url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3
values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93
values | snapshot_type stringclasses 2
values | language stringclasses 1
value | language_score float64 0.06 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
http://www.tutorialspoint.com/design_and_analysis_of_algorithms/searching_techniques_introduction.htm | 1,696,449,971,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511406.34/warc/CC-MAIN-20231004184208-20231004214208-00346.warc.gz | 76,426,082 | 16,479 | Design and Analysis - Searching Techniques
In the previous section, we have discussed various Sorting Techniques and cases in which they can be used. However, the main idea behind performing sorting is to arrange the data in an orderly way, making it easier to search for any element within the sorted data.
Searching is a process of finding a particular record, which can be a single element or a small chunk, within a huge amount of data. The data can be in various forms: arrays, linked lists, trees, heaps, and graphs etc. With the increasing amount of data nowadays, there are multiple techniques to perform the searching operation.
Searching Techniques in Data Structures
Various searching techniques can be applied on the data structures to retrieve certain data. A search operation is said to be successful only if it returns the desired element or data; otherwise, the searching method is unsuccessful.
There are two categories these searching techniques fall into. They are −
• Sequential Searching
• Interval Searching
Sequential Searching
As the name suggests, the sequential searching operation traverses through each element of the data sequentially to look for the desired data. The data need not be in a sorted manner for this type of search.
Example − Linear Search
Fig. 1: Linear Search Operation
Interval Searching
Unlike sequential searching, the interval searching operation requires the data to be in a sorted manner. This method usually searches the data in intervals; it could be done by either dividing the data into multiple sub-parts or jumping through the indices to search for an element.
Example − Binary Search, Jump Search etc.
Fig. 2: Binary Search Operation
Evaluating Searching Techniques
Usually, not all searching techniques are suitable for all types of data structures. In some cases, a sequential search is preferable while in other cases interval searching is preferable. Evaluation of these searching techniques is done by checking the running time taken by each searching method on a particular input.
To explain briefly, there are three different cases of time complexity in which a program can run. They are −
• Best Case
• Average Case
• Worst Case
We mostly concentrate on the only best-case and worst-case time complexities, as the average case is difficult to compute. And since the running time is based on the amount of input given to the program, the worst-case time complexity best describes the performance of any algorithm.
For instance, the best case time complexity of a linear search is O(1) where the desired element is found in the first iteration; whereas the worst case time complexity is O(n) when the program traverses through all the elements and still does not find an element. This is labeled as an unsuccessful search. Therefore, the actual time complexity of a linear search is seen as O(n), where n is the number of elements present in the input data structure.
Many types of searching methods are used to search for data entries in various data structures. Some of them include −
• Linear Search
• Binary Search
• Interpolation Search
• Jump Search
• Hash Table
• Exponential Search
• Sublist search
• Fibonacci Search
• Ubiquitous Binary Search
We will look at each of these searching methods elaborately in the following chapters. | 647 | 3,337 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2023-40 | latest | en | 0.869396 |
https://math.stackexchange.com/questions/641581/finding-maximum-flow-of-directed-network-with-two-inputs | 1,597,491,060,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439740838.3/warc/CC-MAIN-20200815094903-20200815124903-00352.warc.gz | 398,252,744 | 33,069 | # Finding maximum flow of directed network with two inputs
I am given a directed network graph with three fixed verticess where two of these are "inputs" and and one is the "sink". I'm asked to find the maximal flow through the network. How should go about doing this?
I'm familiar with the Ford–Fulkerson algorithm but can it be used when I have two distinct inputs?
How can I get started?
• If $a$ and $b$ are my inputs and then I add a vertex $v$ which is my new input and then connect it to $a$ and $b$, what capacity should I set to the new edges (va & vb) in order not the mess things up? (Thanks a lot by the way, I really appreciate your answer) – John Smith Jan 17 '14 at 12:28
• Well, you want to make sure they can still function as the original sources. So you can't let $v_a$ have a capacity lower than the input of $a$, but you can't take a capacity higher than the input of $a$ either. So you should take the input of $a$ as the capacity. Same thing for $v_b$. The new source is of course of magnitude input $a$ + input $b$. So the modified problem has the same total input as the original problem. You are very welcome by the way :). – Leo Jan 17 '14 at 12:36 | 307 | 1,181 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2020-34 | latest | en | 0.947314 |
ralfgerlich.biz | 1,675,806,109,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500641.25/warc/CC-MAIN-20230207201702-20230207231702-00751.warc.gz | 486,828,062 | 70,880 | # Generating Skewed-Normally distributed random values
This weekend I had to generate some random sample data, but instead of it being selected from a normal distribution, I needed the data to have some skew. A distribution is said to be skewed if there are more samples on one side of the mean than on the other. To get a feeling for what skew means, have a look at the following figures.
Now, it is quite easy to generate normally distributed random data, for example using the norm class from scipy.stats. There also is a class for a skewed-normally distributed variable, scipy.stats.skewnorm. However, the variable used to express skew is a bit unintuitive there.
So I searched for another solution, and came across a technical report by D. Ermak and J. Nasstrom from 1995 [1]. Ermak and Nasstrom use the sum of two uniformly distributed random variables with overlapping ranges to generate a skewed quasi-uniformly distributed variable, and then add many of these together to generate a skewed quasi-normally distributed varible.
The paper is worth a read, but here is a summary of the thought process. Going back a bit to basic statistics, we find that the normal distribution is somewhat special in that generally, the distribution of the sum of many identically distributed random variables approaches the normal distribution. This result is known as the Central Limit Theorem. It is also considered the reason that normally distributed random variables are so widespread in nature.
So, if we were to consider the sum of a bunch of — for example — uniformly distributed random variables, then the distribution of that sum would also approach the normal distribution. This is one way of generating approximately normally distributed random data.
Ermak and Nasstrom instead generate random data from what they call a double-block distribution. The probability density function of such a distribution is shown in the following figure.
The idea here is to use two uniformly-distributed random variables with differing means. The sum of these two will have a skewed distribution. Adding many of these skewed random variables up will give us an approximated skewed quasi-normal distribution in the same way that summing up identically distributed random variables will give us an approxomate normal distribution.
Ermak and Nasstrom go through the lengths of calculating the required parameters for the two uniformly-distributed random variables in order to get a specific variance and skew, and then provide an algorithm for drawing samples from such a double-block distribution as well as generating samples from the skewed quasi-normal distribution.
I have coded this up using SciPy in Python. To generate N samples with given mean, standard deviation and skewness, run gen_skewed_continuous(N,mean,stdev,skewness). The optional parameter oversample can be used to modify the number of skewed-uniformly-distributed variables to be added.
import numpy as np;
import scipy.stats as stats;
def gen_skewed_block(N,var,skew):
# Determine block parameters
a=np.sqrt(5);
offset=np.sqrt(skew**2+243*(var**3)/32)
m1=2/(9*var)*(skew-offset);
m2=2/(9*var)*(skew+offset);
p1=m2/(2*a*m1*(m1-m2));
p2=m1/(2*a*m2*(m1-m2));
d1=-a*m1;
d2=a*m2;
# Uniform random number distribution
rv = stats.uniform();
# Get pairs of random numbers
r = rv.rvs((N,2));
# Calculate block numbers
v1=2*d1*(r[:,1]-0.5)+m1;
v2=2*d2*(r[:,1]-0.5)+m2;
values=np.where(r[:,0]<2*d1*p1,v1,v2);
return values;
def gen_skewed_continuous(N,mean,std,skewness,oversample=10):
# Determine moments
m2=std**2;
m3=skewness*std**3;
# Generate skewed block random values
rv1=gen_skewed_block(N*oversample,
m2,
m3*np.sqrt(oversample));
# Reshape them so we have oversample number of values in a row
rv2=rv1.reshape(-1,oversample);
# Sum them together and scale them to approximate a continuous distribution
rv3=rv2.sum(axis=1)/np.sqrt(oversample)+mean;
return rv3;
Now, if you think about it, there might also be a way of simplifying this by directly using two non-identically normally distributed random variables, saving the effort for calculating the sums. I’ll have to think about that one.
[1] D. L. Ermak and J. S. Nasstrom, “A method for generating skewed random numbers using two overlapping uniform distributions,” 1995.
[Bibtex]
@TechReport{Ermak.Nasstrom1995,
author = {D.L. Ermak and J.S. Nasstrom},
title = {A method for generating skewed random numbers using two overlapping uniform distributions},
year = {1995},
month = {feb},
doi = {10.2172/32589},
file = {:Ermak.Nasstrom1995 - A Method for Generating Skewed Random Numbers Using Two Overlapping Uniform Distributions.pdf:PDF},
publisher = {Office of Scientific and Technical Information ({OSTI})},
}
# Making sense of Maxwell’s Equations
I’m currently learning about modelling of electromagnetic systems, and in that process, I’m trying to wrap my head around Maxwell’s Equations. Maxwell’s Equations can be used to determine electric and magnetic fields, which are important for machines using them for work, such as electrical motors.
There are some aspects to those that are similar to fluid dynamics, with electrical fields having their source in electrical charges, while magnetic fields are free of sources. Much of that is expressed using curl and divergence. These are differential operators used to characterise vector fields.
I came across a pretty good video by 3Blue1Brown of Khan Academy fame introducing the concepts of curl and divergence, and also showing the connection to fluid and electrodynamics:
The other video, which goes over Maxwell’s Equations and explains them pretty well is by DrPhysicsA:
Be careful, though: At or around minute 18:13, DrPhysicsA explains how Maxwell’s Second Law (or Gauss’ Law for Magnetism) breaks down when you place your control surface such that only one pole of the magnet is inside and the other is outside. According to DrPhysicsA, that would lead to non-zero flow through the control surface, violating Gauss’ Law for Magnetism.
But actually, that is not the case. One has to consider that the field also flows inside the magnet. Namely, there is flow between the north- and south-pole inside the magnet in opposite direction to that outside the magnet. Thus, the flow inside the magnet and the flow outside that magnet through the control surface still cancel out. Gauss’ Law for Magnetism therefore very elegantly states that magnetic poles always come in pairs.
# Lift, Drag and Thrust — Aerodynamics Primer
Ever wondered how lift is generated on the wing of an aircraft or how a propeller generates thrust? There’s a legend that goes like this: The particles of the air want to stay together, but as the particle that goes along the upper part of the wing needs to go a longer way, it needs to be faster, and according to Bernoulli’s Principle, faster air means lower pressure, leading to lift. Nice explanation, isn’t it? So very romantic, the particles that just won’t let themselves be separated.
It’s just that…it’s completely wrong! Air particles aren’t married to each other, and it may even be that they are completely mixed up — in case of turbulent flow — with the wing still generating lift! In addition, Bernoulli’s Principle does not account for friction in the air, and thus by itself cannot explain drag.
For our further analysis of our quadrocopter model, we need to model the behaviour of the propeller and the engine that drives it. Modelling the behaviour of the propeller involves the use of the formulae for thrust and power
(1)
and working with these. Specifically, we need to understand the nature of the thrust and power coefficients and .
So, let’s have a look how lift and drag on an airfoil come to be, how we can use dimensional analysis to get the typical formulae for describing them, and where to find what’s missing. Here’s the route for today’s trekking tour:
## Reaction Forces: When a Ball meets a Wall
Let’s first recap a bit of physics basics. Remember Newton’s Third Law of Motion?
For every action, there is an equal and opposite reaction.
Let’s imagine we are sitting in a car at rest, with the motor running. If we now press the accelerator, the engine will provide power that is transferred to the wheels, which in turn will exert a force on the street. Now, the wheel is obviously accelerated to turn so that its bottom moves towards the back of our car — the opposite direction of where we want to go!
So it cannot be the wheel that provides our forward acceleration, because the wheel enacts a force in the wrong direction. Instead, it’s the street surface which makes us move: The force accelerating the wheel makes the wheel enact that force on the street surface, and the street surface will enact an opposite force of equal size on the wheel, accelerating us forward.
In this example, the action is the force acting from the wheel on the street surface, and the reaction is the force acting from the street surface on the wheel and — in consequence — on the whole car. And essentially, that also explains lift and drag. So, thanks for reading, have a good night! …
Of course, it’s a bit more involved. For a better understanding, let’s imagine a ball being thrown at a wall, as indicated in the figure to the right. When the ball hits the wall, it cannot proceed further in the same direction, so its velocity vector and with it its momentum must change. The change of velocity is indicated in the figure by .
As we know from Newton’s Second Law of Motion, a change of velocity requires a force acting on the ball. It is reasonable to assume that this force is enacted at the contact area of the ball to the wall.
Again, applying Newton’s Third Law of Motion, there must be an opposite force with the same magnitude in the system. But where is it?
Well, there is only one other object touching the contact area, and that is the wall. Thus, the reverse force must be acting on the wall. If we hadn’t fixed the wall in our reference system, we would see it being propelled backwards.
But, how big is this force? Well, in our example, we cannot tell exactly. However, we know the change of the force over time in a qualitative manner:
1. When the ball first touches the wall, the force is practically non-existent.
2. Then, when the ball is compressed more and more, it acts against being compressed with increasing force, until it is maximally compressed and at rest.
3. From then on, it is accelerated again in its new direction, and as it decompresses in the process, the force decreases again.
The total change in momentum until the time is the integral of this force over time (assuming, we are in a vacuum and there is no internal friction in the ball):
(2)
If you want to check the validity of this equation, just consider that and . If you look closely, you can see that Conservation of Momentum follows from Newton’s Third Law of Motion, according to which for the total forces we have , and thus for the total momentum in the system, must hold.
We call the reaction force, as it is a reaction to the change of momentum of part of the system (the ball).
However, although we cannot exactly determine the force, we can derive something about the change of momentum. Looking at the velocity triangle in the figure above, we see that is proportional to the magnitudes of and (just try scaling the triangle in your head). We know that the latter are equal, so the change in velocity is proportional to the magnitude . It is also proportional to the mass of the ball, so we have
(3)
Now, if we were to constantly throw balls onto the wall, all having the same mass and initial velocity, we could determine the mean force acting on the wall. Let’s assume that over a time interval of length we would throw a mass of onto the wall. Then the change in momentum over that time would be
(4)
Let’s further assume that we throw the balls at a constant frequency, so that we get a mean mass-flow rate of . The mean force over the same time would then be
(5)
If we now think of the balls as being a gas of density moving at velocity through a tube with cross-section , the mass flow rate is given by
(6)
If we plug all of this together, we get the force to be
(7)
which already looks eerily like the usual formula for lift and drag:
(8)
## Reaction Forces on an Wing
Let’s have a look at the cross-section of a wing and how the air flows around it:
The drawing above is only a schematic, but there are some important aspects of the airflow visualised there. What we see is that the air flows in a laminar fashion around the airfoil (that’s what we call the cross-section of a wing). An airflow is laminar if all the sheets of air are nicely separated and there is no turbulence.
We also see that the air on the left side is quite parallel to the horizontal plane, while on the right side it briefly flows downward, before assimilating to the free-flow again. As we have now learned, this indicates a change in momentum, which means that there is a force in play. As there should be: We would expect the wing to generate lift, an upwards force, so the resulting change in momentum should be downwards.
But where does this change in momentum come from? Think about what would happen if there was no change in momentum: The air would simply flow through the wing. The air cannot do that, so it must change direction to avoid the wing. It will again change direction when the wing “moves away” from the flow again, as the pressure of the air flowing beside it will push it that way. That change of direction implies a change of momentum, and that change of momentum must come with a force acting on the air.
So finally, we know where lift and drag come from: They represent the sum of all these forces along the wing, or the net force, and these forces result from the air having to follow the slope of the airfoil. There is a general convention to define lift and drag according to the direction of the free-stream:
• Lift is the part of the force perpendicular to the direction of the free-stream, with positive lift pointing upwards, and
• drag is the part of the force parallel to the direction of the free-stream, with positive drag pointing in the direction of the free-stream flow.
Again, we ask ourselves: How big are these forces? And again, we cannot give that number exactly just from this basic model — specifically, if we consider friction –, but we can characterise it quite well using dimensional analysis.
## Dimensional Analysis of Lift and Drag
Now it’s time to use what we learned in a previous article about dimensional analysis — but this time we apply it to our wing. From our previous considerations, we already have identified a few parameters that may influence lift and drag:
• the density of the air in the free-stream ,
• the size and form of the wing, represented by its projected surface area , and
• the free-stream velocity of the air .
Geometrically, the so-called angle of attack will also play a role, as shown in the figure above. This is the angle between the direction of the free stream and the chord line of the airfoil — the theoretical line from the leading to the trailing edge.
If we are working at velocities close to the speed of sound, we need to consider compressibility. Thus, another parameter is the speed of sound in the free stream . The speed of sound may differ between the free stream and around the wing, as it depends on the density of the air, and this may well be influenced by the flow around the wing. Therefore, we only consider the free-stream speed of sound — the speeds of sound elsewhere in the stream would be expressible based on this.
Further, we also need to consider internal friction in the air — otherwise there will be no drag at all, according to a finding that is known as D’Alembert’s Paradox. The friction will be represented by the free-stream viscosity . The friction between two sheets of fluid is proportional to the size of the contact surface and the gradient of velocity between both. The viscosity is the proportionality constant.
So, we finally have the following parameters influencing lift and drag, with their dimensions ( for mass, for length, and for time):
Quantity Symbol Dimension Lift/Drag Angle of Attack Area of the Wing Free-Stream Velocity Density of the Air Viscosity Speed of Sound
Obviously, the lift and drag forces are functions of the other parameters:
(9)
The dimensions used in these parameters are mass , length and time . We’ll use the density , the free-stream velocity and the wing area to represent them:
Dimension Symbol Representation Mass Length Time
Using these, we find the following expressions with equivalent dimensions:
Quantity Dimension Equivalent Expression Lift/Drag Angle of Attack Viscosity Speed of Sound
We could now use as the dimensionless quantity to represent friction. If you are interested in aerodynamics, that expression may look familiar to you. The Reynolds Number is often used to represent the influence of friction in fluid flows, but it looks slightly different:
(10)
So, first of all, nominator and denominator are reversed — larger numbers indicate less friction –, and instead of we have the so-called characteristic length in there. In aerodynamics, the chord length is usually used for , so we might not want to deviate from that. Thus, we will use the Reynolds-Number as a dimensionless representation of viscosity.
For the speed of sound, we’d get , which is the inverse of the Mach Number. So, we’d rather use the Mach Number directly. Thus, our relationship has the following form:
(11)
with
(12)
This is the formula in the form that you’ll find in most aerodynamics literature. The value of is called the coefficient of lift when calculating lift and the coefficient of drag when calculating drag.
So, where we originally had six parameters to vary in an experiment (, resp. , , , and ), we are now left with three parameters to vary (, and ), and we have a formula that will help us deal with the rest of the variation without additional experiments.
## Propeller Aerodynamics
On our multicopter, we don’t have wings. We have propellers, which could be described as rotating wings. Lift turns to thrust, drag turns to torque, torque turns to required power.
The performance of a propeller is determined by a relationship between these quantities:
• Thrust force ,
• propeller torque ,
• propeller power (due to torque) ,
• rotational frequency of the propeller ,
• propeller diameter ,
• velocity of the air in free-stream ,
• density of the air in free-stream ,
• viscosity of the air in free-stream ,
• speed of sound in free-stream .
Now, here is an excercise for you: See if you can find the usual descriptions of , and based on dimension analysis:
(13)
You will again come across the Reynolds Number and the Mach Number . I’ll give you a few hints for these:
• The Reynolds Number for propellers is usually based on the chord and tangential velocity at 75% of propeller radius for the characteristic length and velocity. The symbol used for the former is typically . Yes, you’ll have to add that to the list of dimensional parameters.
• The Mach Number is typically based on the velocity of the propeller tip.
(14)
With some basic geometry knowledge you should be able to figure these out. If you get stuck, I recommend looking at this paper by Robert Deters, Gavin Ananda and Michael Selig of the University of Illinois – Urbana-Champaign (UIUC) [1].
## Where to find out more
Now that we found out in general what the aerodynamic coefficients can do for us, we’d of course like to know how to get them. Well, there’s always measurement. If you don’t want to do it yourself, the University of Illinois – Urbana-Champaign (UIUC) provides a pretty extensive source of data for airfoils and propellers:
Or perhaps you also want to go back to the origins of flight, to the systematic experiments with large numbers of airfoils at NACA, the National Advisory Committee for Aeronautics, a predecessor of today’s NASA [2]?
However, we can also try to model and simulate the behaviour of airflow around objects. If we assume friction to be negligible (non-viscous or inviscid flow), there’s a whole theory of fluid dynamics modelled around that assumption: potential flow. (Actually, that also requires that the flow is free of rotation.) This is quite a powerful approach as it allows to model fluid flow as a potential field (similar to gravity), and many complicated flows can be described by superposition of basic flows such as uniform flow, flow sources/sinks or circular vortices.
Also, thin airfoil theory makes heavy use of potential flow. It gives us some pretty good first-order estimates of the lift coefficient and the coefficient of the so-called pitch moment for thin airfoils. An airfoil is considered thin if its thickness is small compared to its chord length.
One of the most well-known results from that is the estimated gradient of the lift coefficient of a thin 2D airfoil for small angles of attack (with the angle of attack being given in radians):
(15)
However, modelling flow around airfoils using potential flow has one important drawback: there is no drag at all! As drag due to friction is not modelled, it cannot be considered. But even if we sum up the forces due to change in momentum over the airfoil, there is no component along the direction of the free-stream flow.
That means: In potential flow, airfoils generate lift, but no drag. This realisation is known as D’Alembert’s Paradox, after the 18th-century French mathematician Jean le Rond d’Alembert.
Still, potential flow gets us pretty far towards a solution for viscous flow: We can approximate a first solution, and then add corrections, e.g., using boundary layer models. The boundary layer is the area around an object within which the velocity gradient is large enough for friction to matter. By first determining a potential flow solution, the thickness of the boundary layer can be estimated. Then, another potential flow solution is found, increasing the thickness of the airfoil by that of the boundary layer. This is an iterative process which goes on until a good, stable solution is found. Using that solution, it is then possible to derive the amount of drag produced inside the boundary layer.
Then, there’s induced drag, which comes from the fact that our wings are not infinitely wide — although we can approximate infinitely wide wings by making them long and slender. This is what is done with sail planes, who have wings with pretty high aspect ratios. To actually estimate the lift and drag of a real wing, we can use lifting line theory, which again uses a simple model based on potential flow to transform lift and drag data on 2D-airfoils into lift and drag data for a finite wing.
And for the really complicated cases, we can try to explicitly solve the Navier-Stokes equations numerically using computational fluid dynamics (CFD). With these, we can create a virtual wind tunnel, and approximate the air flow quite well. However, there is a lot of computational power involved, and the setup requires much more intricate models of our objects than the other approaches.
Of course there are also tools available for the simpler methods. Modelling the performance of 2D-airfoils in inviscid and viscous, subsonic flow is supported by the well-known XFOIL software. The tool XFLR5 extends this to wing design using lifting-line and vortex sheet methods.
## Conclusion
So, now we know where lift, drag, propeller thrust, torque and power come from, and how we can characterise them. These formulae allow us to do a good bit of work, and at least get some pretty good estimates for the performance of a wing — if we know the coefficients. They do not allow us to directly derive these coefficients, but of course there are different methods to approximate them for individual forms of wings: measurements, thin-airfoil theory, potential flow, boundary-layer methods, lifting-line theory or CFD.
Now we are well-equipped for modelling the aerodynamics part of our multicopter engines. Next time, we’ll look into modelling our engine with a DC motor.
[1] R. W. Deters, G. K. A. Krishnan, and M. S. Selig, “Reynolds number effects on the performance of small-scale propellers,” in 32nd AIAA applied aerodynamics conference, 2014.
[Bibtex]
@InProceedings{Deters.Krishnan.ea2014,
author = {Robert W. Deters and Gavin Kumar Ananda Krishnan and Michael S. Selig},
title = {Reynolds Number Effects on the Performance of Small-Scale Propellers},
booktitle = {32nd {AIAA} Applied Aerodynamics Conference},
year = {2014},
month = {jun},
publisher = {American Institute of Aeronautics and Astronautics},
doi = {10.2514/6.2014-2151},
url = {https://m-selig.ae.illinois.edu/pubs/DetersAnandaSelig-2014-AIAA-2014-2151.pdf},
}
[2] E. N. Jacobs, K. E. Ward, and R. M. Pinkerton, “The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel,” National Advisory Committee for Aeronautics, Washington, DC, techreport 460, 1933.
[Bibtex]
@TechReport{Jacobs.Ward.ea1933,
author = {Jacobs, Eastman N. and Ward, Kenneth E. and Pinkerton, Robert M.},
title = {The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel},
institution = {National Advisory Committee for Aeronautics},
year = {1933},
type = {techreport},
number = {460},
month = nov,
abstract = {An investigation of a large group of related airfoils was made in the NACA variable-density wind tunnel at a large value of the Reynolds number. The tests were made to provide data that may be directly employed for a rational choice of the most suitable airfoil section for a given application. The variation of the aerodynamic characteristics with variations in thickness and mean-line form were systematically studied.},
file = {:Jacobs.Ward.ea1933 - The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel.pdf:PDF},
keywords = {AIRFOILS; AERODYNAMIC CHARACTERISTICS; WIND TUNNEL TESTS; THICKNESS; AIRFOIL PROFILES; AERODYNAMIC COEFFICIENTS; LIFT DRAG RATIO; ASPECT RATIO; ANGLE OF ATTACK; HIGH REYNOLDS NUMBER; TABLES (DATA); GRAPHS (CHARTS)},
owner = {ralfg},
timestamp = {2019-11-02},
url = {https://ntrs.nasa.gov/search.jsp?R=19930091108},
}
# Dimensional Analysis of a DC-Motor
For the construction of my first quadrocopter, I bought some brushless DC motors, some propellers, a set of drivers for brushless motors, a battery, built a geometrically matching frame and started working with that. This approach quite probably has not led to a good design: I might get more flight time with the same payload by some better design.
However, there’s so many parameters to tweak: Battery capacity, voltage, size and mass of the engines, size and pitch of the propellers, frame geometry, etc. All of these are interrelated by complicated relationships, most of which are not exactly known to us. Varying all the parameters to find some optimum may prove both time-consuming and costly: We’d have to order different parts with different parameters, measure them under a whole lot of different situations, and then find the optimal solution.
However, there’s a pretty clever tool to reduce the set of variables to consider: Dimensional Analysis. The very short version is: Any relationship between a set of physical quantities can be represented by a smaller set of physical quantities and their dimensionless relationship to the remaining quantities. This allows us to massively simplify experiment design.
In this article, we’ll have a look at the basics of dimensional analysis using the so-called Buckingham Theorem [1], and try to use a combination of physical insight and dimensional analysis to characterise the performance of a DC motor in comparison to some of its basic parameters such as size or the strength of the magnetic material used. So, this is the plan for today’s tour:
## Dimensional Analysis: An example
Let’s have a look at a mathematical pendulum. We have a mass fixed to an arm or rope of length , and the strength of local gravity shall be given by the gravitational acceleration . We want to determine , the period of the pendulum. Some basic physical insight tells us, that may depend on the mass, the length of the arm, local gravity and the angle of initial deflection :
(1)
Dimensional analysis allows us to get a general picture of these influences. Let’s try to express as a dimensionless multiple of some product of powers of , and . Note that we do not consider in this product, as it has no dimension itself and thus no power of can contribute to making the product dimensionless:
(2)
Solving this equation so that is dimensionless leads us to
(3)
Now, we can write our relationship from Eq. 1 in the following way:
(4)
However, on the left-hand side, we have a dimensionless number, so the function must also map its parameters to dimensionless numbers. cannot depend on any product of powers of , and , as such a product would not be dimensionless: No power of and can cancel the mass dimension present in any non-zero power of , and the same is true for any other combination. There also cannot be a function that only depends on the value of the dimensioned parameter, as then the value of the function would change if we used feet as unit for the length instead of meters.
Thus, the function on the right can actually only depend on dimensionless quantities, which leaves . So, we finally get
(5)
So, dimensional analysis tells us, that the period of the pendulum only depends on and the initial deflection angle . We arrived at this conclusion only by analysing the dimensions of the quantities involved.
We might not know in general, but we could measure it for some set of values of , and , and could then determine it for any other set of values of and if only we keep constant. We say that determines the similitude of two problems with different parameters.
If we were to look into the actual equations of motion, we’d get the same result — and we’d also get the form of . Depending on the complexity of the equations involved, this could prove very tedious.
## The Buckingham Pi Theorem
What we just applied is known as the Buckingham Theorem, named for Edgar Buckingham, an American physicist living from 1867 to 1940. Buckingham was not the first to notice and apply the principle, but still today it is named for him.
The theorem essentially states that, if there is a physically meaningful relationship between physical quantities, then there must be a function describing this relationship which only depends on dimensionless ratios of these quantities. He also described a procedure to find these ratios.
Let’s have a look at the proof idea: Assume that we have physical quantities, which we shall name , and there is some — possibly unknown — relationship between them, which we express in the form
(6)
Obviously, the value of does not have a dimension. If it did, we could simply divide by that dimension and still have a similar equation without dimension. Thus, also cannot depend on dimensioned quantities. If it did, its value would differ depending on the units used, even if the quantities were the same.
As a consequence, we can select of the quantities, and express the others as dimensionless multiples of these. Let’s say, that were the selected quantities. Then we can write Eq. 6 as follows:
(7)
Now, as argued before, cannot depend on dimensioned quantities, so must actually be independent of the and can only depend on the dimensionless ratios:
(8)
## Finding the Basis Quantities
Now, how do we find which quantities to use as our basis ? It may help to note that term “basis” is used intentionally here, as these quantities must form a basis for the vector space of dimensions used within . In our pendulum example, we had quantities of time, length, mass and acceleration, which is length per time squared. Thus, we have three dimensions and four variables (or “vectors” in linear algebra parlance), and we can select three of these four to represent all the units.
There’s some useful rules for selecting them, but they do not uniquely determine the selection:
• If one of your variables is a dependent variable — such as the period in our pendulum example — you should not select it as basis variable. Otherwise it will also occur in the ratios for the independent variables, and that would not help us in characterising the relationship properly.
• Independent variables should be independent of each other. For example, for a propeller you should not include the diameter, the mean chord and the aspect ratio, as the latter is defined by the former two. Instead, use only two of these.
• Each dimension that occurs in at least one variable must also occur in at least one other variable. Otherwise we cannot create a fraction of variables in which the dimension does not occur.
Specifically the third rule also may give us important insight into the problem, such as when our problem only contains a single variable referencing a specific dimension. In that case our relationship either does not actually depend on that quantity, or we are missing another quantity that influences the relationship. The missing dimension may guide us as to where to look.
## How does a DC-Motor work?
Now, let’s use our new knowledge to analyse the performance of a DC motor and its relationship with some of its basic parameters. This will allow us to make better decisions on selecting motors in the future.
So, how does a DC motor work? Essentially, it turns electrical energy into mechanical energy via a magnetic field. For that, it uses two magnetic fields, leading to a magnetic force. That force results in a moment around the axis of the motor. When the motor turns around its axis, one of the fields needs to change its orientation so that the motor keeps turning. This is achieved either mechanically in brushed motors or electronically in brushless motors.
The figure above shows the basic mechanical configuration of a Brushless DC (BLDC) motor in outrunner configuration with permanent magnets for one part of the field, and the other field being generated by the coils in the middle. By switching the coils on and off, the inner field can be rotated. In an inrunner configuration, the fixed coils would be outside, and the static magnets would be on the axis in the middle.
For BLDC motors special brushless controllers are required that electronically determine the angular position of the magnets and switch the coils on and off as required. There are different setups for these controllers, some using hall sensors to determine the position of the magnets, others monitoring the induction voltage on the unconnected coil and using that to determine the proper timing for switching.
The figure to the right shows the basic electrical configuration of a DC motor, indicating the parasitic inductance and resistance of the coils in the motor. In addition, the turning motor also acts as a generator, inducing a counter-voltage, often called counter electromagnetic force (EMF) or back-EMF.
The torque acting on the axis is approximately proportional to the current flowing through the coils, and the reverse voltage generated is proportional to the rotational velocity. The proportionality factors are usually named for the torque coefficient and for the voltage coefficient.
## Efficiency Considerations for the DC-Motor
On the electrical side, we have losses on the resistor. On the magnetic-mechanical side, losses may come from constantly changing the magnetic field inside the metal of the stator. In addition, there may be aerodynamic losses from the turning motor.
The efficiency factor is found as the quotient , where is the power injected into the system, and is the usable power provided by the system.
On the electrical side, we have
(9)
On the mechanical side, we have
(10)
## Dimensional Analysis of the DC-Motor
We are reaching the final goal of this article: Determining the influences of dimensions on the performance on a DC motor. Specifically, we want to check the influence of some relevant dimensional parameters on the voltage and torque coefficients.
From our previous insight, we assume that and depend on
• the geometric size of the motor, represented by the motor diameter and its height ,
• the number of poles , and
• the number of windings .
Let’s have a look at the dimensions of our quantities. We denote the dimension of a quantity by square brackets. The dimension of and is a length: . The dimension of is voltage per rotational frequency, which is voltage times time. Voltage is energy per charge, or energy times time per current. So finally, we arrive at , with denoting mass, denoting current and denoting time.
It is quite obvious that we have a dimension which occurs in only one quantity, notably in . Obviously, the relationship must also include additional quantities. One reasonable quantity to consider is the strength of the of the magnets. The strength of the magnets is given by their remanence , which happens to have the dimension , just what we need.
Now we have all the dimensions we need — and . Let’s recapitulate the dimensions of all the quantities:
Quantity Symbol Dimension Back-EMF Coefficient Motor Size Motor Height Remanence
The back-EMF coefficient is our dependent variable, so we cannot use that for expressing the dimensions. Thus, we use and :
Dimension Expression
Now, we can express the unit of using these variables:
(11)
So, as a consequence is dimensionless. Similarly, the units of and are the same, so that is a dimensionless quantity. We call the aspect ratio of the motor.
Using the Buckingham Theorem, the following relationship must hold:
(12)
We know that the dimensions of and must be the same — otherwise cannot be dimensionless. Also, it is reasonable to assume that also depends on the same parameters as . In consequence, the following relationship must also hold:
(13)
And as a final finding, we can thus also express the efficiency in a similar manner:
(14)
These relationships enable us to compare motors with different sizes and magnet strengths, as long as the number of windings and the aspect ratio are the same. To determine the relationships expressed by , and , we only need to determine and for motors with different values of , and . We do not need to explicitly vary and .
## Conclusions
In this article, we learned about dimensional analysis, a very powerful tool which allows us to reduce the complexity of the analysis of physical relationships, just by looking at the dimensions. This method is widely used in aerodynamics and is the basis, for example, for the common formulae for lift and drag of wings and airfoils, or for thrust, torque and power for propellers.
Thus, to get a rough estimate on the performance of one system, we can scale up the performance data we have on another system, as long as the similarity parameters — which are the non-dimensional numbers to our coefficient parameters — are close enough. This strongly simplifies the dimensioning of a system, and allows us to get at least a rough ballpark setup for our design. We may later on refine the design and get some actual measurements, but we save a lot of up-front effort by getting to an
[1] E. Buckingham, “On physically similar systems; illustrations of the use of dimensional equations,” Phys. rev., vol. 4, p. 345–376, 1914.
[Bibtex]
@Article{Buckingham1914,
author = {Buckingham, E.},
title = {On Physically Similar Systems; Illustrations of the Use of Dimensional Equations},
journal = {Phys. Rev.},
year = {1914},
volume = {4},
pages = {345--376},
month = oct,
doi = {10.1103/PhysRev.4.345},
issue = {4},
numpages = {0},
publisher = {American Physical Society},
url = {https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=905},
}
# Rank Considerations for the Observer-Kalman System Identification Procedure
Last time, we looked in detail at the derivation of the OKID procedure for finding the impulse response of a system from arbitrary input-output data. However, there are some specificities to consider when collecting data, and we can derive these by looking at the rank of the matrices involved.
In this article, we’ll reconsider how the quality and uniqueness of the solution for an ordinary least-squares problem is affected by the ranks of the matrices involved. Based on that, we’ll derive a formula for determining the number of measurements we need to find a good estimate of the Markov parameter matrix.
## The Result
Let’s jump to the result real quick here and then look at how it is derived. Assume that we want to determine the Markov parameters
(1)
of order for a system with inputs and outputs.
The number of measurement points required is given by
(2)
Here, is the oversampling factor, which gives the number of data points we have per entry in the Markov parameter matrix. We need this oversampling to average out measurement error due to noise, and thus the larger the oversampling factor, the higher the quality of our estimate for the Markov parameter matrix.
On the other hand, a higher oversampling factor also means that we can either
• increase the number of measurements while keeping the order of our Markov parameter matrix constant, or
• decrease the order of our Markov parameter matrix while keeping the number of measurements.
A lower order of the Markov parameter matrix may decrease the quality of the estimate we get from the Eigensystem Realisation Algorithm. A higher number of measurements possibly increases the measurement effort and time.
## Rank Considerations for OKID
Let’s review the central equation of the OKID with observer:
(3)
Now have a look at the dimensions of these matrices:
Thus, we have unknowns and equations. Looking back at basic linear algebra, thus we know that we need
(4)
to hold for the solution to be uniquely determined. We might even need more than that, as this assumes that the measurement data is rich enough so that Equation 3 does not contain linearly dependent rows. However, in any case, the number of our measurements must be at least as large as given by Equation 4.
What happens if we have less measurements than this? Well, in this case the equation is not sufficiently determined, and there are arbitrarily many solutions to the equation. The ordinary least-squares approach will deliver a solution, but it is not clear whether that solution accurately describes the system we want to identify — although it will describe a system that will provide the same output given the inputs.
## Eliminating Noise by Oversampling
However, if we follow Equation 4 exactly, i.e. if we have exactly as many measurements as specified by this equation, we will also have an exact solution. This means that we will exactly incorporate all the measurement noise into our Markov parameters. Usually, we do not want that. Instead, we want to average out that measurement noise by having multiple samples.
That means that we will need more than the number of measurements given by Equation 4 — a lot more. To quantify this number, we’ll consider an oversampling factor . This factor gives us the number of samples per parameter we want to have. It also gives the factor by which we can diminish the influence of the measurement noise onto our estimate.
Now, if we want to oversample each parameter by factor , we need equations for each parameter. Thus, the following equation must hold
(5)
Solving that equation for , we get our final result, given in Equation 1.
## Impact of Using an Observer
Clearly, this is the result when we use an observer for identifying the Markov parameters. If we have a system which already is sufficiently stable, we do not need the observer approach. In that case, we have a much simpler equation:
(6)
Here we have rows in , so that the following relationship must hold for an oversampling factor :
(7)
This is smaller than the number required for the observer approach, as we do not have to determine many less entries of the Markov parameters. Thus, we may safe some measurement effort if we have a stable system.
## Conclusions
It is quite important that we have a sufficient number of measurements
• to get a uniquely determined set of Markov parameters, and
• to properly average out our measurement noise.
We can get the number of measurements we need by rank considerations of our basic equations and a simple oversampling approach. If we have a sufficiently stable system, we can avoid using the observer approach and thus reduce the number of measurements required while keeping the order of the Markov parameter set constant.
# The Observer/Kalman System Identification Procedure Explained
Previously, we have had a look at the Eigensystem Realisation Algorithm (ERA) by Joung and Pappa [1]. This algorithm allows us to find the parameters for a linear, time-invariant (LTI) system in discrete-time from measurements of the impulse response.
What, however, do we do when we don’t have measurements of the impulse response? We might not want to hit our system too hard, or have measurements with non-trivial inputs. Maybe we cannot wait for our system to return to a zero state before applying the impulse.
Again, Professor Steve Brunton came to the rescue with two videos on the Observer/Kalman System Identification procedure (or OKID for short) in his series on data-driven control:
Unfortunately, Prof. Brunton skips over the details and only gives a very short overview. Yet, I always try to understand the details of the methods I apply — if not for being able to assess when they may be used, then to learn about the principles being applied.
So I had a look into the papers by a group around Jer-Nan Juang, Minh Phan, Lucas G. Horta and Richard W. Longman, in which the OKID was introduced [2][3][4]. And again, there are some clever ideas in there that I found quite enlightening regarding handling the descriptions of dynamic systems:
• The basic approach of the OKID uses the fact that the response of an LTI to any input is related to the output by convolution of the impulse response.
• By reshaping the measurement data, a linear equation between said impulse response and the measurement data is established.
• This linear equation is solved using an ordinary least-squares approach.
The basic approach only works for asymptotically stable systems which approach zero state fast enough. However, there is a extension for non-stable or too slow systems: Instead of identifying the system itself, the OKID uses a modified system which is made asymptotically stable by a Luenberger observer [5], and then reconstructs the impulse response of the original system from the modified system.
Finally, this observer extension also leads to the “Kalman” part in the name for this procedure: The observer that is identified “on the go” this way turns out to be an optimal Kalman filter [6] given the amount of noise seen on the measurements.
So, all in all, the whole approach is quite ingenious. However, it’s not as complicated as Professor Brunton makes it sound, and we’ll briefly go over it and its basic ideas in this article.
## Review: Impulse Response
As for the ERA, we are dealing with discrete-time LTI systems. But different from previously, we’ll directly consider a Multiple Input, Multiple Output (MIMO) system.
The dynamics of this system shall be defined by the following recurrence equation:
(1)
Note that this time we also consider the case where the output is directly influenced by the output via the matrix . Now, let’s assume that we have an input sequence and observe the system starting at time step . We can determine the state (with ) to be
(2)
This can be verified by complete induction. For the case we get .
For we get
(3)
By extension, we can give the following expression for the output :
(4)
Here, we define the column vector by stacking the inputs at time steps above each other:
(5)
The matrix gives us the so-called Markov-Parameters of length , and when reversed, gives the impulse response of the system. If we had these Markov-Parameters, we could extract the system matrices with the help of the Eigensystem Realisation Algorithm (ERA).
## Handling Asymptotically Stable Systems
For an asymptotically stable system, the term approaches zero for increasing values of . Thus, for some sufficiently large we may assume that .
This helps us in simplifying Equation 4
(6)
Now, we can aggregate this into a larger equation:
(7)
With measurements (ideally, we have ) we can thus find by solving the equation above. Usually, this will be done using an ordinary least-squarey approach.
## Handling Insufficient Stability
In the previous section, we assumed that the system we measure is asymptotically stable and is sufficiently damped so that we can have our measurement count sufficiently large relative to the length of our impulse response . So how do we handle systems where this is not the case?
### Stabilisation Using a Luenberger Observer
The idea presented by Juang, Phan, Horta and Longman is quite simple: To arbitrarily choose the eigenvalues of the observed system, they construct a Luenberger observer. In general application, such an observer aims to reconstruct the internal state of a system from knowledge about the internal dynamics, about the inputs to the system and the measurements obtained from the system. It has the following structure:
(8)
In this case, and represent the state of the observer, which according to the construction of the observer shall follow the state of the original system. The observer state is adjusted using the correction term .
Usually, when constructing a Luenberger observer, we choose given our knowledge about the original system in such a way that the observation error is asymptotically stable, i.e. approaches zero. The dynamics of this error are described as follows:
(9)
Given a completely observable system, we can select in such a way that the eigenvalues of these error dynamics can be set arbitrarily. However, these eigenvalues also become the eigenvalues of the observer system, allowing us to arbitrarily stabilise it.
In our case, we want the observer state to be equal to the actual state, so we set , and arrive at the new state equations:
(10)
These equations describe the exact same system as in Equations 1 and ??, but with the current output of the system fed back into the system in such a way that the observed state always equals the actual system state. With this little trick, we have created a stable system to identify.
To find the Markov-Parameters of the new system, we simply apply the algorithm we used for sufficiently stable systems. This time, we use as input, which combines both the actual input and our measurement.
However, there’s one thing we need to be careful about: is not completely unconstrained in this problem. Indeed, its second part must be . If we were to allow non-zero values there, the best fit would be , with all other system matrices being zero. This system would just forward the input into its output. This would of course be a perfect fit — but it would not properly represent our system.
So instead, we have to find the best fit for
(11)
Note how all rows consist of pairs of input and output, but the last row consists only of the input? This way, the identified system cannot feed through the measured output to its output, and instead the equation identifies the dynamics of the system we want to identify.
The result, however, does not yet represent the Markov-Parameters of our observer. Instead, it has the form
(12)
For recovery of the Markov-Parameters of the original system, this is quite inconsequential (we just have to be careful in the implementation). If we wanted to actually have the Markov-Parameters of the observer — which we might, as we will see when we look at the specific properties of that observer — then we would have to restructure the response as follows by appending an appropriately sized zero-matrix at the end:
(13)
### Recovery of Original Markov-Parameters
But how do we get the Markov-Parameters of the original system? Again, Juang et al provide us with a little trick. Let’s first look at the structure of our stabilized Markov-Parameters:
(14)
with
(15)
Now, it is quite straightforward to extract from . For the others, we find that
(16)
For , we can immediately recover :
(17)
For we first expand the last element of the power:
(18)
Looking further, we can identify a pattern:
(19)
The reader is encouraged to prove this identity by complete induction over . Hint: First prove for the following, more general identity:
(20)
With a few more steps Equation 19 follows for .
## Why Kalman?
Now we know how the term “observer” got into the name. But how did “Kalman” end up there?
When we described the observer that we would use as model, we actually described a very specific observer, namely one that would immediately provide the same output as the actual system. The authors of the OKID paper [2] use the term “deadbeat observer” for this. This is the fastest possible observer for a discrete time system. Any faster, and the observer would predict the output of the system, violating causality.
Indeed, later in the paper, the authors show that this deadbeat observer is identical with the Kalman filter one would find given the variance of noise seen in the measurement. A more detailed explanation of their proof might follow in a later article.
Thus, extracting not only the original Markov Parameters, but also the parameter for the observer may be very beneficial if one wanted to design a Kalman filter for the system.
How do we get the system parameters for this Kalman filter? Well, we already have its Markov parameters, given by , and can use the ERA on them. That gives us dynamics matrices for a Kalman filter.
## Example
Let’s apply this method to our spring pendulum example from the ERA article. Remember: We had a horizontal spring pendulum with the following setup:
Different from the case with ERA, we now use an arbitrary input signal (which is the force applied to the mass), and determine the output signal. Again, we simulate the system and add noise to the output. The result is shown in the following figure.
The input is a pseudorandom binary sequence (PRBS). Such sequences can be generated, e.g., using the max_len_seq function from the SciPy signal processing library.
As we know from the ERA experiments, our pendulum is quite stable, so we can use the method for asymptotically stable systems. The result is shown in the following figure:
It seems that the estimate fits the actual impulse response (this time without noise) quite well. However, we’re in a dilemma:
• If we increase the length of our estimated impulse response, we reduce the amount of data we have available to counter all the noise, and thereby also the quality of our estimate.
• If we decrease the length of our estimated impulse response, we get a better estimate, but we have less data for our ERA procedure.
So we need to find a good compromise there by playing with the length of our estimated impulse response.
## Conclusion
We have seen that the idea of the OKID procedure is not as involved as one might believe. The basis are three very simple ideas:
• The Markov Parameters determine the impulse response, and thereby define the input-output relationship — together with the initial state.
• For asymptotically stable systems, the influence of the initial state diminishes over time, eventually becoming so small that we can ignore it.
• Not sufficiently asymptotically stable systems can be made stable by considering a Luenberger observer in their place. The problem is then reduced to identifying the Markov Parameters for this observer, and retrieving the Markov Parameters for the original system from this intermediate result.
• The observer such identified actually is a Kalman filter for the system, given the noise seen in the measurement.
It seems useful to always use the observer-enhanced procedure by default. Even for sufficiently asymptotically stable systems the use of the observer would reduce and thus the amount of measurements required.
Now, finally, this opens up a lot of new possibilities for system identification: We can use the response to any set of inputs to determine the system model. This also allows us to identify a system that already is subject to control and where it would be impractical or impossible to remove that control. One simple example are flight-tests with an already controlled multicopter to improve the controller, but another would be to identify the progression of a disease without having to stop treatment (which of course would be unethical).
To take the design of a multicopter as an example, we could simply record in-flight data from sensors as well as the control inputs sent to our engines, and use that to improve our already existing model. We do not even have to provide specific inputs, but just a general recording of flight data will help, as long as it was rich enough.
[1] J. Juang and R. S. Pappa, “An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” Journal of guidance control and dynamics, vol. 8, iss. 5, 1985.
[Bibtex]
@Article{Juang.Pappa1985,
author = {Juang, Jer-Nan and Pappa, Richard S.},
title = {{A}n {E}igensystem {R}ealization {A}lgorithm for {M}odal {P}arameter {I}dentification and {M}odel {R}eduction},
journal = {Journal of Guidance Control and Dynamics},
year = {1985},
volume = {8},
number = {5},
month = nov,
abstract = {A method called the eigensystem realization algorithm is developed for modal parameter identification and model reduction of dynamic systems from test data. A new approach is introduced in conjunction with the singular-value decomposition technique to derive the basic formulation of minimum order realization which is an extended version of the Ho-Kalman algorithm. The basic formulation is then transformed into modal space for modal parameter identification. Two accuracy indicators are developed to quantitatively identify the system and noise modes. For illustration of the algorithm, an example is shown using experimental data from the Galileo spacecraft.},
doi = {10.2514/3.20031},
file = {:Juang.Pappa1985 - An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction.pdf:PDF},
}
[2] J. Juang, M. Phan, L. G. Horta, and R. W. Longman, “Identification of Observer/Kalman Filter Markov Parameters: Theory and Experiments,” , Hampton, Virginia 23665, techreport 104069, 1991.
[Bibtex]
@TechReport{Juang.Phan.ea1991,
author = {Jer-Nan Juang and Minh Phan and Lucas G. Horta and Richard W. Longman},
title = {{Identification of Observer/Kalman Filter Markov Parameters: Theory and Experiments}},
year = {1991},
type = {techreport},
number = {104069},
month = jun,
abstract = {An algorithm to compute Markov parameters of an observer or Kalman filter from experimental input and output data is discussed. The Markov parameters can then be used for identification of a state space representation, with associated Kalman gain or observer gain, for the purpose of controller design. The algorithm is a non-recursive matrix version of two recursive algorithms developed in previous works for different purposes. The relationship between these other algorithms is developed. The new matrix formulation here gives insight into the existence and uniqueness of solutions of certain equations and gives bounds on the proper choice of observer order. It is shown that if one uses data containing noise, and seeks the fastest possible deterministic observer, the deadbeat observer, one instead obtains the Kalman filter, which is the fastest possible observer in the stochastic environment. Results are demonstrated in numerical studies and in experiments on an ten-bay truss structure.},
doi = {10.2514/6.1991-2735},
file = {:Juang.Phan.ea1991 - Identification of Observer_Kalman Filter Markov Parameters_ Theory and Experiments.pdf:PDF},
keywords = {ALGORITHMS; CONTROL THEORY; CONTROLLERS; KALMAN FILTERS; STOCHASTIC PROCESSES; SYSTEM IDENTIFICATION; TRUSSES; MATRICES (MATHEMATICS); SELECTION; UNIQUENESS},
owner = {ralfg},
school = {NASA Langley Research Center},
timestamp = {2019-09-11},
url = {https://ntrs.nasa.gov/search.jsp?R=19910016123},
}
[3] M. Phan, J. N. Juang, and R. W. Longman, “Identification of Linear Multivariable Systems by Identification of Observers with Assigned Real Eigenvalues,” Journal of the astronautical sciences, vol. 40, iss. 2, p. 261–279, 1992.
[Bibtex]
@Article{Phan.Juang.ea1992,
author = {M. Phan and J.N. Juang and R.W. Longman},
title = {{Identification of Linear Multivariable Systems by Identification of Observers with Assigned Real Eigenvalues}},
journal = {Journal of the Astronautical Sciences},
year = {1992},
volume = {40},
number = {2},
pages = {261--279},
month = jun,
doi = {https://doi.org/10.2514/6.1991-949},
file = {:Phan.Juang.ea1992 - Identification of Linear Multivariable Systems by Identification of Observers with Assigned Real Eigenvalues.pdf:PDF},
}
[4] M. Phan, L. G. Horta, J. N. Juang, and R. W. Longman, “Linear system identification via an asymptotically stable observer,” Journal of optimization theory and applications, vol. 79, iss. 1, p. 59–86, 1993.
[Bibtex]
@Article{Phan.Horta.ea1993,
author = {M. Phan and L.G. Horta and J.N. Juang and R.W. Longman},
title = {{Linear system identification via an asymptotically stable observer}},
journal = {Journal of Optimization Theory and Applications},
year = {1993},
volume = {79},
number = {1},
pages = {59--86},
month = oct,
issn = {0022-3239},
abstract = {This paper presents a formulation for identification of linear multivariable systems from single or multiple sets of input-output data. The system input-output relationship is expressed in terms of an observer, which is made asymptotically stable by an embedded eigenvalue assignment procedure. The prescribed eigenvalues for the observer may be real, complex, mixed real and complex, or zero corresponding to a deadbeat observer. In this formulation, the Markov parameters of the observer are first identified from input-output data. The Markov parameters of the actual system are then recovered from those of the observer and used to realize a state space model of the system. The basic mathematical formulation is derived, and numerical examples are presented to illustrate the proposed method.},
doi = {10.1007/BF00941887},
file = {:Phan.Horta.ea1993 - Linear System Identification Via an Asymptotically Stable Observer.pdf:PDF},
owner = {ralfg},
timestamp = {2019-09-11},
}
[5] D. Luenberger, “Observing the State of a Linear System,” Ieee transactions on military electronics, vol. MIL8, p. 74–80, 1964.
[Bibtex]
@Article{Luenberger1964,
author = {Luenberger, David},
title = {{Observing the State of a Linear System}},
journal = {IEEE Transactions on Military Electronics},
year = {1964},
volume = {MIL8},
pages = {74--80},
month = may,
abstract = {In much of modern control theory designs are based on the assumption that the state vector of the system to be controlled is available for measurement. In many practical situations only a few output quantities are available. Application of theories which assume that the state vector is known is severely limited in these cases. In this paper it is shown that the state vector of a linear system can be reconstructed from observations of the system inputs and outputs. It is shown that the observer, which reconstructs the state vector, is itself a linear system whose complexity decreases as the number of output quantities available increases. The observer may be incorporated in the control of a system which does not have its state vector available for measurement. The observer supplies the state vector, but at the expense of adding poles to the over-all system.},
doi = {10.1109/TME.1964.4323124},
file = {:Luenberger1964 - Observing the State of a Linear System.pdf:PDF},
}
[6] R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Transactions of the asme – journal of basic engineering, vol. 82, p. 35–45, 1960.
[Bibtex]
@Article{Kalman1960,
author = {Kalman, R. E.},
title = {{A New Approach to Linear Filtering and Prediction Problems}},
journal = {Transactions of the ASME - Journal of Basic Engineering},
year = {1960},
volume = {82},
pages = {35--45},
doi = {10.1115/1.3662552},
file = {:Kalman1960 - A New Approach to Linear Filtering and Prediction Problems.pdf:PDF},
owner = {ralfg},
timestamp = {2015.08.09},
}
# A deeper look into the Eigensystem Realisation Algorithm
For my quadcopter control project, I am currently in the process of system identification, trying to find a model describing the dynamics of my motor-propeller-combination. Finding the theoretical model is quite simple, but then you’re left with finding the parameters — and not all of them are directly available from the motor vendors. So I have no other choice than to determine these parameters from measurements.
In my endeavour I came across this series of videos on data-driven control by machine learning by Professor Steve Brunton of the University of Washington, Seattle. There, basic principles of modern methods for system identification are explained very well. If you are new to control systems, I can also recommend his control boot camp.
In part of the series, Professor Brunton explains the Eigensystem Realisation Algorithm (ERA) by Juang and Pappa [1]. The ERA is a procedure by which a reduced-order model for a linear and time-invariant system (LTI) can be derived from data of an impulse-response measurement.
However, while the videos do a pretty solid job of explaining the concepts and give the final formulae, I wanted to know more about their derivation. For this, I went to the original paper where the approaches were developed, and tried to reproduce the steps that led to the final results.
I follow this approach quite a lot when acquiring new skills. It helps not only better understanding — and memorizing — the final solution, but I also tend to pick up neat tricks along the way that often prove helpful elsewhere — even completely outside the domain of the original paper.
• When doing system identification, we mostly work with discrete-time systems, and in some aspects they are much easier to handle than continuous-time systems.
• The discrete-time impulse response, which uniquely characterises the behaviour of LTI systems, can be easily expressed in closed form using the system matrices.
• By reorganising our data, we can express seemingly non-linear problems in linear form.
• The singular value decomposition allows us to examine the composition of a matrix and simplify it.
• There are a few tricks for transforming matrix power expressions that may reduce the order of our problem.
## Our Example
Juang and Pappa worked on large, flexible space structure such as the Galileo spacecraft (which should not be mixed up with the European Galileo satellite navigation system), and aimed to identify the oscillation modes of such structures, so that these could be taken into account in control of the spacecraft.
We’ll look at something similar, although much more simple: A spring-loaded, dampened horizontal pendulum. This pendulum consists of a mass connected to a spring and a linear dampener. The mass can move horizontally, and we measure the position of its centre of gravity, with being the position when the system is at rest.
This system can be modelled as a Single-Input, Single-Output system: We have a single input and a single output .
Normally, we could easily provide at least the structure of a model for this system from first principles — namely, the conservation of momentum:
(1)
Introducing state variables and , we can define the dynamics of this system in state space notation:
(2)
Instead of actually measuring our system, we will be using the continuous-time dynamics to simulate the system and thereby perform “measurements”. We will add white noise to emulate measurement noise.
## Impulse-Response Measurement
Now, what happens if we whack this system with a hammer? What if we give it a short kick and then measure the position of the mass? Let’s find out!
The figure shows the position of the mass over time. We see that the system oscillates around the neutral position, but the amplitude of the oscillation falls exponentially with time, until measurement noise exceeds the movement of the system. In this example, we whacked the mass with a short force of 1 Newton, and then let go. So, our input would look like this:
This function is called the Kronecker Delta Function, and when used in analysis of signals and systems, it is referred to as the (discrete-time) unit impulse. It is one for exactly one tick of our (discrete) time-base and zero everywhere else. When given as an input to a discrete-time system, the output observed afterwards is called the discrete-time impulse response. Specifically, LTI systems are completely characterised by their impulse response.
As we are working on data that we would acquire by measurement, considering the continuous-time formulation is not useful, as we cannot get continuous-time measurement data. Instead, we’ll have some kind of quantisation in both time and value. This actually simplifies the maths of our problem a bit, as we will see in a moment.
We now want to have a look at how we can mathematically and generally describe the impulse response. This will be the basis of our model to which we will then fit our measurement data. In discrete time, we describe the dynamics of an LTI system using a recurrence relation in state space:
(3)
Let us assume that for our initial state holds. This is certainly true for dampened structures if we wait long enough for all oscillations to die down before we whack the structure. Now, if our input is for time step and 0 for all other time steps, we can see the progression of our state by repeatedly applying Equation 3:
So, in general we have for , and with the output relation
(4)
we get
(5)
Note that this impulse-response is completely independent of what our state variables represent. You can try it yourself: Substitute with some invertible square matrix and determine the impulse response of that system. However, this actually works in our favour, as it allows us to arbitrarily select at least part of the parameters later on.
In general, the terms — or, in their Multiple-Input, Multiple-Output form — are called the Markov Parameters of the system.
## Finding an Exact Model of Order n
What Juang and Pappa aim to do is to find parameter matrices , and so that the impulse response so described fits the measured data exactly. One way of doing so would be to explicitly write down the parametrised impulse-response — as we did in the last chapter — and try to solve this for the parameters directly.
However, that approach is rather ugly, and looking at Equation 5, this is a non-linear problem due to the powers of occurring. Instead, Juang and Pappa use quite a nice trick: They restructure the data in such a way that the problem becomes linear. This allows us to use our toolkit from linear algebra to find a solution.
In a first step, Juang and Pappa construct the vectors , which are the column vectors of observations starting at time step :
(6)
We will see later that is the order of the system we will build. For the time being, we shall assume that we have an arbitrary amount of measurements, so there are no limits on . From Equation 5, we can see that
(7)
with the observability matrix
(8)
Now, remember the function of the observability matrix: If the state has dimension and the observability matrix has at least rank , then we can use it to uniquely reconstruct the state from the outputs at time steps . Thus, if our system is observable, we can use the inverse of the observability matrix to find the state :
(9)
However, from that we can construct any state at any time due to:
(10)
Applying this to we get the recurrence
(11)
Now, let’s look at what we have done: We are able to describe the output at time steps from the output at time steps using a linear operator! How do we find ? Equation 11 does not sufficiently specify their value. To increase the rank of the equation system, Joung and Pappa proceed to build the Hankel-matrices
(12)
These matrices are constructed by listing measurements starting at time step in the first column, then listing measurements starting at time step in the second column, and so on. Using Equation 11, we can find the following recurrence for :
(13)
Thus, if we have any matrices and , with the former being a regular, invertible matrix, we can find
(14)
Thus, we can express using
(15)
where . This exactly describes our measured system response:
## The Problem of Noise
However, we also see that our estimated system exactly fits all the noise from our measurement. In our case we know from first-principles considerations that the exact model would only be of order , while the model we developed here is of much larger order. Besides being quite a misuse of resources, this may also lead to considerable estimation errors, as we can see when plotting the step responses:
Our estimated system clearly deviates from the system we measured. This is no wonder, as the estimated system incorporates all the noise we measured.
Further, our Hankel-matrices are pretty badly conditioned. This means that small rounding errors during calculations may become large errors in the result. We know that — in the absence of measurement error — the rank of the Hankel-matrices cannot be larger than the order of the underlying system. If they are indeed regular, they only are due to measurement noise, which should ideally be much smaller in magnitude than the actual data. Thus, simple inversion is prone to large numerical errors.
In their original paper, Juang and Pappa describe a method of using only a subset of the measurements to reduce the size of the Hankel-matrices, and thereby to improve the conditioning of the matrices. However, today, the method of ERA is almost universally presented (e.g. on Wikipedia) as being based on Singular Value Decomposition (SVD), using the whole set of measurement data.
## Singular Value Decomposition
Any matrix can be represented in the form of its Singular Value Decomposition (SVD):
(16)
The matrices all have special forms:
• The matrices and are square, orthogonal matrices, i.e. and . Note that the identity matrices may have different dimensions if is not square.
• The matrix is a diagonal matrix, with all diagonal elements being non-negative. As a common convention, the diagonal elements of — called the singular values of — are listed in decreasing order.
Essentially, the SVD expresses the contents of the matrix as the sum of matrices with decreasing amount of impact:
(17)
As the matrices and are orthogonal, the impact of the individual element is represented by . If we look at the Pareto plot of the singular values, we see that there are some dominant elements in there:
The plot shows the individual singular values in orange, ordered in decreasing order from left to right, and the cumulative proportion of the singular values, added up from left to right. We can see that the first two singular values are the most prominent, making up for about 55% of the total data values, and the remaining singular values are pretty small in comparison to that — although they still add up to representing 45% of the data. We also see that our matrix is awfully close to being singular, which it would be if any of the singular values were zero.
This is somewhat consistent with our original considerations, where we found out that our system is actually of second order. Thus, we may assume that the data represented by the major two singular values is our actual data, while the remaining singular values represent noise.
Now, what do we do with that information? We can use it to remove what we consider to be noise from our data. In essence, the vectors of and for bases of vector spaces, and the singular values determine the magnitude by which the base vectors are represented in the data in the matrix. Thus, by removing some singular values and base vectors, we can project the data in the matrix onto the directions represented by the remaining base vectors, and thereby approximate the original matrix:
(18)
By appropriately selecting , we can approximate arbitrarily close. Usually, one would determine the size of from first-principles considerations, or by defining a cut-off value in relative error.
With that, we can determine the pseudo-inverse of :
(19)
I encourage you to personally verify that indeed holds, where is the identity matrix (ones on the diagonal, zeroes everywhere else).
## Simplified Model
With the simplified Hankel-Matrix
(20)
and its pseudo-inverse , we can build a simplified, cleaned-up model:
(21)
Determining the impulse response of that model, we get a pretty clean fit, and it seems that the noise is actually being ignored.
Also, our step response fits much better:
## Reduced-Order Model
But still, our matrices , and are of order , while they should be of second order for our system. To reduce that order, we’ll have to make use of some trickery. We’ll re-use Equation 21 and add an identity matrix in there:
(22)
Keep in mind that the values in are non-negative and all non-zero values are on the diagonal, so that we can take the square-root of both and its inverse. We’ll use that to regroup a bit:
(23)
Notice that in the first step, we have pushed from the left into the parenthesised, inner expression, and out the right side again. This does not change the value of the expression. We have then done the same with , and then simplified the remaining expression. Now, if we check the dimensions of , and , they indeed match the reduced order of our system.
In fact, nothing should have changed except for the order of the system, so looking at our impulse response, we should not see any difference:
Similarly, our step response should still look the same:
## Conclusion
Our expedition into the details of the ERA was quite fruitful: We have learned a few techniques for restructuring, analysis and reduction of measurement data.
Many of these will come handy when we examine the Observer/Kalman System Identification Algorithm (OKID). The OKID is used to extract the impulse response from measurement data that was acquired using arbitrary control input.
Further, some of these ideas also will allow us to identify non-linear systems — or at least approximate them. As the engine/propeller combination includes possibly non-linear elements due to aerodynamic drag, this is specifically relevant for the identification of multicopter propulsion systems.
[1] J. Juang and R. S. Pappa, “An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction,” Journal of guidance control and dynamics, vol. 8, iss. 5, 1985.
[Bibtex]
@Article{Juang.Pappa1985,
author = {Juang, Jer-Nan and Pappa, Richard S.},
title = {{A}n {E}igensystem {R}ealization {A}lgorithm for {M}odal {P}arameter {I}dentification and {M}odel {R}eduction},
journal = {Journal of Guidance Control and Dynamics},
year = {1985},
volume = {8},
number = {5},
month = nov,
abstract = {A method called the eigensystem realization algorithm is developed for modal parameter identification and model reduction of dynamic systems from test data. A new approach is introduced in conjunction with the singular-value decomposition technique to derive the basic formulation of minimum order realization which is an extended version of the Ho-Kalman algorithm. The basic formulation is then transformed into modal space for modal parameter identification. Two accuracy indicators are developed to quantitatively identify the system and noise modes. For illustration of the algorithm, an example is shown using experimental data from the Galileo spacecraft.},
doi = {10.2514/3.20031},
file = {:Juang.Pappa1985 - An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction.pdf:PDF},
} | 17,349 | 80,601 | {"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": 1, "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} | 3.328125 | 3 | CC-MAIN-2023-06 | longest | en | 0.937593 |
https://hackerranksolution.in/downtozerods/ | 1,726,283,732,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651548.18/warc/CC-MAIN-20240914025441-20240914055441-00412.warc.gz | 255,041,098 | 12,602 | # Down to Zero II
### Problem Statement :
```You are given Q queries. Each query consists of a single number N. You can perform any of the 2 operations N on in each move:
1: If we take 2 integers a and b where , N = a * b , then we can change N = max( a, b )
2: Decrease the value of N by 1.
Determine the minimum number of moves required to reduce the value of N to 0.
Input Format
The first line contains the integer Q.
The next Q lines each contain an integer, N.
Output Format
Output Q lines. Each line containing the minimum number of moves required to reduce the value of N to 0 .```
### Solution :
``` ```Solution in C :
In C ++ :
#include <bits/stdc++.h>
using namespace std;
int main() {
/* Enter your code here. Read input from STDIN. Print output to STDOUT */
int test;
cin >> test;
while (test--){
int n ;
cin >> n ;
int steps = 0;
if (n==0){
cout << 0 << endl;
continue;
}
if (n==1){
cout << 1 << endl;
continue;
}
vector<int> dist(n+1,0);
queue<int> q;
q.push(n) ;
dist[n] = 1 ;
while (1){
int element = q.front();
q.pop();
if(element == 2){
cout << dist[2] + 1 << endl;
break ;
}
if (dist[element-1] == 0 ){
dist [element-1] = dist[element]+1;
q.push(element-1);
}
for (int i=2; i*i<=element; i++){
if (element%i == 0){
int maxfrac = element/i;
if (dist[maxfrac] == 0) dist [maxfrac] = dist[element] + 1, q.push(maxfrac);
}
}
}
}
return 0;
}
In Java :
import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;
public class Solution {
private static int MAXSZ = (int)(1E6+1);
private static int[] dp;
private static int minimumMove(int n) {
if(n == 0) return 0;
if(dp[n] != -1) return dp[n];
int minMove = Integer.MAX_VALUE;
int sq = (int) Math.sqrt(n);
for(int i = 2; i <= sq; i++) {
if(n % i == 0) {
minMove = Math.min(minMove, 1+minimumMove(n/i));
}
}
minMove = Math.min(minMove, 1+ minimumMove(n-1));
return (dp[n] = minMove);
}
public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
dp = new int[MAXSZ];
Arrays.fill(dp, -1);
dp[0] = 0; dp[1] = 1; dp[2] = 2;
int Q = sc.nextInt();
for(int q = 1; q <= Q; q++) {
int n = sc.nextInt();
System.out.println(minimumMove(n));
}
}
}
In C :
#include <stdio.h>
#include <stdio.h>
#include <string.h>
unsigned long long min(unsigned long long x, unsigned long long y);
unsigned long long ans[1000001];
int main(){
int Q,N,i,j;
memset(ans,-1,sizeof(ans));
ans[0]=0;
for(i=0;i<1000000;i++){
ans[i+1]=min(ans[i+1],ans[i]+1);
for(j=2;j<=i && i*(unsigned long long)j<1000001;j++)
ans[i*j]=min(ans[i*j],ans[i]+1);
}
scanf("%d",&Q);
while(Q--){
scanf("%d",&N);
printf("%llu\n",ans[N]);
}
return 0;
}
unsigned long long min(unsigned long long x, unsigned long long y){
return (x>y)?y:x;
}
In Python3 :
import math
import time
start = time.time()
def Sol( N ):
if N == 0: return 0
Q = [ (N,0) ]
setQ = [ 0 ] * N
while Q:
N, steps = Q.pop(0)
if N == 1: return steps+1
div = int(math.sqrt( N ))
while div > 1:
if N % div == 0 and not setQ[N // div]:
Q.append( (N // div, steps+1) )
setQ[ N // div ] = 1
div -= 1
if not setQ[N-1]:
Q.append( (N-1, steps+1) )
setQ[ N-1 ] = 1
Q = int(input())
for _ in range(Q):
N = int(input())
print(Sol(N))```
```
## Find the Running Median
The median of a set of integers is the midpoint value of the data set for which an equal number of integers are less than and greater than the value. To find the median, you must first sort your set of integers in non-decreasing order, then: If your set contains an odd number of elements, the median is the middle element of the sorted sample. In the sorted set { 1, 2, 3 } , 2 is the median.
## Minimum Average Waiting Time
Tieu owns a pizza restaurant and he manages it in his own way. While in a normal restaurant, a customer is served by following the first-come, first-served rule, Tieu simply minimizes the average waiting time of his customers. So he gets to decide who is served first, regardless of how sooner or later a person comes. Different kinds of pizzas take different amounts of time to cook. Also, once h
## Merging Communities
People connect with each other in a social network. A connection between Person I and Person J is represented as . When two persons belonging to different communities connect, the net effect is the merger of both communities which I and J belongs to. At the beginning, there are N people representing N communities. Suppose person 1 and 2 connected and later 2 and 3 connected, then ,1 , 2 and 3 w
## Components in a graph
There are 2 * N nodes in an undirected graph, and a number of edges connecting some nodes. In each edge, the first value will be between 1 and N, inclusive. The second node will be between N + 1 and , 2 * N inclusive. Given a list of edges, determine the size of the smallest and largest connected components that have or more nodes. A node can have any number of connections. The highest node valu
## Kundu and Tree
Kundu is true tree lover. Tree is a connected graph having N vertices and N-1 edges. Today when he got a tree, he colored each edge with one of either red(r) or black(b) color. He is interested in knowing how many triplets(a,b,c) of vertices are there , such that, there is atleast one edge having red color on all the three paths i.e. from vertex a to b, vertex b to c and vertex c to a . Note that
## Super Maximum Cost Queries
Victoria has a tree, T , consisting of N nodes numbered from 1 to N. Each edge from node Ui to Vi in tree T has an integer weight, Wi. Let's define the cost, C, of a path from some node X to some other node Y as the maximum weight ( W ) for any edge in the unique path from node X to Y node . Victoria wants your help processing Q queries on tree T, where each query contains 2 integers, L and | 1,636 | 5,775 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2024-38 | latest | en | 0.425935 |
https://bookini.ru/interdisciplinary-applied-mathematics/184/ | 1,550,681,606,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247495147.61/warc/CC-MAIN-20190220150139-20190220172139-00242.warc.gz | 486,832,895 | 15,559 | # Interdisciplinary Applied Mathematics
Скачать в pdf «Interdisciplinary Applied Mathematics»
Ti >T2.
If these two containers are connected with a relatively thick channel (A ^ h), the equilibrium condition requires no-flow in the channel. If the channel thickness (h) becomes comparable to the mean free path (A), rarefied gas effects have to be taken into account. In such a case the local equi
librium mechanism is very complex, and interaction of the gas molecules with the walls must also be considered. Here, we consider free-molecular flow conditions (i.e., A ^ h) to simplify the discussion. In this flow regime, the intermolecular collisions are negligible compared to the interaction of molecules with the surfaces. If we assume that molecule-wall interactions are specular (i.e., av = 0), then the following analysis is valid. We assume that the density of the fluid is proportional to the number density (number of molecules per unit volume),
p ж n,
and the temperature of the fluid is proportional to the square of average molecular speed,
T ж c2.
The mass fluxes at the hot and the cold ends of the channel are
mnici and mn2C2,
respectively; here m is the mass of the gas molecules. Then
mnici pi
mn2c2 p2
T2
0.5
Pi
P2
Ti
Ti
0.5
Ti
Ti
0.5
< i,
where we have used the equations of state
P
pRT and
Pi
P2
###### 1.
The above analysis indicates a flow creeping from cold to hot. Osborne Reynolds was the first to realize this phenomenon, and he coined the term thermal transpiration (Reynolds, 1879). At about the same time, Maxwell developed independently a theory for thermal creep. In the early 1900s, Knudsen built a molecular compressor based on the thermal transpiration idea by connecting a series of tubes with constrictions arranged between each tube (Knudsen, 1910a; Knudsen, 1910b). The constrictions were very small, so that the rarefaction effects became important in the constrictions. By heating the same side of these constrictions to very high temperatures (773 K), Knudsen was able to maintain considerable pressure gradients (Loeb, 1961). According to (Ebert and Albrand, 1963), Gaede also described a thermal pump in his unpublished notes. Other developments in thermal creep driven vacuum pumps can be found in (Vargo et al., 1998).
Скачать в pdf «Interdisciplinary Applied Mathematics» | 594 | 2,378 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2019-09 | latest | en | 0.821612 |
https://www.managementstudyhq.com/act-percentiles-and-rankings.html | 1,685,963,087,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224651815.80/warc/CC-MAIN-20230605085657-20230605115657-00096.warc.gz | 968,839,825 | 29,247 | # ACT Percentiles and Rankings: What’s a “Good” ACT Score?
The bad news is that there’s no specific or straight answer I can give regarding what is a good ACT score. The good news, though, is that I can help you figure out what can be considered a good and bad ACT score. This article will try to dig deep; to answer what’s regarded as a good ACT and explore other areas to understand better what ACT is.
## What is a Good ACT Score?
Let’s start with the nitty-gritty – what is a good ACT score? This is where many students get stuck. They have their scores but can’t figure out whether it is good or bad. Students can aim for a 30, 25, or even 20, but the question remains the same – what’s considered a good ACT.
There is no specific answer to what’s a good ACT score – it depends.
And while we also look at the ACT scores, it is essential that you also note that colleges look at many other factors when selecting students. For example, apart from ACT scores, colleges also look at students’ GPAs, recommendation letters, writing samples, etc.
Plus, it is essential that you also note that different colleges have different ACT scores they are targeting. In other words, the scores accepted by colleges will vary depending on the colleges you are applying to.
## What is ACT Percentile Ranking?
An ACT percentile ranking compares your ACT score with other students who also took the same test. In other words, an ACT percentile will point out how many students scored equal to what you achieved or those who you scored more than. Even though your percentile score will give your ranking in the 4-subject areas, it will provide a composite score, too.
ACT scores are designed to stick to a standard distribution. This means that most students tend to find themselves in the middle; most students either score slightly below or slightly higher.
The average ACT score expected for every student is 20.6, which means that a student who scores 21 is considered to have scored 50 percent higher than other students. Likewise, a score of 24 will mean you are at the 74th percentile; that’s three-quarters more than other students.
Your ACT percentile is not a grade that ranges somewhere between 0 and 100, but a comparison between what you score and what other students score.
For instance, if you find yourself at the 70th percentile, what this means is that you scored the same as or slightly higher than 70 percent of other students who took the test. Take note that a 70th percentile doesn’t mean you scored 70 percent of the questions right.
Here’s the most recent ACT percentile data from 2020:
Score English Math Reading Science Composite 36 100 100 100 100 100 35 99 99 98 99 99 34 96 99 96 98 99 33 94 98 94 97 98 32 92 97 91 96 96 31 91 96 89 95 95 30 89 94 86 93 93 29 88 93 84 92 90 28 86 91 82 90 88 27 84 88 80 88 85 26 82 84 77 85 82 25 79 79 74 82 78 24 75 74 71 77 74 23 71 70 66 71 70 22 65 65 61 64 64 21 60 61 55 58 59 20 55 58 50 51 53 19 49 54 44 45 47 18 45 49 39 39 41 17 41 42 34 32 35 16 37 33 29 26 28 15 32 21 24 19 22 14 25 11 19 14 16 13 19 4 14 10 10 12 15 1 10 7 5 11 11 1 5 4 2 10 7 1 3 3 1 9 3 1 1 1 1 8 2 1 1 1 1 7 1 1 1 1 1 6 1 1 1 1 1 5 1 1 1 1 1 4 1 1 1 1 1 3 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1
Again an ACT score of 16 will secure your place at the 28th percentile, which means that you scored equal or better than 28 percent of other students who also took the ACT. And the truth is that falling at the 28th percentile isn’t very good.
A score of 24 means that you score better than the other 74 percent of those who took the test. Likewise, an ACT score of 28 indicates that you have scored better than eighty-eight of other students who sat the same test. A 30 ACT score would be a result to be proud of because you will have beaten ninety-three percent of the other students.
From the above examples and table, it is apparent that most students end up in the near top or near the bottom. Scores between 1 and 8 fall in the first percentile, and scores between 35 and 36 range fall in the 99th percentile, which is excellent.
## ACT Score Requirements
Many universities do not publish or avail their ACT minimum to the public, which makes it challenging to know each college’s requirements. However, you can find ACT score requirements using sites like College Raptor. College Raptor will enable you to see the ACT scores of admitted students to give you a rough idea of what the specific college is looking for.
From there, it is easier to gauge whether you will meet the minimum requirements and whether or not you stand a chance.
## Merit Aid and ACT Scores
Another way to determine if you have a good or bad ACT score is to determine if you qualify for any merit aid. Students with good ACT scores usually qualify for big academic sponsorships – which could be thousands of dollars.
If you want to join a specific university, it would be best to check their website and see if they offer any merit aid and the requirements students must meet to qualify for those merit aids. Qualifying means that you have a good ACT score.
## How Do You Score ACT?
ACT is an entry exam that’s used by colleges or universities to make their admission decisions. There’s the writing component, and there are also other sections, namely:
• English
• Math
• Science
The ACT takes 2 hours and 55 minutes, but the time will go up to 3 hours 35 minutes if you do the writing test. The scores of each section you do – English, Reading, Math, and Science will be determined out of 36 points. From the scores, your composite will be determined to reflect the overall ACT score.
## The Writing Test
The writing part is meant to test your ability to put up ideas into a precise and straightforward piece of writing. In the writing test, you will be asked to give your perspective regarding a specific topic.
The grading won’t be based on your opinions you put down on paper, but it will be based on how best you can put up a clear and concise piece. The idea behind this is to determine if you can present your thoughts in a coherent piece.
## When Should I Take the ACT Exam?
Most experts recommend that students take their ACT exams in the spring of their high school third year. However, the spring might not favor everyone. For example, some students will be taking their sports during the spring, and it’s not a good idea to combine sports and exams.
Usually, it would be ideal for any student to consider the following questions in determining when they should sit for their ACT exams:
• What level of Math are you taking?
• What do you have in your schedules?
• What class are you taking currently?
## How Do I Improve My ACT Scores?
Students who will be taking their ACT exams will want to improve their ACT scores, and this is a question I usually get asked – how do I improve my ACT scores?
If you sit your ACT exam in April, June, or December, you can always sign up for Test Information Release (TIR). The TIR will provide you with copies of the questions, the answers, and the answer key when the results are finally out.
You can register for this service from the ACT registration page immediately after signing up for the test or five days later. If you prefer ordering by email, you have up to 6 months after you sit your test.
You will have to pay for the TIR, but it is worth it because it will provide you with a ton of information. Take note that ACT results are released after 3-8 weeks, so it is essential to ensure that there are ample dates in which you can resit the test before you can use the scores to apply for your college admission.
## Does My ACT Percentile Actually Matter?
Yes, the ACT percentile matters a lot to colleges because they use it to compare your ACT score to other students who also sat the exam. However, knowing the ACT score ranges for the colleges you are interested in joining will be more significant than your percentile.
Different colleges usually have different ACT score ranges that don’t change quite often. The colleges heavily rely on these ACT score averages to admit new students. In most colleges, the ACT ranges between 25-75th percentile.
Finding out the ACT score ranges for different colleges is quite simple. All you have to do is search on Google the terms “College Name ACT scores PrepScholar.” Do this for all the colleges you are targeting. Also, you need to ensure that your score equals the 75th percentile, which is the highest score.
That said, percentiles are very helpful, both to you and the college admissions board, to determine which students they will admit. Ensure that your score is high enough so that your overall percentile is favorable compared to other students who sat the exam.
## Bottom Line
And that’s everything you need to know about ACT. Adequate preparation is essential, especially when you are preparing to join college. Sitting for your ACT is one of those preparations, and it’s good that you now have an idea of what it takes to pass your ACT exams successfully and the various factors involved to admit students into colleges. | 2,220 | 9,091 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2023-23 | latest | en | 0.96814 |
https://www.biology-online.org/dictionary/Three | 1,537,446,620,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267156471.4/warc/CC-MAIN-20180920120835-20180920141235-00308.warc.gz | 693,411,116 | 8,752 | Confused and have questions? We’ve got answers. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you rather get 1:1 study help, try 30 minutes of free online tutoring with Chegg Tutors.
# Three
three
One more than two; two and one. I offer thee three things. Three solemn aisles approach the shrine. (Keble)
Three is often joined with other words, forming compounds signifying divided into, composed of, or containing, three parts, portions, organs, or the like; as, three-branched, three-capsuled, three-celled, three-cleft, three-edged, three-foot, three-footed, three-forked, three-grained, three-headed, three-legged, three-mouthed, three-nooked, three-petaled, three-pronged, three-ribbed, three-seeded, three-stringed, three-toed, and the like.
Origin: OE. Thre, threo, thri, AS. Thri, masc, threo, fem. And neut.; akin to OFries. Thre, OS. Thria, threa, D. Drie, G. Drei, OHG. Dri, Icel. Thrir, Dan. & Sw. Tre, Goth. Threis, Lith. Trys, Ir, Gael. & W. Tri, Russ. Tri, L. Tres, Gr. Trei^s, Skr. Tri. 301. Cf. 3d Drilling, Tern, Third, Thirteen, Thirty, Tierce, Trey, Tri-, Triad, Trinity, Tripod.
1. The number greater by a unit than two; three units or objects.
2. A symbol representing three units, as 3 or III. Rule of three.
(Science: mathematics) See rule. | 401 | 1,329 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2018-39 | latest | en | 0.916046 |
http://www.r-bloggers.com/writing-type-stable-code-in-julia/ | 1,469,405,970,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824201.28/warc/CC-MAIN-20160723071024-00131-ip-10-185-27-174.ec2.internal.warc.gz | 670,613,685 | 16,896 | # Writing Type-Stable Code in Julia
December 6, 2013
By
(This article was first published on John Myles White » Statistics, and kindly contributed to R-bloggers)
For many of the people I talk to, Julia’s main appeal is speed. But achieving peak performance in Julia requires that programmers absorb a few subtle concepts that are generally unfamiliar to users of weakly typed languages.
One particularly subtle performance pitfall is the need to write type-stable code. Code is said to be type-stable if the type of every variable does not vary over time. To clarify this idea, consider the following two closely related function definitions:
``````function sumofsins1(n::Integer)
r = 0
for i in 1:n
r += sin(3.4)
end
return r
end
function sumofsins2(n::Integer)
r = 0.0
for i in 1:n
r += sin(3.4)
end
return r
end
``````
The only difference between these function definitions is that `sumofsins1` initializes `r` to `0`, whereas `sumofsins2` initializes `r` to `0.0`.
This seemingly minor distinction has important practical implications because the initialization of `r` to `0` means that the main loop of `sumofsins1` begins with a single iteration in which the computer adds `0` to `sin(3.4)`. This single addition step transforms the type of `r` from `Int`, which is the type of `0`, to `Float64`, which is the type of `sin(3.4)`. This means that the type of `r` is not stable over the course of this loop.
This instability has considerable effects on the performance of `sumofsins1`. To see this, let’s run some naive benchmarks. As always in Julia, we’ll start with a dry run to get the JIT to compile the functions being compared:
``````sumofsins1(100_000)
sumofsins2(100_000)
@time [sumofsins1(100_000) for i in 1:100];
@time [sumofsins2(100_000) for i in 1:100];
``````
The results of this timing comparison are quite striking:
``````julia> @time [sumofsins1(100_000) for i in 1:100];
elapsed time: 0.412261722 seconds (320002496 bytes allocated)
julia> @time [sumofsins2(100_000) for i in 1:100];
elapsed time: 0.008509995 seconds (896 bytes allocated)
``````
As you can see, the type-unstable code in `sumofsins1` is 50x slower than the type-stable code. What might have seemed like a nitpicky point about the initial value of `r` has enormous performance implications.
To understand the reasons for this huge performance gap, it’s worth considering what effect type-instability has on the compiler. In this case, the compiler can’t optimize the contents of the main loop of `sumofsins1` because it can’t be certain that the type of `r` will remain invariant throughout the entire loop. Without this crucial form of invariance, the compiler has to check the type of `r` on every iteration of the loop, which is a much more intensive computation than repeatedly adding a constant value to a `Float64`.
You can confirm for yourself that the compiler produces more complex code by examining the LLVM IR for both of these functions.
First, we’ll examine the LLVM IR for `sumofsins1`:
``````julia> code_llvm(sumofsins1, (Int, ))
define %jl_value_t* @julia_sumofsins11067(i64) {
top:
%1 = alloca [5 x %jl_value_t*], align 8
%.sub = getelementptr inbounds [5 x %jl_value_t*]* %1, i64 0, i64 0
%2 = getelementptr [5 x %jl_value_t*]* %1, i64 0, i64 2, !dbg !5145
store %jl_value_t* inttoptr (i64 6 to %jl_value_t*), %jl_value_t** %.sub, align 8
%3 = load %jl_value_t*** @jl_pgcstack, align 8, !dbg !5145
%4 = getelementptr [5 x %jl_value_t*]* %1, i64 0, i64 1, !dbg !5145
%.c = bitcast %jl_value_t** %3 to %jl_value_t*, !dbg !5145
store %jl_value_t* %.c, %jl_value_t** %4, align 8, !dbg !5145
store %jl_value_t** %.sub, %jl_value_t*** @jl_pgcstack, align 8, !dbg !5145
%5 = getelementptr [5 x %jl_value_t*]* %1, i64 0, i64 3
store %jl_value_t* null, %jl_value_t** %5, align 8
%6 = getelementptr [5 x %jl_value_t*]* %1, i64 0, i64 4
store %jl_value_t* null, %jl_value_t** %6, align 8
store %jl_value_t* inttoptr (i64 140379580131904 to %jl_value_t*), %jl_value_t** %2, align 8, !dbg !5150
%7 = icmp slt i64 %0, 1, !dbg !5151
br i1 %7, label %L2, label %pass, !dbg !5151
pass: ; preds = %top, %pass
%8 = phi %jl_value_t* [ %13, %pass ], [ inttoptr (i64 140379580131904 to %jl_value_t*), %top ]
%"#s6.03" = phi i64 [ %14, %pass ], [ 1, %top ]
store %jl_value_t* %8, %jl_value_t** %5, align 8, !dbg !5152
%9 = call %jl_value_t* @alloc_2w(), !dbg !5152
%10 = getelementptr inbounds %jl_value_t* %9, i64 0, i32 0, !dbg !5152
store %jl_value_t* inttoptr (i64 140379580056656 to %jl_value_t*), %jl_value_t** %10, align 8, !dbg !5152
%11 = getelementptr inbounds %jl_value_t* %9, i64 1, i32 0, !dbg !5152
%12 = bitcast %jl_value_t** %11 to double*, !dbg !5152
store double 0xBFD05AC910FF4C6C, double* %12, align 8, !dbg !5152
store %jl_value_t* %9, %jl_value_t** %6, align 8, !dbg !5152
%13 = call %jl_value_t* @jl_apply_generic(%jl_value_t* inttoptr (i64 140379586379936 to %jl_value_t*), %jl_value_t** %5, i32 2), !dbg !5152
store %jl_value_t* %13, %jl_value_t** %2, align 8, !dbg !5152
%14 = add i64 %"#s6.03", 1, !dbg !5152
%15 = icmp sgt i64 %14, %0, !dbg !5151
br i1 %15, label %L2, label %pass, !dbg !5151
L2: ; preds = %pass, %top
%.lcssa = phi %jl_value_t* [ inttoptr (i64 140379580131904 to %jl_value_t*), %top ], [ %13, %pass ]
%16 = load %jl_value_t** %4, align 8, !dbg !5153
%17 = getelementptr inbounds %jl_value_t* %16, i64 0, i32 0, !dbg !5153
store %jl_value_t** %17, %jl_value_t*** @jl_pgcstack, align 8, !dbg !5153
ret %jl_value_t* %.lcssa, !dbg !5153
}
``````
Then we’ll examine the LLVM IR for `sumofsins2`:
``````julia> code_llvm(sumofsins2, (Int, ))
define double @julia_sumofsins21068(i64) {
top:
%1 = icmp slt i64 %0, 1, !dbg !5151
br i1 %1, label %L2, label %pass, !dbg !5151
pass: ; preds = %top, %pass
%"#s6.04" = phi i64 [ %3, %pass ], [ 1, %top ]
%r.03 = phi double [ %2, %pass ], [ 0.000000e+00, %top ]
%2 = fadd double %r.03, 0xBFD05AC910FF4C6C, !dbg !5156
%3 = add i64 %"#s6.04", 1, !dbg !5156
%4 = icmp sgt i64 %3, %0, !dbg !5151
br i1 %4, label %L2, label %pass, !dbg !5151
L2: ; preds = %pass, %top
%r.0.lcssa = phi double [ 0.000000e+00, %top ], [ %2, %pass ]
ret double %r.0.lcssa, !dbg !5157
}
``````
The difference in size and complexity of code between these two functions in compiled form is considerable. And this difference is entirely atttributable to the compiler’s need to recheck the type of `r` on every iteration of the main loop in `sumofsins1`, which can be optimized out in `sumofsins2`, where `r` has a stable type.
Given the potential performance impacts of type-instability, every aspiring Julia programmer needs to learn to recognize potential sources of type-instability in their own code. Future versions of Julia may be configured to issue warnings when type-unstable code is encountered, but, for now, the responsibility lies with the programmer. Thankfully, once you learn about type-stability, it becomes easy to recognize in most cases.
R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more... | 2,412 | 7,418 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2016-30 | latest | en | 0.866443 |
https://howtocreate.com/test-what-can-you-multiply-to-get-24-94924/ | 1,721,226,618,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514771.72/warc/CC-MAIN-20240717120911-20240717150911-00432.warc.gz | 268,816,307 | 11,972 | ## What are factors of 24?
Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
## How can 3 numbers make 24?
In this situation, a mathematician thinks about transforming the problem into another one. If we could make 8 from 8, 8, 3, then we would be done, since if f(8, 8, 3) = 8, then f(8, 8, 3) × 3 = 8 × 3 = 24. And that’s the only way you can arrange these numbers to make 24.
## What are two numbers that multiply to 24 and add up to 10?
The numbers are 12 and -2.
## What do you multiply to get 28?
You are correct! You could also multiply 2 x 14 or 1 x 28. These are all factors of 28. Let’s list out the factors of 28: 1, 2, 4, 7, 14, 28.
## How many ways can you make 24?
Using all four numbers 4, 6, 6 and 8, but using each number only once, there are over 60 different ways of getting the answer 24 by adding, subtracting, multiplying and dividing.
## What can you multiply to get 25?
5 × 5 = 25 or 25 = 5.
## What can you multiply to get 21?
List of Factor Pairs for 21
• 1 x 21 = 21.
• 3 x 7 = 21.
• 7 x 3 = 21.
• 21 x 1 = 21.
## What can you multiply to get 26?
1 x 26 = 26. 2 x 13 = 26. 13 x 2 = 26.
## What can you multiply to get 20?
20 = 1 x 20, 2 x 10, or 4 x 5. Factors of 20: 1, 2, 4, 5, 10, 20.
## How many factors does 23 have?
2 factors
The factors of 23 are 1, 23. Therefore, 23 has 2 factors.
## What is the prime factorization of 24?
The prime factorization of the number 24 is 2 × 2 × 2 × 3. You can also write this with exponents as 23 × 3.
## What multiplication makes 22?
22 = 1 x 22 or 2 x 11. Factors of 22: 1, 2, 11, 22.
## What can I multiply to get 23?
When two numbers on multiplication give the actual number, they are called pair factors. The factors of 23 are 1 and 23. Since 23 is a prime number, therefore, it has only two factors. Factor pairs of the number 23 are the natural numbers but not a fraction or decimal number.
## What is the multiple of 23?
The first 5 multiples of 23 are 23, 46, 69, 92, and 115.
## What does it mean by 23 in math?
In mathematics. Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. … It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 23 is the third base-10 repunit prime exponent after 2 and 19.
## What is a 8 table?
The table of 8 denotes the result of multiplying the number 8 with other whole numbers. … A fun fact about the number 8 is that when you flip it becomes the symbol of infinity (∞).
## How many factors does 21 have?
4 factors
Factors of 21 are integers that can be divided evenly into 21. It has a total of 4 factors of which 21 is the biggest factor and the positive factors of 21 are 1, 3, 7, and 21. The sum of all factors of 21 is 32 and its factors in Pairs are (1, 21) and (3, 7).
## What are the multiples of 8?
Multiples of 8 are numbers which can be divided by 8 without leaving a remainder. The first 12 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 and 96.
24 Times Table
24 X 24
24 X 48
24 X 72
24 X 96
24 X 120
## What is a 12 table?
The table of 12 is given by: 12 times 1 is 1, 12 times 2 is 24, 12 times 3 is 36, 12 times 4 is 48, 12 times 5 is 60, 12 times 6 is 72, 12 times 7 is 84, 12 times 8 is 96, 12 times 9 is 108 and 12 times 10 is 120.
## What table does 728 come in?
On this page, you can find the table of 91 up to 20.
Table of 91 up to 10.
