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Calculating Elasticity and Percentage Changes
Learning Objectives
• Mathematically differentiate between elastic, inelastic, and unitary elasticities of demand
• Calculate percentage changes, or growth rates
• Differentiate between the midpoint elasticity approach and the point elasticity approach in calculating elasticity
Calculating Elasticity
The formula for calculating elasticity is:
[latex]\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}[/latex].
Let’s look at the practical example mentioned earlier about cigarettes. Certain groups of cigarette smokers, such as teenage, minority, low-income, and casual smokers, are somewhat sensitive to
changes in price: for every 10 percent increase in the price of a pack of cigarettes, the smoking rates drop about 7 percent. Plugging those numbers into the formula, we get
[latex]\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}=\frac{-7\%}{10\%}=-0.7[/latex]
Inelastic, Elastic, and Unitary Demand
So what does the number -0.7 tell us about the elasticity of demand? The negative sign reflects the law of demand: at a higher price, the quantity demanded for cigarettes declines. All price
elasticities of demand have a negative sign, so it’s easiest to think about elasticity in absolute value, ignoring the negative sign. The fact that the result is less than one is more important than
the negative sign. It tells us that the size of the quantity change is less than the size of the price change (i.e. the numerator in the elasticity formula is less than the denominator). This tells
us that it would take a relatively large price change in order to cause a relatively small change in quantity demanded. In other words, consumer responsiveness to a change in price is relatively
small. Therefore, when the elasticity is less than 1, we say that demand is inelastic.
The data above indicate that the demand for cigarettes by teenagers, minority, low income and casual smokers is relatively inelastic. Addicted adult smokers, though, are even less sensitive to
changes in the price—most are willing to pay whatever it takes to support their smoking habit. We can say that their demand is even more inelastic than low income or casual smokers.
Different products have different price elasticities of demand. If the absolute value of the elasticity of some product is greater than one, it means that the change in the quantity demanded is
greater than the change in price. This indicates a larger reaction to price change, which we describe as elastic. If the elasticity is equal to one, it means that the change in the quantity demanded
is exactly equal to the change in price, so the demand response is exactly proportional to the change in price. We call this unitary elasticity, because unitary means one.
Watch It
Watch this video carefully to understand how to solve for elasticity and to see what the numerical values for elasticity mean when applied to economic situations.
You can view the transcript for “Episode 16: Elasticity of Demand” here (opens in new window).
Calculating Percentage Changes and Growth Rates
Before we dive deeper into solving for elasticity, let’s first make sure we are comfortable calculating percentage changes, also known as a growth rates. The formula for computing a growth rate is
[latex]\text{Percentage change}=\frac{\text{Change in quantity}}{\text{Quantity}}[/latex]
Suppose that a job pays $10 per hour. At some point, the individual doing the job is given a $2-per-hour raise. The percentage change (or growth rate) in pay is
[latex]\frac{\$2}{\$10}=0.20\text{ or }20\%[/latex].
Now to solve for elasticity, we use the growth rate, or percentage change, of the quantity demanded as well as the percentage change in price in order to to examine how these two variables are
related. The price elasticity of demand is the ratio between the percentage change in the quantity demanded (Qd) and the corresponding percent change in price:
[latex]\text{Price elasticity of demand}=\frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}[/latex]
There are two general methods for calculating elasticities: the point elasticity approach and the midpoint (or arc) elasticity approach. Elasticity looks at the percentage change in quantity demanded
divided by the percentage change in price, but which quantity and which price should be the denominator in the percentage calculation? The point approach uses the initial price and initial quantity
to measure percent change. This makes the math easier, but the more accurate approach is the midpoint approach, which uses the average price and average quantity over the price and quantity change.
(These are the price and quantity halfway between the initial point and the final point.) Let’s compare the two approaches. Suppose the quantity demanded of a product was 100 at one point on the
demand curve, and then it moved to 103 at another point. The growth rate, or percentage change in quantity demanded, would be the change in quantity demanded [latex]{(103-100)}[/latex] divided by the
average of the two quantities demanded:
In other words, the growth rate:
[latex]\begin{array}{r}{\frac{103-100}{(103+100)/2}}\\{=\frac{3}{101.5}}\\{=0.0296}\\{=2.96\%\text{ growth}}\end{array}[/latex]
Note that if we used the point approach, the calculation would be:
[latex]\frac{(103–100)}{100}=3\%\text{ growth}[/latex]
This produces nearly the same result as the slightly more complicated midpoint method (3% vs. 2.96%). If you need a rough approximation, use the point method. If you need accuracy, use the
midpoint method. Note: as the two points become closer together, the point elasticity becomes a closer approximation to the arc elasticity.
In this module you will often be asked to calculate the percentage change in the quantity. Keep in mind that this is same as the the growth rate of the quantity. As you work through the course and
find other applications for calculate growth rates, you will be well prepared.
Try It
These next questions allow you to get as much practice as you need, as you can click the link at the top of the questions (“Try another version of these questions”) to get a new version of the
questions. Practice until you feel comfortable with this concept.
elastic demand:
when the calculated elasticity of demand is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price
elastic supply:
when the calculated elasticity of either supply is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price
inelastic demand:
when the calculated elasticity of demand is less than one, indicating that a 1 percent increase in price paid by the consumer leads to less than a 1 percent change in purchases (and vice versa);
this indicates a low responsiveness by consumers to price changes
inelastic supply:
when the calculated elasticity of supply is less than one, indicating that a 1 percent increase in price paid to the firm will result in a less than 1 percent increase in production by the firm;
this indicates a low responsiveness of the firm to price increases (and vice versa if prices drop)
midpoint elasticity approach:
Most accurate approach to solving for elasticity in which the percent changes in quantity demanded and price are measured relative to the average quantity demanded and price; the initial quantity
demand is subtracted from the new quantity demanded; then divided by the average of the two quantities demanded; similarly, the initial price is subtracted from the new price, then divided by the
average of the two prices
point elasticity approach:
approximate method for solving for elasticity in which the percent changes are measured relative to the initial quantity demanded and price; the initial quantity demanded is subtracted from the
new quantity demanded, then divided by the initial quantity demanded; similarly, the initial price is subtracted from the new price, then divided by the initial price.
unitary elasticity:
when the calculated elasticity is equal to one indicating that a change in the price of the good or service results in a proportional change in the quantity demanded or supplied | {"url":"https://courses.lumenlearning.com/wm-microeconomics/chapter/calculating-elasticity-and-percentage-changes/","timestamp":"2024-11-10T12:40:39Z","content_type":"text/html","content_length":"59139","record_id":"<urn:uuid:cd5f3307-b6d3-490c-92c2-fc9588cd7790>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00538.warc.gz"} |
2023 CATHLEEN SYNGE MORAWETZ PRIZE AWARDED TO DR. STEFANOS ARETAKIS
Media Release – April 3, 2023
Canadian Mathematical Society
2023 CATHLEEN SYNGE MORAWETZ PRIZE AWARDED TO DR. STEFANOS ARETAKIS
OTTAWA, ON – The Canadian Mathematical Society (CMS) is pleased to announce that Dr. Stefanos Aretakis (University of Toronto) has been named the recipient of the 2023 Cathleen Synge Morawetz Prize.
This prize was awarded for an outstanding research publication, or a series of closely related publications on the topic of Applied Mathematics for Dr. Aretakis’ groundbreaking work on instability in
extremal black holes (what has come to be known as Aretakis instability), conservation laws for wave equations, and their long-term behaviour in asymptotically flat backgrounds. Highlights of this
work include:
1. Aretakis, Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I. Commun. Math. Phys. (2011) 307, 17–63.
2. Angelopoulos, S. Aretakis, and D. Gajic Horizon Hair of Extremal Black Holes and Measurements at Null Infinity, Phys. Rev. Lett. 121, 131102 (2018).
3. Aretakis, The Characteristic Gluing Problem and Conservation Laws for the Wave Equation on Null Hypersurfaces, Annals of PDE (2017), 3:3.
4. Angelopoulos, S. Aretakis & D. Gajic A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes, Annals of PDE 4: 15 (2018).
The first in this series of notable contributions is the influential, single-author publication in 2011 (and another in 2015) where Dr. Aretakis discovered a surprising instability mechanism in
extremal black holes, which he established using conceptually and technically novel methods. This resolved a longstanding open question in General Relativity, and has had a major impact on research
in the field. Coincidentally, Cathleen Synge Morawetz herself had studied an analogous mathematical question in R^n. Dr. Aretakis and his team subsequently used asymptotics of solutions of the wave
equation to propose a new observational signature for extremal black holes, published as editor’s selection in Phys. Rev Lett (2018) and later in full mathematical detail in Adv. