turn everyone into mush) So to use that method you would need some kind of impact dampening. like a long tube facing away from the explosion that would allow some of the force through and a hydraulic system to control how fast the cockpit/living area would move relative to the outer shell. We do have the ability to construct a massive ship, but we don't have the technology to launch one. Your best bet is to construct the ship in orbit around the moon. That saves you the large amount of fuel you would need to get it off the earth. The diameter of most payloads on existing rockets is limited to just a few meters. Any ship you could construct in orbit using conventional rockets would have to be made of hundreds of small modules, either put together like a honey comb (which would actually be pretty rugged in case of hull breaches), or a series of segments like the ISS. Also, if you want everyone to survive the radiation, the hull would need to be metal that is several feet thick. Also having a hull that thick would be necessary to survive thousands of years of colliding with space dust and micro-meteors. That would be way to heavy to launch from the ground. Secondly shipping raw materials from the earth even to low earth orbit is about$2000 a pound at todays commercial prices. The moon is actually pretty rich in Titanium, which you could use to build the hull. It could be much cheaper to set-up a mining and casting operation on the moon that could fabricate the largest metal pieces of the ships hull.
You could use a space-elevator going from the moon's surface to lunar orbit to get the large metal parts from the surface of the moon into orbit around the moon where you are building the ship. That would be significantly cheaper and easier than rockets. A space elevator from the earth is not possible with todays technology because carbon-nano-tube ropes with sufficient fiber length are not yet mass-producible. But a space elevator from the moon is possible with conventional steel cable due to the lower gravity. Companies like LiftPort (www.LiftPort.com) have already developed every part of the space elevator technology and are only awaiting more money, and a customer.
Your ship will need to be completely self-sufficient once its flying, so you will need machine shops on board that can make or repair every part of the ship, and fabricate every type of micro-chip or circuit card.
As it happens, you can save a lot of money by fabricating the machine shops on the moon as a module, and then using them to initially build the rest of the ship. At some point in the construction, the fabrication facilities are moved from the lunar surface to become part of the ship itself. That means that you get most of the cost of the moon base for free since it was really just part of the ship.
Also, very importantly, by using the ships own machine shops to initially build the ship you prove that you can really repair and replace any part of the ship using the tools and processes available in those machine shops. It eliminates the possibility of forgetting something until after launch.
Over time things on the ship will break. Those machine shop modules will need to make or repair every part of the ship, and fabricate every type of micro-chip or circuit card. Also you should have multiples (at least 3) of every tool or system, so that if one breaks there is a backup. Also, its better if each backup system is not identical so they don't all fail at exactly the same time due to the same disaster.
You will need a nuclear reactor to power the ship. A solar array won't generate much power as you move away from the sun.
The trip time could be just a few years or decades if the planet is in our solar system, or perhaps one of Jupiter's large moons. The trip time will most likely be 1000s of years not 100s if the planet is outside our solar system. The ship design would be vastly different depending on which scenario you choose.
If you just need to travel a few years/decades you could get away with just recycling water and air, with a huge storehouse of dried/canned food. If you have to leave the solar system then you will need to recycle everything.
We do have recycling technology. The toilet on the international space station for example recycles all the urine back into water for the astronauts. But of course plants will do that also if you like.
I already mentioned some price cutting measures, but its worth noting that one can't really estimate the cost of this design by comparing it to historical rocket prices which are based on the way the US government builds things by bidding them out to way over-priced defense contractors. The reason Space-X was founded was because Elon Musk realized that the price that Boeing/Lockheed were charging the U.S. gov for rockets was about 100x the cost of the raw materials.
Secondly, building a much larger ship would probably get a further price break due to economies of scale.
Thirdly, the 500 people who are going on the ship need to know how to build it anyways in case something breaks. So they might as well build it themselves the first time (so you save a lot on labor costs).
A good portion of the people you bring will have to be PHD level scientists and engineers who are capable of doing the ship design. Elon Musk, founder of Space-X wants to make mankind a multi-planet species. So someone like him might be willing to pay for the construction of the moon base, in exchange for your group doing the design and construction work for free and then giving him the rights to sell access to the base, or license the plans to others for profit. It really would be a great deal for him because most of the cost of developing anything is the labor cost to all the scientists and engineers, and techs who build it.
In fact every part of the ship design could be license or sold for profit to help finance further construction. There are plenty of private billionaires and governments who might be interested.
You may be able to lease access to the moon base to venture capitalists interested in mining precious metals from the moon.
Private billionaires are often motivated, adventurous people, and maybe some of them will want to go too. They may see great opportunity in being in on the ground floor of the founding of a new planet. So if you could convince a few dozen of them to go along that could provides some funding too.
Step 0: Fund a space elevator.
Step 1: Build the space elevator.
Step 2: Make money with your space elevator.
Step 3: Build your ship in geostationary orbit using your space elevator.
The biggest cost factor for any space mission is getting it into orbit. And there is just no way around wasting tons and tons of fuel for each ton you deliver to low earth orbit. And that's only for low earth orbit. With a space elevator, you just get so much more efficient.
But it's not just that you get efficient, you'll open a new, very cost efficient route to space. People will love it. And they'll throw their money at you, just to get a glimpse at the earth from geostationary orbit. Or to deliver payloads into other orbits cheaply and safely. Or launch missions to other planets from the far end of your elevator.
With this kind of steady inflow of money, you can start building stuff that makes life in space easier. Start with some serious stations along your elevator, especially a huge one in geostationary orbit. The beauty is, that you can build step by step, with each step slightly enhancing the lifes of your guests, or expanding your resources to host more guests, host them for longer periods, feed them with space grown food, etc.
Only when your ship is ready to leave will the rest of humanity realize that they'll be left behind...
## protected by Monica Cellio♦Aug 25 '16 at 1:03
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
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2019-06-24 11:51:00
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https://se.mathworks.com/help/deeplearning/ug/define-custom-deep-learning-layers.html
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## Define Custom Deep Learning Layers
Tip
This topic explains how to define custom deep learning layers for your problems. For a list of built-in layers in Deep Learning Toolbox™, see List of Deep Learning Layers.
This topic explains the architecture of deep learning layers and how to define custom layers to use for your problems.
TypeDescription
Layer
Define a custom deep learning layer and specify optional learnable parameters.
For an example showing how to define a custom layer with learnable parameters, see Define Custom Deep Learning Layer with Learnable Parameters. For an example showing how to define a custom layer with multiple inputs, see Define Custom Deep Learning Layer with Multiple Inputs.
Classification Output Layer
Define a custom classification output layer and specify a loss function.
For an example showing how to define a custom classification output layer and specify a loss function, see Define Custom Classification Output Layer.
Regression Output Layer
Define a custom regression output layer and specify a loss function.
For an example showing how to define a custom regression output layer and specify a loss function, see Define Custom Regression Output Layer.
### Layer Templates
You can use the following templates to define new layers.
### Intermediate Layer Architecture
During training, the software iteratively performs forward and backward passes through the network.
When making a forward pass through the network, each layer takes the outputs of the previous layers, applies a function, and then outputs (forward propagates) the results to the next layers.
Layers can have multiple inputs or outputs. For example, a layer can take X1, …, Xn from multiple previous layers and forward propagate the outputs Z1, …, Zm to the next layers.
At the end of a forward pass of the network, the output layer calculates the loss L between the predictions Y and the true targets T.
During the backward pass of a network, each layer takes the derivatives of the loss with respect to the outputs of the layer, computes the derivatives of the loss L with respect to the inputs, and then backward propagates the results. If the layer has learnable parameters, then the layer also computes the derivatives of the layer weights (learnable parameters). The layer uses the derivatives of the weights to update the learnable parameters.
The following figure describes the flow of data through a deep neural network and highlights the data flow through a layer with a single input X, a single output Z, and a learnable parameter W.
#### Intermediate Layer Properties
Declare the layer properties in the properties section of the class definition.
By default, custom intermediate layers have these properties.
PropertyDescription
NameLayer name, specified as a character vector or a string scalar. To include a layer in a layer graph, you must specify a nonempty, unique layer name. If you train a series network with the layer and Name is set to '', then the software automatically assigns a name to the layer at training time.
Description
One-line description of the layer, specified as a character vector or a string scalar. This description appears when the layer is displayed in a Layer array. If you do not specify a layer description, then the software displays the layer class name.
TypeType of the layer, specified as a character vector or a string scalar. The value of Type appears when the layer is displayed in a Layer array. If you do not specify a layer type, then the software displays the layer class name.
NumInputsNumber of inputs of the layer, specified as a positive integer. If you do not specify this value, then the software automatically sets NumInputs to the number of names in InputNames. The default value is 1.
InputNamesInput names of the layer, specified as a cell array of character vectors. If you do not specify this value and NumInputs is greater than 1, then the software automatically sets InputNames to {'in1',...,'inN'}, where N is equal to NumInputs. The default value is {'in'}.
NumOutputsNumber of outputs of the layer, specified as a positive integer. If you do not specify this value, then the software automatically sets NumOutputs to the number of names in OutputNames. The default value is 1.
OutputNamesOutput names of the layer, specified as a cell array of character vectors. If you do not specify this value and NumOutputs is greater than 1, then the software automatically sets OutputNames to {'out1',...,'outM'}, where M is equal to NumOutputs. The default value is {'out'}.
If the layer has no other properties, then you can omit the properties section.
Tip
If you are creating a layer with multiple inputs, then you must set either the NumInputs or InputNames properties in the layer constructor. If you are creating a layer with multiple outputs, then you must set either the NumOutputs or OutputNames properties in the layer constructor. For an example, see Define Custom Deep Learning Layer with Multiple Inputs.
#### Learnable Parameters
Declare the layer learnable parameters in the properties (Learnable) section of the class definition. You can specify numeric arrays or dlnetwork objects as learnable parameters. If the layer has no learnable parameters, then you can omit the properties (Learnable) section.
Optionally, you can specify the learning rate factor and the L2 factor of the learnable parameters. By default, each learnable parameter has its learning rate factor and L2 factor set to 1.
For both built-in and custom layers, you can set and get the learn rate factors and L2 regularization factors using the following functions.
FunctionDescription
setLearnRateFactorSet the learn rate factor of a learnable parameter.
setL2FactorSet the L2 regularization factor of a learnable parameter.
getLearnRateFactorGet the learn rate factor of a learnable parameter.
getL2FactorGet the L2 regularization factor of a learnable parameter.
To specify the learning rate factor and the L2 factor of a learnable parameter, use the syntaxes layer = setLearnRateFactor(layer,'MyParameterName',value) and layer = setL2Factor(layer,parameterName,value), respectively.
To get the value of the learning rate factor and the L2 factor of a learnable parameter, use the syntaxes getLearnRateFactor(layer,'MyParameterName') and getL2Factor(layer,parameterName) respectively.
For example, this syntax sets the learn rate factor of the learnable parameter with the name 'Alpha' to 0.1.
layer = setLearnRateFactor(layer,'Alpha',0.1);
#### Forward Functions
Some layers behave differently during training and during prediction. For example, a dropout layer performs dropout only during training and has no effect during prediction. A layer uses one of two functions to perform a forward pass: predict or forward. If the forward pass is at prediction time, then the layer uses the predict function. If the forward pass is at training time, then the layer uses the forward function. If you do not require two different functions for prediction time and training time, then you can omit the forward function. In this case, the layer uses predict at training time.
If you define the function forward and custom backward function, then you must assign a value to the argument memory, which you can use during backward propagation.
The syntax for predict is [Z1,…,Zm] = predict(layer,X1,…,Xn), where X1,…,Xn are the n layer inputs and Z1,…,Zm are the m layer outputs. The values n and m must correspond to the NumInputs and NumOutputs properties of the layer.
Tip
If the number of inputs to predict can vary, then use varargin instead of X1,…,Xn. In this case, varargin is a cell array of the inputs, where varargin{i} corresponds to Xi. If the number of outputs can vary, then use varargout instead of Z1,…,Zm. In this case, varargout is a cell array of the outputs, where varargout{j} corresponds to Zj.
Tip
If the custom layer has a dlnetwork object for a learnable parameter, then in the predict function of the custom layer, use the predict function for the dlnetwork. Using the dlnetwork object predict function ensures that the software uses the correct layer operations for prediction.
The syntax for forward is [Z1,…,Zm,memory] = forward(layer,X1,…,Xn), where X1,…,Xn are the n layer inputs, Z1,…,Zm are the m layer outputs, and memory is the memory of the layer.
Tip
If the number of inputs to forward can vary, then use varargin instead of X1,…,Xn. In this case, varargin is a cell array of the inputs, where varargin{i} corresponds to Xi. If the number of outputs can vary, then use varargout instead of Z1,…,Zm. In this case, varargout is a cell array of the outputs, where varargout{j} corresponds to Zj for j = 1,…,NumOutputs and varargout{NumOutputs + 1} corresponds to memory.
Tip
If the custom layer has a dlnetwork object for a learnable parameter, then in the forward function of the custom layer, use the forward function of the dlnetwork object. Using the dlnetwork object forward function ensures that the software uses the correct layer operations for training.
The dimensions of the inputs depend on the type of data and the output of the connected layers:
Layer InputInput SizeObservation Dimension
2-D imagesh-by-w-by-c-by-N, where h, w, and c correspond to the height, width, and number of channels of the images, respectively, and N is the number of observations.4
3-D imagesh-by-w-by-d-by-c-by-N, where h, w, d, and c correspond to the height, width, depth, and number of channels of the 3-D images, respectively, and N is the number of observations.5
Vector sequencesc-by-N-by-S, where c is the number of features of the sequences, N is the number of observations, and S is the sequence length.2
2-D image sequencesh-by-w-by-c-by-N-by-S, where h, w, and c correspond to the height, width, and number of channels of the images, respectively, N is the number of observations, and S is the sequence length.4
3-D image sequencesh-by-w-by-d-by-c-by-N-by-S, where h, w, d, and c correspond to the height, width, depth, and number of channels of the 3-D images, respectively, N is the number of observations, and S is the sequence length.5
For layers that output sequences, the layers can output sequences of any length or output data with no time dimension. Note that when training a network that outputs sequences using the trainNetwork function, the lengths of the input and output sequences must match.
#### Backward Function
The layer backward function computes the derivatives of the loss with respect to the input data and then outputs (backward propagates) results to the previous layer. If the layer has learnable parameters (for example, layer weights), then backward also computes the derivatives of the learnable parameters. When using the trainNetwork function, the layer automatically updates the learnable parameters using these derivatives during the backward pass.
Defining the backward function is optional. If you do not specify a backward function, and the layer forward functions support dlarray objects, then the software automatically determines the backward function using automatic differentiation. For a list of functions that support dlarray objects, see List of Functions with dlarray Support. Define a custom backward function when you want to:
• Use a specific algorithm to compute the derivatives.
• Use operations in the forward functions that do not support dlarray objects.
Custom layers with learnable dlnetwork objects do not support custom backward functions.
To define a custom backward function, create a function named backward.
The syntax for backward is [dLdX1,…,dLdXn,dLdW1,…,dLdWk] = backward(layer,X1,…,Xn,Z1,…,Zm,dLdZ1,…,dLdZm,memory), where:
• X1,…,Xn are the n layer inputs
• Z1,…,Zm are the m outputs of the layer forward functions
• dLdZ1,…,dLdZm are the gradients backward propagated from the next layer
• memory is the memory output of forward if forward is defined, otherwise, memory is [].
For the outputs, dLdX1,…,dLdXn are the derivatives of the loss with respect to the layer inputs and dLdW1,…,dLdWk are the derivatives of the loss with respect to the k learnable parameters. To reduce memory usage by preventing unused variables being saved between the forward and backward pass, replace the corresponding input arguments with ~.
Tip
If the number of inputs to backward can vary, then use varargin instead of the input arguments after layer. In this case, varargin is a cell array of the inputs, where varargin{i} corresponds to Xi for i=1,…,NumInputs, varargin{NumInputs+j} and varargin{NumInputs+NumOutputs+j} correspond to Zj and dLdZj, respectively, for j=1,…,NumOutputs, and varargin{end} corresponds to memory.
If the number of outputs can vary, then use varargout instead of the output arguments. In this case, varargout is a cell array of the outputs, where varargout{i} corresponds to dLdXi for i=1,…,NumInputs and varargout{NumInputs+t} corresponds to dLdWt for t=1,…,k, where k is the number of learnable parameters.
The values of X1,…,Xn and Z1,…,Zm are the same as in the forward functions. The dimensions of dLdZ1,…,dLdZm are the same as the dimensions of Z1,…,Zm, respectively.
The dimensions and data type of dLdX1,…,dLdxn are the same as the dimensions and data type of X1,…,Xn, respectively. The dimensions and data types of dLdW1,…,dLdWk are the same as the dimensions and data types of W1,…,Wk, respectively.
To calculate the derivatives of the loss, you can use the chain rule:
$\frac{\partial L}{\partial {X}^{\left(i\right)}}=\sum _{j}^{}\frac{\partial L}{\partial {z}_{j}}\frac{\partial {z}_{j}}{\partial {X}^{\left(i\right)}}$
$\frac{\partial L}{\partial {W}_{i}}=\sum _{j}\frac{\partial L}{\partial {Z}_{j}}\frac{\partial {Z}_{j}}{\partial {W}_{i}}$
When using the trainNetwork function, the layer automatically updates the learnable parameters using the derivatives dLdW1,…,dLdWk during the backward pass.
For an example showing how to define a custom backward function, see Specify Custom Layer Backward Function.
#### GPU Compatibility
If the layer forward functions fully support dlarray objects, then the layer is GPU compatible. Otherwise, to be GPU compatible, the layer functions must support inputs and return outputs of type gpuArray (Parallel Computing Toolbox).
Many MATLAB® built-in functions support gpuArray (Parallel Computing Toolbox) and dlarray input arguments. For a list of functions that support dlarray objects, see List of Functions with dlarray Support. For a list of functions that execute on a GPU, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox). To use a GPU for deep learning, you must also have a supported GPU device. For information on supported devices, see GPU Support by Release (Parallel Computing Toolbox). For more information on working with GPUs in MATLAB, see GPU Computing in MATLAB (Parallel Computing Toolbox).
#### Code Generation Compatibility
To create a custom layer that supports code generation:
• The layer must specify the pragma %#codegen in the layer definition.
• The inputs of predict must be:
• Consistent in dimension. Each input must have the same number of dimensions.
• Consistent in batch size. Each input must have the same batch size.
• The outputs of predict must be consistent in dimension and batch size with the layer inputs.
• Nonscalar properties must have type single, double, or character array.
• Scalar properties must have type numeric, logical, or string.
Code generation supports intermediate layers with 2-D image input only.
For an example showing how to create a custom layer that supports code generation, see Define Custom Deep Learning Layer for Code Generation.
#### Network Composition
To create a custom layer that itself defines a layer graph, you can specify a dlnetwork object as a learnable parameter. This method is known as network composition. You can use network composition to:
• Create a single custom layer that represents a block of learnable layers, for example, a residual block.
• Create a network with control flow, for example, a network with a section that can dynamically change depending on the input data.
• Create a network with loops, for example, a network with sections that feed the output back into itself.
### Check Validity of Layer
If you create a custom deep learning layer, then you can use the checkLayer function to check that the layer is valid. The function checks layers for validity, GPU compatibility, correctly defined gradients, and code generation compatibility. To check that a layer is valid, run the following command:
checkLayer(layer,validInputSize,'ObservationDimension',dim)
where layer is an instance of the layer, validInputSize is a vector or cell array specifying the valid input sizes to the layer, and dim specifies the dimension of the observations in the layer input data. For large input sizes, the gradient checks take longer to run. To speed up the tests, specify a smaller valid input size.
#### Check Validity of Layer Using checkLayer
Check the layer validity of the custom layer preluLayer.
Define a custom PReLU layer. To create this layer, save the file preluLayer.m in the current folder.
Create an instance of the layer and check its validity using checkLayer. Specify the valid input size to be the size of a single observation of typical input to the layer. The layer expects 4-D array inputs, where the first three dimensions correspond to the height, width, and number of channels of the previous layer output, and the fourth dimension corresponds to the observations.
Specify the typical size of the input of an observation and set 'ObservationDimension' to 4.
layer = preluLayer(20,'prelu');
validInputSize = [24 24 20];
checkLayer(layer,validInputSize,'ObservationDimension',4)
Skipping GPU tests. No compatible GPU device found.
Skipping code generation compatibility tests. To check validity of the layer for code generation, specify the 'CheckCodegenCompatibility' and 'ObservationDimension' options.
Running nnet.checklayer.TestLayerWithoutBackward
.......... ...
Done nnet.checklayer.TestLayerWithoutBackward
__________
Test Summary:
13 Passed, 0 Failed, 0 Incomplete, 9 Skipped.
Time elapsed: 0.18046 seconds.
Here, the function does not detect any issues with the layer.
### Include Layer in Network
You can use a custom layer in the same way as any other layer in Deep Learning Toolbox.
Define a custom PReLU layer. To create this layer, save the file preluLayer.m in the current folder.
Create a layer array that includes the custom layer preluLayer.
layers = [
imageInputLayer([28 28 1])
convolution2dLayer(5,20)
batchNormalizationLayer
preluLayer(20,'prelu')
fullyConnectedLayer(10)
softmaxLayer
classificationLayer];
### Output Layer Architecture
At the end of a forward pass at training time, an output layer takes the predictions (outputs) y of the previous layer and calculates the loss L between these predictions and the training targets. The output layer computes the derivatives of the loss L with respect to the predictions y and outputs (backward propagates) results to the previous layer.
The following figure describes the flow of data through a convolutional neural network and an output layer.
#### Output Layer Properties
Declare the layer properties in the properties section of the class definition.
By default, custom output layers have the following properties:
• NameLayer name, specified as a character vector or a string scalar. To include a layer in a layer graph, you must specify a nonempty, unique layer name. If you train a series network with the layer and Name is set to '', then the software automatically assigns a name to the layer at training time.
• Description – One-line description of the layer, specified as a character vector or a string scalar. This description appears when the layer is displayed in a Layer array. If you do not specify a layer description, then the software displays "Classification Output" or "Regression Output".
• Type – Type of the layer, specified as a character vector or a string scalar. The value of Type appears when the layer is displayed in a Layer array. If you do not specify a layer type, then the software displays the layer class name.
Custom classification layers also have the following property:
• ClassesClasses of the output layer, specified as a categorical vector, string array, cell array of character vectors, or 'auto'. If Classes is 'auto', then the software automatically sets the classes at training time. If you specify the string array or cell array of character vectors str, then the software sets the classes of the output layer to categorical(str,str).
Custom regression layers also have the following property:
• ResponseNamesNames of the responses, specified a cell array of character vectors or a string array. At training time, the software automatically sets the response names according to the training data. The default is {}.
If the layer has no other properties, then you can omit the properties section.
#### Loss Functions
The output layer computes the loss L between predictions and targets using the forward loss function and computes the derivatives of the loss with respect to the predictions using the backward loss function.
The syntax for forwardLoss is loss = forwardLoss(layer, Y, T). The input Y corresponds to the predictions made by the network. These predictions are the output of the previous layer. The input T corresponds to the training targets. The output loss is the loss between Y and T according to the specified loss function. The output loss must be scalar.
If the layer forward loss function supports dlarray objects, then the software automatically determines the backward loss function. For a list of functions that support dlarray objects, see List of Functions with dlarray Support. Alternatively, to define a custom backward loss function, create a function named backwardLoss. For an example showing how to define a custom backward loss function, see Specify Custom Output Layer Backward Loss Function.
The syntax for backwardLoss is dLdY = backwardLoss(layer, Y, T). The input Y contains the predictions made by the network and T contains the training targets. The output dLdY is the derivative of the loss with respect to the predictions Y. The output dLdY must be the same size as the layer input Y.
For classification problems, the dimensions of T depend on the type of problem.
2-D image classification1-by-1-by-K-by-N, where K is the number of classes and N is the number of observations.4
3-D image classification1-by-1-by-1-by-K-by-N, where K is the number of classes and N is the number of observations.5
Sequence-to-label classificationK-by-N, where K is the number of classes and N is the number of observations.2
Sequence-to-sequence classificationK-by-N-by-S, where K is the number of classes, N is the number of observations, and S is the sequence length.2
The size of Y depends on the output of the previous layer. To ensure that Y is the same size as T, you must include a layer that outputs the correct size before the output layer. For example, to ensure that Y is a 4-D array of prediction scores for K classes, you can include a fully connected layer of size K followed by a softmax layer before the output layer.
For regression problems, the dimensions of T also depend on the type of problem.
2-D image regression1-by-1-by-R-by-N, where R is the number of responses and N is the number of observations.4
2-D Image-to-image regressionh-by-w-by-c-by-N, where h, w, and c are the height, width, and number of channels of the output respectively, and N is the number of observations.4
3-D image regression1-by-1-by-1-by-R-by-N, where R is the number of responses and N is the number of observations.5
3-D Image-to-image regressionh-by-w-by-d-by-c-by-N, where h, w, d, and c are the height, width, depth, and number of channels of the output respectively, and N is the number of observations.5
Sequence-to-one regressionR-by-N, where R is the number of responses and N is the number of observations.2
Sequence-to-sequence regressionR-by-N-by-S, where R is the number of responses, N is the number of observations, and S is the sequence length.2
For example, if the network defines an image regression network with one response and has mini-batches of size 50, then T is a 4-D array of size 1-by-1-by-1-by-50.
The size of Y depends on the output of the previous layer. To ensure that Y is the same size as T, you must include a layer that outputs the correct size before the output layer. For example, for image regression with R responses, to ensure that Y is a 4-D array of the correct size, you can include a fully connected layer of size R before the output layer.
The forwardLoss and backwardLoss functions have the following output arguments.
FunctionOutput ArgumentDescription
forwardLosslossCalculated loss between the predictions Y and the true target T.
backwardLossdLdYDerivative of the loss with respect to the predictions Y.
The backwardLoss must output dLdY with the size expected by the previous layer and dLdY to be the same size as Y.
#### GPU Compatibility
If the layer forward functions fully support dlarray objects, then the layer is GPU compatible. Otherwise, to be GPU compatible, the layer functions must support inputs and return outputs of type gpuArray (Parallel Computing Toolbox).
Many MATLAB built-in functions support gpuArray (Parallel Computing Toolbox) and dlarray input arguments. For a list of functions that support dlarray objects, see List of Functions with dlarray Support. For a list of functions that execute on a GPU, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox). To use a GPU for deep learning, you must also have a supported GPU device. For information on supported devices, see GPU Support by Release (Parallel Computing Toolbox). For more information on working with GPUs in MATLAB, see GPU Computing in MATLAB (Parallel Computing Toolbox).
#### Include Custom Regression Output Layer in Network
You can use a custom output layer in the same way as any other output layer in Deep Learning Toolbox. This section shows how to create and train a network for regression using a custom output layer.
The example constructs a convolutional neural network architecture, trains a network, and uses the trained network to predict angles of rotated, handwritten digits. These predictions are useful for optical character recognition.
Define a custom mean absolute error regression layer. To create this layer, save the file maeRegressionLayer.m in the current folder.
[XTrain,~,YTrain] = digitTrain4DArrayData;
Create a layer array and include the custom regression output layer maeRegressionLayer.
layers = [
imageInputLayer([28 28 1])
convolution2dLayer(5,20)
batchNormalizationLayer
reluLayer
fullyConnectedLayer(1)
maeRegressionLayer('mae')]
layers =
6x1 Layer array with layers:
1 '' Image Input 28x28x1 images with 'zerocenter' normalization
2 '' Convolution 20 5x5 convolutions with stride [1 1] and padding [0 0 0 0]
3 '' Batch Normalization Batch normalization
4 '' ReLU ReLU
5 '' Fully Connected 1 fully connected layer
6 'mae' Regression Output Mean absolute error
Set the training options and train the network.
options = trainingOptions('sgdm','Verbose',false);
net = trainNetwork(XTrain,YTrain,layers,options);
Evaluate the network performance by calculating the prediction error between the predicted and actual angles of rotation.
[XTest,~,YTest] = digitTest4DArrayData;
YPred = predict(net,XTest);
predictionError = YTest - YPred;
Calculate the number of predictions within an acceptable error margin from the true angles. Set the threshold to 10 degrees and calculate the percentage of predictions within this threshold.
thr = 10;
numCorrect = sum(abs(predictionError) < thr);
numTestImages = size(XTest,4);
accuracy = numCorrect/numTestImages
accuracy = 0.7524
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2021-05-13 00:36:17
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https://docs.digitalearthafrica.org/en/latest/sandbox/notebooks/Real_world_examples/Vegetation_anomalies_seasonal.html
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# Cropland seasonal vegetation condition anomalies
Keywords: data used; landsat 8, vegetation; anomalies, band index; NDVI;
## Background
Understanding how the vegetated landscape responds to longer-term environmental drivers such as the El Nino Southern Oscillation (ENSO) or climate change, requires the calculation of seasonal anomalies. Standardised seasonal anomalies subtract the long-term seasonal mean from a time-series and then divide the result by the long-term standard deviation, thus removing seasonal variability and highlighting change related to longer-term drivers.
### Description
This notebook will calculate seasonal standardised NDVI anomalies for any given season and year. The long-term seasonal climatologies (both mean and standard deviation) are calculated on-the-fly.
$$\text{Standardised anomaly }=\frac{x-m}{s}$$
$$x$$ is the seasonal mean, $$m$$ is the long-term mean, and $$s$$ is the long-term standard deviation.
Note: It is a convention to establish climatologies based on a 30-year time range to account for inter-annual and inter-decadal modes of climate variability (often 1980–2010 or 1960–1990). As the Landsat archive over Africa is not consistent before 2000, the climatologies here have been calculated using the date-range 1999–2019 (inclusive). While this is not ideal, a 20-year climatology should suffice to capture the bulk of inter-annual and inter-decadal variability; for example, both a major El Nino (2015/16) and a major La Nina (2011) are captured by this time-range.
The following steps are taken in the notebook:
1. Load cloud-masked Landsat data over the region of interest for the years over which the climatologies will be computed
2. Create a map of clear-pixel counts
3. Calculate the NDVI long-term mean and long-term standard deviation
4. Calculate standardised NDVI anomalies
5. Focus on agricultural regions using the DE Africa cropland extent map
6. Plot the results
## Getting started
To run this analysis, run all the cells in the notebook, starting with the “Load packages” cell.
After finishing the analysis, return to the “Analysis parameters” cell, modify some values (e.g. choose a different location or time period to analyse) and re-run the analysis.
Load key Python packages and supporting functions for the analysis.
[1]:
%matplotlib inline
import datacube
import xarray as xr
import numpy as np
import matplotlib.pyplot as plt
from deafrica_tools.bandindices import calculate_indices
from deafrica_tools.plotting import display_map
/env/lib/python3.8/site-packages/geopandas/_compat.py:106: UserWarning: The Shapely GEOS version (3.8.0-CAPI-1.13.1 ) is incompatible with the GEOS version PyGEOS was compiled with (3.9.1-CAPI-1.14.2). Conversions between both will be slow.
warnings.warn(
## Set up a Dask cluster
Dask can be used to better manage memory use and conduct the analysis in parallel. For an introduction to using Dask with Digital Earth Africa, see the Dask notebook.
Note: We recommend opening the Dask processing window to view the different computations that are being executed; to do this, see the Dask dashboard in DE Africa section of the Dask notebook.
To use Dask, set up the local computing cluster using the cell below.
[2]:
### Cluster
• Workers: 1
• Cores: 3
• Memory: 28.14 GB
### Connect to the datacube
Activate the datacube database, which provides functionality for loading and displaying stored Earth observation data.
[3]:
dc = datacube.Datacube(app="Vegetation_anomalies")
### Analysis parameters
The following cell sets important parameters for the analysis:
• lat: The central latitude to analyse (e.g. 14.283).
• lon: The central longitude to analyse (e.g. -16.921).
• buffer: The number of square degrees to load around the central latitude and longitude. For reasonable loading times, set this as 0.2 or lower.
• time_range: The date range over which the NDVI climatolgies will be calculated (e.g. ('1999', '2019'))
• year: The year for which we will calculate the standardised anomaly, e.g. '2021'
• season: The season for which we will calculate the standardised anomaly, e.g 'DJF','JFM', 'FMA' etc.
• resolution: The x and y cell resolution of the satellite data, e.g. (-30, 30) will load Landsat data at its native 30 x 30m resolution
• dask_chunks: the size of the dask chunks, dask breaks data into manageable chunks that can be easily stored in memory, e.g. dict(x=1000,y=1000)
If running the notebook for the first time, keep the default settings below. This will demonstrate how the analysis works and provide meaningful results. The example explores coastal change in Ponto, Senegal.
To run the notebook for a different area, make sure Landsat data is available for the new location, which you can check at the DE Africa Explorer (use the drop-down menu to view all Landsat products).
[4]:
# Define the area of interest
lat, lon = 12.4917, 37.7649 #near Lake Tana, Ethiopia
buffer = 0.2
# Set the range of dates for the climatology
time_range = ('1999', '2019')
year = '2021'
season = 'MJJ'
resolution = (-30, 30)
# Combine central lat,lon with buffer to get area of interest
lat_range = (lat-buffer, lat+buffer)
lon_range = (lon-buffer, lon+buffer)
## View the selected location
The next cell will display the selected area on an interactive map. Feel free to zoom in and out to get a better understanding of the area you’ll be analysing. Clicking on any point of the map will reveal the latitude and longitude coordinates of that point.
[5]:
display_map(x=lon_range, y=lat_range)
[5]:
### Define a function for filtering data to the season of interest
[6]:
quarter = {'JFM': [1,2,3],
'FMA': [2,3,4],
'MAM': [3,4,5],
'AMJ': [4,5,6],
'MJJ': [5,6,7],
'JJA': [6,7,8],
'JAS': [7,8,9],
'ASO': [8,9,10],
'SON': [9,10,11],
'OND': [10,11,12],
'NDJ': [11,12,1],
'DJF': [12,1,2]
}
def filter_season(dataset):
dss = []
if dataset.time.begin.month in quarter[season]:
dss.append(dataset)
return dss
The first step in this analysis is to load in Landsat data for the lat_range, lon_range and time_range we provided above. The code below uses the load_ard function to load in data from the Landsat 5, 7 and 8 satellites for the area and time specified. For more information, see the Using load_ard notebook.
The function will also automatically mask out clouds from the dataset, allowing us to focus on pixels that contain useful data:
[7]:
# Create the 'query' dictionary object, which contains the longitudes,
# latitudes and time provided above
query = {
'x': lon_range,
'y': lat_range,
'time': time_range,
'measurements': ['red','nir','pixel_quality'],
'resolution': resolution,
'output_crs':'epsg:6933',
}
# Load available data Landsat 8
products=['ls7_sr','ls8_sr'],
group_by='solar_day',
predicate=filter_season,
**query)
Using pixel quality parameters for USGS Collection 2
Finding datasets
ls7_sr
ls8_sr
Filtering datasets using filter function
Applying morphological filters to pq mask [('opening', 3), ('dilation', 3)]
Re-scaling Landsat C2 data
Returning 181 time steps as a dask array
## Generate a clear pixel count summary
This will help us understand how many observations are going into the NDVI climatology calculations. Too few observations might indicate a bias in the climatology. Remember, the role of the climatology is to define ‘average’ conditions, too few observations will not provide a realistic estimation of the typical conditions.
[8]:
#create bit flag dictionary
flags_def = ds.pixel_quality.attrs["flags_definition"]
quality_flags = dict(
cloud="high_confidence", # True where there is cloud
)
#calculate the number of clear observation per pixel
[9]:
pq_count.plot(figsize=(10,10))
plt.title('Clear pixel count for NDVI climatologies');
## Calculate NDVI climatologies
This will take a few minutes to run because we will bring the climatologies into memory. Check the Dask dashboard to see progress. Access the dashboard by clicking on the Dashboard link generated when you created the cluster.
[10]:
#calculate NDVI
ndvi = calculate_indices(ds, 'NDVI', collection='c2', drop=True)
ndvi = ndvi.persist()
#calculate the climatologies and bring into memory
climatology_mean = ndvi.mean("time").NDVI.compute()
climatology_std = ndvi.std("time").NDVI.compute()
Dropping bands ['red', 'nir', 'pixel_quality', 'cloud_mask']
## Calculate standardised anomalies
Step 1: Load data from the season we’re analysing
Step 2. Calculate the mean NDVI during the season
[11]:
season_query = {
'x': lon_range,
'y': lat_range,
'time': (year+'-'+str(quarter[season][0]), year+'-'+str(quarter[season][-1])),
'measurements': ['red','nir'],
'resolution': resolution,
'output_crs':'epsg:6933',
}
# Load available data, Landsat 8 only
products=['ls8_sr'],
group_by='solar_day',
predicate=filter_season,
**season_query)
#calculate mean NDVI
season_ndvi = calculate_indices(season_ds, 'NDVI', collection='c2', drop=True)
seasonal_mean = season_ndvi.mean('time').NDVI.compute()
Using pixel quality parameters for USGS Collection 2
Finding datasets
ls8_sr
Filtering datasets using filter function
Re-scaling Landsat C2 data
Returning 10 time steps as a dask array
Dropping bands ['red', 'nir']
Step 3: Now we can calculate the standardised anomalies by subtracting the long-term mean and dividing by the long-term standard deviation
[12]:
stand_anomalies = xr.apply_ufunc(
lambda x, m, s: (x - m) / s,
seasonal_mean,
climatology_mean,
climatology_std,
output_dtypes=[ds.red.dtype],
)
## Incorporating DE Africa’s cropland extent map
Load the cropland mask over the region of interest. The default analysed here is in Ethiopia, so we need to load the crop_mask_eastern product, which cover the countries of Ethiopia, Kenya, Tanzania, Rwanda, and Burundi.
[13]:
time=('2019'),
measurements='filtered',
resampling='nearest',
like=ds.geobox).filtered.squeeze()
plt.title('Cropland Extent');
### Plot NDVI climatolgies, seasonal mean, and anomalies for cropping regions only
Below we mask out the regions that aren’t cropping, revealing only the condition of the cropped regions.
[14]:
climatology_mean=climatology_mean.where(cm, np.nan)
climatology_std=climatology_std.where(cm, np.nan)
seasonal_mean=seasonal_mean.where(cm, np.nan)
stand_anomalies=stand_anomalies.where(cm, np.nan)
[15]:
#plot al layers
plt.rcParams['axes.facecolor'] = 'gray' # makes transparent pixels obvious
fig,ax = plt.subplots(2,2, sharey=True, sharex=True, figsize=(15,15))
climatology_mean.plot.imshow(ax=ax[0,0], cmap='YlGn' ,vmin=0, vmax=0.75)
ax[0,0].set_title('NDVI: '+season+'mean climatology')
climatology_std.plot.imshow(ax=ax[0,1], vmin=0, vmax=0.25)
ax[0,1].set_title('NDVI: '+season+'std dev climatology')
seasonal_mean.plot.imshow(ax=ax[1,0], cmap='YlGn', vmin=0, vmax=0.75)
ax[1,0].set_title('NDVI: '+year+" "+season+' mean')
stand_anomalies.plot.imshow(ax=ax[1,1], cmap='BrBG',vmin=-2, vmax=2)
ax[1,1].set_title('NDVI: '+year+" "+season+' standardised anomaly')
plt.tight_layout();
## Drawing Conclusions
Here are some questions to think about:
1. How does the seasonal NDVI mean compare with the long term mean of NDVI?
2. Do the cropping regions tend to have high or low standard deviations in NDVI?
3. Looking at the map of standardised anomalies, are the crops fairing better or worse than average? And how unusual are the anomalies compared with average?
4. What are other environmental data might help us confirm the drivers of the changes in NDVI?
Contact: If you need assistance, please post a question on the Open Data Cube Slack channel or on the GIS Stack Exchange using the open-data-cube tag (you can view previously asked questions here). If you would like to report an issue with this notebook, you can file one on Github.
Compatible datacube version:
[16]:
print(datacube.__version__)
1.8.5
Last Tested:
[17]:
from datetime import datetime
datetime.today().strftime('%Y-%m-%d')
[17]:
'2021-09-22'
|
2021-10-27 12:44:19
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.27045613527297974, "perplexity": 11238.267086372574}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588153.7/warc/CC-MAIN-20211027115745-20211027145745-00335.warc.gz"}
|
https://www.deepdyve.com/lp/springer_journal/additional-aspects-of-the-generalized-linear-fractional-branching-memESqwUzM
|
# Additional aspects of the generalized linear-fractional branching process
Additional aspects of the generalized linear-fractional branching process We derive some additional results on the Bienyamé–Galton–Watson-branching process with $$\theta$$ θ -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the $$\theta$$ θ process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the $$\theta$$ θ -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the $$\theta$$ θ process, together with its powers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Institute of Statistical Mathematics Springer Journals
# Additional aspects of the generalized linear-fractional branching process
, Volume 69 (5) – Jul 19, 2016
23 pages
Publisher
Springer Journals
Subject
Statistics; Statistics, general; Statistics for Business/Economics/Mathematical Finance/Insurance
ISSN
0020-3157
eISSN
1572-9052
D.O.I.
10.1007/s10463-016-0573-x
Publisher site
See Article on Publisher Site
### Abstract
We derive some additional results on the Bienyamé–Galton–Watson-branching process with $$\theta$$ θ -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the $$\theta$$ θ process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the $$\theta$$ θ -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the $$\theta$$ θ process, together with its powers.
### Journal
Annals of the Institute of Statistical MathematicsSpringer Journals
Published: Jul 19, 2016
## You’re reading a free preview. Subscribe to read the entire article.
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|
2018-11-15 00:09:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6596124172210693, "perplexity": 1636.7568131247474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742322.51/warc/CC-MAIN-20181114232605-20181115014605-00314.warc.gz"}
|
https://codereview.stackexchange.com/questions/167013/detecting-ui-thread-hanging-and-logging-stacktrace
|
# Detecting UI thread hanging and logging stacktrace
I have an application that needs to always be responsive. I've written a class that is designed to monitor the UI thread. The goal is to provide useful information in the logs to be able to understand when the UI thread becomes unresponsive and to determine which code is the cause.
To do this I check how long it takes to process something on the UI thread, if its longer than a certain threshold, I log a warning. If this happens multiple times in succession, it prints the stacktrace.
I've had this in production for about 2 weeks now, and so far it is working as expected. However, given that it uses deprecated methods to get the stacktrace, I'd like to know if this might end up causing more problems than it solves.
public class ThreadMonitor
{
private static readonly ILog Log = LogManager.GetLogger(MethodBase.GetCurrentMethod().DeclaringType);
{
this.pollingFrequency = pollingFrequency;
this.delayThreshold = delayThreshold;
this.stackTraceIterations = stackTraceIterations;
}
public void Run()
{
{
while (true)
{
var task = dispatcher.InvokeAsync(() => { });
for (var i = 0; i < stackTraceIterations; i++)
{
{
Log.Debug($"{(i + 1) * 100}ms Delay on thread {thread.Name} ({task.Status})"); } else { break; } } if (task.Status == DispatcherOperationStatus.Completed) continue; var stackTrace = GetStackTrace(thread); Log.Debug($"StackTrace of UI Thread: {stackTrace}");
}
});
}
#pragma warning disable 0618
{
StackTrace stackTrace = null;
{
try { targetThread.Resume(); } catch { }
}).Start();
try { stackTrace = new StackTrace(targetThread, true); }
catch { /* Deadlock */ }
finally
{
catch { stackTrace = null; /* Deadlock */ }
}
return stackTrace;
}
#pragma warning restore 0618
}
• Typo in parameter name in ThreadMonitor and ThreadMonitor field name -- pollingFrequencey, I think you mean pollingFrequency. – jrh Jul 13 '17 at 21:06
Great idea! That makes it possible to get information about hanging / lacking GUIs. That kind of (probably) valuable information are only available via customer feedback otherwise :). It would be interesting to know if you already use that ThreadMonitor, if you already evaluated the log files and if you got some interesting information out of it ;).
• There is no need to use a Task for the infinity loop. You can just use a Thread here.
• However, when using a Task, I would start it with TaskCreationOptions.LongRunning. Otherwise, a thread from the thread pool will be used (and blocked). Thread pool threads are usually hold available for short running actions.
• There is no way to get notified if an error occurs within the endless loop. To avoid that, put the endless loop in a try catch block or add a ContinueWith for the failure case to the created task.
• The class allows to start the ThreadMonitor multiple times. That is actually not desired, therefore I would throw an exception when trying to call Run twice.
|
2019-05-23 00:00:49
|
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|
http://claesjohnson.blogspot.com/2019/08/einsteins-biggest-mistake-mixing-of.html
|
## måndag 26 augusti 2019
### Einstein's Biggest Blunder: Mixing of Space into Time
Einstein "biggest scientific blunder" in his own view was the introduction of a zero-order term with coefficient $\Lambda$ named the cosmological constant in his cosmological field equations as a fix to get a stationary universe.
But an even bigger mistake/blunder was to change the view of the mixing of space into time expressed in the Lorentz transformation from that of Lorentz as a formality without true physical meaning, into a reality of space-time with space and time on equal footing in his 1905 article presenting his special theory of relativity SR to the world. Einstein took this step in sharp contradiction with the view of Leibniz with
• space = order of coexistence
• time = order of succession.
Einstein had a poor understanding of mathematics, which apparently allowed him to believe that just because (1d) space can be ordered along a spatial coordinate axis with space coordinate $x$ as a real number, and time can be ordered along a temporal coordinate axis with time coordinate $t$ as a real number, and the coordinate axes of real numbers for $x$ and $t$ superficially look the same, (1d) space cannot be distinguished from time. Einstein expressed this revelation as
• Time and space are modes by which we think and not conditions in which we live.
• Space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind union of the two will preserve an independent reality.
Einstein thereby left the physics of Leibniz with a sharp distinction between space and time, into a world of imagination without distinction and thus physics. This was Einstein's biggest mistake!
It led Einstein to work with "events" without spatial extension for which the essential aspect of coexistence had no meaning and physics was lost. In particular, it led Einstein to a special theory of relativity without physics based on a confused derivation of the Lorentz transformation incorrectly assuming that two different light pulses with spatial extension emitted in two different systems are one and the same light pulse without spatial extension.
What is so completely amazing is that Einstein's mistake of mixing space and time has become a religion for modern physicists. The twin paradox shows that this is confusing fake physics. No wonder that modern physics is in state of deep crisis, caused by Einstein in particular.
|
2021-07-23 17:04:57
|
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|
https://math.stackexchange.com/questions/1088135/prove-that-the-curvature-of-gamma-is-frac-kappa-alpha-sin2-theta
|
# Prove that the curvature of $\gamma$ is $\frac{\kappa_{\alpha}}{\sin^2\theta}$
Let $\alpha:I\to {\mathbb R}^3$ be a cylindrical helix with a unit vector $u$ such that $u\cdot T_{\alpha}$ is a constant for all $t\in I$. For $t_0\in I$, the curve $$\gamma(t)=\alpha(t)-((\alpha(t)-\alpha(t_0))\cdot u)u$$ is called a cross-sectional curve of the cyliner on which $\alpha$ lies. Prove that the curvature of $\gamma$ is $\frac{\kappa_{\alpha}}{sin^2\theta}$, where $\kappa_{\alpha}$ is the curvature of $\alpha$, and $\theta$ is the angle between $T_{\alpha}$ and $u$
I've proved that:
$$\sin\theta=\frac{\kappa_{\alpha}}{\sqrt{{\kappa_{\alpha}}^2+\tau_{\alpha}^2}}$$
so we're trying to show that:
$$\kappa_{\gamma}=\kappa_{\alpha}+\frac{\tau_{\alpha}^2}{\kappa_{\alpha}}$$
Expressing both sides in terms of their derivatives, we have:
$$\frac{|\gamma'\times\gamma''|}{|\gamma'|^3}=\frac{|\alpha'\times\alpha''|}{|\alpha'|^3}+\frac{|\alpha'|^3}{|\alpha'\times\alpha''|}\frac{((\alpha'\times\alpha")\cdot{\alpha'''})^2}{|\alpha'\times\alpha"|^4}$$.
Since $\gamma'=\alpha'-(\alpha'\cdot u)u$ and $\gamma''=\alpha''-(\alpha''\cdot u)u$, we get:
$$\frac{|(\alpha'-(\alpha'\cdot u)u)\times{\alpha''}-(\alpha''\cdot u)\alpha'\times u|}{|\alpha'-(\alpha'\cdot u)u|^3}=\frac{|\alpha'\times\alpha''|}{|\alpha'|^3}+\frac{|\alpha'|^3}{|\alpha'\times\alpha''|}\frac{((\alpha'\times\alpha")\cdot{\alpha'''})^2}{|\alpha'\times\alpha"|^4}$$
Because of the complexity of the equation, I think I should approach to it at some other perspective. I also believe that the conclusion $\sin \theta=\frac{\kappa_{\alpha}}{\sqrt{{\kappa_{\alpha}}^2+\tau_{\alpha}^2}}$ should still be utilized. I hope someone could give me a clue.
• Just want to confirm that t is NOT a arc-length parameter? – Xipan Xiao Jan 2 '15 at 16:43
• If it is arc-length parameter, the curvature is just $|\alpha''|$ and $\theta$ is constant. – Xipan Xiao Jan 2 '15 at 17:19
• @XipanXiao The fact that $\theta$ is constant is not implied by assuming that $t$ is an arc length parameter, instead, this is implied by the fact that $\alpha$ is a cylindrical helix. Also why is the curvature just $|\alpha''|$ when assuming that $t$ is an arc length paramenter? I tried and get $|\alpha''-(\alpha''\cdot u)u|$ – pxc3110 Jan 2 '15 at 17:29
• $const=u\cdot \alpha'=1\cdot 1\cdot \cos\theta=\cos\theta$ and $k_\alpha=|\alpha''|$ – Xipan Xiao Jan 2 '15 at 17:36
• And you need to state it explicitly in your post whether t is an arc-length parameter or not. It makes much difference. – Xipan Xiao Jan 2 '15 at 17:38
If t is an arc-length parameter, $\gamma'=\alpha'-(\alpha'\cdot u)u=\alpha'-cu$ where $c=\alpha'\cdot u=\cos\theta$ is constant as assumed. So $\gamma''=\alpha''$ and $|\gamma'|^2=|\alpha'|^2+c^2-2c|\alpha'|\cos\theta=|\alpha'|^2-c^2=1-\cos^2\theta=\sin^2\theta$
So let $\beta(s)=\gamma(s/\sin\theta)$, $s$ is an arc-length parameter and the curvature is $|\beta''|=|\gamma''/\sin^2\theta|=|\alpha''/\sin^2\theta|=k_\alpha/\sin^2\theta$
• By the way, do you know that a curve is called a cylindrical curve if an only if $\frac{\kappa}{\tau}$ is a constant, which in this case equals to $tan\theta$? – pxc3110 Jan 3 '15 at 1:08
|
2020-01-22 08:11:58
|
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|
http://mathoverflow.net/feeds/question/105679
|
Weierstrass transform in complex variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T02:37:10Z http://mathoverflow.net/feeds/question/105679 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105679/weierstrass-transform-in-complex-variable Weierstrass transform in complex variable Hu Yi Chen 2012-08-28T03:54:57Z 2012-08-28T03:54:57Z <p>The usual Weierstrass transform of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as: $$e^{D^2/2}f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-yD}f(x)e^{-y^2/2} dy=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-y^2/2}dy$$ where $D=\frac{d}{dx}$.</p> <p>Now if $D$ is with respect to complex variable $z$, how will the Weierstrass transform be different from the one above?</p>
|
2013-05-21 02:37:12
|
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|
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