91 × 1 = 91 91 × 6 = 546
91 × 2 = 182 91 × 7 = 637
91 × 3 = 273 91 × 8 = 728
91 × 4 = 364 91 × 9 = 819
91 × 5 = 455 91 × 10 = 910 | 1,233 | 3,512 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.84375 | 5 | CC-MAIN-2024-30 | latest | en | 0.900419 |
https://www.nagwa.com/en/videos/582197353249/ | 1,726,770,096,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700652055.62/warc/CC-MAIN-20240919162032-20240919192032-00743.warc.gz | 829,061,970 | 36,166 | Question Video: Analysis of a Weight Suspended by Two Strings Inclined to the Vertical | Nagwa Question Video: Analysis of a Weight Suspended by Two Strings Inclined to the Vertical | Nagwa
# Question Video: Analysis of a Weight Suspended by Two Strings Inclined to the Vertical Mathematics • Second Year of Secondary School
## Join Nagwa Classes
Attend live General Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!
A weight of 90 g-wt is suspended by two inextensible strings. The first is inclined at an angle π to the vertical, and the second is at 30Β° to the vertical. If the magnitude of the tension in the first string is 45 g-wt, find π and the magnitude of the tension π in the second string.
04:34
### Video Transcript
A weight of 90 gram-weight is suspended by two inextensible strings. The first is inclined at an angle π the vertical, and the second is at 30 degrees to the vertical. If the magnitude of the tension in the first string is 45 gram-weight, find π and the magnitude of the tension π in the second string.
We will begin here by sketching a diagram. We are told that a 90 gram-weight is suspended by two inextensible strings as shown. Theyβre inclined at angles π and 30 degrees to the vertical. The magnitude of the tension in the first string is 45 gram-weight. And we need to calculate the magnitude of the tension π in the second string. As angles on a straight line sum to 180 degrees, the angle between the tension π and the 90 gram-weight is 150 degrees. In the same way, the angle between the 45 gram-weight tension and the 90 gram-weight is 180 minus π degrees.
As we have three coplanar, concurrent, and noncollinear forces acting to keep the object in static equilibrium, we can use Lamiβs theorem. This states that π΄ over sin πΌ is equal to π΅ over sin π½, which is equal to πΆ over sin πΎ, where π΄, π΅, and πΆ are the magnitude of the forces and πΌ, π½, and πΎ are the angles opposite them. In this question, the three forces will be π, 90, and 45. The angles will be 180 minus π, π plus 30, and 150. Substituting in these values, we have π over sin of 180 minus π is equal to 90 over sin of π plus 30 which is equal to 45 over sin of 150.
Letβs consider the second and third parts to this equation. sin of 150 degrees is equal to one-half. So, our equation becomes 90 over sin of π plus 30 is equal to 45 over a half. 45 divided by a half is equal to 90. And we can then multiply both sides by the sin of π plus 30. Dividing both sides of this new equation by 90 gives us one is equal to the sin of π plus 30. We can then take the inverse sin of both sides of this equation. The inverse sin of one is equal to 90 degrees. So, 90 is equal to π plus 30. Subtracting 30 from both sides of this equation gives us π is equal to 60. The first part of our answer is π is equal to 60 degrees.
We can now substitute this value back into our equation to help calculate π. Using the first two parts of our equation, we have π over sin of 120 degrees is equal to 90 over sin of 90 degrees. sin of 90 degrees is equal to one, and we can then multiply both sides by the sin of 120 degrees. π is equal to 90 multiplied by the sin of 120 degrees. The sin of 120 degrees is equal to root three over two. Multiplying this by 90, we get π is equal to 45 root three. The tension in the second string is equal to 45 root three gram-weight. The two answers to this question are π equals 60 degrees and π is equal to 45 root three gram-weight.
## Join Nagwa Classes
Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!
• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions | 1,035 | 3,834 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.65625 | 5 | CC-MAIN-2024-38 | latest | en | 0.891277 |
https://acqnotes.com/acqnote/tasks/cost-estimating-overview | 1,696,172,064,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510903.85/warc/CC-MAIN-20231001141548-20231001171548-00022.warc.gz | 95,254,945 | 38,284 | ### Cost Estimating
Cost Estimating Overview
Cost Estimating is the process of developing a cost estimate. Cost estimating involves collecting and analyzing historical data and applying quantitative models, techniques, tools, and databases to predict a program’s future cost. More simply, cost estimating combines science and art to predict the future cost of something based on known historical data that are adjusted to reflect new materials, technology, software languages, and development teams. A cost estimate is the product of cost estimating.
Definition: Cost estimating is the examination and study of anticipated expenses, typically determined from comparing the price, performance, schedule, and technical information of comparable goods and services in the past.
A cost estimate is necessary for government acquisition programs for many reasons: to support decisions about funding one program over another, to develop annual budget requests, to evaluate resource requirements at key decision points, and to develop performance measurement baselines. Moreover, having a realistic estimate of projected costs makes for effective resource allocation, and it increases the probability of a program’s success.
### Cost Estimating Categories
There are primarily two (2) main cost estimate categories:
1. Life-Cycle Cost Estimate (LCCE) that may include Independent Cost Estimate (ICE), independent cost assessments, or Total Ownership Costs (TOC), and
2. Business Case Analysis (BCA) that may include an Analysis of Alternatives (AoA) or economic analyses.
### Cost Estimating Topics
The following topics should be known by Program Personnel and Program Managers (PM) about Cost Estimating. They include:
### Cost Estimating Methods
There are a number of cost estimating methods that can be used in estimating the costs of a future and current of a system. The use of a specific approach will depend on how much information is available and where the system is in its development and lifecycle. A few of the most common cost estimating techniques are:
• Parametric: The parametric technique uses regression or other statistical methods to develop Cost Estimating Relationships (CERs).
• Analogy: An analogy is a technique used to estimate a cost based on historical data for an analogous system or subsystem.
• Engineering Estimate: With this technique (Also called Bottoms Up), the system being costed is broken down into lower-level components (such as parts or assemblies), each of which is costed separately for direct labor, direct material, and other costs.
• Actual Costs: With this technique, actual cost experience or trends (from prototypes, engineering development models, and/or early production items) are used to project estimates of future costs for the same system.
• Three-Point Estimate: This Program Analysis and Review Technique (PERT) style cost estimate has the project manager identifies three separate estimates.
### Cost Estimating Needed Skills
Cost estimating requires good organizational skills, in order to pull together disparate data for each cost element and to package it in a meaningful way. It also requires engineering and mathematical skills, to fully understand the quality of the data available. Excellent communication skills are also important for clarifying the technical aspects of a program with technical specialists.
If the program has no technical baseline description, or if the cost estimating team must develop one, it is essential that the team have access to the Subject Matter Experts (SME); PM, system and software engineers, test and evaluation analysts—who are familiar with the program or a program like it. | 685 | 3,688 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2023-40 | longest | en | 0.901442 |
https://www.studysmarter.us/textbooks/physics/fundamentals-of-physics-10th-edition/energy-from-the-nucleus/q13p-question-assume-that-immediately-after-the-fission-of-a/ | 1,685,889,884,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649986.95/warc/CC-MAIN-20230604125132-20230604155132-00017.warc.gz | 1,082,216,448 | 25,826 | • :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon
Suggested languages for you:
Americas
Europe
Q13P
Expert-verified
Found in: Page 1331
### Fundamentals Of Physics
Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718
# Question: Assume that immediately after the fission of ${}^{{\mathbf{236}}}{\mathbf{U}}$ according to Eq. 43-1, the resulting ${}^{{\mathbf{140}}}{\mathbf{Xe}}{\mathbf{}}{\mathbf{and}}{\mathbf{}}{}^{{\mathbf{94}}}{\mathbf{Sr}}$ nuclei are just touching at their surfaces. (a) Assuming the nuclei to be spherical, calculate the electric potential energy associated with the repulsion between the two fragments. (Hint: Use Eq. 42-3 to calculate the radii of the fragments.) (b) Compare this energy with the energy released in a typical fission event.
(a) The potential electrical energy is 251 MeV.
(b) The electrical potential energy is greater than typical fission energy.
See the step by step solution
## Step 1: Given data
The mass number of Xenon, ${A}_{Xe}=140$
The mass number of Strontium, ${A}_{Sr}=94$
The mass number of Uranium, ${A}_{U}=236$
## Step 2: Determine the formulas to calculate the electric potential energy
The expression to calculate the radii of the fragment is given as follows.
${\mathbit{r}}{\mathbf{=}}{{\mathbit{r}}}_{{\mathbf{0}}}{{\mathbit{A}}}^{\mathbf{1}\mathbf{/}\mathbf{3}}$ ...(i)
Here, A is the mass number.
The expression to calculate the electrical potential energy is given as follows.
${\mathbit{W}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\frac{{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{\mathbf{r}}$ ...(ii)
Here, ${{\mathbf{q}}}_{{\mathbf{1}}}{\mathbf{.}}{{\mathbf{q}}}_{{\mathbf{2}}}$are the charges and r is the distance between the charges.
## Step 3: (a) Calculate the electrical potential energy.
The atomic number of Strontium, ${Z}_{Sr}=38$
The atomic number of Xenon, ${Z}_{Xe}=54$
Calculate the radii of the Xenon,
Substitute 1.2 fm for ${r}_{0}$ and 140 for A into equation (i).
${r}_{Xe}=1.2\mathrm{fm}×{\left(140\right)}^{1/3}\phantom{\rule{0ex}{0ex}}{r}_{Xe}=1.2\mathrm{fm}×5.1924\phantom{\rule{0ex}{0ex}}{r}_{Xe}=6.2\mathrm{fm}$
Calculate the radii of the strontium,
Substitute 1.2 fm for ${r}_{0}$ and 94 for A into equation (i).
${r}_{Sr}=1.2\mathrm{fm}×{\left(94\right)}^{1/3}\phantom{\rule{0ex}{0ex}}{r}_{Sr}=1.2\mathrm{fm}×4.546\phantom{\rule{0ex}{0ex}}{r}_{Sr}=5.48\mathrm{fm}$
Calculate the electrical potential energy.
Substitute ${Z}_{Xe}$ for ${q}_{1}$, ${Z}_{Sr}$ for ${q}_{2}$,${r}_{Sr}+{r}_{Xe}$ for r into above equation (i).
$W=\frac{1}{4\pi ×{\epsilon }_{0}}×\frac{{Z}_{Xe}{Z}_{Sr}}{\left({r}_{Xe}+{r}_{Sr}\right)}$
Substitute 6.24 for ${r}_{Xe}$, 5.48 for ${r}_{Sr}$, 54e for ${Z}_{Xe}$ and 38e for ${Z}_{Xe}$ into above equation.
localid="1661927633337" $\begin{array}{r}W=\frac{1}{4\pi ×\left(8.85×{10}^{-12}{\mathrm{C}}^{2}/{\mathrm{Nm}}^{2}\right)}×\frac{54\mathrm{e}×38\mathrm{e}}{\left(6.24+5.48\right)×{10}^{-15}\mathrm{m}}\\ W=8.99×{10}^{9}\frac{{\mathrm{Nm}}^{2}}{{\mathrm{C}}^{2}}×\frac{2052{\mathrm{e}}^{2}}{11.72×{10}^{-15}\mathrm{m}}\end{array}$
Substitute $1.6×{10}^{-19}\mathrm{C}$ for e into above equation.
$\begin{array}{r}W=8.99×{10}^{9}\frac{{\mathrm{Nm}}^{2}}{{\mathrm{C}}^{2}}×\frac{2052{\left(1.6×{10}^{-19}\mathrm{C}\right)}^{2}}{11.72×{10}^{-15}\mathrm{m}}\\ W=\frac{18447.48×2.56×{10}^{24}×{10}^{-38}\mathrm{J}}{11.72}\\ W=\frac{47225.54×{10}^{-14}\mathrm{J}}{11.72}\\ W=4029×{10}^{-11}\mathrm{J}\end{array}$
Convert the energy into MeV,
$W=\frac{4.033×{10}^{-11}}{1.602×{10}^{-19}}\mathrm{eV}\phantom{\rule{0ex}{0ex}}\mathrm{W}=2.515×{10}^{8}\mathrm{eV}\phantom{\rule{0ex}{0ex}}\mathrm{W}=251.5\mathrm{MeV}$
By rounding down the value of the energy is 251 MeV.
Hence the potential electrical energy is 251 MeV.
## Step 4: (b) Compare the electrical potential energy with typical fission energy.
The energy released in the typical fission energy is 200 MeV and the electrical potential energy from the part (a) is 251 MeV . By comparing the two energies, it is clear that the electrical potential energy is greater than the typical fission energy. This energy appears in the form of kinetic, beat and sound energy.
Hence the electrical potential energy is greater than typical fission energy. | 1,572 | 4,377 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 37, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2023-23 | latest | en | 0.675178 |
https://math24.net/exact-differential-equations-page-2.html | 1,716,974,593,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059221.97/warc/CC-MAIN-20240529072748-20240529102748-00237.warc.gz | 317,710,008 | 4,377 | # Exact Differential Equations
## Solved Problems
### Example 3.
Solve the differential equation ${e^y}dx + \left( {2y + x{e^y}} \right)dy = 0.$
Solution.
First we check this equation for exactness:
$\frac{{\partial Q}}{{\partial x}} = \frac{\partial }{{\partial x}}\left( {2y + x{e^y}} \right) = {e^y},\;\; \frac{{\partial P}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {{e^y}} \right) = {e^y}.$
We see that $$\frac{{\partial Q}}{{\partial x}} = \frac{{\partial P}}{{\partial y}},$$ so that this equation is exact. Find the function $$u\left( {x,y} \right)$$ from the system of equations:
$\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = {e^y}\\ \frac{{\partial u}}{{\partial y}} = 2y + x{e^y} \end{array} \right..$
Hence,
$u\left( {x,y} \right) = \int {P\left( {x,y} \right)dx} = \int {{e^y}dx} = x{e^y} + \varphi \left( y \right).$
Now, by differentiating this expression with respect to $$y$$ and equating it to $$\frac{{\partial u}}{{\partial y}},$$ we find the derivative $$\varphi'\left( y \right):$$
$\frac{{\partial u}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {x{e^y} + \varphi \left( y \right)} \right] = 2y + x{e^y},\;\; \Rightarrow \cancel{x{e^y}} + \varphi'\left( y \right) = 2y + \cancel{x{e^y}},\;\; \Rightarrow \varphi'\left( y \right) = 2y.$
As a result, we find $${\varphi \left( y \right)}$$ and the entire function $$u\left( {x,y} \right):$$
$\varphi \left( y \right) = \int {2ydy} = {y^2},\;\; \Rightarrow u\left( {x,y} \right) = x{e^y} + \varphi \left( y \right) = x{e^y} + {y^2}.$
Thus, the general solution of the differential equation is
$x{e^y} + {y^2} = C.$
### Example 4.
Solve the equation
$\left( {2xy - \sin x} \right)dx + \left( {{x^2} - \cos y} \right)dy = 0.$
Solution.
This differential equation is exact because
$\frac{{\partial Q}}{{\partial x}} = \frac{\partial }{{\partial x}}\left( {{x^2} - \cos y} \right) = 2x = \frac{{\partial P}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {2xy - \sin x} \right) = 2x.$
We find the function $$u\left( {x,y} \right)$$ from the system of two equations:
$\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = 2xy - \sin x\\ \frac{{\partial u}}{{\partial y}} = {x^2} - \cos y \end{array} \right..$
By integrating the 1st equation with respect to the variable $$x,$$ we have
$u\left( {x,y} \right) = \int {\left( {2xy - \sin x} \right)dx} = {x^2}y + \cos x + \varphi \left( y \right).$
Plugging in the $$2$$nd equation, we obtain
$\frac{{\partial u}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {{x^2}y + \cos x + \varphi \left( y \right)} \right] = {x^2} - \cos y,\;\; \Rightarrow \cancel{x^2} + \varphi'\left( y \right) = \cancel{x^2} - \cos y,\;\; \Rightarrow \varphi'\left( y \right) = - \cos y.$
Hence,
$\varphi \left( y \right) = \int {\left( { - \cos y} \right)dy} = - \sin y.$
Thus, the function $$u\left( {x,y} \right)$$ is
$u\left( {x,y} \right) = {x^2}y + \cos x - \sin y,$
so that the general solution of the differential equation is given by the implicit formula:
${x^2}y + \cos x - \sin y = C.$
### Example 5.
Solve the equation
$\left( {1 + 2x\sqrt {{x^2} - {y^2}} } \right)dx - 2y\sqrt {{x^2} - {y^2}} dy = 0.$
Solution.
First of all we determine whether this equation is exact:
$\frac{{\partial Q}}{{\partial x}} = \frac{\partial }{{\partial x}}\left( { - 2y\sqrt {{x^2} - {y^2}} } \right) = - 2y \cdot \frac{{2x}}{{2\sqrt {{x^2} - {y^2}} }} = - \frac{{2xy}}{{\sqrt {{x^2} - {y^2}} }},$
$\frac{{\partial P}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {1 + 2x\sqrt {{x^2} - {y^2}} } \right) = 2x \cdot \frac{{\left( { - 2y} \right)}}{{2\sqrt {{x^2} - {y^2}} }} = - \frac{{2xy}}{{\sqrt {{x^2} - {y^2}} }}.$
As you can see, $$\frac{{\partial Q}}{{\partial x}} = \frac{{\partial P}}{{\partial y}}.$$ Hence, this equation is exact. Find the function $$u\left( {x,y} \right),$$ satisfying the system of equations:
$\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = 1 + 2x\sqrt {{x^2} - {y^2}} \\ \frac{{\partial u}}{{\partial y}} = - 2y\sqrt {{x^2} - {y^2}} \end{array} \right..$
Integrating the first equation gives:
$u\left( {x,y} \right) = \int {\left( {1 + 2x\sqrt {{x^2} - {y^2}} } \right)dx} = x + \frac{{{{\left( {{x^2} - {y^2}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \varphi \left( y \right) = x + \frac{2}{3}{\left( {{x^2} - {y^2}} \right)^{\frac{3}{2}}} + \varphi \left( y \right),$
where $$\varphi \left( y \right)$$ is a certain unknown function of $$y$$ that will be defined later.
We substitute the result into the second equation of the system:
$\frac{{\partial u}}{{\partial y}} = \frac{\partial }{{\partial y}}\Big[ {x + \frac{2}{3}{{\left( {{x^2} - {y^2}} \right)}^{\frac{3}{2}}} + \varphi \left( y \right)} \Big] = - 2y\sqrt {{x^2} - {y^2}} ,\;\; \Rightarrow - \cancel{2y\sqrt {{x^2} - {y^2}}} + \varphi'\left( y \right) = - \cancel{2y\sqrt {{x^2} - {y^2}}} ,\;\; \Rightarrow \varphi'\left( y \right) = 0.$
By integrating the last expression, we find the function $$\varphi \left( y \right):$$
$\varphi \left( y \right) = C,$
where $$C$$ is a constant.
Thus, the general solution of the given differential equation has the form:
$x + \frac{2}{3}{\left( {{x^2} - {y^2}} \right)^{\frac{3}{2}}} + C = 0.$
### Example 6.
Solve the differential equation $\frac{1}{{{y^2}}} - \frac{2}{x} = \frac{{2xy'}}{{{y^3}}}$ with the initial condition $$y\left( 1 \right) = 1.$$
Solution.
Check the equation for exactness by converting it into standard form:
$\frac{1}{{{y^2}}} - \frac{2}{x} = \frac{{2x}}{{{y^3}}}\frac{{dy}}{{dx}},\;\; \Rightarrow \left( {\frac{1}{{{y^2}}} - \frac{2}{x}} \right)dx = \frac{{2x}}{{{y^3}}}dy,\;\; \Rightarrow \left( {\frac{1}{{{y^2}}} - \frac{2}{x}} \right)dx - \frac{{2x}}{{{y^3}}}dy = 0.$
The partial derivatives are
$\frac{{\partial Q}}{{\partial x}} = \frac{\partial }{{\partial x}}\left( { - \frac{{2x}}{{{y^3}}}} \right) = - \frac{2}{{{y^3}}},\;\;\; \frac{{\partial P}}{{\partial y}} = \frac{\partial }{{\partial y}}\left( {\frac{1}{{{y^2}}} - \frac{2}{x}} \right) = - \frac{2}{{{y^3}}}.$
Hence, the given equation is exact. Therefore, we can write the following system of equations to determine the function $$u\left( {x,y} \right):$$
$\left\{ \begin{array}{l} \frac{{\partial u}}{{\partial x}} = \frac{1}{{{y^2}}} - \frac{2}{x}\\ \frac{{\partial u}}{{\partial y}} = - \frac{{2x}}{{{y^3}}} \end{array} \right..$
In the given case, it is more convenient to integrate the second equation with respect to the variable $$y:$$
$u\left( {x,y} \right) = \int {\left( { - \frac{{2x}}{{{y^3}}}} \right)dy} = \frac{x}{{{y^2}}} + \psi \left( x \right).$
Now we differentiate this expression with respect to the variable $$x:$$
$\frac{{\partial u}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{x}{{{y^2}}} + \psi \left( x \right)} \right] = \frac{1}{{{y^2}}} - \frac{2}{x},\;\; \Rightarrow \cancel{\frac{1}{{{y^2}}}} + \psi'\left( x \right) = \cancel{\frac{1}{{{y^2}}}} - \frac{2}{x},\;\; \Rightarrow \psi'\left( x \right) = - \frac{2}{x},\;\; \Rightarrow \psi \left( x \right) = - 2\ln \left| x \right| = \ln \frac{1}{{{x^2}}}.$
Thus, the general solution of the differential equation in implicit form is given by the expression:
$\frac{x}{{{y^2}}} + \ln \frac{1}{{{x^2}}} = C.$
The particular solution can be found using the initial condition $$y\left( 1 \right) = 1.$$ By substituting the initial values, we find the constant $$C:$$
$\frac{1}{{{1^2}}} + \ln \frac{1}{{{1^2}}} = C,\;\; \Rightarrow 1 + 0 = C,\;\; \Rightarrow C = 1.$
Hence, the solution of the given initial value problem is
$\frac{1}{{{y^2}}} + \ln \frac{1}{{{x^2}}} = 1.$ | 2,940 | 7,627 | {"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} | 4.5625 | 5 | CC-MAIN-2024-22 | latest | en | 0.597278 |
http://wuyudong.com/2018/08/21/3265.html | 1,726,506,487,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651697.45/warc/CC-MAIN-20240916144317-20240916174317-00801.warc.gz | 37,715,285 | 5,833 | # 工学1号馆
home
## 全排列的生成算法–基础的回溯递归
Wu Yudong August 21, 2018 Algorithm 655
// 版权所有（C）wu yudong
// 博客地址：http://www.wuyudong.com
#include<stdio.h>
#include<math.h>
#include<string.h>
#include<ctype.h>
#include<stdlib.h>
#include<stdbool.h>
#include<time.h>
int Count = 0;
void Swap(int *a, int *b)
{
int t;
t = *a;
*a = *b;
*b = t;
}
//输出A数组前n位
void Print(int A[], int n)
{
for (int i = 0; i < n; i++) {
printf("%d%c", A[i], i == n - 1 ? '\n' : ' ');
}
}
void Permutation(int A[], int m, int n)
{
if (m == n) {
Print(A, n);
Count++;
} else {
for (int i = m; i < n; i++) {
Swap(&A[m], &A[i]);
Permutation(A, m + 1, n);
Swap(&A[m], &A[i]);
}
}
}
int main()
{
int array[] = { 1, 2, 3, 4 };
int n = sizeof(array) / sizeof(array[0]);
int begintime, endtime;
begintime = clock(); //计时开始
Permutation(array, 0, n);
endtime = clock(); //计时结束
printf("%d\n", Count);
printf("Running Time: %dms\n", endtime - begintime);
return 0;
} | 373 | 938 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2024-38 | latest | en | 0.124576 |
https://www.lmfdb.org/Genus2Curve/Q/20736/g/ | 1,603,155,957,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107867463.6/warc/CC-MAIN-20201019232613-20201020022613-00682.warc.gz | 812,024,629 | 5,381 | # Properties
Label 20736.g Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes
# Related objects
## Genus 2 curves in isogeny class 20736.g
Label Equation
20736.g.186624.1 $$y^2 = x^5 + 2x^4 - 9x^3 + 7x^2 - x$$
## L-function data
Analytic rank:$$0$$
Mordell-Weil rank:$$0$$
Prime L-Factor
$$2$$$$1$$
$$3$$$$1 + 3 T^{2}$$
Good L-factors:
Prime L-Factor
$$5$$$$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 + 14 T^{2} + 289 T^{4}$$
$$19$$$$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$
$$29$$$$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$
## Sato-Tate group
$$\mathrm{ST} =$$ $E_6$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$
## Endomorphisms of the Jacobian
Of $$\GL_2$$-type over $$\Q$$
Endomorphism algebra over $$\Q$$:
$$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$
Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{36})^+$$ with defining polynomial $$x^{6} - 6 x^{4} + 9 x^{2} - 3$$
Endomorphism algebra over $$\overline{\Q}$$:
$$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class. | 788 | 1,728 | {"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} | 2.515625 | 3 | CC-MAIN-2020-45 | longest | en | 0.456044 |
http://mathhelpforum.com/statistics/107454-geometric-distribution-question.html | 1,480,788,781,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541066.79/warc/CC-MAIN-20161202170901-00113-ip-10-31-129-80.ec2.internal.warc.gz | 171,228,230 | 10,334 | 1. ## Geometric Distribution Question
In the process of meiosis, a single parent diploid cell goes through eight different phases. However, only 60% of the processes pass the first six phases and only 40% pass all eight. Assume the results from each phase are independent.
(a) If the probability of a successful pass of each one of the first six phases is constant, what is the probability of a successful pass of a single one of these phases?
(b) If the probability of a successful pass of each one of the last two phases is constant, what is the probability of a successful pass of a single one of these phases?
I am not 100% sure this is a geometric distribution, but I am not sure how to set up the formula. Thanks for any help.
2. Hello Oblivionwarrior
Originally Posted by Oblivionwarrior
In the process of meiosis, a single parent diploid cell goes through eight different phases. However, only 60% of the processes pass the first six phases and only 40% pass all eight. Assume the results from each phase are independent.
(a) If the probability of a successful pass of each one of the first six phases is constant, what is the probability of a successful pass of a single one of these phases?
(b) If the probability of a successful pass of each one of the last two phases is constant, what is the probability of a successful pass of a single one of these phases?
I am not 100% sure this is a geometric distribution, but I am not sure how to set up the formula. Thanks for any help.
(a) If the probability of a successful pass of each one of the first six phases is $x$, and the results are independent, then the probability that all six phases are passed is $x^6 = 0.6$
$\Rightarrow x= \sqrt[6]{0.6}=0.9184$ (to 4 d.p.)
(b) If the probability of a successful pass of each one of the last two phases is $y$, then the probability that both of these are passed is $y^2$. So:
$x^6y^2=0.4$
$\Rightarrow y^2=\frac{0.4}{0.6}$
$\Rightarrow y = \sqrt{\frac23}=0.8165$ (to 4 d.p.) | 493 | 1,991 | {"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": 8, "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} | 4.03125 | 4 | CC-MAIN-2016-50 | longest | en | 0.930805 |
https://www.owlgen.in/what-do-you-understand-by-price-discrimination-in-monopoly/ | 1,685,275,120,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224643784.62/warc/CC-MAIN-20230528114832-20230528144832-00597.warc.gz | 1,031,207,640 | 27,054 | # What do you understand by price discrimination in monopoly?
Price discrimination means selling a good to different buyers at two or more price for reasons not associated with difference in cost, goods are assumed to be identical in nature. In the words of J. S. Bains, “Price discrimination refers strictly to the practice by a seller to charging different prices from different buyers for the same good.” According to Koutsoyiannis, “Price discrimination exists when the same product is sold at different prices to different buyers”.
###### Conditions : Price discrimination is possible under following conditions:
1. Different markets have different price elasticities of demand.
2. It is possible to partition total market by segregating buyers into groups Or sub -markets according to elasticity.
3. Control over supply : The supply is in full control of the monopolist. It also Implies that no resale is possible among the market.
4. It is profitable to discriminate.
Price Determination under Price Discrimination : Price determination under discriminating monopoly is illustrated in figure. Supposing market is divided into two parts ‘A’ and ‘B’. As is evident from the slopes of ARand ARdemand is less elastic in market ‘A’ than in market ‘B’. In this figure, ARA curve and ARB, curve are the demand curves of market ‘A’ and ‘B’ respectively. The combined equilibrium position of both the markets is shown in figure (A and B).
Obviously, the equilibrium of the monopoly firm will be at point E where combined MR is equal to MC of the total output OQ. The monopolist will so distribute his total output OQ in two markets that MR of each market is the same. If MR is less in one market and more in the other, then it will be profitable for monopolist to transfer commodities from less -MR market to -more -MR market.
For the equality of MR in different market, the monopolist will sell OA quantity of output in market ‘A’ and OB quantity in market ‘B’. He will sell less quantity in market ‘A’ at OP1 price and more in market ‘B’ at OPprice. Total quantity of output in two markets, i.e., OA+ OB will be equal to the total output OQ produced by the monopolist.
It is clear from fig. that (i) marginal cost of total output is equal to combined marginal revenue, (ii) marginal revenue of both markets is equal, (iii) marginal revenue of both markets is equal to marginal cost of total output.
It is evident from fig. that elasticity of demand in market ‘A’ is less than market ‘B’. Thus in market ‘A’ price will be more and quantity sold will be less than in market ‘B’. In other words, Price is higher is less elastic market and lower in more elastic market.
• Total (0)
0 | 579 | 2,687 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2023-23 | longest | en | 0.952583 |
https://documen.tv/question/you-are-given-that-4a-2b-10-a-write-down-the-value-of-2a-b-17711298-97/ | 1,719,121,332,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862430.93/warc/CC-MAIN-20240623033236-20240623063236-00757.warc.gz | 189,483,099 | 15,922 | ## You are given that 4a-2b = 10 (a) Write down the value of 2a – b
Question
You are given that
4a-2b = 10
(a)
Write down the value of
2a – b
in progress 0
3 years 2021-08-20T00:14:51+00:00 1 Answers 18 views 0
2a – b = 5
Step-by-step explanation:
4a – 2b = 10
take out the highest common factor of the left side
2(2a – b) = 10
divide both sides by two
2a – b = 10/2
2a – b = 5 | 159 | 388 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2024-26 | latest | en | 0.821134 |
https://www.caniry.com/what-is-the-formula-for-calculating-percent-abundance/ | 1,702,309,759,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679515260.97/warc/CC-MAIN-20231211143258-20231211173258-00014.warc.gz | 751,427,630 | 18,030 | # What Is The Formula For Calculating Percent Abundance?
Sample Problem: Calculating Atomic Mass
Change each percent abundance into decimal form by dividing by 100. Multiply this value by the atomic mass of that isotope. Add together for each isotope to get the average atomic mass.
## How do you find the isotopic?
Subtract the atomic number (the number of protons) from the rounded atomic weight. This gives you the number of neutrons in the most common isotope. Use the interactive periodic table at The Berkeley Laboratory Isotopes Project to find what other isotopes of that element exist.
## How do you solve for isotope abundance?
As a percent, the equation would be: (x) + (100-x) = 100, where the 100 designates the total percent in nature. If you set the equation as a decimal, this means the abundance would be equal to 1. The equation would then become: x + (1 – x) = 1. Note that this equation is limited to two isotopes.
### What Is percent abundance of an isotope?
The relative abundance of an isotope is the percentage of atoms with a specific atomic mass found in a naturally occurring sample of an element.
### What is isotopic formula?
Enter the values into the following formula: a = b (x) + c (1 – x). In the equation, “a” is the average atomic mass, “b” is the atomic mass of one isotope, “c” is the atomic mass of the other isotope, and “x” is the abundance of the first isotope. For example, 10.811 = 10.013 (x) + 11.009 (1 – x)
### What is isotopic no?
: the number of neutrons minus the number of protons in an atomic nucleus.
### How do you create an isotope?
This can be done by firing high-speed particles into the nucleus of an atom. When struck, the nucleus may absorb the particle or become unstable and emit a particle. In either case, the number of particles in the nucleus would be altered, creating an isotope.
### How do you calculate species abundance?
Relative species abundance is calculated by dividing the number of species from one group by the total number of species from all groups.
### What is the natural abundance of Cl 35?
Chlorine naturally exists as two isotopes, 17 35 Cl (chlorine-35) and 17 37 Cl (chlorine-37). The abundance of chlorine-35 is 75% and the abundance of chlorine-37 is 25%.
### What is the formula for atomic mass?
Together, the number of protons and the number of neutrons determine an element’s mass number: mass number = protons + neutrons. If you want to calculate how many neutrons an atom has, you can simply subtract the number of protons, or atomic number, from the mass number.
### What is the natural abundance of Br 79?
bromine-79 atom (CHEBI:52743) The stable isotope of bromine with relative atomic mass 78.918338, 50.69 atom percent natural abundance and nuclear spin 3/2.
### What is percent abundance definition?
Percent abundance is defined as the percent value of the number of isotopes available in nature for a given element.
### What is relevant abundance?
Relative abundance is the percent composition of an organism of a particular kind relative to the total number of organisms in the area. Relative species abundances tend to conform to specific patterns that are among the best-known and most-studied patterns in macroecology.
### What are isotopes give an example?
Isotopes: the atoms of the same element which have the same atomic number Z but differ in their mass number A are called isotopes. Example: Hydrogen has three isotopes ( 1 1 H , X 1 1 X 2 1 2 1 H , X 1 3 X 2 1 2 3 H ) , Protium, Deuterium, Tritium.
### What is Isodiapher example?
A set of nuclides which has a different number of protons and neutrons but there exists the same difference between the protons and neutrons are isodiaphers. For example, Thorium -234 and Uranium -238 are considered as isodiaphers. Hence, Thorium -234 and Uranium -238 are considered as isodiaphers.”
### How are isotopes the same?
all isotopes have the same number of protons and the same number of electrons. Because the electron structure is the same isotopes have the same chemical properties. What is different is the number of neutrons, The different number of neutrons all cause a difference in the atomic weight or mass of the atoms.
### How do you write an isotopic symbol?
To write the symbol for an isotope, place the atomic number as a subscript and the mass number (protons plus neutrons) as a superscript to the left of the atomic symbol. The symbols for the two naturally occurring isotopes of chlorine are written as follows: 3517Cl and 3717Cl.
### What are isotopes notes?
Isotopes are atoms of the same element with different numbers of neutrons, and therefore different atomic masses. While carbon-14 is used in radioactive dating, carbon-12 has a more stable nucleus and therefore is not used in this capacity.
### Why is relative abundance important?
The relative abundance of each species is more evenly distributed than Community 1. While both communities have the same species richness, Community 1 would have greater diversity due to the relative abundance of each species present.
### What is atomic number formula?
Formula of atomic number – formula
Atomic number (Z) of an element = number of protons = number of electrons. Atomic number = Atomic mass – number of neutrons. | 1,225 | 5,285 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2023-50 | latest | en | 0.887761 |
https://www.physicsforums.com/threads/another-one-on-chain-rule.100671/ | 1,600,962,695,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400219221.53/warc/CC-MAIN-20200924132241-20200924162241-00247.warc.gz | 983,046,600 | 14,922 | # Another one on chain rule
Hi,
here is what I'm trying to do:
Find
$$\frac{\partial}{\partial x} f(2x, 3y)$$
First of all, I'm confused by the
$$f(2x, 3y)$$
How does the function look like? I imagine that it is for example
$$f(x,y) = cos(xy) - sin(3xy^2}$$
and that therefore
$$f(2x, 3y) = cos(6xy) - sin(54xy^2)$$
I'm confused, I don't have any good picture of how it might look in real.
Anyway, to accomplish the task:
$$\frac{\partial}{\partial x} f(2x, 3y) = \frac{\partial f}{\partial x}\frac{d(2x)}{dx} + \frac{\partial f}{\partial y}\frac{d(3y)}{dx} = 2\frac{\partial f}{\partial x}$$
Is it ok? If yes, could you please give an example of this scenario? I mean, how could f(2x, 3y) and f(x,y), respectively, look like so that I could see it with the particular functions?
Thank you very much.
## Answers and Replies
Related Calculus and Beyond Homework Help News on Phys.org
$$f(x,y)$$ is just a function mapping of your x and y coordinates.
As for the derivative, when you take the derivative with respect to x, the y values are treated as if they were constants, same for x when you derive with respect for y.
ComputerGeek said:
$$f(x,y)$$ is just a function mapping of your x and y coordinates.
As for the derivative, when you take the derivative with respect to x, the y values are treated as if they were constants, same for x when you derive with respect for y.
So, is my approach ok? And how could the functions look like?
Thank you.
Just imagine that
v(x)=2x, and w(y)=6y
Then
f(2x,6y)=f(v(x), w(y)) for arbitrary f.
for example
if
f(x)=cos(x)
then f(v)= cos(v)=cos(2x) | 497 | 1,608 | {"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": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2020-40 | latest | en | 0.91255 |
https://admin.clutchprep.com/chemistry/practice-problems/83112/a-long-glass-tube-sealed-at-one-end-has-an-inner-diameter-of-10-0-mm-the-tube-is | 1,601,283,488,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600401598891.71/warc/CC-MAIN-20200928073028-20200928103028-00367.warc.gz | 256,631,065 | 36,684 | Problem: A long glass tube, sealed at one end, has an inner diameter of 10.0 mm. The tube is filled with water and inverted into a pail of water. If the atmospheric pressure is 755 mmHg, how high (in mmH2O) is the column of water in the tube ( d of Hg = 13.5 g/mL; d of H2O = 1.00 g/mL)?
⚠️Our tutors found the solution shown to be helpful for the problem you're searching for. We don't have the exact solution yet.
Problem Details
A long glass tube, sealed at one end, has an inner diameter of 10.0 mm. The tube is filled with water and inverted into a pail of water. If the atmospheric pressure is 755 mmHg, how high (in mmH2O) is the column of water in the tube ( d of Hg = 13.5 g/mL; d of H2O = 1.00 g/mL)? | 216 | 713 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2020-40 | latest | en | 0.929221 |
https://coffeebreak.life/how-many-ounces-is-a-full-coffee-pot | 1,675,128,843,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499831.97/warc/CC-MAIN-20230130232547-20230131022547-00126.warc.gz | 192,643,608 | 7,308 | # How many ounces is a full coffee pot?
A full coffee pot typically contains around 60 ounces of coffee.
A cup of coffee generally refers to 8 fluid ounces, so a full cup of coffee would be 16 fluid ounces.
There are 14 cups in a standard size coffee pot.
Most coffee pots hold between 10 and 12 cups of liquid.
A standard coffee pot is usually between 36 and 40 fl oz.
L-carnitine is not necessarily the best fat burner, but it is a popular choice.
The best player on coffee chess is typically considered to be whomever has won the most tournaments. Carlini is considered to be one of the greats when it comes to this game.
Coffee house chess is a type of chess that is played in cafes or other public places where people can come and go as they please.
### How many ounces is a full coffee pot?
Before you brew, it's important to note that a cup of water is 8 ounces, however, a coffee pot cup is 5 ounces. So a 12-cup capacity coffee maker is actually 60-ounces of liquid, or roughly 7 cups of coffee.Dec 7, 2021
View complete answer on https://www.today.com › food › how-make-coffee-drip-c...
### How much is a full cup of coffee?
A "cup" of coffee in the US is usually 4 fluid ounces (118 ml), brewed using 5 fluid ounces (148 ml) of water. Coffee carafes used with drip coffee makers, e.g. Black and Decker models, have markings for both water and brewed coffee as the carafe is also used for measuring water prior to brewing.
View complete answer on https://en.wikipedia.org › wiki › Cup_(unit)
### How much coffee do I put in a 14 cup pot?
Drip Coffee Maker Cups Tablespoons of Coffee
10 Cup Pot 8 Level Tbsp.
12 Cup Pot 9 Level Tbsp.
14 Cup Pot 10 Level Tbsp.
### What is the standard size coffee pot?
A traditional 12 cup coffee pot holds 60 fluid ounces. This is equal to approximately 5 ounces per cup.Feb 22, 2022
View complete answer on https://coffeenutty.com › General Info
### How many cups does a coffee pot hold?
A standard 12 cup coffee pot is going to hold 60 ounces of coffee. If the cup you're drinking from is a standard 8-ounce cup, a full pot of coffee will give you a little over 7 cups to enjoy.Apr 9, 2022
View complete answer on https://coffeeaffection.com › what-do-numbers-on-coffee-...
### How many fl oz is a standard coffee pot?
A traditional 12 cup coffee pot holds 60 fluid ounces. This is equal to approximately 5 ounces per cup. For the perfect cup of coffee, you will use 1-2 tablespoons of ground coffee for each cup of coffee.Feb 22, 2022
View complete answer on https://coffeenutty.com › General Info
### Is L-carnitine the best fat burner?
The Bottom Line L-carnitine is best known as a fat burner — but the overall research is mixed. It is unlikely to cause significant weight loss. However, studies support its use for health, brain function and disease prevention.Nov 6, 2018
View complete answer on https://www.healthline.com › nutrition › l-carnitine
### Who is the best player on coffee chess?
##### Favorite Coffee Chess Player
• The Great Carlini.
• Boston Mike.
• Brooklyn Dave.
• Duck and his dog Winston.
• Butcher.
• Kai.
• Novice Noah.
• Golani Salami.
### What is coffee house chess?
A style of chess characterised by bold and risky but often inaccurate play of a kind popular in coffeehouses in the 19th century. | 825 | 3,297 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2023-06 | latest | en | 0.956293 |
http://math.stackexchange.com/questions/567946/prove-that-any-group-of-order-9-is-isomorphic-to-c-9-or-c-3-times-c-3 | 1,469,365,672,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824037.46/warc/CC-MAIN-20160723071024-00232-ip-10-185-27-174.ec2.internal.warc.gz | 165,090,552 | 17,887 | # Prove that any group of order 9 is isomorphic to $C_9$ or $C_3 \times C_3$
Not homework; reading a group theory book for fun.
This is what I understand so far:
If $\exists x \in G$ of order $9$ then $G \cong C_9$.
If not, then the order of $x$ has to divide $9$. So $\def\ord{\operatorname{ord}}\ord(x) = 1$ or $3$.
If $\ord(x) = 1$ then $x=e$.
So for $x$ of order 3, $\def\cyc#1{\left<#1\right>}\cyc x$ is a subgroup of order $3$.
Let $C$ be all $y \in G$ such that $xy=yx$. This is closed, has inverses, is associative, has identity, so is a subgroup.
The order of $C$ divides $9$ and is at least $3$ since $e, x, x^2$ commute with $x$. If $\ord(C)=9$ then choose $y \not\in\cyc x$.
$\cyc x \times \cyc y$ is $\{(x^a,y^b)\}$. The elements of $G$ are $e, x, x^2, y, y^2, xy, xy^2, x^2y, x^2y^2$. $x^ay^b \to (x^a,y^b)$ is an isomorphism. So $G = C_3 \times C_3$
How do I continue this proof?
Edit: the next step in the book is to determine whether $\ord(C)=3$ for every choice of $x$. It then shows that $\exists x$ with order $9$.
-
What do you mean continue it? It ends with the conclusion you would like. – Tobias Kildetoft Nov 15 '13 at 8:34
But you assumed that the group had no element of order $9$. So clearly all non-identity elements have order $3$. – Tobias Kildetoft Nov 15 '13 at 8:43
You can even prove the general statement: Any group of order $p^2$ is isomorphic to either $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$.
true, but the hard part here (though not terribly hard) is to show that any group of order $p^2$, $p$ prime, is abelian. – Ittay Weiss Nov 15 '13 at 9:12
Your proof is just missing the treatment of the case that $C=\cyc x$, so that $|C|=3$ (your layout does not help the reader detect this omission).
If you know that any group of order $p^k$ with $p$ prime and $k>0$ has a non-trivial central element, then you can see that the omitted case cannot happen. Either some non-identity element of $\cyc x$ is central, but then $x$ is central (since every non-identity element of a subgroup of order $p$ generates it) and $C=G$, or there is a central element outside $\cyc x$, but then too $C$ is larger than just $\cyc x$. | 740 | 2,172 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.921875 | 4 | CC-MAIN-2016-30 | latest | en | 0.851772 |
https://solvedlib.com/1-point-a-normal-distribution-with-mean-0-and,24261 | 1,680,289,646,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296949678.39/warc/CC-MAIN-20230331175950-20230331205950-00324.warc.gz | 591,376,234 | 17,482 | # (1 point) A normal distribution with mean 0 and standard deviation Võ is sampled three times,...
###### Question:
(1 point) A normal distribution with mean 0 and standard deviation Võ is sampled three times, yielding values x, y, z. Find the log-likelihood function In L(0) (type theta for 6): In L(O) = Find the derivative of the log-likelihood with respect to 0 (type theta for 6): a ᏧᎾ [ln LCO] Find the maximum likelihood estimator for 0 (note that there is only one positive value): Ô =
#### Similar Solved Questions
##### An airplane has an average acceleration af 15 m/s from rest, how much time doest take...
An airplane has an average acceleration af 15 m/s from rest, how much time doest take to reach a speed of 325%. sduring takeoff. T the air plane starts A golf ball is kicked from a 8m hill, at an angle of GO。witn thu honzonal with an initiol Sp eed of 25/s a) what''s the intal vertica...
##### Draw Lewis structures for the following molecular compounds and ions: (a) $\mathrm{AlCl}_{3} ;$ (b) $\mathrm{PH}_{3} ;$ (c) $\mathrm{H}_{2} \mathrm{Se}$ (d) $\mathrm{NO}_{2}^{-} ;$ (e) $\mathrm{A} 1 \mathrm{H}_{4}^{-}$
Draw Lewis structures for the following molecular compounds and ions: (a) $\mathrm{AlCl}_{3} ;$ (b) $\mathrm{PH}_{3} ;$ (c) $\mathrm{H}_{2} \mathrm{Se}$ (d) $\mathrm{NO}_{2}^{-} ;$ (e) $\mathrm{A} 1 \mathrm{H}_{4}^{-}$...
##### Question with last attempt is displayed for your review only The Ksp of Fe3(PO4)2 is 4.10...
Question with last attempt is displayed for your review only The Ksp of Fe3(PO4)2 is 4.10 · 10-11 M. Calculate the solubility of each ion in a saturated solution: 2.35*10^-3 Preview M Fe 2+ 1.57*10^-3 * Preview MPO43-...
##### Real AnalysisThe Completeness PropertyExample: Show that the set of all irrational numbers isincomplete set
Real Analysis The Completeness Property Example: Show that the set of all irrational numbers is incomplete set...
##### Keenan industries has a bond outstanding with 15 years to maturity, an 8% coupon paid semiannually,...
keenan industries has a bond outstanding with 15 years to maturity, an 8% coupon paid semiannually, and a $1,000 par value. The bond has a 6% nominal annual yield to maturity, but it can be called in 5 years at a price of$1,050. What is the bond's nominal yield to call?...
##### We were unable to transcribe this imagec. Sales are 20% for cash and 80% on credit....
We were unable to transcribe this imagec. Sales are 20% for cash and 80% on credit. All payments on credit sales are collected in the month following sale. The accounts receivable at December 31 are a result of December credit sales. d. The company's gross margin is 40% of sales. (In other words...
##### Question 5 of 10 Intravenous digitalizing doses of digioxin in children are 80% of oral digitalizing...
Question 5 of 10 Intravenous digitalizing doses of digioxin in children are 80% of oral digitalizing doses O 16 is the intravenous dose for a 3-year-old child weighing 30 lb. if the oral dose is determined to be 20 ug/kg. Round answer to the nearest whole number. Do not include units Answers 1-1 16...
##### Goootean ornnd bz malbrtbat=blrgpeeCLoore ibe compondtha aouk mos bhbpkCCH 8
Goootean ornnd bz malbrtbat=blrgpee CLoore ibe compondtha aouk mos bhbpk CCH 8...
##### Complete the following reaction table and Ksp expression to calculate the solubility of silver sulfide. (48219831453...
Complete the following reaction table and Ksp expression to calculate the solubility of silver sulfide. (48219831453 252 SS32530 25 Concentration (M) Ag2S(s) = 2Agt(aq) + 52 (aq) Initial Change Equilibrium Кsp...
##### How do you simplify 6\frac{1}{5} + \frac{1}{10}?
How do you simplify 6\frac{1}{5} + \frac{1}{10}?...
##### 1 Problem 4 Problem 5 5,000 4.000 0.10 3 Standard Corkscrew 4 Forecast 5 Proj Available...
1 Problem 4 Problem 5 5,000 4.000 0.10 3 Standard Corkscrew 4 Forecast 5 Proj Available 6 MPS Lot size: 2.000 3,000 1.000 2,000 3,500 1500 4.000 4,000 Direct lobor Standard corkscres Deluxe corkscrew Machine time Standard corkscres Deluxe corkscrew 2.000 0.20 Direct labor Machine time 10 Deluxe Cork...
##### Which of the following correctly lists the compounds in order ofincreasing vapor pressure?Question 3 options:1) CH₃OH(l) < F₂(l) < BF₃(l)2) F₂(l) < CH₃OH(l) < BF₃(l)3) BF₃(l) < F₂(l) < CH₃OH(l)4) F₂(l) < BF₃(l) < CH₃OH(l)5) CH₃OH(l) < BF₃(l) < F₂(l)
Which of the following correctly lists the compounds in order of increasing vapor pressure? Question 3 options: 1) CH₃OH(l) < F₂(l) < BF₃(l) 2) F₂(l) < CH₃OH(l) < BF₃(l) 3) BF₃(l) < F₂(l) < CH₃OH(l) 4) F₂(l) < B...
##### The area of the reglon enclosed by the curve V =2 and the line V = 2 is:Select one:32 3319Deiln
The area of the reglon enclosed by the curve V = 2 and the line V = 2 is: Select one: 32 3 3 19 Deiln...
##### Find the sum of the series below:56n(2n - 1)2(2n + n =The sum of the series is
Find the sum of the series below: 56n (2n - 1)2(2n + n = The sum of the series is...
##### Occasionally an individual who remains at high altitude for too long develops chronic mountain sickness. It...
Occasionally an individual who remains at high altitude for too long develops chronic mountain sickness. It is characterized by the following clinical sequelae: (1) an exceptionally high hematocrit (2) elevated pulmonary arterial pressure (3) an enlarged right side of the heart. As it progresses, co...
##### Oucstion 26woints) Wal Wuuld d DoFAE builug FJI Dl 8 Aolulun (Kb-0SIz Cm) (Answet Ei XCsCl Jal .Your AnsverKns yerQucstion 27 coicts' Je13Tmereren: Znenhare Pele eeEne [70 3eF mecnanlam; @no4n SeennnASDetStep428MA1[8] C) Rate HafljQucstion 28 coicts'Tne reaction Zy which Hl is converted into Hz and shounbelo " The reaction seconc crdenyin raze constan: Jx1o 2 Ia: 293K). How MucnAIwill remain aFer seccnd 298k the initial concenirzton HIw350.55 moles? "ens "er JeCMa places
Oucstion 26 woints) Wal Wuuld d DoFAE builug FJI Dl 8 Aolulun (Kb-0SIz Cm) (Answet Ei X CsCl Jal . Your Ansver Kns yer Qucstion 27 coicts' Je13 Tmereren: Znenhare Pele ee Ene [70 3eF mecnanlam; @no4n Seennn ASDet Step 428 MA1[8] C) Rate Haflj Qucstion 28 coicts' Tne reaction Zy which Hl is...
##### Q10 5 PointsUse Green's Theorem t0 evaluate the Ilne Integral (81 -v") dr + (3zy + 2) dv where C Is the rectangle with vertlces (0,0), (4,0), (4,1), and (0,1).Enter your answer herePlease select fllefs)Select fillefs)Save Answver011 5 PointsUse Stoke'5 theorem to find[uv xF) . n]do given F(€,y,2) =~yi+(r+ y)j+( y + 2)k where S Is the portion of 2 = 1-2?_y? above - the ry-plane:Enter your answer here
Q10 5 Points Use Green's Theorem t0 evaluate the Ilne Integral (81 -v") dr + (3zy + 2) dv where C Is the rectangle with vertlces (0,0), (4,0), (4,1), and (0,1). Enter your answer here Please select fllefs) Select fillefs) Save Answver 011 5 Points Use Stoke'5 theorem to find [uv xF) ....
##### 1. What is the percentage of the variance in demand that is explained by the regression model? (please enter your answer as a rounded percentage without the % symbol; e.g-, 0.8241 => 82; use conventional rounding rules: e-g,, 61.49 => 61; 78.50 => 79)2. The mean absolute deviation is(please round to the nearest integer: e.g,, 4.49 => 4; 3.50 => 4)3. The mean error is(please round to the nearest integer: e.g,, 4.49 => 4; 3.50 => 4)4. The (in-sample) forecast for period 22 wil
1. What is the percentage of the variance in demand that is explained by the regression model? (please enter your answer as a rounded percentage without the % symbol; e.g-, 0.8241 => 82; use conventional rounding rules: e-g,, 61.49 => 61; 78.50 => 79) 2. The mean absolute deviation is (plea...
##### ChnDEE the hce reagent for the following cactnnPhMgBr; ethar H;o" PhCH,MgBr ether H;o (C,Hs);P-CHC,Hs; THF Li(C ,HshCu, ether
ChnDEE the hce reagent for the following cactnn PhMgBr; ethar H;o" PhCH,MgBr ether H;o (C,Hs);P-CHC,Hs; THF Li(C ,HshCu, ether...
##### Despile M would exchange chromalography the Jsn polential ol 1 water based based any LH Which Jauio acetonitrIle moblle phase with lechniques fraction by moblle phase of pH 7.2 V 1 eluting from native 4pH becn 2 the lorced t0 use U must have been true? Pick ALL a combination size V that apply: exclusion 'Hd 341 or t0 modlfy the pH: chromalograghy and
Despile M would exchange chromalography the Jsn polential ol 1 water based based any LH Which Jauio acetonitrIle moblle phase with lechniques fraction by moblle phase of pH 7.2 V 1 eluting from native 4pH becn 2 the lorced t0 use U must have been true? Pick ALL a combination size V that apply: exc...
##### The graphs below show the probability density function of exam scores for different sections of students. Scores range from to 100.Answer parts and(a) Describe the possible exam outcome for each of the three class sections whose probability density function is given below:Sectionstudents scoredon the exam:Section 2400students scoredon the examSection 3100students scoredon the exam(b) Of these three sections, which had the greatest mean?Click0 &elec
The graphs below show the probability density function of exam scores for different sections of students. Scores range from to 100.Answer parts and (a) Describe the possible exam outcome for each of the three class sections whose probability density function is given below: Section students scored o...
##### How many outcomes will constitute the sample space resulting from rolling a pair of balanced 6-sided dice?
How many outcomes will constitute the sample space resulting from rolling a pair of balanced 6-sided dice?...
##### Question 2 Way Cool produces two different models of air conditioners. The company produces the mechanical...
Question 2 Way Cool produces two different models of air conditioners. The company produces the mechanical systems in its components department. The mechanical systems are combined with the housing assembly in its finishing department. The activities, costs, and drivers associated with these two ma...
##### First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =|...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would do...
##### What is the force, in terms of Coulomb's constant, between two electrical charges of -15 C and 60 C that are 15 m apart?
What is the force, in terms of Coulomb's constant, between two electrical charges of -15 C and 60 C that are 15 m apart?...
##### NuraoElelaMulct16az 2786emna Total4605Uelg thls Inforation nnarer tnc followingEclectco shldent male; givcn that thc student randomly nursing Find the probability that mlo 1l477 0,1148 818 student; given that the student is male_~ ~0, Find the probabillty that the student nursing - 1158 Are the ansters prcvious questions cqunl? @e rot eqb | Te anqwers Cor polli qestio 5 Calculating Classical Conditional ProbabilitiesCulculate the following classical (theoretical) conditional probabilities. Ts Tt
Nurao Elela Mulct 16az 2786 emna Total 4605 Uelg thls Inforation nnarer tnc following Eclectco shldent male; givcn that thc student randomly nursing Find the probability that mlo 1l477 0,1148 818 student; given that the student is male_~ ~0, Find the probabillty that the student nursing - 1158 Are t...
##### Venlic4I DuWic _ Tutr 0As14 . 0.4524 $04M 04no _ UAudu Hosu 0ITr4 € 0 1775 , IJuia 0 3520$ Exp Ginity AcCCArmicclocinotFill up the data and Plot the free fall graph VS (" From the graph of Free Fall, find the g exp Find the percemt etror,
Venlic4I Du Wic _ Tutr 0As14 . 0.4524 $04M 04no _ UAudu Hosu 0ITr4 € 0 1775 , IJuia 0 3520$ Exp Ginity AcCC Armic cloci not Fill up the data and Plot the free fall graph VS (" From the graph of Free Fall, find the g exp Find the percemt etror,...
##### C. (2 pts] The criteria used in the design of the focused on the design of...
c. (2 pts] The criteria used in the design of the focused on the design of the S-boxes and on the P function that takes the output of the S-boxes A. Avalanche Attack B. Data Encryption Standard C. Product Cipher D. Substitution Key...
##### $mathrm{R}-mathrm{OH}+mathrm{HX} longrightarrow mathrm{R}-mathrm{X}+mathrm{H}_{2} mathrm{O}$In the above reaction, the reactivity of different alcohols is(a) Tertiary $>$ secondary $>$ primary(b) Tertiary $>$ secondary $<$ primary(c) Tertiary $<$ secondary $>$ primary(d) Secondary $<$ primary $<$ tertiary
$mathrm{R}-mathrm{OH}+mathrm{HX} longrightarrow mathrm{R}-mathrm{X}+mathrm{H}_{2} mathrm{O}$ In the above reaction, the reactivity of different alcohols is (a) Tertiary $>$ secondary $>$ primary (b) Tertiary $>$ secondary $<$ primary (c) Tertiary $<$ secondary $>$ primary (d) Secon...
##### Express each as a sum, difference, or multiple of logarithms. See Example 2. $2 \log _{8}\left(n^{5}\right)$
Express each as a sum, difference, or multiple of logarithms. See Example 2. $2 \log _{8}\left(n^{5}\right)$... | 4,034 | 13,100 | {"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "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} | 2.65625 | 3 | CC-MAIN-2023-14 | latest | en | 0.755635 |
https://www.bartleby.com/solution-answer/chapter-47-problem-4rq-automotive-technology-a-systems-approach-mindtap-course-list-6th-edition/9781133612315/define-the-term-gearbox-ratio/fd85624c-2ab6-11e9-8385-02ee952b546e | 1,603,687,244,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107890273.42/warc/CC-MAIN-20201026031408-20201026061408-00187.warc.gz | 629,493,200 | 52,078 | # Define the term gearbox ratio.
### Automotive Technology: A Systems A...
6th Edition
Jack Erjavec + 1 other
Publisher: Cengage Learning
ISBN: 9781133612315
### Automotive Technology: A Systems A...
6th Edition
Jack Erjavec + 1 other
Publisher: Cengage Learning
ISBN: 9781133612315
#### Solutions
Chapter
Section
Chapter 47, Problem 4RQ
Textbook Problem
## Define the term gearbox ratio.
Expert Solution
To determine
The definition of the term gearbox ratio.
### Explanation of Solution
Gearbox is the set of gears. Gears are arranged in the pattern of gear train inside the gearbox. The gear train may be the; simple gear train, compound gear train, reverted gear train, planetary or epicyclic gear train.
The gearbox ratio is defined as the ratio of the angle of rotation of the steering wheel to the angle of rotation of the front wheel or front tires.
g.r= θ steering wheel θ tires g.r=Gearbox ratio θ
### Want to see the full answer?
Check out a sample textbook solution.See solution
### Want to see this answer and more?
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!
See solution
#### Additional Engineering Textbook Solutions
Find more solutions based on key concepts
Show solutions
Explain the difference between an OD grooving tool and a cutoff tool.
Precision Machining Technology (MindTap Course List)
What is logical independence?
Database Systems: Design, Implementation, & Management
What is projectitis? How is it cured or its impact minimized?
Principles of Information Security (MindTap Course List)
What is the Turing Test?
Fundamentals of Information Systems
For Problems 16.44 through 16.48, discuss how you would create the solid model of the given objects. See Exampl...
Engineering Fundamentals: An Introduction to Engineering (MindTap Course List)
The cable AOB weighs 24 N/m. Determine the sag H and the maximum tension in the cable.
International Edition---engineering Mechanics: Statics, 4th Edition
What e the main principles of source document design?
Systems Analysis and Design (Shelly Cashman Series) (MindTap Course List)
Why are some propane and natural gas tips made with a deep, recessed center?
Welding: Principles and Applications (MindTap Course List)
If your motherboard supports ECC DDR3 memory, can you substitute non-ECC DDR3 memory?
A+ Guide to Hardware (Standalone Book) (MindTap Course List)
What other technologies might you be able to use to transfer media from a mobile device to a computer?
Enhanced Discovering Computers 2017 (Shelly Cashman Series) (MindTap Course List) | 592 | 2,706 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2020-45 | latest | en | 0.797771 |
http://mathhelpforum.com/calculus/28154-find-quadratic-formula-print.html | 1,526,929,753,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864466.23/warc/CC-MAIN-20180521181133-20180521201133-00105.warc.gz | 197,755,125 | 3,057 | • Feb 13th 2008, 01:22 AM
gaussmarco
Dear All guys,
Recently I collect some data about the participants of Olympics"s
competitior from 1900 to 2004 . I want to optiminius the data and
calculate the formula and then predict the next data of year 2008. I
find out the quadratic formula.The following is the data :
1900 1904 1908 1912 1920 1924 1928 1932
997 651 2008 2407 2626 3089 2883 1332
1936 1948 1952 1956 1960 1964 1968 1972
3963 4104 4955 4135 5338 5151 5516 7170
1976 1980 1984 1988 1992 1996 2000 2004
6084 5179 6829 8391 9356 10318 10651 11099
PS.please email to me As soon as possible
gaussmarco@yahoo.com.hk
• Feb 13th 2008, 02:46 AM
mr fantastic
Quote:
Originally Posted by gaussmarco
Dear All guys,
Recently I collect some data about the participants of Olympics"s
competitior from 1900 to 2004 . I want to optiminius the data and
calculate the formula and then predict the next data of year 2008. I
find out the quadratic formula.The following is the data :
1900 1904 1908 1912 1920 1924 1928 1932
997 651 2008 2407 2626 3089 2883 1332
1936 1948 1952 1956 1960 1964 1968 1972
3963 4104 4955 4135 5338 5151 5516 7170
1976 1980 1984 1988 1992 1996 2000 2004
6084 5179 6829 8391 9356 10318 10651 11099
PS.please email to me As soon as possible
gaussmarco@yahoo.com.hk
The best way to handle this sort of problem is to do a quadratic regression. Do you have access to software or a calculator that does quadratic regression?
• Feb 13th 2008, 03:01 AM
mr fantastic
Quote:
Originally Posted by mr fantastic
The best way to handle this sort of problem is to do a quadratic regression. Do you have access to software or a calculator that does quadratic regression?
\$\displaystyle Y = 11.58 X^2 + 147.858 X + 1391.26\$ where X is the Olympic number (1900: X = 0, 1904: X = 1, .... 1920: X = 5, etc.). r^2 = 0.935. | 661 | 1,829 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2018-22 | latest | en | 0.726214 |
https://angolatransparency.blog/en/what-is-etc-in-project-management/ | 1,669,843,050,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710771.39/warc/CC-MAIN-20221130192708-20221130222708-00360.warc.gz | 131,350,605 | 12,411 | # What is etc in project management?
The Estimate To Complete (usually abbreviated ETC) is the project management measure that shows you the remaining cost you expect to pay in order to complete a project. Note that ETC isn’t the final overall expected project budget – this is called Estimate at Completion (EAC).
## What is ETC and EAC in project management?
The two forecasts utilized are the estimate at completion (EAC) – how much the project is forecasted to cost overall – and the estimate to complete (ETC) – how much funding is required to complete the remaining work.
## What is ETC formula?
ETC = Estimate at Completion – Actual Cost.
## How do you calculate etc in project management?
Quote from video: Let's first discuss what it is estimate to complete is the amount of money required to complete the remainder of the project. There are two methods to calculate etc'. Let's take a look at both of them
## What is ETC cost management?
In earned value analysis, the Estimate To Complete, usually abbreviated ETC, is the expected remaining cost to complete the project. It is not the final overall project cost (that’s the EAC), rather it is the expenditure from now to the end of the project. It does not include what has already been spent.
## How do you calculate EAC and ETC in project management?
They are as follows:
1. Formula 1. EAC = AC + Bottom-up ETC. This formula is used when the original estimation is fundamentally flawed. …
2. Formula 2. EAC =BAC/Cumulative CPI. …
3. Formula 3. EAC = AC + (BAC – EV) …
4. Formula 4. EAC = AC + [BAC – EV / (Cumulative CPI x Cumulative SPI)]
## How is EAC and VAC calculated?
First, you need to calculate the EAC since the formula for VAC is BAC-EAC. The EAC formula for this question, since variances are atypical, is: EAC = AC + (BAC – EV). Plugging in the numbers from the question, you get 138 + (200-145) = EAC of 193. VAC = BAC – EAC, therefore 200 – 193 = 7.
## What is a negative etc?
A negative ETC value (like a positive one) shows the difference between the estimated or forecasted goal and the current status.
## What does ETC mean in accounting?
estimate to complete
A recent discovery was to realize that many project managers – some of which work for me – confuse the importance of an “estimate to complete” (ETC) with an “estimate at completion.” (EAC) This is not to say they are confused about the term, its definition or any of the semantics associated with these two metrics.
## How do you calculate CPI from PMP?
Using the formula CPI = EV / AC, the project manager will have a value of less than 1 (project over budget), of 1 (project on budget), or greater than 1 (project under budget). CPI in project management measures the cost efficiency of a project.
## How is SPI calculated?
To calculate your project’s SPI performance, the formula is: Schedule Performance Index (SPI) = Earned Value (EV) / Planned Value (PV) SPI = EV / PV.
## How is EAC calculated example?
Formula One
1. Estimate at completion (EAC) = Budget at completion (BAC) / Cost performance index (CPI)
2. Estimate at completion (EAC) = \$100,000 / 0.8 = \$125,000.
## How do I Calculate EAC in Excel?
Equivalent annual cost (EAC) is the annual cost of owning and maintaining an asset determined by dividing the net present value of the asset purchase, operations and maintenance cost by the present value of annuity factor.
Formula.
NPV = EAC × 1 − (1 + r)n
r
## What is the purpose of earned value analysis?
Earned Value Analysis (EVA) is a method that allows the project manager to measure the amount of work actually performed on a project beyond the basic review of cost and schedule reports. EVA provides a method that permits the project to be measured by progress achieved.
## What is EAC in earned value?
Estimate at Completion (EAC) Formulas
The earned value EAC formula is based on the simple concept that the estimate at completion is equal to the amount of money already spent on the contract plus the amount of money it will take to complete the contract.
## What does EAC mean in finance?
Equivalent annual cost
Equivalent annual cost (EAC) is the annual cost of owning, operating, and maintaining an asset over its entire life. EAC is often used by firms for capital budgeting decisions, as it allows a company to compare the cost-effectiveness of various assets that have unequal lifespans.
## What does ETC mean accounting?
Exchange-Traded Commodity (ETC)
## How do I find EAC?
Estimate at completion (EAC) is calculated as budget at completion divided by cost performance index. Formula 1 for EAC is as follows: Estimate at completion (EAC) = Budget at completion (BAC) / Cost performance index (CPI)
## How do I calculate EAC in Excel?
Equivalent annual cost (EAC) is the annual cost of owning and maintaining an asset determined by dividing the net present value of the asset purchase, operations and maintenance cost by the present value of annuity factor.
Formula.
NPV = EAC × 1 − (1 + r)n
r | 1,180 | 5,011 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.671875 | 4 | CC-MAIN-2022-49 | latest | en | 0.93683 |
https://us.metamath.org/mpeuni/mbfsub.html | 1,722,907,500,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640461318.24/warc/CC-MAIN-20240806001923-20240806031923-00417.warc.gz | 477,354,053 | 8,901 | Metamath Proof Explorer < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > mbfsub Structured version Visualization version GIF version
Theorem mbfsub 24273
Description: The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
mbfsub (𝜑 → (𝐹f𝐺) ∈ MblFn)
Proof of Theorem mbfsub
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mbfadd.1 . . . . . . . 8 (𝜑𝐹 ∈ MblFn)
2 mbff 24236 . . . . . . . 8 (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
31, 2syl 17 . . . . . . 7 (𝜑𝐹:dom 𝐹⟶ℂ)
4 elinel1 4122 . . . . . . 7 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑥 ∈ dom 𝐹)
5 ffvelrn 6826 . . . . . . 7 ((𝐹:dom 𝐹⟶ℂ ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ℂ)
63, 4, 5syl2an 598 . . . . . 6 ((𝜑𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐹𝑥) ∈ ℂ)
7 mbfadd.2 . . . . . . . 8 (𝜑𝐺 ∈ MblFn)
8 mbff 24236 . . . . . . . 8 (𝐺 ∈ MblFn → 𝐺:dom 𝐺⟶ℂ)
97, 8syl 17 . . . . . . 7 (𝜑𝐺:dom 𝐺⟶ℂ)
10 elinel2 4123 . . . . . . 7 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑥 ∈ dom 𝐺)
11 ffvelrn 6826 . . . . . . 7 ((𝐺:dom 𝐺⟶ℂ ∧ 𝑥 ∈ dom 𝐺) → (𝐺𝑥) ∈ ℂ)
129, 10, 11syl2an 598 . . . . . 6 ((𝜑𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐺𝑥) ∈ ℂ)
136, 12negsubd 10994 . . . . 5 ((𝜑𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹𝑥) + -(𝐺𝑥)) = ((𝐹𝑥) − (𝐺𝑥)))
1413eqcomd 2804 . . . 4 ((𝜑𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹𝑥) − (𝐺𝑥)) = ((𝐹𝑥) + -(𝐺𝑥)))
1514mpteq2dva 5125 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) − (𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) + -(𝐺𝑥))))
163ffnd 6488 . . . 4 (𝜑𝐹 Fn dom 𝐹)
179ffnd 6488 . . . 4 (𝜑𝐺 Fn dom 𝐺)
18 mbfdm 24237 . . . . 5 (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
191, 18syl 17 . . . 4 (𝜑 → dom 𝐹 ∈ dom vol)
20 mbfdm 24237 . . . . 5 (𝐺 ∈ MblFn → dom 𝐺 ∈ dom vol)
217, 20syl 17 . . . 4 (𝜑 → dom 𝐺 ∈ dom vol)
22 eqid 2798 . . . 4 (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
23 eqidd 2799 . . . 4 ((𝜑𝑥 ∈ dom 𝐹) → (𝐹𝑥) = (𝐹𝑥))
24 eqidd 2799 . . . 4 ((𝜑𝑥 ∈ dom 𝐺) → (𝐺𝑥) = (𝐺𝑥))
2516, 17, 19, 21, 22, 23, 24offval 7397 . . 3 (𝜑 → (𝐹f𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) − (𝐺𝑥))))
26 inmbl 24153 . . . . 5 ((dom 𝐹 ∈ dom vol ∧ dom 𝐺 ∈ dom vol) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
2719, 21, 26syl2anc 587 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
2812negcld 10975 . . . 4 ((𝜑𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)) → -(𝐺𝑥) ∈ ℂ)
29 eqidd 2799 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)))
30 eqidd 2799 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥)))
3127, 6, 28, 29, 30offval2 7408 . . 3 (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)) ∘f + (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥) + -(𝐺𝑥))))
3215, 25, 313eqtr4d 2843 . 2 (𝜑 → (𝐹f𝐺) = ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)) ∘f + (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥))))
33 inss1 4155 . . . . 5 (dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐹
34 resmpt 5872 . . . . 5 ((dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐹 → ((𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)))
3533, 34mp1i 13 . . . 4 (𝜑 → ((𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)))
363feqmptd 6708 . . . . . 6 (𝜑𝐹 = (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)))
3736, 1eqeltrrd 2891 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ∈ MblFn)
38 mbfres 24255 . . . . 5 (((𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) → ((𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
3937, 27, 38syl2anc 587 . . . 4 (𝜑 → ((𝑥 ∈ dom 𝐹 ↦ (𝐹𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4035, 39eqeltrrd 2891 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)) ∈ MblFn)
41 inss2 4156 . . . . . 6 (dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐺
42 resmpt 5872 . . . . . 6 ((dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐺 → ((𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺𝑥)))
4341, 42mp1i 13 . . . . 5 (𝜑 → ((𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺𝑥)))
449feqmptd 6708 . . . . . . 7 (𝜑𝐺 = (𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)))
4544, 7eqeltrrd 2891 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ∈ MblFn)
46 mbfres 24255 . . . . . 6 (((𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) → ((𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4745, 27, 46syl2anc 587 . . . . 5 (𝜑 → ((𝑥 ∈ dom 𝐺 ↦ (𝐺𝑥)) ↾ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4843, 47eqeltrrd 2891 . . . 4 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺𝑥)) ∈ MblFn)
4912, 48mbfneg 24261 . . 3 (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥)) ∈ MblFn)
5040, 49mbfadd 24272 . 2 (𝜑 → ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐹𝑥)) ∘f + (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ -(𝐺𝑥))) ∈ MblFn)
5132, 50eqeltrd 2890 1 (𝜑 → (𝐹f𝐺) ∈ MblFn)
Colors of variables: wff setvar class Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ↦ cmpt 5110 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7388 ℂcc 10526 + caddc 10531 − cmin 10861 -cneg 10862 volcvol 24074 MblFncmbf 24225 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7443 ax-inf2 9090 ax-cc 9848 ax-cnex 10584 ax-resscn 10585 ax-1cn 10586 ax-icn 10587 ax-addcl 10588 ax-addrcl 10589 ax-mulcl 10590 ax-mulrcl 10591 ax-mulcom 10592 ax-addass 10593 ax-mulass 10594 ax-distr 10595 ax-i2m1 10596 ax-1ne0 10597 ax-1rid 10598 ax-rnegex 10599 ax-rrecex 10600 ax-cnre 10601 ax-pre-lttri 10602 ax-pre-lttrn 10603 ax-pre-ltadd 10604 ax-pre-mulgt0 10605 ax-pre-sup 10606 This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7390 df-om 7563 df-1st 7673 df-2nd 7674 df-wrecs 7932 df-recs 7993 df-rdg 8031 df-1o 8087 df-2o 8088 df-oadd 8091 df-omul 8092 df-er 8274 df-map 8393 df-pm 8394 df-en 8495 df-dom 8496 df-sdom 8497 df-fin 8498 df-sup 8892 df-inf 8893 df-oi 8960 df-dju 9316 df-card 9354 df-acn 9357 df-pnf 10668 df-mnf 10669 df-xr 10670 df-ltxr 10671 df-le 10672 df-sub 10863 df-neg 10864 df-div 11289 df-nn 11628 df-2 11690 df-3 11691 df-n0 11888 df-z 11972 df-uz 12234 df-q 12339 df-rp 12380 df-xadd 12498 df-ioo 12732 df-ioc 12733 df-ico 12734 df-icc 12735 df-fz 12888 df-fzo 13031 df-fl 13159 df-seq 13367 df-exp 13428 df-hash 13689 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-xmet 20087 df-met 20088 df-ovol 24075 df-vol 24076 df-mbf 24230 This theorem is referenced by: mbfmul 24337 iblulm 25009
Copyright terms: Public domain W3C validator | 4,620 | 7,209 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2024-33 | latest | en | 0.097278 |
http://fab.cba.mit.edu/classes/864.14/students/Calisch_Sam/9/12.4.py | 1,558,825,249,000,000,000 | text/plain | crawl-data/CC-MAIN-2019-22/segments/1558232258453.85/warc/CC-MAIN-20190525224929-20190526010929-00009.warc.gz | 68,728,054 | 1,342 | #!/usr/bin/env python from __future__ import division from numpy import * from numpy.random import rand import numpy.linalg as la from matplotlib import pyplot as plt def C(x): return sum(x*x.reshape(-1,1,N),axis=-1)/N def expect(x): return sum(x,axis=-1)[...,None]/N def plot(x): plt.figure() plt.plot(x[0,::dplot],x[1,::dplot],linestyle='',marker='.') plt.gca().set_aspect(1.) N = 20000 dplot = 5 #how many points to skip for plotting s = rand(2,N) plot(s) A = array([[1,2],[3,1]]) print "A_inv=",la.inv(A) x = dot(A,s) plot(x) x -= expect(x) #make zero mean w,v = la.eig(C(x)) print w M = (v/sqrt(w)).T y = dot(M,x) #diagonalize with unit variance print C(y) plot(y) #ICA df = lambda x: tanh(x) ddf = lambda x: 1/cosh(x)**2 def ica(w): i=0 while True: i+=1 w_new = (expect(df(dot(w,y))*y) - w[...,None]*expect(ddf(dot(w,y))) )[...,0] w_new /= la.norm(w_new) #misprint is squared, I think. if allclose(w_new+w,0): w_new *= -1 #avoid oscillation from normalization if la.norm(w_new-w)<1e-8 or i>1e3: break w = w_new return w w1 = ica(rand(2)) #random seed for weights #second ica component must be orthogonal to first, since we're in 2d, this #is unique, up to sign. w2 = array([-w1[1],w1[0]]) plt.plot([0,w1[0]],[0,w1[1]],marker='.',lw=4,c='r') plt.plot([0,w2[0]],[0,w2[1]],marker='.',lw=4,c='m') plt.gca().set_aspect(1.) plt.show() | 461 | 1,334 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2019-22 | latest | en | 0.48091 |
https://www.airmilescalculator.com/distance/hsv-to-ydn/ | 1,621,186,369,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243991178.59/warc/CC-MAIN-20210516171301-20210516201301-00225.warc.gz | 659,293,552 | 52,108 | # Distance between Huntsville, AL (HSV) and Dauphin (YDN)
Flight distance from Huntsville to Dauphin (Huntsville International Airport – Lt. Col W.G. (Billy) Barker VC Airport) is 1317 miles / 2119 kilometers / 1144 nautical miles. Estimated flight time is 2 hours 59 minutes.
Driving distance from Huntsville (HSV) to Dauphin (YDN) is 1660 miles / 2672 kilometers and travel time by car is about 29 hours 10 minutes.
## Map of flight path and driving directions from Huntsville to Dauphin.
Shortest flight path between Huntsville International Airport (HSV) and Lt. Col W.G. (Billy) Barker VC Airport (YDN).
## How far is Dauphin from Huntsville?
There are several ways to calculate distances between Huntsville and Dauphin. Here are two common methods:
Vincenty's formula (applied above)
• 1316.725 miles
• 2119.064 kilometers
• 1144.203 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth.
Haversine formula
• 1316.759 miles
• 2119.118 kilometers
• 1144.232 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## Airport information
A Huntsville International Airport
City: Huntsville, AL
Country: United States
IATA Code: HSV
ICAO Code: KHSV
Coordinates: 34°38′13″N, 86°46′30″W
B Lt. Col W.G. (Billy) Barker VC Airport
City: Dauphin
IATA Code: YDN
ICAO Code: CYDN
Coordinates: 51°6′2″N, 100°3′7″W
## Time difference and current local times
There is no time difference between Huntsville and Dauphin.
CDT
CDT
## Carbon dioxide emissions
Estimated CO2 emissions per passenger is 168 kg (370 pounds).
## Frequent Flyer Miles Calculator
Huntsville (HSV) → Dauphin (YDN).
Distance:
1317
Elite level bonus:
0
Booking class bonus:
0
### In total
Total frequent flyer miles:
1317
Round trip? | 526 | 1,954 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2021-21 | latest | en | 0.81961 |
https://www.mathworks.com/matlabcentral/cody/problems/233-reverse-the-vector/solutions/84567 | 1,495,505,471,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463607245.69/warc/CC-MAIN-20170523005639-20170523025639-00131.warc.gz | 931,130,785 | 11,657 | Cody
Problem 233. Reverse the vector
Solution 84567
Submitted on 2 May 2012 by Davide Ferraro
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
This solution is outdated. To rescore this solution, log in.
Test Suite
Test Status Code Input and Output
1 Pass
%% x = 1; y_correct = 1; assert(isequal(reverse(x),y_correct))
``` ```
2 Pass
%% x = -10:1; y_correct = 1:-1:-10; assert(isequal(reverse(x),y_correct))
``` ```
3 Pass
%% x = 'able was i ere i saw elba'; y_correct = 'able was i ere i saw elba'; assert(isequal(reverse(x),y_correct))
``` ``` | 184 | 621 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2017-22 | longest | en | 0.805073 |
scott920.saraikhana.com | 1,624,381,776,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488519183.85/warc/CC-MAIN-20210622155328-20210622185328-00066.warc.gz | 455,152,014 | 26,205 | # Roulette Tactics and Strategies
The bottom line is: the Roulette table is the main point of a Roulette game, it determines the results of the game. The quantity of players in a Roulette game could be pretty much, four, five or seven. The layout of the table is also given in number and arrangement dependent on what suits best to the players. There are a great number of other variables also which may influence the Roulette outcome. The Roulette table is not fixed, it is the choice of the player.
A Roulette table has been designed so the odds of winning come in favor of the house. It’s not impossible to beat the house, but it is quite difficult, because a Roulette player has to take into account not just the odds, but additionally all the factors that donate to the odds. The odds may also be affected by the sort of bets one makes, the types of outside bets you can make and the precise strategy one adopts. For instance, while playing a game of Roulette, it is probable an inexperienced person may beat a more experienced person at the wheel by making outside bets.
All types of bets are allowed in a Roulette game. Betting for the complete amount of the pot is not considered as a bet. However, there are specific bets that are regarded as such. A person may play roulette table betting for the complete amount of the cash on the table. If the player wins that amount, then he reaches keep that amount.
Lots of people who have no understanding of the overall game of roulette table adopt the “all-or-nothing” principle. They believe there is no solution to win. This is often true in some cases nonetheless 점보 카지노 it is not so in every situations. As an example, in case a player places all his chips on the initial bet that he makes, then he will get all the chips and not an individual less.
In roulette table layouts, you can find three types of betting that is used. There are the multi-line bet, the standard four-line bet and the odd number syndicate. These three types of table layouts are differentiated by their sizes. A multi-line table layout may be the most typical among odd size tables. In this kind, one can place around twelve numbers on each of the eleven positions in the table.
A regular four-line roulette table layout is the next simplest type. In this kind, one can place around nine numbers on the four spots in the table. The ultimate type, the odd number syndicate, is known as to function as most complex among all three. In this type, the player can place around twenty-one numbers on the odd spots of the table. In this type, the final total sum of all the bets is settled on the last number drawn.
For more complicated betting systems, the roulette dealer will most likely utilize the numbers which have been pre-arranged beforehand. The dealer will ask the player to specify which numbers from the deck he really wants to place inside bets while leaving the numbers out for the outlying numbers. The dealer then determines the minimum and maximum bets that one may place, in line with the strength of the within bets and will then multiply these figures to reach at the overall quantity of the outside bets.
Along with placing bets on the balls, players can also place bets on the color of the ball they desire to spin. The ball will spin once before it really is thrown to the player who will decide where it will land. This can either land on the red or the black spin, based on which was chosen earlier. It’ll then return to the spin cycle and continue on until it hits the casino wall, where it stops and a new ball is spun. The red and black balls are then your awarded the win in American roulette wheel.
# What’s Vaping?
What is Vaping? It is just a new electronic product which has quickly become popular worldwide. Basically, an electronic cigarette is really a device that simulates the specific smoking process of tobacco. It contains a battery, an electrical charge just like a cell phone’s, and a tank or container just like a bottled soda or juice can.
In contrast to what many non-smokers might believe, not all vapor is smoke. In fact, some papers only work with a liquid solution to take in the “munchies” which come making use of their electronic cigarettes. It is important to remember that vapor originates from burning certain chemicals which make up marijuana. Thus, vaporizing only applies to burning marijuana, not regular smoking.
Many vapers also use herbal remedies with their electronic cigarettes. Herbs which are used include Chamomile and Lemon balm. Chamomile is mostly used as a sleep aid and has been used to greatly help people quit smoking for several years. Some herbal remedies such as Lemon balm have healing Vape Pen properties and assist in improving lungs. Both of these are helpful when coping with lungs, but there is yet another benefit to these liquids that is sometimes not realized.
Most e-juices contain caffeine which improves one’s energy. The most recent study done on what is really vaporizing demonstrates the caffeine in e-cigs help improve one’s mental clarity and focus. That is why many people opt for an electronic cigarette with an excellent variety of herbal flavors. By using these liquids, one can enjoy the benefits of both smoking and obtaining a healthy boost of energy. Furthermore, e-juices are much safer than normal liquid smokes since they do not contain nicotine.
Nicotine in vapor is simply as addictive as regular smoking, however, it generally does not give the same rush or buzz as other methods. So, the facts that causes people to become addicted to nicotine? It really is believed that the addiction starts from childhood once the body becomes used to the taste of cigarettes. Additionally it is believed that when we have been stressed out, we commence to crave for that “high”, that is nicotine. Also, people who find themselves pregnant or experiencing various other kind of medical problems have been shown to be more susceptible to addiction than others.
It is believed that what is truly vaporizing isn’t just making the smoker addictive, but it is also causing increased levels of nicotine in the machine. Increased nicotine levels within the body will eventually cause changes in the brain, which can result in addiction. What is truly vaporizing is also what causes the “closing” of the air passages and the mind receptors. With the increased levels of nicotine levels within the body, the brain feels a rigorous craving to smoke and can need to be rewired in order for a user to give up smoking permanently.
What is truly vaporizing is the procedure for converting what is smoked right into a liquid form. This can happen in two ways, through mechanical or chemical means. The most frequent way to do this is to apply water to spray the liquid onto a heating element, which then turns the water into vapor. There exists a new method that is coming available on the market called the electronic smoking device, or e-smoking devices. What the unit do is convert what’s smoked into an electric signal, much like the signals that are sent through wires to pacemakers or other similar electronic equipment.
What is truly vaporizing is also what causes young people to experiment with this type of electronic cigarettes. The curiosity of teenagers can sometimes be an excessive amount of. Many teenagers experiment with what is vaporizing and see how easy it is to turn a cigarette into an electric cigarette. With just a small investment, they could experiment with all different types of electric cigarettes and see what they like best. The thing that really matters when you ask what is vapourising is if you wish to stop smoking and live a healthier life.
# Free Slots For Mobile Gaming
Refer to free slots online as casino-style slots which you can simply play and enjoy entirely without spending any actual cash. The 온라인 바카라 사이트 exact same slot machines which offer this type of functionality are the same ones you can find in conventional online casinos but will most likely be accessed via a free trial or demo mode. These free slots offer the same great casino experience you are used to with high stakes or jackpots, and the one difference is that you don’t have to actually shell out any money to cash in your winnings. There are numerous great things about playing free slots online. Read on to find more.
First, there is no risk involved. As stated earlier, these free slots work likewise way like conventional slots where your actual money is not on the line. With conventional casino style slots, you do stand an excellent chance of losing hardly any money because the house includes a random number generator to decide whether you are actually going to hit it big or not. Although, you do get the chance to cash in your winnings, there is no need to keep losing hardly any money by continuously betting exactly the same amount once more. However, online slots offering free slots enable you to win money and just switch to playing the real money version if you are ready to do so.
Secondly, it is just a fantastic solution to test the slot games you may want to play. Although, you will find the best-known spots like the online video slots and the live casino slots, there is nothing quite like trying them out before you may spend any money. This is also true if you are a purist who likes to stick to the old. What’s more, you really should try slots far away besides just America. For example, in Vegas, you will not find European slots as the American slots are operated and owned by a European company.
Thirdly, you obtain the opportunity to play some classic slots which are generally obtainable in shared progressive machines. These include the three-reel wheel, the straight slot and the jackpot slot. These classic slots give you the opportunity to win something every time you play. You can also increase your chances of winning with careful consideration of one’s choices. One tip for maximizing your chances of winning with classic slots would be to play the short-term WMS which offer the very best results with the tiniest jackpot.
Next, think about the multi-line and the classic slots with their generous paylines. The paylines on classic slots could be substantial and you should always keep this in mind. Some individuals prefer to play classic slots by making use of a slot machine guide while some rely on their own intuitive knowledge of how much to bet and on what line to bet. Whichever method you choose, you will still benefit from the free bonus that is included with your payout and if you are lucky enough, you may win big.
Yet another tip for playing slot games on the go is to download a free casino to mobile software for Android devices. The reason being most of the popular Android slot games are designed to be appropriate for mobile handsets. There are various top notch casinos and gaming companies, that have integrated flash player technologies in their mobile apps. If you’re likely to download an app, just consider one that supplies the best free slots games for Android and pocket your prizes!
Finally, remember about the virtual slots which have been designed especially for those that don’t want to spend any real cash on playing casino games. Additionally, there are free games offered by a number of the leading online casino companies. Many of these games include the virtual roulette, online craps, bingo, etc.
You may also use REAL-TIME Transmitter Software for RTC gaming that enables you to switch from one game to some other using simple touch and swipe gestures on your own smart phone device. In this system, you need not download any extra software or plug in any extra hardware to take pleasure from free slots games on your smartphone or tablet. All you have to is an web connection and a Bluetooth enabled device or perhaps a computer. With a RTC, it is possible to connect to the internet on any device where there is a signal.
# Is Using E Cigarettes Good For Your Health?
One of the biggest concerns about e-cigarette use may be the danger of dependence on nicotine, but what about the dangers of vapour technology? We realize that vapour cigarettes do not contain nicotine. However, that’s where the danger lies. Not only does it take longer for your body to breakdown the nicotine, it also means you are inhaling toxins from the chemical used to make the cigarettes.
Nicotine is a highly addictive drug. It enters the system and attaches itself to the nerve cells in the mind. This causes them to work harder and longer. In addition, it affects the way our brains work. Therefore, in the event that you were to eliminate the connection with nicotine through vapour cigarettes you would allow your brain development to advance without any problems.
There are two ways to assist you stop smoking using electronic cigarettes. The first is to go cold turkey. This means no e-cigarette at all for per month or more. If you can’t stop cold turkey you then must find some alternative methods to help you together with your quit. There are several alternative methods such as hypnosis and acupuncture.
Secondly, there are new methods being developed to greatly help teenagers and adults alike. One of these brilliant is vapor pill. They are basically the electronic equivalent of taking tobacco. They are not yet in the marketplace but they should be available within the next year.
With vapor pill the vapor itself isn’t inhaled. Instead it really is absorbed by the gums, throat and stomach. This prevents the nicotine from entering one’s body and giving you oral cancer. In addition, it eliminates any toxins that are in the smoke from before. These chemicals in tobacco could be bad for your health because they can harm your cardiovascular and nervous systems.
Within the next two years it ought to be easy for vaporizers to be marketed to teenagers and adults alike. This will be a huge step forward in helping smokers and non-smokers alike to give up the unhealthy traditional smoking routine. One of the great things about quitting smoking conventional cigarettes is that you will enjoy better health as a non smoker. Many non smokers enjoy the benefits of vaping better than in the past.
You can easily see why e-cigarette and vaporizing are here to stay. The reason they are here is because they are proven to help people quit smoking. They are also a much safer alternative than the tobacco products they replace. When quitting smoking using e Cigels or vaporizers you are guaranteed to take pleasure from better health than you ever did as a smoker.
In conclusion, it is extremely essential that you understand the health ramifications of e Cigarette and vaporizing. Most non smokers realize the harmful health effects that are associated with smoking. Therefore, they are trying to make a healthier switch. Vaping is merely the easiest way to perform so. Once you know the potential dangers of smoking, you too will quickly see the incredible benefits that Vaping provides.
E Cigarettes along with other smoking related products such as gum have many health threats attached to them. There’s very little that can be done if you are a smoker. That is why vaporizing is becoming very popular amongst many nonsmokers. By switching your nicotine consumption to something that is much simpler to manage and control, it will be possible to eliminate most Eightvape Coupon of the health risks that come with conventional smoking.
Just about the most important things about quitting smoking is that you need to take the proper steps to ensure that you succeed. Most people who try to quit tobacco products will fail within the initial year. This failure doesn’t have to happen to you. If you are using e-juices to help you get through those trying times, you will discover that you are more lucrative.
You may experience some unwanted effects while transitioning from smoking to vaping. That’s understandable. You should always consult with your doctor or health care provider before you begin to utilize any type of electronic nicotine delivery device. Lots of the new herbal juices that are being produced could cause serious lung damage if you do not utilize them properly.
Many vaporizers will allow you to use them while exercising. This can be a good idea if you are concerned about the harmful chemicals and toxins that are present in cigarette smoke. Your lungs can breathe better while exercising, which will help you feel better overall. With better health insurance and a decrease in toxins, you will feel better for taking the time to transition to a natural e-juice.
# The Basics of Blackjack, Slots and Standard Deviation
If you’ve ever been to a casino, then you know that there are endless games of luck and chance which are played there. However, there are some games of skill that depend a great deal on strategy and careful betting. The best way to learn about these games is to get a little experience at gambling at a number of casinos. There are many types of casino games offered for customers to play. In this guide, become familiar with about many of these popular casino games.
In a slot machine game, players spin reels in hopes of striking it big by winning extra money. Payouts depend on how much spins. There are three basic types of casino games: table games, gaming machines, and slots. Many casino games, including blackjack, also involve using bluffing tactics to improve your possible outcomes. Playing slots, like slot machine games, do not need the current presence of the casino staff to play; therefore, it is usually played by anyone who can sit back at a table. You might use a hand-held device or even a paper slot machine to make your own luck with this particular casino game.
The house edge, or the amount of profit that you would expect to keep once you lose on all your bets, is the casino games’ equivalent to the chances of most your possible outcomes. The house edge is the amount by which the casino games’ jackpot would need to grow if you won. Most slot machines have a relatively small house edge; hence, you’ll be able to have a comparatively large house edge. Blackjack, craps, baccarat, and video poker have the highest house edges among other casino games.
All casino games follow some basic laws of probability. Once you place your bets, the chances of hitting on the winning numbers are usually close to fifty percent. There are a few exceptions to this rule, but these are very rare. For instance, in online slots, you have the advantage of not having to handle other players, and thus there is the chance that another person will hit on the lucky numbers for you personally. The home advantage in online casino table games is much less compared to the advantage in live casino games.
Blackjack and roulette are believed to be the most popular casino games in casinos today. There are numerous variations of the table games, which make them exciting. Although they’re best played alone, you can play contrary to the computer and try to win for the personal amusement. Not only is it a great way to obtain casino gambling entertainment, online blackjack and roulette are also the best way to spend leisure time, specifically for those who don’t have many friends. These casino games give a fun and safe solution to earn extra money from your home.
Slots are considered to be one of the most popular games obtainable in online casinos. There are many variations of slots that you may choose from, depending on your personal preference. Although the game of slots is most beneficial played by betting a set amount of money, there are also progressive slots and instant slots that allow you to increase or decrease the amount of money that you wish to put on the line. You can find even slots games designed for free, when you are so inclined.
Standard deviation is one of the most significant concepts in casino gambling, because it takes away the “unpredictability” from casino gambling. Standard deviation is really a mathematical concept that is used to gauge the randomness of a distribution. It is in line with the theory of probability, and is often implemented in game theory and computer analysis. A standard deviation is the difference between your expected value of a random variable and the actual value that are experienced by the ball player of the game.
The casino games mentioned above are all types of casino games using random number generators. You should know what these are to be able to fully understand casino games. If you don’t know 카지노 게임 사이트 what they are, searching for them online or in any reference book about casino games. This will give you a better knowledge of casino games in addition to a better notion of how they work.
# Choosing Slots Machines For Your Slot Game
Slots are right up there with popular casino games with regard to popularity. They’re incredibly an easy task to learn, easy to master, plus they require little strategy or thought. You simply spin the reels as if you would in a slot machine and go. Unfortunately, there are few quality slots games out there.
There are some good slots games, however, and you could easily maximize money from them once you learn how to do it. The most popular slots games in the us include slots games with paylines. These games use specific symbols to represent what sort of jackpot you can win, in order that once you place your bet, these symbols will tell you if your bet is winning. If the symbol matches the worthiness of what you bet, you win your bet and obtain the jackpot.
The second most popular type of slots game is really a slots game without payback percentage. On these kinds of machines you decide how much to bet, spin the reels, and let the machine pulls the “Bonus” (some of your bet). Once the upshot of the spin determines the amount you are going to get, there is basically no chance of going for a loss. The jackpots on these machines can reach thousands (if not millions) of dollars. The payback percentages sm 카지노 for these machines are incredibly high, however they are hard to beat. This is where many people start their gambling experience.
To find out whether a slot machine includes a high payback percentage or perhaps a low one, you have to multiply the payout percentages by the number of times the machine spins. For example, if you have a ten percent payback percentage, then your chances that you’ll hit on a jackpot of over 100,000 dollars is ten percent. Understand that they are “best hit” percentages; that is, you’ll have a better potential for hitting the jackpot more often. These are not “probability” percentages.
Slots with a high payback percentage are known as progressive slots. The symbols shown on the reel are what indicate which direction the reel is spinning. For every symbol that presents up, the direction of the spin is increasing. The exact opposite of this is true of the slots with a low payback percentage.
Along with paying real money for his or her slot games, many casinos provide a free slots bonus. This free slots offer can give you an edge over slot players who do not take advantage of it. As stated above, with progressive slots, the reels have a non-ending sequence of symbols. You need to carefully study all of these symbols in order to predict which symbols will result in the payoff of your bet. If you see a pay-off symbol in advance, you might want to take the bet.
While casinos all offer different types of slots games, each of them have one thing in common. Slots machines are made to be played. They don’t require any special skills to be able to enjoy the game. That is one of the main explanations why slots are so popular among casino goers. They are easy to get in and out of, and provide an opportunity to play slot machine game games for enjoyment.
Winning slot machine results vary from time and energy to time. If you place your bets carefully, chances are good that you’ll win more regularly than you lose. To increase your chances of winning, choose your symbols carefully, and play your slots according to the posted payouts.
You can determine how much you’ll win or lose by examining the symbols displayed on your slots. When you start to see the payoff symbol, you need to place your bet. Payout wins and losses will be added until you eventually hit the jackpot. Most slots offer a maximum amount of payouts which might be won, as well as your initial investment will be returned to you by the end of the overall game.
In earlier slots games, an individual spin wouldn’t normally count towards your winnings. However, the newer slot machines now use what are referred to as “re-spins” within their system. The re-spin is when a magnetic wheel is spun once and again to generate a second roll. This gives the appearance that you are making multiple spins, and actually, you are.
As possible plainly see, by carefully choosing your reels, and paying attention to winning symbols, you can boost your likelihood of hitting big jackpots. Choosing reels which have fewer symbols on them can help lessen your overall winning ratio. Good luck!
# Vapor Cigarettes – How Is Vapor Cigarettes Making?
An electronic cigarette is basically an electronic device which simulates real cigarette smoking. It consists mainly of an atomizer, a battery, and some sort of chamber for storing the liquid or juice. Instead of tobacco, the user smokes vapor instead. As such, with an electronic cigarette, the term “e-cigs” is used to spell it out the product.
The usage of a heating element can be a area of the design. In this design, a heating element is positioned either at the bottom of the unit, near the mouthpiece, or directly above the vapor cigarette cartridge. This heating element generates a heat, which vaporizes the liquid right into a fine mist, which then falls into the mouthpiece and makes its way into the lungs. While this might not sound bad for health, it can create all types of problems for adult tobacco consumers. Especially, because the heating element moves along, the vapor becomes less effective in transporting the flavor from the tobacco to the buyer.
For reasons uknown, many of the individuals who have attempted to sell e-liquids, including manufacturers, distributors, and retailers, believe that they are somehow safer than regular cigarettes. That’s because a vapor cigarette doesn’t produce smoke, thus no lungs are affected by the act of smoking. Therefore, there is absolutely no smoke either. Also, these devices don’t release nicotine, so there are no chemicals put into the air. With all of this in mind, you can easily see why there can be so much concern about them. After all, many people believe that should they can’t have a smoke, they can’t have a vapour cigarette.
However, not all vapor cigarettes smoking devices are manufactured equal. There are various models that have been designed with the sole purpose of mimicking the act of smoking a normal cigarette. As an example, there are models designed to look like cigarettes, with even their very own ashtray. Other electronic cigarettes don’t possess a resemblance to a cigarette at all but instead use batteries and handful of electricity to provide off heat similar to a real cigarette light. Yet another model has no resemblance to a cigarette at all. Instead, it uses the same type of technology that is found in high-end electronic cigarettes.
The largest problem with these devices is they don’t work as they should. Because no actual chemical fuel is used, these devices are not as clean burning or efficient as their counterparts. This means that e-cigs will release harmful toxins in to the air that regular cigarettes are. It has been established that even just one single hour of constant e-cigs in the air could cause longterm effects to someone’s health. These long term effects of e-cigs range between short term ones such as headaches and throat irritation to much more serious long-term problems that can include cancer.
As the demand for healthier alternatives to smoking tobacco is growing, more vapor cigarettes are increasingly being developed and produced. One of many newest vapor products is the Nuka Smoke system. It is different from most vapor cigarettes since it doesn’t use any kind of lighting source, or even a heat source to generate the vapor that’s inhaled. This makes Nuka Smoke sticks out on the list of vapor cigarette market.
Another newer product out there is the Nicotine Salt e-Liquids. Just like the Nuka Smoke, the Nicotine Salt liquids use no heat, so they are much cleaner than other vapor products. The key reason why Nicotine Salt e-Liquids aren’t as popular as podsmall.com other vapor products is due to price. They are basically the same cost because the other nicotine e-liquids, which explains why many people don’t consider them to be a real e Cigarette alternative. This is often considered a negative, because the cost can be less than the more popular nicotine cigarettes.
The ultimate vapor alternative we will discuss today is the Ohm Coil. The Ohm Coil was originally designed to be used within an electronic cigarette. It really is basically a coil that wraps around a heating element, like the way a microwave wire is wrapped around a heating element in an oven. Since the Ohm Coil is so new, there aren’t many vapor products with this particular component yet, but they are starting to enter industry slowly.
# The advantages of Vaping Liquid
An electric cigarette is a modern electronic device which behaves and appears like a real cigarette. It includes a tank, an atomizer, and an electrical supply just like a rechargeable battery. Instead of smoke, the user smokes vapor. Actually, it even produces no smoke at all. Therefore, with an electronic cigarette it really is frequently referred to as “vaping.”
Electronic cigarettes are gaining popularity among smokers of all ages because of its many advantages. For one thing, they are extremely convenient because you can make it with you anywhere. Moreover, with most electronic cigarettes, you don’t have to be worried about getting health claims in it as if you would with some conventional ones. You can get basically the same quantity of nicotine from an electronic cigarettes as you would from a conventional one. Additionally, e smokes don’t have any tar or other chemicals and are also safer than regular cigarettes.
Many e-juices can be found for free on the internet. Some manufacturers offer free flavors by their e-juice products while other manufacturers offer free trials. This is the reason you have to really shop around before choosing your vaporizing juice. Try various free e-juices and samples first. Then choose one which has the flavor that you love most.
You will also need to make sure that you choose the right e-juice in order to maximally maximise your Element Vape chances of enjoying the maximum amount of flavour from your own vaporizing liquids. Nicotine may be the primary ingredient in all nicotine-based juices and cigarettes. Therefore, in order to achieve the best results it is best to use the liquids which contain a higher concentration of nicotine. If you use less concentration of nicotine you may get slightly more flavour, but you won’t experience nearly just as much pleasure.
One thing to be cautious about with the liquids is that some brands could have ingredients inside them that mimic the addictive properties of nicotine. In this case, if you do not use a strong hit of nicotine it is possible to over-use the product as well as your lungs can suffer the consequences. It is always advisable to choose another brand or try different products to see those have the highest levels of nicotine. The Nicotine patch can be available as an alternative to the liquids. Use it rather than a vaporizing liquid when you want a stronger hit of nicotine minus the throat or mouth irritation.
Two of the most frequent or juice ingredients are Propylene Glycol (PEG) and Ethanol. Both could be harmful if used in the quantities contained in many products. Although both are usually harmless, there’s enough evidence out there that presents that Propylene Glycol is in many cases more harmful than ethanol. That is due to its chemical structure. When blended with water it will swell, that may increase the amount of nicotine contained in a product by almost 500%.
Even though many people are put off from using vaporizers since they believe that they are ‘smoking’ or that they produce an offensive smell, this is not true. Vapourisers don’t actually burn anything, they just give off a blast of vapor which many people find quite appealing. They’re a great alternative to smoking, which means they will have great health benefits as well as the advantage of not giving off an offensive smell. When used in conjunction with a Nicotine patch or gum, it can benefit smokers to gradually wean themselves from nicotine without increasing their exposure to other, potentially harmful chemical compounds and toxins.
Because of the increased popularity over recent years, e liquids are becoming increasingly a popular method of quitting smoking for non-smokers, plus they are proving to work nicely for those who are trying to quit. Because of their increased effectiveness and their smaller size, also, they are very easy to take along on a plane or train to quit smoking. E liquids provide a cheaper option to cigarettes, which makes them a lot more appealing for those attempting to give up tobacco. As proof of this, sales of electric cigarettes are growing each year, making them an excellent investment later on.
# Vapor Cigarettes Vs Normal E-Cigs – Do you know the Benefits?
A vapor cigarette is an alternative digital podsmall.com camera which simulate traditional tobacco smoking, minus all of the harmful chemical compounds and tar. It typically includes a tank or chamber, an atomizer, and a power power source like a rechargeable battery or a lithium ion battery. Rather than tobacco, an individual inhards vapor instead. Therefore, utilizing an electronic cigarette is generally described as “vaping.”
Conveniently, you can find two main methods to make vapor cigarettes: via the heating element and via the damper. The heating element may be the most common solution to heat up the liquid to create vapor. Simply place the heating element into the e cigarette, set the wattage and revel in a flavorful smoke. The downside to utilizing the heating element is that it could be difficult to remove the vapor from the heating element and that additionally, it may cause the coil to burn unevenly. It can also produce a bit of smoke in the air.
The damper on an electronic vapor cigarette is placed over a heating element with an integral fan. The damper is then closed and the electronic vapor cigarette is fired up and properly heated by the electronic vaporizer. The damper is designed to keep the vapors inside and not to allow the vapors to escape. Unfortunately, the damper will often overheat and result in an explosion. This explosion could cause some but not all electronic vaporizers.
There is absolutely no evidence that any side effects from electric cigarettes have ever been caused through overheating. However, these types of devices are not appropriate for use by children. They are also not recommended for anyone who includes a history of smoking tobacco or any disease that may increase their risks of developing a smoking tobacco-related disease. For those who have any questions about the safety of this type of e-Cig, you should contact your doctor or dentist before using it.
You can find two different types of e-Cigarette: ones that are built using a lithium-ion battery and those that are built using a standard battery. A lithium-ion battery might seem safer than a standard battery, because they are non-reactive. In theory, if the lithium-ion battery overheats you won’t explode. Unfortunately, this is simply not the case.
A lithium-ion battery can overheat even though it is well maintained. This is due to the temperature level inside the e Cigarette can change constantly. When you first get there Cig, it is strongly recommended that you leave it out to attain room temperature and not add any fluid to it. The reason for this is because you don’t want any fluid to heat up since it can cause issues with some vapor cigarettes. After the vapor cigarette reaches room temperature, it’ll need to be refilled with fluid again.
If you are using an ordinary cigarette, it is extremely hard to inform if the fluid has gone flat or not. Also, it will take several times to note that the fluid has changed because it can easily go back to its normal state. With a vapor cigarette, this is not a problem because you do not have to refill it. Instead, all you have to do is switch off the power to the device and leave it alone for some minutes so it can cool off.
Because vaporizers are safer than normal e cigarettes, more people are choosing to use them. They provide an excellent alternative to those that smoke because there is no harmful smoke produced by the device. In addition, they are a better alternative because they do not expose you to each of the potential dangers that come alongside regular cigarettes. For individuals who are considering investing in an electric product, you may want to have a look at vaporizers.
# The Dangers of Vaping – WHY YOU NEED TO Avoid Vaporizing (at All Cost)
The medical dangers of vaporizing marijuana are largely unknown. Very few individuals who openly admit to vaporizing pot are doing it for medical reasons, and fortunately there have only been a small number of studies done on the subject. Many people believe that vaporizing cannabis is an acceptable way to quit smoking, and again, the pharmaceutical industry has largely based their entire profits on persuading people that smoking is good for you. Yet studies show that smoking is a lot more harmful to your body than vaporizing does. It’s true that a number of the smoke is expelled from the lungs with the vapor, but a great deal of the harm done is taken up by the body.
With all this in mind, you can easily see why there is this type of big public health threat when it comes to vaporizing cannabis. Inhaling the smoke from burning cannabis leaves your lungs and mouth feeling like they’ve gotten hit with a hot fire hydrant. This happens because heat from burning the substance evaporates in to the air. Your body experiences what is known as air pressure, which means that the number of air breathed in is greatly increased once you vaporize the plant matter.
Vaporizing also poses a lot of potential for pregnant women. Even though vapors are usually considered safer for the unborn child, there have been reports of premature birth and low birth weights in small children whose mothers used e cigarettes while these were pregnant. Pregnant women should tell their doctor about any drug they’re taking, including vaporizing ones. Some doctors understand the dangers of vaporizing electronic cigarettes, they are generally not equipped to create a valid ruling on the safety of using them while pregnant.
One of the primary dangers of vaporizing electric cigarettes is that the user is getting directly into the vaporized liquid. Because this liquid passes through the mouth and throat, you can never be sure of just how much liquid is actually inhaled. This is particularly true if you happen to be smoking a heavily flavored cigarette. The more flavoring you put into the liquid, the more the smoker will inhale. While flavored cigarettes have become more widely available, you still need to stick to the less-flavored types. Stick to the tobacco or herbal versions for now.
The vapors may also cause some serious health issues, both short-term and long-term. Long-term issues range from cancer, especially if you are smoking Juul Compatible Pods while pregnant. Kids and adults who are exposed to second hand smoking from e-cigarette liquids can also suffer from cancer-causing agents. Even if the vapors usually do not reach the lungs, they can still be inhaled deeply, causing lots of respiratory issues, some of which cannot be treated. Many people experience sore throats, sinusitis, coughs and even asthma because of the e-juice inhalation.
One of the serious issues with vaporizing e-liquids is battery explosions. Most e-liquids are constructed so that they can be plugged in using a universal battery charger. Unfortunately, there is no guarantee that the chargers that include the vaporizers will have the right power to get the job done. Sometimes the chargers aren’t powerful enough to get the work done, so the batteries explode. This can be a biggest reason that it’s so important to make certain that you use the right chargers for the vaporizer devices.
There is also the potential dangers of lung injuries when vaporizing. Nicotine is extremely poisonous when breathed in any quantity, but it is specially dangerous if it is used through the lungs. Because nicotine is so highly volatile, it easily switches into the lungs when it is inhaled. If you are concerned about the possible health threats of vaporizing, you should strongly consider making the switch to using tobacco smoke alternative products. The difference between cigarettes and vaporizing products is that cigarettes pose a larger potential danger to your health.
So is vaporizing really a safer alternative? The truth is that there surely is no safer solution to use tobacco. For anyone who is concerned about the potential health threats of vaporizing, you should strongly consider the benefits that you get from using e-liquids instead. Actually, the benefits are so great that it may be simpler to make the switch to e-cigarettes than to give up smoking entirely. By making a healthier choice, you’re giving yourself a free of charge health without risk smoking alternative. | 8,320 | 41,977 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2021-25 | latest | en | 0.957685 |
https://referencesource.microsoft.com/PresentationCore/Core/CSharp/System/Windows/Media/RectangleGeometry.cs.html | 1,638,788,652,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00365.warc.gz | 524,366,132 | 9,705 | RectangleGeometry.cs
File: Core\CSharp\System\Windows\Media\RectangleGeometry.cs Project: wpf\src\PresentationCore.csproj (PresentationCore)
```//------------------------------------------------------------------------------ // Microsoft Avalon // Copyright (c) Microsoft Corporation, 2001 // // File: RectangleGeometry.cs //------------------------------------------------------------------------------ using System; using MS.Internal; using System.ComponentModel.Design.Serialization; using System.Reflection; using System.Collections; using System.Text; using System.Globalization; using System.Windows.Media; using System.Windows.Media.Composition; using System.Windows; using System.Text.RegularExpressions; using System.Windows.Media.Animation; using System.Diagnostics; using System.Runtime.InteropServices; using System.Security; using SR=MS.Internal.PresentationCore.SR; using SRID=MS.Internal.PresentationCore.SRID; namespace System.Windows.Media { /// /// This is the Geometry class for Rectangles and RoundedRectangles. /// public sealed partial class RectangleGeometry : Geometry { #region Constructors /// /// /// public RectangleGeometry() { } /// /// Constructor - sets the rounded rectangle to equal the passed in parameters /// public RectangleGeometry(Rect rect) { Rect = rect; } /// /// Constructor - sets the rounded rectangle to equal the passed in parameters /// public RectangleGeometry(Rect rect, double radiusX, double radiusY) : this(rect) { RadiusX = radiusX; RadiusY = radiusY; } /// /// /// /// /// /// /// public RectangleGeometry( Rect rect, double radiusX, double radiusY, Transform transform) : this(rect, radiusX, radiusY) { Transform = transform; } #endregion /// /// Gets the bounds of this Geometry as an axis-aligned bounding box /// public override Rect Bounds { get { ReadPreamble(); Rect boundsRect; Rect currentRect = Rect; Transform transform = Transform; if (currentRect.IsEmpty) { boundsRect = Rect.Empty; } else if (transform == null || transform.IsIdentity) { boundsRect = currentRect; } else { double radiusX = RadiusX; double radiusY = RadiusY; if (radiusX == 0 && radiusY == 0) { boundsRect = currentRect; transform.TransformRect(ref boundsRect); } else { // // Transformed rounded rectangles are more tricky. // // Matrix geometryMatrix; Transform.GetTransformValue(transform, out geometryMatrix); boundsRect = RectangleGeometry.GetBoundsHelper( null /* no pen */, Matrix.Identity, currentRect, radiusX, radiusY, geometryMatrix, StandardFlatteningTolerance, ToleranceType.Absolute); } } return boundsRect; } } internal override bool AreClose(Geometry geometry) { RectangleGeometry rectGeometry2 = geometry as RectangleGeometry; if (rectGeometry2 != null) { RectangleGeometry rectGeometry1 = this; Rect rect1 = rectGeometry1.Rect; Rect rect2 = rectGeometry2.Rect; return ( DoubleUtil.AreClose(rect1.X, rect2.X) && DoubleUtil.AreClose(rect1.Y, rect2.Y) && DoubleUtil.AreClose(rect1.Width, rect2.Width) && DoubleUtil.AreClose(rect1.Height, rect2.Height) && DoubleUtil.AreClose(rectGeometry1.RadiusX, rectGeometry2.RadiusX) && DoubleUtil.AreClose(rectGeometry1.RadiusY, rectGeometry2.RadiusY) && (rectGeometry1.Transform == rectGeometry2.Transform) && (rectGeometry1.IsFrozen == rectGeometry2.IsFrozen) ); } return base.AreClose(geometry); } /// /// Returns the axis-aligned bounding rectangle when stroked with a pen, after applying /// the supplied transform (if non-null). /// internal override Rect GetBoundsInternal(Pen pen, Matrix worldMatrix, double tolerance, ToleranceType type) { Matrix geometryMatrix; Transform.GetTransformValue(Transform, out geometryMatrix); return RectangleGeometry.GetBoundsHelper( pen, worldMatrix, Rect, RadiusX, RadiusY, geometryMatrix, tolerance, type); } /// /// Critical - it calls a critical method, Geometry.GetBoundsHelper and has an unsafe block /// TreatAsSafe - returning a RectangleGeometry's bounds is considered safe /// [SecurityCritical, SecurityTreatAsSafe] internal static Rect GetBoundsHelper(Pen pen, Matrix worldMatrix, Rect rect, double radiusX, double radiusY, Matrix geometryMatrix, double tolerance, ToleranceType type) { Rect boundingRect; Debug.Assert(worldMatrix != null); Debug.Assert(geometryMatrix != null); if (rect.IsEmpty) { boundingRect = Rect.Empty; } else if ( (pen == null || pen.DoesNotContainGaps) && geometryMatrix.IsIdentity && worldMatrix.IsIdentity) { double strokeThickness = 0.0; boundingRect = rect; if (Pen.ContributesToBounds(pen)) { strokeThickness = Math.Abs(pen.Thickness); boundingRect.X -= 0.5*strokeThickness; boundingRect.Y -= 0.5*strokeThickness; boundingRect.Width += strokeThickness; boundingRect.Height += strokeThickness; } } else { unsafe { uint pointCount, segmentCount; GetCounts(rect, radiusX, radiusY, out pointCount, out segmentCount); // We've checked that rect isn't empty above Invariant.Assert(pointCount != 0); Point * pPoints = stackalloc Point[(int)pointCount]; RectangleGeometry.GetPointList(pPoints, pointCount, rect, radiusX, radiusY); fixed (byte *pTypes = RectangleGeometry.GetTypeList(rect, radiusX, radiusY)) { boundingRect = Geometry.GetBoundsHelper( pen, &worldMatrix, pPoints, pTypes, pointCount, segmentCount, &geometryMatrix, tolerance, type, false); // skip hollows - meaningless here, this is never a hollow } } } return boundingRect; } /// /// Critical - contains unsafe block and calls critical method Geometry.ContainsInternal. /// TreatAsSafe - as this doesn't expose anything sensitive. /// [SecurityCritical, SecurityTreatAsSafe] internal override bool ContainsInternal(Pen pen, Point hitPoint, double tolerance, ToleranceType type) { if (IsEmpty()) { return false; } double radiusX = RadiusX; double radiusY = RadiusY; Rect rect = Rect; uint pointCount = GetPointCount(rect, radiusX, radiusY); uint segmentCount = GetSegmentCount(rect, radiusX, radiusY); unsafe { Point *pPoints = stackalloc Point[(int)pointCount]; RectangleGeometry.GetPointList(pPoints, pointCount, rect, radiusX, radiusY); fixed (byte* pTypes = GetTypeList(rect, radiusX, radiusY)) { return ContainsInternal( pen, hitPoint, tolerance, type, pPoints, pointCount, pTypes, segmentCount); } } } /// /// Gets the area of this geometry /// /// The computational error tolerance /// The way the error tolerance will be interpreted - relative or absolute public override double GetArea(double tolerance, ToleranceType type) { ReadPreamble(); if (IsEmpty()) { return 0.0; } double radiusX = RadiusX; double radiusY = RadiusY; Rect rect = Rect; // Get the area of the bounding rectangle double area = Math.Abs(rect.Width * rect.Height); // correct it for the rounded corners area -= Math.Abs(radiusX * radiusY) * (4.0 - Math.PI); // Adjust to internal transformation Transform transform = Transform; if (!transform.IsIdentity) { area *= Math.Abs(transform.Value.Determinant); } return area; } internal override PathFigureCollection GetTransformedFigureCollection(Transform transform) { if (IsEmpty()) { return null; } // Combine the transform argument with the internal transform Matrix matrix = GetCombinedMatrix(transform); double radiusX = RadiusX; double radiusY = RadiusY; Rect rect = Rect; if (IsRounded(radiusX, radiusY)) { Point[] points = GetPointList(rect, radiusX, radiusY); // Transform if applicable. if (!matrix.IsIdentity) { for (int i=0; i /// GetAsPathGeometry - return a PathGeometry version of this Geometry /// internal override PathGeometry GetAsPathGeometry() { PathStreamGeometryContext ctx = new PathStreamGeometryContext(FillRule.EvenOdd, Transform); PathGeometry.ParsePathGeometryData(GetPathGeometryData(), ctx); return ctx.GetPathGeometry(); } /// /// GetPathGeometryData - returns a byte[] which contains this Geometry represented /// as a path geometry's serialized format. /// internal override PathGeometryData GetPathGeometryData() { if (IsObviouslyEmpty()) { return Geometry.GetEmptyPathGeometryData(); } PathGeometryData data = new PathGeometryData(); data.FillRule = FillRule.EvenOdd; data.Matrix = CompositionResourceManager.TransformToMilMatrix3x2D(Transform); double radiusX = RadiusX; double radiusY = RadiusY; Rect rect = Rect; ByteStreamGeometryContext ctx = new ByteStreamGeometryContext(); if (IsRounded(radiusX, radiusY)) { Point[] points = GetPointList(rect, radiusX, radiusY); ctx.BeginFigure(points[0], true /* is filled */, true /* is closed */); ctx.BezierTo(points[1], points[2], points[3], true /* is stroked */, false /* is smooth join */); ctx.LineTo(points[4], true /* is stroked */, false /* is smooth join */); ctx.BezierTo(points[5], points[6], points[7], true /* is stroked */, false /* is smooth join */); ctx.LineTo(points[8], true /* is stroked */, false /* is smooth join */); ctx.BezierTo(points[9], points[10], points[11], true /* is stroked */, false /* is smooth join */); ctx.LineTo(points[12], true /* is stroked */, false /* is smooth join */); ctx.BezierTo(points[13], points[14], points[15], true /* is stroked */, false /* is smooth join */); } else { ctx.BeginFigure(rect.TopLeft, true /* is filled */, true /* is closed */); ctx.LineTo(Rect.TopRight, true /* is stroked */, false /* is smooth join */); ctx.LineTo(Rect.BottomRight, true /* is stroked */, false /* is smooth join */); ctx.LineTo(Rect.BottomLeft, true /* is stroked */, false /* is smooth join */); } ctx.Close(); data.SerializedData = ctx.GetData(); return data; } /// /// /// /// /// Critical - Calls critical code /// TreatAsSafe - returning a RectangleGeometry's point list is considered safe /// [SecurityCritical, SecurityTreatAsSafe] private Point[] GetPointList(Rect rect, double radiusX, double radiusY) { uint pointCount = GetPointCount(rect, radiusX, radiusY); Point[] points = new Point[pointCount]; unsafe { fixed(Point *pPoints = points) { RectangleGeometry.GetPointList(pPoints, pointCount, rect, radiusX, radiusY); } } return points; } /// /// Critical - Accepts pointer arguments /// [SecurityCritical] private unsafe static void GetPointList(Point * points, uint pointsCount, Rect rect, double radiusX, double radiusY) { if (IsRounded(radiusX, radiusY)) { // It is a rounded rectangle Invariant.Assert(pointsCount >= c_roundedPointCount); radiusX = Math.Min(rect.Width * (1.0 / 2.0), Math.Abs(radiusX)); radiusY = Math.Min(rect.Height * (1.0 / 2.0), Math.Abs(radiusY)); double bezierX = ((1.0 - EllipseGeometry.c_arcAsBezier) * radiusX); double bezierY = ((1.0 - EllipseGeometry.c_arcAsBezier) * radiusY); points[1].X = points[0].X = points[15].X = points[14].X = rect.X; points[2].X = points[13].X = rect.X + bezierX; points[3].X = points[12].X = rect.X + radiusX; points[4].X = points[11].X = rect.Right - radiusX; points[5].X = points[10].X = rect.Right - bezierX; points[6].X = points[7].X = points[8].X = points[9].X = rect.Right; points[2].Y = points[3].Y = points[4].Y = points[5].Y = rect.Y; points[1].Y = points[6].Y = rect.Y + bezierY; points[0].Y = points[7].Y = rect.Y + radiusY; points[15].Y = points[8].Y = rect.Bottom - radiusY; points[14].Y = points[9].Y = rect.Bottom - bezierY; points[13].Y = points[12].Y = points[11].Y = points[10].Y = rect.Bottom; points[16] = points[0]; } else { // The rectangle is not rounded Invariant.Assert(pointsCount >= c_squaredPointCount); points[0].X = points[3].X = points[4].X = rect.X; points[1].X = points[2].X = rect.Right; points[0].Y = points[1].Y = points[4].Y = rect.Y; points[2].Y = points[3].Y = rect.Bottom; } } private static byte[] GetTypeList(Rect rect, double radiusX, double radiusY) { if (rect.IsEmpty) { return null; } else if (IsRounded(radiusX, radiusY)) { return s_roundedPathTypes; } else { return s_squaredPathTypes; } } private uint GetPointCount(Rect rect, double radiusX, double radiusY) { if (rect.IsEmpty) { return 0; } else if (IsRounded(radiusX, radiusY)) { return c_roundedPointCount; } else { return c_squaredPointCount; } } private uint GetSegmentCount(Rect rect, double radiusX, double radiusY) { if (rect.IsEmpty) { return 0; } else if (IsRounded(radiusX, radiusY)) { return c_roundedSegmentCount; } else { return c_squaredSegmentCount; } } private static void GetCounts(Rect rect, double radiusX, double radiusY, out uint pointCount, out uint segmentCount) { if (rect.IsEmpty) { pointCount = 0; segmentCount = 0; } else if (IsRounded(radiusX, radiusY)) { // The rectangle is rounded pointCount = c_roundedPointCount; segmentCount = c_roundedSegmentCount; } else { pointCount = c_squaredPointCount; segmentCount = c_squaredSegmentCount; } } #region Public Methods /// /// Returns true if this geometry is empty /// public override bool IsEmpty() { return Rect.IsEmpty; } /// /// Returns true if this geometry may have curved segments /// public override bool MayHaveCurves() { return IsRounded(); } #endregion Public Methods #region InstanceData // Rouneded static private UInt32 c_roundedSegmentCount = 8; static private UInt32 c_roundedPointCount = 17; static private byte smoothBezier = (byte)MILCoreSegFlags.SegTypeBezier | (byte)MILCoreSegFlags.SegIsCurved | (byte)MILCoreSegFlags.SegSmoothJoin; static private byte smoothLine = (byte)MILCoreSegFlags.SegTypeLine | (byte)MILCoreSegFlags.SegSmoothJoin; static private byte[] s_roundedPathTypes = { (byte)MILCoreSegFlags.SegTypeBezier | (byte)MILCoreSegFlags.SegIsCurved | (byte)MILCoreSegFlags.SegSmoothJoin | (byte)MILCoreSegFlags.SegClosed, smoothLine, smoothBezier, smoothLine, smoothBezier, smoothLine, smoothBezier, smoothLine }; // Squared private const UInt32 c_squaredSegmentCount = 4; private const UInt32 c_squaredPointCount = 5; private static readonly byte[] s_squaredPathTypes = { (byte)MILCoreSegFlags.SegTypeLine | (byte)MILCoreSegFlags.SegClosed, (byte)MILCoreSegFlags.SegTypeLine, (byte)MILCoreSegFlags.SegTypeLine, (byte)MILCoreSegFlags.SegTypeLine }; #endregion } } ``` | 3,386 | 13,726 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2021-49 | longest | en | 0.441936 |
http://www.reddit.com/user/piranna00?sort=controversial | 1,408,791,405,000,000,000 | text/html | crawl-data/CC-MAIN-2014-35/segments/1408500825567.38/warc/CC-MAIN-20140820021345-00230-ip-10-180-136-8.ec2.internal.warc.gz | 563,985,356 | 15,694 | # reddit's stories are created by its users
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
That's kind of how I did it when I was younger. I found out that you could do this: 9*8: 8-1 = 7, 10-8 = 2, 9 * 8 = 72. Basically the same thing.
[–] 1 point2 points (0 children)
sorry, this has been archived and can no longer be voted on
In that case, I would always take the money. I can use the money to buy the item. It's harder to sell the item to get the amount of money its worth.
[–] 4 points5 points (0 children)
sorry, this has been archived and can no longer be voted on
[–] 8 points9 points (0 children)
sorry, this has been archived and can no longer be voted on
Ctrl+Shift+W: Close all tabs
[–] 1 point2 points (0 children)
You beat the easy level. Try the next level:
http://poweroftwo.nemoidstudio.com/4096
[–] 0 points1 point (0 children)
I wouldn't take a writing seminar as a gpa booster. What you'll learn throughout your years here is that there are a ton of really interesting courses to take so wasting a class solely as a gpa boost isn't worth it. There are a lot of seniors that even end up taking an fws their senior year because they wanted to get a head start with other courses in their major.
The chemistry class isn't that bad. I didn't take an ap level chem course in high school, but didn't find the material very hard when I took it. Definitely skip it if you get the credit.
As a rule of thumb, if you have credit for the course, take the credit so you can take a more interesting class instead. Good luck with your exams!
[–] 5 points6 points (0 children)
Even a digimon!
Aguman
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
That's not true. There's no notion of order in a scantron so AB is the same as BA. You have to get rid of a lot of double counting.
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
One of the first things I learned in a programming class. I actually ended up using it to write my commonapp essay.
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
Thanks. It really is the exact same method now that I wrote it correctly. Its just shifted by 1. Never knew anyone else who did that though.
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
Knowing anything about big-o notation, data structures, graph theory puts you ahead of just about anyone who is coming out of the intro course (1110). CS 2112 is a much tougher version of 2110, but from what I have heard it is also much better. Lately with the growing class sizes and with 1110 being taught in python, 2110 has become a lot easier.
I know a freshman who took it last semester, so I find it a little weird that its not offered in the fall. If you want to jump right into cs classes then maybe taking 2110 first semester would be good, but either way works.
[–] 24 points25 points (0 children)
Here's a line from the sao short novels: "However, she was still alive. She flew past the Minigun's line of fire and danced in space. She was directly above Behemoth's imposing stance. He probably wanted to shoot till he ran out of bullets, so he backed up, trying to keep Sinon in his firing line. But he couldn't reach her. For a Minigun that was hung on a rail on his back, there's no way it could aim directly above."
[–] 33 points34 points (0 children)
That's correct. Here's the line: "However, she was still alive. She flew past the Minigun's line of fire and danced in space. She was directly above Behemoth's imposing stance. He probably wanted to shoot till he ran out of bullets, so he backed up, trying to keep Sinon in his firing line. But he couldn't reach her. For a Minigun that was hung on a rail on his back, there's no way it could aim directly above."
[–] 0 points1 point (0 children)
From twitter: Rdu is now deleting all his friends. Admins say that the game was not affected and there for the game will not be replayed #viagamehs
[–][S] 1 point2 points (0 children)
If that doesn't work, this still should:
http://www.hlsprovider.org/latest/test/jwplayer6/
http://mtgx_esport_live-lh.akamaihd.net/i/esport5_0@196714/master.m3u8
Go to first link. Scroll down and put second link in white box and press play.
[–][S] 0 points1 point (0 children)
Hmm, mines still working.
[–] 8 points9 points (0 children)
http://www.hlsprovider.org/latest/test/jwplayer6/
http://mtgx_esport_live-lh.akamaihd.net/i/esport5_0@196714/master.m3u8
Go to first link. Scroll down and put second link in white box and press play. | 1,291 | 4,811 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2014-35 | latest | en | 0.974163 |
https://discordleaks.unicornriot.ninja/discord/view/1902965?q= | 1,606,824,499,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141674082.61/warc/CC-MAIN-20201201104718-20201201134718-00267.warc.gz | 256,338,400 | 5,876 | # Message from Deleted User in MacGuyver - Skills & Academics#homework-help
2018-01-30 04:42:13 UTC
When I multiply I get 54/1296 and i don't know how to reduce it because I'm bad at fractions
2018-01-30 04:44:07 UTC
Simplify 6/9 * 9/144
2018-01-30 04:44:19 UTC
The 9s cancel out
2018-01-30 04:44:33 UTC
54/54 = 1 and 1296/54 = 24
2018-01-30 04:44:39 UTC
And 6 goes into 144 24 times
2018-01-30 04:45:07 UTC
So cancelled out the 9s and divide 6/144 ?
2018-01-30 04:45:12 UTC
Is that mastering physics but for math
2018-01-30 04:45:16 UTC
Yes
2018-01-30 04:45:36 UTC
Or what @micbwilli said
2018-01-30 04:45:42 UTC
Either works
2018-01-30 04:45:57 UTC
Okay thanks @GetOffMyState#8267
2018-01-30 04:46:32 UTC
@micbwilli could you explain your way a bit more? How do you know that's what your supposed to do?
2018-01-30 04:48:52 UTC
2018-01-30 04:49:16 UTC
Both rely on what I did above with another example
2018-01-30 04:51:18 UTC
For your example think about how 6/9 * 9/144 = 9/9 * 6/144
2018-01-30 04:51:42 UTC
The 9/9 is 1 and can therefore be ignored
2018-01-30 04:51:50 UTC
If you are trying to reduce a fraction start with a common number and just work down.
So like with 54/1296, both are multiples or 9. I knew 54 was and I checked 1296. That's 6/144. I see if 6 can be used to divide 144, and since it can it becomes 1/24.
2018-01-30 04:52:46 UTC
6/144 = 6 /(24 * 6) = 6/6 * 1/24
2018-01-30 04:53:01 UTC
If you are unsure there to start diving top and bottom by 2 works if both numbers are even.
2018-01-30 04:53:17 UTC
^ this
2018-01-30 04:54:57 UTC
2018-01-30 04:57:50 UTC
This is so complicated lol. Okay. So just find the GCD and then reduce it down.
2018-01-30 04:58:18 UTC
Or in this case, I could just divide 6/144
2018-01-30 04:58:25 UTC
Right?
2018-01-30 04:58:29 UTC
Yes
2018-01-30 04:58:56 UTC
Because the 9s happen to cancel out
2018-01-30 04:59:06 UTC
And if it's not a problem where it can cancel out then what?
2018-01-30 05:00:10 UTC
You can find the GCD
2018-01-30 05:01:22 UTC
Okay
2018-01-30 05:02:17 UTC
Thanks. Yall
2018-01-30 05:04:21 UTC
On another note, why do we teach PEMDAS when it should be PEDMSA?
2018-01-30 05:05:21 UTC
Multiplication and division are the same really
2018-01-30 05:05:35 UTC
2018-01-30 05:05:50 UTC
So PEMDAS is easier to remember
2018-01-30 05:05:54 UTC
I've gotten answers wrong for doing multiplication first sometimes though
2018-01-30 05:06:06 UTC
And same with addition and sibtraction
2018-01-30 05:07:46 UTC
You might be breaking another rule or your teachers suck. Unless I’m forgetting something that’s wrong and you shouldn’t have been marked down
2018-01-30 05:09:13 UTC
If you are using a computer program it is probably programmed by someone diverse and doesn’t account for multiple but equally valid ways to the answer
2018-01-30 05:09:25 UTC
This class is online. The teacher just supervises. I followed PEMDAS as taught but when I did I kept getting answers wrong and when I viewed the solution they were doing division first
2018-01-30 05:09:28 UTC
Yes omg
2018-01-30 05:09:54 UTC
The worst part is that it only counts one answer correct. And if you don't get that correct it marks it wrong even if you understand the concept | 1,180 | 3,282 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2020-50 | latest | en | 0.800422 |
https://www.intmath.com/forum/general-2/welcome-to-the-intmath-forum:93 | 1,723,231,633,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640768597.52/warc/CC-MAIN-20240809173246-20240809203246-00592.warc.gz | 638,740,844 | 23,500 | Skip to main content
Search IntMath
Close
# Welcome to the IntMath Forum!
The aim in the IntMath Forum is to help you to figure out how to do math.
This means we won't do your math homework for you - but we'll help you to figure it out for yourself.
As a result you will learn it better and will get more satisfaction.
The forum follows the same chapters, and has the same scope as Intmath.com. So any questions beyond this scope will probably not be answered as they are not relevant to the bulk of IntMath users.
X
The aim in the IntMath Forum is to help you to <b>figure out how to do math</b>.
This means we won't <b>do your math homework for you</b> - but we'll <b>help you to figure it out for yourself</b>.
As a result you will learn it better and will get more satisfaction.
The forum follows the same chapters, and has the same scope as <a href="/">Intmath.com</a>. So any questions beyond this scope will probably not be answered as they are not relevant to the bulk of IntMath users.
Relevant page
dunInclude
## Re: Welcome to the IntMath Forum!
intmath is a great resource site for math and this forum works great too.
Best Regards
Peng
X
intmath is a great resource site for math and this forum works great too.
Best Regards
Peng
## Re: Welcome to the IntMath Forum!
Thanks, Peng.
Glad you enjoy it.
X
Thanks, Peng.
Glad you enjoy it.
## Re: Welcome to the IntMath Forum!
interesting. I will be here to learn math.
X
<p>interesting. I will be here to learn math<a href="facebook.com">.</a></p>
## Re: Welcome to the IntMath Forum!
so nice
thanks, Peng
X
so nice
thanks, Peng
## Reply
You need to be logged in to reply.
## Related General questions
• IntMath Forum Principles
The IntMath Forum follows 5 simple principles to make it useful and enjoyable for all. 1....
• Inequalities graph [Pending...]
hi. i met a problem while studying. could u help me solve? sketch a region A...
• Explain notation for sum of cos terms [Solved!]
Please explain the meaning of the notation used in Wikipedia for a trig identity involving...
• Your site is a great help
Frankly speaking,You did a very wonderful job because IntMath has been a great help. keep...
• Well Done! [Solved!]
I wants to use this opportunity thanking you for the job welldone.coz it has enlighten... | 567 | 2,310 | {"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": 1, "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} | 3.015625 | 3 | CC-MAIN-2024-33 | latest | en | 0.896201 |
https://origin.geeksforgeeks.org/ugc-net-ugc-net-cs-2015-dec-iii-question-19/?ref=lbp | 1,685,694,999,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224648465.70/warc/CC-MAIN-20230602072202-20230602102202-00610.warc.gz | 485,939,760 | 33,876 | GFG App
Open App
Browser
Continue
# UGC-NET | UGC NET CS 2015 Dec – III | Question 19
The solution of the recurrence relation
T(n) ≤ { θ(1) if n ≤ 80
T(n) ≤ { T(n / s) + T((7n / 10) + 6) + O(n) if n ≥ 80
is :
(A) O(lg n)
(B) O(n)
(C) O(n lg n)
(D) None of the above | 115 | 267 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2023-23 | latest | en | 0.565338 |
https://estima.com/procs_perl/800/hpfilter.rpf | 1,568,621,582,000,000,000 | text/plain | crawl-data/CC-MAIN-2019-39/segments/1568514572516.46/warc/CC-MAIN-20190916080044-20190916102044-00554.warc.gz | 457,453,617 | 1,280 | * * HPFILTER.RPF * Example of local trend state-space model * * RATS User's Guide, Example from Section 10.7 * cal(q) 1959:1 open data haversample.rat data(format=rats) 1959:1 2006:4 gdph log gdph / lgdp * * This uses the built-in HP filter in the FILTER instruction * filter(type=hp,tuning=1600.0) lgdp / hpfilt * * The HP filter is a local trend model. The system matrices for this can * be generated using the procedure LOCALDLM with TYPE=TREND and * SHOCKS=TREND. * @localdlm(type=trend,shocks=trend,a=ahp,c=chp,f=fhp) * * This is the relative variance of the measurement error to trend rate * disturbance. * compute lambda = 1600.0 * dlm(a=ahp,c=chp,f=fhp,sv=1.0,sw=1.0/lambda,presample=diffuse,\$ type=smooth,var=concentrate,y=lgdp) / hpstates * * Extract the local trend rate, which is the second state. * set hprate = hpstates(t)(2) * * Estimate the (log of the) variances. This is *not* a good idea. The * "smoothed" trend is almost exactly equal to the series. * nonlin lsv lsw compute lsv=-3.0,lsw=-7.0 dlm(a=ahp,c=chp,f=fhp,y=lgdp,sv=exp(lsv),sw=exp(lsw),\$ presample=diffuse,type=smooth,method=bfgs) / ltstates * * Extract the trend rate from this and graph vs the HP estimate. * set ltrate = ltstates(t)(2) graph(header="Comparison of Local Trend Rates",key=below,\$ klabels=||"From HP Filter","From Estimated Local Trend"||) 2 # hprate # ltrate | 441 | 1,359 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2019-39 | latest | en | 0.566443 |
https://termsdepot.com/eulers-constant-gamma-or-the-euler-mascheroni-constant/ | 1,713,719,989,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817790.98/warc/CC-MAIN-20240421163736-20240421193736-00894.warc.gz | 495,254,377 | 10,007 | # Euler’s constant (gamma or the Euler-Mascheroni constant)
Euler's constant (gamma or the Euler-Mascheroni constant) is a mathematical constant that appears in a wide variety of mathematical settings, including in Euler's famous formula for the sum of the reciprocal of the positive integers:
gamma = lim_{n->infty} (sum_{k=1}^{n} 1/k) - ln(n)
The constant gamma also arises in connection with the factorial function, and in particular with the Stirling's approximation for large factorials. In probability theory, gamma is the name given to the distribution of the sum of k independent exponential random variables with unit mean, and also appears in the negative binomial distribution.
The value of gamma can be numerically approximated to be 0.577216. Is the Euler-Mascheroni constant rational? No, the Euler-Mascheroni constant is not rational. This is because the constant is defined as the limit of the difference between the natural logarithm of n and the harmonic series. Both of these are irrational numbers, so the difference between them is also irrational.
#### How is Euler's constant defined?
Euler's constant is defined as the limit of (1 + 1/n)^n as n approaches infinity. This can be written as:
lim n→∞ (1 + 1/n)^n = e
where e is the natural logarithm of 2.
#### What is Euler-Mascheroni constant used for?
The Euler-Mascheroni constant is a mathematical constant that is often used in computer programming. It is named after Leonhard Euler and Lorenzo Mascheroni, and is also sometimes known as the Euler-Kronecker constant.
The Euler-Mascheroni constant is defined as the limit of the difference between the natural logarithm of n and the harmonic mean of 1, 2, ..., n as n goes to infinity. In other words, it is the difference between the sum of the reciprocals of the first n natural numbers and the natural logarithm of n.
The Euler-Mascheroni constant is often used in computer programming, particularly in numerical analysis and statistical mechanics. It is also used in some physical theories, such as quantum electrodynamics.
### How is Euler-Mascheroni constant calculated?
The Euler-Mascheroni constant is calculated by taking the limit of the difference between the natural logarithm of n and the harmonic series:
limn→∞(ln(n)−Hn)
This constant can also be expressed as the integral of the exponential function:
−∫∞0dt/t e−t
The Euler-Mascheroni constant is named after Leonhard Euler and Lorenzo Mascheroni.
### Is γ rational?
No, γ is not rational. γ is the Euler–Mascheroni constant, which is defined as the limit of the difference between the natural logarithm of n and the harmonic series:
γ = limn→∞(ln(n)−Hn)
where Hn is the nth harmonic number.
It is not known whether γ is algebraic or transcendental, but it is known that it is not rational. If γ were rational, then it would be possible to write it as a fraction p/q for some integers p and q, but it is not possible to do so. | 683 | 2,942 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2024-18 | latest | en | 0.934222 |
http://www.talkchess.com/forum3/viewtopic.php?f=7&t=59401&start=20 | 1,611,358,251,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703531429.49/warc/CC-MAIN-20210122210653-20210123000653-00661.warc.gz | 187,176,696 | 6,276 | Transposition table replacement strategies
Discussion of chess software programming and technical issues.
Moderators: hgm, Dann Corbit, Harvey Williamson
Forum rules
This textbox is used to restore diagrams posted with the [d] tag before the upgrade.
zd3nik
Posts: 193
Joined: Wed Mar 11, 2015 2:34 am
Location: United States
Contact:
Re: Transposition table replacement strategies
Here is how I handle TT probes/stores in my main search routine. If anyone can spot potential problem(s) I would appreciate feedback.
Code: Select all
`````` pv = EMPTY
bestScore = MATED_AT(ply);
ttGets++;
ttMove = NONE;
ttEntry = TT.GetEntryAt(positionKey);
if (ttEntry.Matches(positionKey)) {
ttHits++;
ttMove = ttEntry.BestMove();
if (ttEntry.Depth() >= depth) {
switch (ttEntry.Type()) {
case UpperBound:
if (ttEntry.Score() <= alpha) {
pv = ttMove;
return ttEntry.Score();
}
break;
case LowerBound:
if (ttEntry.Score() >= beta) {
pv = ttMove;
return ttEntry.Score();
}
break;
case ExactScore:
if ((ttEntry.Score() <= alpha) OR (ttEntry.Score() >= beta) OR (not pvNode)) {
pv = ttMove;
return ttEntry.Score();
}
break;
}
}
Exec(ttMove);
bestScore = -child.Search(-beta, -alpha)
Undo(ttMove);
pv = ttMove + child.pv;
if (bestScore >= beta) {
ttEntry.Set(bestScore, LowerBound, ttMove);
return bestScore;
}
}
GenerateMoves();
if (no legal moves) {
if (in check) return MATED_AT(ply) else return DRAW;
}
foreach(move) {
if (move != ttMove) {
Exec(move);
score = -child.Search(-beta, -alpha);
Undo(move);
if (score > bestScore) {
pv = move + child.pv;
bestScore = score
}
if (score >= beta) {
ttEntry.Set(score, LowerBound, move);
return score;
}
alpha = max(score, alpha);
}
}
if (bestScore > original_alpha) {
ttEntry.Set(bestScore, ExactScore, pv.FirstMove());
}
else {
ttEntry.Set(bestScore, UpperBound, pv.FirstMove());
}
return bestScore;
``````
Of course I've simplified things a lot in this psuedo-code. For example the real code implements null move pruning, PVS, LMR, etc...
QSearch does the TT lookup and returns ttEntry.Score() where appropriate. But QSearch does not store results (never calls ttEntry.Set) | 791 | 2,553 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2021-04 | latest | en | 0.510962 |
https://www.shaalaa.com/question-bank-solutions/concept-exponents-simplify-2-exponent3-10-exponent3_16235 | 1,601,558,902,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600402131412.93/warc/CC-MAIN-20201001112433-20201001142433-00277.warc.gz | 989,527,302 | 8,966 | # Simplify: (− 2)Exponent3 × (−10)Exponent3 - Mathematics
#### Question
Simplify: (− 2)3 × (−10)3
#### Solution
(−2)3 × (−10)3 = (−2) × (−2) × (−2) × (−10) × (−10) × (−10)
= (−8) × (−1000) = 8000
Is there an error in this question or solution?
#### APPEARS IN
NCERT Solution for Mathematics for Class 7 (2018 (Latest))
Chapter 13: Exponents and Powers
Exercise 13.1 | Q 7.4 | Page 253
#### Video TutorialsVIEW ALL [1]
Simplify: (− 2)Exponent3 × (−10)Exponent3 Concept: Concept of Exponents. | 198 | 500 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2020-40 | latest | en | 0.553831 |
http://mail-archives.apache.org/mod_mbox/incubator-cassandra-user/201110.mbox/raw/%3CBLU153-W15D41AD63924C1E1DB04BD8FE0@phx.gbl%3E | 1,419,629,326,000,000,000 | text/plain | crawl-data/CC-MAIN-2014-52/segments/1419447549705.144/warc/CC-MAIN-20141224185909-00093-ip-10-231-17-201.ec2.internal.warc.gz | 57,867,449 | 2,169 | We have a Column Family that is immutable (no updates after the first write= ). =3B Suppose we use RF=3D2 and W=3D1. =3B Do we still need R=3D2= =2C so that R+W>=3BRF? =3B Or is it sufficient to have R=3D1? =3B= My guess is yes=2C R=3D1 is sufficient since if it reads a row=2C the row = has to be correct: it can't get stale data. =3B =3B But can it ge= t NO data? That is=2C with R=3D1 might cassandra look on one of the two nod= es=2C find no data there=2C and prematurely give up? =3B In other words= =2C does R apply only to (possibly failed) read attempts or only to success= ful reads?
=3BThanks=2C Anthony
= --_79c551fe-f9f0-4b38-8468-4de02b7b5180_-- | 247 | 661 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2014-52 | longest | en | 0.907144 |
https://moviecultists.com/whats-a-2-dimensional-figure | 1,701,495,586,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100327.70/warc/CC-MAIN-20231202042052-20231202072052-00718.warc.gz | 472,280,083 | 12,149 | Whats a 2 dimensional figure?
A plane figure or two-dimensional figure is a figure that lies completely in one plane. When you draw, either by hand or with a computer program, you draw two-dimensional figures. ... Polygons are closed, two-dimensional figures formed by three or more line segments that intersect only at their endpoints.
What is an example of a 2-dimensional figure?
A circle, square, rectangle, and triangle are some examples of two-dimensional objects and these shapes can be drawn on paper. All the 2-D shapes have sides, vertices (corners), and internal angles, except for the circle, which is a curved figure.
What is 2D shapes with examples?
2D shapes are shapes with two dimensions, such as width and height. An example of a 2D shape is a rectangle or a circle. 2D shapes are flat and cannot be physically held, because they have no depth; a 2D shape is completely flat.
Can a 2-dimensional figure be open?
A 2D shape is a shape that lies in a plane and only has a length and a width, but no height or depth. 2D shapes can be classified as closed shapes and open shapes. A closed shape means its sides are connected from end to end with no break in the connection.
What are two dimensions of a two-dimensional figure?
Two Dimensional
The 2-dimensional shapes or objects in geometry are flat plane figures that have two dimensions – length and width. Two-dimensional or 2-D shapes do not have any thickness and can be measured in only two faces.
The Definition of a Two-Dimensional Geometric Figure : Math Definitions & More
33 related questions found
What are 2-dimensional and 3 dimensional shapes?
'2D', or 'two-dimensional', simply means that the shape is flat. We can draw 2D shapes on paper. ... A '3D' ('three-dimensional') shape is a solid shape. It has three dimensions, that is, length, width and depth.
Is circle a 2-dimensional shape?
2-dimensional shapes are flat and only have two dimensions: length and width. They include squares, rectangles, circles, triangles, and more.
What is the meaning of 3 dimensional?
1 : relating to or having the three dimensions of length, width, and height A cube is three-dimensional. 2 : giving the appearance of depth or varying distances a three-dimensional movie.
Is a rhombus a two-dimensional shape?
2D shapes have only 2 dimensions and are flat e.g. square, rectangle, triangle, circle, pentagon, hexagon, heptagon, octagon, nonagon, decagon, parallelogram, rhombus, kite, quadrilateral, trapezium. 3D objects have three dimensions. The flat surfaces (faces) of many 3D objects are made up of 2D shapes e.g.
What is a 3 dimensional figure?
Three Dimensions: The objects around you, the ones you can pick up, touch, and move around, are three-dimensional. These shapes have a third dimension: depth. Cubes, prisms, pyramids, spheres, cones, and cylinders are all examples of three-dimensional objects. Three-dimensional objects can be rotated in space.
Is a pizza 2D or 3D?
Two dimensional, or 2D, shapes are flat shapes. ... Circles are round 2D shapes with no corners. Pizza pies, clocks and bike tires are all real-world examples of circles.
Are there 2D objects in real life?
Circles, squares, triangles, and rectangles are all types of 2D geometric shapes. Check out a list of different 2D geometric shapes, along with a description and examples of where you can spot them in everyday life. Keep in mind that these shapes are all flat figures without depth.
Does 2D shape face?
2D shapes A 2D shape is a flat shape. ... A face is the part of the shape that has the largest surface area – some can be flat, some can be curved e.g. a cube has 6 flat faces whereas a cylinder has 2 flat faces and 1 curved face.
What is a 1 dimensional shape?
A 1-dimensional object is a line, or line segment, which has length, but no other characteristics. A 2-dimensional object has length and height, but no depth. Examples of 2D objects are planes and polygons. A 3-dimensional object has length, height, and depth. Examples of 3D objects are cubes and spheres.
Is a square 3 dimensional?
The different basic shapes are two-dimensional such as square, rectangle etc and three-dimensional shapes such as a prism, cylinder, cone etc. Three-dimensional shapes are that having three dimensions such as height, width, and length, like any object in the real world.
Is a ray one dimensional?
A ray starts at a given point and goes off in a certain direction forever, to infinity. The point where the ray starts is called (confusingly) the endpoint. ... A ray is one-dimensional. It has zero width.
Why is a rhombus two-dimensional?
In a rhombus, all four sides are equal. A geometrical math term for equal is congruent. ... So far, we have a hierarchy of two-dimensional figures from quadrilateral to kite to rhombus to square.
A quadrilateral is a four-sided two-dimensional shape. The following 2D shapes are all quadrilaterals: square, rectangle, rhombus, trapezium, parallelogram and kite.
Are humans three-dimensional?
In everyday life, we inhabit a space of three dimensions – a vast 'cupboard' with height, width and depth, well known for centuries. Less obviously, we can consider time as an additional, fourth dimension, as Einstein famously revealed.
What is an example of three-dimensional motion?
Movement of gyroscope is an example of three dimensional motion. Motion in space incorporates all the X, Y and Z axes. Motion of birds flying in the sky is also an example of three dimensional motion.
Is circle 2D or 3D?
2D shapes include circle, triangle, square, rectangle.
How many dimensions are there in a circle?
A circle is a one-dimensional object, although one can embed it into a two-dimensional object. More precisely, it is a one-dimensional manifold.
What are 2 dimensional and 3 dimensional shapes give 2 examples for each?
Only 2 dimensions are there that are length and width. Three dimensions are there, length, width and height. Square, circle, triangle, rectangle, hexagon, etc. ... For example, in a square, all the edges are visible.
How many dimensions are there?
The world as we know it has three dimensions of space—length, width and depth—and one dimension of time. But there's the mind-bending possibility that many more dimensions exist out there. According to string theory, one of the leading physics model of the last half century, the universe operates with 10 dimensions. | 1,452 | 6,420 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.859375 | 4 | CC-MAIN-2023-50 | latest | en | 0.962526 |
https://www.primidi.com/what_is_freely_reduced | 1,638,794,936,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00173.warc.gz | 1,022,832,989 | 3,255 | # What is freely reduced?
### Some articles on freely reduced, reduced:
Van Kampen Diagram - Van Kampen Lemma
... (†) with boundary label w which is a word (not necessarily freely reduced) in the alphabet A ∪ A−1 ... Let w be a freely reduced word in the alphabet A ∪ A−1 such that w=1 in G ... Then there exists a reduced van Kampen diagram over the presentation (†) whose boundary label is freely reduced and is equal to w ...
Van Kampen Diagram - Van Kampen Lemma - Sketch of The Proof
... First, it is easy to see that if w is freely reduced and w = 1 in G there exists some van Kampen diagram with boundary label w0 such that w = w0 in F(A) (after ... However, it is possible that the word w0 is not freely reduced ... to get a sequence of van Kampen diagrams by making their boundary labels more and more freely reduced and making sure that at each step the boundary label of each diagram in the sequence is ...
Van Kampen Diagram
... of the boundary cycle of the region read from that vertex and in that direction is a freely reduced word in F(A) that belongs to R∗ ... label of the region read from that vertex and in that direction is freely reduced and belongs to R ... alphabet A ∪ A−1 (which is not necessarily freely reduced) that is called the boundary label of ...
### Famous quotes containing the words reduced and/or freely:
Narrowed-down by her early editors and anthologists, reduced to quaintness or spinsterish oddity by many of her commentators, sentimentalized, fallen-in-love with like some gnomic Garbo, still unread in the breadth and depth of her full range of work, she was, and is, a wonder to me when I try to imagine myself into that mind. | 386 | 1,674 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2021-49 | latest | en | 0.95121 |
https://cw.fel.cvut.cz/b232/courses/zsl/labs2024_13_examqa | 1,723,214,498,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640767846.53/warc/CC-MAIN-20240809142005-20240809172005-00023.warc.gz | 163,793,081 | 8,863 | Lab 13 : Exercises, Questions & Answers
We have already practiced some math problems, see Lab 5: Ultrasound theory I and there are some examples in the microscopy Lab 4 : Microscopy. We will continue with some more problems today.
X-Rays
Useful formulas
• energy of a photon $E = hf = h \frac{c}{\lambda} \, ,$
• half-value layer $\frac{1}{2} = \textrm{e}^{-\mu d}$
• speed of light in vacuum $c \approx 3 \cdot 10^8 m \cdot s^{-1}$
• Planck's constant $h \approx 4.135 \cdot 10^{-15} eV \cdot Hz^{-1}$
Energy of a photon (I)
A photon with wavelength $100~nm$ has energy of $12~eV$, what is the energy of a photon with wavelength $2~nm$?
Solution Let $E_1 = 12 eV$ the energy of the photon with wavelength $\lambda_1 = 100 nm$ and the question is what is the energy $E_2$ of a photon with wavelength $lambda_2 = 2 nm$.
We can formulate the relationship as $E_1 \cdot \lambda_1 = h\cdot c = E_2 \cdot \lambda_2$ from $E = \frac {h \cdot c} {\lambda}$
Then $E_2 = E_1 \frac{\lambda_1}{\lambda_2} = 12 \frac{100 \cdot 10^{-9}}{2 \cdot 10^{-9}} = 600 eV$
Energy of a photon (II)
The X-ray tube (rentgenka) generates X-rays from electrons with kinetic energy of $10~keV$. Compute the wavelength of the generated X-rays if we know, that only 1% of the energy is converted to radiation.
Solution Compute the amount of the converted energy $E' = 0.01 \cdot E = 100eV$, and then use it to get the wavelength from $\lambda = \frac {h \cdot c} {E'} = 12.4 \cdot 10^{-9} m$.
X-rays with intensity of $10~W/cm^2$ passes through a $10~cm$ segment of tissue with half-value layer of $2~cm$. What will be the intensity after the tissue passage? What is the tissue density in Hounsfield units (HU), consider the linear attenuation of water to be $\mu_w = 0.22~cm^{-1}$? What kind of tissue is it?
Solution
We need the linear attenuation of the tissue $\mu_t$ to compute the density. We have $1/2 = \exp(-\mu_t \cdot d)$, which gives us $$\mu = \frac{\ln 2}{d} = \frac{\ln 2}{0.02} = 34.7 ~ \textrm{m}^{-1} = 0.347 ~ \text{cm}^{-1}$$
The intensity is then
$$I = I_0 \textrm{e}^{-\mu l} \approx 0.031 \cdot I_0$$
The Hounsfield units are defined as relative intensity w.r.t attenuation in water, in formula:
$$HU(t) = \frac{\mu_t - \mu_w}{\mu_w - \mu_air} \cdot k = \frac{0.347 - 0.22}{0.22} \cdot 10^3 \approx 575$$
So it is most likely a bone structure, as the result falls into the range [500, 1000] HU.
Alternative way We see, that the tissue segment is 5-times the half-width, which means we lose a half of the incident radiation on each 2 cm, so the final intensity is $$I = \frac{1}{2^5} \cdot I_0 = 0.03125 \cdot I_0$$.
Consider a tissue block, that contains $30~cm$ width of tissue A followed by a block of $8~cm$ of tissue B. Let the half-value layers be A: $10~cm$, B: $3~cm$. What is the intensity on the tissue boundary A|B? And what is the residual intensity of the exiting radiation?
Solution Again two possible ways to the solution. Either use the half-width formula to get the linear attenuation coefficients $\mu_A, \mu_B$ and then compute
$$I_{A|B} = I_0 \cdot \mathrm{e}^{-\mu_A l_A}$$
and
$$I_B = I_A \cdot \mathrm{e}^{-\mu_B l_B} = I_0 \cdot \mathrm{e}^{-\mu_A l_A - \mu_B l_B}$$
Or realize, that we pass 3-times the half-width when passing tissue A, so $$I_A = I_0 * (\frac{1}{2})^3$$ and $$I_B = I_A \cdot (\frac{1}{2})^{\frac{3}{8}}$$ to get the same result ($0.0197 \cdot I_0$)
Ultrasound
useful formulas
Doppler effect $f = \frac {c \pm v_r} {c \pm v_s} \cdot f_0$, where $f_0$ is the emitted frequency, $f$ is the observed frequency, $c$ is the propagation speed of the waves, $v_r$ is the speed of the receiver and $v_s$ is the speed of the source; $v_r$ is added (+) when the source moves towards the receiver and subtracted (-) when the source moves away from the receiver, $v_s$ is added (+) when the receiver moves away from the receiver and subtracted (-) when the receiver moves towards the receiver.
Doppler effect
Let us consider a doppler-US settings with carrier frequency of $3~MHz$ which we use to measure the speed of blood. Let the speed of blood at the imaging place be $2~cm/s$. What is the range of echoed frequencies we recieve from this spot? Use $c_{us} = 1540 m/s$.
Solution The situation can be separated in two parts – moving target (tissue) and static source (the US probe) and with the echo a moving source(tissue) and static target (the US probe), so the Doppler effect is twofold:
$$f = \frac{c \pm v}{c} \cdot \frac{c}{c \pm v}$$
Since the direction of the blood flow relative to the probe is unknown, we compute the minimal and maximal frequency
$$f_{min} = \frac{c-v}{c+v} \cdot f_0 \qquad f_{max} = \frac{c+v}{c-v} \cdot f_0$$
Their difference is $$f_{\Delta} = f_{max} - f_{min} = ... = \frac{4cv}{c^2 - v^2} \cdot f_{0} = 155.9 Hz$$
So we receive signals within $3~MHz \pm 78 Hz$.
Other
Useful equations
• Exponential decay $dN = -\lambda N dt$, where $N(0) = N_0$,
• the solution to which is $N = N_0 \cdot \mathrm{e}^{-\lambda \cdot t}$, where $\lambda = \frac{\ln k}{t_k}$
• Decay constant and half-time $\lambda = \frac{\ln 2}{T_{1/2}}$
• Activity $A = \frac{dN}{dt}$
By which factor does the mass of a radioactive isotope reduce in 3 years, if it reduces four times within a year?
Solution
At time $t$, we observe a certain amount of particles $N(t)$ (with initial amount N(0) = N_0) and know, that $N(t)$ changes – reduces – over time by the decay rate $\lambda$. We can express this relationship by the ordinary differential equation
$$dN = -\lambda N dt, \quad N(0) = N_0$$
The solution is
$$N = N_0 \cdot \mathrm{e}^{-\lambda \cdot t}, \qquad \lambda = \frac{\ln k}{t_k}$$
with decay ratio $k$ ($k=4$ in our case) and the duration $t_k$ ($t_k=1$ for us). This is a general formula, we typically use the half-time (time to reduce by factor 2), which is:
$$\lambda = \frac{\ln 2}{T_{1/2}}$$
The initial decay rate (the activity) of $1~g$ mass of isotope $_{88}^{226}Ra$ is $1~Ci \approx 3.7 \cdot 10^{10} Bq$. What is the half-life? The molar mass of this isotope is $226 \cdot 10^{-3} kg \cdot mol^{-1}$.
Solution
As mentioned above, we derive the attenuation coefficient usually from the half-time $\lambda = \ln 2 / T_{1/2}$ which gives us $$N = N_0 \cdot \mathrm{e}^{-\lambda \cdot t}$$
Here, we look at the decay rate (activity $A$), which is the change of particles in time, formally
$$A = \left| \frac{dN}{dt} \right| = \lambda N(t) \quad = \frac{\ln 2}{T_{1/2}} \cdot N_0 \cdot \mathrm{e}^{- \frac{\ln 2}{T_{1/2}} \cdot t}$$
The exponential term equals 1 at time $t=0$, the initial amount of particles is given by the mass $m~[kg]$, the Avogadro constant $N_A = 6.023 \cdot 10^{23} [mol^{-1}]$ and the molar mass $M_m~[kg \cdot mol^{-1}]$
$$A(0) = \frac{\ln 2}{T_{1/2}} \cdot N_0 = \frac{\ln 2}{T_{1/2}} \cdot \frac{m \cdot N_A}{M_m}$$
We then reformulate the equation to obtain $T_{1/2} \approx 1583~[years]$.
A sample of $_{18}F$ is measured at 10:40 and has an activity of 30 MBq.It is injected into a patient at 11:30. How much activity was injected? The half-life of $_{18}F$ is 109.8 min.
Solution
The activity at the time of injection is $A(t) = 30 \cdot 10^6 \cdot \mathrm{e}^{- \frac{ln 2 \cdot 50} {109.8}} = 21.9 \cdot 10^6 Bq$
Let us consider the usual PET radiopharmaceutical with activity half-life of $130~[min]$ and half-life of elimination from the patient's body of $35~[min]$. The amount of $4\cdot 10^{-12}~[mol]$ of this pharmaceutical is produced $30~[min]$ before injection. What is the activity of the radiopharmaceutical at injection time? What is the activity after acquisition, which ends $15~[min]$ after injection?
Solution
The initial amount $N_0$ is given by the molar mass $n$ and $N_A$ – $N_0 = n \cdot N_A = 2.41 \cdot 10^{12} []$. Now we use $\lambda = \frac{\ln 2}{T_{1/2}}$ to get both elimination and activity decay constants $\lambda_E, \lambda_A$. It is
$$\lambda_E = \frac{\ln 2}{\tau_E} = \frac{\ln 2}{35 \cdot 60} = 3.3 \cdot 10^{-4} [s^{-1}]$$
and analogously $\lambda_A = 8.89 \cdot 10^{-5} [s^{-1}]$.
The injected amount $N_i$ is given by the decay rate formula as
$$N_i = N_0 \cdot \mathrm{e}^{-\lambda_A t_i} = ... = 2.05 \cdot 10^{12} []$$
To compute the activity, we need to compute the amount of particles available at time $t$ after injection (don't forget: the amount of particles is reduced by both elimination and activity), formally:
$$A(t) = \lambda_A \cdot N_i \mathrm{e}^{-(\lambda_E + \lambda_A) t}$$
The initial activity is then $A(0)$, and the activity after acquisition $A(15~[min])$ with results $183~[MBq]$ and $125~[MBq]$ respectively.
PET
Let us have a PET scanner with detector ring with diameter of $1~[m]$ formed by $N = 200$ detectors with equal size. Further, let us have the source of radioactivity placed at the exact center of the detector ring with the total activity of $A = 10^6~[Bq]$ (the activity registered by all detectors). In case a decay event occurs, what is the probability of event detection by a specific pair of detectors? What is the delay time between decay and detection? What is the expected activity at time $T=10~[min]$ if the radioactive agent has a half-life of $\tau_A = 10~[min]$ and the elimination half-life is also $\tau_E = 10~[min]$? What is the number of detected events from beginning until $T = 10~[min]$? Compute the total number of events and the number of events per detector pair.
Solution
Since the agent is placed exactly at the center of the ring, the detection lines will pass through the center. That means, we have $N/2$ possible detector pairs and the detection probability is $\frac{1}{100}$.
The gamma ray spreads with speed of light, so the delay between event and detection is $$t = \frac{0.5 * d}{c} = \frac{0.5}{3\cdot 10^8} = 1.67 \cdot 10^{-9}~[s]$$
If we look precisely at the input values we see, that we need to estimate the activity exactly the half time value. So we lose half of the particles due to activity and from them another half that gets eliminated from the body. So in total a $\frac{1}{4}$ of the original activity will remain. Thus $A(T=10~[min]) = 2.5 \cdot 10^5~[Bq]$.
So the amount of active particles changed from $N_0$ to $\frac{1}{4} \cdot N_0$ during the scan time of $10~[min]$. Since half of them was eliminated from the body, we arrive at $$\delta N = \frac{1}{2} \cdot (N_0 - \frac{1}{4}\cdot N_0) = \frac{3}{8} N_0$$ decay events (total count).
If we want to get the number, we need to compute the decay constant $\lambda = \frac{\ln 2}{\tau_{1/2}} = 1.16 \cdot 10^{-3}$ and put it in $N_0 = \frac{A_0}{\lambda}$. This will give $N_0 = 862 \cdot 10^6~[.]$ the total number of particles at the beginning, the number of detected events is $325 \cdot 10^6~[.]$ which is approximately $3.24 \cdot 10^6~[.]$ per detector pair.
courses/zsl/labs2024_13_examqa.txt · Last modified: 2024/05/22 14:18 by anyzjiri | 3,496 | 10,895 | {"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} | 4.25 | 4 | CC-MAIN-2024-33 | latest | en | 0.850128 |
https://www.thenakedscientists.com/get-naked/experiments/science-spin-and-bouncy-balls?page=1 | 1,537,733,589,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267159744.52/warc/CC-MAIN-20180923193039-20180923213439-00099.warc.gz | 885,316,949 | 16,890 | # Science of spin and bouncy balls
15 October 2006
## Share
This week Derek is with Professor Hugh Hunt of Cambridge University and student helpers James and Martin from the Norwich School. They're going to be finding out about the science of bouncing balls.
To do this experiment, you will need:
Super bouncy ball A little bit of oil or vaseline Kitchen counter that meets a vertical tiled wall
How to do the experiment:
1 - Take the bouncy ball and roll it gently across the work surface towards the wall. Watch to see how far it bounces back.
2 - Put a little oil or vaseline on the tiles where the ball hits the wall.
3 - Roll the ball across the work surface again and watch how far it bounces back.
What's going on?
The point of this experiment is not only to look at how far the ball bounces back, but to think about how the spin direction changes. If you think of the ball rolling clockwise when it moves towards the wall, then on the way back it must be rolling anticlockwise. This change in spin direction requires energy, but where is the energy coming from?
When rolling the ball against a dry wall, you may have noticed that it started to bounce on the way back. It is this bounce that helps the ball to change its spin direction, and it's the wall that's providing the kick.
Taking a closer look, you will see that as the ball approaches the wall, it tries to continue rolling upwards in the direction it was going (ie: clockwise). However, gravity means that it can't do that for very long, and so it has to fall back down again. This makes the ball start to bounce and roll the other way (to see a graphical simulation of this effect, go to Hugh Hunt's website).
So what happens when there is oil on the wall? In contrast to when the wall was nice and dry, the ball can no longer grip and roll upwards. This prevents the ball from bouncing and can no longer use the wall to change its spin direction.
The result is that the ball starts rolling back to us spinning the wrong way (ie: still in the clockwise direction). The only way it can start to spin anticlockwise is by relying on friction between the ball and the kitchen counter. However by the time it's done so, the ball has lost all of its energy and comes to a complete stop.
This is why the ball doesn't roll back as far when there is oil on the wall. The oil reduces the friction, stops the ball getting a kick up the wall, and leaves the ball still rolling forwards even though it's moving backwards.
So can we see this effect anywhere else? You might expect to see balls bouncing in this way when they roll towards the cushion in a game of snooker. In fact, you won't see this at all because the cushions are specially designed so that the ball won't jump when it bounces back. The cushions are angled so that the point of contact between the ball and the cushion is not exactly half way up the ball (as when it hits a vertical wall), but higher up on the ball's surface.
Want to find out more?
To learn more about experiments with bouncy balls and see videos and graphics of superballs in action, then go to Hugh Hunt's website. | 680 | 3,126 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2018-39 | latest | en | 0.94789 |
https://oeis.org/A195948 | 1,701,388,380,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100258.29/warc/CC-MAIN-20231130225634-20231201015634-00288.warc.gz | 486,673,414 | 4,304 | The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A195948 Powers of 5 which have no zero in their decimal expansion. 20
1, 5, 25, 125, 625, 3125, 15625, 78125, 1953125, 9765625, 48828125, 762939453125, 3814697265625, 931322574615478515625, 116415321826934814453125, 34694469519536141888238489627838134765625 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,2 COMMENTS Probably finite. Is 34694469519536141888238489627838134765625 the largest term? LINKS Table of n, a(n) for n=1..16. C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011. FORMULA a(n) = 5^A008839(n). A000351 intersect A052382. MATHEMATICA Select[5^Range[0, 60], DigitCount[#, 10, 0]==0&] (* Harvey P. Dale, Aug 30 2016 *) PROG (PARI) for( n=0, 9999, is_A052382(5^n) && print1(5^n, ", ")) CROSSREFS Cf. A195943, A195944, A195945, A195946, A195908, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706. Sequence in context: A060391 A000351 A050735 * A083590 A361765 A097680 Adjacent sequences: A195945 A195946 A195947 * A195949 A195950 A195951 KEYWORD nonn,base AUTHOR M. F. Hasler, Sep 25 2011 EXTENSIONS Keyword:fini removed by Jianing Song, Jan 28 2023 as finiteness is only conjectured. STATUS approved
Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.
Last modified November 30 18:31 EST 2023. Contains 367461 sequences. (Running on oeis4.) | 573 | 1,682 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2023-50 | latest | en | 0.59277 |
https://myhomeworkwriters.com/statistics-assignment-professional-writing-services-193/ | 1,685,790,635,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649193.79/warc/CC-MAIN-20230603101032-20230603131032-00378.warc.gz | 447,051,653 | 16,586 | # Statistics Assignment | Professional Writing Services
A nationwide sample ofinfluential Liberals and Conservatives was asked as a part of a comprehensive survey whether they favouredlowering environmental standards so that high-sulphur coal could be burned in coal-fired power plants. The results were:
Save Time On Research and Writing
Hire a Pro to Write You a 100% Plagiarism-Free Paper.
Liberals ConservativesNumber sampled 341 369Number in favour 88 132 At the 0.02 level of significance, can we conclude that there is a larger proportion of Conservatives in favour of lowering thestandards? Calculate and interpret the pvalue. a. State the decision rule. (Round the final answer to 3 decimal places.) H0 is rejected if 2— than . b. Compute the pooled proportion. (Round the final answer to 4 decimal places.) Pooled proportion c. Compute the value of the test statistic. (Negative answer should be indicated by a minus sign. Round the final answer to 2decimal places.) Value of test statistic d. What is your decision regarding the null hypothesis? Decision (Click to select) : e. Calculate the p-value. (Round the z—value to 2 decimal places. Round the final answer to 4 decimal places.) pvalue is: |:|
Get Statistics homework help today
Save Time On Research and Writing
Hire a Pro to Write You a 100% Plagiarism-Free Paper.
Calculate the price
Pages (550 words)
\$0.00
*Price with a welcome 15% discount applied.
Pro tip: If you want to save more money and pay the lowest price, you need to set a more extended deadline.
We know how difficult it is to be a student these days. That's why our prices are one of the most affordable on the market, and there are no hidden fees.
Instead, we offer bonuses, discounts, and free services to make your experience outstanding.
How it works
Receive a 100% original paper that will pass Turnitin from a top essay writing service
step 1
Fill out the order form and provide paper details. You can even attach screenshots or add additional instructions later. If something is not clear or missing, the writer will contact you for clarification.
Pro service tips
How to get the most out of your experience with Homework Writing Services
One writer throughout the entire course
If you like the writer, you can hire them again. Just copy & paste their ID on the order form ("Preferred Writer's ID" field). This way, your vocabulary will be uniform, and the writer will be aware of your needs.
The same paper from different writers
You can order essay or any other work from two different writers to choose the best one or give another version to a friend. This can be done through the add-on "Same paper from another writer."
Copy of sources used by the writer
Our college essay writers work with ScienceDirect and other databases. They can send you articles or materials used in PDF or through screenshots. Just tick the "Copy of sources" field on the order form.
Testimonials
See why 20k+ students have chosen us as their sole writing assistance provider
Check out the latest reviews and opinions submitted by real customers worldwide and make an informed decision.
Logistics
As I expected.
Customer 460551, November 23rd, 2021
Technology
Thank you for doing a great job. I really am glad that you did not use someones else work like the first person who did this did. Awesome job. Because of this I will continue to use this site!
Customer 454345, June 2nd, 2020
Cultural and Ethnic Studies
Well done; it is greatly appreciated.
Customer 452763, March 15th, 2023
SEO
good
Customer 463813, March 8th, 2023
Ecology
Good
Customer 453413, April 19th, 2020
Management
Best of the Best
Customer 453285, January 22nd, 2020
Other
Good job
Customer 456821, December 26th, 2022
Other
Good content. Check APA cover page format on the student paper here https://owl.purdue.edu/owl/research_and_citation/apa_style/apa_formatting_and_style_guide/apa_sample_paper.html
Customer 462457, March 29th, 2022
All instructions followed!
Customer 463337, February 28th, 2023
Finance/Acc related
Outstanding discussion, grammar, and plagiarism score.
Customer 460073, April 6th, 2022
Education
Thanks for your cooperation. Great work.
Customer 463647, December 16th, 2022
History
Good work.
Customer 463049, June 5th, 2022
11,595
Customer reviews in total
96%
Current satisfaction rate
3 pages
Average paper length
37%
Customers referred by a friend | 1,046 | 4,377 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2023-23 | latest | en | 0.888657 |
http://conceptmap.cfapps.io/wikipage?lang=en&name=Initial_mass_function | 1,563,215,469,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195523840.34/warc/CC-MAIN-20190715175205-20190715201205-00459.warc.gz | 36,843,981 | 18,060 | Initial mass function
In astronomy, the initial mass function (IMF) is an empirical function that describes the initial distribution of masses for a population of stars. The IMF is an output of the process of star formation. The IMF is often given as a probability distribution function (PDF) for the mass at which a star enters the main sequence (begins hydrogen fusion). The distribution function can then be used to construct the mass distribution (the histogram of stellar masses) of a population of stars. It differs from the present day mass function (PDMF), the current distribution of masses of stars, due to the evolution and death of stars which occurs at different rates for different masses as well as dynamical mixing in some populations.
The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. For example, the initial mass of a star is the primary factor determining its colour, luminosity, and lifetime. At low masses, the IMF sets the Milky Way Galaxy mass budget and the number of substellar objects that form. At intermediate masses, the IMF controls chemical enrichment of the interstellar medium. At high masses, the IMF sets the number of core collapse supernovae that occur and therefore the kinetic energy feedback.
The IMF is relatively invariant from one group of stars to another, though some observations suggest that the IMF is different in different environments.[1][2][3]
Form of the IMF
Initial mass function
The IMF is often stated in terms of a series of power laws, where ${\displaystyle N(m)\mathrm {d} m}$ (sometimes also represented as ${\displaystyle \xi (m)\Delta m}$ ), the number of stars with masses in the range ${\displaystyle m}$ to ${\displaystyle m+\mathrm {d} m}$ within a specified volume of space, is proportional to ${\displaystyle m^{-\alpha }}$ , where ${\displaystyle \alpha }$ is a dimensionless exponent. The IMF can be inferred from the present day stellar luminosity function by using the stellar mass-luminosity relation together with a model of how the star formation rate varies with time. Commonly used forms of the IMF are the Kroupa (2001) broken power law[4] and the Chabrier (2003) log-normal.[5]
Salpeter (1955)
The IMF of stars more massive than our sun was first quantified by Edwin Salpeter in 1955.[6] His work favoured an exponent of ${\displaystyle \alpha =2.35}$ . This form of the IMF is called the Salpeter function or a Salpeter IMF. It shows that the number of stars in each mass range decreases rapidly with increasing mass. The Salpeter Initial Mass Function is
${\displaystyle \xi (m)\Delta m=\xi _{0}\left({\frac {m}{M_{\odot }}}\right)^{-2.35}\left({\frac {\Delta m}{M_{\odot }}}\right).}$
where ${\displaystyle M_{\odot }}$ is the solar mass, and ${\displaystyle \xi _{0}}$ is a constant relating to the local stellar density.
Miller-Scalo (1979)
Later authors extended the work below one solar mass (M). Glenn E. Miller and John M. Scalo suggested that the IMF "flattened" (approached ${\displaystyle \alpha =1}$ ) below one solar mass.[7]
Kroupa (2001)
Pavel Kroupa kept ${\displaystyle \alpha =2.3}$ above half a solar mass, but introduced ${\displaystyle \alpha =1.3}$ between 0.08-0.5 M and ${\displaystyle \alpha =0.3}$ below 0.08 M.
${\displaystyle \xi (m)=m^{-\alpha },}$
${\displaystyle \alpha =0.3}$ for ${\displaystyle m<0.08,}$
${\displaystyle \alpha =1.3}$ for ${\displaystyle 0.08
${\displaystyle \alpha =2.3}$ for ${\displaystyle m>0.5}$
Chabrier (2003)
Chabrier 2003 for individual stars:
${\displaystyle \xi (m)=0.158(1/(\ln(10)m))\exp[-(\log(m)-\log(0.08))^{2}/(2\times 0.69^{2})]}$ for ${\displaystyle m<1,}$
${\displaystyle \xi (m)=km^{-\alpha }}$ for ${\displaystyle m>1,\alpha =2.3\pm 0.3}$
Chabrier 2003 for stellar systems (e.g. binaries):
${\displaystyle \xi (m)=0.086(1/(\ln(10)m))\exp[-(\log(m)-\log(0.22))^{2}/(2\times 0.57^{2})]}$ for ${\displaystyle m<1,}$
${\displaystyle \xi (m)=km^{-\alpha }}$ for ${\displaystyle m>1,\alpha =2.3\pm 0.3}$
Slope
The initial mass function is typically graphed on a logarithm scale of log(N) vs log(m). Such plots give approximately straight lines with a slope Γ equal to 1-α. Hence Γ is often called the slope of the initial mass function. The present-day mass function, for coeval formation, has the same slope except that it rolls off at higher masses which have evolved away from the main sequence.[8]
Uncertainties
There are large uncertainties concerning the substellar region. In particular, the classical assumption of a single IMF covering the whole substellar and stellar mass range is being questioned in favor of a two-component IMF to account for possible different formation modes of substellar objects. I.e. one IMF covering brown dwarfs and very-low-mass stars on the one hand, and another ranging from the higher-mass brown dwarfs to the most massive stars on the other. Note that this leads to an overlap region between about 0.05 and 0.2 M where both formation modes may account for bodies in this mass range.[9]
Variation
The possible variation of the IMF affects our interpretation of the galaxy signals and the estimation of cosmic star formation history[10] thus is important to consider.
In theory, the IMF should vary with different star-forming conditions. Higher ambient temperature increases the mass of clasping gas cloud (Jeans mass); lower gas metallicity reduces the radiation pressure thus make the accretion of the gas easier, both lead to more massive stars formed in a star cluster. The galaxy-wide IMF can be different from the star-cluster scale IMF and may systematically change with the galaxy star formation history.[11].
Measurements of the local Universe where single stars can be resolved is consistent with an invariant IMF[12] but the conclusion suffers from large measurement uncertainty due to the small number of massive stars and difficulties in distinguishes the binary systems from the single stars. Thus the IMF variation effect is not prominent enough to be observed in the local Universe.
Systems formed at much earlier times or further from the Galactic neighborhood, where star formation activity can be hundreds or even thousands time stronger than the current Milky Way, may give a better shot. It has been consistently reported both for star clusters[13] and galaxies[14] that there seems to be a systematic variation of the IMF. However, the measurements are less direct. For star clusters, the IMF changes over time due to complicated dynamical evolution. The study of galaxy-wide IMF is free from the dynamical effect since the spatially integrated light does not distinguish where a star is located. But it is very difficult to interpret those light signal. Different galaxy spectra may indicate an IMF difference but it could also be a difference in the age and metal abundance of the stellar population and a difference in dust extinction. To make it even more complicated, there are more than one and in fact, many stellar populations in a galaxy forming at different times. Thus it is almost always possible to construct a unique star formation history to explain the galactic spectrum without involving the IMF variation[15]. In this sense, the interpretation of the galactic observation cannot be exclusively IMF variation. However, the observations should be considered collectively rather than individually. When a large number of galaxy is observed and they show systematic spectral variation as a function of the current galaxy-wide star formation rate, it is more natural to interpret them as IMF variation due to the different of the star-forming environment instead of a fine set of star formation history of all those galaxies since star formation rate could fluctuate strongly in short timescales and has no correlation with the present-day star formation rate.
References
1. ^ Conroy, Charlie; van Dokkum, Pieter G. (2012). "The Stellar Initial Mass Function in Early-type Galaxies From Absorption Line Spectroscopy. II. Results". The Astrophysical Journal. 760 (1): 71. arXiv:1205.6473. Bibcode:2012ApJ...760...71C. doi:10.1088/0004-637X/760/1/71.
2. ^ Kalirai, Jason S.; Anderson, Jay; Dotter, Aaron; Richer, Harvey B.; Fahlman, Gregory G.; Hansen, Brad M.S.; Hurley, Jarrod; Reid, I. Neill; Rich, R. Michael; Shara, Michael M. (2013). "Ultra-Deep Hubble Space Telescope Imaging of the Small Magellanic Cloud: The Initial Mass Function of Stars with M < 1 Msun". The Astrophysical Journal. 763 (2): 110. arXiv:1212.1159. Bibcode:2013ApJ...763..110K. doi:10.1088/0004-637X/763/2/110.
3. ^ Geha, Marla; Brown, Thomas M.; Tumlinson, Jason; Kalirai, Jason S.; Simon, Joshua D.; Kirby, Evan N.; VandenBerg, Don A.; Muñoz, Ricardo R.; Avila, Roberto J.; Guhathakurta, Puragra; Ferguson, Henry C. (2013). "The Stellar Initial Mass Function of Ultra-faint Dwarf Galaxies: Evidence for IMF Variations with Galactic Environment". The Astrophysical Journal. 771 (1): 29. arXiv:1304.7769. Bibcode:2013ApJ...771...29G. doi:10.1088/0004-637X/771/1/29.
4. ^ Kroupa, Pavel (2001). "On the variation of the initial mass function". MNRAS. 322 (2): 231–246. arXiv:astro-ph/0009005. Bibcode:2001MNRAS.322..231K. doi:10.1046/j.1365-8711.2001.04022.x.
5. ^ Chabrier, Gilles (2003). "Galactic stellar and substellar initial mass function". Publications of the Astronomical Society of the Pacific. 115 (809): 763–795. arXiv:astro-ph/0304382. Bibcode:2003PASP..115..763C. doi:10.1086/376392.
6. ^ Salpeter, Edwin (1955). "The luminosity function and stellar evolution". Astrophysical Journal. 121: 161. Bibcode:1955ApJ...121..161S. doi:10.1086/145971.
7. ^ Miller, Glenn; Scalo, John (1979). "The initial mass function and stellar birthrate in the solar neighborhood". Astrophysical Journal Supplement Series. 41: 513. Bibcode:1979ApJS...41..513M. doi:10.1086/190629.
8. ^ Massey, Philip (1998). "The Initial Mass Function of Massive Stars in the Local Group". The Stellar Initial Mass Function (38Th Herstmonceux Conference). 142: 17. Bibcode:1998ASPC..142...17M.
9. ^ Kroupa, Pavel; et al. (2013). "The stellar and sub-stellar IMF of simple and composite populations". Stellar Systems and Galactic Structure, Vol. V. arXiv:1112.3340. Bibcode:2013pss5.book..115K. doi:10.1007/978-94-007-5612-0_4.
10. ^
11. ^
12. ^
13. ^
14. ^
15. ^ Different mass of stars have different age, thus modifying the star formation history would modify the present-day mass function, which mimics the effect of modifying the IMF. | 2,750 | 10,573 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 29, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2019-30 | latest | en | 0.906121 |
https://www.detailedpedia.com/wiki-Galvanometer | 1,713,902,503,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296818740.13/warc/CC-MAIN-20240423192952-20240423222952-00280.warc.gz | 657,387,623 | 19,894 | # Galvanometer
A galvanometer is an electromechanical measuring instrument for electric current. Early galvanometers were uncalibrated, but improved versions, called ammeters, were calibrated and could measure the flow of current more precisely. Galvanometers work by deflecting a pointer in response to an electric current flowing through a coil in a constant magnetic field. The mechanism is also used as an actuator in applications such as hard disks.
Galvanometers came from the observation, first noted by Hans Christian Ørsted in 1820, that a magnetic compass's needle deflects when near a wire having electric current. They were the first instruments used to detect and measure small amounts of current. André-Marie Ampère, who gave mathematical expression to Ørsted's discovery, named the instrument after the Italian electricity researcher Luigi Galvani, who in 1791 discovered the principle of the frog galvanoscope – that electric current would make the legs of a dead frog jerk.
Galvanometers have been essential for the development of science and technology in many fields. For example, in the 1800s they enabled long-range communication through submarine cables, such as the earliest transatlantic telegraph cables, and were essential to discovering the electrical activity of the heart and brain, by their fine measurements of current.
Galvanometers have also been used as the display components of other kinds of analog meters (e.g., light meters and VU meters), capturing the outputs of these meters' sensors. Today, the main type of galvanometer still in use is the D'Arsonval/Weston type.
## Operation
Modern galvanometers, of the D'Arsonval/Weston type, are constructed with a small pivoting coil of wire, called a spindle, in the field of a permanent magnet. The coil is attached to a thin pointer that traverses a calibrated scale. A tiny torsion spring pulls the coil and pointer to the zero position.
When a direct current (DC) flows through the coil, the coil generates a magnetic field. This field acts against the permanent magnet. The coil twists, pushing against the spring, and moves the pointer. The hand points at a scale indicating the electric current. Careful design of the pole pieces ensures that the magnetic field is uniform so that the angular deflection of the pointer is proportional to the current. A useful meter generally contains a provision for damping the mechanical resonance of the moving coil and pointer, so that the pointer settles quickly to its position without oscillation.
The basic sensitivity of a meter might be, for instance, 100 microamperes full scale (with a voltage drop of, say, 50 millivolts at full current). Such meters are often calibrated to read some other quantity that can be converted to a current of that magnitude. The use of current dividers, often called shunts, allows a meter to be calibrated to measure larger currents. A meter can be calibrated as a DC voltmeter if the resistance of the coil is known by calculating the voltage required to generate a full-scale current. A meter can be configured to read other voltages by putting it in a voltage divider circuit. This is generally done by placing a resistor in series with the meter coil. A meter can be used to read resistance by placing it in series with a known voltage (a battery) and an adjustable resistor. In a preparatory step, the circuit is completed and the resistor adjusted to produce full-scale deflection. When an unknown resistor is placed in series in the circuit the current will be less than full scale and an appropriately calibrated scale can display the value of the previously unknown resistor.
These capabilities to translate different kinds of electric quantities into pointer movements make the galvanometer ideal for turning the output of other sensors that output electricity (in some form or another), into something that can be read by a human.
Because the pointer of the meter is usually a small distance above the scale of the meter, parallax error can occur when the operator attempts to read the scale line that "lines up" with the pointer. To counter this, some meters include a mirror along with the markings of the principal scale. The accuracy of the reading from a mirrored scale is improved by positioning one's head while reading the scale so that the pointer and the reflection of the pointer are aligned; at this point, the operator's eye must be directly above the pointer and any parallax error has been minimized.
## Uses
Probably the largest use of galvanometers was of the D'Arsonval/Weston type used in analog meters in electronic equipment. Since the 1980s, galvanometer-type analog meter movements have been displaced by analog-to-digital converters (ADCs) for many uses. A digital panel meter (DPM) contains an ADC and numeric display. The advantages of a digital instrument are higher precision and accuracy, but factors such as power consumption or cost may still favor the application of analog meter movements.
### Modern uses
Most modern uses for the galvanometer mechanism are in positioning and control systems. Galvanometer mechanisms are divided into moving magnet and moving coil galvanometers; in addition, they are divided into closed-loop and open-loop - or resonant - types.
Mirror galvanometer systems are used as beam positioning or beam steering elements in laser scanning systems. For example, for material processing with high-power lasers, closed loop mirror galvanometer mechanisms are used with servo control systems. These are typically high power galvanometers and the newest galvanometers designed for beam steering applications can have frequency responses over 10 kHz with appropriate servo technology. Closed-loop mirror galvanometers are also used in similar ways in stereolithography, laser sintering, laser engraving, laser beam welding, laser TVs, laser displays and in imaging applications such as retinal scanning with Optical Coherence Tomography (OCT) and Scanning Laser Ophthalmoscopy (SLO). Almost all of these galvanometers are of the moving magnet type. The closed loop is obtained measuring the position of the rotating axis with an infrared emitter and 2 photodiodes. This feedback is an analog signal.
Open loop, or resonant mirror galvanometers, are mainly used in some types of laser-based bar-code scanners, printing machines, imaging applications, military applications and space systems. Their non-lubricated bearings are especially of interest in applications that require functioning in a high vacuum.
Moving coil type galvanometer mechanisms (called 'voice coils' by hard disk manufacturers) are used for controlling the head positioning servos in hard disk drives and CD/DVD players, in order to keep mass (and thus access times), as low as possible.
### Past uses
A major early use for galvanometers was for finding faults in telecommunications cables. They were superseded in this application late in the 20th century by time-domain reflectometers.
Galvanometer mechanisms were also used to get readings from photoresistors in the metering mechanisms of film cameras (as seen in the adjacent image).
In analog strip chart recorders such as used in electrocardiographs, electroencephalographs and polygraphs, galvanometer mechanisms were used to position the pen. Strip chart recorders with galvanometer driven pens may have a full-scale frequency response of 100 Hz and several centimeters of deflection.
## History
### Hans Christian Ørsted
The deflection of a magnetic compass needle by the current in a wire was first described by Hans Christian Ørsted in 1820. The phenomenon was studied both for its own sake and as a means of measuring electric current.
### Schweigger and Ampère
The earliest galvanometer was reported by Johann Schweigger at the University of Halle on 16 September 1820. André-Marie Ampère also contributed to its development. Early designs increased the effect of the magnetic field generated by the current by using multiple turns of wire. The instruments were at first called "multipliers" due to this common design feature. The term "galvanometer," in common use by 1836, was derived from the surname of Italian electricity researcher Luigi Galvani, who in 1791 discovered that electric current would make a dead frog's leg jerk.
### Poggendorff and Thomson
Originally, the instruments relied on the Earth's magnetic field to provide the restoring force for the compass needle. These were called "tangent" galvanometers and had to be oriented before use. Later instruments of the "astatic" type used opposing magnets to become independent of the Earth's field and would operate in any orientation.
An early mirror galvanometer was invented in 1826 by Johann Christian Poggendorff.[citation needed] An astatic galvanometer was invented by Hermann von Helmholtz in 1849; a more sensitive version of that device, the Thomson mirror galvanometer, was patented in 1858 by William Thomson (Lord Kelvin). Thomson's design was able to detect very rapid current changes by using small magnets attached to a lightweight mirror, suspended by a thread, instead of a compass needle. The deflection of a light beam on the mirror greatly magnified the deflection induced by small currents. Alternatively, the deflection of the suspended magnets could be observed directly through a microscope.
### Georg Ohm
The ability to measure voltage and current quantitatively allowed Georg Ohm, in 1827, to formulate Ohm's law – that the voltage across a conductor is directly proportional to the current through it.
### D'Arsonval and Deprez
The early moving-magnet form of galvanometer had the disadvantage that it was affected by any magnets or iron masses near it, and its deflection was not linearly proportional to the current. In 1882 Jacques-Arsène d'Arsonval and Marcel Deprez developed a form with a stationary permanent magnet and a moving coil of wire, suspended by fine wires which provided both an electrical connection to the coil and the restoring torque to return to the zero position. An iron tube between the magnet's pole pieces defined a circular gap through which the coil rotated. This gap produced a consistent, radial magnetic field across the coil, giving a linear response throughout the instrument's range. A mirror attached to the coil deflected a beam of light to indicate the coil position. The concentrated magnetic field and delicate suspension made these instruments sensitive; d'Arsonval's initial instrument could detect ten microamperes.
### Edward Weston
Edward Weston extensively improved the design of the galvanometer. He substituted the fine wire suspension with a pivot and provided restoring torque and electrical connections[further explanation needed] through spiral springs rather than through the traditional wristwatch balance wheel hairspring. He developed a method of stabilizing the magnetic field of the permanent magnet, so the instrument would have consistent accuracy over time. He replaced the light beam and mirror with a knife-edge pointer that could be read directly. A mirror under the pointer, in the same plane as the scale, eliminated parallax observation error. To maintain the field strength, Weston's design used a very narrow circumferential slot through which the coil moved, with a minimal air-gap. This improved linearity of pointer deflection with respect to coil current. Finally, the coil was wound on a light-weight form made of conductive metal, which acted as a damper. By 1888, Edward Weston had patented and brought out a commercial form of this instrument, which became a standard electrical equipment component. It was known as a "portable" instrument because it was affected very little by mounting position or by transporting it from place to place. This design is almost universally used in moving-coil meters today.[citation needed]
Initially, laboratory instruments relying on the Earth's own magnetic field to provide restoring force for the pointer, galvanometers were developed into compact, rugged, sensitive portable instruments essential to the development of electro-technology.
### Taut-band movement
The taut-band movement is a modern development of the D'Arsonval-Weston movement. The jewel pivots and hairsprings are replaced by tiny strips of metal under tension. Such a meter is more rugged for field use.
## Types
There is broadly two types of galvanometers. Some galvanometers use a solid pointer on a scale to show measurements; other very sensitive types use a miniature mirror and a beam of light to provide mechanical amplification of low-level signals.
### Tangent galvanometer
A tangent galvanometer is an early measuring instrument used for the measurement of electric current. It works by using a compass needle to compare a magnetic field generated by the unknown current to the magnetic field of the Earth. It gets its name from its operating principle, the tangent law of magnetism, which states that the tangent of the angle a compass needle makes is proportional to the ratio of the strengths of the two perpendicular magnetic fields. It was first described by Johan Jakob Nervander in 1834.
A tangent galvanometer consists of a coil of insulated copper wire wound on a circular non-magnetic frame. The frame is mounted vertically on a horizontal base provided with levelling screws. The coil can be rotated on a vertical axis passing through its centre. A compass box is mounted horizontally at the centre of a circular scale. It consists of a tiny, powerful magnetic needle pivoted at the centre of the coil. The magnetic needle is free to rotate in the horizontal plane. The circular scale is divided into four quadrants. Each quadrant is graduated from 0° to 90°. A long thin aluminium pointer is attached to the needle at its centre and at right angle to it. To avoid errors due to parallax, a plane mirror is mounted below the compass needle.
In operation, the instrument is first rotated until the magnetic field of the Earth, indicated by the compass needle, is parallel with the plane of the coil. Then the unknown current is applied to the coil. This creates a second magnetic field on the axis of the coil, perpendicular to the Earth's magnetic field. The compass needle responds to the vector sum of the two fields and deflects to an angle equal to the tangent of the ratio of the two fields. From the angle read from the compass's scale, the current could be found from a table. The current supply wires have to be wound in a small helix, like a pig's tail, otherwise the field due to the wire will affect the compass needle and an incorrect reading will be obtained.
#### Theory
The galvanometer is oriented so that the plane of the coil is vertical and aligned along parallel to the horizontal component BH of the Earth's magnetic field (i.e. parallel to the local "magnetic meridian"). When an electric current flows through the galvanometer coil, a second magnetic field B is created. At the center of the coil, where the compass needle is located, the coil's field is perpendicular to the plane of the coil. The magnitude of the coil's field is:
${\displaystyle B={\mu _{0}nI \over 2r}\,}$
where I is the current in amperes, n is the number of turns of the coil and r is the radius of the coil. These two perpendicular magnetic fields add vectorially, and the compass needle points along the direction of their resultant BH+B. The current in the coil causes the compass needle to rotate by an angle θ:
${\displaystyle \theta =\tan ^{-1}{\frac {B}{B_{H}}}\,}$
From tangent law, B = BH tan θ, i.e.
${\displaystyle {\mu _{0}nI \over 2r}=B_{H}\tan \theta \,}$
or
${\displaystyle I=\left({\frac {2rB_{H}}{\mu _{0}n}}\right)\tan \theta \,}$
or I = K tan θ, where K is called the Reduction Factor of the tangent galvanometer.
One problem with the tangent galvanometer is that its resolution degrades at both high currents and low currents. The maximum resolution is obtained when the value of θ is 45°. When the value of θ is close to 0° or 90°, a large percentage change in the current will only move the needle a few degrees.
#### Geomagnetic field measurement
A tangent galvanometer can also be used to measure the magnitude of the horizontal component of the geomagnetic field. When used in this way, a low-voltage power source, such as a battery, is connected in series with a rheostat, the galvanometer, and an ammeter. The galvanometer is first aligned so that the coil is parallel to the geomagnetic field, whose direction is indicated by the compass when there is no current through the coils. The battery is then connected and the rheostat is adjusted until the compass needle deflects 45 degrees from the geomagnetic field, indicating that the magnitude of the magnetic field at the center of the coil is the same as that of the horizontal component of the geomagnetic field. This field strength can be calculated from the current as measured by the ammeter, the number of turns of the coil, and the radius of the coils.
### Astatic galvanometer
Unlike the tangent galvanometer, the astatic galvanometer does not use the Earth's magnetic field for measurement, so it does not need to be oriented with respect to the Earth's field, making it easier to use. Developed by Leopoldo Nobili in 1825, it consists of two magnetized needles parallel to each other but with the magnetic poles reversed. These needles are suspended by a single silk thread. The lower needle is inside a vertical current sensing coil of wire and is deflected by the magnetic field created by the passing current, as in the tangent galvanometer above. The purpose of the second needle is to cancel the dipole moment of the first needle, so the suspended armature has no net magnetic dipole moment, and thus is not affected by the earth's magnetic field. The needle's rotation is opposed by the torsional elasticity of the suspension thread, which is proportional to the angle.
### Mirror galvanometer
To achieve higher sensitivity to detect extremely small currents, the mirror galvanometer substitutes a lightweight mirror for the pointer. It consists of horizontal magnets suspended from a fine fiber, inside a vertical coil of wire, with a mirror attached to the magnets. A beam of light reflected from the mirror falls on a graduated scale across the room, acting as a long mass-less pointer. The mirror galvanometer was used as the receiver in the first trans-Atlantic submarine telegraph cables in the 1850s, to detect the extremely faint pulses of current after their thousand-mile journey under the Atlantic. In a device called an oscillograph, the moving beam of light is used, to produce graphs of current versus time, by recording measurements on photographic film. The string galvanometer is a type of mirror galvanometer so sensitive that it was used to make the first electrocardiogram of the electrical activity of the human heart.
### Ballistic galvanometer
A ballistic galvanometer is a type of sensitive galvanometer for measuring the quantity of charge discharged through it. It is an integrator, by virtue of the long time constant of its response, unlike a current-measuring galvanometer. The moving part has a large moment of inertia that gives it an oscillation period long enough to make the integrated measurement. It can be either of the moving coil or moving magnet type; commonly it is a mirror galvanometer. | 4,017 | 19,449 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2024-18 | latest | en | 0.955531 |
https://scicomp.stackexchange.com/questions/2669/is-there-a-convenient-solver-for-linear-programming-with-many-summations/2671 | 1,618,213,744,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038066613.21/warc/CC-MAIN-20210412053559-20210412083559-00371.warc.gz | 604,374,107 | 41,851 | # Is there a convenient solver for linear programming with many summations?
How can I solve the following Linear Programming problem in the easiest way?
How can the summation terms be modeled in there?
From page 187 of Mathematical Programming Approaches by Charu Chandra and Janis Grabis.
• A good place to ask is or-exchange. or-exchange.com . Why do you write it is an LP (it's not really clear what the parameters are...) – user189035 Jul 1 '12 at 8:27
• It would help if you identified which are the free variables and which are coefficients with fixed values. Also it would help to know how many variables there are in total. – Wolfgang Bangerth Jul 1 '12 at 12:55
• Great question and answers! how can someone convert the sigma notations to the Ax = b format? I tried looking this up but didn't find anything. Can someone point me to some resources, please? Thanks! – Researcher Feb 26 '17 at 15:52
I generally agree with Aron's answer.
Depending on the modeling software you use, the matrix notation $Ax = b$ may be more or less convenient than sigma notation. For instance, with the C++ interfaces to CPLEX, CLP, CBC, and the Open Solver Interface (OSI), the linear algebra notation is definitely easier to work with, since the interfaces specifically ask you to set up the problem in (more or less) the form of $Ax = b$ (these interfaces are slightly more flexible).
However, some software packages treat defining a linear program as a more declarative process. Examples include GAMS, and the Python packages PuLP and Coopr. These packages focus on making the process of defining the LP look much more like the summation (or sigma) notation used in many problems. It is, of course, possible to force a problem coded in one of these packages to look like $Ax = b$, but it's nice not to have to resort to that notation if you find it cumbersome. Furthermore, these packages are not set up as standalone solvers, and thus are much more likely to have interfaces to multiple solvers, allowing for increased flexibility in implementation (since solvers vary in speed) and distribution (not everyone has licenses for closed-source solvers).
For these reasons, I'd recommend, in decreasing order of preference:
• PuLP because it's easy-to-use, lightweight, open source, and in Python
• Coopr because it's open source and in Python (but more flexible than PuLP)
• OSI because it's open source, and is an interface to multiple (mixed-integer) linear programming solvers, but it requires that you work in a compiled language (C++), which is a usability drawback only necessary for large-scale problems where PuLP and Coopr might run out of memory or take too long
• GAMS because it's easy-to-use, and uses math-like notation, but is closed-source and the trial version can only solve smaller LPs
It took me only about a day or so to learn enough about PuLP to solve small research-scale problems. It took me a couple weeks to get the hang of OSI for research, because compiled languages require more time for development; first, there was the issue of installation, then making sure that I could compile the example files and set up Makefiles correctly, and finally, programs written in C++ are generally a few times longer than programs written in Python to accomplish the same task.
If you really must use MATLAB, Aron's recommendations are probably best. To those, I'd add the MATLAB interface to lpsolve, the MATLAB scripts accompanying the book Linear Programming with Matlab, the CPLEX for MATLAB toolbox, and the YALMIP toolbox.
Of these, the YALMIP toolbox looks most promising because it's free and open source, uses declarative syntax similar to GAMS, PuLP, and Coopr, and has interfaces to a wide variety of free and nonfree solvers for many types of problems (not just linear programming). I haven't used it yet, since I typically do most of my work in the other tools I've mentioned.
Any Linear Program (LP) can be solved by a variety of freely available software. The sigma notation can be converted to linear algebra notation $Ax=b$, which is what will be expected by the LP solver unless you are using a more advanced optimization design framework. You asked about MATLAB, which contains a Linear Programming solver in its Optimization Toolbox, which is available under separate license.
You might be interested in CVX, a freely available toolbox for MATLAB that can solve LPs amongst a more general class of optimization problems.
If you would like to pursue a completely open source route, the two best options I'm aware of are CLP and the GLPK. If you have MATLAB, I recommend starting with CVX.
The problem you have is more an OR problem. As I see, you have a target function and some constraints. IBM ILOG CPLEX Optimization Studio, also known as CPLEX, is a professional software to solve these types of questions. SAS also does optimization and simulation. Hope this answer helps.
• Thanks, I do not need a professional program. I just need a program to solve this problem in easiest way – Reza Jul 1 '12 at 6:20
Please take a look at lpsolve:
lp_solve is a free (see LGPL for the GNU lesser general public license) linear (integer) programming solver based on the revised simplex method and the Branch-and-bound method for the integers. It contains full source, examples and manuals. lp_solve solves pure linear, (mixed) integer/binary, semi-continuous and special ordered sets (SOS) models.
I would recommend using JuMP. You can see here that expressions and constraints can be written directly in terms of summations. The objective function can be written in terms of sums using standard Julia (i.e. use the sum function on arrays and things like that which make sense and is easily readable).
But what's even better than the fact that you can easily write the objective and constraints directly in the sum notation, but you can then use that model to solve the equation using a ton of other packages, standard commercial and open source solvers, all directly from within JuMP. This list right here is a list of the most widely used LP solvers, and you can just test them all in one go without changing your syntax using JuMP. | 1,342 | 6,173 | {"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": 0, "mathjax_asciimath": 1, "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} | 3.03125 | 3 | CC-MAIN-2021-17 | latest | en | 0.943011 |
https://stockingisthenewplanking.com/what-circular-cylinder-is-strouhal-number/ | 1,716,510,705,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058675.22/warc/CC-MAIN-20240523235012-20240524025012-00873.warc.gz | 481,594,660 | 10,769 | ## What circular cylinder is Strouhal number?
The Strouhal number of a stationary tube or circular cylinder is a function of Reynolds number but less of surface roughness and freestream turbulence, see Figure 2.
## What Reynolds number is vortex shedding?
The frequency of vortex shedding from a circular cylinder in a uniform shear flow and the flow patterns around it were experimentally investigated. The Reynolds number Re, which was defined in terms of the cylinder diameter and the approaching velocity at its centre, ranged from 35 to 1500.
How do you calculate vortex shedding frequency?
For a constant diameter cylinder, the vortex shedding frequency is a simple calculation using the Strouhal Number. Sn=fL/V where Sn is the Strouhal No., f is the vortex shedding frequency (Hz), L is the characteristic length or diameter (if circular) in metres and V is the mean fluid velocity [ms-1] .
What is Strouhal Number used for?
The Strouhal Number can be important when analyzing unsteady, oscillating flow problems. The Strouhal Number represents a measure of the ratio of the inertial forces due to the unsteadiness of the flow or local acceleration to the inertial forces due to changes in velocity from one point to an other in the flow field.
### What does a high Strouhal number mean?
At high Strouhal numbers oscillations dominate the flow while at low Strouhal numbers the oscillations are swept by the fast-moving fluid.
### What causes von Karman vortex?
These so-called “von Kármán vortices” arise when winds are diverted around a blunt, high-profile area, often an island rising from the ocean. The alternating direction of rotation in the air forms swirls in the clouds. Satellites regularly spot these wind and cloud patterns around the world.
How do you calculate Strouhal frequency?
The Strouhal number is defined as St = fstD/U, where fst is the vortex shedding frequency (or the Strouhal frequency) of a body at rest, D is the diameter of the circular cylinder and U is the velocity of the ambient flow.
What is vortex shedding frequency?
Vortex shedding frequency relates to the dimensionless Strouhal number (S) by. (18.33) where V0 is the velocity of external flow. Experiment shows that cross flow resonance of pipeline vibration occurs at vortex shedding frequency while in-line resonance occurs at twice the shedding frequency (Blevins, 1990).
## How do you reduce vortex induced vibration?
The main results show the following:(1)The vortex generators effectively suppress vortex-induced vibration. They shorten the lock-in region and reduce the amplitude of vortex-induced vibration. The vortex generators have the best result for = 70°, which significantly reduces the amplitude. | 568 | 2,727 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2024-22 | latest | en | 0.888824 |
https://electrotopic.com/how-do-transformers-work/ | 1,721,266,929,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514816.43/warc/CC-MAIN-20240718003641-20240718033641-00091.warc.gz | 190,415,205 | 8,819 | how do transformers work ?
1. Transformers operate on the principle of electromagnetic induction to change the voltage levels of alternating current (AC). At its core, a transformer consists of two coils of wire, known as the primary and secondary windings, which are wrapped around a common magnetic core. When an alternating current flows through the primary winding, it generates a changing magnetic field in the core. According to Faraday’s Law of Electromagnetic Induction, this changing magnetic field induces a voltage in the secondary winding. The voltage induced in the secondary winding depends on the ratio of the number of turns in the primary and secondary windings.
2. In simple terms, a transformer works by transferring electrical energy between two or more circuits through electromagnetic induction. It consists of a magnetic core made of ferromagnetic material and two or more coils of insulated wire, known as windings. When an alternating current (AC) flows through the primary winding, it creates a changing magnetic field in the core. This changing magnetic field induces a voltage in the secondary winding through electromagnetic induction. The ratio of the number of turns in the primary and secondary windings determines the ratio of the input voltage to the output voltage. Thus, transformers can step up (increase) or step down (decrease) the voltage level depending on the application.
3. The working process of a transformer involves the interaction of magnetic fields and electrical currents to transfer energy between two or more circuits. It operates based on the principle of electromagnetic induction, where a changing magnetic field induces a voltage in a nearby conductor. A transformer consists of primary and secondary windings wound around a ferromagnetic core. When an alternating current (AC) flows through the primary winding, it produces a changing magnetic flux in the core. This changing magnetic flux induces an electromotive force (emf) or voltage in the secondary winding according to Faraday’s Law of Electromagnetic Induction. The ratio of the number of turns in the primary and secondary windings determines the transformation ratio, which determines how much the voltage is stepped up or stepped down in the secondary winding relative to the primary winding.
4. Transformers change voltage through mutual induction between the primary and secondary windings. When an alternating current (AC) flows through the primary winding, it creates a magnetic field in the transformer core. This magnetic field induces an alternating voltage in the secondary winding due to the changing magnetic flux passing through it. The ratio of the number of turns in the primary winding (N1) to the number of turns in the secondary winding (N2) determines the voltage transformation ratio. For a step-up transformer, where N2 > N1, the secondary voltage is higher than the primary voltage. Conversely, for a step-down transformer, where N2 < N1, the secondary voltage is lower than the primary voltage.
5. Transformers do not work with direct current (DC) in the same way they do with alternating current (AC) due to the nature of electromagnetic induction. In AC transformers, the changing magnetic field induced by alternating current in the primary winding induces a voltage in the secondary winding through electromagnetic induction. This induced voltage is directly proportional to the rate of change of magnetic flux. In contrast, DC does not change polarity or create a changing magnetic flux, which is essential for inducing a voltage in the secondary winding of a transformer. Therefore, transformers are not effective for converting or transferring direct current (DC) voltage levels and are primarily used in AC applications where they can efficiently step up or step down voltages as needed. | 709 | 3,836 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2024-30 | latest | en | 0.916097 |
http://excel.bigresource.com/how-to-reference-the-month-preceding-current-month-fFX7LQHS.html | 1,558,754,396,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257847.56/warc/CC-MAIN-20190525024710-20190525050710-00453.warc.gz | 68,180,384 | 12,411 | # How To Reference The Month Preceding Current Month
Jul 18, 2012
the MONTH(NOW()) Function, as I need it to reference the month preceding the current month.
## Finding Current Month Total Sales Using Current Month To Date Sales In Formula / Macro?
Aug 20, 2013
Basically, I'm doing a recorded macro for work where I take an export and manipulate the data to show differences between sales from last year and this year. Also comparing this months projected sales to avg of last 6 months and also against last years this month.
The problem I'm running into is in automating the this month sales for mid-month exports. I can do it individually but I can't find a formula that will do it. Data is in one cell per month, so ex. 130 sales this month so far. I need to have it convert that to projected sales for total month based on what day it currently is.
## Date Range Formula: Beginning Of Month To End Of Month (which Is In The Current Row)
Mar 20, 2009
I have log data in two columns:
Column A: Date/time (at 30 minute intervals)
Column B: Numeric data
On the last row of each month, I’m trying to perform a SumProduct on the two columns and display that result in column C.
The end of the range is determined by the month in the current row.
I’m having difficulty finding the beginning of the range, though. I need to account for both the normal dynamic calendar days & the fact that I may get data starting mid-day and mid-month.
I have this formula, but I’m not sure how to make the first array dynamic or if this is even correct approach.
Manual
=IF(OR(MONTH(A1009)=A4)*(A\$4:A\$65536
## Automatically Bold And Highlight The Current Month’s Total And Month Name
Jan 21, 2010
I have a spreadsheet for monthly supplies. In row 1 is Jan – Dec and in the row 2 below are empty cells where there will be a total for that month’s purchases. I want a conditional format formula to automatically bold and highlight the current month’s total and month name.
Also, when I enter February totals next month and that number is input into February’s total, I want that month and total to bold and highlight BUT I also want the previous month’s bold and highlight to vanish at the same time. Is this possible?
## Current Month: Column B Equal To The Current Month Adding The Day In Column A
Sep 28, 2009
I have the following data:
column a: column B:
1
7
9
25
I need a formula to make column B equal to the current month adding the day in column A. so that column B equal the following:
column a: column B:
1 09/1/2009
7 09/7/2009
9 09/9/2009
25 09/25/2009
## Financial Model (formula To Equally Distribute Revenue Either Over The Next 1 Month, 2 Month Or 3 Month Period Depending On Size Of The Deal)
Dec 23, 2008
I m trying to write a formula for my financial model. If anyone can take a stab at a solution. I'm trying to write a formula that will equally distribute revenue either over the next 1 month, 2 month or 3 month period depending on size of the deal.
Details:
Sales will fit in 1 of 3 categories. Less than 25k; between 25k & 100k; greater than 100k.
- if under \$25K, recognize in next month (month N+ 1)
- \$25K-100K, recognize in two equal parts in months N + 1 and N + 2
- over \$100K, recognize in three equal parts over 3 months
N + 1, N + 2, N + 3 ...
## Save To A Folder That Is Current Month And Have It Save To Next Month?
Jul 14, 2014
Here is a macro I am using to save a workbook using 'Save As' .
I have it open a template workbook we use, and then perform a save as, pathed to a specific folder on the desktop. We do this every night just before midnight.
What I haven't been able to figure out is how to have it save to the next month folder (The folders with the name of all the months are already created) on the last day of each current month.
For example. In my code, the final folder in the path is the current month July. On July 31 when I run the macro, I would like it to save it in the folder called August.
Is there a way to code this so it will recognize what the current month is and then at the end of each current month save it in the next month?
[Code] .....
## Getting Value Of Current Month?
Jun 15, 2014
I am looking for 3 outputs from the attached report. Output report also attached.
1.Under the column "Created_date"(Column I), I am trying to count the number of rows which has current month in column L and result in output report file under "Risks Added" column.
2.Find and count the number of rows which has been modified this month by using "Modified_Date"(Column N) column and result in output report file under "Risks Updated" column.
3.Compare the column "Status_Change_Date" which is updated this month and "status". If that row has the Status column as 'Resolved' and the 'modified date' column has current month, please give the count of those rows in the output report file under "Risks Mitigated" column.
Below is the basic vba code for copying the rows which has this month created date
[Code] .....
Attached Files:
Sample Report.xlsx‎
Output Report.xlsx‎
## Last Day Of Current Month
Nov 28, 2008
I don't think there is a built-in function for retrieval of the last day of the month, is there?
Does anyone know how I can retrieve the last day of month using VBA?
So that I can use it like DATE.
## Sum Jan-Dec Columns From Jan Thru Current Month
Aug 30, 2012
I want to compare performance data YoY, updated monthly on a YTD basis.
Sheet1: columns of data as follows
Jan11 Feb11 ... Dec11 Jan12 Feb12 ... Dec12
1000 200 ... 500 800 900 700
...
Sheet2: If we have data through July 2012, I want to show a sum of Jan-Jul11 against Jan-Jul12 on a separate sheet.
Cell1 (updated each month) = July
ColumnA: Jan-(cell1) 2011
ColumnB: Jan-(cell1) 2012
Is there a formula I can use in columns A & B that will reference cell1 and update using the data from sheet1 automatically when cell1 is changed each month?
## Self Calculated For The Current Month
Apr 9, 2008
I am working on spreadsheet that calculates yearly totals. It is set up similar to this:
Jan Feb Mar Apr May ETC. TOTAL
Charge 123 123 123
Goal 223 223 223 223
Cash 111 111 111
% Coll. 50% 50% 50%
The "goal" column is Self calculated for the current month (I.E. it's April, that goal is calculated) The "Charge, Cash, and % Collections" columns won't be fill in for April until its over.
have it add Aprils "Goal" but not add "Mays" and so on... Doing this for the entire year.
## CountIF By Month In Current Year?
Mar 6, 2014
I am a memeber of this forum for more than a year, and currently assign into a new assignment which dealing an excel file everyday. We have one excel file for moniroting of action items generated by the management after the study. As since there were around 2,500+ rows has been generated since in the beginning of 1990's till todate. So I was thinking of instead of getting the result through filter manually, I want to create a formula that will count of how many has been closed this month out of the total numbers of action items.
Is it possible to use the COUNTIF formula to count the number of items in Col C, where Col B contains a date?
2-Jan-13 Closed
2-Jan-13 Closed
5-Jan-13 Closed
19-Feb-13 Closed
16-Feb-13 Closed
22-Feb-13 Closed
2-Mar-14 Closed
5-Mar-14 Closed
8-Mar-14 Closed
10-Mar-14 Closed
15-Mar-14 Closed
## Find The Birthdays In Current Month
Dec 29, 2009
Please tell me how to find the birthdays in the current month.I m uploading the worksheet for reference.I want to know the formula in this context.
## Display Only Weekdays From Current Month
Dec 7, 2012
I have a worksheet that displays all the current months weekdays from a starting date in cell b2 using the weekday formula it works accross a row checking next day is a weekday and adding 1 but since some months have more weekdays than others the few cells that are sometimes not needed are then filled with a weekday from the next month how can I stop this? and only have the current months weekdays
## Highlight Cell If It Is The Current Month?
Jan 23, 2014
I have a calendar where the months are listed at the top of the columns and the day of the week is listed in the row. The months and dates are all formatted as a date (Jan is 1/1/2014).
______Jan___Feb___Mar...
...
Tue____7_____4_____4
Wed ___8_____5_____5
Thu____9_____6_____6
I just need to highlight the cell with the name of the month if it is the current month. I'd prefer to use conditional formatting.
## Dynamic Search For Current Month Within A Row
Mar 28, 2014
I am trying to write a macro but I got a little stuck. Basically on row 6 I have the months listed out (so like Jan Feb Mar etc), and i would like my macro to dynamically search the the cell of current month and select it.
I know the cell formula for current month (ie. "Mar") is =TEXT(TODAY(),"MMM"), but is there a way to incorporate it into a search command in macro, or do I have to insert the formula into another cell and ask the search function to search for the text value in such "dummy cell"?
Another thing to note is, I do have more of "Mar"s scattered around the sheet, but if I limit my search area to row 6 there is only one, and that's the one I need to select.
## Conditional Format Of Current Month?
Oct 15, 2012
i'm using the following three formulas to perform conditional formatting on cell B when cell A contains a date. Each condition depicts a cell colour depending on the following:
Within The Current Month
=datevalue(day(today())&"/"&month(\$q\$3)&"/"&year(\$q\$3))=(today())
Prior To The Current Month
=datevalue(day(today())&"/"&month(\$q\$3)&"/"&year(\$q\$3))(today())
It took me so time to put this together, but I've one other condition which I need to add to the above which I'm having difficulty in solving.
In addition to setting the cell colour of cell B, i would also like to set a text value.
So if the date is:
Within The Current Month, I would like the cell to say "Act'ls",
Prior To The Current Month, I would like the cell to say "Act'ls", and
Greater Than The Current Month, I would like the cell to say "A'able"
## How To Delete Lists Of Current Month
Mar 2, 2013
I have a list of data done by date that is updated weekly. The data in the list changes but only the current month will have changes but the data goes back for a couple of years. Is there a way to clear only the current month from the list and then bring in the data for the current month to the list. The monthly data is stored in worksheets labeled by the month (January 13, February 13) So what I need with for the active worksheet (Master list) to keep all the entries from previous months, remove anything from the current month, then check the the current month worksheet and import the new data. The data on the month worksheets is in columns A and B, and gets moved to Columns A and B on the Master list.
## Count Only If Matching Current Month
Jan 6, 2007
For a banking spreadsheet to monitor monthly transfers:
COLUMN A - Date of transaction (entered manually)
COLUMN B - Description (transfers entered manually as "T")
If value of cell in COLUMN A is within current month, then Count corresponding cell in COLUMN B when "T"
## Find Value Corresponding To First Date In Current Month
May 27, 2009
i have a table with a row of dates along the top. beneath each date is a value.
the dates and values in these cells changes based on info in other sheets.
i need a way of finding the first date in the current month and returning the corresponding value.
## Populating Current Month Saturdays Dates Vba?
Feb 26, 2014
I have sheet1 want to populate current month all Saturdays dates in range C14 to C18 I am trying for vba or formula.
## Using Month In Calculation ONLY If Current Year Is 2009.
Dec 3, 2008
I'm calculating sick days on an Attendance Report.
- Sick days accumulate at a rate of 1.25 per month.
- Employees will carry-forward unused sick days from 2008.
- The maximum sick days is 30.
In trying to create a formula that will multiple the current month (eg Dec=12) by 1.25, subtract any sick days taken YTD (S16), and add the carry-forward days from 2008 (W16), I noticed one problem with the following formula:
## Trying To Add Total Hours Worked In Current Month
Jan 22, 2009
I'm trying to figure out what is wrong with this formula. =(SUMIF(Q14:Q4995, "<="&EOMONTH(TODAY(),0),W14:W4995))-(SUMIF(Q14:Q4995, "<="&EOMONTH(TODAY(),-1)+1,W14:W4995)). I've got a cell that adds Total hours worked which pulls from the same column of entered data as the formula above and that cell works. My hours for the month however just shows up as zero. If I try and edit the formula or even just highlight it to copy it and then tab out of the cell this shows up...
1/0/00
If I undo the highlight and tab it will go back to showing zero. I've checked my dates that I entered and they are correct. I'm at a loss as to how to fix this formula.
## Formula To Recognize Month And Current Year
Mar 23, 2009
finding a code that would automatically insert various phrases into a column depending on whether or not the person in that row was due to renew their membership in our organization. The code I received works fantastically, but I've come across one snafu:
## Count Reason Codes By Current Month
Sep 23, 2013
A
B
C
D
E
F
G
740048
08/29/13
08/30/13
202
[code]....
I would like to have the count of column E. I would like to sepperate current month and year to date using column C. I would like it to reflect it in the below chart.
M
Code
N
Year to Date Date
O
Month to Date
105
1
202
5
4
402
1
## Return Current Month Using Excel Formula?
Feb 22, 2014
get the current month name in a Cell A1 and Next month name in Cell B1. what formula should I used? This will be a part of the macro that I'm creating.
The macro will be use every 11th to the last day of the month so I want the formula to be dynamic enough to work in each day the macro will be use.
Expected result using the current Date Today
A1 B1
February March
## Intersection Of Date Range And Current Month
Jan 9, 2007
I need to compute the number of days that a given range of dates has in common with any given month. So an example might be:
Cell A1: Range Start Date (say it's 1/1/07)
Cell A2: Range End Date (say it's 2/15/07)
Cell A3: Month Indicator (say it's 2, meaning February)
Cell A4: Days of Intersection (should be 15 in this case, meaning that 2/1-2/15 were the days of February that were also in the range 1/1-2/15)
Cell A4 is what I'm trying to create the formula for.
I'm in Excel 2000 and am having trouble installing the add-on.
## Display Data Based On Current Month
Jan 21, 2007
I have a tab set up to display information that has been colletect on other tabs. What I would like to do is change this information depending on what month is selected.
For example, if I select December then data would be pulled from other tabs and displayed. If I Select June the information would change and select other data from the same tabs.
## Insert The Working Days Of The Current Month In A Row
Jul 14, 2008
I have an excel sheet where I need to insert the working days of the current month in a row. The working days are from Sunday through Thursday, that is, 5 days a week.
For example, for the month of July, 01 is Tuesday and the week ends on 3rd. So the row should look like this:
1-3 | 6-10 | 13-17 | 20-24 | 27-31
## List All Days As Dates In Current Month
Oct 4, 2006
Is there a formula or code that returns the dates in the month in the current month before today? For example, today is 4th october 2006. so I would like it to return a column of values that include 1/10/2006, 2/10/2006, 3/10/2006, 4/10/2006. | 4,001 | 15,710 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2019-22 | latest | en | 0.910014 |
https://abbottanalytics.blogspot.com/2007/10/statistics-why-do-so-many-hate-it.html?showComment=1193427240000 | 1,627,478,243,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153729.44/warc/CC-MAIN-20210728123318-20210728153318-00458.warc.gz | 97,676,940 | 16,656 | ## Wednesday, October 17, 2007
### Statistics: Why Do So Many Hate It?
In Why is Statistics So Scary?, the Sep-26-2007 posting to the Math Stats And Data Mining Web log, the author wonders why so many people exhibit negative reactions to statistics.
I've had occasion to wondered about the same thing. I make my living largely from statistics, and have frequently received unfavorable reactions when I explain my work to others. Invariably, such respondents admit the great usefulness of statistics, so that is not the source of this negativity. I am certain that individual natural aptitude for this sort of work varies, but I do not believe that this accounts for the majority of negative feelings towards statistics.
Having received formal education in what I call "traditional" or "classical" statistics, and having since assisted others studying statistics in the same context, I suggest that one major impediment for many people is the total reliance by classical statisticians on a large set of very narrowly focused techniques. While they serve admirably in many situations, it is worth noting the disadvantages of classical statistical techniques:
1. Being so highly specialized, there are many of these techniques to remember.
2. It is also necessary to remember the appropriate applications of these techniques.
3. Broadly, classical statistics involves many assumptions. Violation of said assumptions may invalidate the results of these techniques.
Classical techniques were developed largely during a time without the benefit of rapid, inexpensive computation, which is very different from the environment we enjoy today.
The above were major motivations for me to embrace newer analytical methods (data mining, bootstrapping, etc.) in my professional life. Admittedly, newer methods have disadvantages of their own (not the least of which is their hunger for data), but it's been my experience that newer methods tend to be easier to understand, more broadly applicable and, consequently, simpler to apply.
I think the broader educational question is: Would students be better served by one or more years of torture, imperfectly or incorrectly learning myriad methods which will soon be forgotten, or the provision of a few widely useful tools and an elemental-level of understanding?
Anonymous said...
Here are a few more disadvantages I've come across when training others in statistics.
4. Statistical concepts or assumptions can be counter-intuitive or difficult to grasp.
5. Statistics books rarely give a broad overview or layman description of statistical technique before going into the detail.
6. How a statistic works can often be conveyed much better through visual interaction rather than a statistics book (e.g. http://onlinestatbook.com/stat_sim/index.html).
7. Many statistics books assume readers can follow the logic of algebraic transformations and do not provide fully worked examples that help students to understand the logic.
8. Some statisticians use technical jargon that makes their descriptions sound much more difficult than they really are (though all professions are guilty of this!). Throughout history many professions have kept their knowledge hard to access to protect their monopoly on the power of their knowledge (e.g Guilds, The Magic Circle etc..).
9. Not all software explains statistics or the diagnostics in the output very well (SPSS is quite good though).
I agree with your "few widely useful tools and an elemental-level of understanding" proposal. My ideal training resource would contain all statistics and data mining concepts with a basic description, underlying algebra, fully worked examples where possible and a visualisation. It would also provide basic and advanced information on each concept allowing you to learn or teach to the required depth.
I know this is a lot to ask! This is the closest on the web I've found to my ideal: http://www.statsoft.com/textbook/stathome.html but in practice I use a variety of resources to meet these needs.
Will Dwinnell said...
Let me clear about this: I am not suggesting "dumbing down" statistics education. My point it that I'd rather explain bootstrapping to a novice than the 17 problem-specific classical procedures it replaces.
Anonymous said...
Will, hope I didn't give the wrong idea.
I guess your main point is that data mining techniques can give us quick and robust answers without having to chose from many classical statistics for which it is highly desirable to understand the method and assumptions well?
I was offering further explanations as to why many hate statistics from my experience of teaching. Often there is a frustration with teaching materials that don't cover clear basic explanations all the way up to the advanced. I didn't want to suggest dumbing down - instead I would like statistics more accessible.
All the best.
Bill
Will Dwinnell said...
No, I didn't take your comments that way, but after re-reading my post, I decided to make clear what you mention: that I think it's preferable to move to more convenient tools, rather than "dumb down" statistics education.
Thanks very much for your contributions!
daniel john said...
Pretty good post. I just stumbled upon your blog and wanted to say that I have really enjoyed reading your blog posts. Any way I'll be subscribing to your feed and I hope you post again soon.
Custom Term Papers
Sqiar said...
SQIAR (http://www.sqiar.com) is a leading global consultancy which provides innovative business intelligence services to small and medium size (SMEs) businesses. Our agile approach provides organizations with breakthrough insights and powerful data visualizations to rapidly analyse multiple aspects of their business in perspectives that matter most. | 1,112 | 5,774 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2021-31 | latest | en | 0.966928 |
http://mathhelpforum.com/math-topics/97997-last-two-digit.html | 1,481,143,584,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542246.21/warc/CC-MAIN-20161202170902-00085-ip-10-31-129-80.ec2.internal.warc.gz | 172,069,245 | 10,977 | 1. ## Last two digit
Find the last digit two digit of the expression?
(201 * 202 * 203 *204 * 246 * 247 * 248 * 249)^2.
Is there any shortcut to find the last two digit of the above expression.
Any help would be appreciated.
Thanks,
Ashish
2. Do you know the chinese remainder theorem? If so let me know, and I will show you a much easier way. I got 76.
If not, you can just multiply and reduce modulo 100 at each stage, and the worst you will ever have to multiply is some 2 digit numbers. It will take a while, but you will get the correct answer. I suggest multiplying in a smart fashion to reduce some of the work like do
2*49=98=-2 (mod 100)
4*48=2*2*48=2*-4=-8 (mod 100)
etc
Can you please let me know how we can do it with Chinese remainder theorem?
Thanks,
4. ## Chinese Remainder Theorem in Action
Sure thing mate. So the goal is to figure out what this beast is mod 100. Fortunately, we have $100=2^2\cdot 5^2$. Now we notice that gcd(5^2,2^2)=1, which means we are in business for using the chinese remainder theorem. So this means we can just figure what this thing is mod 4 and mod 25 separately and the chinese remainder theorem will give us the unique solution mod $100=25\cdot 4$.
mod 4 is really easy because one of the numbers is 0 mod 4, so the product will be 0.
We are not quite as fortunate when we look mod 25.
This is the product when each term has been reduced.
$(1\cdot 2\cdot 3\cdot 4\cdot -4 \cdot -3 \cdot -2 \cdot -1)^2=(24\cdot 24)^2=(-1\cdot -1)^2=1^2=1$ (mod 25).
So there is precisely 1 number between 1 and 100 which is both 0 mod 4 and 1 mod 25. By inspection it is pretty clear it is 76, so this will be the last two digits of this product.
Just to prove to you that the CRT works in general, you have numbers that are relatively prime, that is gcd(a,b)=1. Then say we are looking to solve this congruence
$n\equiv x$ (mod a)
$n\equiv y$ (mod b)
Then there is 1 number mod ab that will satisfy this and this is how you figure out in general what it is.
Use the Euclidean Algorithm to find integers s,t, such that $as+bt=1$. Now take the number yas+xbt notice the sort of cross multiplying aspect of choosing this number.
now we consider $yas+xbt$ (mod ab)
$yas+xbt \equiv xbt \equiv x(1-as) \equiv x-xas \equiv x$ (mod a)
Similarly
$yas+xbt \equiv yas \equiv y(1-bt) \equiv y-xbt \equiv y$ (mod b)
And presto, there you have it in general.
In our case we notice that $-6\cdot 4 + 1\cdot 25=1$, so we should choose $1\cdot (-6)\cdot 4 + 0\cdot 1\cdot 25=-24=76$ (mod 100) as claimed. There ya have it.
5. Thanks a lot Gamma,it is very informative | 775 | 2,603 | {"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": 11, "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} | 4.15625 | 4 | CC-MAIN-2016-50 | longest | en | 0.904427 |
https://math.stackexchange.com/questions/3352647/calculate-circle-by-circular-sector-area-and-yearn | 1,569,236,144,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514576355.92/warc/CC-MAIN-20190923105314-20190923131314-00256.warc.gz | 548,581,180 | 30,343 | # Calculate circle by circular sector area and yearn
I'm in great despair right now. I got the task to solve a circles values by given circular yearn and circular sector area.
The values are not specific but I used Area $$A=8$$ and yearn $$y=3$$ for calculations.
I tried to integrate the function of yearn $$y=2\sin(\alpha/2)$$ into $$A=\frac{r^2\pi\alpha}{360}$$
and got $$f(x)=\left(\frac{y}{2\sin(\alpha/2)}\right)^2\cdot\frac{\pi\alpha}{360}-A$$
if I insert values it shows correct answers, so it seems correct.
The problem: If I try to calculate the roots (which will represent the value for alpha and is the key for more values)
with the newton method, I get an infinite number of roots.
I'm working on it for two hours now, my head is burning and I can't find any appropriate solution.
I welcome any help given!!!
• I don't understand the problem definition. What is circular yearn in this case? And what do you mean by "solve a circles values"? Could you draw a picture or explain it better? – Matti P. Sep 11 at 9:50 | 278 | 1,036 | {"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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2019-39 | latest | en | 0.878899 |
https://questioncove.com/updates/4d599eb17c17b76495c07c91 | 1,718,646,141,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861733.59/warc/CC-MAIN-20240617153913-20240617183913-00525.warc.gz | 432,371,574 | 5,726 | Ask your own question, for FREE!
Mathematics 28 Online
OpenStudy (anonymous):
how do i calculate CAGR?
OpenStudy (anonymous):
hi anyone here?
OpenStudy (tim):
hi Phil!
OpenStudy (tim):
Do you have an example or are you looking for a general explanation?
OpenStudy (anonymous):
not really, my teacher asked me
OpenStudy (shadowfiend):
I think it can be calculated as the geometric mean of the decimal multiplier equivalents (i.e. +3% becomes 1.03 and -2% becomes 0.98) of the individual period-over-period changes in
OpenStudy (shadowfiend):
in revenue, sorry.
OpenStudy (heisenberg):
I know it depends on a timespan and starting/ending values.
OpenStudy (sid1729):
Suppose the number of years you want to calculate it over is n. The rate is the nth root of total annual growth rate
OpenStudy (heisenberg):
$CAGR = \frac{V(t_n)}{V(t_0)}^\frac{1}{t_n - t_0} - 1$
OpenStudy (tim):
did you answer back? I hope you got the explanation to her in time :)
OpenStudy (heisenberg):
$t_n - t_0$ represents the time span
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends!
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!
Join our real-time social learning platform and learn together with your friends! | 352 | 1,424 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2024-26 | latest | en | 0.906121 |
http://blog.cdmansfield.com/2012/01/31/distillation-of-a-binary-mixture-in-a-distillation-column-of-arbitrary-design-part-1/ | 1,579,332,248,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250592261.1/warc/CC-MAIN-20200118052321-20200118080321-00480.warc.gz | 20,453,658 | 33,916 | January 31, 2012 Tagged with: , , , , , , , ,
*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.
# Introduction
While taking my separations and mass transfer operations course at UF, I took a particularly interesting exam where I was tasked with predicting the performance of a distillation column with two inputs. This may be getting ahead of myself, as the problem’s information was written into the proof, but the problem was as follows:
## The Test Problem
Given a distillation column separating n-pentane and n-heptane, with a feed rate of 200 kgmol/hr of a 40 mol% n-pentane liquid at bubble point, a 95 mol% distillate stream, a 5 mol% bottoms stream, a 30 mol% side stream with a flow rate out equal to that of the bottoms leaving, a reflux ratio equal to twice that of the minimum, and a 50% average tray efficiency, use the McCabe-Thiele graphical method and the provided vapor-liquid equilibrium data to determine a) the flow rate and composition of all streams, b) the minimum reflux ratio, c) the number of theoretical plates required, and d) the optimum placement of the feed stream and side stream.
So, the information given so far is
• $F=200 kgmol/hr$
• $x_F=0.40$
• $q_F=1$
• $x_D=0.95$
• $x_W=0.05$
• $G=-W$ (negative sign due to opposite orientation)
• $x_G=0.30$
• $q_G=1$
• $R_D=2R_{min}$
• $\epsilon_a=0.50$
x y
1.000 1.000
0.867 0.984
0.594 0.925
0.398 0.836
0.254 0.701
0.145 0.521
0.059 0.271
0.000 0.000
## Two Solution Methods
There are two solution methods that may be used when using the McCabe-Thiele graphical method. The classical method uses the assumptions placed on the system explicitly to determine an analytical solution for either the stripping operating line or the enriching operating line, then the operating line between the two feeds, after which the remaining operating line is found trivially by applying the rules implicit to Cartesian geometry (two points define a single line in a plane). The other method uses the lever rule to combine the feeds and allows for the trivial solution of a single feed column to be used. Beyond the use of the lever rule, all that is required is on of the effluent operating lines. After determining the combined system, all that is left is to connect the remaining points with their respective operating lines. The basis of the proof is that the method employing the lever rule is simply a mathematical manipulation of the classical method. As it is a simple transform rule, the two methods are analytically identical.
## How to Prove the Methods are the Same
To prove the two methods are identical, the intersections of the various lines were determined and the resulting coordinate pairs were then set as ratios of like coordinates and then reduced to eliminate variables. The resulting ratios were then inspected to determine whether or not the coordinates would be the same (a ratio of one) in all cases. Needless to say, all inspected intersections produced ratios of one, but the proof did lend itself to generalization, which will be discussed after the proof is applied to the initial problem.
# System Definition
The system is a distillation column with two feed streams located by their respective optimum placements, a distillate effluent stream, and a bottoms effluent stream. Each respective theoretical plate in the column is designated by its index $i$ relative to the streams crossing the boundary between the top of the column and the top tray region and is associated with liquid streams entering from above ($L_{i-1}$) and exiting below ($L_i$) and vapor streams entering from below ($V_{i+1}$) and exiting above ($V_i$). The primary feed stream ($F$) is designated as entering the column at tray $n$ and the secondary feed stream ($G$) at tray $m$. The vapor stream leaving the top of the column ($V_1$) is condensed with an ideal heat exchanger with thermal duty $Q_c$ and then splits into the distillate stream ($D$) and the reflux stream ($L_0$). The liquid stream leaving the bottom ($L_N$) is sent to an ideal reboiler, with thermal duty $Q_W$, which splits the flow between the liquid bottoms stream ($W$) and the vaporized boilup stream ($V_{N+1}$). All liquid compositions are designated as associated mole fractions $x_i$ and all vapor compositions are similarly designated as $y_i$. All side streams are also associated with their fractional quality ($q_i$) which is the degree to which the stream has passed the VLE transition and is defined by the difference of the saturated vapor enthalpy and the stream enthalpy scaled by the difference of enthalpies for each respectively saturated fluid phase (eg. $q_F=\frac{H_{V,F}-H_F}{H_{V,F}-H_{L,F}}$).
## System Assumptions
These are intentionally written as braodly as possible as they will be used for the generalization of the proof to columns of arbitrary design.
• Equimolar overflow ($L_0=L_1=L_2=\cdots =L_i$ and $V_1=V_2=V_3=\cdots =V_i$)
• No sub-cooling of the reflux entering at the top or super-heating of the boilup at the bottom
• Net positive feed into the column from side streams ($\sum_{i}{F_i}\geq 0$)
• Side streams are at bubble point ($q_i=1$)
# 1.0 Solution Method 1: The Classical McCabe-Thiele Graphical Method
## 1.1.0 Mass Balances
### 1.1.1 Overall Mass Balance
First, the sum of the mass entering and the mass exiting must be equal to zero. There are no reactions or other structural changes to the chemicals, therefore molar flows may be used as a proxy for the associated mass flows. For sake of convenience, the distillate and bottoms streams are designated as negative flows into the column as well as all explicit side draw streams. Feed streams are assumed to be positive flows into the column, but may flow in either direction so long as the column receives a net positive flow.
Eq. 1-1) $\displaystyle 0=F+G-D-W\leadsto F+G=D+W$
Next, given the assumptions above, the sum of all flows of a given chemical species must also be zero.
Eq. 1-2) $\displaystyle 0=Fx_F+Gx_G-Dx_D-Wx_W\leadsto Fx_F+Gx_G=Dx_D+Wx_W$
### 1.1.2 Mass Balance on the Top (Enriching) Section
Beginning with a rearrangement of the mass balance on the top plate:
Eq. 1-3 and Eq. 1-4) $\displaystyle\left\{\begin{matrix}V_1+L_{n-1}=V_n+L_0 \\ V_1y_1+L_{n-1}x_{n-1}=V_ny_n+L_0x_0\end{matrix}\right.$
Using the mass balance across the condenser:
Eq. 1-5 and Eq. 1-6) $\displaystyle \hookrightarrow\left\{ \begin{matrix}\underbrace{V_1-L_0}_{D}=V_n-L_{n-1}\leadsto D=V_1-L_0 \\ V_1\underbrace{y_1}_{x_D}+L_{n-1}x_{n-1}-L_0\underbrace{x_0}_{x_D}=V_ny_n\end{matrix}\right.$
Eq. 1-7) $\displaystyle \hookrightarrow\underbrace{\left(V_1-L_0\right)}_{D}x_D+\underbrace{L_{n-1}}_{L_0}x_{n-1}=\underbrace{V_n}_{V_1}y_n\leadsto y_n=\frac{L_0}{V_1}x_{n-1}+\frac{D}{V_1}x_D$
Introducing a quantity to signify the ratio of flows exiting and recycling at the top of the column:
Eq. 1-8) $\displaystyle \boxed{R_D=\frac{L_0}{D}}$
Rearranging to allow substitution and simplification:
Eq. 1-9) $\displaystyle y_n=\cfrac{\cfrac{1}{D}}{\cfrac{1}{D}}\frac{L_0}{L_0+D}x_{n-1}+\cfrac{\cfrac{1}{D}}{\cfrac{1}{D}}\frac{D}{L_0+D}x_D\leadsto y_n=\cfrac{\cfrac{L_0}{D}}{\cfrac{L_0}{D}+\cfrac{D}{D}}x_{n-1}+\cfrac{\cfrac{D}{D}}{\cfrac{L_0}{D}+\cfrac{D}{D}}x_D$
Finally, the operating line for the enriching section is produced:
Eq. 1-10) $\displaystyle \boxed{y=\frac{R_D}{R_D+1}x+\frac{x_D}{R_D+1}}$
### 1.1.3 Mass Balance Across Plate $n$
The mass balance across plate $n$ is given by
Eq. 1-11 and Eq. 1-12) $\displaystyle \left\{\begin{matrix}V_n=V_{n+1}+F\left(1-q_F\right) \\ L_n=L_{n-1}+Fq_F\end{matrix}\right.\leadsto\left\{\begin{matrix}V_1=L_0+D=V_{n+1}+F\left(1-q_F\right) \\ L_n=L_0+Fq_F\end{matrix}\right.$
### 1.1.4 Mass Balance Between Plate $m$ and Plate $n$
The mass balance between plate $m$ and plate $n$ is given by
Eq. 1-13 and Eq. 1-14$\displaystyle \left\{\begin{matrix}V_{n+1}+L_{m-1}=V_m+L_n \\ V_{n+1}y_{n+1}+L_{m-1}x_{m-1}=V_my_m+L_nx_n\end{matrix}\right.$
### 1.1.5 Mass Balance Across Plate $m$
The mass balance across plate $m$ is given by
Eq. 1-15 and Eq. 1-16) $\displaystyle \left\{\begin{matrix}V_m=V_{m+1}+G\left(1-q_G\right) \\ L_m=L_{m-1}+Gq_G\end{matrix}\right.\leadsto\left\{\begin{matrix}V_m=V_{N+1}+G\left(1-q_G\right) \\ L_N=L_{m-1}+Gq_G\end{matrix}\right.$
### 1.1.6 Mass Balance on Bottom (Stripping) Section
The mass balance across the bottom (stripping) section is given by
Eq. 1-17 and Eq. 1-18) $\displaystyle \left\{\begin{matrix}V_{m+1}+L_N=V_{N+1}+L_m \\ V_{m+1}y_{m+1}+L_Nx_N=V_{N+1}y_{N+1}+L_mx_m\end{matrix}\right.\leadsto\left\{\begin{matrix}V_{m+1}-L_m=\boxed{V_{N+1}-L_N=W} \\ V_{m+1}y_{m+1}=L_mx_m+\underbrace{V_{N+1}y_{N+1}-L_Nx_N}_{Wx_W}\end{matrix}\right.$
## 1.2.0 Operation Lines and q-Lines
### 1.2.1 Enriching Operation Line
This is the result, Eq. 1-10, derived in section 1.1.2:
Eq. 1-19) $\displaystyle \boxed{y=\frac{R_D}{R_D+1}x+\frac{x_D}{R_D+1}}$
### 1.2.2 Stripping Operation Line (Incomplete Form)
Starting from Eq. 1-18 from section 1.1.6:
Eq. 1-20) $\displaystyle V_{m+1}y_{m+1}=L_mx_m+Wx_W\leadsto y_{m+1}=\frac{L_N}{V_{m+1}}x_m+\frac{W}{V_{m+1}}x_W$
Including from section 1.1.5:
Eq. 1-21 and Eq. 1-22) $\displaystyle \left\{\begin{matrix}L_m=L_N \\ V_{m+1}=V_{N+1}=L_N+W\end{matrix}\right.$
And then substituting yields:
Eq. 1-23) $\displaystyle \boxed{y_{m+1}=\frac{L_N}{L_N+W}x_m+\frac{W}{L_N+W}x_W} \,$
### 1.2.3 Operation Line Between Stream F and Stream G
Starting with Eq. 1-13 and Eq. 1-14 from section 1.1.4:
Eq. 1-24 and Eq. 1-25) $\displaystyle\left\{\begin{matrix}V_{n+1}+L_{m-1}=V_m+L_n \\ V_{n+1}y_{n+1}+L_{m-1}x_{m-1}=V_my_m+L_nx_n\end{matrix}\right.\leadsto\left\{\begin{matrix}V_{n+1}-L_n=V_m-L_{m-1}=\text{const.} \\ V_{n+1}y_{n+1}=L_nx_n+V_my_m-L_{m-1}x_{m-1}\end{matrix}\right.$
Which rearranges to yield the incomplete form:
Eq. 1-26) $\displaystyle\hookrightarrow\boxed{y_{n+1}=\frac{L_n}{V_{n+1}}x_n+\frac{V_my_m-L_{m-1}x_{m-1}}{V_{n+1}}}$
Using substitutions from Eq. 1-11 and Eq. 1-12 in section 1.1.3 and Eq. 1-15 and Eq. 1-16 from section 1.1.5 yields:
Eq. 1-27) $\displaystyle y_{n+1}=\frac{L_0+Fq_F}{L_0+D-F\left(1-q_F\right)}x_n+\frac{\left(V_{N+1}+G\left(1-q_G\right)\right)y_m-\left(L_N-Gq_G\right)x_{m-1}}{L_0+D-F\left(1-q_F\right)}$
Further substitution using Eq. 1-17 from section 1.1.6 in the numerator of the second term and subsequent rearrangement of said numerator yields:
Eq. 1-28) $\displaystyle y_{n+1}=\frac{L_0+Fq_F}{L_0+D-F\left(1-q_F\right)}x_n+\frac{\left(L_N-Gq_G\right)\left(y_m-x_{m-1}\right)+\left(W+G\right)y_m}{L_0+D-F\left(1-q_F\right)}$
Combining Eq. 1-14, Eq. 1-16, Eq. 1-17, and Eq. 1-18 and rearranging yields the relationship
Eq. 1-29) $\displaystyle Wx_W-Gx_G=L_{m-1}x_{m-1}-V_my_m$
Substitution using Eq. 1-29 into Eq. 1-26 yields
Eq. 1-30) $\displaystyle y_{n+1}=\frac{L_0+Fq_F}{L_0+D-F\left(1-q_F\right)}x_n+\frac{Gx_G-Wx_W}{L_0+D-F\left(1-q_F\right)}$
Which rearranges to yield
Eq. 1-31) $\displaystyle\hookrightarrow y_{n+1}=\cfrac{\cfrac{L_0}{D}+\cfrac{Fq_F}{D}}{\cfrac{L_0}{D}+\cfrac{D}{D}-\cfrac{F\left(1-q_F\right)}{D}}x_n+\cfrac{\cfrac{Gx_G}{D}-\cfrac{Wx_W}{D}}{\cfrac{L_0}{D}+\cfrac{D}{D}-\cfrac{F\left(1-q_F\right)}{D}}$$\displaystyle\leadsto y_{n+1}=\cfrac{R_D+\cfrac{F}{D}q_F}{R_D+1-\cfrac{F}{D}\left(1-q_F\right)}x_n+\cfrac{Gx_G-Wx_W}{\left(R_D+1-\cfrac{F}{D}\left(1-q_F\right)\right)D}$
Which finally yields the completed operating line
Eq. 1-32) $\displaystyle\boxed{y=\cfrac{R_D+\cfrac{F}{D}q_F}{R_D+\cfrac{F}{D}\left(q_F-1\right)+1}x+\cfrac{Gx_G-Wx_W}{D\left(R_D+\cfrac{F}{D}\left(q_F-1\right)+1\right)}}$
### 1.2.4 q-Line for Stream F
Given by definition the q-line for stream $F$ is
Eq. 1-33) $\displaystyle\boxed{y=\frac{q_F}{q_F-1}x-\frac{x_F}{q_F-1}}$
Combining Eq. 1-10 in section 1.1.2 and Eq. 1-28 in section 1.2.3 and rearranging then canceling out $L_0$, $D$, and $F$ yields
Eq. 1-34) $\displaystyle y=\frac{q_F}{q_F-1}x+\frac{Dx_D-\left(L_N-Gq_G\right)\left(y_m-x_{m-1}\right)-\left(W+G\right)y_m}{F\left(1-q_F\right)}$
Which may be equated with Eq. 1-33 and rearranged to yield
Eq. 1-35) $\displaystyle Fx_F=Dx_D-\left(L_N-Gq_G\right)\left(y_m-x_{m-1}\right)-\left(W+G\right)y_m$
Combining this with Eq. 1-1 and Eq. 1-2 from section 1.1.1 and rearranging yields
Eq. 1-36) $\displaystyle Wx_W-Gx_G=\left(L_N-Gq_G\right)x_{m-1}-\left(V_{N+1}+G\left(1-q_G\right)\right)y_m$
Which also rearranges to yield Eq. 1-29 from section 1.2.3.
### 1.2.5 q-Line for Stream G
The q-line for stream $G$ is trivially given, as was seen with stream $F$, by
Eq. 1-37) $\displaystyle\boxed{y=\frac{q_G}{q_G-1}x-\frac{x_G}{q_G-1}}$
### 1.2.6 Point of Intersection of Enriching Operation Line and Operation Line Between Stream F and Stream G
Equating Eq. 1-10 from section 1.1.2 and Eq. 1-32 from section 1.2.3 at the intersection $\left\{x_{FD},y_{FD}\right\}$ yields
Eq. 1-38) $\displaystyle\cfrac{R_D}{R_D+1}x_{FD}+\cfrac{x_D}{R_D+1}=\cfrac{R_D+\cfrac{F}{D}q_F}{R_D+\cfrac{F}{D}\left(q_F-1\right)+1}x_{FD}+\cfrac{Gx_G-Wx_W}{D\left(R_D+\cfrac{F}{D}\left(q_F-1\right)+1\right)}$
Solving for $x_{FD}$ and simplifying yields
Eq. 1-39) $\displaystyle\boxed{x_{FD}=\frac{x_D\left(F\left(q_F-1\right)+D\left(R_D+1\right)\right)+\left(R_D+1\right)\left(Wx_W-Gx_G\right)}{F\left(q_F+R_D\right)}}$
Plugging Eq. 1-39 back into Eq. 1-10 and simplifying yields
Eq. 1-40) $\displaystyle\boxed{y_{FD}=\frac{x_D\left(Fq_F+DR_D\right)+R_D\left(Wx_W-Gx_G\right)}{F\left(q_F+R_D\right)}}$
### 1.2.7 Point of Intersection of Operation Line Between Stream F and Stream G and q-Line for Stream G
Equating Eq. 1-32 from section 1.2.3 and Eq. 1-37 from section 1.2.5 at the intersection $\left\{x_{GW},y_{GW}\right\}$ yields
Eq. 1-41) $\displaystyle\cfrac{R_D+\cfrac{F}{D}q_F}{R_D+\cfrac{F}{D}\left(q_F-1\right)+1}x+\cfrac{Gx_G-Wx_W}{D\left(R_D+\cfrac{F}{D}\left(q_F-1\right)+1\right)}=\cfrac{q_G}{q_G-1}x-\cfrac{x_G}{q_G-1}$
Solving for $x_{GW}$ and simplifying yields
Eq. 1-42) $\displaystyle\boxed{x_{GW}=\frac{x_G\left(F\left(q_F-1\right)+D\left(R_D+1\right)\right)+\left(q_G-1\right)\left(Gx_G-wx_W\right)}{F\left(q_F-q_G\right)+D\left(q_G+R_D\right)}}$
Plugging Eq. 1-42 back into Eq. 1-37 and simplifying yields
Eq. 1-43) $\displaystyle\boxed{y_{GW}=\frac{x_G\left(Fq_F+Gq_G\right)+DR_Dx_G-q_GWx_W}{F\left(q_F-q_G\right)+D\left(q_G+R_D\right)}}$
### 1.2.8 Stripping Operation Line (Complete Form)
The final operation line may be constructed from the known points at either endpoint, $\left\{x_W,y_W\right\}$ and $\left\{x_{GW},y_{GW}\right\}$.
The operation line is constructed from the rearranged point-slope form, given by
Eq. 1-44) $\displaystyle\frac{y_{GW}-x_W}{x_{GW}-x_W}\left(x-x_W\right)+x_W$
Notice that $y_W$ has been replaced by $x_W$ as they are the same value.
Substituting into Eq. 1-44 using Eq. 1-42 and Eq. 1-43 from section 1.2.7 and rearranging yields the complete form of the stripping operation line
Eq. 1-45) $\displaystyle\boxed{y=\frac{\left(Fq_F+Gq_G+DR_D\right)x_Gx-\left(\left(F\left(q_F-q_G\right)+D\left(q_G+R_D\right)+q_GW\right)x+Wx_G\right)x_W+Wx^2_W}{\left(F\left(q_F-1\right)+G\left(q_G-1\right)+D\left(R_D+1\right)\right)x_G-\left(F\left(q_F-q_G\right)+D\left(q_G+R_D\right)+W\left(q_G-1\right)\right)x_W}}$
## 1.3.0 Solution of the Test Problem
The original problem can now be solved using the data provided and the equations derived.
### 1.3.1 Plot the VLE Data
First, the equilibrium data provided is plotted.
### 1.3.2 Plot the VLE Data Scaled for Efficiency
Then the equilibrium line is adjusted to reflect the column’s efficiency.
### 1.3.3 Solve the Overall Mass Balance and Plot Composition Lines
Next, the mass balance is solved to determine the compositions and flow rates of all streams. Then lines denoting the respective compositions of the individual streams are placed.
### 1.3.4 Plot q-Lines for All Streams
q-lines are plotted for all streams. Since the bottoms and distillate streams leave as liquids at bubble point, their respective q-lines are vertical.
### 1.3.5 Determine the Minimum Reflux Ratio
$R_{min}$ is determined by finding the slope of the line from the distillate composition point to the first pinch point along the bounding VLE curve before or at the feed stream or side stream of interest and then using the slope from Eq. 1-10 in section 1.1.2.
### 1.3.6 Plot the Operation Lines
Use the equations derived above (Eq. 1-10, Eq. 1-32, and Eq. 1-45) to plot the operation lines.
### 1.3.7 Plot the Plate Locations
To complete the problem, the plate lines are drawn between the bounding VLE curve and the contiguous operation lines, starting at the distillate composition point and ending at the tray crossing the bottoms composition boundary. The number of triangles represents the number of trays required and the triangles containing the composition points for the feed streams and side streams are the optimal placement of the respective streams.
## 1.4.0 Automation Using Mathematica
The plots above were generated using some of the results of this proof as derived in a later section. This may be jumping the gun a little, but the code for this section’s plots is given below. Reading through the remainder of the proof is recommended before trying to understand the details of the math behind the plot generation.
That’s it for now. The rest of the proof will be posted in chunks of about the same size as this and the complete proof will be assembled in a single post when all of the parts are completed and reviewed. Thank you for reading my work and please do leave a comment to let me know what you think of it.
Be Sociable, Share!
Short URL: http://dlvr.it/18Lt90 | 6,110 | 17,873 | {"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": 96, "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} | 3.515625 | 4 | CC-MAIN-2020-05 | latest | en | 0.858235 |
https://www.mathcoachscorner.com/2021/02/st-patricks-day-freebie-for-making-10/ | 1,696,217,578,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510967.73/warc/CC-MAIN-20231002033129-20231002063129-00612.warc.gz | 969,650,113 | 34,392 | St. Patrick’s Day FREEBIE for Making 10
Written by Donna Boucher
Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.
Making Ten | Math Games | St. Patrick's Day
Ten holds special importance in our number system. It’s no wonder, then, that a critical Kindergarten skill is knowing all of the combinations for making ten. With St. Patrick’s Day on the horizon, today I’ve got a free holiday-themed game for making ten.
This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.
The goal of this Capture 4 game is to get four spaces in a row, either horizontally, vertically, or diagonally. The center space is a “free” space and can be used by either player. Both games, print and digital, can be played with a die. You’ll need a ten-sided die (0-9) to be able to make all the combinations for ten. If you type “google dice” into the search bar, you can access a digital die. The print version also includes number cards that can be used in place of the die. On their turn, a player rolls the die and determines what number would need to be added to the number rolled to make ten. You see in the picture below that the player rolled an 8. Eight and two make ten, so the player claims and marks one of the spaces containing a 2. Players take turns until one player gets four spaces in a row.
You can grab your freebie HERE! Be sure to check this post for additional FREE games for making 10!
1 Comment
1. Can’t wait to try. | 418 | 1,865 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2023-40 | longest | en | 0.92961 |
https://electronics.stackexchange.com/questions/237670/controlling-the-frequency-of-ac-used-to-energize-an-electromagnet/240623 | 1,726,467,202,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651676.3/warc/CC-MAIN-20240916044225-20240916074225-00094.warc.gz | 197,836,053 | 48,661 | # Controlling the frequency of AC used to energize an electromagnet
I am trying to build an electromagnetic pendulum which essentially has a pendulum arm with a permanent magnet at its end which is 'kicked' by an electromagnet placed at the mean position to drive the pendulum without letting it damp. Now I was wondering how I can change the frequency of the AC to correspondingly change the magnetic field strength and obtain different amplitudes doing so. I came across VFDs and this build http://blog.hardcore.lt/mic/archives/011040.html.
I am fairly new to electronics. I am more of a hobbyist. This seems way too complicated, at least for me and i can't help but feel whether I am overthinking this. I already have my reasons for not using a 555 timer and just DC pulses to energize the electromagnet so that won't do. In case VFD is the best way to do this, can I please implore any one of you to dumb this build down for me or give me a simpler way of doing it which I am guessing is possible since I am not really using this for an AC motor?
• do you want bi-directional magnetic fields, or will the magnetic field always be in one direction with varying strength (e.g. do you need both positive and negative voltages across the electromagnet)? Commented Jun 1, 2016 at 18:33
• Yes. That is my reason for not using just timed DC pulses. I want bidirectional fields flipping at the frequency of the AC. Commented Jun 1, 2016 at 18:39
• I would think that you use a magnetic switch (reed or Hall) to sense the pendulum passing and then pulse the electromagnet. This will automatically compensate for varying pendulum length and frequency. Commented Jun 1, 2016 at 20:56
• But really cmon. Doubochinski's pendulum. Bethenod's pendulum. Or any Argumental pendulum. Google it! PLEASE!! There is even one of many technical papers on the subject - Cintra and Argoul 2016 - nonlinear argumental oscillations: a few examples of modulation via spatial position. This will tell you how you can build 6 variations of such a pendulum! Commented Jun 12, 2016 at 5:35
• It is easy enough to drive an electromagnet with a power amplifier (or a "power oscillator", for that matter) Yes, you appear to be over-thinking this. It appears to be a wacky laboratory curiosity. Is there any benefit to this experiment beyond the ability to say that you reproduced the phenomenon? Commented Jun 12, 2016 at 22:42
If it is difficult to answer your question that is because it is a puzzle to understand it in the first place. A pendulum is a RESONANT device in which the freqqquency of its swing is controlled principally by its effectiove length. The only effect of your electromagnetic "kick" is to replace thev energy lost in air-resistance to its movement. Any attempt to affect the dynamics of your pendulum must rerquire an enormous amount of power. Whatever may be your purpose you might be more successful with a conical pendulum.
• The aim is to not change the frequency of the pendulum but change the frequency of AC and I don't wish to change the dynamics but merely drive it with the magnetic field to overcome the drag resistance, friction etc actually. Google Doubochinski's pendulum and you'll find what you need Commented Jun 8, 2016 at 10:56
According the Wikipedia, Doubochinski's pendulum does not vary the ac frequency. Instead, the starting conditions dictate the steady-state amplitude.
If you merely want to keep the pendulum swinging uniformly using an electromagnet, the simplest way is to drive the electromagnet with a negative-resistance circuit. This will automatically sense the swinging of the pendulum and adjust the current appropriately to keep the pendulum swinging uniformly.
A fine and elegant early example of this is a tunnel diode motor (also a tunnel diode pendulum) that appeared in C. L. Strong's "The Amateur Scientist" column in "Scientific American" magazine for October 1965 on page 112. Besides the swinging magnet, it uses just a coil of wire for the electromagnet, a tunnel diode for the negative resistance, two resistors to set the appropriate bias voltage for the tunnel diode, and a 1.5 volt battery. The tunnel diode and swinging magnet apply a combination of dc and ac to the coil. (Mr. Strong says he also successfully made a pendulum with the tunnel diode powered with a battery improvised with blotting paper, moistened with saliva, and sandwiched between a nickel and a penny.)
These days, tunnel diodes are scarce, but the tunnel diode can be replaced with a simple negative resistance circuit made of a couple of FETs or a couple of bipolar transistors with a few resistors. Google "lambda circuit" for various examples. I have had good results using either the tunnel diode or a lambda circuit. You can merely change the bias voltage to change the amplitude of the swings.
If you are set on demonstrating Doubochinski's pendulum, you can use most any signal generator, as suggested by another responder. As long as you are using a typical pendulum, hanging from a low-friction pivot or on a thread, and swinging in air, there will be very little energy loss per swing, and a signal generator should provide plenty of power. If you don't already have a signal generator, you could use an el cheapo model such as a "1HZ-500KHz DDS Function Signal Generator" that you can get from a number of on-line vendors. It is settable in 1-Hz steps, giving good frequency resolution at all but the lowest frequencies. You said you don't want to use a 555, but if you don't object to an IC, you could use one of the resistance-controlled clock chips, such as from Linear Technology. With any of these schemes, if you want to drive a huge pendulum and you find you do need more power, you could merely add a buffer amplifier chip having a low-impedance output, or you could use an audio amplifier from a sound system.
• The steady state amplitude is brought about by velocity modulation by means of the frequency of AC depending on the phase relationship. You see it says a larger array of amplitudes become available with the frequency? That's what I want. How I will get it is by the initial conditions yes, but to be able to maintain the pendulum at some particular amplitude I need to be able to give different frequencies of AC. Supposing I want 59 degrees amplitude, I need to let go at 59 degrees but in order to be able to keep it there, I need to give 50 Hertz AC. I cannot go for less and expect the same amp. Commented Jun 12, 2016 at 5:46
Keeping a pendulum going is not about just producing the right drive frequency. The phase of the drive signal is critically important.
The point of a pendulum is usually to free-swing as much as possible. These devices are also designed to loose very little energy each swing. Put another way, they are very high Q filters. As such, you don't want to, and don't need to, drive the pendulum all the time. You only need to give it just enough boost each cycle to match the little bit of energy lost that cycle. This is easiest done with a small pulse once or twice per cycle. A electronic analogy is a class C amplifier.
To add mechanical energy, the pendulum needs to be pushed in the same direction as it is moving. Since it is moving most at the bottom (middle) of the swing, this is the logical place to add the energy. A simple way to do that is to turn on a magnet shortly before the pendulum reaches the bottom, then turn it off when it does reach the bottom. This is assuming a simple magnet that attracts the pendulum.
The tricky part is knowing when to turn the magnet on. You could use some kind of position sensor, like a light beam. However, these tend to take significant power. A cute trick is to use the electro-magnet in the bottom as a sensing device during one half-swing, then the driving device during the other. Since the pendulum period can be accurately known, measuring its phase once per cycle is enough to tell you where the pendulum is at all times.
You can even measure the period between detections to correct for minor variations in the pendulum's frequency. I did that in a for-fun project that used the pendulum period to measure temperature (the pendulum rod was copper tubing). It worked very well, and the thermometer was amazingly sensitive and accurate.
• The point is to not just build a driven oscillator although the driving is really important to me. Google Doubochinski's pendulum. It's a non linear oscillator which is what I'm trying to build here. The frequency of AC modulates the velocity of the pendulum to get discrete amplitudes. I shouldn't or need not influence the phase since the pendulum apparently, does it on its own, adjusting it's phase to get accelerated in the positive half cycle, decelerated in the negative half cycle. if there whole number full cycle, they won't get accelerated but anything else and the net accn is non-zero. Commented Jun 8, 2016 at 11:32
• @K.CKarthik, If you are making Doubochinski's pendulum (never heard of it) then you should put that in your question. Is there some reason you can't use a commercial signal generator? Commented Jun 8, 2016 at 13:22
• It was a deliberate choice to not name it for a reason you yourself are a prime example of. Names can only confuse or deflect. The objective has been outlined but obviously not well enough and I'll take the blame for it. A signal generator would be perfect if I could only amplify power? How should I go about doing that? Commented Jun 8, 2016 at 13:39
There are any number of ways of driving an electromagnet from a variable frequency. If you want to just use a bench signal generator, you would need a simple AMPLIFIER to drive the electromagnet. This is exactly how speakers, headphones, earbuds, etc. operate. Depending on the impedance of your electromagnet, a common audio amplifier would possibly be a simple and cost-effective solution.
To amplify the power of the output from a signal generator you simply add a power-amplifier: i.e one which affords the ability to raise both the voltage and the current simultaneously. Are you asking for advice on how to buy, or how to design, such a unit?
• In a manner of speaking, yes. How should I put together say a signal generator and a power amplifier circuit together? A lot of cheap Audio power amplifiers I have checked out seem to have a limited power out put range but if I built an electromagnet accordingly, will I be able to use the audio power amplifier? Commented Jun 12, 2016 at 18:38
... can I please implore any one of you to dumb this build down for me or give me a simpler way of doing it ...
simulate this circuit – Schematic created using CircuitLab
Figure 1. Simple experimental setup.
I would be inclined to build a test setup as shown in Figure 1.
• Find a signal generator app for your mobile phone.
• Connect the headphone jack to the input of a suitably powerful audio amplifier.
• Find / wind a solenoid with enough resistance to meet the minimum required by the amplifier - typically 4 to 8 $\Omega$. If you feel the amplifier is complaining about the highly inductive load you can add a few ohms series resistance to damp it down.
• Set the signal generator app to 'sine' output and turn up the volume gradually monitoring coil and amplifier temperature for problems.
You may wish to measure inductor voltage or current to ensure consistency as you vary the frequency. | 2,560 | 11,395 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2024-38 | latest | en | 0.91131 |
https://media4math.com/TEKS-8-11B | 1,601,408,611,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600402088830.87/warc/CC-MAIN-20200929190110-20200929220110-00461.warc.gz | 516,660,912 | 9,241 | ## TEKS-8-11B
There are 49 resources.
Title Description Thumbnail Image Curriculum Topics
Data Analysis
Data Analysis
Data Analysis
Data Analysis
Data Analysis
Data Analysis
Data Analysis
Data Analysis
Data Analysis
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## Related Resources
### To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
## INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphing a Scatterplot
### In this Slide Show, learn how graph data in a scatterplot using the graph window. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.
Graphs of Linear Functions, Slope-Intercept Form, Data Analysis
Data Gathering
Data Gathering
Data Gathering
Data Gathering
Data Gathering | 540 | 2,250 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2020-40 | latest | en | 0.759371 |
https://physics.icalculator.com/optics/equation-of-lenses/revision-notes.html | 1,726,050,675,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651383.5/warc/CC-MAIN-20240911084051-20240911114051-00393.warc.gz | 423,289,386 | 7,334 | # Lenses. Equation of Lenses. Image Formation of Lenses Revision Notes
In addition to the revision notes for Lenses. Equation of Lenses. Image Formation of Lenses on this page, you can also access the following Optics learning resources for Lenses. Equation of Lenses. Image Formation of Lenses
Optics Learning Material
Tutorial IDTitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
12.9Lenses. Equation of Lenses. Image Formation of Lenses
In these revision notes for Lenses. Equation of Lenses. Image Formation of Lenses, we cover the following key points:
• What are lenses?
• How many types of lenses are there?
• What are the main features of lenses?
• How the image is formed in converging and diverging lenses?
• What is the equation of lenses?
• How to find the magnification produced by lenses?
• What are aberrations in lenses?
• How to combine various optical tools in an optical system?
## Lenses. Equation of Lenses. Image Formation of Lenses Revision Notes
Lenses are optical tools used to enlarge or reduce the size of images by means of refraction of light. A lens is a piece of transparent material, usually circular in shape, with two polished surfaces, either or both of which is/are curved.
There are two main categories of lenses: converging (concave) and diverging (convex). Converging lenses are thicker at middle and thinner in extremities, while diverging lenses are thinner at middle and thicker in extremities.
All lenses except plane-concave and plane-convex ones have two foci, because they are formed by joining two spherical parts. As a result, they have two centres (one in each side) as well.
Like in spherical mirrors, we have to use the special rays to build up the image in lenses. However, unlike in curved mirrors, two special rays are enough to build the image in lenses. They are:
1. The ray that originates from the top of object, is incident to the lens in parallel to the principal axis and after refracting through lens, it passes through focus in converging lenses while in diverging lenses the extension passes through focus.
2. The ray that originates from the top of object, passes through the middle of lens (point O) then moves away without changing direction, as is normally incident to the lens surface.
Converging lenses are very similar in concept to concave mirrors. Therefore, we have again six possible cases of image formation in converging lenses depending to the position of object in respect to the lens.
1. The object is beyond the centre of curvature. In this case, the image is formed at the other side of lens, between focus and centre of curvature; it is diminished and inverted. The image is also real because it is produced from the two reflected rays and not from their extensions.
2. The object is at centre of curvature. In this case, the image will form at the other centre of curvature; it has the same size as the object and is inverted. The image is real because it is produced from the two reflected rays and not from their extensions.
3. The object is between centre of curvature and focus. The image is formed at the other side of lens, beyond the other centre of curvature. The image is magnified and inverted; it is real because is produced from the two reflected rays and not from their extensions.
4. The object is at focus. In this case, there is no image as the refracted rays are parallel.
5. The object is close to the lens than focus. In this case, the reflected rays diverge and therefore, we take their extension to build up the image. As a result, the image is at the same side to the object; it is enlarged, erect and virtual, as it is obtained by considering the rays' extensions.
6. The object is at infinity. In this case, the image has no dimensions. It is a bright point at focus as all parallel rays coming from the object converge at focus.
Like in convex mirrors, the image formation in diverging lenses has only one case. The image is formed closer to the mirror than focus. It is erect and diminished. Since the image is obtained from rays extensions, it is virtual.
The equation of lenses is identical to that of curved mirrors. Thus, if we denote by do the position of object in respect to the lens, by dîthe position of image and by F the focus (focal length), we obtain
1/do + 1/dî = 1/F
The sign rules are identical to those used in spherical mirrors, i.e.
1. The object's distance do is always taken as positive.
2. The image's distance dîis taken as positive when the image is real, otherwise it is negative.
3. Focal length F is positive for converging lenses and negative for diverging ones.
The most important feature of lenses (for which they are produced) is the magnification they provide. The approach is the same as magnification in curved mirrors. This means we can use two formulae for the calculation of magnification:
M = himage/hobject
where h stands for height, and
M = dimage/dobject
Aberration is the non-regular deviation of light rays through lenses due to non-uniform thickness, causing images of objects to be blurred.
In an ideal system, every point on the object will focus to a point of zero size on the image. However, in reality this does not occur, because lenses are not ideal optical tools. As a result, parallel rays do not converge at a single dimensionless point as assumed earlier, but in a zone around focus.
In curved mirrors, aberrations are less visible as light does not enter inside the glass but it is reflected by the mirroring surface. Therefore, curved mirrors are more preferable than lenses to be used in powerful optical systems such as telescopes and microscopes.
We can combine optical tools such as plane and curved mirror with lenses to produce new optical systems.
## Whats next?
Enjoy the "Lenses. Equation of Lenses. Image Formation of Lenses" revision notes? People who liked the "Lenses. Equation of Lenses. Image Formation of Lenses" revision notes found the following resources useful:
1. Revision Notes Feedback. Helps other - Leave a rating for this revision notes (see below)
2. Optics Physics tutorial: Lenses. Equation of Lenses. Image Formation of Lenses. Read the Lenses. Equation of Lenses. Image Formation of Lenses physics tutorial and build your physics knowledge of Optics
3. Optics Practice Questions: Lenses. Equation of Lenses. Image Formation of Lenses. Test and improve your knowledge of Lenses. Equation of Lenses. Image Formation of Lenses with example questins and answers
4. Check your calculations for Optics questions with our excellent Optics calculators which contain full equations and calculations clearly displayed line by line. See the Optics Calculators by iCalculator™ below.
5. Continuing learning optics - read our next physics tutorial: Power of Lenses. The Human Eye
## Help others Learning Physics just like you
We hope you found this Physics tutorial "Lenses. Equation of Lenses. Image Formation of Lenses" useful. If you did it would be great if you could spare the time to rate this physics tutorial (simply click on the number of stars that match your assessment of this physics learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of physics and other disciplines. | 1,587 | 7,378 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2024-38 | latest | en | 0.939511 |
https://www.physicsforums.com/threads/quantum-mechanics-three-spin-observables.798204/ | 1,529,331,363,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267860557.7/warc/CC-MAIN-20180618125242-20180618145242-00405.warc.gz | 867,500,826 | 14,529 | # Quantum Mechanics: Three Spin Observables
1. Feb 16, 2015
### Robben
When trying to solve $\mathbb{S}^2 =\hbar^2s(s+1)\mathbb{I},$
I got that $\mathbb{S}^2 = \mathbb{S}^2 _x+\mathbb{S}^2_y+\mathbb{S}^2_z = \frac{3\hbar^2}{4} \left[\begin{array}{ c c }1 & 0\\0 & 1\end{array} \right] = \frac{3\hbar^2}{4}\mathbb{I},$ but how does $\frac{3\hbar^2}{4} = \hbar^2s(s+1)?$
2. Feb 16, 2015
### dextercioby
I don't understand, It's a matrix equality => 4 equalities.
3. Feb 16, 2015
### Robben
4. Feb 16, 2015
### dextercioby
Well, S^2 is equal to a 2x2 matrix in 2 different ways. One is through the eigenvalue equation, and the other is through the sum of the squares of the 3 Pauli matrices. So you can equal these 2 matrices and obtain 4 equalities *one for each matrix element*.
5. Feb 16, 2015
### Robben
Hm, so how does $\frac{3\hbar^2}{4} = \hbar^2s(s+1)?$
6. Feb 16, 2015
### Strilanc
Can you explain what $\mathbb{S}$ and $s$ are supposed to be? Is $\mathbb{S}$ required to be unitary? Hermitian? A linear combination of Pauli matrices? Is $s$ a scaling factor? Real? Complex? Known ahead of time?
If $\mathbb{S}$ is a linear combination of Pauli matrices like $\hat{v} \cdot \vec{\sigma}$, where $\hat{v}$ is a unit vector, then it is unitary and Hermitian and thus its own inverse. So $\mathbb{S}^2$ will be just $\mathbb{I}$, meaning $\hbar^2 s(s+1)$ must equal 1, and all that's left is a quadratic equation.
7. Feb 16, 2015
### Robben
$s$ is the eigenvalues and $\mathbb{S}$ is Hermitian.
8. Feb 17, 2015
### Strilanc
In that case I think it reduces to $\left( s \hat{v} \cdot \vec{\sigma} \right)^2 = \hbar^2 s(s+1) \mathbb{I}$, where $\hat{v}$ is an arbitrary unit vector and $s$ is a root of $s^2 (1 - \hbar^2) - s \hbar^2 = 0$.
9. Feb 17, 2015
### samalkhaiat
Because $s = \frac{1}{2}$:
1) Don't you have $\frac{3}{4} \hbar^{2} = \frac{1}{2} ( \frac{1}{2} + 1 ) \hbar^{2} , \ \ \Rightarrow s = \frac{1}{2}$ ?
2) For what $s > 0$, do you have $s^{2} + s - \frac{3}{4} = ( s - \frac{1}{2} ) ( s + \frac{3}{2} ) = 0$ ?
10. Feb 17, 2015
### Robben
I see, thank you very much guys! | 836 | 2,119 | {"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": 2, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.4375 | 3 | CC-MAIN-2018-26 | latest | en | 0.717351 |
http://www.peggychow.com/what-about-computations-with-challenging-framework/ | 1,680,426,357,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296950422.77/warc/CC-MAIN-20230402074255-20230402104255-00282.warc.gz | 87,521,447 | 8,462 | # What about computations with challenging framework?
What about computations with challenging framework?
In the event your nesting development cannot be utilized, we want an effective way to merge the brand new options that come with several or more monads in one formula. This is accomplished by doing calculations contained in this a monad where the values is actually themselves monadic philosophy in another monad. Such, we possibly may would computations regarding Continuation monad out-of kind of Cont (IO Sequence) a great if we need certainly to perform We/O into the computation on the Continuation monad. We are able to have fun with a beneficial monad out of method of Condition (Often Err an effective) a to merge the features of State and Error monads in a single calculation.
Envision a slight amendment to your example where i carry out the same I/O at first, but we possibly may wanted even more input in the middle of new calculation regarding Continuation monad. In this situation, we’ll let the user so you’re able to establish a portion of the yields worth in the event the type in worth is actually a certain variety. As We/O depends on part of the formula from the Continuation monad and you will area of the formula from the Continuation monad utilizes the result of the brand new I/O, we cannot make use of the nested monad trend.
As an alternative, we result in the formula on the Continuation monad have fun with viewpoints away from the new IO monad. Just what had previously been Int and you can Sequence values are actually away from form of IO Int and you can IO Sequence . We can not pull beliefs regarding IO monad – it’s a single-means monad – so we may prefer to nest nothing perform-reduces of your own IO monad inside the Extension monad to control the values. I fool around with a helper function toIO to make it better whenever we are performing opinions from the IO monad nested when you look at the Extension monad.
## Example 20
Also which superficial analogy has gotten complicated and you may unappealing when we tried to merge some other monads in identical computation. It works, however it isn’t really prettyparing this new code top-by-top reveals the amount to which the fresh guidelines monad combination approach pollutes the fresh code. | 475 | 2,297 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2023-14 | longest | en | 0.920675 |
https://climlab.readthedocs.io/en/latest/api/climlab.radiation.transmissivity.html | 1,590,781,209,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347406365.40/warc/CC-MAIN-20200529183529-20200529213529-00279.warc.gz | 294,064,082 | 5,823 | # transmissivity¶
class climlab.radiation.transmissivity.Transmissivity(absorptivity, reflectivity=None, axis=0)[source]
Bases: object
Class for calculating and store transmissivity between levels, and computing radiative fluxes between levels.
Input: numpy array of absorptivities. It is assumed that the last dimension is vertical levels.
Attributes: (all stored as numpy arrays):
• N: number of levels
• absorptivity: level absorptivity (N)
• transmissivity: level transmissivity (N)
• Tup: transmissivity matrix for upwelling beam (N+1, N+1)
• Tdown: transmissivity matrix for downwelling beam (N+1, N+1)
Example for N = 3 atmospheric layers:
tau is a vector of transmissivities
$\tau = \left[ 1, \tau_0, \tau_1, \tau_2 \right]$
A is a matrix
$\begin{split}A= \left[ \begin{array}{cccc} 1 & 1 & 1 & 1 \\ \tau_0 & 1 & 1 & 1 \\ \tau_0 & \tau_1 & 1 & 1 \\ \tau_0 & \tau_1 & \tau_2 & 1 \\ \end{array} \right]\end{split}$
We then take the cumulative product along columns, and finally take the lower triangle of the result to get
$\begin{split}Tup= \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ \tau_0 & 1 & 0 & 0 \\ \tau_1 \tau_0 & \tau_1 & 1 & 0 \\ \tau_2 \tau_1 \tau_00 & \tau_2 \tau_1 & \tau_2 & 1 \\ \end{array} \right]\end{split}$
and Tdown = transpose(Tup)
Construct an emission vector for the downwelling beam:
Edown = [E0, E1, E2, fromspace]
Now we can get the downwelling beam by matrix multiplication:
D = Tdown * Edown
For the upwelling beam, we start by adding the reflected part at the surface to the surface emissions:
Eup = [emit_sfc + albedo_sfc*D[0], E0, E1, E2]
So that the upwelling flux is
U = Tup * Eup
The total flux, positive up is thus
F = U - D
The absorbed radiation at the surface is then -F[0] The absorbed radiation in the atmosphere is the flux convergence:
-diff(F)
Methods
flux_down(self, fluxDownTop[, emission]) Compute upwelling radiative flux at interfaces between layers. flux_up(self, fluxUpBottom[, emission]) Compute downwelling radiative flux at interfaces between layers.
flux_reflected_up
flux_down(self, fluxDownTop, emission=None)[source]
Compute upwelling radiative flux at interfaces between layers.
Inputs:
• fluxUpBottom: flux up from bottom
• emission: emission from atmospheric levels (N) defaults to zero if not given
Returns:
• vector of upwelling radiative flux between levels (N+1) element N is the flux up to space.
flux_reflected_up(self, fluxDown, albedo_sfc=0.0)[source]
flux_up(self, fluxUpBottom, emission=None)[source]
Compute downwelling radiative flux at interfaces between layers.
Inputs:
• fluxDownTop: flux down at top
• emission: emission from atmospheric levels (N) defaults to zero if not given
Returns:
• vector of downwelling radiative flux between levels (N+1) element 0 is the flux down to the surface.
climlab.radiation.transmissivity.compute_T_vectorized(transmissivity)[source]
climlab.radiation.transmissivity.tril(array, k=0)[source]
Lower triangle of an array. Return a copy of an array with elements above the k-th diagonal zeroed. Need a multi-dimensional version here because numpy.tril does not broadcast for numpy verison < 1.9. | 916 | 3,159 | {"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": 0, "mathjax_asciimath": 1, "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} | 2.9375 | 3 | CC-MAIN-2020-24 | longest | en | 0.775666 |
https://www.educalc.net/?p=1246481&r=29133 | 1,620,372,625,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988775.25/warc/CC-MAIN-20210507060253-20210507090253-00435.warc.gz | 728,971,147 | 5,334 | Tutorial for Users » hp 10bII - Tutorial » NPV Problem on my new 10bII
# NPV Problem on my new 10bII
Brian wrote:
I am facing problem with the calculation of NPV on my new 10bII.
For example to calculate NPV of the following cash flows: - -240,000 + 200,000 / 1.08 + 100,000 / (1.08)2 + 20,000 / (1.08)3
I should get and did get the answer +46796 when I used the calculator. But yesterday I noticed the calculation was wrong and I got 76,960.65 instead.
These were what I keyed in:-
Shift c all
240000 (+ / -) CFj
200,000 CFj
100,000 CFj
20,000 CFj
8 I / YR
Shift NPV
Answer I got is 76,960.65 which is wrong.
Previously when I did the above calculation following the same process, I got the correct answer. I think there is something wrong. When I checked this against the slower method of working out each discounted rate for each cash flow, I got the correct answer 46796. Why?
---
Hi Brian,
You need to set the P / YR to 1. The default is 12.
1 Shift P / Yr
Shift c all "Screen should display 1"
240000 (+ / -) CFj
200,000 CFj
100,000 CFj
20,000 CFj
8 I / YR
Shift NPV "Screen should display 46,795.71"
Here is a primer to setting up your HP 10bII.
Updated On: 07.09.25
1. On 07-May-2018, Nokuthula wrote:
I m having a problem using the calculator its an app which I have bought similar to a calculator. all other calculations are correct but, when I keying in the cfj it takes me to the window where I suppose to key in cash flows. so where do I key in the CFj again after each cash flow because the calculator does not give me the option to go back to the first screen.
It only allows me to key in the first cash flow once how do I go back to the window where I will key in CFj. please help
2. On 11-Apr-2016, Helen wrote:
Good day
I am also having a problem with something similar , I did the above calculations step by step but I get an answer 80,000 then it says x TVM on the corner of the screen. How do I clear this? | 568 | 1,957 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2021-21 | latest | en | 0.912779 |
http://www.advancedconverter.com/unit-conversions/power-conversion/btu-per-hour-to-watts | 1,519,417,731,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891814833.62/warc/CC-MAIN-20180223194145-20180223214145-00234.warc.gz | 380,012,756 | 7,609 | Converter Btu per hour to Watts
Conversion Btu per hour to Watt
The Btu per hour is a unit of power used in the power industry and heating/cooling applications.
This tool converts btu per hour to watts (btu/h to w) and vice versa. 1 btu per hour ≈ 0.2931 watts. The user must fill one of the two fields and the conversion will become automatically.
btu per hour watts <=> precision:auto0 decimal1 decimal2 decimals3 decimals4 decimals5 decimals6 decimals7 decimals8 decimals9 decimals10 decimals11 decimals12 decimals
1 btu per hour = 0.2931 watts
Formula btu per hour in watts (btu/h in w). W ≈ btu per hour*0.2931
Conversions btu per hour to other units
Btu per hour to Kilowatt Btu per hour to Watt Btu per hour to Volt-ampere Btu per hour to Btu per minute Btu per hour to Horsepower Btu per hour to Ton
Table btu/hour to watt
1 btu/hour = 0.2931 watt11 btu/hour = 3.2238 watt21 btu/hour = 6.1545 watt
2 btu/hour = 0.5861 watt12 btu/hour = 3.5169 watt22 btu/hour = 6.4476 watt
3 btu/hour = 0.8792 watt13 btu/hour = 3.8099 watt23 btu/hour = 6.7406 watt
4 btu/hour = 1.1723 watt14 btu/hour = 4.103 watt24 btu/hour = 7.0337 watt
5 btu/hour = 1.4654 watt15 btu/hour = 4.3961 watt25 btu/hour = 7.3268 watt
6 btu/hour = 1.7584 watt16 btu/hour = 4.6891 watt26 btu/hour = 7.6198 watt
7 btu/hour = 2.0515 watt17 btu/hour = 4.9822 watt27 btu/hour = 7.9129 watt
8 btu/hour = 2.3446 watt18 btu/hour = 5.2753 watt28 btu/hour = 8.206 watt
9 btu/hour = 2.6376 watt19 btu/hour = 5.5684 watt29 btu/hour = 8.4991 watt
10 btu/hour = 2.9307 watt20 btu/hour = 5.8614 watt30 btu/hour = 8.7921 watt
40 btu/hour = 11.7228 watt70 btu/hour = 20.515 watt100 btu/hour = 29.3071 watt
50 btu/hour = 14.6536 watt80 btu/hour = 23.4457 watt110 btu/hour = 32.2378 watt
60 btu/hour = 17.5843 watt90 btu/hour = 26.3764 watt120 btu/hour = 35.1685 watt
200 btu/hour = 58.6142 watt500 btu/hour = 146.536 watt800 btu/hour = 234.457 watt
300 btu/hour = 87.9213 watt600 btu/hour = 175.843 watt900 btu/hour = 263.764 watt
400 btu/hour = 117.228 watt700 btu/hour = 205.15 watt1000 btu/hour = 293.071 watt
Power Conversions
Kilowatt to Watt Kilowatt to Volt-ampere Kilowatt to Btu/hour Kilowatt to Btu/minute Kilowatt to Horsepower Kilowatt to Ton Watt to Kilowatt Watt to Volt-ampere Watt to Btu/hour Watt to Btu/minute Watt to Horsepower Watt to Ton Volt-ampere to Kilowatt Volt-ampere to Watt Volt-ampere to Btu/hour Volt-ampere to Btu/minute Volt-ampere to Horsepower Volt-ampere to Ton Btu/minute to Kilowatt Btu/minute to Watt Btu/minute to Volt-ampere Btu/minute to Btu/hour Btu/minute to Horsepower Btu/minute to Ton Horsepower to Kilowatt Horsepower to Watt Horsepower to Volt-ampere Horsepower to Btu/hour Horsepower to Btu/minute Horsepower to Ton Ton to Kilowatt Ton to Watt Ton to Volt-ampere Ton to Btu/hour Ton to Btu/minute Ton to Horsepower | 1,040 | 2,832 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2018-09 | latest | en | 0.4623 |
http://builtonadare.blogspot.com/2009/10/lesson-11-from-c-to-shining-c.html | 1,508,791,230,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187826642.70/warc/CC-MAIN-20171023202120-20171023222120-00811.warc.gz | 56,508,477 | 11,652 | ## Wednesday, October 28, 2009
### Lesson 11: From C To Shining C
Welcome to the penultimate post on strongly asteroidal graphs. We've seen category A constructions, in which a neighbor is added to one of the asteroidal vertices of S3, and category B constructions, which modify the path between two asteroidal vertices. Today, we will look at category C constructions, which do both.
In a category C construction, a vertex c1 is added which is adjacent to a1, b1, b2 and b3, and a category B construction is added to the path X1. In addition, edges are added between b1 and every vertex in the category B construction. The modified category B construction still creates an a1-light path, because there will not be consecutive vertices adjacent to c1. Equivalently, this can be thought of as adding a regular category B construction, and then making c1 adjacent to every vertex except a2 and a3. However, it is more convenient, in terms of adding later constructions, to modify the category B construction so that all of the vertices are adjacent to b1. Each category B construction has a corresponding category C construction, so the category C constructions can intuitively be labeled C1-C4. The constructions are in the figure below.
Because c1 is adjacent to b2 and b3, removing b1 from construction Cn will make a graph isomorphic to one with construction Bn applied. Therefore, in order to create a new minimal sAT, it is necessary for b1 to be part of another light path.
In construction C1, this means that a category B or C construction must be applied to either a2 or a3, but in constructions C2 and C4, b1 is already part of at least one light path, which includes one vertex from the category B construction (it is difficult to see, but these paths are colored red and blue in the diagram).
Construction C1 does not combine with construction A3 (a parasol is created), but we do get new minimal sATs by applying construction B1 or B2 to the path X2, and applying constructions A1, A2, or B1 to a3 (or X3); these are graphs 37-41. In addition, construction C1 can be combined with constructions B1 and B3 to create graph 42; construction B4 can be used instead of B3 to create graph 44, which we'll get to in a second. In the above figure, note that several of these graphs have been rotated/reflected.
Applying C1 a second time can lead to several new graphs. If the vertices c1 and c2 are not adjacent to each other, then the vertices v1 and v2 become unnecessary. If c1 and c2 are adjacent, but v1 and c2 are not, then v2 is still unnecessary. Finally, we can allow c1 to be adjacent to c2 and v2, and c2 to be additionally adjacent to v1.
Adding construction B1 or B2 to any of these combinations, or B3 to either of the latter two combinations, will create a new minimal strongly asteroidal graph. Graphs 43-45 are those that occur using construction B2, graphs 46-48 use B2, and graphs 49 and 50 use B3. The category A constructions and construction B4 do not lead to new minimal graphs.
Finally, there is one way to apply construction C1 to all three asteroidal vertices. The vertices c1, c2, and c3 are adjacent to each other, and only one additional vertex v is necessary; it is adjacent to b1, b2, and b3. This creates graph 51. Other modifications of the adjacencies between c1, c2, c3, and v1, v2, and v3 do not create new minimal graphs; the most common result is a 3-sun.
Only a few minimal graphs arise from the other category C constructions. Construction C2 actually is a minimal sAT by itself, as the adjacencies between b1 and the B2 portion of the construction create a2- and a3-light paths. This is graph 52. Construction C3 does not lead to any new minimal graphs; because the only added vertex in construction B3 is already adjacent to b1, the addition of vertex c1 is unnecessary to create an a1-light path. Finally, construction C4, like construction C2, contains an a2-light path. Category B constructions that modify the path X3 do not combine with this, for the same reason they do not combine with construction B4. However, constructions A1 and A2 can be applied to a3 to make graphs 53 and 54.
These are the final minimal sATs that arise from the graph S3. Next time, we'll finish up by looking at the infinite families of asteroidal graphs, and seeing how to modify those to create strongly asteroidal graphs. | 1,041 | 4,362 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2017-43 | longest | en | 0.934088 |
https://stat.ethz.ch/pipermail/r-sig-geo/2019-October/027707.html | 1,579,657,781,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250606269.37/warc/CC-MAIN-20200122012204-20200122041204-00362.warc.gz | 669,109,773 | 3,788 | # [R-sig-Geo] Alternate statistical test to linear regression?
r@i@1290 m@iii@g oii @im@com r@i@1290 m@iii@g oii @im@com
Wed Oct 23 22:52:34 CEST 2019
```Hi Greg and others,
Thank you for these explanations and clarifications, as they are much appreciated!
Indeed, I do have some datasets that exhibit some distinct skewness. Simple scatter plots do show at least some linearity between my x and y variables (albeit weak, given the scattered nature of data points), but could this be sufficient to try simple linear regression? Also, if the data is overly skewed, could transforming it (such as logarithmically) justify the use of simple linear regression and/or correlation, if it causes the data to become mildly skewed in distribution? I have large sample sizes for all of my datasets, and the variables are continuous.
That would pretty much cover all of my questions concerning this!
Thank you, once again, for your time!
-----Original Message-----
From: Greg Snow <538280 using gmail.com>
To: rain1290 <rain1290 using aim.com>
Cc: r-sig-geo <r-sig-geo using r-project.org>
Sent: Wed, Oct 23, 2019 3:49 pm
Subject: Re: [R-sig-Geo] Alternate statistical test to linear regression?
First, please expunge the "(N>30)" concept from your mind. This is an
oversimplified rule of thumb used in introductory statistics courses
(I am guilty of doing this in intro stat as well, but I try to
emphasize to my students that it is only a rule of thumb for that
class and the truth is more complex once you are in the real world, so
consult with a statistician). There is nothing magical about a sample
size of 30, I have seen cases where n=6 is large enough for the CLT
and cases where n=10,000 was not big enough.
If the data is not overly skewed and your sample size is large then
you can just use regression as is and the inference will be
approximately correct (with a really good approximation). But with
skewness we often prefer the median over the mean and least squares
regression is equivalent to fitting a mean, some of the robust
regression options are equivalent to fitting a median, so they may be
preferable on that count.
Note that Pearson's correlation does not test linearity, it assumes
linearity (and bivariate normality). Most issues with regression will
be the same for the correlation.
On Wed, Oct 23, 2019 at 11:25 AM <rain1290 using aim.com> wrote:
>
> Hi Greg and others,
>
> Thank you for your very informative response! I actually made a mistake in my initial message, in that I was actually testing for the y variable, not the x. I will also look into those packages on CRAN, but even if there is some skewness on the y, because my sample size is much larger than 30 (N>30), it might be safe to apply a linear regression analysis, if we can assume linearity?
>
> A useful alternative would be to use correlation coefficients to test the degree of association between the x and y variables; specifically, the Pearson correlation coefficient, since both x and y variables are quantitative. Does that make sense?
>
> Thanks again,
>
>
> -----Original Message-----
> From: Greg Snow <538280 using gmail.com>
> To: rain1290 <rain1290 using aim.com>
> Cc: r-sig-geo <r-sig-geo using r-project.org>
> Sent: Wed, Oct 23, 2019 1:00 pm
> Subject: Re: [R-sig-Geo] Alternate statistical test to linear regression?
>
> Note that the normality assumptions are about the residuals (or about
> y conditional on x), not on the x variable(s) or all of y
> (non-conditional). If x is highly skewed and the residuals are normal
> then diagnostics just on y will also show skewness (if there is a
> relationship between x and y).
>
> Also, the normality assumptions are about the tests and confidence
> intervals, the least squares fit is legitimate (but possibly not the
> most interesting fit) whether the residuals are normal or not. The
> Central Limit Theorem also applies in regression, so if the residuals
> are non-normal, but you have a large sample size then the tests and
> intervals will still be approximately correct (with the quality of the
> approximation depending on the degree of non-normality and sample
> size).
>
> There are many alternative tools. There is a task view on CRAN for
> Robust Statistical Methods that gives summaries of many packages and
> tools for robust regression (and other things as well) which does not
> depend on the normality assumptions.
>
>
> On Wed, Oct 23, 2019 at 9:21 AM rain1290--- via R-sig-Geo
> <r-sig-geo using r-project.org> wrote:
> >
> > Greetings,
> > I am testing to see if linear relationships exist between my x and y variables. I conducted various diagnoses in R to test for normality of the x variable data by using qqnorm, qqline and histograms that show the distribution of the data. If the data is shown to be normally distributed in either normal quantile plots or in the histograms (i.e. a bell curve-shaped distribution), I would assume normality and apply the linear regression model, using "lm". However, in some cases, my distributions do not satisfy the normality criteria, and so I feel that using the linear regression model, in those cases, would not be appropriate. For that reason, would you be able to suggest an alternate test to the linear regression model in R? Maybe a non-parametric counterpart to it?
> > Thank you, and any help would be greatly appreciated!
> > [[alternative HTML version deleted]]
> >
> > _______________________________________________
> > R-sig-Geo mailing list
> > R-sig-Geo using r-project.org
> > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>
>
>
>
> --
> Gregory (Greg) L. Snow Ph.D.
> 538280 using gmail.com
--
Gregory (Greg) L. Snow Ph.D.
538280 using gmail.com
[[alternative HTML version deleted]]
``` | 1,399 | 5,744 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.34375 | 3 | CC-MAIN-2020-05 | latest | en | 0.930105 |
http://www.tensornetworktheory.org/documentation/a00058_source.html | 1,537,361,390,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267156224.9/warc/CC-MAIN-20180919122227-20180919142227-00509.warc.gz | 405,427,217 | 6,109 | Tensor Network Theory Library Beta release 1.2.1 A library of routines for performing TNT-based operations
tntMpsCreateBosonOp.c
1 /*
2 Authors: Sarah Al-Assam, Stephen Clark and Dieter Jaksch
3 $LastChangedDate$
4 (c) University of Oxford 2014
5 */
6
12 /* Include the header for the TNT library */
13 #include "tntMpsInternal.h"
14
29 void tntMpsCreateBosonOp(unsigned n_max,
30 tntNode *b,
31 tntNode *bdag,
32 tntNode *n,
33 tntNode *os_int,
34 tntNode *eye)
35 {
36 tntComplexArray barr; /* Arrays to create the operators: only b is created from elements, and the remaining arrays created from this */
37 unsigned physdim = n_max+1; /* Physical dimension is 2 for a spin half system */
38 unsigned j; /* Row number */
39 tntNode tnA, tnB, tnC, tnD; /* Used for contracting nodes */
40 tntNode bl, bdagl; /* Local copy of nodes */
41 tntIntArray qninf; /* Quantum number information for the physical legs */
42
43 /* Quantum number information to set for invariant operators only */
44 if (TNT_SYMM_U1 == tntSymmTypeGet()) {
45 qninf = tntIntArrayAlloc(physdim);
46 for (j = 1; j < physdim; j++) qninf.vals[j] = j;
47 }
48
49 /* Check whether each node pointer is NULL, before creating each node
50 (apart from b and bdag, which need to be created to create all the other nodes, but will be freed and set back to NULL at the end
51 of the routine if not required). */
52
53 /* \b = [ 0 1 0 0 0 ...]
54 [ 0 0 sqrt(2) 0 0 ...]
55 [ 0 0 0 sqrt(3) 0 ...]
56 [ 0 0 0 0 2 ...]
57 [ 0 0 0 0 0 ...]
58 [ ..... ] */
59 /* Create empty array */
60 barr = tntComplexArrayAlloc(physdim, physdim);
61
62 /* Put elements in row j-1, col j */
63 for (j = 1; j < physdim; j++) {
64 barr.vals[j-1 + j*physdim].re = sqrt((double) j);
65 }
66
67 /* Create node from elements */
68 bl = tntNodeCreate(&barr, "DU", physdim, physdim);
69
70 /* Free the array - it is no longer needed */
71 tntComplexArrayFree(&barr);
72
73 /* take transpose of b to create bdag by swapping legs */
74 bdagl = tntNodeCopy(bl);
75 tntNodeMapLegs(bdagl,"UD=DU");
76
77 /* n = bdag*b (U(1) invariant)
78 Make node through contracting annihilation and creation operators */
79 if (NULL != n) {
80 tnA = tntNodeCopy(bdagl);
81 tnB = tntNodeCopy(bl);
82 tntNodeJoin(tnA,"U",tnB,"D");
83
84 *n = tntNodeContract(tnA, tnB);
85
86 if (TNT_SYMM_U1 == tntSymmTypeGet()) {
87 tntNodeSetQN(*n,"D",&qninf,TNT_QN_IN);
88 tntNodeSetQN(*n,"U",&qninf,TNT_QN_OUT);
89 }
90 }
91
92 /* os_int = bdag*bdag*b*b (U(1) invariant)
93 Make node through contracting annihilation and creation operators */
94 if (NULL != os_int) {
95 /* Make a copy of all required operators */
96 tnA = tntNodeCopy(bl);
97 tnB = tntNodeCopy(bl);
98 tnC = tntNodeCopy(bdagl);
99 tnD = tntNodeCopy(bdagl);
100
101 /* Join them together in sequence i.e. s.t. such that annihilation operator is closest to MPS (i.e. the top node) */
102 tntNodeJoin(tnA,"D",tnB,"U");
103 tntNodeJoin(tnB,"D",tnC,"U");
104 tntNodeJoin(tnC,"D",tnD,"U");
105
106 *os_int = tntNodeListContract(NULL, tnA, tnB, tnC, tnD);
107
108 if (TNT_SYMM_U1 == tntSymmTypeGet()) {
109 tntNodeSetQN(*os_int,"D",&qninf,TNT_QN_IN);
110 tntNodeSetQN(*os_int,"U",&qninf,TNT_QN_OUT);
111 }
112 }
113
114 /* eye = [ 1 0]
115 [ 0 1] (U(1) invariant) */
116 if (NULL != eye) {
117 *eye = tntNodeCreateEyeOp(bl);
118
119 if (TNT_SYMM_U1 == tntSymmTypeGet()) {
120 tntNodeSetQN(*eye,"D",&qninf,TNT_QN_IN);
121 tntNodeSetQN(*eye,"U",&qninf,TNT_QN_OUT);
122 }
123 }
124
125 /* Free the qn information if it was used */
126 if (TNT_SYMM_U1 == tntSymmTypeGet()) tntIntArrayFree(&qninf);
127
128 /* Free the nodes for b and bdag if not required, otherwise assign to pointer arguments */
129 if (NULL == b) tntNodeFree(&bl);
130 else *b = bl;
131
132 if (NULL == bdag) tntNodeFree(&bdagl);
133 else *bdag = bdagl;
134
135 return;
136 }
tntNode tntNodeCopy(tntNode A)
Definition: tntNodeUtil.c:304
void tntNodeSetQN(tntNode A, tntLegLabel legA, tntIntArray *qvals, int legdir)
Definition: tntNodeQN.c:37
void tntNodeJoin(tntNode A, tntLegLabel legA, tntNode B, tntLegLabel legB)
Definition: tntNodeConn.c:52
tntNode tntNodeCreateEyeOp(tntNode A)
Definition: tntNodeUtil.c:195
void tntNodeMapLegs(tntNode A, tntLegLabel legmap)
Definition: tntNodeConn.c:125
void tntMpsCreateBosonOp(unsigned n_max, tntNode *b, tntNode *bdag, tntNode *n, tntNode *os_int, tntNode *eye)
void tntNodeFree(tntNode *A)
Definition: tntNodeUtil.c:275
tntComplexArray tntComplexArrayAlloc(unsigned numrows, unsigned numcols)
Definition: tntArray.c:392
void tntIntArrayFree(tntIntArray *arr)
Definition: tntArray.c:620
tntIntArray tntIntArrayAlloc(unsigned numrows, unsigned numcols)
Definition: tntArray.c:29
tntNode tntNodeContract(tntNode A, tntNode B, tntLegLabel legMapAC, tntLegLabel legMapBC)
tntNode tntNodeCreate(tntComplexArray *nodeVals, tntLegLabel leglabels, unsigned dimleg1, unsigned dim...)
Definition: tntNodeUtil.c:354
void tntComplexArrayFree(tntComplexArray *arr)
Definition: tntArray.c:650
int tntSymmTypeGet(void)
Definition: tntSys.c:498 | 1,795 | 5,084 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2018-39 | latest | en | 0.443537 |
https://www.chegg.com/homework-help/intermediate-algebra-5th-edition-chapter-11-solutions-9781111567675 | 1,529,389,940,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267861980.33/warc/CC-MAIN-20180619060647-20180619080647-00080.warc.gz | 791,380,407 | 21,359 | Skip Navigation
# Intermediate Algebra (5th Edition) View more editions Solutions for Chapter 11
• 9128 step-by-step solutions
• Solved by professors & experts
• iOS, Android, & web
Chapter: Problem:
Give the elements of the set that belong to each of the following sets. [Section 1.2]
a. Whole numbers
b. Rational numbers
c. Irrational numbers
d. Real numbers
Sample Solution
Chapter: Problem:
• Step 1 of 4
(a) Whole numbers are those that belong to the set . Of the numbers in the set , only 0 is a whole number.
• Step 2 of 4
(b) Rational numbers are those that can be expressed as a fraction , where a and b are integers and b is not 0. Of the numbers in the set , the rational numbers are .
• Step 3 of 4
(c) Irrational numbers cannot be expressed as fractions with an integer numerator and nonzero integer denominator. They are non-repeating, non-terminating decimals. The numbers in the set that are irrational are .
• Step 4 of 4
(d) Real numbers are either rational or irrational. They do not include imaginary numbers, which are expressed in terms of . The real numbers in the set are .
Corresponding Textbook
Intermediate Algebra | 5th Edition
9781111567675ISBN-13: 1111567670ISBN: Authors:
Alternate ISBN: 9781111987589, 9781133289586, 9781133289609, 9781133289616, 9781133423768, 9781133714378 | 347 | 1,324 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2018-26 | latest | en | 0.829266 |
http://www.stat.umn.edu/macanova/htmlhelp/node492.htm | 1,709,193,808,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474795.48/warc/CC-MAIN-20240229071243-20240229101243-00633.warc.gz | 63,913,647 | 2,960 | Next: glscrit() Up: Multivariate Macros Help File Previous: facanal() Contents
# forstep()
Usage:
```forstep(i,H,E,fh,fe), integer i > 0, fh > 0, fe > 0, REAL symmetric matrices H and E with no MISSING values ```
Keywords: factor analysis, iteration
```NOTE: This macro is OBSOLETE and is retained only for backward
compatibility because it was in file MacAnova.mac in earlier versions of
MacAnova. For doing stepwise variable selection in discriminant
analysis you should use newer macros dastepsetup(), daentervar(),
daremovevar(), dastepstatus() and dasteplook().
Macro forstep() performs a variable inclusion step in forward stepwise
variable selection in linear discriminant analysis.
forstep() is intended to be used after you have used manova("y =
groups"), where y is a data matrix and groups is a factor, to compute
hypothesis and error matrices H = matrix(SS[2,,]) and E =
matrix(SS[3,,]), with fh = DF[2] and fe = DF[3] degrees of freedom
respectively.
Status information about the variables currently "in" and "out' is
maintained in integer vectors INS and OUTS containing numbers of
variables currently included and currently excluded. When no variables
are "in", INS = 0; when all variables are "in", OUTS = NULL. INS must
be initialized, usually to 0, before forstep() can be used.
forstep(j,H,E,fh,fe), where j is the number of a variable not currently
"in", adds j to INS and removes it from OUTS, and then uses macro
compf() to compute F-to-enter for all variables included in the updated
INS. The Fs-to-enter are the analysis of covariance Fs for each "out"
variable, with the "in" variables being used as covariates. See topic
compf(). When no variables are "in", the Fs-to-enter are the ordinary
ANOVA F-statistics for each variable.
The value returned (which will normally be printed if not assigned) is
structure(f:F_to_enter, df:vector(fh,fe-k), ins:INS,outs:OUTS), where
F_to_enter is the vector of F-to-enter statistics, one for each
variable not in INS, INS and OUTs are copies of the status vectors INS
and OUTS. k is the number of variables currently "in".
The F-to-enter statistics have nominal degrees of freedom fh and fe -
k. The next variable to be entered, if any, is normally the variable
with the largest F-to-enter. The decision to enter it is based on the
size of F-to-enter.
You can somewhat automate the start of this process as follows:
Cmd> manova("y = groups", silent:T) # response matrix y, factor groups
Cmd> H <- matrix(SS[2,,]); E <- matrix(SS[3,,])
Cmd> fh <- DF[2]; fe <- DF[3]
Cmd> INS <- 0; stuff <- compf(H,E,fh,fe)
The last step can be repeated to bring "in" variables. Of course, in
practice, you want to examine the computed F-to-enter statistics to see
if another variable *should* be entered.
You can do a backward step (variable deletion) using macro backstep().
One difference between backstep() and forstep() is that backstep()
determines the variable to eliminate, and then updates INS and OUTS; you
must tell forstep() which variable to include. See backstep() for | 789 | 3,058 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2024-10 | latest | en | 0.855037 |
https://mattermodeling.stackexchange.com/questions/3505/determining-fermi-energy-in-density-functional-theory | 1,642,820,028,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320303729.69/warc/CC-MAIN-20220122012907-20220122042907-00331.warc.gz | 406,051,417 | 33,571 | # Determining Fermi energy in density functional theory
How is the Fermi energy determined in DFT calculations and how reliable are the obtained values? I know this is affected by the temperature and whether the material is an insulator/semiconductor/metal, but its not clear to me the actual value is calculated.
• I made a small edit, I dont think there is a need to get different answers for different flavors of DFT. I think that will come naturally as people talk about improvements or problems. Oct 14 '20 at 13:59
• @TristanMaxson alright, makes sense. Oct 14 '20 at 14:00
• @Jack The fermi energy in semiconductors is not well defined in DFT. I do think we have a question somewhere on this but I don't have the time to look for it Jan 15 at 19:29
The Fermi energy is defined implicitly, by requiring that the total of the band occupancies $$f$$ is the number of electrons, $$N$$:
$$\sum_{b=0}^{N_b}f_b=N\tag{1}\label{occ}$$
where $$N_b$$ is the number of bands in the calculation. The occupancy is computed from
$$f_b = f\left(\frac{\epsilon_b-E_F}{\sigma}\right)\tag{2}\label{smearing}$$
where $$f$$ is the smearing function (e.g. Gaussian, Fermi-Dirac, Methfessel-Paxton), $$\sigma$$ is the smearing width, $$\epsilon_b$$ is the band eigenvalue, and $$E_F$$ is the Fermi energy. Only the latter of these is unknown.
A simple approach is to use a bisection search. To start with, we need to bracket the true $$E_F$$; as an example, you could try setting $$E_F$$ to the minimum computed Kohn-Sham eigenvalue, and then to the maximum computed Kohn-Sham eigenvalue. There are much better choices of course, but this is a simple approach and it does work (it just takes more steps than a better choice). Once you've bracketed the real answer, you use a standard bisection search to narrow the bracket until you've determined $$E_F$$ to your desired accuracy.
Note that this method also demonstrates why $$E_F$$ is not well-defined for semiconductors or insulators: any value in the band-gap gives the same occupancies (to within machine precision), apart from a small region where the smearing function has pushed some weight into the band-gap.
Usually, it is assumed that the Fermi energy is the energy of the highest occupied electronic state. It is well defined for metals with partially filled bands. For semiconductors and dielectrics, however, there may be a discrepancy between different software packages, since the Fermi energy can correspond to the center of the bandgap at $$\pu{0 K}$$ in the case of an undoped semiconductor or it can be at the top of the valence band. The former convention pins the Fermi energy to the energy of a state which has the occupation probability of $$0.5$$ even at $$T>\pu{0 K}$$.
• But in metals one has to use fractional occupations. Is it even well defined which orbitals are occupied and which are not? Oct 15 '20 at 23:06
This has been answered very well already but just to add a practical note: For semiconductors, Vasp places the Fermi level at the valence band maximum ($$\pm$$ smearing energy) whereas Quantum Espresso or GPAW place it at the center of the gap. If you use another code then it's advisable to double-check what convention/algorithm it uses.
New contributor
Peter Schindler is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct. | 830 | 3,391 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2022-05 | longest | en | 0.918852 |
https://progettovoce.com/k0f2hu4k/truth-of-sentences-in-mathematics-98d5a5 | 1,621,058,794,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243989812.47/warc/CC-MAIN-20210515035645-20210515065645-00063.warc.gz | 493,336,945 | 16,425 | Validity 7 2.3. And figure out the sixth for yourself! Topics include sentences and statements, logical connectors, conditionals, biconditionals, equivalence and tautologies. One part of elementary mathematics consists of learning how to solve equations. The symbol for this is $$Λ$$. Interactive simulation the most controversial math riddle ever! The truth value of a mathematical statement can be determined by application of known rules, axioms and laws of mathematics. See if you can figure out if the third ever halts, and then run the program for yourself to see! Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both. One also talks of model-theoretic semantics of natural languages, which is a way of describing the meanings of natural language sentences, not a way of giving them meanings. Note: The word 'then' is optional, and a conditional will often omit the word 'then'. The translation slightly differently depending on whether the quantifier is universal or existential: Note that the second input needs to be a function; reflecting that it’s a sentence with free variables. If we run the above program on a Turing machine equipped with a halting oracle, what will we get? Statement: If we do not go to school on Memorial Day and Memorial day is a holiday, then we do not work on Memorial Day. Mathematical platonism can be defined as the conjunction of thefollowing three theses: Some representative definitions of ‘mathematicalplatonism’ are listed in the supplement Some Definitions of Platonism and document that the above definition is fairly standard. We now move up a level in the hierarchy, by adding unbounded quantifiers. This gives some sense of just how hard math is. If it does, then ∃y (x > y) must be true, and if not, then it must be false. II. The table shows what type of machine suffices to decide the truth value of a sentence, depending on where on the arithmetic hierarchy the sentence falls and whether the sentence is true or false. Arguments 5 2.2. Drawing up Truth Tables []. Mathematical logic is introduced in this unit. One probable reason for this is that if ′ is any other sentence which is equivalent to its unprovability, then and ′ are equivalent (see, e.g., Lindström, 1996). Can you speak in English? Using truth tables we can systematically verify that two statements are indeed logically equivalent. Philosophers of religion are religious. ( ∧ )∨~ ∧ ~ ( ∧ )∨~ T T T F T Truth is important. How uncomputable are the Busy Beaver numbers? The negation of statement p is " not p", symbolized by "~p". To tell the truth, I did it because I was pissed off at him over my losing Annie. On the other hand, if our sentence was true, then we would be faced with the familiar feature of universal quantifiers: we’d run forever looking for a counterexample and never find one. Now we can quite easily translate our example sentences as programs: The first is a true Σ1 sentence, so it terminates and returns True. What would an oracle for the truth value of Σ1 sentences be like? Now we have a false Π1 sentence rather than a false Π2 sentence, and as such we can find a counterexample and halt. Definition: truth set of an open sentence with one variable The truth set of an open sentence with one variable is the collection of objects in the universal set that can be substituted for the variable to make the predicate a true statement. Platonism in general (as opposed to platonism about mathematicsspecifically) is any view that arises from the above three claims byreplacing the adjective ‘mathematical’ by any otheradjective. 2. In other words A(E(Φ)) only halts if A finds out that E(Φ) is false; but E(Φ) never halts if it’s false! A sentence that can be judged to be true or false is called a statement, or a closed sentence. What kind of truths are we striving for in Physics? atautology, if it is always true. The fourth is a true Π1 sentence, which means that it will never halt (it will keep looking for a counterexample and failing to find one forever). The truth value of theses sentences depends upon the value replacing the variable. This could be done by specifying a specific substitution, for example, “$$3+x = 12$$ where $$x = 9\text{,}$$” which is a true statement. So, the first row naturally follows this definition. Concept: Mathematical Logic - Truth Value of … As such we are concerned with sentences that are either true or false. Truth, in metaphysics and the philosophy of language, the property of sentences, assertions, beliefs, thoughts, or propositions that are said, in ordinary discourse, to agree with the facts or to state what is the case.. What we’ll discuss is a way to convert sentences of Peano arithmetic to computer programs. Statement: Memorial Day is a holiday and we do not work on Memorial Day. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. So far, we’ve only talked about the simplest kinds of sentences, with no unbounded quantifiers. A disjunction is true if either statement is true or if both statements are true! Example: p _:p. acontradiction, if it always false. Same for Π1 sentences: we just ask if A(Φ) ever halts and return False if so, and True otherwise. If you’ve only been introduced to the semantic version of the hierarchy, what you see here might differ a bit from what you recognize. MAT 17: Introduction to Mathematics Truth Tables for Compound Logical Statements and Propositions – Answers Directions: Complete a truth table for each exercise. Previously I talked about the arithmetic hierarchy for sets, and how it relates to the decidability of sets. And the set of all sentences is in some sense infinitely uncomputable (you’ll see in a bit in what sense exactly this is). Which of the following sentence is a statement? Justify your answer if it is a statement. aimed at demonstrating the truth of an assertion. A ... Be prepared to express each statement symbolically, then state the truth value of each mathematical statement. How to use analytic in a sentence. Consider the sentence (H & I) → H.We consider all the possible combinations of true and false for H and I, which gives us four rows. Real World Math Horror Stories from Real encounters. So " n is an even number " may be true or false. is false because when the "if" clause is true, the 'then' clause is false. He spoke the truth, just as her father lied to her. We can talk about a sentence’s essential level on the arithmetic hierarchy, which is the lowest level of the logically equivalent sentence. It is easier to determine the truth value of such an elaborate compound statement when a truth … Examples: • Is the following statement True or False? The first two claims are tolerably clear for present pu… 6. For instance…. Here I will be primarily interested in the entirely-syntactic version of the arithmetic hierarchy. A preposition is a definition sentence which is true or false but not both. A proposition is a declarative sentence that declares a fact that is either true or false, but not both. I encourage you to think about these functions for a few minutes until you’re satisfied that not only do they capture the unbounded universal and existential quantifiers, but that there’s no better way to define them. In this respect, STT is one of the most influential ideas in contemporary analytic philosophy. Some sentences that do not have a truth value or may have more than one truth value are not propositions. Jane is a computer science major. The reason the sentence “$$3 + x = 12$$” is not a statement is that it contains a variable. Truth Value of a Statement. TM = Ordinary Turing MachineTM2 = TM + oracle for TMTM3 = TM + oracle for TM2. Learn more. And that says nothing about the second-order truths of arithmetic! The fifth is a false Π1 sentence, so it does halt at the first moment it finds a value of x and y whose sum is 10. Important Solutions 3108. 4. One part of elementary mathematics consists of learning how to solve equations. 1. 3 Back to the Truth of the Gödel Sentence. Truth values that are between 0 and 1 indicate varying degrees of truth. Deductive Systems 12 2.4. This translation works, because y + y = x is only going to be true if y is less than or equal to x. Is quantum mechanics simpler than classical physics? Add your answer and earn points. Πn sentences start with a block of universal quantifiers, alternates quantifiers n – 1 times, and then ends in a Σ0 sentence. New questions in Math. 155. The simplest types of sentences have no quantifiers at all. 135. 137. Second, it is also a philosophical doctrine which elaborates the notion of truth investigated by philosophers since antiquity. Can an irrational number raised to an irrational power be rational? A closed sentence is an objective statement which is either true or false. Syntax: The statements given in a problem are represented via propositional symbols. How to use proof in a sentence. Σ2 sentences: ∃x1 ∃x2 … ∃xk Φ(x1, x2, …, xk), where Φ is Π1.Π2 sentences: ∀x1 ∀x2 … ∀xk Φ(x1, x2, …, xk), where Φ is Σ1. Truth value here and everywhere else in this post refers to truth value in the standard model of arithmetic. However, while they are uncomputable, they would become computable if we had a stronger Turing machine. Truth is usually held to be the opposite of falsehood.The concept of truth is discussed and debated in various contexts, including philosophy, art, theology, and science. 70. Explanation: The if clause is always false (humans are not cats), and the then clause is always true (squares always have corners). Example: p. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Statement: We work on Memorial Day or Memorial Day is a holiday. Hopping Midpoints and Mathematical Snowflakes, Firing Squads and The Fine Tuning Argument, Measurement without interaction in quantum mechanics. When can we say that the truth value of mathematics sentence or english sentence can be determined reslieestacio9 is waiting for your help. A statement is said to have truth value T or F according to whether the statement considered is true or false. For example, the conditional "If you are on time, then you are late." Textbook Solutions 9842. Rephrasing a mathematical statement can often lends insight into what it is saying, or how to prove or refute it. Truth tables are constructed throughout this unit. Truth is the aim of belief; falsity is a fault. And the entire statement is true. But if there is no such example (i.e. (whenever you see $$ν$$ read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p $$ν$$ q. Traditionally, the sentence is called the Gödel sentence of the theory in consideration (say, of ). An "extreme anti-objectivist" is someone who answers "none". Give your own expression and sentences that conform with the stated type and truth value. To represent propositions, propositional variables are used. Question Papers 219. So we can generate these sentences by searching for PA proofs of equivalence and keeping track of the lowest level of the arithmetic hierarchy attained so far. Provide details and share your research! The characteristic truth table for conjunction, for example, gives the truth conditions for any sentence of the form (A & B).Even if the conjuncts A and B are long, complicated sentences, the conjunction is true if and only if both A and B are true. Example 3.1.3. But we didn't say what value n has! THEREFORE, the entire statement is false. In Example 1, each of the first four sentences is represented by a conditional statement in symbolic form. For example: i. x × 5 = 20 This is an open sentence as its truth depends Submitted by Prerana Jain, on August 31, 2018 . Statement: We work on Memorial Day or Memorial Day is a holiday. The same goes for a sentence like ∃x ∀y (x > y): for this program to halt, it would require that ∀y (x > y) is found to be true for some value of x, But ∀y (x > y) will never be found true, because universally quantified sentences can only be found false! We can translate sentences with bounded quantifiers into programs by converting each bounded quantifier to a for loop. So now we’re allowed sentences with a block of one type of unbounded quantifier followed by a block of the other type of unbounded quantifier, and ending with a Σ0 sentence. Number sentences that are inequalities also have truth values. These sentences are essentially uncomputable; not just uncomputable in virtue of their form, but truly uncomputable in all of their logical equivalents. Maharashtra State Board HSC Science (General) 12th Board Exam. Dialogue: Why you should one-box in Newcomb’s problem. Thanks for contributing an answer to Mathematics Stack Exchange! Mathematics is the only instructional material that can be presented in an entirely undogmatic way. Stanford Libraries' official online search tool for books, media, journals, databases, government documents and more. Is The Fundamental Postulate of Statistical Mechanics A Priori? Tautologies and Contraction. Truth Value of a Statement. A mathematical theory of truth and an application to the regress problem S. Heikkil a Department of Mathematical Sciences, University of Oulu BOX 3000, FIN-90014, Oulu, Finland E-mail: sheikki@cc.oulu. What time is it? In case of a statement, write down the truth value. [and would raise no objections to an -inconsistent extension?] We’re now ready to generalize. They are the model theory of truth and the proof theory of truth. The soundness and completeness of first order logic, and the recursive nature of the axioms of PA, tells us that the set of sentences that are logically equivalent to a given sentence of PA is recursively enumerable. Not all mathematical sentences are statements. Introduction to Mathematical Logic 4 1. True and false are called truth values. With an unbounded existential quantifier, all one needs to do is find a single example where the statement is true and then return True. No Turing machine can decide the truth values of Σ2 and Π2 sentences. The practice problems below cover the truth values of conditionals, disjunction, conjunction, and negation. This reflects the nature of unbounded quantifiers. Try running some examples of Σ2 or Π2 sentences and see what happens. The Central Paradox of Statistical Mechanics: The Problem of The Past. Sentential Logic 24 1. The Necessity of Statistical Mechanics for Getting Macro From Micro, Logic, Theism, and Boltzmann Brains: On Cognitively Unstable Beliefs. Open sentence An open sentence is a sentence whose truth can vary according to some conditions, which are not stated in the sentence. Truth value in the sense of “being true in all models of PA” is a much simpler matter; PA is recursively axiomatizable and first order logic is sound and complete, so any sentence that’s true in all models of PA can be eventually proven by a program that enumerates all the theorems of PA. Π n sentences start with a block of universal quantifiers, alternates quantifiers n – 1 times, and then ends in a Σ 0 sentence. Here are some examples of axioms in mathematics: 1. Mathematics is the science of what is clear by itself. 70. Introduction to Mathematical Logic (Part 4: Zermelo-Fraenkel Set Theory), The Weirdest Consequence of the Axiom of Choice, Introduction to Mathematical Logic (Part 3: Reconciling Gödel’s Completeness And Incompleteness Theorems), Introduction to Mathematical Logic (Part 2: The Natural Numbers), Introduction to Mathematical Logic (Part 1). collection of declarative statements that has either a truth value \"true” or a truth value \"false But avoid … Asking for help, clarification, or responding to other answers. 209 6.2 Sentences are not indeterminate. (whenever you see $$Λ$$ , just read 'and') When two simple sentences, p and q, are joined in a conjunction statement, the conjunction is expressed symbolically as p $$Λ$$ q. He played with truth, as he had done before. These are called propositions. A(Φ) never returns True, and E(Φ) never returns False. Example: p ^q. Algebra Q&A Library B. a. In this article, we will learn about the basic operations and the truth table of the preposition logic in discrete mathematics. If a human is a cat, then squares have corners. Here, a proposition is a statement that can be shown to be either true or false but not both. So, of the three sentences above, only the first one is a statement in the mathematical sense. Every mathematical state-ment is either true or false. The method for drawing up a truth table for any compound expression is described below, and four examples then follow. 92. The square of every real number is positive. This he says comes down to asking: "Which undecidable mathematical sentences have determinate truth values?". Concept: Mathematical Logic - Truth Value of Statement in Logic. State which of the following sentence is a statement. The assertion at the end of the sequence is called the conclusion, and the preceding statements are called ... sentences. How will quantum computing impact the world? This has nothing to do with the (x > y) being quantified over, it’s entirely about the structure of the quantifiers. And as you move up the arithmetic hierarchy, it requires more and more powerful halting oracles to decide whether sentences are true: If we define Σω to be the union of all the Σ classes in the hierarchy, and Πω the union of the Π classes, then deciding the truth value of Σω ⋃ Πω (the set of all arithmetic sentences) would require a TMω – a Turing machine with an oracle for TM, TM2, TM3, and so on. A result on the incompleteness of mathematics, Proving the Completeness of Propositional Logic, Four Pre-Gödelian Limitations on Mathematics, In defense of collateralized debt obligations (CDOs), Six Case Studies in Consequentialist Reasoning, The laugh-hospital of constructive mathematics, For Loops and Bounded Quantifiers in Lambda Calculus. So our goal in this section is to separate the formulas of $$\mathcal{L}$$ into one of two classes: the sentences (like the second example above) and the nonsentences. Question Bank Solutions 10695. For example: It runs forever! So to determine that this sentence is true, we’d need an oracle for the halting problem for this new more powerful Turing machine! So now you know how to write a program that determines the truth value of any Σ0/Π0 sentence! I If U is the integers then 9x P(x) is true. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". So if a sentence is true in all models of PA, then there’s an algorithm that will tell you that in a finite amount of time (though it will run forever on an input that’s false in some models). Let c represent "We work on Memorial Day.". In this article, we will learn about the basic operations and the truth table of the preposition logic in discrete mathematics. Row 3: p is false, q is true. Now we can quite easily translate each of the examples, using lambda notation to more conveniently define the necessary functions. G teaches Math or Mr. G teaches Science' is true if Mr. G is teaches science classes as well as math classes! Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. These quantifiers must all appear out front and be the same type of quantifier (all universal or all existential). n is an even number. 6. Each of these programs, when run, determines whether or not the sentence is true. A mathematical sentence is a sentence that states a fact or contains a complete idea. Σ1 sentences: ∃x1 ∃x2 … ∃xk Phi(x1, x2, …, xk), where Phi is Π0.Π1 sentences: ∀x1 ∀x2 … ∀xk Phi(x1, x2, …, xk), where Phi is Σ0. The conditional statement is saying that if p is true, then q will immediately follow and thus be true. If Jane is a math major or Jane is a computer science major, then Jane will take Math 150. Let’s think about that for a minute more. Pneumonic: the way to remember the symbol for disjunction is that, this symbol ν looks like the 'r' in or, the keyword of disjunction statements. Submitted by Prerana Jain, on August 31, 2018 . The Formal Language L+ S 24 2.2. 5. What’s the probability that an election winner leads throughout the entire vote? Are the Busy Beaver numbers independent of mathematics? If we were to look into the structure of this program, we’d see that A(Φ) only halts if it finds a counterexample to Φ, and E(Φ) only halts if it finds an example of Φ. For example, the statement ‘2 plus 2 is four’ has truth value T, whereas the statement ‘2 plus 2 is five’ has truth value F. The first is seen in mathematical (and philosophical) logic. A mathematical sentence is a sentence that states a fact or contains a complete idea. A statement is said to have truth value T or F according to whether the statement considered is true or false. Let p : 2 × 0 = 2, q : 2 + 0 = 2. ∃x ∃y (x⋅x = y)∃x (x⋅x = 5)∃x ∀y < x (x+y > x⋅y), ∀x (x + 0 = x)∀x ∀y (x + y < 10)∀x ∃y < 10 (y⋅y + y = x). This should suggest to us that adding bounded quantifiers doesn’t actually increase the computational difficulty. Uniquely among Khmer Rouge leaders, he … Soundness and Completeness 17 Chapter 2. Statement: If we go to school on Memorial Day, then we work on Memorial Day. Introduction 24 2. assertion or declarative sentence which is true or false, but not both. It’s important to note here that “logically equivalent sentence” is a cross-model notion: A and B are logically equivalent if and only if they have the same truth values in every model of PA, not just the standard model. Each sentence consists of a single propositional symbol. mathematics definition: 1. the study of numbers, shapes, and space using reason and usually a special system of symbols and…. By contrast,the axiomatic approach … Analytic definition is - of or relating to analysis or analytics; especially : separating something into component parts or constituent elements. Informal settings satisfying certain natural conditions, Tarski’stheorem on the undefinability of the truth predicate shows that adefinition of a truth predicate requires resources that go beyondthose of the formal language for which truth is going to be defined.In these cases definitional approaches to truth have to fail. For example, the statement ‘2 plus 2 is four’ has truth value T, whereas the statement ‘2 plus 2 is five’ has truth value F. In general, Σ n sentences start with a block of existential quantifiers, and then alternate between blocks of existential and universal quantifiers n – 1 times before ending in a Σ 0 sentence. Now we can evaluate the inner existential quantifier for any given value of x. Please be sure to answer the question. For example, 3 < 4, 6 + 8 > 15 > 12, and (15 + 3) 2 < 1000 - 32 are all true number sentences, while the sentence 9 > 3(4) is false. We’ve found a counterexample, so our program will terminate and return False. The method of truth tables illustrated above is provably correct – the truth table for a tautology will end in a column with only T, while the truth table for a sentence that is not a tautology will contain a row whose final column is F, and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. One way to make the sentence into a statement is to specify the value of the variable in some way. Could there be truth to Mary's suspicions. Think of the following statement. Who would win in a fight, logic or computation? The example above could have been expressed: If you are absent, you have a make up assignment to complete. Whenever all of the truth values in the final column are true, the statement is a tautology. Looking at the final column in the truth table, you can see that all the truth values are T (for true). truth. Another way would be if we could simply ask whether E(Φ) ever halts! 215 6.3 A formula which is NOT logically valid (but could be mistaken for one) 217 6.4 Some logically valid formulae; checking truth with ∨,→, and ∃ … But if the universally quantified statement is true of all numbers, then the function will have to keep searching through the numbers forever, hoping to find a counterexample. However, it is far from clear that truth is a definable notion. Denition 1.1. Depending on what $$x$$ is, the sentence is either true or false, but right now it is neither. Be prepared to express each statement symbolically, then state the truth value of each mathematical statement. where appropriate. Therefore, Jane will take Math 150. Even when we do this, we will still find sentences that have no logical equivalents below Σ2 or Π2. So let’s look at them individually. Thinking in first order: Are the natural numbers countable? Proof definition is - the cogency of evidence that compels acceptance by the mind of a truth or a fact. Why? Let b represent "Memorial Day is a holiday." Learning ExperiencesA. In each of these examples, the bounded quantifier could in principle be expanded out, leaving us with a finite quantifier-free sentence. 261. The truth value depends not only on P, but also on the domain U. Just understanding the first-order truths of arithmetic requires an infinity of halting oracles, each more powerful than the last. 2, 1983 MAX DEHN Chapter 1 Introduction The purpose of this booklet is to give you a number of exercises on proposi-tional, first order and modal logics to complement the topics and exercises Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent? 7.2 Truth Tables for Negation, Conjunction, and Disjunction Introduction to Truth Tables Construct a truth table for a statement with a conjunction and/or a negation and determine its truth value Construct a truth table for a statement with a disjunction and/or a negation and determine its truth value Most human activities depend upon the concept, where its nature as a concept is assumed rather than being a subject of discussion; these include most of the sciences, law, journalism, and everyday life. Show Answer. — Isaac Barrow. In general, Σn sentences start with a block of existential quantifiers, and then alternate between blocks of existential and universal quantifiers n – 1 times before ending in a Σ0 sentence. In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. Making statements based on opinion; back them up with references or personal experience. 45. As we’ll see, whether a sentence evaluates to true in the standard model of arithmetic turns out to be much more difficult to determine in general. In mathematics we are in the business of proving or disproving certain types of sentences. In other words, the statement 'The clock is slow or the time is correct' is a false statement only if both parts are false! Of learning how to solve equations truth of sentences in mathematics only talked about the basic operations the... Table for any given value of statement p is false because when the if clause is,. Equivalence and tautologies one is a statement axiomatic approach … Syntax and semantics define a way to the... I.E., not, then state the truth of truth of sentences in mathematics truth value the. Is claiming something to be proved true sentences to be true or false is called a statement that can translated... Allow all true sentences to be true or false, but also on the domain U involves alternating.... In principle be expanded out, leaving us with a finite quantifier-free sentence statement... In relation to arithmetic with the answer all '' = TM + oracle for =. To have truth values? by a conditional statement in logic, a is. The decidability of sets... ' can decide the truth value of statement p is p! Hopefully it ’ s the probability that an election winner leads throughout the entire?... We can translate any sentence with bounded quantifiers into programs by converting each bounded quantifier could principle. Connectors,, and the then clause is false, q is true or false but not both throughout entire. 0 '' to this class: Π0 and Σ0 mathematics definition: the... A conditional will often omit the word 'then ' clause is true or false, q is true natural countable. Sentences have determinate truth values that are between 0 and 1 indicate varying degrees of.... That truth is a way to make the sentence mathematical logic - truth value here and everywhere in! The given statement is a statement which contains a complete idea according to whether the statement considered is or... Sentences can be judged to be proved conditional if '' clause is false 2 0. Problems below cover the truth, I did it because truth of sentences in mathematics was pissed off at him over losing... Is someone who answers None '' then ∃y ( x > y ) is, the.! Usually employing the word 'then ' true and the Fine Tuning Argument, Measurement without interaction in quantum Mechanics 5. Says comes down to Asking: which undecidable mathematical sentences have determinate values! Caveat here, a proposition is a statement which is true Mr. G teaches Math Mr.. Table of the variable in some way you can see that all the truth values are T ( true... Of or relating to analysis or analytics ; especially: separating something into component parts constituent! Y ) must be false - it all depends on the value a. Machinetm2 = TM + oracle for TM2 Day, then state the truth, as he had before... 207 6.1 Tarski ’ s think about that for a minute more what. True sentences to be true decide their truth sets Π2 sentence, so our program will and. Lambda notation to more conveniently define the necessary truth of sentences in mathematics simplest kinds of have... English sentence can be embedded in ℚ and ℝ 4 a statement is false. Notion of truth is optional, and a conditional statement in symbolic form of the Past Firing! Always false ), then squares have truth of sentences in mathematics is waiting for your help oracles, each more than. All '' it is neither a tautology nor a contradiction special system of symbols and… truth of sentences in mathematics statement is precise! The integers then 9x p ( x > y ) must be true shapes, E... And equivalent basic statements ( i.e., not, then state the truth values of Σ1 be! Could have been expressed: if we go to school on Memorial Day implies that we work on Day. Unbounded universal quantifier is claiming something to be proved make more complicated sentences. Which are not propositions unshaken Foundation of Sciences, and as such, brought together this! Unstable Beliefs answer all '' or to join two simple sentences truth by! Convert sentences of arithmetic, etc. relation to arithmetic with the answer all.! Via propositional symbols of that statement is p ∧ q not p '', symbolized ! Means to say that the truth value of theses sentences depends upon the value of any sentence. Or contains a complete idea selection puzzle: Why is BIC ≠ AIC s clear we! Only if we had a stronger Turing machine fight, logic, a proposition is a declarative that. So in particular, for x = 0, we will find that Ey ( >... Simply declare to be true or false in a problem are represented via propositional symbols 0... To express each statement symbolically, then we work on Memorial Day truth of sentences in mathematics mathematics – the unshaken Foundation of,! Their form, but also on the domain U more powerful than last! 0 = 2, q: 2 + 0 = 2, q: 2 × 0 = 2 q. We are truth of sentences in mathematics with sentences that do not have a truth table for given! Of axioms in mathematics simple sentences the standard model of arithmetic requires an of! Unshaken Foundation of Sciences, and as such, brought together like this they never halt on input. If 1x2 =, then q truth of sentences in mathematics immediately follow and thus there are two names both. What \ ( y\ ) represents of \ ( x\ ) is true if Mr. G is teaches classes. Not only on p, but not both of rules would be we... A false Π1 sentence rather than a false statement is always false units... One is a computer science major, then q will immediately follow thus! Saying, or: Math is hard stated in the mathematical sense sample below! Just uncomputable truth of sentences in mathematics virtue of their form, but not both philosophical doctrine which the. P. acontradiction, if it is also an arbitrary constant, which are not stated in mathematical... Assertion or declarative sentence that is either true or false, q: 2 × 0 =,! Because when the if '' clause is true or false, but not both Past...: p. acontingency, if 1x2 =, then q will immediately follow and thus there are two approaches... 1. the study of numbers, shapes, and ~ real number x, if it always false,! Of all numbers, and as such, brought together like this they never halt on any input on. The key words 'If.... then... ' in example 1, each of these,. Striving for in Physics i. x × 5 = 20 this is an objective statement is. A fault 1x =−, which can be judged to be either unsound—that is, the sentence an! Into a program of this form any tautologies and equivalent basic statements i.e.! He says comes down to Asking: which undecidable mathematical sentences have no quantifiers at all = ordinary MachineTM2. Are essentially uncomputable ; not just uncomputable in virtue of their form, but also the. Is - of or relating to analysis or analytics ; especially: separating something into component parts or elements... Does, then it must be true, or it might be or! Compound sentence formed by the word and to join two simple sentences “ true ” or false! Word or to join two simple sentences since antiquity any set of rules would be unsound—that. Statements are indeed logically equivalent UK truth of sentences in mathematics Discrete mathematics what we ’ ve talked. Are absent, you can return false ( say, of ) open sentences ( ! ~p '' is a sentence that states truth of sentences in mathematics fact that is true or false that says about. Since +, ⋅, =, then state the truth value of statement in sentence! Mathematical sense program of this form propositional symbol begins with an uppercase letter and may be true and! Something to be either true or false provide an example only on p, but also the. Lends insight into what it means to say that this logical system a! A statement in symbolic form what happens 'then ' is true to more conveniently define necessary... Is to specify the value of each mathematical statement influential ideas in contemporary analytic philosophy if is. A definition sentence which is true thinking in first order: are the model theory of truth value or have. Quantifier could in principle be expanded out, leaving us with a finite quantifier-free.. Central Paradox of Statistical Mechanics: the statements given in a fight, logic or computation clear itself. Jane will take Math 150, Measurement without interaction in quantum Mechanics B . Be primarily interested in the entirely-syntactic version of the variable classes as well Math. Usually expressed with the stated type and truth value of mathematical statements investigate their truth of. We are concerned with sentences that are between 0 and 1 indicate varying degrees of truth opposed and. Expanded out, leaving us with a block of universal quantifiers, alternates quantifiers –! Learning how to prove or refute it times, and the proof theory of truth the. Uncomputable, they would become computable if we had a stronger Turing machine connectors, and! Philosophical ) logic uncomputable ; not just uncomputable in virtue of their form, but on. Of Edinburgh, UK ) Discrete mathematics major, then truth of sentences in mathematics = or 1x =− while! Saying that if p is false only the first one is a fault winner leads throughout the entire?! Midpoints and mathematical Snowflakes, Firing Squads and the Fine Tuning Argument, Measurement without interaction quantum... | 8,109 | 36,249 | {"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": 1, "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} | 3.84375 | 4 | CC-MAIN-2021-21 | longest | en | 0.885189 |
http://bubuko.com/infodetail-3676273.html | 1,603,852,898,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107896048.53/warc/CC-MAIN-20201028014458-20201028044458-00202.warc.gz | 19,173,953 | 13,086 | # Codeforces Round #672 (Div. 2)
#### T1
A题给我们一个数列，每次可以交换相邻元素的位置，问我们能不能在 $$\frac{n(n-1)}{2}-1$$ 次操作内将元素调整为不下降序列。
#### T2
B题也是给出一个数列，要求求出满足 $$i < j$$$$a_i\ \oplus\ a_j < a_i\ \&\ a_j$$ 的数对的个数。
#### T3——1
C题分为 $$\text{easy version}$$$$\text{hard version}$$,我只会 $$\text{easy version}$$ \kk
$$f_i = \max(a_i,a_i+\max\limits_{j=1}^{j<i}g_j)$$
$$g_i = -a_i+\max\limits_{j=1}^{j<i}f_j$$
$$f_i = \max(a_i,f_{i-1},a_i+g_{i-1})$$
$$g_i = \max(g_{i-1},-a_i+f_{i-1})$$
$$f = \max(a_i,f,a_i+g)$$
$$g = \max(g,f-a_i)$$
#### T4
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <queue>
#define ll long long
using namespace std;
int read()
{
int a = 0,x = 1;
char ch = getchar();
while(ch > ‘9‘ || ch < ‘0‘){
if(ch == ‘-‘) x = -1;
ch = getchar();
}
while(ch >= ‘0‘ && ch <= ‘9‘){
a = a*10 + ch-‘0‘;
ch = getchar();
}
return a*x;
}
const int P=998244353,N = 1e6+7;
struct node {
int l,r;
friend bool operator < (node a,node b) {return a.r > b.r;}
}arr[N];
bool cmp(node a,node b)
{
return a.l < b.l;
}
int n,m;
priority_queue<node>q;
ll fpow(ll a,ll x)
{
if(x == 0) return 1;
if(x == 1) return a;
ll tmp = fpow(a,x/2);
if(x&1) return tmp*tmp%P*a%P;
else return tmp*tmp%P;
}
ll f[N];
ll C(ll a,ll b)
{
return f[a]*fpow(f[b],P-2)%P*fpow(f[a-b],P-2)%P;
}
int main()
{
n = read(),m = read();
f[0] = 1;
for(int i = 1;i <= n;i ++) f[i] = f[i-1]*i%P;
for(int i = 1;i <= n;i ++) {
arr[i].l = read(),arr[i].r = read();
}
ll ans = 0;
sort(arr+1,arr+1+n,cmp);
for(int i = 1;i <= n;i ++) {
while(!q.empty() && q.top().r < arr[i].l) q.pop();
int tmp = q.size();
// printf("%d \n",tmp);
if(tmp+1>=m) {
(ans += C(tmp,m-1)) %= P;
}
q.push(arr[i]);
}
printf("%lld\n",ans);
}
Codeforces Round #672 (Div. 2)
(0)
(0)
© 2014 bubuko.com 版权所有 鲁ICP备09046678号-4 | 781 | 1,771 | {"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": 1, "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} | 2.828125 | 3 | CC-MAIN-2020-45 | latest | en | 0.179526 |
https://www.elitetrader.com/et/threads/activate-deactivate-system.36083/page-2 | 1,511,145,085,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934805894.15/warc/CC-MAIN-20171120013853-20171120033853-00471.warc.gz | 756,758,882 | 14,449 | General Topics
Markets
Technical Topics
Brokerage Firms
Community Lounge
Site Support
# Activate/Deactivate System?
Discussion in 'Journals' started by EricP, Jul 29, 2004.
1. ### abogdan
Marvelous! Now that I had a chance to think about it, it makes a lot of sense! Thank you for your contribution, very nice!
#11 Jul 29, 2004
2. ### shorty_mcshort
Just a quick question. For example, if you were trading stocks and pyramid your profits. Would this test be valid or would you have to test it on the same # of shares or same dollar amounts on each trade?
#12 Jul 29, 2004
3. ### EricP
I consider each trade to be an independant event, regardless of whether it is 200 shares or 1500 shares. I think it would still be worthwhile and useful for you, although I would avoid the word 'valid,' as this might imply a sense of statistical accuracy that we do not have or require for our need (in light of distribution not being bell curve). Being less than perfect can be very profitable, while taking the effort to have a statistically 100% supported analysis might be academically interesting (not!) but is a major waste of time for a trader, IMO. We are seeking to get 90% of the results with 10% of the effort, which I think this method achieves nicely for me.
-Eric
#13 Jul 29, 2004
4. ### shorty_mcshort
Very interesting. Thanks for sharing. If someone is interested in learning more statistics that you could apply to trading where could I find that information?
#14 Jul 29, 2004
5. ### Hofferino
How does one come up with a value for standard deviation per trade (such as your \$150 in the example)?
#15 Jul 29, 2004
6. ### EricP
Microsoft Excel has a function for the Standard Deviation.
STDEV(a1:a51), for example to calculate the standard deviation for the numbers in cells a1 to a51.
Many calculators also have a function for this, and I'm sure you can search the internet for the actual equation for calculating the standard deviation if needed for some custom coding.
-Eric
#16 Jul 29, 2004
7. ### EricP
Hmmmm. Good question. Hopefully, someone will have a link or book recommendation for us. Acrary's recent thread has some very good information, in case you missed it.
-Eric
#17 Jul 29, 2004
8. ### shorty_mcshort
Hoff,
If you have Excel then you can go to Insert/Function type in Standard Deviation in the search box and click on STDEVP.
Also note that the original post talks about standard deviation of profits not per trade.
#18 Jul 29, 2004
9. ### EricP
I think you both have the right idea. I am talking about the standard deviation of the profits from the trades. Using the first example posted:
+\$2000
+\$350
(\$300)
+\$1400
(\$3000)
+\$750
And plugging into Excel for STDEV, I get a standard deviation of \$2652. Also, you could use the STDEVP function, which estimates the Standard Deviation over an entire population of data, and I get \$1875 as the Standard Deviation. Either way should be fine. Recognize again that we are not seeking 100% perfection in accuracy, but a rigid and structural way to activate and deactivate trading systems.
-Eric
P.S. For what it's worth, the values for STDEV and STDEVP will much more closely approximate one another when you are using a larger number of datapoints in your analyis.
#19 Jul 29, 2004
10. ### TheStudent
EricP,
Thanks for your contributions - this is very nicely done.
Question :
you mentioned testing for profit confidence as follows
[ p * n ^0.5 ] / sigma
where
p = net profit per trade
sigma = std dev per trade
we can factor out n so, it's basically
[ n^0.5] * [p / sigma]
It is clear that assuming p > 0, as n increases, your score is going to increase, and that you can theoretically seek for any score you want by increasing n.
I guess to bring this a step further, would you say that for any given n, the corresponding score will tell you the confidence of being profitable for a period of length n?
Thus : if n = 100 and the score corresponds to 98%, then we can say that the system has a 98% confidence in being profitable every 100 trades. If the same system has a score of 80% at n = 50, then we can say that at 50 trades, the confidence of being profitable drops to 80%.
Thus, the question becomes - say if we are tracking the last 100 trades of a system and working out the corresponding score based on n = 100, and then dropping the system if the score falls below 80% or whatever benchmark we set.
How do we know 100 is the appropriate length to use? How do we know 80% is the appropriate cut-off to use?
Should we just be arbitrary based on our "judgement" or is there a rigourous way of approaching this problem?
#20 Jul 29, 2004
ET IS FREE BECAUSE OF THE FINANCIAL SUPPORT FROM THESE COMPANIES: | 1,191 | 4,771 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2017-47 | longest | en | 0.919138 |
https://www.physicsforums.com/threads/u-substitution-on-sin-x-5-dx.633333/ | 1,591,526,996,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590348526471.98/warc/CC-MAIN-20200607075929-20200607105929-00312.warc.gz | 844,709,895 | 18,328 | # U substitution on ∫sin(x^5)dx
## Homework Statement
Use Part 2 of the Fundamental Theorem of Calculus to find the derivative.
$$\int_3^x sin(t^{5}) \, dt$$
## The Attempt at a Solution
I know the general idea of what I'm supposed to do as far as evaluate the indefinate integral and then do a subtraction of the upper limit and lower limits...but I can't even get to the point of finding the indefinate integral. (maybe it's that I just "think" I know what i'm supposed to do..)
$$\int_3^x sin(t^{5}) \, dt$$
I'm letting u = $t^{5}$
so du = $5t^{4}$
then it looks like dt can be replaced by $\frac{1}{5t^{4}}$
so that $$\int_3^x \frac{1}{5t^4} sin(u) \, du$$
However, our professor has instructed that mixing variables within the integral is not allowed because it can't be evaluated.
So how to I do u-substitution on this integral?
Related Calculus and Beyond Homework Help News on Phys.org
vela
Staff Emeritus
Homework Helper
You don't. Note that the problem says to use the fundamental theorem of calculus. Don't ignore that bit of info.
SammyS
Staff Emeritus
Homework Helper
Gold Member
## Homework Statement
Use Part 2 of the Fundamental Theorem of Calculus to find the derivative.
$$\int_3^x sin(t^{5}) \, dt$$
## The Attempt at a Solution
I know the general idea of what I'm supposed to do as far as evaluate the indefinate integral and then do a subtraction of the upper limit and lower limits...but I can't even get to the point of finding the indefinate integral. (maybe it's that I just "think" I know what i'm supposed to do..)
$$\int_3^x sin(t^{5}) \, dt$$
I'm letting u = $t^{5}$
so du = $5t^{4}$
then it looks like dt can be replaced by $\frac{1}{5t^{4}}$
so that $$\int_3^x \frac{1}{5t^4} sin(u) \, du$$
However, our professor has instructed that mixing variables within the integral is not allowed because it can't be evaluated.
So how to I do u-substitution on this integral?
What vela said, plus ... you're to find the derivative.
oh, it says find the derivative...lol. I just saw the integral sign and began trying to integrate. I didn't even notice that there is a d/dt notation to the right of the integral clear as day. And so I figured the fundamental theorem being referred to was the property that is actually FTC1. Anyway, Thanks :) sin(x^5) duh. I turned a 30 second problem into a 20 minute headache. Thanks Vela and Sammy.
SammyS
Staff Emeritus
Homework Helper
Gold Member
oh, it says find the derivative...lol. I just saw the integral sign and began trying to integrate. I didn't even notice that there is a d/dt notation to the right of the integral clear as day. And so I figured the fundamental theorem being referred to was the property that is actually FTC1. Anyway, Thanks :) sin(x^5) duh. I turned a 30 second problem into a 20 minute headache. Thanks Vela and Sammy.
The limits od the integral are 3 and x, so the integral is a function of x, not t.
The derivative of this integral is with respect to x.
I.e.
Find
$\displaystyle \frac{d}{dx}\ \left(\ \int_3^x sin(t^{5}) \, dt\ \right)\ .$
The limits od the integral are 3 and x, so the integral is a function of x, not t.
The derivative of this integral is with respect to x.
I.e.
Find
$\displaystyle \frac{d}{dx}\ \left(\ \int_3^x sin(t^{5}) \, dt\ \right)\ .$
Thanks, I noticed that right after I submitted the post and editted with the correct notation right afterwards.
Is this just $5x^4 sin(x^{5})$ ?
SammyS
Staff Emeritus
Homework Helper
Gold Member
Is this just $5x^4 sin(x^{5})$ ?
No.
The anti-derivative of $5x^4 \sin(x^{5})$ is $-\cos(x^{5}) +C$,
not $\displaystyle \int \sin(x^{5}) dx\ .$
So this should be $sin(x^{5})$, correct?
HallsofIvy
Homework Helper
No, it isn't. What is the derivative of sin(u(x)) with respect to x. What is the derivative of cos(u(x)) with respect to x?
cos(u(x)) u'(x) and -sin(u(x)) u'(x).
Chestermiller
Mentor
So this should be $sin(x^{5})$, correct?
This result looks correct to me. There is no x under the integral sign. I don't understand what Halls of Ivy is saying.
Chet
SammyS
Staff Emeritus
Homework Helper
Gold Member
So this should be $sin(x^{5})$, correct?
If you mean:
Is $\displaystyle \frac{d}{dx}\ \left(\ \int_3^x sin(t^{5}) \, dt\ \right)=\sin(x^5)\ ?$
Then I agree with Chestermiller that, "Yes it is."
Here's how I arrive at that:
Let F(t) be an anti-derivative of sin(t5). In other words, let $\displaystyle F(t)=\int sin(t^{5}) \, dt\ .$
Then $\displaystyle \int_3^x sin(t^{5}) \, dt=F(x)-F(3)\ .$
Therefore, $\displaystyle \frac{d}{dx}\ \left(\ \int_3^x sin(t^{5}) \, dt\ \right)=\frac{d}{dx}\left(F(x)-F(3)\right)=\sin(x^5)-0\ .$
Mark44
Mentor
oh, it says find the derivative...lol. I just saw the integral sign and began trying to integrate. I didn't even notice that there is a d/dt notation to the right of the integral clear as day. | 1,423 | 4,844 | {"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} | 3.984375 | 4 | CC-MAIN-2020-24 | longest | en | 0.917238 |
https://toolslick.com/capitalization-rate-calculator | 1,529,632,358,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864337.41/warc/CC-MAIN-20180622010629-20180622030629-00394.warc.gz | 729,340,474 | 9,102 | # Capitalization Rate Calculator
Capitalization Rate Calculator is used to compute the capitalization rate - an estimate of the investor's potential return on his or her investment
Capitalization Rate (%):
``..``
#### Background Information
What does Capitalization Rate mean?
The capitalization rate, or cap rate, helps in evaluating a real estate investment. It is a useful tool for people interested in real estate.
Importance of Bond Yield
The annual percentage yield is the rate of interest earned or accrued over the course of a year or 365 days. It takes into account the effect of compounding. The annual percentage yield gives you a better sense of an investment's true return, and of a loan's true cost.
Formula
Capitalization Rate = Net Operating Income / Current Market Value (Sales Price) of the asset
Example
Let's assume a man wants to buy an investment property worth \$900,000. He expects that it will generate \$125,000/year after the operating costs. Using the above formula, the cap rate will be:-
13.89% (\$125,000 / \$900,000 = 0.1389 = 13.89%)
Feedback | 241 | 1,084 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2018-26 | latest | en | 0.891802 |
https://byjus.com/maths/euclidean-geometry/ | 1,723,333,859,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640826253.62/warc/CC-MAIN-20240810221853-20240811011853-00823.warc.gz | 116,891,564 | 110,186 | # Euclidean Geometry
Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different axioms and theorems. It is basically introduced for flat surfaces or plane surfaces. Geometry is derived from the Greek words ‘geo’ which means earth and ‘metrein’ which means ‘to measure’.
Euclidean geometry is better explained especially for the shapes of geometrical figures and planes. This part of geometry was employed by the Greek mathematician Euclid, who has also described it in his book, Elements. Therefore, this geometry is also called Euclid geometry
The axioms or postulates are the assumptions that are obvious universal truths, they are not proved. Euclid has introduced the geometry fundamentals like geometric shapes and figures in his book elements and has stated 5 main axioms or postulates. Here, we are going to discuss the definition of euclidean geometry, its elements, axioms and five important postulates.
## History of Euclid Geometry
The excavations at Harappa and Mohenjo-Daro depict the extremely well-planned towns of Indus Valley Civilization (about 3300-1300 BC). The flawless construction of Pyramids by the Egyptians is yet another example of extensive use of geometrical techniques used by the people back then. In India, the Sulba Sutras, textbooks on Geometry depict that the Indian Vedic Period had a tradition of Geometry.
The development of geometry was taking place gradually, when Euclid, a teacher of mathematics, at Alexandria in Egypt, collected most of these evolutions in geometry and compiled it into his famous treatise, which he named ‘Elements’.
## What is Euclidean Geometry?
Euclidean Geometry is considered an axiomatic system, where all the theorems are derived from a small number of simple axioms. Since the term “Geometry” deals with things like points, lines, angles, squares, triangles, and other shapes, Euclidean Geometry is also known as “plane geometry”. It deals with the properties and relationships between all things.
Plane Geometry Solid Geometry Congruence of triangles Similarity of triangles Areas Pythagorean theorem Circles Regular polygons Conic sections Volume Regular solids
### Examples of Euclidean Geometry
The two common examples of Euclidean geometry are angles and circles. Angles are said as the inclination of two straight lines. A circle is a plane figure, that has all the points at a constant distance (called the radius) from the center.
## Euclidean and Non-Euclidean Geometry
There is a difference between Euclidean and non-Euclidean geometry in the nature of parallel lines. In Euclidean geometry, for the given point and line, there is exactly a single line that passes through the given points in the same plane and it never intersects.
Non-Euclidean is different from Euclidean geometry. The spherical geometry is an example of non-Euclidean geometry because lines are not straight here.
## Properties of Euclidean Geometry
• It is the study of plane geometry and solid geometry
• It defined point, line and a plane
• A solid has shape, size, position, and can be moved from one place to another.
• The interior angles of a triangle add up to 180 degrees
• Two parallel lines never cross each other
• The shortest distance between two points is always a straight line
## Elements in Euclidean Geometry
In Euclidean geometry, Euclid’s Elements is a mathematical and geometrical work consisting of 13 books written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Further, the ‘Elements’ was divided into thirteen books that popularized geometry all over the world. As a whole, these Elements is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.
Book 1 to 4th and 6th discuss plane geometry. He gave five postulates for plane geometry known as Euclid’s Postulates and the geometry is known as Euclidean geometry. It was through his works, we have a collective source for learning geometry; it lays the foundation for geometry as we know it now.
## What were Euclidean Axioms?
Here are the seven axioms are given by Euclid for geometry.
1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another
## What were Euclid’s Five Postulates?
Before discussing Postulates in Euclidean geometry, let us discuss a few terms as listed by Euclid in his book 1 of the ‘Elements’. The postulated statements of these are:
• Assume the three steps from solids to points as solids-surface-lines-points. In each step, one dimension is lost.
• A solid has 3 dimensions, the surface has 2, the line has 1 and the point is dimensionless.
• A point is anything that has no part, a breadthless length is a line and the ends of a line point.
• A surface is something that has length and breadth only.
It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail.
### Euclid’s Postulate 1
“A straight line can be drawn from any one point to another point.”
This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points.
### Euclid’s Postulate 2
“A terminated line can be further produced indefinitely.”
In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line.
### Euclid’s Postulate 3
“A circle can be drawn with any centre and any radius.”
Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment.
### Euclid’s Postulate 4
“All right angles are equal to one another.”
All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations.
### Euclid’s Postulate 5
“If a straight line falling on two other straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is less than two right angles.”
Further, these Postulates and axioms were used by him to prove other geometrical concepts using deductive reasoning. No doubt the foundation of present-day geometry was laid by him and his book the ‘Elements’.
## Euclid Geometry Worksheet
1. How many dimensions do solids, points and surfaces have?
2. What is the shape of a pyramid’s base?
3. If a + b =10 and a = c, then prove that c + b =10.
4. Can two distinct intersecting lines be parallel to each other at the same time? Justify.
5. Read the following sentence and mention which of Euclid’s axiom is followed: “X’s salary is equal to Y’s salary. Due to the recession, the salaries of X and y are reduced to half. Now the final salary of X will still be equal to Y.”
## Related Articles
Keep visiting BYJU’S to get more such maths topics explained in an easy way. Also, register now and access numerous video lessons on different maths concepts.
## Frequently Asked Questions – FAQs
Q1
### What is Euclidean Geometry?
Euclidean geometry is the study of flat shapes or figures of flat surfaces and straight lines in two dimensions.
Q2
### What is the difference between Euclidean and non-Euclidean Geometry?
Euclidean geometry deals with figures of flat surfaces but all other figures which do not fall under this category come under non-Euclidean geometry. For example, a curved shape or spherical shape is a part of non-Euclidean geometry.
Q3
### What are the five postulates of Euclid’s geometry?
1. A straight line may be drawn from any point to another point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any centre and radius.
4. All right angles are equal to each other.
5. If a straight line falls on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
Q4
### What are the three types of geometry?
In the two-dimensional plane, there are majorly three types of geometries.
Euclidean (for flat surfaces)
Spherical (for curved surfaces)
Hyperbolic
Q5
### What is the use of euclidean geometry?
Euclidean geometry is majorly used in the field of architecture to build a variety of structures and buildings. Designing is the huge application of this geometry. Also, in surveying, it is used to do the levelling of the ground.
Q6
### Mention three axioms that are given by Euclid.
1. Things which are equal to the same thing are equal to one another
2. The whole is greater than the part
3. Things which coincide with one another are equal to one another | 2,054 | 9,510 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2024-33 | latest | en | 0.960035 |
https://www.shaalaa.com/question-bank-solutions/find-following-parameter-dc-diamond-cubic-structure-no-atoms-unit-cell-co-ordination-number-nearest-atomic-distance-atomic-radius-apf-study-characteristics-unit-cell-diamond-zns-nacl-hcp_57431 | 1,656,306,630,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103328647.18/warc/CC-MAIN-20220627043200-20220627073200-00788.warc.gz | 1,039,024,125 | 12,691 | # Find the following parameter for DC(Diamond Cubic) structure:- ⦁ No. of atoms per unit cell.⦁ Co-ordination number.⦁ Nearest atomic distance.⦁ Atomic radius.⦁ APF. - Applied Physics 1
Find the following parameter for DC(Diamond Cubic) structure:-
⦁ No. of atoms per unit cell.
⦁ Co-ordination number.
⦁ Nearest atomic distance.
⦁ APF.
#### Solution
This can be thought of as a combination of two FCC sublattice as if one FCC sublattice is translated along a body diagonal of the other sublattice through a distance √3a/4 from one of its corners. Therefore in addition to regular FCC atoms additional atoms, one on each of the four body diagonals are found in this structure. Two FCC sublattice can form the DC structure.
⦁ NUMBER OF ATOMS/ UNIT CELL(n):-
Total number of atoms/ unit cell = (1/8 ×8 corners) +(1/2xx6face cornes ) +(4 × 1 on each body diagonal)
∴ n = 8
⦁ CO-ORDINATION NUMBER(N)
Every atom with co-ordination (a/4, a/4, a/4) has four nearest neighbour, one at the nearest corner and three at the face centres of the three adjacent faces passing through the nearest corner. For example, the atom A, J, K and L are the nearest neighbours of the additional atom P. the five atoms A, J, K and P from a tetrahedron
∴ N = 4
The distance between two nearest neighbours is
AP =( √3a)/4 = 2r ∴ r = (√3a)/8
⦁ ATOMIC PACKING FACTOR(APF):-
APF = (8xx(4π)/3(sqrt(3a)/8)^3)/(a^3)=0.34
Packing efficiency = 34%
Hence , it is a loosely packed structure. But as all the atoms are attached with covalent bonds this is a strong structure.
⦁ VOID SPACE:-
The void space in a DC unit cell is 66% of the entire cell structure.
Concept: Study of characteristics of unit cell of Diamond, ZnS, NaCl and HCP;
Is there an error in this question or solution? | 518 | 1,770 | {"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": 1, "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} | 3.6875 | 4 | CC-MAIN-2022-27 | latest | en | 0.875785 |
https://paulfranceschi.com/page/3/ | 1,508,404,096,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187823260.52/warc/CC-MAIN-20171019084246-20171019104246-00360.warc.gz | 764,809,831 | 127,565 | Posprint in English (with additional illustrations) of a paper published in French in Dialogue Vol. 40, Winter 2001, pp. 99-123 under the title “Une Solution pour le Paradoxe de Goodman”.
In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.
Paul Franceschi
p.franceschi@univ-corse.fr
originally published in Dialogue, winter 2001, vol. 40, pp. 99-123
ABSTRACT: In the classical version of Goodman’s paradox, the universe where the problem takes place is ambiguous. The conditions of induction being accurately described, I define then a framework of n-universes, allowing the distinction, among the criteria of a given n-universe, between constants and variables. Within this framework, I distinguish between two versions of the problem, respectively taking place: (i) in an n-universe the variables of which are colour and time; (ii) in an n-universe the variables of which are colour, time and space. Finally, I show that each of these versions admits a specific resolution.
1. The problem
Nelson Goodman
Goodman’s Paradox (thereafter GP) has been described by Nelson Goodman (1946).i Goodman exposes his paradox as follows.ii Consider an urn containing 100 balls. A ball is drawn each day from the urn, during 99 days, until today. At each time, the ball extracted from the urn is red. Intuitively, one expects that the 100th ball drawn from the urn will also be red. This prediction is based on the generalisation according to which all the balls in the urn are red. However, if one considers the property S “drawn before today and red or drawn after today and non-red”, one notes that this property is also satisfied by the 99 instances already observed. But the prediction which now ensue, based on the generalisation according to which all the balls are S, is that the 100th ball will be non-red. And this contradicts the preceding conclusion, which however conforms with our intuition.iii
Goodman expresses GP with the help of an enumerative induction. And one can model GP in terms of the straight rule (SR). If one takes (D) for the definition of the “red” predicate, (I) for the enumeration of the instances, (H) for the ensuing generalisation, and (P) for the corresponding prediction, one has then:
(D) R = red
(I) Rb1·Rb2·Rb3·…·Rb99
(H) Rb1·Rb2·Rb3·…·Rb99·Rb100
(P) Rb100
And also, with the predicate S:
(D*) S = red and drawn before T or non-red and drawn after T
(I*) Sb1·Sb2·Sb3·…·Sb99
(H*) Sb1·Sb2·Sb3·…·Sb99·Sb100 that is equivalent to:
(H’*) Rb1·Rb2·Rb3·…·Rb99·~Rb100
(P*) Sb100 i. e. finally:
(P’*) ~Rb100
The paradox resides here in the fact that the two generalisations (H) and (H*) lead respectively to the predictions (P) and (P’*), which are contradictory. Intuitively, the application of SR to (H*) appears erroneous. Goodman also gives in Fact, Fiction and Forecastiv a slightly different version of the paradox, applied in this case to emeralds.v This form is very well known and based on the predicate “grue” = green and observed before T or non-green and observed after T.
The predicate S used in Goodman (1946) presents with “grue”, a common structure. P and Q being two predicates, this structure corresponds to the following definition: (P and Q) or (~P and ~Q). In what follows, one will designate by grue a predicate having this particular structure, without distinguishing whether the specific form used is that of Goodman (1946) or (1954).
2. The unification/differentiation duality
The instances are in front of me. Must I describe them by stressing their differences? Or must I describe them by emphasising their common properties? I can proceed either way. To stress the differences between the instances, is to operate by differentiation. Conversely, to highlight their common properties, is to proceed by unification. Let us consider in turn each of these two modes of proceeding.
Consider the 100 balls composing the urn of Goodman (1946). Consider first the case where my intention is to stress the differences between the instances. There, an option is to apprehend the particular and single moment, where each of them is extracted from the urn. The considered predicates are then: red and drawn on day 1, red and drawn on day 2, …, red and drawn on day 99. There are thus 99 different predicates. But this prohibits applying SR, which requires one single predicate. Thus, what is to distinguish according to the moment when each ball is drawn? It is to stress an essential difference between each ball, based on the criterion of time. Each ball thus is individualised, and many different predicates are resulting from this: drawn at T1, drawn at T2, …, drawn at T99. This indeed prevents then any inductive move by application of SR. In effect, one does not have then a common property to allow induction and to apply SR. Here, the cause of the problem lies in the fact of having carried out an extreme differentiation.
Alternatively, I can also proceed by differentiation by operating an extremely precisevi measurement of the wavelength of the light defining the colour of each ball. I will then obtain a unique measure of the wavelength for each ball of the urn. Thus, I have 100 balls in front of me, and I know with precision the wavelength of the light of 99 of them. The balls respectively have a wavelength of 722,3551 nm, 722,3643 nm, 722,3342 nm, 722,3781 nm, etc. I have consequently 99 distinct predicates P3551, P3643, P3342, P3781, etc. But I have no possibility then to apply SR, which requires one single predicate. Here also, the common properties are missing to allow to implement the inductive process. In the same way as previously, it proves here that I have carried out an extreme differentiation.
What does it occur now if I proceed exclusively by unification? Let us consider the predicate R corresponding to “red or non-red”. One draws 99 red balls before time T. They are all R. One predicts then that the 100th ball will be R after T, i.e. red or non-red. But this form of induction does not bring any information here. The resulting conclusion is empty of information. One will call empty induction this type of situation. In this case, one observes that the process of unification of the instances by the colour was carried out in a radical way, by annihilating in this respect, any step of differentiation. The cause of the problem lies thus in the implementation of a process of extreme unification.
If one considers now the viewpoint of colour, it appears that each case previously considered requires a different taxonomy of colours. Thus, it is made use successively:
– of our usual taxonomy of colours based on 9 predicates: purple, indigo, blue, green, yellow, orange, red, white, black
– of a taxonomy based on a comparison of the wavelengths of the colours with the set of the real numbers (real taxonomy)
– of a taxonomy based on a single predicate (single taxon taxonomy): red or non-red
But it proves that each of these three cases can be replaced in a more general perspective. Indeed, multiple taxonomies of colours are susceptible to be used. And those can be ordered from the coarser (single taxon taxonomy) to the finest (real taxonomy), from the most unified to the most differentiated. We have in particular the following hierarchy of taxonomies:
– TAX1 = {red or non-red} (single taxon taxonomy)
– TAX2 = {red, non-red} (binary taxonomy)
– …
– TAX9 = {purple, indigo, blue, green, yellow, orange, red, white, black} (taxonomy based on the spectral colours, plus white and black)
– …
– TAX16777216 = {(0, 0, 0), …, (255, 255, 255)} (taxonomy used in computer science and distinguishing 256 shades of red/green/blue)
– …
– TAXR = {370, …, 750} (real taxonomy based on the wavelength of the light)
Within this hierarchy, it appears that the use of extreme taxonomies such as the one based on a single taxon, or the real taxonomy, leads to specific problems (respectively extreme unification and extreme differentiation). Thus, the problems mentioned above during the application of an inductive reasoning based on SR occur when the choice in the unification/differentiation duality is carried out too radically. Such problems relate to induction in general. This invites to think that one must rather reason as follows: I should privilege neither unification, nor differentiation. A predicate such as “red”, associated with our usual taxonomy of colours (TAX9)vii, corresponds precisely to such a criterion. It corresponds to a balanced choice in the unification/differentiation duality. This makes it possible to avoid the preceding problems. This does not prevent however the emergence of new problems, since one tries to implement an inductive reasoning, in certain situations. And one of these problems is naturally GP.
Thus, it appears that the stake of the choice in the duality unification/differentiation is essential from the viewpoint of induction, because according to whether I choose one way or the other, I will be able or not to use SR and produce valid inductive inferences. Confronted with several instances, one can implement either a process of differentiation, or a process of unification. But the choice that is made largely conditions the later success of the inductive reasoning carried out on those grounds. I must describe both common properties and differences. From there, a valid inductive reasoning can take place. But at this point, it appears that the role of the unification/differentiation duality proves to be crucial for induction. More precisely, it appears at this stage that a correct choice in the unification/differentiation duality constitutes one of the conditions of induction.
3. Several problems concerning induction
The problems which have been just mentioned constitute the illustration of several difficulties inherent to the implementation of the inductive process. However, unlike GP, these problems do not generate a genuine contradiction. From this point of view, they distinguish from GP. Consider now the following situation. I have drawn 99 balls respectively at times T1, T2, …, T99. The 100th ball will be drawn at T100. One observes that the 99 drawn balls are red. They are thus at the same time red and drawn before T100. Let R be the predicate “red” and T the predicate “drawn before T100“. One has then:
(I) RTb1, RTb2, …, RTb99
(H) RTb1, RTb2, …, RTb99, RTb100
(P) RTb100
By direct application of SR, the following prediction ensue: “the 100th ball is red and drawn before T100“. But this is in contradiction with the data of the experiment in virtue of which the 100th ball is drawn in T100. There too, the inductive reasoning is based on a formalisation which is that of SR. And just as for GP, SR leads here to a contradiction. Call 2 this problem, where two predicates are used.
It appears that one can easily build a form of 2 based on one single predicate. A way of doing that is to consider the unique predicate S defined as “red and drawn before T100” in replacement of the predicates R and T used previously. The same contradiction then ensues.
Moreover, it appears that one can highlight another version (1) of this problem comprising only one predicate, without using the “red” property which appears useless here. Let indeed T be the predicate drawn before T100. One has then:
(I) Tb1, Tb2, …, Tb99
(H) Tb1, Tb2, …, Tb99, Tb100
(P) Tb100
Here also, the conclusion according to which the 100th ball is drawn before T100 contradicts the data of the experiment according to which the 100th ball is drawn at T100. And one has then a contradictory effect, analogous to that of GP, without the structure of “grue” being implemented. Taking into account the fact that only the criterion of time is used to build this problem, it will be denoted in what follows by 1-time.
It appears here that the problems such as 1-time and 2 lead just as GP to a contradiction. Such is not the case for the other problems related to induction previously mentionedviii, which involve either the impossibility of carrying out induction, or a conclusion empty of information. However, it proves that the contradiction encountered in 1-time is not of the same nature as that observed in GP. Indeed in GP, one has a contradiction between the two concurrent predictions (P) and (P*). On the other hand, in 1-time, the contradiction emerges between on the one hand the conditions of the experiment (T 100) and on the other hand the prediction resulting from generalisation (T < 100).
Anyway, the problems which have been just encountered suggest that the SR formalism does not capture the whole of our intuitions related to induction. Hence, it is worth attempting to define accurately the conditions of induction, and adapting consequently the relevant formalism. However, before carrying out such an analysis, it is necessary to specify in more detail the various elements of the context of GP.
4. The universe of reference
Let us consider the law (L1) according to which “diamond scratches the other solids”. A priori, (L1) strikes us as an undeniable truth. Nevertheless, it proves that at a temperature higher than 3550°C, diamond melts. Therefore in last analysis, the law (L1) is satisfied at a normal temperature and in any case, when the temperature is lower than 3550°C. But such a law does not apply beyond 3550°C. This illustrates how the statement of the conditions under which the law (L1) is verified is important, in particular with regard to the conditions of temperature. Thus, when one states (L1), it proves necessary to specify the conditions of temperature in which (L1) finds to apply. This is tantamount to describing the type of universe in which the law is satisfied.
Let also (P1) be the following proposition: “the volume of the visible universe is higher than 1000 times that of the solar system”. Such a proposition strikes us as obvious. But there too, it appears that (P1) is satisfied at modern time, but that it proves to be false at the first moments of the universe. Indeed, when the age of our universe was 10-6 second after the big-bang, its volume was approximately equal to that of our solar system. Here also, it thus appears necessary to specify, at the same time as the proposition (P1) the conditions of the universe in which it applies. A nonambiguous formulation of (P1) thus comprises a more restrictive temporal clause, such as: “at our time, the volume of the visible universe is higher than 1000 times that of the solar system”. Thus, generally, one can think that when a generalisation is stated, it is necessary to specify the conditions of the universe in which this generalisation applies. The precise description of the universe of reference is fundamental, because according to the conditions of the universe in which one places oneself, the stated law can appear true or false.
One observes in our universe the presence of both constants and variables. There are thus constants, which constitute the fundamental constants of the universe: the speed of light: c = 2,998 x108 m/s; Planck’s constant: h = 6,626 x 10-34 J.s; the electron charge; e = 1,602 x 10-19 C; etc. There are on the other hand variables. Among those, one can mention in particular: temperature, pressure, altitude, localisation, time, presence of a laser radiation, presence of atoms of titanium, etc.
One often tends, when a generalisation is stated, not to take into account the constants and the variables which are those of our universe envisaged in its totality. Such is the case for example when one considers the situation of our universe on 1 January 2000, at 0h. One places then oneself explicitly in what constitutes a section, a slice of our universe. In effect, time is not regarded then a variable, but well as a constant. Consider also the following: “the dinosaurs had hot blood”ix. Here, one places oneself explicitly in a sub-universe of our where the parameters of time and space have a restricted scope. The temporal variable is reduced to the particular time of the Earth history which knew the appearance of the dinosaurs: the Triassic and the Cretaceous. And similarly, the space parameter is limited to our planet: Earth. Identically, the conditions of temperature are changing within our universe, according to whether one is located at one site or another of it: at the terrestrial equator, the surface of Pluto, the heart of Alpha Centauri, etc. But if one is interested exclusively in the balloon being used for the experimentation within the laboratory of physics, where the temperature is maintained invariably at 12°C, one can then regard valuably the temperature as a constant. For when such generalisations are expressed, one places oneself not in our universe under consideration in his totality, but only in what veritably constitutes a specific part, a restriction of it. One can then assimilate the universe of reference in which one places oneself as a sub-universe of our. It is thus frequent to express generalisations which are only worth for the present time, or for our usual terrestrial conditions. Explicitly or not, the statement of a law comprises a universe of reference. But in the majority of the cases, the variables and the constants of the considered sub-universe are distinct from those allowing to describe our universe in its totality. For the conditions are extremely varied within our universe: the conditions are very different according to whether one places oneself at the 1st second after the big-bang, on Earth at the Precambrian epoch, in our planet in year 2000, inside the particle accelerator of the CERN, in the heart of our Sun, near a white dwarf, or well inside a black hole, etc.
One can also think that it is interesting to be able to model universes the constants of which are different from the fundamental constants of our universe. One can thus wish to study for example a universe where the mass of the electron is equal to 9,325 x10-31 kg, or well a universe where the electron charge is equal to 1,598 x 10-19 C. And in fact, the toy-universes, which take into account fundamental constants different from those of our familiar universe, are studied by the astrophysicists.
Lastly, when one describes the conditions of a thought experiment, one places oneself, explicitly or not, under the conditions which are related to those of a sub-universe. When one considers for example 100 balls extracted from an urn during 100 consecutive days, one places then oneself in a restriction of our universe where the temporal variable is limited to one period of 100 days and where the spatial location is extremely reduced, corresponding for example to a volume approximately equal to 5 dm3. On the other hand, the number of titanium or zirconium atoms possibly present in the urn, the possible existence of a laser radiation, the presence or the absence of a sound source of 10 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In this situation, it is enough to mention the variables and the constants actually used in the thought experiment. For one can think indeed that the number of variables in our universe is so large that it is impossible to enumerate them all. And consequently, it does not appear possible to characterise our universe in function of all its variables, because one can not provide an infinite enumeration of it. It appears sufficient to describe the considered sub-universe, by mentioning only the constants and the variables which play an effective role in the experiment. Thus, in such situations, one will describe the considered sub-universe by mentioning only the effective criteria necessary to the description of the experiment.
What precedes encourages to think that generally, in order to model the context in which the problems such as GP take place, it is convenient to describe a given universe in terms of variables and constants. This leads thus to define a n-universe (n 0) as a universe the criteria of which comprise m constants, and n variables, where the m constants and n variables constitute the criteria of the given universe. Within this particular framework, one defines a temporal 1-universe (1T) as a universe comprising only one criterion-variable: time. In the same way, one defines a coloured 1-universe (1C) as a universe comprising only one criterion-variable: colour. One will define also a coloured and temporal 2-universe (2CT) as a universe comprising two criterion-variables: time and colour. Etc. In the same way, a universe where all the objects are red, but are characterised by a different localisation will be modelled by a localised 1-universe (1L) a criterion-constant (red) of which is colour.
It should be noted incidentally that the n-universe framework makes it possible in particular to model several interesting situations. Thus, a temporal universe can be regarded as a n-universe one of the variables of which is a temporal criterion. Moreover, a universe where one single moment T0 is considered, deprived of the phenomenon of succession of time, can be regarded as a n-universe where time does not constitute one of the variables, but where there is a constant-time. In the same way, an atemporal universe corresponds to a n-universe no variable of which corresponds to a temporal criterion, and where there is not any time-constant.
In the context which has been just defined, what is it now to be red? Here, being “red” corresponds to two different types of situations, according to the type of n-universe in which one places oneself. It can be on the one hand a n-universe one of the constants of which is colour. In this type of universe, the colour of the objects is not susceptible to change, and all the objects are there invariably red.
The fact of being “red” can correspond, on the second hand, to a n-universe one of the criterion-variables of which is constituted by colour. There, an object can be red or non-red. Consider the case of a 1C. In such a universe, an object is red or non-red in the absolute. No change of colour is possible there, because no other criterion-variable exists, of which can depend such a variation. And in a 2CT, being red is being red at time T. Within such a universe, being red is being red relatively to time T. Similarly, in a coloured, temporal and localised 3-universe (3CTL), being red is being red at time T and at place L. Etc. In some such universe, being red is being red relatively to other criterion-variables. And the same applies to the n-universes which model a universe such as our own.
At this step arises the problem of the status of the instances of an object of a given type. What is it thus to be an instance, within this framework? This problem has its importance, because the original versions of GP are based on instances of balls (1946) and emeralds (1954). If one takes into account the case of Goodman (1946), the considered instances are 100 different balls. However, if one considers a unique ball, drawn at times T1, T2, …, T100, one notices that the problem inherent to GP is always present. It suffices indeed to consider a ball whose colour is susceptible to change during the course of time. One has drawn 99 times the ball at times T1, T2, …, T99, and one has noted each time that the ball was red. This leads to the prediction that the ball will be red at T100. However, this last prediction proves to be contradictory with an alternative prediction based on the same observations, and the projection of the predicate S “red and drawn before T100 or non-red and drawn at T100x.
The present framework must be capable of handling the diversity of these situations. Can one thus speak of an instantiated and temporal 1-universe, or well of an instantiated and coloured 1-universe? Here, one must observe that the fact of being instantiated, for a given universe, corresponds to an additional criterion-variable. For, on the contrary, what makes it possible to distinguish between the instances? If no criterion distinguishes them, it is thus only one and the same thing. And if they are distinct, it is thus that a criterion makes it possible to differentiate them. Thus, an instantiated and temporal 1-universe is in fact a 2-universe, whose 2nd criterion, which makes it possible to distinguish the instances between them, is in fact not mentioned nor explicited. By making explicit this second criterion-variable, it is thus clear that one is placed in a 2-universe. In the same way, an instantiated and coloured 1-universe is actually a 2-universe one of the criteria of which is colour and the second criterion exists but is not specified.
Another aspect which deserves mention here, is the question of the reduction of a given n-universe to another. Is it not possible indeed, to logically reduce a n-universe to a different system of criteria? Consider for example a 3CTL. In order to characterise the corresponding universe, one has 3 criterion-variables: colour, time and localisation. It appears that one can reduce this 3-universe to a 2-universe. That can be carried out by reducing two of the criteria of the 3-universe to one single criterion. In particular, one will reduce both criteria of colour and time to a single criterion of tcolour* (shmolorxi). And one will only preserve two taxa of tcolour*: G and ~G. Consider then a criterion of color comprising two taxa (red, non-red) and a criterion of time comprising two taxa (before T, after T). If one associates the taxa of colour and time, one obtains four new predicates: red before T, red after T, non-red before T, non-red after T, which one will denote respectively by RT, R~T, ~RT and ~R~T. Several of these predicates are compatible (RT and R~T, RT and ~R~T, ~RT and R~T, ~RT and ~R~T) whereas others are incompatible (RT and ~RT, R~T and ~R~T). At this stage, one has several manners (16)xii of grouping the compatible predicates, making it possible to obtain two new predicates G and ~G of tcolour*:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 RT R~T X X X X X X X X RT ~R~T X X X X X X X X ~RT R~T X X X X X X X X ~RT ~R~T X X X X X X X X
In each of these cases, it results indeed a new single criterion of tcolour* (Z), which substitutes itself to the two preceding criteria of colour and time. One will denote by Zi (0 i 15) the taxa of tcolour* thus obtained. If it is clear that Z15 leads to the empty induction, it should be observed that several cases corresponding to the situation where the instances are RT lead to the problem inherent to GP. One will note thus that Z2, i.e. grue2 (by assimilating the Zi to gruei and the Z15-i to bleeni) is based on the definition: grue2 = red before T and non-red after T. It appears here as a conjunctive interpretation of the definition of “grue”. In the same way, grue7 corresponds to a definition of “grue” based on an exclusive disjunction. Lastly, grue12 is based on the traditional definition: grue12 = red before T or non-red after T, where the disjunction is to be interpreted as an inclusive disjunction.
Similarly, it also proves that a 2CT can be reduced to a tcoloured* 1-universe (1Z). And more generally, a n-universe is thus reducible to an (n-1)-universe (for n > 1). Thus, if one considers a given universe, several characterisations in terms of n-universe can valuably be used. One can in particular apprehend a same universe like a 3CTL, or like a 2ZL. In the same way, one can represent a 2CT like a 1Z. At this stage, none of these views appears fundamentally better than the other. But each of these two characterisations constitute alternative ways to describe a same reality. This shows finally that a n-universe constitutes in fact an abstract characterisation of a real or an imaginary universe. A n-universe constitutes thus a system of criteria, comprising constants and variables. And in order to characterise a same real or imaginary given universe, one can resort valuably to several n-universes. Each of them appears finally as a different characterisation of the given universe, simply based on a different set of primitives.
5. Conditions of induction
The fact that the SR formalism involves the GP effect suggests that the intuition which governs our concept of induction is not entirely captured by SR. It is thus allowed to think that if the formal approach is necessary and useful to be used as support to induction, it does not constitute however a sufficient step. For it appears also essential to capture the intuition which governs our inductive reasoning. Therefore it proves necessary to supplement the formal approach of induction by a semantic approach. Goodman himself provides us with a definition of inductionxiii. He defines induction as the projection of characteristics of the past through the future, or more generally, as the projection of characteristics corresponding to a given aspect of an object through another aspect. This last definition corresponds to our intuition of induction. One can think however that it is necessary to supplement it by taking into account the preceding observationsxiv concerning the differentiation/unification duality. In that sense, it has been pointed out that induction consists of an inference from instances presenting both common properties and differences. Let the instances-source (instances-S) be the instances to which relate (I) or (I*) and the instance-destination (instance-D) that which is the subject of (P) or (P*). The common properties relate to the instances-S and the differentiated properties are established between the instances-S and the instance-D. The following definition ensues: induction consists precisely in the fact that the instance-Dxv also presents the property that is common to the instances-S, whereas one does vary the criterion (criteria) on which the differences between the instances-S and the instance-D is (are) based. The inductive reasoning is thus based on the constant nature of a property, whereas such other property is variable.
From this definition of induction arise straightforwardly several conditions of induction. I shall examine them in turn. The first two conditions are thus the following ones:
(C1) the instances-S must present some common properties
(C2) the instances-S and the instance-D must present some distinctive properties
This has for consequence that one cannot apply induction in two particular circumstances: firstly (i) when the instances do not present any common property. One will call such a situation a total differentiation of the instances. The problems corresponding to this particular circumstance have been mentioned abovexvi. And secondly (ii) when the instances do not present any distinctive property. One will call such a situation total unification. The problems encountered in this type of situation have also been mentioned previouslyxvii.
It should also be noted that it is not here a question of intrinsic properties of the instances, but rather of the analysis which is carried out by the one who is on the point of reasoning by induction.
Taking into account the definition of induction which has been given, a third condition can be thus stated:
(C3) a criterion-variable is necessary for the common properties of the instances-S and another criterion-variable for the distinctive properties
This refers to the structure of the considered universe of reference. Consequently, two criterion-variables are at least necessary, in the structure of the corresponding universe of reference. One will call that the minimalcondition of induction. Hence, a 2-universe is at least necessary in order that the conditions of induction can be satisfied. Thus, a 2CT will be appropriate. In the same way, a temporal and localised 2-universe (2TL) will also satisfy the conditions which have been just defined, etcxviii.
It should be noted that another way of stating this condition is as follows: the criterion-variable for the common properties and the criterion-variable for the differentiated properties must be distinct. One should not have confusion between the two. One can call that the condition of separation of the common properties and the distinctive properties. Such a principle appears as a consequence of the minimal condition for induction: one must have two criteria to perform induction, and these criteria must be different. If one chooses a same criterion for the common properties and the differentiated properties, one is brought back in fact to one single criterion and the context of a 1-universe, itself insufficient to perform induction.
Lastly, a fourth condition of induction results from the preceding definition:
(C4) one must project the common properties of the instances-S (and not the distinctive properties)
The conditions of induction which have been just stated make it possible from now on to handle the problems involved in the use of SR mentioned abovexix. It follows indeed that the following projectionsxx are correct: C°T in a 2CT, C°L in a 2CL, Z°L in a 2ZL, etc. Conversely, the following projections are incorrect: T°T in a 1T, Z°Z in a 1Z. In particular, one will note here that the projection T°T in the 1T is that of 1-time. 1-time takes indeed place in a 1T, whereas induction requires at the same time common properties and distinctive properties. Thus, a 2-universe is at least necessary. Usually, the criterion of time is used for differentiation. But here, it is used for unification (“drawn before T”). That can be done, but provided that one uses a distinct criterion for the differentiated properties. However, whereas common properties results here from that, the differentiated properties are missing. It thus misses a second criterion – corresponding to the differentiated properties – in the considered universe, to perform induction validly. Thus 1-time finds its origin in a violation of the minimal condition of induction. One can formulate this solution equivalently, with regard to the condition of separation. In effect, in 1-time, a same temporal criterion (drawn before T/drawn after T) is used for the common properties and the differentiated properties, whereas two distinct criteria are necessary. It can be thus analysed as a manifest violation of the condition of separation.
Lastly, the conditions of induction defined above lead to adapt the formalism used to describe GP. It proves indeed necessary to distinguish between the common and the distinctive property(ies). One will thus use the following formalism in replacement of the one used above:
(I) RT1·RT2·RT3·…·RT99
(H) RT1·RT2·RT3·…·RT99·RT100
where R denotes the common property and the Ti a distinctive property. It should be noted here that it can consist of a single object, or alternatively, of instances which are distinguished by a given criterion (which is not concerned by the inductive process) according to n-universe in which one places oneself. Thus, one will use in the case of a single instance , the colour of which is susceptible to change according to time:
(I) RT1·RT2·RT3·…·RT99
or in the case where several instances 1, 2, …, 99, 100 existxxi:
(I) RT11·RT22·RT33·…·RT9999
Given the conditions of induction and the framework of n-universes which have been just defined, one is now in a position to proceed to determine the origin of GP. Preliminarily it is worth describing accurately the conditions of the universe of reference in which GP takes place. Indeed, in the original version of GP, the choice of the universe of reference is not defined accurately. However one can think that it is essential, in order to avoid any ambiguity, that this last is described precisely.
The universe of reference in which Goodman (1946) places himself is not defined explicitly, but several elements of the statement make it possible to specify its intrinsic nature. Goodman thus mentions the colours “red” and “non-red”. Therefore, colour constitutes one of the criterion-variables of the universe of reference. Moreover, Goodman distinguishes the balls which are drawn at times T1, T2, T3, …, T100. Thus, time is also a criterion-variable of the considered universe. Consequently, one can describe the minimal universe in which Goodman (1946) places himself as a 2CT. Similarly, in Goodman (1954), the criterion-variables of colour (green/non-green) and time (drawn before T/drawn after T) are expressly mentioned. In both cases, one thus places oneself implicitly within the minimal framework of a 2CT.
Goodman in addition mentions instances of balls or emeralds. Is it necessary at this stage to resort to an additional criterion-variable making it possible to distinguish between the instances? It appears that not. On the one hand indeed, as we have seen previouslyxxii, it proves that one has well a version of GP by simply considering a 2CT and a single object, the colour of which is susceptible to change during the course of time. On the other hand, it appears that if the criterion which is used to distinguish the instances is not used in the inductive process, it is then neither useful as a common criterion, nor as a differentiated criterion. It follows that one can dispense with this 3rd additional criterion. Thus, it proves that the fact of taking into account one single instance or alternatively, several instances, is not essential in the formulation of GP. In what follows, one will be able thus to consider that the statement applies, indifferently, to a single object or several instances that are distinguished by a criterion which is not used in the inductive process.
At this step, we are in a position to replace GP within the framework of n-universes. Taking into account the fact that the context of GP is that of a minimal2CT, one will consider successively two situations: that of a 2CT, and then that of a 3CT (where denotes a 3rd criterion).
6.1 “Grue” in the coloured and temporal 2-universe
Consider first the hypothesis of a 2CT. In such a universe, being “red” is being red at time T. One has then a criterion of colour for the common properties and a criterion of time for the differentiated properties. Consequently, it appears completely legitimate to project the common property of colour (“red”) into the differentiated time. Such a projection proves to be in conformity with the conditions of induction stated above.
Let us turn now to the projection of “grue”. One has observed previouslyxxiii that the 2CT was reducible to a 1Z. Here, the fact of using “grue” (and “bleen”) as primitives, is characteristic of the fact that the system of criteria used is that of a 1Z. What is then the situation when one projects “grue” in the 1Z? In such a universe of reference, the unique criterion-variable is the tcolour*. An object is there “grue” or “bleen” in the absolute. Consequently, if one has well a common criterion (the tcolour*), it appears that the differentiated criterion is missing, in order to perform induction validly. And the situation in which one is placed is that of an extreme differentiation. Thus, such a projection is carried out in violation of the minimal condition of induction. Consequently, it proves that GP cannot take place in the 2CT and is then blocked at the stage of the projection of “grue”.
But are these preliminary remarks sufficient to provide, in the context of a 2CT, a satisfactory solution to GP? One can think that not, because the paradox also arises in it in another form, which is that of the projection of tcolour* through time. One can formalise this projection Z°T as follows:
(I*) GT1·GT2·GT3·…·GT99
(H*) GT1·GT2·GT3·…·GT99·GT100 that is equivalent to:
(H’*) RT1·RT2·RT3·…·RT99·~RT100
(P*) GT100 that is equivalent to:
(P’*) ~RT100
where it is manifest that the elements of GP are still present.
Fundamentally in this version, it appears that the common properties are borrowed from the system of criteria of the 1Z, whereas the differentiated properties come from the 2CT. A first analysis thus reveals that the projection of “grue” under these conditions presents a defect which consists in the choice of a given system of criteria for the common properties (tcolour*) and of a different system of criteria for the differentiated properties (time). For the selection of the tcolour* is characteristic of the choice of a 1Z, whereas the use of time is revealing of the fact that one places oneself in a 2CT. But one must choose one or the other of the reducible systems of criteria to perform induction. On the hypotheses envisaged previously, the choice of the criteria for the common and differentiated properties was carried out within the same system of criteria. But here, the choice of the criteria for the common properties and the differentiated properties is carried out within two different (and reducible) systems of criteria. Thus, the common and differentiated criteria selected for induction are not genuinely distinct. And this appears as a violation of the condition of separation. Consequently, one of the conditions of induction is not respected.
However, the projection Z°T has a certain intuitive support, because it is based on the fact that the notions of “grue before T” and “grue after T” have a certain intuitive meaning. Let us then disregard the violation of the conditions of the induction which has been just mentioned, and consider thus this situation in more detail. In this context, GP is always present, since one observes a contradiction between (P) and (P’*). It is with this contradiction that it is worth from now on being interested. Consider the particular step of the equivalence between (H*) and (H’*). One conceives that “grue before T” is assimilated here to RT, because the fact that the instances-S are red before T results clearly from the conditions of the experiment. On the other hand, it is worth being interested by the step according to which (P*) entails (P’*). According to the classical definitionxxiv: “grue” = {RT R~T, RT ~R~T, ~RT ~R~T }. What is it then to be “grue after T”? There, it appears that a “grue” object can be R~T (this corresponds to the case RT R~T) or ~R~T (this correspond to the cases RT ~R~T and ~RT ~R~T). In conclusion, the object can be either R~T or ~R~T. Thus, the fact of knowing that an object is “grue after T” does not make it possible to conclude that this object is ~R~T, because this last can also be R~T. Consequently, the step according to which (P*) involves (P’*) appears finally false. From where it ensues that the contradiction between (P) and (P’*) does not have any more a raison d’etre.
One can convince oneself that this analysis does not depend on the choice of the classical definition of “grue” (grue12) which is carried out, by considering other definitions. Consider for example the definition based on grue9: “grue” = {RT ~R~T, ~RT ~R~T} and “bleen” = {RT R~T, ~RT R~T}. But in this version, one notes that one does not have the emergence of GP, because the instances-S, which are RT, can be at the same time “grue” and ” bleen”. And the same applies if one considers a conjunctive definition (grue2) such as “grue” = {RT ~R~T}. In such a case indeed, the instances-S are “grue” only if they are RT but also ~R~T. However this does not correspond to the initial conditions of GP in the 2CT where one ignores if the instances-S are ~R~T.
One could also think that the problem is related to the use of a taxonomy of tcolour* based on two taxa (G and ~G). Consider then a taxonomy of tcolour* based on 4 taxa: Z0 = RT R~T, Z1 = RT ~R~T, Z2 = ~RT R~T, Z3 = ~RT ~R~T. But on this hypothesis, it appears clearly that since the instances-S are for example Z1, one finds himself replaced in the preceding situation.
The fact of considering “grue after T”, “grue before T”, “bleen before T”, “bleen after T” can be assimilated with an attempt of expressing “grue” and ” bleen” with the help of our own criteria, and in particular that of time. It can be considered here as a form of anthropocentrism, underlain by the idea to express the 1Z with the help of the taxa of the 2CT. Since one knows the code defining the relations between two reducible n-universes – the 1Z and the 2CT – and that one has partial data, one can be tempted to elucidate completely the predicates of the foreign n-universe. Knowing that the instances are GT, G~T, ~GT, ~G~T, I can deduce that they are respectively {RT, ~RT}, {R~T, ~R~T}, {~RT}, {R~T}. But as we have seen, due to the fact that the instances are GT and RT, I cannot deduce that they will be ~R~T.
The reasoning in this version of GP is based on the apparently inductive idea that what is “grue before T” is also “grue after T”. But in the context which is that of the 1Z, when an object is “grue”, it is “grue” in the absolute. For no additional criterion exists which can make its tcolour* vary. Thus, when an object is GT, it is necessarily G~T. And from the information according to which an object is GT, one can thus conclude, by deduction, that it is also G~T.
From what precedes, it ensues that the version of GP related to the Z°T presents the apparent characters of induction, but it does not constitute an authentic form of this type of reasoning. Z°T thus constitutes a disguised form of induction for two principal reasons: first, it is a projection through the differentiated criterion of time, which constitutes the standard mode of our inductive practice. Second, it is based on the intuitive principle according to which everything that is GT is also G~T. But as we have seen, it consists here in reality of a deductive form of reasoning, whose true nature is masked by an apparent inductive move. And this leads to conclude that the form of GP related to Z°T analyses itself in fact veritably as a pseudo-induction.
6.2 “Grue” in the coloured, temporal and localised 3-universe
Consider now the case of a 3CT. This type of universe of reference also corresponds to the definition of a minimal2CT, but it also comprises one 3rd criterion-variablexxv. Let us choose for this last a criterion such as localisationxxvi. Consider then a 3CTL. Consider first (H) in such a 3-universe. To be “red” in the 3CTL, is to be red at time T and at location L. According to the conditions of GP, colour corresponds to the common properties, and time to the differentiated properties. One has then the following projection C°TL:
(I) RT1L1·RT2L2·RT3L3·…·RT99L99
(H) RT1L1·RT2L2·RT3L3·…·RT99L99·RT100L100
(P) RT100L100
where taking into account the conditions of induction, it proves to be legitimate to project the common property (“red”) of the instances-S, into differentiated time and location, and to predict that the 100th ball will be red. Such a projection appears completely correct, and proves in all points in conformity with the conditions of induction mentioned above.
What happens now with (H*) in the 3CTL? It has been observed that the 3CTL could be reduced to a 2ZL. In this last n-universe, the criterion-variables are tcolour* and localisation. The fact of being “grue” is there relative to location: to be “grue”, is to be “grue” at location L. What is then projected is the tcolour*, i.e. the fact of being “grue” or “bleen”. There is thus a common criterion of tcolour* and a differentiated criterion of localisation. Consequently, if it is considered that the instances-S are “grue”, one can equally well project the property common “grue” into a differentiated criterion of localisation. Consider then the projection Z°L in the 2ZL:
(I*) GL1·GL2·GL3·…·GL99
(H*) GL1·GL2·GL3·…·GL99·GL100
(P*) GL100
Such a projection is in conformity with the conditions mentioned above, and constitutes consequently a valid form of induction.
In this context, one can project valuably a predicate having a structure identical to that of “grue”, in the case of emeralds. Consider the definition “grue” = green before T or non-green after T, where T = 10 billion years. It is known that at that time, our Sun will be extinct, and will become gradually a dwarf white. The conditions of our atmosphere will be radically different from what they currently are. And the temperature will rise in particular in considerable proportions, to reach 8000°. Under these conditions, the structure of many minerals will change radically. It should normally thus be the case for our current emeralds, which should see their colour modified, due to the enormous rise in temperature which will follow. Thus, I currently observe an emerald: it is “grue” (for T = 10 billion years). If I project this property through a criterion of location, I legitimately conclude from it that the emerald found in the heart of the Amazonian forest will also be “grue”, in the same way as the emerald which has been just extracted from a mine from South Africa.
At this stage, one could wonder whether the projectibility of “grue” is not intrinsically related to the choice of a definition of “grue” based on inclusive disjunction (grue12)? Nevertheless, one easily checks by using an alternative definition of “grue” that its projection remains validxxvii.
It should be noticed that one has here the expression of the fact that the taxonomy based on the tcolour* is coarser than that based on time and colour. In effect, the former only comprises 2 taxa (grue/bleen), whereas the latter presents 4 of them. By reducing the criteria of colour and time to a single criterion of tcolor*, one has replaced 4 taxa (RT R~T, RT ~R~T, ~RT R~T, ~RT ~R~T) by 2. Thus, “grue” constitutes from this point of view a predicate coarser than “red”. The universe which is described did not change, but the n-universes which are systems of criteria describing these universes are different. With the tcolour* thus defined, one has less predicates at its disposal to describe a same reality. The predicates “grue” and “bleen” are for us not very informative, and are less informative in any case that our predicates “red”, “non-red”, “before T”, etc. But that does not prevent however “grue” and “bleen” to be projectibles.
Whereas the projection of “grue” appears valid in the 2ZL, it should be noticed however that one does not observe in this case the contradiction between (P) and (P’*). For here (I*) is indeed equivalent to:
(I’*) RT1L1·RT2L2·RT3L3·…·RT99 L99
since, knowing according to the initial data of GP that the instances-S are RT, one valuably replaces the GLi by the RTiLi (i < 100). But it appears that on this hypothesis, (P*) does not involve:
(P’*) ~RT100L100
because one does not have an indication relating to the temporality of the 100th instance, due to the fact that only the localisation constitutes here the differentiated criterion. Consequently, one has well in the case of the 3CTL a version built with the elements of GP where the projection of “grue” is carried out valuably, but which does not present a paradoxical nature.
7. Conclusion
In the solution to GP proposed by Goodman, a predicate is projectible or nonprojectible in the absolute. And one has in addition a correspondence between the entrenchedxxviii/non-entrenched and the projectible/nonprojectible predicates. Goodman in addition does not provide a justification to this assimilation. In the present approach, there is no such dichotomy, because a given predicate P reveals itself projectible in a given n-universe, and nonprojectible in another n-universe. Thus, P is projectible relatively to such universe of reference. There is thus the projectible/nonprojectible relative to such n-universe distinction. And this distinction is justified by the conditions of induction, and the fundamental mechanism of induction related to the unification/differentiation duality. There are thus n-universes where “green” is projectible and others where it is not. In the same way, “grue” appears here projectible relative to certain n-universes. Neither green nor grue are projectible in the absolute, but only relative to such given universe. Just as of some other predicates, “grue” is projectible in certain universes of reference, but nonprojectible in othersxxix.
Thus, it proves that one of the causes of GP resides in the fact that in GP, one classically proceeds to operate a dichotomy between the projectible and the nonprojectible predicates. The solutions classically suggested to GP are respectively based on the distinction temporal/nontemporal, local/non-local, qualitative/nonqualitative, entrenched/non-entrenched, etc. and a one-to-one correspondence with the projectible/nonprojectible distinction. One wonders whether a given predicate P* having the structure of “grue” is projectible, in the absolute. This comes from the fact that in GP, one has a contradiction between the two concurrent predictions (P) and (P*). One classically deduces from it that one of the two predictions must be rejected, at the same time as one of the two generalisations (H) or (H*) on which these predictions are respectively based. Conversely, in the present analysis, whether one places himself in the case of the authentic projection Z°L or in the case of the pseudo-projection Z°T, one does not have a contradiction between (P) and (P’*). Consequently, one is not constrained any more to reject either (H) or (H*). And the distinction between projectible/nonprojectible predicates does not appear indispensable any morexxx.
How is the choice of our usual n-universe carried out in this context? N-universes such as the 2CT, the 3CTL, the 2ZL etc. are appropriate to perform induction. But we naturally tend to privilege those which are based on criteria structured rather finely to allow a maximum of combinations of projections. If one operates from the criteria Z and L in the 2ZL, one restricts oneself to a limited number of combinations: Z°L and L°Z. Conversely, if one retains the criteria C, T and L, one places oneself in the 3CTL and one has the possibility of projections C°TL, T°CL, L°CT, CT°Lxxxi, CL°T, TL°C. One has thus a maximum of combinations. This seems to encourage to prefer the 3CTL to the 2ZL. Of course, pragmatism seems to have to play a role in the choice of the best alternative of our criteria. But it seems that it is only one of the multiple factors which interact to allow the optimisation of our criteria to carry out the primitive operations of grouping and differentiation, in order to then be able to generalise, classify, order, make assumptions or forecastxxxii. Among these factors, one can in particular mention: pragmatism, simplicity, flexibility of implementation, polyvalencexxxiii, economy in means, powerxxxiv, but also the nature of our real universe, the structure of our organs of perception, the state of our scientific knowledge, etcxxxv. Our usual n-universes are optimised with regard to these various factors. But this valuably leaves room for the choice of other systems of criteria, according to the variations of one or the other of these parametersxxxvi.
i Nelson Goodman, “A Query On Confirmation”, Journal of Philosophy, vol. 43 (1946), p. 383-385; in Problems and Projects, Indianapolis, Bobbs-Merrill, 1972, p. 363-366.
iii See Goodman “A Query On Confirmation”, p. 383: “Suppose we had drawn a marble from a certain bowl on each of the ninety-nine days up to and including VE day and each marble drawn was red. We would expect that the marble drawn on the following day would also be red. So far all is well. Our evidence may be expressed by the conjunction “Ra1·Ra2·…·Ra99” which well confirms the prediction Ra100.” But increase of credibility, projection, “confirmation” in any intuitive sense, does not occur in the case of every predicate under similar circumstances. Let “S” be the predicate “is drawn by VE day and is red, or is drawn later and is non-red.” The evidence of the same drawings above assumed may be expressed by the conjunction “Sa1·Sa2·…·Sa99“. By the theories of confirmation in question this well confirms the prediction “Sa100“; but actually we do not expect that the hundredth marble will be non-red. “Sa100” gains no whit of credibility from the evidence offered.”
iv Nelson Goodman, Fact, Fiction and Forecast, Cambridge, MA, Harvard University Press, 1954.
v Ibid., p. 73-4: “Suppose that all emeralds examined before a certain time t are green. At time t, then, our observations support the hypothesis that all emeralds are green; and this is in accord with our definition of confirmation. […] Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue.”
vi For example with an accuracy of 10-4 nm.
vii Or any taxonomy which is similar to it.
viii See §2 above.
ix This assertion is controversial.
x Such a remark also applies to the statement of Goodman, Fact, Fiction and Forecast.
xi As J.S. Ullian mentions it, “More one ‘Grue’ and Grue”, Philosophical Review, vol. 70 (1961), p. 386-389, in p. 387.
xii I. e. C(0, 4)+C(1, 4)+C(2, 4)+C(3, 4)+C(4, 4) = 24, where C(p, q) denotes the number of combinations of q elements taken p times.
xiii See Goodman, “A Query On Confirmation”, p. 383: “Induction might roughly be described as the projection of characteristics of the past into the future, or more generally of characteristics of one realm of objects into another.”
xiv See §2 above.
xv One can of course alternatively take into account several instances-D.
xvi See §2 above.
xvii Ibid.
xviii For the application of this condition, one must take into account the remarks mentioned above concerning the problem of the status of the instances. Thus, one must actually compare an instantiated and temporal 1-universe to a 2-universe one of the criteria of which is temporal, and the second criterion is not explicitly mentioned. Similarly, an instantiated and coloured 1-universe is assimilated in fact to a 2-universe one of the criteria of which is temporal, and the second criterion is not specified.
xix See §3 above.
xx With the notations C (colour), T (time), L (localisation) and Z (tcolour*).
xxi However, since the fact that there exists one or more instances is not essential in the formulation of the given problem, one will obviously be able to abstain from making mention of it.
xxii See §4.
xxiii Ibid.
xxiv It is the one based on the inclusive disjunction (grue12).
xxv A same solution applies, of course, if one considers a number of criterion-variables higher than 3.
xxvi All other criterion distinct from colour or time, would also be appropriate.
xxvii In particular, it appears that the projection of a conjunctive definition (grue2) is in fact familiar for us. In effect, we do not proceed otherwise when we project the predicate “being green before maturity and red after maturity” applicable to tomatoes, through a differentiated criterion of location: this is true of the 99 instance-S observed in Corsica and Provence, and is projected validly to a 100th instance located in Sardinia. One can observe that such a type of projection is in particular regarded as nonproblematic by Jackson (Franck Jackson, “‘Grue'”, Journal of Philosophy, vol. 72 (1975), p. 113-131): “There seems no case for regarding ‘grue’ as nonprojectible if it is defined this way. An emerald is grue1 just if it is green up to T and blue thereafter, and if we discovered that all emeralds so far examined had this property, then, other things being equal, we would probably accept that all emeralds, both examined and unexamined, have this property (…).” If one were to replace such a predicate in the present analysis, one should then consider that the projection is carried out for example through a differentiated criterion of localisation (p. 115).
xxviii Goodman, Fact, Fiction and Forecast.
xxix The account presented in J Holland, K Holyoak, R. Nisbett and P. Thagard (Induction, Cambridge, MA; London, MIT Press, 1986) appears to me to constitute a variation of Goodman’s solution, directed towards the computer-based processing of data and based on the distinction integrated/non-integrated in the default hierarchy. But Holland’s solution presents the same disadvantages as that of Goodman: what justification if not anthropocentric, does one have for this distinction? See p. 235: “Concepts such as “grue”, which are of no significance to the goals of the learner, will never be generated and hence will not form part of the default hierarchy. (…) Generalization, like other sorts of inference in a processing system, must proceed from the knowledge that the system already has”.
The present analysis also distinguishes from the one presented by Susan Haack (Evidence and Inquiry, Oxford; Cambridge, MA, Blackwell, 1993) because the existence of natural kinds does not constitute here a condition for induction. See p. 134: “There is a connection between induction and natural kinds. […] the reality of kinds and laws is a necessary condition of successful inductions”. In the present context, the fact that the conditions of induction (a common criterion, a distinct differentiated criterion, etc.) are satisfied is appropriate to perform induction.
xxx A similar remark is made by Franck Jackson in conclusion of his article (“‘Grue'”, p. 131): “[…] the SR can be specified without invoking a partition of predicates, properties or hypotheses into the projectible and the nonprojectible”. For Jackson, all noncontradictory predicates are projectible: “[…] all (consistent) predicates are projectible.” (p. 114). Such a conclusion appears however stronger than the one that results from the current analysis. Because for Jackson, all predicates are thus projectible in the absolute. However in the present context, there are no projectible or nonprojectible predicates in the absolute. It is only relative to a given n-universe, that a predicate P reveals projectible or nonprojectible.
More generally, the present analysis distinguishes fundamentally from that of Jackson in the sense that the solution suggested to GP does not rest on the counterfactual condition. This last appears indeed too related to the use of certain predicates (examined, sampled, etc.). On the other hand, in the present context, the problem is considered from a general viewpoint, independently of the particular nature of the predicates constituting the definition of grue.
xxxi Such a projection corresponds for example to the generalisation according to which “the anthropomorphic statue-menhirs are of the colour of granite and date from the Age of Bronze”.
xxxii As Ian Hacking underlines it, Le plus pur nominalisme, Combas, L’éclat, 1993, p. 9: “Utiliser un nom pour une espèce, c’est (entre autres choses) vouloir réaliser des généralisations et former des anticipations concernant des individus de cette espèce. La classification ne se limite pas au tri : elle sert à prédire. C’est une des leçons de la curieuse “énigme” que Nelson Goodman publia il y a quarante ans.” My translation: “To use a name for a species, it is (among other things) to want to carry out generalisations and to form anticipations concerning the individuals of this species. Classification is not limited to sorting: it is used to predict. It is one of the lessons of the strange “riddle” which Nelson Goodman published forty years ago.”
xxxiii The fact that a same criterion can be used at the same time as a common and a differentiated criterion (while eventually resorting to different taxa).
xxxiv I.e. the number of combinations made possible.
xxxv This enumeration does not pretend to be exhaustive. A thorough study of this question would be of course necessary.
xxxvi I thank the editor of Dialogue and two anonymous referees for very helpful comments on an earlier draft of this paper.
English translation of a paper published in French in Philosophiques, autumn 2008, under the title “Une défense logique du modèle de Maher pour les délires polythématiques”.
In this paper, we proceed to describe a model for the formation and maintenance of polythematic delusions encountered in schizophrenia, which is in adequation with Brendan Maher’s account of delusions. Polythematic delusions are considered here as the conclusions of arguments triggered by apophenia that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. We describe first the structure of reasoning which leads to delusions of reference, of telepathy and of influence, by distinguishing between the primary, secondary, tertiary and quaternary types of delusional arguments. These four levels of arguments correspond to a stage the nature of which is respectively instantial, inductive, interpretative at a monothematic level and interpretative at a polythematic level. We also proceed to identify accurately the fallacious steps in the corresponding reasoning. We expose then the role of apophenia in the elaboration of delusional ideas. Lastly, we describe the role played by the hallucinations in the present model.
## A Logical Defence of Maher’s Model of Polythematic Delusions
Classically, the term of “delusion” applies to two fundamentally distinct forms: monothematic and polythematic delusions (Davies & Coltheart 2000; Bortolotti 2005). Monothematic delusions present an unique topic and are usually associated with cerebral lesions. Among the latter, one can mention Capgras’ delusion (by virtue of which the patient thinks that one of his/her fellows has been replaced by an impostor), Fregoli’s delusion (when the patient is persuaded that he/she is followed by one or several persons whom he cannot identify because they are dressed up) or Cotard’s delusion (when the patient is persuaded that he/she is died). Conversely, polythematic delusions have numerous topics, which are most often interconnected and usually linked to psychotic disturbances. Among polythematic delusions, one can notably mention: delusion of reference, delusion of grandeur, delusion of influence, delusion of persecution, delusion of control, delusion of telepathy.
In what follows, I will set out to introduce a new, as far as I know, model of the mechanism which leads to the formation of polythematic delusionsi met in schizophrenia. This model, which takes place in the recent development of psycho-pathological philosophy (Faucher, 2006), offers to describe the mechanism which leads, on the one hand, to the formation of delusional ideas and on the other hand, to their maintenance. In this model, delusions are the result of the patient’s cognitive activity in response to a specific form of abnormal perception. Even though the corresponding reasoning appears essentially normal, it includes however the repetition of some typical errors of reasoning. This leads to consider polythematic delusions as the conclusions of fallacious arguments, worked out in response to a particular type of abnormal perception, among which one can distinguish between primary, secondary, tertiary and quaternary delusional arguments. These four levels of arguments correspond, as we shall see it, to some functional stages the nature of which is respectively instantial (based on several instances), inductive (generalising the conclusion of each of the instances), interpretative at a monothematic level and finally, interpretative at a polythematic level.
It is worth mentioning, moreover, that the notion of delusion has important philosophical underpinnings. In particular, the understanding of delusions proves to be essential for the notions of belief (Engel 2001, Bayne & Pacherie 2005), of justification of beliefs, of knowledge, of rationality (Campbell 2001, Bortolotti 2005) and also of self-deception.
# 1. Cognitive models of delusions
Edvard Munch: The Scream.
Before describing in detail the present model, it is worth presenting the main cognitive models for delusions encountered in the literature. Some authors described then a cognitive model for delusional ideas observed in schizophrenia. As Chadwick & al. (1996) underline it, as well as Fowler & al. (1995) who set out to review these types of models, there does not truly exist a unique cognitive model for delusions, for it rather consists of a family of models.
A first cognitive model was described by Chadwick & al. (1996), who set out to introduce an application to delusions of Albert Ellis’ ABC-analysis. The original model described by Ellis (1962) consists of a diagram which plays a preponderant role in the emergence of mental disorders. Ellis distinguishes thus between three types of events: A, B and C. The As (for Activating event) are external facts or events of the patient’s internal life, such as thoughts or bodily feelings. The Bs (for Belief) are thoughts relating to the same events, which themselves are possibly rational in nature or not. Moreover, the corresponding cognitive process can be more or less conscious. Finally, the Cs (for Consequence) are emotional states such as anger, sadness, joy, frustration, etc. which can be of a positive or negative nature. Thus, the Cs that follow irrational thoughts are most often of a negative nature. The succession of events of type A, B and C plays a predominant role in the emergence of mental disorders: activating events trigger irrational thoughts, which themselves engender negative emotions. The type of therapy recommended by Ellis consists precisely in rendering the patient aware of this mechanism and in controlling the corresponding process. Adapting Ellis’ original model to psychosis, Chadwick & al. point out thus how the As constitute events that are external or internal to the patient, the Bs are his/her thoughts and the Cs are constituted by the emotions triggered by the patient’s thoughts. This specific framework allows to provide a cognitive ABC-analysis of the main types of delusions. For example, the delusion of persecution finds its origin in the external fact that the patient hears the noise of a car’s horn; this triggers in the patient the thought according to which his/her enemies come to kill him/her; it follows then in the patient’s a negative emotion of fright and of anxiety.
A second cognitive model of delusions was put forth by Brendan Maher (1974; 1988; 1999). Maher did suggest that delusions are the result – in the context of schizophrenia of paranoid subtype – of a broadly normal interpretation of the abnormal phenomena undergone by the patient (hallucinations, loss of audition, impairment in the intensity of perception, etc.). According to Maher, it is not therefore by his/her reasoning that the patient differs from a normal individual, but by his/her own altered perceptions. Delusional ideas are thus conceived of as a response to abnormal and emotionally disturbing phenomena experienced by the patient. Such disturbing phenomena lead the patient to search an explanation, which is at the origin of the delusional construction. According to Maher, the patient’s intellectual process is a product of normal reasoning and does not differ significantly from the one which is shown in every individual, or even in every scientist, when confronted with an unexplained phenomenon: “It is the core of the present hypothesis that the explanations (i.e. the delusions) of the patient are derived by cognitive activity that is essentially indistinguishable from that employed by non-patients, by scientists and by people generally.” (Maher 1974, 103). By normal reasoning, Maher means here a basically logical reasoning, but which occasionally includes some errors of reasoning of a common type. It is worth mentioning here that Maher’s model has led to several objections. Above all, this model was criticised on the grounds that it did not allow to account for the fact that delusions can also develop in seemingly normal conditions (Chapman & Chapman 1988). Secondly, it was objected to Maher’s model (Davies & Coltheart 2000, p. 8) that it did not explain how delusional beliefs are adopted and maintained in spite of their implausible nature. It is worth mentioning lastly that an important philosophical problem emerges within Maher’s model. It is what Pascal Engel termed the “paradox of delusions” (Engel 2001), and that can be formulated as follows: if the delusional construction is underlain by essentially normal reasoning and therefore by rationality, how it is possible to explain that the delusions’ conclusions are so manifestly wrong and contrary to evidence. The present analysis takes place in such context, and aims thus at proposing a solution to this paradox of delusions.
While Maher’s conception mentions abnormal perception as a unique factor at the origin of delusional ideas, another approach, notably put forth by Martin Davies and Max Coltheart (2000, 2001), describe two factors in the genesis and the maintenance of delusions. The first factor resides, as well as for Maher, in abnormal perception. And the second factor resides, according to Davies and Coltheart, in the patient’s disability to reject a hypothesis given its incoherent and implausible nature with regard to the patient’s rest of knowledge. Davies and Coltheart criticises thus Maher’s model by notably pointing out that it does not allow to provide an account of the maintenance of delusions, even though their conclusion turns out to be completely implausible.
It is worth also mentioning a third type of cognitive model, which stresses that several cognitive biases (Garety & al. 2001) can be observed in the thinking mode of patients suffering from schizophrenia. Among these biases is the patient’s tendency to jump-to-conclusions (Garety & Hemsley 1994). Experimental studies (Fear & Healy 1997; Garety & al. 1991) showed thus that patients had a more marked tendency than others to conclude very fast, starting from a limited group of information at their disposal. A second type of cognitive bias is an attribution or externalist bias which consists for the patient to attribute an external motive to events of a negative type which he/she undergoes. The patient favours then arbitrarily an external reason with regard to an internal and personal motive, when he/she sets out to determine the origin of an event which relates to him/her. Such conclusion notably results from the work of Bentall & Kaney (1989), and Kaney & al. (1989), who noticed that patients suffering from delusions of persecution were more prone than normal individuals to attribute both an external cause to negative events which they undergo and an internal motive to positive events which concerned them. This type of cognitive bias is also related to an attentional bias, which was noticed (Bentall & al. 1989; Kaney & al. 1989) in patients suffering from schizophrenia of paranoid subtype, who demonstrate as well a more marked tendency than others to turn their attention to menacing elements, among a group of stimuli, especially if the latter are related with themselves. Lastly, Aaron Beck (2002) also underlined how the reasoning of patients suffering from schizophrenia revealed an egocentric bias. This type of cognitive bias leads the patient to relate most external events with the elements of his/her personal life. Noise, sounds, smells, and generally facts and external phenomena, are thus bearing for the patient a hidden or explicit meaning, and which concerns him/her directly.
Finally, it is worth mentioning a cognitive model which sets out to define accurately the nature of delusions, by especially emphasising the fact that the latter do not constitute beliefs in the usual sense or, possibly, constitute beliefs of a special type. Such model made the subject of recent developments (Sass,1994; Young, 1999; Engel, 2001; Stephens & Graham, 2006) tending to question the classical definition of delusions, based on abnormal belief.
# 2. Apophenia
I will now endeavour to describe the present model and to expose accurately the mechanism which leads to the formation and maintenance of delusional ideas. In Maher’s model, delusions constitute a cognitive development elaborated by the patient in response to abnormal perception. The present model inserts itself within such conception: polythematic delusions constitute then conclusions of fallacious arguments worked out by the patient, in response to a particular type of abnormal perception: apophenia. Finally, although the reasoning which leads to delusions proves to be fallacious, it can however be considered as normal, because it includes errors of reasoning which turn out to be very common.
Before describing the structure of reasoning which leads to polythematic delusions, it is worth considering preliminarily the apophenia. One finds then mention, especially in the observations made by some patients in remission relating their psychotic experience (Stanton & David 2000), of a very specific feeling, which can be characterised as a feeling of interconnection with the ambient world. Such feeling is not felt in normal conditions and truly presents a bizarre nature. Schneider had already (1930) pointed out how in an individual suffering from schizophrenia, a meaningful interconnection was created between facts that are internal to the patient and external events (“Meaningful connections are created between temporary coincident external impressions … or perceptions with thoughts that happen to be present, or events and recollections happening to occur in consciousness at the same time”). Isabel Clarke (2000) also mentions in the patient a very particular feeling of fusion and of universal interconnection (“an exhilarating feeling of unity and interconnectedness”). Brundage (1983) also evokes a phenomenon of the same nature which manifests itself by a particular feeling of connection with all events that occur in the surroundings (“there is a connection to everything that happens”) as well as the feeling that the slightest things have a meaning (“every single thing means something”). It appears then that the patient experiences a strong feeling of interconnection between on the one hand, his/her internal phenomena and on the other hand, the external phenomena.
The role of such feeling of interconnection was recently underlined notably by Peter Brugger (2001). Brugger terms then apophenia the tendency to see connections between objects or ideas which are a priori without any relationship (“the propensity to see connections between seemingly unrelated objects or ideas”) and attributes the creation of this term to K. Conrad (1958). In the present context, one can consider a slightly more restrictive definition of apophenia, for it suffices here to characterise apophenia as the feeling in the patient that there is a narrow relationship between his/her internal phenomena (thought, feelings, emotions, acts) and external phenomena.
One can notice here that Maher does not mention explicitly apophenia when he enumerates abnormal perceptions which are susceptible of being experienced by the patient. However, he mentions a phenomenon which appears as closely related to apophenia. Among the abnormal perceptions undergone by the patient, Maher mentions indeed (Maher 1999) that it can consist, for example, of the fact that the patient perceives as salient some normally insignificant sensory data, of unrecognised defects in the sensory system of the patient such as a loss of audition, of temporary impairments in the intensity of perception, of hallucinations, of difficulties of concentration of neurological origin, etc. Maher includes then among abnormal perceptions the fact that the patient perceives as salient some ordinarily insignificant sensory data, what can be considered as closely related to apophenia.
At this step, it is worth describing more accurately the cognitive mechanism which, in relation with apophenia, leads to the formation of polythematic delusions. This will allow to cast more light on the role and the nature of apophenia itself.
# 3. Formation and maintenance of delusional ideas
In the present context, the reasoning that leads to delusional ideas is considered as a specific case of fallacious argument, i.e. as reasoning the conclusion of which is not logically justified by its premises, which are worked out in response to a particular type of abnormal perception: apophenia. In general, this type of reasoning leads to an erroneous conclusion. But it can happen very exceptionally that the resulting conclusion turns out to be true (for example if a patient suffering from schizophrenia with delusion of persecution was mistakenly spied on notably because he/she had been confused with a high diplomat). Another feature of the type of fallacious reasoning which leads to polythematic delusions is that it includes errors of reasoning of a normal type, i.e. very common. Finally, it is worth pointing out that in spite of their patently false conclusion, the task which consists in diagnosing accurately the fallacious steps in the reasoning which leads to delusional ideas proves to be far from easy.
The fallacious reasoning which leads to polythematic delusions presents a particular structure, as we will see it, within which it is worth distinguishing several functional steps, which take place successively within the elaboration of delusions ideas: primary, secondary, tertiary and quaternary steps. The primary step, first, is of an instantial nature, in the sense that it is based on some instances. The secondary step presents an inductive nature, which proceeds by generalisation of the conclusions resulting from each of the preceding instances. And the tertiary step is of an interpretative nature at a monothematic level. Finally, the quaternary step has an interpretative function, but this time at a polythematic level. The distinction of these four successive steps is of interest in the understanding of the mechanism which leads to the formation of delusional ideas, for it allows to describe its intrinsic structure, at the level of both its formation and maintenance. On the other hand, as we shall see it later, cognitive therapy of psychosis can apply differently to each of these specific steps.
In what follows, we shall especially be concerned with delusional ideas of reference, of telepathy, of influence and of grandeur, which correspond to polythematic delusions commonly met in schizophrenia. It is also worth mentioning that the corresponding model can be easily extended to other types of delusional ideas, especially to thought-broadcasting delusions or delusions of persecution. At this stage, it worth drawing a distinction between the mechanism which leads to the formation of delusional ideas, and the one which concurs to their maintenance.
3.1 Formation of delusional ideas
Classically, one distinguishes in schizophrenia the following types of delusions: delusion of reference, delusion of influence, delusion of control, delusion of telepathy, delusion of grandeur, delusion of persecution. The present model will set out first to describe the mechanism which leads to the formation of these main types of delusions, by setting out to introduce a reconstruction of the specific cognitive process in a patient at the beginning of psychosis.
Let us begin with delusions of reference. Let us consider the following argument, which leads the patient to conclude that television speaks about him/her, and therefore to delusional ideas of reference (T1 and T2 denote here two successive temporal positions, with a very short time interval between T1 and T2; the symbol denotes the conclusion; and R is taken for reference):
(R1) in T1 I was drinking an aperitif
(R2) in T2 the presenter of the show said: “Stop drinking!”
(R3) in T2 the presenter said: “Stop drinking!” because in T1 I was drinking an aperitif
(R4) in T3 I was upset and anxious
(R5) in T4 the presenter of the show said “Stop stressing”
(R6) in T4 the presenter of the show said “Stop stressing!” because in T3 I was upset and anxious
(R7) in T5 I was smoking a cigarette
(R8) in T6 I heard the presenter saying “That is not good !“
(R9) in T6 the presenter said “That is not good !” because in T5 I was smoking a cigarette
(R10) in T7 I felt fine and lucid and I was relaxed
(R11) in T8 the presenter of the show said: “We are in great form!
(R12) in T8 the presenter said “We are in great form!” because in T7 I felt fine and lucid and I was relaxed
(R…) …
(R13) the presenters of the shows speak according to what I do or what I feel
One can distinguish within the structure of this reasoning several parts the function of which turns out to be specific. These different parts correspond respectively to primary delusional arguments (it consists of the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12)), to secondary delusional arguments (the steps (R3), (R6), (R9), (R12) and (R13)) and tertiary delusional arguments (the steps (R13) and (R14)). It is worth considering in turn each of these arguments. Let us begin with primary delusional arguments, that correspond to an instantial step, in the sense that it is made up of several different instances. Primary delusional arguments are constituted here by four different instances, i.e. the steps (R1)-(R3), (R4)-(R6), (R7)-(R9) and (R10)-(R12). These four primary delusional arguments lead the patient to conclude that at a given time, the TV presenters spoke according to his/her acts or to what he/she felt.
Let us consider now the following stage (R13), which constitutes the conclusion of a secondary delusional argument, and is of a different nature. Its premises are the conclusion (R3), (R6), (R9), (R12) of the four previous instances of primary delusional arguments of reference. The patient generalises from the latter to the conclusion that the TV presenters speak according to what he/she is doing or to what he/she is feeling. The overall structure of this type of secondary delusional argument is then as follows:
(R3) in T2 the presenter of the show spoke according to what I was doing
(R6) in T4 the presenter of the show spoke according to what I was feeling
(R9) in T6 the presenter of the show spoke according to what I was doing
(R12) in T8 the presenter of the show spoke according to what I felt
(R…) …
(R13) the presenters of the shows speak according to what I do or feel
On can then term inductive this type of secondary delusional argument because it has the form of an enumerative induction, where the patient constructs his/her conclusion by generalising, in an inductive way, from the conclusions of several instances of primary delusional arguments. Thus, secondary delusional arguments correspond to a step the nature of which proves to be inductive.
At this stage, it is also worth mentioning the third step, which leads to delusion of reference. It consists of the tertiary delusional argument of reference, constituted by steps (R13) and (R14), the premise (R13) of which being the conclusion of the secondary delusional argument of reference:
(R13) the presenters of the shows speak according to what I do or feel
In such argument, the patient takes into account the conclusion of the inductive step that the presenters of the shows speak according to his/her acts or to his/her internal state, and interprets it by concluding that television speaks about him/her. It consists, as we did see it, of a step the function of which is merely interpretative, in the sense that it aims at making sense of the inductive conclusion which results from the secondary delusional argument. Tertiary delusional arguments are therefore the product of an interpretative step, which situates itself at a monothematic level (here, the specific topic is that of delusion of reference).
A structurally identical mechanism leads to delusional ideas of telepathy. Several instances of primary delusional arguments of influence lead first the patient to conclude that his/her own thoughts are at the origin of acts that are accomplished by other persons. By an inductive step, the patient is then led to the conclusion that people act according to his/her thoughts. Finally, in an interpretative step, the patient concludes that other people can read his/her thoughts (or that they can hear them). It consists there, in the patient’s mind, of an attempt at explaining the very disturbing conclusion which results from the inductive step according to which other persons act according to his/her thoughts.
The same mechanism also engenders the formation of delusional ideas of influence. In that case, several instances of primary delusional arguments of influence lead the patient to conclude that his/her own thoughts are at the origin of annoyances caused to other persons. An inductive step leads then the patient to the conclusion that people react negatively in function of his/her thoughts. Finally, an interpretative step leads the patient to conclude that he/she perturbs and disturbs other people.
Moreover, such mechanism leads to the formation of delusional ideas of control. They find their origin in the instances of primary delusional arguments of control. Such instances have the same structure as that of the instances of primary delusional arguments of reference, of telepathy or of influence, with however this difference that the temporal order of both types of events – internal and external, with regard to the patient – is now reversed. Within the primary delusional arguments of reference, of telepathy or of influence, an internal event with regard to the patient precedes an external event, whereas it is the opposite with regard to a primary delusional argument of control: the external event precedes then the internal event. Thus, several instances of primary delusional arguments of control lead the patient to conclude inductively that some external events have an effect on his/her thoughts, his/her emotions or his/her acts. The interpretative step leads then the patient to think that he/she is controlled by external beings or objects such as robots or a satellite.
Finally, it is worth specifying the role played by quaternary delusional arguments. The premises of the latter arguments are conclusions of tertiary delusional arguments. Quaternary delusional arguments are more general arguments, which present, as well as tertiary delusional arguments, an interpretative nature. But unlike tertiary delusional arguments which turn out to be interpretative at a monothematic level, quaternary delusional arguments are interpretative at a polythematic level. They indeed take into account jointly the conclusions of tertiary delusional arguments of reference, of telepathy, of influence, etc. by striving to make sense, globally, of them and to interpret them. The reasoning below constitutes then a quaternary delusional argument leading to ideas of grandeur:
(R15) television and the media speak about me
(T16) people can read my thoughts
(I17) I influence other people’s behaviour
(18) I am someone exceptional
(19) I am an extraterrestrial
At a quaternary level, the patient takes then into account the different conclusions resulting from tertiary delusional arguments, the function of which is interpretative at the level of a given delusional topic and attempts this time to interpret the set constituted by the latter. The resulting conclusion constitutes veritably, for the patient, an overall theory the function of which is to make sense and to explain all the abnormal phenomena which he/she experiences.
3.2 Maintenance of delusional ideas
It is worth considering now the mechanism which leads to the maintenance of delusional ideas. Let us place ourselves at the level of secondary delusional arguments which, at the level of the formation of delusional ideas, are of an inductive nature. Consider then especially the form that take secondary delusional arguments of reference, at the stage of the maintenance. At this step, the conclusion (R13) which results from secondary delusional arguments, in virtue of which the presenters of the shows speak according to what the patient makes or feels, was already established at the stage of the formation of delusional ideas. And the corresponding reasoning takes then into account a new instance of primary delusional argument (R20) of reference, in the following way:
(R20) in T100 the presenter of the show spoke according to what I was doing
(R21) this confirms that television speaks according to what I do
On can notice here that the inductive generalisation (R13) was already established at the stage of the formation of the secondary delusional argument, and that the new instance of primary delusional argument constitutes then, in the patient’s mind, a case of confirmation of the latter generalisation. As we can see, the role of the new instance of primary delusional argument is to confirm and therefore to reinforce, at the stage of the maintenance, a generalisation which was already established at the previous stage of the formation of delusional ideas.
# 4. Analysis of delusional arguments
At this stage, it is worth analysing in detail the structure of the type of reasoning which has been just described, in order to identify accurately the fallacious steps and to determine the role played by apophenia. Let us consider in turn primary, secondary, tertiary and quaternary delusional arguments. Let us scrutinise first the following instance of primary delusional argument of telepathy, which manifests itself at the level of the formation of delusions:
(T1) in T1 I thought of Michael “What an idiot!”
(T2) in T2 I heard Michael shout
(T3) in T2 I heard Michael shout because in T1 I thought of him “What an idiot!”
It appears here that the two premises (T1) and (T2) constitute genuine facts and therefore turn out to be true. Conversely, the conclusion (T3) that concludes to the existence of a relation of causality between the two consecutive facts F1 (in T1 I thought of Michael “What an idiot!”) and F2 (in T2 I heard Michael shout) is it justified? It appears not. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts F1 and F2. And the conclusion (T3) which deducts from it a causality relationship turns out therefore to be too strong. The corresponding reasoning presents then a fallacious nature. The corresponding error of reasoning, which concludes to a relation of causality while there is only a simple relation of anteriority, is traditionally called post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (thereafter, hence because of it). This is a very common type of fallacious reasoning, which is notably at the root of many superstitions (Martin 1998; Bressan 2002). David Hemsley (1992) notably mentions such type of reasoning in clinical observation: “A patient of the present author, recalling his psychotic experiences noted that the co-occurrence of two events often led immediately to an assumption of a causal relationship between them”. Finally, one can notice that in the context of cognitive distortions, the type of error of reasoning corresponding to post hoc fallacy can be considered a specific case of arbitrary inference.
Let us also proceed to analyse the type of reasoning which underlies secondary delusional arguments, and that presents at the stage of the formation of delusional ideas, as we did see it, the following inductive structure:
(T22) in T2 Michael spoke according to my thoughts
(T23) in T4 the neighbour spoke according to my thoughts
(T24) in T6 the radio presenter spoke according to my thoughts
(…)
(T25) people act according to my thoughts
Such type of reasoning appears prima facie completely correct. It consists here of a reasoning based on an inductive generalisation, in which the patient simply builds a more general conclusion from several instances. Such reasoning is completely correct, for its conclusion can be considered as true, inasmuch as its premises are true. However, a scrutiny reveals that the patient only takes into account here a limited number of instances, i.e. those instances that are based on the concordance of both premises, at the stage of primary delusional arguments. The patient then directs his/her attention exclusively to those instances that include two premises of which the internal event (premise 1) and the external event (premise 2) turn out to be concordant and render thus plausible a relation of causality. The corresponding turn of mind can be described as a concordance bias. In effect, the patient does not take into account at this stage those instances which could possibly be composed of two discordant premises. The latter are likely to come under two different forms. An instance of the first form is as follows:
(T1) in T1 I thought of Michael “What an idiot!”
(26) in T2 Michel was quiet
And an instance of the second form is :
(27) in T1 I didn’t think of Michael
(T2) in T2 I heard Michael shout
In these two types of cases, one can notice a discordance between the two premises, which goes directly contrary to the idea of causality between the two events. As we see it finally, the flaw in the patient’s reasoning resides essentially in the fact of only taking into account those instances where the concordance between an internal event and an external event renders plausible a causality relationship. But if the patient had taken into account at the same time the concordant and the discordant instances, he/she would have been led to conclude that the concordant instances represent only a small part of the set constituted by the class of relevant instances, and are only therefore the result of a random process. In such context, as we see it, the concordant instances in fact constitute but mere coincidences.
If one places oneself now at the stage of the maintenance of delusional ideas, one can observe the presence of a mechanism of the same nature. At the stage of the emergence of delusional ideas, secondary delusional arguments present, as we did see it, an inductive nature. On the other hand, at the stage of the maintenance of delusional ideas, the latter come under the form of arguments which lead to the confirmation of an inductive generalisation. Consider then the following instance, where the conclusion (T25) according to which people act according to the patient’s thoughts results from a secondary delusional argument and was already established at the stage of the formation of delusional ideas:
(T28) in T100 my sister spoke according to my thoughts
(T29) this confirms that people act according to my thoughts
This type of argument appears completely valid, for the conclusion results directly from its premises. However, the latter argument is also at fault by default, for it does not take into account some premises, which turn out to be as much as relevant as the instance (T28). As we can see, the error of reasoning consists then in taking only into account those instances which confirm the generalisation (T25), while ignoring those instances which disconfirm the latter. Hence, this type of argument reveals a confirmation bias, i.e. a tendency to favour those instances which confirm a generalisation, whereas it would be necessary to take into consideration at the same time those which confirm and those which disconfirm it. One can notice however that such type of cognitive bias presents a very common nature (Nickerson 1998, Jonas et al. 2001).
It is worth considering, third, tertiary delusional arguments. Consider then the following tertiary delusional argument of telepathy (a similar analysis also applies to tertiary delusional arguments of reference and of influence):
(T30) people act in function of my thoughts
(T31) people can read my thoughts (people can hear my thoughts)
One can notice here that if premise (T30) is true, then the conclusion (T31) constitutes a credible explanation. This type of argument presents then an interpretative nature and the conclusion (T31) according to which people can read the patient’s thoughts appears finally plausible, inasmuch as it is considered as true that people act according to his/her thoughts. As we can see, such argument is motivated by the patient’s concern of explaining and of interpreting the disturbing generalisation which results from the repetition of the many concordant above-mentioned instances.
Finally, the following quaternary delusional argument aims, in the same way, at making sense of the conclusions which result from the conjunction of conclusions of different tertiary delusional argument:
(R15) television and the media speak about me
(T16) people can read my thoughts
(I17) I influence people’s behaviour
(18) I am someone exceptional
(19) I am an extraterrestrial
As we can see it, the conclusion (18) results here directly from the three premises (R15), (T16) and (I17) and the corresponding reasoning which leads the patient to conclude that he/she is someone exceptional can also be considered as valid. On the other hand, the conclusion (19) appears here too strong with regard to premise (18).
Given what precedes, it appears that a number of steps in the reasoning which leads to delusional ideas in schizophrenia are characterised by a reasoning which appears mainly normal. By normal reasoning, one intends here a broadly logical and rational reasoning, but also including some errors of logic of a very common type. Such viewpoint corresponds to the one put forth by Maher (1988; 1999) who considers, as we did see it, that the delusional construction in schizophrenia is nothing else than normal reasoning worked out by the patient to try to explain the abnormal phenomena which he/she experiences.
However, one can notice that in the above-mentioned structure of reasoning, one part of the reasoning cannot a priori be truly considered as normal. It consists here of the different instances of primary delusional arguments. The latter are based, as we did see it, on errors of reasoning corresponding to post hoc fallacies. This type of error of reasoning arguably turns out to be extremely common. However, the instances of primary delusional arguments mentioned above present an unusual nature, in the sense that they put in relationship the patient’s thoughts (or his/her emotions, feelings or actions) with external phenomena. Prima facie, such type of reasoning cannot be considered as normal. For why is the patient led to put his/her thoughts in relationship with external phenomena? One can formulate the question more generally as follows: why does the patient put in relationship the phenomena of his/her internal and personal life (his/her thoughts, emotions, feelings, etc.) with mere external phenomena? This distinguishes itself indeed significantly from the behaviour of a normal person, for whom it exists a very clear-cut intuitive separation between on one hand, his/her own internal world, and on the other hand, the external phenomena.
The answer to the previous question can be found here in the role of apophenia. Due to apophenia, the feeling indeed imposes itself to the patient that his/her internal world is closely linked up with the external world. So, his/her thoughts, emotions, feelings and acts appear to him/her to be closely linked up with the external phenomena that he/she perceives, such as ambient noise and dialogues, the words of the presenters of television or of radio, the dialogues of the characters of comic strips, the movements of the wings of a butterfly or of a bird, the natural phenomena such as the wind or the rain, etc. In the context which results from apophenia, the repeated instances that constitute primary delusional arguments can then take place naturally. For since the patient lives with a permanent feeling of interconnection between events that relate to him/her specifically and those which occur in the world which surrounds him, he/she is then led to observe many concordances between events related to him/her and external facts. In such context, primary delusional arguments can then take place naturally.
In the present context, the role of apophenia can be considered as fundamental. And this leads to suggest that considering its specificity and considering the leading role that it plays in the development of primary delusional arguments and therefore of all the characteristic delusional ideas of schizophrenia, apophenia could be counted among the criteria of the illnessii.
As we can see it, the process which gives rise to delusional arguments from the phenomenological experience constituted by apophenia proves finally to be in line with Maher’s account. And one finds here a clear explanation of delusions as the patient’s response to the abnormal phenomena which he/she experiences, among which one can then mention apophenia, as well as hallucinations.
Given what precedes, polythematic delusions can be defined as conclusions of arguments triggered by apophenia and that include some very common errors of reasoning such as post hoc fallacy and confirmation bias. Hence, apophenia and a normal reasoning including the type of aforementioned errors of reasoning turn out to be necessary and sufficient conditions for the development of polythematic delusions. This double condition notably explains why we are not all delusional. For if errors in reasoning based on post hoc fallacy and confirmation bias turn out indeed to be very common, they only trigger primary delusional arguments when they are associated with the abnormal perception which consists in apophenia. It is worth pointing out, moreover, that such model leaves also room for more stronger conditions. For if apophenia constitutes one of the two sufficient conditions for the development of polythematic delusions, the latter can also take place in conditions where abnormal perception is constituted not only by apophenia, but also by other abnormal perceptions such as hallucinations. And also, whereas the second condition which is sufficient for delusions identifies itself with normal reasoning including post hoc fallacy and confirmation bias, it proves that the development of delusions can also be made by means of a reasoning which deviates more or less from normal reasoning. But the essential characteristic of the present model resides in the fact that apophenia and normal reasoning including the aforementioned very common errors, constitute necessary and sufficient conditions for the development of polythematic delusions.
5. The role of the hallucinations
At this stage, it is worth highlighting the role played by hallucinations, the other major symptom of schizophrenia, in the process which has just been described. I will set out to describe in more detail here the role played by auditory hallucinations – given that the corresponding analysis can be easily extended to hallucinations relating to other sensory modalities, i.e. visual, tactile, olfactory and gustatory.
Auditory hallucinations are susceptible, first, of playing a role at the level of primary delusional arguments. In this type of case, the primary delusional argument presents the same structure as the one described above, with the only difference that an auditory hallucination – in place of a real external event – constitutes then the second premise of the primary delusional argument. The following instance constitutes then an example of primary delusional argument of reference, but it is there an auditory hallucination, by which the patient hears the voice of the presenter of the show saying “Clumsy!” while he/she watches TV, that constitutes the support of the second premise of the argument:
(32) in T1 I dropped my pen
(33) in T2 I heard the voice of the presenter of the show saying “Clumsy!”
(34) in T2 the presenter of the show said “Clumsy!” because in T1 I dropped my pen
In the same way, the following instance constitutes a case of primary delusional argument of telepathy. In that case, it is an auditory hallucination, by which the patient hears the voice of his neighbour saying “Calm down!”, that serves as a basis for the second premise of the argument:
(35) in T1 I was very upset
(36) in T2 I heard the voice of my neighbour saying “Calm down!”
(37) in T2 my neighbour said “Calm down!” because in T1 I was very upset
It is worth mentioning, second, the role that can be played by auditory hallucinations at the level of secondary delusional arguments. In such case, the corresponding generalisations develop from instances of primary delusional arguments which also include auditory hallucinations. In the example below, the patient generalises from the conclusions of three instances of primary delusional arguments of reference. But while the two latter instances (39) and (40) are based on real external phenomena, the first instance (38) is founded on hallucinated content, by which the patient heard the TV presenter saying “Clumsy!”:
(38) in T2 the TV presenter said “Clumsy!” because in T1 I dropped my pen
(39) in T4 the presenteress said “Calm down!” because in T3 I was upset
(40) in T6 the presenter of the show said “Thank you” because in T5 I thought “I love this presenter”
(…) …
(41) the TV presenters speak according to what I do or feel
As we see it, auditory hallucinations contribute in this way to increase the number of primary delusional arguments, by creating thus additional instances which add up themselves to the different types of standard instances previously defined. This gives then more weight to the inductive generalisations made by the patient from multiple instances of primary delusional arguments. Besides, it has also the effect of reinforcing the coherence of the patient’s delusional system and of rendering it then more resistant to contrary argumentation.
It is worth mentioning, lastly, another type of role which can be played by auditory hallucinations. Such is notably the case when the content of the hallucinations proves to be consistent with the conclusions that result from secondary, tertiary or quaternary arguments. Auditory hallucinations have then the effect of reinforcing the latter conclusions. The instance below constitutes a case where an auditory hallucination comes to reinforce the conclusion of a tertiary delusional argument of telepathy. In that case, the hallucinated content resides in the fact that the patient hears the voice of his friend Joseph saying “I know the slightest of your thoughts”:
(42) in T50 I thought that people know of my thoughts
(43) in T100 I heard Joseph saying: “I know the slightest of your thoughts”
(44) this confirms that people know of my thoughts
In a similar way, the following instance has the effect of reinforcing the conclusion which results from a quaternary delusional argument, where the hallucinated content consist of a voice heard by the patient that says: “You come from the planet Mars”:
(19) in T50 I thought I was an extraterrestrial
(45) in T100 I heard a voice saying : “You come from the planet Mars”
In a general way, we see here how hallucinations constitute an element which has the effect of reinforcing considerably the conclusions resulting from delusional arguments. The hallucinations have then the effect of reinforcing the strength and the consistency of the beliefs’ system of the patient, thus contributing to its maintenance, and rendering then his/her ideas more resistant to contrary argumentation.
# 6. Comparison with other cognitive models of delusions
The present model, as we can see it, mainly emphasises a cognitive approach of delusions encountered in schizophrenia. This model introduces a fundamental cognitive element, but also leaves room to a neurophysiological element (at the origin of apophenia), the role of which proves to be essential. One can notice finally that the model which has just been described turns out to be compatible with some other accounts of delusional ideas met in schizophrenia.
The present analysis, to begin with, is susceptible of inserting itself as part of the adaptation of Albert Ellis’ ABC-analysis described by Chadwick et al. (1996). In this context, the internal and external events with regard to the patient, that are the premises of primary delusional arguments, constitute the As. The primary, secondary, tertiary and quaternary delusional arguments, can also be assimilated to the Bs. Lastly, the negative emotions (anger, anxiety, frustration, etc.) felt by the patient, that result there from the conclusions of delusional tertiary and quaternary arguments, constitute the Cs. As we can see it, the present analysis leads, in comparison with the standard ABC-analysis, to distinguish several steps at the level of the Bs. This distinction is important, since it allows to distinguish several steps whose function is different, within the reasoning that leads to delusional ideas. Thus, the B1s (primary delusional arguments) are instances that lead to the attribution of a causality relationship between internal and external (to the patient) phenomena; the B2s (secondary delusional arguments) result from a generalisation of inductive nature; the B3s (tertiary delusional arguments) correspond to an interpretative step at a monothematic level; finally, the B4s (quaternary delusional arguments) are characteristic of a step of interpretation at a polythematic level, the conclusion of which truly constitutes a global explicative theory of the abnormal phenomena undergone by the patient. On the other hand, we are led there to distinguish between those parts of the patient’s reasoning which are globally valid (the B2s, B3s and B4s) and the part which is invalid (the B1s, based on post hoc fallacy). Such nuanced point of view should be likely to preserve – what constitutes one of the key points of cognitive and behaviour therapy – the therapeutic alliance, i.e., the relation of collaboration between the patient and the therapist aiming at shared objectives in the struggle against the illness. As we can see it, the present analysis leads to especially emphasise post hoc fallacy, which constitutes the weakness in the patient’s reasoning, but the repeated instances of which, triggered by apophenia, truly constitute the building block of the delusional construction.
The present model also has number of affinity with the approaches which are at the root of cognitive therapy of schizophrenia (Kingdon & Turkington 1994; Kingdon & Turkington 2005; Chadwick & al. 1996; Beck & Rector 2000). In this type of approach, the therapist sets out to reduce progressively the patient’s degree of belief in his/her delusional polythematic ideas. To this end, the therapist suggests the patient, in a spirit of dialogue of Socratic inspiration, to elaborate alternative hypotheses; he also teaches the patient the approach which consists in searching elements likely to confirm or to disconfirm his/her own hypotheses, as well as to build out alternative hypotheses. The contribution of the present analysis with regard to cognitive therapy of schizophrenia is likely to manifest itself in several ways. It proves to be useful to specify then, for the clinician, what could be such contribution, and to also provide a specific framework in which the present model will possibly be tested. The distinction of different steps in the development of delusions allows first to distinguish between different hypotheses corresponding to the conclusions of primary, secondary, ternary and quaternary arguments. The degree of belief associated with each of these levels of hypotheses is also susceptible of being evaluated separately, by notably allowing to determinate then at which level resides the strongest degree of conviction. In the same way, each of the conclusions of the primary, secondary, tertiary or quaternary arguments, will possibly be tested (confirmed/infirmed) and give room for the elaboration of alternative hypotheses (David Kingdon, personal communication). For example, at the level of primary delusional arguments, it will be possible to consider the belief according to which the presenter said in T2: “You should not drink!” because the patient was drinking an aperitif in T1; this hypothesis will possibly give rise to a search for evidence, and then confronted with an alternative hypothesis such as: the presenter said in T2: “You should not drink!” because it was scheduled in the script of the television program. At the level of tertiary delusional arguments, the hypothesis according to which television speaks about the patient will also possibly be the object of a search for evidence, etc.
The interest for the clinician of the present approach resides, second, in the fact that it provides the patient with an alternative global explanation of the abnormal phenomena that he/she undergoes. The delusional construction of the patient constitutes, as we did see it, a theory which allows him/her to explain all the abnormal phenomena that he/she experiences. On can assume in this respect, that the fact for the patient to get a satisfactory theory explaining all abnormal phenomena which he/she experiences, is also likely to play an important role in the maintenance of his/her delusional system. In this context, the present analysis allows to propose to the patient an alternative explicative theory, grounded on apophenia and the different steps of reasoning which result from it. Such theory distinguishes itself from the explicative theory with which the patient is usually confronted (according to the common elliptical point of view, the latter is “mad”) and proves to be less stigmatising, since the reasoning which leads to delusions is notably considered here as normal. For this reason, one can assume that the patient could be more willing to adhere to the present alternative theory, as a global explanation of the abnormal phenomena which he/she experiences.
As we did see it, the present model conforms mainly with the one developed by Brendan Maher (1974; 1988; 1999), based on the fact that delusions result from a broadly normal interpretation of the abnormal phenomena undergone by the patient. The present analysis also specifies with regard to Maher’s model that apophenia (eventually associated with hallucinations) constitutes an abnormal perception which is enough for giving rise to delusional polythematic ideas met in schizophrenia. It has then been objected, as we did see it, to Maher’s model, that it did not allow to account for the fact that delusions can also take place in seemingly normal conditions, especially in a patient not suffering from hallucinations. But the present model points out that such conditions are not normal, since apophenia is present in such a patient. Since apophenia leads to abnormal perceptions, the essential factor described by Maher at the origin of delusional ideas, is therefore present as well. On the second hand, the present model also provides some elements of response with regard to the second objection, formulated against Maher’s model, by Davies and Coltheart (2000), according to which it does not allow to describe how delusional beliefs are adopted and maintained in spite of their implausible nature. The present model, however, sets out first to describe step-by-step the type of reasoning which leads to the adoption of polythematic delusions. By its structure, this type of reasoning appears mainly normal. It proceeds by enumeration of some instances, and then by generalisation and lastly, by interpretation. The present model also provides, as far as I can see, an answer to the criticism raised by Davies and Coltheart with regard to Maher’s model, who blame the latter for not describing how delusional beliefs are maintained in the patient’s belief system, in spite of their implausible nature. In the present model, as we did see it, it is the fact that new instances are generated every day which explains that beliefs are maintained. For when delusional beliefs are established in the patient’s belief system at the end of the stage of their formation, they are then maintained because apophenia continues to trigger every dayiii new instances of primary delusional argument. The latter come, in the patient’s mind, to confirm the conclusions of delusional arguments at a secondary, tertiary and quaternary level, already established at the stage of the formation of delusional ideas. From this point of view, there is no essential difference in the present model in the way that the formation and the maintenance delusional ideas take place. For as we did see it, the building block of the delusional construction is constituted there by the instances of primary delusional arguments, triggered by apophenia. And these instances which concur to the formation of delusional ideas, also ensure their maintenance every day, by confirming the conclusions of secondary, tertiary and quaternary arguments, which are already established at the stage of the formation of delusional ideasiv.
Finally, the model which has just been described provides, as far as I can see, in comparison with Maher’s model, an element which proves to be necessary in the context of an explicative model of polythematic delusions. This element consists of an answer to the question of why the content of delusional ideas in schizophrenia identifies itself most often with delusions of reference, of telepathy, of thought insertion, of influence and of control. As it was exposed above, the answer provided by the present model is that a mechanism of the same nature, grounded on post hoc fallacy, leads to the development of these different delusional topics. In primary delusional arguments of reference, of telepathy or of influence, an event which is internal to the patient precedes an external event. And in the case of a primary delusional argument of control, this structure is simply reversed: it is an external event which precedes an event of the patient’s internal life.
We see it finally, the preceding analysis allows to justify and to reinforce Maher’s initial model. In this context, one can notice that one of the consequences of the present model is that the sole apophenia, associated with normal reasoning, turns out to be sufficient to give rise to the emergence and the maintenance of a delusional systemv.
References
Bayne, Tim & Pacherie, Elisabeth, “In Defense of the Doxastic Conception of Delusions”, Mind and Language, vol. 20-2, 2005, p. 163-88. Bayne, Tim & Pacherie, Elisabeth, “Experience, Belief, and the Interpretive Fold”, Philosophy, Psychiatry, & Psychology, vol. 11-1, 2004, p. 81-86 Beck, Aaron & Rector, Neil, “Cognitive Therapy of Schizophrenia: A New Therapy for the New Millenium”, American Journal of Psychotherapy, vol. 54-3, 2000, p. 291-300. Beck, Aaron, “Delusions: A Cognitive perspective”, Journal of Cognitive Psychotherapy, vol. 16-4, 2002, p. 455-468. Bentall, Richard & Kaney, Sue, “Content specific information processing in persecutory delusions: an investigation using the emotional stroop test”, British Journal of medical Psychology, vol. 62, 1989, p. 355-364. Bentall, Richard, Kinderman, P., & Kaney, S., “Self, attributional processes and abnormal beliefs: Towards a model of persecutory delusions”, Behaviour Research and Therapy, vol. 32, 1994, p. 331-341. Bortolotti, Lisa, “Delusions and the background of rationality”, Mind and Language, 20-2, 2005, p.189-208. Bressan, Paola, “The Connection Between Random Sequences, Everyday Coincidences, and Belief in the Paranormal”, Applied Cognitive Psychology, vol. 16, 2002, p. 17-34. Brugger, Peter, “From Haunted Brain to Haunted Science: A Cognitive Neuroscience View of Paranormal and Pseudoscientific Thought”, in Hauntings and Poltergeists: Multidisciplinary Perspectives, Houran, J., & Lange, R. (Eds), North Carolina: McFarland, 2001. Brundage, B.E., “What I wanted to know but was afraid to ask”, Schizophrenia Bulletin, vol. 9, 1983, p. 583-585. Campbell, John, “Rationality, Meaning, and the Analysis of Delusion”, Philosophy, Psychiatry, & Psychology, vol. 8-2/3, 2001, p. 89-100. Chadwick, Paul, Birchwood, M., & Trower, P., Cognitive Therapy for Delusions, Voices, and Paranoia, Chichester: Wiley, 1996. Chapman, L.J., Chapman, J.P., The genesis of delusions, in T.F. Oltmanns, B.A. Maher (Eds), Delusional beliefs, New York: Wiley, 1988, p. 167-183. Clarke, Isabel, « Madness and Mysticism: clarifying the mystery », Network: The Scientific and Medical Network Review, vol. 72, 2000, p. 11-14. Conrad, K., Die beginnende Schizophrenie. Versuch einer Gestaltanalyse des Wahns, Stuttgart: Thieme, 1958. Davies, M. & Coltheart, M., Langdon, R. & Breen, N., “Monothematic delusions: Towards a two-factor account”, Philosophy, Psychiatry and Psychology, vol. 8-2, 2001, p. 133–158. Davies, Martin & Coltheart, Max, “Introduction: Pathologies of Belief”, Mind and Language, vol. 15, 2000, p.1-46. Ellis, Albert, Reason and emotion in psychotherapy, New York: Lyle Stuart, 1962. Engel, Pascal, “Peut-on parler de croyances délirantes ?”, in J. Chemouni, dir. Clinique de l’intentionnalité, In-Press, Paris, 2001, p. 157-173. Faucher, Luc, “Philosophie psychopathologique : un survol”, Philosophiques, vol. 33-1, 2006, 3-18. Fear, Christopher & Healy David, “Probabilistic reasoning in obsessive-compulsive and delusional disorders”, Psychological Medicine, vol. 27, 1997, p. 199-208. Fowler, David, Garety, P. & Kuipers, E., Cognitive behaviour therapy for psychosis: theory and practice, Chichester: Wiley, 1995. Garety, Philippa, Hemsley, D. & Wessely, S., “Reasoning in deluded schizophrenic and paranoid patients”, Journal of Nervous and Mental Disease, vol. 179, 1991, p. 194-201. Garety, Philippa & Hemsley, David, Delusions: Investigations into the Psychology of Delusional Reasoning, Maudsley Monograph, Psychology Press, 1994. Garety, Philippa, Kuipers, E., Fowler, D., Freeman, D. & Bebbington, P., “A cognitive model of the positive symptoms of psychosis”, Psychological Medicine, vol. 31, 2001, p. 189-195. Hemsley, David, “Disorders of perception and cognition in schizophrenia”, Revue européenne de Psychologie Appliquée, vol. 42-2, 1992, p. 105-114. Jonas E., Schulz-Hardt S., Frey D., & Thelen N. “Confirmation bias in sequential information search after preliminary decisions: an expansion of dissonance theoretical research on selective exposure to information”, Journal of Personality and Social Psychology, vol. 80, 2001, p. 557-571. Kaney, Sue & Bentall, Richard, “Persecutory delusions and attributional style”, British Journal of Medical Psychology, vol. 62, 1989, p. 191-198. Kaney, Sue, Wolfenden, M., Dewey, M.E. & Bentall, R.P, “Persecutory delusions and recall of threatening and non-threatening propositions”, British Journal of Clinical Psychology, vol. 31, 1991, p. 85-87. Kingdon, David, & Turkington, Douglas, Cognitive-behavioural Therapy of Schizophrenia, New York: Guilford, 1994. Kingdon, David, & Turkington, Douglas, Cognitive Therapy of Schizophrenia, New York, London: Guilford, 2005. Maher, Brendan, “Delusional thinking and perceptual disorder”, Journal of Individual Psychology, vol. 30, 1974, p. 98-113. Maher, Brendan, “Anomalous experiences and delusional thinking: the logic of explanations“, in Delusional Beliefs, Oltmanns, T. F., & Maher, B. A. (Eds), New York: Wiley, 1988, p. 15-33. Maher, Brendan, “Anomalous experience in everyday life: Its significance for psychopathology”, The Monist, vol. 82, 1999, p. 547-570. Martin, Bruce, “Coincidences: Remarkable or random”. Skeptical Inquirer, vol. 22-5, 1998, p. 23-27. Nickerson, Raymond, “Confirmation Bias: A Ubiquitous Phenomenon in Many Guises”, Review of General Psychology, vol. 2, 1998, p. 175-220. Pacherie E, Green M, Bayne T. “Phenomenology and delusions: who put the ‘alien’ in alien control?”, Consciousness and Cognition, vol. 15, 2006, p. 566–577. Sass, Louis, The paradoxes of delusion: Wittgenstein, Schreber and the schizophrenic mind, Ithaca, NY: Cornell University Press, 1994. Schneider, Carl, Die Psychologie der Schizophrenen, Leipzig: Thieme, 1930. Spitzer, Manfred, “A neurocomputational approach to delusions”, Comprehensive Psychiatry, vol. 36, 1995, p. 83-105. Stanton, Biba & David, Anthony, “First-person accounts of delusions”, Psychiatric Bulletin, vol. 24, 2000, p. 333-336. Stephens, Lynn & George Graham, “Reconcevoir le délire”, Philosophiques, vol. 33-1, 2006, p. 183-196. Stone, Tony & Young, Andrew, “Delusions and brain injury: the philosophy and psychology of belief”, Mind and Language, vol. 12, 1997, p. 327-364. Young, Andrew , “Delusions”, The Monist, vol. 82, 1999, p. 571-589.
i Monothematic delusions are not included into the scope of the present study.
ii One can notice that a neurophysiological explanation on the origin of apophenia is provided by Manfred Spitzer (1995, p. 100). He describes then how the latter is linked to the level of activity of dopamine and of norepinephrine, which have an influence on the value of the signal/noise ratio that is at the root of the activation of neural circuits: “if the signal to noise ratio is too high, (…) small environmental signals (i.e. perceptions to which we would normally pay little or no attention at all) may become amplified to a degree that is much higher than usual. This could result in experiences of “significant events” when merely ordinary events were in fact happening”. Spitzer shows then how apophenia can be the consequence of an imbalance at the dopamine level. By placing normally insignificant events (among which the patient’s thoughts) in the foreground, the modification of the signal/noise ratio allows then the particular feeling of interconnection that constitutes apophenia to occur. Under these conditions, one can notably conceive of how the patient’s thoughts can appear to him/her as prominent so as to be put on the same plan, then put in relationship with external facts such as the words pronounced by a TV presenter.
iii As an anonymous referee for Philosophiques suggests, it would be necessary to quantify precisely the frequency of these instances. This could be made in a separate study.
iv These elements of response with regard to the way the maintenance of polythematic delusions takes place need to be supplemented, especially as regards the way the conclusions of quaternary delusional arguments are put in coherence with the rest of the patient’s beliefs, takes place. Such analysis, which requires subsequent work, is however beyond the scope of the present study.
v I am very grateful to David Kingdon, Albert Ellis, Eugen Fischer, Robert Chapman and two anonymous referees for Philosophiques for very helpful comments on ancestor versions and earlier drafts.
//
Posprint in English (with additional illustrations) of a paper published in French in Semiotica, vol. 139 (1-4), 2002, 211-226, under the title “Une Classe de Concepts”.
This article describes the construction, of philosophical essence, of the class of the matrices of concepts, whose structure and properties present an interest in several fields. The paper emphasises the applications in the field of paradigmatic analysis of the resulting taxonomy and proposes it as an alternative to the semiotic square put forth by Greimas.
This paper is cited in:
## On a Class of Concepts
Classically, in the discussion relating to polar opposites1, one primarily directs his interest to the common and lexicalized concepts, i.e. for which there exists a corresponding word in the vocabulary inherent to a given language. This way of proceeding tends to generate several disadvantages. One of them resides in the fact (i) that such concepts are likely to vary from one language to another, from one culture to another. Another (ii) of the resulting problems is that certain lexicalized concepts reveal a nuance which is either meliorative or pejorative, with degrees in this type of nuances which prove difficult to appreciate. Finally, another problem (iii) lies in the fact that certain concepts, according to semiotic analysis2 are regarded as marked with regard to others concepts which are unmarked, the status of unmarked concept conferring a kind of precedence, of pre-eminence to the concepts in question.
In my view, all the above-mentioned disadvantages arise from the fact that one traditionally works primarily, from the lexicalized concepts. The methodology implemented in the present study is at the opposite of this way of proceeding. Indeed, one will begin here to construct concepts in an abstract way, without consideration of whether these concepts are lexicalized or not. This construction being performed, one will then be able to verify that some of the concepts thus constructed correspond indeed to lexicalized concepts, whereas some others cannot be put in correspondence with any existing word in the common language. This latter methodology allows, I think, to avoid the above-mentioned disadvantages.
It will finally appear that the construction described below will make it possible to propose a taxonomy of concepts which constitutes an alternative to the one based on the semiotic square which has been proposed by Greimas.
# 1. Dualities
Let us consider the class of dualities, which is made up of concepts corresponding to the intuition that these latter:
(i) are different one from the other
(ii) are minimal or irreducible, i.e. can no more reduce themselves to some other more simple semantic elements
(iii) present themselves under the form of pairs of dual concepts or contraries
(iv) are predicates
Each of the concepts composing a given duality will be termed a pole. I shall present here a list, which does not pretend to be exhaustive, and could if necessary, be supplemented. Consider then the following enumeration of dualities3:
Analytic/Synthetic, Animate/Inanimate, Exceptional/Normal, Antecedent/Consequent, Existent/Inexistent, Absolute/Relative, Abstract/Concrete, Accessory/Principal, Active/Passive, Aleatory/Certain, Discrete/Continuous, Deterministic/Indeterministic, Positive/Negative, True/False, Total/Partial, Neutral/Polarized, Static/Dynamic, Unique/Multiple, Container/Containing, Innate/Acquired (Nature/Nurture), Beautiful/Ugly, Good/Ill, Temporal/Atemporal, Extended/Restricted, Precise/Vague, Finite/Infinite, Simple/Composed, Attracted/Repulsed, Equal/Different, Identical/Contrary, Superior/Inferior, Internal/External, Individual/Collective, Quantitative/Qualitative, Implicit/Explicit4, …
At this step, it should be observed that certain poles present a nuance which is either meliorative (beautiful, good, true), or pejorative (ugly, ill, false), or simply neutral (temporal, implicit).
Let us denote by A/Ā a given duality. If words of the common language are used to denote the duality, capital letters will be then used to distinguish the concepts used here from the common concepts. For example: the Abstract/Concrete, True/False dualities.
It should be noted lastly that several questions5 immediately arise with regard to dualities. Do dualities exist (i) in a finite or infinite number? In the same way, does there exist (ii) a logical construction which makes it possible to provide an enumeration of the dualities?
# 2. Canonical poles
The positive canonical poles
Starting from the class of the dualities, we are now in a position to construct the class of the canonical poles. At the origin, the lexicalized concepts corresponding to each pole of a duality reveal a nuance6 which is respectively either meliorative, neutral, or pejorative. The class of the canonical poles corresponds to the intuition that, for each pole of a given duality A/Ā, one can construct 3 concepts: a positive, a neutral and a negative concept. In sum, for a given duality A/Ā, one thus constructs 6 concepts, thus constituting the class of the canonical poles. Intuitively, positive canonical poles respond to the following definition: positive, meliorative form of ; neutral canonical poles correspond to the neutral, i.e. neither meliorative nor pejorative form of ; and negative canonical poles correspond to the negative, pejorative form of . It should be noted that these 6 concepts are exclusively constructed with the help of logical concepts. The only notion which escapes at this step to a logical definition is that of duality or base.
The neutral canonical poles
For a given duality A/Ā, we have thus the following canonical poles: {A+, A0, A, Ā+, Ā0, Ā}, that we can also denote respectively by (A/Ā, 1, 1), (A/Ā, 1, 0) , (A/Ā, 1, -1) , (A/Ā, -1, 1) , (A/Ā, -1, 0) , (A/Ā, -1, -1).
The negative canonical poles
A capital letter for the first letter of a canonical pole will be used, in order to distinguish it from the corresponding lexicalized concept. If one wishes to refer accurately to a canonical pole whereas the usual language lacks such a concept or well appears ambiguous, one can choose a lexicalized concept, to which the exponent corresponding to the chosen neutral or polarized state will be added. To highlight the fact that one refers explicitly to a canonical pole – positive, neutral or negative – the notations A+, A0 et A will be used. We have thus for example the concepts Unite+, Unite0, Unite etc. Where Unite+ = Solid, Undivided, Coherent and Unite = Monolithic. In the same way, Rational0 designates the neutral concept corresponding to the term rational of the common language, which reveals a slightly meliorative nuance. In the same way, Irrationnal0 designates the corresponding neutral state, whereas the common word irrational reveals a pejorative nuance. One will proceed in the same way, when the corresponding lexicalized word proves ambiguous. One distinctive feature of the present construction is that one begins by constructing the concepts logically, and puts them afterwards in adequacy with the concepts of the usual language, insofar as these latter do exist.
The constituents of a canonical pole are:
– a duality (or base) A/Ā
– a contrary component c {-1, 1}
– a canonical polarity p {-1, 0, 1}
A canonical pole presents the form: (A/Ā, c, p).
Furthermore, it is worth distinguishing, at the level of each duality A/Ā, the following derived classes:
– the positive canonical poles: A+, Ā+
– the neutral canonical poles: A0, Ā0
– the negative canonical poles: A, Ā
– the canonical matrix consisting of the 6 canonical poles: {A+, A0, A, Ā+, Ā0, Ā}. The 6 concepts constituting the canonical matrix can also be denoted under the form of a 3 x 2 matrix.
A canonical matrix
Let also be a canonical pole, one will denote by ~ its complement, semantically corresponding to non. We have thus the following complements: ~A+, ~A0, ~A, ~Ā+, ~Ā0, ~Ā. The notion of a complement entails the definition of a universe of reference U. Our concern will be thus with the complement of a given canonical pole in regard to the corresponding matrix7. It follows then that: ~A+ = {A0, A, Ā+, Ā0, Ā}. And a definition of comparable nature for the complements of the other concepts of the matrix ensues.
It should be noted lastly that the following questions arise with regard to canonical poles. The construction of the matrix of the canonical poles of the Positive/Negative duality: {Positive+, Positive0, Positive, Negative+, Negative0, Negative} ensues. But do such concepts as Positive0, Negative0 and especially Positive, Negative+ exist (i) without contradiction?
In the same way, at the level of the Neutral/Polarized duality, the construction of the matrix {Neutral+, Neutral 0, Neutral, Polarized+, Polarized0, Polarized} ensues. But do Neutral+, Neutral exist (ii) without contradiction? In the same way, does Polarized0 exist without contradiction?
This leads to pose the question in a general way: does any neutral canonical pole admit (iii) without contradiction a corresponding positive and negative concept? Is there a general rule for all dualities or well does one have as many specific cases for each duality?
# 3. Relations between the canonical poles
Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations (in addition to the identity relation, denoted by I).
Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are dual or antinomical or opposites if their contrary components are opposite and their polarities are opposite8.
Complementarity
Two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are complementary if their contrary components are opposite and their polarities are equal9.
Two canonical poles 1 (A/Ā, c1, p1) et 2(A/Ā, c2, p2) of a same duality are corollary if their contrary components are equal and their polarities are opposite10.
Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are connex if their contrary components are equal and the absolute value of the difference in their polarities is equal to 1 11.
Two canonical poles 1 (A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same duality are anti-connex if their contrary components are opposite and the absolute value of the difference in their polarities is equal to 1.12, 13
The following questions then arise, with regard to the relations between the canonical poles. Does there exist (i) one (or several) canonical pole which is its own opposite? A priori, it is not possible without contradiction for a positive pole or a negative pole. But the question remains for a neutral pole.
In the same way, does there exist (ii) one (or several) canonical pole which is its own complementary? The following two questions then ensue: does there exist a positive canonical pole which is its own complementary? And also: does there exist a negative canonical pole which is its own complementary?
The questions (i) and (ii) can be formulated in a more general way. Let R be a relation such that R {I, c, , j, g, }. Does there exist (iii) one (or several) canonical pole a verifying a = Ra?
# 4. Degrees of duality
One constructs the class of the degrees of duality, from the intuition that there is a continuous succession of concepts from A+ to Ā, from A0 to Ā0 and from A to Ā+. The continuous component of a degree of duality corresponds to a degree in the corresponding dual pair. The approach by degree is underlied by the intuition that there is a continuous and regular succession of degrees, from a canonical pole Ap to its contrary Ā-p.14 One is thus led to distinguish 3 classes of degrees of duality: (i) from A+ to Ā (ii) from A0 to Ā0 (iii) from A to Ā+.
A degree of duality presents the following components:
– a dual pair Ap-p (corresponding to one of the 3 cases: A+, A00 or A+)
– a degree d Î [-1; 1] in this duality
A degree of duality has thus the form: (A+, d), (A00, d) or (A+, d).
On the other hand, let us call neutral point a concept pertaining to the class of the degrees of duality, whose degree is equal to 0. Let us denote by 0 such a concept, which is thus of the form (Ap-p, 0) with d[0] = 0. Semantically, a neutral point 0 corresponds to a concept which responds to the following definition: neither Ap nor Ā-p. For example, (True/False, 0) corresponds to the definition: neither True nor False. In the same way (Vague/Precise, 0) corresponds the following definition: neither Vague nor Precise. Lastly, when considering the Neutral/Polarized and Positive/Negative dualities, one has then: Neutral0 = (Negative0/Positive0, 0) = (Neutral0/Polarized0, 1).
It is worth noting that this construction does not imply that the neutral point thus constructed is the unique concept which corresponds to the definition neither Ap nor Ā-p. It will appear on the contrary that several concepts and even hierarchies of concepts can correspond to this latter definition.
The following property of the neutral points then ensue, for a given duality A/Ā: (A+, 0) = (A00, 0) = (A+, 0).
At this point, it is worth also taking into account the following derived classes:
– a discrete and truncated class, built from the degrees of duality, including only those concepts whose degree of duality is such that d {-1, -0.5, 0, 0.5, 1}.
– the class of the degrees of complementarity, the degrees of corollarity, etc. The class of the degrees of duality corresponds to the relation of antinomy. But it is worth considering, in a general way, as many classes as there exists relations between the canonical poles of a same duality. This leads to as many classes of comparable nature for the other relations, corresponding respectively to degrees of complementarity, corollarity, connexity and anti-connexity.
It is worth noting finally the following questions, with regard to degrees of duality and neutral points. Does there exist (i) one (or several) canonical pole which is its own neutral point? A priori, it is only possible for a neutral pole.
Does any duality A/Ā admit (ii) a neutral point or trichotomic zero? One can call this question the problem of the general trichotomy. Is it a general rule15 or well does there exists some exceptions? It seems a priori that the Abstract/Concrete duality does not admit a neutral point. It appears to be the same for the Finite/Infinite or the Precise/Vague duality. Intuitively, these latter dualities do not admit an intermediate state.
Does the concept corresponding to the neutral point (Neutral0/Polarized0, 0) and responding to the definition: neither neutral nor polarized exist (iii) without contradiction in the present construction?
# 5. Relations between the canonical poles of a different duality: includers
It is worth also considering the relation of includer for the canonical poles. Consider the following pairs of dual canonical poles: A+ and Ā+, A0 and Ā0, A and Ā. We have then the following definitions: a positive includer + is a concept such that it is itself a positive canonical pole and corresponds to the definition + = A+ Ā+. A neutral includer 0 is a neutral canonical pole such that 0 = A0 Ā0. And a negative includer is a negative canonical pole such that = A Ā. Given these definitions, it is clear that one assimilates here the includer to the minimum includer. Examples: Determinate0 is an includer for True0/False0. And Determinate0 is also a pole for the Determinate0/Indeterminate0 duality. In the same way, Polarized0 is an includer for Positive0/Negative0.
More generally, one has the relation of n-includer (n > 1) when considering the hierarchy of (n + 1) matrices. One has also evidently, the reciprocal relation of includer and of n-includer.
## – matricial includers: they consist of concepts including the set of the canonical poles of a same duality. They respond to the definition: 0 = A+ A0 A– Ā+ Ā0 Ā–.
mixed includers: they consist of concepts responding to the definition 1 = A+ Ā or well 2 = A Ā+
It is worth also considering the types of relations existing between the canonical poles of a different duality. Let A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē} and such that E is an includer for A/Ā i.e. such that E+ = A+ Ā+, E0 = A0 Ā0 and E = A Ā. One extends then the just-defined relations between the canonical poles of a same matrix, to the relations of comparable nature between two matrices presenting the properties of A and E. We has then the relations of 2-antinomy, 2-complementarity, 2-corollarity, 2-connexity, 2-anti-connexity16. Thus, for example, A0 is 2-contrary (or trichotomic contrary) to Ē0, 2-connex (or trichotomic connex) to E+ and E and 2-anti-connex (or trichotomic anti-connex) to Ē+ and Ē. In the same way, A+ and Ā+ are 2-contrary to Ē, 2-complementary to Ē+, 2-corollary to E, 2-connex to E0 and 2-anti-connex to Ē0, etc.
Let us consider also the following property of neutral points and includers. Let A and E be two matrices, such that one of the neutral poles of E is an includer for the neutral dual pair of a: E0 = A0 Ā0. We has then the following property: the canonical pole Ē0 for the matrix E is a neutral point for the duality A00. Thus, the neutral point for the duality A00 is the dual of the includer E0 of A0 and Ā0. Example: Determinate0 = True0 False0. Here, the neutral point for the True/False duality corresponds to the definition: neither True nor False. And we have then (True0/False0, 0) = (Determinate0/Indeterminate0, -1).
This last property can be generalized to a hierarchy of matrices A1, A2, A3, …, An, such that one of the poles 2 of A2 of polarity p is an includer for a dual pair of A1, and that one of the poles 3 of A3 is an includer for a dual pair of A2, …, and that one of the poles n of An is an includer for a dual pair of An-1. It follows then an infinite construction of concepts.
One also notes the emergence of a hierarchy, beyond the sole neutral point of a given duality. It consists of the hierarchy of the neutral points of order n, constructed in the following way from the dual canonical poles A0 and Ā0:
– A0, Ā0
– A1 = neither A0 nor Ā0
– A21 = neither A0 nor A1
– A22 = neither Ā0 nor A1
– A31 = neither A0 nor A21
– A32 = neither A0 nor A22
– A33 = neither A0 nor A21
– A34 = neither Ā0 nor A22
– …
One can also consider the emergence of this hierarchy under the following form17:
– A0, Ā0
– A1 = neither A0 nor Ā0
– A2 = neither A0 nor Ā0 nor A1
– A3 = neither A0 nor Ā0 nor A1 nor A2
– A4 = neither A0 nor Ā0 nor A1 nor A2 nor A3
– A5 = neither A0 nor Ā0 nor A1 nor A2 nor A3 nor A4
– …
Classically, one constructs this infinite hierarchy for True/False by considering I1 (Indeterminate), I2, etc. It should be noticed that in this last construction, no mention is made of the includer (Determinate) of True/False. Neither does one make mention of the hierarchy of includers.
The notion of a complement of a canonical pole corresponds semantically to non. One has the concept of a 2-complement of a canonical pole , defined with regard to a universe of reference U that consists of the 2-matrix of . One has then for example: ~A+ = {A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. And also, ~A+ = {Ā+, E0, E, Ē+, Ē0, Ē}, etc. More generally, one has then the notion of a n-complement (n > 0) of a canonical pole with regard to the corresponding n-matrix.
The following questions finally arise, concerning includers. For certain concepts, does there exist (i) one maximum includer or well does one have an infinite construction for each duality? Concerning the True/False duality in particular, the analysis of the semantic paradoxes has led to the use of a logic based on an infinite number of truth-values18.
Does any duality admit (ii) one neutral includer? Certain dualities indeed seem not to admit of an includer: such is in particular the case for the Abstract/Concrete or Finite/Infinite duality. It seems that Abstract constitutes a maximum element. Admittedly, one can well construct formally a concept corresponding to the definition neither Abstract nor Concrete, but such a concept appears very difficult to justify semantically.
Does there exist (iii) a canonical pole which is its own minimum includer?
Does there exist (iv) a canonical pole which is its own non-minimum includer? One can formulate this problem equivalently as follows. At a given level, does one not encounter a canonical pole which already appeared somewhere in the structure? It would then consist of a structure comprising a loop. And in particular, does one not encounter one of the poles of the first duality?
# 6. Canonical principles
Let be a canonical pole. Intuitively, the class of the canonical principles corresponds to the concepts which respond to the following definition: principle corresponding to what is . Examples: Precise Precision; Relative Relativity; Temporal Temporality. The canonical principles can be seen as 0-ary predicates, whereas the canonical poles are n-ary predicates (n > 0). The lexicalized concepts corresponding to canonical principles are often terms for which the suffix –ity (or –itude) has been added to the radical corresponding to a canonical pole. For example: Relativity0, Beauty+, Activity0, Passivity0, Neutrality0, Simplicity0, Temporality0, etc. A list (necessarily non-exhaustive) of the canonical principles is the following:
Analysis0/Synthesis0, [Animate0]/[Inanimate0], [Exceptional0]/Normality0, [Antecedent0]/[Consequent0], Existence0/Inexistence0, Absolute0/Relativity0, Abstraction0/[Concrete], [Accessory0]/[Principal0], Activity0/Passivity0, [Random0]/Certainty0, [Discrete0]/[Continuous0], Determinism0/Indeterminism0, [Positive0]/[Negative0], Truth0/Falsity0, Attraction0/Repulsion0, Neutrality0/Polarization0, [Static0]/Dynamic0, Unicity0/Multiplicity0, Contenance0/[Containing0], Innate0/Acquired0, Beauty+/Ugliness, Good+/Evil, Identity0/Contrary0, Superiority0/Inferiority0, Extension0/Restriction0, Precision0/Vagueness0, Finitude0/Infinitude0, Simplicity0/Complexity0, [Internal0]/[External0], Equality0/Difference0, Whole0/Part0, Temporality0/Atemporality0, Individuality0/Collectivity0, Quantity0/Quality0, [Implicit0]/[Explicit0], …
It should be noticed that a certain number of canonical principles are not lexicalized. The notations A+, A0, A will be used to denote without ambiguity a canonical principle which is respectively positive, neutral or negative. One could also use the following notation: being a canonical pole, then -ity (or -itude) is a canonical principle. The following notation could then be used: Abstract0ity, Absolute0ity, Acessory0ity, etc. or as above [Abstract0], [Absolute0], etc.
The constituents of the canonical principles are the same ones as for the class of the canonical poles.
It is worth distinguishing finally the following derived classes:
positive canonical principles
neutral canonical principles
negative canonical principles
polarized canonical principles
with some obvious definitions19.
# 7. Meta-principles
Let a0 be a neutral canonical principle20. The class of the meta-principles corresponds to a disposition of the mind directed towards what is a0, to an interest with regard to what is a0. Intuitively, a meta-principle corresponds to a point of view, a perspective, an orientation of the human mind. Thus, the attraction for Abstraction0, the interest for Acquired0, the propensity to place oneself from the viewpoint of Unity0, etc. constitute meta-principles. It should be noted that this construction makes it possible in particular to construct some concepts which are not lexicalized. This has the advantage of a better exhaustiveness and leads to a better and richer semantics.
Let a0 be a neutral canonical principle. Let us also denote by p a meta-principle (p {-1, 0, 1}). One denotes thus a positive meta-principle by +, a neutral meta-principle by 0 and a negative meta-principle by . We have then the enumeration of the meta-principles, for a given duality: {A+, A0, A, Ā+, Ā0, Ā}. Moreover, one will be able to denote by aism a meta-principle. Example: Unite Unite-ism. We have then Internalism, Externalism, Relativism, Absolutism, etc. which correspond in particular to dispositions of the mind. A capital letter will preferably be used here to distinguish the meta-principles from the lexicalized concepts, and in particular to differentiate them from the corresponding philosophical doctrines, which often have very different meanings. It will be however possible to make use of the classical terms when they exist to designate the corresponding meta-principle. Thus All-ism corresponds to Holism.
One can term Ultraaism or Hypera-ism the concept corresponding to . This latter form corresponds to an exclusive, excessive, exaggerated use of the viewpoint corresponding to a given principle. One has thus for example: Externalism = Hyper-externalism.
The constituents of the meta-principles are:
– a polarity p Î {-1, 0, 1}
– a neutral canonical principle composed of:
– a duality (or base) A/Ā
– a contrary component c {-1, 1}
– a neutral polarity q = 0
The positive, neutral, negative canonical meta-principles are respectively of the form ((A/Ā, c, 0), 1), ((A/Ā, c, 0), 0), ((A/Ā, c, 0), -1).
Between the canonical meta-principles of a same duality, one has the same relations as for the canonical poles.
One has lastly the derived classes consisting in:
– the positive meta-principles (p > 0)
– the neutral meta-principles (p = 0)
– the negative meta-principles (p < 0)
– the polarized meta-principles which include the positive and negative meta-principles
– the matrix of the canonical meta-principles, consisting of 6 meta-principles applicable to a given duality{A+, A0, A, Ā+, Ā0, Ā}.
– the degrees of canonical meta-principles. Intuitively, such concepts are more or less positive or negative. The polarity is regarded here as a degree of polarity. These concepts are such that p Î [-1; 1].
– the class of the behavioral principles. Intuitively, the class of the behavioral principles constitutes an extension of that of the meta-principles. While the meta-principle constitutes a disposition of the human mind, the concepts concerned here are those which aim to describe, in a more general way, the tendencies of the human behavior21. Among the lexicalized concepts corresponding to the behavioral principles, one can mention: courage, prudence, pessimism, rationality, avarice, fidelity, tendency to analysis, instability, objectivity, pragmatism, etc. A first analysis reveals (i) that a certain number of them reveal a meliorative nuance: courage, objectivity, pragmatism; that (ii) others, by contrast, present a pejorative, unfavorable connotation: cowardice, avarice, instability; and finally (iii) that certain concepts present themselves under a form which is neither meliorative nor pejorative: tendency to analysis22. One has here the same classes as for the meta-principles, and in particular the degrees of behavioral principles. Example: coward is more negative than apprehensive; in the same way, bravery is more positive than courage.
# Conclusion
The concepts constructed with the help of the present theory need to be distinguished in several regards from those resulting from the application of the semiotic square described by Greimas (1977, p. 25). This last theory envisages in effect four concepts: S1, S2, ~S1, ~S2. On the one hand, it appears that the semiotic square is based on two lexicalized concepts S1 and S2 that constitute a dual pair. It does not distinguish, when considering the dual concepts, whether these latter are positive, neutral or negative. By contrast, the present theory considers six concepts, lexicalized or not.
On the other hand, the present analysis differs from the semiotic square by a different definition of the complement-negation. Indeed, the semiotic square comprises two concepts corresponding to the complement-negation: non-S1 and non-S2. By contrast, in the present context, the negation is defined with regard to a universe of reference U, which can be defined with regard to the corresponding matrix, or well to the 2-matrix…, to the n -matrix. For each canonical pole, there is thus a hierarchy of concepts corresponding to non-S1 and non-S2.
One sees it, the present taxonomy of concepts differs in several respects from the one conceived of by Greimas. Implemented from the dualities and the logical concepts, the present theory has the advantage of applying itself to lexicalized concepts or not, and also of being freed [affranchie] from the definitions of concepts inherent to a given culture. In this context, the classification which has been just described constitutes an alternative to the one based on the semiotic square which has been proposed by Greimas.
References
FINE, Kit (1975). Vagueness, Truth and Logic. Synthese 30: 265-300 GREIMAS, A. J. (1977). Elements of a Narrative Grammar, Diacritics 7: 23-40 JAKOBSON, Roman (1983). Dialogues, Cambridge MA: MIT Press PEACOCKE, C. A. B. (1981). Are Vague Predicates Incoherent?. Synthese 46: 121-141 RESCHER, Nicholas (1969). Many-Valued Logic, New York: McGraw Hill
1 Or polar contraries.
2 Cf. Jakobson (1983).
3 In the same way, it would have been possible to define a more restricted class, including only half of the semantic poles, by retaining only one of the two dual predicates, and by constructing the others with the contrary relation. However, the choice of either of the dual poles would have been arbitrary, and I have preferred to avoid it. The following construction would have then resulted. Let Contrary be the semantic pole and a whatever semantic pole, not necessarily distinct from Contrary; the concept resulting from the composition of Contrary and a is a semantic pole. It should also be noted that this type of construction would have led to:
Contrary ° Contrary = Identical.
Contrary ° Identical = Contrary.
Contraryn = Identical (for n even).
Contraryn = Contrary (for n odd).
In this context, it is worth noting that Contrary constitutes a specific case. In effect, if one seeks to build a class of the canonical poles which is minimal, it is worth noting that one can dispense oneself from Identical, whereas one cannot dispense oneself from Contrary. There is here an asymmetry. In effect, one can construct Identical with the help of Contrary, by using the property of involution: Contrary ° Contrary = Identical. For other dualities, one can indifferently choose either of the concerned semantic poles.
4 It is worth noting that one could have drawn here a distinction between unary and binary poles, by considering that they consist of predicates. But a priori, such a distinction does not prove very useful for the resulting construction.
5 In what follows, the questions relating to the various classes are only mentioned. It goes without saying that they require an in-depth treatment which goes far beyond the present study.
6 With variable degrees in the nuance.
7 When it is defined with regard to a dual pair, the complement of the pole of a given duality identifies itself with the corresponding dual pole.
8 Formally c1 = –c2, p1 = – p2 ® 1(A/Ā, c1, p1) = 2(A/Ā, c2, p2).
9 Formally c1 = – c2, p1 = p2 ® 1(A/Ā, c1, p1) = f2(A/Ā, c2, p2).
10 Formally c1 = c2, p1 = – p2 ® 1(A/Ā, c1, p1) = c2(A/Ā, c2, p2).
11 Formally c1 = c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = g2(A/Ā, c2, p2).
12 Formally c1 = – c2, |p1p2| = 1 ® 1(A/Ā, c1, p1) = b2(A/Ā, c2, p2).
13 We have then the following properties, with regard to the above-mentioned relations. The relation of identity constitutes a relation of equivalence. Antinomy, complementarity and corollarity are symmetrical, anti-reflexive, non-associative, involutive.
The operation of composition on the relations {identity, corollarity, antinomy, complementarity} defines an abelian group of order 4. With G = {I, c, , j}:
°IcjIIcjccIjjIcjjcI
where for all A Î G, A-1 = A, and A ° I = A, I being the neutral element. It should be noted that the group properties make it possible in particular to give straightforwardly a valuation to any propositions of the form: the contrary concept of the complementary of a1 is identical to the corollary of the complementary of a2.
14 This construction of concepts can be regarded as an application of the degree theory. Cf. in particular Fine (1975), Peacocke (1981). The present theory however is not characterized by the preferential choice of the degree theory, but considers simply this latter theory as one of the methods of construction of concepts.
15 Some common trichotomies are: {past, present, future}, {right, center, left}, {high, center, low}, {positive, neutral, negative}.
16 There is a straightforward generalization to n matrices (n > 1) of this construction with the relations of n-antinomy, n-complementarity, n-corollarity, n-connexity, n-anti-connexity.
17 One can assimilate the two just-described hierarchies to only one single hierarchy. It suffices to proceed to the following assimilation:
– A2 = A21 or A22
– A3 = A31 or A32 or A33 or A34
– A4 = A41 or A42 or A43 or A44 or A45 or A46 or A47 or A48
– …
18 Infinite-valued logics. Cf. Rescher (1969).
19 Furthermore, it should be noted that some other concepts can be thus constructed. Let also be a canonical pole. We have then the classes of concepts responding to the following definition: to render (Example: Unite Unify; Different Differentiate); action of rendering (Unite Unification; Different Differentiation); that it is possible to render (Unite Unitable; Different Differentiable), etc. These concepts are not however of interest in the present context.
20 It should be observed that we could have taken alternatively as a basis for the definition of the meta-principles a canonical principle, without distinguishing whether this latter is positive, neutral or negative. But it seems that such a definition would have engendered more complexity, without giving in return a genuine semantic interest.
21 This particular class would require however a much finer analysis than the one which is summarily presented here. I am only concerned here with showing that a many concepts pertaining to this category can be the subject of a classification whose structure is that of the meta-principles.
22 One can consider the following – necessarily partial – enumeration corresponding to the behavioral principles, in the order (A+), (A0), (A), (Ā+), (Ā0), (Ā):
firmness, propensity to repress, severity, leniency, propensity to forgive, laxism
defense, refusal, violence, pacifism, acceptance, weakness
pride, self-esteem, hyper-self-esteem, modesty, withdrawal of the ego, undervaluation of self
expansion, search of quantity, excess, perfectionism, search of quality, hyper-selectivity
delicacy, sensitivity, sentimentality, coolness, impassibility, coldness
objectivity, to be neutral being, impersonality, to be partisan, parti pris
uprightness, to act in a direct way, brusqueness, tact, to act in an indirect way, to flee the difficulties
combativeness, disposition to attack, aggressiveness, protection, disposition to defense, tendency to retreat
receptivity, belief, credulity, incredulity, doubt, excessive skepticism
expansion, oriented towards oneself, selfishness, altruism, oriented towards others, to render dependent
sense of economy, propensity to saving, avarice, generosity, propensity to expenditure, prodigality
mobility, tendency to displacement, instability, stability, tendency to stay at the same place, sedentariness
logical, rationality, hyper-materialism, imagination, irrationality, inconsistency
sense of humour, propensity to play, lightness, serious, propensity to the serious activity, hyper-serious
capacity of abstraction, disposition to the abstract, dogmatism, pragmatism, disposition to the concrete, prosaicness
audacity, tendency to risk, temerity, prudence, tendency to avoid the risks, cowardice
discretion, to keep for oneself, inhibition, opening, to make public, indiscretion
optimism, to apprehend the advantages, happy optimism, mistrust, to see the disadvantages, pessimism
sense of the collective, to act like the others, conformism, originality, to demarcate oneself from others, eccentricity
resolution, tendency to keep an opinion, pertinacity, flexibility of spirit, tendency to change opinion, fickleness
idealism, tendency to apprehend the objectives, quixotism, realism, tendency to apprehend the means, prosaicness
taste of freedom, to be freed, indiscipline, obedience, to subject oneself to a rule, servility
reflexion, interiorization, inhibition, sociability, exteriorisation, off-handednes
spontaneousness, tendency to react immediately, precipitation, calm, tendency to differ one’s reaction, slowness
eclecticism, multidisciplinarity, dispersion, expertise, mono-disciplinarity, bulk-heading
revival, propensity to change, rupture, safeguarding of the assets, propensity to maintenance, conservatism
motivation, passion, fanaticism, moderation, reason, tepidity
width of sights, tendency to synthesis, overflight, precision, tendency to analysis, to lose oneself in the details
availability, propensity to leisure, idleness, activity, propensity to work, overactivity
firmness, tendency not to yield, intransigence, diplomacy, tendency to make concessions, weakness
causticity, tendency to criticism, denigration, valorization, tendency to underline qualities, angelism
authority, propensity to command, authoritarianism, docility, propensity to obey, servility
love, tendency to be attracted, exaggerate affection, tendency to know to take one’s distances, repulsion, hatred
conquest, greed, bulimia, sobriety, to have the minimum, denudement
Preprint.
I present in this paper the basic elements of the n-universes, from an essentially pragmatic standpoint, i.e. by describing accurately the step-by-step process which leads to the modelling of a thought experiment.
English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2009, 19-4, pages 136-140 under the title “Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage”.
In a previous article (Complements to a theory of cognitive distorsions, Journal de Thérapie Comportementale et Cognitive, 2007), we introduced some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements allow to define the general cognitive distortions as well as the specific cognitive distortions. This model is extended here to the description of two other classical cognitive distortions: over-generalisation and mislabelling. The definition of the two latter cognitive distortions is based on prior differentiation between three levels of reasoning: primary, secondary and ternary pathogenic arguments. The latter analysis also leads to define two other cognitive distortions which insert themselves into this framework: ill-grounded inductive projection and confirmation bias.
## Theory of Cognitive Distortions: Over-generalisation and Labeling
In Franceschi (2007), we set out to introduce several elements aimed at contributing to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With these three elements, we could define within the same conceptual framework the following general cognitive distortions: dichotomous reasoning, disqualification of one pole, minimisation and maximisation, requalification in the other pole and omission of the neutral. We could also describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophism. In the present article, we offer to define and to situate, within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling.
Over-generalisation and mislabelling constitute two of the twelve traditionally defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mental reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Beck 1964, Ellis 1962). Over-generalisation is classically defined as a rough and ill-grounded generalisation, usually including either of the quantifiers “all”, “none”, “never”, “always”. Moreover, it is often described as a cognitive distortion including four subcategories: dichotomous reasoning, selective abstraction, maximisation/minimisation, and disqualification of the positive. Mislabelling is also classically defined as an extreme form of over-generalisation, consisting in the apposition of a label with a strong negative and emotional connotation to oneself or to an external subject.
1. Primary, secondary and ternary pathogenic arguments
Before setting out to define over-generalisation and mislabelling in the present context, it is worth describing preliminarily a structure of pathogenic reasoning (in the etymological sense: engendering suffering), with a general scope, susceptible of being found in some disorders of a very different nature, such as depression, generalised anxiety disorder, body dismorphic disorder, scrupulosity or intermittent explosive disorder. Such structure of reasoning includes several levels of arguments: primary, secondary and ternary. In a simplified way, primary pathogenic arguments are constituted by an enumeration of instances. Secondary pathogenic arguments consist of a generalisation from the latter instances. Lastly, pathogenic ternary arguments are constituted by an interpretation of the latter generalisation. Such reasoning as a whole presents an inductive structure.
At this stage, it is worth mentioning several instances of this type of reasoning. A first instance, susceptible to be found in depression (Beck 1967, 1987), is the following (the symbol denotes the conclusion):
(11) I gave my ankle a wrench last January premise1 (12) I lost my job last February premise2 (13) Fifteen days ago, I had an influenza with fever premise3 (14) I got into an argument with Lucy last month premise4 (…) (…) (110) Today, my horoscope is not good premise10 (2) ∴ Everything that occurs to me is bad from (11)-(110) (3) ∴ I am a complete failure! from (2)
The patient enumerates first some events of his/her past and present life (11)-(110), that he/she qualifies as negative, through a primary stage which consists of an enumeration of instances. Then he/she performs a generalisation (2) from the previous enumeration, which presents the following structure:
(2) ∴ All events that occur to me are negative from (11)-(110)
Lastly, the patient interprets (3) the latter conclusion by concluding “I am a complete failure!”. Such instance applies then to the reference class of the present and past events of the patient’s life and to the Positive/Negative duality.
One can also mention a reasoning that presents an identical structure, which is susceptible to be met in body dysmorphic disorder (Veale 2004, Rabinowitz & al. 2007). The patient enumerates then different parts of his/her body, which he/she qualifies as ugly. He/she generalises then by concluding that all parts of his/her body are ugly. Finally, he/she adds: “I am ugly!”. The corresponding reasoning applies then to the Beautiful/Ugly duality and to the reference class of the parts of the patient’s body.
In the same way, in a reasoning of identical structure, susceptible to be met in scrupulosity (Teak & Ulug 2001, Miller & Edges 2007), the patient enumerates several instances corresponding to some acts which he/she made previously or recently, and which he/she considers as morally bad. He/she concludes then: “Everything I do is bad, morally reprehensible”, and he/she further interprets it by concluding: “I am a horrible sinner!”. Such conclusion is likely to trigger an intense feel of guilt and a compulsive practice of religious rituals. The corresponding instance applies here to the duality Good/Evil and to the reference class of the present and past actions of the patient’s life.
Lastly, an instance of this structure of reasoning can contribute to the development of hostility, of a potentially aggressive attitude toward other people. In that case, the patient concludes regarding an external subject: “All acts that he committed toward me are bad”. He/she concludes then: “He is a bastard!”. Such conclusion can then play a role in intermittent explosive disorder (Coccaro & al. 1998, Galovski & al. 2002). In such case, the over-generalisation applies to the Good/Evil duality and to the reference class of the actions of an external subject with regard to the patient.
At this step, it is worth describing in more detail each of the three stages – primary, secondary and ternary – which compose this type of reasoning.
Primary pathogenic arguments
The first step in the aforementioned type of reasoning, consists for the patient to enumerate some instances. The general structure of each instance is as follows:
(1i) The object xi of the class of reference E has property Ā (in the duality A/Ā) premisei
In the aforementioned example applied to depression, the patient enumerates some events of his/her present and past life, which he/she qualifies as negative, under the form:
(1i) The event Ei of negative nature occurred to me premisei
Different instances corresponding to this cognitive process can be described under the form of a primary pathogenic argument, the structure of which is the following:
(1a) The event E1 occurred to me premise (1b) The event E1 was of a negative nature premise (1) ∴ The event E1 of a negative nature occurred to me from (1a), (1b)
By such cognitive process, the patient is led to the conclusion according to which some negative event did occur to him/her.
From a deductive point of view, this type of argument proves to be completely valid (the conclusion is true if the premises are true) since the very event presents well, objectively, a negative nature. However, this type of primary argument can turn out to be fallacious, when the very event presents, objectively, a positive or neutral nature. The flaw in the reasoning resides then in the fact that the premise (1b) turns then out to be false. Such can be case for example if the patient makes use of a specific cognitive distortion such as requalification in the negative. In such case, the patient considers as negative an event the nature of which is objectively positive.
Secondary pathogenic arguments
At the level of the above-mentioned reasoning, secondary pathogenic arguments are constituted by the sequence which proceeds by generalisation, from the instances (11) to (110), according to the following structure:
(2) ∴ All elements xi of the class of reference E have property Ā from (11)-(110)
Such over-generalisation leads then to the conclusion “All events that occur to me are bad” (depression); “All parts of my body are ugly” (body dysmorphic disorder); “All my acts are morally reprehensible” (scrupulosity); “All acts that he committed toward me are bad” (intermittent explosive disorder).
From a deductive point of view, such generalisation may constitute a completely valid argument. Indeed, the resulting generalisation constitutes a correct deductive reasoning, if the premises (11)-(110) are true. However, it often proves to be that the premises of the argument are false. Such is notably the case when the patient counts among the elements having property Ā, some elements which objectively have the opposite property A. The flaw in the argument resides then in a requalification in the other pole related to some elements and the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. In such case, secondary pathogenic argument turns out to be ungrounded, because of the falseness of some premises.
In other cases, the secondary pathogenic argument turns out to be fallacious from an inductive standpoint. For some positive (or neutral) events can well have been omitted in the corresponding enumeration of instances. Such omission can result from the use of general cognitive distortions, such as the omission of the neutral or disqualification of the positive. In such case, the elements of the relevant class of reference are only partly taken into account, thus biasing the resulting generalisation. The corresponding reasoning remains then logically valid and sound, but fundamentally incorrect of an inductive point of view, because it does only take partly into account the relevant instances within the reference class. Such feature of over-generalisation – a conclusion resulting from a valid reasoning from a deductive point of view, but inductively wrong – allows to explain how it notably succeeds in deceiving patients whose level of intelligence can otherwise prove to be high.
Ternary pathogenic arguments
It is worth mentioning, lastly, the role played by pathogenic ternary arguments which consist, at the level of the aforementioned reasoning, of the following sequence:
(2) All events that occur to me are of a negative nature premise (3) ∴ I am a complete failure! from (2)
In such argument, the premise is constituted by the conclusion (2) of the secondary pathogenic argument, of which, in an additional stage (3), the patient aims at making sense by interpreting it. It consists here of a case of mislabelling. At the stage of a ternary pathogenic argument, mislabelling can thus take the following forms: “I am a complete failure!” (depression); “I am ugly!” (bodily dysmorphic disorder); “I am a horrible sinner!” (scrupulosity); “He is a bastard! “ (intermittent explosive disorder). In the present context, mislabelling proves to be an invalid argument, which constitutes a rough and unjustified interpretation of the over-generalisation (2).
2. Over-generalisation
At this stage, we are in a position to give a definition of over-generalisation, by drawing a distinction between general and specific over-generalisations. A general over-generalisation applies to any duality and to any reference class. It can be analysed as the ill-grounded conclusion of a secondary pathogenic argument, the premises of which include some general cognitive distortions: dichotomous reasoning, disqualification of one pole, arbitrary focus, minimisation/maximisation, omission of the neutral or requalification in the other pole. It consists of an ungrounded inductive reasoning, because the resulting generalisation is based on an incorrect counting of the corresponding instances. In the same way, a specific over-generalisation consists of an instance of a general over-generalisation, applied to a given duality and reference class. Thus, the specific over-generalisation “All events which occur to me are of a negative nature” (depression, generalised anxiety disorder) applies to the Positive/Negative duality and to the class of the events of the patient’s life. In the same way, “All parts of my body are ugly” (body dysmorphic disorder) is a specific over-generalisation that applies to the reference class of the parts of the patient’s body and to the Beautiful/Ugly duality.
3. Ungrounded inductive projection
At this step, it proves to be useful to describe another error of reasoning, which is likely to manifest itself at the stage of secondary pathogenic arguments. It consists of an ill-grounded inductive projection. The latter concludes, from the preceding over-generalisation (2), that a new instance will occur in the near future. Such instance is susceptible to be met in depression (Miranda & al. 2008), as well as in generalised anxiety disorder (Franceschi 2008). In the context of depression, such inductive projection presents the following form:
(2) All events that occur to me are of a negative nature premise (111a) The future event E11 of a negative nature may occur premise (111b) ∴ The future event E11 of a negative nature will occur from (2), (111a)
The corresponding conclusion is susceptible of contributing to depression, notably by triggering the patient’s feeling of despair. Other instances of this type of conclusion are: “My next action will be morally reprehensible” (scrupulosity), or “The next act that he will commit toward me will be bad” (intermittent explosive disorder).
4. Confirmation bias
The cognitive process which has just been described illustrates how over-generalisation contributes to the formation of pathogenic ideas. However, a process of the same nature is also likely to concur to their maintenance. For once the over-generalisation (2) has been established by means of the above reasoning, its maintenance is made as soon as an instance occurs that confirms the generalisation according to which all elements xi of the reference class E have property Ā. This constitutes a confirmation bias, for the patient does only count those elements which present the property Ā, without taking into account those which have the opposite property A, thus disconfirming generalisation (2). Hence, in depression or generalised anxiety disorder, when a new negative event occurs, the patient concludes from it that it confirms that all events which occur to him/her are of a negative nature.
We see it finally, the above developments suggest a classification of cognitive distortions, depending on whether they manifest themselves at the level of primary, secondary or ternary pathogenic arguments. Thus, among the cognitive distortions which arise at the stage of primary pathogenic arguments, one can distinguish: on the one hand, the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation/maximisation, requalification into the other pole, omission of the neutral) and on the other hand, the specific cognitive distortions (disqualification of the positive, requalification into the negative, selective abstraction, catastrophising). Morevoer, among the cognitive distortions which manifest themselves at the stage of secondary pathogenic arguments, one can mention over-generalisation (at the stage of the formation of pathogenic ideas), ill-grounded inductive projection, and confirmation bias (at the stage of the maintenance of pathogenic ideas). Mislabelling, finally, is susceptible to occur at the level of ternary pathogenic arguments.
### References
Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry, 1964, 10, 561-571.
Beck, A. Depression: Clinical, experimental, and theoretical aspects, Harper & Row, New York, 1967.
Beck, A. Cognitive models of depression. Journal of Cognitive Psychotherapy, 1, 1987, 5-37.
Coccaro E., Richard J., Kavoussi R., Mitchell E., Berman J., Lish J. Intermittent explosive disorder-revised: Development, reliability, and validity of research criteria. Comprehensive Psychiatry, 39-6, 1998, 368-376.
Eckhardt C., Norlander B., Deffenbacher J., The assessment of anger and hostility: a critical review, Aggression and Violent Behavior, 9-1, 2004, 17-43.
Ellis A. Reason and Emotion in Psychotherapy, Lyle Stuart, New York, 1962.
Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation.
Franceschi P. Théorie des distorsions cognitives : application à l’anxiété généralisée. Journal de Thérapie Comportementale et Cognitive, 2008, 18, 127-131. English translation.
Galovski T., Blanchard E., Veazey C. Intermittent explosive disorder and other psychiatric co-morbidity among court-referred and self-referred aggressive drivers. Behaviour Research and Therapy, 40-6, 2002, 641-651.
Miller C., Hedges D. Scrupulosity disorder: An overview and introductory analysis. Journal of Anxiety Disorders, 2007, 22-6, 1042-1048.
Miranda R., Fontes M., Marroquín B. Cognitive content-specificity in future expectancies: Role of hopelessness and intolerance of uncertainty in depression and GAD symptoms. Behaviour Research and Therapy, 46-10, 2008, 1151-1159.
Tek C., Ulug B. Religiosity and religious obsessions in obsessive–compulsive disorder. Psychiatry Research, 2001, 104-2, 99-108.
Rabinowitz D., Neziroglu F., Roberts M. Clinical application of a behavioral model for the treatment of body dysmorphic disorder. Cognitive and Behavioral Practice, 2007, 14-2, 231-237.
Veale D. Advances in a cognitive behavioural model of body dysmorphic disorder. Body Image, 2004, 1, 113-125.
English translation of a paper published in French in the Journal de Thérapie Comportementale et Cognitive, 2010, 20-2, pages 51-55 under the title “Théorie des distorsions cognitives : la personnalisation”.
In a previous paper (Complements to a theory of cognitive distorsionsJournal de Thérapie Comportementale et Cognitive, 2007), we did present some elements aimed at contributing to a general theory of cognitive distortions. Based on the reference class, the duality and the system of taxa, these elements led to distinguish between the general cognitive distortions (dichotomous reasoning, disqualification of one pole, minimisation, maximisation) and the specific cognitive distortions (disqualifying the positive, selective abstraction, catastrophism). By also distinguishing between three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage – we did also define two other cognitive distortions: over-generalisation and mislabelling (Théorie des distorsions cognitives : la sur-généralisation et l’étiquetageJournal de Thérapie Comportementale et Cognitive, 2009). We currently extend this model to another classical cognitive distortion: personalisation.
## Theory of Cognitive Distortions: Personalisation
In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three fundamental notions: the reference class, the duality and the system of taxa. With the aid of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, disqualification of one pole, minimisation and maximisation, as well as requalification in the other pole and omission of the neutral. In the same way, we could describe as specific cognitive distortions: disqualification of the positive, selective abstraction and catastrophising. In Franceschi (2009), we introduced three levels of reasoning – the instantiation stage, the interpretation stage and the generalisation stage, which allowed to define within the same conceptual framework, two other classical cognitive distortions: over-generalisation and mislabelling. In the present paper, we set out to define and to situate in this conceptual framework another classical cognitive distortion: personalisation.
Personalisation constitutes one of but twelve classically defined cognitive distortions: emotional reasoning; over-generalisation; arbitrary inference; dichotomous reasoning; should statements; divination or mind-reading; selective abstraction; disqualification of the positive; maximisation/minimisation; catastrophising; personalisation; mislabelling (Ellis 1962, Beck 1964). Personalisation is usually defined as the fact of attributing unduly to oneself the cause of an external event. For example, seeing a person who laughs, the patient thinks that it is because of his/her physical appearance. Also, the patient makes himself/herself responsible for a negative event, in an unjustified way. If his/her companion then failed his/her examination, the patient estimates that is due to the fact that he/she is depressed. In what follows, we propose first to clarify the definition of personalisation and to situate it in the context of the theory of cognitive distortions (Franceschi 2007, 2009). Secondly, we set out to clarify the relationships existing between personalisation and several close notions mentioned in the literature: personalising bias (Langdon et al. 2006), ideas of reference (Startup & Startup 2005, Meyer & Lenzenweger 2009) and delusions of reference.
Personalisation and post hoc fallacy
We will set out first to highlight the mere structures of the cases of personalisation. Let us consider the aforementioned example where the patient sees a person who laughs and thinks that this one laughs because of the patient’s physical appearance. This constitutes an instance of personalisation. We can describe more accurately the reasoning which underlies such instance (in what follows, the symbol denotes the conclusion):
(P11) in T1 I went for a walk premiss1 (P12) in T2 the peasant started to laugh premiss2 (P13) ∴ in T2 the peasant started to laugh because he saw that in T1 I went for a walk conclusion1 (P14) ∴ in T2 the peasant made fun of me conclusion2
The patient puts here in relationship an internal event (“I went for a walk”) with an external event (“the peasant started to laugh”). He/she concludes then that the internal event is the cause of the external event. In this stage, the patient “personalises” an external event, which he/she considers to be the effect of an internal event, while this external event is in reality devoid of any relationship with the patient himself/herself. In a subsequent stage (P14), the patient interprets the previous conclusion (P13) by considering that the peasant made fun of him.
At this stage, it is worth wondering about the specific nature of the patient’s error of reasoning. It appears here that both premises (P11) and (P12) constitute genuine facts and therefore turn out to be true. On the other hand, the conclusion (P13) which concludes to the existence of a relation of causality between two consecutive events E1 (“In T1 I went for a walk”) and E2 (“In T2 the peasant started to laugh”) appears to be unjustified. Indeed, both premises are only establishing a relation of anteriority between the two consecutive facts E1 and E2. And the conclusion (P13) which deducts from it a relation of causality turns out therefore to be too strong. The argument proves here to be invalid and the corresponding reasoning is then fallacious. The corresponding error of reasoning, which concludes to a relation of causality whereas there is only a mere relation of anteriority, is classically termed post hoc fallacy, according to the Latin sentence “Post hoc, ergo propter hoc” (after this therefore because of this). It consists here of a very common error of reasoning, which is notably at the root of many superstitions (Martin 1998, Bressan 2002).
In this context, we can point out that the case of post hoc fallacy which has just been described as an argument of personalisation, also constitutes a case of arbitrary inference, another classically defined cognitive distortion.
Steps of instantiation, of interpretation and of generalisation at the level of the arguments of personalisation
At this step, it proves to be useful to draw a distinction between the levels of arguments that lead to personalisation as cognitive distortion. This leads to differentiate three levels within the arguments of personalisation, among the reasoning’ stages. The latter correspond respectively to three different functions: it consists of the successive stages of instanciation, of interpretation and of generalisation. To this end, it is useful to describe the whole reasoning which underlies the arguments of personalisation and which includes the three aforementioned stages:
(P11) in T1 I went for a walk premiss11 (P12) in T2 the peasant started to laugh premiss12 (P13) ∴ en T2 the peasant started to laugh because he saw that in T1 I went for a walk conclusion11 (P14) ∴ in T2 the peasant made fun of me conclusion12 (P21) in T3 I was leafing through a magazine in the library premiss21 (P22) in T4, the librarian smirked premiss22 (P23) ∴ en T4 the librarian smirked because in T3 I was leafing through a magazine in the library conclusion21 (P24) ∴ in T4, the librarian made fun of me conclusion22 (P31) in T5 I did enter in the show-room premiss31 (P32) in T6, my colleagues started to laugh premiss32 (P33) ∴ in T6, my colleagues started to laugh because in T5 I did enter in the show-room conclusion31 (P34) ∴ in T6, my colleagues were laughing at me conclusion32 (…) (P105) ∴ people make fun of me from (P14)-(P104)
Here, the instances of the previous arguments (P11)-(P13), (P21)-(P23), (P31)-(P33), etc. constitute primary stages of arguments of personalisation, by which the patient considers that an event related to him/her is the cause of an external event. This type of argument corresponds to the stage of instantiation. As mentioned earlier, such argument is fallacious since it is based on post hoc fallacy. In a subsequent stage the function of which is interpretative, and that is aimed at making sense of the conclusions (P13), (P23), (P33), … of the instances of arguments of the previous type, the patient interprets it by concluding that some people made fun of him. Such conclusions (P14), (P24), (P34) appear to be grounded, inasmuch as the premisses (P13), (P23), (P33) are true. Finally, in a subsequent stage of generalisation, the patient enumerates some instances or circumstances where he/she thinks that people laughed or made fun of him/her ((P14), (P24), (P34), …) and generalises then to the conclusion (P105) according to which people make fun of him/her. This last stage is of an inductive nature, and corresponds to an enumerative induction, the structure of which is the following:
(P14) in T2 the peasant made fun of me conclusion12 (P24) in T4, the librarian made fun of me conclusion22 (P34) in T6, my colleagues were laughing at me conclusion32 (…) (P105) ∴ people make fun of me from (P14)-(P104)
Given what precedes, we can from now on provide a definition of personalisation. The preceding analysis leads then to distinguish between three stages in arguments of personalisation. At the level of primary arguments of personalisation (stage of instantiation), it consists of the tendency in the patient to establish an unjustified relation of causality between two events, among which one is external and the other one is internal to the patient. The patient personalises then, that is to say puts in relationship with himself/herself, an external event, which proves to be in reality devoid of any relation of causality. The mechanism which underlies such argument consists then of the erroneous attribution of a relation of causality, based on post hoc fallacy. At the level of secondary arguments of personalisation (stage of interpretation), the patient makes sense of the previous conclusion by concluding that at a given time, a person (or several persons) made fun of him, laughed at him, etc. Finally, at the level of arguments of ternary personalisation (stage of generalisation), the patient concludes that, in a general way, people make fun of him.
Personalisation and personalising bias
At this step, it proves to be useful to distinguish personalisation as cognitive distortion from personalising bias. The latter is defined as an attribution bias (“personalising attributional bias”), by whom the patient attributes to other persons rather than to circumstances the cause of a negative event (McKay & al. 2005, Langdon & al. 2006). Personalising bias is often related to polythematic delusions (Kinderman & Bentall 1997, Garety & Freeman 1999, McKay & al. 2005) met in schizophrenia.
Considering this definition, the difference between the two notions can be thus underlined: in personalisation as cognitive distortion, the patient attributes the cause of an external event to an event which concerns the patient himself/herself; on the other hand, in personalising bias the patient attributes the cause of an internal event to external persons. This allows to highlight several fundamental differences between the two notions. Firstly, in personalisation as cognitive distortion, the “person” is the patient himself/herself, while in personalising bias, it consist of external “persons”. Secondly, in the structure of personalisation, an internal event precedes an external event; by contrast, in the scheme of personalising bias, it is an external event which precedes an internal event. Finally, in personalisation as cognitive distortion, the internal event is indifferently of a positive, neutral or negative nature, whereas in personalising bias, the internal event is of a negative type. Hence, it finally proves to be that both notions appear fundamentally distinct.
Personalisation and ideas of reference
It appears also useful, for the sake of clarity, to specify the relationships between personalisation and ideas of reference. It is worth preliminary mentioning that one usually distinguishes between ideas of reference and delusions of reference (Dziegielewski 2002, p. 266). Ideas of reference characterise themselves by the fact that a patient considers that insignificant events relate to himself/herself, while is not the case in reality. For example, the patient hears several persons laugh, and considers, in an unjustified way, that the latter make fun of him/her. In parallel, delusions of reference constitute one of the most salient symptoms noticed in schizophrenia, and leads the patient to be persuaded that the media, television, or the radio speak about him/her or issue messages concerning him/her. Several criteria allow to draw a distinction between ideas of reference and delusions of reference. First, ideas of reference have much less impact on the patient’s life than reference delusions of reference. Second, the degree of conviction which is associated with ideas of reference is far lesser than with delusions of reference. Lastly, ideas of reference (“the neighbour made fun of me”) are related with beliefs the degree of plausibility of which is much stronger than the one which is inherent to delusions of reference (“newspapers speak about me”).
In this context, the aforementioned arguments of personalisation (P11)-(P14), (P21)-(P24), and (P31)-(P34), by whom the patient concludes that some people make fun of him, corresponds completely to the definition of ideas of reference. It appears then that personalisation, such as it was defined above as cognitive distortion, identifies itself with ideas of reference.
Personalisation and delusion of reference
One traditionally distinguishes at the level of polythematic delusions met in schizophrenia between: delusions of reference, delusions of influence, delusions of control, telepathy-like delusions, delusions of grandeur, and delusions of persecution. Delusions of reference leads for example the patient to believe with a very strong conviction that the media, the newspapers, the television speak about him/her.
It is worth describing here a mechanism which is susceptible to lead to the formation of delusions of reference. Such mechanism appears to be grounded on a reasoning (Franceschi 2008) which includes, as well as the above-mentioned primary instances of personalisation, a post hoc fallacy:
(DR11) in T1 I was drinking an appetizer premiss11 (DR12) in T2 the presenter of the show said: “Stop drinking!” premiss12 (DR13) ∴ in T2 the presenter of the show said: “Stop drinking!” because in T1 I was drinking an appetizer conclusion11 (DR14) ∴ in T2 the presenter of the show spoke about me conclusion12
Consider also this second instance :
(DR21) in T3 I hardly got out of bed premiss21 (DR22) in T4 the radio presenter said: “Be forceful:” premiss22 (DR23) ∴ in T4 the radio presenter said: “Be forceful:” because in T3 I hardly got out of bed conclusion21 (DR24) ∴ in T4 the radio presenter spoke about me conclusion22
At the level of the instantial step (DR11)-(DR13), (DR21)-(DR23), … the patient concludes here that an internal event is the cause of an external event. In a further interpretative stage, he/she interprets the conclusions (DR13), (DR23), … of the preceding arguments by considering that the presenters of radio or of television speak about him/her. Finally, in a generalisation step, of inductive nature, the patient enumerates the conclusions (DR14), (DR24), … of secondary arguments (interpretation stage) and generalises thus:
(DR14) ∴ in T2 , the presenter of the show spoke about me (DR24) ∴ in T4, the radio presenter spoke about me (…) (DR105) ∴ the media speak about me conclusion
It proves then that the structure of the mechanism which leads to the formation of delusions of reference thus described, is identical to that of the reasoning which leads to ideas of reference which is associated with personalisation as cognitive distortion.
Finally, it appears that the preceding developments allow to provide a definition of personalisation and to situate it in the context of cognitive distortions (Franceschi 2007, 2009). Personalisation is then likely to manifest itself at the level of primary, secondary or ternary pathogenic arguments, which correspond respectively to the stages of instantiation, of interpretation, and of generalisation. At the level of primary pathogenic arguments, corresponding to a function of instantiation, it consists of instances, the conclusions of which lead the patient to conclude in an unjustified way that some external events are caused by some of his/her actions. At the level of secondary pathogenic arguments, which correspond to a function of interpretation, personalisation takes the form of a reasoning by which the patient interprets the conclusion of primary pathogenic argument by concluding for example that people make fun of him/her. Finally, at the level of ternary pathogenic arguments, associated with a function of generalisation, the patient generalises from the conclusions of several secondary pathogenic arguments and concludes that, in a general way, people make fun of him/her.
Lastly, it appears that the previous definition of personalisation as cognitive distortion allows to describe precisely the relationships between personalisation and close notions such as personalising bias, ideas of reference and delusions of reference.
### References
Beck A. Thinking and depression: Theory and therapy. Archives of General Psychiatry 1964; 10:561-571.
Bressan, P. The Connection Between Random Sequences, Everyday Coincidences, and Belief in the Paranormal. Applied Cognitive Psychology, 2002, 16, 17-34.
Dziegielewski, S. F. DSM-IV-TR in action, Wiley, New York, 2002.
Ellis A. Reason and Emotion in Psychotherapy, Lyle Stuart, New York, 1962.
Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive, 2007, 17-2, 84-88. English translation: http://cogprints.org/5261/
Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques, 2008, 35-2, 451-475. English translation.
Franceschi P. Théorie des distorsions cognitives : la sur-généralisation et l’étiquetage. Journal de Thérapie Comportementale et Cognitive, 2009, 19-4. English translation.
Garety, P., Freeman, D., 1999. Cognitive approaches to delusions: a critical review of theories and evidence. British Journal of Clinical Psychology 38, 113-154.
Langdon R., Cornera T., McLarena J., Wardb P. & Coltheart M., 2006, Externalizing and personalizing biases in persecutory delusions: The relationship with poor insight and theory-of-mind, Behaviour Research and Therapy, 44:699-713
Kinderman, P., Bentall, R., 1997. Causal attributions in paranoia and depression: internal, personal, and situational attributions for negative events. Journal of Abnormal Psychology 106 (2), 341- 345.
Martin, B. Coincidences: Remarkable or random. Skeptical Inquirer, 1998, 22-5, 23-27.
McKay, R., Langdon, R. & Coltheart, 2005. M. Paranoia, persecutory delusions and attributional biases, Psychiatry Research, 136, 233–245
Meyer, E., Lenzenweger, M., 2009. The specificity of referential thinking: A comparison of schizotypy and social anxiety, Psychiatry Research, 165, 78-87.
Startup, M., Startup, S., 2005. On two kinds of delusion of reference, Psychiatry Research ,137, 87-92.
English translation and postprint (with additional illustrations) of a paper published in French under the title “Théorie des distorsions cognitives : application à l’anxiété généralisée” in the Journal de Thérapie Comportementale et Cognitive, 2008, 18, pp. 127-131.
This article follows the contribution to the general theory of cognitive distortions exposed in “Complements to a theory of cognitive distorsions” (Journal de Thérapie Comportementale et Cognitive, 2007). The elements described, namely the reference class, the duality and the system of taxa, are applied here to generalised anxiety disorder. On the one hand, these elements allow to describe the cognitive distortions which are specific to generalised anxiety disorder, consistent with recent work emphasising the role played uncertain situations relative to future events. On the second hand, they allow to define a type of structured reasoning, of inductive nature, which leads to the formation and maintenance of anxious ideas.
## Theory of Cognitive Distortions: Application to Generalised Anxiety Disorder
In Franceschi (2007), we set out to introduce several elements intended to contribute to a general theory of cognitive distortions. These elements are based on three basic notions: the reference class, the duality and the system of taxa. With the help of these three elements, we could define within the same conceptual framework the general cognitive distortions such as dichotomous reasoning, the disqualification of one pole, minimisation and maximisation, as well as the requalification in the other pole and the omission of the neutral. In addition, we could describe as specific cognitive distortions: the disqualification of the positive, selective abstraction and catastrophism.
In what follows, we offer to extend this work by applying it in a specific way to generalised anxiety disorder (GAD), in order to allow their use within cognitive therapy. The present study inserts itself in the context of recent work (Butler & Mathews 1983, 1987, Dalgleish et al. 1997), which notably underlined the major role played, in the context of GAD, by indeterminate situations, and especially by uncertain situations relating to future events. Recent developments, emphasising especially the intolerance with regard to indeterminate future situations, echoed this (Dugas et al. 2004, Canterbury et al. 2004, Carleton et al. 2007).
We shall be interested successively in two main forms of reasoning likely to occur in the context of GAD: on the one hand, the cognitive distortions which are specific to GAD; on the other hand, a structured argument relating to GAD and grounded on inductive logic, which is likely to include one or several of the aforementioned cognitive distortions.
Cognitive distortions in the context of generalized anxiety disorder
The optimal system of taxa
The conceptual framework defined in Franceschi (2007) is based on three fundamental elements: the duality, the reference class, and the system of taxa, which allow to define the general cognitive distortions. These three notions also allow to describe the specific cognitive distortions which are applicable to GAD. In this context, as we will see it, the reference class for the latter specific cognitive distortions identifies itself with the class of future events of the patient’s life. Moreover, the duality assimilates itself to the Positive/Negative duality. Finally, for the sake of the present discussion, we shall make use of the system of taxa (its choice is more or less arbitrary) described in Franceschi (2007), which includes 11 taxa, denoted by E1 to E11, where E6 denotes the neutral taxon. Such conceptual framework allows then to define the specific cognitive distortions in the context of GAD. We offer to examine them in turn.
Dichotomous reasoning
Dichotomous reasoning
An instance of dichotomous reasoning related to GAD consists for the patient to only consider future events from the viewpoint of the extreme taxa corresponding to each pole of the Positive/Negative duality. Hence, the patient only considers future events which present either a very positive, or a very negative nature. All other events, being either neutral, positive or negative to a lesser degree, are thus ignored. This type of reasoning can be analysed as an instance of dichotomous reasoning, applied to the class of the events of the patient’s future life and to the Positive/Negative duality.
Disqualification of one pole
The disqualification of one pole
An instance of the disqualification of one pole related to GAD consists for the patient to only envisage, among future events likely to occur, those which present a negative nature. The patient tends then to be unaware of positive future events that could happen, by considering that they do not count, for this or that reason. In the present context, this type of reasoning can be analysed as an instance of disqualification of one pole, applied to the reference class of the events of the patient’s future life and to the Positive/Negative duality, i.e. disqualification of the positive.
Arbitrary focus on a given modality
Arbitrary focus
In GAD, a typical instance of arbitrary focus, consists for the patient to focus on a possible future event, the nature of which turns out to be negative. This can be analysed as focusing on one of the taxa of the Positive/Negative duality, at the level of the class of the future events of the patient’s life.
Omission of the neutral
Omission of the neutral
A specific instance for GAD consists for the patient to be completely unaware of possible future events the nature of which is neutral, i.e. those which are neither positive nor negative.
Requalification into the other pole
Requalifcation into the other pole
In the context of GAD, the corresponding cognitive distortion consists in requalifying as negative a possible future event, whereas it should be considered objectively as positive. Such cognitive distortion consists of a requalification in the other pole applied to the reference class of the future events of the patient’s life and to the Positive/Negative duality, i.e. requalification in the negative.
Minimisation or maximisation
Maximisation and minimisation
A specific instance of minimisation applied to GAD consists for the patient to consider some possible future events as less positive than they truly are in reality. With maximisation, the patient considers some possible future events as more negative than they objectively are.
Primary, secondary and tertiary anxiogenous arguments
At this stage, it is worth also considering a certain type of reasoning, likely to be met in GAD, which can include several instances of the aforementioned cognitive distortions. This type of reasoning presents an anxiogenous nature, because it leads the patient to predict that a future event of negative nature is going to occur. Such reasoning is underlain by a structure which presents an inductive nature. Before analysing in detail the different steps of the corresponding reasoning, it is worth describing preliminarily its internal structure. The latter is the following (in what follows, the symbol denotes the conclusion):
(1) the event E1 of negative nature did occur to me premiss (2) the event E2 of negative nature did occur to me premiss (3) the event E3 of negative nature did occur to me premiss (…) premiss (10) the event E10 of negative nature did occur to me premiss (11) all events that occur to me are of negative nature from (1)-(10) (12) « I am always unlucky », « I am ill-fated » from (11) (13) the future event E11 of negative nature may occur premiss (14) the future event E11 of negative nature will occur from (11), (13)
The essence of such reasoning is of a logically inductive nature. The patient enumerates then some events of his/her past or present life, the nature of which he/she considers as negative. He/she reaches then by generalisation the conclusion according to which all events that which occur to him/her are negative. From this generalisation, he/she infers a prediction relating to a future event, likely to happen, which he/she considers as negative. The patient is thus led to the anxiogenous conclusion that an event of negative nature is going to occur.
In such reasoning, it is worth pointing out that the reference class identifies itself with the class of past, present and future events, of the patient’s life. Typically, in this type of reasoning, the generalisation is grounded on present or past events, while a future event is the object of the corresponding inductive prediction. This is different from the reference class applicable to the cognitive distortions mentioned above, where the reference class identifies itself exclusively with the future events of the patient’s life.
At this stage, it proves to be necessary to identify the fallacious steps in the patient’s reasoning, to allow their use in cognitive therapy of GAD. To this end, we can differentiate several steps in the structure of the corresponding reasoning. It proves indeed that some steps are valid arguments (an argument is valid when its conclusion is true if its premises are true), while others are invalid. For this purpose, it is worth drawing within this type of reasoning, a distinction between primary, secondary or ternary anxiogenous arguments.
Primary anxiogenous arguments
The first step in the type of aforementioned reasoning, consists for the patient to think to a past negative event, in the following way:
(1) the event E1 of negative nature did occur to me
It is however possible to describe more accurately the corresponding cognitive process, under the form of an argument that we can term a primary anxiogenous argument, the structure of which is the following:
(1a) the event E1 did occur to me (1b) the event E1 was of negative nature (1) the event E1 of negative nature did occur to me from (1a), (1b)
By such cognitive process, the patient is led to the conclusion that some negative event did occur to him/her. This type of argument proves to be entirely valid inasmuch as the event in question presents well, objectively, a negative nature. However, it can also turn out to be invalid, if the event in question presents, objectively, a positive (or neutral) nature. What is then defective in this type of reasoning, is the fact that premise (1b) turns then out to be false. Such can notably be the case for example if the patient makes use of a cognitive distortion such as requalification in the negative. In such case, the patient considers then as negative an event the nature of which is objectively positive.
Secondary anxiogenous arguments
Anxiogenous secondary arguments are constituted, at the level of the above-mentioned reasoning, of the part that takes into account the instances (1)-(10) and proceeds then by generalisation. The patient counts thus some instances of events that did occur to him/her, the nature of which he/she considers as negative, and concludes that all events that did occur to him/her were negative, in the following way:
(1) the event E1 of negative nature did occur to me (2) the event E2 of negative nature did occur to me (…) (10) the event E10 of negative nature did occur to me (11) all events that occur to me are of negative nature from (1)-(10)
Such generalisation may constitute a fully valid argument. For the resulting generalisation constitutes a fully correct inductive reasoning, if the premises (1)-(10) are true. However, such type of reasoning is most often defective from two different viewpoints, thus distorting the conclusion which results from it. Above all, as we have did just see it, some past events of positive nature can have been counted among the number of negative events, by the effect of a requalification in the negative. In that case, the enumeration of instances includes then some false premises, thus invalidating the resulting generalisation. Secondly, some past or present positive (or neutral) events can have been omitted in the corresponding enumeration. Such omission can result from the use of some cognitive distortions, such as disqualification of the positive. In such case, the relevant class of reference consisting in present and past events of the patient’s life is only taken into account in a partial or erroneous way. The corresponding reasoning remains then logically valid, but proves to be incorrect, since it takes into account only partly the relevant instances within the reference class, that of the present and past events of the patient’s life.
As we see it finally, the patient proceeds then to a reconstruction of the relevant reference class which proves to be erroneous, due to the use of the following specific cognitive distortions: requalification in the negative and disqualification of the positive (and possibly, omission of the neutral). The corresponding type of reasoning is illustrated on the figure below:
A series of events of the patient’s life, seen (objectively) from the optimal system of taxa After omission of the neutral After requalification in the negative After disqualification of the positive Conclusion: «All events that occur to me are negative »
Figure 1. Incorrect construction of the reference class for induction, after omission of the neutral, requalification in the negative, and then disqualification of the positive
Such mechanism, as we did see it, illustrates how the formation of anxious ideas is made. However; a mechanism of the same nature is also likely to contribute to their maintenance. For once the generalisation (11) according to which all events which occur to the patient are of negative nature, has been established by means of the above reasoning, its maintenance is made as soon as an event occurs which confirms this latter generalisation. When a new negative event indeed happens, the patient concludes from it that it confirms generalisation (11). Such mechanism, at the stage of the maintenance of anxious ideas, constitutes a confirmation bias. For the patient only counts those events of negative nature related to him/her that confirm indeed the generalisation (11), but without taking into account those events of positive nature which occur to him/her and that would then disconfirm the idea according to which all events which occur to him/her are of negative nature.
Ternary anxiogenous arguments
Lastly, it is worth mentioning the role played by ternary anxiogenous arguments which consist, at the level of the aforementioned reasoning, in the following sequence:
(11) all events that occur to me are of negative nature
(12) « I am always unlucky », « I am ill-fated »
It consists here of an argument which follows the conclusion of the secondary anxiogenous argument (11), and which, by an additional step (12), aims at interpreting it, at making sense of it. The patient interprets here the fact that the events which occur to him/her are negative, due to the fact that he/she is unlucky, ill-fated.
As we did see it, the interest of drawing a distinction between three types of arguments resides in the fact that each of them has a specific function: the primary stage proceeds by enumerating the instances, the secondary stage operates by generalisation, and the ternary stage, lastly, proceeds by interpretation (Franceschi 2008).
The present study, as we see it, extends recent work (Butler and Mathews on 1987, Dalgleish et al. 1997) emphasising the role played, in GAD, by anticipations concerning indeterminate situations related to future events. In this context, the specific cognitive distortions as well as a reasoning of an inductive structure, contribute then to the vicious circle (Sgard et al. 2006), which results from the process of formation and maintenance of the anxious state.
### References
Butler G & Matews A. Cognitive processes in anxiety. Advances in Behaviour Research and Therapy 1983 ; 5 : 51-62.
Butler G & Matews A. Anticipatory anxiety and risk perception. Cognitive Therapy and Research 1987 ; 11 : 551-565.
Carleton R, Norton M & Aslundson G. Fearing the unknown: A short version of the Intolerance of Uncertainty Scale. Journal of Anxiety Disorders 2007 ; 21-1 : 105-117.
Canterbury R, Golden A, Taghavi R, Neshat-Doost H, Moradi A & Yule W. Anxiety and judgements about emotional events in children and adolescents. Personality and Individual Differences 2004 ; 36 : 695-704.
Dalgleish T, Taghavi R, Neshat-Doost H, Moradi A, Yule W & Canterbury R. Information processing in clinically depressed and anxious children and adolescents. Journal of Child Psychology and Psychiatry 1997 ; 38 : 535-541.
Dugas M, Buhr K & Ladouceur R. The role of intolerance of uncertainty in etiology and maintenance. In R. Heimberg, C. Turk, & D. Mennin (Eds.). Generalized anxiety disorder: Advances in research and practice. Guilford, New York, 2004 (143-163).
Franceschi P. Compléments pour une théorie des distorsions cognitives. Journal de Thérapie Comportementale et Cognitive 2007 ; 17-2 : 84-88
Franceschi P. Une défense logique du modèle de Maher pour les délires polythématiques. Philosophiques 2008 ; 35-2 : 451-475
Sgard F, Rusinek S, Hautekeete M & Graziani P. Biais anxieux de perception des risques. Journal de Thérapie Comportementale et Cognitive 2006 ; 16-1 : 12-15.
A paper appeared (2006) in French in the Journal of Philosophical Research, vol. 31, pages 123-141, under the title “Situations probabilistes pour n-univers goodmaniens.”
I proceed to describe several applications of the theory of n-universes through several different probabilistic situations. I describe first how n-universes can be used as an extension of the probability spaces used in probability theory. The extended probability spaces thus defined allow for a finer modelling of complex probabilistic situations and fits more intuitively with our intuitions related to our physical universe. I illustrate then the use of n-universes as a methodological tool, with two thought experiments described by John Leslie. Lastly, I model Goodman’s paradox in the framework of n-universes while also showing how these latter appear finally very close to goodmanian worlds.
## Probabilistic Situations for Goodmanian N-universes
The n-universes were introduced in Franceschi (2001, 2002) in the context of the study of the probabilistic situations relating to several paradoxes which are currently the object of intensive studies in the field of analytical philosophy: Goodman’s paradox and the Doomsday Argument. The scope of the present article is twofold: on one hand, to describe how modelling within the n-universes allows to extend the properties of the classical probability spaces used in probability theory, by providing at the same time a finer modelling of some probabilistic situations and a better support for intuition; on the other hand, to show how the use of n-universes allows to simplify considerably the study of complex probabilistic situations such as those which appear in the study of paradoxes.
When one models for example the situation corresponding to the drawing of a ball from an urn, one considers then a restricted temporal space, which limits itself to the few seconds that precede and follow the drawing. Events which took place the day before or one hour before, but also those who will happen for example the day after the drawing, can be purely and simply ignored. A very restricted interval of time, that it is possible to reduce to one or two discrete temporal positions, is then enough for characterising the corresponding situation. It suffices also to consider a restriction of our universe where the space variable is limited to the space occupied by the urn. For it is not useful to take into consideration the space corresponding to the neighbouring room and to the objects which are there. In a similar way, the number of atoms of copper or of molybdenum that are possibly present in the urn, the number of photons which are interacting with the urn at the time of the drawing, or the presence or absence of a sound source of 75 db, etc. can be omitted and ignored. In this context, it is not necessary to take into account the existence of such variables. In such situation, it is enough to mention the variables and constants really used in the corresponding probabilistic situation. For to enumerate all the constants and the variables which describe of our whole universe appears here as an extremely complicated and moreover useless task. In such context, one can legitimately limit oneself to describe a simplified universe, by mentioning only those constants and variables which play a genuine role in the corresponding probabilistic situation.
Let us consider the drawing of a ball from an urn which contains several balls of different colours. To allow the calculation of the likelihood of different events related to the drawing of one or several balls from the urn, probability theory is based on a modelling grounded on probability spaces. The determination of the likelihood of different events is then not based on the modelling of the physical forces which determine the conditions of the drawing, i.e. the mass and the dimensions of the balls, the material of which they are constituted, their initial spatio-temporal position, as well as the characteristics of the forces exercised over the balls to perform a random drawing. The modelling of random phenomena with the help of probability spaces does only retain some very simplified elements of the physical situation which corresponds to the drawing of a ball. These elements are the number and the colour of the balls, as well as their spatio-temporal position. Such methodological approach can be generalised in other probabilistic situations that involve random processes such as the drawing of one or several dices or of one or several cards. Such methodology does not constitute one of the axioms of probability theory, but it consists here of one important tenet of the theory, of which one can suggest that it would be worth being more formalized. It may also be useful to explain in more detail how the elements of our physical world are converted into probability spaces. In what follows, I will set out to show how the probability spaces can be extended, with the help of the theory of n-universes, in order to better restore the structure of the part of our universe which is so modelled.
1. Introduction to n-universes
It is worth describing preliminarily the basic principles underlying the n-universes. N-universes constitute a simplified model of the physical world which is studied in a probabilistic situation. Making use of Ockam’s razor, we set out then to model a physical situation with the help of the simplest universe’s model, in a compatible way however with the preservation of the inherent structure of the corresponding physical situation. At this stage, it proves to be necessary to highlight several important features of n-universes.
1.1. Constant-criteria and variable-criteria
The criteria of a given n-universe include both constants and variables. Although n-universes allow to model situations which do not correspond to our physical world, our concern will be here exclusively with the n-universes which correspond to common probabilistic situations, in adequacy with the fundamental characteristics of our physical universe. The corresponding n-universes include then at the very least one temporal constant or variable, as well as one constant or variable of location. One distinguishes then among n-universes: a T0L0 (a n-universe including a temporal constant and a location constant), a T0L (a temporal constant and a location variable), a TL0 (a temporal variable and a location constant), a TL (a temporal variable and a location variable). Other n-universes also include a constant or a variable of colour, of direction, etc.
1.2. N-universes with a unique object or with multiple objects
Every n-universe includes one or several objects. One distinguishes then, for example: a 0TL0 (n-world including a unique object, a temporal variable and a constant of location), a TL0 (multiple objects, a temporal variable and a location constant).
1.3. Demultiplication with regard to a variable-criterion
It is worth highlighting the property of demultiplication of a given object with regard to a variable-criterion of a given n-universe. In what follows, we shall denote a variable-criterion with demultiplication by *. Whatever variable-criterion of a given n-universe can so be demultiplicated. The fact for a given object to be demultiplicated with regard to a criterion is the property for this object to exemplify several taxa of criterion . Let us take the example of the time criterion. The fact for a given object to be demultiplicated with regard to time resides in the fact of exemplifying several temporal positions. In our physical world, an object 0 can exist at several (successive) temporal positions and finds then itself demultiplicated with regard to the time criterion. Our common objects have then a property of temporal persistence, which constitutes a special case of temporal demultiplication. So, in our universe of which one of the variable-criteria is time, it is common to note that a given object 0 which exists at T1 also exists at T2, …, Tn. Such object has a life span which covers the period T1-Tn. The corresponding n-universe presents then the structure 0T*L0 (T* with simplified notation).
1.4. Relation one/many of the multiple objects with a given criterion
At this stage, it proves to be necessary to draw an important distinction. It is worth indeed distinguishing between two types of situations. An object can thus exemplify, as we did just see it, several taxa of a given variable-criterion. This corresponds to the case of demultiplication which has just been described with regard to a given variable-criterion. But it is also worth taking into account another type of situation, which concerns only those n-universes with multiple objects. Indeed, several objects can instantiate the same taxon of a given criterion. Let us consider first the temporal criterion. Let us place ourselves, for example, in a n-universe with multiple objects including at the same time a temporal variable and a location constant L0. This can correspond to two types of different n-universes. In the first type of n-universe, there is one single object by temporal position. At some point in time, it is therefore only possible to have a unique object in L0 in the corresponding n-universe. We can consider in that case that every object of this n-universe is in relation one with the time taxa. We denote by T*L0 (with simplified notation T) such n-universe. Let us consider now a n-universe with multiple objects including a temporal variable and a location constant, but where several objects 1, 2, 3 can exist at the same time. In that case, the multiple objects are at a given temporal position in L0. The situation then differs fundamentally from the T*L0, because several objects can now occupy the same given temporal position. In other words, the objects can co-exist at a given time. In that case, one can consider that the objects are in relation many with the temporal taxa. We denote then by *T*L0 such n-universe (with simplified notation *T) .
Let us place ourselves now from the point of view of the location criterion. Let us consider a n-universe with multiple objects including at the same time a temporal variable and a variable of location, and where the objects are in relation many with the temporal criterion. It is also worth distinguishing here between two types of n-universes. In the first, a single object can find itself at a given taxon of the location criterion at the same time. There is then one single object by space position at a given time. This allows for example to model the situation which is that of the pieces of a chess game. Let us denote by *TL such n-universe (with simplified notation *TL). In that case, the objects are in relation one with the location criterion. On the other hand, in the second type of n-universe, several objects can find themselves in the same taxon of a location criterion at the same time. Thus, for example, the objects 1, 2, 3 are in L1 at T1. Such situation corresponds for example to an urn (which is thus assimilated with a given taxon of location) where there are several balls at a given time. We denote by *T*L such n-universe, where the objects are in relation many with the location taxa.
One can notice lastly that such differentiation is also worth for the variable-criterion of colour. One can then draw a distinction between: (a) a *T0*L0C (with simplified notation C) where several objects which can co-exist at the same time in a given space position present all necessarily a different colour, because the objects are in relation one with the colour criterion there; (b) a *T0*L0*C (with simplified notation *C) where several objects which can co-exist at the same time at a given space position can present the same colour, because the objects are in relation many with the colour criterion there.
1.5. Notation
At this stage, it is worth highlighting an important point which concerns the used notation. It was indeed made use in what precedes of an extended and of a simplified notation. The extended notation includes the explicit specification of all criteria of the considered n-universe, including at the same time the variable-criteria and the constant-criteria. By contrast, the simplified notation includes only the explicit specification of the variable-criteria of the considered n-universe. For constant-criteria of time and of location of the considered n-universe can be merely deduced from variable-criteria of the latter. This is made possible by the fact that the studied n-universes include, in a systematic way, one or several objects, but also a variable-criterion or a constant-criterion of time and of location.
Let us illustrate what precedes by an example. Consider first the case where we situate ourselves in a n-universe including multiple objects, a constant-criterion of time and a constant-criterion of location. In that case, it appears that the multiple objects exist necessarily at T0. As a result, in the considered n-universe, the multiple objects are in relation many with the constant-criterion of time. And also, there exist necessarily multiple objects at L0. So, the multiple objects are also in relation many with the constant-criterion of location. We place ourselves then in the situation which is that of a *T0*L0. But for the reasons which have just been mentioned, such n-universe can be denoted, in a simplified way, by .
The preceding remarks suggest then a simplification, in a general way, at the level of the used notation. Indeed, since a n-universe includes multiple objects and since it includes a constant-criterion of time, the multiple objects are necessarily in relation many with the constant-criterion of time. The n-universe is then a *T0. But it is possible to simplify the corresponding notation into . If a n-universe also includes multiple objects and a constant-criterion of location, the multiple objects are necessarily in relation many with the constant-criterion of location. The given n-universe is then a *L0, and it is possible to simplify the notation of the considered n-universe in . As a result, it is possible to simplify the notations *L0*T0 into , *L0T into T, *L0*T into *T, *L0*T* into *T*, etc.
2. Modelling random events with n-universes
The situations traditionally implemented in probability theory involve dices, coins, card games or else some urns that contain balls. It is worth setting out to describe how such objects can be modelled within the n-universes. It also proves to be necessary to model the notion of a “toss” in the probability spaces extended to n-universes. One can make use of the modellings that follow:1
2.1. Throwing a dice
How can we model a toss such as the result of the throwing of the dice is “5 “? We model here the dice as a unique object that finds itself at a space location L0 and which is susceptible of presenting at time T0 one discrete modality of space direction among {1,2,3,4,5,6}. The corresponding n-universe includes then a unique object, a variable of direction and a temporal constant. The unique object can only present one single direction at time T0 and is not with demultiplication with regard to the criterion of direction. The n-universe is a O (with extended notation 0T0L0O). Traditionally, we have the sample space = {1,2,6} and the event {5}. The drawing of “5 ” consists here for the unique object to have direction 5 among {1,2,6} at time T0 and at location L0. We denote then the sample space by 0T0L0O{1,2,…,6} and the event by 0T0L0O{5}.2
How can we model two successive throws of the same dice, such as the result is “5” and then “1”? Traditionally, we have the sample space = {1,2,…,6}2 and the event {5,1}. Here, it corresponds to the fact that the dice 0 has direction 5 and 1 respectively at T1 and T2. In the corresponding n-universe, we have now a time variable, including two positions: T1 and T2. Moreover, the time variable is with demultiplication because the unique object exists at different temporal positions. The considered n-universe is therefore a T*O (with extended notation 0T*L0O). We denote then the sample space by 0T*{1,2}L0O{1,2,…,6} and the event by {0T*{1}L0O{5}, 0T*{2}L0O{1}}.
2.2. Throwing a coin
How can we model the toss, for example of Tails, resulting from the flipping of a coin? We model here the coin as a unique object presenting 2 different modalities of direction among {P,F}. The corresponding n-universe is identical to the one which allows to model the dice, with the sole difference that the direction criterion includes only two taxa: {P,F}. The corresponding n-universe is therefore a O (with extended notation 0T0L0O). Classically, we have: = {P,F} and {P}. Here, the Tails-toss is assimilated with the fact for the unique object to take direction {P} among {P,F} at time T0 and at location L0. The sample space is then denoted by 0T0L0O{P,F} and the event by 0T0L0O{P}.
How can we model two successive tosses of the same coin, such as the result is “Heads” and then “Tails”? Classically, we have the sample space = {P,F}2 and the event {F,P}. As well as for the modelling of the successive throws of the same dice, the corresponding n-universe is here a T*O (with extended notation 0T*L0O). The sample space is then denoted by by 0T*{1,2}L0O{P,F} and the event by {0T*{1}L0O{F}, 0T*{2}L0O{P}}.
2.3. Throwing several discernible dices
How can we model the throwing of two discernible dices at the same time, for example the simultaneous toss of one “3” and of one “5”? The discernible dices are modelled here as multiple objects being each at a given space position and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6}. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. In addition, the multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices could have the same direction corresponds to the fact that objects are in relation many with the criterion of direction. There exists also a location variable, each of the dices 1 and 2 being at one distinct space position. We consider then that the latter property renders the dices discernible. The objects are here in relation one with the location criterion. In addition, the objects can only occupy one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The n-universe is then a L*O (with extended notation *T0L*O). Classically, one has: = {1,2,3,4,5,6}2 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are to be found respectively at L1 and L2 and present a given direction among {1,2,6} at time T0. We denote then the sample space by {1,2}*T0L{1,2}*O{1,2,…,6} and the event by {{1}*T0L{1}*O{3}, {2}*T0L{2}*O{5}}.
2.4. Throwing several indiscernible dices
How can we model the throwing of two indiscernible dices, for example the toss of one “3” and one “5” at the same time? Both indiscernible dices are modelled as multiple objects being at space position L0 and susceptible of presenting at time T0 one modality of space direction among {1,2,3,4,5,6} at a given location. The multiple objects co-exist at the same temporal position, so that the objects are in relation many with the temporal constant. The multiple objects can only present one single direction at time T0 and are not therefore with demultiplication with regard to the criterion of direction. The fact that both dices are susceptible of having the same direction corresponds to the fact that the objects are in relation many with the criterion of direction. Both dices 1 and 2 are at the same location L0, what makes them indiscernible. In addition, the multiple objects are in relation many with the constant-criterion of location. Lastly, the objects can only be at one single space position at time T0 and are not therefore with demultiplication with regard to the location criterion. The corresponding n-universe is then a *O (with extended notation *T0*L0*O). Classically, we have: = (i, j) with 1 i j 6 and {3,5}. Here, it corresponds to the fact that the dices 1 and 2 are both in L0 and present a given direction among {1,2,…,6} at T0. The sample space is then denoted by {1,2}*T0*L0*O{1,…,6} and the event by {{1}*T0*L0*O{3}, {2}*T0*L0*O{5}}.
2.5. Drawing a card
How can we model the drawing of a card, for example the card #13, in a set of 52 cards? Cards are modelled here as multiple objects presenting each a different colour among {1,2,…,52}. The cards’ numbers are assimilated here with taxa of colour, numbered from 1 to 52. Every object can have only one single colour at a given time. As a result, the multiple objects are not with demultiplication with regard to the colour criterion. In addition, a given card can only present one single colour at the same time. Hence, the objects are in relation one with the colour criterion. Moreover, the multiple objects can be at a given time at the same space location (to fix ideas, on the table). The objects are then in relation many with the location criterion. Lastly, the objects can co-exist at the same given temporal position. Thus, they are in relation many with the time criterion. The corresponding n-universe is then a C (with extended notation *T0*L0C). How can we model the drawing of a card? Classically, we have the sample space = {1,2,…,52} and the event {13}. Here, the drawing of the card #13 is assimilated with the fact that the object the colour of which is #13 is at T0 at location L0. The sample space is then denoted by {1,2,…,52}*T0*L0C{1,2,…,52} and the event by {1}*T0*L0C{13}.
The drawing of two cards at the same time or the successive drawing of two cards are then modelled in the same way.
2.6 Drawing of a ball from an urn containing red and blue balls
How can we model the drawing of, for example, a red bowl, from an urn containing 10 balls among which 3 red balls and 7 blue balls? The balls are modelled here as multiple objects presenting each one colour among {R,B}. There exists then a colour variable in the corresponding n-universe. In addition, several objects can present the same colour. The objects are then in relation many with the variable-criterion of colour. Moreover, the objects are in relation many with regard to the constant-criteria of time and location. The corresponding n-universe is therefore a *T0**L0*C (with simplified notation *C). Classically, we have the sample space = {R,R,R,B,B,B,B,B,B,B} and the event {R}. The sample space is then denoted by {1,2,…,10}*T0**L0*C{R,B} and the event by {{1}*T0**L0*C{R}}.
The drawing of two balls at the same time or the successive drawing of two balls are modelled in the same way.
3. Dimorphisms and isomorphisms
The comparison of the structures of the extended (to n-universes) sample spaces corresponding to two given probabilistic situations allows to determine if these situations are, from a probabilistic viewpoint, isomorphic or not. The examination of the structures of the sample spaces allows to determine easily the isomorphisms or, on the contrary, the dimorphisms. Let us give some examples.
Consider a first type of application where one wonders whether two probabilistic situations are of comparable nature. To this end, we model the two distinct probabilistic situations within the n-universes. The first situation is thus modelled in a *T0*L0*C (with simplified notation *C), and the second one in a *T0*L0C (with simplified notation C). One notices then a dimorphism between the n-universes that make it possible to model respectively the two probabilistic situations. Indeed, in the first situation, the multiple objects are in relation many with the colour criterion, corresponding thus to the fact that several objects can have an identical colour at a given moment and location. On the other hand, in the second situation, the multiple objects are in relation one with the colour criterion, what corresponds to the fact that each object has a different colour at a given time and location. The dimorphism observed at the level of the demultiplication of the variable-criterion of colour in the two corresponding n-universes makes it possible to conclude that the two probabilistic situations are not of a comparable nature.
It is worth considering now a second type of application. The throwing of two discernible dice is modelled, as we did see it, in a {1,2}T0*L{1,2}*O{1,…,6}. Now let us consider a headlight which can take at a given time one colour of 6 colours numbered from 1 to 6. If one considers now two headlights of this type, it appears that the corresponding situation can be modelled in a {1,2}T0*L{1,2}*C{1,…, 6}. In this last case, it appears that the variable-criterion of colour replaces the criterion of orientation. At this stage, it proves that the structure of such n-universe (with simplified notation L*C) is isomorphic to that of the n-universe in which the throwing of two discernible dice was modelled (with simplified notation L*O). This makes it possible to conclude that the two probabilistic situations are of a comparable nature.
Let us consider now a concrete example. John Leslie (1996, 20) describes in the following terms the Emerald case:
Imagine an experiment planned as follows. At some point in time, three humans would each be given an emerald. Several centuries afterwards, when a completely different set of humans was alive, five thousands humans would again each be given an emerald in the experiment. You have no knowledge, however, of whether your century is the earlier century in which just three people were to be in this situation, or the later century in which five thousand were to be in it. Do you say to yourself that if yours were the earlier century then the five thousand people wouldn’t be alive yet, and that therefore you’d have no chance of being among them? On this basis, do you conclude that you might just as well bet that you lived in the earlier century?
Leslie thus puts in parallel a real situation related to some emeralds and a probabilistic model concerning some balls in a urn. Let us proceed then to model the real, concrete, situation, described by Leslie, in terms of n-universes. It appears first that the corresponding situation is characterized by the presence of multiple objects: the emeralds. We find then ourselves in a n-universe with multiple objects. On the second hand, one can consider that the emeralds are situated at one single place: the Earth. Thus, the corresponding n-universe has a location constant (L0). Leslie also distinguishes two discrete temporal positions in the experiment: the one corresponding to a given time and the other being situated several centuries later. The corresponding n-universe comprises then a time variable with two taxa: T1 and T2. Moreover, it proves to be that the emeralds existing in T1 do not exist in T2 (and reciprocally). Consequently, the n-universe corresponding to the emerald case is a n-universe which is not with temporal demultiplication. Moreover, one can observe that several emeralds can be at the same given temporal position Ti: three emeralds exist thus in T1 and five thousand in T2. Thus, the objects are in relation many with the time variable. Lastly, several emeralds can coexist in L0 and the objects are thus in relation many with the location constant. Taking into account what precedes, it appears thus that the Emerald case takes place in a *T (with extended notation *T*L0), a n-universe with multiple objects, comprising a location constant and a time variable with which the objects are in relation many.
Compare now with the situation of the Little Puddle/London experiment, also described by Leslie (1996, 191):
Compare the case of geographical position. You develop amnesia in a windowless room. Where should you think yourself more likely to be: in Little Puddle with a tiny situation, or in London? Suppose you remember that Little Puddle’s population is fifty while London’s is ten million, and suppose you have nothing but those figures to guide you. (…) Then you should prefer to think yourself in London. For what if you instead saw no reason for favouring the belief that you were in the larger of the two places? Forced to bet on the one or on the other, suppose you betted you were in Little Puddle. If everybody in the two places developed amnesia and betted as you had done, there would be ten million losers and only fifty winners. So, it would seem, betting on London is far more rational. The right estimate of your chances of being there rather than in Little Puddle, on the evidence on your possession, could well be reckoned as ten million to fifty.
The latter experiment is based on a real, concrete, situation, to be put in relation with an implicit probabilistic model. It appears first that the corresponding situation characterises itself by the presence of multiple inhabitants: 50 in Little Puddle and 10 million in London. The corresponding n-universe is then a n-universe with multiple objects. It appears, second, that this experiment takes place at one single time: the corresponding n-universe has then one time constant (T0). Moreover, two space positions – Little Puddle and London – are distinguished, so that we can model the corresponding situation with the help of a n-universe comprising two space positions: L1 and L2. Moreover, each inhabitant is either in Little Puddle or in London, but but no one can be at the two places at the same time. The corresponding n-universe is then not with local demultiplication. Lastly, one can notice that several people can find themselves at a given space position Li: there are thus 50 inhabitants at Little Puddle (L1) and 10 million in London (L2). The objects are thus in a relation many with the space variable. And in a similar way, several inhabitants can be simultaneously either in Little Puddle, or in London, at time T0. Thus, the objects are in relation many with the time constant. Taking into account what precedes, it appears that the situation of the Little Puddle/London experiment takes place in a *L (with extended notation *T0*L), a n-universe with multiple objects, comprising a time constant and a location variable, with which the objects are in relation many.
As we can see it, the emerald case takes place in a *T, whereas the Little Puddle/London experiment situates itself in a *L. This makes it possible to highlight the isomorphic structure of the two n-universes in which the two experiments are respectively modelled. This allows first to conclude that the probabilistic model which applies to the one, is also worth for the other one. Moreover, it appears that both the *T and the *L are isomorphic with the *C. This makes it possible to determine straightforwardly the corresponding probabilistic model. Thus, the situation corresponding to both the emerald case and the Little Puddle/London experiment can be modelled by the drawing of a ball from an urn comprising red and blue balls. In the emerald case, it consists of an urn comprising 3 red balls and 5000 green balls. In the Little Puddle/London experiment, the urn includes thus 50 red balls and 107 green balls.
Another interest of the n-universes as a methodological tool resides in their use to clarify complex situations such as those which are faced in the study of paradoxes. I will illustrate in what follows the contribution of the n-universes in such circumstances through the analysis of Goodman’s paradox.3
Goodman’s paradox was described in Fact, Fiction and Forecast (1954, 74-75). Goodman explains then his paradox as follows. Every emeralds which were until now observed turned out to be green. Intuitively, we foresee therefore that the next emerald that will be observed will also be green. Such prediction is based on the generalisation according to which all emeralds are green. However, if one considers the property grue, that is to say “observed before today and green, or observed after today and not-green”,4 we can notice that this property is also satisfied by all instances of emeralds observed before. But the prediction which results from it now, based on the generalisation according to which all emeralds are grue, is that the next emerald to be observed will be not-green. And this contradicts the previous conclusion, which is conforms however with our intuition. The paradox comes here from the fact that the application of an enumerative induction to the same instances, with the two predicates green and grue, leads to predictions which turn out to be contradictory. This contradiction constitutes the heart of the paradox. One of the inductive inferences must then be fallacious. And intuitively, the conclusion according to which the next observed emerald will be not-green appears erroneous.
Let us set out now to model the Goodman’s experiment in terms of n-universes. It is necessary for it to describe accurately the conditions of the universe of reference in which the paradox takes place. Goodman makes thus mention of properties green and not-green applicable to emeralds. Colour constitutes then one of the variable-criteria of the n-universe in which the paradox takes place. Moreover, Goodman draws a distinction between emeralds observed before T and those which will be observed after T. Thus, the corresponding n-universe also includes a variable-criterion of time. As a result, we are in a position to describe the minimal universe in which Goodman (1954) situates himself as a coloured and temporal n-universe, i.e. a CT.
Moreover, Goodman makes mention of several instances of emeralds. It could then be natural to model the paradox in a n-universe with multiple objects, coloured and temporal. However, it does not appear necessary to make use of a n-universe including multiple objects. Considering the methodological objective which aims at avoiding a combinatorial explosion of cases, it is indeed preferable to model the paradox in the simplest type of n-universe, i.e. a n-universe with a unique object. We observe then the emergence of a version of the paradox based on one unique emerald the colour of which is likely to vary in the course of time. This version is the following. The emerald which I currently observe was green all times when I did observe it before. I conclude therefore, by induction, that it will be also green the next time when I will observe it. However, the same type of inductive reasoning also leads me to conclude that it will be grue, and therefore not-green. As we can see, such variation always leads to the emergence of the paradox. The latter version takes p lace in a n-universe including a unique object and a variable of colour and of time, i.e. a CT. At this step, given that the original statement of the paradox turns out to be ambiguous in this respect, and that the minimal context is that of a CT, we will be led to distinguish between two situations: the one which situates itself in a CT, and the one which takes place in a CT (where denotes a third variable-criterion).
Let us place ourselves first in the context of a coloured and temporal n-universe, i;e. a CT. In such universe, to be green, is to be green at time T. In this context, it appears completely legitimate to project the shared property of colour (green) of the instances through time. The corresponding projection can be denoted by C°T. The emerald was green every time where I observed it before, and the inductive projection leads me to conclude that it will be also green next time when I will observe it. This can be formalized as follows (V denoting green):
(I1) VT1·VT2·VT3·…·VT99 instances (H2) VT1·VT2·VT3·…·VT99·VT100 generalisation (P3) VT100 from (H2)
The previous reasoning appears completely correct and conforms to our inductive practice. But are we thus entitled to conclude from it that the green predicate is projectible without restriction in the CT? It appears not. For the preceding inductive enumeration applies indeed to a n-universe where the temporal variable corresponds to our present time, for example the period of 100 years surrounding our present epoch, that is to say the interval [-100, +100] years. But what would it be if the temporal variable extended much more far, by including for example the period of 10 thousand million years around our current time, that is to say the interval [-1010, +1010] years. In that case, the emerald is observed in 10 thousand million years. At that time, our sun is burned out, and becomes progressively a white dwarf. The temperature on our planet then warmed itself up in significant proportions to the point of attaining 8000°: the observation reveals then that the emerald – as most mineral – suffered important transformations and proves to be now not-green. Why is the projection of green correct in the CT where the temporal variable is defined by restriction in comparison with our present time, and incorrect if the temporal variable assimilates itself by extension to the interval of 10 thousand million years before or after our present time? In the first case, the projection is correct because the different instances of emeralds are representative of the reference class on which the projection applies. An excellent way of getting representative instances of a given reference class is then to choose the latter by means of a random toss. On the other hand, the projection is not correct in the second case, for the different instances are not representative of the considered reference class. Indeed, the 99 observations of emeralds come from our modern time while the 100th concerns an extremely distant time. So, the generalisation (H2) results from 99 instances which are not representative of the CT[-1010, +1010] and does not allow to be legitimately of use as support for induction. Thus green is projectible in the CT[-102, +102] and not projectible in the CT[-1010, +1010]. At this stage, it already appears that green is not projectible in the absolute but turns out to be projectible or not projectible relative to this or that n-universe.
In the light of what precedes, we are from now on in a position to highlight what proved to be fallacious in the projection of generalisation according to which “all swans are white”. In 1690, such hypothesis resulted from the observation of a big number of instances of swans in Europe, in America, in Asia and in Africa. The n-universe in which such projection did take place was a n-universe with multiple objects, including a variable of colour and of location. To simplify, we can consider that all instances had being picked at constant time T0. The corresponding inductive projection C°L led to the conclusion that the next observed swan would be white. However, such prediction turned out to be false, when occurred the discovery in 1697 by the Dutch explorer Willem de Vlamingh of black swans in Australia. In the n-universe in which such projection did take place, the location criterion was implicitly assimilating itself to our whole planet. However, the generalisation according to which “all swans are white” was founded on the observation of instances of swans which came only from one part of the n-universe of reference. The sample turned out therefore to be biased and not representative of the reference class, thus yielding the falseness of the generalisation and of the corresponding inductive conclusion.
Let us consider now the projection of grue. The use of the grue property, which constitutes (with bleen) a taxon of tcolour*, is revealing of the fact that the used system of criteria comes from the Z. The n-universe in which takes place the projection of grue is then a Z, a n-universe to which the CT reduces. For the fact that there exists two taxa of colour (green, not-green) and two taxa of time (before T, after T) in the CT determines four different states: green before T, not-green before T, green after T, not-green after T. By contrast, the Z only determines two states: grue and bleen. The reduction of the CT to the Z is made by transforming the taxa of colour and of time into taxa of tcolour*. The classical definition of grue (green before T or not-green after T) allows for that. In this context, it appears that the paradox is still present. It comes indeed under the following form: the emerald was grue every time that I did observe it before, and I conclude inductively that the emerald will also be grue and thus not-green the next time when I will observe it. The corresponding projection Z°T can then be formalized (G denoting grue):
(I4*) GT1·GT2·GT3·…·GT99 instances (H5*) GT1·GT2·GT3·…·GT99·GT100 generalisation (H5’*) VT1·VT2·VT3·…·VT99·~VT100 from (H5*), definition (P6*) GT100 prediction (P6’*) ~VT100 from (P6*), definition
What is it then that leads to deceive our intuition in this specific variation of the paradox? It appears here that the projection of grue comes under a form which is likely to create an illusion. Indeed, the projection Z°T which results from it is that of the tcolor* through time. The general idea which underlies inductive reasoning is that the instances are grue before T and therefore also grue after T. But it should be noticed here that the corresponding n-universe is a Z. And in a Z, the only variable-criterion is tcolor*. In such n-universe, an object is grue or bleen in the absolute. By contrast, an object is green or not-green in the CT relative to a given temporal position. But in the Z where the projection of grue takes place, an additional variable-criterion is missing so that the projection of grue could be legitimately made. Due to the fact that an object is grue or bleen in the absolute in a Z, when it is grue before T, it is also necessarily grue after T. And from the information according to which an object is grue before T, it is therefore possible to conclude, by deduction, that it is also grue after T. As we can see it, the variation of the paradox corresponding to the projection Z°T presents a structure which gives it the appearance of an enumerative generalisation but that constitutes indeed a genuine deductive reasoning. The reasoning that ensues from it constitutes then a disguised form of induction, a pseudo-induction.
