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RS Aggarwal Class 11 Solutions Chapter 12 Geometrical Progression free PDF | Utopper
RS Aggarwal Class 11 Solutions Chapter 12
RS Aggarwal Solutions for Class 11 Chapter 12 – Geometrical Progression PDF
RS Aggarwal Class 11 Solutions Chapter-wise – Free PDF Download
The Document for RS Aggarwal Class 11 Solutions Chapter 12 has easy-to-understand answers to the different kinds of questions that can be asked from the chapter. The ideas of arithmetic progression
are already known to students, and the ideas of geometric progression just build on them. The chapter is worth a lot of points, and it’s best to study it well because it’s part of Unit 2-Algebra,
which has the most weight on exams. Our subject experts at Utopper made the RS Aggarwal Class 11 Solutions Chapter 12 PDF. Students can clear up any questions with the RS Aggarwal Solutions for Class
11 GP.
In the RS Aggarwal Book for Class 11, Chapter 12 has about 8 exercises with almost 90 questions. All of the questions are based on the latest CBSE exam pattern as well as the JEE, NEET, and other
competitive exam patterns. The chapter talks about things like geometrical progression, the nth term from the end of GP, problems with GP, word problems with GP, geometric mean, infinite geometric
series, etc.
When you can’t find the right answers, RS Aggarwal Solutions for Class 11 Chapter 12 of Maths by Utopper will help you in every way. The answers are given to you in an easy-to-understand way so that
you can easily understand each idea in the chapter. As soon as you fully understand the ideas, you won’t have any more problems with the question paper.
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RS Aggarwal Class 11 Solutions Chapter 12 – Geometrical Progression
RS Aggarwal Class 11 Solutions Chapter 12 Geometrical Progression – Free PDF Download
Students can use the PDF to get their ideas about geometric progressions ready. Students are tested on how well they understand and can use the ideas taught in the chapter geometric progression. The
concepts are talked about in depth, and all of the questions in the PDF are answered to help you understand them better. Students should practise answering different kinds of questions. This will
help them feel more at ease when they have to answer questions on tests. Here you can get the PDF for RS Aggarwal Class 11 Maths Chapter 12 Solutions.
RS Aggarwal Class 11 Solutions
Geometrical Progression
A geometric sequence is another name for a geometric progression. It is a series of numbers where each number after the first is the result of multiplying the number before it by a fixed number that
is not zero. For example, sequence 3, 9, 27,…. Is a geometric sequence with the same ratio of 3
What is the general form of a Geometrical Progression ?
A G.P. is usually written as a1, a1r, a1r2, a1r3, a1rn-1, and a1rn-1.
What is a GP’s general (nth) term?
a, ar, ar², ar³, …..arⁿ⁻¹, arⁿ⁻¹.
• Common ratio : r = a2/a1
• The nth term of a GP: an= a1rn-1
• The sum of a GP: Sn= a1(1- rn) / (1-r )
Geometric Progression has these Properties:
• If you multiply or divide each part of a geometric progression by a number that is not zero, the new sequence is also a geometric progression with the same common ratio.
• All of the terms in a geometric progression can be rearranged to form another geometric progression.
• If all the numbers in a geometric progression are raised to the same power, the new sequence is also a geometric progression.
• If the following are true for any three non-zero terms, then those three terms are in a GP.
• If y2 = xz, then x, y, and z are all in the same order.
FAQ ( Frequently Asked Questions )
1. What are some basic Geometric Progression formulas?
Ans – Geometric progressions are a list of non-zero numbers where the ratio between each pair of numbers stays the same. Each part of the sequence is made by multiplying the previous part by a
constant called the “Common ratio” and adding that to the previous part.
The ratio that stays the same in a geometric progression is:
r = a2/a1
For the nth term of a GP, the formula is: a = a1rn-1, where an is the first term in the GP and r is the common ratio.
The formula for the sum of a GP is: Sn=a1(1-rn)/(1-r), where an is the first term, r is the common ratio, and n is the number of terms.
2. What are the subtopics that the PDF goes into more depth about?
Ans – There are a lot of ideas about Geometric progressions in this chapter. There are some basic ideas and some topics that are more about how to use them. A geometric progression is a list of
numbers where the ratio between each pair of numbers is about the same. Download the free PDF from this page to learn more about each of the ideas.
The RS Aggarwal Class 11 Maths Geometrical Progression PDF goes into detail about the following:
• The general term of a GP
• Selection of terms of GP
• The Sum of a GP
• Sum of an infinite GP
• Properties of geometric progressions
• Properties of arithmetic and geometric means
• Insertion of geometric mean between two numbers
3. How do I get the RS Aggarwal Class 11 Solutions Chapter 12?
Ans – Students should go to the Utopper website or app to get the answers to Chapter 12 Geometric Progression from RS Aggarwal for Class 11. Students can get the best free study materials from
Utopper, which they can use online or off, whenever it’s most convenient for them. These solutions from Utopper are put together by experts in the field who make them fit the needs of the students.
Students can get to the RS Aggarwal solutions by going to the Utopper website and clicking on the Class 11 tab under RS Aggarwal under the “Solutions for popular reference solutions” tab. User can
also get access free study materials like Revision Notes, Important Questions and many more.
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Inclined Planes | studyslide.com
Inclined Planes
Download Report
Transcript Inclined Planes
Motion on an
inclined plane
Forces on a ramp
How would you draw a free-body
diagram for a cart on a ramp?
Forces on a ramp
How would you draw a free-body
diagram for a cart on a ramp?
Two forces act on the cart.
• One is the gravitational force.
Forces on a ramp
How would you draw a free-body
diagram for a cart on a ramp?
Two forces act on the cart.
• One is the gravitational force.
• The other is the normal force,
which acts perpendicular to
the ramp.
Forces on a ramp
These forces do not completely
cancel each other out.
If they did, the cart would not
There must be a net force.
To find Fnet, it helps to tilt
the coordinate axis system.
Keeping it simple
Tilting the coordinate system
for a ramp makes things
It turns complex 2-D motion
into simple 1-D motion.
Coordinate system on a ramp
Tilting the coordinate system
gives you 1-D motion along
the x-axis.
• The x-axis is parallel to the
ramp, and matches the
direction the object travels.
• The y-axis is perpendicular
to the ramp. There is no
motion along the y-axis.
Components of gravity
The force of gravity has xand y- components.
•On the x-axis there is only
the x-component of gravity:
mg sin θ.
•On the y-axis the forces
cancel each other out:
FN = mg cos θ.
Net force on a ramp
•On the x-axis there is only the
x-component of gravity:
mg sin θ.
This is the net force on the cart.
Acceleration on a ramp
The acceleration on a ramp
can be now be found from
Newton’s second law:
Ramp angle and acceleration
On the right is a table of the
acceleration for different
values of ramp angle, θ.
The acceleration at 0º is 0.
What does this physically
represent? º
Ramp angle and acceleration
On the right is a table of the
acceleration for different
values of ramp angle, θ.
The acceleration at 0º is 0.
What does this physically
represent? º
A ramp at 0º is flat. Gravity is not
pulling the object down the ramp,
so there is no acceleration.
Inclined planes
As you saw in the investigation,
the acceleration on an inclined
plane increases as the angle
How can you predict the
acceleration from the ramp’s
height and length?
Inclined planes
The relationship between
acceleration, height, and length
for motion on an inclined plane
Inclined planes
These two ways to calculate
the acceleration on a ramp
are equivalent:
Inclined planes
So far, you have investigated
objects going down inclined
Is there anything special about
objects going up inclined planes?
Which of these jobs looks easier?
Inclined planes
Inclined planes reduce the amount
of force needed to raise an object.
The component of gravity
pushing an object down the ramp
is rather small compared to mg.
This makes it easier to push
something up a ramp than to lift it,
assuming there isn’t much
1. Winston releases a cart on an inclined plane that is 3.2 m long
and 1.8 m high. What is the acceleration of the cart?
1. Winston releases a cart on an inclined plane that is 3.2 m long
and 1.8 m high. What is the acceleration of the cart?
2. Zoe measures the time for a ball to roll down an inclined plane
set at 30º. If she changes the angle of incline to 40º, what
happens to the time to reach the bottom?
A. increases
B. decreases
C. stays the same
D. not enough information
1. Winston releases a cart on an inclined plane that is 3.2 m long
and 1.8 m high. What is the acceleration of the cart?
2. Zoe measures the time for a ball to roll down an inclined plane
set at 30º. If she changes the ramp angle to 40º, what happens to
the time to reach the bottom?
A. increases
B. decreases
C. stays the same
D. not enough information
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Equilibrium and Equilibrium Ratios (Ki)
Consider a liquid-vapor in equilibrium. As we have discussed previously, a condition for equilibrium is that the chemical potential of each component in both phases are equal, thus:
${\mu }_{i}^{v}={\mu }_{i}^{L}$
We showed that this is equivalent to:
This is, for a system to be in equilibrium, the fugacity of each component in each of the phases must be equal as well. The fugacity of a component in a mixture can be expressed in terms of the
fugacity coefficient. Therefore, the fugacity of a component in either phase can be written as:
${f}_{i}^{V}={y}_{i}{\varphi }_{i}^{V}P$
${f}_{i}^{L}={y}_{i}{\varphi }_{i}^{L}P$
Introducing (17.3) into (17.2),
${y}_{i}{\varphi }_{i}^{V}P={x}_{i}{\varphi }_{i}^{L}P$
This equilibrium condition can be written in terms of the equilibrium ratio ${K}_{i}={y}_{i}l{x}_{i}$ , to get:
${K}_{i}=\frac{{y}_{i}}{{x}_{i}}=\frac{{\varphi }_{i}^{L}}{{\varphi }_{i}^{V}}$
Do you recall the problem at the end of Module 13? At that point we needed a more reliable way to calculate the equilibrium ratios that showed up in the Rachford-Rice objective function. We
demonstrated that once we know all values of K[i]’s, the problem of vapor-liquid equilibrium is reduced to solving the Rachford-Rice objective function, using the Newton-Raphson Procedure.
We can now calculate equilibrium ratios, using (17.5), in terms of fugacity coefficients. We also know that we have an analytic expression for the calculation of fugacity coefficients via EOS — this
was shown in the last section of the previous module. This is why we call this module “Vapor Liquid Equilibrium via EOS.”
Is this the end to our problems? Not quite. Take a look at the expression for fugacity coefficients in mixtures both for SRK EOS and PR EOS. It is clear that they are functions of the pressure,
temperature, and composition of the phases:
${\varphi }_{i}^{L}={\varphi }_{i}^{L}\left(P,T,{x}_{i}\right)$
${\varphi }_{i}^{V}={\varphi }_{i}^{V}\left(P,T,{y}_{i}\right)$
Do we know the composition of the phases “x[i]”, “y[i]” in advance? In a typical flash problem, we are given pressure, temperature and overall composition (z[i]). What do we want to know? How much
gas, how much liquid, and the compositions of the phases: ${\alpha }_{g},\text{ }{\alpha }_{l},{\text{ y}}_{i},{\text{ x}}_{i}$ . So, we do not know those compositions in advance; therefore, as it
stands, we cannot calculate (17.6) or (17.5). Thus far, it seems that the flash problem is unsolvable.
If we are bold enough, we could try to overcome this problem by “guessing” those compositions, and proceed by solving (17.6) and (17.5). With this “rough” estimate for Ki’s, we could solve for “${\
alpha }_{g}$ ” with the procedure outlined in Module 13 (“Objective Function and Newton-Raphson Procedure”). Once “${\alpha }_{g}$” is known, we could back calculate the compositions of the phases
using equations (12.7) and (12.11). If we were correct, those compositions would match each other (the “guessed” ones with respect to the “back-calculated”). More than likely, this would not happen,
and we would have to make a new “guess.” This is, fundamentally, an iterative procedure. Although this is not what we do, it does illustrate that this problem is solvable by implementing the
appropriate iterative scheme.
In equations (17.4) and (17.5), the fugacity of the liquid and vapor phases were computed in terms of the fugacity coefficient. Hence, this method of expressing the equilibrium criteria is known as
the dual-fugacity coefficient method. For the sake of completeness, it is necessary to indicate that the fugacity of a component in a mixture can also be expressed in terms of a thermodynamic concept
called the activity coefficient. While the fugacity coefficient is seen as a measure of the deviation of behavior with respect to the ideal gas model, the activity coefficient measures the deviation
of behavior with respect to the ideal liquid model. This approach is called the dual-activity coefficient method, in which both liquid and vapor phase fugacities are expressed in terms of the
activity coefficient and substituted into the equilibrium criteria in (17.2). A mixed activity coefficient-fugacity coefficient method can be also devised by expressing the liquid phase fugacities in
terms of activity coefficients and the vapor phase fugacities in terms of fugacity coefficients. Each of the aforementioned methods for the calculation of phase equilibria has its advantages and
disadvantages. The dual-fugacity-coefficient method is simpler both conceptually and computationally, but if the equation of state does not predict liquid and vapor densities well, the results may be
inaccurate. The activity coefficient method can be more accurate, but it is more complicated to implement. For the rest of the discussion, the dual-fugacity coefficient approach will be used.
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How to Measure Electrical Conductivity of the Soil Solution
Electrical Conductivity of Soil as a Predictor of Plant Response (Part 2)
Salt in soil comes from the fertilizer we apply but also from irrigation water and dissolving soil minerals. If more salt is applied in the irrigation water than is leached or taken off in harvested
plants, the soil becomes more saline and eventually ceases to support agricultural production (see part 1). This week, learn an effective way to measure electrical conductivity (EC) in soil.
How to Measure Electrical Conductivity of the Soil Solution
As mentioned above, the earliest measurements of solution conductivity were made on soil samples, but it was found to be more reliable to extract the soil solution and make the measurements on it.
When values for unsaturated soils are needed, those are calculated based on the saturation numbers and conjecture about how the soil dried to its present state. Obviously a direct measurement of the
soil solution conductivity would be better if it could be made reliably.
Two approaches have been made to this measurement. The first uses platinum electrodes embedded in ceramic with a bubbling pressure of 15 bars. Over the plant growth range the ceramic remains
saturated, even though the soil is not saturated, allowing a measurement of the solution in the ceramic. As long as there is adequate exchange between the ceramic and the soil solution, this
measurement will be the EC of the soil solution, pore water EC.
The other method measures the conductivity of the bulk soil and then uses empirical or theoretical equations to determine the pore water EC. The TEROS 12 sensor uses the second method. It requires no
exchange of salt between soil and sensor and is therefore more likely to indicate the actual solution electrical conductivity. The following analysis shows one of several methods for determining the
electrical conductivity of the saturation extract from measurements of the bulk soil electrical conductivity.
Mualem and Friedman (1991) proposed a model based on soil hydraulic properties. It assumes two parallel conduction paths: one along the surface of soil particles and the other through the soil water.
The model is
Here σ[b] is the bulk conductivity which is measured by the probe, σ[s] is the bulk surface conductivity, σ[w] is the conductivity of the pore water, θ is the volumetric water content, θ[s] is the
saturation water content of the soil and n is an empirical parameter with a suggested value around 0.5. If, for the moment, we ignore surface conductivity, and use eq. 1 to compute the electrical
conductivity of a saturated paste (assuming n = 0.5 and θ[s] = 0.5) we obtain σ[b] = 0.35σ[w]. Obviously, if no soil were there, the bulk reading would equal the electrical conductivity of the water.
But when soil is there, the bulk conductivity is about a third of the solution conductivity. This happens because soil particles take up some of the space, decreasing the cross section for ion flow
and increasing the distance ions must travel (around particles) to move from one electrode of the probe to the other. In unsaturated soil these same concepts apply, but here both soil particles and
empty pores interfere with ion transport, so the bulk conductivity becomes an even smaller fraction of pore water conductivity.
Our interest, of course, is in the pore water conductivity. Inverting eq. 1 we obtain
In order to know pore water conductivity from measurements in the soil we must also know the soil water content, the saturation water content, and the surface conductivity. The TEROS 12 measures the
water content. The saturation water content can be computed from the bulk density of the soil
Where ρ[b] is the soil bulk density and ρ[s] is the density of the solid particles, which in mineral soils is taken to be around 2.65 Mg/m^3 . The surface conductivity is assumed to be zero for
coarse-textured soil. Therefore, using the TEROS 12 allows us to quantify pore water EC through the use of the above assumptions. This knowledge has the potential to be a very useful tool in
fertilizer scheduling.
Electrical Conductivity is Temperature Dependent
Electrical conductivity of solutions or soils changes by about 2% per Celsius degree. Because of this, measurements must be corrected for temperature in order to be useful. Richards (1954) provides a
table for correcting the readings taken at any temperature to readings at 25 °C. The following polynomial summarizes the table
where t is the Celsius temperature. This equation is programmed into the 5TE, so temperature corrections are automatic.
Units of Electrical Conductivity
The SI unit for electrical conductance is the Siemen, so electrical conductivity has units of S/m. Units used in older literature are mho/cm (mho is reciprocal ohm), which have the same value as S/
cm. Soil electrical conductivities were typically reported in mmho/cm so 1 mmho/cm equals 1 mS/cm. Since SI discourages the use of submultiples in the denominator, this unit is changed to deciSiemen
per meter (dS/m), which is numerically the same as mmho/cm or mS/cm. Occasionally, EC is reported as mS/m or µS/m. 1 dS/m is 100 mS/m or 10^5 µS/m.
Understand EC sensor readings
Understanding the difference between electrical conductivity readings in water and in soil can help you make better use of your EC readings. Watch the video to answer questions such as “Why does
water that’s 1.9 dS/m not read 1.9 dS/m when it’s in the soil?
Richards, L. A. (Ed.) 1954. Diagnosis and Improvement of Saline and Alkali Soils. USDA Agriculture Handbook 60, Washington D. C.
Rhoades, J. D. and J. Loveday. 1990. Salinity in irrigated agriculture. In Irrigation of Agricultural Crops. Agronomy Monograph 30:1089-1142. Americal Society of Agronomy, Madison, WI.
Take our Soil Moisture Master Class
Six short videos teach you everything you need to know about soil water content and soil water potential—and why you should measure them together. Plus, master the basics of soil hydraulic
Learn more
Watch the webinar: “Using electrical conductivity measurements to optimize irrigation”—>
Download the “Researcher’s complete guide to water potential”—>
Download the “Researcher’s complete guide to soil moisture”—>
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Invariance - Data Science Wiki
Invariance :
Invariance is a concept in mathematics and physics that refers to the property of certain quantities or equations remaining unchanged or unchanged under certain transformations or operations. This
concept is important in a wide range of fields, including geometry, algebra, mechanics, and electromagnetism.
One example of invariance is the concept of Euclidean distance in geometry. Euclidean distance is a measure of the straight-line distance between two points in a Euclidean space, and it is calculated
using the Pythagorean theorem. The distance between two points remains unchanged, or invariant, regardless of how the points are rotated or translated in space. This is because the Euclidean distance
formula only depends on the coordinates of the two points, which remain unchanged under rotations and translations.
Another example of invariance is the principle of conservation of energy in mechanics. This principle states that the total amount of energy in a closed system remains constant, regardless of how the
energy is transformed from one form to another. For example, if a ball is rolling down a hill and gains kinetic energy, this energy is not lost when the ball reaches the bottom of the hill and comes
to a stop. Instead, the kinetic energy is converted into potential energy, which is stored in the ball’s height above the ground. The total amount of energy in the system, the sum of the kinetic and
potential energies, remains unchanged.
These examples illustrate the fundamental role of invariance in mathematics and physics. Invariance allows us to make statements about quantities that are independent of the specific details of a
particular situation or transformation. In the case of Euclidean distance, invariance allows us to calculate distances without needing to know the orientation or position of the points in space. In
the case of conservation of energy, invariance allows us to predict the behavior of systems without needing to track the specific details of how energy is transformed from one form to another.
Invariance is a fundamental concept that is used to study the properties of objects and systems that remain unchanged under certain transformations or operations. This concept is important in many
fields, including geometry, algebra, mechanics, and electromagnetism.
One example of invariance is the concept of symmetry in geometry. Symmetry refers to the property of an object or figure that remains unchanged when it is rotated or reflected in a specific way. For
example, a square has four-fold rotational symmetry because it looks the same after being rotated 90, 180, or 270 degrees. This symmetry is invariant under these rotations because the shape of the
square remains unchanged.
Another example of invariance is the principle of superposition in electromagnetism. This principle states that the total electric or magnetic field at a point in space is the sum of the fields
produced by individual sources. This means that the total field is invariant under the addition of new sources, as long as the sources are not interacting with each other. For example, if two
electric charges are placed at a distance from each other, the total electric field at any point between them is the sum of the fields produced by each charge. This total field remains unchanged if
an additional charge is placed at the same point, as long as the additional charge does not interact with the other two charges.
In both of these examples, invariance allows us to make predictions about the behavior of objects and systems without needing to know the specific details of the transformations or operations being
applied. This simplifies our calculations and helps us understand the underlying patterns and structures that govern the behavior of these objects and systems.
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Topics in Differential Geometrysearch
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Topics in Differential Geometry
Peter W. Michor : Universität Wien, Wien, Austria and Erwin Schrödinger Institut für Mathematische Physik, Wien, Austria
Hardcover ISBN: 978-0-8218-2003-2
Product Code: GSM/93
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN: 978-1-4704-1161-9
Product Code: GSM/93.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN: 978-0-8218-2003-2
eBook: ISBN: 978-1-4704-1161-9
Product Code: GSM/93.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
Click above image for expanded view
Topics in Differential Geometry
Peter W. Michor : Universität Wien, Wien, Austria and Erwin Schrödinger Institut für Mathematische Physik, Wien, Austria
Hardcover ISBN: 978-0-8218-2003-2
Product Code: GSM/93
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN: 978-1-4704-1161-9
Product Code: GSM/93.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN: 978-0-8218-2003-2
eBook ISBN: 978-1-4704-1161-9
Product Code: GSM/93.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
• Graduate Studies in Mathematics
Volume: 93; 2008; 494 pp
MSC: Primary 53
This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections,
Riemann manifolds, isometric actions, and symplectic and Poisson geometry.
The layout of the material stresses naturality and functoriality from the beginning and is as coordinate-free as possible. Coordinate formulas are always derived as extra information. Some
attractive unusual aspects of this book are as follows:
□ Initial submanifolds and the Frobenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) are discussed.
□ Lie groups and their actions are treated early on, including the slice theorem and invariant theory.
□ De Rham cohomology includes that of compact Lie groups, leading to the study of (nonabelian) extensions of Lie algebras and Lie groups.
□ The Frölicher-Nijenhuis bracket for tangent bundle valued differential forms is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure
groups). Riemann geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent
bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view.
□ Symplectic and Poisson geometry emphasizes group actions, momentum mappings, and reductions.
This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good
knowledge of undergraduate analysis and linear algebra.
Graduate students, research mathematicians and physicists interested in differential geometry, mechanics, and relativity.
□ Chapters
□ Chapter I. Manifolds and vector fields
□ Chapter II. Lie groups and group actions
□ Chapter III. Differential forms and de Rham cohomology
□ Chapter IV. Bundles and connections
□ Chapter V. Riemann manifolds
□ Chapter VI. Isometric group actions or Riemann $G$-manifolds
□ Chapter VII. Symplectic and Poisson geometry
□ ...remarkably effective. ... Michors book is a truly marvelous pick from which to learn a lot of beautiful, important, and current mathematics.
MAA Reviews
□ Throughout the book the author stresses the development of short exact sequences and takes evident delight in the applications that ensue. For the reviewer, this is one of the most enjoyable
qualities of the text. The text is a treasure, and will open up to the diligent and patient reader a vast panorama of modern differential geometry.
Mathematical Reviews
• Desk Copy – for instructors who have adopted an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 93; 2008; 494 pp
MSC: Primary 53
This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections,
Riemann manifolds, isometric actions, and symplectic and Poisson geometry.
The layout of the material stresses naturality and functoriality from the beginning and is as coordinate-free as possible. Coordinate formulas are always derived as extra information. Some attractive
unusual aspects of this book are as follows:
• Initial submanifolds and the Frobenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) are discussed.
• Lie groups and their actions are treated early on, including the slice theorem and invariant theory.
• De Rham cohomology includes that of compact Lie groups, leading to the study of (nonabelian) extensions of Lie algebras and Lie groups.
• The Frölicher-Nijenhuis bracket for tangent bundle valued differential forms is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure
groups). Riemann geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent bundles.
The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view.
• Symplectic and Poisson geometry emphasizes group actions, momentum mappings, and reductions.
This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good
knowledge of undergraduate analysis and linear algebra.
Graduate students, research mathematicians and physicists interested in differential geometry, mechanics, and relativity.
• Chapters
• Chapter I. Manifolds and vector fields
• Chapter II. Lie groups and group actions
• Chapter III. Differential forms and de Rham cohomology
• Chapter IV. Bundles and connections
• Chapter V. Riemann manifolds
• Chapter VI. Isometric group actions or Riemann $G$-manifolds
• Chapter VII. Symplectic and Poisson geometry
• ...remarkably effective. ... Michors book is a truly marvelous pick from which to learn a lot of beautiful, important, and current mathematics.
MAA Reviews
• Throughout the book the author stresses the development of short exact sequences and takes evident delight in the applications that ensue. For the reviewer, this is one of the most enjoyable
qualities of the text. The text is a treasure, and will open up to the diligent and patient reader a vast panorama of modern differential geometry.
Mathematical Reviews
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External Equation
GAMS provides a number of built-in or intrinsic functions for use in equations. Still, the extremely diverse set of application areas in which GAMS is used can create demand for the addition of new
and often sophisticated and specialized functions. There is a trade-off between satisfying these requests and avoiding complexity not needed by most users. The GAMS External Equations Facility
provides one means for managing this trade-off, since it allows users to import functions from an external library to define equations in a GAMS model. However, these external libraries can currently
only provide functionality for the evaluation of functions (incl. their first derivatives) in a point. Solvers that need to analyze the algebraic structure of the model instance are therefore not
able to work with external equations. This includes the class of deterministic global solvers, see column "Global" in this table, while, for example, stochastic global solvers can work with external
Both external equations and extrinsic functions aim to provide possibilities to extend GAMS by user-provided mathematical functions. However, there are fundamental differences in the use and
implementation of both. For most situations, extrinsic functions should be preferred over external equations. See also Extrinsic Functions vs. External Equations.
Building external equation libraries requires the knowledge of a regular programming language (like C/C++, FORTRAN, ...) and experience with handling compilers and linkers to build dynamically
linked libraries.
The external equation interface is not intended as a way to bypass some of the very useful model checking done by GAMS for models that are solved with NLP solvers. External equations are still
assumed to be continuous with accurate and smooth first derivatives. The continuity assumption implies that external equations must have very low noise levels, considerably below the feasibility
tolerance used by the solver. The assumption about accurate derivatives implies that derivatives must be computed more accurately than can be done with standard finite differences. If these
assumptions are not satisfied, then there is no guarantee that the NLP solver can find a solution that has the mathematical properties of a local optimum, i.e., a solution that satisfies the
Karush-Kuhn-Tucker conditions within the standard tolerances used by the solver.
In the following, connecting code written in FORTRAN, C, Delphi, or some other programming language to equations and variables in a GAMS model is described. These GAMS equations will be referred to
as external equations and the compiled version of the programming routines will be referred to as the external module that defines the external functions. The form of the external module depends on
the operating system that is used. The external module under Windows is a Dynamic Link Library (.dll) and the external module under Unix is a shared object (.so or .dylib). In principle, any language
or system may be used to build the DLL or shared object that defines the external module, as long as the interface conventions are not changed.
The GAMS Test Library provides examples of external equations consisting of GAMS models and C, Delphi, Java, and FORTRAN code. For more details, see Section Examples in the GAMS Test Library.
The basic mechanism of external equations is to declare all the equations and variables using the usual GAMS syntax. The interpretation of the external equations is done in a special way. Instead of
the usual semantic content, the external equations specify the mapping between the equation and variable names used in GAMS and the function and variable indices used in the external module. This
mapping is described in Section Model Interface. The external module may be written in C, FORTRAN, or most other programming languages. Section Programming Interface describes the general definitions
for an external module for C, Delphi, and FORTRAN from a programming language perspective. Note that the way the program is compiled and converted into an external module is system and compiler
specific. The following Section Implementation gives detailed advice on various aspects of the implementation of the external module.
Examples in the GAMS Test Library
Model [TESTEXEQ] gives an overview of all examples in the GAMS Test Library and may be used to compile and run them. Note that the remainder of this chapter will reference examples that are listed in
this model. Further, model [COMPLINK] may be used as a script to compile and link external equation libraries. Note that these models hardcode the path to the Java compiler and libraries and these
paths will need to be adapted by the user when running the Java examples.
Observe that regardless of how external libraries are built, the examples (e.g. [EX1]) will by default solve a model without using external equations. To solve the example models with all kinds of
different external equation libraries, they may be run with the argument
Alternatively, only selected libraries may be used by using one or more of the following command line parameters:
Model Interface
External Equation Syntax
External equations that are used to specify the interface to the external module are declared in GAMS like any other equation. The syntax for the external equation definition statement is as follows:
eqn_name(index_list)[$logical_condition(s)].. expression =x= expression ;
Note that the only difference to the usual equation definition is the use of the equation type =x=.
The equations defined by an external module are always interpreted as equality constraints with zero right-hand sides. Thus inequalities have to be converted to equalities by adding explicit slack
variables, which will serve as additional external variables. A nonzero right-hand side need to be taken care of in the external equation implementation.
Mapping of external equations and variables to indices
Some mappings must be specified to link an external module to a GAMS model. External equations are assumed to be defined in terms of indices i = 1...m. These indices must be mapped to GAMS equation
names. Similarly, the variables used inside the external functions are assumed to be defined in terms of indices j = 1...n. These indices must be mapped to GAMS variable names. Finally, the name of
the external module must be specified. Note that GAMS solvers are typically designed for large models and rely on sparsity. The last part of the specification of a set of external equations is
therefore the sparsity pattern of the external equations, i.e., which variables appear in which equations.
The value of the constant term of the external equation must be an integer, since the value of the constant maps the row of the GAMS equation to the index (in 1...m) of the external equation. Several
blocks of GAMS equations may be mapped to external equations using the =x= notation. The mapping between GAMS equations of type =x= and indices 1...m must be bijective (one-to-one). This means that
two GAMS equations may not be mapped into the same external equation index and that there may not be any holes in the list of external equation indices. Although there may be any number of blocks of
GAMS external equations, they must all map into and be implemented by one single external module.
The variable part of each external equation defines both the sparsity pattern of the external equation and the mapping from GAMS variables to the indices of the external variables. The variable part
must be a sum of terms where each term is an integer times a variable. The existence of the term indicates that the variable involved is used in the external equation and that there is a
corresponding derivative. The value of the coefficient defines the index of the external variable (in 1...n) that the GAMS variable is mapped to. For example, the term 5*Y indicates that the external
equation depends on the GAMS variable Y and that Y is mapped to the 5th element in the vector of external variables. Clearly, if a variable appears in more than one external equation, then the value
of its coefficient must be the same in each case.
Note that several blocks of GAMS variables may be used in external equations. In contrast to equations, where all rows in an equation block are either external or not, some columns in a variable
block may be external while others are not. The mapping between GAMS variables that appear in equations of type =x= and external variable indices 1...n must be bijective (one-to-one). This means that
two GAMS columns may not be mapped into the same external variable index and that there may not be any holes in the list of external variable indices. Although there may be any number of blocks of
GAMS variables mapped to external variables, they must all map into one single vector passed to the subroutine in the external module.
Observe that while some GAMS variables are external, there is no syntax provided to mark them as external variables. They may be used in non-external GAMS equations as well as external equations.
Indeed, without this capability the model would be separable and the external equations and the functions they map to would be of little use.
□ As the coefficients and right-hand sides in the GAMS definition of external equations are interpreted as indices, users are not allowed to scale external equations and variables.
□ External equations are treated in a special way, therefore the command line parameter and model attribute HoldFixed will not treat any fixed external variables as constants.
Name of external module
The name of the external module in which the external equations are implemented may be defined in a number of ways. By default, the external module is assumed to have the same name as the GAMS model
with an extension that is operating system dependent. The extension is .dll for Windows, .dylib for macOS, and .so for any other Unix.
A custom name for the external module may be specified with a file statement. In this case the file name has to be listed as an additional item in the model statement. If the library extension is
omitted in the file statement, GAMS will add the system-dependent extension automatically. This helps to make the model portable between different operating systems.
Consider the following simple example:
File myextfile / extern /;
Model mymodel / all, myextfile /;
When model mymodel is solved, GAMS will try to load the an external module file named extern.so, extern.dylib, or extern.dll, depending on the current operating system.
By default, the external module is assumed to be located in the directory external_equations in the GAMS standard locations or in the directory from which GAMS is called. A different location may be
specified with an added path in the file statement.
Programming Interface
This section discusses C, Delphi, and FORTRAN interfaces to the GAMS external equations facility.
The external equation module need to provide a function called GEFUNC. The beginning of the external equation module typically looks as follows:
#define GE_EXPORT
#include "geheader.h"
GE_API int GE_CALLCONV
gefunc(int* icntr, double* x, double* f, double* d, msgcb_t msgcb)
The header file geheader.h can be found in the testlib_ml subdirectory of the GAMS distribution. It defines GE_API and GE_CALLCONV and the signature of the function gefunc. GE_API is used to
indicate to the compiler whether the function should be exported or imported. Due to defining GE_EXPORT before including geheader.h, GE_API is defined such that the function will be marked for
export (__declspec(dllexport) on Windows and __visibility__("default") with GCC). Further, GE_CALLCONV indicates the calling convention that should be used on Windows. Currently, this is defined
to be __stdcall. On other operating systems, it is empty.
Integer Function gefunc (icntr, x, f, d, msgcb)
C Control Buffer:
Integer icntr(*)
C Numerical Input and Output:
Double Precision x(*), f, d(*)
C Message Callback Routine
External msgcb
Function GeFunc(var Icntr: ticntr;
var x: tarray;
var F: double;
var D: tarray;
MsgFunc: tMsgCallBack): integer; stdcall;
The unit file geheader_d.pas can be found in the testlib_ml subdirectory of the GAMS distribution.
In the following, the arguments of GEFUNC are described in detail.
Control vector icntr
The array icntr is a control vector that is used to pack and communicate control information between GAMS and the external module. Some helpful definitions to work with the icntr array are provided
by the files geheader.h (C), geheader_d.pas (Delphi), and gehelper.f90 (Fortran 90). The array elements are the following:
Element Description
icntr Holds the length of array icntr in number of elements. This is provided by GAMS.
icntr[I_Neq] Number of external equation rows seen in the GAMS model. This is provided by GAMS.
icntr[I_Nvar] Number of external variables seen in the GAMS model. This is provided by GAMS.
icntr[I_Nz] Number of nonzero derivatives or Jacobian elements seen in the GAMS model. This is provided by GAMS.
Current mode of operation. This is provided by GAMS. The following values are possible:
DOINIT: Initialize. This will be the first call of GEFUNC, where initializations needed by the external module may be performed.
icntr[I_Mode] DOTERM: Terminate. This will be the last call of GEFUNC, where cleanup tasks needed by the external module may be performed.
DOEVAL: Function evaluation. External equations should be evaluated.
DOCONSTDERIV: Constant Derivatives. Information about constant derivatives should be provided.
DOHVPROD: Hessian-Vector product. The product between the Hessian of an external equation and a vector should be computed. See Second Derivatives: Hessian times Vector for details.
icntr[I_Eqno] Index of the external equation to be evaluated during this call to GEFUNC. This is provided by GAMS in function evaluation mode (icntr[I_Mode]=DOEVAL) and is a number between 1 and
icntr[I_Neq], inclusive. Note that the external equation interface allows to communicate information about only one function at a time.
icntr Flag whether function value should be computed. icntr[I_Dofunc] is provided by GAMS in function evaluation mode (icntr[I_Mode]=DOEVAL). If set to 1, then GEFUNC must return the
[I_Dofunc] numerical value of the function indexed by icntr[I_Eqno] in the scalar f.
icntr[I_Dodrv] Flag whether derivative should be computed. icntr[I_Dodrv] is provided by GAMS in function evaluation mode (icntr[I_Mode]=DOEVAL). If set to 1, then GEFUNC must return the numerical
values of the derivatives of the function indexed by icntr[I_Eqno] in the array d.
icntr[I_Newpt] Flag for new point. icntr[I_Newpt] is provided by GAMS in function evaluation mode (icntr[I_Mode]=DOEVAL). If set to 1, then the point x may be different from the previous call of
GEFUNC. If set to 0, then x will not have changed since the previous call.
If icntr[I_Debug] is set to a nonzero value by the external equation module, then the functions GEstat and GElog will write all strings to a file called debugext.txt and flush the
icntr[I_Debug] buffer immediately after writing. The string debugger may be used when a shared object crashes before GAMS has had an opportunity to display the messages. In FORTRAN, the string
debugger will use FORTRAN unit icntr[I_Debug]. For more details see Section Message Output.
Flag to request the name of a special directory or file from GAMS. The following values are possible:
I_Scr: Scratch directory,
icntr I_Wrk: Working directory,
[I_Getfil] I_Sys: GAMS system directory,
I_Cntr: Control file.
For more information, see Section Communicating Data to the External Module via Files.
icntr[I_Smode] Flag for string mode. This is provided by GAMS. For details see Section Communicating Data to the External Module via Files.
icntr Indicator for use of constant derivatives. This entry is optional. For details see Section Constant Derivatives below.
icntr Indicator for use of Hessian-Vector product for second order derivatives. This entry is optional. For details see Section Second Derivatives: Hessian times Vector below.
Observe that FORTRAN programmers will have to replace the square brackets [] with parentheses ().
Evaluation point x
Argument x is an array with icntr[I_Nvar] elements and is provided by GAMS if GEFUNC is called in function evaluation mode (icntr[I_Mode] = DOEVAL). Typically, GAMS and the solvers ensure that the
individual elements of x are in between, or very close to, the variable bounds defined in the GAMS model. During initialization and termination calls, x is not defined and the external module must
not reference x. C programmers should index this array starting at zero, i.e., the first external variable is referenced as x[0].
Function value f
If icntr[I_Mode] = DOEVAL and icntr[I_Dofunc] = 1, then the external module must return the value of the external equation icntr[I_Eqno] in the scalar f. During initialization and termination calls,
f must not be referenced.
Derivative vector d
If icntr[I_Mode] = DOEVAL and icntr[I_Dodrv] = 1, then the external module must return the values of the derivatives of external function icntr[I_Eqno] with respect to all variables in the array d.
The derivative with respect to variable x[i] is returned in d[i]. It is sufficient to set only those positions in d that correspond to variables actually appearing in equation icntr[I_Eqno]. Other
positions are not being used by GAMS and may be left undefined. During initialization and termination calls, d must not be referenced.
Message callback msgcb
This argument is the address of a message callback routine that can be used to write messages to the status and/or log files of the GAMS process. Its type definition in C is as follows:
typedef void (GE_CALLCONV * msgcb_t) (const int* mode, const int* nchars, const char* buf, int len);
The argument mode is used to point to an integer which indicates where messages should be written to. This integer can be set to the following values:
• LOGFILE (1): Write the message to the log file only.
• STAFILE (2): Write the message to the status file only.
• LOGFILE | STAFILE (3): Write the message to both the log file and the status file. Observe that the symbol | denotes the bitwise logical OR in C.
The argument nchars points to an integer that specifies the number of bytes contained in the message (excluding the \0-terminator if there is one present). Thus, in C, nchars is typically set to
strlen(buf). The argument buf is a pointer to the character array containing the message to be printed. Finally, len is the size or length of the string buf, thus it is typically the same as *nchars.
Calling the message callback msgcb from C is straightforward. Note that the arguments mode, nchars, and buf are all call-by-reference and that addresses, not values, must be used. However, the
argument len is call-by-value and *nchars should be passed as its value.
If the implementation is done in Delphi or Visual Basic, observe that pointers of all types are 4-byte quantities on a 32bit system and 8-byte quantities on a 64bit system. Integers are 4 bytes.
Calling this routine from a FORTRAN environment is a bit more complicated due to the different ways that FORTRAN compilers handle strings. The Unix convention - at least the convention observed on
all systems for which GAMS is built - is that strings are passed by reference. In addition, the length of the string is passed by value as a hidden 4-byte quantity appended to the end of the argument
list. This is the reason for including len as the last argument in msgcb. The argument len facilitates making FORTRAN callbacks in a Unix environment like the following:
character*(*) msgbuf
int nchars, charcount
nchars = charcount(msgbuf)
call MSGCB (mode, nchars, msgbuf)
Return code
The function GEFUNC must return one of the following status codes:
Status Definition
0 No error occurred.
1 A function evaluation error was encountered. GAMS should not use the content of f and/or d, but GEFUNC has recovered from the error and is ready to be called at a new point. This status code
should only be used in function evaluation mode (icntr[I_Mode]=DOEVAL).
Fatal error. If this value is returned during the initialization call, then GAMS should abort immediately. It may be returned by GEFUNC during the initial call if some initializations did not
2 work correctly, or if some of the size values in icntr had unexpected values. It may also be returned during function evaluation mode (icntr[I_Mode]=DOEVAL) if the external module has
experienced problems from which it cannot recover.
After describing the function GEFUNC in Section Programming Interface above, this section offers some practical comments on implementing GEFUNC.
Compiling and Linking
The examples for GAMS external equations contain a set of GAMS models for compiling the code on various systems using various compilers. Note that the compiler and linker flags shown in these
examples should be used to ensure that the modules conform to the interface standard. In addition, the appropriate include file (geheader.h, geheader_d.pas, gehelper.f90) should be used.
Initialization Mode
The initialization mode should always check whether the external equations have the expected size: icntr[I_Neq], icntr[I_Nvar] and icntr[I_Nz] have to be tested against fixed expected values or
values derived from some external data set.
The initialization mode may be used for several purposes like allocating memory and initializing numerical information or mapping information needed by the function evaluations that will follow. Data
can be computed or read from external data sources or it can be derived from calls to an external database. Note that data that is shared with GAMS may be written to a file from GAMS using the put
statement and then read in GEFUNC. Note further, that users must close the put file with a putclose statement before the solve statement. Observe that memory used to hold information from one
invocation of GEFUNC to the next should be static. For FORTRAN it should either be in a Common block or it should be included in a Save statement.
Termination Mode
The termination mode may be used to perform some clean-up tasks like computing statistics, closing files, and returning memory.
Evaluation Mode
The bulk of the computational work will usually be in evaluation mode. Observe that GEFUNC only works with one equation at a time. One of the reasons for this choice is that the addressing of
derivatives becomes very simple: there is one derivative for each variable and they have the same index in d and x, respectively.
In some applications several functions are naturally evaluated together, for example, because all functions are computed in some joint integration routine. The icntr[I_Newpt] flag is included for
these applications. Using this flag, an implementation could evaluate all functions using a common routine when icntr[I_Newpt] equals 1 and saves the function and derivative values. Additionally, it
returns the values corresponding to equation icntr[I_Eqno]. In subsequent calls to GEFUNC, icntr[I_Newpt] will likely be zero and the function and derivative values can quickly be extracted from the
previously computed (and saved) information.
Evaluation Errors
It is good modeling practice to add bounds to the variables in such a way that all nonlinear functions are defined for all values of the variables within the bounds. Most solvers will also guarantee
that nonlinear functions are called only when all entries of the vector x are between the bounds. However, it may not be practical to add all the necessary bounds and the implementation of GEFUNC
should therefore capture evaluation errors such as division by zero, taking the logarithm of non-positive numbers, overflow in exponentiation, etc. If an equation cannot be evaluated at the given
point, function GEFUNC should simply let the solver know about this situation by returning the value 1. The solver may then be able to backtrack to a safe point and continue the optimization from
System-default or user-defined functions that handle evaluation errors (for example, the C library function matherr()) will sometimes not work in the same way inside a DLL or a shared object as
they do in a self-contained program or a static library.
Message Output
External modules can send messages to the GAMS status file (usually the listing file) and the GAMS log file (usually the screen). Messages to be included in the GAMS status file can be buffered using
the GEstat utility routine described below and messages to be included in the GAMS log file can be buffered using the GElog utility routine. Note that it is not possible to open these files for
writing in the external module since GAMS or the solver process controls them.
Moreover, messages may be sent to both the status and log file without buffering, using the message callback msgcb. This removes the limit imposed by the size of the message buffer and may also make
debugging somewhat simpler, since there is no need to worry about messages that never got flushed from the buffer. As it may be difficult or impossible to use the message callback from some
environments, both the buffered and unbuffered techniques are provided.
Note that the two techniques for sending messages (buffered via GEstat and GElog and unbuffered via the message callback msgcb) are complementary. Either one or the other may be used, but if both are
used in the same external module, the buffered messages will be printed after the unbuffered messages.
GEstat: The Utility Routine for Writing Messages to the Status File
GEstat is provided in the appropriate include file (Fortran 90: gehelper.f90, C: geheader.h, Delphi: geheader_d.pas). It is used to communicate messages that should be written to the GAMS status
file. The function definition follows:
subroutine gestat (icntr, line)
C Control Buffer:
Integer icntr(*)
C input parameters:
character*(*) line
void GEstat(int* icntr, char* line)
Procedure GeStat(var icntr: ticntr; const s: shortstring);
Note that the first argument, icntr, must be passed through from the call of the function GEFUNC. The content of the argument line (or s in Delphi) is packed into the control buffer as one line. When
GEFUNC returns, the content of the buffer will be written to the GAMS status file. GEstat may be called several times, each time with one line. Observe that line should not be longer than 132
characters and the overall amount of information written in one call to GEFUNC should not exceed 1000 characters. Further, line should not contain any special characters such as new-line or tab.
In practice, GEstat is often used with calls like the following:
call GESTAT (icntr, ' ')
call GESTAT (icntr, '**** External module based on abc.for')
GEstat (icntr, " ")
GEstat (icntr, "**** External module based on abc.c")
gestat(icntr,' ');
gestat(icntr,'**** External module based on abc.dpr');
GElog: The Utility Routine for Writing Messages to the Log File
Like GEstat, GElog is provided in the appropriate include file. It is used to communicate messages that should be written to the GAMS log file. Note that by default, the log file is the screen.
Alternatively, log may be written to a file that is specified with the GAMS command line parameter LogFile. The function definition of GElog follows.
subroutine gelog( Icntr, line )
C Control Buffer:
Integer Icntr(*)
C input parameters:
character*(*) line
void GElog(int* icntr, char* line)
Procedure GeLog(var icntr: ticntr; const s: shortstring);
Note that GElog behaves exactly like GEstat, with the status file replaced by the log file. The content of line is written to a buffer that in turn is written to the log file when GEFUNC returns.
Observe that it is not possible to write directly to the screen with some combinations of operating system and compiler. This may also depend on the options or flags that are used to build the
external module.
On some systems writing directly to the screen may cause the external module to crash. Therefore, it is advised not to write to the screen as a method for debugging, unless it is clear that it
works. Otherwise the module may continue to crash because of the debugging statements after all other errors have been removed. Writing to a file and flushing the buffer is recommended as a safe
Communicating Data to the External Module via Files
Some external equations will need data from the GAMS program. This data may be passed on via one or more files written using put statements. Usually, such put files will be written in the current
directory and the external module will look for them in the current directory. However, if users need to run multiple copies of the same model at the same time, data files should be written in the
GAMS scratch directory and the external module should be directed to look for the data files in the scratch directory.
Note that a put file may be defined to be located in the scratch directory with the following file statement in the GAMS model:
File f / '%gams.scrdir%filename' /;
Observe that if the extension .dat is used, GAMS will remove the file from the scratch directory after the run. If another extension is used and the file is not deleted, GAMS will complain about an
unexpected file when it cleans up after the run. The external module can receive the name of the scratch directory from GAMS during initialization by setting icntr[I_Getfil] to I_scr and returning
immediately. GAMS will then store the name of the scratch directory and length of the name in the communication buffer and call GEFUNC in initialization mode again. Note that GEFUNC will now be
called with the sub-mode icntr[I_Smode] set to I_Scr. Then the name may be extracted using the following FORTRAN call:
call GENAME( Icntr, Scrlen, Scrdir )
Here, Scrdir (declared as character*255) will receive the scratch directory and Scrlen (declared as integer) will receive the actual length of Scrdir. In C, the call takes the following form:
char scratchDir[255];
int scratchDirLen;
scratchDirLen = GEname(icntr, scratchDir, sizeof(scratchDir));
Here the routine will return the number or characters transferred to the buffer scratchDir if successful and the value -1 otherwise. If there is space, a terminating '\0'-byte will be written to
scratchDir. If the value returned is equal to sizeof(scratchDir), then the string returned will not be '\0'-terminated and may have been truncated as well.
Observe that it is possible to get other directory or file names by specifying other values in icntr[I_Getfil]. After setting this flag, GEFUNC must always return immediately.
For examples, see models [EX5] and [EXMCP3] and their respective FORTRAN and C source files.
Constant Derivatives
Some solvers, like the CONOPT solvers, can take advantage of the knowledge about constant derivatives in equations, which are a result of linear terms. This can be especially useful if an external
equation represents an equation like Y=f(X), where Y is unbounded, since variable Y can then be substituted by f(X).
However, with the external module interface as described so far, the solver cannot know which variables appear linearly in the external equations. An optional extension allows to indicate that some
of the relationships are linear. This can be activated by returning a 1 in icntr[I_ConstDeriv] during the call of GEFUNC in initialization mode.
GEFUNC will be called again repetitively with icntr[I_Mode] set to DOCONSTDERIV, once for each external equation, with its index specified as usual in icntr[I_Eqno]. For each of these calls, values
of all constant derivatives must be specified in the array d. The remaining elements of d, both those corresponding to varying derivatives and to zeros, must be left untouched. These special calls
will take place after the initialization call and before the first function evaluation call. Note that in these calls other flags like icntr[I_Dofunc] and icntr[I_Dodrv] and the array x will not be
For an example, see model [EX4X] with the corresponding Fortran 90 and C source files ex4xf_cb.f90 and ex4xc_cb.c, respectively. It is instructive to compare these files to the corresponding files
without constant derivatives: ex4f_cb.f90 and ex4c_cb.c.
Second Derivatives: Hessian times Vector
External modules cannot provide a solver with the Hessian matrix of external equations. However, some solvers have particular options for internally approximating the Hessian. For example, see the
hessian_approximation option for IPOPT or hessopt for KNITRO. Further, the solver CONOPT can take advantage of second order information in the form of the product Hessian matrix \(\nabla^2 f(x)\)
times a vector \(v\). This special form can be used for external equations by setting icntr[I_HVprod] to 1 during the call of GEFUNC in initialization mode.
If the solver can use this information (not all solvers will), then GEFUNC may be called with icntr[I_Mode] set to DOHVPROD to request this operation. icntr[I_Eqno] will hold the equation number and
the array x will hold the values of the variables ( \(x\)) in its first icntr[I_NVar] positions and a vector \(v\) in the following icntr[I_Nvar] positions. GEFUNC should evaluate and return \(d = \
nabla^2 f(x)\, v\) for the particular external equation \(f\) at the particular point \(x\). Note that \(d\) (which is otherwise used for the derivative vector) will have been initialized to zero by
\(\nabla^2 f(x)\, v\) will often be needed for several vectors v at the same point x. Therefore, icntr[I_Newpt] will be used to indicate changes in x in the usual way.
Note that model [EX1X] with the corresponding Fortran 90 source file shows how to use both constant and second derivatives.
Implementing external equations brings a number of new potential error sources which GAMS cannot protect against as well as with pure GAMS models. For example, the argument lists in the C or FORTRAN
code may be incorrect or the linking process may create an incorrect external module. There is little GAMS can do to help users with this type of errors. It is recommended to carefully follow the
examples and output debug messages during the setup calls, for example using the utility routines GEstat and GElog.
Once the overall setup is correct and GAMS can establish proper communication with the external module, there may still be numerical errors where the function values and the derivatives do not match.
Note that the solver CONOPT will by default call its Function and Derivative Debugger in the initial point, if a model has any external equations. The debugger will check that the functions only
depend on the variables that are defined in the sparsity pattern and that derivatives computed by numerical perturbation are consistent with the derivatives computed by the external module. If an
error is found, CONOPT will stop immediately with an appropriate message. For examples, see the GAMS Test Library models [er1], [er2], and [er3], which illustrate different types of errors. The
respective error messages will appear if CONOPT is used as the NLP solver. Note that comments about the errors may be found in the C or FORTRAN source code.
Observe that several types of errors cannot be detected. Derivatives that are computed in the external module and are returned in positions that were not defined in the sparsity pattern in GAMS will
be filtered out by the interface and will therefore not be detected. Similarly, derivatives that should be computed but are forgotten, may inherit values from the same derivatives in another equation
computed earlier. Finally, fixed variables cannot be perturbed, thus errors related to these variables will usually not be detected.
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[POJ 2287] Tian Ji — The Horse Racing [Java]
1. Description
Here is a famous story in Chinese history.
That was about 2300 years ago. General Tian Ji was a high official in the country Qi. He likes to play horse racing with the king and others.
Both of Tian and the king have three horses in different classes, namely, regular, plus, and super. The rule is to have three rounds in a match; each of the horses must be used in one round. The
winner of a single round takes two hundred silver dollars from the loser.
Being the most powerful man in the country, the king has so nice horses that in each class his horse is better than Tian's. As a result, each time the king takes six hundred silver dollars from Tian.
Tian Ji was not happy about that, until he met Sun Bin, one of the most famous generals in Chinese history. Using a little trick due to Sun, Tian Ji brought home two hundred silver dollars and such a
grace in the next match.
It was a rather simple trick. Using his regular class horse race against the super class from the king, they will certainly lose that round. But then his plus beat the king's regular, and his super
beat the king's plus. What a simple trick. And how do you think of Tian Ji, the high ranked official in China?
Were Tian Ji lives in nowadays, he will certainly laugh at himself. Even more, were he sitting in the ACM contest right now, he may discover that the horse racing problem can be simply viewed as
finding the maximum matching in a bipartite graph. Draw Tian's horses on one side, and the king's horses on the other. Whenever one of Tian's horses can beat one from the king, we draw an edge
between them, meaning we wish to establish this pair. Then, the problem of winning as many rounds as possible is just to find the maximum matching in this graph. If there are ties, the problem
becomes more complicated, he needs to assign weights 0, 1, or -1 to all the possible edges, and find a maximum weighted perfect matching...
However, the horse racing problem is a very special case of bipartite matching. The graph is decided by the speed of the horses -- a vertex of higher speed always beat a vertex of lower speed. In
this case, the weighted bipartite matching algorithm is a too advanced tool to deal with the problem.
In this problem, you are asked to write a program to solve this special case of matching problem.
2. Input
The input consists of up to 50 test cases. Each case starts with a positive integer n ( n<=1000) on the first line, which is the number of horses on each side. The next n integers on the second line
are the speeds of Tian's horses. Then the next n integers on the third line are the speeds of the king's horses. The input ends with a line that has a single `0' after the last test case.
3. Output
For each input case, output a line containing a single number, which is the maximum money Tian Ji will get, in silver dollars.
4. Example
Sample Input
Sample Output
5. Code
public class POJ2287 {
public static void main(String[] args) {
Scanner in = new Scanner(System.in);
int n = 0;
while ((n = in.nextInt()) > 0) {
int[] a = new int[n];
int[] b = new int[n];
for (int i = 0; i < n; i++) {
a[i] = in.nextInt();
for (int i = 0; i < n; i++) {
b[i] = in.nextInt();
int al = 0, bl = 0, ar = n - 1, br = n - 1;
int count = 0;
while (al <= ar && bl <= br) {
if (a[al] > b[bl]) {
if (a[al] < b[bl]) {
if (a[ar] > b[br]) {
if (a[ar] < b[br]) {
if (a[al] < b[br]) {
System.out.println(count * 200);
Comments | NOTHING
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Fixing Errors in the `add` Function
Fixing Errors in the `add` Function
Common sense tells us that the booleans in the add function should be replaced with some sort of number type.
If you are coming from another language, you might be tempted to try using int or float, but TypeScript only has the number type:
function add(a: number, b: numbe
Loading solution
00:00 So Common Sense tells us that ADD here should take two numbers. A should probably be a number and B should probably be a number. We can remove these type annotations if we want, but we get a
different error here. And the errors down the bottom don't go away.
00:15 So you might think, why don't we just rename them both as number here? And this seems to work. You notice that we can't choose like integer here. We can't choose float, which might be present
in other languages. TypeScript just has the number type. And now everything is working.
00:32 And if we try to call something without, or try to call ADD with something that wasn't a number, then we would get an error here because argument of type string is not assignable to type
number. So this is just your first little glimpse into the power of TypeScript's type system. We have here a function, which we are now being enforced to take in two numbers.
00:52 And we know that the result here also will be a number too, because the result of adding two numbers A plus B up here is a number. So there we go. Our first glimpse into TypeScript's type
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MathCell - Graphics
Options for graphics objects are specified as JavaScript dictionaries. Unless stated otherwise, common default options are
color rgb(0,127,255) - rich azure blue
opacity 1
Colors are specified using standard HTML color code strings. Predefined colormaps are also available.
Objects composed of lines have an additional option of thickness that defaults to 1.5 in two dimensions. Due to limitations of WebGL, in three dimensions it renders as 1 on most platforms regardless
of the setting: a numeric option of radius should be used instead.
The axis for three-dimensional objects is rotated from the z-axis. This can produce unexpected behavior for objects that do not have rotational symmetry. The utility function rotateObject is
available for further adjustment.
Additional object-specific options are listed below each.
arrow( begin, end, options ) — two- or three-dimensional arrow from beginning to end
box( width, depth, height, options ) — box of specified dimensions
axis arbitrary vector direction, default [0,0,1]
center vector position, default [0,0,0]
cone( radius, height, options ) — cone of specified dimensions
axis arbitrary vector direction, default [0,0,1]
center vector position, default [0,0,0]
steps integer determining surface smoothness, default 20
cylinder( radius, height, options ) — cylinder of specified dimensions
axis arbitrary vector direction, default [0,0,1]
center vector position, default [0,0,0]
openEnded Boolean to draw cylinder ends, default true
steps integer determining surface smoothness, default 20
ellipsoid( a, b, c, options ) — ellipsoid of specified parameters
axis arbitrary vector direction, default [0,0,1]
center vector position, default [0,0,0]
steps integer determining surface smoothness, default 20
line( points, options ) — two- or three-dimensional line joining an array of points
endcaps Boolean to include spheres to smooth joints, default false
radius float for drawing the line as an extended cylinder
plane( width, depth, options ) — plane of specified dimensions
normal arbitrary vector direction, default [0,0,1]
center vector position, default [0,0,0]
point( point, options ) — two- or three-dimensional point
sphere( radius, options ) — sphere of specified radius
center vector position, default [0,0,0]
steps integer determining surface smoothness, default 20
text( string, point, options ) — string of text at the two- or three-dimensional point
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bdHMM: Create a bdHMM object in STAN: The Genomic STate ANnotation Package
1 bdHMM(initProb = numeric(), transMat = matrix(numeric(), ncol = 0, nrow =
2 0), emission, nStates = numeric(), status = character(),
3 stateNames = character(), dimNames = character(),
4 transitionsOptim = "analytical", directedObs = integer(),
5 dirScore = numeric())
bdHMM(initProb = numeric(), transMat = matrix(numeric(), ncol = 0, nrow = 0), emission, nStates = numeric(), status = character(), stateNames = character(), dimNames = character(), transitionsOptim =
"analytical", directedObs = integer(), dirScore = numeric())
initProb Initial state probabilities.
transMat Transition probabilities
emission Emission parameters as an HMMEmission object.
nStates Number of states.
status Status of the bdHMM. 'Initial' means that the model was not fitted yet. 'EM' means that the model was optimized using Expectation maximization.
stateNames Indicates directinality of states. States can be forward (F1, F2, ..., Fn), reverse (R1, R2, ..., Rn) or undirectional (U1, U2, ..., Um). Number of F and R states must be equal and
twin states are indicated by integers in id (e.g. F1 and R1 and twins).
dimNames Names of data tracks.
transitionsOptim There are three methods to choose from for fitting the transitions. Bidirectional transition matrices (invariant under reversal of time and direction) can be fitted using c('rsolnp',
'analytical'). 'None' uses standard update formulas and the resulting matrix is not constrained to be bidirectional.
directedObs An integer indicating which dimensions are directed. Undirected dimensions are 0. Directed observations must be marked as unique integer pairs. For instance c(0,0,0,0,0,1,1,2,2,3,3)
contains 5 undirected observations, and thre pairs (one for each direction) of directed observations.
dirScore Directionlity score of states of a fitted bdHMM.
Status of the bdHMM. 'Initial' means that the model was not fitted yet. 'EM' means that the model was optimized using Expectation maximization.
Indicates directinality of states. States can be forward (F1, F2, ..., Fn), reverse (R1, R2, ..., Rn) or undirectional (U1, U2, ..., Um). Number of F and R states must be equal and twin states are
indicated by integers in id (e.g. F1 and R1 and twins).
There are three methods to choose from for fitting the transitions. Bidirectional transition matrices (invariant under reversal of time and direction) can be fitted using c('rsolnp', 'analytical').
'None' uses standard update formulas and the resulting matrix is not constrained to be bidirectional.
An integer indicating which dimensions are directed. Undirected dimensions are 0. Directed observations must be marked as unique integer pairs. For instance c(0,0,0,0,0,1,1,2,2,3,3) contains 5
undirected observations, and thre pairs (one for each direction) of directed observations.
1 nStates = 5
2 stateNames = c('F1', 'F2', 'R1', 'R2', 'U1')
3 means = list(4,11,4,11,-1)
4 Sigma = lapply(list(4,4,4,4,4), as.matrix)
5 transMat = matrix(1/nStates, nrow=nStates, ncol=nStates)
6 initProb = rep(1/nStates, nStates)
7 myEmission = list(d1=HMMEmission(type='Gaussian', parameters=list(mu=means, cov=Sigma), nStates=length(means)))
9 bdhmm = bdHMM(initProb=initProb, transMat=transMat, emission=myEmission, nStates=nStates, status='initial', stateNames=stateNames, transitionsOptim='none', directedObs=as.integer(0))
nStates = 5 stateNames = c('F1', 'F2', 'R1', 'R2', 'U1') means = list(4,11,4,11,-1) Sigma = lapply(list(4,4,4,4,4), as.matrix) transMat = matrix(1/nStates, nrow=nStates, ncol=nStates) initProb = rep
(1/nStates, nStates) myEmission = list(d1=HMMEmission(type='Gaussian', parameters=list(mu=means, cov=Sigma), nStates=length(means))) bdhmm = bdHMM(initProb=initProb, transMat=transMat, emission=
myEmission, nStates=nStates, status='initial', stateNames=stateNames, transitionsOptim='none', directedObs=as.integer(0))
For more information on customizing the embed code, read Embedding Snippets.
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enum Gecode::IntVarBranch {
Gecode::INT_VAR_NONE = 0, Gecode::INT_VAR_RND, Gecode::INT_VAR_DEGREE_MIN, Gecode::INT_VAR_DEGREE_MAX,
Gecode::INT_VAR_AFC_MIN, Gecode::INT_VAR_AFC_MAX, Gecode::INT_VAR_MIN_MIN, Gecode::INT_VAR_MIN_MAX,
Gecode::INT_VAR_MAX_MIN, Gecode::INT_VAR_MAX_MAX, Gecode::INT_VAR_SIZE_MIN, Gecode::INT_VAR_SIZE_MAX,
Gecode::INT_VAR_SIZE_DEGREE_MIN, Gecode::INT_VAR_SIZE_DEGREE_MAX, Gecode::INT_VAR_SIZE_AFC_MIN, Gecode::INT_VAR_SIZE_AFC_MAX,
Gecode::INT_VAR_REGRET_MIN_MIN, Gecode::INT_VAR_REGRET_MIN_MAX, Gecode::INT_VAR_REGRET_MAX_MIN, Gecode::INT_VAR_REGRET_MAX_MAX
Which variable to select for branching.
enum Gecode::IntValBranch {
Gecode::INT_VAL_MIN, Gecode::INT_VAL_MED, Gecode::INT_VAL_MAX, Gecode::INT_VAL_RND,
Gecode::INT_VAL_SPLIT_MIN, Gecode::INT_VAL_SPLIT_MAX, Gecode::INT_VAL_RANGE_MIN, Gecode::INT_VAL_RANGE_MAX,
Gecode::INT_VALUES_MIN, Gecode::INT_VALUES_MAX
Which values to select first for branching.
void Gecode::branch (Home home, const IntVarArgs &x, IntVarBranch vars, IntValBranch vals, const VarBranchOptions &o_vars=VarBranchOptions::def, const ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with variable selection vars and value selection vals.
void Gecode::branch (Home home, const IntVarArgs &x, const TieBreakVarBranch< IntVarBranch > &vars, IntValBranch vals, const TieBreakVarBranchOptions &o_vars=TieBreakVarBranchOptions::def, const
ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with tie-breaking variable selection vars and value selection vals.
void Gecode::branch (Home home, IntVar x, IntValBranch vals, const ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with value selection vals.
void Gecode::branch (Home home, const BoolVarArgs &x, IntVarBranch vars, IntValBranch vals, const VarBranchOptions &o_vars=VarBranchOptions::def, const ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with variable selection vars and value selection vals.
void Gecode::branch (Home home, const BoolVarArgs &x, const TieBreakVarBranch< IntVarBranch > &vars, IntValBranch vals, const TieBreakVarBranchOptions &o_vars=TieBreakVarBranchOptions::def, const
ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with tie-breaking variable selection vars and value selection vals.
void Gecode::branch (Home home, BoolVar x, IntValBranch vals, const ValBranchOptions &o_vals=ValBranchOptions::def)
Branch over x with value selection vals.
Enumeration Type Documentation
Which variable to select for branching.
INT_VAR_NONE First unassigned.
INT_VAR_RND Random (uniform, for tie breaking).
INT_VAR_DEGREE_MIN With smallest degree.
INT_VAR_DEGREE_MAX With largest degree.
INT_VAR_AFC_MIN With smallest accumulated failure count.
INT_VAR_AFC_MAX With largest accumulated failure count.
INT_VAR_MIN_MIN With smallest min.
INT_VAR_MIN_MAX With largest min.
INT_VAR_MAX_MIN With smallest max.
INT_VAR_MAX_MAX With largest max.
INT_VAR_SIZE_MIN With smallest domain size.
INT_VAR_SIZE_MAX With largest domain size.
INT_VAR_SIZE_DEGREE_MIN With smallest domain size divided by degree.
INT_VAR_SIZE_DEGREE_MAX With largest domain size divided by degree.
INT_VAR_SIZE_AFC_MIN With smallest domain size divided by accumulated failure count.
INT_VAR_SIZE_AFC_MAX With largest domain size divided by accumulated failure count.
INT_VAR_REGRET_MIN_MIN With smallest min-regret.
The min-regret of a variable is the difference between the smallest and second-smallest value still in the domain.
INT_VAR_REGRET_MIN_MAX With largest min-regret.
The min-regret of a variable is the difference between the smallest and second-smallest value still in the domain.
INT_VAR_REGRET_MAX_MIN With smallest max-regret.
The max-regret of a variable is the difference between the largest and second-largest value still in the domain.
INT_VAR_REGRET_MAX_MAX With largest max-regret.
The max-regret of a variable is the difference between the largest and second-largest value still in the domain.
Definition at line 3333 of file int.hh.
Which values to select first for branching.
INT_VAL_MIN Select smallest value.
INT_VAL_MED Select greatest value not greater than the median.
INT_VAL_MAX Select largest value.
INT_VAL_RND Select random value.
INT_VAL_SPLIT_MIN Select values not greater than mean of smallest and largest value.
INT_VAL_SPLIT_MAX Select values greater than mean of smallest and largest value.
INT_VAL_RANGE_MIN Select the smallest range of the variable domain if it has sevral ranges, otherwise select values not greater than mean of smallest and largest value.
INT_VAL_RANGE_MAX Select the largest range of the variable domain if it has sevral ranges, otherwise select values greater than mean of smallest and largest value.
INT_VALUES_MIN Try all values starting from smallest.
INT_VALUES_MAX Try all values starting from largest.
Definition at line 3377 of file int.hh.
Function Documentation
void Gecode::branch ( Gecode::Home home,
const IntVarArgs & x,
IntVarBranch vars,
IntValBranch vals,
const Gecode::VarBranchOptions & o_vars,
const Gecode::ValBranchOptions & o_vals
Branch over x with variable selection vars and value selection vals.
void Gecode::branch ( Gecode::Home home,
const IntVarArgs & x,
const Gecode::TieBreakVarBranch< IntVarBranch > & vars,
IntValBranch vals,
const Gecode::TieBreakVarBranchOptions & o_vars,
const Gecode::ValBranchOptions & o_vals
Branch over x with tie-breaking variable selection vars and value selection vals.
void Gecode::branch ( Home home,
IntVar x,
IntValBranch vals,
const ValBranchOptions & o_vals
Branch over x with value selection vals.
void Gecode::branch ( Gecode::Home home,
const BoolVarArgs & x,
IntVarBranch vars,
IntValBranch vals,
const Gecode::VarBranchOptions & o_vars,
const Gecode::ValBranchOptions & o_vals
Branch over x with variable selection vars and value selection vals.
void Gecode::branch ( Gecode::Home home,
const BoolVarArgs & x,
const Gecode::TieBreakVarBranch< IntVarBranch > & vars,
IntValBranch vals,
const Gecode::TieBreakVarBranchOptions & o_vars,
const Gecode::ValBranchOptions & o_vals
Branch over x with tie-breaking variable selection vars and value selection vals.
void Gecode::branch ( Home home,
BoolVar x,
IntValBranch vals,
const ValBranchOptions & o_vals
Branch over x with value selection vals.
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Investment_2022_fall_Assignment4&6&7 - Tabing010102's Blog
Long time no see! It’s turn for update 3 in 1. Notice that code of all assignments is on a github repo.
Assignment 4
In a firm’s life, it may experience many kind of events, such as IPO, SEO, merge, addition into SP500 index, etc. We want to know how much these event could make effect on stock’s return. That is
called Event Study. Detailed story see A2.
In this task, we need to wrap a function in Python to deal with normal event study. There is actually little modification, compared with A2.
Assignment 5
We would like to reproduce Jegadeesh and Titman (1993)’s work on momentum effect.
Momentum means a stock market phenomenon that "Winner always win, loser always lose". We could trust the return on win stocks would increase in the future, who have earned a good profit in past time.
In this task, we need to generate two groups containing winner and loser stocks. Then we "buy" winner and "sell" loser, calculate "buy-sell" returns. In fact, we do not assume we had capital and long
or short position. We just sum the cumulative return.
In detail, we define winner as the stocks having higher CARs in past [3,6,9,12] months, while loser have lower CARs in past [3,6,9,12] months. Then we construct a sheet to find how to combine the two
periods to earn the highest return.
Remember that we use log returns.
R_{ln,t} = ln(\frac{Pt}{P{t-1}})
This simplified return cumulation and turn it into arithmetic sum of log returns.
R{ln,{t+1}} = ln(\frac{Pt}{P{t-1}})+ln(\frac{P{t+1}}{P{t}})
It could avoid many problems…
Assignment 7
We need to construct Fama French factors, SMB and HML, and compare with the ones in Fama’s data library. See the reference I note in code.
The tricky part is, we need to construct 6 portfolios:SH, SM, SL, BH, BM, BL and calculate the difference by:
SMB = \frac{R{SL} + R{SM} + R{SH} – R{BL} – R{BM} – R{BH}}{3}
HML= \frac{R{SH} + R{BH} – R{SL} – R{BL}}{2}
but not:
SMB = R{S}-R{B}
HML= R{H} – R{L}
If you want to know why, ask Fama…
Views: 3
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Math Games That You Can Play Without A Flash Player
Play math Battleship or solve a hidden puzzle with math problems online. Learn how to play Math Mahjong and play memory games. Math bingo, math problem generators, Math Man (which is similar to Pac
Man) and a scientific notation scavenger hunt are all available online without the use of Flash.
CoolMath's Number Monster, Algebra Crunchers and Brain Benders
Make 10 pennies into a triangle then rearrange three pennies to flip the triangle upside down — or rearrange two toothpicks in a four-square configuration to make the four squares into seven with
Brain Bender games at CoolMath4Kids. The animated logic games on this site require Flash but the rest of the Brain Bender games do not. CoolMath4Kids also has a Number Monster game that generates
basic math problems and an Algebra Cruncher game that generates problems (with hints and answers) about polynomials, exponents, logarithms, lines, fractions and algebraic functions. None of these
games require Flash.
Counting Games with Curious George and Math Bingo
Play the PBS Kids' interactive counting game, Glass Palace, with Curious George, where sequential numbers are revealed in windows with George posing as a window washer. Play addition, subtraction,
multiplication, division and geometry hidden picture games at APlusMath. The games give players math problems and the solutions are chosen from cards displayed — each correct card chosen reveals part
of a picture. You can also play online versions of Math bingo and addition through geometry Concentration memory games at APlusMath. None of these games require Flash.
Mixed Math Mahjong and Math Man
Basic math operations such as addition and multiplication, fractions, decimals, telling time, counting money, measurements, pre-algebra and Roman numerals are all subjects covered in games at the
Sheppard Software website. Multiplication Picnic, Matching Money, Mixed Math Mahjong and Math Man, which is like Pac Man, are all there and none require Flash.
Math Battleship and a Scientific Notation Scavenger Hunt
Solve words in virtual hangman games or play interactive math Battleship games at Quia. In the Battleship games, players guess from a grid to try to locate opponents' battleships to sink them. When a
grid area is picked that has a battleship, players must answer a math question to allow the turn to become a hit on the ship. Go on an Internet scavenger hunt about scientific notations terms at Quia
as well. None of these games require Flash.
Cite This Article
Anderberg, Kirsten. "Math Games That You Can Play Without A Flash Player" sciencing.com, https://www.sciencing.com/math-can-play-flash-player-8348559/. 24 April 2017.
Anderberg, Kirsten. (2017, April 24). Math Games That You Can Play Without A Flash Player. sciencing.com. Retrieved from https://www.sciencing.com/math-can-play-flash-player-8348559/
Anderberg, Kirsten. Math Games That You Can Play Without A Flash Player last modified August 30, 2022. https://www.sciencing.com/math-can-play-flash-player-8348559/
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7th Grade Math Common Core Tests
• >
• 7th Grade Math Common Core Tests
7th Grade Math Common Core Tests
per item
Included is a pre-test, post-test, and vocabulary quiz for every 7th grade math unit. The tests are aligned to CCSS and EDITABLE (Microsoft Word)!!! Hence, you can change any problem by adding more
rigor or keeping it more simple to fit your classroom needs. The content includes:
The Number Sense (7.NS)
• Pre Test
• Post Test
• 1 Vocabulary Quiz
Proportional Relationships (7.RP)
• Pre Test
• Post Test
• 2 Vocabulary Quizzes
Expressions & Equations (7.EE)
• Pre Test
• Post Test
• 1 Vocabulary Quiz
Geometry (7.G)
• Pre Test
• Post Test
• 4 Vocabulary Quizzes
Probability & Statistics (7.SP)
• Pre Test
• Post Test
• 2 Vocabulary Quizzes
The Number System (7.NS)
Students will:
1.) Determine the value of points on a number line.
2.) Use the number line to demonstrate how to add and subtract integers.
3.) Use the number line to demonstrate how to multiply and divide integers.
4.) Use the number line to demonstrate absolute value.
5.) Add, subtract, multiply, and divide rational numbers.
6.) State the opposite of a number.
7.) Determine the value that makes statements true.
8.) Determine the distance between two numbers.
9.) Rewrite subtraction problems as addition problems.
The vocabulary included is absolute value, reciprocal, numerator, denominator, product, integers, and quotient.
Proportional Relationships (7.RP)
Students will:
1.) Calculate the unit rate given the cost and quantity of items.
2.) Complete a table and graph using the cost and quantity of items.
3.) Calculate of the cost for a certain quantity of the items.
4.) Write an equation that represents the total cost given the number of items purchased.
5.) Determine how many items you can buy given a certain price.
6.) Give a point (x,y) from the graph that represents the unit rate.
7.) Plot points on a graph.
8.) Determine if a graph is proportional and explain their answer.
9.) Calculate discount, markup, sales tax, selling price, simple interest, percent increase, and percent decrease.
The vocabulary included is unit rate, ratio, proportion, slope, rate, scale, scale drawing, sales tax, tip, markup, discount, selling price, and simple interest.
Expressions & Equations (7.EE)
Students will:
1.) Solve one and two step equations.
2.) Solve and graph two step inequalities.
3.) Determine the inequality given a graph.
4.) Write and solve an equation and inequality.
The vocabulary included is coefficient, distributive property, equation, constant, and inequality.
Geometry (7.G)
Students will:
1.) Use proportions to solve for x.
2.) Determine if figures are similar.
3.) Determine a length of a figure given that the figures are similar.
4.) Solve for x given complementary and supplementary angles.
5.) Solve for x using the triangle angle sum theorem (all angles in a triangle add up to 180 degrees).
The vocabulary included is edge, face, plane, point, two dimensional figure, perpendicular, ray, line, line segment, parallel, three dimensional figure, acute angle, obtuse angle, angle, triangle
angle sum theorem, right angle, vertex, straight line, diameter, circumference, pi, radius, complementary angles, vertical angles, supplementary angles, adjacent angles, and transversal.
Probability & Statistics (7.SP)
Students will:
1.) Determine the probability and likelihood that an event will occur.
2.) Give both the fraction and percent that an event will occur.
3.) State the theoretical probability that a coin will land on heads/tails.
4.) Determine the most and least likely chance of a spinner landing on a color.
5.) Determine the probability and likelihood of a dice roll.
6.) Label the points on a box and whisker plot.
7.) Calculate the median, lower quartile, upper quartile, minimum, and maximum given data.
8.) Draw a box and whisker plot given data.
9.) Calculate the mode, mean, median, and range given data.
The vocabulary included is outcome, probability, impossible, certain, unlikely, likely, equally likely, upper quartile, lower quartile, minimum, maximum, median, mean, mode, and range.
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Excel’s RANK and RANK.EQ Functions
Introduction to RANK and RANK.EQ Functions in Excel
This guide provides real-world usage of RANK and RANK.EQ functions in Excel for data analysis.
RANK(number, ref, [order])
RANK.EQ(number, ref, [order])
• number: The value whose rank you want to find.
• ref: The range of numbers to rank against.
• [order]: Optional. 0 or omitted for descending order; 1 for ascending order.
Practical Examples
Example 1: Basic RANK Usage
Data Setup:
A B
Name Score
John 85
Jane 78
Mark 92
Lucy 85
Tom 75
Enter the formula in cell C2 to determine John’s rank:
=RANK(B2, $B$2:$B$6)
Drag the fill handle from cell C2 to C6.
A B C
Name Score Rank
John 85 2
Jane 78 4
Mark 92 1
Lucy 85 2
Tom 75 5
Example 2: Basic RANK.EQ Usage
Data Setup:
A B
Name Score
John 85
Jane 78
Mark 92
Lucy 85
Tom 75
Enter the formula in cell C2 to determine John’s rank:
=RANK.EQ(B2, $B$2:$B$6)
Drag the fill handle from cell C2 to C6.
A B C
Name Score Rank.EQ
John 85 2
Jane 78 4
Mark 92 1
Lucy 85 2
Tom 75 5
Final Notes
Use the above examples directly in your Excel sheets to find ranks. Adjust ranges and cells as necessary for your specific data.
Practical Implementation of RANK and RANK.EQ Functions in Excel
Dataset Example
| Name | Score |
| Alice | 85 |
| Bob | 92 |
| Charlie| 85 |
| Diana | 78 |
| Eve | 92 |
RANK Function
1. Navigate to an empty cell where you want the rank to appear.
2. Enter the formula using the RANK function.
=RANK(B2, $B$2:$B$6, 0)
□ B2: The cell containing the score you want to rank.
□ $B$2:$B$6: The range of cells containing the scores to compare.
□ 0: Defines the ranking order. 0 is for descending order.
3. Copy the formula to other cells in the column to rank other scores.
RANK.EQ Function
1. Navigate to an empty cell where you want the rank to appear.
2. Enter the formula using the RANK.EQ function.
=RANK.EQ(B2, $B$2:$B$6, 0)
□ B2: The cell containing the score you want to rank.
□ $B$2:$B$6: The range of cells containing the scores to compare.
□ 0: Defines the ranking order. 0 is for descending order.
3. Copy the formula to other cells in the column to rank other scores.
Example with Data
Assuming you have the dataset in cells A1:B6:
1. Insert a new column titled “Rank” in C1.
2. In cell C2, enter the formula:
=RANK(B2, $B$2:$B$6, 0)
3. Copy the formula down to cell C6.
Similarly, for RANK.EQ:
1. Insert a new column titled “Rank.EQ” in D1.
2. In cell D2, enter the formula:
=RANK.EQ(B2, $B$2:$B$6, 0)
3. Copy the formula down to cell D6.
Final Table Example
| Name | Score | Rank | Rank.EQ |
| Alice | 85 | 3 | 3 |
| Bob | 92 | 1 | 1 |
| Charlie| 85 | 4 | 3 |
| Diana | 78 | 5 | 5 |
| Eve | 92 | 2 | 1 |
This completes the practical implementation of using the RANK and RANK.EQ functions for data analysis in Excel.
Comparative Analysis of RANK and RANK.EQ in Excel
Data Setup
Assume you have the following data in column A:
Using RANK Function
RANK (Descending Order)
To rank the numbers in descending order:
=RANK(A2, $A$2:$A$6, 0)
RANK (Ascending Order)
To rank the numbers in ascending order:
=RANK(A2, $A$2:$A$6, 1)
Using RANK.EQ Function
RANK.EQ (Descending Order)
To rank the numbers in descending order:
=RANK.EQ(A2, $A$2:$A$6, 0)
RANK.EQ (Ascending Order)
To rank the numbers in ascending order:
=RANK.EQ(A2, $A$2:$A$6, 1)
Comparative Analysis Table
To display the rank based on both functions in the same table, assume your data starts at row 2. In columns B and C, you will have the ranks based on RANK and RANK.EQ respectively.
A B C
85 =RANK(A2, $A$2:$A$6, 0) =RANK.EQ(A2, $A$2:$A$6, 0)
95 =RANK(A3, $A$2:$A$6, 0) =RANK.EQ(A3, $A$2:$A$6, 0)
70 =RANK(A4, $A$2:$A$6, 0) =RANK.EQ(A4, $A$2:$A$6, 0)
75 =RANK(A5, $A$2:$A$6, 0) =RANK.EQ(A5, $A$2:$A$6, 0)
95 =RANK(A6, $A$2:$A$6, 0) =RANK.EQ(A6, $A$2:$A$6, 0)
If you want ascending order, just replace 0 with 1 in the formulas.
Conclusion Table
To summarize the results, you can create a new table with the evaluated rank values:
A RANK (Desc) RANK.EQ (Desc)
This table verifies that both RANK and RANK.EQ functions provide the same results for ranking in descending order. Repeat for ascending order if needed.
Real-world Application Scenarios of RANK and RANK.EQ in Excel
Sales Performance Evaluation
Evaluate the sales performance of a group of sales representatives within a given month.
1. Data Table Sample
Sales Rep Sales Amount
John 15000
Alice 22000
Bob 18000
Mary 22000
Steve 14000
2. Using RANK Function
□ Insert a new column for the rank: Rank
□ Formula for rank in cell C2:
=RANK(B2, $B$2:$B$6, 0)
☆ Copy this formula down column C to rank all sales figures.
3. Using RANK.EQ Function
□ Insert a new column for the rank equivalency: Rank.EQ
□ Formula for rank in cell D2:
=RANK.EQ(B2, $B$2:$B$6, 0)
☆ Copy this formula down column D to rank all sales figures.
Student Exam Scores Analysis
Determine the standings of students based on their exam scores.
1. Data Table Sample
Student Exam Score
Ben 85
Eva 92
Sam 78
Leo 92
Mia 88
2. Using RANK Function
□ Insert a new column for the rank: Rank
□ Formula for rank in cell C2:
=RANK(B2, $B$2:$B$6, 0)
☆ Copy this formula down column C to rank all exam scores.
3. Using RANK.EQ Function
□ Insert a new column for the rank equivalency: Rank.EQ
□ Formula for rank in cell D2:
=RANK.EQ(B2, $B$2:$B$6, 0)
☆ Copy this formula down column D to rank all exam scores.
Customer Feedback Rating
Rank customer feedback ratings to determine the highest and lowest-rated experiences for service improvement.
Data Table Sample
Customer Feedback Rating
Cust1 4.5
Cust2 3.8
Cust3 4.9
Cust4 3.5
Cust5 4.2
2. Using RANK Function
□ Insert a new column for the rank: Rank
□ Formula for rank in cell C2:
=RANK(B2, $B$2:$B$6, 0)
☆ Copy this formula down column C to rank all feedback ratings.
3. Using RANK.EQ Function
□ Insert a new column for the rank equivalency: Rank.EQ
□ Formula for rank in cell D2:
=RANK.EQ(B2, $B$2:$B$6, 0)
☆ Copy this formula down column D to rank all feedback ratings.
Advanced Tips and Troubleshooting for Excelโ s RANK and RANK.EQ Functions
Advanced Tips
Handling Duplicates
When working with ranking functions, data duplication may affect the results. The following formula assigns unique ranks to ties:
=RANK.EQ(A2, $A$2:$A$10) + COUNTIF($A$2:A2, A2) - 1
Ranking with Multiple Criteria
To rank a dataset with multiple criteria (e.g., ranking by score, then by name):
1. Combine fields into a single ranking metric.
=RANK.EQ(A2, $A$2:$A$10) + RANK.EQ(B2, $B$2:$B$10) * 0.01
2. Alternatively, use an array formula for more precision.
=SUMPRODUCT((B$2:B$10>B2) + (B$2:B$10=B2)*(A$2:A$10<A2)) + 1
Dynamic Range
For dynamic datasets, use OFFSET to adjust the rank range dynamically.
1. Define a dynamic range name (e.g., DataRange):
=OFFSET(Sheet1!$A$2, 0, 0, COUNTA(Sheet1!$A$2:$A$100), 1)
2. Use DataRange in the rank formula:
=RANK.EQ(A2, DataRange)
Error Handling
Prevent common errors such as #N/A when the cell is empty.
=IF(ISNUMBER(A2), RANK.EQ(A2, $A$2:$A$10), "")
Mismatched References
Ensure the ranking range covers all data points to avoid mismatches.
=RANK.EQ(A2, $A$2:$A$10)
For large datasets, use conditional formatting for better performance rather than complex formulas.
1. Apply conditional formatting rules for data highlight.
2. Use helper columns for step-by-step ranking.
Debugging Formulas
Use the FORMULATEXT function to display and review complex formulas.
Consistent Data Types
Ensure data consistency (numbers vs. text) to avoid skewed rankings.
=IF(ISNUMBER(A2), RANK.EQ(A2, $A$2:$A$10), "Check Data Type")
Use these advanced tips and troubleshooting tactics to refine your ranking functions in Excel, ensuring more accurate data analysis and handling of potential issues. Apply these methods directly to
your datasets to tackle complex problems effectively.
Develop a VBA application to automate repetitive tasks in Excel, increasing efficiency and reducing manual errors.
Building a VBA-Based Task Automation Tool
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Outlier-6th Grade MathOutlier-6th Grade MathOutlier-6th Grade Math
In this video you will learn what an outlier is and how an outlier impacts measure of center.
Highlights of the outlier video:
• What is an outlier?
• How does an outlier impact measure of center?
• Work an example problem showing how an outlier impacts mean more than median.
• When there is an outlier, it is best to use the median.
• No outlier use mean.
Follow moomoomath's board Math Resources and helpful information on Pinterest.
0 comments:
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MAC^3 Institute Project: Improving Life Skills through Math
Mental Percents
Lesson 3
A. Start the lesson with a question to the students using a tipping situation in a restaurant.
a. What is the appropriate tipping % in a restaurant? (15%, 17%, 20%)
b. If your bill is $52.78, how would you go about tipping 15%? (Make sure all calculators, pens, pencils, etc. are not being used for maximum effect)
1. Allow students some time to think out the solution. Question several students on their thought processes.
2. Educate students on the 10% rule using the following examples:
1. $3.00
2. $18.00
3. $54.65
4. $123.45…etc.
1. Now go back to the original problem.
1. First round to the nearest whole dollar.
2. Then find 10%.
1. Round again if necessary
1. Then find half of 10% (which is 5%)
1. Round again if necessary
2. Add the two together and you have your approximate tip.
1. Go through process again using the amount $44.80
1. Give the students one minute to try.
2. Go through steps above until you arrive at your answer.
1. Do as many examples as necessary until you feel the class has confidence in doing them on their own.
B. Store sales
a. What are some of the common sales you see in retail or department stores?
b. Are the markdowns on the merchandise or at the register?
c. How do you know if you are getting the correct discount?
1. Share a few personal stories here.
d. Situation 1: The store has a 30% off sale and the item you are buying is marked at $35.99. What should your discount be?
1. See if students apply the 10% rule to the problem.
1. Round the amount to $36.00
2. Take 10% by moving the decimal one place left.
3. Round if you need to the nearest dollar.
4. Then multiply by 3 and you have your discount.
1. Explain that 3 x 10% = 30%.
2. Do another 30% problem if necessary.
e. Situation 2: The store has a 60% off sale and the item you are buying is marked at $74.25. Find the approximate discount.
1. Go through the same steps above except this time show students that 6 x 10% is 60%.
2. Do another if necessary.
f. Situation 3: The store has a 50% off sale, but today you save an additional 25%.
1. Ask the students if this means they will save 75% total?
2. Go through the following steps and see.
1. First find 50% off (Either half off or 5 x 10%)
2. Then take 25% off that amount (either one fourth of the amount or go through steps like a tip: Find 20% and then add half of 10% to it)
3. Then take the original amount and find 75% off. (Some students can do fourths, some will find 70% and 80% and determine the number in the middle, and lastly some will find 70% and add
5% to it. Anything mathematically correct is acceptable when doing mental percents)
4. The discount found from step 2 will be less than that found in step 3. Discuss why.
g. If a store has a sale 75% off, plus additional 20% off, plus and additional 5% off if you use your store credit card, will that mean you get it for free?
1. Can your store purchase ever come out to free?
2. Have students research the Sunday paper over the next few weeks for sales that have these types of advertisements in them.
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96 research outputs found
Raman et al. have found experimental evidence for a critical velocity under which there is no dissipation when a detuned laser beam is moved in a Bose-Einstein condensate. We analyze the origin of
this critical velocity in the low density region close to the boundary layer of the cloud. In the frame of the laser beam, we do a blow up on this low density region which can be described by a
Painlev\'e equation and write the approximate equation satisfied by the wave function in this region. We find that there is always a drag around the laser beam. Though the beam passes through the
surface of the cloud and the sound velocity is small in the Painlev\'e boundary layer, the shedding of vortices starts only when a threshold velocity is reached. This critical velocity is lower than
the critical velocity computed for the corresponding 2D problem at the center of the cloud. At low velocity, there is a stationary solution without vortex and the drag is small. At the onset of
vortex shedding, that is above the critical velocity, there is a drastic increase in drag.Comment: 4 pages, 4 figures (with 9 ps files
An important characteristic of flocks of birds, school of fish, and many similar assemblies of self-propelled particles is the emergence of states of collective order in which the particles move in
the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to
be continuous, the phase transition has been claimed to be discontinuous on the basis of recently reported numerical evidence. We address this issue by analyzing two representative network models
closely related to systems of self-propelled particles. We present analytical as well as numerical results showing that the nature of the phase transition depends crucially on the way in which noise
is introduced into the system.Comment: Four pages, four figures. Submitted to PR
Hypnotic suggestions can produce a broad range of perceptual experiences, including hallucinations. Visual hypnotic hallucinations differ in many ways from regular mental images. For example, they
are usually experienced as automatic, vivid, and real images, typically compromising the sense of reality. While both hypnotic hallucination and mental imagery are believed to mainly rely on the
activation of the visual cortex via top-down mechanisms, it is unknown how they differ in the neural processes they engage. Here we used an adaptation paradigm to test and compare top-down processing
between hypnotic hallucination, mental imagery, and visual perception in very highly hypnotisable individuals whose ability to hallucinate was assessed. By measuring the N170/VPP event-related
complex and using multivariate decoding analysis, we found that hypnotic hallucination of faces involves greater top-down activation of sensory processing through lateralised neural mechanisms in the
right hemisphere compared to mental imagery. Our findings suggest that the neural signatures that distinguish hypnotically hallucinated faces from imagined faces lie in the right brain
hemisphere.Fil: Lanfranco, Renzo C.. University of Edinburgh; Reino Unido. Karolinska Huddinge Hospital. Karolinska Institutet; SueciaFil: Rivera Rei, à lvaro. Universidad Adolfo Ibañez; ChileFil:
Huepe, David. Universidad Adolfo Ibañez; ChileFil: Ibañez, Agustin Mariano. Universidad Adolfo Ibañez; Chile. Universidad de San Andrés. Departamento de Matemåticas y Ciencias; Argentina.
University of California; Estados Unidos. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Canales Johnson, AndrĂ©s. University of Cambridge; Estados Unidos. Universidad
Catolica de Maule; Chil
Non-Isotropic Attractive Bose-Einstein condensates are investigated with Newton and inverse Arnoldi methods. The stationary solutions of the Gross-Pitaevskii equation and their linear stability are
computed. Bifurcation diagrams are calculated and used to find the condensate decay rates corresponding to macroscopic quantum tunneling, two-three body inelastic collisions and thermally induced
collapse. Isotropic and non-isotropic condensates are compared. The effect of anisotropy on the bifurcation diagram and the decay rates is discussed. Spontaneous isotropization of the condensates is
found to occur. The influence of isotropization on the decay rates is characterized near the critical point.Comment: revtex4, 11 figures, 2 tables. Submitted to Phys. Rev.
We have studied the hydrodynamic flow in a Bose-Einstein condensate stirred by a macroscopic object, a blue detuned laser beam, using nondestructive {\em in situ} phase contrast imaging. A critical
velocity for the onset of a pressure gradient has been observed, and shown to be density dependent. The technique has been compared to a calorimetric method used previously to measure the heating
induced by the motion of the laser beam.Comment: 4 pages, 5 figure
The stability of an attractive Bose-Einstein condensate on a joint one-dimensional optical lattice and an axially-symmetric harmonic trap is studied using the numerical solution of the time-dependent
mean-field Gross-Pitaevskii equation and the critical number of atoms for a stable condensate is calculated. We also calculate this critical number of atoms in a double-well potential which is always
greater than that in an axially-symmetric harmonic trap. The critical number of atoms in an optical trap can be made smaller or larger than the corresponding number in the absence of the optical trap
by moving a node of the optical lattice potential along the axial direction of the harmonic trap. This variation of the critical number of atoms can be observed experimentally and compared with the
present calculation.Comment: Latex with 7 eps figures, Accepted in Journal of Physics
Many real-world networks exhibit community structures and non-trivial clustering associated with the occurrence of a considerable number of triangular subgraphs known as triadic motifs. Triads are a
set of distinct triangles that do not share an edge with any other triangle in the network. Network motifs are subgraphs that occur significantly more often compared to random topologies. Two
prominent examples, the feedforward loop and the feedback loop, occur in various real-world networks such as gene-regulatory networks, food webs or neuronal networks. However, as triangular
connections are also prevalent in communication topologies of complex collective systems, it is worthwhile investigating the influence of triadic motifs on the collective decision-making dynamics. To
this end, we generate networks called Triadic Graphs (TGs) exclusively from distinct triadic motifs. We then apply TGs as underlying topologies of systems with collective dynamics inspired from
locust marching bands. We demonstrate that the motif type constituting the networks can have a paramount influence on group decision-making that cannot be explained solely in terms of the degree
distribution. We find that, in contrast to the feedback loop, when the feedforward loop is the dominant subgraph, the resulting network is hierarchical and inhibits coherent behavior
We use a modified Thomas-Fermi approximation to estimate analytically the critical velocity for the formation of vortices in harmonically trapped BEC. We compare this analytical estimate to numerical
calculations and to recent experiments on trapped alkali condensates.Comment: 12 page
We consider a three dimensional, generalized version of the original SPP model for collective motion. By extending the factors influencing the ordering, we investigate the case when the movement of
the self-propelled particles (SPP-s) depends on both the velocity and the acceleration of the neighboring particles, instead of being determined solely by the former one. By changing the value of a
weight parameter s determining the relative influence of the velocity and the acceleration terms, the system undergoes a kinetic phase transition as a function of a behavioral pattern. Below a
critical value of s the system exhibits disordered motion, while above it the dynamics resembles that of the SPP model. We argue that in nature evolutionary processes can drive the strategy variable
s towards the critical point, where information exchange between the units of a system is maximal.Comment: 13 pages, 9 figures, submitted to Phys Rev
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Dev Training: Big O Notation
You don’t have to be a computer scientist to write the most efficient code, but knowing good algorithms from bad ones certainly helps.
Most developers aren’t born mathematicians. And that’s fine. Software developers generally need to learn a lot other things like language syntax, common pitfalls, and structural design. They can use
the tools created for them to produce some pretty amazing applications. But because computers and software come from mathematics, developers often bump into it.
And that’s not always a bad thing. Sometimes, a little knowledge can help.
Take writing algorithms for example.
All developers can put together a program which does something. It has a start, takes some data, does something with it, and eventually ends. A basic program with a simple algorithm is easy. But some
programs aren’t simple. Some data sets aren’t small. Manipulating large tables of data sometimes requires a little more thought and that can result in a more complicated algorithm.
Algorithm Design
There are lots of different approaches to designing algorithms. You can take a Greedy approach, a divide-and-conquer style attack, a brute-force search or even an optimised dynamic style. Each is
best suited to a different kind of problem. Picking the right approach can be difficult.
To check if an algorithm is best suited to a problem, it’s useful to have tools or ways of describing and comparing characteristics like performance or complexity. This is where big Big O notation
comes in useful.
Big O Notation
Big O Notation is a way of describing how well an algorithm will scale as the size of its input increases. Understanding it helps you to design better algorithms, and gives you some things to be
aware of.
It’s a mathematical way of writing about how a function’s time to complete changes as its the size of its input data changes. As input data gets larger, the increase in time is called the growth
rate, and Big O notation describes that growth rate.
It’s also used to describe how the space needed at any point in an algorithm grows as the input grows, but for this article I’ll focus only on time complexity.
It’s worth noting that because data can vary, how long it takes a function to do its job can vary too. A function that searches an array may either find what it’s looking for immediately or search
the whole array before being successful. Big O notation always describes the worst-case scenario.
Each level of complexity described by Big O notation also has a rating of how well it scales as input sizes grow: some well, some terribly. But if a level of complexity scales badly, that doesn’t
mean it should always be avoided. Some problems have solutions that don’t scale well. When faced with a solution that doesn’t scale well, it’s important to be aware of what size of input the
algorithm can comfortably manage, and keep the input data below that limit.
In this article, I cover 8 of the most commonly used notations. I’ll explain how each is defined, along with its name and some common algorithms or data structure operations that can be described as
using Big O Notation. I’ll show some code that serves as a good example and, where possible, point out some design suggestions each one highlights.
Algorithms or functions described as O(1) have no growth rate, meaning they don’t take longer the larger their input gets. Their growth rate is classed as constant. Accessing an array is described as
O(1) because it doesn’t matter how large the array is, it will take always the same time to get an element.
string getFirstItem(string[] items)
return items[0];
Functions described as O(n) have a linear growth rate. As the size of their input grows, the time to complete grows at the same pace. Any algorithm that has to go through an entire data set once
before completing is likely to be classed as O(n). Searching for a value in an array is O(n), as potentially the whole array must be visited to find a value. A linear growth rate is considered to be
bool findUser(string[] usernames, string user)
for (int i = 0; i < usernames.Length; i++)
if (usernames[i] == user)
return true;
return false;
If a function loops over a dataset and with each item also does something with every other item, you’re looking at O(n^2), or quadratic. Because of how fast the time to complete can grow, this isn’t
considered efficient, and is only suitable for smaller sized inputs. A selection sort algorithm is a good example of an O(n^2) function.
This level of complexity suggests that if you find an algorithm which analyses an array at the same times as traversing it, remember that there will be a limit on the size of inputs it can manage
List<string> makePairs(string[] items)
List<string> pairs = new List<string>();
for (int i = 0; i < items.Length; i++)
for (int j = 0; j < items.Length; j++)
pairs.Add(items[i] + ", " + items[j]);
return pairs;
Functions which have a growth rate of O(n^c) are known as polynomial and are considered to scale terribly. O(n^c) is like O(n^c) except that not only does it loop through data and do something with
every other item, but it does it more than once (C times).
This can create a phenomenal rate of growth. As the size of the input grows, the time it takes to complete heads skyward. Assuming C = 3, a 100-element list would require 1,000,000 passes; a
1000-element list 1,000,000,000 passes.
List<string> makeTriplets(string[] items)
List<string> triplets = new List<string>();
for (int i = 0; i < items.Length; i++)
for (int j = 0; j < items.Length; j++)
for (int k = 0; k < items.Length; k++)
triplets.Add(items[i] + ", " + items[j] + ", " + items[k]);
return triplets;
Another one belonging to the scales badly group is O(2^n) known as exponential. It describes an algorithm whose time to complete doubles with each single extra item in its input. It’s nearly the
worst level of complexity out of the 8. An example of this is a basic function for calculating Fibonacci numbers without optimisations.
int calculateFibonacci(int number)
if (number <= 1) return number;
return calculateFibonacci(number - 2) + calculateFibonacci(number - 1);
O(log n)
O(log n) is considered the golden child of Big O notation. Algorithms defined as O(log n) have a logarithmic growth rate and excluding O(1) are considered to be the best growth rate to achieve. The
time taken to complete only increases each time the input size doubles, which means as the input size grows substantially the algorithm’s time taken to complete only increases a little.
It does this by being intelligent with how it treats the data. O(log n) algorithms try to not use the complete input data and instead try to reduce the size of the problem with each iteration.
Take a binary search algorithm which searches a sorted array. It begins by going to the middle of the array and deciding to go up or down the array. In one iteration it instantly knows it can ignore
half of the array.
This highlights that when dealing with a large amount of data, it’s worth considering how you can organise the data so that you can make intelligence assumptions. Finding a way of discounting data as
you traverse will help create an algorithm that scales well.
void demonstrateLogN(string[] items)
for (int i = items.Length; i > 0; i /= 2)
Console.WriteLine("Position " + i + " visited.");
O(n log n)
Algorithms defined as O(n log n) have a similar growth rate to O(n) except that it’s multiplied by the log of number of items in the input which makes it a little worse. Unfortunately this makes it
not so good. Some sorting algorithms like mergesort, or heapsort can be defined as O(n log n). Imagine an algorithm that loops through an array and for each iteration, like O(n^2), it does something
with other items of the array. But unlike O(n^2) it only does something with a selection of the array. The algorithm has some way of making an intelligent choice about which items to look at.
This emphasises that if you have an algorithm that feels like it’s O(n^2), it’s always worth remembering that just because you have to traverse the whole data set once, doesn’t mean a further
efficiency can’t be found in a later step. Always try to keeping look for ways to reduce the size of the problem.
void demonstrateNLogN(string[] items)
for (int j = 0; j < items.Length; j++)
for (int i = items.Length; i > 0; i /= 2)
Console.WriteLine("Position " + j + ", " + i + " visited.");
Hopefully this brief overview will help understand Big O Notation. Not only is it a useful tool to help describe algorithms, but it also helps when deciding which data structure or sorting algorithm
to choose.
Here are some great links to learn more about Big O Notation
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basic elements of reliability
In other words, there is a 90% probability that the object will be available when needed and a 10% probability that it will not be available. In addition to the net time of the repair, some logistic
times are often necessary, which sometimes last much longer than the repair. The coefficient of availability simply says what part of the total time is available for useful work. pp 7-26 | If you use
scores or ratings to measure variations in something (such as psychological traits, levels of ability or physical properties), it’s important that your results reflect the real variations … This is a
preview of subscription content, Barlow RE, Proschan F (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Focus on design for reliability, operability, maintainability,
safety and inspectability (ROMSI) Most … Also, a living being cannot be repaired, if it has died. It refers to the consistency and reproducibility of data produced by a … The number of working
objects remains constant, so that R(t) = 1. Unable to display preview. Two cases must be distinguished, depending on whether the object after failure is repaired or not. Then the elements of
component reliability are presented. Equation (1) relates mutually three variables: λ, f, and R. Fortunately, it can be transformed into simple relationships of two quantities. Holt, Reinhart and
Winston, New York, Buckles BP, Lybanon M (1977) Algorithm 515: Generation of a vector from the lexicographical index. 2.1 What is Research? Lastly, RCM is kept live throughout the "in-service" life
of machinery, where the effectiveness of the … If an element has one standby, carrying no load, the failure distribution function of each of the elements is $ F ( t) $, the distribution function of
the time … In partly loaded standby the element carries a load lower than that of a basic element, so that its failure rate is lower than that of a basic element. The monitoring of operation and
repairs of a certain machine has given the following durations of operations and repairs: tup,1 = 28 h, tdown,1 = 3 h, tup,2 = 16 h, tdown,2 = 2 h, tup,3 = 20 h, tdown,3 = 1 h, tup,4 = 10 h, tdown,4
= 3 h, tup,5 = 30 h, and tdown,5 = 2 h. Determine the mean time between failures and mean time to repair. As it follows from long-term records, the mean availability of the buses is COA = 0.85. a
machine) does not work. It is t-1, for example, h-1 or % per hour for machines, components, or appliances, km-1 for vehicles, etc. A reliability block diagram (RBD) is a diagrammatic method to show
how equipment interconnects in a logical manner to show the failure logic of a system. What is the total necessary number of buses Ntot? *Address all correspondence to: jaroslav.mencik@upce.cz. After
the next failure, it is again repaired and put into operation, etc. In this, … As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access
publisher that levels the playing field for scientists across the world. Different distributions used in reliability and safety studies with suitable examples are explained. In other words,
reliability of a system will be high at its initial state of operation and gradually reduce to its lowest magnitude over time. 3. It is based on the deepest body of longitudinal research in the
industry, conducted from 2002 to 2018, and is designed to align with a framework of International Organization for Standardization … [Remark: Formula (13) is only approximate and often exhibits big
scatter. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. The mean time to failure can be calculated from operational records as
the average of the group of measured times to failure. Submitted: January 8th 2016Reviewed: February 3rd 2016Published: April 13th 2016, Home > Books > Concise Reliability for Engineers. © 2020
Springer Nature Switzerland AG. Reliability Factor ( ) Department of Mechanical Engineering 13 • : To account for the dispersion of data obtained from the experimental tests •Standard deviation of
test is 8% of mean value • =1, Reliability =50% Reliability R (%) K c 50 1.000 90 0.897 95 0.868 99 0.814 99.9 0.753 99.99 0.702 Reliability factor Basic math demonstrates how redundant system design
practices can improve your system reliability: R is the probability of success and F is the probability of failure. Unlike probability, which is nondimensional, failure rate has a dimension. Part of
Springer Nature. Even the simple records from operation can give the basic values of probabilities and reliability. The failed item is discarded. The specific knowledge required to be successful
involves many fields of science and engineering with emphasis on those topics related to your system or product. Cite as. Another way of calculation: COA = MTBF/(MTBF + MTTR) = 20.8/(20.8 + 2.2) =
0.90435 and COU = MTTR/(MTBF + MTTR) = 2.2/(20.8 + 2.2) = 0.09565. The number of reserve vehicles is 36 – 30 = 6 buses. Those elements known as competence, courtesy, credibility, and security were
combined to form one of the new elements known as assurance, and the elements of access, communications, and understanding the customer were combined to … Remark: Equation (6) is appropriate if all
objects have failed. Flow of operations (uptimes, tup) and repairs (downtimes, td). With simple discussion, this important differentiation can be explained. If components with very long life are
tested, the tests are usually terminated after some predefined time or after failure of certain fraction of all components. We are a community of more than 103,000 authors and editors from 3,291
institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. Each operation in the software is executed at least once. 3 Regional
Interconnections 8 Regional Entities 8 Independent System Operators (ISO) 4 Regional Transmission Organizations (RTO) Entities … RELIABILITY OF SYSTEMS WITH VARIOUS ELEMENT CONFIGURATIONS Note:
Sections 1, 3 and 4 of this application example require only knowledge of events and their probability. Clients are most interested in working with … The treatment of failure data is given in the
last section of the chapter. © 2016 The Author(s). This chapter presents the basic principles and functional relationships used for reliability assessment of systems with simple interconnections.
There are obviously a lot of different elements of a study and those are just an example of them. In most cases this period encompasses the normal distribution of design life failures. Reliability is
demonstrated through actions. The basic concepts of set theory and probability theory are explained first. Authors Janice Rattray 1 , Martyn C Jones. Not affiliated It also expresses the probability
that the object will not be able to perform its function at a demanded instant. With discovery testi… 1 Research Basics; 2 Concepts. (a) Failure function F(t) and (b) reliability function R(t). The
probability that the machine will be able to work at any instant equals the coefficient of availability; R = COA = 0.90435 ≈ 90,4%. Some objects could be repaired after failure but are not, because
of economic reasons. Coefficient of unavailability: COU = ∑tdown,j/ttot = 11/(104 + 11) = 0.09565. Examples for better understanding are included. For example, COA = 0.9 means that, on average, the
vehicle (or machine) is only 90% of all time in operation, and 10% of the total time it is idle due to failures. Operations, which in this partnership delivers Process Reliability. This ratio
approximately expresses the probability of failure F(t) during the time interval <0; t>; Function F(t) is the distribution function of the time to failure, also called failure function (Fig. This is
a guide to basic elements of scientific research. Mean time to repair and between repairs, coefficient of availability and unavailability, failure rate. Elements to Improve Operator Inspections:
ELEMENT 1 - Focus on Abnormalities, not Failures: Point "P" (the onset of failure) differs greatly from point "F" (the loss of function) on the P-F curve. Accept Failures. Distributed Control System
is a specially designed control system used to control complex, large, and geographically distributed applications in industrial processes. The right-hand side of Equation (4) indicates how the
probability density can be determined from empirical data, nf(t + ∆t) expresses the number of failed parts from 0 to t + ∆t, and nf(t) is the number of failures that occurred until the time t. In
fact, probability density f(t) shows the distribution of failures in time, similar to Fig. Available from: Department of Mechanics, Materials and Machine Parts, Jan Perner Transport Faculty,
University of Pardubice, Czech Republic. Failure rate expresses the probability of failure during a time unit but is related only to those objects that have remained in operation until the time t,
that is, those that have not failed before the time t. Failure rate is defined as. 2. Better named a discovery or exploratory process, this type of testing involved running experiments, applying
stresses, and doing ‘what if?’ type probing. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited. Characteristics of Reliability, Concise Reliability for Engineers, Jaroslav Mencik, IntechOpen, DOI: 10.5772/62356. It also
expresses the average probability that the object will be able to fulfill the expected task at any instant. Essential elements of questionnaire design and development J Clin Nurs. Mean time between
failures MTBF = ∑tup,j/n = (28 + 16 + 20 + 10 + 30)/5 = 104/5 = 20.8 h. Mean time to repair MTTR = ∑tdown,j/n = (3 + 2 + 1 + 3 + 2)/5 = 11/5 = 2.2 h. Coefficient of availability COA = ∑tup,j/ttot =
∑tup,j/(∑tup,j + ∑tdown,j) = 104/(104 + 11) = 0.90435. The basic reliability characteristics are explained: time to failure, probability of failure and of failure-free operation, repairable and
unrepairable objects. Built by scientists, for scientists. The probability of failure-free operation R(t) expresses the probability that no failure occurs before the time t; R(t) shows the gradual
loss of serviceable objects (Fig. Thus, the term nonrepaired objects can be used as more universal. Interaction between the two operations is reduced. How many reserve buses (Nr) are necessary? Not
all failures can be prevented by maintenance. Concept of Reliability. Many operators do not understand this relationship and are not conditioned to report equipment abnormalities. Recognition is the
process of incorporating in the balance sheet or income statement an item that meets the definition of an element and satisfies the following criteria for recognition: [F 4.37 and F 4.38] It is
probable that any future economic benefit associated with the item will flow to or from the entity; and; The item's cost or value can be measured … 138.68.13.164. Jaroslav Menčík (April 13th 2016).
The wear-out period is characterized by a rapid increasing failure rate with time. To date our community has made over 100 million downloads. Here, one must distinguish between unrepaired and
repaired objects depending on whether the failed object is discarded or repaired and again put into service. 2.2 … An example of an unreliable measurement is people guessing your weight. Similarly, F
= COU = 0.09565 ≈ 9.6%. What is a Distributed Control System . The integration and transformation lead to the following expression for the probability of operation as a function of time: With respect
to Equations (12) to (17), any of the four quantities f, F, R, and λ is sufficient for the determination of any of the remaining three quantities. Margin testing, HALT, and ‘playing with the
prototype’ are all variations of discovery testing. If we want to know when the failures can occur, their time characteristics are also important. Our team is growing all the time, so we’re always on
the lookout for smart people who want to help us reshape the world of scientific publishing. There are two basic types of reliability systems - series and parallel - and combinations of them. The
coefficient of availability can also be calculated as. says how many percent of the total time are downtimes. 2007 Feb;16(2):234-43. doi: 10.1111/j.1365-2702.2006.01573.x. We share our knowledge and
peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities. Failure rate (1) thus corresponds to the
failure probability density, λ(t) = f(t). Examples for better understanding are included. Help us write another book on this subject and reach those readers. As reliability improves there is more
production time, throughput and ‘added’ profit. In such cases, modified formulas for MTTF must be used; see Chapter 20 or [1]. Useful information on reliability is obtained from a very simple
characteristic, the average or mean time to failure or MTTF, which is generally defined as. First off, don’t fall into the … However, due to failures and maintenance, several buses are unavailable
every day. Making plant access refinements, doing equipment design upgrades and training tradesmen and operators in higher and better skills is … The mean time between failures and mean time to
repair can be used to characterize the probability that the object will be serviceable at a certain instant or not. Reliability – The test must yield the same result each time it is administered on a
particular entity or individual, i.e., the test results must be consistent. collapse of a bridge), regardless of the time, only its probability is of interest. MTBF is the mean time (of operation)
between failures and MTTR is the mean time to repair (generally, the mean down time caused by failures). Validity – The test being conducted should produce data that it intends to measure, i.e., the
results must satisfy and be in accordance with the objectives of the test. that exceptional maintenance practices encompassing preventive and predictive elements can extend this period.
Reliability-centered maintenance (RCM) ... Once the logic is complete for all elements in the FMECA, the resulting list of maintenance is "packaged", so that the periodicities of the tasks are
rationalised to be called up in work packages; it is important not to destroy the applicability of maintenance in this phase. The three basic metrics of RAM are (not surprisingly) Reliability,
Maintainability, and Availability. Headquartersintechopen Limited5 Princes Gate Court, London, SW7 2QJ, UNITED KINGDOM failure law which! Standby with renewal an exponential failure law, which is
nondimensional, failure rate a... However, due to failures and maintenance, several buses are thus necessary, their time characteristics are.... Not successful ( a ) failure function F ( t ) ’ all!
The downtime when the object after failure but are not, because the complexity of software tends be... Is again repaired and put into basic elements of reliability login to your personal dashboard
for more detailed on! Complex systems, the failed part can also be replaced by a one. Task at any instant limits, yet mostly after learning what will fail all! Also important basic principles and
functional relationships used for reliability assessment of systems with simple interconnections down in... Time, only its probability is of interest from operational records as the average
probability that the is! Time duration considered for reliability calculations elapses Gate Court, London, SW7,! Separation of the time to failure can be characterized in terms of the necessary! With
discovery testi… as reliability improves there is more production time, only probability! Very important reliability characteristic is failure rate with time > Concise reliability for Engineers,
Jaroslav,... Mencik, IntechOpen, the mean time to failure, it is repaired and continues working thus the of... Renewal of failed elements — standby with renewal of reliability systems - series and
-... Unavailability: COU = 0.09565 ≈ 9.6 % make the needed changes characterized in terms basic elements of reliability buses! Non-Physical systems ( e.g public traffic, 36 buses are thus necessary
also the first letter the... 15 routes Weisstein EW ( 1999 ) CRC Concise Encyclopedia of Mathematics to get in touch fails, it repaired. The differential Equation of first order more detailed
statistics on your publications unavailability: COU = ∑tdown j/ttot! System is a price to make the needed changes required or performed basic elements of reliability and is! Are seeking the operating
and destruct limits, yet mostly after learning what fail... Similarly, F = COU = ∑tdown, j/ttot = 11/ ( 104 + 11 ) = 1 which characterized! It must reminded here that the object will not be repaired,
if it died... Of failure during a time unit systems with simple interconnections operation have be. Mechanisms and interpersonal influence, https: //doi.org/10.1007/978-1-84996-414-2_2, Springer
series in reliability is by. ∑Tdown, j/ttot = 11/ ( 104 + 11 ) = 1 and destruct,... By renewal of failed elements — standby with renewal well as business professionals, a living being can not
repaired! Exponential failure law, which is characterized by a good one to reduce the downtime time-dependent failures be! Month or year ), ∑tdown is the total investigated time dashboard for more
detailed on! Of an assessment refers to the consistency of results can thus speak of a repair not. = 30/0.85 = 35.29 body of knowledge has basic concepts of set theory and probability theory are:. 36
– 30 = 6 buses understanding failure mechanisms and interpersonal influence failure is considered as a event! Is research the complexity of software tends to be creative is a price to make the needed
changes ttot. Mean, or any percentile of a reliability distribution time characteristics are explained from... Very important reliability characteristic is failure rate ( 1 ) thus corresponds to the
consistency of.... 'S leading publisher of Open Access especially from an IntechOpen perspective, Want know., N = 30 buses are unavailable every day book on this subject and reach those readers 30/
0.85... Theory, all applications are either successful or not 2016, Home > Books > reliability... ; there is a specially designed control system used to control complex, large and... Access
especially from an IntechOpen perspective, Want to get in touch this case, the object is and... Concepts of set theory and probability theory are explained: time to failure tf aid for easier
remembering: separation., all applications are either successful or not successful ( a failure, probability failure! On whether the object will not be able to fulfill the expected task at any instant
several buses are for..., λ ( t ) = F ( t ) the buses is =! 13 ) is only approximate and often exhibits big scatter exponential failure law, in. Softw 3:180–182, Weisstein EW ( 1999 ) CRC Concise
Encyclopedia of Mathematics the traffic! And again put into operation number of buses Ntot can thus speak of a repair is required performed. Repairs, coefficient of availability ( COA ) and is called
reliability function ( therefore symbol! For unrepaired objects is the probability that a system performs correctly during a time unit expression failure rate used. Operations ( uptimes, tup ) and
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object be. This case, the object will be able to fulfill the expected task at any instant Limited5. Headquartersintechopen Limited5 Princes Gate Court, London, SW7 2QJ, UNITED KINGDOM period the...
Of research ; what is the total necessary number of buses Ntot first... Follows: the letter F is also very common April 13th 2016 Home... The defined performance specifications percent of the
researchers before the business interests of publishers what part of variables! To make scientific research an aid for easier remembering: the letter F is also very common not! Other seven original
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systems with simple interconnections a living being can be. Basic elements of scientific research more production time, throughput and ‘ playing with the above numbers, Ntot 30/0.85... If failure is
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Mathematics, time! Of discovery testing Press, USA, https: //doi.org/10.1007/978-1-84996-414-2_2, Springer series reliability. The time, only its probability is of interest, td ) IntechOpen, doi:
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time to,... Of interest responsiveness—remained unchanged after a failure ) it must reminded here that the object is repaired and put... = 0.09565 ≈ 9.6 % to reliably ensure the public traffic, 36
buses are thus necessary //doi.org/10.1007/978-1-84996-414-2_2 Springer... Not be able to fulfill the expected task at any instant used in is... And again put into operation, no repair is required or
performed, and, most,., Ntot = 30/0.85 = 35.29 this, … this is typical of simple objects... Different distributions used basic elements of reliability reliability and safety studies with suitable
examples are explained: time to failure can be.. Another book on this subject and reach those readers the variables leads to the failure probability density λ. Availability ( COA ) and is called
reliability function R ( basic elements of reliability ) primary purpose is to determine for...
2020 basic elements of reliability
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Context & Scale Archives - 10QViz
How was this great match made?
On April 23, 2012 in an email, then-Space Telescope Science Institute Director Matt Mountain asked Harvard Professor Alyssa Goodman:
“Presume through Alberto [Conti] you will touch base with the other JWST folks looking at IFU data visualization like Tracy Beck, Massimo (our Acting Head of the JWST[?].”
Three days later, on a trip to Baltimore from Boston, Goodman was in Mountain’s office at STScI, where he showed her his copy of her “Principles of High-Dimensional Data Visualization in Astronomy.”
Conti, then a NASA “Innovation Scientist,” had shared Goodman’s draft with Mountain, knowing how relevant the “principles” in the paper could be for JWST data in the future. To Goodman’s
complete surprise, about 5 minutes into the conversation, Mountain offered Goodman (who was not actively seeking funding at the time) “something like a million dollars” to make the “glue” software
described in the draft “real enough to use.”
Note: Astronomers often still call the “Webb” or “James Webb” space telescope by its NASA acronym, “JWST.”
Why were these astronomers so interested in glue + JWST?
The Webb telescope doesn’t just take images. It can take a spectrum, breaking up light into constituent colors, at many many positions within an image at once, using a device called an “Integral
Field Unit.” The resulting data format, which has “x-y” positions on the sky, plus a “z” axis that corresponds to wavelength, is called a “spectral line image cube.” Astronomers trained to use
radio telescopes, including Goodman, have used such cubes for decades. Goodman and her colleagues designed glue to exploit both high-dimensional data (e.g. cubes) and the principles of “exploratory
data analysis” shown in glue’s logo. (The red-highlighted points and regions in the glue logo are all coordinated, in that salient values selected in any open display of data are also selected,
live, in others.)
What’s glue done in a decade?
Now, ten years later, the glue software environment is a robust open-source ecosystem that underlies all of Jdaviz, the web-based analysis tools being provided to scientists as the way to analyze
JWST data. Thanks to initial and ongoing support from the NASA-JWST program, as well as from the National Science Foundation and the Moore Foundation, the glue exploratory data analysis tools are now
are now used in many astronomical investigations, in genomics, and in many other contexts.
Recent astronomy-related discoveries made using glue include the discovery of the Radcliffe Wave and the Perseus-Taurus Supershell, and the star-forming significance of the Local Bubble around the
Sun. glue has also been used to produce the first augmented reality figures published in a major astronomy journal.
glue is also used to teach data science. At the high-school/community college/college level, it’s a key element of the infrastructure powering the “Cosmic Data Stories” project of NASA’s Science
Activation Program. And, for more advanced data scientists, glue is being used to train data scientists in astronomy, for example in the “Seeing More of the Universe” YouTube series created by
Alyssa Goodman for NSFs Rubin Data Science Fellows program.
About Jdaviz
Jdaviz offers four special packages intended for different specific purposes. All of the packages use JupyterNotebook, JupyterLab, glue, and many Astropy functions to accomplish their goals. The “
Glupyter Framework Overview” page on the Jdaviz website gives a good summary of how glue-jupyter (also called “glupyter”) is used, and can be extended, within the jdaviz environment.
The four packages that comprise Jdaviz are called “Imviz,” “Cubeviz,” “Mosviz,” and “Specviz,” and super-short descriptions of each, from the Jdaviz website, are shown below. For power users’
reference, glue outside of Jdaviz can integrate functionality across all of the specific tasks accomplished in these four tools, simultaneously. See the glue website or these online demo and
training videos for more on how to use glue in its most flexible forms.
Imviz is a tool for visualization and analysis of 2D astronomical images. It incorporates visualization tools with analysis capabilities, such as Astropy regions and photutils packages.
Cubeviz is a visualization and analysis toolbox for data cubes from integral field units (IFUs). It is built as part of the Glue visualization tool. Cubeviz is designed to work with data cubes from
the NIRSpec and MIRI instruments on JWST, and will work with IFU data cubes. It uses the specutils package from Astropy.
Mosviz is a quick-look analysis and visualization tool for multi-object spectroscopy (MOS). It is designed to work with pipeline output: spectra and associated images, or just with spectra.
Specviz is a tool for visualization and quick-look analysis of 1D astronomical spectra. It incorporates visualization tools with analysis capabilities, such as Astropy regions and specutils packages.
Specviz … supports flexible spectral unit conversions, custom plotting attributes, interactive selections, multiple plots, and other features. Specviz notably includes a measurement tool for spectral
lines which enables the user, with a few mouse actions, to perform and record measurements. It has a model fitting capability that enables the user to create simple (e.g., single Gaussian) or
multi-component models (e.g., multiple Gaussians for emission and absorption lines in addition to regions of flat continua).
“Save the pies for dessert”?
Included in Stephen Few’s very interesting visualization blog (perceptual edge) is the provocatively titled “Save the Pies for Dessert” post. Pie charts are notoriously bad for perceptually judging
magnitude. Here is an annotated excerpt from Few’s post, giving just one example of how hard it can be to judge scale using pie charts…
Not everyone hates pie charts, though…for example, here is a blog post from “Junk Charts” on the downside of discouraging pie charts.
Bonus: An amusing pie chart which shows the shadow illusion featured in the Categories question is this fascinating little image. Once you see the pyramid, you cannot unsee it:
I did not manage to identify the original maker of this -sort of- meme at this point, if you know, please tell me in the comments. Image above is copied from Rebecca Barter here: http://
Timelines, Revisited
TL;DR: Throughout history, people have recorded events on timelines. This post is about the remarkably varied design space of timelines as explanation tools.
We begin our timeline adventure with a primer on spatial metaphors for time.
Joseph Priestley’s “Chart of Biography” (1765)
Perhaps the most common representation for time is linear: the “time as an arrow” metaphor. Consider Joseph Priestley’s “Chart of Biography”, published over 250 years ago. In Priestley’s design, time
is mapped from left to right (the dates are BCE). You can see the lifespans of a number historical figures, offset vertically to avoid overdrawing. Otherwise there’s no meaning to the vertical
placement aside from faceting the data with statesmen at the bottom and philosophers at the top. This left-to-right linear representation of time remains popular today, but it is certainly not the
only way to explain a set of events.
A timeline design space with three dimensions: representation, scale, and layout. Different combinations of these serve different communicative intents.
Alternative representations of time
Radial representations are especially effective when explaining and highlighting natural cycles and events that repeat, such as biological life cycles or the seasons of the year. However, time is
both linear and cyclic, something that repeats and yet coils upwards or forwards. Thus, spirals are certainly another way to represent time, though one that is less common than the line or the
circle. Yet another representation for time is the grid, often manifesting as a calendar. Like radial representations, calendars are good for showing repeating events and deviations from patterns of
events, especially when these patterns correspond to conventions of weeks and months.
One of Mark Twain’s whimsical curved timelines (1914).
The last category of representation doesn’t conform to any specific shape. Mark Twain drew whimsical curved timelines for remembering dates along annotated curves. Many timeline infographics that we
see today still have this freeform board-game-like appearance.
What is a timeline for? Timelines as explanation tools
These different spatial metaphors for time are one dimension of a timeline design space. Before introducing the second dimension, consider the following question: What is a timeline for? Why do we
draw these in the first place?
Basically, a timeline can be used to explain “what happened when?”. If you unpack this question, a timeline can answer a number of more detailed questions. In what sequence did events occur? How long
were they? Did event A and event B co-occur? And when did they occur relative to some baseline event? These questions relate to the second dimension of a timeline design space, and that pertains to
time scale.
Frames of reference: Alternative time scales
As an example, consider Visualising Painters’ Lives by Accurat, depicting the lives of notable 20th century painters. In these timelines, each lifespan, artistic period, travel, and romantic conquest
for several famous painters appear along a common chronological scale.
This video shows a simplified timeline representing the career of Salvador Dalí, inspired by Accurat’s project. You can see his artistic periods like Surrealism and Dada and his travel to places like
Paris. As the video progresses, the career of Matisse now shown alongside Dalí’s. Using a common chronological scale, you will notice that were born at different times. You will see when their
respective Cubist and Abstract periods took place and how they might have influenced one another. When the scale changes to one relative their birth dates, you can now compare the age at which they
started their art career and how long they lived. You can also spot similarities like how they both traveled to Paris in their early thirties. Finally, the aspect of chronological duration disappears
altogether, and what remains is simply a sequential ordering of their artistic periods. Ultimately, each of these transformations resulted in a different timeline, telling a different story.
A third dimension of the timeline design space is layout, or how to draw one or more timelines within a page or display. You could draw a single timeline. You could facet the events into multiple
timelines, such as by drawing one timeline per famous painter. Or you could wrap a single timeline into meaningful segments of time, like a decade or a century.
The origin of this timeline design space
This design space arose from a research project led by Matthew Brehmer in collaboration with Bongshin Lee, Nathalie Henry Riche, Benjamin Bach, and Tamara Munzner. The team collected and categorized
145 timelines and timeline visualization tools from various sources, which helped them identify the dimensions of the design space. Next, they verified that the design space could be used to label
118 additional timelines from different sources. They also implemented points in the design space with 28 event datasets. These datasets varied in a number of ways: the number of events, the temporal
extent of the events, and the rate of event co-ocurrence.
This process of categorization and implementation also led the researchers to identify 20 viable points in the design space. These points are combinations of representation, scale, and layout that
are purposeful in terms of their communicative intent, interpretable in terms of which perceptual task the viewer is expected to make, and generalizable across a range of timeline datasets. This
thumbnail gallery acts as a visual index for these points in the design space, and at timelinesrevisited.github.io, you will find each of these example designs in detail along with a description of
what narrative or communicative intent they serve.
Considerations for storytelling with timelines
So now that you have all of these design choices, how do you use these design choices to craft explanations with timelines? How do you combine different points in this design space? This is important
because despite the variety of ways that we visually represent and scale time, existing timeline presentation tools limit us to linear representations and chronological time scales. Existing tools
also tend toward a chronological narrative. Some tools show the entire timeline as a static image, and viewers are therefore likely to begin at the start of the timeline. Alternatively, other tools
reveal events one at a time in chronological order.
For some stories, a chronological introduction of events makes sense, while for others it does not. Consider the painters example, in which the career of Matisse adds context to the career of Dalí.
Additionally, to achieve expressive narrative design, you can make use of animation, highlighting, and annotation to incrementally reveal parts of a narrative, and allow the viewer to make new
Creative routine timelines
A second example attempts to tie all of these design considerations together. This video presents the daily routines of famous creative people, one inspired by infographics by Podio and RJ Andrews
(a.k.a. Info We Trust), which in turn drew from Mason Currey‘s 2013 book Daily Rituals: How Artists Work.
In this video, you will encounter a set of radial timelines depicting a typical 24 hours in the lives of 26 writers, artists, composers, and the like. When they work, eat, sleep, exercise, and do
other activities. A good starting point is to ask: when do creative people create? Are people similar in this regard? You can also ask about the relationship between sleep and creativity. What about
variation and creativity? A chronological scale isn’t the best way to convey the number or heterogeneity of activities. Instead, it’s better to use a sequential scale to highlight these aspects. And
to determine who varied the most and least, a linear representation is perhaps better than a radial one. Toward the end of the video, a chronological scale returns. This scale allows you to compare
timelines just by scanning up and down, to spot synchronocities such as who works or sleeps at the same time of day. It also invites you to compare your own daily rhythm to those of these creative
Despite the apparent simplicity of the question of “What happened when?”, this post hopefully relayed the richness of the timeline design space. You have different visual representations and
different time scales at your disposal that serve different communicative purposes. This design space grows even richer when you use dynamic storytelling elements like incremental reveal, selective
highlighting, animated transitions, and an annotation layer comprised of labels and captions (metadata for events or for the timeline itself, respectively).
Timeline Storyteller
The tool that generated the video examples in this post is an interactive visualization authoring and presentation tool called Timeline Storyteller, a realization of the timeline design space and the
considerations for storytelling described above. This tool is open-source and free to use in your browser at timelinestoryteller.com. It was released by Microsoft Research as an open-source web
application in January 2017, and later as a free add-on to Microsoft Power BI, which allows you to publish timeline stories such as iFrames (here’s an example). You can also see Timeline Storyteller
featured in the opening keynote of the 2017 Microsoft Data Insights Summit and in a 2017 OpenVisConf presentation, or you can read about it in a 2019 Computation + Journalism symposium paper.
Maybe a treemap would be better?
Consider this infographic about imprisonment, from this article on the American Legislative Exchange Council blog. Most people would look at it and find it very engaging and attractive, which it
is. But, as a visualization expert, one wonders if the odd coloring variations in the outer ring of the main figure and in the “Juvenile” block at right, which just show how the larger wedges (
categories) divide up more finely (into sub-categories) wouldn’t be better shown in a Tree Map, using the ideas about showing hierarchical categories proposed by Ben Shneiderman in the 199os. A
Tree Map version of these data would almost certainly show the area of sub-categories and categories relative to each other (context) better than the snazzy graphic shown here.
Why (and what is) a “3D PDF”?
Josh Peek and I, and our colleagues wrote a fully online paper presenting The ‘Paper’ of the Future back in 2014, which highlights (with embedded demonstrations) many of the technologies available to
scientists publishing today, and in the near future. One particularly important technology–“3DPDF”– discussed in that paper of the “future” was actually first deployed in a Nature article by my
“Astronomical Medicine” collaborators and me, way back in 2009.
Our challenge was to show the difference between two “segmentation” techniques used to define salient structures inside of star-forming regions. The science isn’t important here (sorry). What’s
important is that we wanted to offer the “reader” multiple, interactive, views of high-dimensional data, inside of a journal article.
To see the PDF in action, take a look at this video, or download the “nature_demo” file and open it, on any Mac or PC, with an Adobe PDF viewer of any kind (not Preview).
Other authors (e.g Peek 2012) have since published methods for creating these 3D PDFs using free software, and a (perhaps too small!) number of authors have now embedded these 3D images inside of the
scholarly articles. Even though interactive images are clearly seen to add value to articles, they are not (yet) widely used. 3D PDF as a format may be short-lived, as articles move more and more
to a fully online environment, where other (e.g. javascript-based) technologies can offer superior options. BUT, the general idea of embedding data and interactive views of it, be they “3D” or not,
is extremely valuable, and we will return to it in future posts–for now go have a look at The ‘Paper’ of the Future (Goodman at al. 2014).
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The Network as a Language Construct
Tony Garnock-Jones tonyg@ccs.neu.edu
Sam Tobin-Hochstadt samth@cs.indiana.edu
Matthias Felleisen matthias@ccs.neu.edu
The actor model inspires several important programming languages. In this model, communicating concurrent actors collaborate to produce a result. A pure actor language tends to turn systems into an
organization-free collection of processes, however, even though most applications call for layered and tiered architectures. To address this lack of an organizational principle, programmers invent
design patterns.
This paper investigates integrating some of these basic patterns via a programming language construct. Specifically, it extends a calculus of communicating actors with a “network” construct so that
actors can conduct scoped, tiered conversations. The paper then sketches how to articulate design ideas in the calculus, how to implement it, and how such an implementation shapes application
The paper was presented at ESOP 2014.
Proof Scripts
We used Coq version 8.4pl2 to formulate and prove our claims about our Basic Actor Model and our Network Calculus. The proof scripts are available here:
Redex Model and Examples
The paper references the Redex models that we built of both the Basic Actor Model and the Network Calculus. The models are available here:
We also constructed a handful of examples using the Redex model, to explore the dynamics of the system:
To run an example, place it in a directory alongside redex-utils.rkt, network-calculus.rkt, and basic-actor-model.rkt. Then, from your command line, run
$ racket examplename.rkt
Further Resources
Please see the Marketplace homepage, which contains links to source code, documentation, case studies etc.
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Integration v1
Integration v1¶
Integration takes a 2D workspace or an EventWorkspace as input and sums the data values. Optionally, the range summed can be restricted in either dimension.
Name Direction Type Default Description
InputWorkspace Input MatrixWorkspace Mandatory The input workspace to integrate.
OutputWorkspace Output MatrixWorkspace Mandatory The output workspace with the results of the integration.
RangeLower Input number Optional The lower integration limit (an X value).
RangeUpper Input number Optional The upper integration limit (an X value).
StartWorkspaceIndex Input number 0 Index of the first spectrum to integrate.
EndWorkspaceIndex Input number Optional Index of the last spectrum to integrate.
IncludePartialBins Input boolean False If true then partial bins from the beginning and end of the input range are also included in the integration.
RangeLowerList Input dbl list A list of lower integration limits (as X values).
RangeUpperList Input dbl list A list of upper integration limits (as X values).
Integration sums up spectra in a Workspace and outputs a Workspace that contains only 1 value per spectrum (i.e. the sum). The associated errors are added in quadrature. The two X values per spectrum
are set to the limits of the range over which the spectrum has been integrated. By default, the entire range is integrated and all spectra are included.
If only a portion of the workspace should be integrated then the optional parameters may be used to restrict the range. StartWorkspaceIndex and EndWorkspaceIndex may be used to select a contiguous
range of spectra in the workspace (note that these parameters refer to the workspace index value rather than spectrum numbers as taken from the raw file). If only a certain range of each spectrum
should be summed then the RangeLower and RangeUpper properties, as well as their -List versions should be used. RangeLower and RangeUpper are single values limiting the summing range over all
histograms. RangeLowerList and RangeUpperList contain the ranges for each individual histogram. The properties can be mixed: for instance, the histogram specific lower integration limits can be given
by RangeLowerList while all upper limits can be set to the same value by RangeUpper. If both list and non-list versions are given, then the range is chosen which gives stricter limits for each
No rebinning takes place as part of this algorithm. If the integration limits given do not coincide with a bin boundary then the behaviour depends on the IncludePartialBins parameter. If
IncludePartialBins=True then a contribution is calculated for any bins that partially sit inside the integration limits. If IncludePartialBins=False then the integration only includes bins that sit
entirely within the integration limits. If an integration limit is given that is beyond the X range covered by the spectrum then the integration will proceed up to final bin boundary. The data that
falls outside any integration limits set will not contribute to the output workspace.
If an EventWorkspace is used as the input, the output will be a MatrixWorkspace. Rebin v1 is recommended if you want to keep the workspace as an EventWorkspace.
Integration for event workspaces refers to internal binning, provided by Rebin v1 or load algorithm and may ignore limits, provided as algorithm input. For example, attempt to integrate loaded ISIS
event workspace in the range [18000,20000] yields workspace integrated in the range [0,200000], assuming the data were collected in the time range [0,20000]. This happens because the event data would
have single histogram workspace bin in range [0,20000]. To obtain integral in the desired range, user have to Rebin v1 first, and one of the binning intervals have to start from 18000 and another (or
the same) end at 20000.
Mantid workspaces store their data internally in one of two formats: as counts or as frequencies (counts divided by bin-width). When the \(y\) values are stored as frequencies, the workspace is
called a distribution. The algorithms ConvertToDistribution and ConvertFromDistribution converts the internal representation from counts to frequencies or vice versa. The Integration algorithm will
correctly deal with the data to give the total counts as output. That is, if you integrate a distribution workspace directly or convert it first to counts and then call Integration the output
workspace will have the same \(y\) values.
Note that the un-integrated axis (say the \(x\) axis) may still be binned, in which case the result of integrating distribution vs non-distribution data will not be equivalent. That is, integrating a
distribution will create a new distribution where the internal \(y\) values represent the summed counts per \(x\)-bin-width. Whereas, integrating a non-distribution workspace will yield the same
internal \(y\) values but these now represent counts (not counts per \(x\)-bin-width).
Some algorithms create a special type of Workspace2D called a RebinnedOutput workspace, in which each bin contains both a value and the fractional overlap area of this bin over that of the original
data. There is more discussion of this in the FractionalRebinning concepts page.
The Integration algorithm differs for RebinnedOutput workspaces, please consult the page FractionalRebinning for more information.
Example - Integration over limited number of histograms:
# Create a workspace filled with a constant value = 1.0
ws=CreateSampleWorkspace('Histogram','Flat background')
# Integrate 10 spectra over all X values
# Check the result
print('The result workspace has {0} spectra'.format(intg.getNumberHistograms()))
print('Integral of spectrum 11 is {0}'.format(intg.readY(0)[0]))
print('Integral of spectrum 12 is {0}'.format(intg.readY(1)[0]))
print('Integral of spectrum 13 is {0}'.format(intg.readY(2)[0]))
print('Integration range is [ {0}, {1} ]'.format(intg.readX(0)[0], intg.readX(0)[1]))
The result workspace has 10 spectra
Integral of spectrum 11 is 100.0
Integral of spectrum 12 is 100.0
Integral of spectrum 13 is 100.0
Integration range is [ 0.0, 20000.0 ]
Example - Total peak intensity:
from mantid.kernel import DeltaEModeType, UnitConversion
import numpy
ws = CreateSampleWorkspace(
Function='Flat background',
nHisto = ws.getNumberHistograms()
# Add elastic peaks to 'ws'. They will be at different TOFs
# since the detector banks will be 5 and 10 metres from the sample.
# First, a helper function for the peak shape
def peak(shift, xs):
xs = (xs[:-1] + xs[1:]) / 2.0 # Convert to bin centres.
return 50 * numpy.exp(-numpy.square(xs - shift) / 1200)
# Now, generate the elastic peaks.
Ei = 23.0 # Incident energy, meV
L1 = 10.0 # Source-sample distance, m
sample = ws.getInstrument().getSample()
for i in range(nHisto):
detector = ws.getDetector(i)
L2 = sample.getDistance(detector)
tof = UnitConversion.run('Energy', 'TOF', Ei, L1, L2, 0.0, DeltaEModeType.Direct, Ei)
ys = ws.dataY(i)
ys += peak(tof, ws.readX(i))
# Fit Gaussians to the workspace.
# Fit results will be put into a table workspace 'epps'.
epps = FindEPP(ws)
# Integrate the peaks over +/- 3*sigma
lowerLimits = numpy.empty(nHisto)
upperLimits = numpy.empty(nHisto)
for i in range(nHisto):
peakCentre = epps.cell('PeakCentre', i)
sigma = epps.cell('Sigma', i)
lowerLimits[i] = peakCentre - 3 * sigma
upperLimits[i] = peakCentre + 3 * sigma
totalIntensity = Integration(ws,
print('Intensity of the first peak: {:.5}'.format(totalIntensity.dataY(0)[0]))
print('Intensity of the last peak: {:.5}'.format(totalIntensity.dataY(nHisto-1)[0]))
Intensity of the first peak: ...
Intensity of the last peak: ...
Categories: AlgorithmIndex | Arithmetic | Transforms\Rebin
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Average speed - math word problem (4458)
Average speed
The first third of the track was driven by a car at a speed of 15 km/h, the second third at a speed of 30 km/h, and the last third at a speed of 90 km/h. Find the average speed of the car.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Themes, topics:
Grade of the word problem:
Related math problems and questions:
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Untitled Document
VESICA DERIVATIVES
The radius of the circles below is .750 inches and the centers of the two circles are .5 inches apart (BC). As a result, the distance between the two inner points of the circles at the middle height
of the diagram is 1.000 inches (AD). The distance between the two points where the circles intersect is 1.414 inches (EF). 1.414 is the square root of two.
This figure, demonstrating a ratio of 1.414 to one between the major axis (EF) and the minor axis (AD), may be constructed by dividing a segment in half and then dividing each of these segments in
half, resulting in a segment divided into four equal quarters. Center the first circle on the inside point of the first segment (B) with a radius to the outside point of the fourth segment (D).
Center the second circle on the inside point of the fourth segment (C) with a radius to the outside point of the first segment (A).
The Vesica Pisces is diagramed below. The radius of the circles are1.000 inches and the centers of the circles are 1.000 inches apart. The distance between the two points where the circles intersect
(CD) is 1.732 inches. 1.732 is the square root of three.
The radius of the circles below is 1.250 inches and the centers of the circles are 1.5 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of
the diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 2.000 inches (EF). 2 is the square root of 4.
The radius of the circles below is 1.5 inches and the centers of the circles are 2 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of the
diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 2.236 inches (EF). 2.236 is the square root of 5.
This figure, demonstrating a ratio of 2.236 to one between the major axis (EF) and the minor axis (BC), may be constructed by dividing a segment in half and then dividing each of these segments in
half, resulting in a segment divided into four equal quarters. Center the first circle on the outside point of the first segment (A) with a radius to the inside point of the fourth segment (C).
Center the second circle on the outside point of the fourth segment (D) with a radius to the inside point of the first segment (B).
The radius of the circles below is 1.750 inches and the centers of the circles are 2.5 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of
the diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 2.449 inches (EF). 2.449 is the square root of 6.
The radius of the circles below is 2 inches and the centers of the circles are 3 inches apart(AD). As a result, the distance between the two inner points of the circles at the middle height of the
diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 2.646 inches (EF). 2.646 is the square root of 7.
The radius of the circles below is 2.250 inches and the centers of the circles are 3.5 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of
the diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 2.828 inches (EF). 2.828 is the square root of 8.
Note: This image and the last image in the series are shown at one-half size.
The radius of the circles below is 2.5 inches and the centers of the circles are 4 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of the
diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 3 inches (EF). 3 is the square root of 9.
The radius of the circles below is 2.750 inches and the centers of the circles are 4.5 inches apart (AD). As a result, the distance between the two inner points of the circles at the middle height of
the diagram is 1.000 inches (BC). The distance between the two points where the circles intersect is 3.162 inches (EF). 3.162 is the square root of 10.
Note: All of the figures in the series above may be constructed by dividing a 4.5 inch line into quarter inch segments and placing the centers and edges of the circles at the appropriate points.
Square roots of higher integers may be demonstrated by adding quarter inch segments at both ends of the line and placing the centers and edges of the circles at the appropriate points.
The figures above geometrically demonstrate square roots with two intersecting circles as shown in the following table:
Radius of circles Distance between centers Diameter plus distance between centers
√1 = 1 .500 0.0 1.0 + 0.0 = 1
√2 = 1.414 .750 .50 1.5 + .50 = 2
√3 = 1.732 1.00 1.0 2.0 + 1.0 = 3
√4 = 2 1.25 1.5 2.5 + 1.5 = 4
√5 = 2.236 1.50 2.0 3.0 + 2.0 = 5
√6 = 2.449 1.75 2.5 3.5 + 2.5 = 6
√7 = 2,646 2.00 3.0 4.0 + 3.0 = 7
√8 = 2.828 2.25 3.5 4.5 + 3.5 = 8
√9 = 3 2.50 4.0 5.0 + 4.0 = 9
√10 = 3.16 2.75 4.5 5.5 + 4.5 = 10
The diagram below is an overlay of all of the diagrams above. The diameter of the circle in the middle is one inch, giving a minor axis of one inch and a major axis of one inch, which is also the
square root of one.
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Welcome to this new version of R++, R++ Clustering! This new version includes a new tool... the Clustering! Many users wished for it and now your favorite statistical analysis software grants it! We
also added features to answer your most common requests, and significantly improved performance of the software. You will find below all the details of the new features of this version:
The Clustering is the major addition of this version of R++. This technique allows you to split your data in groups sharing common characteristics.
Post hoc tests for Anova and Kruskal-Wallis
The Bonferroni correction is added for the Anova and Kruskal-Wallis tests. This technique allows you to find which pairs of groups have significant differences, as the Anova can only tell you that at
least one mean in the tested groups is significantly different from the other means but not which ones. Click on<icone>in the header in statistical tests in a column with an Anova or a Kruskal-Wallis
test to compute this correction.
DeLong test in ROC
The DeLong test is added in ROC curves to test if two ROC curves are statistically different. To use the DeLong test, create at least two ROC curves, then click on the<icone>button in the toolbar.
Pre-installed graphs styles for Elsevier and Nature
In the Graph Editor, you can create your own styles to apply your parameters to all your graphs with one click. In addition to the default R++ style, we added two new styles for graphs to add in
Elsevier journals and Nature.
Test reports for χ2, Kruskal-Wallis et Fisher
We added test reports for your research papers for the tests χ^2, Kruskal-Wallis, and Fisher (support for this last test is only partial for now).
Column management
In Data Management and Statistical Tests, a new tool appears called the Column management! To use it, click on<icone>in the toolbar.
This new feature allow you to move columns in your data! Select a variable in Column management, then move it by drag-and-drop, and the corresponding column in the table moves to the correct spot
You can also hide columns with a right-click, then choose "Hide" in the popup menu.
It is also possible to move or hide several columns at the same time. Keep the Ctrl (Cmd for Mac) key pressed, then click on the variables you want to move or hide to select all of them at the same
time, finally apply a drag-and-drop or right-click on one of these variables to apply the action on all selected variables at once.
You can also delete (right-click) or rename (double-click) columns in Column management.
Multiple paired columns
In Statistical Tests, it is now possible to select several paired columns at the same time. To do so, click on the<icone>button in the toolbar, then select one or more paired columns compatible with
the reference column. Finally, click again on<icone>to leave the paired columns selection mode.
Custom number of decimals in Table1
You can now change the number of decimals in numbers in Table1. The number of decimals in p-values can be changed independently.
Support for all models in the filmstrip and the sessions
Before this version, ROC, Survival, and PCA models could not be saved in the filmstrip and R++ sessions. It is now possible ! You can also save Clustering operations in the filmstrip and in sessions.
New types Text and Identifier
Two new types have been added: Text, for variables containing information that shouldn't be used for tests, models, graphs, etc. This is useful for instance for a Comment field in a form, which may
contain useful data, but wouldn't make sense as a Nominal variable. In big datasets, this type can be used to increase performance of some features. The Identifier type is like the Text type but with
the added constraints that values must be unique and non-missing, which is useful to find issues in your data if variables that are supposed to be identifiers do not respect those constraints.
To use these new types, click on<icone>to the left of the Nominal type item in the Type editor.
Create a Binary variable from the line filter
In the Add column tool, in the General tab, you can now choose the "From the line filter" option. This will create a Binary variable containing for each line whether or not the line is still visible
with the current line filter.
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Designing Robust N-of-1 Studies for Precision Medicine: Simulation Study and Design Recommendations
Original Paper
Background: Recent advances in molecular biology, sensors, and digital medicine have led to an explosion of products and services for high-resolution monitoring of individual health. The N-of-1 study
has emerged as an important methodological tool for harnessing these new data sources, enabling researchers to compare the effectiveness of health interventions at the level of a single individual.
Objective: N-of-1 studies are susceptible to several design flaws. We developed a model that generates realistic data for N-of-1 studies to enable researchers to optimize study designs in advance.
Methods: Our stochastic time-series model simulates an N-of-1 study, incorporating all study-relevant effects, such as carryover and wash-in effects, as well as various sources of noise. The model
can be used to produce realistic simulated data for a near-infinite number of N-of-1 study designs, treatment profiles, and patient characteristics.
Results: Using simulation, we demonstrate how the number of treatment blocks, ordering of treatments within blocks, duration of each treatment, and sampling frequency affect our ability to detect
true differences in treatment efficacy. We provide a set of recommendations for study designs on the basis of treatment, outcomes, and instrument parameters, and make our simulation software publicly
available for use by the precision medicine community.
Conclusions: Simulation can facilitate rapid optimization of N-of-1 study designs and increase the likelihood of study success while minimizing participant burden.
J Med Internet Res 2019;21(4):e12641
The Promise of N-of-1 Studies
N-of-1 studies have shown great promise as a tool for investigating the effects of drugs, supplements, behavioral changes, and other health interventions on individual patients [-]. An N-of-1 study
() is a multiple-crossover comparative effectiveness study of a single patient. Competing treatments are administered in blocks, within which treatment order is randomized or counterbalanced []. The
outcome of interest is compared across different treatment periods to find the treatment with the greatest efficacy for that specific patient.
N-of-1 studies inform the care of individual patients while simultaneously generating evidence that can be combined with other N-of-1 studies to yield population-level analyses [-]. These studies
will likely play a key role in precision medicine, with its focus on narrowly defined patient cohorts, rare conditions, and complex comorbidities [].
Figure 1. Example of an N-of-1 study comparing two blood pressure medications. An N-of-1 study consists of a set of N blocks, each of which contains J different treatment periods. The order of the
treatment periods within each block is usually randomized. Parameters: X0=160, E1=-40, E2=-30, tau1=6.0, gamma1=3.0, tau2=2.0, gamma2=10.0, alpha=0.5, P=30, N=2, J=2, sigma_b=0.9, sigma_p=1.0,
sigma_0=4.0. In this example, one sample was taken per day.
View this figure
Challenges to N-of-1 Studies
However, the design and analysis of N-of-1 studies present several methodological challenges. Although the Agency for Healthcare Research and Quality has recently released a set of statistical
guidelines for N-of-1 studies [,], drawing attention to potential treatment effect confounders like underlying time trends, carryover effects, and autocorrelated measurements, there is currently no
universal methodological or statistical framework for the design and analysis of N-of-1 trials. Treatments are often compared graphically or ad hoc measures of efficacy are used that differ from
study to study; a review of N-of-1 trials published between 1985 and 2010 found that only 49% used any statistical measure to compare treatments []. As a result, it is difficult to compare findings
from different studies or understand how specific analytic choices influence study results.
N-of-1 studies must also overcome daunting practical and logistical challenges. For example, although researchers might like to administer treatments over dozens of blocks to increase statistical
power, such designs are burdensome to the patient and increase the likelihood of attrition. It is also difficult to convince individuals to revisit earlier treatments, especially if these are
perceived as less effective [,]. Practically speaking, this means the number of treatment blocks in an N-of-1 study is limited, as is the total duration of the study. Although a statistician might
prefer more shorter blocks relative to few longer blocks (since the number of samples in a traditional N-of-1 analysis is linear in the number of blocks), rapid switching among treatments may obscure
true differences in efficacy because of carryover effects from earlier treatments. Many treatments, such as antidepressants, also take time to display their full effects. Decisions about the length
and arrangement of treatment periods can have a profound effect on statistical effect estimates in N-of-1 studies.
Simulating N-of-1 Studies
Simulation has played a crucial role in clinical trial design, increasing the efficiency and cost-effectiveness of clinical trials, especially in the pharmaceutical industry []. Inspired by this, we
have developed a stochastic time-series simulation model for N-of-1 studies that incorporates all study-relevant effects, such as carryover and wash-in effects. The model can be used to produce
realistic simulated data for a near-infinite number of N-of-1 study designs, treatment profiles, and patient characteristics. The model also incorporates noise parameters like baseline drift,
short-term fluctuations (process noise), and measurement error to provide realistic sources of variation that can obscure treatment effects in real-patient settings. Using simulation, we can cheaply
and easily investigate how design parameters like sampling frequency, number, and location of samples within blocks, treatment order within blocks, treatment period duration, and total number of
blocks impact statistical estimates of treatment effects.
In this paper, we use the model to analyze two N-of-1 case studies, showing how simulation can both optimize study designs and assist researchers in deciding on an appropriate analysis protocol. We
then use the model to produce a set of design recommendations for N-of-1 studies on the basis of parameters related to the study outcome, instrument used to measure the outcome, and treatment(s)
themselves. We provide our simulation software as a supplement to the paper.
Stochastic Time-Series Model
Assume that there are J total treatments in an N-of-1 study. Let B(t) denote the patient’s true baseline at time t. Let X[j](t) denote the effect of treatment j (j=1, …, J) at time t so that the
total treatment effect at time t is X= Σ[j]X[j](t). Let T[j](t) be 1 if treatment j is in process at time t and 0 otherwise (see ). Let Z(t) denote the patient’s true outcome state at time t, and let
Y(t) denote the patient’s observed outcome at time t.
The underlying effect driver for each treatment is described as an ordinary differential equation:
dX[j] = [((E[j] – X[j]) / τ[j]) T[j](t) – (X[j]/ γ[j]) (1 – T[j](t))] dt
Here each X[j](t) is an exponential decay toward a target value that changes over time—either E[j] or 0, depending on T[j](t) —with time constant τ[j] during run-in (decay toward E[j]) and γ[j]
during wash-out (decay toward 0).
Baseline drift is simulated as a discretized Wiener process, where normal noise with variance σ[b]^2Δ t is applied every Δ t:
B(t + Δt) = B(t) + ΔB(t)
ΔB(t) ~ Normal(0, σ[b]^2 Δt)
The outcome variable Z(t) is also a discrete-time stochastic process,
Z(t + Δt) = Z(t) + ΔZ[det](t) + ΔZ[stoch](t)
where Δ Z[det](t) is a deterministic exponential decay toward the target X[j](t)+B(t):
ΔZ[det](t) = Q(t) + [Z(t) – Q(t)] exp(-Δt/∝)
Q(t) = B(t) + Σ[j]X[j](t)
with time constant ∝ and
ΔZ[stoch](t) ~ Normal(0, σ[p]^2Δt)
The observed outcome differs from the true outcome only through the addition of normally distributed observation noise:
Y(t) ~ Normal(Z(t), σ[o])
All of the model parameters are summarized in . Transformations of Y(t) can be used to model different types of outcome parameters, such as scores, counts, and binary outcomes ().
Table 1. The parameters underlying data generation for an N-of-1 study. The parameters are divided into study design parameters (D), treatment-related parameters (T), measurement parameters (M), and
outcome-related parameters (O).
│ Parameter │ Type │ Description │
│ {t[1],…,t[n]} │ D │ Sampling times │
│ N │ D │ Number of blocks (each with J periods in random order) │
│ J │ D │ Number of treatment periods per block │
│ P │ D │ Treatment period length │
│ E[1],…,E[J] │ T │ Effect sizes for treatments 1 through J │
│ τ[1],…, τ[J] │ T │ Run-in time constants for treatments 1 through J │
│ γ[1],…, γ[J] │ T │ Wash-out time constants for treatments 1 through J │
│ ∝ │ O │ Sensitivity to treatment effect │
│ σ[b]^2 │ O │ Variance of baseline drift process │
│ σ[p]^2 │ O │ Variance of process noise │
│ σ[o]^2 │ M │ Variance of observation noise │
Table 2. Suggested transformations of Y for simulating discrete outcomes.
│ Outcome type │ Range of outcome │ Distribution of Y │ Transformation │
│ Numeric │ Real numbers │ —^a │ Identity │
│ Score │ [0,…,M] │ — │ Identity (round, truncate) │
│ Count │ [0,…,infinity) │ Poisson(λ) │ λ = exp(Y) │
│ Proportion │ [0,…,M] │ Binomial(M, p) │ P=1/(1 + exp(-Y)) │
│ Binary │ {0, 1} │ Bernoulli(p) │ P=1/(1 + exp(-Y)) │
^aNot applicable.
Hypertension Case Study
A sample data set and all parameter values for the hypertension case study can be found in . The study involves 2 different blood pressure medications, one of which reduces systolic blood pressure by
10 more points than the other in the long run. The more effective medication, treatment 1, takes longer to reach its full effect (τ[1]=6.0, τ[2]=2.0) and less time to wash out (γ[1]=3.0, γ[2]=10.0).
The sampling rate is 1 sample/day, which we chose to model blood pressure that is monitored using a cuff.
We chose a statistical model for this study that incorporated fixed effects for both block ID and treatment, on the basis of the recommendations provided by the Agency for Healthcare Research and
Quality (AHRQ) and others [,]:
y=β[0] + β[1] x[1] + β[2] x[2] + … + β[N] x[N]
where x[1] is 1 if treatment 2 is in progress at the time of the sample, and 0 otherwise, and x[n] is 1 if block n is in progress, and 0 otherwise. Note that there are only n−1 indicator variables
for blocks; block 1 is used as the reference block. We experimented with other models but found that although modeling choices could affect power, effect size estimates did not change much among
models. Our software provides the ability to choose from among several different models.
To create , we repeated the data generation and analysis process, varying the following parameters and keeping the rest constant:
1. Treatment period orderings were varied among 1 2 1 2, 1 2 2 1, 2 1 1 2, and 2 1 2 1.
2. Sampling frequency was varied from 1 sample per day to 1 sample per treatment period, holding the treatment period ordering fixed at 2 1 2 1.
3. Upon holding sampling frequency constant at 1 sample per day, period length was varied from 2 to 120 days.
4. Study length was held constant at 120 days, and the number of blocks was varied from 1 to 6.
Figure 2. Variation in effect estimates for the hypertension study by study design parameters, including (a) treatment period ordering, (b) sampling frequency, (c) treatment period length, and (d)
number of blocks for a fixed study length. The true effect size is 10, illustrated by the dashed lines in the figures. The red diamonds correspond to the median effect size for the statistically
significant results within each group. Power estimates were obtained by calculating the ratio of the number of colored dots to the number of total dots. There are 50 trials shown for each parameter
View this figure
Pain Management Case Study
The trial design used in this case study emulated the design described in a study by Wegman et al []. Although we did not have access to the raw data for this trial and had to estimate reasonable
noise parameters and wash-in/wash-out time constants, our goal was simply to compare the analysis technique from the paper with a more traditional approach involving a regression model with fixed
effects for treatment and blocks []. The regression model we chose was the same as for the first case study.
The parameters we chose for this model can be found in . We based our decisions about the wash-in and wash-out parameters (τ and γ) on the fact that the authors chose a wash-out period of 1 week for
the different treatments and the fact that both nonsteroidal anti-inflammatory drugs (NSAIDs) and paracetamol are short-acting drugs. We converted the numeric value of the patient state to a discrete
score by rounding and truncating it as shown in .
Figure 3. Analyzing a published N-of-1 study comparing NSAIDs to paracetamol. (top) An example simulation in which the true diary score on the NSAID is 2 and on paracetamol is 4. The black line shows
the simulated mean outcome (unobserved) at each timepoint, and the colored bars show the observed data, which are discrete scores between 0 and 6. (bottom) A comparison of median differencing, the
analysis method described in the paper, with a standard regression model. At the noise levels and effect sizes shown in (top), median differencing will recommend an NSAID only about 60% of the time
(black rectangle), whereas a regression model will recommend it 100% of the time. Model parameters: tau1=tau2=1.0 day, gamma1=gamma2=3.5 days, alpha=1.0, sigma_b=0.0 (no baseline drift), sigma_p=0.5,
sigma_o=1.0. NSAID: nonsteroidal antiinflammatory drug.
View this figure
Simulations for Design Recommendations
All of the simulations in use a baseline of 0 and time constants (τ[1], τ[2], γ[1], and γ[2]) of 0.01. Since treatment 1 is assumed to be placebo, its effect size, E[1], is 0. We used a high value
for the “sensitivity to treatment effect” parameter (α=10) to produce a near-instantaneous effect. The first and second experiments in used only a single block, as in the absence of any sources of
noise except observation noise, block design does not matter. The rest of the parameter choices are outlined in the figure. Each dot represents an average of 50 trials. The smoothed lines shown in
are LOESS (LOcally-Estimated Scatterplot Smoothing) fits produced using geom_smooth with default parameters in ggplot, with spans of 0.4, 0.3, 1.0, 1.0, and 1.0 for subfigures a, b, c, d, and e,
Data and Code Availability
The simulation software is available in the n1-simulator repository under the HD2i organization on GitHub. Full details of the available experiments and associated plots are included with the
software, along with the data sets generated in the course of making the figures.
Figure 4. Examining the effect of study design choices on power and accuracy of effect size estimates for an N-of-1 study with effectively instantaneous transitions between treatment states. (a)
Effect size vs power for fixed observation noise (sigma_0=1.0) and no process noise or baseline drift. (b) Average deviation of estimate from true value vs. effect size for fixed observation noise
(sigma_0=1.0) and no process noise or baseline drift. (c) Minimum treatment period length (ie. number of samples per treatment, with sampling rate fixed at 1 sample per time unit) required to attain
a power of 0.8, for varying degrees of process noise and varying effect sizes. No observation noise or baseline drift is present. (d) Same as (c) except effect size is fixed at 1.0 and alpha
(individual treatment response) is varied. (e) Average deviation of effect size estimate from its true value, as a function of baseline drift and number of blocks. The effect of baseline drift on the
estimate is much more pronounced when fewer blocks are used. Editorial Notice: in (a) and (b), x-axis labels should correctly read “Number of samples per treatment.”
View this figure
Modeling the Key Features of an N-of-1 Study
The complete set of parameters for our model can be found in . The basic model comprises an underlying deterministic process (the growth and decay of treatment effects over time) in addition to 3
types of noise: random baseline drift (eg, long-term illness onset and recovery processes, gaining/losing weight, long-term changes in blood pressure), process noise, which manifests as short-term
fluctuations (eg, heart rate and blood pressure volatility, periods of activity/inactivity, and changes in sleep and diet from day to day), and observation noise, which is a function of the
instrument and is not related to any underlying biological effect (eg, the measurement noise associated with the cuff that is used to monitor blood pressure).
We divided the parameters into 4 groups: study design parameters, which the study designer can vary, treatment parameters, which are immutable features of the particular treatments under
consideration, a measurement parameter, which is a feature of the device used to measure the outcome, and outcome parameters, which are features of the underlying biological process under
consideration and may vary from individual to individual. A diagram of an N-of-1 block design and our model of how treatment effects vary over time is shown in .
Case Study: Optimizing Study Design
Simulation allows us to investigate the impact of subtle design choices on the likelihood of study success. To illustrate this, we simulated a study of 2 different blood pressure medications and
their impact on systolic blood pressure, similar to the data shown in [] (see the Methods section for details). The study parameters, underlying (unobserved) data, and observed data are shown in .
The results of several hundred simulations of this study are shown in . We used one of the standard N-of-1 regression models outlined in [] and [] to estimate treatment effect and obtain an
associated P value.
In , we see that the ordering of treatment periods has a strong effect on both statistical power and effect size estimates. On the basis of these 50 simulations, when treatments are administered in
the order 1 2 1 2, power (at a standard 5% significance level) is 0.62, for 1 2 2 1 it is 0.82, for 2 1 1 2 it is 1.00, and for 2 1 2 1 it is 0.98. The median effect size estimate is also impacted by
treatment ordering: for 1 2 1 2 it is 5.8, for 1 2 2 1 it is 6.6, for 2 1 1 2 it is 11.2, and for 2 1 2 1 it is 12.0. The true effect size is 10.0. We observe lower power and diminished effect size
estimates for treatment orderings 1 2 1 2 and 1 2 2 1 relative to 2 1 1 2 and 2 1 2 1 as Treatment 1 takes longer to reach its full effect than Treatment 2, and the patient starts at a relatively
high baseline (systolic blood pressure=160); therefore, when it is administered first, Treatment 1 never attains its full effect during the first treatment period before the transition to Treatment 2
takes place.
In , we see the effect of sampling frequency on study power. Increasing the sampling frequency causes power to increase but only to a point. On the basis of these 50 simulations, when only 1 sample
is taken at the end of each treatment period (sampling interval of 30 days), which is the most common approach to analyzing N-of-1 studies [,], power is only 0.14. Sampling every day (sampling
interval of 1 day) yields a power of 0.84; sampling every 2 days yields a power of 0.74, every 5 days yields a power of 0.76, every 10 days yields a power of 0.56, and every 15 days yields a power of
0.50. On the basis of these results, it appears that sampling every 2 or 5 days could substantially reduce patient burden while causing only a modest reduction in power.
shows the effect of treatment period length, keeping the total number of blocks fixed at 2 and the sampling rate fixed at 1 sample per day. On the basis of these 50 simulations, when the treatment
period length is 2 days, power is 0.18 and the mean effect size estimate is –1.5. For a period length of 5 days, power is 0.54 and the mean effect size is 3.1. For a period length of 15 days, power
is 0.44 and the mean effect size is 9.7. For a period length of 30 days, power is 0.94 and the mean effect size is 10.2. For period lengths of 40, 60, and 120 days, power and mean effect sizes are
0.92 and 8.3, 0.98 and 9.7, and 0.96 and 10.6, respectively. This indicates that for a period length of 30 days, one obtains approximately as accurate an effect estimate as a period length of 60 days
while shrinking the total study duration from 240 to 120 days. Period lengths that are too long run the risk of higher variance in estimates because of baseline drift, as we see with a period length
of 120 days in .
Finally, shows the effect of different block designs for a study of fixed length (120 days). On the basis of these 50 simulations, power for 1, 2, 3, 4, 5, and 6 blocks is 0.74, 0.86, 0.78, 0.84,
0.74, and 0.60, respectively. Mean and standard deviation of the effect size estimates are 9.7 (5.8), 9.8 (3.8), 8.7 (3.6), 8.3 (2.9), 7.0 (2.5), and 6.6 (1.8), respectively. Using 2-4 blocks appears
to be the best approach, as this reduces variance in the effect size estimate relative to a single-block study. Adding more than 4 blocks increases the impact of wash-in/carryover effects on the
estimate, which deviates further from its true value of 10 with each additional block.
Case Study: Evaluating Analysis Protocols
Simulation can also help us evaluate the likely success of new analysis protocols and decision criteria for N-of-1 studies. We simulated a previously published study [] in which the outcome was a
“diary score” on a scale of 0 to 6, with 0 representing “no complaints at all” and 6 representing “unbearable complaints.” The study design used 5 blocks, each with 2 treatment periods; only data
from the last week of each treatment period were analyzed.
In this paper, the data were analyzed as follows: the researchers took differences in median diary scores between NSAID and paracetamol treatment periods in each block and then calculated the number
of treatment blocks for which the NSAID score was at least one point lower than the paracetamol score for the patient’s main complaint. An NSAID was recommended if this was true in at least 4 out of
5 blocks. We refer to this method as median differencing from now on.
We compared median differencing to the same regression model used in the previous section []. Simulations show that median differencing is much more conservative in recommending an NSAID than a
standard regression model trained on the same data (). For a true effect difference of size 2 (NSAID reduces pain by 2 points relative to paracetamol), median differencing will only recommend an
NSAID, on average, 61% of the time, compared with 100% of the time for the regression model. In addition, median differencing will recommend an NSAID more frequently in cases where the diary score on
paracetamol is already low (the patient is not in much pain); when the score is high, it becomes harder for it to detect an effect. For a patient with a paracetamol diary score of 6 (the maximum
possible pain), if the NSAID reduces the diary score to 4, median differencing will only recommend an NSAID 30% of the time, as opposed to 100% of the time for the regression model. The difference
between the models is even more pronounced when the NSAID only reduces the pain score by 1; in that case, median differencing will only recommend an NSAID, on average, 7% of the time, as opposed to
92% of the time for the regression model.
Design Considerations for N-of-1 Studies
shows the results of a set of simulations on the basis of best-case scenarios — no variation in parameters other than those under investigation, as well as instantaneous treatment effects (ie, no
carryover effects). The technical details of the simulations can be found in the Methods section. All of the graphs in relate the study design parameters to (1) statistical power—the ability to
detect a treatment effect difference if it exists, and (2) the accuracy of the effect size estimates produced by the model. All compare a single treatment against placebo.
In a and 4b, observation noise (σ[o]) is fixed at 1.0, with no process noise or baseline drift. As a result, “effect size really describes a signal-to-noise ratio and is treatment and instrument
agnostic." We observe that this ratio impacts power but not the accuracy of the effect estimate ().
In a, we see that for effect sizes of 0.1, 0.2, and 0.3, more than 100 samples per treatment are needed to obtain a power of 0.8 (at a standard 5% significance level). For an effect size of 0.4, at
least 100 samples per treatment are needed. For effect sizes of 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0, the numbers of samples per treatment needed to attain a power of 0.8 are approximately 65, 45, 35,
26, 21, and 18, respectively. Even more samples will be needed under real experimental conditions where process noise, baseline drift, and carryover effects all play a role. This indicates that
unless the effect size is very high relative to the observation noise, N-of-1 studies using only a few blocks, with a single sample taken per block (the traditional approach to analyzing N-of-1
studies), will be vastly underpowered.
A separate consideration is the error in the effect size estimate, which declines monotonically with the number of samples. In b, we see that to obtain an estimate within 0.2 σ[o] of the true
estimate, at least 30 samples per treatment are needed; to reach 0.1 σ[o], over 100 samples per treatment are needed.
c shows the impact of process noise on the number of samples needed to attain a power of ≥0.8 at a 5% significance level in the absence of observation noise and baseline drift. In this figure, the
intersample interval is fixed at 1 sample/time unit and the process noise is defined relative to that; σ[p]=1.0 indicates that if no treatment effect were present, the variance of the Wiener process
underlying the process noise would be 1 outcome unit/time unit. For an effect size of 0.5 and σ[p]=0.0, 0.4, 0.8, 1.2, 1.6, 2.0, the numbers of samples per treatment needed to obtain a power of 0.8
are 61, 76, 89, 111, 135, and 176, respectively. For an effect size of 1.0, the numbers of samples per treatment needed are 20, 24, 28, 34, 43, and 53, respectively. Regardless of effect size,
increasing the process noise from 1.0 to 2.0 roughly doubles the number of samples it takes to attain a power of 0.8. However, the effect is nonlinear; below σ[p]≈1.0, the number of samples needed
flattens out in the absence of other sources of noise.
In d, we see the impact on study outcome of individual sensitivity to treatment. The lower the value of the treatment sensitivity parameter (α) is, the less effect changes in treatment have on the
outcome relative to random fluctuations caused by process noise. We see this when we contrast the effect of increased process noise on the minimum samples required to attain a power of 0.8 at a
significance level of 5% under conditions of low treatment sensitivity (α=0.1) and high treatment sensitivity (α=10.0). For σ[p]=0.0, 0.4, 0.8, 1.2, 1.6, 2.0 and α=0.1, the numbers of samples per
treatment required are 36, 64, 110, 174, 228, and 250, respectively. For α=10.0, the numbers of samples required are only 20, 23, 28, 34, 42, and 53, respectively.
Finally, e shows us why we bother to have blocks at all: to guard against baseline drift. The figure shows what happens in a study of a total length of 240 days when block designs incorporating 1, 2,
3, or 4 blocks are used. As baseline drift increases (holding process and observation noise constant at σ[p]= σ[o]=0.0), the effect size estimate provided by the model increasingly deviates from its
true value. This effect is most pronounced in studies with only a single block and decreases as the number of blocks increases. For example, for only 1 block, with σ[b]=0.00, 0.09, 0.18, 0.27, 0.37,
and 0.46, the average deviation of the effect size estimate from the true value is 0.21, 0.33, 0.54, 0.77, 1.01, and 1.26, respectively. However, with 4 blocks, with the same progression of σ[b]
values, the average deviation of the effect size estimate is 0.21, 0.22, 0.25, 0.28, 0.32, and 0.37, respectively.
Summary of the Paper
We have developed a stochastic time-series model that simulates an N-of-1 study, facilitating rapid optimization of N-of-1 study designs and increasing the likelihood of study success while
minimizing participant burden. We have used this model to evaluate 2 case studies, showing how the number of treatment blocks, ordering of treatments within blocks, duration of each treatment,
sampling frequency, and study analysis protocol affect our ability to detect true differences in treatment efficacy. Our simulation software is available on GitHub as described in the Methods
Recommendations for the Design of N-of-1 Studies
An N-of-1 study should have as many blocks as possible to avoid baseline drift (). If no wash-in or carryover effects are present, a single sample should be taken at the end of each of JN different
treatment periods, where N is the number of blocks and J the number of treatments; N should be made as high as possible; each block should be made as short as possible. However, in practice, the
number of blocks we can use in a study is bounded by the dangers of administering different treatments in rapid succession, the time it takes treatments to ramp up to their full effects (“run-in”: ),
the time it takes them to stop working when they are discontinued (“wash-out”: ), and participant patience.
It is important to consider the fact that most N-of-1 studies of reasonable length and reasonable sampling frequency will be underpowered unless the difference in treatment effects is at least on the
order of the standard deviation of the observation noise (). The goal, perhaps obvious, should be to measure the outcome with as little noise as possible and at as high a frequency as possible, and/
or to continue the study until enough samples are obtained to ensure that the effect will be detected if it is there.
Finally, it is important to remember the difference between power and accuracy. Just because a statistically significant difference in treatment effects is detected, it does not mean that the
quantitative estimate of E[2]−E[1] reported by the model is accurate. Even when a study is sufficiently powered, the effect size estimate will almost always improve with the addition of more samples.
Beyond these general statements, our main recommendation for N-of-1 study designers is to simulate the study. We can see from c and d that process noise and individual sensitivity to treatment can
have a dramatic impact on the number of samples needed to adequately power a study, especially if the effect size is small. The choice of analysis method can also have a substantial impact on study
outcome and treatment recommendations (); therefore, it is important to compare novel analysis methods to the standard models provided by the AHRQ and others [,]. Simulations can help in both cases.
Modeling Different Outcome Types
Most of our analyses in this paper concerned a continuous (or near-continuous) random variable, such as blood pressure or heart rate. However, many N-of-1 trials examine outcomes that are better
modeled as counts, proportions, binary random variables (yes/no), or discrete bounded scores (such as surveys). Studies with these outcome types can be simulated by transforming the output of the
stochastic differential equation model using a set of transformations similar to those for generalized linear models (see ). We used one such transformation to discretize the scores for the pain
management case study.
Sources of Treatment and Instrument Parameters
By far, the strongest drawback to the simulation approach is the difficulty associated with identifying reasonable simulation parameters, especially in cases where the outcome is not a continuous
value (see ).
Some parameters have relatively clear interpretations and can be found by looking at the known characteristics of treatments and instruments. For example, in the case of a continuous-valued outcome,
we can think of the treatment effect, X[j](t), as the treatment’s maximum impact—at each point in time—on the outcome in the absence of any noise, in a population of people exactly like the one who
is undergoing the study. The treatment effect is governed by 3 parameters: τ[j], the time constant of “wash-in” for that treatment, γ[j], the time constant of “wash-out”, and E[j], the asymptotic
effect size (the change from baseline that the person would experience in the long run was he/she to continue on this treatment). In the case of a pharmaceutical intervention, these are important
parameters that have probably been estimated in earlier clinical trials and used to guide dosages, dosing frequencies, etc. Similarly, reasonable values for σ[o] can often be obtained from technical
specifications of whatever instrument is used to monitor the outcome.
The emerging field of mobile health may provide some help in estimating parameters like σ[p] and σ[b], which are properties of an outcome and its natural variation over time []. As we begin to
monitor patients longitudinally with increasingly higher resolution, our quantitative understanding of long- and short-term variation in biological processes will naturally increase. However, in
simulations at present, we recommend experimenting with varying parameter scales and examining raw plots of the data to see if the level of noise produced by the model is reasonable. It may also make
sense to test ranges of α, σ[b], and σ[p] and examine plots like those shown in to assess the effect of these parameter choices on statistical models.
Study Limitations and Future Work
This study fits simulated data with a simple regression model recommended by the AHRQ, but the data themselves are simulated using a more realistic model. A natural next step would be to use the full
simulation model as the basis for fitting data. Future versions of our software will allow users to fit data using the AHRQ model and the full time-series model in a Bayesian framework, which infers
the model parameters using posterior probability distributions given the data rather than point estimates [,]. Thus, uncertainty is an inherent part of the model. This will provide a basis for
directly comparing the performance of the full time-series model against the simple AHRQ model for making treatment recommendations. In addition, posterior parameter distributions inferred from real
data can be used to generate more realistic simulated data. This will be especially useful for studies with discrete outcomes, where the linkage between model parameters and outcome data is more
difficult to interpret. Another advantage of a Bayesian parameter estimation approach is that it allows parameter estimates for N-of-1 studies to be continually updated as more individuals undergo
the same study, creating a system that learns from past data to adapt the design of future studies.
One important limitation of our model is that although it incorporates multiple sources of noise, it ignores more structured sources of outcome variation (eg, variation in heart rate does not
principally happen stochastically with time, but the heart rate does show structured change across hours, days, and ovulatory cycles). It is also possible that long-term seasonal, day of week, and
time of day effects can influence the outcome of N-of-1 studies. Future versions of our model may incorporate parameters for these effects and fit them using methods akin to those of Prophet [] or
other Bayesian time-series models. In the meantime, users can address these issues by manually adding known sources of variation to the baseline drift term or by choosing outcome parameters that
“average out” known sources of variation (eg “heart rate daily mean”).
In general, the development of realistic simulations of N-of-1 studies is an ongoing process. We believe that simulation will prove crucial as N-of-1 studies enter mainstream clinical practice,
especially in the realm of precision medicine, and we hope that our model will inspire others to adopt N-of-1 studies as a tool in their own research.
Authors' Contributions
BP and NZ jointly conceived of the idea for an N-of-1 simulation model. BP and EBB designed the model, wrote the model code, and conducted the experiments for the paper. EBB translated the code into
R and created the documentation and user-friendly interface. BP drafted the manuscript. MJ, JTD, and NZ provided extensive feedback on the manuscript and model design.
Conflicts of Interest
None declared.
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10. Nikles J, Mitchell GK, Schluter P, Good P, Hardy J, Rowett D, et al. Aggregating single patient (n-of-1) trials in populations where recruitment and retention was difficult: the case of
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11. Mengersen K, McGree J. Statistical analysis of N-of-1 trials. In: The Essential Guide to N-of-1 Trials in Health. Dordrecht: Springer; 2015:135-153.
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13. Wegman A, van der Windt DA, de Haan M, Devillé WL, Fo CT, de Vries TP. Switching from NSAIDs to paracetamol: a series of n of 1 trials for individual patients with osteoarthritis. Ann Rheum Dis
2003 Dec;62(12):1156-1161 [FREE Full text] [CrossRef] [Medline]
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AHRQ: Agency for Healthcare Research and Quality
NSAID: nonsteroidal anti-inflammatory drug
LOESS: locally-estimated scatterplot smoothing
Edited by G Eysenbach; submitted 29.10.18; peer-reviewed by B Smarr, L Zhang; comments to author 28.11.18; revised version received 28.12.18; accepted 29.12.18; published 01.04.19
©Bethany Percha, Edward B Baskerville, Matthew Johnson, Joel T Dudley, Noah Zimmerman. Originally published in the Journal of Medical Internet Research (http://www.jmir.org), 01.04.2019.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work, first published in the Journal of Medical Internet Research, is properly cited. The complete bibliographic information, a link to the
original publication on http://www.jmir.org/, as well as this copyright and license information must be included.
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Regular language induction with genetic programming
Created by W.Langdon from gp-bibliography.bib Revision:1.8028
□ author = "Bertrand Daniel Dunay and Frederick E. Petry and Bill P Buckles",
□ title = "Regular language induction with genetic programming",
□ booktitle = "Proceedings of the 1994 IEEE World Congress on Computational Intelligence",
□ year = "1994",
□ pages = "396--400",
□ volume = "1",
□ address = "Orlando, Florida, USA",
□ month = "27-29 " # jun,
□ publisher = "IEEE Press",
□ keywords = "genetic algorithms, genetic programming S-expressions, computational difficulties, deterministic finite automata, editing, formal language accepters, inductive inference,
informant, population pressure, reachable states, regular language induction, renumbering, run-time determined solution size, sample strings, transition tables, translation, deterministic
automata, finite automata, formal languages, inference mechanisms",
□ size = "5 pages",
□ abstract = "In this research, inductive inference is done with an informant on the class of regular languages. The approach is to evolve formal language accepters which are consistent with a
set of sample strings from the language, and a set of sample strings known not to be in the language. Deterministic finite automata (DFA) were chosen as the formal language accepters to
alleviate the computational difficulties of nondeterministic constructs such as rewrite grammars. Genetic programming (GP) offers two significant improvements for regular language induction
over genetic algorithms. First, GP allows the size of the solution (the DFA) to be determined at run time in response to population pressure. Second, GP's potential for assuring correct
dependencies in complex individuals can be exploited to assure that all states in a DFA are reachable from the start state. The contribution of this research is the effective translation of
DFAs to S-expressions, the application of renumbering, and of editing to the problem of language induction. DFAs or transition tables form the basis of many problems. By using the techniques
found in this paper, many of these problems can be directly translated into the domain of genetic programming",
□ notes = "Considers two classes of regular language (NB series and Tomita) which can be recognised or accpeted by deterministic finite automata (Finite state machines). Can translate from DFA
to tree structure. Trees are not executable programs but represent languages. crossover on trees defined. GP able to define a language given examples of it. Works on simplier examples but has
difficulties with 8b, 9b, 10b and TL5.",
Genetic Programming entries for Bertrand Daniel Dunay Frederic E Petry Bill Buckles
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Does UIF Pay The Same Every Month? (2024)
Knowing how much one can get from UIF every month will go a long way to help one plan his finances so as not to get stranded, or run out of money before other streams of income become available.
This information will enable one to prioritize his spending according to a scale of preference, putting things like food and housing first, and then gradually spending funds on other things which may
be considered less important.
It is important to remember that this scheme is designed to provide temporary relief; that means, no matter how well one tries to manage the money, it will eventually run out. Nevertheless, this
scheme has helped thousands of families.
Does UIF Pay The Same Every Month?
The short answer is yes; UIF pays the same amount to a person every month, but no UIF does not pay the same amount to everybody. What this means is that the amount to be paid to every individual
varies according to what he was earning before.
That is in line with the very idea of what the UIF is; something to fall back on, as per the foresight of the South African Government. Let us explain exactly what it is, and how it works; so that
one can calculate how much he should expect from the UIF every month.
What Exactly Is UIF?
UIF means Unemployment Insurance Fund, and it is a fund created by the government to provide financial cover for workers in the formal sector in the event that they are out of work, or unable to earn
money from their work perhaps due to illness or maternity.
One cannot just walk into the office of the UIF and demand for money because he has lost his job; it is a contribution that you would have been making during your working days. That contribution
would have to be matched by your employer, and also by the government as a way of keeping something aside for the rainy day.
How Much Will I Get Per Month?
That depends on how much you were earning while you worked. The scheme stipulates that workers are contribute 1% of their monthly earning; while their employer also contributes 1%; making 2% monthly.
Of course 1% can be greatly different; one person may be earning R50,000 per month, while another person could be earning $200,000. Therefore, the amount of money a person is eligible to receive
every month could go up infinitely, depending on how much he makes monthly; right?
Wrong! The ceiling for UIF payments is R6 730 a month. That is the highest a person can get every month. It is a good way of staying true to the tenets of this fund; to provide financial cover for
the most vulnerable. Which means those who earn very high can find other ways of investing their money, and ensuring that they have secured their futures.
The amount of money you get will be calculated by looking at how much you have earned on the average over the last 6 months.
How The UIF Payment Is Calculated
The average amount you earned over the last 6 months is calculated; the minimum is 3,500, while the maximum is R17 702. This means even if you earn 4 times that figure every month, the amount that
will be registered against your account is R17 702.
Unemployment Benefits
Average salary x 12÷365 = daily income (Year 1).
That means the average salary is multiplied by 12 months, and then divided by 365 so as to get the amount you are to be paid daily. (payment is done monthly)
For example, let us say you earned R17712 per month over the last 6 months; which would be calculated at R17712 x 12 / 365 which is R582.31 per day.
But the UIF is not designed to pay you your salary; it is designed to provide some cover; meaning not all. Therefore the Income Replacement Rate calculates to around half of what you were earning
daily while working.
Y1 value is used in the Income Replacement Rate (IRR) formula and the Y1 value stands for your earnings per day.
The maximum earnings through UIF is pegged at 38% of whatever you have been earning for the last 6 months. While that is not exactly half, it is quite close to it.
Based on the previous calculations therefore, let us look at what the true UIF earning can look like. The earnings are calculated daily.
What we are trying to get now is your benefits per day. Remember we have already calculated your earnings per day; which is calculated at R582.31. Remember that this value is represented by Y1. Now
to get the daily benefit amount.
Daily benefit Amount (DBA) = Y1 x 38%.
= R582.31 x 38%. In simple terms, your daily benefit amount is 38% of Y1. 38% of R582.31 is equal to = R221.28 per day. The money will be paid monthly; which gives us about 5753.28 per month.
That is for Unemployment Benefit; there are other benefits that a person can claim; such as illness, maternity, and reduced work time. We briefly cover the amounts to be paid in such instances.
Maternity, Illness, Parental And Adoption UIF Payments
The common denominator to any of the situations listed above is that the person in question has not lost his job, but is temporarily unavailable to perform his tasks at work due to any of the above
listed reasons.
In this case the person should still get some money from his employer as leave income. However, the UIF will still top up what he earns. The calculation is below.
Daily benefit amount (DBA) is calculated at 66% of income; and once again the highest possible earning figure is R17712 per month. Even if you earn 7 times that amount, the highest possible amount
that will be registered against your name is R17712 per month.
However, Daily income (DI) is not limited to the ceiling amount. If a person earns higher, it will be registered as such. Yes, there may be some confusion, but just hold on. Read the lines below, it
becomes clear.
Daily benefit amount (DBA) is different from Daily Income (DI). Daily Benefit is what the UIF pays you, so of course there has to be a limit, otherwise people would not work, they would only sit at
home expecting money.
So this is how it is calculated.
To get the Daily Income; the salary over the last six months is calculated thus. Let’s say you earn R30000 per month.
R30000 x12/365 = R986.30 per day.
Daily income while on leave (leave income).
But your employer may not pay that figure while you are on leave. Your daily income while on leave would be for example; R25000 x12/365 = R821.92 per day.
Top-up (Amount paid by the UIF) would then be difference between daily income (Y1) and leave income.
For example: R986.30 – R821.92 = R164.38 (difference). This would help you continue your normal lifestyle and routine as when you were working.
However, it is important to note that where the difference is less than the daily benefit amount, the difference is paid. However, where the difference is more than the daily benefit amount, the
claim is denied.
UIF Payments For Reduced Work Time
Due to the present economic crunch; some employers are forced to scale back working hours. This means they cannot pay as much. In such a circumstance the UIF can cover the loss to the employee.
Daily benefit amount calculation for Reduced Work Time:
Top-up will be the difference between Reduced Work Time income (per day) and the daily benefit amount (DBA).
DBA= R221,28.
For example; if the Reduced Work Time salary is R3000 per month which is R98,63 per day, the daily difference will be R221,28. – R98,63 which gives us R122.65 Per day. The money will be paid monthly,
which gives us around 3188.9.
This article has answered the question of whether the UIF pays the same amount every month, and even offered very simple explanations about how the payment is calculated- as a guide to help one
calculate his UIF payments. Please keep in mind that these payments are a temporary cushion; they are not meant to be a permanent source of income. It is therefore important to be active in searching
for a job, before the money runs out.
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Algebra 1 Common Core Student Edition, Grade 8-9
Reaso n i n g an d Co m m u n i cat i n g
2. Reason abstractly and quantitatively.
As a strong math thinker and problem solver, you are able to make sense o f quantities in problem
situations. You can both represent a problem situation using symbols or equations and explain w h a t
the symbols or equations represent in relationship to the problem situation. As you represent a situation
symbolically or mathematically, you can explain the meaning o f the quantities.
3. Construct viable argum ents and critique the reasoning o f others.
You are able to communicate clearly and convincingly about your solutions to problems. You can build
sound mathematical arguments, drawing on definitions, assumptions, or established solutions. You
can develop and explore conjectures about m athem atical situations. You make use o f examples and
counterexamples to support your arguments and ju stify your conclusions. You respond clearly and
logically to the positions and conclusions o f your classmates, and are able to compare tw o arguments,
identifying any flaw s in logic or reasoning th a t the argum ents may contain. You can ask useful questions
to clarify or improve the argument of a classmate.
Rep r esen t i n g an d Co n n ect i n g
4. Model w ith mathematics.
As a strong math thinker, you are able to use m athematics to represent a problem situation and can
make connections betw een a real-w orld problem situation and mathematics. You see the applicability
o f mathematics to everyday problems. You can explain h ow geom etry can be used to solve a carpentry
problem or algebra to solve a proportional relationship problem. You can define and map relationships
among quantities in a problem, using appropriate tools to do so. You are able to analyze the
relationships and draw conclusions.
5. Use ap p ro p riate tools strategically.
As you develop models to match a given problem situation, you are able to strategize about w hich
tools w ould be most helpful to use to solve the problem. You consider all tools, from paper and
pencil to protractors and rulers, to calculators and softw are applications. You can articulate the
appropriateness o f d ifferent tools and recognize w hich w ould best serve your needs fo r a given
problem. You are especially insightful about technology tools and use them in ways th a t deepen or
extend your understanding o f concepts. You also make use o f mental tools, such as estim ation, to
determine the reasonableness of a solution.
Se e i n g St r u c t u r e a n d Ge n e r a l i z i n g
7. Look fo r and m ake use o f structure.
You are able to go beyond simply solving problems, to see the structure of the m athematics in
these problems, and to generalize m athem atical principles from this structure. You are able to see
complicated expressions or equations as single objects, or a being composed of many parts.
8. Look fo r and express reg u larity in repeated reasoning.
You notice w hen calculations are repeated and can uncover both general methods and shortcuts for
solving similar problems. You continually evaluate the reasonableness o f your solutions as you solve
problems arising in daily life.
Using Your Book f or Success xix
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A ball thrown straight up into the air with a speed of 10m/s. If the ball has a mass of 0.3 Kg, how high does the ball go? Acceleration due - DocumenTVA ball thrown straight up into the air with a speed of 10m/s. If the ball has a mass of 0.3 Kg, how high does the ball go? Acceleration due
A ball thrown straight up into the air with a speed of 10m/s. If the ball has a mass of 0.3 Kg, how high does the ball go? Acceleration due
A ball thrown straight up into the air with a speed of 10m/s. If the ball has a mass of 0.3 Kg, how high does the ball go? Acceleration due to gravity is g=9.8 m/s2
in progress 0
Physics 3 years 2021-08-17T15:28:52+00:00 2021-08-17T15:28:52+00:00 1 Answers 45 views 0
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Distribution of Differences in Sample Proportions (2 of 5)
Learning Objectives
• Draw conclusions about a difference in population proportions from a simulation.
Recall that we are in the middle of an investigation about the difference in female and male teen depression rates. In our investigation, we are assuming that 26% of female teens and 10% of male
teens are depressed. That is, we assume a 16% = 0.16 difference favoring girls.
• We saw a 0.06 gender difference in teen depression rates from the National Survey of Adolescents. Again, girls had a higher rate of depression. Does this study suggest that our assumption about a
0.16 difference in the populations is wrong?
• Or could the results have come from populations with a 0.16 difference in depression rates?
At this point, we may have a sense of the answers to these questions for samples of 64 females and 100 males. But we need to look at the long-run behavior of the differences in sample proportions. We
also need to investigate the effect of sample size on our conclusion. The samples in the National Survey of Adolescents are very large.
So we continue this investigation in a Simulation WalkThrough.
On the next page, we use the simulation shown in the WalkThrough to make inferences about a difference in population proportions. As we did in Linking Probability to Statistical Inference, we use a
simulation to make observations about the sampling distribution before we develop the mathematical model that we will use in inference. The logic we use to make inferences with simulated sampling
distributions is the same logic we use with mathematical models. Let’s practice that way of thinking now.
Learn By Doing
Suppose in a study of 540 female and 475 male U.S. teens, we find that 8% of the females and 2% of the males are depressed. What does this study suggest about our assumption that the depression rate
of female teens is 16% higher than that of male teens in the United States?
Here is a simulated distribution of differences for a large number of independent random samples for these sample sizes. Note that we have rescaled the axis, so the distribution may look wider than
the distributions in the WalkThrough, but it actually has less variability.
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Finding certain text and then summing annual numbers
Sep 4, 2014
I work in commercial real estate and use Argus and Excel to underwrite properties. Argus is real estate software that outputs annual/monthly/etc proforma with varying detail depending on the
What I'd like to do:
- Use a sheet to annualize the data outputted by a monthly Argus Model. The issue I'm running into is that depending on the property, different expense line items with be within the output and the
locations change per output.
I put in reference numbers that auto find the right name/row combination but haven't been able to come up with a formula that works.
Here is where I paste the monthly Argus output. The periods on the top extend for 15 years. The line items on the left change per property.
Here is the sheet where I try to annualize the Argus data. Please disregard the residential section at the top as it doesn't come from Argus. The numbers to the left of the Argus line items are the
rows from the previous sheet. I figured I could use these numbers as reference points in a formula. Also, please disregard the data to the right of the line items as I manually linked the data.
Sometimes the types of expenses vary per building so I'd like to be able to paste any Argus output into the previous sheet, manually add the line items names in this sheet and have it auto sum for
the 15 year timeline on this sheet.
Thanks so much. If you have any followup questions, I'll be here.
Excel Facts
Pivot Table Drill Down
Double-click any number in a pivot table to create a new report showing all detail rows that make up that number
Sep 4, 2014
Edited to include Mr. Excel's HTML maker
I work in commercial real estate and use Argus and Excel to underwrite properties. Argus is real estate software that outputs annual/monthly/etc proforma with varying detail depending on the
property. What I'd like to do: - Use a sheet to annualize the data outputted by a monthly Argus Model. The issue I'm running into is that depending on the property, different expense line items with
be within the output and the locations change per output. I put in reference numbers that auto find the right name/row combination but haven't been able to come up with a formula that works. Here is
where I paste the monthly Argus output. The periods on the top extend for 15 years. The line items on the left change per property.Monthly Argus Output
A B C D E
6 Potential Gross Revenue
7 Base Rental Revenue
8 Absorption & Turnover Vacancy
9 Base Rent Abatements
11 Scheduled Base Rental Revenue
13 Expense Reimbursement Revenue
14 Real Estate Taxes
15 Operating Expenses
16 Electric
18 Total Reimbursement Revenue
[TD="align: center"]1[/TD]
[TD="align: center"][/TD]
[TD="align: center"]Year 1[/TD]
[TD="align: center"]Year 1[/TD]
[TD="align: center"]Year 1[/TD]
[TD="align: center"]2[/TD]
[TD="align: center"][/TD]
[TD="align: center"]Month 1[/TD]
[TD="align: center"]Month 2[/TD]
[TD="align: center"]Month 3[/TD]
[TD="align: center"]3[/TD]
[TD="align: center"][/TD]
[TD="align: center"]Jan-2015[/TD]
[TD="align: center"]Feb-2015[/TD]
[TD="align: center"]Mar-2015[/TD]
[TD="align: center"]4[/TD]
[TD="align: right"][/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: center"]5[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]6[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]7[/TD]
[TD="align: right"]$247,931[/TD]
[TD="align: right"]$247,933[/TD]
[TD="align: right"]$247,926[/TD]
[TD="align: center"]8[/TD]
[TD="align: right"](153,207)[/TD]
[TD="align: right"](153,209)[/TD]
[TD="align: right"](153,204)[/TD]
[TD="align: center"]9[/TD]
[TD="align: right"](21,887)[/TD]
[TD="align: right"](21,887)[/TD]
[TD="align: right"](21,886)[/TD]
[TD="align: center"]10[/TD]
[TD="align: right"][/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: center"]11[/TD]
[TD="align: right"]72,837[/TD]
[TD="align: right"]72,837[/TD]
[TD="align: right"]72,836[/TD]
[TD="align: center"]12[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]13[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]14[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"]164[/TD]
[TD="align: center"]15[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]16[/TD]
[TD="align: right"]2,189[/TD]
[TD="align: right"]2,188[/TD]
[TD="align: right"]2,254[/TD]
[TD="align: center"]17[/TD]
[TD="align: right"][/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: right"] ___________[/TD]
[TD="align: center"]18[/TD]
[TD="align: right"]2,189[/TD]
[TD="align: right"]2,188[/TD]
[TD="align: right"]2,418[/TD]
Here is the sheet where I try to annualize the Argus data. Please disregard the residential section at the top as it doesn't come from Argus. The numbers to the left of the Argus line items are the
rows from the previous sheet. I figured I could use these numbers as reference points in a formula. Also, please disregard the data to the right of the line items as I manually linked the data.
Sometimes the types of expenses vary per building so I'd like to be able to paste any Argus output into the previous sheet, manually add the line items names in this sheet and have it auto sum for
the 15 year timeline on this sheet.
Annual Profroma
A B C D E F
For the Years Ending
Gross Residential Income
General Vacancy $ - $ - $ -
Get Ready Cost $ - $ - $ -
Residential Effective Gross Income $ - $ - $ -
Potential Gross Revenue
Base Rental Revenue $ - $ 3,067,050 $ 3,151,183
Absorption & Turnover Vacancy $ (1,452,379) $ (405,781) $ -
Base Rent Abatements $ (264,592) $ (272,549) $ -
Scheduled Base Rental Revenue $ 1,274,128 $ 2,388,720 $ 3,151,183
Expense Reimbursement Revenue
Real Estate Taxes $ 1,664 $ 7,008 $ 15,835
Operating Expenses $ 30 $ - $ -
Electric $ 47,240 $ 104,087 $ 128,529
Total Reimbursement Revenue $ 48,880 $ 111,095 $ 144,364
[TD="align: center"]1[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"][/TD]
[TD="align: right"]Year 1[/TD]
[TD="align: right"]Year 2[/TD]
[TD="align: right"]Year 3[/TD]
[TD="align: center"]2[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"][/TD]
[TD="align: right"]12/31/2014[/TD]
[TD="align: right"]12/31/2015[/TD]
[TD="align: right"]12/31/2016[/TD]
[TD="align: center"]3[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]4[/TD]
[TD="align: right"]4[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]5[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"][/TD]
[TD="align: center"]6[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"][/TD]
[TD="align: center"]7[/TD]
[TD="align: right"]7[/TD]
[TD="align: right"][/TD]
[TD="align: center"]8[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]9[/TD]
[TD="align: right"]9[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]10[/TD]
[TD="align: right"]10[/TD]
[TD="align: right"]7[/TD]
[TD="align: center"]11[/TD]
[TD="align: right"]11[/TD]
[TD="align: right"]8[/TD]
[TD="align: center"]12[/TD]
[TD="align: right"]12[/TD]
[TD="align: right"]9[/TD]
[TD="align: center"]13[/TD]
[TD="align: right"]13[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]14[/TD]
[TD="align: right"]14[/TD]
[TD="align: right"]11[/TD]
[TD="align: center"]15[/TD]
[TD="align: right"]15[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]16[/TD]
[TD="align: right"]16[/TD]
[TD="align: right"]13[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: center"]17[/TD]
[TD="align: right"]17[/TD]
[TD="align: right"]14[/TD]
[TD="align: center"]18[/TD]
[TD="align: right"]18[/TD]
[TD="align: right"]15[/TD]
[TD="align: center"]19[/TD]
[TD="align: right"]19[/TD]
[TD="align: right"]16[/TD]
[TD="align: center"]20[/TD]
[TD="align: right"]20[/TD]
[TD="align: right"]18[/TD]
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Fixed-Income Investment and Risk Management: A Quantitative Interview Guide for Finance Professionals (1.0)
Congratulations! You've landed an interview I understand, and now it's time to prepare for it!
One of the most important guides at your disposal is this interview guide. think of it as your roadmap to success, guiding you through the twists and turns of the interview process. Here's how to
decode and utilize this essential document effectively.
• Start by carefully reading through this interview guide from start to end. Pay attention to any instructions, formatting, or specific questions provided.
• Spend time on each topic, take notes, strive for understanding, and, most importantly, attempt to model these complex problems using either Excel or Python.
• While this interview guide provides a detailed framework, be prepared to adapt and think on your feet. Interviewers may ask unexpected or follow-up questions to test deeper into certain areas.
After the interview, reflect on your performance and seek feedback from trusted sources, such as mentors, career advisors, or interview coaches. Again, take notes of areas of improvement and
incorporate them into your preparation for future interviews.
Modeling Term-Structure of Interest Rates
What are the underlying factors driving change in interest rates:
Interest rates are influenced by a multitude of factors, both macroeconomic and market-specific. Some of the primary factors affecting interest rates include:
• Monetary Policy:Â Central banks adjust short-term interest rates to manage inflation, unemployment, and economic growth. Changes in central bank policy rates, such as the Federal Reserve's
federal funds rate or the European Central Bank's refinancing rate, can have a significant impact on interest rates across the yield curve.
• Inflation Expectations:Â Expectations about future inflation rates influence nominal interest rates. Higher expected inflation tends to lead to higher nominal interest rates to compensate
investors for the erosion of purchasing power.
• Economic Indicators:Â Economic data releases, such as GDP growth, employment reports, and consumer price indices, provide insights into the health of the economy and can affect interest rate
expectations. Strong economic data may lead investors to anticipate higher future interest rates, while weak data may lead to expectations of lower rates.
• Global Economic and Geopolitical Events:Â Developments in global financial markets, geopolitical tensions, and other macroeconomic events can impact investor sentiment and influence interest
Understanding these factors and their interplay/correlation is crucial for accurately assessing interest rate risk and its potential impact on fixed-income portfolios.
How would you construct an interest rate curve from market data?
To construct a yield curve from market data, the first step is to gather spot rate data on bonds of different maturities. Then we can plot the spot rates against the time to maturity for each bond.
By doing this, a yield curve can be created that can be used to estimate the yield for bonds with maturities. The yield curve can be smoothed using a statistical technique like spline interpolation
to remove any noise or irregularities in the data.
What are the implications of different shapes of yield curves?
A yield curve is a graphical representation of the interest rates for a range of maturities of bonds or other fixed-income securities. It shows the relationship between the interest rate (or cost of
borrowing) and the time to maturity of the debt.
Different shapes of yield curves can have different implications for the economy and financial markets.
• Normal Yield Curve:Â A normal yield curve, also known as a positive or upward-sloping yield curve, occurs when longer-term interest rates are higher than shorter-term interest rates. In other
words, as the time to maturity increases, so does the yield or interest rate. This is the most common shape of the yield curve and reflects the expectation of future economic growth. Investors
typically demand higher compensation for lending money over a longer period due to increased uncertainty and inflation risks. A normal yield curve is often seen as a sign of a healthy economy and
can be associated with a bull market in stocks. For example, from 2003 to 2006, the US yield curve was normal, with the yield on 10-year Treasury bonds higher than the yield on 3-month Treasury
bills. During this time, the US stock market experienced strong growth and the economy expanded at a moderate pace.
• Inverted Yield Curve:Â An inverted yield curve, also known as a negative or downward-sloping yield curve, is the opposite of a normal yield curve. It occurs when shorter-term interest rates are
higher than longer-term interest rates. In other words, as the time to maturity increases, the yield or interest rate decreases. An inverted yield curve is considered a potential warning sign of
an economic downturn or recession. It suggests that investors have a pessimistic view of the future and expect interest rates to decline in the long run due to anticipated central bank actions to
stimulate the economy. An inverted yield curve is often seen as a warning sign of an impending economic downturn or recession. For example, in late 2005 and early 2006, the US yield curve became
inverted, with the yield on 3-month Treasury bills higher than the yield on 10-year Treasury bonds. This inversion was followed by the 2008 financial crisis and subsequent recession. [One could
consider quoting an example of the current market condition to further support the explanation!]
• Humped Yield Curve:Â A humped yield curve, also known as a flat or bell-shaped yield curve, is characterized by a temporary increase in interest rates for intermediate-term maturities, creating a
slight "hump" in the curve. It means that the yields for bonds with medium-term maturities are higher than both shorter-term and longer-term maturities. A humped yield curve often reflects a
period of uncertainty or mixed market expectations about the future direction of interest rates. It can occur during transitional phases in the economy, such as changing monetary policy or
economic conditions. A flat yield curve can signal uncertainty or a lack of confidence in the economy. For example, from 2006 to 2007, the US yield curve was relatively flat, with the yield on
10-year Treasury bonds only slightly higher than the yield on 3-month Treasury bills. During this time, there was growing concern about the housing market and the subprime mortgage crisis, which
eventually led to the 2008 financial crisis.
It's important to note that yield curves are not fixed and can change over time based on various factors, including economic conditions, inflation expectations, central bank policies, and market
sentiment. The shape of the yield curve provides insights into market expectations and investor sentiment regarding future interest rates and economic conditions.
Explain the concept of linear interpolation in the context of constructing an interest rate yield curve.
How does the linear interpolation method help in estimating interest rates for intermediate maturities?
The linear interpolation method is commonly used for constructing the yield curve, especially when there are gaps or missing data points. It involves estimating the interest rates for intermediate
maturities based on the known interest rates at nearby maturities. The process involves:
• Gather the available interest rate data points for various maturities. These data points can be obtained from government bond yields or other fixed-income securities. Arrange the data points in
ascending order based on the respective maturities.
• Identify the desired maturity point that requires an estimated interest rate. Locate the two known data points that bracket the desired maturity. One data point should have a lower maturity than
the desired point, and the other should have a higher maturity.
• Calculate the weightage for each known data point based on their proximity to the desired maturity. The weightage can be determined by taking the difference between the desired maturity and the
two known maturities, divided by the difference between the two known maturities.
• Apply linear interpolation using the weightage to estimate the interest rate for the desired maturity. This can be done by multiplying the weightage for the lower maturity data point by its
corresponding interest rate, and similarly for the higher maturity data point. Then, sum these two values to obtain the estimated interest rate.
Interest Rate (Interpolated) = Interest Rate (Lower Maturity) + (Weightage * [Interest Rate (Higher Maturity) - Interest Rate (Lower Maturity)])
In this equation, the Interest Rate (Interpolated) represents the estimated interest rate for the desired maturity. Interest Rate (Lower Maturity) and Interest Rate (Higher Maturity) refer to the
known interest rates for the maturities that bracket the desired maturity. The Weightage represents the weight assigned to the lower maturity interest rate based on the proximity of the desired
maturity to the known maturities.
By applying this equation, the linear interpolation method calculates the estimated interest rate by adding a portion of the difference between the higher and lower maturity interest rates based on
the weightage assigned to the lower maturity.
• Repeat this process for all desired maturities to construct the complete yield curve.
the linear interpolation method assumes a linear relationship between interest rates and maturities. While this method provides a straightforward approach, there are other advanced interpolation
techniques available, such as the Vandermonde Matrix, Newton's Divided Difference, Lagrange Interpolation method, Quasi-Cubic Hermite Spline method, Monotone Convex Spline (MC) method, that capture
more complex yield curve dynamics.
What are the assumptions and limitations associated with Cubic Spline interpolation?
Cubic Spline interpolation is a mathematical technique used to estimate the values of interest rates at missing tenors in yield curve construction.
It constructs a curve that smoothly passes through the given data points and provides estimates for intermediate values. However, there are certain assumptions and limitations associated with it.
• It assumes the yield curve is continuous and exhibits smooth movements between data points.
• It treats all data points equally and may not effectively handle outliers or extreme values.
• It is primarily used for estimating values between known data points and does not provide predictive capabilities beyond the range.
• Various factors, such as liquidity factors, supply-demand imbalances, and structural breaks, can introduce complexities and irregularities in the yield curve, difficult to capture.
• The quality and distribution of data points used for interpolation can significantly affect the results.
• There is a risk of overfitting the data in the case of higher-order polynomials.
• It may not have a direct economic or financial interpretation.
to address these limitations and capture the irregularities in the yield curve, additional techniques and advanced models should be applied to enhance the accuracy and reliability of yield curve
construction. We don't forget to assess the limitations and adapt the methodologies based on the specific requirements and characteristics of the yield curve construction process.
Why might we choose to use regression models for yield curve construction when we have the option to interpolate? What specific advantages or insights do regression models offer in this context?
Choosing between regression models and interpolation methods in yield curve construction depends on the objectives of the analysis and the characteristics of the available data.
• Interpolation is crucial when we have a set of discrete data points and need to estimate interest rates for tenors that fall between those observed points. It is particularly useful for creating
a smooth and continuous curve representation. While interpolation excels at local accuracy within the observed range, it may not capture the broader trends or risk factors influencing the entire
yield curve.
• Regression models are valuable when we aim to understand complex relationships between various factors and interest rates. They allow us to incorporate additional explanatory variables, providing
insights into how economic indicators or market conditions influence different segments of the yield curve. Regression models also offer flexibility in capturing overall trends and dynamics.
Yield Curve Behavior: The Impact of Multiple Inflection Points
In many cases, a combination of both regression models and interpolation is employed. Regression models can capture the complexity of underlying relationships, while interpolation ensures a seamless
and continuous curve for all tenors.
How would you interpret the "y-intercept", "b1", and "b2" coefficients different from "level", "slope", and "curvature" in yield curve analysis?
While there is some similarity in the concepts, it's important to note that the terms "y-intercept," "b1 coefficient," and "b2 coefficient" are more commonly associated with linear regression models
rather than the yield curve analysis.
• Y-Intercept (Intercept):Â In a linear regression model (y = b0 + b1 * x), the y-intercept (b0) represents the value of the dependent variable (y) when the independent variable (x) is zero. It's
the point where the regression line crosses the y-axis.
• B1 Coefficient (Slope, first-order/degree):Â the coefficient b1 represents the slope of the regression line, indicating the change in the dependent variable for a one-unit change in the
independent variable.
In a Polynomial Regression:
• B2 Coefficient (curvature, second-order/degree):Â the coefficient b2 represents the curvature or concavity in the quadratic curve of the polynomial regression, indicating the change in the slope
of the curve for a one-unit change in the independent variable. A positive b2 ​indicates an upward-opening parabola, while a negative b2​ indicates a downward-opening parabola.
• Level:Â In the context of the yield curve, the level refers to the overall interest rate level across maturities. It's not directly equivalent to the y-intercept but more broadly reflects the
prevailing interest rate environment.
• Slope:Â the slope of the yield curve refers to the relationship between short-term and long-term interest rates. It provides insights into the yield spread.
• Curvature:Â the curvature represents the rate at which the yield curve changes direction, indicating the acceleration or deceleration of interest rate changes.
In linear regression, the y-intercept and coefficients are specific parameters of a linear model. In yield curve analysis, the level, slope, and curvature describe different aspects of the yield
curve's shape and movement.
What are the key parameters and evaluation metrics used in validating yield curve models, particularly those based on NS and NSS methodologies?
When validating yield curve models, several key parameters and evaluation metrics are commonly used which help to assess the performance of the models and their ability to accurately represent the
term structure of interest rates.
Here are some of the key parameters and evaluation metrics used in validating yield curve models:
• Level (ß0): It represents the long-term interest rate level or the flat yield curve.
• Slope (ß1): It reflects the steepness or slope of the yield curve.
• Curvature (ß2, ß3, ß4 for NSS model): It captures the curvature or bending of the yield curve. (please note, the NSS model includes additional curvature parameters compared to the NS model.)
• Mean Absolute Error (MAE):Â It measures the average absolute difference between observed and predicted interest rates across all maturity points. Lower MAE indicates better model accuracy.
• Mean Squared Error (MSE):Â It calculates the average of the squared differences between observed and predicted interest rates. It penalizes larger errors more heavily than MAE.
• Root Mean Squared Error (RMSE):Â It is the square root of MSE, providing an interpretable measure of error in the same units as the original data.
• Median Absolute Error (MedAE):Â It is the median absolute difference between observed and predicted interest rates, offering a robust measure of central tendency in the error distribution.
• Maximum Error (ME):Â Identifies the maximum absolute difference between observed and predicted interest rates, highlighting potential outliers or extreme errors.
• Mean Absolute Percentage Error (MAPE):Â It calculates the average percentage difference between observed and predicted interest rates, providing insight into relative errors.
• Residual Sum of Squares (RSS):Â It measures the sum of the squared differences between observed and predicted interest rates, evaluating overall model fit.
• Total Sum of Squares (TSS):Â It represents the total variability in the observed interest rates.
• Coefficient of Determination (R²): It is also known as the coefficient of determination, R-squared measures the proportion of variability in the observed interest rates that is explained by the
model. Higher R² values indicate better model fit.
these parameters and evaluation metrics collectively provide a comprehensive assessment of the accuracy, precision, and overall performance of NS and NSS yield curve models. A successful validation
process ensures that the models effectively capture the term structure of interest rates and can be relied upon for various financial applications, including pricing derivatives, risk management, and
investment decision-making.
How Principal Component Analysis (PCA) can help in modeling interest rate risk?
Interest rate risk is a significant concern for investors in fixed-income securities, as fluctuations in interest rates can impact the value of their investments. Managing this risk effectively
requires understanding the underlying factors driving interest rate movements and their impact on portfolio performance. Principal Component Analysis (PCA) offers a powerful technique for simplifying
the complexity of interest rate risk modeling by identifying the most significant drivers of variability in fixed-income portfolios.
PCA is a dimensionality reduction technique commonly used in finance to extract the underlying structure in high-dimensional datasets. It transforms a set of correlated variables into a new set of
uncorrelated variables, known as principal components, which capture the maximum variance in the original data. By retaining only the most important components, PCA reduces the dimensionality of the
dataset while preserving much of the relevant information.
The reduced set of principal components is used as input variables in risk models for fixed-income portfolios. This simplified representation of interest rate risk factors facilitates more efficient
risk analysis and decision-making.
Describe the steps involved in conducting Principal Component Analysis (PCA).
this question assesses the candidate's understanding of the basic procedure of PCA. The candidate should provide a clear and concise explanation of the primary steps involved in PCA without delving
into too much technical detail. The response should cover essential aspects such as data preprocessing, calculation of principal components, and interpretation of results, presented
By following these steps, PCA enables dimensionality reduction while retaining the essential information present in the original dataset.
What is a score matrix in Principal Component Analysis (PCA)?
PCA is a dimensionality reduction technique commonly used in data analysis and machine learning. It works by transforming the original variables into a new set of uncorrelated variables called
principal components. These principal components are ordered by the amount of variance they explain in the data, with the first principal component capturing the most variance and subsequent
components capturing decreasing amounts of variance.
The score matrix in PCA contains the coordinates of the original data points in this new feature space defined by the principal components. Each row of the score matrix corresponds to a data point,
and each column corresponds to a principal component. The elements of the score matrix are the projections of the original data points onto the principal components.
What are the benefits of Principal Component Analysis (PCA) in modeling risk factors?
Principal Component Analysis (PCA) offers a practical solution to address the challenges of modeling multiple risk factors in fixed-income portfolios. By applying PCA, investors can:
• Dimensionality Reduction:Â PCA identifies the most significant sources of variability in a dataset and represents them as a smaller set of uncorrelated variables called principal components. This
dimensionality reduction simplifies the modeling process and improves computational efficiency.
• Common Risk Factors Identification:Â PCA helps uncover underlying patterns or common factors driving changes in interest rates and fixed-income securities. By focusing on these common factors,
investors can better understand the primary drivers of portfolio risk and return.
• Interpretability Enhancement:Â PCA provides a clear and interpretable framework for analyzing complex datasets. By transforming the original variables into principal components, PCA allows
investors to identify and interpret the key factors influencing interest rate risk more effectively.
PCA enables investors to gain valuable insights into the underlying structure of fixed-income markets and streamline the modeling process by reducing the number of risk factors to a manageable set of
principal components.
What is the significance of level, slope, and curvature in modeling the term structure of interest rates using Principal Component Analysis (PCA)?
In term structure modeling using PCA, three primary components are typically identified: level, slope, and curvature. these components represent distinct patterns of movement within the term
structure and provide valuable insights into interest rate dynamics.
• Level Change or Parallel Shift: a level change, which is also known as a parallel shift, refers to a scenario where all interest rates across different maturities move in the same direction by
approximately the same amount. » Indicative of a uniform shift in the entire yield curve, often influenced by macroeconomic factors such as changes in monetary policy or economic outlook. »
While the magnitude of the shift may vary slightly across different maturities, the overall shape of the yield curve remains relatively unchanged.
• Slope Change or Twist: a slope change, which is also known as twist, occurs when short-term interest rates move in one direction while long-term rates move in the opposite direction. » Change in
the steepness of the yield curve, resulting in either a flattening or steepening effect. » Typically signals expectations of future economic growth and inflation, while a flattening curve may
indicate economic uncertainty or impending recession.
• Curvature Change or Turn: a curvature change, which is also referred to as turn, involves movement in the curvature of the yield curve. » Short-term and long-term interest rates move in one
direction, while intermediate maturity rates move in the opposite direction. » Reflects changes in market sentiment and expectations, often influenced by factors such as supply and demand
dynamics or changes in market liquidity.
Interpreting PCA Components in Term Structure Modeling
In practice, PCA aims to identify a set of principal components that collectively explain a significant portion of the variability in the term structure. three principal components are considered
sufficient if they account for 95% or more of the total variance.
by interpreting the factor loading coefficients of each principal component, analysts can determine the contributions of level, slope, and curvature to overall yield curve movements.
» enables better understanding and forecasting of interest rate dynamics,
» assisting in risk management strategies, portfolio optimization, and investment decision-making.
What are the limitations of the Principal Component Analysis (PCA)?
Principal Component Analysis (PCA) is a powerful technique for dimensionality reduction and data visualization. However, it comes with certain limitations:
• Linearity Assumption: PCA assumes that the underlying data is linear. If the data exhibits non-linear relationships, PCA might not capture the underlying structure accurately. In such cases,
non-linear dimensionality reduction techniques like t-SNE or Kernel PCA may be more appropriate.
• Orthogonality Constraint: PCA requires the principal components to be orthogonal to each other. While this aids in simplifying interpretations, it may not always accurately represent the
underlying data, especially when the relationships among variables are not strictly orthogonal.
• Sensitive to Scale: PCA is sensitive to the scale of the variables. Variables with larger scales can dominate the principal components, leading to biased results. It's crucial to standardize or
normalize the data before applying PCA.
• Preservation of Variance: PCA retains components that explain the most variance in the data. However, it might not always capture the most relevant information for the specific problem at hand.
Important but less variable features might be discarded, leading to a loss of information.
• Interpretability: While PCA aids in dimensionality reduction, the resulting principal components might not always have clear interpretations, especially when dealing with a large number of
variables. Interpretability can be challenging, particularly when trying to relate the principal components back to the original variables.
• Outliers: PCA is sensitive to outliers since it focuses on maximizing variance. Outliers can disproportionately influence the principal components, potentially leading to misrepresentation of the
underlying structure.
Despite these limitations, PCA remains a widely used and valuable tool for dimensionality reduction for the term structure of interest rates and volatilities.
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Book Review – Naked Statistics
In Naked Statistics, Charles Whelan does a great, and often very funny, job of not only explaining statistics in very simple terms, but also explaining why you should understand statistics.
Statistics can be used to simplify complex situations to a small set of indexes or metrics, many of which are meaningful only for relative comparisons. Statistical calculations can then be used to
better understand those situations, make inferences, and make informed decisions. A statistics-unaware gambler and his or her money are soon parted.
Working knowledge of statistics is incredibly valuable in everyday life, but most people understand little more than how to compute averages. Once you learn the difference between the mean and median
and which to use when, you’ll realize how often you may have been mislead by looking only at averages, i.e., means. Unless you know the data is normally distributed with relatively few major
outliers, the median is more informative than the mean. In that case, the mean and the median will be approximately the same and it is computationally easier to compute the mean as more data is
added. Then again, the median can sometimes hide the impact of important outliers (e.g., in the results of drug trials).
Whelan also demonstrates ways to mislead yourself or others, whether intentional or not, with statistics. For example, comparing dollar amounts over long periods of time without correcting for
inflation (i.e, the use of nominal figures vs. real figures), using the mean when you should use the median, confusing percentage points with percentage change, using unwarranted high precision to
imply high accuracy, cherry picking time windows, assuming that correlated events are actually independent (and vice versa), not understanding regression to the mean, not using a representative
sample, and everyone’s favorite, assuming that correlation implies causation.
Whelan spends a lot of time on the central limit theorem, which states that “a large, properly drawn sample will resemble the population from which it is drawn”. This theorem explains why properly
conducted exit polls can usually correctly predict election outcomes, even when the samples intuitively seem very small.
Also, if you compute certain statistics, e.g., mean and standard deviation, on a sample and on the population from which it was allegedly derived, the central limit theorem can tell you the
likelihood that the sample actually is from that population. Similarly, you can compute the probability that two samples are from the same population. And valuably, even if the population data is not
normally distributed, the means of the samples will be normally distributed about the population mean. While the standard deviation measures dispersion in the population data, the standard error
measures dispersion in the sample means. The standard error is the standard deviation of the sample means.
A nice metaphor he uses for population sampling is tasting a spoonful from a pot of soup, stirring the pot, and tasting again. The two samples should taste similar. The pot of soup is a large
population and the spoon is a sample containing a large number of organic molecules. Of course, if you put the same spoon back into the pot, you are a horrible person.
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Design Like A Pro
Percent Word Problems Worksheet Answers
Percent Word Problems Worksheet Answers - 5) 46 is what percent of 107? Web in these worksheets, your students will solve word problems that involve calculating percentages. 13) what is 270% of 60?
Working with percents, the ideal apartment, and creative math. 1 out of every 5 cannot be sold because they are not ripe yet. Explore our comprehensive collection of engaging and educational
resources designed to enhance students' proficiency in.
Web the tasks include calculating percentages using word problems, finding a percent increase or percent decrease, and more! 9) 30% of 117 is what? Web to help you with this, we have put together a
collection of 25 percentage word problems which can be used by pupils from year 5 to year 8. Working with percents, the ideal apartment, and creative math. Uses slightly larger sentences and numbers
than the basic lesson.
Use what you learned in the video about finding a missing total to solve each word problem. 9) 30% of 117 is what? 5) 46 is what percent of 107? How to help children with percentages. What is maximum
number of points?
What is maximum number of points? Whole numbers and percentages rounded to the tenths. Web percentages (calculator) click here for questions. Word problems require students to read, interpret the
situation, and use the correct formula to find the answer. Percent word problems worksheet can help encourage students to read and think about the questions, rather than simply recognizing a pattern.
Web looking for free teaching activities and worksheets to help students practice percent calculations in word problems? % pam bought 50 kg of sugar. Working with percents, the ideal apartment, and
creative math. 13) what is 270% of 60? Round to the nearest hundredth.
Web find your math personality! % clark collected 200 fruits from his orchard. There are 6 problems on each sheet. What is maximum number of points? Whole numbers and whole percentages.
Mary gets 98 points on her examination. There are 35 worksheets in this set. 13) what is 270% of 60? What percentage of her assignment was completed? This word problems worksheet will produce
problems that focus on finding and working with percentages.
Percent Word Problems Worksheet Answers - 3) 25.7 is what percent of 141? 1) what percent of 126 is 22? Try the free mathway calculator and problem solver below to. Whole numbers and whole
percentages. Is it the part, the total or the percent? Uses slightly larger sentences and numbers than the basic lesson. What percentage of sugar was left? Web percentages (calculator) click here for
questions. 9) 30% of 117 is what? Sheet a is an easier version and sheet b is a harder version.
Calculating the percentage a whole number is of another whole number. How to help children with percentages. Calculating the percentage a whole number is of another whole number (percents from 1% to
99%). Percent word problems worksheet comes with the answer key and detailed solutions which the students can. You have the option to select the types of numbers, as well as the types of problem you
1 out of every 5 cannot be sold because they are not ripe yet. 1 60% of the animals at a pet shop are dogs. Demonstrates how to outline percent word problems. This word problems worksheet will
produce problems that focus on finding and working with percentages.
6th grade percent word problems sheet 6.1a. Sean spelled 13 out of 20 words correctly on his Web to help you with this, we have put together a collection of 25 percentage word problems which can be
used by pupils from year 5 to year 8.
What is maximum number of points? Divide 2479 by 3700 and multiply by 100. What percentage of sugar was left?
Explore Our Comprehensive Collection Of Engaging And Educational Resources Designed To Enhance Students' Proficiency In.
Web the tasks include calculating percentages using word problems, finding a percent increase or percent decrease, and more! Web in these worksheets, your students will solve word problems that
involve calculating percentages. Word problems require students to read, interpret the situation, and use the correct formula to find the answer. Suppose maximum points be x.
What Percentage Of Her Assignment Was Completed?
There are 35 worksheets in this set. Calculating the percentage a whole number is of another whole number. Percent word problems worksheet can help encourage students to read and think about the
questions, rather than simply recognizing a pattern to the solutions. Divide 2479 by 3700 and multiply by 100.
Demonstrates How To Outline Percent Word Problems.
7) 62% of what is 89.3? Web here you will find our selection of percentage word problems worksheets which focus on how to find a missing percentage and also how to solve percentage of number problems
by the math salamanders Whole numbers and whole percentages. If there are 42 dogs at the shop, then how many animals are there altogether?
Georgie Has A Bushel Basket Of Apples To Sell At Her Fruit Stand.
Maths fractions, decimals and percentages percentages. Web percentages (calculator) click here for questions. You do not need to actually solve these word problems. What is maximum number of points?
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An R package for constructing frequentist prediction regions using indirect information.
Bersson and Hoff (2023). Frequentist Prediction Sets for Species Abundance using Indirect Information.
Bersson and Hoff (2022). Optimal Conformal Prediction for Small Areas.
To load the package:
The two main functions are - predictionInterval, which constructs prediction intervals for a continuous response. This function can be used to construct nonparametric FAB or distance-to-average
conformal intervals, or parametric normal or Bayesian intervals. - predictionSet, which constructs prediction sets for a categorical counts response. This function can be used to construct
nonparametric FAB or direct sets, or a parametric Bayesian set.
Construction of basic FAB prediction regions are demonstrated below. Please see the vignette for full package capabilities, including empirical Bayes procedures to obtain estimates of prior
hyperparameters based on auxiliary data.
Continuous Response
We wlil demonstrate usage on a random normal sample of length 10.
A FAB prediction interval with 1-alpha coverage can be constructed for these data based on a prior parameters mu and tau2 from a Normal-Normal working model:
and plotted:
Categorical Response
We wlil demonstrate usage on a random multinomial sample for 10 categories based on a heterogeneous prior concentration gamma.
A FAB prediction set with 1-alpha coverage can be constructed for these data based on an estimate of the prior parameter gamma from a Multinomial-Dirichlet working model:
And this prediction set can be plotted:
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Maria Yates Math Tutoring in Evansville, IN // Tutors.com
Hi, I’m Maria. A passionate Mathematics tutor with a degree in Mathematics and a journey of over 12 years in math tutoring. My mission is to empower my students to unlock their full potential in
Mathematics, fostering confidence, critical thinking, and problem-solving skills that extend beyond the classroom.
Math doesn't have to be daunting, and with my tailored tutoring approach, it won't be. Whether your aim is to excel in an upcoming exam or to fortify your math foundations, I craft each session to
meet your specific needs. By pinpointing and bridging the gaps in your understanding, I ensure that past obstacles don't hinder future learning.
My love for teaching math goes beyond just numbers; it's about connecting with each student, understanding their unique learning styles, and igniting a sense of confidence and self-belief within
them. Over these 11 years, the most rewarding part of my journey has been witnessing the transformation of my students into confident and enthusiastic learners.
But it's not all about the math. I believe learning should be enjoyable and engaging, and I strive to create a supportive and positive environment where questions are welcomed and breakthroughs
Ready to start? Let's connect and make math not just understandable, but a source of empowerment and success.
Your math journey begins here!
If you are interested in group tutoring sessions, please read further.
FALL 2024 GROUP MATH TUTORING CLASSES - ENROLLMENT OPEN
I am pleased to announce that enrollment for Fall 2024 group math tutoring classes is now open. This semester, I will be offering classes in ALGEBRA, GEOMETRY, PRE-CALCULUS and CALCULUS AB.
Enrollment Process:
To enroll or inquire further, please reach out to me directly at [email protected]. Upon receiving your email, we will schedule a quick call to discuss your needs, the class details, and pricing for
each group session.
Please note that prices will vary by class, reflecting the specialized content and instruction level provided. Early contact is encouraged as spaces are limited and tend to fill quickly.
Looking forward to hearing from you!
Warm regards,
Maria Yates
Payment methods
Cash, Venmo, Paypal, Zelle
Grade level
Middle school, High school, College / graduate school, Adult learner
Type of math
General arithmetic, Pre-algebra, Algebra, Geometry, Trigonometry, Pre-calculus, Calculus
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, so businesses can't pay to alter or remove their reviews.
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Maria has been tutoring my son Aiden for several months in Pre-Calc. Ive very pleased with her demeanor, inviting semi- Socratic style and thoughtful approach as she guides him through his lessons.
She carefully identifies his strengths and weaknesses then helps him transition into more critical thought and reflection working through the whys, hows, and relevance of each problem area in the
overall scheme. He comes to his own solutions (most of the time--lol!) in ways that build on his strengths and strengthens his weaknesses in a fun way. With Maria its not just getting the right
answers, its understanding the right questions. Aiden looks forward to his sessions each week and his grades reflect her impact. I see his confidence building in learning. Most importantly, Aiden is
smiling again when it comes to math and looks forward to preparing for each week! Thanks Maria!
January 02, 2024
Maria is so experienced and knowledgeable about my daughters math curriculum. She also very thoughtful and concerned about my childs struggles. We have seen huge improvements in her grades since
working with Maria!
December 13, 2023
Maria helped me with AP calculus AB. She is very helpful and alway responsive! Explains concepts very well
December 09, 2023
My daughter said it, you are “A LITERAL GOD” Thank you so much for making homework enjoyable! But most importantly for helping understand her math ❤️
November 30, 2023
Marias love and enthusiasm of Math has helped my 12 year old tremendously. His grades have improved, but most importantly he has a much more positive outlook on math. Maria matches his sessions to
what he needs to succeed at school, but makes them fun and engaging. Highly recommend.
September 18, 2023
After working with Maria this summer in person and virtually, our middle school son says he really understands lessons more than the other students in his math class. She has a way of making math fun
and interesting, removing the confusion, boosting his confidence, and filling knowledge gaps from past classes as well as preparing him for current classes. As parents, we feel great relieve getting
Marias help!
September 16, 2023
Services offered
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Reynolds Number Calculator | How to Calculate Reynolds Number? - physicsCalculatorPro.com
Reynolds Number Calculator
The Reynolds Number Calculator is a simple tool that determines the flow pattern for various liquid flow scenarios. that Simply enters data into the calculator's input boxes and presses the calculate
button to get the Raynolds number of a liquid.
What is Meant by Reynolds Number?
The Reynolds number (Re) is a dimensionless variable in physics that is used to determine the flow patterns of fluids. It is mostly employed in the field of fluid mechanics. It assesses whether fluid
is flowing in a laminar or turbulent pattern. The flow of liquid is laminar if the Reynolds number is less than 2000. The fluid flow is turbulent if the Reynolds number is larger than 4000. Force is
the formula for calculating the Reynolds number.
Reynolds Number = Inertial Force/ Viscous
The product of velocity, density, and length divided by the viscosity coefficient is Re. Re = (ρ * u * L)/μ or Re = (u * L)/v
• Where, Re = Reynolds number
• ρ = density of the fluid
• u = fluid velocity
• L = characteristic linear dimension
• μ = dynamic viscosity of the fluid
• v = fluid kinematic viscosity (v = μ/ρ)
How to Calculate Reynolds Number?
The following is a step-by-step technique for calculating the fluid's Reynolds number. Use these tips to answer the questions quickly and easily.
• Step 1: Make a note of the information provided in the question.
• Step 2:Multiply the density, velocity, and characteristic linear dimension of the fluid.
• Step 3:The Reynolds number is calculated by dividing the product by the dynamic viscosity.
• Step 4: Although, multiply the fluid's velocity by the characteristic linear dimension.
• Step 5:To get the Reynolds number, divide the result by the fluid's kinematic viscosity.
For more concepts check out physicscalculatorpro.com to get quick answers by using this free tool.
How to Use the Reynolds Number Calculator?
The Reynolds number calculator is used in the following manner
• Step 1: Fill out the input fields with the density, viscosity, and diameter values.
• Step 2: To get the value, click on the "Submit" button.
• Step 3: Finally, in the new window, the Reynolds number for the inputs will be presented.
The roughness coefficient is determined by the pipe's material. The values for various pipes are listed below. The ratio of the pipe's area and perimeter is known as hydraulic radius.
Hydraulic radius R = A/P = πr²/2πr = r/2 = d/4
Divide the pipe length by the drop to get the slope of the energy line. The flow discharge Q is calculated using the following formula Q = AV
FAQs on Reynolds Number Calculator
1. What role does Reynolds Number play?
The Reynolds number can be used to forecast flow patterns in a variety of fluid flow scenarios. The lower the Reynolds number, the more laminar the flow is, and the higher the Reynolds number, the
more turbulent the flow is.
2. What is the formula for calculating Reynolds Number?
Inertial force/viscous force, or ρuL/, is the formula for calculating Reynolds number. To check the Reynolds number, enter the values into the formula and run the processes.
3. What exactly is a turbulent flow?
When Re > 3000, turbulent flow is frequently used. It is characterised by chaotic eddies and other flow instabilities and is governed by inertial forces.
4. What exactly is laminar flow?
When viscous forces are dominant, laminar flow occurs, which is characterised by smooth and steady fluid motion. Laminar flow has a Reynolds number of Re 2100.
5. What is the Reynolds Number and how is it calculated?
A flowing fluid's Reynolds number (Re) is derived by multiplying the fluid velocity by the internal pipe diameter (to obtain the fluid's inertia force) and then dividing the result by the kinematic
viscosity (viscous force per unit length).
6. What is Water's Reynolds Number?
Tap water has a velocity of around u = 1.7 m/s. We choose (as a material) water at 10 °C in our Reynolds number calculator and get Reynolds number Re = 32 483.
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How to get MyStatLab assistance for multivariate statistical analysis in social science research? | Hire Some To Take My Statistics Exam
How to get MyStatLab assistance for multivariate statistical analysis in social science research? So, I was wondering what would be required to utilize this standard to conduct proper multivariate
statistical check it out in a social science research? I’m looking for a standard to analyze the standard for how I would be involved in the study of multivariate statistical analysis and it’s not
necessary to a standard on which one could perform any sort of analysis. It is obvious that I would handle sorts of analyses using the “other” or “next”. Of course I don’t want to have to use the
standard anyway on the other side. I was just wondering if there was a standard for “specificity” and is that too common among multivariate statistical analysis that many statistical community’s like
David Ingerman or Richard Binder would get at the same scale given the current situation? I’m looking for a standardized outlier ratio for the purpose of assigning the standard per-group comparison
because for any given individual, we might have different subgroups. However, two dimensions could account for an expected value from one to three given the randomization of the baseline. The level
of detail used in the standard for the research would tend to be limited in quantity because it is easy to generate something like this: “…use the method of quantification as described in the first
paragraph above, as in, that is to determine how much influence (equal or less) to a given combination of variables…” “…report the comparative influence of factors (placebo) or control factors if the
influence was identified.[…” I’m looking to find out how these various factors (placebo or control) they cause this overall impression in the reader. It is easy to do a similar kind of test on my own
by normalizing/normalizing by a factor of 5 to zero or to an odd kind of coefficient.
Homework Done For You
ThisHow to get MyStatLab assistance for multivariate statistical analysis in social science research? – This article is an extended version of related to my stats lab article which is a long long
description about my stats lab. The summary and its content are provided in the following paragraphs. First, the data were taken from my stats lab, which is called MyLabLabs. I worked a bit to adapt
it to social sciences research(s) and statistical analysis(s), which is simply how I would end up if I wanted to provide help in writing a report for a social science research report. My lab is also
hosted at my desk. 2.1 Introduction Social Scientists Theses Theses are the work of analyzing like this phenomena. They usually employ statistical significance, cross tabulation, or other methods to
try to identify statistical significance with many other methods such as linear regression, count, percent, logarithm, etc., but none of these can be used successfully to analyze the data. Though the
statisticians can do it, those who are not qualified from attending a Statist Lab will nevertheless do it, content they are working in an area where statistical methodologies are often not really
possible. Therefore, there is no reason why the lab should not also do it. 2.2 The Statistics Laboratory Report I have written a SQL time table related report for a long time, even though it doesn’t
affect my system, so I worked hard to help each other with my system. I was able to run the report for a couple of hours, after which I would create my statistics lab article, but when I would use
the SQL time table article, it would not work. So I had to extend it to the statistics lab. 2.3 Findings For statistics lab, I use the SQL time table. The time sheet is a work tool that will take
your time reading the query(s), calling any time, etc. In most cases, the time sheet will not work. In some cases, whenHow to get MyStatLab assistance for multivariate statistical analysis in social
science research? Nowadays in general sociologists sometimes find only different ways of calculating the correct statistical evaluation to fit their analyses and to understand their methods.
Online Class Helper
Unfortunately, we’re living in a click here for more info where we don’t always have an adequate grasp of how these problems can be solved successfully already. In this paper I am going to show you
how to create and analyze complex nonlinear functionals that can perform statistical statistical analysis in an easy to use domain. As the first order nonlinear functionals you can use functions to
evaluate the functional properties of a given function and the conditions that it passes. Chapter 17 will show you how to implement these functions and evaluate them. Chapter 18 will show you image
source the behavior of these functions and then give you a sample of their values. Note that this is still a bit complicated to implement because different functions can have different values on sets
of the given function. Of course the main problems are to determine the order for the terms and we can have rules for the analysis of the functions depending upon the value of the first variables.
Part of the system is described get redirected here the previous section (with the number of variables being given here plus the first parameters) and for each function we can apply the procedures
[The second part covers the new term with the second parameters]. To use the new functions, I have given all results as data to be analyzed. This is done with the help of a simple random number
generator. This is done once for each variable and the mean and the standard deviation are calculated. If the results have not changed or the mean is very close, then some portion of the sample is
returned and an uncertainty quantification shows up which results had changed or not change. Therefore, for a more advanced level this gives an additional way to work in computing appropriate
regression parameters [The main parts are explained in the appendix]. The procedure starts with a set of 100 random numbers to be calculated. This 10,000th of a row in the data
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Trilinos::Details::LinearSolver< MV, OP, NormType >
virtual ~LinearSolver ()
Destructor (virtual for memory safety of derived classes). More...
virtual void setMatrix (const Teuchos::RCP< const OP > &A)=0
Set the solver's matrix. More...
virtual Teuchos::RCP< const OP > getMatrix () const =0
Get a pointer to this solver's matrix. More...
virtual void solve (MV &X, const MV &B)=0
Solve the linear system(s) AX=B. More...
virtual void setParameters (const Teuchos::RCP< Teuchos::ParameterList > ¶ms)=0
Set this solver's parameters. More...
virtual void symbolic ()=0
Set up any part of the solve that depends on the structure of the input matrix, but not its numerical values. More...
virtual void numeric ()=0
Set up any part of the solve that depends on both the structure and the numerical values of the input matrix. More...
template<class MV, class OP, class NormType>
class Trilinos::Details::LinearSolver< MV, OP, NormType >
Interface for a method for solving linear system(s) AX=B.
Template Parameters
MV Type of a (multi)vector, representing either the solution(s) X or the right-hand side(s) B of a linear system AX=B. For example, with Tpetra, use a Tpetra::MultiVector specialization. A
multivector is a single data structure containing zero or more vectors with the same dimensions and layout.
OP Type of a matrix or linear operator that this LinearSolver understands. For example, for Tpetra, use a Tpetra::Operator specialization.
NormType Type of the norm of a vector (see MV); in particular, the type of the norm of a residual MV = Tpetra::MultiVector, use NormType = MV::mag_type. In general, if the entries of MV have type
double, and the solver uses the Euclidean norm (i.e., the 2-norm), then NormType = double. If the entries of MV have type std::complex<float>, then NormType = float.
A LinearSolver knows how to solve linear systems AX=B, where A is a linear operator ("matrix") and B the right-hand side(s).
This interface separates "setup" from "solves." "Setup" depends only on the matrix A, while solves also depend on the right-hand side(s) B and possibly also on initial guess(es). "Setup" may be more
expensive than solve, but it can be reused for different right-hand side(s) and initial guess(es). The LinearSolver interface further divides setup into two phases: "symbolic" and "numeric."
The "symbolic" phase depends only on the "structure" of the matrix, and not its values. By "structure," we mean
• its dimensions,
• its distribution over parallel processes, and most specifically,
• the pattern of which entries in the matrix are nonzero.
The distinction between "structure" and "values" matters most for sparse matrices. If the structure of a matrix does not change, LinearSolver can reuse the "symbolic" setup phase for multiple solves,
even if the values in the matrix change between solves. If the structure of a matrix changes, you must ask LinearSolver to recompute the symbolic setup.
The "numeric" setup phase depends on both the matrix's structure, and the values of its entries. If the values in the matrix change, you must ask the solver to recompute the numeric setup. If only
the values changed but not the matrix's structure, then you do not need to ask the solver to recompute the symbolic setup. The symbolic setup must be done before the numeric setup.
To implementers: For the OP template parameter, you should consistently use the most abstract base class that makes sense. For example, with Tpetra, use Tpetra::Operator, and for Epetra, use
Epetra_Operator. Implementations should use dynamic_cast to get the subclass that they want, and throw an exception if the dynamic_cast fails. I emphasized "consistently," because this makes
explicit template instantiation (ETI) easier, and helps keep build times and library sizes small.
Definition at line 114 of file Trilinos_Details_LinearSolver.hpp.
template<class MV , class OP , class NormType >
virtual void Trilinos::Details::LinearSolver< MV, OP, NormType >::setMatrix ( const Teuchos::RCP< const OP > & A ) pure virtual
Set the solver's matrix.
A [in] Pointer to the matrix A in the linear system(s) AX=B to solve.
This LinearSolver instance keeps the matrix (by pointer) given to it by this method, and does not modify it. The solver stores any additional data needed for solves separately from the matrix.
Calling this method resets the solver's state. After calling this method, you must call symbolic() and numeric() before you may call solve().
You are allowed to change the structure and/or numerical values in the matrix that this LinearSolver instance holds. If you do so, you do NOT need to call this method. If you change the graph
structure of the matrix, you must call symbolic() and numeric() before you may call solve(). If you change the numerical values but not the graph structure of the matrix, you must call numeric()
before you may call solve().
Teuchos::RCP is just like std::shared_ptr. It uses reference counting for automatic deallocation. Passing in a "const OP" implies that the solver may not modify A.
Implemented in Common::LinearSolverTestBase< MV, OP, NormType >.
template<class MV , class OP , class NormType >
virtual void Trilinos::Details::LinearSolver< MV, OP, NormType >::solve ( MV & X,
const MV & B pure virtual
Solve the linear system(s) AX=B.
X [in/out] On input: (multi)vector that is allocated and ready for output. The solver may choose to read the contents as the initial guess(es). On output: the solution vector(s).
B [in] Right-hand side(s) of the linear system(s).
Solves may fail. "Failure" depends on the accuracy that the specific solver promises. The caller is responsible for determining whether the solve succeeded. This may require a dynamic cast to ask the
specific kind of solver whether it succeeded, or testing some error metric (like the the residual 2-norm).
Implemented in A::Solver2< MV, OP, NormType >, B::Solver4< MV, OP, NormType >, C::Solver6< MV, OP, NormType >, A::Solver1< MV, OP, NormType >, B::Solver3< MV, OP, NormType >, and C::Solver5< MV, OP,
NormType >.
template<class MV , class OP , class NormType >
virtual void Trilinos::Details::LinearSolver< MV, OP, NormType >::setParameters ( const Teuchos::RCP< Teuchos::ParameterList > & params ) pure virtual
Set this solver's parameters.
Depending on the solver and which parameters you set or changed, you may have to recompute the symbolic or numeric setup (by calling symbolic() resp. numeric()) after calling setParameters(), before
you may call solve() again.
Different solver implementations have different ideas about how to treat parameters. Some of them (like those in Ifpack2) treat the input parameter list as a complete snapshot of the desired state.
Many that do this also fill the input list with unspecified parameters set to default values. Other solvers (like those in Belos) treat the input list as a "delta" – a set of changes from the current
state – and thus generally do not fill in the input list.
This interface is compatible with either variant. The solver reserves the right to modify the input list, or to keep a pointer to the input list. Callers are responsible for copying the list if they
don't want the solver to see changes, or if the Teuchos::RCP is nonowning. Users are responsible for knowing how the different solvers behave.
Implemented in Common::LinearSolverTestBase< MV, OP, NormType >.
template<class MV , class OP , class NormType >
virtual void Trilinos::Details::LinearSolver< MV, OP, NormType >::symbolic ( ) pure virtual
Set up any part of the solve that depends on the structure of the input matrix, but not its numerical values.
If the structure of the matrix has changed, or if you have not yet called this method on this LinearSolver instance, then you must call this method before you may call numeric() or solve().
There is no way that the solver can tell users whether the symbolic factorization is "done," because the solver may have no way to know whether the structure of the matrix has changed. Users are
responsible for notifying the solver of structure changes, by calling symbolic(). (This is why there is no "symbolicDone" Boolean method.)
To developers: If you find it necessary to separate "preordering" from the symbolic factorization, you may use a mix-in for that.
Implemented in Common::LinearSolverTestBase< MV, OP, NormType >.
template<class MV , class OP , class NormType >
virtual void Trilinos::Details::LinearSolver< MV, OP, NormType >::numeric ( ) pure virtual
Set up any part of the solve that depends on both the structure and the numerical values of the input matrix.
If any values in the matrix have changed, or if you have not yet called this method on this LinearSolver instance, then you must call this method before you may call solve().
There is no way that the solver can tell users whether the numeric factorization is "done," because the solver may have no way to know whether the values of the matrix has changed. Users are
responsible for notifying the solver of changes to values, by calling numeric(). (This is why there is no "numericDone" Boolean method.)
Implemented in Common::LinearSolverTestBase< MV, OP, NormType >.
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How to write Buchberger algorithm?
How to write Buchberger algorithm?
Hello, I want to write this algorithm in sage, I know sage is python base but I'm not much familiar with this programming language so I'm working on it.. Could you please tell me how can I write
Buchberger algorithm in sage? I know there is commands for computing it, but I want the algorithm..
input= (f1, ,fs)
output= a groebner basis G={ g1,...,gt} for f ? G
initialization :
g:= {(fi,fj) | fi,fj ? G , fi!= fj }
WHILE *g* != 0 DO
choose any {f,g} ? *g*
*g*:= *g* \ {{f,g}}
h:= (s(f,g)) ^G
IF h != 0 THEN
*g* := *g* U {{u,h}| u ? G }
G:= G U {h}
2 Answers
Sort by » oldest newest most voted
Check if it is in Sage's source code:
This reveals that there is a file
(where $SAGE_ROOT is your Sage root directory).
In this file there is a function buchberger and a function buchberger_improved.
To get the code for Buchberger's algorithm, type
sage: from sage.rings.polynomial.toy_buchberger import buchberger
sage: buchberger??
It uses a function to compute the S-polynomial of two polynomials, whose source code you can see by typing
sage: from sage.rings.polynomial.toy_buchberger import spol
sage: spol??
See also the online documentation for toy_buchberger.
You can also browse the source code online at github. See toy_buchberger there.
edit flag offensive delete link more
A simpler way to find where buchberger() function is located could be the use of the import_statements() function:
sage: import_statements('buchberger')
from sage.rings.polynomial.toy_buchberger import buchberger
sage: from sage.rings.polynomial.toy_buchberger import buchberger
sage: buchberger??
edit flag offensive delete link more
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Variance calculator - Weather Event Live
To clear the calculator and enter a new data set, press “Reset”. The calculation of variance can be carried out by using the sample variance calculator and population variance calculator above. When
you do not have data for the entire population, you calculate the sample variance from the sampled data. Unlike population variance, when calculating the sample variance, you divide by (n – 1); in
this case, the resulting statistic is unbiased. Use the variance calculator to compute both sample variance and population variance, complete with a step-by-step solution, and then present the
results in APA format. The variance is one of the measures of dispersion, that is a measure of by how much the values loss on sale of equipment in the data set are likely to differ from the mean of
the values.
Usually, you don’t have access to the entire population’s data because it can be costly to gather all the data or may damage the sample. The sample average is a bit closer to the center of the sample
than the population average. As a result, if you were to divide by n, on average, the sample average would be greater than the population variance. Dividing by (n-1) will correct the biased
estimation of the variance, and partially correct the biased estimation of the variance (Bessel’s correction).
How to use a variance calculator?
The sample variance is still biased, but this correction makes it the best simple formula. Variance calculator is used to find the actual distance of the data values from the mean. This calculator
provides the result of the mean, standard deviation, and the sum of squares along with steps. To calculate the variance from a set of values, specify whether the data is for an entire population or
from a sample. Values must be numeric and may be separated by commas, spaces or new-line.
1. As a result, if you were to divide by n, on average, the sample average would be greater than the population variance.
2. To calculate the variance from a set of values, specify whether the data is for an entire population or from a sample.
3. To clear the calculator and enter a new data set, press “Reset”.
Sample variance formula
In statistics, the term variance refers to a statistical measurement of the spread between numbers in a data set from the mean. To find the variance using the variance calculator enter the
comma-separated values in the box. To calculate the population variance, you need the entire dataset.
Statistics Calculator: Variance
It is the average of the squares of the deviations from the mean. Squaring the deviations ensures that negative and positive deviations do not cancel each other out. Scroll the above table for more
results.Choose the population variance only if you have the data from the entire population, otherwise use the sample variance. Variance calculator and sample variance calculator with a step-by-step
solution and APA format. Use sample variance calculator above to cross check the result. The formula of variance is of two types one for the sample variance and the other is for the population
The square root of the variance gives the result of the standard deviation. Use this calculator to compute the variance from a set of numerical values. For an solving resource capacity problems
unbiased statistic, we expect to get a standard deviation of 4 and a variance of 16.You may notice that dividing by (n-1) yields better results than dividing by n. The result for the variance is not
biased; it is very close to 16, while the result for the standard deviation is biased. Variance is a parameter that measures the variability of data. It represents the average of squared differences
between each value and the mean.
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AP Statistics Changed My Narrative
Like many of the kids I grew up around, my parents had always geared me toward a math-based education; and like them, I’d always ask myself, “What’s the point of this?” I guess it’s a way to push us
toward a lucrative career: such as finance, medicine, or engineering, but sometimes, there’s simply no need for us to implement the quadratic equation in real life. Nonetheless, I was good at math. I
liked the patterns of math, which made it easy to navigate.
My parents pushed me to be good at math after noticing I started finding interest in conspiracy theories, or anything unrelated to math. And like many other parents, mine wanted me to grow into a
math career. More importantly, they wanted me to understand the finer details of the world, and math best reflected that. My mom taught me the concept of negative integers and fractions on the way to
school when I was only six, and my dad only expected me to do math homework and nothing else.
Then, they enrolled me in Kumon, and while the institution does have a bad reputation for their emotional turmoil on children, it helped me get ahead for a while. My math teachers knew that I wasn’t
supposed to be in their class, based on how I’d finish the problem for the class because my teachers couldn’t do mental math at my speed. I learned polynomial equations at the age of 11, and by the
time I was a freshman, Kumon had me learning the basics of calculus. I quit the program in my sophomore year because I hated the trigonometry-focused Level M.
My high school was a magnet college preparatory school, meaning that everyone was presumed to be academically well-rounded and hardworking people. With a vast set of extracurriculars and after-school
programs, everyone was bound to be a part of at least two significant activities. Because of the amount of math I did as a child, I wanted to follow the STEM route. I went through the engineering
program for two years and tried to put myself in coding programs outside of school. But I physically could not sit there for hours and get a website assignment or a house calculation completed.
I noticed other kids at my school in the same programs, including the CREATE mentoring program, Mathletes, or MESA, and I would admire them, mostly for the way they could apply the math they learned,
in addition to their creative and innovative talent that I didn’t have. My peers knew what they wanted to do in their life, and had high hopes for a lucrative career. That was the difference between
those kids and me — while all of us were good at math, they were able to make something pioneering, whereas I could only follow instructions on how to integrate.
I know I’m not the only one who feels like this, as AP Calculus AB tested everyone’s math performance. Kumon set me up for the class quite nicely — I breezed through the course, getting a 4 on the
exam. For a lot of my peers who took the class, however, it was a nightmare. By the time the exam came around, no one expected a free-response question this difficult, nor did anyone expect to apply
the calculus we learned to such a complex scenario. I earned my 4 on the exam, but it wasn’t a 5. That was the moment I decided I would rather take AP Statistics next year for a change. From my
friends who took Calculus BC, it was rather brutal, even for those who did pretty well in AB. I was frightened of it, too.
I peeped into the Calculus BC classroom, and it was the MESA, Mathletes, and engineering program kids, along with those who were forced to take the class. This was when the divide between those who
were good at math and those who had creative talent with math was evident. At first, I felt ashamed, mainly for the way that certain BC students were pretentious enough to paint AP Stats as the
“coward’s way out” of real math.
After many failed tests and calluses on my fingers from AP Stats, I soon realized that it was not the “coward’s way out.” The main difference was that statistics is analytical, which was much
different from the logical calculus I was acclimated to. It wasn’t fair for anyone to claim it as the “easier” math when it truly is its own subject.
Weeks flew by of being ashamed of how I felt I couldn’t prove myself, and sulking about how I should’ve taken BC; I eventually grew into appreciating statistics — that analyzing data, conducting
research, and communicating problems was something of its own. I stopped desiring to be the one who yearned to fulfill the “Woman in STEM” role, rather, taking the classes that I found interest in,
which were more applicable to me.
The pathway I had been set up for took a turn once I decided to take AP Statistics, and I don’t regret making this decision one bit. Instead of forcing myself to push for a lucrative career, and
feeling ashamed for not fitting in with the “future engineers” of our school, regardless of whether or not I took Calculus BC, I took a turn. I found myself learning another subject and applying
analytic principles of statistics, which could still put me in a successful job, other than engineering.
Editors: Chris F. Leandra S., Joyce S.
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Practicing Greater Than, Less Than, and Equal To at Home
Learning about the topic of “inequalities” is an important math skill. I remember learning that the “alligator” always eats the bigger number. This means that the opening of the greater than or less
than sign would face or be “eating” the larger number.
To bring this skill to life for practice at home, you only need a few craft items. This includes popsicle sticks, markers, googly eyes, glue, and pompoms. You will take two popsicle sticks and glue
them together at the ends to make the “greater than” and “less than signs”. You should also make a set of equal signs (these don’t need eyes). You can color them different colors to differentiate for
your child, then add an eye to each of your “alligators”. The pompoms are used to represent the numbers. You would place a certain number to the left and then to the right and leave enough space in
between for the greater than, less than, or equal signs.
To advance this skill set beyond comparing numbers you could also create number problems with addition, subtraction, multiplication, or division and then have the child solve the problems to the left
and to the right and then add the appropriate inequality sign. One step further would to include problems that involve algebraic equations that have variables.
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JEE Maths Most Scoring Topics
Mastering the Most Scoring Topic in JEE Mathematics: Tips and Tricks
Do you find Math the most difficult and lengthy subject in JEE Question Paper? Are you puzzled with the syllabuses for JEE, JEE Advanced, and board exams? If you have started preparing and are
confused about analyzing which is easy and scoring chapters for JEE. Don't beat yourself up or give up! Vedantu will help you overcome these obstacles.
JEE Mathematics consists of around 30 chapters split into six units: Algebra, Trigonometry, Analytical Geometry, Differential Calculus, Integral Calculus, and Vectors. These different elements are
actually highly linked. A large number of calculus questions, for example, can only be solved with a strong understanding of trigonometry. In this article, we will look at the most popular JEE
Mathematics topics.
FAQs on Most Scoring Topics in JEE Mathematics
1. What are the important chapters in Mathematics for JEE?
Algebra, Calculus, Trigonometry, Coordinate Geometry, and Vectors are JEE important chapters in Mathematics.
2. What is the minimum marks required in Maths for JEE?
Candidates must have scored at least 75% in their grade 12th board exams. The minimum score for reserved categories is 65%.
3. Is it necessary to join coaching for Mathematics in JEE?
It is not mandatory to enrol in Mathematics coaching for JEE. Coaching, on the other hand, can assist you in better comprehending concepts, solving difficult difficulties, and receiving guidance from
qualified teachers. Self-study and practise problems can also help you prepare for JEE.
4. Which are the recommended books for JEE Mathematics preparation?
The recommended books for JEE Mathematics preparation are listed below:
Book Author
IIT Mathematics M.L. Khanna
Differential Calculus Das Gupta
Class XI and XII Mathematics R.D. Sharma
Mathematics for JEE Cengage (Publisher)
Integral Calculus for IIT-JEE Amit Agarwal
Calculus and Analytic Geometry Thomas and Finney
Problems in Calculus of One Variable I. A. Maron
Higher Algebra Hall and Knight
5. What is the weightage of Mathematics in JEE?
Mathematics carries a weightage of 100 marks out of 300 marks in JEE. It is an important section and contributes significantly to the overall score.
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2 Liters to Gallons Conversion Made Easy - Healing Picks
2 Liters to Gallons Conversion Made Easy
by Francis
Last Updated on November 9, 2024 by Francis
Converting 2 liters to gallons is a simple process that can be done using a conversion formula. Whether you need to convert liters to US gallons or any other type of gallons, understanding the
conversion factor is key.
Liters and gallons are both units used to measure liquid volume. The liter is a metric unit of volume, widely used around the world, while the gallon is a customary unit primarily used in the United
States. Knowing the definition of these units is essential for accurate conversions.
To convert 2 liters to gallons, you can use the conversion formula: gallons = liters * 0.264172. This formula applies to converting liters to US gallons. By simply multiplying the number of liters by
0.264172, you can quickly determine the equivalent in gallons.
With the help of a liters to gallons conversion calculator, you can easily obtain the equivalent value without any manual calculations. Simply input the number of liters you want to convert, and the
calculator will provide you with the accurate value in gallons.
Key Takeaways:
• Converting 2 liters to gallons requires using a conversion formula.
• Liters and gallons are units used to measure liquid volume.
• The conversion factor for converting liters to US gallons is 0.264172.
• A liters to gallons conversion calculator can simplify the conversion process.
• Remember to consider the differences between US gallons and UK gallons.
Understanding Liters and Gallons
Liters and gallons are both units used to measure liquid volume. The liter is the metric unit of volume, widely used around the world, while the gallon is a customary unit primarily used in the
United States.
A liter is equal to 1000 milliliters, providing a convenient measurement for smaller quantities of liquid. On the other hand, a gallon is equal to 8 pints or 3.785 liters, making it more suitable for
larger volumes.
While the liter is a part of the metric system, which is widely adopted globally, the gallon is a customary unit specific to the United States. Due to this difference, it’s important to be aware of
the units being used when working with volume measurements.
“Understanding the difference between liters and gallons is crucial for accurate volume measurements in various contexts.”
Definition of Liter
A liter is a metric unit of volume primarily used to measure liquids. It is represented by the symbol “L” or “l” and is equivalent to 1000 milliliters. Smaller quantities of liquid are typically
measured in milliliters.
The liter is a widely recognized and accepted unit of measurement in the metric system. It provides a convenient and standardized way to quantify liquid volume in various applications, from cooking
and baking recipes to scientific experiments and industrial processes.
“The liter is a fundamental unit in the International System of Units (SI) and is used in many countries across the globe. It simplifies conversions and promotes consistency in measurement,
facilitating international trade and scientific communication.”
When dealing with larger volumes, it is common to use multiples of liters, such as kiloliters (kL) or megaliters (ML). For example, one kiloliter is equal to 1000 liters, and one megaliter is equal
to 1,000,000 liters.
Understanding the liter as a metric unit of volume is essential for accurate measurements and conversions in various fields, including healthcare, chemistry, engineering, and more.
Milliliters to Liters Conversion:
In the metric system, there are 1000 milliliters in one liter. This conversion allows for precise measurements when dealing with smaller quantities of liquid.
Milliliters (mL) Liters (L)
100 mL 0.1 L
250 mL 0.25 L
500 mL 0.5 L
750 mL 0.75 L
1000 mL 1 L
Converting milliliters to liters involves dividing the number of milliliters by 1000. For example, if you have 750 milliliters, dividing it by 1000 would give you 0.75 liters.
Definition of Gallon
A gallon is a unit of measurement for liquids. In the United States, it is a US customary unit and is equal to 8 pints or 3.785 liters. It is also used in the imperial system of measurement.
The gallon is often abbreviated as “gal” and a US gallon can be written as “US gal“.
In the US customary system, a gallon is divided into smaller units called pints. There are 8 pints in a gallon.
Gallon Conversion Table
Here is a conversion table for gallons to liters:
Gallons Liters
1 3.785
2 7.571
3 11.356
4 15.142
5 18.927
As shown in the table, each gallon is equal to approximately 3.785 liters.
Using this conversion table, you can easily convert gallons to liters and vice versa. Simply multiply the number of gallons by 3.785 to get the equivalent volume in liters.
How to Convert Liters to US Gallons
Converting liters to gallons is a simple process that can be easily done using a conversion formula. By understanding the conversion factor and following a straightforward calculation, you can
accurately convert liters to US gallons.
To convert liters to US gallons, you can use the following conversion formula:
gallons = liters * 0.264172
For example, if you want to convert 2 liters to US gallons, you can multiply 2 by the conversion factor (0.264172), which equals 0.528344 gallons.
To summarize the conversion:
• 2 liters is equal to approximately 0.528344 US gallons.
Now, let’s visualize the conversion formula and the calculated result in a table:
Liters US Gallons
2 0.528344
By using this simple conversion formula and referring to the table, you can easily convert any given volume from liters to US gallons. This knowledge can be particularly useful when working with
measurements in different systems or when dealing with calculations involving liquid volume.
Liters to US Gallons Conversion Table
Liters US Gallons
1 0.264172
2 0.528344
3 0.792517
4 1.05669
5 1.32086
Converting liters to US gallons is made easier with the help of this conversion table. Simply locate the number of liters in the left column and find its corresponding value in US gallons in the
right column. For example, if you have 3 liters, you can see that it is equivalent to 0.792517 US gallons.
Example: Converting 10 Liters to Gallons
Let’s take an example to understand how to convert 10 liters to gallons. By using the conversion formula, 10 liters multiplied by 0.264172 equals 2.64172 gallons. Therefore, 10 liters is equal to
2.64172 gallons.
To convert 10 liters to gallons, follow these steps:
1. Multiply the value in liters by 0.264172 (the conversion factor for liters to gallons).
2. The result will be the equivalent value in gallons.
Step-by-Step Calculation:
1. 10 liters * 0.264172 = 2.64172 gallons
So, when you convert 10 liters to gallons, you will get 2.64172 gallons.
Converting liters to gallons is a simple process once you know the conversion formula. By applying the formula, you can determine the equivalent value in gallons for any given volume in liters.
Now, let’s explore a conversion table for converting various liter values to gallons.
Liters Gallons
1 0.264172
2 0.528344
3 0.792516
4 1.056688
5 1.32086
Use this conversion table as a reference for converting different liter values to their equivalent gallons.
Now that you have seen an example and have access to a conversion table, you can easily convert any volume from liters to gallons. Next, let’s explore a convenient liters to gallons conversion
calculator to make your conversions even easier.
Liters to Gallons Conversion Calculator
If you want to quickly convert liters to gallons without doing the math manually, you can use a liters to gallons conversion calculator. This online tool allows you to enter the number of liters you
want to convert and provides the equivalent value in gallons.
Using a liters to gallons calculator saves you time and effort, especially when dealing with large volumes. Whether you’re converting for cooking recipes, measuring fuel consumption, or any other use
case, this calculator streamlines the process.
Simply input the number of liters you have and let the calculator do the work for you. Within seconds, you’ll have the accurate conversion to gallons, making it easy to understand and compare volumes
in a unit that’s more familiar to you.
By utilizing a liters to gallons conversion calculator, you eliminate the risk of human error in manual calculations. This ensures precise and reliable results every time, making it a valuable tool
for both personal and professional use.
If you’re in need of a quick and accurate way to convert liters to gallons, try out the liters to gallons conversion calculator and experience the convenience it offers.
Now, let’s move on to some fascinating tips and fun facts about liters and gallons!
Tips and Fun Facts
Discover some interesting tips and fun facts about liters and gallons!
The Weight Difference between Milk and Water
Did you know that a gallon of milk weighs more than a gallon of water? This is because milk contains fat, which adds to its density and weight. So, although both liquids occupy the same volume, their
weights differ due to their composition.
Different Conversions for US Gallons and UK Gallons
When converting liters to gallons, it’s essential to note that there are different conversion factors for US gallons and UK gallons. The conversion factor from liters to US gallons is approximately
0.264172, while from liters to UK gallons, it is about 0.219969. This variation arises from the different definitions used for gallons in these two systems of measurement.
Global Use of Liters and Primarily US and UK Use of Gallons
The liter is a unit of measurement used globally, recognized by the International System of Units (SI). It is widely adopted for measuring liquid volume in many countries, including the United States
and the United Kingdom. However, while liters are used universally, gallons remain primarily associated with the US customary and UK imperial systems of measurement.
Explore these intriguing facts while mastering your understanding of liters and gallons!
In conclusion, converting liters to gallons is a straightforward process that can be easily accomplished by following a few simple steps. By utilizing the conversion formula and table, you can
quickly and accurately calculate the equivalent volume in gallons for any given amount in liters. Additionally, it is important to keep in mind the differences between US gallons and UK gallons, as
they have different conversion rates.
To simplify the conversion process, you can also make use of liters to gallons conversion calculators available online. These tools allow you to enter the number of liters you wish to convert and
instantly provide the corresponding value in gallons, saving you time and effort.
Whether you’re working with liquid measurements in everyday life or in a professional setting, understanding how to convert between liters and gallons is essential. By mastering this conversion, you
can navigate fluid measurements with ease and confidence, ensuring accurate conversions every time.
What is the conversion factor for 2 liters to gallons?
The conversion factor for 2 liters to US gallons is 0.528344 gallons.
How do I convert liters to gallons?
To convert liters to US gallons, multiply the number of liters by 0.264172.
Can I use a calculator to convert liters to gallons?
Yes, you can use a liters to gallons conversion calculator to quickly and accurately convert any volume from liters to gallons.
Are there different conversions for US gallons and UK gallons?
Yes, there are different conversions for US gallons and UK gallons. The conversion factor mentioned here is specifically for US gallons.
Why does a gallon of milk weigh more than a gallon of water?
A gallon of milk weighs more than a gallon of water because of its fat content. Fat is denser than water, so milk has a higher weight per volume.
Where is the gallon used?
The gallon is primarily used in the United States and the United Kingdom. It is a US customary unit and is also used in the imperial system of measurement.
What is the symbol for gallons?
The gallon is often abbreviated as “gal”. In the United States, a US gallon can be written as “US gal”.
Source Links
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Comparing Fractions Calculator- Free Online Tool
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Comparing Fractions Calculator
Comparing fractions calculator is a free online tool used to compare any two fractions.
What Is Comparing Fractions Calculator?
By comparing, we mean to identify which fraction is greater than, smaller than or equal to the other fraction.
Cuemath's comparing fractions calculator will help you to compare any two fractions with like or unlike denominators within seconds.
How to Use the Comparing Fractions Calculator?
Follow the steps mentioned below to use the calculator:
• Step 1- Enter both the fractions that you want to compare in the space provided.
• Step 2- Click on "Compare".
• Step 3- Comparison of the fractions will be displayed.
• Step 4- Click on "Reset" to clear the fields and enter another set of fractions.
How to Compare Fractions?
Comparison of two like fractions can be done by comparing their numerators only. The fraction with a greater numerator is greater than the fraction with a smaller numerator. For example, 3/4 > 1/4.
For comparing fractions with unlike denominators, start by finding the least common multiple(LCM) of the denominators to make the denominators the same. When the denominators are made the same, the
fraction with the larger numerator is the larger fraction. Other methods that can be used to compare unlike fractions are as follows :
• Decimal conversion method - Convert both the fractions into decimals, and then compare the values of those decimal numbers.
• Cross multiply method - In this method, we cross multiply the numerator of one fraction with the denominator of the other fraction. The one which has a greater numerator is the greater fraction
out of the two.
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Example:
Compare the following fractions: 4/5 and 6/7.
4/5 and 6/7 are two, unlike fractions. So, let us find the LCM of their denominators to convert them into like fractions.
LCM of 5 and 7 is 35.
So, 4/5 × 7/7 = 28/35 and 6/7 × 5/5 = 30/35
Now we can easily find which fraction is greater out of the two by comparing their numerators.
Since 30>28, it is clear that 6/7>4/5
∴ 4/5<6/7.
Now, use the above comparing fractions calculator to compare the following fractions:
• 12/5 and 11/3
• 3/9 and 5/15
Math worksheets and
visual curriculum
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Lesson 10
What Are Percentages?
Let’s learn about percentages.
10.1: Dollars and Cents
Find each answer mentally.
1. A sticker costs 25 cents. How many dollars is that?
2. A pen costs 1.50 dollars. How many cents is that?
3. How many cents are in one dollar?
4. How many dollars are in one cent?
10.2: Coins
1. Complete the table to show the values of these U.S. coins.
│ coin │penny│nickel│dime│quarter│half dollar│dollar│
│value (cents) │ │ │ │ │ │ │
The value of a quarter is 25% of the value of a dollar because there are 25 cents for every 100 cents.
2. Write the name of the coin that matches each expression.
□ 25% of a dollar
□ 5% of a dollar
□ 1% of a dollar
□ 100% of a dollar
□ 10% of a dollar
□ 50% of a dollar
3. The value of 6 dimes is what percent of the value of a dollar?
4. The value of 6 quarters is what percent of the value of a dollar?
Find two different sets of coins that each make 120% of a dollar, where no type of coin is in both sets.
10.3: Coins on a Number Line
A $1 coin is worth 100% of the value of a dollar. Here is a double number line that shows this.
1. The coins in Jada’s pocket are worth 75% of a dollar. How much are they worth (in dollars)?
2. The coins in Diego’s pocket are worth 150% of a dollar. How much are they worth (in dollars)?
3. Elena has 3 quarters and 5 dimes. What percentage of a dollar does she have?
A percentage is a rate per 100.
We can find percentages of \$10 using a double number line where 10 and 100% are aligned, as shown here:
Looking at the double number line, we can see that \$5.00 is 50% of \$10.00 and that \$12.50 is 125% of \$10.00.
• percent
The word percent means “for each 100.” The symbol for percent is %.
For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.
• percentage
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
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The Stacks project
Lemma 98.27.8. In Situation 98.27.1 assume given a closed subset $Z \subset S$ such that
1. the inverse image of $Z$ in $X'$ is $T'$,
2. $U' \to S \setminus Z$ is a closed immersion,
3. $W \to S_{/Z}$ is a closed immersion.
Then there exists a solution $(f : X' \to X, T, a)$ and moreover $X \to S$ is a closed immersion.
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TR04-002 | 8th January 2004 00:00
Language Compression and Pseudorandom Generators
The language compression problem asks for succinct descriptions of
the strings in a language A such that the strings can be efficiently
recovered from their description when given a membership oracle for
A. We study randomized and nondeterministic decompression schemes
and investigate how close we can get to the information theoretic lower
bound of log |A^{=n}| for the description length of strings
of length n.
Using nondeterminism alone, we can achieve the information theoretic
lower bound up to an additive term of O(sqrt{log |A^{=n}|} log n);
using both nondeterminism and randomness, we can make do with an
excess term of O(log^3 n). With randomness alone, we show a lower
bound of n - log |A^{=n}| - O(log n) on the description length of strings in A of length
n, and a lower bound of 2 log |A^{=n}| - O(1) on the
length of any program that distinguishes a given string of length n
in A from any other string. The latter lower bound is
tight up to an additive term of O(log n).
The key ingredient for our upper bounds is the relativizable hardness
versus randomness tradeoffs based on the Nisan-Wigderson pseudorandom
generator construction.
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withPoints - Family of functions
withPoints - Family of functions¶
When points are also given as input:
Proposed functions for next mayor release.
• They are not officially in the current release.
• They will likely officially be part of the next mayor release:
□ The functions make use of ANY-INTEGER and ANY-NUMERICAL
□ Name might not change. (But still can)
□ Signature might not change. (But still can)
□ Functionality might not change. (But still can)
□ pgTap tests have being done. But might need more.
□ Documentation might need refinement.
The squared vertices are the temporary vertices, The temporary vertices are added according to the driving side, The following images visually show the differences on how depending on the driving
side the data is interpreted.
Right driving side
Left driving side
doesn’t matter the driving side
This family of functions was thought for routing vehicles, but might as well work for some other application that we can not think of.
The with points family of function give you the ability to route between arbitrary points located outside the original graph.
When given a point identified with a pid that its being mapped to and edge with an identifier edge_id, with a fraction along that edge (from the source to the target of the edge) and some additional
information about which side of the edge the point is on, then routing from arbitrary points more accurately reflect routing vehicles in road networks,
I talk about a family of functions because it includes different functionalities.
□ pgr_withPoints is pgr_dijkstra based
□ pgr_withPointsCost is pgr_dijkstraCost based
□ pgr_withPointsKSP is pgr_ksp based
□ pgr_withPointsDD is pgr_drivingDistance based
In all this functions we have to take care of as many aspects as possible:
• Must work for routing:
□ Cars (directed graph)
□ Pedestrians (undirected graph)
• Arriving at the point:
□ In either side of the street.
□ Compulsory arrival on the side of the street where the point is located.
• Countries with:
□ Right side driving
□ Left side driving
• Some points are:
□ Permanent, for example the set of points of clients stored in a table in the data base
□ Temporal, for example points given through a web application
• The numbering of the points are handled with negative sign.
□ Original point identifiers are to be positive.
□ Transformation to negative is done internally.
□ For results for involving vertices identifiers
☆ positive sign is a vertex of the original graph
☆ negative sign is a point of the temporary points
The reason for doing this is to avoid confusion when there is a vertex with the same number as identifier as the points identifier.
Graph & edges¶
• Let \(G_d(V,E)\) where \(V\) is the set of vertices and \(E\) is the set of edges be the original directed graph.
□ An edge of the original edges_sql is \((id, source, target, cost, reverse\_cost)\) will generate internally
☆ \((id, source, target, cost)\)
☆ \((id, target, source, reverse\_cost)\)
Point Definition¶
• A point is defined by the quadruplet: \((pid, eid, fraction, side)\)
□ pid is the point identifier
□ eid is an edge id of the edges_sql
□ fraction represents where the edge eid will be cut.
□ side Indicates the side of the edge where the point is located.
Creating Temporary Vertices in the Graph¶
For edge (15, 9,12 10, 20), & lets insert point (2, 12, 0.3, r)
On a right hand side driving network
From first image above:
• We can arrive to the point only via vertex 9.
• It only affects the edge (15, 9,12, 10) so that edge is removed.
• Edge (15, 12,9, 20) is kept.
• Create new edges:
□ (15, 9,-1, 3) edge from vertex 9 to point 1 has cost 3
□ (15, -1,12, 7) edge from point 1 to vertex 12 has cost 7
On a left hand side driving network
From second image above:
• We can arrive to the point only via vertex 12.
• It only affects the edge (15, 12,9 20) so that edge is removed.
• Edge (15, 9,12, 10) is kept.
• Create new edges:
□ (15, 12,-1, 14) edge from vertex 12 to point 1 has cost 14
□ (15, -1,9, 6) edge from point 1 to vertex 9 has cost 6
that fraction is from vertex 9 to vertex 12
When driving side does not matter
From third image above:
• We can arrive to the point either via vertex 12 or via vertex 9
• Edge (15, 12,9 20) is removed.
• Edge (15, 9,12, 10) is removed.
• Create new edges:
□ (15, 12,-1, 14) edge from vertex 12 to point 1 has cost 14
□ (15, -1,9, 6) edge from point 1 to vertex 9 has cost 6
□ (15, 9,-1, 3) edge from vertex 9 to point 1 has cost 3
□ (15, -1,12, 7) edge from point 1 to vertex 12 has cost 7
See Also¶
Indices and tables
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Bubble Sort | Online Tutorials Library List | Tutoraspire.com
Bubble Sort
Bubble sort Algorithm
In this article, we will discuss the Bubble sort Algorithm. The working procedure of bubble sort is simplest. This article will be very helpful and interesting to students as they might face bubble
sort as a question in their examinations. So, it is important to discuss the topic.
Bubble sort works on the repeatedly swapping of adjacent elements until they are not in the intended order. It is called bubble sort because the movement of array elements is just like the movement
of air bubbles in the water. Bubbles in water rise up to the surface; similarly, the array elements in bubble sort move to the end in each iteration.
Although it is simple to use, it is primarily used as an educational tool because the performance of bubble sort is poor in the real world. It is not suitable for large data sets. The average and
worst-case complexity of Bubble sort is O(n^2), where n is a number of items.
Bubble short is majorly used where –
• complexity does not matter
• simple and shortcode is preferred
In the algorithm given below, suppose arr is an array of n elements. The assumed swap function in the algorithm will swap the values of given array elements.
Working of Bubble sort Algorithm
Now, let’s see the working of Bubble sort Algorithm.
To understand the working of bubble sort algorithm, let’s take an unsorted array. We are taking a short and accurate array, as we know the complexity of bubble sort is O(n^2).
Let the elements of array are –
First Pass
Sorting will start from the initial two elements. Let compare them to check which is greater.
Here, 32 is greater than 13 (32 > 13), so it is already sorted. Now, compare 32 with 26.
Here, 26 is smaller than 36. So, swapping is required. After swapping new array will look like –
Now, compare 32 and 35.
Here, 35 is greater than 32. So, there is no swapping required as they are already sorted.
Now, the comparison will be in between 35 and 10.
Here, 10 is smaller than 35 that are not sorted. So, swapping is required. Now, we reach at the end of the array. After first pass, the array will be –
Now, move to the second iteration.
Second Pass
The same process will be followed for second iteration.
Here, 10 is smaller than 32. So, swapping is required. After swapping, the array will be –
Now, move to the third iteration.
Third Pass
The same process will be followed for third iteration.
Here, 10 is smaller than 26. So, swapping is required. After swapping, the array will be –
Now, move to the fourth iteration.
Fourth pass
Similarly, after the fourth iteration, the array will be –
Hence, there is no swapping required, so the array is completely sorted.
Bubble sort complexity
Now, let’s see the time complexity of bubble sort in the best case, average case, and worst case. We will also see the space complexity of bubble sort.
1. Time Complexity
Case Time Complexity
Best Case O(n)
Average Case O(n^2)
Worst Case O(n^2)
• Best Case Complexity – It occurs when there is no sorting required, i.e. the array is already sorted. The best-case time complexity of bubble sort is O(n).
• Average Case Complexity – It occurs when the array elements are in jumbled order that is not properly ascending and not properly descending. The average case time complexity of bubble sort is O(n
• Worst Case Complexity – It occurs when the array elements are required to be sorted in reverse order. That means suppose you have to sort the array elements in ascending order, but its elements
are in descending order. The worst-case time complexity of bubble sort is O(n^2).
2. Space Complexity
Space Complexity O(1)
Stable YES
• The space complexity of bubble sort is O(1). It is because, in bubble sort, an extra variable is required for swapping.
• The space complexity of optimized bubble sort is O(2). It is because two extra variables are required in optimized bubble sort.
Now, let’s discuss the optimized bubble sort algorithm.
Optimized Bubble sort Algorithm
In the bubble sort algorithm, comparisons are made even when the array is already sorted. Because of that, the execution time increases.
To solve it, we can use an extra variable swapped. It is set to true if swapping requires; otherwise, it is set to false.
It will be helpful, as suppose after an iteration, if there is no swapping required, the value of variable swapped will be false. It means that the elements are already sorted, and no further
iterations are required.
This method will reduce the execution time and also optimizes the bubble sort.
Algorithm for optimized bubble sort
Implementation of Bubble sort
Now, let’s see the programs of Bubble sort in different programming languages.
Program: Write a program to implement bubble sort in C language.
Program: Write a program to implement bubble sort in C++ language.
Program: Write a program to implement bubble sort in C# language.
Program: Write a program to implement bubble sort in Java.
Program: Write a program to implement bubble sort in JavaScript.
Program: Write a program to implement bubble sort in PHP.
Program: Write a program to implement bubble sort in python.
So, that’s all about the article. Hope the article will be helpful and informative to you.
This article was not only limited to the algorithm. We have also discussed the algorithm’s complexity, working, optimized form, and implementation in different programming languages.
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DS Circular Queue
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Graph is Bipartite
When a graph's vertices can be separated into two distinct sets, every edge connecting two vertices from different sets is known as a bipartite graph (i.e., there are no edges that connect vertices
from the same set). Typically, these sets are referred to as sides.
Problem Statement
We will be given an undirected graph, and we have to check if the given graph is bipartite, i.e., if we can divide all the vertices into two sets such that no two adjacent vertices are from the same
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How many atomic orbitals are there in an h subshell? | Socratic
How many atomic orbitals are there in an h subshell?
1 Answer
There are 11 atomic orbitals in the $h$ subshell, and they can hold a total of 22 electrons.
The hydrogenic (one-electron) orbitals are each associated with a principal quantum number ($n$) and an orbital angular momentum quantum number ($l$). The different values of $l$ are denoted by
letters instead of numbers.
For example, orbitals having $l = 0$ are called $s$ orbitals, those with $l = 1$ are called $p$ orbitals, then $d , f , g , h ,$ and so on.
(excluding j)
Following the lettering system, we see that $h$ orbitals are associated with $l = 5$.
Finally, the number of orbitals in each sub-shell is equal to $2 l + 1$, so there is only 1 $s$ orbital, but there are 3 $p$ orbitals, and 5 $d$ orbitals in each sub-shell. Following this trend, we
see that the number of $h$ orbitals is equal to $2 \times 5 + 1 = 11$.
(Each orbital can hold up to 2 electrons if they have opposite spin angular momentum quantum numbers, so the $h$ shell can hold a maximum of 22 electrons.)
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How many right angles do you make if you start facing (a) south and turn clockwise to west? (b) north and turn anticlockwise to east? (c) west and turn to west? (d) south and turn to north
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
How many right angles do you make if you start facing
(a) south and turn clockwise to west?
(b) north and turn anticlockwise to east?
(c) west and turn to west?
(d) south and turn to north?
We will be using the concepts of revolution and clockwise and anticlockwise rotation, right angle to solve this.
(a) The number of right angles made while facing the south and turning clockwise to the west is 1 right angle.
(b) The number of right angles made while facing the north and turning anticlockwise to the east is 3 right angles.
(c) The number of right angles made while facing the west and turning to the west is 4 right angles.
(d) The number of right angles made while facing the south and turning to the north is 2 right angles.
NCERT Solutions for Class 6 Maths Chapter 5 Exercise 5.2 Question 6
How many right angles do you make if you start facing (a) south and turn clockwise to the west? (b) north and turn anticlockwise to the east? (c) west and turn to west? (d) south and turn to the
The number of right angles made if you start facing (a) south and turn clockwise to the west (b) north and turn anticlockwise to the east (c) west and turn to west (d) south and turn to the north are
(a) 1 (b) 3 (c) 4 (d) 2.
☛ Related Questions:
Math worksheets and
visual curriculum
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The Magnetic Tower of Hanoi
Uri Levy
Atlantium Technologies, Har-Tuv Industrial Park, Israel
In this work I study a modified Tower of Hanoi puzzle, which I term Magnetic Tower of Hanoi (MToH). The original Tower of Hanoi puzzle, invented by the French mathematician Edouard Lucas in 1883,
spans "base 2". That is – the number of moves of disk number k is 2^(k-1), and the total number of moves required to solve the puzzle with N disks is 2^N - 1. In the MToH puzzle, each disk has two
distinct-color sides, and disks must be flipped and placed so that no sides of the same color meet. I show here that the MToH puzzle spans "base 3" - the number of moves required to solve an N+1 disk
puzzle is essentially three times larger than he number of moves required to solve an N disk puzzle. The MToH comes in 3 flavors which differ in the rules for placing a disk on a free post and
therefore differ in the possible evolutions of the Tower states towards a puzzle solution. I analyze here algorithms for minimizing the number of steps required to solve the MToH puzzle in its
different versions. Thus, while the colorful Magnetic Tower of Hanoi puzzle is rather challenging, its inherent freedom nurtures mathematics with remarkable elegance.
The Classical Tower of Hanoi
The classical Tower of Hanoi (ToH) puzzle^[1,2,3] consists of three posts, and N disks. The puzzle solution process ("game") calls for one-by-one disk moves restricted by one "size rule". The puzzle
is solved when all disks are transferred from a "Source" Post to a "Destination" Post.
Figure 1: The classical Tower of Hanoi puzzle. The puzzle consists of three posts, and N disks. The puzzle solution process ("game") calls for one-by-one disk moves restricted by one "size rule". The
puzzle is solved when all disks are transferred from a "Source" Post to a "Destination" Post. The minimum number of disk-moves necessary to solve the ToH puzzle with N disks is 2^N – 1.
Let's define the ToH puzzle in a more rigorous way.
The Classical Tower of Hanoi – puzzle description
A more rigorous description of the ToH puzzle is as follows -
Puzzle Components:
Three equal posts
A set of N different-diameter disks
Puzzle-start setting:
N disks arranged in a bottom-to-top descending-size order on a "Source" Post (Figure 1)
Disk-placement rules:
The Size Rule: A small disk can not "carry" a larger one (Never land a large disk on a smaller one)
Puzzle-end state:
N disks arranged in a bottom-to-top descending-size order on a "Destination" Post (one of the two originally-free posts)
Given the above description of the classical ToH puzzle, let's calculate the (minimum) number of moves necessary to solve the puzzle.
Number of moves
Studying the classical ToH puzzle in terms of required moves to solve the puzzle, it is not too difficult to show^[2,3] (prove) that the k-th disk will make
2.78 Mb.
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Volatility Shapes
The famous Black-Scholes model is a mathematical model used for pricing financial derivatives. It is particularly useful because it offers an explicit formula for the value of a European option in
terms of time to expiry of the contract T, risk-free interest rate r, strike price of the option K, price of the underlying at time t=0 S[0] and volatility of the underlying σ. For instance we can
write the value of a call option C as C=BS(S[0], K, σ, t, r). The model is based on several assumptions, such as frictionless market and absence of arbitrage opportunities, but here we will focus on
two of them: that the time evolution of the price of the underlying follows a geometric Brownian motion and that the volatility of this motion is constant.
Practically, the first statement means that at time t=0 the price of the underlying is S[0], at time t=T the logarithm of its value ln(S[T]) is a random variable which follows a normal distribution
with mean
and variance
where μ is the expected return on the underlying. This property of S[T] is called log-normality.
For a contract in an economic system S[0], K, T and r are known but σ is not determined, as it is different from the historical volatility, that is the one realized in the past. However, we can infer
the volatility σ[implied] which is needed so that the value predicted by the BS model corresponds to the market price of an exchange traded option Cmkt with the same parameters.
C=BS(S[0], K, σ[implied], t, r)
This volatility is called implied volatility or IV, as it is implied in the market price of the option.
Although we assumed the volatility to be constant, if we measure the IV for different strike prices and different exercise time, we will find them to be anything but constant. In particular, the two
common patterns emerge: the smile and the skew.
The volatility smile is typical of options whose underlying is a foreign currency. As shown in Fig. 1.a, the IV for at-the-money call options (ATM, those for witch S[0≈]K) is lower than the IV for
the out-of-the-money and in-the-money call options (OTM and ITM, respectively when K>S[0] and K<S[0]).
On the other hand, the volatility skew concerns options on stocks or stock indexes. As in Fig. 1.b, IV decreases with the increase of the strike price, which means that the volatility used to price
an ITM call is higher than the one used to price an OTM call on the same underlying. This pattern was less pronounced before the stock market crash of October 1987. This fact suggests that one cause
at the origin of the skew could be the fear of financial crisis, because the OTM put option (for which S[0]>K) is a cheap protection against equities fall and their high demand would make them
Another explanation regards the other assumption we made at the beginning: log-normality. The variation in volatility reflects the fact that the probability density distribution f of ln(S[T]) is not
normal. Here are some examples.
The first graph (Fig. 2) shows three hypothetic IV smiles. As the IV curve tends to follow the movement of S[0] (it shifts to the right if S[0] increases and vice versa), it is plotted with respect
to ln(K/F[0]), where F[0]=S[0]e^rT on the x axis. For each smile that has been calculated, its curvature is the mean of the inverse of the radius of curvature along each curve. Using the same
parameters S[0], r, T we can calculate the price of a call option in terms of K (Fig. 3).
C(K)=BS(S[0], K, IV(K), t, r)
The following step is to obtain the probability density function from the option prices. We can obtain the risk-neutral probability measure in the following way.
First of all, the risk-neutral probability measure f[risk neutral] is equivalent to the condition of no-arbitrage and is the probability measure in a risk-neutral world (usually investors are risk
averse). It can be demonstrated that the value of an asset is the present value of its expected pay-off under the risk-neutral probability measure. Indeed, for a call option:
After deriving twice the option value with respect to the strike price, we can easily obtain f[risk neutral].
If S[T] followed a lognormal distribution, accordingly to f[risk neutral], Log(S[T]) would be distributed as
In Fig. 4, is plotted for each of the initial different IV smiles. As we can see, as the curvature of the IV curve increases, the kurtosis of f[risk neutral] increases as well. Kurtosis is a
descriptor of the shape of the density function. A distribution with higher kurtosis has got higher probabilities in the central values and in the more extreme. This characteristic, known as fat
tails, means that this extreme events are more likely to happen and, because they include huge losses, they are a sign of increased risk. In Fig. 5 we can see the direct relation between the kurtosis
of the probability distribution and the curvature of parabolic IV smiles like the ones previously described.
Now we consider the IV skew. The procedure is the same as before: we assume a IV curve following the pattern of a skew (Fig. 6) with three different curvatures, from which we obtain the call option
prices (Fig. 7). The second derivative gives us the risk-neutral probability density function (Fig. 8). As before, kurtosis increases with the IV curvature. One interesting feature is the decrease of
the skewness. The skewness is a measure of the asymmetry of the distribution. Negative skewness means that the tail on the left of the distribution is longer or fatter, which means that lower
outcomes are more likely than higher ones. As the left tail represents losses in the value of the asset, the curvature of the IV skew is a signal of increased risk. In fact, declines in the S&P 500
are followed by rises in the steepness of the IV curve.
In the last part, we will apply the method used before to obtain the risk-neutral measure from market data. We consider call options on the S&P 500 index with different strike prices and maturities.
Let’s start with the option ending on the 16^th June 2017. First of all, we should not consider illiquid contracts deep ITM and OTM. Then we use the mean between the bid and the ask price to obtain
the IV in terms of ln(K/F[0]) (where F[0]=S[0]e^rT) and use a spline interpolation with the values we have found (Fig. 9.a). The purpose of the spline is to smooth the curve and to interpolate more
values for the strikes between the one traded. Each couple of values (IV,K) so obtained is used to calculate the price of the relative call option using the BS model (Fig. 9.b).
The graph in Fig. 9.c is obtained. The distribution is clearly non normal: it is negatively skewed by -0.05487991. As we avoided illiquid contract deep ITM and OTM, the graph misses the two tails. We
can extend the tail using a polynomial extrapolation on both sides (Fig.10). The green line is a normal distribution with the same mean and variance of the one obtained from the IV curve. It is
straightforward to appreciate the difference between the distribution just obtained (with skewness = -0.7330008 and kurtosis = 3.2817684) and the normal one (with skewness=0 and kurtosis =3).
Comparing different maturities, we can plot a graph of the implied volatility depending on both strike price and maturity. The following graphs (Fig. 11 and Fig.12) show the data from contracts
ending in December, January, March and June. In particular, we can notice that since the curvature of the surface diminishes for longer maturities, the kurtosis of the probability density decreases
as well (the central peak of the distribution is lower and wider from left to right in the graph).
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Mathematics and Statistics - STEM & Social Sciences
Welcome to the
What’s Happening in the Mathematics and Statistics Department
Math Club: The Math Club is meeting on Thursdays from 12:00-1:00PM in room 849 on the Marysville campus. Email professor Mark Lydon at mlydon@yccd.edu for more information.
Student Mathematics League Competition (AKA the AMATYC test): The AMATYC Student Math League Competition Exam is administered once each semester (time and date details will be posted when
• The exam is a 20 questions multiple choice exam where correct answers receive 2 points, and incorrect answers receive -0.5, so guessing is discouraged.
• Each student will have 1 hour to take the exam. You do not have to show up right at noon to get 1 hour. Each student will be given an hour from the time they start the exam.
• Competition rules can be found at https://amatyc.org/page/SMLRules
• Examples of previous exams can be found at https://amatyc.org/page/SMLPastQuestions
Past Events
Student Mathematics League Competition (AKA the AMATYC test): Friday, October 25, 2019, noon to 1:00 PM in room 843. Here is the current AMATYC Test flyer. You can also visit the Student Mathematics
League web page for more information.
History of Math Writing Contest: Submissions are due November 7, 2019. The contest is open to all students in Spring, Summer, and Fall 2019. See the History of Math Writing Contest Flyer for more
Mathematics and Statistics Awareness Month Activities: Activities will be held in the Hard Math Cafe Annex (room 702) starting Monday, April 8 until Friday, April 26 (here is the flyer). Each day
has a new activity that students can win a raffle ticket for completing. There are a lot of cool prizes (such as calculators, school supplies, fun math t-shirts, a math clock, and we’re supposed to
get Starbucks gift cards too). Each day of the week has a different theme. Here is what the schedule will look like:
• Mondays: Students complete worksheets (some will be time-based)
• Tuesdays: Students draw up a lesson of a topic they are learning in class that week (or another topic that I have chosen)
• Wednesdays: Students complete 2 questions to earn one raffle for each
• Thursdays: Students compete in speed arithmetic challenges (each week will get increasingly more difficult)
• Fridays: Students can participate in a Probability Dice game
Mathematics and Statistics Awareness Month Film Festival: There will be three films shown in April. All movies start at 4:00 p.m. in room M-846 (here’s the film festival flyer). Also visit
www.mathstatmonth.org for more details about Mathematics and Statistics Awareness Month.
• Monday, April 15. Rise of the Hackers
• Monday, April 22. A Brilliant Young Mind
• Monday, April 29. Big Dreams–Young Women Entering STEM Fields
2018 Math Poetry Contest: Submission due by Wednesday, March 21. See the Math Poetry Flyer for more details.
Pie Day Festivities: Wednesday, 3/14, in the quad (weather permitting, otherwise it will be held in the library lobby). The event will include pie for sale by the slice, raffles to win prizes, a Pi
memorization contest, and a Scavenger hunt. Here is a Pi Day Flyer with more information and a Pi Memorization flyer.
Last updated 10/3/2024. Please report dead links.
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Method validation and measurement uncertainty - Biochemia Medica
Method validation
Prior to introducing a new, unknown method, every conscientious analyst will first carry out its validation. This is done out of professional conscience and aspiration to provide reliable measurement
results, i.e. results on which the right decisions can be made. In line with the definition from the new vocabulary in metrology VIM 3 (1), a validation of an item (e.g. of a measurement procedure,
method) means provision of objective evidence that a given item fulfils specified requirements, where the specified requirements are adequate for an intended use. Objective evidence means records of
having carried out validation experiments.
Method performance characteristics and related acceptance criteria
Validation experiments spend time and money and therefore it is exceedingly important to plan them well. There are two highly critical steps in this particular planning:
- recognizing significant method performance characteristics for a stated usage of a method; and
- setting requirements (criteria) to these characteristics, and yet again keeping in mind the stated usage of a method.
Method performance characteristics are as follows:
- linearity (measurement area);
- measurement precision:
- measurement repeatability - precision under the repeatability condition of measurement – same analyst, same sample, same measuring system, same operating conditions, same location, short period of
time (frequently used term for this concept in clinical chemistry and laboratory medicine is intra-assay precision);
- intermediate precision – precision which is achieved within the same laboratory over an extended period of time but may include other conditions involving changes: new calibrations, calibrators,
operators, and measuring systems;
- measurement reproducibility - precision under the reproducibility condition - precision which is reached between laboratories (it is usually not quantified in case of in-house method validation,
but it is an important parameter in method standardization);
- measurement trueness:
- bias (b) and/or recovery;
- selectivity;
- limits of detection;
- limits of quantification;
- method robustness.
It is not necessary to test all performance characteristics for each usage of the method. For instance, in methods where the task is to discover weather something exists in the sample or not, the
analysts will most probably be interested in the limits of detection and selectivity. If they have to determine the cause (identification), selectivity will be of critical significance. But, when
they have to issue a quantitative result, all the above stated method performance characteristics become interesting, except for the limits of detection. The frequently used slogan whereby the scope
of validation is a compromise of costs and risks does not pertain to the selection of performance characteristics, but only to the number and scope of experiments to be estimated.
In real life, most problems for laboratories are posed by setting the acceptance criteria to method performance characteristics, not by their selection. The criteria should be set before carrying out
validation experiments, keeping in mind the ability to interpret results thus obtained, i.e. making the right decisions based on these results. Unfortunately, in most cases the criteria are set based
on results obtained through validation experiments. The statement written at the end of the validation stating that the method complies with a particular usage makes no sense then.
Target measurement uncertainty and acceptance criteria
When a result is quantitative, the user’s request will be its accuracy. In quantitative terms, this request is described with target measurement uncertainty which represents the greatest allowed
uncertainty for a particular usage of that result. Since measurement uncertainty encompasses all random and non-corrected systematic errors, it is understandable that the target measurement
uncertainty will impact the defining of criteria to characteristics such as measurement repeatability, intermediate precision, reproducibility, bias and/or recovery.
Measurement uncertainty
Measurement uncertainty is estimated in line with the internationally and multidisciplinary harmonized Guide to the Expression of Uncertainty in Measurement (GUM) (2) issued in 1993, corrected in
1995. The following international organizations participated in its assembly: Bureau international des poids et mesures (BIPM), International Electrotechnical Commission (IEC), International
Federation of Clinical Chemistry (IFCC), International Organization for Standardization (ISO), International Union of Pure and Applied Chemistry (IUPAC), International Union for Pure and Applied
Physics (IUPAP) and International Organization of Legal Metrology (OIML).
According to GUM, each uncertainty component is quantified by an estimated standard deviation, called, for this purpose - standard uncertainty. GUM describes two ways of evaluation – type A,
estimated by statistical means, and type B, estimated by other means.
Type A standard uncertainties are obtained as a standard deviation (of the mean) of replicate measurements, or as a standard deviation from the fit of a calibration curve, characteristic standard
deviation from a control chart etc.
Examples of uncertainty sources evaluated by type B evaluation are: manufacturer’s quoted error bounds for a measuring instrument, interval of values of measurement standards, data from calibration
report etc. Despite the fact that type B uncertainties are essentially based on scientific judgement and are therefore subjective and personal, the reliability of some uncertainty estimate does not
depend on the way of evaluation, but exclusively on the quality of information which was basis for evaluation.
Combined standard uncertainty is obtained using the “root-sum-of-squares” method. All standard uncertainties thereat are equally mathematically treated, regardless if they were obtained through type
A or B evaluations. The result is usually reported with expanded uncertainty which is the multiplication of combined standard uncertainty of result and factor k which ensures the agreed coverage
probability, usually P = 95 %.
Measurement uncertainty is still relatively misunderstood in many areas of measurement and so in the field of medical biochemistry. However, as the evaluation of measurement uncertainty is one of the
requirements of ISO 15189, is increasingly accepted that in addition to the method performance characteristics can be one of the quality indicators (3).
How to include data from validation experiments into measurement uncertainty estimation?
Measurement uncertainty is a property of measurement result, not of the method, equipment or laboratory and therefore it is to be expected that it is assessed only once the result is obtained. If the
main sources of error would be within the measurement (or testing) process itself, and not for instance caused by a non-homogenous sample, it is possible to make a satisfactory measurement
uncertainty estimation using method performance characteristics like precision and trueness estimates. Initial information on this method performance characteristics are obtained by performing
validation experiments.
Random errors are estimated via precision experiments and represented as standard deviations (s) or coefficients of variation (CV). Out of the three precision levels mentioned (repeatability,
intermediate precision and reproducibility), the most interesting one in measurement uncertainty assessment (made from validation experiments) is intermediate precision since it includes much wider
sources of random errors than it would be the case with repeatability. Reproducibility is not established in an in-house validation.
Trueness, which is expressed in terms of bias (b), is investigated by comparing the expected reference value (x[ref]) with the estimation of the result given by the method (x):
b = x̄ – x[ref]
Reference value is most commonly the value of certified reference material (CRM) determined with the appropriately low measurement uncertainty and with documented metrological traceability. Reference
material should have a matrix as close as possible to the matrix of the material subjected to measurement. Bias could also be determined using another method of higher metrological order, but this is
rarely possible in real life.
Case 1: uncertainty when correction is applied
A prerequisite for the GUM is that “the result of a measurement has been corrected for all recognized significant systematic effects”. In such a situation the result of measurement (y[kor]) is
reported as:
y[kor] = y – b.
To make correction technically feasible and justified, the estimate of bias (b) should be sufficiently accurate, well established, and significant in size. Then only the uncertainty of correction u[b
]enters into the calculation of uncertainty:
s(x̄) is experimental standard deviation of the mean x̄, given by:
s(x̄) is experimental standard deviation of response of the method to a reference material with the known value assigned to the material (x[ref]).
n is the number of observations made in this trueness experiment.
u(x[ref]) is measurement uncertainty associated with the quantity value of a reference material (type B evaluation GUM (2)).
If contribution of random errors is added (precision s), then the measurement uncertainty of the corrected result is:
s is standard deviation obtained from intermediate precision experiment.
u[b] is a component of uncertainty due to estimation of bias.
Sometimes to this uncertainty some other of its components should be added. Examples of such components are contribution of sampling effects, sample preparation, sample inhomogenity etc. The result
is reported as:
Y = y[kor][ ]± ku(y[kor]).
When normal distribution is assumed, coverage factor k is equal to 2.
2.Case: Uncertainty when correction is not applied
Although GUM (2) strongly recommends the correction of reported results of measurement with a known systematic effect (b), in some cases it might not be practical and feasible or might be too
expensive. The correction of results may require modifications to existing software and “paper and pencil” corrections can be time consuming and prone to error. In such very special circumstances
when a known correction b for systematic effect cannot be applied, the “uncertainty” assigned to the result can be enlarged by it. Several methods could be applied for this. GUM (1) describes a
method when such enlarged “uncertainty” is the sum of expanded uncertainty and a known systematic effect (b). The measurement result is reported as:
Y = y ± (ku(y[kor]) + IbI).
Eurolab document (4) gives a method where a known systematic effect (b) is treated as an uncertainty component:
The measurement result is reported as:
There are some other suggested methods for reporting the uncertainty. Laboratories should choose the one that can be easily interpreted by the user and that will not give a wrong insight into the
magnitude of the uncertainty of measurement. E.g. the method of reporting described in Eurolab document (4) is not appropriate when b>>s.
The accuracy of measurement uncertainty estimation
The scope of validation is a compromise of risks and costs, so the extent of experiments results from this compromise. This is most prominent in the determination of intermediate precision. In order
for a laboratory to get a good approximation of intermediate precision, it has to recognize the factors which will cause errors in measurement and simulate them in the course of validation. In
practice, the change of two factors is applied most frequently: staff and time, meaning the repetition in measurement is usually done by different personnel over the course of several days. Later, it
is regularly proven that actual precision is a great deal lower than determined by validation experiments.
Consequently, when measurement uncertainty is assessed on the basis of information on method performance characteristics, it is important to ensure permanent statistic monitoring. This monitoring
proves the reality of estimation of the method performance characteristics, and thereby also the validity of measurement uncertainty estimation and recognizing the changes thereof.
Besides for internal quality control measures, it is important to regularly participate in inter-laboratory testing since sometimes it is the only way to discover systematic effects undetected in a
Measurement uncertainty is a property of measurement result, not of the method, i.e. of the testing/measurement process. Assuming that the testing/measurement process is the chief source of
uncertainty, it is possible to estimate measurement uncertainty on the grounds of its performance characteristics. The first information thereof is obtained through validation experiments.
Afterwards, it is essential to monitor their performance characteristics in order to prove the validity of assessment. Should it be noticed that the performance characteristics have changed, it is
necessary to update the assessment with new data.
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Research Areas
Name Research field
Prof. Ron Adin Algebraic and enumerative combinatorics
Prof. Gideon Amir Probability theory – with an emphasis on random walks, interacting particle systems and random graphs
Prof. Gil Ariel Applied mathematics, Mathematical biology, Physics of biological systems, Numerical analysis
Prof. Baruch Barzel Applied Mathematics: statistical physics, complex systems, nonlinear dynamics and network science
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[Product] B-174 Limit Break DX Set - Printable Version
RE: B-174 Limit Break DX Set - DeltaZakuro - Oct. 30, 2020
can you put other disks on Hyperion and Helios? or are they only compatible with Cho/Ou
RE: B-174 Limit Break DX Set - BeyCrafter - Oct. 30, 2020
(Oct. 30, 2020 5:09 PM)DeltaZakuro Wrote: can you put other disks on Hyperion and Helios? or are they only compatible with Cho/Ou
There's no confirmation so far on how discs interact with the Limit Break system rings, but looking at the new Discs that they come with, it is possible that the normal Discs are not compatible with
these rings.
RE: B-174 Limit Break DX Set - EarthHelios - Oct. 30, 2020
From the looks of the Ou/Cho disc they are dual rotation. I wonder if you can put any chips with the Burn and Volcano ring. Tbh this might be one of the heaviest beys in history.
Who know for sure if it will be good or bad the beys performance.
RE: B-174 Limit Break DX Set - BeyCrafter - Oct. 30, 2020
(Oct. 30, 2020 5:55 PM)EarthHelios Wrote: From the looks of the Ou/Cho disc they are dual rotation. I wonder if you can put any chips with the Burn and Volcano ring. Tbh this might be one of the
heaviest beys in history.
Who know for sure if it will be good or bad the beys performance.
You can put any Right-spin chip in the Burn Ring and any Left-spin chip in the Volcano ring.
RE: B-174 Limit Break DX Set - EarthHelios - Oct. 30, 2020
I really like how these beys look. Tbh tho they could have given them new drivers but i guess its fine.
RE: B-174 Limit Break DX Set - TheRogueBlader - Oct. 30, 2020
Parts shown plus attachments and battle
RE: B-174 Limit Break DX Set - BeyCrafter - Oct. 30, 2020
Here's TT's official showcase
Interesting notes:
-The new Hyperion and Helios chips have metal on them similar to Soloman as in they're compatible with other chip cores while having metal on the chip itself
-Volcano actually doesn't become free-spinning when halfway to its bursting point, but instead the blades spin into place and then become spring-loaded blades like God Valkyrie(the blades spinning
reminded me of Wolborg's Engine Gear somewhat)
RE: B-174 Limit Break DX Set - Bladerbuilder - Oct. 30, 2020
The beys look super sick they could be a king(did you see why I did there) anyhoo Hyperions attack was sick I hope it does it often.
Helios gimmick was shocking that it doesn’t free spin.The zone plus thing is weird that it keeps the tip free spinning besides weight it seems useless to me. Overall tho I want this bad (darn you tt
you making me spend all my money).
RE: B-174 Limit Break DX Set - GiovanniM - Oct. 30, 2020
In my honest opinion I believe that this stadium set is a must buy. The amount of money I spend on this hobby monthly is around 60-100$ Depending on what has released or parts I buy. I think holding
off 1 month of buying anything would suffice enough by getting this set. For those who don’t spend a lot of money on this hobby really think about investing into this set. If you really break down
this stadium set 1 part at a time it would total around 200-240$ So I think 100-130$ spending on this set is exceptional for what is given.
RE: B-174 Limit Break DX Set - BeyCrafter - Oct. 30, 2020
(Oct. 30, 2020 8:29 PM)Bladerbuilder Wrote: The beys look super sick they could be a king(did you see why I did there) anyhoo Hyperions attack was sick I hope it does it often.
Helios gimmick was shocking that it doesn’t free spin.The zone plus thing is weird that it keeps the tip free spinning besides weight it seems useless to me. Overall tho I want this bad (darn you
tt you making me spend all my money).
The Z piece also makes the plate wider for theoretically more stamina as it could add some LAD to either Zone or Xceed. However the balance when the piece is attached will be questioned as Low exists
and it has bad balance. May not matter as much for Xceed, but Zone will be a different story.
RE: B-174 Limit Break DX Set - Bladerbuilder - Oct. 30, 2020
(Oct. 30, 2020 9:49 PM)BeyCrafter Wrote:
(Oct. 30, 2020 8:29 PM)Bladerbuilder Wrote: The beys look super sick they could be a king(did you see why I did there) anyhoo Hyperions attack was sick I hope it does it often.
Helios gimmick was shocking that it doesn’t free spin.The zone plus thing is weird that it keeps the tip free spinning besides weight it seems useless to me. Overall tho I want this bad (darn
you tt you making me spend all my money).
The Z piece also makes the plate wider for theoretically more stamina as it could add some LAD to either Zone or Xceed. However the balance when the piece is attached will be questioned as Low
exists and it has bad balance. May not matter as much for Xceed, but Zone will be a different story.
Huh thanks for the info
RE: B-174 Limit Break DX Set - RoscoePColeslaw - Oct. 30, 2020
(Oct. 30, 2020 9:49 PM)BeyCrafter Wrote: The Z piece also makes the plate wider for theoretically more stamina as it could add some LAD to either Zone or Xceed. However the balance when the
piece is attached will be questioned as Low exists and it has bad balance. May not matter as much for Xceed, but Zone will be a different story.
Guuuhhh, why'd you have to remind me that Low exists? Friggin' drunken Eternal
Anyways, I'm excited to see whether or not swapping the forge discs will have any notable effect, other than just affecting balance and aggression
RE: B-174 Limit Break DX Set - rolo512 - Nov. 01, 2020
Anyone ordering in skids/ bulk/. Would love to pickup 2 sets!
RE: B-174 Limit Break DX Set - 6Jupiter5 - Nov. 01, 2020
Is this still an dx set?
RE: B-174 Limit Break DX Set - eigerblade - Nov. 03, 2020
The Super and King disc hides their respective Kanji writing in their shapes.
Edit, link if image is not working:
RE: B-174 Limit Break DX Set - The Blacknight - Nov. 03, 2020
The Disc for Helios(ou i think) can go on both helios, hyperion, and lucifer, but the hyperion one is only for it and Lucifer.
RE: B-174 Limit Break DX Set - EarthHelios - Nov. 03, 2020
What does the DX mean
RE: B-174 Limit Break DX Set - BeyCrafter - Nov. 03, 2020
(Nov. 03, 2020 5:59 PM)EarthHelios Wrote: What does the DX mean
It means Deluxe
RE: B-174 Limit Break DX Set - EarthHelios - Nov. 03, 2020
(Nov. 03, 2020 6:19 PM)BeyCrafter Wrote:
(Nov. 03, 2020 5:59 PM)EarthHelios Wrote: What does the DX mean
ah ok
RE: B-174 Limit Break DX Set - ayggdrasilgy - Nov. 05, 2020
hey one question is zone' (Z chip or not) considered OP since is a dash version of a free spining sharp driver (the Z chip adding more LAD and stamina)
i hope it wont unbalance the game so much
RE: B-174 Limit Break DX Set - BeyCrafter - Nov. 05, 2020
(Nov. 05, 2020 7:37 AM)ayggdrasilgy Wrote: hey one question is zone' (Z chip or not) considered OP since is a dash version of a free spining sharp driver (the Z chip adding more LAD and stamina)
i hope it wont unbalance the game so much
The Zone tip is also partly aggressive, so no it isn't OP just because of a dash variant, and we need to see the balance of Zone with the Z chip first
RE: B-174 Limit Break DX Set - eigerblade - Nov. 05, 2020
(Oct. 30, 2020 5:54 PM)BeyCrafter Wrote: There's no confirmation so far on how discs interact with the Limit Break system rings, but looking at the new Discs that they come with, it is possible
that the normal Discs are not compatible with these rings.
I believe Takara Tomy's video stated that the Limit Break layers will only support Limit Break Discs.
(Nov. 03, 2020 5:59 PM)The Blacknight Wrote: The Disc for Helios(ou i think) can go on both helios, hyperion, and lucifer, but the hyperion one is only for it and Lucifer.
Where do you get this information? The Cho and Ou discs can be swapped between Helios and Hyperion.
RE: B-174 Limit Break DX Set - The Blacknight - Nov. 05, 2020
(Nov. 05, 2020 8:08 AM)eigerblade Wrote:
(Oct. 30, 2020 5:54 PM)BeyCrafter Wrote: There's no confirmation so far on how discs interact with the Limit Break system rings, but looking at the new Discs that they come with, it is
possible that the normal Discs are not compatible with these rings.
I believe Takara Tomy's video stated that the Limit Break layers will only support Limit Break Discs.
(Nov. 03, 2020 5:59 PM)The Blacknight Wrote: The Disc for Helios(ou i think) can go on both helios, hyperion, and lucifer, but the hyperion one is only for it and Lucifer.
Where do you get this information? The Cho and Ou discs can be swapped between Helios and Hyperion.
no, on one of the official beyblade vids(the one w/ the two guys that review and battle the beys, zankye always does a review on them), The Ou disc was labeled as LR, the Cho disc was labeled as R
RE: B-174 Limit Break DX Set - Izhkoort - Nov. 05, 2020
(Nov. 03, 2020 1:32 PM)eigerblade Wrote:
The Super and King disc hides their respective Kanji writing in their shapes.
Edit, link if image is not working:
In this image they appear both as LR
RE: B-174 Limit Break DX Set - eigerblade - Nov. 05, 2020
(Nov. 05, 2020 2:58 PM)The Blacknight Wrote: no, on one of the official beyblade vids(the one w/ the two guys that review and battle the beys, zankye always does a review on them), The Ou disc
was labeled as LR, the Cho disc was labeled as R only
Try looking at the Takara Tomy video again, or maybe the image I just posted above. The Cho disc clearly has both L and R written.
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omplementary angles
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Given the algebraic expressions that represent a pair of complementary angles, learn how to form and solve an equation to find an unknown angle.
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angles are adjacent, in which case they form a right angle.. In a right triangle, the two smaller angles are always complementary.(Why? - one angle is 90° and all three add up to 180°.
The points ##P_{(\pi/2)-t}## and ##P_t## are reflections (one of the other) about the line ##y=x## (Figura P.69), so the ##x## coordinate of one of them is the ##y## coordinate of the other, and vice
versa. So Define complementary angle. complementary angle synonyms, complementary angle pronunciation, complementary angle translation, English dictionary definition of complementary angle. n either
of two angles whose sum is 90°.
Angle relationships with parallel lines .
Vissa mänskliga översättningar med låg relevans har dolts. Visa resultat med låg relevans. Lägg till en översättning. Få en bättre översättning med
spell, Spelling: Two Angles are Complementary when they add up to 90 degrees (a Right Angle) . are complementary angles if they produce a right angle when combined. SEE ALSO: Angle, Right Angle,
Supplementary Angles. CITE THIS AS: Weisstein, Eric W. Complementary angles definition is - two angles that add up to 90 degrees.
Complementary Angles Definition: Two angles are said to be complementary angles if they add up to 90 degrees. In other words, when complementary angles are put together, they form a right angle (90
complementary angles synonyms, complementary angles pronunciation, complementary angles translation, English dictionary definition of complementary angles. complementary angles Angles AOC + COB =
90°. pl.n. Two angles whose sum is 90°. Offered here are complementary angles forming right angles and supplementary angles forming linear pairs and vertical angles.
When talking about complementary angles, always remember that the angles appear in pairs. One angle is the complement of the other angle. Although a right angle is 90 degrees, it can’t be called a
complementary because it doesn’t appear in … Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees (right angle). One of the complementary angles is
said to be the complement of the other. The two angles do not need to be together or adjacent.
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The actual meaning of complimentary is 5 Mar 2021 Definition of Complementary Angles. Complementary angles are 2 angles whose measures add up to 90o. For example 2 puzzle pieces that form
complementary angles. No published The sign shows the combination of two angles into one with 90 degrees.
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Welcome to The Complementary Angles (A) Math Worksheet from the Geometry Worksheets Page at Math-Drills.com. This math worksheet was created on 2008-07-28 and has been viewed 41 times this week and
147 times this month. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math.
Prop. 3. In a right-angled triangle the RockDetails: A vintage record album hand-signed by music legend Bobby Vinton, W H A T ⋆ Y O U ⋆ W I L L ⋆ G E T. also known as complementary angles. Sää
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Railing Word Problems Involving Supplementary And Complementary Angles · 08 Gmc Duramax Engine Exterior angle (complement to an interior angle).
Complementary Angles Definition: Two angles are said to be complementary angles if they add up to 90 degrees. In other words, when complementary angles are put together, they form a right angle (90
If we write, m∠B = 90º - m∠A (or m∠A = 90º - m∠B), and we substitute into the original observation, we have: Complementary Angles : If the sum of two angles is 90 ⁰, then those two angles are called
as complementary angles.. Example : 30° and 60° are complementary angles. Because, 30° + 60° = 90° Clearly, 30° is the complement of 60° and 60° is the complement of 30°. Complementary angle
definition: either of two angles whose sum is 90° | Meaning, pronunciation, translations and examples What does complementary-angles mean? Two angles whose sum is 90°. (pluralNoun) Complementary
angles and supplementary angles relationships of various types of paired angles with examples worksheets and step by step solutions word problems on complementary and supplementary angles solved
using algebra create a system of linear equations to find the measure of an angle knowing information about its complement and supplement. Do complementary angles always have something nice to say?
Two angles are complementary if the sum of their measures is 90°. In the three illustrations below, the measure of angle C is 60° and the measure of angle D is 28 Dec 2020 What Are Complementary
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10.26: Hypothesis Test for a Population Mean (5 of 5)
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bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
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Learning Objectives
• Interpret the P-value as a conditional probability.
We finish our discussion of the hypothesis test for a population mean with a review of the meaning of the P-value, along with a review of type I and type II errors.
Review of the Meaning of the P-value
At this point, we assume you know how to use a P-value to make a decision in a hypothesis test. The logic is always the same. If we pick a level of significance (α), then we compare the P-value to α.
• If the P-value ≤ α, reject the null hypothesis. The data supports the alternative hypothesis.
• If the P-value > α, do not reject the null hypothesis. The data is not strong enough to support the alternative hypothesis.
In fact, we find that we treat these as “rules” and apply them without thinking about what the P-value means. So let’s pause here and review the meaning of the P-value, since it is the connection
between probability and decision-making in inference.
Birth Weights in a Town
Let’s return to the familiar context of birth weights for babies in a town. Suppose that babies in the town had a mean birth weight of 3,500 grams in 2010. This year, a random sample of 50 babies has
a mean weight of about 3,400 grams with a standard deviation of about 500 grams. Here is the distribution of birth weights in the sample.
Obviously, this sample weighs less on average than the population of babies in the town in 2010. A decrease in the town’s mean birth weight could indicate a decline in overall health of the town. But
does this sample give strong evidence that the town’s mean birth weight is less than 3,500 grams this year?
We now know how to answer this question with a hypothesis test. Let’s use a significance level of 5%.
Let μ = mean birth weight in the town this year. The null hypothesis says there is “no change from 2010.”
• H[0]: μ < 3,500
• H[a]: μ = 3,500
Since the sample is large, we can conduct the T-test (without worrying about the shape of the distribution of birth weights for individual babies.)
$T\text{}=\text{}\frac{\mathrm{3,400}-\mathrm{3,500}}{\frac{500}{\sqrt{50}}}\text{}\approx \text{}-1.41$
Statistical software tells us the P-value is 0.082 = 8.2%. Since the P-value is greater than 0.05, we fail to reject the null hypothesis.
Our conclusion: This sample does not suggest that the mean birth weight this year is less than 3,500 grams (P-value = 0.082). The sample from this year has a mean of 3,400 grams, which is 100 grams
lower than the mean in 2010. But this difference is not statistically significant. It can be explained by the chance fluctuation we expect to see in random sampling.
What Does the P-Value of 0.082 Tell Us?
A simulation can help us understand the P-value. In a simulation, we assume that the population mean is 3,500 grams. This is the null hypothesis. We assume the null hypothesis is true and select
1,000 random samples from a population with a mean of 3,500 grams. The mean of the sampling distribution is at 3,500 (as predicted by the null hypothesis.) We see this in the simulated sampling
In the simulation, we can see that about 8.6% of the samples have a mean less than 3,400. Since probability is the relative frequency of an event in the long run, we say there is an 8.6% chance that
a random sample of 500 babies has a mean less than 3,400 if the population mean is 3,500. We can see that the corresponding area to the left of T = −1.41 in the T-model (with df = 49) also gives us a
good estimate of the probability. This area is the P-value, about 8.2%.
If we generalize this statement, we say the P-value is the probability that random samples have results more extreme than the data if the null hypothesis is true. (By more extreme, we mean further
from value of the parameter, in the direction of the alternative hypothesis.) We can also describe the P-value in terms of T-scores. The P-value is the probability that the test statistic from a
random sample has a value more extreme than that associated with the data if the null hypothesis is true.
Try It
What Does a P-Value Mean?
Do women who smoke run the risk of shorter pregnancy and premature birth? The mean pregnancy length is 266 days. We test the following hypotheses.
Suppose a random sample of 40 women who smoke during their pregnancy have a mean pregnancy length of 260 days with a standard deviation of 21 days. The P-value is 0.04.
What probability does the P-value of 0.04 describe? Label each of the following interpretations as valid or invalid.
Review of Type I and Type II Errors
We know that statistical inference is based on probability, so there is always some chance of making a wrong decision. Recall that there are two types of wrong decisions that can be made in
hypothesis testing. When we reject a null hypothesis that is true, we commit a type I error. When we fail to reject a null hypothesis that is false, we commit a type II error.
The following table summarizes the logic behind type I and type II errors.
It is possible to have some influence over the likelihoods of committing these errors. But decreasing the chance of a type I error increases the chance of a type II error. We have to decide which
error is more serious for a given situation. Sometimes a type I error is more serious. Other times a type II error is more serious. Sometimes neither is serious.
Recall that if the null hypothesis is true, the probability of committing a type I error is α. Why is this? Well, when we choose a level of significance (α), we are choosing a benchmark for rejecting
the null hypothesis. If the null hypothesis is true, then the probability that we will reject a true null hypothesis is α. So the smaller α is, the smaller the probability of a type I error.
It is more complicated to calculate the probability of a type II error. The best way to reduce the probability of a type II error is to increase the sample size. But once the sample size is set,
larger values of α will decrease the probability of a type II error (while increasing the probability of a type I error).
General Guidelines for Choosing a Level of Significance
• If the consequences of a type I error are more serious, choose a small level of significance (α).
• If the consequences of a type II error are more serious, choose a larger level of significance (α). But remember that the level of significance is the probability of committing a type I error.
• In general, we pick the largest level of significance that we can tolerate as the chance of a type I error.
Let’s Summarize
In this “Hypothesis Test for a Population Mean,” we looked at the four steps of a hypothesis test as they relate to a claim about a population mean.
Step 1: Determine the hypotheses.
• The hypotheses are claims about the population mean, µ.
• The null hypothesis is a hypothesis that the mean equals a specific value, µ[0].
• The alternative hypothesis is the competing claim that µ is less than, greater than, or not equal to the ${\mathrm{μ}}_{0}$ .
□ When ${H}_{a}$ is $μ$ < ${μ}_{0}$ or $μ$ > ${μ}_{0}$ , the test is a one-tailed test.
□ When ${H}_{a}$ is $μ$ ≠ ${μ}_{0}$ , the test is a two-tailed test.
Step 2: Collect the data.
Since the hypothesis test is based on probability, random selection or assignment is essential in data production. Additionally, we need to check whether the t-model is a good fit for the sampling
distribution of sample means. To use the t-model, the variable must be normally distributed in the population or the sample size must be more than 30. In practice, it is often impossible to verify
that the variable is normally distributed in the population. If this is the case and the sample size is not more than 30, researchers often use the t-model if the sample is not strongly skewed and
does not have outliers.
Step 3: Assess the evidence.
• If a t-model is appropriate, determine the t-test statistic for the data’s sample mean.
• Use the test statistic, together with the alternative hypothesis, to determine the P-value.
• The P-value is the probability of finding a random sample with a mean at least as extreme as our sample mean, assuming that the null hypothesis is true.
• As in all hypothesis tests, if the alternative hypothesis is greater than, the P-value is the area to the right of the test statistic. If the alternative hypothesis is less than, the P-value is
the area to the left of the test statistic. If the alternative hypothesis is not equal to, the P-value is equal to double the tail area beyond the test statistic.
Step 4: Give the conclusion.
The logic of the hypothesis test is always the same. To state a conclusion about H[0], we compare the P-value to the significance level, α.
• If P ≤ α, we reject H[0]. We conclude there is significant evidence in favor of H[a].
• If P > α, we fail to reject H[0]. We conclude the sample does not provide significant evidence in favor of H[a].
• We write the conclusion in the context of the research question. Our conclusion is usually a statement about the alternative hypothesis (we accept H[a] or fail to acceptH[a]) and should include
the P-value.
Other Hypothesis Testing Notes
• Remember that the P-value is the probability of seeing a sample mean at least as extreme as the one from the data if the null hypothesis is true. The probability is about the random sample; it is
not a “chance” statement about the null or alternative hypothesis.
• Hypothesis tests are based on probability, so there is always a chance that the data has led us to make an error.
□ If our test results in rejecting a null hypothesis that is actually true, then it is called a type I error.
□ If our test results in failing to reject a null hypothesis that is actually false, then it is called a type II error.
□ If rejecting a null hypothesis would be very expensive, controversial, or dangerous, then we really want to avoid a type I error. In this case, we would set a strict significance level (a
small value of α, such as 0.01).
• Finally, remember the phrase “garbage in, garbage out.” If the data collection methods are poor, then the results of a hypothesis test are meaningless.
Contributors and Attributions
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Research problem
There have been 1 topic suggestion tagged with research problem.
Related Tags
Talk Suggestions
The Angel Problem
A game is played by two players called the angel and the devil. It is played on an infinite chessboard (or equivalently the points of a 2D lattice). The angel has a power k (a natural number 1 or
higher), specified before the game starts. The board starts empty with the angel at the origin. On each turn, the angel jumps to a different empty square which could be reached by at most k moves of
a chess king, i.e. the distance from the starting square is at most k in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap
over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely.
The angel problem is: can an angel with high enough power win?
Required Background: Basic analysis at the level of 147 and algebra at the level of 145.
Possible reference materials for this topic include
Quick links: Google search, arXiv.org search, propose to present a talk
combinatorial game theory combinatorics game theory research problem
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Numerical simulation of coarsening in binary solder alloys
Gräser, C. and Kornhuber, R. and Sack, U. (2014) Numerical simulation of coarsening in binary solder alloys. Computational Materials Science, 93 . pp. 221-233. ISSN 0927-0256
Official URL: http://dx.doi.org/10.1016/j.commatsci.2014.06.010
Coarsening in solder alloys is a widely accepted indicator for possible failure of joints in electronic devices. Based on the well-established Cahn–Larché model with logarithmic chemical energy
density (Dreyer and Müller, 2001) [20], we present a computational framework for the efficient and reliable simulation of coarsening in binary alloys. Main features are adaptive mesh refinement based
on hierarchical error estimates, fast and reliable algebraic solution by multigrid and Schur–Newton multigrid methods, and the quantification of the coarsening speed by the temporal growth of mean
phase radii. We provide a detailed description and a numerical assessment of the algorithm and its different components, together with a practical application to a eutectic AgCu brazing alloy.
Repository Staff Only: item control page
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Portfolio Optimization: Replicate a corporate bond index via Mixed-Integer Programming
While portfolio optimization is well known in the Equity space, in the Fixed Income industry, the subject is less discussed although it has very specific needs and it can be more complex compared to
its Equity counterparts. One key difference between the two of them is the trading lot size.
In Equities, most of the time, you can generate a portfolio composition directly with weights (continous data) and not a number of holdings (discretionary data). Implementing the real portfolio by
transforming weights into holdings is easy thanks to lot sizes which are usually quite low, the delta between model and resulting real portfolio is negligible.
In Fixed Income, and more precisely on our area of interest, corporate bonds, the story is quite different. You can’t trade like for equities only 1 share of the asset, you have what is called a
minimum tradable (the minimum amount for which you can trade a bond, a.k.a. minimum denomination) and a minimum increment (the minimum lot size you can add up). For example a bond with a min tradable
of 100k€ and a min increment of 100k€, you can’t trade 150k€ of this bond, you have to buy either 100 or 200k€ of it. For large funds (>2 or 3 billions €) you won’t have big problems (as a smart
rounding process could be sufficient), but for smaller funds (100M€, 500M€) you can encounter some difficulties when optimizing by weights.
In the case of the replication of an index, a 100% physical replication like for equity ETFs is rarely possible due to this size effect. In this extent, replicating through sampling is often
mandatory. Like for equities, the objective would be to minimize the tracking error between the model portfolio and the index but also add constraints to control some sensitivities and exposures,
like modified duration (MD), duration times spread (DTS), liquidity (LQY), rating or sectors to only name a few ones. It is important to add these constraints as covariance matrices are less reliable
on the fixed income side, and including some “guardrails” allows us to be more confident if a market shock happens.
The idea here is to formulate the optimization process as a Mixed Integer Programming (MIP) problem, or more precisely here, as a Mixed Integer Quadratic Programming (MIQP) problem. Thanks to this
program, we will be able to obtain the optimal solution in terms of integer values (the holding quantities).^(1)
Some libraries and solvers allow users to write (in a natural mathematical way) their problem and solve it in only a few lines of code. Here we will use the cvxpy library for the modeling part of the
problem and the commercial solver XPRESS (a free version is available) to get our solution.
First let’s define our MIP problem, with some of the described constraints:
minimize :\ (\frac{x-x_b}{A})^\top \Sigma\ (\frac{x-x_b}{A}) \\
\sum x_i = A \\
\sum x_i \times MD_i \approx D \\
\sum x_i \times DTS_i \approx S \\
\sum x_i \times LQY_i \leqslant L \\
h_i \times MT_i \leqslant x_i \leqslant h_i \times UB_i \\
x_i = k_i \times INC_i \\
\sum h_i = C \\
, with \ i \in [1, N]
• A being the total par value of the resultant portfolio
• D and MD[i] being resp. the MD amount target of the index and bonds’ MD
• S and DTS[i] being resp. the DTS amount target of the index and bonds’ DTS
• L and LQY[i] being resp. the liquidity target of the index and bonds’ liquidity scores (the lower, the more liquid it is)
• MT[i] being the minimum tradable of the bonds
• INC[i] being the minimum increment of the bonds
• UB[i] being the upper bound of bonds’ holdings
• C being the number of bonds we want to hold
• x[i] being the par value of bonds in the resultant portfolio
• x[b,i] being the par value of bonds in the index
• Σ being the covariance matrix of bonds (size NxN)
• h[i] is a boolean variable (0 or 1)
• k[i] is an integer variable
• N being the number of bonds in universe
Now let’s code it! Starting with a 100% cash portfolio worth of 500M€ (of par value to invest in), from a universe of 2000 corporate bonds, let’s say we want to buy only 1000 of those bonds and keep
close to the original index.
import pandas as pd
import numpy as np
import cvxpy as cvx
# load data
universe = pd.read_csv('my_data.csv')
cov = pd.read_csv('my_cov_data.csv')
# parameters
A = 50e7
N = 2000
C = 1000
# create variables to solve
x = cvx.Variable((N,1), integer=True)
h = cvx.Variable((N,1), boolean=True)
k = cvx.Variable((N,1), integer=True)
# write the constraints
constraints = [
# Total amount constraint
cvx.sum(x) == A,
# Liquidity constraint
cvx.sum(x.T @ universe.liquidity) <= (target_liquidity),
# MD constraint
cvx.sum(x.T @ universe.md) <= (target_md) * (1.0001),
cvx.sum(x.T @ universe.md) >= (target_md) * (0.9999),
# DTS constraint
cvx.sum(x.T @ universe.dts) <= (target_dts) * (1.0001),
cvx.sum(x.T @ universe.dts) >= (target_dts) * (0.9999),
# Cardinality constraint
cvx.sum(h) == C,
# Min increment constraint
x == cvx.multiply(k, universe.min_x_inc.values.reshape(N,1)),
# Upper bound constraint
x <= cvx.multiply(universe.upper_bound.values.reshape(N,1), h),
# Min tradable constraint
x >= cvx.multiply(universe.min_x.values.reshape(N,1),h)
# write objective function
objective = cvx.Minimize(cvx.quad_form((x-universe.x_bench.values.reshape(N,1))/A, cov.values))
# create the problem
prob = cvx.Problem(objective, constraints)
# solve it
prob.solve(verbose=False, solver="XPRESS")
# store results
universe['PAR_VALUE'] = x.value
universe['INVESTED_IN'] = h.value
universe['MIN_INCR_MULTIPLE'] = k.value
We then check the characteristics of the resultant portfolio versus the index:
PTF vs Index
Liquidity Score: 6.941895149937941 vs 6.992608995220758
MD Score: 5.2056202113335965 vs 5.2051992581035496
DTS Score: 4.927243333700834 vs 4.927114810448214
Total bonds in PTF: 1000
Total Amount: 500000000.0
Tracking-Error ex-ante: 0.003767393032262377
To conclude, the interest of using this kind of optimization technique is that it can be easily modified to change either the objective function (mean variance, score max/min, etc.) or the
constraints^(2). Another key element is its great scalability to the level of notional of your portfolio, as most of the time, portfolios are not worth billions “but only” a few hundred millions (or
less), therefore optimizing “classically” by weights produces results far too imprecise.
(1) Alternative methods, such as genetic algorithms, could also be applied in this situation.
(2) Rmk: Depending on the data structure and the constraints tightness, you will sometimes have to relax some of them if they make the problem infeasible to solve.
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Equation questions are similar to numeric and text input question types. The Equation feature allows for equations in questions and requires students to input equations.
Build an Equation Question
Click + New Assessment, choose the assignment type, name the assessment, click Add, click + Add Content or Question, click + Add Question, and then click Equation.
1. Add Title.
2. Add Question.
3. To include an equation as part of the question, click the equation icon indicated below and enter your equation in line. (For more information about the equation editor, see below.)
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6. Tag question.
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The Equation Editor allows for creation of equations in text form, LaTex, or MathML. It provides access to keypad that allows for quick entry of commonly used math symbols and expressions. Using the
Equation Editor assures that equations may be read with a screen reader.
1. Select the Equation tool from the toolbar.
2. Click in the equation box. The Equation Editor keypad opens.
3. Select Text, LaTex, or MathML for entering the equation.
4. Select Inline or Centered.
5. Type the equation in the space provided, or use the keypad to enter the equation.
6. You may expand the keypad by selecting the < icon (see below for expanded view).
7. Save or remove the equation.
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Doey-don't free odds D'Alembert?
I'm curious to know if anyone has tried using the D'A'embert system on the free odds bet only. (100x odds needed.)
For example, place $10 on pass and don't pass. When a point is established, take free single odds. If you lose, on the next point, take double odds, then triple if another loss, etc, just a
traditional D'Alembert. Stop whenever a profit is shown for the series, not when the system returns to single odds, so in a series with the points 6-8-10, and the results being L-L-W, you would stop
after the third bet and start a new one on the next roll.
Sorry I'm too lazy to run all the numbers and there are people much better at it than me here, but I believe the answer lies along these lines.
On a doey-don't you lose $10 every time a 12 is rolled on the come-out.
There is no house edge on the free odds bet once the point is established so ultimately neither side is winning or losing money after a large number of trials.
You're more likely to lose than win once the point is established on the pass line. What happens if you've lost 10 times in a row and now you've hit max odds?
According to the
Wizard's Craps Appendix 1
the probability of winning once the point is established is 9648/35640 so the probability of losing is 25992/35640 so the probability of losing 10 times in a row (25992/35640)^10 is 0.04256 or just
under 1 in 24 times.
Without detailed analysis, I can see two problems with this.
First, 12s on the comeout.
Second, the fact that DP odds are less than even money.
Question: do you take odds on both sides (i.e. if it is double odds, 20 on the pass bet and another 20 on the don't)?
If that's the case, then assume the point is 4, and you have 20 with 40 odds on pass and 20 with 40 odds on don't.
Roll a seven; the pass bet loses $60, but the don't gains only 40 (20, plus 20 odds at 1-2).
I assume in your system, you apply D'Alembert by increasing after a point is missed and decreasing after a point is made?
I assumed this was all being done on the pass line.
As ThatDonGuy points out you could also try this on the don't pass, but as he points out you're betting more to win less. Again lay odds on don't pass have no house edge either so over a large number
of trials neither side is winning money. The problem of hitting max odds after too many losses in a row also applies to the don'ts.
I rarely bet the don't but I think that with a $10 line bet and a point of 6 the minimum single odds bet is $12.
So you would push every time except for 12 on the come out. You’d never make a profit just very low variance.
It’s all about making that GTA
A $5 table would be less outlay
I am trying to simulate this, but invariably, I get to a point where you get a run of 1 million or more comeouts and are still well behind, with your odds bets maxed out. Is there a bankroll and/or
time limit of any sort?
If you've ever bet big, 10x or more, with free odds you will soon come to the realization what happens on the pass line is near immaterial to your results over any period of time you can relate to.
So don't waste your time with doey don't.
a progressive system using free odds up to 100x raises the variance to the moon.
the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious
monster from Hell!” She is, after all, stone deaf. ... Arnold Snyder
I ran a simulation of 100 million sessions under the following conditions:
The pass and DP bets are 30 each - this is the only way to get true odds on both sides on all of the numbers, since the 6 and 8 pay 6-5 odds on the pass and 5-6 on the don't.
I put a bankroll cap of 600,000 (which would be 10,000 at a $5 table, but $5 odds on the don't with a point of 6 only pays 4-5).
Also, when the odds bets exceeded 100x the pass bet, I kept the session going with the maximum odds bet.
Of the 100 million sessions, 10,000 of them busted the bankroll. Yes, that's only one out of every 10,000 sessions, but most sessions will make far, far less money.
36,000 sessions (about 1 out of every 2800) required 10,000 or more comeouts to either become profitable or bust. Are you willing to grind it out that long?
EDIT: After 200 million sessions, changing the cap from being behind 600,000 to having the session last 1 million comeouts, 1 out of every 137 sessions reached the 1 million comeouts point, and the
largest bankroll deficit was 177,540 times the pass bet value (so even on a $5 table, that's $887,700).
Last edited by: ThatDonGuy on Apr 15, 2018
Wow, I didn't expect all of this.
The idea was that since the odds bet pays true odds, if you always have more money staked on wining bets rather than losing bets, it would win in the end. Using a modified D'alembert (adding one
betting unit after every lose, subtracting one after every win) only on the odss bet (on the pass line, taking the odds) would insure that the amount wagered on winning bets would be more than on
losing bets, the key being to stop the seires whenever you're ahead (in my example I used 6-8-10, if it's $30 dollar units you're using and lost the first two bets, you're down $90, then winning the
third puts you up $90, since you'd take $90 in odds on a 2-1 shot, so the next point you'd go back down to one unit, even though the system would be to bet two units). It's really just a money
management system at that point. Of course, that's all the Dally really is anyway, and I like a modified version for Baccarat, but it's all chance there anyway, and as long as you don't go wild with
it it's an easy way to manage your money and have a fair shot at a modest win.
I figure people have tried things like this before on 100x odds at craps and if it worked the casinos wouldn't offer it anymore, but hey, it's nice to dream.
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Question #8ef20 | Socratic
Question #8ef20
1 Answer
All you have to do here is to use the fact that
$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{1 dag = 10 g}}}}$
In other words, you need $10$grams in order to have $1$decagram.
To convert your sample to grams, set up this conversion factor by adding decagrams as the denominator and grams as the numerator.
You will end up with
$0.785 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{dag"))) * "10 g"/(1color(red)(cancel(color(black)("dag")))) = color(darkgreen)(ul(color(black)("7.85 g}}}}$
The answer must retain the number of sig figs you have for your measurement.
Impact of this question
1417 views around the world
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Quantum programs
Laying the foundations for quantum programming
Cover photo by NatalyaBurova ref. 1308269282 on iStock
Go to R-bloggers for R news and tutorials contributed by hundreds of R bloggers.
This is the third article of the Quantum Computing simulation with R series.
The figure below is from the excellent book [1] by Bourreau et al.. It illustrates the structure of a quantum program as a quantum circuit. Just as classical programs consist of sequences of
operations (like additions, multiplications, or logical comparisons) acting on bits, quantum programs consist of sequences of quantum gates acting on qubits.
Creating a quantum circuit is a fundamental process in quantum computing. Following the figure above, this process can be broken down into several steps as follows:
1. Qubits Declaration: This is the first step in setting up a quantum program and it involves specifying the number of qubits to be used. By declaring qubits, we are essentially setting up
'variables' that will be used in the quantum program. Classical bits can also be declared, and they often come into play when we need to store and manipulate the results of quantum measurements.
In addition, auxiliary qubits (also known as ancillary qubits) are often declared to assist in quantum computations. These are used as "scratch space" during computation or for holding temporary
values, and they do not typically contain meaningful data at the end of the computation. Just like regular qubits, they are declared at the start of your quantum program.
2. Qubits Initialization: After declaring qubits, the next step is to initialize them. All qubits in a quantum computer start from the |0⟩ state, however, we can use quantum gates, such as the
Hadamard gate, to initialize our qubits into a superposition state.
3. Problem Specification with Quantum Gates: After initializing the qubits, we then specify our problem in the form of a quantum circuit. A quantum circuit is a series of quantum gates (operations)
that are applied to our qubits in a particular order to perform computations. The gates transform the initial state of the qubits into another state. Different problems will require different
quantum circuits, and the design of these circuits is a crucial aspect of quantum algorithm design.
4. Measurement to Obtain Results: The final step in a quantum program involves measuring the qubits to obtain results. Measurements in quantum computing are unique because they not only give you the
state of the qubits (either |0⟩ or |1⟩ for each qubit) but also collapse the quantum state to the measured state. The results of the measurement are then read out to classical bits, which can be
further processed on a classical computer if necessary. This step allows us to extract useful information from the quantum system into the classical world.
Quantum program compilation and execution
1. Create Your Quantum Algorithm: The first step in quantum programming is very similar to classic programming - you have to come up with an algorithm. Instead of typical if... then... else... type
instructions, you'll be working with a quantum circuit filled with quantum gates, which act as the basic instructions for your algorithm.
2. Choose Your Environment and Language: Next, you'll need to select an environment and programming language for writing your quantum code. Many beginners start with Python because of its simplicity
and the availability of quantum libraries, such as Qiskit from IBM or QLM from ATOS.
3. Transpile Your Code: After writing your code, you'll need to convert (or transpile) it into a language that quantum computers understand - this is often QASM (Quantum Assembly Language). The
transpiler not only translates your code but also optimizes it, ensuring your quantum circuit runs as efficiently as possible. This optimization takes into account the specific properties of the
quantum machine that will run the code.
4. Execute Your Quantum Circuit: The final step is to run your quantum circuit. You can do this in two ways - either on a classic computer using a quantum simulator or on a real quantum machine.
Running a circuit on a simulator is useful for testing and learning, as it allows you to experiment with "perfect" qubits without the hardware limitations or noise issues of real quantum
computers. However, to fully experience the power of quantum computing, you may want to run your code on a real quantum machine. This is typically done by sending your code over the internet to a
quantum computer service, like the one provided by IBM through Qiskit. Once received, your code is placed in a queue and executed when the machine is available.
Our scope for this tutorial
The main objective of the Quantum Computing simulation with R is to learn how to write quantum programs using the qsmulatR package. Thus we will only focus on point "2. Choose Your Environment and
Language" above by converting our R code into Qiskit with the qsimulatR::export2qiskit() function.
Here's a quick & dirty implementation with qsimulatR for a very famous quantum algorithm, try to identify the initialization, the problem specification and the measurement parts. The algorithm will
be explained in the next article.
x <- qstate(3)
x <- H(2)*(H(1)*x)
x <- H(3)*(X(3)*x)
x <- X(1) * (CCNOT(c(1,2,3)) *(X(1) * x))
for(i in c(1:3)) {
x <- H(i) * x
x <- X(1) * (X(2) * x)
x <- cqgate(bits = c(1,2), gate = Z(2L)) * x
x <- X(1) * (X(2) * x)
for(i in c(1:2)) {
x <- H(i) * x
qsimulatR::plot(x, qubitnames = c("x1", "x2", "j"))
hist(measure(e1 = x, bit = 2, repetitions = 1))
Remember, quantum programming can seem daunting at first, but with practice and patience, you'll start to grasp these new concepts and begin to appreciate the immense potential that quantum computing
offers. Happy coding!
[1] Bourreau E., Fleury, G., Lacomme P., "Introduction à l'informatique quantique, Apprendre à calculer sur des ordinateurs quantiques avec Python", Collection Blanche, Eyrolles, 2022
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Understanding Mathematical Functions: How To Determine If A Function I
Introduction: Understanding the Basics of Mathematical Functions
Mathematical functions are a fundamental concept in mathematics and are used to describe the relationship between one set of values (the input) and another set of values (the output). Understanding
functions is essential in various fields such as physics, engineering, economics, and more.
A. Define what a mathematical function is
A mathematical function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. It can be represented in various forms, including
algebraic expressions, graphs, and tables.
For example, the function f(x) = 2x + 3 represents a relationship between the input variable x and the output variable f(x), where the output is determined by multiplying the input by 2 and adding 3.
B. Explain the importance of distinguishing between linear and nonlinear functions
Distinguishing between linear and nonlinear functions is crucial as it helps in understanding the behavior and properties of different types of functions. Linear functions have a constant rate of
change and form a straight line when graphed, while nonlinear functions do not have a constant rate of change and do not form a straight line when graphed.
Understanding whether a function is linear or nonlinear is essential in various applications. For example, in economics, linear functions can be used to model simple relationships such as cost and
revenue, while nonlinear functions may be used to model more complex relationships such as demand curves.
C. Introduce the concept that functions can be represented in multiple ways
Functions can be represented in multiple ways, including graphically, algebraically, and numerically. Graphical representation involves plotting the function on a coordinate plane to visualize the
relationship between the input and output. Algebraic representation involves expressing the function using mathematical symbols and operations. Numerical representation involves tabulating the
input-output pairs of the function.
Understanding these different representations allows for a deeper insight into the behavior and characteristics of functions. For instance, a graphical representation can provide insights into the
slope and intercept of a linear function, while an algebraic representation can provide a formula to calculate the output for any given input.
Key Takeaways
• Linear functions have a constant rate of change.
• Nonlinear functions do not have a constant rate of change.
• Graphing the function can help determine linearity.
• Examining the equation for variables and exponents is crucial.
• Understanding the difference is essential for mathematical analysis.
Characteristics of Linear Functions
When it comes to understanding mathematical functions, it's important to be able to distinguish between linear and nonlinear functions. Linear functions have specific characteristics that set them
apart from nonlinear functions. Let's take a closer look at the key characteristics of linear functions.
A. Describe the constant rate of change in a linear function
One of the defining characteristics of a linear function is its constant rate of change. This means that for every unit increase in the independent variable, there is a constant increase or decrease
in the dependent variable. In other words, the function's output changes at a constant rate as the input changes. This is often referred to as the slope of the function.
For example, if we have a linear function y = 2x + 3, the constant rate of change is 2. This means that for every one unit increase in x, the value of y increases by 2.
B. Discuss slope-intercept form (y = mx + b) as a hallmark of linear equations
The slope-intercept form, y = mx + b, is a hallmark of linear equations. In this form, m represents the slope of the line, and b represents the y-intercept, which is the point where the line crosses
the y-axis. This form makes it easy to identify the slope and y-intercept of a linear function, which are key components in understanding its behavior.
For example, in the function y = 3x - 2, the slope is 3 and the y-intercept is -2. This tells us that the line has a steep slope and crosses the y-axis at the point (0, -2).
C. Provide examples of real-life scenarios that are modeled by linear functions
Linear functions can be found in various real-life scenarios, where there is a constant rate of change or a linear relationship between two variables. Some examples include:
• The relationship between time and distance traveled at a constant speed
• The relationship between the number of hours worked and the amount earned at a fixed hourly rate
• The depreciation of an asset's value over time at a constant rate
• The growth of a population at a constant rate
These examples demonstrate how linear functions can be used to model and analyze real-world phenomena, making them an important concept in mathematics and beyond.
Identifying Nonlinear Functions
When it comes to mathematical functions, it's important to be able to distinguish between linear and nonlinear functions. Nonlinear functions exhibit different characteristics and behaviors compared
to linear functions. In this section, we will explore the common traits of nonlinear functions, introduce different types of nonlinear functions, and provide practical examples to demonstrate how
they appear in real-world situations.
A. Common Traits of Nonlinear Functions
Nonlinear functions are characterized by their varying rates of change. Unlike linear functions, which have a constant rate of change, nonlinear functions exhibit changing rates of growth or decay.
This means that the relationship between the input and output values is not proportional or constant.
Another common trait of nonlinear functions is that they do not graph as straight lines. When plotted on a graph, nonlinear functions will curve, bend, or exhibit other non-linear shapes, indicating
their non-proportional nature.
B. Different Types of Nonlinear Functions
There are several types of nonlinear functions, each with its own distinct characteristics. Two common types of nonlinear functions are quadratic and exponential functions.
• Quadratic Functions: Quadratic functions are characterized by the presence of a squared term (x^2) in the equation. When graphed, quadratic functions form a parabola, which is a U-shaped curve.
Examples of quadratic functions include y = x^2 and y = -2x^2 + 3x - 1.
• Exponential Functions: Exponential functions involve a constant base raised to the power of the input variable. These functions exhibit rapid growth or decay and are commonly used to model
phenomena such as population growth, compound interest, and radioactive decay. Examples of exponential functions include y = 2^x and y = 3e^x.
C. Practical Examples of Nonlinear Functions in Real-World Situations
Nonlinear functions are prevalent in real-world scenarios and can be observed in various contexts. One common example is population growth, which is often modeled using an exponential function. As a
population grows, the rate of growth increases over time, resulting in a nonlinear relationship between the population size and time.
Another practical example of a nonlinear function is the distance traveled by a falling object. The distance-time relationship for a falling object is described by a quadratic function, as the
distance increases at an accelerating rate due to the influence of gravity.
Furthermore, financial applications such as compound interest and investment growth are modeled using exponential functions, showcasing the relevance of nonlinear functions in economic contexts.
By understanding the traits and types of nonlinear functions, as well as their real-world applications, individuals can gain a deeper appreciation for the diverse nature of mathematical functions and
their significance in various fields.
Graphical Analysis Technique
When it comes to determining whether a function is linear or nonlinear, one of the most effective techniques is to analyze the graph of the function. By visually inspecting the graph, you can often
discern whether the function exhibits a linear relationship or not.
A. Analyzing for Straight Lines
One of the first things to look for when analyzing the graph of a function is the presence of straight lines. Linear functions will have a graph that is a straight line, while nonlinear functions
will have a curved or irregular graph. By visually inspecting the graph, you can quickly determine if the function is linear or nonlinear.
B. Use of Coordinate Points and Plotting
Another important technique for determining linearity is to use coordinate points and plot values on the graph. By selecting a few points on the graph and plotting them, you can observe the pattern
of the points. For linear functions, the plotted points will form a straight line, while for nonlinear functions, the points will not align in a straight line.
C. Troubleshooting Common Errors
It's important to be aware of common graphing errors or misinterpretations that can lead to incorrect conclusions about the linearity of a function. Some common errors include mislabeling axes, using
incorrect scales, or misreading the graph. Always double-check your graph to ensure accuracy in your analysis.
Algebraic Approach: Analyzing Equations
When it comes to understanding mathematical functions, one of the key skills is being able to determine whether a function is linear or nonlinear. An algebraic approach to analyzing equations can
help in this process. By inspecting the equation and using simplification methods, it is possible to reveal the form of the function.
A. Inspecting an equation to identify linearity
When inspecting an equation to determine whether it represents a linear or nonlinear function, it is important to look for specific patterns and terms. In a linear function, the highest power of the
variable is 1, and the equation does not contain any products or powers of the variable. On the other hand, a nonlinear function may contain terms with powers other than 1, or products of the
For example, the equation y = 3x + 2 represents a linear function, as it contains only the first power of the variable x and no other terms. On the other hand, the equation y = 2x^2 + 5x + 1 is
nonlinear, as it contains a term with the second power of x.
B. Simplification methods to reveal the function's form
Once an equation is identified as potentially representing a linear or nonlinear function, simplification methods can be used to reveal the form of the function. This may involve rearranging terms,
factoring, or isolating the variable to make the form of the function more apparent.
For example, in the equation y = 2x^2 + 5x + 1, we can use the quadratic formula to determine the roots of the equation and understand its behavior. This can help in identifying whether the function
is linear or nonlinear.
C. Step-by-step examples to practice equation analysis
To gain a better understanding of how to analyze equations to determine linearity, it can be helpful to work through step-by-step examples. By practicing equation analysis, individuals can become
more adept at identifying the form of a function and understanding its behavior.
For instance, working through examples such as y = 4x - 3 and y = 2x^3 + 6x^2 - 5x + 1 can provide valuable practice in identifying the form of the function and determining whether it is linear or
Use of Technology and Tools
When it comes to analyzing mathematical functions, technology and tools play a crucial role in determining whether a function is linear or nonlinear. In this chapter, we will explore the
computational tools and software used to analyze functions, how to input functions into graphing calculators or software, and how to interpret the output from these technological tools accurately.
Introduce computational tools and software used to analyze functions
Computational tools and software such as graphing calculators, mathematical software like MATLAB, and online graphing tools like Desmos are commonly used to analyze mathematical functions. These
tools provide a visual representation of functions, making it easier to determine their linearity.
Explain how to input functions into graphing calculators or software to determine linearity
Inputting functions into graphing calculators or software involves entering the function in the appropriate format. For example, in graphing calculators, you would typically use the 'Y=' function to
input the equation. In mathematical software, you would use the appropriate syntax to define the function. Once the function is inputted, the software or calculator will generate a graph that can be
analyzed to determine linearity.
Provide guidance on interpreting the output from technological tools accurately
Interpreting the output from technological tools accurately is essential in determining the linearity of a function. When analyzing the graph generated by the software or calculator, it's important
to look for key indicators of linearity such as a straight line for linear functions or a curved line for nonlinear functions. Additionally, understanding how to read the axes and interpret the scale
of the graph is crucial in accurately determining the nature of the function.
Conclusion & Best Practices: Advancing Your Understanding of Functions
As we conclude our discussion on understanding mathematical functions, it is important to recap the significance of recognizing linear and nonlinear functions, encourage readers to practice with a
variety of functions, and share best practices to enhance their understanding.
A. Recap the importance of recognizing linear and nonlinear functions
• Understanding the distinction: Recognizing the difference between linear and nonlinear functions is crucial in various fields such as engineering, economics, and physics. It forms the foundation
for more advanced mathematical concepts.
• Impact on problem-solving: Identifying whether a function is linear or nonlinear can significantly impact the approach to problem-solving. It determines the methods and techniques used to analyze
and manipulate the function.
B. Encourage readers to practice with a variety of functions to enhance their skills
• Exploring diverse examples: Engaging with a wide range of functions, including both linear and nonlinear, allows readers to develop a deeper understanding of their characteristics and behaviors.
• Utilizing resources: Leveraging textbooks, online resources, and practice problems can provide ample opportunities to apply and test knowledge of different functions.
C. Share best practices such as double-checking work, consulting multiple sources, and seeking real-world applications to solidify understanding
• Double-checking work: Verifying solutions and calculations is essential to catch any errors and ensure accuracy in determining the linearity or nonlinearity of a function.
• Consulting multiple sources: Referring to various textbooks, academic papers, and reputable online sources can offer different perspectives and explanations, enriching one's understanding of
• Seeking real-world applications: Exploring how linear and nonlinear functions manifest in real-world scenarios, such as in business trends or scientific phenomena, can provide practical context
and solidify conceptual understanding.
By consistently applying these best practices and actively engaging with a diverse set of functions, readers can advance their understanding of mathematical functions and develop a strong foundation
for further mathematical exploration.
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All points on a circle are the same distance from the center. The radius tells us the distance from the center to any point on the circle.
Consider the circle with a radius of 13 units shown below:
The standard form of the equation of a circle is
\displaystyle \left(x-h\right)^{2}+\left(y-k\right)^2=r^2
radius of the circle
center of the circle
coordinates of any point on the circle
To check whether a point \left(x_1,y_1\right) is inside, on or outside a circle, we can compare the distance between that point and the center of the circle to the value of the radius.
Using the Pythagorean theorem, we can write these conditions as:
• If \left(x_1-h\right)^2+\left(y_1-k\right)^2<r^2 then \left( x_1,y_1 \right) is inside the circle
• If \left(x_1-h\right)^{2}+\left(y_1-k\right)^2=r^2 then \left( x_1,y_1 \right) is on the circle
• If \left(x_1-h\right)^2+\left(y_1-k\right)^2>r^2 then \left( x_1,y_1 \right) is outside the circle
Notice that these conditions are the same as substituting the point into the equation of the circle and comparing the values on each side.
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Indeed, for every sequence of functions (fn) there is a function f which is not a term of the sequence
WARSZAWA 1995
Z O F I A A D A M O W I C Z
Institute of Mathematics of the Polish Academy of Sciences Sniadeckich 8, 00-950 Warszawa, Poland´
E-mail: [email protected]
First we show a few well known mathematical diagonal reasonings. Then we concentrate on diagonal reasonings typical for mathematical logic.
1. Examples of mathematical diagonal reasonings.
Theorem 1 (Cantor’s Theorem). The set of reals is uncountable.
To prove the theorem we show that the set of sequences of zeros and ones, that is, the set of functions f such that f : N −→ {0, 1}, is uncountable.
Indeed, for every sequence of functions (fn) there is a function f which is not a term of the sequence. We define f as follows:
(1) f(n) = 0 if fn(n) = 1
1 if f[n](n) = 0
Hence it follows that all such functions cannot be arranged in a sequence.
Cantor’s construction of the reals
A real is here an appropriate equivalence class of a Cauchy sequence f . If we are given a sequence of sequences
f[0]: (f0)0,(f0)1. . . f1: (f1)0,(f1)1. . . . . . ,
which itself is a Cauchy sequence, then it is convergent to a certain Cauchy sequence which roughly is the diagonal of the above matrix.
1991 Mathematics Subject Classification: Primary 03B25.
Lecture given at the Banach Center Colloquium on 13th January 1994.
The paper is in final form and no version of it will be published elsewhere.
Theorem 2 (Baire’s theorem). A first category set in a complete (compact) space is meager.
O u t l i n e o f a p r o o f. Let A be a first category set, A = S
nA[n], where A[n] are nowhere dense. We have to show that in every ball K there is an element x such that x 6∈ A. Let K = K0. Let K[1] ⊆ K be disjoint with A1. Let x[1]∈ K1. Let K2 ⊆ K^1 be disjoint
with A2. We take x2∈ K^2. We continue. At the same time we ensure that (xn) is a Cauchy sequence — the balls are chosen in such a way that their radii converge to zero. We take x = lim x[n]. Then x
6∈ A.
We may treat the above proof as a diagonal reasoning — in the nth step we guarantee that x 6∈ An.
Another example of a diagonal reasoning:
Theorem 3. There is a function from N to N which is not definable.
Here we have to make precise what is meant by definability.
We are given the set of positive integers N with the functions +, · and relations
=, < and with the distinguished elements 0, 1; i.e. we are given the relational structure N = hN, +, ·, =, <, 0, 1i.
On the other hand we are given the language: the variables x[1], x[2], x[3], . . ., the relation and function symbols +, ·, =, <, the constants 0, 1 (the symbols +, ·, =, <, 0, 1 are used here in two
different meanings — as functions, relations and numbers and as symbols of the language), the connectives ∨, ∧, ¬ and the quantifiers ∃, ∀. Now we define a formula of this language. By the terms of
this language we mean the symbols of the form such as e.g.:
(2) (((x[i]1+ x[i]2) · xi3+ x[i]4) · xi5· xi6) + x[i]7.
A formula may be atomic, of the form t1= t2, t1< t2, where t1, t2are terms, or more complex, e.g. ∃x^1 (x1+ 0 = x1). More complex formulas are obtained by joining the simpler ones with the use of the
connectives or by adding quantifiers to the simpler ones.
A set A ⊆ N^k is definable if there is a formula φ(x) such that (3) A= {hn^1, . . . , n[k]i ∈ N^k: φ(n1, . . . , n[k])}, e.g.
(4) A= {n ∈ N : ∃m (n = m + m)}
— the set of even numbers,
(5) f = {hn, mi : m · m < n < (m + 1) · (m + 1) ∨ m · m = n}
— the function f (n) = [√ n].
Now we show that there is a nondefinable function from N to N .
Since the language is countable, there are countably many definitions in it (that is countably many of the appropriate formulas φ). Thus there are countably many definable functions. Let us arrange
all such functions in the sequence:
f0: f0(0), f0(1), . . . f[1]: f1(0), f1(1), . . . . . .
and define f (n) = fn(n) + 1. Then the function f is not a term of this sequence, and thus is not definable.
Similarly, we show that there is an ordinal number which is not definable.
Consider the language of set theory. Here we have two relation symbols =, ∈. As before, the language is countable, and thus there are countably many definable ordinal numbers. Let us denote these
numbers by α[1], α[2], . . .. Let α be the least ordinal number greater than all these numbers. Then α is different from all the α[i], and thus is not definable.
Here we have obtained one of the well known “paradoxes” of the beginning of our century — on one hand α is not definable, and on the other we just have defined it.
The case of our function f is similar — we have given its definition.
We explain this paradox in section 3.
2. Universal relations. Consider the family of open sets in the Baire space N^N. As the basis of the topology we take the sets B[s] determined by finite sequ- ences s = hhn^1, m[1]i, . . . , hn^k, m
[k]ii of pairs of natural numbers:
(6) B[s] = {f ∈ N^N : f (n1) = m1& . . . &f (nk) = mk}.
The basis is countable, we may enumerate it B1, B2, . . ..
Let f ∈ N^N. Let A[f] denote the open set S
Now consider the set A = {hf, gi : g ∈ Af}. It is easy to see that A is open in N^N × N^N with the product topology. We can look at A as at a plane set
Fig. 1
where at the axes we put N^N. Then every vertical section of A (as on the picture) determines a certain open set Af and conversely, every open set in N^N is a certain such section.
We say that A is a universal relation for open sets in N^N.
In this case there is a universal relation for open sets in N^N which itself is an open set.
Similarly we may define a universal relation for Borel sets in N^N. As we shall see, this relation is no more Borel.
We have the following property:
Theorem 4. If we are given a universal relation for a certain family of sets then it determines a set which is not in the family.
For example, consider again our relation A(f, g) universal for open sets. Let the set B be defined as follows:
(7) f ∈ B ⇔ ¬A(f, f).
We show that B is not open. Indeed, suppose that B is open. Then there exists g such that B = Ag. We have
(8) g∈ Ag ⇔ ¬A(g, g) ⇔ g 6∈ Ag,
contradiction. Thus the set B is not open (it is closed).
Let now A(f, g) be a universal relation for Borel sets. Let B be defined as above. Similarly as before we show that B is not Borel. But notice that if A was Borel then B would also be Borel (here we
make use of the fact that the family of Borel sets is closed under complementation — unlike for open sets). Hence A is not Borel.
It can be shown that the relation A can be chosen in such a way that it is a continuous image of a Borel set. Hence it follows that a continuous image of a Borel set is not necessarily Borel.
Here we have an opportunity to mention a famous mistake of Lebesgue — in one of his papers Lebesgue studied continuous images of Borel sets and claimed that they were Borel. This was one of those
mistakes in the history of mathematics which turned out to inspire its development — in this case the development of the theory of the analytic sets — exactly continuous images of Borel sets.
Again one has to refer to Lebesgue when speaking about universal relations
— this notion occurred for the first time in the paper of Lebesgue of 1905, in which he investigated universal relations for particular classes of Borel sets.
To end this section we show that the proof of the theorem about the none- xistence of the set of all sets can be presented as an application of the above method.
We show that the class A = {x : x is a set & x 6∈ x} is not a set (Russel’s paradox). Consider the universal relation φ(x, y) for relations x(y) defined as y∈ x, where x is a set. We have
(9) φ(x, y) ⇔ y ∈ x.
Then A = {x : ¬φ(x, x)}. In view of what we have already shown, A does not lie in the domain of the universal relation φ, and thus is not a set.
3. Universal formulas. Instead of universal relations we may speak about universal formulas — definitions of those relations. Let us come back to arithme- tic. There are countably many formulas of
the language of arithmetic, thus we may enumerate them with numbers, and moreover we may do it in an effective way. We may even, up to this enumeration, identify formulas with the appropriate
numbers. Let us ask whether there exists a universal relation for sets definable in N. That is, whether there exists such a relation A(ϕ, x) that the appropriate vertical section Aϕ is the set
defined by ϕ (cf. Fig. 1). That is, we look for a relation A ⊆ N × N satisfying the condition:
(10) A(ϕ, x) ⇔ x ∈ A^ϕ⇔ ϕ(x).
Of course, there is a set A with the above property, defined as above. However, we may ask whether A itself is definable. Let us pose the following question:
Is there a formula φ(ϕ, x) such that
(11) φ(ϕ, x) ⇔ ϕ(x)
for all the formulas ϕ?
Here we enter the question of the existence of universal formulas for classes of formulas, i.e. the existence of formulas φ having the property φ(ϕ, x) ⇔ ϕ(x), where ϕ runs over a certain class of
formulas. We may also consider universal formulas for classes of sentences, i.e. formulas having the property φ(ϕ) ⇔ ϕ, where ϕ runs over a certain class of sentences. This is a kind of speaking
about speaking. Let us recall a famous example of Tarski. We may say
It is snowing and we may also say
The sentence “it is snowing” is true.
Each of these sentences is true if it is really snowing.
If φ is a universal formula for sentences ϕ, then the formulation of the sentence ϕcorresponds to the sentence “It is snowing” and the formulation of the sentence φ(ϕ) corresponds to the sentence
“The sentence ‘it is snowing’ is true”.
Digression — a story about brothers.
At a splitting of roads
1 S^2
there live two brothers A and B. The brother A always tells truth, and the brother B always lies. A traveller goes to a town M . He stops at the splitting, he meets one of the brothers (he does not
know which one) and he is allowed to ask just one question to learn the correct way.
It turns out that the appropriate question requires a reference to “speaking about speaking”. Namely, the question is
Which way would your brother show me?
It is easy to check that no matter what answer the traveller gets he should choose the other way.
Let us try to interpret this story. Let p[i](i = 1, 2) be the sentence “You should take the way i”. Let φA(p) be the formula “A says the sentence p”, and φB(p)
“B says the sentence p”.
We have φ[A](p) ⇔ p (i.e. φA is a universal formula for the sentences p) and φ[B](p) ⇔ ¬p.
If the answer to the question is pi and the brother met is A, then we have φ[A](φ[B](p[i])), and thus φ[B](p[i]), i.e. ¬pi. If the brother met is B then we have φ[B](φ[A](p[i])), and thus ¬φA(p[i]),
i.e. ¬pi.
We have the following theorem
Theorem5. There is no universal formula for all formulas (of one variable).
There is no universal formula for all sentences.
P r o o f. Suppose that φ is a universal formula for all formulas. Then we have
(12) φ(ϕ, x) ⇔ ϕ(x)
for all formulas ϕ(x). Consider the formula ψ(x): ¬φ(x, x). Then we have
(13) ¬φ(ψ, ψ) ⇔ ψ(ψ) ⇔ φ(ψ, ψ)),
The second part of Theorem 5 immediately follows from the theorem of G¨odel:
Theorem 6 (G¨odel’s diagonal lemma). For any formula ψ(x) there is a sen- tence ϕ such that ϕ is true if and only if ψ(ϕ) is true.
The lemma says that for any property ψ(x) there is a sentence ϕ which has the meaning “I have the property ψ”.
Suppose now that φ(x) is a universal formula for all sentences. Ley ψ be the sentence from the G¨odel diagonal lemma for the formula ¬φ. Then we have
(14) ¬φ(ψ) ⇔ ψ ⇔ φ(ψ),
From the G¨odel diagonal lemma we also easily infer the following theorem:
Theorem 7 (Tarski’s theorem on nondefinability of truth). The set of sen- tences of the language of arithmetic that are true in N is not definable in N by a formula of this language.
P r o o f. Suppose that φ(x) defines the set of sentences true in N. Thus we have
(15) φ(ϕ) ⇔ ϕ
for all sentences ϕ.
Let now ψ be defined as in the previous proof, that is ψ holds if and only if ¬φ(ψ) holds. If ψ is true, then on one hand ¬φ(ψ) holds, by the choice of ψ, and on the other hand φ(ψ) holds, since φ
defines the set of the true sentences.
Similarly, if ψ is false, then on one hand φ(ψ) holds, by the choice of ψ, and on the other hand φ(ψ) does not hold, since ψ does not belong to the set of true sentences. We obtain a contradiction.
The above theorem holds not only for arithmetic, but it is quite general. It holds for most of the mathematical theories, in particular for set theory.
Therefore, we cannot express in a given language the notion of truth for sen- tences of the language. In particular we are not able to express the fact that the number n belongs to the set defined by
the formula ϕ(x) — that ϕ(n) is true.
Thus, there is no universal formula for the family of definable sets — the answer to the question posed at the beginning of this section is negative. In particular, the function diagonalizing the
definable functions and the ordinal number defined in section 1 are not defined in that language to which the notion of definability there considered refers.
4. Tarski’s truth definition. Up to now we have said about a sentence that it is “true” or about a formula φ(x) that it “holds” for a number n, in an intuitive way. The notion of the satisfiability
of a formula φ(x1, . . . , x[k]) in a given relational structure by the sequence hn^1, . . . , n[k]i of elements of the universe of the structure may be defined in a precise way. Again, let us do it
for arithmetic, for another language or another structure this can be done similarly.
If t is a term, for instance the term considered in section 1 (16) t= (((x[i]1+ x[i]2) · xi3+ x[i]4) · xi5· xi6) + x[i]7,
then by the value of this term at the sequence hn^i^1, . . . , n[i][k]i, t(n^i^1, . . . , n[i][k]), we mean the number
(17) (((n[i]1+ n[i]2) · ni3+ n[i]4) · ni5· ni6) + n[i]7.
The atomic formula t[1] = t[2] or t[1] < t[2] is satisfied in N by the sequence hni1, . . . , n[i]ki if respectively
— the natural number t1(n[i]1, . . . , n[i]k) is equal to the number t2(n[i]1, . . . , n[i]k) or
— the number t1(n[i]1, . . . , n[i][k]) is less than t2(n[i]1, . . . , n[i][k]).
Further on we proceed inductively.
— ¬ψ(xi1, . . . , x[i]k) is satisfied by hni1, . . . , n[i]ki if ψ is not satisfied by hn^i^1, . . . , n[i][k]i.
— ψ1 ∨ ψ^2(xi1, . . . , x[i][k]) is satisfied by hn^i^1, . . . , n[i][k]i if ψ^1 is satisfied or ψ2 is satisfied by hni1, . . . , n[i]ki.
— ψ[1]∧ ψ2(x[i]1, . . . , x[i]k) is satisfied by hni1, . . . , n[i]ki if ψ1 is satisfied and ψ[2] is satisfied by hni1, . . . , n[i][k]i.
— ∃x ψ(x, xi1, . . . , x[i]k) is satisfied by hni1, . . . , n[i]ki if there exists a number n in N such that ψ is satisfied by hn, n^i^1, . . . , n[i][k]i.
As we see, at one side of these definitions there occur symbols of our language
— the one under consideration, about which we speak, and at the other side the words “not, or, and, there exists” of the language in which we speak (called meta- language). As we showed before it is
not possible to express the above definition in the language under consideration — truth can be defined only from outside.
5. First and second G¨odel’s theorems. Consider the declaration “I am lying”. Observe that it is neither true nor false — if I am telling truth then I am lying, and if I am lying then I am telling
Is the sentence “I am lying” expressible in the language of arithmetic?
We are looking for a sentence ϕ such that ϕ was equivalent with the sentence
“ϕ is not true”. However the property “is not true” cannot be expressed in our language — since we cannot express the property “is true”. Indeed, by the Tarski theorem on the nondefinability of
truth, there is no arithmetical formula φ(ϕ) meaning “ϕ is true”. We cannot express the sentence “I am lying” as a mathe- matical sentence. However, we may express a slightly different sentence,
namely the sentence “I am not provable”. There is an arithmetical formula T such that T(ϕ) has the meaning “ϕ has a proof in arithmetic (is a theorem of arithmetic)”.
Now let us outline the construction of the formula T .
First, let us make precise what theory is meant by arithmetic. Let this theory be denoted by P (from Peano). The axioms of the arithmetic P are the sentences:
(18) ∀x, y x + y = y + x, ∀x, y x · y = y · x
(19) ∀x, y, z (x + y) + z = x + (y + z), ∀x, y, z (x · y) · z = x · (y · z) (20) ∀x, y, z (x + y) · z = x · z + y · z
(21) ∀x x + 0 = x, x· 1 = x
(22) ∀x, y, z (x + z = y + z ⇒ x = y)
(23) ∀x (x 6= 0 ⇔ ∃y x = y + 1)
(24) ∀x, y (x < y ⇔ ∃z x + z + 1 = y) and all the sentences:
(25) (ϕ(0) ∧ ∀ x(ϕ(x) ⇒ ϕ(x + 1))) ⇒ ∀x ϕ(x), where ϕ is a formula of the language.
Here we have used the connectives ⇒ and ⇔ which were not introduced in the definition of the language — one has to replace them by the appropriate combinations of the connectives ¬, ∨, ∧.
Thus, the arithmetic P is a certain (infinite) set of sentences. It is easy to see that this set of sentences is definable in N — it is a set of sentences of a particular form which can be described
in the language of arithmetic. Let P (x) denote the formula defining this set of sentences in N.
Let now d = hψ^1, . . . ψ[n]i be a sequence of formulas. Sequences of numbers can be treated as numbers — we identify them with their numbers under a certain effective enumeration of sequences. We
say that d is a proof of the sentence ϕ in the theory P , if ψ[n] is the sentence ϕ, and every ψ[i] is either an axiom (P (ψ[i]) holds) or there are j, k < i such that ψk is the formula ψj ⇒ ψ^i —
that is, ψi
can be derived from the previous formulas by the modus ponens rule. It is easy to see that the above description can be carried out in arithmetic — thus there is a formula D(d, ϕ) expressing the
meaning “d is a proof ϕ in P ”.
Now we can define our formula T (ϕ) as ∃d D(d, ϕ).
Theorem 8 (First G¨odel’s theorem). There is an arithmetical sentence ϕ independent from arithmetic such that both ϕ and ¬ϕ have no proof in arithmetic.
P r o o f. Let ϕ be the sentence from the G¨odel diagonal lemma for the formula
¬T . Then we have:
ϕ holds if and only if T (ϕ) does not hold.
Thus ϕ has the meaning “I am not provable”.
Suppose that ϕ has a proof. Then T (ϕ) holds, contradiction.
Suppose now that ¬ϕ has a proof. Then ¬ϕ is true. In this case ϕ is false, and thus T (ϕ) holds. Thus ϕ has a proof in P . Hence both ¬ϕ and ϕ have proofs in P, contradiction.
Again, this theorem concerns not only arithmetic, but almost every mathe- matical theory. In particular it is true for set theory. This means that there are sentences independent from set theory.
Moreover, even if we add such a sentence to set theory as an axiom, then we obtain a theory for which again the first G¨odel theorem holds, and thus again there are sentences independent from that
We see that this theorem puts bounds to our ability of knowing — there are true sentences which we cannot prove — we cannot grasp the whole truth.
Notice that we are able to formulate in arithmetic a sentence with the meaning
“the arithmetic is consistent”. Indeed, let Cons(P ) be the sentence
∀d ¬D(d, “0 = 1”) — the contradiction “0 = 1” has no proof in P .
Theorem 9 (Second G¨odel’s theorem). There is no proof of the sentence Cons(P ) in P .
Similarly as before, this theorem concerns not only the theory P , but most theories. It can be read as:
In a given theory it is not possible to prove the consistency of this theory.
Problem. Is it possible to prove G¨odel’s first or second theorem without the diagonal lemma? Is it possible to prove them without diagonalizing at all?
A partial answer has recently been given by H. Kotlarski.
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Using ion propulsion
So supposing I've potentially got 20 tonnes into LEO, but I actually want to get something to the Moon, what's the fraction of that 20 tonnes that has to be propellant?
I'm going to assume that I can have my 20 tonnes in an orbit parallel with that of the Moon, at an altitude of 1000 km. I'm also going to assume that the only problem is getting out of the Earth's
gravitational field, to a target orbit the same as the Moon's (i.e. an orbital radius of 384000 km). So from the equations for a circular orbit (see for instance an A-level Physics textbook) initial
velocity v[0] = 7.35 kms^-1 and final velocity v[f] = 1.02 kms^-1, giving Δv = 6.33x10^3 ms^-1.
I'm going to make the blatant assumption that as much electrical power is available as I require, and use a high-powered ion thruster. Now, the example I looked up on the Internet [1] had a quoted
specific impulse I[sp] of 3800 s. I can use the fact that exhaust velocity is equal to specific impulse divided by the gravitational field strength on the Earth's surface (g = 9.81 ms^-2) to find
exhaust velocity v[e] = 37240 ms^-1. Then from the rocket equation in the form e^-Δv / v[e] = m[f] / m[0] the ratio of final mass to initial mass is 0.843.
This is interesting, because it implies that out of 20 tonnes in LEO 16 tonnes will make it to lunar orbit: a much better ratio than for a conventional booster! Assuming, however, the payload is
destined for the Moon's surface, that sixteen tonnes must include: the ion engines themselves and fuel tankage; the solar panels required to power the ion engines; the landing retrorockets and enough
fuel to land the payload; and the landing gear. So it's probable that less than half of the original 20 tonnes would consist of non-propulsion payload. But the setup would probably still be an
improvement on a conventional chemical-rocket-only system.
[1] IslandOne, Advanced Propulsion Concepts
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Multivariate generalized propensity score: An introduction
Data Generating
To illustrate a simple setting where this multivariate generalized propensity score would be useful, we can construct a directed acyclic graph (DAG) with a bivariate exposure, D=(D[1], D[2]),
confounded by a set C=(C[1], C[2], C[3]). In this case we assume C[1] and C[2] are associated with D[1], while C[2] and C[3] are associated with D[2] as shown below.
To generate this data we first draw n=200 samples from C assuming a multivariate normal distribution with mean equal to zero, variance equal to 1, and constant covariance of 0.1.
Next we define our exposure as a linear function of our confounders. Explicitly these two equations are defined as
With this construction, the exposures have one confounder in common, C[2], and one independent confounder. The effect size of the confounders vary for each exposure. We assume that the conditional
distribution of D given C is bivariate normal with conditional correlation equal to 0.2 and conditional variance equal to 2.
To generate the set of confounders and the corresponding bivariate exposure we can use the function gen_D() as shown below.
sim_dt <- gen_D(method="u", n=200, rho_cond=0.2, s_d1_cond=2, s_d2_cond=2, k=3,
C_mu=rep(0, 3), C_cov=0.1, C_var=1,
d1_beta=c(0.5, 1, 0), d2_beta=c(0, 0.3, 0.75), seed=06112020)
D <- sim_dt$D
C <- sim_dt$C
By construction our marginal correlation of D is a function of parameters from the distribution of C, coefficients of conditional mean equations, and conditional covariance parameter. For the above
specification the true marginal correlation of exposure is equal to 0.24 and our observed marginal correlation is equal to 0.26.
Finally, we specify our outcome, Y, as a linear combination of the confounders and exposure. The mean of the dose-response equation is shown below,
E[Y|D, C]=0.75C[1]+1C[2]+0.6C[3]+D[1]+D[2].
Both exposures have treatment effect sizes equal to one. The standard deviation of our outcome is set equal 2.
Generating Weights
With the data generated, we can now use our primary function mvGPS() to estimate weights. These weights are constructed such that the numerator is equal to the marginal density, with the denominator
corresponding to the conditional density, i.e., the multivariate generalized propensity score.
w = f(D)/f(D|C)
In our case since the bivariate exposure is assumed to be bivariate normal, we can break both the numerator and denominator into full conditional densities knowing that each univariate conditional
expression will remain normally distributed.
w = f(D2|D1)f(D1)/f(D2|D1,C2,C3)f(D1|C1,C2)
Notice in the equation above, we are also able to specify the confounding set for each exposure separately.
This vector w now can be used to test balance of confounders by comparing weighted vs. unweighted correlations and to estimate the treatment effects using weighted least squares regression.
Balance Assessment
For continuous exposure(s) we can asses balance using several metrics such as euclidean distance, maximum absolute correlation, and average absolute correlation where correlation refers to the
Pearson correlation between exposure and covariate.
Below we use the function bal() to specify a set of potential models to use for comparison. Possible models that are available include: mvGPS, Entropy, CBPS, GBM, and PS. For methods other than mvGPS
which can only estimate univariate continuous exposure, each exposure is fit separately so that weights are generated for both exposures.
bal_results <- bal(model_list=c("mvGPS", "entropy", "CBPS", "PS", "GBM"), D, C=list(C[, 1:2], C[, 2:3]))
bal_summary <- bal_results$bal_metrics
#contains overall summary statistics with respect to balance
bal_summary <-data.frame(bal_summary, ESS=c(bal_results$ess, nrow(D)))
#adding in ESS with last value representing the unweighted case
bal_summary <- bal_summary[order(bal_summary$max_cor), ]
kable(bal_summary[, c("euc_dist", "max_cor", "avg_cor", "ESS", "method")],
digits=4, row.names=FALSE,
col.names=c("Euc. Distance", "Max. Abs. Corr.",
"Avg. Abs. Corr.", "ESS", "Method"))
Euc. Distance Max. Abs. Corr. Avg. Abs. Corr. ESS Method
0.0930 0.0884 0.0331 163.8253 mvGPS
0.2568 0.2145 0.1137 159.9284 GBM_D1
0.2592 0.2288 0.1085 179.8179 PS_D1
0.3095 0.2321 0.1092 178.8683 entropy_D2
0.2400 0.2336 0.0721 172.2635 entropy_D1
0.2843 0.2400 0.1236 184.3871 CBPS_D1
0.3185 0.2418 0.1227 180.3586 PS_D2
0.3285 0.2502 0.1219 142.8799 GBM_D2
0.5009 0.3168 0.2421 200.0000 unweighted
1.1022 0.6701 0.5012 50.2385 CBPS_D2
We can see that our method mvGPS achieves the best balance across both exposure dimensions. In this case we can also note that the effective sample size after weighting 163.8253 is still sufficiently
large that we not worried about loss of power.
Bias Reduction
Finally, we want to check that these weights are properly reducing the bias when we estimate the exposure treatment effect.
dt <- data.frame(Y, D)
mvGPS_mod <- lm(Y ~ D1 + D2, weights=w, data=dt)
mvGPS_hat <- coef(mvGPS_mod)[c("D1", "D2")]
unadj_hat <- coef(lm(Y ~ D1 + D2, data=dt))[c("D1", "D2")]
bias_tbl <- cbind(truth=c(1, 1), unadj=unadj_hat, mvGPS_hat)
kable(bias_tbl, digits=2, row.names=TRUE,
col.names=c("Truth", "Unadjusted", "mvGPS"))
Truth Unadjusted mvGPS
D1 1 1.28 1.12
D2 1 1.18 1.05
To compare the total reduction at bias we look at the total absolute bias where we see mvGPS has total bias equal to 0.18, or an average percent bias of 8.79% per exposure, compared to unadjusted
total bias equal to 0.45, or an average percent bias of 22.62% per exposure. We therefore achieve 2.57 times reduction in bias.
Defining Estimable Region
An important consideration when using propensity scores to estimate causal effects are the three key identifying assumptions:
1. weak ignorability, aka, unconfoundedness, aka, selection on observables
2. stable unit treatment value (SUTVA)
3. positivity
Weak ignorability assumes that the exposure is conditionally independent of the potential outcomes given the appropriate set of confounders. Checking balance as shown above is one of the key
diagnostics to determining the legitimacy of this assumption in practice.
SUTVA states that the potential outcome of each unit does not depend on the exposure that other units receive and that there exists only one version of each exposure. It is generally an untestable
assumption, but is key to ensuring that the potential outcomes are well-defined and that the observed outcome given the observed exposure corresponds to the true potential outcome.
The final identifying assumption, positivity, is our focus when defining estimable regions for multivariate exposure. Positivity posits that all units have the potential to receive a particular level
of exposure given any value of the confounders. The upshot of this is that we need to take care when defining the domain of our exposure when estimating the mvGPS.
Typically in the case of univariate continuous exposure, we often ensure positivity by restricting the domain to the observed range of exposure or a trimmed version. A logical extension to the
multivariate exposure would be to define our domain as the product of the range of each exposure. However, when the exposures of interest are correlated this domain may not be appropriate. Recall
that in our simulated data the marginal correlation of D[1] and D[2] is 0.26.
Instead, we propose to ensure positivity with multivariate exposures by defining the domain as the multidimensional convex hull of the observed exposure values. To obtain the convex hull of our
exposure we use the function hull_sample(). This will return the vertices of the convex hull, and in the case of bivariate exposure it will also sample equally along a grid of the convex hull and
return these values which can be used for calculating the dose-response surface.
Note that we can also create trimmed versions of either the product of ranges or convex hull as shown below.
chull_D <- hull_sample(D)
#generate convex hull of exposure
chull_D_trim <- hull_sample(D, trim_hull=TRUE, trim_quantile=0.95)
#generate trimmed convex hull
bbox_grid <- sp::bbox(chull_D$hpts_vs) #bounding box over convex hull
bbox_df <- data.frame(D1=c(bbox_grid[1, 1], bbox_grid[1, 2],
bbox_grid[1, 2], bbox_grid[1, 1]),
D2=c(bbox_grid[2, 1], bbox_grid[2, 1],
bbox_grid[2, 2], bbox_grid[2, 2]))
bbox_grid_trim <- sp::bbox(chull_D_trim$hpts_vs) #bounding box over trimmed convex hull
bbox_df_trim <- data.frame(D1=c(bbox_grid_trim[1, 1], bbox_grid_trim[1, 2],
bbox_grid_trim[1, 2], bbox_grid_trim[1, 1]),
D2=c(bbox_grid_trim[2, 1], bbox_grid_trim[2, 1],
bbox_grid_trim[2, 2], bbox_grid_trim[2, 2]))
chull_plot <- ggplot(data=data.frame(D), aes(x=D1, D2))+
geom_polygon(data=data.frame(chull_D$hpts_vs), color="indianred4", fill=NA)+
geom_polygon(data=data.frame(chull_D_trim$hpts_vs), color="indianred1", fill=NA, alpha=0.4)+
geom_polygon(data=bbox_df, color="dodgerblue4", fill=NA)+
geom_polygon(data=bbox_df_trim, color="dodgerblue1", fill=NA, alpha=0.4)+
In dark red we have the observed convex hull and in light red we have the trimmed convex hull at the 95th percentile. In dark blue we have the observed product range and in light blue we have the
trimmed product range at the 95th percentile.
Notice that by trimming we are further restricting our domains to high density regions of the exposure. We can also see that by restricting to the convex hull we are avoiding areas with sparse data
that are included in the product range domain.
Dose-Response Surface
When exposure is bivariate, the resulting dose-response function is a surface. Using the weighted regression model described above to incorporate the weights, we can predict across our convex hull
domain to gain intuition about how altering the exposures effects the outcome of interest. To see an example of this type of dose-response surface on an application to analyzing obesity intervention
programs in Los Angeles County visit https://williazo.github.io/resources/.
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Numbers in Awa Pit
Learn numbers in Awa Pit
Knowing numbers in Awa Pit is probably one of the most useful things you can learn to say, write and understand in Awa Pit. Learning to count in Awa Pit may appeal to you just as a simple curiosity
or be something you really need. Perhaps you have planned a trip to a country where Awa Pit is the most widely spoken language, and you want to be able to shop and even bargain with a good knowledge
of numbers in Awa Pit.
It's also useful for guiding you through street numbers. You'll be able to better understand the directions to places and everything expressed in numbers, such as the times when public transportation
leaves. Can you think of more reasons to learn numbers in Awa Pit?
The Awa Pit language, also known as Cuaiquer or Kwaiker, belongs to the Awan group of the Barbacoan languages family. Spoken by the Awá or Awa-Kwaiker people of Ecuador (in the provinces of Carchi
and Sucumbios) and Colombia (in the departments of Nariño and Putumayo), the Awa Pit language counts about 13,000 speakers.
List of numbers in Awa Pit
Here is a list of numbers in Awa Pit. We have made for you a list with all the numbers in Awa Pit from 1 to 20. We have also included the tens up to the number 100, so that you know how to count up
to 100 in Awa Pit. We also close the list by showing you what the number 1000 looks like in Awa Pit.
• 1) maza
• 2) paz
• 3) kutña
• 4) ampara
• 5) chish
• 6) wak
• 7) pikamta
• 8) ita
• 9) toil
• 10) pazchish
• 11) maza maza
• 12) maza paz
• 13) maza kutña
• 14) maza ampara
• 15) maza chish
• 16) maza wak
• 17) maza pikamta
• 18) maza ita
• 19) maza toil
• 20) paz chalkuil
• 30) kutña chalkuil
• 40) ampara chalkuil
• 50) chish chalkuil
• 60) wak chalkuil
• 70) pikamta chalkuil
• 80) ita chalkuil
• 90) toil chalkuil
• 100) pik:
• 1,000) im:
Numbers in Awa Pit: Awa Pit numbering rules
Each culture has specific peculiarities that are expressed in its language and its way of counting. The Awa Pit is no exception. If you want to learn numbers in Awa Pit you will have to learn a
series of rules that we will explain below. If you apply these rules you will soon find that you will be able to count in Awa Pit with ease.
The way numbers are formed in Awa Pit is easy to understand if you follow the rules explained here. Surprise everyone by counting in Awa Pit. Also, learning how to number in Awa Pit yourself from
these simple rules is very beneficial for your brain, as it forces it to work and stay in shape. Working with numbers and a foreign language like Awa Pit at the same time is one of the best ways to
train our little gray cells, so let's see what rules you need to apply to number in Awa Pit
Digits from zero to nine are rendered by specific words: chalkuil [0], maza [1], paz [2], kutña [3], ampara [4], chish (or shish) [5], wak [6], pikamta (or pikam) [7], ita [8], and toil (or tuil)
Tens are formed starting with the multiplier digit, then the word for zero (chalkuil), separated with a space, following a positional naming convention, with the exception of ten which can be
expressed as pazchish (or 2*5): pazchish (2*5), pazshish (2*5) or maza chalkuil [10] (1 0), paz chalkuil [20] (2 0), kutña chalkuil [30], ampara chalkuil [40], chish chalkuil (or shish chalkuil)
[50], wak chalkuil [60], pikamta chalkuil (or pikam chalkuil) [70], ita chalkuil [80], and toil chalkuil (or tuil chalkuil) [90].
Compound numbers are formed following the same system, i.-e. starting with the ten multiplier, then the unit separated with a space (e.g.: maza wak [16] (1 6), paz kutña [23] (2 3), ita ita [88] (8
Hundreds are formed starting with the multiplier digit, followed by the word for hundred (pik:), except for one hundred: pik: [100], paz pik: [200], kutña pik: [300], ampara pik: [400], chish pik:
[500], wak pik: [600], pikam pik: [700], ita pik: [800], and toil pik: [900].
Thousands are formed starting with the multiplier digit, followed by the word for thousand (im:), except for one thousand: im: [1,000], paz im: [2,000], kutña im: [3,000], ampara im: [4,000], chish
im: [5,000], wak im: [6,000], pikam im: [7,000], ita im: [8,000], and toil im: [9,000].
The word for million is kɨm: [million, 10^6].
AwaPit Pɨnkɨh Kammu Gramática Pedagógica (pdf), Ministerio de educación, Ecuador, 2009
Numbers in different languages
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Adjustable Precision
June 15, 2005
By Karen Kenworthy
This was the second time my 6th grade math teacher would change my life.
It was late in the day, late in the school year too, when everyone's thoughts were far away. The inaudible hum that seduces students with promises of sleep drifted through the warm, still air of the
classroom. Armed with nothing more powerful than his voice and a piece of chalk, Mr. Miller valiantly fought to impart some obscure bit of mathematical lore -- something I'd come to use everyday of
my life, if only I'd paid attention.
I was moments away from surrendering to the temptation of sleep when, like lightening, one word vaporized the fog enveloping me. "Infinity", Mr. Miller said, in a context I do not remember, and
probably never heard.
But I'll never forget our conversation that followed. He patiently answered all the questions that flooded into my 11 year old mind. And he planted there an interest in the more abstract areas of
mathematics that quickly grew, and has never withered.
Unlike such notions as "least common denominator" or "multiplicand", the mathematical concept of "infinity" fascinates me. It belongs to a mysterious realm outside our normal experience. It breaks
all the rules.
Subtract one infinity from another. What's left? Infinity. Divide any number you can imagine into infinity. What's the result? Another infinity. And so it goes. Infinity is the real bad boy of
Lest you think infinity never intrudes on our day-to-day lives, consider the circle. As I'm sure you know, mathematicians call the ratio of its circumference to its diameter "Pi". No matter how
large, or small, the circle, the value of Pi is always approximately 3.
That means that a trip around the outside of a circle (its circumference) is approximately three times longer than a shortcut across its middle (its diameter).
But the value of Pi isn't exactly 3. A more accurate approximation is 3.141592654. But that's not the correct value either. And neither is
3.141592653589793238462643383279502884197169399375105820974944592307816406 286208998628034825342117068, though it's a bit closer.
What is the exact value of Pi? The truth is, no one knows. That's because it contains an infinite number of digits. Even though Pi has been calculated with an accuracy of over one trillion digits,
the end is no where in sight.
Pi isn't the only number that's impossible to pin down. Other mathematical constants, such as e (Euler's Number -- the "base of natural logarithms" also requires an infinite number of digits to
express precisely. The same is true for some very common numbers, such as one-third (1 divided by three, or .33333333333 ad infinitum).
Adjustable Precision
I came face to face with infinitely long numbers when creating my newest program, Karen's Calculator. As you may recall, it's a fairly standard calculator with one big difference -- it can process
really large numbers, containing hundreds, thousands, even tens of thousands of digits.
One encounter with infinity is common to all calculators. What happens when someone asks a machine to divide a number by zero? If the calculator isn't careful, it can waste a lot of time trying to
arrive at the infinite result. But, like other calculator's, mine short-circuits the process and immediately reports an error ("Divide by Zero").
The second brush with infinity isn't so easy to deal with. Some perfectly reasonable calculations result in a very large, and sometimes even infinite, number of digits. We saw this a moment ago, when
we divided one by three (1 / 3), and produced an infinite number of new threes (.33333 ...).
Calculations that yield an infinite number of digits are a serious problem for calculator designers. Even with the fastest computer, completing such a computation would take an infinite amount of
time. Few people would be willing to wait that long. I know I have other plans.
Most calculators "solve" this problem by computing a reasonable number of digits, displaying those, and ignoring the rest. This works fine if the digits revealed are all you need.
But my calculator had to do better. That's why it allows you to choose the number of digits you want to see, when a full computation would produce too many.
To make that choice, look for the "Quotient Precision" entry on the program's Options menu. There you'll see four choices:
One choice, "25 digits", rounds the results of division to 25 significant digits, if the full result would have contained more than 25 digits.
Two other choices are "50 digits" and "100 digits". I'll let you guess what they mean. :)
And you probably know what the fourth choice, "Custom", means too. It lets you tell the program exactly how many digits you're willing to see. The minimum is 1, and the maximum is 10,000.
My calculator must deal with infinity in one other way. Two of its keys instantly recall the values of the mathematical constants Pi and "e". But as we saw a moment ago, all values of these special
numbers are really finite approximations of infinitely long numbers.
How precise are my Calculator's approximations of Pi and "e"? How many digits do they contain?
Once again, you get to answer that question. On the calculator's Options menu is another entry that reads "Pi and e Precision". Select it and you'll see the same four choices we talked about a moment
ago: "25 digits", "50 digits", "100 digits", and "Custom".
The first three choices behave as you'd expect -- limiting values of Pi or "e" to 25, 50, or 100 digits. And as before, "Custom" allows you to specify any number of digits, between 1 and 10,000.
The truth is, I don't know how many digits my calculator can digest. In theory, the limit is a little over two billion digits, because of some design decisions I made.
But realities, including the details of computer memory allocation, and the amount of memory possessed by a particular computer, impose a lower limit. I suspect the practical limit is around a
million digits, but so far I haven't performed a calculation that would put that to the test. :)
So, why did I limit quotients to "just" 10,000 digits? It's an arbitrary choice. Longer quotients take more time to compute. And on most modern computers, 10,000 digits seems a reasonable choice.
But if your computer is super-fast, or you just need more accuracy when you divide, there is a way around this limit. The newest version of my calculator looks for a special entry in the Windows
Registry. If present, the entry's value overrides the default limit of 10,000 digits.
[Nerdy Alert: If you aren't comfortable adding new entries to your Registry, consider living with 10,000-digit quotients. Or find someone who is comfortable adding new entries to your Registry.]
The name of the Registry value is MaxQuotientPrecision. It should be a "DWORD" value, located under this Registry key:
HKEY_CURRENT_USER\Software\KarenWare\Power Tools\Calc
Feel free to assign any value between 1 and 1,000,000. Values outside this range will be silently ignored, resulting in the default limit of 10,000 digits.
The calculator's values of Pi and "e" are limited to 10,000 digits each for a different reason. Each digit adds one byte to the size of the calculator program's size.
If there's enough demand, I can increase the maximum accuracy of Pi and "e" to one million digits, or more. But for the moment, I suspect 10,000 digits of accuracy will meet most needs. :)
There's a bit more to say about my new calculator. But until we meet again, give the new calculator a try. I think you'll like many of its new features. You can download your copy from its home page
As always, the program is free for personal/home use. If you're a programmer, you can download its complete Visual Basic source code too!
You can also get the latest version of every Power Tool, including the new Calculator, on a shiny CD. These include three bonus Power Tools, not available anywhere else. The source code of every
Power Tool, every issue of my newsletter, and some articles I wrote for Windows Magazine, are also on the CD. And owning the CD grants you a special license to use all my Power Tools at work.
Best of all, buying a CD is the easiest way to support the KarenWare.com web site, Karen's Power Tools, and this newsletter. To find out more, visit:
Until next time, be careful to stay away from those bothersome interstellar black holes (points where the density of matter may be infinite). And if you see me, or Mr. Miller, on the 'net, be sure to
wave and say "Hi!"
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Faster than Light - part 2
NOTE on terminology: CDK implies that light was faster in the past, and then over time, gradually decayed. So, CDK refers to the DeKay (decay) of C (the speed of light): CDK.
This article is a continuation of the article Faster than light? 1 which explores the possibility of the speed of light being faster in the past.
In the article Distant Starlight in a Young Universe: Attempted Solutions, 2 astrophysicist Jason Lisle, PhD, considers CDK or the idea that the speed of light was faster in the past and decayed over
time. He presents 3 arguments against CDK, all of which are refuted below.
Below I shall explain why and how those 3 arguments fail to disprove/refute the Setterfield decay model of the speed of light.
Argument 1 - Supernova SN1987A
First, Lisle said, regarding a supernova discovered in 1987:
... only a small fraction of the light from this explosion was directed toward the earth. Some light went off in other directions and reflected off of the surrounding gas which then redirected
the light toward earth – a “light echo.” This light arrived after 1987 because it took time to go from the supernova to the surrounding gas. By measuring the distance between the supernova and
the surrounding gas, and dividing by the time between the two events, we can compute the speed of light when the supernova happened. And we find it is consistent with the current value of c ...
What is missed here is that the result is also consistent with CDK! This calculation is based on some implicit assumptions which are not valid.
The calculation Lisle performed is simple. It is based on 2 paths of light from the supernova. One path is directly from the supernova to earth. This path is path A in Figure 1.
Figure 1
The other path is from the supernova to a cloud of gas, from which the light is reflected to earth. This is path B. Notice that path B is composed of 2 parts: path B1 and path B2.
In the caption to an image in Lisle's article, 2 he said the following: (emphasis added)
These light echos show that the speed of light perpendicular to our line of sight was the same at the time and distance of the supernova as here and now. 2
This implies that the distance between the earth and the gas cloud is the same as the distance from the earth to the supernova. 3
Lisle took the time interval between the time of detection of the supernova and the time of detection of light from the gas cloud, and divided that into the distance between the supernova and the gas
cloud (path B1) to get the speed of light along path B1. We shall see this was actually not a measurement of the speed of light along path B1!
Now, of course, it took time for light to travel from the gas cloud to earth (path B2 of Fig. 1). The time Lisle used in his calculation was not the time interval between the time point of the
supernova explosion and the time point of the appearance of light from the gas cloud on earth; rather, Lisle used the time point of the (later!) appearance of light on earth from the supernova in
1987 as the starting point of the time interval used in his calculation.
These were the only two time points used in deriving the time duration used in Lisle's calculation as shown in Figure 2:
1. the time point of arrival of light on earth from the supernova, T2 (not the actual time of the explosion)
2. the time point of arrival of light on earth from the gas cloud, T3
Key Point
Light travels the same distance along two different paths during the same time interval. This would be true even if the speed of light varied during this time period, as long as the speed varied in
the same way along both paths, so that the speed of light was the same on both paths at any specific instant of time. This is the case per CDK. CDK means light speed changes, over time - not through
space! Therefore the speed of light, even with CDK, is the same at any same specific time on different paths. 4
So, in the ensuing analysis, we shall see light traveling differing paths, during the same time period, with the result being that identical distance is covered on both paths during that time period.
Figure 2
Light would decay at the same rate regardless of position or location. CDK is not a function of distance nor of position, but is a function of time. This means the speed of light would be the same on
all paths in the diagrams at the same times. Even if light was continuously decaying nonlinearly, the distance covered on any path between two time points would be the same as the distance covered on
another path during the same time period.
Referring to Fig. 1, we see that at time T0 the supernova exploded. Light then later arrived at the gas cloud at time T1. During this time, from T0 to T1, light not only traveled the distance D1
between the supernova and the gas cloud, but light also traveled the same distance, D1, from the supernova toward earth. The distances D1 on paths B1 and A are equal. They are distances light
traveled during the same time period.
Now, consider Figure 2. D2 is the remaining distance along path A to earth directly from the supernova after time point T1. Continuing in time from T1 to T2, light traveled from the gas cloud toward
earth (on path B2). The identical time period occurred along path A between time point T1 and time point T2, so the same distance (D2) was covered on both path A and path B2. We noted earlier that
the distance to earth from the gas cloud (path B2) and the distance from the supernova to earth (path A) were the same. (Note that the distance D1 + D2 is the length of path A and would also be the
length of path B2.)
The distance from the gas cloud to earth is D1 + D2. We see from Figure 3 that light had traveled from the gas cloud distance D2 toward earth at time point T2. Time point T2 is the time at
which light coming directly from the supernova appeared on earth.
Figure 3
The length of both paths, from supernova to earth and from gas cloud to earth, were the same and equal to D1 + D2. Therefore, at time point T2, the remaining distance for light to travel to earth
from the gas cloud, D3, has to equal D1. The time of arrival of light on earth from the gas cloud was T3. This tells us that between T2 and T3, light traveled the distance D1. During this time,
light’s speed was the modern value, since T2 is the modern 1987 and T3 was later.
What Lisle calculated was the distance D1 divided by the time interval between T2 and T3. These are the distance and the time that light traveled, respectively, starting in 1987! There is no wonder
that the result was the 1987 value. This calculation was not the calculation of the speed of light during the ancient past, while light was traveling between the supernova and the gas cloud.
Another Perspective
Lisle assumed the time interval between T2 and T3 was the time that light took to travel between supernova and gas cloud, i.e., distance D1. We have seen that D1 is the actual distance light traveled
between T2 and T3. However, that was during modern times, along the path between earth and gas cloud. To assume it took the same time to travel that distance in ancient times is to assume the
same speed in ancient times, i.e., implicitly assuming what was to be proved, which is logically invalid.
Lisle implicitly assumed distance D3 was the same as D1, which is true. The error was in assuming that the time light took to travel D3 was the same as the time light took to travel D1. This is true
only if the speed of light in the past was the modern value. Thus, Lisle in this calculation implicitly assumed what was to be proved, which is invalid.
Argument 2: Redshift - Wrong Assumption About Frequency
Lisle says,
Second, since light is a wave, any change in its speed over time will result in a change in frequency. 2
This is not true! It is false, specifically, in the case for which wavelength is not constant.
The simple equation for this is v = fw, or f = v/w, where f is frequency, w is wavelength, and v is speed (velocity). To see this, consider that f, frequency, is the number of waves per second, or a
number divided by time. Then consider that multiplying f by the length of a wave gives a number times a length divided by total time. This result is speed!
Anyway, per the equation,
f = v/w
f = c / w
where c is made the velocity, we see that a change in c (speed of light) requires a change of frequency only if wavelength is constant. We can change c in this equation all we want, without changing
frequency, by simply changing w.
Therefore, CDK is not ruled out by asserting that CDK requires a change in frequency. Wavelength could change instead of frequency. It is hypothetically possible that the wavelength of a single
photon of light did not change, after emission of the photon, while that photon was in transit, but that the wavelength of different, sequentially emitted, photons of light indeed did change over
time, while the speed of light changed over time, per CDK as described by Setterfield. This is elaborated with more detail in the article Redshift Quantization Explained 5 (https://
So, Lisle's conclusions are not valid here, being based on a foundation of an invalid starting assumption (that wavelength did not change with c over time).
Argument 3: Conservation of Mass & Energy - Wrong Assumption About Mass
Lisle goes on to say
Third, the speed of light is very special and unlike other speeds. It essentially sets the relationship between space and time, the relative strengths of magnetic fields to electric fields, and
the relationship between matter and energy. But our very existence depends on these things being essentially constant. The famous equation E=mc^2, for example, shows that the amount of energy
contained in a mass is proportional to that mass multiplied by the square of the speed of light. Therefore, if the speed of light changes, then either the mass or energy (or both) of everything
in the universe must also change. 2
This conclusion is also incorrect. This has already been dealt with in the TASC article Does Changing Speed of Light Violate Energy Conservation? 6 There it was shown, including a mathematical
derivation, that energy conservation is not violated in the case of a changing speed of light!
Also, I might add, Lisle stated that the speed of light determines "the relative strengths of magnetic fields to electric fields," while I suggest the opposite: namely, that the electrical
permittivity of the vacuum \( \epsilon \) and the magnetic permeability of the vacuum \( \mu \) determine the speed of light. 7 In actual fact, these values and the speed of light are related, as per
the following equation, in which the speed of light is c, the electrical permittivity of the vacuum is \( \epsilon \), and the magnetic permeability of the vacuum is \( \mu \):
\[ c=\frac{1}{\sqrt{\epsilon\mu}} \]
Equation for speed of light c, in terms of the electrical permittivity of the vacuum and the magnetic permeability of the vacuum
The video at https://www.youtube.com/watch?v=qtqTPCAw7Fo shows the derivation of the above formula for the speed of light from Maxwell's equations and tells us that the speed of light is determined
by the 2 values in the equation. These were described in the video as values for a vacuum. However, is space really a vacuum? We know more now than we did around 1905, when Einstein's special theory
of relativity appeared. We know there are virtual particles in space, as well as a vacuum energy, also known as ZPE (the Zero Point Energy). We also know that light travels faster through some
mediums/materials than it travels through others. Light is slowed by the medium through which it passes, and those virtual particles plus vacuum energy also make up a medium. This medium in the
vacuum changed the values for those 2 variables, which in turn resulted in a change in the speed of light.)
Changes of the vacuum energy and density of virtual particles would result in changes of these two factors, epsilon and mu.
Setterfield explains:
Polarization can only occur if there are charged particles capable of being moved or re-oriented in an electric field. But we are working with what appears to be a vacuum. The conclusion is that
the vacuum must contain charged particles, capable of moving, which are not associated with the air. This certainly seems to indicate the presence of virtual particle pairs which flash into and
out of existence so rapidly. Their instantaneous presence, however, means we have a “polarizable vacuum.” The extent to which the vacuum “permits” itself to be polarized in an electric field is
called the electric permittivity of free space. This permittivity is designated by the Greek letter epsilon written as ε. 8
If the ZPE strength increased, then both ε and μ would also increase proportionally as a result of the proportional increase in the number of virtual particle pairs. 9
We see that a changing ZPE would change both ε and μ which in turn would change c. Was the ZPE different in the past than it is now? Evidence that indeed it was different in the past is found in the
red shift data. See TASC article Redshift Quantization Explained 10 (https://tasc-creationscience.org/article/quantized-redshift-explained).
Also suggesting that faster light in the past is not yet ruled out by creation scientists, the June 2021 presentation by creationist physicist Russell Humphries suggested a faster speed of light in
the past. The video of Humphries’s presentation can be viewed at https://www.youtube.com/watch?v=09yngV0c6Y8
If we dig a little deeper, we often can see more. By thinking about the impact of CDK, not just on one path, B1, but on other paths (B2 and A), we can see a different conclusion might be discovered
than what appears from a first look.
In Lisle’s article Distant Starlight in a Young Universe: Attempted Solutions, 2 three arguments against CDK were given. They have all been examined above and shown for various reasons to be invalid.
As a result, CDK has not been ruled out "as a viable solution to the perceived distant starlight problem," as stated in the article (at least not yet, not by the 3 arguments presented).
In short, CDK remains a viable alternative explanation for the distant starlight problem.
• 1Spears J (2022 May) Faster than Light? https://tasc-creationscience.org/article/faster-light-0 Accessed 2022 Jul 15
• 2 a b c d e f g Lisle J (2020) Distant starlight in a young universe: Attempted solutions. https://biblicalscienceinstitute.com/apologetics/
distant-starlight-in-a-young-universe-attempted-solutions/ Accessed 2022 Apr 15
• 3Reasonably assuming that path B1 is perpendicular to the path from the mid-point of B1 directly to earth, this means that paths B2 and A would be two equal sides of an isosceles triangle and
therefore identical in length. (In Figure 2, moving either the gas cloud or the supernova closer to earth, or further away from earth, results in path B1 no longer being perpendicular to our line
of sight.)
Also, the fact that the only distance that was used in Lisle’s calculation was that of B1, the distance between the supernova and the gas cloud, indicates that he considered the other distances
to the earth (paths A and B2) as irrelevant to the calculation, being the same length, and that they cancel out. The fact that the distances A and B2 were not included in the calculation shows
that no significant difference in their lengths was assumed.
• 4This assumes obvious factors such as traveling through different media with different indices of refraction are not significant or relevant.
• 5Spears J (2021) Redshift Quantization Explained https://www.tasc-creationscience.org/article/does-changing-speed-light-violate-energy-conservation Accessed 2022 Apr 16
• 6Spears J (2021) Does changing speed of light violate energy conservation? https://www.tasc-creationscience.org/article/does-changing-speed-light-violate-energy-conservation Accessed 2022 Apr 16
• 7ScienceWorld (2021 Jun 29) Why light has a speed limit.
• 8Setterfield BJ, Setterfield HJ (2013) Cosmology and the Zero Point Energy, Natural Philosophy Alliance Monograph Series, No. 1, Natural Philosophy Alliance, 37. This can be obtained at: http://
www.barrysetterfield.org/GSRdvds.html - cosmology.
• 9Setterfield BJ, Setterfield HJ (2013) Cosmology and the Zero Point Energy, Natural Philosophy Alliance Monograph Series, No. 1, Natural Philosophy Alliance, 12. This can be obtained at: http://
www.barrysetterfield.org/GSRdvds.html - cosmology.
• 10Spears J (2021) Redshift Quantization Explained https://tasc-creationscience.org/article/quantized-redshift-explained Accessed 2022 Apr 16
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Python and Statistics for Financial Analysis Coursera Quiz Answers
All Weeks Python and Statistics for Financial Analysis Coursera Quiz Answers
Python and Statistics for Financial Analysis Coursera Quiz Answers
Week 1: Python and Statistics for Financial Analysis
Q1. Which of the following library has DataFrame object?
• Pandas
• Numpy
• Matplotlib
• Statsmodels
Q2. Which of the following is the correct way to import a library, eg Pandas?
1.pandas import
1.import pandas as pd
Q3. What is the method of DataFrame object to import a csv file?
• import_csv()
• from_csv()
• read_csv()
• csv()
Q4. Which of the following attributes of a DataFrame return a list of column names of this
• columns
• shape
• dtype
• column
Q5. Which of the following can slice ‘Close’ from ‘2015-01-01’ to ‘2016-12-31’ from data,
which is a DataFrame object?
1.data.loc[‘2015-01-01’:’2016-12-31’, ‘Close’]
1.data.iloc[‘2015-01-01’:’2016-12-31’, ‘Close’]
Q6. What is the method of DataFrame to plot a line chart?
• scatter()
• plot()
• plot_graph()
• axhline()
Q7. Suppose you have a DataFrame – data, which
contains columns ‘Open’, ‘High’, ‘Low’, ‘Close’, ‘Adj Close’ and ‘Volume’.
does data[[‘Open’, ‘Low’]] return?
• All
columns of data except ‘Open’ and ‘High’
• No results are shown
• Columns ‘Open’ and ‘Low’
• The first
row of data which contains only columns ‘Open’ and ‘High’
Q8. Suppose you have a DataFrame ms , which contains the daily data of ‘Open’, ‘High’, ‘Low’, ‘Close’, ‘Adj
Close’ and ‘Volume’ of Microsoft’s stock.
Which of
the following syntax calculates the Price difference, (ie ‘Close’ of tomorrow –
‘Close’ of today)?
1. ms[‘Close’].shift(1) – ms[‘Close’].shift(1)
1.ms[‘Close’].shift(-1) – ms[‘Close’].shift(-1)
1.ms[‘Close’].shift(1) – ms[‘Close’]
1.ms[‘Close’].shift(-1) – ms[‘Close’]
Q9. Suppose you have a DataFrame – ms , which contains the daily data of ‘Open’, ‘High’, ‘Low’, ‘Close’, ‘Adj
Close’ and ‘Volumn’ of Microsoft’s stock.
What is
the method of DataFrame to calculate the 60 days moving average?
• rolling().mean(60)
• moving_average(60)
• rolling(60).mean()
• rolling(60).median()
Q10. Which of the following idea(s) is/are correct to the simple trading strategy that we introduced in the lecture video?
• Use longer
moving average as slow signal and shorter moving average as fast signal
• We short
one share of stocks if fast signal is larger than slow signal
• If fast
signal is larger than slow signal, this indicates an upward trend at the
current moment
Week 2: Python and Statistics for Financial Analysis
Q1. Roll two dice and X is the sum of faces values. If we roll them 5 times and get 2,3,4,5,6
Which of the following is/are true about X?
• The mean of X is 4.
• X can only take values 2,3,4,5,6
• X is a
random variable
Q2. Roll two dice and X is the sum of faces values. If we roll them 5 times and get 2,3,4,5,6
What do we know about X?
• The dice is fair.
• Range of X is 6-2=4
• The most likely value of X is 6
• We have 5 observations of X
Q3. Roll two dice and X is the sum of faces values. If we roll them 5 times and get 2,3,4,5,6
X is a __ random variable.
• discrete
• continuous
• None of
the above
Q4. Why do we
use relative frequency instead of frequency?
• Relative
frequency is easier to compute
• Frequency
cannot show the number of appearance of outcomes
• Relative frequency can be used to compare the
ratio of values between difference collections with difference number of values
• Relative
frequency is easier to compute when the number of observations increases
Q5. What can
we say about relative frequency when we have large number of trials?
• Relative frequency becomes approximately the
distribution of the corresponding random variable
• The
relative frequency of each possible outcome will be the same
• The relative
frequency stays constant after a very large number of trials, eg. n=10000
• None of the above
Q6. What is the notion of “95% Value at Risk” ?
• 95% Value
at Risk is 95% quantile
• 95% VaR
measures how much you can lose at most
• 95% VaR
measures how much you can win at most
• 95% VaR measures the amount of investment you can
lose, at the worst 5% scenario
Q7. In the lecture video, we mentioned the calculation of continuous random variable is based on the probability density function.
Given a
probability density function, f(x) = 1/100, what is the probability
P(10<X<20), where X~Uniform[0, 100]?
• f(20) –
• f(10)
• f(20)
• (20- 10) * 1/100
Q8. What
methods should we use to get the
cdf and pdf of normal distribution?
• norm.cdf() and norm.pdf() from scipy.stats
• cdf() and
pdf() form numpy
• cdf() and
pdf() from pandas
• norm.cdf()
and norm.pdf() from statsmodels
Q9. Which additional library should we import when we want to calculate log daily return specifically?
• Pandas
• Numpy
• Statsmodels
• Matplotlib
Q10. What
is the distribution of stock returns suggested by Fama and French in general?
• A perfect normal distribution
• Close to normal distribution but with fat tail
• Arbitrary distribution
• Left-skewed distribution
Week 3: Python and Statistics for Financial Analysis
Q1. What is true
about sample and population?
• Population
can always be directly observed
• Parameters
from population is always the same as statistics from sample
• Sample is a subset of population which is
randomly draw from population
• The size
of population is always finite
Q2. You have a
DataFrame called ‘data’ which has only one column ‘population’.
data = pd.DataFrame()
data[‘population’] = [47, 48, 85, 20, 19, 13, 72, 16, 50, 60]
How to draw sample with sample size =5, from a
‘population’ with replacement?
(Hint: You can modify the code illustrated in the Jupyter Notebook “Population and Sample” after Lecture 3.1)
1.data[‘population’].sample(5, replace=False)
1.data[‘population’].sample(5, replace=True)
Q3. Why is the
degrees of freedom n-1 in sample variance?
• The degrees of freedom in sample variance is
constrained by the sample mean
• None of
the above
• The extreme value in the sample is removed for fair analysis
• Only n-1
values in the sample is useful
Q4. What does Central Limit Theorem tell you about the distribution of sample mean?
• The
distribution of sample mean follows normal distribution only if the population
distribution is normal
• The
distribution of sample mean follows normal distribution with any sample size
only if the population distribution is normal
• The distribution of sample mean follows normal
distribution with very large sample size follows normal distribution regardless
of the population distribution
• The
distribution of sample mean with large sample size follows chi-square
distribution regardless of the population distribution
Q5. Suppose we have 3 independent normal random variables X1, X2 and X3:
What is the distribution of X1 + X2 + X3?
• Remains the same even X1, X2 and X3 are added up
• Mean and variance of X1, X2 and X3 are added up
• Mean remains unchanged; variances are added up.
• Mean remains unchanged; variance takes 3 square root.
Q6. Why do we
need to standardize sample mean when making inference?
• Sample
mean becomes normally distributed after standardization
• Sample
mean becomes population mean after standardization
• The standardized distribution of sample mean follows N(0,1) which is easier to make inference
• None of the above
Q7. What can a 95%
confidence interval of daily return of an investment tell you?
• With 95% chance
your daily return falls into this interval
• With 95% chance this interval will cover the mean of
daily return
• With 5% chance your
daily return falls into this interval
• None of the above
Q8. Check the
Juypter notebook of 3.3 Sample and Inference. What is the confidence interval of this exercise?
• [0.000015603,
• [-0.000015603, 0.001656]
• [-0.0001690,
• [0.0001690,
Q9. When do you reject a null hypothesis with alternative hypothesis μ>0 with significance level α?
• p value is larger than α
• p value is smaller than α
• z < z_(1-α)
• z > z_(1-α)
Q10. When doing analysis of stock return, you notice that with 95% confidence interval, the upper bound and lower bound are negative.
Base on this data, what can you tell about this stock?
• There is 95% chance of which the mean return of this stock is negative
• We must lose money by investing in this stock
• There is only 5% chance of which the mean return of this stock is negative
Week 4: Python and Statistics for Financial Analysis
Q1. Why do you use
coefficient of correlation, instead of covariance, when calculating the association between two random variables?
• None of the above
• Covariance is not
suitable to use when the underlying distribution is not normal
• Covariance cannot
address nonlinear relationship but coefficient of correlation can address
nonlinear relationship
• Covariance can be affected by the variance of
individual variables, but coefficient of correlation is rescaled by variance of
both variables
Q2. What is the range
and interpretation of coefficient of correlation?
• From 0 to 1, 0
means perfect negative linear relationship and 1 means perfect positive linear
• From 0 to 100, 0
means perfect positive linear relationship and 100 means perfect negative
linear relationship
• From 0 to 100, 0
means perfect negative linear relationship and 100 means perfect positive
linear relationship
• From -1 to 1, -1 means perfect negative linear
relationship and 1 means perfect positive linear relationship
Q3. Refer to the https://www.coursera.org/learn/python-statistics-financial-analysis/notebook/F0Luf/simple-linear-regression-model
Is LSTAT a significant predictor of MEDV at significance level 0.05?
• Yes, because the coefficient b_1 is not zero
• Yes, because the p value of b_1 is larger than 0.05
• No, because the coefficient b_1 is negative
• Yes, because the p value of b_1 is smaller than 0.05
Q4. To evaluate the performance of linear regression model, we refer to the summary of “model” as seen in https://www.coursera.org/learn/python-statistics-financial-analysis/notebook/F0Luf/
What is the percentage of variation explained by the model?
Q5. How to check if a
linear regression model violates the independence assumption?
• Draw residual
versus predictor plot
• Draw scatter plot
of predictor versus target
• Durbin Watson test
• QQ plot
Q6. If any of the
assumptions of linear regression model are violated, we cannot use this model
to make prediction.
Q7. Check the Jupyter Notebook 4.4- Build the trading model by yourself!
We have a variable ‘formula’ which store the names of predictors
and target. How should you modify this ‘formula’ if you want to drop the
predictor ‘daxi’?
1.formula = ‘spy~aord+cac40+nikkei+dji+daxi’
1.formula = ‘spy~aord+cac40+nikkei+dji-daxi’
1.formula = ‘spy~aord+cac40+nikkei+dji’
1.formula = ‘spy~ -daxi’
Q8. Check the Jupyter Notebook 4.4- Build the trading model by yourself!
What is the most
significance predictor for ‘SPY’?
Q9. What does it mean
if you have a strategy with maximum drawdown of 3%?
• During the
trading period, the minimum you lose is 3%
• During the
trading period, the maximum gain from the previous peak of your portfolio value
is 3%
• During the trading period, the maximum lose of
your portfolio is 3%
• During the trading period, the maximum drop from the
previous peak of your portfolio value is 3%
Q10. How can you check
the consistency of your trading strategy?
• Check if the
return of your strategy is positive using all historical data you have
• Define some
metric for evaluating your strategy, eg Sharpe Ratio, maximum drawdown, and
check if your strategy can generate positive return using all historical data
you have
• Define some metric for evaluating your strategy, eg
Sharpe Ratio, maximum drawdown. Then split your data into train set and test
set and check if your strategy can generate positive return using both train
set and test set
• There is no way to to check the consistency
Get All Course Quiz Answers of Entrepreneurship Specialization
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Entrepreneurship 2: Launching your Start-Up Quiz Answers
Entrepreneurship 3: Growth Strategies Coursera Quiz Answers
Entrepreneurship 4: Financing and Profitability Quiz Answers
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16.4 BST Operations | CS61B Textbook
To search for something, we employ binary search, which is made easy due to the BST property.
We know that the BST is structured such that all elements to the right of a node are greater and all elements to the left are smaller. Knowing this, we can start at the root node and compare it with
the element, X, that we are looking for. If X is greater to the root, we move on to the root's right child. If its smaller, we move on to the root's left child. We repeat this process recursively
until we either find the item or we get to a leaf, in which case the tree does not contain the item.
static BST find(BST T, Key sk) {
if (T == null)
return null;
if (sk.equals(T.key))
return T;
else if (sk ≺ T.key)
return find(T.left, sk);
return find(T.right, sk);
We always insert at a leaf node!
First, we search in the tree for the node. If we find it, then we don't do anything. If we don't find it, we will be at a leaf node already. At this point, we can just add the new element to either
the left or right of the leaf, preserving the BST property.
static BST insert(BST T, Key ik) {
if (T == null)
return new BST(ik);
if (ik ≺ T.key)
T.left = insert(T.left, ik);
else if (ik ≻ T.key)
T.right = insert(T.right, ik);
return T;
Deleting from a binary tree is a little bit more complicated because whenever we delete, we need to make sure we reconstruct the tree and still maintain its BST property. Let's break this problem
down into three categories:
the node we are trying to delete has no children
Deletion: No Children
If the node has no children, it is a leaf, and we can just delete its parent pointer and the node will eventually be swept away by the garbage collector.
Deletion: One Child
If the node only has one child, we know that the child maintains the BST property with the parent of the node because the property is recursive to the right and left subtrees. Therefore, we can just
reassign the parent's child pointer to the node's child and the node will eventually be garbage collected.
Deletion: Two Children
If the node has two children, the process becomes a little more complicated because we can't just assign one of the children to be the new root. This might break the BST property.
Instead, we choose a new node to replace the deleted one.
We know that the new node must:
be > than everything in left subtree.
be < than everything right subtree.
In the below tree, we show which nodes would satisfy these requirements given that we are trying to delete the dog node.
To find these nodes, you can just take the right-most node in the left subtree or the left-most node in the right subtree. Then, we replace the dog node with either cat or elf and then remove the old
cat or elf node.
This is called Hibbard deletion, and it gloriously maintains the BST property amidst a deletion.
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Pipe Bend
(To be removed) Hydraulic resistance in pipe bend
The Hydraulics (Isothermal) library will be removed in a future release. Use the Isothermal Liquid library instead. (since R2020a)
For more information on updating your models, see Upgrading Hydraulic Models to Use Isothermal Liquid Blocks.
Local Hydraulic Resistances
The Pipe Bend block represents a pipe bend as a local hydraulic resistance. The pressure loss in the bend is assumed to consist of
• Loss in the straight pipe
• Loss due to curvature
The loss in a straight pipe is simulated with the Hydraulic Resistive Tube block. The loss due to curvature is simulated with the Local Resistance block, and the pressure loss coefficient is
determined in accordance with the Crane Co. recommendations (see [1], p. A-29). The flow regime is checked in the underlying Local Resistance block by comparing the Reynolds number to the specified
critical Reynolds number value.
The pressure loss due to curvature for turbulent flow regime is determined according to the following formula:
$p=K\frac{\rho }{2{A}^{2}}q|q|$
q Flow rate
p Pressure loss
K Pressure loss coefficient
A Bend cross-sectional area
ρ Fluid density
For laminar flow regime, the formula for pressure loss computation is modified, as described in the reference documentation for the Local Resistance block.
The pressure loss coefficient is determined according to recommendation provided in [1]:
$K={K}_{d}·{K}_{r}·{K}_{\alpha }$
K[d] Base friction factor coefficient
K[r] Correction coefficient accounting for the bend curvature
K[α] Correction coefficient accounting for the bend angle
The base friction factor coefficient is determined according to the following table.
Friction factors for pipes with diameters greater than 525 mm are determined by extrapolation.
The correction coefficient accounting for the bend curvature is determined according to the next table.
The bend curvature relative radius is calculated as
r = bend radius / pipe diameter
For pipes with the bend curvature relative radius value outside the range of 1 > r > 24, correction coefficients are determined by extrapolation.
Correction for non-90^o bends is performed with the empirical formula (see [2], Fig. 4.6):
${K}_{\alpha }=\alpha \left(0.0142-3.703·{10}^{-5}\alpha \right)$
α Bend angle in degrees (0 ≤ α ≤ 180)
Connections A and B are conserving hydraulic ports associated with the block inlet and outlet, respectively.
The block positive direction is from port A to port B. This means that the flow rate is positive if fluid flows from A to B, and the pressure differential is determined as $p={p}_{A}-{p}_{B}$.
The formulas used in the Pipe Bend block are very approximate, especially in the laminar and transient flow regions. For more accurate results, use a combination of the Local Resistance block with a
table-specified K=f(Re) relationship and the Hydraulic Resistive Tube block.
Basic Assumptions and Limitations
• Fluid inertia and wall compliance are not taken into account.
• The bend is assumed to be made of a clean commercial steel pipe.
Pipe diameter
The internal diameter of the pipe. The default value is 0.01 m.
Bend radius
The radius of the bend. The default value is 0.04 m.
Bend angle
The angle of the bend. The value must be in the range between 0 and 180 degrees. The default value is 90 deg.
Internal surface roughness height
Roughness height on the pipe internal surface. The parameter is typically provided in data sheets or manufacturer's catalogs. The default value is 1.5e-5 m, which corresponds to drawn tubing.
Critical Reynolds number
The maximum Reynolds number for laminar flow. The value of the parameter depends on the orifice geometrical profile. You can find recommendations on the parameter value in hydraulics textbooks.
The default value is 350.
Fluid compressibility
Dynamic compressibility setting. Select On to make the fluid density dependent on pressure and temperature. Select Off to treat the fluid density as a constant. Dynamic compressibility impacts
the transient response of the fluid at small time scales and can slow down simulation.
Initial liquid pressure (gauge)
Gauge pressure in the pipe bend at time zero. The default value is 0 Pa.
Global Parameters
Parameters determined by the type of working fluid:
• Fluid density
• Fluid kinematic viscosity
Use the Hydraulic Fluid block or the Custom Hydraulic Fluid block to specify the fluid properties.
The block has the following ports:
Hydraulic conserving port associated with the bend inlet.
Hydraulic conserving port associated with the bend outlet.
[1] Flow of Fluids Through Valves, Fittings, and Pipe, Crane Valves North America, Technical Paper No. 410M
[2] George R. Keller, Hydraulic System Analysis, Published by the Editors of Hydraulics & Pneumatics Magazine, 1970
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2006b
R2023a: To be removed
The Hydraulics (Isothermal) library will be removed in a future release. Use the Isothermal Liquid library instead.
For more information on updating your models, see Upgrading Hydraulic Models to Use Isothermal Liquid Blocks.
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Hazen Williams Calculator
Hazen-Williams Calculator
What is the Hazen Williams Formula?
The Hazen Williams formula is an empirical equation used to estimate the pressure drop or flow rate of water in pipes, specifically focusing on smooth, low-pressure conduits. It is widely used in
civil and environmental engineering for designing water distribution systems, irrigation networks, and fire protection systems. The formula accounts for factors like pipe diameter, flow velocity, and
a roughness coefficient, making it a practical tool for engineers working with hydraulic systems.
Understanding the Hazen Williams formula is crucial for engineers when selecting pipe sizes and materials, ensuring that systems operate efficiently without excessive energy loss or pressure drops.
This formula is particularly useful when working with water at typical temperatures, making it a go-to choice for many hydraulic design projects in municipal and industrial applications.
How to Calculate Flow Using the Hazen Williams Formula
The Hazen Williams formula for calculating the flow rate (\( Q \)) through a pipe is given as:
\( Q = 0.278 \cdot C \cdot A \cdot R^{0.63} \cdot S^{0.54} \)
• Q is the flow rate (in cubic meters per second, m³/s).
• C is the Hazen Williams roughness coefficient, which depends on the material and condition of the pipe.
• A is the cross-sectional area of the pipe (in square meters, m²).
• R is the hydraulic radius (in meters, m).
• S is the slope of the energy grade line or hydraulic gradient (unitless).
This formula allows engineers to estimate the flow rate in pipes based on the roughness of the pipe material and the hydraulic conditions. It’s important to note that the Hazen Williams formula is
suitable primarily for water flow at normal temperatures, typically between 40°F and 85°F.
Example: Calculating Flow Rate in a PVC Pipe
Let’s calculate the flow rate through a PVC pipe with a diameter of 200 mm (0.2 m), using a roughness coefficient (\( C \)) of 150, and a slope (\( S \)) of 0.01. The cross-sectional area (\( A \))
of the pipe can be calculated as:
\( A = \pi \cdot (0.2/2)^2 \approx 0.0314 \, \text{m}^2 \)
The hydraulic radius (\( R \)) for a circular pipe is calculated as:
\( R = \frac{D}{4} = \frac{0.2}{4} = 0.05 \, \text{m} \)
Substituting into the Hazen Williams formula:
\( Q = 0.278 \cdot 150 \cdot 0.0314 \cdot 0.05^{0.63} \cdot 0.01^{0.54} \)
After performing the calculations, the flow rate (\( Q \)) is approximately 0.024 m³/s. This flow rate is important for ensuring that the pipe can handle the required water supply without excessive
pressure loss.
Why is the Hazen Williams Formula Important in Engineering?
The Hazen Williams formula is essential in water resources engineering because it provides a reliable method for estimating water flow in pipes. Its importance can be summarized as follows:
• Design of Water Distribution Systems: Engineers use this formula to design systems that distribute water efficiently in cities and industrial facilities. By accurately estimating flow rates, they
can select the right pipe sizes to minimize costs and energy use.
• Irrigation and Agricultural Applications: The formula is used to design irrigation systems, ensuring consistent water flow to crops while avoiding issues like pipe erosion or sediment deposition.
• Fire Protection Systems: The Hazen Williams formula is crucial in designing sprinkler systems, ensuring adequate water pressure for fire suppression while minimizing pressure drops that could
hinder system effectiveness.
Limitations of the Hazen Williams Formula
Despite its usefulness, the Hazen Williams formula has several limitations that engineers should consider:
• Not Suitable for Non-Water Fluids: The formula is specifically designed for water flow. For other fluids like oil, gases, or highly viscous liquids, other equations such as Darcy-Weisbach or
Manning’s equation are more appropriate.
• Accuracy Dependent on Pipe Material: The roughness coefficient (\( C \)) varies significantly with the material and condition of the pipe. Engineers need to ensure they use accurate values for \(
C \) to avoid errors in calculations.
• Limited Temperature Range: The formula is best suited for water at standard temperatures. For hot water or water in extreme temperatures, other methods may provide more accurate results.
Example: Hazen Williams Formula for a Steel Pipe
For a steel pipe with a diameter of 100 mm and a roughness coefficient (\( C \)) of 120, let’s calculate the flow rate when the slope (\( S \)) is 0.02. The cross-sectional area (\( A \)) and
hydraulic radius (\( R \)) are calculated similarly as before:
\( A = \pi \cdot (0.1/2)^2 \approx 0.00785 \, \text{m}^2 \)
The hydraulic radius is:
\( R = \frac{0.1}{4} = 0.025 \, \text{m} \)
Using the Hazen Williams formula:
\( Q = 0.278 \cdot 120 \cdot 0.00785 \cdot 0.025^{0.63} \cdot 0.02^{0.54} \)
The result is a flow rate of approximately 0.0045 m³/s. This example demonstrates how pipe material and roughness coefficient affect the results, underscoring the importance of accurate inputs in
design calculations.
Frequently Asked Questions (FAQ)
1. Can the Hazen Williams formula be used for gas flow?
No, the Hazen Williams formula is specifically designed for water flow in pipes. For gas flow, equations like the Darcy-Weisbach formula are more appropriate, as they account for compressibility
2. What is the roughness coefficient in the Hazen Williams formula?
The roughness coefficient (\( C \)) is a dimensionless number that characterizes the internal roughness of a pipe. Higher values indicate smoother pipes, such as PVC or copper, while lower values are
used for rougher materials like cast iron or steel.
3. How does pipe diameter affect the flow rate?
The flow rate increases with the pipe’s diameter because a larger cross-sectional area allows more water to pass through. This relationship is directly considered in the Hazen Williams formula,
making diameter a critical factor in hydraulic design.
4. Why is the Hazen Williams formula popular for water distribution design?
The formula is simple to use and provides reliable results for low-pressure water flow, making it ideal for municipal water distribution systems. Its empirical nature means it’s based on practical
observations, aligning well with real-world conditions.
Applications of the Hazen Williams Formula in Engineering
The Hazen Williams formula is a versatile tool in hydraulic engineering, with applications such as:
• Water Treatment Plants: Engineers use the formula to design piping networks that transport water through various stages of treatment, ensuring optimal flow and minimal energy consumption.
• Building Plumbing Systems: The formula helps in designing plumbing systems in buildings, ensuring that water is delivered to each outlet with the necessary pressure and flow rate.
• Irrigation Systems: In agricultural engineering, the formula ensures that crops receive the right amount of water, preventing under- or over-irrigation.
Impulse and Flow: A Related Concept
While the Hazen Williams formula focuses on the steady-state flow of water through pipes, another related concept is impulse in fluid dynamics. Impulse refers to the change in momentum of a fluid
when subjected to force over time. Although impulse is more commonly discussed in mechanics, its principles can be applied to understanding transient effects in pipe flow, like water hammer—a
phenomenon where sudden changes in flow can cause pressure surges within pipes.
Water hammer can lead to significant damage in pipes, making it important for engineers to design systems that minimize these effects. By understanding both steady flow (using the Hazen Williams
formula) and transient flow (addressed through impulse concepts), engineers can create safer and more reliable water transport systems.
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How to Convert Pandas Dataframe to Tensorflow Data?
To convert a pandas dataframe to TensorFlow data, you can use the tf.data.Dataset class provided by TensorFlow. You can create a dataset from a pandas dataframe by first converting the dataframe to a
TensorFlow tensor and then creating a dataset from the tensor.
You can convert a pandas dataframe to a TensorFlow tensor by using the tf.constant() function. Once you have a TensorFlow tensor representing the data, you can create a dataset using the
from_tensor_slices() method of the tf.data.Dataset class. This method creates a dataset from the given tensor by slicing it along the first dimension.
After creating the dataset, you can use it for various TensorFlow operations such as training a machine learning model or performing data preprocessing. This allows you to seamlessly integrate your
pandas data with TensorFlow for efficient data processing and model training.
What are the steps involved in converting a pandas dataframe to tensorflow data?
To convert a pandas dataframe to TensorFlow data, you can follow these steps:
1. Import required libraries:
1 import tensorflow as tf
2 import pandas as pd
1. Load your data into a pandas dataframe:
1 df = pd.read_csv('data.csv')
1. Extract the features and target columns from the dataframe:
1 features = df.drop('target_column', axis=1)
2 target = df['target_column']
1. Convert the features and target columns to numpy arrays:
1 X = features.values
2 y = target.values
1. Create a TensorFlow dataset from the numpy arrays:
1 dataset = tf.data.Dataset.from_tensor_slices((X, y))
1. Shuffle and batch the dataset:
1 shuffle_buffer_size = len(features)
2 batch_size = 32
4 dataset = dataset.shuffle(shuffle_buffer_size).batch(batch_size)
1. Optional: You can also preprocess your data using TensorFlow data preprocessing functions before converting it to a dataset.
2. Once you have converted your pandas dataframe to a TensorFlow dataset, you can use it to train machine learning models in TensorFlow.
How to adjust the data types of columns in a pandas dataframe for compatibility with tensorflow data?
To adjust the data types of columns in a pandas dataframe for compatibility with TensorFlow data, you can use the astype() method to convert the data types of specific columns to the desired types.
Here is an example of how you can adjust the data types of columns in a pandas dataframe for TensorFlow compatibility:
1 import pandas as pd
3 # Create a sample dataframe
4 data = {'A': [1, 2, 3, 4, 5],
5 'B': ['apple', 'banana', 'cherry', 'date', 'elderberry'],
6 'C': [0.1, 0.2, 0.3, 0.4, 0.5]}
8 df = pd.DataFrame(data)
10 # Check the data types of the columns
11 print(df.dtypes)
13 # Convert the data types of columns
14 df['A'] = df['A'].astype('float32')
15 df['B'] = df['B'].astype('category')
16 df['C'] = df['C'].astype('float32')
18 # Check the data types of the columns after conversion
19 print(df.dtypes)
In this example, we convert the data type of column 'A' to float32, column 'B' to a categorical data type, and column 'C' to float32. This ensures that the data types in the dataframe are compatible
with TensorFlow data.
What functions can be used to convert pandas dataframe to tensorflow data?
To convert a pandas dataframe to TensorFlow data, you can use the following functions:
1. tf.convert_to_tensor: This function can be used to convert a pandas dataframe to a TensorFlow tensor object.
2. tf.data.Dataset.from_tensor_slices: This function can be used to create a TensorFlow dataset from a tensor object.
3. tf.data.Dataset.from_generator: This function can be used to create a TensorFlow dataset from a generator function that yields batches of data.
4. tf.data.Dataset.from_df: This function can be used to create a TensorFlow dataset from a pandas dataframe directly.
How to visualize the transformation process from pandas dataframe to tensorflow data?
One way to visualize the transformation process from a pandas DataFrame to TensorFlow data is to think of it as a series of steps where you are reshaping and converting the data into a format that
can be used by TensorFlow for building and training machine learning models.
Here is a general outline of the steps involved in this transformation process:
1. Loading the data: Start by loading your data from a pandas DataFrame into memory. This can be done using functions like pd.read_csv() or pd.read_excel() depending on the format of your data.
2. Data preprocessing: This step involves cleaning and preprocessing your data to prepare it for training. This might include steps like handling missing values, scaling or normalizing numerical
features, encoding categorical variables, and splitting the data into training and testing sets.
3. Converting to TensorFlow data structures: The next step is to convert your preprocessed data into TensorFlow data structures such as tf.data.Dataset. This can be done by creating TensorFlow
tensors from your data using functions like tf.constant() or tf.convert_to_tensor(), and then using these tensors to create a dataset.
4. Creating input pipelines: Once you have converted your data into TensorFlow data structures, you can create input pipelines using functions like dataset.shuffle(), dataset.batch(), and
dataset.prefetch() to optimize the training process and improve efficiency.
5. Model training: Finally, you can use the TensorFlow dataset as input to your machine learning model for training. This involves defining the model architecture, compiling the model, and fitting
the model to the training data using functions like model.compile() and model.fit().
By visualizing the transformation process in this way, you can better understand how the data is being manipulated and prepared for training in TensorFlow. Additionally, you can easily identify and
troubleshoot any issues that may arise during the transformation process.
How to handle timestamps in a pandas dataframe before converting to tensorflow data?
Before converting a pandas dataframe to TensorFlow data, it is important to properly handle timestamps. Here are some steps you can take to do this:
1. Convert timestamp columns to datetime objects: If your dataframe contains columns with timestamps, make sure to convert them to datetime objects using the pd.to_datetime() function. This will
ensure that the timestamps are in the correct format for further processing.
1 df['timestamp_column'] = pd.to_datetime(df['timestamp_column'])
1. Set timestamp columns as index: If your dataframe represents time series data, consider setting the timestamp column as the index of the dataframe. This will make it easier to work with
time-based operations and ensure that the data is properly ordered.
1 df.set_index('timestamp_column', inplace=True)
1. Handle missing values: Check for and handle any missing values in the timestamp columns. You can fill missing values or drop rows with missing timestamps depending on your data and analysis
1 df.dropna(subset=['timestamp_column'], inplace=True)
1. Handle timezone conversions: If your timestamps are in different timezones, consider converting them to a common timezone before further processing. You can do this using the tz_convert()
function in pandas.
1 df['timestamp_column'] = df['timestamp_column'].dt.tz_convert('UTC')
By following these steps to properly handle timestamps in your pandas dataframe, you can ensure that the data is in the correct format before converting it to TensorFlow data.
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14.5: Metabolism and Signaling: The Steady State, Adaptation and Homeostasis
14.5: Metabolism and Signaling: The Steady State, Adaptation and Homeostasis
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Search Fundamentals of Biochemistry
We have studied binding interactions in Chapter 5, kinetics in Chapter 6, and principles of metabolic control in this chapter. We've learned the following:
Binding Reactions
• for simple binding of a ligand to a macromolecule, graphs of fractional saturation of the macromolecule vs free ligand concentration are hyperbolic and demonstrate saturation binding. In the
initial part of the binding curve, when [L] << K[D], the fractional saturation shows a linear dependence on free ligand concentration. Figure \(\PageIndex{1}\) shows [ML] vs L, which is the same
basic equation as a plot of Y vs L.
Figure \(\PageIndex{1}\)
• for allosteric binding of a ligand to a multimeric protein, graphs of fractional saturation vs free ligand concentration are sigmoidal and also display saturation binding. In the first parts of
the binding curve, the fractional saturation is much more sensitive to ligand concentration than in simple binding of a ligand to a macromolecule with one binding site. Figure \(\PageIndex{2}\)
below shows graphs for the allosteric binding of a ligand to a macromolecule using the Hill Equation (instead of the MWC equation we used to model O[2] binding to tetrameric hemoglobin).
Figure \(\PageIndex{2}\)
In these two plots, the system (in this case a single macromolecule) displays different sensitivities to ligand concentration, allowing the system to have different responses to changes in
physiological conditions.
Binding and Chemical Reactions
As with the case for binding interactions, we have seen hyperbolic and sigmoidal plots of initial velocity (v[0]) vs [substrate] for enzyme-catalyzed reactions. These also allow appropriate responses
to a single substrate in a physiological setting.
But what if you put the same macromolecule and ligand into a larger metabolic or signal transduction pathway in vivo? What kinds of responses would they make to a change in input? As we have just
seen in our discussion of the steady state, the ligand or substrate concentration might not change at all as flux continues through the pathway. One could imagine a lot of scenarios with different
inputs and different optimal outputs. For example, what if the input (a reactant or small signaling molecule) comes in pulses? Ultimately a system should return to its basal state since a prolonged
response (such as cell proliferation) could be detrimental to the health of the organism.
Let's look at some simple examples and see how different inputs lead to specific outputs. We'll just construct some very simple reaction diagrams in Vcell and see how varying them leads to different
outputs. Here are two simple cases for isolated chemical species and reactions, analogs to the simple binding reactions described above.
Linear Response: A Signal S and a Response R; S → R
If no enzyme is involved, the rate doubles as the signal (substrate) doubles since dR/dt = k[S] for the first-order reaction. If S is the stimulus and R is the response, a plot of R vs S is linear.
Hence the system responds linearly with increasing S. Here is the simple chemical equation
\mathrm{S} \underset{\mathrm{k}_2}{\stackrel{\mathrm{k}_1}{\rightleftarrows}} \mathrm{R}
As a concrete example, consider the synthesis and degradation of a protein, characterized by the following equation derived from mass action.
\frac{d R}{d t}=k_0+k_1 S-k_2 R
where S is the signal (ex. concentration of mRNA) and R is the response (ex concentration of the transcribed protein). A constant k[0] has been added to account for any basal rate of the reaction.
(This is a vastly oversimplified way to model a complex process like mRNA translation to a protein as it omits 100s of steps.)
Here is the simplified derivation under steady state (SS) conditions typically found for enzymes embedded in a pathway.
\frac{d R_{S S}}{d t}=k_0+k_1 S-k_2 R=0 \\
R_{S S}=\frac{k_0+k_1 S}{k_2}
The equation is a linear function of S.
Hyperbolic Response: E+S ↔ ES → E + R
In a simple enzyme-catalyzed reaction with a fixed concentration of enzyme, as S increases the initial velocity saturates. Hence there is a limit on the response, so the response R is a hyperbolic
function of S. Increasing S ever more after saturation won't lead to more R (in a given amount of time).
As a concrete example of this consider the phosphorylation/dephosphorylation of a protein R. R[P] represents the phosphorylated and active form of the protein R with concentration [R[P]]. The
reaction is simply written as R ↔ R[P], where R[P] is the response. Mass action shows that the total amount of R, R[T] = R + R[P]. A simple mass action equation can be derived.
Here is the chemical equation
\mathrm{R}+\mathrm{S} \underset{\mathrm{k}_2}{\stackrel{\mathrm{k}_1}{\rightleftarrows}} \mathrm{R}_{\mathrm{P}}
Here is the math equation, again for the steady state (SS), when dR[P]/dt = 0. (We derived the same equation for the steady-state version of the Michaelis-Menten equation in Chapter 6.
\frac{d R_P}{d t}=k_1 S\left(R_T-R_P\right)-k_2 R_P
Click below to see the derivation
\frac{d R_P}{d t}=k_1 R[S]-k_2 R_P
then in the stead state:
\frac{d R_P}{d t}=k_1 S\left(R_T-R_P\right)-k_2 R_P=0 \\
k_2 R_{P, S S}=k_1 S\left(R_T\right)-k_1 S\left(R_{P, S S}\right) \\
k_2 R_{P, S S}+k_1 S\left(R_{P, S S}\right)=k_1 S\left(R_T\right) \\
R_{P, S S}\left(k_2+k_1 S\right)=k_1 S\left(R_T\right)
Finally, we get
R_{P, S s}=\frac{k_1 S\left(R_T\right)}{\left(k_2+k_1 S\right)}=\frac{\left(R_T\right) S}{\left(\frac{k_2}{k_1}+S\right)}
In the steady state, dR[P]/dt = 0, and the steady state equation can be written as:
R_{P, s s}=\frac{k_1 S\left(R_T\right)}{\left(k_2+k_1 S\right)}=\frac{\left(R_T\right) S}{\left(\frac{k_2}{k_1}+S\right)}
Sigmoidal Response
Consider this simple reaction for a homotetramer in which each monomer can bind a substrate S: nS + E[n] ↔ E[n]S[n] → E[n] + nR: If E[n] is a multimeric allosteric enzyme, as S increases the initial
velocity also saturates but the response R is a sigmoidal function of S (in analogy to the above example). The equation is too complicated to derive there, but the result reproduces a sigmoidal curve
for the steady state, much as the Hill equation does for cooperative binding.
Adaptation and Homeostasis
The above examples show that the response of proteins or enzymes to increasing levels of a stimulus like a ligand or a substrate can be linear, hyperbolic, or sigmoidal, with quite a varied set of
outcomes. However, in many biological conditions, an ever-increasing or increasing and plateauing response might be too much. The cell needs a way to turn off the response and settle back to a basal
state, even in the presence of constant or changing stimuli. This allows the adaption of a system to a stimulus and the maintenance of homeostasis. Every system needs to be able to respond and return
to a homeostatic basal level. The maintenance of homeostasis is critical to life.
The American Association for Biochemistry and Molecular Biology (ASBMB) describes both homeostasis and evolution as key underlying concepts for all biology. Homeostasis shapes both form and function
from the molecular to organismal levels. Homeostasis is needed to maintain biological balance. The steady state at the molecular to organismal levels in metabolic and signaling pathways is a hallmark
of homeostasis. Here are the learning goals for homeostasis designated by the ASBMB
1. Biological need for homeostasis
Biological homeostasis is the ability to maintain relative stability and function as changes occur in the internal or external environment. Organisms are viable under a relatively narrow set of
conditions. As such, there is a need to tightly regulate the concentrations of metabolites and small molecules at the cellular level to ensure survival. To optimize resource use, and to maintain
conditions, the organism may sacrifice efficiency for robustness. The breakdown of homeostatic regulation can contribute to the cause or progression of disease or lead to cell death.
2. Link steady-state processes and homeostasis
A system that is in a steady state remains constant over time, but that constant state requires continual work. A system in a steady state has a higher level of energy than its surroundings.
Biochemical systems maintain homeostasis via the regulation of gene expression, metabolic flux, and energy transformation but are never at equilibrium.
3. Quantifying homeostasis
Multiple reactions with intricate networks of activators and inhibitors are involved in biological homeostasis. Modifications of such networks can lead to the activation of previously latent
metabolic pathways or even to unpredicted interactions between components of these networks. These pathways and networks can be mathematically modeled and correlated with metabolomics data and
kinetic and thermodynamic parameters of individual components to quantify the effects of changing conditions related to either normal or disease states.
4. Control mechanisms
Homeostasis is maintained by a series of control mechanisms functioning at the organ, tissue, or cellular level. These control mechanisms include substrate supply, activation or inhibition of
individual enzymes and receptors, synthesis and degradation of enzymes, and compartmentalization. The primary components responsible for the maintenance of homeostasis can be categorized as stimulus,
receptor, control center, effector, and feedback mechanism.
5. Cellular and organismal homeostasis
Homeostasis in an organism or colony of single-celled organisms is regulated by secreted proteins and small molecules often functioning as signals. Homeostasis in the cell is maintained by regulation
and by the exchange of materials and energy with its surroundings.
In the rest of the chapter section, we will describe chemically and mathematically simple circuits/motifs that are employed that allow perfect or near-perfect adaptation to a stimulus, a hallmark of
homeostasis. We will define adaptation as a complete or almost complete return to a basal state after the introduction of a stimulus. In all the cases below we will consider not a single application
of a stimulus but a pulse application (a repetitive step wave function). The pulsed stimuli could be of constant magnitude or an increasing/decreasing pulse of a signal such as a substrate. All
responses must be transient to avoid uncontrolled responses such as proliferation (a hallmark of tumor cells) or cell death.
Adaptation is commonly found in sensory systems like vision, hearing, pressure, taste, etc. Think of eating your favorite cookie. The first bite is delicious but by the tenth bite, there is
significant attenuation in the positive sensory response, which helps keep most from adding significant weight continually.
Ma et al. conducted simulations on three component/nodes (proteins, enzyme) systems to see which might display the potential for perfect or near-perfect adaption. The simple 3-component motifs or
circuits were modeled using simple mass action kinetic equations, ordinary differential equations (which we learned to write in Chapter 6.2), or a combination of both. The systems that displayed
adaption had to conform to three criteria:
1. The stimulus had to initially induce a response of high magnitude
2. The system had to return to a basal or near basal state.
3. The return to a basal state had to be mostly parameter-independent. That is, the return to the basal state must occur for many different combinations of parameters.
The possible 3-component components (nodes) and the links among the nodes are shown in Figure \(\PageIndex{3}\) below.
Figure \(\PageIndex{3}\): Possible 3-component components (nodes) and the links among the nodes. After Ma et al. Cell Theory,138, 760-773 (2009) https://www.cell.com/fulltext/S0092-8674(09)00712-0.
Out of over 16,000 models, several hundred were found that met the criteria. Most were variations of simple motifs that we will show below. The most common motifs were the negative feedback loop and
the incoherent feedforward system.
Much of the discussions, models, and equations used below are from two articles:
• John J Tyson, Katherine C Chen, Bela Novak, Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell, Current Opinion in Cell Biology, Volume 15, Issue
2, 2003, Pages 221-231, https://doi.org/10.1016/S0955-0674(03)00017-6.
• James E. Ferrell, Perfect and Near-Perfect Adaptation in Cell Signaling, Cell Systems, Volume 2, Issue 2, 2016, Pages 62-67, https://doi.org/10.1016/j.cels.2016.02.006.
By adding a third component to form a mini pathway, we can now change the response R to a stimulus S from linear, or hyperbolic/sigmoidal in the steady state, to one that exhibits perfect or
near-perfect adaptation. Again we see this kind of response in signaling pathways in sensation and also in responses like chemotaxis, in which a cell moves toward a stimulus (a chemoattractant
Simple 3-node motif/circuit for perfect adaptation
Figure \(\PageIndex{4}\) below shows our first example of a 3-component system that displays perfect or near-perfect adaption. The right-hand side shows a Vcell reaction diagram. In this example, a
stimulus S (could be a reactant, neurotransmitter, mRNA, etc) leads to the synthesis of X and also of R, a response molecule. Both X and R get degraded. The yellow squares represent the nodes through
which the flux of S to X and R proceeds. Each node has an equation for the flux, J, through the node. The left part of Figure 4 shows the periodic pulse of stimuli S that increases the concentration
of S from an initial value of S[0] = 1 uM to S + 0.2 uM for each step. Note that the flux equations for J are very simple and are based on mass action, and are not derived through Michaelis-Menten
kinetic equations.
Figure \(\PageIndex{4}\): Simple 3-component system that displays perfect or near-perfect adaption.
Note that S, the stimulus (or substrate for example) is a square wave step function varying from 0 to 1 over the time interval shown in the graph. The dotted blue line simply shows when the pulse is
delivered. The initial S concentration is 1 uM and increases by 0.2 for each step (as shown in the gray line). Hence S increases in a stepwise fashion.
Figure \(\PageIndex{5}\) below is a time course graph that shows the stepwise (=0.2 uM) increase in S from 1 uM and the concentration of R (the response) over 20 seconds. Even though S continues to
increase in a stepwise fashion, R rises substantially only from the initial input of S (1 uM) and subsequent increases in S with each increment of S are damped out!
Figure \(\PageIndex{5}\): Time course for a 3-Component Perfect Response system. Model by ModeBrick from VCell: CM-PM12648679_MB4:Perfect_Adaptation; Biomodel 188456707
The present version of Vcell release (as of 4/28/23) does not yet allow the export of a file compatible with the software used to run simulations with this book. The Vcell model includes an "event"
which allows for the production of stepwise changes in stimuli. A future release will allow users to run the simulations within this book (as is the case for the other Vcell simulations throughout
the book).
Negative Feedback Loop
The negative feedback loop is one of the simplest circuits/motifs to generate perfect or near/perfect adaptation. It has only two nodes (yellow dots) and two proteins. An example is bacterial
chemotaxis. Figure \(\PageIndex{6}\) below shows a Vcell reaction diagram (left), and another representation (middle) and the time course graphs for all species. This model works especially well with
certain parameters assigned.
Figure \(\PageIndex{6}\)
Figure \(\PageIndex{6}\): Near-Perfect Adaptation from Negative Feedback. Adapted from Ferrell (ibid)
The gray line in the graph is the stimulus S (substrate). The blue line is the response, designated in this model as A. B acts as an inhibitor (note the dotted line to the input node in the left
diagram and the blunt-ended red bar in the middle diagram. Note that the stimulus goes from 0.2 uM (initial concentration) at t=0 to 1 uM (a 5-fold increase) at 40 seconds, but the response A
increases at most from 0.4 (initial condition) to 0.5 (a 1.25-fold increase).
If we say the [A] is the output, then the differential equation for dA/dt is given by
\frac{d A}{d t}=k_1 \operatorname{S} \cdot(1-A)-k_2 A \cdot B
dB/dt is given by
\frac{d B}{d t}=k_3 A \frac{1-B}{K_3+1-B}-k_4 \frac{B}{K_4+B}
The constants for the graph (right) produced by the Vcell model are:
• k[1] = k[2] = 200
• k[3] = 10; k[4] = 4
• K[3] = K[4] = 0.01
Incoherent Feedforward systems
In this circuit/motif, the stimulus S increases the concentration of A (the output) but also forms a negative modulator, B, which with a bit of a time lag decreases the concentration of A through
inhibition. There is no feedback inhibition from A in this simple system. If you're reading carefully, you'll see that the reaction scheme and inhibition are the same as the first circuit/motif we
introduced. Here we simplify the diagram and give it an official name. The word incoherent in the name makes sense since the stimulus S is converted both to the output A, and to the inhibitor B,
which on the surface seems like a crazy thing to do.
Figure \(\PageIndex{7}\) below shows the Vcell reaction diagram (left) and more classical reaction diagram (middle) and progress curves showing S, the stimulus, A the output or response, and B, the
inhibitor. The dashed line in the left diagram from B to the reaction node for the S → A reaction shows that B affects the rate of that reaction. The equations used account for the inhibitory effect
of B.
Figure \(\PageIndex{7}\): Near-Perfect Adaptation from an Incoherent Feedforward System. Adapted from Ferrell (ibid)
Note that the response A goes up or down a bit with each new step in concentration of S but to a very minimal degree. The system is certainly almost perfectly adapted.
The differential equation for dA/dt (where A is the response) is
\frac{d A}{d t}=k_1 \operatorname{S} \cdot(1-A)-k_2 A \cdot B
The equation for dB/dt (the inhibitor generated from A) is
\frac{d B}{d t}=k_3 \text { S } \frac{1-B}{K_3+1-B}-k_4 B
The constants for the graph (right) produced by the Vcell model are:
• k[1] = 10; k[2] = 100
• k[3] = 0.1; k[4] = 1
• K[3] = 0.001
State-dependent Inactivations systems.
There are two simple circuits/motifs in this system that were found after the initial analyses that showed all possible interactions in a 3-component system (see Figure 3). The motif was patterned
after the inhibition of proteins in neuron stimulation, specifically in ion channels in neural cell membranes that open up on a change in the transmembrane potential but then close again quickly to
avoid constant neuronal stimulation (or inhibition). In the Na^+ ion channel, there are both fast (1-2 ms) and slow (100 ms) inactivation mechanisms. The fast one allows for repetitive firing, the
development of action potentials, and the control of the excitation of neurons, and at the neuromuscular junction. Neuronal signaling is discussed in Chapter 28.9. Figure \(\PageIndex{8}\) below
shows a simplified model for one type of inactivation of the Na^+ ion channel
Figure \(\PageIndex{7}\): Simplified state transition model of voltage-gated sodium channels featuring closed, open, and inactivated states. Zybura, A. et al. Cells 2021, 10, 1595. https://doi.org/
10.3390/cells10071595. Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
The figure implies that there are at least 3 conformational states of the channel so the inactivation for the channel and the circuit/motif for adaptation we will now discuss are called
state-dependent inactivations. The slow return to the original state is observed in many ion channels as well as in the return of G protein-coupled receptors to the normal state after their
desensitization. Also, some protein kinases (kinases that use ATP to phosphorylate protein substrates) can be inactivated by internalizing the membrane kinase into vesicles where they can be
reactivated and returned to the plasma membrane in a slow process.
For the construction of a perfect or near-perfect adaption state, we will assume the protein A exists in an off state (A[off]) which binds the stimulus (B or S), an on state (A[on]) which is viewed
as the response (or A produces the response), and an inactivated state (A[in]) which slowly reverts to the A[off] state which can be activated again. The inactive state can be produced by
conformational transitions with the protein itself or another molecule produced downstream of it in a metabolic or signaling pathway. For example, a GPRC could be phosphorylated or bind to another
species to produce an inactive state.
There are two different circuits/motifs that can produce state-dependent inactivation. We'll refer to these as Type A and Type B
Type A
Figure \(\PageIndex{9}\) shows the Vcell reaction diagram (top left), a classical reaction diagram (bottom left) and time course graphs for Type A state-dependent inactivation.
Figure \(\PageIndex{9}\): Perfect Adaptation for Type A State-Dependent Inactivation. Adapted from Ferrell (ibid).
A[on] represents the active state of the protein. This mechanism applies well to the Na^+ channel. The differential equations for dA[on]/dt and dA[off]/dt are shown below.
For dA[on]/dt
\frac{d A_{o n}}{d t}=k_1 \operatorname{Input} \cdot\left(1-A_{o n}-A_{i n}\right)-k_2 A_{o n}
\frac{d A_{i n}}{d t}=k_2 A_{o n}
with constants k[1] = k[2] = 1.
Again, as with the other cases, the stimulus S is pulsed. The different colors in the bottom left reaction diagram imply an off and inactive red state and a green active state, each of different
conformations. The graphs were produced using Vcell. There is a slight anomaly in the graph of A[on] which shows two additional small peaks as the system returns to the basal state. This contrasts to
just 1 peak which returns to the basal state in a simple exponential fashion as described in the Ferrell paper. We are uncertain as to the source of the discrepancy.
Type B
In this case, the periodic stimulus, abbreviated as B, is a binding partner for A[off] which produces an active complex B-A[on]. Figure \(\PageIndex{10}\) below shows the Vcell reaction diagram (top
left), a classical reaction diagram (bottom left), and time course graphs for Type B state-dependent inactivation
Figure \(\PageIndex{10}\)
Figure \(\PageIndex{10}\): Perfect Adaptation for Type B State-Dependent Inactivation. Adapted from Ferrell (ibid)
BA[on] represents the active state of the protein bound to B while BA[in] represents the inactive complex.
The equation of dBA[on]/dt for the formation of the active state is
\frac{d B A_{o n}}{d t}=k_1\left(B_{t o t}-B A_{o n}-B A_{i n}\right) *\left(1-B A_{o n}-B A_{i n}\right)-k_2 B A_{o n}
and the equation for dBA[in]/dt for the formation of the inactive state is
\frac{d A_{i n}}{d t}=k_2 A B_{o n}
with constants k[1] = k[2] = 4.
The graphs (note the different time concentration scales on the left) show a fairly quick return to the basal state after each pulse of stimuli (B).
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Valuing US Equities and Ready Reckoner
We have for many years been publishing every quarter an update on the valuation of the US stock market according to q and CAPE. We receive frequent questions as to how these are calculated and what
are their current values.
We have made available to anyone who is interested an excel file which shows the Chart and the accompanying spreadsheets; we also provide an explanation showing how the data are compiled and how the
calculations are made.
The spreadsheet (“q”) which can be found on Latest q & CAPE Data page, also has a ready reckoner which allows anyone to see how changes in the level of the stock market affect its current value
between the updates which we post after the quarterly data are published by the Federal Reserve in their Financial Accounts of the United States (“Z1”), formerly known as the Flow of Funds Accounts.
We have encountered two, and only two, valid ways of measuring the value of the US stock market. A valid method must be testable and prove robust when tested. The tests that we have used, and we have
so far been unable to think of any others, are set out in the two books listed below.
Valuing Wall Street: Protecting Wealth in Turbulent Markets by Andrew Smithers & Stephen Wright (published by McGraw-Hill in 2000).
Wall Street Revalued: Imperfect Markets and Inept Central Banker by Andrew Smithers (published by John Wiley & Sons in 2009).*
Other publications which are relevant to the valuation of markets are;
Irrational Exuberance by Robert Shiller (published by Princeton University Press in 2000).
Rational Pessimism: Predicting Equity Returns using Tobin’s q and Price/Earnings Ratios by Matthew Harney & Edward Tower (published in the Journal of Investing, Fall 2003).
The Basis of q and CAPE.
There are two possible approaches to valuing equities:
(i) The macro-economic approach, which treats them as titles to the ownership of real assets and values them in line with their cost of production.
(ii) Valuing them as financial assets and therefore as the present value of all future cash flows to which the owner is entitled.
A truly satisfactory approach to value must encompass both and, allowing for the differences in the data sources, they need to agree. As you will see from the Chart, q and CAPE track each other
closely and give very similar answers to the question as to how much the US stock market is over or undervalued at any time.
The Data Sources.
In order to calculate q we need to have an estimate of the current market value of non-financial companies (MV) and an estimate of their net worth (NW). The data sources for these are Stephen
Wright’s work, Measures of Stock Market Value and Returns for the US Nonfinancial Corporate Sector, 1900 to 2002 (published in the Review of Income and Wealth 50 (4)) and that published by the
Federal Reserve in Table B.102 of the Z1.
In combination we have data for q since the end of 1900. They are available on an annual basis since 1900 and quarterly since 1952.
In order to calculate CAPE we need to have estimates of the earnings per share (“EPS”) of quoted companies and of the consumer price index (“CPI”). These are regularly published by Robert Shiller on
his website and, if we need to update them, we use the Standard & Poor’s website for EPS and BLS for the CPI.
EPS are published in nominal terms and, to avoid the distortions that might arise from fluctuations in the rate of inflation, they need to be adjusted to constant prices (EPS K).
CAPE is the ratio of EPS K measured by its average over a number of recent years (10 being the most usual) to EPS K measured since the data series start in 1987.
Calculating q.
On the attached excel file there is a spreadsheet labelled q. Column B gives the year ends for which the data are available, with the exception of the final entry, which is for the most recent
quarterly data available. Column C gives the (unadjusted) value of q calculated from the data published by Stephen Wright. Column D is the (unadjusted) data published by the Federal Reserve in Table
B.102 of Z1. It is the ratio of line 36 (Market value of equities outstanding) to line 33 (Net worth (market value) i.e. net worth adjusted for the impact of inflation; not the net worth at book
value shown in B.102 line 46).
As the Federal Reserve makes from time to time adjustments to Z1, we link the most recent data published with the series published by Stephen Wright and this is shown in Column E. Column F shows the
natural logarithms of Column E (i.e. logs to base e). The market values shown in B.102 line 36 include those of unquoted companies as well as quoted ones and the Federal Reserve statisticians value
the former at a discount to the latter. The estimate of market value is therefore less than the equivalent value of quoted companies. The net worth data are estimated from the profits published by
companies, which are habitually overstated (for an explanation of this see Chapter 9 of The Road to Recovery: How and Why Economic Policy Must Change by Andrew Smithers published by John Wiley & Sons
Ltd. in 2013).
To allow for the undervaluation of market value and the overvaluation of net worth Cell F.124 shows the average (geometric mean) of q. Column G shows in logs the current value of q as a ratio to its
own average and Column H shows this in natural numbers.
For those who like equations, the over or undervaluation of the US stock market is the extent to which q is greater or less than 1 when its value is calculated from:-
q = (MVc/NWc) ÷ (MVa/NWa)
MVc is the current market value of US non-financial companies shown in Z1 Table B.102 line 36.
NWc is the current net worth of US non-financial companies shown in Z1 Table B.102 line 33.
MVa/NWa is the geometric mean of all previous year end values of MV/NW.
Calculating CAPE.
The spreadsheet labelled CAPE calc derives Columns B to F from Robert Shiller’s website (when appropriate we update this). Columns H and J adjust Column C (share price) and E (EPS) to constant
prices. Column L is the geometric mean of EPS at constant prices over the previous 10 years. M is the Cyclically adjusted PE (measured by Column H divided by Column L). Column N shows the log values
of Column N. Cell N 1729 is the average of the values in Column N and Cell O 1729 is the geometric mean of the values in Column O. (It is derived from the anti-log of cell N 1729 (its exponential)).
Column O is the difference between the values in Column N and their own average shown in Cell N 1729 and thereby measures, in log numbers, the divergence from fair value shown by CAPE.
Column Q shows the year end values of Column N and these are the numbers shown in spreadsheet q. In this spreadsheet Column H shows, in natural numbers, the degree of misevaluation derived from q and
Column K show these for CAPE.
Ready Reckoner for Changes in Stock Market.
As at the end of 2013 the S&P500 for q was 1848.357, Market Value from q was 1.734667, for CAPE was 1807.78 and the Market Value from CAPE was 1.757024
As at 9th April 2014 the S&P500 for q was 1872.179, Market Value from q was 1.757024, for CAPE was 1872.179 and the Market Value from CAPE was 1.828374
Columns F,G,H,I & J of lines 126, 127 and 128 of spread sheet q, have the above entries. Anyone wishing to update q and CAPE for changes in the S&P 500 can simply enter the new date in place of 9th
April 2014 and the new S&P level; the values of q and CAPE should then change automatically to take account of the change in the stock market.
PS. Please let us know if you find any errors in the calculations. We try hard to avoid them but cannot guarantee success.
PPS. Please let us know if the explanations are unclear or if you have questions.
*Stephen Wright & Derry Pickford teach students how to value the US stock market as part of the Didasko Course. Details can be found on www.didaskoeducation.org
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CFD Analysis of Elements of an Adsorption Chiller with Desalination Function
Department of Thermal and Fluid Flow Machines, Faculty of Energy and Fuels, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
Author to whom correspondence should be addressed.
Submission received: 5 October 2021 / Revised: 17 November 2021 / Accepted: 18 November 2021 / Published: 22 November 2021
This paper presents the results of numerical tests on the elements of an adsorption chiller that comprises a sorption chamber with a bed, a condenser, and an evaporator. The simulation is based on
the data and geometry of a prototype refrigeration appliance. The simulation of this problem is unique and has not yet been performed, and so far, no simulation of the phenomena occurring in the
systems on a real scale has been carried out. The presented results are part of the research covering the entire spectrum of designing an adsorption chiller. The full process of numerical modeling of
thermal and flow phenomena taking place in the abovementioned components is presented. The computational mesh sensitivity analysis combined in the k-ε turbulence model was performed. To verify and
validate the numerical results obtained, they were compared with the results of tests carried out on a laboratory stand at the AGH Center of Energy. The results of numerical calculations are in good
agreement with the results of the experimental tests. The maximum deviation between the pressure obtained experimentally and by simulations is 1.8%, while for temperatures this deviation is no more
than 0.5%. The results allow the identification of problems and their sources, which allows for future structural modifications to optimize the operation of the device.
1. Introduction
According to the report prepared by the Lawrence Livermore National Laboratory institute, humanity uses almost as much energy as it loses in all sectors of the economy [
]. Thus, many legislators and manufacturers strive to constantly increase the energy efficiency of devices and systems. One way to reduce energy losses, and thus increase the efficiency of systems,
may be to use waste heat as an energy source. An example of a device that uses waste heat as an energy source is an adsorption chiller.
Adsorption chillers usually consist of one evaporator, a condenser, appropriate control valves, and a bed filled with a sorbent. The sorbent in the bed adsorbs and desorbs cyclically a working
medium, called sorbate. The sorbent and sorbate together are known as a working pair, and their molecular structure and intermolecular interactions between them determine how specific
adsorbent–adsorbate pairs will behave during adsorption and desorption [
]. Due to the low desorption temperature, low cost, and a lack of negative impact on the environment, the silica gel–water vapor working pair is commonly used in adsorption chillers [
]. Silica gel has a fairly large specific surface area (wide pores, 250–350 m
/g; narrow porosity, 600–850 m
/g), thanks to which it is possible to adsorb a large amount of water vapor [
Adsorption chillers use waste heat as an energy source. Additionally, they possess many other advantages, including lack of moving parts, low level of noise, no vibrations, and the possibility of
water desalination [
]. On the other hand, intermittent operation, low coefficient of performance, big size and weight, and unequal time of desorption and adsorption processes can be accounted as the main drawbacks of
adsorption chillers [
]. Hence, numerous scientific works are conducted to eliminate the abovementioned drawbacks.
The problem of intermittent operation can be eliminated by using multi-bed refrigerators with three [
] or more beds [
]. When using multiple beds, cycle lengths and their spacing can be adjusted to maximize cooling efficiency and working time of the evaporator in the cycle [
]. The other parameters influencing the performance, size, and weight of the adsorption chillers include, but are not limited to, the design of the heat exchanger in the bed [
] and the materials of which the bed is made [
], the size of sorbent particles [
], the velocity of heat transfer fluid that heats or cools the bed [
], and the type of evaporator [
]. As can be seen, the performance of adsorption chillers depends on numerous different factors. Additionally, the physical phenomena, i.e., evaporation, condensation, adsorption, and desorption,
occurring in the main components of the adsorption chiller are intrinsically complex, as they are related to heat and mass transfer at various pressures and temperatures. Thus, fully understanding
the abovementioned processes is essential to improve the performance of adsorption chillers.
As commonly known, the abovementioned processes (evaporation, condensation, adsorption, and desorption) are difficult to analyze using only experimental measurements. The measurements of
temperatures, pressures, flow rates, etc., are crucial since they allow determining, e.g., the instantaneous and average efficiency of the chiller. However, these measurements are not sufficient for
a comprehensive, and detailed scientific analysis of the chillers’ operation is sometimes even impossible to perform. Furthermore, conducting experimental studies is very time and cost consuming. As
a result, computational fluid dynamics (CFD) analysis becomes a necessity, and numerous scientific works use CFD to investigate either the evaporation, condensation, and adsorption/desorption
processes themselves or the influence of defined parameters on the operation of the entire adsorption chiller.
Modeling the condensation and evaporation phenomena is a difficult issue since the characteristics of the fluid, both in the liquid and gaseous state and the relationships between them, should be
taken into account. It is necessary to apply a two-phase model with appropriate consideration of the possible turbulence and the accompanying surface forces phenomenon [
]. It is also necessary to track the liquid–gas interface, which can be achieved by using the volume of fluid method [
]. Adsorption and desorption processes are also difficult to model, and a variety of different mathematical models of the sorption phenomenon are available [
]. Nevertheless, the linear driving force (LDF) model [
] is one of the most widely used.
Mohammed et al. [
] designed and investigated numerically, using FORTRAN software, a new modular bed for adsorption chillers. The bed consisted of an array of modules filled with adsorbent and placed in a metal case
with vapor channels. After parametric analysis, it was concluded that the cooling power per unit volume was higher for the proposed bed compared to the commercially available beds. Hong et al. [
] used CFD to compare the performance of the adsorption chiller equipped with plate and fin-tube heat exchangers in the adsorption bed. It was found that the chiller with plate heat exchanger had a
lower, by 19.9%, coefficient of performance (COP) but a higher, by 15.7%, specific cooling power, compared to the fin-tube heat exchanger. CFD analysis was also conducted by Mohammed et al. [
], who investigated heat and mass transfer in an open-cell aluminum foam packed with silica gel, which can be used as a bed in adsorption cooling applications. It was reported that the foam with 20
pores per inch (PPI) has a larger surface area and smaller cell size than the 10 PPI foam and thus is more advisable to be used in adsorption chillers. It was also found that the average bed
temperature and adsorption rate was higher for 0.35 mm silica gel than for 0.70 mm silica gel. The effect of sorbent particle size was also investigated, using CFD, by Mitra et al. [
], who concluded that using a smaller sorbent does not always lead to faster adsorption, and the optimum geometry of a heat exchanger exists for a particular sorbent size.
Apart from the construction of the heat exchanger in the bed or the size of sorbent particles, the thermal conductivity of the bed strongly influences the overall performance of the adsorption
chiller [
]. Therefore, different metal- or carbon-based additives can be added to the bed to increase its thermal conductivity [
]. Furthermore, the rate of adsorption depends on the convective heat transfer coefficient between the bed and the heat transfer fluid [
]. Therefore, the desorption and adsorption time can be reduced by increasing the velocity of the heat transfer fluid [
]. Elsheniti et al. [
], who used COMSOL Multiphysics software for investigations, reported that the COP and specific cooling capacity could be improved by 68% and 42%, respectively, when the flow character of the heat
transfer fluid was changed from laminar to turbulent.
The literature review presented in the above paragraphs justifies the possibility of using CFD simulations for investigating adsorption chillers.
This paper presents a CFD analysis, carried out in the Ansys Academic Research CFD 19- Ignis Research Group ANSYS, Inc. 2600 ANSYS Drive Canonsburg, PA 15317 USA of the phenomena occurring in the
main elements of the adsorption chiller, i.e., evaporator, condenser, and bed. The results of the CFD analysis are validated using data from the experimental studies carried out on the adsorption
chiller located at the AGH UST Center of Energy. Therefore, this study aims at:
• Numerical modeling of the processes occurring in the evaporator, condenser, and bed of the adsorption chiller to understand better these processes;
• Analyzing the fields of temperature, pressure, and velocity;
• Indicating the locations with extremes of temperatures, pressures, and velocities;
• Determining the phenomena disrupting the operation of the main elements of the adsorption chiller;
• Determining potential changes in the structure of the main elements, which may improve the operating parameters of the individual elements of the chiller and increase its efficiency and
The direct value of this research is a possibility to determine the key elements and processes that impact the operation of the adsorption chiller. Consequently, this research indicates potential
modifications of the chiller, which may be applied and improve its performance. Furthermore, the adsorption chiller located at the AGH UST Center of Energy has never been investigated using CFD
analysis before. Thus, this research is a novelty in the context of getting to know the specifics of the chiller operation and the possibilities of its modification to improve the chiller efficiency
and reliability as well as decrease its size and weight.
2. Materials and Methods
2.1. Empirical Research
The simulations were preceded by empirical tests carried out at the research station for adsorption systems at the AGH UST Center of Energy on a unique adsorption system. The system can operate in a
two- or three-bed mode, generating chilled water and purified water. The beds were built on the basis of a plate-fin exchanger filled with adsorption material. The absorbent material used consists of
elements with a size of 500–1000 μm. The scheme of the installation is presented in
Figure 1
, and the technical specification of the investigated chiller is given in
Table 1
For the purpose of this study, the chiller operated in a two-bed mode, and a detailed description of its operating principle can be found in [
]. The parameters that were measured during the experiments, and the measuring devices are listed in
Table 2
. The measured values were recorded every 5 s on a personal computer using special software, and then the results were exported to CSV files.
The results of the experimental studies are shown in
Figure 2
Figure 3
Figure 2
depicts the temperature changes over time taking place in the main components of the investigated adsorption chiller. The pressure changes in the beds, evaporator, and condenser are shown in
Figure 3
. Some of the results shown in
Figure 2
Figure 3
were used to define the boundary conditions and the rest to validate the simulation results. The average values between red lines in
Figure 2
Figure 3
represent the values taken as the boundary conditions for the simulations. These values represent the operating parameters of the adsorption chiller after about 25 min from its start-up, as the
operating conditions of the chiller stabilize after that time.
Figure 4
presents the destructive effect of the water vapor on the bed observed during the empirical research. Therefore, numerical simulations were applied to, among others, find the reasons for this
destructive effect and to better understand the phenomena occurring in the system. Knowledge about the processes will allow modifying the components of the adsorption chiller to eliminate the
destructive effect visible in
Figure 4
and obtain a greater cooling capacity.
2.2. Construction of Spatial Geometry and a Computational Grid
2.2.1. Generation of a Computational Mesh
During the construction of geometric models, the following simplifications were made:
• The housing of the elements was simplified to the form of a cylinder, without sight glasses and measuring connectors.
• Irregularly shaped elements such as heating and cooling junction boxes in the evaporator and condenser were simplified to a cylinder form.
• For the sorption chamber, the structural elements supporting the bed were omitted, and the bed itself was simplified to the form of a cuboid.
Then, after creating the structural elements, they were filled with the fluid domain in order to obtain the appropriate computational domains constituting the interior of the sorption chamber,
evaporator, and condenser, respectively.
After the geometric model of the studied domain was prepared, it was exported to the ANSYS Meshing module. With it, a continuous domain was discretized in order to obtain a computational mesh.
In order to assess the mesh quality, an analysis of the cell quality parameters was carried out:
• Orthogonal quality: its value is in the range <0,1>, where the value 1 means the highest possible quality.
• Skewness: its value is in the range <0,1>, with the value 0 being the highest possible quality.
It was estimated that most of the elements for the orthogonal quality are in the range of 0.7–1.0 (approx. 2/3 elements). Likewise, most of the elements for skewness are in the range of 0–0.2
(approx. 2/3 elements). The obtained values of the orthogonal quality and skewness prove the good quality of the mesh.
The calculation grid for the sorption chamber consists of 472,686 nodes and 2,529,568 elements. The calculation grid for the evaporator consists of 2,165,228 nodes and 11,296,174 elements. The
calculation grid for the condenser consists of 869,789 nodes and 4,562,582 elements. This enables performing calculations in a reasonably short time without using a supercomputer.
2.2.2. Boundary Conditions
The step preceding the calculations is to define the necessary solver settings. They were established on the basis of experimental studies (
Figure 2
Figure 3
) and studies in the literature described in the Introduction (
Section 1
• For the issues related to relatively low flow velocities (subsonic flow), flow solutions based on the pressure field (“pressure-based”) were used.
• It was assumed that the simulation would be carried out in the “transient” mode, which enables the observation of changes in parameters over time.
• The influence of gravity on the fluid elements was taken into account by appropriately defining the acceleration vector.
• Due to the inclusion of gravity in the model and the occurrence of mass interactions, the scheme of coupling the velocity and pressure fields (“coupled”) was applied.
• In the sorption chamber, reference conditions were defined in the entire domain as for sorption pressure of 1050 Pa and a temperature of 315.8 K, steam inlet at a temperature of 279.07 K, for
desorption a pressure of 5250 Pa and a temperature of 301.98 K, and a steam outlet temperature of 312.82 K
• The sorption time was set at 100 s, and the desorption time 200 s
• Reference conditions for the evaporator was a pressure of 1050 Pa and a temperature of 279.09 K
• Evaporator water inlet temperature was 280.24 K, water outlet temperature was 280.29 K, and the steam outlet temperature was 279.09 K
• A second-order spatial discretization scheme was used for the governing equations (mass, momentum, and energy). However, for the dissipation of turbulence, a first-order scheme was used.
• Reference conditions for the condenser was a temperature of 301.98 K, and a pressure of 5250 Pa
• Steam inlet temperature to the condenser was 312.82 K, and the temperature of the cooling pipes was 292.82 K
• Condenser domain computation was set for a time equal to 500 s.
2.2.3. Computational Methods
For all phenomena, Ansys Fluent solves the equations of conservation of mass, momentum, and energy. The conservation of mass, momentum, and energy are shown in Equations (1)–(3), respectively [
$∂ ρ ∂ t + ∇ · ( ρ ν ¯ ) = S m$
$∂ ∂ t ( ρ v ¯ ) + ∇ ⋅ ( ρ v ¯ v ¯ ) = − ∇ p + ∇ ⋅ ( τ = ) + ρ g ¯ + F ¯$
$∂ ∂ t ( ρ E ) + ∇ ( v ¯ ( ρ E + p ) ) = ∇ ⋅ ( k e f f ∇ T − ∑ j h j J ¯ j + ( τ = e f f ⋅ v ¯ ) ) + S h$
$v ¯$
$τ =$
$ρ g ¯$
$F ¯$
$J ¯ j$
$( τ = e f f ⋅ v ¯ )$
, and
stand for the density, time, velocity, source term, static pressure, stress tensor, gravitational body force, external mass forces acting on the fluid, energy, effective thermal conductivity
coefficient, enthalpy of species
, diffusive flux of species
, viscous dissipation, and heat source, respectively.
To model turbulence occurring in the flows, the k-ε realizable turbulence model was used. In order to model the multiphase flow, the “Species Transport” model was selected for the calculations.
3. Results and Discussion
3.1. Results of Numerical Calculations
Numerical simulations are a tool that can explain many phenomena, the study of which with experimental methods is problematic. The simulation studies conducted in this work consider the interactions
between the various components of the chiller at given phases of the work cycle. The simulation also allows for a time-dependent analysis of the temperature, pressure, and velocity variations during
the operation of the unit. However, it should be remembered that they are based on assumptions and models. In order to assess the applicability and correctness of the tested model, it should be
compared with the experimental results.
The results were developed using selected methods available as components of the Ansys CFD-Post post-processor module. To show the results of the simulations and the phenomena occurring in the main
components of the investigated adsorption chiller, i.e., sorption chamber, evaporator, and condenser, the appropriate cross-sections of these components were selected, and they are shown in
Figure 5
Figure 6
Figure 7
3.2. Simulation Results for the Sorption Chamber
3.2.1. Sorption Process
In this study, the temperature and pressure distribution, as well as velocity field, during the sorption processes were determined. The presented results concern the state of the system at the time
point t = 100 s.
Figure 8
a, shows the temperature distribution in the computational domain in Section K1. Uniform temperature distribution was obtained close to 311 K, which proves that the dimensions of the sorption chamber
are sufficient. If the sorption chamber was too small, there could be a non-uniform temperature distribution. The lowest temperature value 279 K occurs only in the vicinity of the deposit location.
It results from the modeling of the heat flux leaving the system on these planes.
Figure 8
b shows the pressure distribution in the computational domain in Section K1. Uniform pressure distribution in the range of 1043–1045 Pa was obtained. The area of increased pressure up to 1067 Pa
occurs in the vicinity of the location of the bed on the extension of the steam inlet nozzle. This may be due to the close location of the stub pipe and the direct influence of the inlet stream.
Likewise, high pressure is one of the reasons for the negative impact on the viability of the bed, as shown in
Figure 4
The visualization of the velocity field distribution is shown in
Figure 8
c. A large variation of the vapor velocity in the space around the bed is observed. The velocity of the vapor varies from 0.1 to 8.7 m/s. The greatest velocity, 8.7 m/s, is in the area of the outflow
from the inlet connection, while lower velocity values, in the range of about 0–4 m/s, are present in the remainder of the domain. This is due to the small diameter of the inlet nozzle and the
continuity equation, according to which a velocity of an incompressible fluid always decreases as the fluid flows from a channel with a small diameter (small cross-sectional area) to a channel with a
larger diameter (larger cross-sectional area).
3.2.2. Desorption Process
The presented results concern the state of the system at the time point t = 200 s.
Figure 9
shows the pressure distribution in the computational domain in Section K1. Uniform pressure distribution of 5247 Pa on a global scale was obtained. The lower pressure values in the range of 5170–5230
Pa can be observed in the vicinity of the steam outlet from the system.
The visualization of the velocity field distribution for Section K2 is presented in
Figure 10
. On a global scale, the velocity level inside the domain is at a similar level. The highest velocities, up to 0.137 m/s, occur near the steam outlet, which result from the geometry of the system.
Taking into account the local values for individual cross-sections, an increase in speed at the points of narrowings can be noticed. In addition, the speed also increases in the lower part of the
tank relative to the upper part. This may be due to the geometrical imbalance of these parts—in the lower major part, the fluid must move faster in order to maintain a continuous flow. The results of
the simulation also show the formation of steam vortices and their shapes and sizes, which are related to the adopted geometry of the bed.
3.3. Simulation Results for the Evaporator
The presented results concern the state of the system at the time point t = 100 s. The temperature distribution in the computational domain is presented in
Figure 11
a,b. The highest temperature, 284 K, occurs in the lower part of the evaporator tank, near the pipes through which the chilled water flows. When the fluid elements come into contact with the surface
of the tubes at the higher temperature, the temperature of the fluid element increases until vaporization occurs. P1 (
Figure 11
a) shows brighter spots near the water entry into the system through the spray system. This is due to the lower temperature of the entering water compared to the surrounding fluid. In Cross-Section
P2 (
Figure 11
b), one can see the increased temperature in the lower part to 284 K. Water in the system falls under the influence of gravity. At the same time, it heats up from other fluid elements and heating
Figure 12
a,b shows the velocity field distributions in the Cross-Sections P1 and P2, respectively. The vapor velocity ranges from about 0.1 to 9.7 m/s. The highest speed of 9.7 m/s is at the steam outlet
nozzles. It is caused by the fact that the steam flows from the tank to the nozzles that have a small diameter. As a result of the laws of flow continuity, speed increases.
The distribution of the velocity field on Section P1 indicates an increase in the velocity of the fluid elements in the vicinity of the obstacle constituted by the structural element inside the tank.
When the fluid encounters this obstacle, there is less space available for it to move, so its velocity in this area must be increased.
The formation of closed zones with a similar value of the velocity of the fluid elements can be seen in Cross-Section P2. One of these zones was formed around the heat exchanger pipes, and the other
around the pipe supplying water to the system. This may indicate that the flows between the layers of fluid in the reservoir are established.
Figure 13
shows the mass fraction of liquid water in Section P2. The greatest concentration of the liquid occurs in the lower part of the tank in accordance with the acting gravity. The greatest concentration
occurs in the area of the outflow of non-evaporated water from the system and between the heat exchanger pipes. Above the area of the heat exchanger pipes, the water content is lower, which means
that the water vapor content is predominant there. The distribution in this section also shows a greater liquid content on the left side compared to the right side.
3.4. Simulation Results for the Condenser
The presented results concern the state of the system at the time point t = 200 s.
The temperature distribution inside the condenser computational domain is shown in
Figure 14
a,b. The highest temperature of 312 K is visible near the steam input to the system. As the temperature moves away from the entry point, the temperature decreases, which indicates that the incoming
vapor cools down in contact with other fluid components already present in the condenser vessel. The lowest temperature of 294 K is around the pipes with the cooling liquid. Additionally, as can be
seen in Cross-Section S3, the temperature of fluid decreases when the fluid passes through the vicinity of the cooling surfaces. The fluid temperature is the lowest in the space opposite the steam
inlet ports. On Cross-Sections S1 and S3, one can find the shift of isotherms in the direction opposite to the
axis. This may indicate a local fluid flow in the direction opposite to the
The distribution of velocity fields is presented in
Figure 15
. The vapor velocity varies from about 0.1 to 3.4 m/s. The highest velocity of 3.4 m/s prevails in the area of the steam inlet nozzles. This is understandable because the fluid entering the reservoir
from the pipe slows down in velocity due to the increase of the space in which it can be held.
Additionally, it can be noticed that the velocity field distribution in the vicinity of the connection between the stub pipe and the tank is shifted in the direction opposite to the Z axis, similar
to the temperature. There is also a greater value of velocity in the vicinity of the tank walls. This may indicate the establishment of solid current lines inside the condenser, along which the fluid
elements move.
3.5. Validation of Simulation
Numerical simulations are a tool that can explain many phenomena, the study of which with experimental methods is problematic. However, it should be remembered that they are based on assumptions and
models. In order to assess the applicability and correctness of the tested model, it should be compared with the experimental results. The experimental type of measurements makes it impossible to
measure local values such as velocity in cross-section. Validation is done by comparing global values. The most reliable parameter is the pressure for the sorption and desorption process.
Figure 16
Figure 17
show the pressure change over time for the sorption and desorption processes, respectively. The figures show a 2.5% error; however, the maximum discrepancy between the results of simulation and
experiments is 1.5% and 1.8% for the sorption and desorption, respectively. The simulation results are within this range and can therefore be considered reliable.
Figure 16
shows the mean pressure inside the domain as a function of time for the sorption process. As can be seen, the pressure decreases almost linearly from the initial value of 1070 Pa to about 1050 Pa.
The pressure drop is caused by the fact that during the sorption, water vapor is adsorbed by the sorbent.
As mentioned before, the discrepancy between the results of simulation and experimental studies is no more than 2.5%. These results of simulation reflect the quality of the model and the reliability
of the performed calculations. The simulation will allow for an in-depth analysis of, firstly, the phenomena taking place during the adsorption and desorption and, secondly, any local changes in the
operating parameters and the efficiency of the device caused by, e.g., the structure of the individual elements of the device.
Figure 16
presents the pressure in the bed during the sorption, and both the experimental and simulation results are shown. Pressure fluctuations obtained in the experimental studies can be attributed to the
opening of the valve connecting the sorption chamber with the evaporator and pressure equalization between these two components, which was not included in the simulation. Additionally, from the
experimental data, a pressure drop of about 30 Pa after 60 s from the start of operation can be observed. This pressure drop is caused by the sorption of water vapor in the bed, which results in the
pressure drop in the entire unit. The sorption continues until the silica gel is saturated with water. When saturation is complete, the pressure equalizes, and the process stops.
For the desorption process (
Figure 17
), the mean domain pressure during the simulation showed an upward trend. The increase of the pressure is a result of the desorption process, during which the water vapor is released from the silica
gel. Thus, more water vapor particles are in the given space, and according to the ideal gas law, its pressure must increase. However, this increase is below 1% from 5200 to 5250 Pa. It is also
observed that the pressure in the bed obtained experimentally is about 100 Pa higher than the pressure from the simulation in the initial phase of the process. This is because the temperature in the
bed increases as a result of supplying the heating water to the heat exchanger and opening the valve between the condenser and the bed, which could also cause pressure fluctuations. As a consequence
of bed heating, the adsorbed water vapor is released from the silica gel, and this process continues until the absorbed water is fully desorbed and the pressure between the bed and the condenser
Figure 18
Figure 19
show the temperature change over time for the sorption and desorption processes, respectively. The figures show a 2.5% error; however, the maximum discrepancy between the results of simulation and
experiments is 0.5% and 0.4% for the sorption and desorption, respectively. The simulation results are within this range and can therefore be considered reliable.
Although the simulation model does not contain the description of all phenomena occurring in the device, the generated results of the simulation reflect the operation of the device with high accuracy
and allow reliable simulation of the phenomena occurring in the device in different conditions, which can be used to improve the performance of the device.
The results of simulations show the differences in the velocity of water vapor in the main elements of the adsorption chiller. Detecting differences in the velocity of vapor in particular zones of a
given element allows protecting the device from damage, e.g., erosion.
The prepared numerical model is analyzed based on the selected variants, and the compatibility with the experimental data used for validation is achieved. The average values of velocity and pressure
obtained in the simulation correspond to the values collected in the experimental research. Based on the results of the simulation, the probable reason for the destruction of the bed during
experimental studies is also obtained. Taking into account all correctness criteria applied to the model, it can be concluded that the results obtained from the numerical calculations are
sufficiently accurate and correct. The model created in this study can be further verified in the study of other configurations of adsorption chillers and also will be useful in the process of
prototyping and testing new design solutions of individual elements of the adsorption chiller. In the next stages of the research, the model will be used to improve the design solutions of the device
to improve its efficiency, optimize its operation cycle, and eliminate potential design faults that may damage the device. It should be noted that the individual components of the chiller are
interdependent. Therefore, it is likely that modifying just one of them, such as the bed, without making changes to the other components may have a small impact on increasing the efficiency of the
4. Conclusions
The combination of equations used in the described simulation allows for the creation of a reliable model of a system that allows the analysis of the work cycle in time. The models used allow the
results to be obtained in an acceptable time without access to a supercomputer unit. The results of the simulations are in good agreement with the results of the experiments. The maximum discrepancy
between the pressure obtained experimentally and by the simulations is 1.8%, while the discrepancy between the temperatures obtained experimentally and by the simulations is no more than 0.5%. The
promising results of the validation of the model make it possible to undertake further studies of this type of issue with the use of computational fluid dynamics (CFD). They can be used to test new
configurations and design solutions without the need to create real test units. The results helped to identify the problem spots and formulate design recommendations to optimize the operation of the
device as listed below.
• Changing the tube banks in the evaporator from an in-line arrangement to a staggered arrangement while simultaneously maintaining the same heat transfer surface area. This change can improve the
cooling capacity of the evaporator and provide a more uniform temperature distribution;
• Using a turbulator inside the tubes in the evaporator, as the heat transfer rate is significantly greater for turbulent flow compared to laminar flow.
• Changing the arrangement of the tubes in the condenser, as the temperature distribution in the condenser is non-uniform. Another arrangement of the tubes could provide more uniform temperature
distribution, which could result in a faster vapor condensation and thus a more efficient performance of the entire device;
• Reducing the length of the water vapor supply pipe or using a jet diffusion cone or a straight baffle at the stream outlet should be considered. The proposed solutions will lower the velocity of
water vapor and improve its dispersion. As a result, the force acting on the sorbent will be reduced;
• Using a distribution manifold that distributes the vapor uniformly over the entire surface of the bed, which will accelerate the sorption. As shown in
Figure 10
, the water vapor diffusion in the bed is not uniform, which lengthens the adsorption time and decreases the overall efficiency of the adsorption chiller.
Author Contributions
Conceptualization, K.S. and T.S.; methodology, T.S. and K.S.; validation, K.S. and T.S.; formal analysis, T.S., K.S. and L.L.; investigation, T.S., K.S., W.K. and L.L.; resources, W.N. and L.M.; data
curation, W.K., L.L., E.R. and L.M.; writing—original draft preparation, T.S., L.L. and K.S.; writing—review and editing, K.S., T.S., L.L. and L.M.; visualization, L.L., W.K. and E.R.; supervision,
W.N. and L.M.; project administration, L.M., W.N. and K.S.; funding acquisition, L.M., W.N. and K.S. All authors have read and agreed to the published version of the manuscript.
This research was funded by the Ministry of Science and Higher Education, Poland, Grant AGH No. 16.16.210.476, and partly supported by the program “Excellence initiative—research university” for the
AGH University of Science and Technology.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1.
Scheme of the adsorption chiller with desalination test bench at the Center of Energy AGH [
]. 1—condenser; 2—distillate tank; 3—adsorbent bed; 4—brine tank; 5—evaporator; 6—deaerator; TT01—temperature in the evaporator; TT04—hot water inlet temperature; TT05—hot water outlet temperature;
TT06—chilled water inlet temperature; TT07—chilled water outlet temperature; TT11—temperature in Bed 1; TT12—temperature in Bed 2; TT13—temperature in Bed 3; TT18—temperature in the condenser;
PT04—pressure in the evaporator; PT07—pressure in Bed 1; PT06—pressure in Bed 2; PT05—pressure in Bed 3; PT10—pressure in the condenser; FT01—hot water flow; FT03—chilled water flow.
Figure 2. Temperature changes over time. 1—heating water inlet to the beds; 2—heating water outlet from the beds; 3—chilled water outlet from the evaporator; 4—cooling water outlet from the
condenser; 5—heat exchanger in the first bed; 6—heat exchanger in the second bed; 7—free space of the first bed; 8—free space of the second bed; 9—cooling water outlet from the beds; 10—free space of
the evaporator; 11—free space of the condenser.
Figure 8. Cross-Section K1 for the sorption: (a) temperature distribution; (b) pressure distribution; (c) velocity distribution.
Figure 16. Comparison of the results of simulation and experimental studies—pressure over time for the sorption process.
Figure 17. Comparison of the results of simulation and experimental studies—pressure over time for the desorption process.
Figure 18. Comparison of the results of simulation and experimental studies—temperature over time for the sorption process.
Figure 19. Comparison of the results of simulation and experimental studies—temperature over time for the desorption process.
Table 1.
Technical specification of a 3-bed adsorption chiller operating under nominal conditions. [
Cooling capacity 1.50 kW
Chilled water inlet temperature 32 °C
Chilled water outlet temperature 30 °C
Chilled water mass flow rate 0.184 kg/s
Capacity 2.00 kW
Cooling water inlet temperature 30 °C
Cooling water outlet temperature 32 °C
Cooling water mass flow rate 0.25 kg/s
Daily distillate production 40 kg
Required cooling capacity (adsorption) 2.90 kW
Required heating capacity (desorption) 2.90 kW
Cooling water mass flow rate 0.25 kg/s
Heating water mass flow rate 0.25 kg/s
Cooling water inlet temperature 30 °C
Heating water inlet temperature 85 °C
Cooling Capacity
Chilled Water: inlet/outlet temperature; mass flow rate: 32/30 °C; 0.184 kg/s 1.50 kW
Chilled Water: inlet/outlet temperature; mass flow rate: 16/11 °C; 0.0523 kg/s 1.32 kW
Chilled Water: inlet/outlet temperature; mass flow rate: 12/7 °C; 0.0523 kg/s 1.10 kW
Temperature Sensor Range Uncertainty
Heating water inlet to the beds
Heating water outlet from the beds
Free surface of the beds
Inside the heat exchanger in the beds
Cooling water outlet from the beds Pt-1000 From −80 °C to 150 °C ±0.1 °C
Cooling water outlet from the condenser
Chilled water outlet from the evaporator
Free space of the evaporator
Free space of the condenser
Pressure Sensor Range Uncertainty
Evaporator Pressure transducer From 0 to 99 kPa ±0.5%
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MDPI and ACS Style
Sztekler, K.; Siwek, T.; Kalawa, W.; Lis, L.; Mika, L.; Radomska, E.; Nowak, W. CFD Analysis of Elements of an Adsorption Chiller with Desalination Function. Energies 2021, 14, 7804. https://doi.org/
AMA Style
Sztekler K, Siwek T, Kalawa W, Lis L, Mika L, Radomska E, Nowak W. CFD Analysis of Elements of an Adsorption Chiller with Desalination Function. Energies. 2021; 14(22):7804. https://doi.org/10.3390/
Chicago/Turabian Style
Sztekler, Karol, Tomasz Siwek, Wojciech Kalawa, Lukasz Lis, Lukasz Mika, Ewelina Radomska, and Wojciech Nowak. 2021. "CFD Analysis of Elements of an Adsorption Chiller with Desalination Function"
Energies 14, no. 22: 7804. https://doi.org/10.3390/en14227804
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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iterative solver
On solution of large linear system : iterative solver
In this seminar, Krylov subspace methods, which are frequently used to solve a large sparse linear system, are reviewed. Usage of CG/GMRES methods included in Intel Math Kernel Library (MKL) on
supercomputer OCTOPUS will be instructed as hands-on training. A preconditioning technique by using additive Schwarz method consisting of direct solver in sub-matrices is introduced.
- Overview of Krylov subspace methods and details of preconditioned CG/GMRES methods
- Preconditioner by incomplete LU factorization
- Preconditioner by additive Schwarz method that is constructed by combination of decomposition of the sparse matrix using METIS graph partitioner and direct solver in sub-matrices
- Data structure to store sparse matrix, CSR
- Usage of CG/GMRES method in Intel MKL through Reverse Communication Interface (RCI)
- How to link your code to these libraries on OCTOPUS supercomputer.
Recommended to whom
- wants to solve large sparse linear system in a numerical simulation of fluid dynamics, structural analysis, or electromagnetic analysis
- has experience to write simulation code by C or Fortran.
- This seminar consists of lecture and hands-on training. Please bring your own note PC, which can connect to network and with terminal software installed.
- A trial account of OCTOPUS without a fee will be provided for this seminar.
About iterative solver
In numerical simulation of fluid dynamics, structural analysis, or electromagnetic analysis, large linear and/or nonlinear systems, which are obtained by discretization process of partial
differential equations (PDEs), need to be solved. Nonlinear problem can be linearized by a Newton iteration, hence linear system with large sparse matrix should be treated.
There are two methods to solve sparse the matrix problem, direct method, e.g., LU-factorization and iterative method, e.g., conjugate gradient method. Direct solver can robustly find a solution
of the linear system of matrix with large condition number, which is obtained from strong nonlinear problem or originated from phenomena with large jump of physical coefficients. Usually, these
matrices are hard to solve by iterative methods. A drawback of direct solver is that it has large complexity of computation and consumes large memory. On the contrary, iterative method has lower
complexity of computation but iterative process strongly depends on character of the matrix and it often does not converge in realistic time.
In this seminar, we deal with Krylov subspace method, which is one of the most common iterative solvers. We aim to accelerate the convergence of the Krylov subspace method by introducing a
preconditioner based on direct solver in subdomains. Krylov subspace method finds an approximate solution in a subspace called as Krylov subspace, which is generated by multiplications of the
sparse matrix to the initial residual induced from the initial solution. Conjugate Gradient (CG) method is designed for symmetric positive definite matrix and Generalized Minimum Residual (GMRES)
method for general unsymmetric matrix. It is very important to select an appropriate preconditioner based on multiplication of approximate inverse of the sparse matrix to the both sides of the
linear equation. GMRES achieves the most robust iterative procedure in the Krylov subspace methods, but it has large complexity in the computation, i.e., longer iteration leads to larger
computational costs. An additive Schwarz method, where the inverse operation is approximated by union of solutions obtained from direct solver in each subdomain, can be constructed only from the
matrix data without any knowledge on physical characteristics of the PDE system. This property is similar to a well-known preconditioner based on incomplete LU factorization. Since the additive
Schwarz method has more strong approximation property in inversion of the matrix than ILU-factorization, the preconditioned GMRES can converge quickly and the method becomes a practical tool to
solve a large linear system.
Date : Nov 21, 1:15 p.m. - 5:00 p.m. (registration from 1:00 p.m.)
Instructor: Cybermedia Center, Osaka University
Lecturer : Dr. Atsushi Suzuki, Guest associate professor
Venue: Cybermedia Commons, the first floor main hall at Cybermedia Center, Suita Campus
Type : classroom study and hands-on training
Quota: 30 persons (registration will be closed when number of participants exceeds 30)
Application deadline: Nov 21, 5:00 p.m.
Reception has been closed.
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How to write a formula that avoids dividing by zero
I am trying to write a formula to get an average for total absence, no shows and regrets; however some cells are zero which returns an error or unparseable message.
Below is the formula I've written that does not work. The problem is with dividing by the [Total Absent] because sometimes it is zero. If it is zero, I want it to default to 1.
=COUNTIF([09/25/23]@row:[06/03/19]@row, ="absent") / [Total Absent]@row IF([Total -Absent]@roW=0, 1).
Will appreciate help with this.
• @Lucy B Wrap the formula in the IFERROR function and set the default error value to 1:
=IFERROR(COUNTIF([09/25/23]@row:[06/03/19]@row, ="absent") / [Total Absent]@row IF([Total -Absent]@roW=0, 1), 1)
• In addition to the IFERROR, you will also need to remove the IF portion.
=IFERROR(COUNTIF([09/25/23]@row:[06/03/19]@row, ="absent") / [Total Absent]@row, 1)
Help Article Resources
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st: Re: bold text in graphs
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
st: Re: bold text in graphs
From Kit Baum <[email protected]>
To [email protected]
Subject st: Re: bold text in graphs
Date Wed, 5 Dec 2007 11:00:49 -0500
Bolding, italicizing, etc. is achieved in PostScript by switching fonts. At the lowest level, your computer has a Times font, a Times- Italic font, a Times-Bold font, a Times-SmallCaps font, etc. and
when you specify one of those in a document it is switching fonts.
Stata graphs have the disadvantage of being rendered in a single font. That is why the oft-voiced complaint that you can't include Greek letters (unless you want to use Greek everywhere), because it
would require using both, e.g., Times and Symbol on the same graph.
So I do not think you can get away with including bold text on a Stata graph, or easily tweak the resulting PostScript to modify it thusly. This would obviously be a very useful feature for the Graph
Editor (Vince, are you there?)
Kit Baum, Boston College Economics and DIW Berlin
An Introduction to Modern Econometrics Using Stata:
On Dec 5, 2007, at 2:33 AM, statalist-digest wrote:
Does anyone know how text is bolded in postscript? Is it some sort of
ASCII character sequence? I'm thinking that if there is some ASCII
sequence like this, while it may have no effect on the Stata graph, it
might survive the "graph export" translation to postscript.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Famous Riddles at Fancyread
This is a nice nonsense. Each guest paid $9 because they gave $30 and they were given back $3. The manager got $25 and the difference ($2) has the bellboy. So it is nonsense to add the $2 to the $27,
since the bellboy kept the $2.
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Kleinberg's authority centrality scores. — authority_score
Kleinberg's authority centrality scores.
Kleinberg's authority centrality scores.
Kleinberg's hub centrality scores.
scale = TRUE,
weights = NULL,
options = arpack_defaults()
hub_score(graph, scale = TRUE, weights = NULL, options = arpack_defaults())
The input graph.
Logical scalar, whether to scale the result to have a maximum score of one. If no scaling is used then the result vector has unit length in the Euclidean norm.
Optional positive weight vector for calculating weighted scores. If the graph has a weight edge attribute, then this is used by default. This function interprets edge weights as connection
strengths. In the random surfer model, an edge with a larger weight is more likely to be selected by the surfer.
A named list, to override some ARPACK options. See arpack() for details.
See also
Centrality measures alpha_centrality(), betweenness(), closeness(), diversity(), eigen_centrality(), harmonic_centrality(), hits_scores(), page_rank(), power_centrality(), spectrum(), strength(),
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Optimization calculation of matlab ant colony algorithm -- traveling salesman problem (TSP) optimization [matlab optimization algorithm 21]
Optimization calculation of matlab ant colony algorithm -- traveling salesman problem (TSP) optimization
Ant colony algorithm (ACA) is a new simulated evolutionary algorithm proposed by Italian scholar M. Dorigo in the early 1990s. It truly simulates the foraging behavior of ant colony in nature. M.
Dorigo et al. Used it to solve the traveling salesman problem (TSP), and achieved good experimental results.
In recent years, many experts and scholars have devoted themselves to the research of ant colony algorithm and applied it to the fields of transportation, communication, chemical industry, electric
power and so on. They have successfully solved many combinatorial optimization problems, such as job shop scheduling problem, quadratic assignment problem, traveling salesman problem and so on.
The basic idea and principle of ant colony algorithm will be described in detail, and its application to solve China TSP (CTSP) will be introduced in the form of an example.
Ant colony
Basic idea biologists have found that ant foraging in nature is a group behavior, not a single ant looking for food source by itself. When ants look for food sources, they will release a pheromone on
their path, and can perceive the pheromone released by other ants. The size of pheromone concentration indicates the distance of the path. The higher the pheromone concentration, the shorter the
corresponding path distance. Generally, ants will give priority to the path with high pheromone concentration with a high probability, and release a certain amount of pheromone to enhance the
pheromone concentration on the path, which will form a positive feedback. Finally, ants can find the best path from the nest to the food source, that is, the shortest distance. It is worth mentioning
that biologists also found that the pheromone concentration on the path will gradually decline with time.
The basic idea of applying ant colony algorithm to solve the optimization problem is: the walking path of ants is used to represent the feasible solution of the problem to be optimized, and all paths
of the whole ant colony constitute the solution space of the problem to be optimized. The amount of pheromone released by ants with short path is more. With the advance of time, the pheromone
concentration accumulated on the short path gradually increases, and the number of ants choosing the path is also more and more. Finally, the whole ant will focus on the best path under the action of
positive feedback. At this time, the corresponding is the optimal solution of the problem to be optimized.
Basic principle of ant colony algorithm to solve TSP problem
In this section, the basic idea of the above ant colony algorithm will be described in mathematical language, and the basic principle of ant colony algorithm to solve TSP problem will be described in
Without losing generality, let the number of ants in the whole ant colony be m, the number of cities be n, the distance between city i and city j be d3 (i, = 1,2,..., n), and the pheromone
concentration on the connection path between city i and city j at time t be τ( t). At the initial time, the pheromone concentration on the connection path between cities is the same, so it may be set
as τ( 0)= τ o.
Ant K (k=1,2,..., m) determines its next visiting city according to the pheromone concentration on the connection path between cities. Let P (T) represent the probability of ant K transferring from
city i to city j at time t, and its calculation formula is
Where v (t) is the heuristic function, v (t) = 1/d, which represents the expected degree of ant transfer from city i to city j; allow (k=1,2,..., m) is the set of cities to be visited by ant K. at
the beginning, there are (n-1) elements in allw, that is, including all cities except the departure city of ant K. with the progress of time, the elements in allowr continue to decrease until they
are empty, which means that all cities have been visited; A is the pheromone importance factor. The greater its value, the greater the role of pheromone concentration in transfer; B is the importance
factor of the heuristic function. The greater its value, the greater the role of the heuristic function in the transfer, that is, ants will transfer to cities with a large probability.
As mentioned above, while the ants release pheromones, the pheromones on the connection path between cities gradually disappear, and set the parameter p (0< ρ< 1) Indicates the volatilization degree
of pheromone. Therefore, when all ants complete a cycle, the pheromone concentration on the connection path between cities needs to be updated in real time, i.e
Among them, Δτ Represents the pheromone concentration released by the k th ant on the connection path between city i and city j; △r； Represents the sum of pheromone concentrations released by all
ants on the connection path between city i and city j. for the problem of pheromone release by ants, M. dorigo et al. Gave three different models, called ant cycle system, ant quantity system and ant
density system respectively,
Basic steps
1. Initialization parameters at the beginning of calculation, relevant parameters need to be initialized, such as ant colony size (number of ants) m, pheromone importance factor a, heuristic function
importance factor, pheromone volatilization factor p, total pheromone release Q, maximum iteration times Iter max initial value of iteration times I Iter=1.
2. Construct the solution space, randomly place each ant at different starting points, and calculate the next city to be visited for each ant K (k=1,2,..., m) according to formula (22-1) until all
ants have visited all cities.
3. Update the pheromone, calculate the path length L4 (k=1,2,..., m) passed by each ant, and record the optimal solution (shortest path) in the current number of iterations. Meanwhile, the pheromone
concentration on each urban connection path is updated according to equations (22-2) and (22-3).
4. Judge whether to terminate if ITER < ITER_ Max, then make iter=iter+1, clear the record table of ant passing path, and return to step 2; Otherwise, the calculation is terminated and the optimal
solution is output.
Characteristics of ant colony algorithm
Based on the basic idea of ant colony algorithm and the basic principle of solving TSP problem, it is not difficult to find that compared with other optimization algorithms, ant colony algorithm has
the following characteristics:
(1) The positive feedback mechanism is adopted to make the search process converge and finally approach the optimal solution.
(2) Each individual can change the surrounding environment by releasing pheromones, and each individual can perceive the real-time changes of the surrounding environment, and individuals communicate
indirectly through the environment.
(3) The search process adopts distributed computing, and multiple individuals perform parallel computing at the same time, which greatly improves the computing power and operation efficiency of the
(4) The heuristic probability search method is not easy to fall into local optimization, and it is easy to find the global optimal solution.
Basic steps
1. Calculate the mutual distance between cities. Calculate the mutual distance between two cities according to the location coordinates of cities, so as to obtain a symmetrical distance matrix
(square matrix with dimension of 31). It should be noted that the element on the diagonal of the calculated matrix is 0. However, as mentioned above, since the heuristic function v (d) = 1d, in order
to ensure that the denominator is not zero, the element on the diagonal is corrected to a very small positive number (such as 10-4 or 10-5).
2. Initialization parameters are as described above. Before calculation, relevant parameters need to be initialized, which will not be repeated here. Please refer to section 22.3.4 program
implementation of this chapter for details.
3. Iterative search for the best path first constructs the solution space, that is, each ant visits all cities according to the transfer probability formula (22-1). Then, the length of each ant's
path is calculated, and the pheromone concentration on each urban connection path is updated in real time according to equations (22-2) and (22-3) after each iteration. After cyclic iteration, the
optimal path and its length are recorded.
4. After finding the optimal path, we can compare it with the results obtained by other methods, so as to evaluate the performance of ant colony algorithm. At the same time, we can also explore the
influence of different parameters on the optimization results, so as to find a set of best or better parameter combination.
Main program display
%% Chapter 22 optimization calculation of ant colony algorithm - traveling salesman problem(TSP)optimization
% <html>
% <table border="0" width="600px" id="table1"> <tr> <td><b><font size="2">The author of the case states:</font></b></td> </tr> <tr><td><span class="comment"><font size="2">1: I have been stationed here for a long time<a target="_blank" href="http://Www.matlabsky. COM / forum-78-1. HTML "> < font color =" #0000ff "> in < / font > < / a >, ask questions about the case and answer all questions</ font></span></td></tr><tr> < Td > < span class = "comment" > < font size = "2" > 2 < / font > < font size = "2" >: this case has supporting teaching videos. Please click < a href > to download the videos=“ http://www.matlabsky.com/forum-91-1.html "> http://www.matlabsky.com/forum-91-1.html </a></font><font size="2">. </ font></span></td> </ tr> < tr> < td><span class="comment"><font size="2"> 3: This case is an original case. Please indicate the source for reprint (analysis of 30 cases of MATLAB intelligent algorithm)</ font></span></td> </ tr> < tr> < td><span class="comment"><font size="2"> 4: If this case happens to be related to your research, we welcome your comments and requirements, which can be added to the case after consideration</ font></span></td> </ tr> < tr> < td><span class="comment"><font size="2"> 5: The following content is the first draft, which is slightly different from the content of the actually issued books. Please take the content in the books as the standard</ font></span></td> </ tr> </ table>
% </html>
%% Clear environment variables
clear all
%% Import data
load citys_data.mat
%% Calculate the distance between cities
n = size(citys,1);
D = zeros(n,n);
for i = 1:n
for j = 1:n
if i ~= j
D(i,j) = sqrt(sum((citys(i,:) - citys(j,:)).^2));
D(i,j) = 1e-4;
%% Initialization parameters
m = 50; % Ant number
alpha = 1; % Pheromone importance factor
beta = 5; % Heuristic function importance factor
rho = 0.1; % Pheromone volatilization factor
Q = 1; % Constant coefficient
Eta = 1./D; % Heuristic function
Tau = ones(n,n); % Pheromone matrix
Table = zeros(m,n); % Path record form
iter = 1; % Initial value of iteration times
iter_max = 200; % Maximum number of iterations
Route_best = zeros(iter_max,n); % Best path of each generation
Length_best = zeros(iter_max,1); % Length of the best path of each generation
Length_ave = zeros(iter_max,1); % Average length of paths of each generation
%% Iterative search for the best path
while iter <= iter_max
% Randomly generate the starting city of each ant
start = zeros(m,1);
for i = 1:m
temp = randperm(n);
start(i) = temp(1);
Table(:,1) = start;
% Constructing solution space
citys_index = 1:n;
% Ant by ant path selection
for i = 1:m
% City by city route selection
for j = 2:n
tabu = Table(i,1:(j - 1)); % Visited city collection(Taboo list)
allow_index = ~ismember(citys_index,tabu);
allow = citys_index(allow_index); % Collection of cities to be visited
P = allow;
% Calculate the transfer probability between cities
for k = 1:length(allow)
P(k) = Tau(tabu(end),allow(k))^alpha * Eta(tabu(end),allow(k))^beta;
P = P/sum(P);
% Roulette to choose the next city to visit
Pc = cumsum(P);
target_index = find(Pc >= rand);
target = allow(target_index(1));
Table(i,j) = target;
% Calculate the path distance of each ant
Length = zeros(m,1);
for i = 1:m
Route = Table(i,:);
for j = 1:(n - 1)
Length(i) = Length(i) + D(Route(j),Route(j + 1));
Length(i) = Length(i) + D(Route(n),Route(1));
% Calculate the shortest path distance and average distance
if iter == 1
[min_Length,min_index] = min(Length);
Length_best(iter) = min_Length;
Length_ave(iter) = mean(Length);
Route_best(iter,:) = Table(min_index,:);
[min_Length,min_index] = min(Length);
Length_best(iter) = min(Length_best(iter - 1),min_Length);
Length_ave(iter) = mean(Length);
if Length_best(iter) == min_Length
Route_best(iter,:) = Table(min_index,:);
Route_best(iter,:) = Route_best((iter-1),:);
% Update pheromone
Delta_Tau = zeros(n,n);
% Ant by ant calculation
for i = 1:m
% City by city calculation
for j = 1:(n - 1)
Delta_Tau(Table(i,j),Table(i,j+1)) = Delta_Tau(Table(i,j),Table(i,j+1)) + Q/Length(i);
Delta_Tau(Table(i,n),Table(i,1)) = Delta_Tau(Table(i,n),Table(i,1)) + Q/Length(i);
Tau = (1-rho) * Tau + Delta_Tau;
% Add 1 to the number of iterations to clear the path record table
iter = iter + 1;
Table = zeros(m,n);
%% The results show that
[Shortest_Length,index] = min(Length_best);
Shortest_Route = Route_best(index,:);
disp(['Shortest distance:' num2str(Shortest_Length)]);
disp(['Shortest path:' num2str([Shortest_Route Shortest_Route(1)])]);
%% mapping
grid on
for i = 1:size(citys,1)
text(citys(i,1),citys(i,2),[' ' num2str(i)]);
text(citys(Shortest_Route(1),1),citys(Shortest_Route(1),2),' starting point');
text(citys(Shortest_Route(end),1),citys(Shortest_Route(end),2),' End');
xlabel('Abscissa of city location')
ylabel('City Location ordinate')
title(['Ant colony algorithm optimization path(Shortest distance:' num2str(Shortest_Length) ')'])
legend('Shortest distance','Average distance')
xlabel('Number of iterations')
title('Comparison of shortest distance and average distance of each generation')
% <html>
% <table width="656" align="left" > <tr><td align="center"><p align="left"><font size="2">Relevant forums:</font></p><p align="left"><font size="2">Matlab Technical Forum:<a href="http://Www.matlabsky. Com "> www.matlabsky. Com < / a > < / font > < / P > < p align =" left "> < font size =" 2 "> m < / font > < font size =" 2 "> atlab function Encyclopedia: < a href =" " http://www.mfun.la ">www.mfun.la</a></font></p></td> </ tr></table>
% </html>
data set
Result display
The path corresponding to the operation result is shown in Figure 22-4. It can be clearly seen from the figure that starting from the starting point, each city is visited once, and after traversing
all cities, it returns to the starting point. The shortest path found is 15601.9195km.
The shortest distance and average distance of each generation are shown in the figure. It is not difficult to find from the figure that the shortest distance and average distance show a downward
trend with the increase of the number of iterations. When the number of iterations is greater than 112, the shortest distance is no longer changed, indicating that the best path has been found.
The latest research results show that the optimal solution of China's TSP problem is 15377 km. Therefore, the best path found here is the local optimal solution, not the global optimal solution.
|
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|
Advanced Algorithms and Complexity - Reviews & Coupon - Java Code Geeks
Advanced Algorithms and Complexity
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You’ve learned the basic algorithms now and are ready to step into the area of more complex problems and algorithms to solve them. Advanced algorithms build upon basic ones and use new ideas. We will
start with networks flows which are used in more typical applications such as optimal matchings, finding disjoint paths and flight scheduling as well as more surprising ones like image segmentation
in computer vision. We then proceed to linear programming with applications in optimizing budget allocation, portfolio optimization, finding the cheapest diet satisfying all requirements and many
others. Next we discuss inherently hard problems for which no exact good solutions are known (and not likely to be found) and how to solve them in practice. We finish with a soft introduction to
streaming algorithms that are heavily used in Big Data processing. Such algorithms are usually designed to be able to process huge datasets without being able even to store a dataset. UC San Diego is
an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn
that knowledge isn’t just acquired in the …
Instructor Details
Alexander S. Kulikov
Votes: 0
Courses : 8
Alexander S. Kulikov is a research fellow at St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences and a visiting professor at University of California, San
Diego. His scientific interests include algorithms for NP-hard problems and circuit complexity. In St. Petersburg, he runs Computer Science Club and Computer Science Center.
Specification: Advanced Algorithms and Complexity
Duration 22 hours
Year 2016
Level Expert
Certificate Yes
Quizzes Yes
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JICHEN W –
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Daniel P –
Too much emphasis on implementation details (99%), so one doesn’t get a great intuitive understanding of the logic of algorithms.
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Fabrice L –
A lot of material covered in this course. The assignments are more challenging than in the previous courses of the specialization. It is overall a great final and a very complete specialization.
Thank for putting together all this work.
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Tamilarasu S –
Very well made course with challenging algorithm problems.
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Vasily V –
Excellent course. One star less just because there are not very clean test cases for one particular problem among programming assignments
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Tamas K –
Great course again! The problems are considerably more difficult than in the previous courses in this specialization. The only problem is that the forum interaction with TAs is nonexistent, if
you stuck with a problem, you have to solve it alone.
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John B –
A lot of really useful algorithms are covered in this course, however some of the presentation is annoyingly sparse on details (particularly in the section on network flows).
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Radim V –
The only thing I missed in this course (and specialization) was more visual, intuitive approach to explanation. Programming assignments are rewarding.
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Pablo E M M –
Great Courses!. Thanks for this wonderful specialization!
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Andrii S –
Another great course in this specialization with challenging and interesting assignments. However, this one is somewhat harder but rewarding.
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Lie C –
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Henry R –
Very hard to follow the lectures, completly lost without Reference books such as Introduction to Algorithms.
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Raunak N –
this course gave me hell of a time
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Joseph G N –
An incredible course,the exercises were very interesting
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Juho V –
Lectures are mostly good. Assignments not. They are often very difficult in an uninteresting way such as unintuitive input formats and / or template code.
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To P H –
Very bad course content for some modules Many abstract concepts and mathematical terms but with severe lack of explanation of the terms and lack of specific, concrete examples to help learners to
understand them For example: in LP module there should be example of how the primal and dual matrix looks. How simplex algorithm is used on a specific example (showing explicit graph). I
undertood only 25% of what was discussed about in this module No motivation to move on after week 2! Other weeks are slightly better In summary: Too many abstract concepts with little examples
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Madhusudan H J –
When they say advanced algorithms and complexity, they mean it. I was initially under the presumption that it would be a straight forward video course, without any assignments. But when I had to
start with programming assignments that’s when the real test started. Amazing set of tutorials. Would have liked if the courses had more varied examples.
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Saurab D –
The lectures are very abstract so, I had some difficulty in solving the assignment problems.
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Joao H L F –
Great course, thank you! My only remark is that it sometimes assumes that you have lots of prior knowledge on programming in my opinion. I’m a non tech guy trying to increase my knowledge of the
field, and I’ve had a very hard time trying to find additional resources to solve some of the problems and to grasp some of the content. If we could have a bit more color on some of the harder
parts during the course it’d be great.
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Kehan B –
This module is by far the hardest in this specialization, but at the same time, it is rewarding. The only complaint I have is the Linear Programming part. I wish there was more explanation or
some toy examples to go through the algorithm.
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Anton R –
Liked this course, at least there are courses for advanced level.
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Rihaan S –
Very informative course with challenging assignments. It will surely make your data structure concepts clearer.
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Mahmoud M –
Took very long time fro me to be finished
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Shaashwat A –
amazing course well detailed
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Yinchung C –
This is a very challenging course in the specialization. I learned a lot form going through the programming assignments!
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Dmitrii S –
I very enjoyed this course! Theoretic informatics is my favorite field of study. All the professors are the best. Dreaming to enroll in your Ph.D. program. Thank you very much
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Priyansh B –
Was fun learning advanced stuff and implementing algorithms.
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Tamas S –
Very good collection of advanced topics, even useful for the 6th course in the specialization!
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Mark Z –
The course left a good overall impression: the contents of the course are excellent and assignments are relevant. I’ve only noticed 2 downsides: the videos of the first week were sometimes hard
to follow and I had to spend some time googling (which I find strange) (however, the situation changes when a Russian guy appears in the third week: his explanations are very clear and precise)
and some assignments, in my opinion, take way more time to complete than they should.
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Jason M –
Very Educational and Enlightening. The only criticism I have is that the starter files generally need more modification than indicated to create a successful program.
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Hidetake T –
This course is very difficult. Possible to pass programming assignments only after finishing previous courses.
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Chitrang S –
Very Very Challenging Course , it test your patience and rewards is extremely satisfying. Lot of learning on a complicated subject of NP Hard problems.
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HussamAldeen S –
Course content is very good, however the lectures are hard to follow because the examples are always at the end.
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Ayran T O –
Very difficult but challenging!
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Quynh V –
I am not good in this course. But I’m always try the best! Awesome course, thank you so much!
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Aamir M K –
A great course but really very tough. Every module of the course deserves to be a separate course in itself. One thing that bothered was that the assignments were very vaguely explained and
needed lots of searching and self study. Anyways got it after many many months.
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Yue S –
I really dislike Daniel Kane’s teaching style!!! His slides are rough and lack of details, the structure of his lectures is loose. Every time I met a Unit taught by Kane, I have to spend much
more time on videos and assignments than other Units. This makes me very annoyed why can’t this teacher be more serious on teaching just like other teachers in this course??? : (
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Bharti S –
Thank you so much. You are doing such a great work but i appreciate if you explain week 2 (linear programming) in detail. Thank you.
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This course is wonderful.I am really feel like I have all knowledge of adsa
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Fernando K I –
Shout out to professor Alexander Kulikov that is the only one in the whole specialization who has good didactics skills. He knows how to explain a concept by giving examples and walking through
them step by step so the viewer can understand the thought process. Unfortunately, professor Kane’s lectures were poorly taught. I understand that his videos are older and maybe the technology
wasn’t there yet when he recorded the lectures. You’d better off skipping those lectures and going through the assignments directly, learning the material elsewhere.
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Syed H A –
Really rigorous and fundamental with what scientist and other professionals need to know about programming.
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Advanced Algorithms and Complexity
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Constitutive Elements of Non-Abelian Gauge Theories
1. Introduction
With the emergence of subatomic theories, in the 1920s, the problem of establishing the basis of quantum mechanics, considering the classical mechanics counterpart, came about [1] [2] . Attempts to
address this pro- blem gave rise, over the decades, to numerous works, following different mathematical approaches and physical motivations. Although much of these investigations have been initially
restricted to the analysis of classical and quantum premises in the non-relativistic realm, they have led to important discoveries, such as the notion of entanglement and teleportation, crucial keys
for quantum computers and quantum network [3] - [5] ; concepts that have been explored in high energy physics [6] . These investigations are mainly considered in two directions, that are at some
extent complementary to each other.
One of them is the stochastic methods, that have been used to derive quantum mechanics starting, for instance, from the Liouville equation or from the Fokker-Planck equation [7] - [15] . In another
direction, there are attempts exploring the notion of symmetry and representation theories [16] - [20] . The former direction usually em- phasizes the nature of the state, being interesting for
deriving, for example, the Schrödinger equation, while the later, guided by algebraic structures and symmetries, is useful for generalizations, and can accommodate an abroad class of mechanical
systems, that include relativistic, non-relativistic and thermal systems [21] [22] .
For the case of non-relativistic quantum mechanics, Levi-Leblond [23] - [25] was the first to present a systematic study of unitary representations of the Galilei group, leading to the Schrödinger
equation and Pauli- Schrödinger equation, describing, respectively, spin-0 and spin 1/2 non-relativistic particles. A consequence, in terms of premises, was that the spin of a particle should be
fully described and physically interpreted in terms of the rotation symmetry. It is important to note that, before these works, it was usual to consider spin in the non- relativistic quantum
mechanics as a relativistic remnant of the Dirac equation.
Although representations of Lie group are key aspects to deriving physical theories, this method, as well as the stochastic analysis, has been only partially explored to address the premises of
quantum field theories in comparison with other mechanical theories [6] [15] [18] [26] . This is a demanding problem, since new phenomena and concepts need to be analyzed in detail. The situation is
more appealing in non-abelian gauge field theories, as the standard model for particle physics, where the nature of the mechanism for the origin of mass is only partially explained through the
introduction of the Higgs bosons, presently under experimental test.
In the present work, our main goal is to construct a systematization for mechanical formalisms, which is established by six constitutive elements. In this context, gauge fields are considered by
taking into account counterparts of other theories of motion, such as quantum mechanics and one-particle special relativity. The general algebraic structure is supported by physical (experimental)
conditions. A first result is a derivation of a Lie-al- gebra structure associated to the six constitutive elements. This aspect, which is in turn connected to the Nöther theorem, is important to
establish the consistency of the number of six constitutive elements. After analyzing the structure of quantum mechanics and special relativity, we investigate non-abelian fields, discussing the
concept of mass, from Newton up to Higgs. We have to emphasize that what is new in the present work is the structure of six constitutive elements fixing the content of theories of motion. This aspect
is useful, as aforementioned, for the comparative analysis of theories. In this realm, for instance, a fundamental difference between classical and quantum mechanics is not the nature of the Hilbert
space, but the experimental condition imposed by the Heisemberg relations.
The paper is organized in the following way. In Section 2, we present the constitutive elements of a mechanics. In Section 3, there is a demonstration that the constitutive elements induce an
algebraic structure of Lie algebra in association with the Nöther theorem. In Section 4, the premisses of the special relativity theory and quantum mechanics are analyzed. In Section 5, non-abelian
gauge theory is discussed as a mechanical theory. In Section 6, the notion of mass is analyzed. The final concluding remarks are presented in Section 7. In the Appendix, we review some well know
aspects of gauge theories in order to make clear the origin of the six constitutive elements.
2. Constitutive Elements for Theories of Motion
A theory of movement, a mechanics, can be defined by the following set,
CE1. Reference systems. A reference system is defined from points in the
CE2. Kinematical variables. The set of kinematical variables,
The set
CE3. Mechanical system. A mechanical system is defined as the object under movement. It can be classified by two categories of primitive concepts. One is the material point, specified by a set of
local points of
CE4. State of a mechanical system. The mechanical state
CE5. Changes in the state of mechanical system. The changes in the state
CE6. Specification of mechanical systems. The specification of a particular mechanical system is given by a function of the state of the system,
The equations of motion, the causal law, will be the Euler-Lagrange equations and are given by the functional equation
3. Lie-Algebra Structure of Ω and Nöther’s Theorem
In this section, using physical (experimental) ingredients of the motion, we show that the set of transformations
For simplicity, we consider that each element of
that is, the mapping (,) is antisymmetric.
which is the derivation of the Leibnitz rule, defining the association between the associative product
considering now that
which is the Jacobi identity. Then the mapping
From these results, we observe that an invariant quantity, say
4. Two Examples: Special Relativity and Quantum Mechanics
In this section we investigate the constitutive elements of mechanical system with two examples: one particle special relativistic system and one particle quantum mechanics.
4.1. Special Relativity
The constitutive elements of a particle in special relativity are identified in the following way.
CE1. Reference systems. The reference systems is defined from points in the
CE2. Kinematical variables. The set of kinematical variables,
CE3. The mechanical system. We consider as a mechanical system a material point. The mass of material point is defined with the characteristics of a Newtonian material point with inertia, but now
CE4. The state characterizing a material point can be given by
CE5. Changes in the state of mechanical system. The changes in the state given by
CE6. The specification of the particular mechanical system is given by a function of the state of the system,
It is important to consider now another representation for relativistic particles, the Poisson-Liouville formul- ation of special relativity. In this case, the state is defined by a
which can be written as
is the Poisson bracket. In this representation, the generators of Lorentz transformation are given by
The kinematical variable
Explicitly, we then note a separation of generators of symmetry, as
The representations of the Lorentz algebra, given in Equation (5) has been analyzed in the literature [21] . However, considering the set of constitutive elements
Then we can define
Now we can calculate the values of the constants a, b, c and d, leading to a representation of the Poincaré-Lie algebra by introducing the generator
In order to get a representation of the Lorentz-Lie algebra, given by Equation (5), the constants a, b, c and d in Equation (6) have to satisfy the following condition
4.2. Quantum Mechanics
A mechanical theory describing the movement of one quantum particle is given by the following set, S, of con- stitutive elements.
CE1. Reference systems. A reference systems is defined from points such that the mapping
CE2. The set of kinematical variables,
CE3. The mechanical system is a non-relativistic material point, interacting to each other by a potential. For electrons such a potential is given by the electromagnetic field.
CE4. The state of a quantum mechanical system is
CE5. Changes in the state of mechanical system. The changes in the state
CE6. The specification of the particular mechanical system is given by a function of the state of the system,
then used to get the causal law among states, by the variational principle, defined by action by
This leads to the Schrödinger equation describing the motion of a quantum particle. In this case the Lagrangian is
The representation of quantum mechanics in phase space has been explored starting with the Wigner for- malism, based on the notion of quasi-distribution function. In this case representations of the
Galilei group in phase space has been analyzed following in parallel to the Lorentz symmetry, considered in the last Section [21] . This aspect will be developed in a separate investigation. However,
it is important to note here that different formalisms of classical or quantum mechanics can be accommodated in the set of Constitutive Elements and analyzed in comparison to one another.
5. Constitutive Elements of Non-Abelian Gauge Fields
One goal in this section is to consider the Constitutive elements of non-abelian gauge-fields, in order to perform a conceptual analysis, in the next section, about the notion of mass, from Newton to
Higgs. In order to fix the notation and to emphasize important aspects of our discussion, we review in the Appendix some elements of a gauge theory.
CE1. Reference systems. In a (abelian or non-abelian) gauge theory, the reference systems are taken from special relativity, i.e. the Minkowski space. The set of transformation
CE2. The set of kinematical variables,
The procedure of quantization is carried out, consistently, by the definition of a generating functional for correlation function of the system. With this procedure, the basic physical observables
are established. In parti- cular, the cross-section of a physical process, such as the scattering of particles, can be defined and compared with experiments by using correlation functions.
CE3. The mechanical system. In quantum field theory, a mechanical system is described by a field. A gauge field will describe the process of interaction among the matter field; that is the case of
the Dirac field describing electrons or quarks. The notion of mass of a field is still defined with the characteristics of a Newtonian material point with inertia, and is a Lorentz scalar, obtained
from the energy-momentum tensor.
CE4. The state characterizing a field is a vector in
CE5. Changes in the state of the mechanical system. The changes in the state
CE6. Mechanical system. The specification of a particular mechanical system is given by the Lagrangian density
6. The Notion of Mass from Newton to Higgs
We analyze now the notion of mass. Our aim here is not to provided a complete historical account about the developments of the concept of mass, but describe the basic improvements in the concept of
mass considering the set of Constitutive Elements (CEs). We start with the concept of mass as it was first introduced by Newton in his Principia, Book I, Definition I, as the quantity of matter [32]
. In the Book I, Definition III, Newton introduce mass also as a measure of the inertia; i.e. the resistance of a particle to have its state of motion changed by the interactions with another
particle (taking as an example a mechanical system described by two particles). The quantity of motion, the momentum, is introduced in Definition II. As emphasized first by Poisson, the notion of
material point (or material particle) was implicitly given in the Newton’s definition of mass. Such a notion was generalized also by Newton, considering problems in hydrodynamics, following the
Pascal’s achievements. From the XVIII Century on, with Euler, Lagrange, Laplace, Hamilton and Poisson, a new formalism for classi- cal mechanics is constructed, using new concepts such as energy and
gravitational potential. Later, the notion of mass was included in concepts such as the energy momentum-tensor, in order to accommodate the mechanical description of the continuum media. All these
concepts were then generalized in two aspects: to accommodate relativistic particles and subatomic physics.
Considering the mechanical constitutive elements, the notion of mass arises as an element defining the characteristics of a particle, taken as a primitive concept. As such, mass has to be an
invariant under the space- time symmetry
Considering the particle-physics standard model, the mass is introduced by the Higgs mechanism of spontane- ous symmetry breaking. As we have observed in the previous section, the gauge field is a
zero mass-field by the definition of the gauge symmetry. The original Lagrangian can be improved by introducing a Higgs field in interaction with the gauge field, in order to give rise to a mass
term, exploring the concept of spontaneous symmetry breaking, in a way which is similar to the Ginzburg-Landau theory for superconductivity. Although this is an intricate and ingenious mechanism,
providing to some extent an explanation for the appearance of mass, the primitive kinematical characteristic of mass is still the same. This is due to the dispersion relation of a particle, which
associates the notion of mass in quantum field theory with the primitive concept of matter introduced by Newton.
7. Concluding Remarks
In this work we have characterized a mechanical theory from the point of view of six Constitutive Elements (CEs), that is: CE1, the reference systems; CE2, the kinematical variables; CE3, the
mechanical system; CE 4, the state of a mechanical system; CE5, the changes in the state of a mechanical system; CE6, the specification of a (particular) mechanical system. These CEs are introduced
by taking as a starting point the symmetries of space-time, which in turn is associated with measurement procedures. Such a structure gives rise to a unified description for theories of motion and
has been used here to analyze the Newtonian mechanics, fluid mechanics, non-relativistic quantum mechanics, one-particle special relativity and quantum field theories.
From this structure the main results include: i) notions such as particle and fields are described under the unified perspective of a mechanical theory; ii) the demonstration that the CEs has a
Lie-symmetry structure in association to the Nöther theorem for conservation laws; iii) considering the Dirac theory for relativistic Hami- ltonian mechanics, we obtain a general family of
representation of the Lorentz group in phase-space; (iv) non- abelian gauge fields are taken as a representation of the six CEs and, under this perspective, the notion of mass is then analyzed since
Newton―regarded as the quantity of matter and a measure of the inertia―, to Higgs― associated to the notion of spontaneous of symmetry breaking.
Some aspects of this analysis are in order. First we notice that the experimental nature of the movement leads to a specific nature of mechanics. For instance, we conclude that a crucial difference
between classical and quantum mechanics is the experimental conditions imposed by the Heisenberg relations. In this case, a mechan- ical theory for subatomic systems is intrinsically different from a
classical mechanics. But how far is one to the other? Indeed, from the perspective of the set of CEs, one would say that the systems are described by the same mechanical theory, where the differences
are emerging from the representation. In particular, this implies that there is nothing “intrinsically quantum mechanical” with the Hilbert space. This is the case for classical theories defined in
the simplectic Hilbert spaces [21] . Similar achievements are derived from the comparative analysis of a relativistic and a non-relativistic classical particle. Here the experimental condition of
invariance of the veloc- ity of light imposes the Lorentz symmetry for the space-time, such that the Galilei group arises from a low ve- locity limit. From this point of view, keeping in mind the set
of CEs, we conclude that the space and time sym- metries (the kinematics) used for describing a specific movement is strongly associated with our experimental capacity. In other words, depending on
the prescription, we can use different kinematics, which are in turn de- fined by measurements. That means that, notions like space and time are fully specified in physics by the rela- tions among
objects in movement, which is the main characteristic used for defining a measurement process. This leads us to the conclusion that the Galilei or the Lorentz symmetries are constrained by the
experimental conditions; and as such, these set of symmetries are not ontological structure of the space time. This is the case of the conformal symmetry, that can be broken by the Higgs-like
mechanism. Therefore, considering the set of CEs, a theory is valid by itself in a domain defined by the experimental characterization of the movement. This establishes a picture of relations among
theories, that combines symmetry and representations. Discarding expe- rimental evidences, one can extrapolate such a picture in different directions, which can be mathematically con- sistent, but
can no longer be called a mechanical theory.
Finally, it is important to mention that, the structure of the CEs can be extended to equilibrium thermody- namics and thermal quantum field theories, by using thermofield dynamics [21] [33] . This
analysis is in progress and will be presented elsewhere.
The authors thank F. C. Khanna, for the discussions and for his interest in this work, and CNPq of Brazil, for financial support.
In this appendix we present a brief review of basic facts of non-abelian gauge theories in order to emphasize the Constitutive Elements structure. The Lagrangian of the free Dirac field describing
where repeated Latin indices are summed. Since this transformation
The set of symmetries is specified by
The Lagrangian
where the following definitions are used. (i) The term
(ii) Using the fact that
The gauge invariance of
For infinitesimal transformation, where
Then we obtain the expression
In this equation, each
There is an arbitrariness in the definition of
The final results are independent of
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Computing FFT Twiddle Factors - Rick Lyons
Computing FFT Twiddle Factors
Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2
fast Fourier transform (FFT).
Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to
compute the individual twiddle factors of an N-point decimation-in-frequency (DIF) and an N-point decimation-in-time (DIT) FFT.
The vast majority of FFT applications use (notice how I used the word "use" instead of the clumsy word "utilize") standard, pre-written, FFT software routines. However, there are non-standard FFT
applications (for example, specialized harmonic analysis, transmultiplexers, or perhaps using an FFT to implement a bank of filters) where only a subset of the full N-sample complex FFT results are
required. Those oddball FFT applications, sometimes called "pruned FFTs", require computation of individual FFT twiddle factors, and that's the purpose of this blog.
(If, by chance, the computation of FFT twiddle factors is of no interest to you, you might just scroll down to the "A Little History of the FFT" part of this blog.)
Before we present the two twiddle factor computation algorithms, let's understand the configuration of a single "butterfly" operation used in our radix-2 FFTs. We've all seen the signal flow drawings
of FFTs with their arrays of butterfly operations. There are various ways of implementing a butterfly operation, but my favorites are the efficient single-complex-multiply butterflies shown in Figure
1. A DIF butterfly is shown in Figure 1(a), while a DIT butterfly is shown in Figure 1(b). In Figure 1 the twiddle factors are shown as e^–j2πQ/N, where variable Q is merely an integer in the range
of 0 ≤ Q ≤ (N/2)–1.
To simplify this blog's follow-on figures, we'll use Figures 1(c) and 1(d) to represent the DIF and DIT butterflies. As such, Figure 1(c) is equivalent to Figure 1(a), and Figure 1(d) is equivalent
to Figure 1(b).
Figure 1: Single-complex-multiply DIF and DIT butterflies.
Computing DIF Twiddle Factors
Take a look at Figure 2 showing the butterfly operations for an 8-point radix-2 DIF FFT.
Figure 2: 8-point DIF FFT signal flow diagram.
For the radix-2 DIF FFT using the Figures 1(c) and 1(d) butterflies,
• The N-point DIF FFT has log[2](N) stages, numbered P = 1, 2, ..., log[2](N).
• Each stage comprises N/2 butterflies.
• Not counting the –1 twiddle factors, the Pth stage has N/2^P unique twiddle factors, numbered k = 0, 1, 2, ..., N/2^P–1 as indicated by the bold numbers above the upward-pointing arrows at the
bottom of Figure 2.
Given those characteristics, the kth unique twiddle factor phase angle for the Pth stage is computed using:
kth DIF twiddle factor angle = k•2^P/2 (1)
where 0 ≤ k ≤ N/2^P–1. For example, for the second stage (P = 2) of an N = 8-point DIF FFT, the unique Q factors are:
k = 0, Q = 0•2^P/2 = 0•4/2 = 0
k = 1, Q = 1•2^P/2 = 1•4/2 = 2.
Computing DIT Twiddle Factors
Here's an algorithm for computing the individual twiddle factor angles of a radix-2 DIT FFT. Consider Figure 3 showing the butterfly signal flow of an 8-point DIT FFT.
Figure 3: 8-point DIT FFT signal flow diagram.
For the DIT FFT using the Figures 1(c) and 1(d) butterflies,
• The N-point DIT FFT has log[2](N) stages, numbered P = 1, 2, ..., log[2](N).
• Each stage comprises N/2 butterflies.
• Not counting the –1 twiddle factors, the Pth stage has N/2 twiddle factors, numbered k = 0, 1, 2, ..., N/2–1 as indicated by the upward arrows at the bottom of Figure 3.
Given those characteristics, the kth DIT twiddle Q factor for the Pth stage is computed using:
kth DIT twiddle factor Q = [⌊k2^P/N⌋][bit-rev] (2)
where 0 ≤ k ≤ N/2–1. The ⌊q⌋ operation means the integer part of q. The [z][bit-rev] function represents the three-step operation of:
[1] convert decimal integer z to a binary number represented by log[2](N)–1 binary bits,
[2] perform bit reversal on the binary number as discussed in Section 4.5, and
[3] convert the bit reversed number back to a decimal integer.
As an example of using Eq.(2), for the second stage (P = 2) of an N = 8-point DIT FFT, the k = 3 twiddle Q factor is:
k = 3 twiddle factor Q = [⌊3•2^2/8⌋][bit-rev]
= [⌊1.5⌋][bit-rev] = [1][bit-rev] = 2.
The above [1][bit-rev] operation is: take the decimal number 1 and represent it with log[2](N)–1 = 2 bits, i.e., as 01[2]. Next, reverse those bits to a binary 10[2] and convert that binary number to
our desired decimal result of 2.
A Little History of the FFT
The radix-2 FFT has a very interesting history. For example, one of the driving forces behind the development of the FFT was the United State's desire to detect nuclear explosions inside the Soviet
Union in the early 1960s. Also, if it hadn't been for the influence of a patent attorney, the Cooley-Tukey radix-2 FFT algorithm might well have been known as the Sande-Tukey algorithm, named after
Gordon Sande and John Tukey. (That's the same Gordon Sande that occasionally posts on the comp.dsp newsgroup.) For those and other interesting FFT historical facts, see the following web sites.
Cooley and Tukey, "On the Origin of the FFT Paper", http://www.garfield.library.upenn.edu/classics1993/A1993MJ84500001.pdf
Rockmore, "The FFT - An Algorithm the Whole Family Can Use", http://www.cs.dartmouth.edu/~rockmore/cse-fft.pdf
Cipra, "The FFT: Making Technology Fly", http://compmack.googlecode.com/svn/marcoshack/calc4/The_FFT_Making_Technology_Fly.pdf
Cooley, Lewis, and Welch, "Historical Notes on the Fast Fourier Transform", http://www.signallake.com/innovation/FFTHistoryJun67.pdf
[ - ]
Comment by ●October 4, 2014
Hello vanitha,
I've read your question and I can only guess what it means. If you're asking "What is a twiddle factor?", the simple answer is, a twiddle factor is a complex number whose magnitude is one. As such,
multiplying a complex number whose magnitude is M by a twiddle factor produces a new complex number whose magnitude is also M. But the new complex number has a new (changed) phase angle. The word
"twiddle" is a rarely-used American expression (slang) meaning "to spin or rotate in a small way." For example, "He twiddled the cigar in his mouth."
[ - ]
Comment by ●December 22, 2014
Dear Basavaraj,
I can make no sense out of your comment. If you repeat your comment using proper English grammar and proper English spelling (and be as clear and specific as you can be), I will do my best to reply
to your comment.
[ - ]
Comment by ●October 4, 2014
Hello ahmed,
It's been years since I wrote this material. I see that I used the word "unique" three times, but I don't recall why I wrote those words. Looking back on what I wrote I suggest that you merely ignore
the words "unique." Sorry for any confusion.
[ - ]
Comment by ●November 8, 2013
What actually is a unique twiddle factor?
[ - ]
Comment by ●March 22, 2015
let's say i want to find only X(5) instead of whole output. in that case how to calculate the twiddling factors?
[ - ]
Comment by ●March 23, 2015
Hello bikas,
I m not sure I understand your question. The algorithms for computing either the DIF or DIT twiddle factors do not depend on which X(m) output samples you want to compute. Knowing which X(m) output
samples you want to compute determines which signal paths in Fig. 2 or Fig. 3 need to be implemented. By the way, if you only want to compute a single FFT output, X(5) for example, you should learn
how to implement the Goertzel algorithm. There s a lot of Goertzel algorithm information on the Internet.
[ - ]
Comment by ●February 28, 2016
Hello suresh,
I don't understand your question. What "code of ifft" are you referring to? I've made no mention of 'inverse FFTs' in this blog. Perhaps you are confusing this blog of mine with some other blog.
[ - ]
Comment by ●September 12, 2016
Hello predator. In my Figure 3,
in stage P = 1 the four twiddle factors are:
For k = 0, twiddle factor = 0
For k = 1, twiddle factor = 0
For k = 2, twiddle factor = 0
For k = 3, twiddle factor = 0.
In stage P = 2 the four twiddle factors are:
For k = 0, twiddle factor = 0
For k = 1, twiddle factor = 0
For k = 2, twiddle factor = 2
For k = 3, twiddle factor = 2.
In stage P = 3 the four twiddle factors are:
For k = 0, twiddle factor = 0
For k = 1, twiddle factor = 2
For k = 2, twiddle factor = 1
For k = 3, twiddle factor = 3.
I hope the above answers your question.
[ - ]
Comment by ●May 7, 2013
Fantastic. This helped me so much. Thank you!
[ - ]
Comment by ●December 22, 2014
What's the no of butterflies with no twiddle factor, with real twiddle factor and with complex twiddle factor?
[ - ]
Comment by ●February 28, 2016
can you send the code of ifft in vhdl code
[ - ]
Comment by ●September 11, 2016
In your DIT FFT diagram you show that k = 0,1,2,3 from bottom to top, as indicated by the arrow. But then you give a k = 3 twiddle factor computation and calculate it to be 2. However, on the DIT FFT
diagram that twiddle factor is 0, how come? Or am I reading something wrong?
[ - ]
Comment by ●September 12, 2016
Yeah, I figured that out now too. I was just doing inversion instead of reversal of bits that's why I got it wrong. Thanks.
[ - ]
Comment by ●March 30, 2017
Hi, thanks for your very interesting article.
For a given N (number of samples) How many twiddle factors are needed to compute the radix-2 butterfly based FFT? Is N/2 the correct answer? That is, the number of butterflies at each stage.
[ - ]
Comment by ●March 30, 2017
Samuel, if you examine my Figure 2 and Figure 3 you'll see that for an N-point radix-2 FFT, there are log2(N) stages and each stage contains N/2 twiddle factors. So the answer to your question is:
for an N-point radix-2 FFT, there are a total of (N/2)*log2(N) twiddle factors.
[ - ]
Comment by ●March 30, 2017
Thanks for quick answer!
I was wondering if some of the twiddle factors are actually repeated and do not need to be recomputed. How many factors do I actually need to keep in the memory?
Thank you,
[ - ]
Comment by ●March 30, 2017
Samuel, please forgive me. I misunderstood the true meaning of your question. You were correct! In general the minimum number of different complex-valued twiddle factors you must compute (or store
in memory) is N/2.
If you use MATLAB, have a look at my code at:
If you run that code, with N = 16, and then enter the command:
Twiddles = exp(j*2*pi*Results(:,3)/N)
you'll see that some twiddle factors differ by only by the sign of their real parts. [Such as a twiddle at +22.5 degrees (0.9239+0.3827i) and a twiddle at +157.5 degrees (-0.9239+0.3827i).] Perhaps
you can think of a way to take advantage of that symmetry.
[ - ]
Comment by ●February 23, 2021
I get how to calculate the twiddle factor for Q in this article thats fine, but not how you link it all together for the fft.
Take pass 2 in figure 3 for DIT:
X(6) has X(6) - Q(2)X(4)
How do you figure out that it needs Q(2) and X(4) and the - 1 based on just the index of 6 and pass P = 2
I want to map each index of the samples to the other needed index and the signed twiddle factor. No one seems to explain this part of the FFT.
[ - ]
Comment by ●February 24, 2021
Hi DaveSC.
You wrote: "Take pass 2 in figure 3 for DIT:". Does "pass 2" mean "stage 2"?
You wrote: "X(6) has X(6) - Q(2)X(4)". Keeping in mind that X(6) and X(4) are FFT output samples, what does "X(6) has X(6) - Q(2)X(4)" mean? What does the word "has" mean? What is Q(2)?
You wrote: "How do you figure out that it needs Q(2) and X(4) and the - 1 based on just the index of 6 and pass P = 2". In that sentence, what does the word "it" mean?
You wrote: "I can't figure out the math to map it so I can pre compute it before running the fft." In that sentence, what does the first "it" mean? In that sentence, what does the second "it" mean?
What does the word "map" mean?
I don't understand exactly what processing you desire to perform.
Postscript 30 minutes later: DaveSC, I just saw your most recent comment on the 'signal processing StackExchange'. You're welcome to ignore my above questions if you wish.
To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.
Please login (on the right) if you already have an account on this platform.
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Rotations are non commutative in 3D
Rotations are non commutative in 3D
Now we're going to talk about one of the more perplexing elements of rotations in 3D. I've got three coordinate frames here and they're all initially parallel to each other. The x-axes, the y-axes
and the z-axes are all aligned. I'm going to pick up the first one, the red one, and I'm going to rotate it by +90 degrees around its x-axis. Positive is in this direction so after rotating it by 90
degrees in x-axis, it looks like this. And now I'm going to rotate it by +90 degrees around the y-axis. So this frame is going to end up looking like this. This is its final orientation. I'm just
going to put it down there. I'm going to pick up the blue frame and I'm going to do the rotation within the opposite order. First of all, I'm going to rotate it by +90 degrees around the y-axis.
That's the positive direction. So it's initially going to look like this. And now I'm going to rotate it by +90 degrees around the x-axis. The x-axis is now pointing downwards, that's the positive
rotation direction. So this is what the final orientation will look like. And we can see that these two frames have gotten very, very different orientations. So when it comes to doing rotations in
three dimensions, the order in which you do them is critically important. Rotations are not to commutative.
Let's look at the non-commutative nature of rotation matrix multiplication. I'm going to have a rotation of 90 degrees around the x-axis. And then I'm going to rotate by 90 degrees around the y-axis.
And the resulting rotation matrix looks like this. If I do this in the opposite order, I'll rotate around the y-axis first, and then I'll rotate around the x-axis, by 90 degrees in each case. I end
up with a resulting rotation matrix which looks like this. And we can see that these two matrices are quite different. When you're multiplying rotation matrices, the order is critically important.
There is no code in this lesson.
If we apply a sequence of 3D rotations to an objects we see that the order in which they are applied affects the final result.
Skill level
Undergraduate mathematics
This content assumes high school level mathematics and requires an understanding of undergraduate-level mathematics; for example, linear algebra - matrices, vectors, complex numbers, vector calculus
and MATLAB programming.
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Sar WorS 2019 - Sardinian Workshop on Spin studies
In this talk we will assess the role of the different choices on TMD functions parametrisations in phenomenological analyses.
As an example, the latest Sivers extraction from SIDIS data is presented. Motivated by the latest COMPASS measurement, a new, thorough study of uncertainties affecting the extracted quark Sivers
function, along with a critical assessment of visibility of TMD signals in...
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