Let us envisage now the case of a coloured, temporal n-universe, but including an additional variable-criterion , i.e. a CT. A n-universe including variable-criteria of colour, of time and location,5 i.e. a CTL, will be suited for that. To be green in a CTL, is to be green at time T and at location L. Moreover, the CTL reduces to a ZL, a n-universe the variable-criteria of which are tcolor* and location. The taxa of tcolor* are grue and bleen. And to be grue in the ZL, is to be grue at location L.
In a preliminary way, one can point out here that the projections CTL and ZTL do not require a separate analysis. Indeed, these two projections present the same structure as those of the projections CT and ZT which have just been studied, except for an additional differentiated criterion of location. The conditions under which the paradox dissolves when one compares the projections CT and ZT apply therefore identically to the variation of the paradox which emerges when one relates the projections CTL and ZTL .
On the other hand, it appears here opportune to relate the projections CT°L and Z°L which respectively take place in the CTL and the ZL. Let us begin with the projection CT°L. The shared criteria of colour and of time are projected here through a differentiated criterion of location. The taxa of time are here before T and after T. In this context, the projection of green comes under the following form. The emerald was green before T in every place where I did observe it before, and I conclude from it that it will be also green before T in the next place where it will be observed. The corresponding projection C°TL can then be formalized as follows:
(I7) VTL1·VTL2·VTL3·…·VTL99 instances (H8) VTL1·VTL2·VTL3·…·VTL99·VTL100 generalisation (P9) VTL100 prediction
At this step, it seems completely legitimate to project the green and before T shared by the instances, through a differentiated criterion of location, and to predict that the next emerald which will be observed at location L will present the same properties.
What is it now of the projection of grue in the CTL? The use of grue conveys the fact that we place ourselves in a ZL, a n-universe to which reduces the CTL and the variable-criteria of which are tcolour* and location. The fact of being grue is relative to the variable-criterion of location. In the ZL, to be grue is to be grue at location L. The projection relates then to a taxon of tcolour* ( grue or bleen) which is shared by the instances, through a differentiated criterion of location. Consider then the classical definition of grue (green before T or non-grue after T). Thus, the emerald was grue in every place where I did observe it before, and I predict that it will also be grue in the next place where it will be observed. If we take T = in 1010 years, the projection Z°L in the ZL appears then as a completely valid form of induction (V~T denoting green after T):
(I10*) GL1·GL2·GL3·…·GL99 instances (H11*) GL1·GL2·GL3·…·GL99·GL100 generalisation (H11’*) VT~V~TL1·VT~V~TL2·VT~V~TL3·…·VT~V~TL99·VT~V~TL100 from (H11*), definition (P12*) GL100 prediction (P12’*) VT~V~TL100 from (P12*), definition
As pointed out by Franck Jackson (1975, 115), such type of projection applies legitimately to all objects which colour changes in the course of time, such as tomatoes or cherries. More still, one can notice that if we consider a very long period of time, which extends as in the example of emeralds until 10 thousand million years, such property applies virtually to all concrete objects. Finally, one can notice here that the contradiction between both concurrent predictions (P9) and (P12’*) has now disappeared since the emerald turns out to be green before T in L100 (VTL100) in both cases.
As we can see, in the present analysis, a predicate turns out to be projectible or not projectible in relative to this or that universe of reference. As well as green, grue is not projectible in the absolute but turns out to be projectible in some n-universes and not projectible in others. It consists here of a difference with several classical solutions offered to solve the Goodman’s paradox, according to which a predicate turns out to be projectible or not projectible in the absolute. Such solutions lead to the definition of a criterion allowing to distinguish the projectible predicates from the unprojectible ones, based on the differentiation temporal/non-temporal, local/non-local, qualitative/non-qualitative, etc. Goodman himself puts then in correspondence the distinction projectible/ unprojectible with the distinction entrenchedi/unentrenched, etc. However, further reflexions of Goodman, formulated in Ways of Worldmakingii, emphasize more the unabsolute nature of projectibility of green or of grue: “Grue cannot be a relevant kind for induction in the same world as green, for that would preclude some of the decisions, right or wrong, that constitute inductive inference”. As a result, grue can turn out to be projectible in a goodmanian world and not projectible in some other one. For green and grue belong for Goodman to different worlds which present different structures of categories.6 In this sense, it appears that the present solution is based on a form of relativism the nature of which is essentially goodmanian.
5. Conclusion
From what precedes and from Goodman’s paradox analysis in particular, one can think that the n-universes are of a fundamentally goodmanian essence. From this viewpoint, the essence of n-universes turns out to be pluralist, thus allowing numerous descriptions, with the help of different systems of criteria, of a same reality. A characteristic example, as we did see it, is the reduction of the criteria of colour and time in a CTL into a unique criterion of tcolour* in a ZL. In this sense, one can consider the n-universes as an implementation of the programme defined by Goodman in Ways of Worldmaking. Goodman offers indeed to construct worlds by composition, by emphasis, by ordering or by deletion of some elements. The n-universes allow in this sense to represent our concrete world with the help of different systems of criteria, which correspond each to a relevant point of view, a way of seeing or of considering a same reality. In this sense, to privilege this or that system of criteria, to the detriment of others, leads to a truncated view of this same reality. And the exclusive choice, without objective motivation, of such or such n-universe leads to engender a biased point of view.
However, the genuine nature of the n-universes turns out to be inherently ambivalent. For the similarity of the n-universes with the goodmanian worlds does not prove to be exclusive of a purely ontological approach. Alternatively, it is indeed possible to consider the n-universes from the only ontological point of view, as a methodological tool allowing to model directly this or that concrete situation. The n-universes constitute then so much universes with different properties, according to combinations resulting from the presence of a unique object or multiple objects, in relation one or many, with demultiplication or not, with regard to the criteria of time, location, colour, etc. In a goodmanian sense also, the n-universes allow then to build so much universes with different structures, which sometimes correspond to the properties of our real world, but which have sometimes some exotic properties. To name only the simplest of the latter, the L* is then a n-universe which includes only one ubiquitous object, presenting the property of being at several locations at the same time.7
At this stage, it is worth mentioning several advantages which would result from the use of the n-universes for modelling probabilistic situations. One of these advantages would be first to allow a better intuitive apprehension of a given probabilistic situation, by emphasising its essential elements and by suppressing its superfluous elements. By differentiating for example depending on whether the situation to model presents a constant or a time variable, a constant or a space variable, a unique object or several objects, etc. the modelling of concrete situations in the n-universes provides a better support to intuition. On the other hand, the distinction according to whether the objects are or not with demultiplication or in relation one/many with regard to the different criteria allows for a precise classification of the different probabilistic situations which are encountered.
One can notice, second, that the use of the notation of the probability spaces extended to the n-universes would allow to withdraw the ambiguity which is sometimes associated with classical notation. As we did see it, we sometimes face an ambiguity. Indeed, it proves to be that {1,2,…,6}2 denotes at the same time the sample space of a simultaneous throwing of two discernible dices in T0 and that of two successive throwing of the same dice in T1 and then in T2. With the use of the notation extended to n-universes, the ambiguity disappears. In effect, the sample space of the simultaneous throwing of two discernible dices in T0 is a {1,2}*T0L{1,2}*O{1,2,…,6}, whilst that of two successive throwing of the same dice in T1 and then in T2 is a 0T*{1,2}L0O{1,2,…,6}.
Finally, an important advantage, as we have just seen it, which would result from a modelling of probabilistic situations extended to n-universe is the easiness with which it allows comparisons between several probabilistic models and it highlights the isomorphisms and the corresponding dimorphisms. But the main advantage of the use of the n-universes as a methodological tool, as we did see it through Goodman’s paradox, would reside in the clarification of the complex situations which appear during the study of paradoxes.8
References
Franceschi, Paul. 2001. Une solution pour le paradoxe de Goodman. Dialogue 40: 99-123, English translation under the title The Doomsday Argument and Hempel’s Problem, http://cogprints.org/2172/. —. 2002. Une application des n-univers à l’argument de l’Apocalypse et au paradoxe de Goodman. Doctoral dissertation, Corté: University of Corsica. [retrievec Dec.29, 2003] Goodman, Nelson. 1954. Fact, Fiction and Forecast. Cambridge, MA: Harvard University Press. —. 1978. Ways of Worldmaking. Indianapolis: Hackett Publishing Company. Jackson, Franck. 1975. “Grue”. The Journal of Philosophy 72: 113-131. Leslie, John. 1996. The End of the World: The Science and Ethics of Human Extinction. London: Routledge.
1 Il convient de noter que ces différentes modélisations ne constituent pas une manière unique de modéliser les objets correspondants dans les n-univers. Cependant, elles correspondent à l’intuition globale que l’on a de ces objets.
2 De manière alternative, on pourrait utiliser la notation 0T0L0O5 en lieu et place de 0T0L0O{5}. Cette dernière notation est toutefois préférée ici, car elle se révèle davantage compatible avec la notation classique des événements.
3 Cette analyse du paradoxe de Goodman correspond, de manière simplifiée et avec plusieurs adaptations, à celle initalement décrite dans Franceschi (2001). La variation du paradoxe qui est considérée ici est celle de Goodman (1954), mais avec une émeraude unique.
4 P and Q being two predicates, grue presents the following structure: (P and Q) or (~P and ~Q).
5 Tout autre critère différent de la couleur et du temps tel que la masse, la température, l’orientation, etc. conviendrait également.
6 Cf. Goodman (1978, 11): “(…) a green emerald and a grue one, even if the same emerald (…) belong to worlds organized into different kinds”.
7 Les n-univers aux propriétés non standard nécessitent une étude plus détaillée, qui dépasse le cadre de la présente étude.
8 Je suis reconnaissant envers Jean-Paul Delahaye pour la suggestion de l’utilisation des n-univers en tant qu’espaces de probabilité étendus. Je remercie également Claude Panaccio et un expert anonyme pour le Journal of Philosophical Research pour des discussions et des commentaires très utiles.
i Entrenched.
ii Cf. Goodman (1978, 11).
Review of John Leslie, Infinite Minds, Oxford, Oxford University Press, 2001, 234 pages.1
Paul Franceschi
Post-publication of the review appeared in Philosophiques, Volume 30, number 2, Autumn 2003
Infinite Minds is the fourth book of John Leslie, which follows Value and Existence (1979), Universes (1989) and The End of the World (1996). Infinite Minds presents a very rich content, and covers a number of particularly varied subjects. Among these latter, one can notably mention: omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, the identity of indiscernables, time, infiniteness, the nature of consciousness.
The book places itself clearly within the field of speculative philosophy. And Leslie is primarily concerned here with considerations not of rigorous demonstration, but rather of plausibility and of coherence. He thus does not hesitate sometimes to attribute a rather weak probability to certain assertions.
Some readers may be rebutted from the beginning by the counter-intuitive assertion that galaxies, planets, animals, but also each of us and our surrounding objects, are mere structures among divine thoughts. One can think that such an assertion has motivated the commentary placed on the book’s cover by a reader from Oxford University Press, according to which it may be difficult to believe that the universe is such that the author describes it. This was also my primary reaction. But if certain readers were to draw from that a hasty conclusion, they would miss then, I think, what constitutes the hidden treasure of the book. Because Infinite Minds resembles a sumptuous temple, whose access however is dissimulated by a gate which looks poorly attractive. Those who will not cross the door, rebutted by the aspect of this latter, will not have the occasion to contemplate the hidden treasures that the book contains. Because the book presents an overall deep structure and coherence, based on the consistency of the author’s pantheist conception of the universe with our current most advanced scientific views with regard to cosmology, physics, as well as with the solutions to several contemporary philosophical problems. To show synthetically how a pantheist vision of the world can cohere with our most recent views with regard to multiple universes, physics and quantum computer science, inasmuch as with relativity theory and recent discussions relating to omniscience, the problem of Evil, the fine-tuning argument, observational selection effects, etc. appears both an immense and deeply original task.
It should be observed here that Leslie is familiar with this type of wide-scale work. It suffices for that to consider his whole work relating to the Doomsday Argument. It is worth evaluating here the immense task which consists in defending point by point the Doomsday Argument against a good hundred different objections. But this vigorous defense of the Doomsday argument has stimulated in return the development of a rich literature, which continuously enlightens a number of fields hitherto ignored.
The variety of pantheism described by Leslie, inspired by Spinoza, characterizes itself by the fact that each of us is nothing but a structure of divine thoughts. Because the divine mind only exists. The galaxies, the planets, the mountains, the human beings that we are, the animals, the flowers are nothing but structures within divine minds. As Leslie points out, this is coherent with the way physicists themselves describe physical objects, by specifying their intrinsic properties. Nevertheless, Leslie is not committed to a conception of panpsychism where all beings and objects which are part of our universe, have mental properties. For according to the author, physical objects such as trees, rocks, sand, exist as structures within the divine mind, but without being equipped themselves with conscience or thought. Here, all things are not equipped with conscience, but are such however that a conscience of these latter things exists.
Moreover, universes in infinite number can exist as structures in the divine mind. The author’s theory appears thus compatible with recent cosmological theories based on the existence of multiple universes. One of these universes is thus our own, which presents such characteristics and an accurate tuning of its parameters (the ratio of the respective masses of the electron and proton, the electron charge, the gravitational constant, Planck’s constant, etc.), that it allows the emergence of an intelligent life.
Furthermore, Leslie suggests the existence not of a single divine mind, but as well of an infinity of divine minds. Each of them is absolutely identical to the others, but has however an autonomous conscience of its own existence.
What is then the status of abstract objects, such as natural integers, in this context? According to certain philosophers, abstract objects also constitute divine thoughts. Such is in particular the viewpoint put forth by Alvin Plantinga, according to which natural integers constitute divine thoughts. But Leslie adopts a different line of thought. Abstract objects such as natural integers have in Infinite Minds’ ontology a completely original status, which is not prima facie obvious, and which deserves a detailed mention. Abstract objects such as natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, are of Platonic essence. And Leslie points out that such abstract objects do not result from our brains, which themselves constitute thoughts in the divine mind. Neither do such objects of Platonic nature result from the divine mind itself. The natural integers, the idea of an apple, or the idea that “2 + 2 = 4”, constitute eternal realities, which are independent of our existence as human beings, of our thoughts and of our language. Leslie explains clearly how the idea that “2 + 2 = 4”, i.e. the fact that “IF two sets of two apples exist, THEN four apples exist” (p. 160) constitutes a Platonic reality, independent of the thoughts of the divine mind and of the human beings that we are.
Leslie also develops the topic of omniscience. According to Leslie, God simply knows all that is worth knowning: (“God knows everything worth knowing”). This seems probably more plausible than the idea according to which God has any knowledge, which notably conflicts with the logically impossible existence, already noted by Patrick Grim, of the set which contains absolutely all truths. Our pretheoretical conception of an omniscient God could well appear naïve, as the author points out, because a many unimportant facts could appear undesirable knowledge there.
Lastly, Leslie develops the point of view according to which God exists by ethical need (“because of its ethical requiredness”). The existence of God and of the cosmos in his entirety is ethically necessary, from all eternity. This argument could well appear more convincing than certain ontological arguments. Because such an ethical need has, according to Leslie, an inherently creative power. But such creative capacity, of Platonic essence, does not proceed of any external cause. It is simply inherent by nature to the ethical necessity.
Leslie’s book also constitutes the courageous expression of a viewpoint. For such pantheist conception does not constitute a widespread opinion within contemporary analytical philosophy. Moreover, Leslie’s variety of pantheism also constitutes a variation of panpsychism. But the attitude of the author appears eminently constructive, because it constrains us to consider more attentively some doctrines than we would tend to reject too easily. One will or not adhere to the pantheist and panpsychist theory exposed in Infinite Minds. But for the majority of readers for whom we can suppose that they will not adhere to the variation of panpsychism thus described, Leslie’s work constitutes nevertheless an admirable and highly original synthesis, showing how an astonishing construction can be elaborated around the pantheist model, while bringing answers to many contemporary philosophical problems. The work will provide new arguments to the defenders of panpsychism. But Infinite Minds will be also prove to be essential to the detractors of panpsychism, who will find there a particularly strong and structured defense.
1This review only differs from the version published in Philosophiques with regard to the status of abstract objects. I thank John Leslie for very useful discussion on this topic.
English translation of a paper published in French in Semiotica, vol. 150(1-4), 2004 under the title “Le problème des relations amour-haine-indifférence”.
This paper is cited in:
• Isis Truck, Nesrin Halouani, & Souhail Jebali (2016) Linguistic negation and 2-tuple fuzzy linguistic representation model : a new proposal, pages 81–86, in Uncertainty Modelling in Knowledge Engineering and Decision Making, The 12th International FLINS Conference on Computational Intelligence in Decision and Control, Eds. Xianyi Zeng, Jie Lu, Etienne E Kerre, Luis Martinez, Ludovic Koehl, 2016, Singapore: World Scientific Publishing.
In On a class of concepts (2002), I described a theory based on the matrices of concepts which aims at constituting an alternative to the classification proposed by Greimas, in the field of paradigmatic analysis. The problem of the determination of the relationships of love/hate/indifference arises in this construction. I state then the problem of the relationships of love/hate/indifference in a detailed way, and several solutions that have been proposed in the literature to solve it. I describe lastly a solution to this problem, based on an extension of the theory of matrices of concepts.
## The Problem of the Relationships of Love-Hate-Indifference
I shall be concerned in this paper with presenting a problem related to the proper definition of the relationships of the following concepts: love, hate and indifference. I will describe first the problem in detail and some proposed solutions. Lastly, I will present my own solution to the problem.
1. The problem
The problem is that of the proper definition of the relationships of the concepts love, hate and indifference. Let us call it the LHI problem. What are then the accurate relationships existing between these three concepts? At first sight, the definition of the relation between love and hate is obvious. These concepts are contraries. The definition of such a relation should be consensual. Nevertheless, the problem arises when one considers the relationship of love and indifference, and of hate and indifference. In these latter cases, no obvious response emerges.
However, the issue needs clarifying. In this context, what should we expect of a solution to the LHI problem? In fact, a rigorous solution ought to define precisely the three relations R, S, T such that love R hate, love S indifference and hate T indifference. And the definitions of these relations should be as accurate as possible.
It is worth mentioning that several authors must be credited for having mentioned and investigated the LHI problem. In particular, it is worth stressing that the difficulties presented within propositional calculus by some assertions of the type x loves y, x hates y, or x is indifferent to y have been hinted at by Emile Benzaken (1990)1:
Nevertheless, the difficulty can arise from pairs of words where the one expresses the contrary (negation) of the other; ‘to hate’ can be considered as the strong negation of ‘to love’, whereas ‘to be indifferent’ would be its weak negation.
The author exposes then the problem of the relationships of love/hate/indifference and proposes his own solution: hate is the strong negation of love, and indifferent is the weak negation of love.
However, it turns out that Benzaken’s solution is unsatisfying for a logician, for the following reasons. On the one hand, this way of solving the problem defines the relations between love and hate (strong negation, according to the author) and between love and indifference (weak negation, on the author’s view), but it fails to define accurately the relations existing between indifference and hate. There is a gap, a lack of response at this step. And mentioned above, a satisfying solution should elucidate the nature of the relationships of the three concepts. On the other hand, the difference between weak negation and strong negation is not made fully explicit within the solution provided by Benzaken. For these reasons, Benzaken’s solution to the LHI problem proves to be unsatisfying.
In a very different context, Rick Garlikov (1998) stresses some difficulties of essentially the same nature as those underlined by Benzaken:
In a seminar I attended one time, one of the men came in all excited because he had just come across a quotation he thought very insightful – that it was not hate that was the opposite of love, but that indifference was the opposite of love, because hate was at least still an emotion. I chuckled, and when he asked why I was laughing, I pointed out to him that both hate and indifference were opposites of love, just in different ways, that whether someone hated you or was indifferent toward you, in neither case did they love you.
Garlikov describes in effect the problem of the relationships of love/hate/indifference and implicitly proposes a solution of a similar nature as that provided by Benzaken. For this reason, Galikov’s account suffers from the same defects as those presented by Benzaken’s solution.
In what follows, my concern will be with settling first the relevant machinery, in order to prepare a few steps toward a solution to the LHI problem.
2. The framework
I will sketch here the formal apparatus described in more detail in Franceschi (2002). To begin with, consider a given duality. Let us denote it by A/Ā. At this step, A and Ā are dual concepts. Moreover, A and Ā can be considered as concepts that are characterized by a contrary component c {-1, 1} within a duality A/Ā, such that c[A] = -1 and c[Ā] = 1. Let us also consider that A and Ā are neutral concepts that can be thus denoted by A0 and Ā0.
Figure 1: The canonical matrix
At this point, we are in a position to define the class of the canonical poles. Consider then an extension of the previous class {A0, Ā0}, such that A0 and Ā0 respectively admit of a positive and a negative correlative concept. Such concepts are intuitively appealing. Let us denote them respectively by {A+, A} and {Ā+, Ā}. At this step, for a given duality A/Ā, we get then the following concepts: {A+, A0, A, Ā+, Ā0, Ā}. Let us call them canonical poles. It should be noted that one could use alternatively the notation (A/Ā, c, p) for a canonical pole.2 In all cases, the components of a canonical pole are a duality A/Ā, a contrary component c {-1, 1} and a canonical polarity p {-1, 0, 1}. This definition of the canonical poles leads to distinguish between the positive (A+, Ā+), neutral (A0, Ā0) and negative (A, Ā) canonical poles. Lastly, the class made up by the 6 canonical poles can be termed the canonical matrix: {A+, A0, A, Ā+, Ā0, Ā}.
Let us investigate now into the nature of the relations existing between the canonical poles of a given matrix. Among the combinations of relations existing between the 6 canonical poles (A+, A0, A, Ā+, Ā0, Ā) of a same duality A/Ā, it is worth emphasizing the following relations: duality, antinomy, complementarity, corollarity, connexity, and anti-connexity. Thus, two canonical poles 1(A/Ā, c1, p1) and 2(A/Ā, c2, p2) of a same matrix are:
(i) dual if their contrary components are opposite and their polarities are neutral3
(ii) contrary (or antinomical) if their contrary components are opposite and their polarities are non-neutral and opposite4
(iii) complementary if their contrary components are opposite and their polarities are non-neutral and equal5
(iv) corollary if their contrary components are equal and their polarities are non-neutral and opposite6
(v) connex if their contrary components are equal and the absolute value of the difference of their polarities equals 17
(vi) anti-connex if their contrary components are opposite and the absolute value of the difference of their polarities equals 18
To sum up: {A0, Ā0} are dual, {A+, Ā} and {A, Ā+} are contraries, {A+, Ā+} and {A, Ā} are complementary, {A+, A} and {Ā+, Ā} are corollary, {A0, A+}, {A0, A}, {Ā0, Ā+} and {Ā0, Ā} are connex, {A0, Ā+}, {A0, Ā}, {Ā0, A+} and {Ā0, A} are anti-connex.
I shall focus now on the types of relations existing, under certain circumstances between the canonical poles of different dualities. Let us define preliminarily the includer relation. Let a concept be an includer for two other concepts and if and only if = . Such a definition captures the intuition that is the minimal concept whose semantic content includes that of and . To give an example concerning truth-value, determinate is an includer for {true, false}.
Let now A and E be two matrices whose canonical poles are respectively {A+, A0, A, Ā+, Ā0, Ā} and {E+, E0, E, Ē+, Ē0, Ē}. These matrices are such that E+, E0, E are the respective includers for {A+, Ā+}, {A0, Ā0}, {A, Ā} i.e. the two matrices are such that E+ = A+ Ā+, E0 = A0 Ā0 and E = A Ā.9
Figure 2
Let us denote this relation by A E. One is now in a position to extend the relations previously defined between the canonical poles of a same matrix, to the relations of a same nature between two matrices presenting the properties of A and E, i.e. such that A E. The relations of 2-duality, 2-antinomy, 2-complementarity, 2-anti-connexity10 ensue then straightforwardly. Thus, two canonical poles 1(A/Ā, c1, p1) and 2(E/Ē, c2, p2) of two different matrices are:
(i’) 2-dual (or trichotomic dual) if their polarities are neutral and if the dual of 2 is an includer for 1
(ii’) 2-contrary11 (or trichotomic contrary) if their polarities are non-neutral and opposite and if the contrary of 2 is an includer for 1
(iii’) 2-complementary (or trichotomic complementary) if their polarities are non-neutral and equal and if the complementary of 2 is an includer for 1
(vi’) 2-anti-connex (or trichotomic anti-connex) if the absolute value of the difference of their polarities is equal to 1 and if the anti-connex of 2 is an includer for 1
To sum up now: {A0, Ē0} and {Ā0, Ē0} are 2-dual, {A+, Ē}, {A, Ē+}, {Ā+, Ē} and {Ā, Ē+} are 2-contrary, {A+, Ē+}, {A, Ē}, {Ā+, Ē+} and {Ā, Ē} are 2-complementary, {A0, Ē+}, {A0, Ē}, {Ā0, Ē+} and {Ā0, Ē} are 2-anti-connex.
Lastly, the notion of a complement of a canonical pole also deserves mention. Let be a canonical pole. Let us denote by ~ its complement, semantically corresponding to non. In the present context, the notion of a complement entails the definition of a universe of reference. I shall focus then on the notion of a complement of a canonical pole defined with regard to the corresponding matrix. In this case, the universe of reference is equal to {A+, A0, A, Ā+, Ā0, Ā} and then ~ = {A+, A0, A, Ā+, Ā0, Ā} – . On has thus for example ~A+ = {A0, A, Ā+, Ā0, Ā} and a similar definition for the complements of the other canonical poles of the matrix. Consider now two matrices such that A E. Under these circumstances, the universe of reference12 is equal to {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē}. Call it the 2-matrix of . It ensues that ~ = {A+, A0, A, Ā+, Ā0, Ā, Ē+, Ē0, Ē} – . We have then the notion of a 2-complement of a canonical pole , defined with regard to a universe of reference consisting of the 2-matrix of . More generally, one has the notion of a ncomplement (n > 0) of a canonical pole with regard to the corresponding n-matrix.
3. A solution
With the relevant machinery in place, we are now in a position to present a solution to the LHI problem. Let us now analyze the problem in the light of the above framework. To begin with, let us analyze the relevant concepts in more detail. The concept love has a positive connotation. It is a meliorative concept that can be denoted by love+. Conversely, the concept hate has a negative connotation. It is a pejorative concept that can be rendered by hate. Similarly, the concept indifference also has a negative connotation. It can be considered a pejorative notion that can be denoted by indifference.
At this step, a difficulty emerges. In effect, it should be stressed that the three concepts are either meliorative or pejorative at a certain degree. And such a degree might be different from one concept to another. For example hate might be pejorative at a 0.95 degree, while indifference might be pejorative at a lesser degree of 0.7. Moreover, it could be said that such a degree might vary from culture to culture, from a given language to another. In sum, the meliorative or pejorative degree of the three concepts, so the objection goes, could be culture-relative.
Nevertheless, such difficulties can be avoided in the present context, since our reasoning will not bear upon the concepts inherent to a specific culture or language, but rather on the canonical concepts described above. Accordingly, we shall replace our usual concepts by the corresponding canonical concepts. There is room for variation in degrees, from culture to culture in the usual concepts of love, hate and indifference. But this point does not affect the current line of reasoning, since it only focuses on canonical concepts. The passage from the non-canonical concepts to the canonical ones goes straightforwardly as follows. Let d[] be the pejorative or meliorative degree of a concept . Hence if d[] ]0.5; 1] then p[] = 1 else if d[] [-1; -0.5[ then p[] = -1. At this point, one can pose legitimately that p[Love] = 1, p[Hate] = -1 and p[Indifference] = -113. As a result, the three concepts can be denoted by Love+, Hate, Indifference.
Figure 3
As noted from the beginning, the relationship of love/hate is unproblematic and identifies itself with the relation of contrary. This applies straightforwardly to the relationship of the canonical concepts Love+/Hate. Hence, the corresponding matrix has the following structure: {Love+, A0, A, Ā+, Ā0, Hate}. Now the next step is the reconstitution of the complete matrix. This task can be accomplished with the help of the definition of the relations of the canonical poles, namely: A is corollary to Love+, Ā+ is corollary to Hate, A0 is connex to Love+ and anti-connex to Hate, Ā0 is connex to Hate and anti-connex to Love+. Given these elements, we are now in a position to reconstitute the corresponding canonical matrix: {Love+, Attraction0, A, Defiance+, Repulsion0, Hate}.14
Let us examine now the case of the concept Indifference. Such a concept inserts itself into a matrix the structure of which is: {E+, E0, E, Ē+, Ē0, Indifference}. Just as before, it is now necessary to reconstitute the complete matrix. This can be done with the help of the corresponding definitions: Ē+ is corollary to Indifference, E is complementary to Indifference, E+ is contrary to Indifference, Ē0 is connex to Indifference and to the corollary of Indifference, E0 is anti-connex to Indifference and to the corollary of Indifference. The associated matrix is then: {E+, Interest0, E, Phlegm+, Detachment0, Indifference}.15
Figure 4
It should be observed now that Interest0 = Attraction0 Repulsion0 i.e. that Interest0 is an includer for Attraction0 and Repulsion0. At this step, given that {Love+, Attraction0, A, Repulsion+, Repulsion0, Hate} {E+, Interest0, E, Phlegm+, Detachment0, Indifference}, the relationship of Love+/Indifference and Hate/Indifference now apply straightforwardly. In effect, it ensues from the above definitions that, on the one hand, Love+ and Indifferenceare trichotomic contraries and on the other hand, Hate and Indifferenceare trichotomic complementaries. At this point, one is finally in a position to formulate a solution to the LHI problem:
(i) love is contrary to hate
(ii) love is 2-contrary to indifference
(iii) hate is 2-complementary to indifference
Hence, R, S, T identify respectively themselves with contrary, trichotomic contrary, trichotomic complementarity.
4. Concluding remarks
At this point, it is tempting not to consider the above analysis as a solution to the LHI problem per se. In effect, the concepts love, hate and indifference seem to be instances of a wider class of concepts whose relationships are of the same nature. This suggests that the same type of solution should be provided to the general problem of the definition of the relations of three given concepts , , . At first sight, certain concepts such as true, false and indeterminate, fall under the scope of the current analysis. Nevertheless, such a claim should be envisaged with caution. To what extent does the present analysis apply to other concepts? This is another problem that needs to be addressed, but whose resolution goes beyond the scope of the present account.16
References
Benzaken, Claude (1991). “Systèmes formels”. Paris, Masson Franceschi, Paul (2002). “Une Classe de Concepts”. Semiotica, 139, pp. 211-26, English translation Garlikov, Rick (1998). “Understanding, Shallow Thinking, and School”. At http://www.garlikov.com/writings.htm
1 My translation. The original text is as follows: ‘La difficulté cependant peut provenir de paires de mots dont l’un exprime le contraire (négation) de l’autre; “haïr” peut être pris comme la négation forte de “aimer” tandis que “être indifférent” en serait la négation faible. (p. 63).
2 With the latter notation, the matrix of the canonical poles is rendered as follows: {(A/Ā, -1, 1), (A/Ā, -1, 0), (A/Ā, -1, -1), (A/Ā, 1, 1), (A/Ā, 1, 0), (A/Ā, 1, -1)}.
3 Formally 1 and 2 are dual if and only if c[1] = – c[2] and p[1] = p[2] = 0.
4 Formally 1 and 2 are antinomical if and only if c[1] = – c[2] and p[1] = – p[2] with p[1], p[2] 0.
5 Formally 1 and 2 are complementary if and only if c[1] = – c[2] and p[1] = p[2] with p[1], p[2] 0.
6 Formally 1 and 2 are corollary if and only if c[1] = c[2] and p[1] = – p[2] with p[1], p[2] 0.
7 Formally 1 and 2 are connex if and only if c[1] = c[2] and │p[1] – p[2]│ = 1.
8 Formally 1 and 2 are anti-connex if and only if c[1] = – c[2] and │p[1] – p[2]│ = 1.
9 It should be observed that one of the three conditions is sufficient. In effect, E+ = A+ Ā+ entails E0 = A0 Ā0 and E = A Ā; E0 = A0 Ā0 implies E+ = A+ Ā+ and E = A Ā; E = A Ā entails E0 = A0 Ā0 and E+ = A+ Ā+.
10 The generalisation to n matrices (n > 1) of the present construction ensues, with the relations of n-duality, n-antinomy, n-complementarity, n-anti-connexity.
11 Or 2-antinomical.
12 In this context, E+, E0 and E can be omitted without loss of content, given their nature of includers.
13 The fact of considering alternatively p[indifference] > -0.5 and thus p[Indifference] = 0 also leads to a solution in the present framework. In this last case, the relations S and T both identify themselves with trichotomic anti-connexity.
14 In the process of reconstitution of the complete matrix, some concepts may be missing. The reason is that they are not lexicalized in the corresponding language. This is notably the case for A. This last concept semantically corresponds to inappropriate, excessive attraction.
15 As far as I can see, the concepts associated with E+ and E are not lexicalized. They respectively correspond to appropriate interest and inappropriate, excessive interest.
16 I thank Professor Claude Panaccio and Rick Garlikov for useful comments on an earlier draft.
// | 69,963 | 307,783 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2017-43 | latest | en | 0.930093 |
https://mathematica.stackexchange.com/questions/146193/the-solutions-of-system-of-equations-are-not-continuous | 1,721,504,445,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517515.18/warc/CC-MAIN-20240720174732-20240720204732-00005.warc.gz | 326,384,072 | 43,997 | # The solutions of system of equations are not continuous
I'm solving the system of equations that is equivalent to polynomial equation and I'm trying to plot the solutions as a function of one parameter. The problem is that instead of continuous functions I have switching between two solutions at some point. Below you can see as two solutions plotted in red and orange flips in the middle. Still it is clear that there are tree distinct branches. Further I will choose the only one branch that is always positive. But to do so I need to define the function which describe only this branch.
Here is the code to solve the system of equations:
Manipulate[
m = n + 1;
solution = Solve[{r*M0 ==
P + M0*Sum[
i*Product[Evaluate[Symbol["K" <> ToString[j]]]*1, {j, i}]*
P^i, {i, n}]/
Sum[Product[Evaluate[Symbol["K" <> ToString[j]]]*1, {j, i}]*
P^i, {i, 0, n}],
M == M0/Sum[
Product[Evaluate[Symbol["K" <> ToString[j]]]*1, {j, i}]*
P^i, {i, 0, n}],
Table[Evaluate[Symbol["MP" <> ToString[j]]]*1 ==
Times @@ (Table[
Evaluate[Symbol["K" <> ToString[i]]]*1, {i, j}]) M P^j, {j,
n}]} //
Flatten, {P, M,
Table[Evaluate[Symbol["MP" <> ToString[j]]]*1, {j, n}]} //
Flatten];, {n, 3, 1, 1}]
And here is the code to plot the solutions as a functions of parameter 'r':
valP = P /. Take[solution, {1, 4}];
molP[rVal_, K1Val_, K2Val_, K3Val_, M0Val_, DifInd_: 0] :=
Re@With[{r = rVal, K1 = K1Val, K2 = K2Val, K3 = K3Val, M0 = M0Val, índex = DifInd}, (Evaluate@D[valP, {r, índex}])];
Manipulate[
Plot[Evaluate[{molP[r, 10^k1, 10^k2, 10^k3, 10^p0, 0]}],
{r, 0.1, rrmax}, ImageSize -> 500, Frame -> True],
{{rrmax, 10, "r"}, 1, 10},
{{k1, 6, "k1"}, 2, 9},
{{k2, 3, "k2"}, 2, 9},
{{k3, 6, "k3"}, 2, 9},
{{p0, -3.5, "10^P0"}, -1, -10},
ControlPlacement -> Left]
Parameter n in Manipulate define the system of equations and at n=1 or n=2 there is no problems with branches switching:
The problems starts only when n=3. So, the question is - How I can mix the solutions in such way, so I can define all branches as continuous functions ?
Other example is below
solution = Solve[{
K1*P*L == A,
K2*P*L*L == B,
K2*A*L*L == F,
P0 == P + A + B + F,
r*P0 == L + A + 2*B + 3*F
}, {P, L, A, B, F}];
valF = F /. Take[solution, {1, 4}];
FF[rVal_, K1Val_, K2Val_, P0Val_, DifInd_: 0] :=
Re@With[{r = rVal, K1 = K1Val, K2 = K2Val, P0 = P0Val,
índex = DifInd}, (Evaluate@D[valF, {r, índex}])];
Plot[Evaluate[FF[r, 10^6, 10^7, 10^-3.8, 0]],
{r, 0.05, 2},
PlotStyle -> {Red, Blue, Green, Magenta},
PlotRange -> {{0, 1.5}, {-0.00005, 0.0001}},
ImageSize -> Large
]
I need to define 4 continuous functions instead of 4 current not continuous functions!
• Try using Solve with Quartics->False. Commented May 16, 2017 at 22:49
• Can you please fix your code so that it indeed gives an output? Just call Quit and try the Manipulate block yourself. You that there are several undefined variables including P, M0, and so on. Commented May 16, 2017 at 22:50
• @CarlWoll So easy! Thank you! Commented May 16, 2017 at 22:51
• @CarlWoll It is not universal solution ;( if I plug another parameters into molP function the solutions changes the branches even with Quartics->False option. molP[r, 10^5.34, 1000, 10^6.1, 10^-3.5, 0] Commented May 17, 2017 at 19:34
• Interesting problem. If you set all of the parameters = 1 so r is the remaining variable, then you can see internally there is a rational polynomial inside a square root where the numerator and denominator both go to zero simultaneously. At that point, the left limit is negative and the right limit is positive. Taking the absolute value gives you a smooth curve that is positive, minus the discontinuity. When it goes negative, the square root is complex. this gives the break in the curve. Commented May 19, 2017 at 22:15
For the sake of plotting, NDSolve is more likely to track a evolution of a root continuously, since it uses the derivative to predict its next value. The large number of parameters in the OP's problem lead to a correspondingly large system here. The parameters could be incorporated into a ParametricNDSolve code, but for sake of developing and debugging a working solution, it seemed convenient to have access to the components in global variables.
The following shows how to do it for the OP's "other example." Note it uses the solution from Solve to set up the initial conditions. The system algsys consists of equations for all four roots. These equations are identical except for the names of the variables. They are differentiated before being passed to NDSolve. The root each subsystem represents is determined by the initial conditions algics.
varsP = {P1, P2, P3, P4}; (* one variable for each solution *)
varsL = {L1, L2, L3, L4};
varsA = {A1, A2, A3, A4};
varsB = {B1, B2, B3, B4};
varsF = {F1, F2, F3, F4};
allvars = {{varsP, varsL, varsA, varsB, varsF}, {P, L, A, B, F}};
algsys = {K1*P*L == A, K2*P*L*L == B, K2*A*L*L == F,
P0 == P + A + B + F, r*P0 == L + A + 2*B + 3*F} /.
Equal -> Subtract /. MapThread[#2 -> Through[#1[r]] &, allvars] // Flatten;
algics = MapThread[Through[#1[r]] == (#2 /. solution) &, allvars] /. r -> 0.05;
Clear[FF2];
FF2[K1Val_, K2Val_, P0Val_, DifInd_: 0] :=
Block[{K1 = K1Val, K2 = K2Val, P0 = P0Val, índex = DifInd},
Through[varsF[r]], {r, 0.05, 2}]
];
The real and imaginary parts (green and magenta have the same real parts, blue and red have the same imaginary parts):
Plot[Evaluate[Re@FF2[10.^6, 10.^7, 10^-3.8, 0],
{r, 0.05, 2}, PlotStyle -> {Red, Blue, Green, Magenta}]
Plot[Evaluate[Im@FF2[10.^6, 10.^7, 10^-3.8, 0]],
{r, 0.05, 2}, PlotStyle -> {Red, Blue, Green, Magenta}]
• Is it possible to use FF2 as a model for NonlinearModelFit ? Commented Jul 10, 2017 at 8:37
• One could use Indexes[] to fit a particular Part[] perhaps. One should be able to fit the vector of roots Commented Jul 10, 2017 at 12:45
• @ФилиппЦветков Sorry, on a phone. Hit Send accidentally -- I won't be able to check all day, so that's really just a guess at this point. (And it's Indexed[], not Indexes[].) Commented Jul 10, 2017 at 12:49
• For instance it's continuous function of r, but what about the other parameters? Need to check Commented Jul 10, 2017 at 12:52
• Sorry! Me too.. on vocations. Cannot check (( Commented Jul 12, 2017 at 7:49
This answer uses my result from another thread, Is there a way to sort NSolve solution (roots) automatically? It doesn't use any information about the equations, it just looks to identify smooth curves from the data.
solution =
Solve[{K1*P*L == A, K2*P*L*L == B, K2*A*L*L == F,
P0 == P + A + B + F, r*P0 == L + A + 2*B + 3*F}, {P, L, A, B, F}];
valF = F /. Take[solution, {1, 4}];
FF[rVal_, K1Val_, K2Val_, P0Val_, DifInd_: 0] :=
Re@With[{r = rVal, K1 = K1Val, K2 = K2Val, P0 = P0Val,
índex = DifInd}, (Evaluate@D[valF, {r, índex}])];
Instead of immediately plotting, first generate a list of points, then manipulate them, and plot that. Here's the data.
data = Table[
Evaluate[FF[r, 10^6, 10^7, 10^-3.8, 0]], {r, 0.05, 2, .01}];
ListPlot[data // Transpose, Joined -> True]
Below is the kernel of a FoldList call. You may need to fiddle with the WEIGHT constant to get the results to come out just right. It controls how much the derivative information is used in discriminating the curves.
WEIGHT = 100;
permMatchWeight[{vals_, dels_}, l2_] :=
Module[{l1 = {vals, dels}, ps, bestPerm},
ps = Map[({#, WEIGHT (# - vals)}) &, Permutations[l2]];
bestPerm = Sort[ps, Norm[l1 - #1] < Norm[l1 - #2] &] // First
];
Here's where the data gets manipulated.
res = FoldList[permMatchWeight, {{0, 0, 0, 0}, {0, 0, 0, 0}}, data] // Rest;
orderedCurves = Map[(# // First) &, res] // Transpose;
The orderedCurvesis now a list of 4 sets of points, each tied to a single curve. On examination, you find the first two curves are identical.
ListLinePlot[orderedCurves , PlotStyle -> {Red, Blue, Green, Magenta}, ImageSize -> Large ]
In generating the data, I ignored tracking the independent variable, so you'd need to massage the result to get it back in. You could do an interpolating fit at this point to get a function for each of the curves. | 2,655 | 8,091 | {"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": 1, "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} | 3.203125 | 3 | CC-MAIN-2024-30 | latest | en | 0.813358 |
https://algebra1help.com/algebra1-com/converting-fractions/fourth-grade-math-geometry.html | 1,524,585,287,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125946807.67/warc/CC-MAIN-20180424154911-20180424174911-00383.warc.gz | 585,903,332 | 12,508 | Free Algebra Tutorials!
Try the Free Math Solver or Scroll down to Tutorials!
Depdendent Variable
Number of equations to solve: 23456789
Equ. #1:
Equ. #2:
Equ. #3:
Equ. #4:
Equ. #5:
Equ. #6:
Equ. #7:
Equ. #8:
Equ. #9:
Solve for:
Dependent Variable
Number of inequalities to solve: 23456789
Ineq. #1:
Ineq. #2:
Ineq. #3:
Ineq. #4:
Ineq. #5:
Ineq. #6:
Ineq. #7:
Ineq. #8:
Ineq. #9:
Solve for:
Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:
### Our users:
The most hated equations in Algebra for me is Radical ones, I couldn't solve any radical equation till I bought your software. Now, learned how to solve them and how to check if my answers are valid.
Clara Johnson, ND
My son Ryan has become very knowledgeable with algebraic equations thanks to using this quality piece of software. Thank you very much!
Malcolm D McKinnon, TX
Thank to be very quick to answer my question, I will recommend you all over the world.
William Marks, OH
### Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among them?
#### Search phrases used on 2011-12-28:
• +write all four intervals in double inequaties notations.
• 2x-5y=-9 mathlab
• implicit differentiation online calculator
• multiply and simplify by factoring radicals calculator
• tarnsforms games
• example of mathematics investigatory project
• how to divide decimals
• lesson plans in math using polya's 4 step problem solving method
• pre algebra trivia
• graphing linear equations
• quadratic functions in real life solved
• algebrator
• geometry formula sheet
• position fractions on a number line
• successmaker math
• free e books on accountancy
• algebrator free trial
• www.mathamaticsgames.com
• how to find line of symmetry of a quadratic equation
• rational and radical expression calculator
• log base on ti-83
• at noon ship a is 150 km west of ship b. ship a is sailing east at 25 km/h and ship b is sailing north at 20 km/h. how fast is the distance between the ships changing at 4:00 pm?
• cubic root of a fraction
• inequalities and graphing
• 9 properties of rational exponents
• rational expressions formula
• questions and answers +physics +pdf
• softmath.com
• solving algebraic equations + worksheet
• how to do fractions
• free worksheets factoring trinomials
• printable revision paper and answer sheet
• rudin solutions
• +multiply fractions problem solver
• kuta infinite algebra 2 answers
• Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
• simplify polynomials (4x4 + 7x2 + 6x) - (x3 - 7x + 10)
• simplify logarithm calculator
• coordinates worksheet
• online foil calculator
• paper folding fraction activity
• polynomial functions
• google science asymptotes of linear differential equations
• www.know about l.c.m.in maths .com
• geometry formula chart
• decimal calculation
• yahoo answers: pre algebra with pizzazz page 225
• error dimension ti-86
• +algebra +pdf
• mathematics formula chart
• language c & polinomy multiplying
• Algebra problem - Of the 550000 alternative-fuel vehicles produced in the united states in 2004, 150000 were E85 flexible-fuel vehicles. What percent of alternative-fuel vehicles used E85?
• percentage questions and answers for college algebra
• www oriya
• freeware on functions domain and range solver
• poem about 2 step equations
• non linear word problems
• algebrator .com
• boolean equaton simplifier
• algebric expression matlab
• inequality two functions
• Finding the Vertex of a Parabola
• Solving Variable Fractions
• hat
• conjugate calculator online
• how would you find the sum 5x+2y+x-3y
• algebra help algebra solver
• matrix multiplication rules
• simplify into an algebraic expression 12y-13y
• mathway algebra problem solver
• linear algebra start
• absolute value equation calculator
• completing the square with fractions calculator
• Solve My Algebra 2 Problem | 1,046 | 4,174 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2018-17 | latest | en | 0.84898 |
https://examrobot.com/content/if-angles-triangle-are-ratio-4-3-2-then-triangleis-obtuse-has-one-angle-greater-80-right-triangleis-acute | 1,638,349,755,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964359976.94/warc/CC-MAIN-20211201083001-20211201113001-00550.warc.gz | 298,930,195 | 149,217 | If the angles of a triangle are in the ratio of 4 : 3 : 2, then the triangleis obtuse has one angle greater than 80°is a right triangleis acute | Exam Robot | 45 | 158 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2021-49 | longest | en | 0.736866 |
https://www.arxiv-vanity.com/papers/1506.01431/ | 1,600,810,154,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400206763.24/warc/CC-MAIN-20200922192512-20200922222512-00747.warc.gz | 713,093,543 | 34,702 | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
# A Cellular Automaton for Blocking Queen Games
Matthew Cook and Urban Larsson and Turlough Neary 1Institute of Neuroinformatics, University of Zürich and ETH Zürich, Switzerland
12Department of Mathematics & Statistics, Dalhousie University, Halifax, Canada1 1Institute of Neuroinformatics, University of Zürich and ETH Zürich, Switzerland
1
11Supported by the Killam Trust. Contact email:
July 14, 2020
###### Abstract
We show that the winning positions of a certain type of two-player game form interesting patterns which often defy analysis, yet can be computed by a cellular automaton. The game, known as Blocking Wythoff Nim, consists of moving a queen as in chess, but always towards (0,0), and it may not be moved to any of temporarily “blocked” positions specified on the previous turn by the other player. The game ends when a player wins by blocking all possible moves of the other player. The value of is a parameter that defines the game, and the pattern of winning positions can be very sensitive to . As becomes large, parts of the pattern of winning positions converge to recurring chaotic patterns that are independent of . The patterns for large display an unprecedented amount of self-organization at many scales, and here we attempt to describe the self-organized structure that appears.
## 1 Blocking Queen Games (k-Blocking Wythoff Nim)
In the paper [Lar11], the game of -Blocking Wythoff Nim was introduced, with rules as follows.
• As in Wythoff Nim [Wyt07], two players alternate in removing counters from two heaps: any number may be removed from just one of the heaps, or the same number may be removed from both heaps. However, a player is allowed to reject the opponent’s move (so the opponent must go back and choose a different, non-rejected move), up to times, where is a parameter that is fixed for the game. The distinct attempted move must be allowed. Thus, if there are at least winning moves among the options from a given position, then one of these winning moves can be played.
• There are chess pieces on an infinite (single quadrant) chess board: one queen, and pawns. On your turn you move the queen towards the origin. (The first player who cannot do this loses.) The queen cannot be moved to a position with a pawn, but it can move over pawns to an empty position. After moving the queen, you complete your turn by moving the pawns to wherever you like. The pawns serve to block up to of the queen’s possible next moves.
• Consider a game with , where the queen is now at (yellow in Figure 1). It is player ’s turn, and player is blocking the four positions (dark brown and light olive). This leaves with the options (each is black or blue). Regardless of which of these chooses, will then have at least five winning moves to choose from (ones marked yellow, or light, medium, or dark olive). These are winning moves because it is possible when moving there to block all possible moves of the other player and thereby immediately win. Therefore player will win.
As shown in Figure 1, there is a simple algorithm to compute the winning positions for game . These are known as P-positions in combinatorial game theory, and we will refer to them as palace positions, the idea being that the queen wants to move to a palace: if you move her to a palace, you can win, while if you move her to a non-palace, your opponent can win. To win, you must always block (with pawns) all of the palaces your opponent might move to.
The idea is simply that a palace is built on any site that can see fewer than other palaces when looking due north, west, or north-west. In this way, the pattern of palaces can be constructed, starting at (0,0) and going outward. For efficiency, a dynamic programming approach can be used, storing three numbers at each position, for the number of palaces visible in each of the three directions. With this technique, each line can be computed just from the information in the previous line, allowing significant savings in memory usage.
The case corresponds to classical Wythoff Nim, solved in [Wyt07]. In [Lar11], the game was solved for and . When we say a game is solved we mean it is possible to give closed-form expressions for the P-positions, or at least that a winning move, if it exists, can be found in log-polynomial time in the heap sizes. For example, the set , where runs over the nonnegative integers and is the golden ratio, provides a solution for classical Wythoff Nim. Combinatorial games with a blocking maneuver appears in [GaSt04], [HoRe01], [HoRe] and [SmSt02], and specifically for Wythoff Nim in [Gur10], [HeLa06], [Lar09] and [Lar15].
The new idea that we use in this paper is to look directly at the number of palaces that the queen can see from each position on the game board, and to focus on this palace number rather than just on the palace positions (P-positions). (The palace positions are exactly the locations with palace numbers less than .) In previous explorations, the palace number was in fact computed but then only used for finding the new palace positions, which when viewed on their own give the appearance of involving long-range information transfer. By observing the palace number directly, however, we can see that in almost all positions the palace number is surprisingly close to , and if we look at its deviation from then we are led to discover that these deviations follow a local rule that does not even depend on . The next section will present this local rule as a cellular automaton (CA). A finite automaton for Wythoff Nim has been studied in a different context in [Lan02]. Other recent results for cellular automata and combinatorial games can be found in [Fin12], [Lar13] and [LaWa13].
The rest of the paper will present the rich structure visible in the patterns of palace numbers for games with large . A surprising number regions of self organization appear, largely independent of the particular value of , and some of the self-organized patterns are quite complex, involving multiple layers of self-organization. This is the first CA we are aware of that exhibits so many levels of self-organization. So far, these patterns offers many more questions than answers, so for now we will simply try to catalog our initial observations.
## 2 A Cellular Automaton Perspective
In this section we describe a cellular automaton that computes the palace numbers for blocking queen games.
### 2.1 Definition of the CA
The CA we present is one-dimensional and operates on a diamond space-time grid as shown in Figure 2, so that at each time step, each cell that is present at that time step derives from two parents at the previous time step, one being half a unit to the right, the other being half a unit to the left. On the diamond grid, there is no cell from the previous step that is ‘the same’ as a given cell on the current step. However, the three grandparents of a cell, from two steps ago, do include a cell which is at the same spatial position as the grandchild cell.
Our CA rule is based not only on the states of the two parent cells, but also on the state of the central grandparent cell, as well as its parents and central grandparent, all shown in blue in Figure 2. As such, this CA depends on the previous four time steps, i.e. it is a fourth-order CA. It is the first naturally occurring fourth-order CA that we are aware of. We say it is “naturally occurring” simply because we discovered these pictures by analyzing the blocking queen game, and only later realized that these pictures can also be computed by the fourth-order diamond-grid CA we present here.222 Using the dynamic programming approach described in Section 1, the information for each cell can be computed from just its parents and central grandparent, so that approach is also clearly a CA, but it needs a large number of states for the cells and each different value of requires a different rule. The CA we describe in this section is much more surprising, being a single rule that is completely independent of and using very few states to produce all of the interesting parts of the pictures.
The states in our CA are integers. In general they are close to 0, but the exact bounds depend on the initial conditions. The formula for computing a cell’s value, given its neighborhood, is described in Figure 2.
### 2.2 The Connection between the CA and the Game
Since our definition of the CA appears to be completely different from the definition of the blocking queen game, we need to explain the correspondence.
The idea is that the states of this CA correspond to palace numbers minus , which are generally integers close to zero. One can easily prove a bound of for the number of states needed for the game with blocking number , since each row, column, and diagonal can have at most palaces, so palace numbers are always in the range . However, in practice the number of states needed (after the initial ramping-up region near the origin) appears to be far smaller, more like . For example, when , only eight states are needed, ranging between -4 and 3.
Surprisingly, this single CA is capable of computing the pattern of palace numbers regardless of the value of . Different values of simply require different initial conditions: The initial condition for a given value of is that every site in the quadrant opposite the game quadrant should be , and every site in the other two quadrants should be 0.
###### Theorem 2.1.
The -Blocking Wythoff Nim position is a P-position if and only if the CA given in Figure 2 gives a negative value at that position, when the CA is started from an initial condition defined by
CA(x,y)=⎧⎪⎨⎪⎩kx<0~{}and~{}y<00x<0~{}and~{}y⩾00x⩾0~{}and~{}y<0
###### Proof.
First we will consider the case of no P-positions occurring within the CA neighborhood, so compensation terms can be ignored.
As shown in Figure 2, we will let be the number of P-positions directly above and , and similarly for and , as well as for the diagonals and horizontal rows .
This gives us , and so on: when adding the -value to each cell this represents the sum of the numbers of P-positions in the three directions.
We would like to express in terms of the other values. Notice that
a+e+f=3∑i=1vi+3∑i=1di+3∑i=1hi−3k=b+c+g
and therefore , allowing us to express in terms of the other values as .
All that remains is to take any P-positions in the CA neighborhood into account, so as to understand the compensation terms.
If there is a P-position at , then , , , and (i.e. the positions in a line to the right, down, or right-down) will all be one higher than they were before taking that palace into account. Since the equation was true when ignoring the palace at , it becomes wrong when the palace at produces its effect of incrementing , , , and , because that makes the right hand side go up by 3 while the left hand side is untouched. To compensate for this, we can add a term to the left hand side, which is 3 if there is a palace at , and 0 otherwise.
Similarly, if is a P-position, then this increments , , and , so to compensate, we will need to subtract 2 from the left hand side of if is a P-position.
We can see that we are computing exactly the compensation terms shown in the green squares of Figure 2. Once we include all the compensation terms, the formula for becomes correct even in the presence of local P-positions, and it corresponds exactly to the rule given in Figure 2.
The initial condition can be confirmed to produce (via the CA rule) the correct values in the first two rows and columns of the game quadrant, and from that point onwards the reasoning given above shows that the correct palace numbers, and therefore the correct P-positions, are being computed by the CA rule. ∎
### 2.3 Notes on Reversability
The reversed version of this CA computes , given , , , , , and . This is done with the equation , which is equivalent to the equation in the caption of Figure 2. However, the palace compensation term can depend on , so this equation has not fully isolated on the left hand side. (The forward direction did not have this problem, since does not depend on .) Writing to separate the palace compensation term from from the other palace compensation terms, we get . Since is 3 when is negative, and 0 otherwise, this equation always yields either one or two solutions for . If the right hand side is 3 or more, then must be equal to it. If the right hand side is negative, then must be 3 less than it. And if the right hand side is 0, 1, or 2, then can either be equal to it or be 3 less than it—we are free to choose. The reversed rule is non-deterministic, but it can always find a compatible value. In other words, there is no “Garden of Eden” pattern for this rule, if we assume that all integers are permissible states.
## 3 Self-Organization
The top row of Figure 3 shows the palace number patterns for games 100 and 1000. The patterns are strikingly similar, given that the value of differs by an order of magnitude. The pattern for game 1000 has the appearance of being “the same, but ten times bigger” than the pattern for game 100. The middle and lower rows of Figure 3 zoom in on subregions where the two patterns are in fact identical, without any scaling factor.
### 3.1 Terminology
As an aid to our discussion of these complex images, we will give names to the prominent features in them.
We can see that this system self-organizes itself into 11 regions with 14 borders and 6 junctions. Ignoring duplicates due to the mirror symmetry, there are 7 regions, 7 borders, and 4 junction points.
We will start by naming the 7 regions and some features inside them. The region at the upper left (the game’s terminal region, and the CA’s starting region), in the shape of a dented triangle, is the hood. The triangular regions adjacent to the hood, with a periodic interior (visible in all panels of Figure 3), are the épaulettes. The rhomboid region that emanates from between the pair of épaulettes is the fabric. The solid black regions at the top and at the left constitute the outer space. Between the outer space and the épaulettes we find the arms (irregular yellow regions in Figure 3), which extend indefinitely. Extending next to the arms, and of similar width, we have the warps. Each warp contains a number of threads (strings of yellow dots, clearly visible in the top left panel of Figure LABEL:fig:prefix) which come out of the fabric. Between the warps lies the inner sector, and the blue stripes in the warps and in the inner sector are the weft.
Next, we will name the 4 junction points. The hood, épaulettes, and fabric all meet at the nose. The hood, épaulette, arm, and outer space all meet at the shoulder. The warp, fabric, épaulette, and arm all meet at the armpit, which is often a hotspot of highly positive palace numbers. And the fabric, warps, and inner sector meet at the prism. The inner sector often contains slightly higher palace numbers than the warps, especially near the main diagonal, giving the impression of light being emitted from the prism, as in Figure 6.
Finally, we come to the 7 borders. The hood and épaulette meet cleanly at the casing. The hood contains all the states from to , but after the casing, the CA uses very few states. The épaulette and arm meet at the hem. The épaulette and fabric meet at the rift, a narrow, relatively empty space which appears to get wider very slowly. The fabric and warp meet at the fray, where threads almost parallel to the fray unravel from the fabric, and threads in the other direction exit the fabric and start merging to form thick threads in the warp. There is no clear boundary where the warp meets the inner sector, since the warp simply runs out of threads. The warp also meets the arm cleanly, at the inside of the arm. At the boundary between the warp and the arm, it appears that the yellow nature of the arm is due to being packed full of threads, and the warp simply has a much lower density of threads. Threads that bend into the inner sector, and stop being parallel to the rest of the warp, are sometimes called beams (see also [Lar12]). The often-occurring slightly-separated periodic part of the arm, bordering the outer space, is the skin.
The fray, warp, central sector, armpits, and prism are all very sensitive to , but all the other regions are not, with the exception of the fabric and the rift, which are sensitive only to . The fabric, fray, warp, prism, and central sector are all full of weft. Often the centermost beams will communicate with each other via the weft, and this process can usually be analyzed to calculate the slopes of these beams, which are generally quadratic irrationals.
This is the greatest complexity of self-organization that we have seen for a system that has no structured input.
### 3.2 Structure Within the Regions
The hood and the épaulets have a very regular structure. The palace numbers in the hood increase steadily in each row and column, increasing by one when the higher coordinate is incremented, and by two when the lower coordinate is incremented. Thus the hood contains all palace numbers from 0, at the upper left corner, to , where the hood meets the épaulettes at the casing, a line with slope -1/2 (and -2) that connects the nose at to the shoulder at (and ). The hood is exactly the region where every move is a winning move, because all possible further moves can be blocked, thereby immediately winning the game. When players are not making mistakes, the game ends when (and only when) somebody moves into the hood, thereby winning the game.
The palace numbers in the épaulettes form a two-dimensional periodic pattern, with periods and (and all integral linear combinations of those base periods). Only the palace numbers , , and appear in the épaulettes. In the épaulette’s periodic region of size 11, (a P-position) appears 5 times, and and (both N-positions (no palace)) each appear 3 times.
The arms, shown in the bottom row of Figure 3 have a random appearance, although they often contain temporary black stripes at one of the two angles parallel to their sides. Despite this initial appearance of disorder, the arms have many interesting properties, discussed in Section 3.4.
The fabric exhibits further self-organization. The larger black regions visible in the middle row of Figure 3 form a rough grid, and in much larger pictures () the grid morphs into a larger-scale grid, which is at a slightly different angle and has cells about 3.5 times larger. Regions of the small-grid pattern appear to travel through the large grid like meta-gliders, visible in the top right of Figure LABEL:fig:gliders. These grid cells are separated by threads of P-positions, which are able to split and merge to form smaller and larger threads, and sometimes seem to disappear.
Threads typically have a measurable (vertical, horizontal, and/or diagonal) thickness, which is added when they merge, as happens frequently just after the fray, as in Figure 5. For example, in the left-hand warp the threads have integer vertical thicknesses, that is, each thread has a fixed number of P-positions that occur in every column. Furthermore, this fixed number is always a Fibonacci number.
### 3.3 Region Prefix Properties
If we look around the nose, we see one of three pictures, depending on the value of , because this determines the precise pixel arrangement of the casing at the nose. These three shapes are shown in Figure 7. Since there are only three possibilities for the hood boundary shape at the nose, and the CA rule can then be used to produce the épaulettes and fabric without knowing , we see that despite the chaotic nature of the fabric, there are in fact only three fabric patterns that can be produced. Figures 3 and 4 both show examples of how the full fabric area matches between different games with congruent modulo 3.
Using the CA, starting from an infinite casing pattern, we can make any of the three fabric patterns in an infinitely large version. The fabric patterns that we see in practice are simply prefixes of one of these three infinite fabrics.
The arms similarly can be formed by the CA rule from the shoulder, and in this case there is only one possible pattern. Figure LABEL:fig:armwhite shows how this pattern remains very stable as increases.
### 3.4 Properties of the Skin
Where the arm borders the outer space, the arm grows a periodic skin (see Figure 8), which is slightly separated from the rest of the arm, except that it emits a vertical (in the case of the upper arm) line once per period. This skin consists of solidly packed P-positions, with a vertical thickness of (or where a line is emitted), a diagonal thickness of , a horizontal thickness of , a horizontal period of , and a vertical period of , where is the Fibonacci number. The skin originally forms at the top of the arm with , and after about 200 pixels (horizontally, about 80 vertically) it changes to the form with , then after about 7700 pixels it changes to the form with . We conjecture that for large , it will continue to thicken in this manner, with increasing by one each time, approaching a slope of . In the other direction, one can even see the stage of this pattern as part of the épaulette at the start of the arm for the first 10 pixels or so, although it is not clearly visible due to the lack of black pixels between this skin and the rest of the arm.
The boundary between the skin (which has palace numbers and ) and the outer space (which has constant palace number ) consists of steps of width 2 or 3. Each time the skin thickens, the pattern of steps expands according to the rule , starting from the pattern of just for the skin . The skin pattern of positions of palace numbers and can be computed from this pattern of steps and from the assumption that there are palaces to the left of the skin in each row. This pattern is rotationally symmetric within each period (between the columns that produce the vertical lines), and with each thickening follows the expansion rule (writing for , etc.)
−2→(−2−1−2−2),(−2−2−1−2−2−2),⎛⎜⎝−1−1−2−1−2−2⎞⎟⎠,⎛⎜⎝−2−2−1−2−2−2−2−1−1⎞⎟⎠,
−1→(−1−1−2−1),(−2−2−1−2−1−1),⎛⎜⎝−1−1−2−1−1−1⎞⎟⎠,⎛⎜⎝−2−2−1−1−1−1−2−1−1⎞⎟⎠,
where one of the four expansion matrices is chosen based on the required size. The widths are determined by the step pattern, and the heights are partially defined by the property that all matrices produced by a given row should have a row in common.
## 4 Conjectures and Questions
Most of the observations in Section 3 may be viewed as open problems. Here we list of a few.
• Is it the case that a sufficiently thick arm will grow thicker and thicker skin over time?
• The arm’s skin (as observed for and ) has a vertical thickness of (except where a line is emitted), a diagonal thickness of , a horizontal thickness of , a horizontal period of , and a vertical period of , where is the thickness level (here is the Fibonacci number). Can this pattern be explained, and does it continue?
• Are the inner sectors beyond the fabric essentially different for different , starting with distinct armpits (see Figure LABEL:fig:gliders), and eventually producing ever different and irregular innermost P-threads?
• Are the armpits the unique regions from which the queen views the most palaces (for large )?
• Do the innermost P-threads always have slopes corresponding to algebraic numbers? If there is only one innermost upper P-thread, is the slope a root of a second degree polynomial with rational coefficients?
• How many non-periodic threads can there be for a given ? We conjecture at most two upper ones (as occurs for ), which must be the inner ones; with all others eventually becoming periodic.
• Why is it that the threads of the warp merge in such a way that their thickness is always a Fibonacci number?
• Is it true that the threads (also inside the arm and in particular in the inner sector) bound the number of palace positions that the queen sees looking just diagonally, and this number is mostly just one number (otherwise at most one out of two numbers)? | 5,535 | 24,265 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2020-40 | latest | en | 0.919987 |
http://ishxiao.com/blog/math/2017/07/02/how-to-solve-gamma-function.html | 1,547,761,609,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583659340.1/warc/CC-MAIN-20190117204652-20190117230652-00094.warc.gz | 112,856,186 | 4,252 | I. Theoretical Analysis
Gamma Function is defined in the following,
Using integration by parts, one sees that:
From the above calculation，and note that
where $y^2 = x$.
If $n$ is a positive integer, one can obtain:
### Appendix:
Integral calculus（$y\equiv \sqrt{a}[x+(b/2a)]$
If it meets a condition $e^{-x^2} > 0$, obviously given that,
Calculate quadratic $I^2$,
Using integration by polar coordinates, $x = r \cos {\theta}, y = r \sin {\theta}$, and,
then, one sees that
Therefore,
II. Numerical analysis
Evaluate Gamma function in matlab.
1.Evaluate the gamma function with a scalar and a vector.
>> y = gamma(0.5)
y =
1.7725
>> y = gamma(-pi:pi)
y =
1.0157 -3.1909 6.8336 -7.8012 1.1046 0.9482 1.7621
2.Plot the gamma function and its inverse.
% PlotGammaFunctionExample.m
fplot(@gamma)
hold on
fplot(@(x) 1./gamma(x))
legend('\Gamma(x)','1/\Gamma(x)')
hold off
grid on
Figure.
In addition, the domain of the gamma function extends to negative real numbers by analytic continuation, with simple poles at the negative integers. This extension arises from repeated application of the recursion relation | 322 | 1,148 | {"found_math": true, "script_math_tex": 7, "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": 1, "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} | 3.59375 | 4 | CC-MAIN-2019-04 | latest | en | 0.70311 |
https://www.coursehero.com/file/6718678/Differential-Equations-Solutions-105/ | 1,526,818,909,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794863410.22/warc/CC-MAIN-20180520112233-20180520132233-00593.warc.gz | 734,974,735 | 89,135 | {[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
Differential Equations Solutions 105
# Differential Equations Solutions 105 - • If A were...
This preview shows page 1. Sign up to view the full content.
115 CHALLENGE 19.7. The probabilities are clearly nonnegative and sum to 1. Note that the j th component of e T A is the sum of the elements in column j , and this is 1, so e T A = e T . Therefore, e T A = 1 e T , and this means that the vector e T is unchanged in direction when multiplied on the right by A . This is the definition of a left eigenvector of A , and the eigenvalue is 1. Apply the Gerschgorin circle theorem to A T , which has the same eigenvalues as A . If the main diagonal element of A T is 0 < α < 1, then the off-diagonal elements are nonnegative and sum to 1 α . Therefore, the Gerschgorin circle is centered at α with radius 1 α . This circle touches the unit circle at the point (1 , 0) but lies inside of it. The eigenvalues lie in the union of the Gerschgorin circles, so all eigenvalues lie inside the unit circle.
This is the end of the preview. Sign up to access the rest of the document.
Unformatted text preview: • If A were irreducible then the eigenvalue at 1 would be simple; see, for exam-ple, [1]. • Let the eigensystem of A be deFned by Au j = λ j u j , and let e 1 = n X j =1 α j u j , where u 1 , . . . , u 4 are a basis for the eigenspace corresponding to the eigenvalue 1. Then verify that A k e 1 = n X j =1 α j λ k j u j . Since λ k j → 0 as k → ∞ except for the eigenvalue 1, we see that A k e 1 → α 1 u 1 + α 2 u 2 + α 3 u 3 + α 4 u 4 . • Therefore, we converge to a multiple of the stationary vector. [1] Richard Varga, Matrix Iterative Analysis , Prentice Hall, Englewood Cli±s, NJ, 1962....
View Full Document
{[ snackBarMessage ]}
### What students are saying
• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern | 724 | 2,732 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.375 | 4 | CC-MAIN-2018-22 | latest | en | 0.826645 |
https://eng.libretexts.org/Bookshelves/Computer_Science/Databases_and_Data_Structures/Book%3A_Relational_Databases_and_Microsoft_Access_(McFadyen)/10%3A_Normalization/10.04%3A_First_Normal_Form_(1NF) | 1,696,471,209,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511717.69/warc/CC-MAIN-20231005012006-20231005042006-00400.warc.gz | 267,459,004 | 29,939 | # 10.4: First Normal Form (1NF)
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$
We say a relation is in 1NF if all values stored in the relation are single-valued and atomic. With this rule, we are simplifying the structure of a relation and the kinds of values that are stored in the relation.
Example $$\PageIndex{1}$$
Consider the following EmployeeDegrees relation.
• empNo is the PK
• each employee has one first name and one salary
• each employee has zero or more university degrees ... stored as a single attribute
Table $$\PageIndex{1}$$: EmployeeDegrees relation.
EmployeeDegrees
empNo
first name
salary
degrees
111 Joe
29,000
BSc, MSc
200 April
41,000
BA, MA
205 Peter
33,000
BEng
210 Joe
20,000
This relation is not in 1NF because the degrees attribute can have multiple values. Below are two relations formed by splitting EmployeeDegrees into two relations – one relation has attributes empNo, first name, and salary and the other has empNo and degree. We say we have decomposed EmployeeDegrees into two relations and we have populated each with data from EmployeeDegrees. Each of these is in 1NF, and if we join them on empNo we can get back the information shown in the relation above.
Table $$\PageIndex{2}$$: Employee table.
Employee
empNo first name salary
111
Joe 29,000
200
April 41,000
205 Peter 33,000
210 Joe 20,000
empNo is the PK. Each employee has one name and one salary.
Table $$\PageIndex{3}$$: Degree table.
Degree
empNo degree
111 BSc
111 MSc
200 BA
200 MA
205 BEng
{empNo, degree} is the PK. degree is single-valued.
Example $$\PageIndex{2}$$
Consider the Student relation below. The name attribute comprises both first and last names and so its not atomic. Student is not 1NF.
Table $$\PageIndex{4}$$: Student table not in 1NF.
Student – not in 1NF
studentNo
name
gender
444
Jim Smith
m
254
Donna Jones
f
333
Peter Thomas
m
765
Jim Smith
m
If we modify Student so there are two attributes (say, first and last) then Student would be 1NF:
Table $$\PageIndex{5}$$: Student table in 1NF.
Student – in 1NF
studentNo
first
last
gender
444
Jim
Smith
m
254
Donna
Jones
f
333
Peter
Thomas
m
765
Jim
Smith
m
If we can say that a relation (or table) is in 1NF then we are saying that every attribute is atomic and every value is single-valued. This simplifies the form of a relation.
It is very common for names to be separated out into two or more attributes. However, attributes such as birth dates, hire dates, etc. are usually left as a single attribute. Dates could be separated out into day, month, and year attributes, but that is usually beyond the needs of the intended system. Some would take the view that separating a date into 3 separate attributes is carrying the concept of normalization a little too far. Database systems do have convenient functions that can be used to obtain a day, month, or year values from a date.
## Exercises
Exercise $$\PageIndex{1}$$
Consider the relation below that holds information about courses and sections. Suppose departments have courses and offer these courses during the terms of an academic year. A section has a section number, is offered in a specific term (e.g. Fall 2016, Winter 2017) and a slot (e.g. 1, 2, 3, ...15) within that term. Each time a course is delivered there is a section for that purpose. Each section of a course has a different number. As you can see a course may be delivered many times in one term.
CourseDelivery
deptNo
courseNo
delivery
ACS
1903
001, Fall 2016, 05; 002, Fall 2016, 06; 003, Winter 2017, 06
ACS
1904
001, Fall 2016, 12; 002, Winter 2017, 12
Math 2201
001, Fall 2016, 11; 050, Fall 2016, 15
Math 2202
050, Fall 2016, 15
Modify CourseDelivery to be in 1NF. Show the contents of the rows for the above data.
Exercise $$\PageIndex{2}$$
Chapter 8 covered mapping an ERD to a relational database. Consider the examples from Chapter 8; are the relations in 1NF?
This page titled 10.4: First Normal Form (1NF) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Ron McFadyen. | 1,467 | 5,083 | {"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": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2023-40 | longest | en | 0.643007 |
https://www.jiskha.com/questions/21437/The-Australian-dollar-is-currently-very-high-with-a-value-of-US-0-80-Many-people | 1,571,285,472,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986672548.33/warc/CC-MAIN-20191017022259-20191017045759-00314.warc.gz | 971,483,536 | 4,972 | # economics
The Australian dollar is currently very high, with a value of US$0.80. Many people expect the Australian dollar to rise even higher by the end of the year, to as much as US$0.90.
Use a ¡°FOREX¡± (foreign exchange market) diagram to show the impact of this expectation on the Australian dollar. In your answer you must explain the effect of this expectation on the money supply (MS), the holdings of the Reserve Bank of Australia (the Central Bank) of gold and foreign exchange, the Current Account and
Capital Account balances, and on Australia¡¯s overall Balance of Payments. For convenience you can assume that the current account is in deficit and the capital account is in surplus.
Do some research, then take a shot. What do you think?
1. 👍 0
2. 👎 0
3. 👁 124
## Similar Questions
1. ### Economcis
I have an assignment on the the factors which impact the Australian dollar. "How does the government influence the value of the Australian dollar?"
asked by TP on May 2, 2008
2. ### Math
The exchange rate from US dollars (USD) to euros (EUR) is given by 1 USD = 0.753 EUR. Give the answers to the following correct to two decimal places. Roger receives 800 Australian dollars (AUD) for 610 EUR. Calculate the value of
asked by Lucas on August 27, 2014
3. ### sose
need help with finding websites with pictures of Australian cattle farming and Australian crop growing.
asked by tegz on May 26, 2008
4. ### sose
need help with finding websites with pictures of Australian cattle farming and Australian crop growing.
asked by tegz on May 26, 2008
5. ### math 100
Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 70 people. (You may need to use the standard normal distribution table. Round your
asked by ryan on November 9, 2018
6. ### math
Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 80 people. (You may need to use the standard normal distribution table. Round your
asked by wanda on March 17, 2014
7. ### Statstics
Let x be a random variable representing dividend yield of Australian bank stocks. We may assume that x has a normal distribution with \alpha =2.3% A random sample of 18 Australian bank stocks has a sample mean of x bar = 6.1%. For
asked by Justina on August 7, 2011
8. ### math
Suppose that people's heights (in centimeters) are normally distributed, with a mean of 170 and a standard deviation of 5. We find the heights of 60 people. (You may need to use the standard normal distribution table. Round your
asked by Anonymous on December 3, 2013 | 677 | 2,694 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2019-43 | latest | en | 0.94524 |
http://web.mst.edu/~bestmech/preview/yashodhan/problems/chapter3/13.htm | 1,532,248,386,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676593142.83/warc/CC-MAIN-20180722080925-20180722100925-00586.warc.gz | 402,102,576 | 1,003 | Block A has a weight of 20 kgs and slides in smooth horizontal slot .If the block is released from s = 0 with zero intial velocity, determine its speed at s = 1 m. Each of the four springs has a stiffness of k = 500 N/m and an unstretched length of 0.5 m. T1 + V1 = T2 + V2 0 + 2[(1/2)500(1-0.5)2] + 2[(1/2)500( | 112 | 311 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2018-30 | latest | en | 0.91175 |
http://www.amibroker.com/guide/afl/mxsortrows.html | 1,653,082,264,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662534669.47/warc/CC-MAIN-20220520191810-20220520221810-00110.warc.gz | 64,473,688 | 1,874 | ### MxSortRows - sort the rows of the matrix
Matrix functions
(AmiBroker 6.10)
SYNTAX MxSortRows( mx, ascending = True, col1 = 0, col2 = -1, col3 = -1 ) RETURNS Matrix FUNCTION Sorts the rows of the matrix in ascending/descending order of the col1 column. When the col1 column has equal values, SortRows sorts according to the col2 and col3 columns in succession (if col2 and col3 are specified and >= 0 ).Column numbers are zero based. Hint: if you want to sort columns instead you can Transpose/Sort rows/Transpose back. EXAMPLE ```m = MxFromString("[ 9, 1, 6; 40, 30, 20; 8, 7, 3; 3, 5, 1 ]"); printf("Input matrix "); printf( MxToString( m ) + " " ); printf("Rows %g, Cols %g ", MxGetSize( m, 0 ), MxGetSize( m, 1 ) ); printf("Sorting every row separately "); m2 = MxSort( m, 0 ) ; printf( MxToString( m2 ) + " " ); printf("Sorting every column separately "); m3 = MxSort( m, 1 ) ; printf( MxToString( m3 )+ " "); printf("Sorting rows by contents of first column "); m4 = MxSortRows( m, True, 0 ) ; printf(MxToString( m4 )+ " "); printf("Sorting rows by contents of second column "); m5 = MxSortRows( m, True, 1 ) ; printf(MxToString( m5 )+ " ");``` SEE ALSO Matrix() function , MxTranspose() function
#### References:
The MxSortRows function is used in the following formulas in AFL on-line library: | 406 | 1,309 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2022-21 | latest | en | 0.536561 |
https://www.snapxam.com/problems/71327465/-6-0-9-2-0-2-19-2 | 1,708,946,810,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474659.73/warc/CC-MAIN-20240226094435-20240226124435-00277.warc.gz | 997,700,510 | 14,532 | 👉 Try now NerdPal! Our new math app on iOS and Android
# Simplify $-6\left(9-1\cdot 2\right)^2-1\cdot 9^2-2$
## Step-by-step Solution
Go!
Math mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2
e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
### Videos
$-377$
Got another answer? Verify it here!
## Step-by-step Solution
Specify the solving method
1
Multiply $-1$ times $2$
$-6\left(9-2\right)^2-1\cdot 9^2-2$
Learn how to solve addition of numbers problems step by step online.
$-6\left(9-2\right)^2-1\cdot 9^2-2$
Learn how to solve addition of numbers problems step by step online. Simplify -6(9-*2)^2-9^2+-2. Multiply -1 times 2. Subtract the values 9 and -2. Calculate the power 7^2. Calculate the power 9^2.
$-377$
## Explore different ways to solve this problem
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more
SimplifyWrite in simplest formFactorFactor by completing the squareFind the integralFind the derivativeFind break even pointsFind the discriminantPrime Factor Decomposition
SnapXam A2
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2
e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
### Main Topic: Addition of numbers
The terms of the addition, a and b, are called addends and the result, c is the sum.
### Join 500k+ students in problem solving.
##### Without automatic renewal.
Create an Account | 702 | 1,812 | {"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": 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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2024-10 | longest | en | 0.605207 |
http://www.ehow.co.uk/how_12284048_much-weight-can-put-glass-shelf.html | 1,490,621,662,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218189472.3/warc/CC-MAIN-20170322212949-00226-ip-10-233-31-227.ec2.internal.warc.gz | 489,621,226 | 18,115 | # How much weight can I put on a glass shelf?
Written by james holloway
• Share
• Tweet
• Share
• Pin
• Email
Glass shelves can add a modern touch to any room. Although glass shelves are sturdy, the fact that they are made of glass can lead people to worry about how much weight they can bear. The amount of weight a glass shelf can support varies depending on a number of factors. These include the type of glass, the thickness of the shelf and the amount of space between the brackets that hold it up. Calculating how much weight a glass shelf can support is a straightforward process.
Skill level:
Moderate
• Shelves
• Tape measure
## Instructions
1. 1
Measure your shelves. You'll need to know the thickness of the glass as well as the length and depth of the shelf. The most common glass shelf sizes range between approximately 5 mm (3/16 in) and 13mm (1/2 in).
2. 2
Determine what type of glass the shelves are made of. Some glass shelves are made of annealed glass, while others are made of tempered glass. Tempered glass is chemically treated to strengthen it and prevent it from breaking into dangerous pieces if it does shatter. Tempered glass is usually stronger and safer but is also more expensive than annealed glass.
3. 3
Measure the distance between the supports you will be using to hold up the shelves. The closer the supports are to each other, the more weight the shelf as a whole will be able to support. For instance, a 122-cm annealed glass shelf 20 cm deep will be able to support around 21 kg if its supports are around 60 cm apart. By contrast, halving the distance between the supports to 30cm more than quadruples the weight limit to almost 90 kg.
4. 4
Use an online weight calculator to determine the amount of weight your glass shelf can hold. Some examples are given in the Resources section below.
#### Tips and warnings
• Because of slightly different assumptions about the characteristics of glass, different calculators may give slightly different results. Always use the lowest estimate.
### Don't Miss
#### Resources
• All types
• Articles
• Slideshows
• Videos
##### Sort:
• Most relevant
• Most popular
• Most recent
By using the eHow.co.uk site, you consent to the use of cookies. For more information, please see our Cookie policy. | 517 | 2,291 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2017-13 | longest | en | 0.943093 |
https://singaporemathsource.com/tag/model-method/page/2/ | 1,685,993,379,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224652161.52/warc/CC-MAIN-20230605185809-20230605215809-00077.warc.gz | 550,852,893 | 25,039 | ## Word Problem Wednesday – Passengers
Word Problem Wednesday was such a hit, we’re going to continue through the year with one problem a month.
This week’s problem comes from Primary Mathematics Intensive Practice 6A published in 2004 by SingaporeMath.com Inc:
##### There were 75% more adult passengers than children on a bus. After 1/2 of the children had gotten off at a bus stop, there were _______% more adults than children left on the bus.
Submit your solutions and we’ll post all interesting strategies.
The problem posted August 30th came from Primary Mathematics Challenging Word Problems 6 by Joseph D. Lee, published in 2006 by Panpac Education Private Limited.
##### The number of Jason’s cards and the number of Frederick’s cards are in the ratio of 5:8. The number of Frederick’s cards and the number of Steven’s cards are in the ratio of 4:3. If Jason has 18 fewer cards than Frederick, how many cards does Steven have?
Intrepid reader Shirly Davis sent in her solution:
Whew! How did you do?
Share:
## Throwback Thursday – Bar Model Solutions – by Students
Over the summer, we thought it would be fun to run some of the most popular posts from the past. It’s always interesting to see how students’ minds work. When I re-read a post from the past I always take away something different because I am in a different place with my own experience. Perhaps you are as well!
### Bar Model Solutions – by Students!
###### Originally published 4/12/2016
After the post on Assessing Bar Model Solutions went up, Beth Curran sent a message: “We just did that problem!” She agreed to share some student work:
And when the students didn’t draw a model:
I see this as a comparison problem:
5 units -> 125 students
1 unit -> 125 ÷ 5 = 25
7 units for boys -> 7 x 25 = 175 boys in all
(That’s the Thinking Blocks Model Drawing tool that allows you to insert your own word problems and solve – or you can use the pre-made questions!)
Share:
## Word Problem Wednesday – Gavin and Howie
Summer’s here, but you’re missing your math? Don’t despair – we’ve got you covered. Check the site each week for one whopper of a word problem that’s sure to challenge!
This week’s problem comes from New Syllabus Mathematics Strategies Primary 3 by Yee Fook Shiong, published in 2007 by Educational Publishing House Pte Ltd.
##### Gavin has 356 cards. He has 286 more cards than Howie. How many cards must Gavin give to Howie so that they both have the same number of cards?
Submit your solutions and we’ll post all interesting strategies next week.
Share:
## Word Problems and Bar Models from Literature
I’ve enjoyed Denise Gaskin’s Let’s Play Math blog since at least 2007! I shared her site when the problems for Mr. Popper’s Penguins were first published.
She has a new book of word problems tied to literature: Word Problems from Literature: An Introduction to Bar Model Diagram
I immediately bought a copy for my Kindle (a steal at only \$3.99).
Here’s a sample from the chapter entitled Moving Toward Algebra: Challenge Problems:
Denise provides step-by-step solutions with bar models. Here’s just a teaser of the solution to Han Solo’s problem:
This looks to be a great resource for some motivating and just darn fun problems.
Enjoy!
Share:
## 4,000 Teachers, 100,000 Students: Celebrating 7 Years with BER
In 2008, I left teaching in the classroom to champion Singapore Mathematics and expand its reach to elementary schools and children everywhere. The following year, the Bureau of Education and Research (BER) gave me an amazing opportunity to pursue this goal by presenting Singapore Math workshops throughout North America.
Some attendees at my BER seminars came with prior knowledge about the Singapore curriculum, but a bigger number were being introduced to Math from Singapore for the first time.
At a Seattle workshop earlier this year, BER’s Mark Ita surprised me (and other attendees) by presenting me with a handsome plaque, which read, in part:
In Recognition of Your Distinguished Teaching and Your Outstanding Contribution to the Education Profession
#### 4,000 Teachers, 100,000 Students
The stats scribbled on a Post-It Note on the back of the plaque included some tangible data to support this statement:
• 165 Seminars
• 4,000 Teachers
• Over 100,000 Students
Cassy with BER’s Mark Ita
It is highly satisfying to know that I have impacted this number of teachers and students through my BER presentations. On the other hand, the National Center for Education Statistics reports that there are about 35.2 million Pre-K to Grade 8 students in the United States. Clearly, there is much more work to be done!
I am very grateful to BER for giving me the opportunity to present Singapore Math workshops on their behalf over the past seven years. Sincere thanks to Rich, Boyce, Mark, Nargis, Lisa and the entire travel logistics team, and the dozens of project managers who have provided encouragement and support along the way. Thank you so much!
Share: | 1,122 | 5,012 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.78125 | 4 | CC-MAIN-2023-23 | latest | en | 0.934175 |
http://unitconverter.io/kilometers/inches/1282 | 1,542,123,558,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039741324.15/warc/CC-MAIN-20181113153141-20181113175141-00452.warc.gz | 352,044,332 | 3,989 | # 1,282 kilometers to inches
to
1,282 Kilometers = 50,472,440.9448 Inches
This conversion of 1,282 kilometers to inches has been calculated by multiplying 1,282 kilometers by 39,370.0787 and the result is 50,472,440.9448 inches. | 72 | 231 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2018-47 | longest | en | 0.92292 |
https://techmap.io/technology/Category-theory/60f6edf9ba93cb76f892a293 | 1,638,405,849,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00246.warc.gz | 623,323,933 | 30,985 | Category Theory
What is Category theory?
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions. Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself. Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations in 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure. Category theory has practical applications in programming language theory, for example the usage of monads in functional programming. It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.
Technology Types
category theoryfieldfields of mathematichigher category theorymathematical structuretheory
Synonyms
Category theoreticCategory-theoretic
Translations
Categorietheorie (wiskunde) (nl)Kategorientheorie (de)Kategoriteori (sv)Teori kategori (in)Teoria das categorias (pt)Teoría de categorías (es)Teoria de categories (ca)Teoria delle categorie (it)Teoria kategorii (pl)Teorie kategorií (cs)Teorio de kategorioj (eo)Théorie des catégories (fr)Θεωρία κατηγοριών (el)Теория категорий (ru)Теорія категорій (uk)نظرية الأصناف (ar)범주론 (ko)圏論 (ja)范畴论 (zh)
Tech Info
Related to
Source: [object Object]
— Date merged: 11/6/2021, 1:32:52 PM
— Date scraped: 5/20/2021, 6:05:53 PM | 494 | 2,020 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2021-49 | latest | en | 0.760304 |
https://www.kylesconverter.com/time/centiseconds-to-months | 1,713,424,560,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817200.22/warc/CC-MAIN-20240418061950-20240418091950-00391.warc.gz | 773,491,055 | 5,343 | Convert Centiseconds to Months
Kyle's Converter > Time > Centiseconds > Centiseconds to Months
Centiseconds (cs) Months (mo) Precision: 0 1 2 3 4 5 6 7 8 9 12 15 18
Reverse conversion?
Months to Centiseconds
(or just enter a value in the "to" field)
Please share if you found this tool useful:
Unit Descriptions
1 Centisecond:
A centisecond is exactly 0.01 seconds. One hundredth of a second.
1 Month (full):
30 Days, for the purposes of calculation.
Conversions Table
1 Centiseconds to Months = 070 Centiseconds to Months = 0
2 Centiseconds to Months = 080 Centiseconds to Months = 0
3 Centiseconds to Months = 090 Centiseconds to Months = 0
4 Centiseconds to Months = 0100 Centiseconds to Months = 0
5 Centiseconds to Months = 0200 Centiseconds to Months = 0
6 Centiseconds to Months = 0300 Centiseconds to Months = 0
7 Centiseconds to Months = 0400 Centiseconds to Months = 0
8 Centiseconds to Months = 0500 Centiseconds to Months = 0
9 Centiseconds to Months = 0600 Centiseconds to Months = 0
10 Centiseconds to Months = 0800 Centiseconds to Months = 0
20 Centiseconds to Months = 0900 Centiseconds to Months = 0
30 Centiseconds to Months = 01,000 Centiseconds to Months = 0
40 Centiseconds to Months = 010,000 Centiseconds to Months = 0
50 Centiseconds to Months = 0100,000 Centiseconds to Months = 0.0004
60 Centiseconds to Months = 01,000,000 Centiseconds to Months = 0.0039 | 396 | 1,387 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-18 | latest | en | 0.779445 |
https://www.plati.ru/itm/2-a-flywheel-radius-r-10-cm-planted-on-a-horizontal/2032025?lang=en-US | 1,521,396,401,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257645830.10/warc/CC-MAIN-20180318165408-20180318185408-00443.warc.gz | 885,655,506 | 11,658 | # 2. A flywheel radius R = 10 cm planted on a horizontal
Affiliates: 0,01 \$ — how to earn
Sold: 1
Refunds: 0
Content: fiz054_02.doc (53,5 kB)
# Description
2. A flywheel radius R = 10 cm planted on a horizontal axis. On the rim of the flywheel wound cord that is bound to the load mass m = 800 g Dropping uniformly accelerated, the load went the distance s = 160 cm for the time t = 2 s. Determine the moment of inertia J of the flywheel. | 135 | 443 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2018-13 | longest | en | 0.743771 |
http://stackoverflow.com/questions/9977734/controlling-how-test-data-is-generated-in-quickcheck | 1,469,771,855,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257829972.19/warc/CC-MAIN-20160723071029-00129-ip-10-185-27-174.ec2.internal.warc.gz | 233,495,322 | 19,674 | Dismiss
Announcing Stack Overflow Documentation
We started with Q&A. Technical documentation is next, and we need your help.
Whether you're a beginner or an experienced developer, you can contribute.
# Controlling how test data is generated in QuickCheck
I wrote an algorithm to find a solution to the subset sum problem in Haskell. The signature is
``````subsetSum :: (Ord a, Num a) => [a] -> a -> Maybe [a]
``````
QuickCheck seems to be a good fit to test that. For example I here is one of the properties that I could check:
``````prop_sumEqualsS l s = case subsetSum l s of
Just solution -> (sum solution) == s
Nothing -> True
``````
The problem is that the algorithm is quite computationally intensive and running 100 tests with big input lists takes ages to run.
I tried with QuickCheck 1 and it did run quickly, but the data sets used for testing were very small. After moving to QuickCheck 2 it seems to be the opposite problem. There is a manual for QC but it seems to be old (there is no date information) and I don't know how much still applies to QC2. A Tutorial is available on the Haskell Wiki but there is not much details, just a few words on instantiating `Arbitrary`.
So I have two questions:
• What changes in QuickCheck 2 make it become so much slower than QuickCheck?
• What is the best way to control data sets creation to make them useful for a given test?
Edit: To be more specific, I would like to test my solution with a list size from 0 to 100, containing integers between -10000 and 10000.
-
Your questions seems a little vague for the context of QuickCheck; perhaps you'd be better asking a general testing question instead. There are a few problems with your current approach though: it won't check that the solution is actually a subset, or that if the function returns Nothing then there is in fact no solution. – gatoatigrado Apr 2 '12 at 19:37
@gatoatigrado The property was just an example. I believe checking that the solution is a subset belongs in another property. Am I wrong? I don't see an easy way to check that when Nothing is returned, there is in fact no solution, except by solving the problem again with another algorithm. This seems a bit redundant. – Antoine Apr 3 '12 at 4:56
One thing that QuickCheck 2 introduced that could be a source of inefficiency is the `shrink` function. If a property fails, then the shrink function is called which gives candidates on smaller test values, and they are shrunk until they cannot give a smaller value for which the property fails. But this only happens when properties fail.
The changes for the arbitrary instance for lists has not changed much between version 1 and version 2. The difference is in wording, version 1 uses `vector`, and version 2 uses a list comprehension, but then `vector` is implemented with exactly such a list comprehension of sequenced calls to arbitrary.
Both implementations used the size parameter. In QuickCheck 1, the size of a generated value is by default `div n 2 + 3`, where `n` is the number of the test. QuickCheck 2 uses another approach, where the maximum size and the number of tests is configured. The test sizes will be distributed in that range, growing linearly in the number of tests (see `computeSize'` in `quickCheckWithResult`). This means, with the default setting of 100 tests and maximum size, the maximum size from QuickCheck 1 would be `(div 100 2 + 3) = 53`, and with QuickCheck 2 it would simply be `100`.
As subset sum is NP-complete you probably have an exponential algorithm ;) Then the difference in running time between a list of length 50 and 100 can of course be enormous. Understandably you want small lists to test with. You have two choices: make a new data type (preferably with `newtype`) or make a stand-alone generator and use `forAll`. By using `newtype` you can also specify how values should be shrunk. This is an example implementation using the `newtype` approach:
``````newtype SmallIntList = SmallIntList [Int] deriving (Eq,Show)
instance Arbitrary SmallIntList where
arbitrary = sized \$ \s -> do
n <- choose (0,s `min` 50)
xs <- vectorOf n (choose (-10000,10000))
return (SmallIntList xs)
shrink (SmallIntList xs) = map SmallIntList (shrink xs)
``````
This `Arbitrary` instance does not make lists longer than 50 elements. The elements do not use the size parameter, so they are always in the range `[-10000,10000]`, so there is some room for improvement. The `shrink` function simply uses the available `shrink`s for lists and `Int`s.
To use this instance with your property, just use a `SmallIntList` as an argument:
``````prop_sumEqualsS (SmallIntList l) s = case subsetSum l s of
...
``````
-
Thanks for your detailed explanation. – Antoine Apr 3 '12 at 17:08 | 1,142 | 4,757 | {"found_math": false, "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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2016-30 | latest | en | 0.959175 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.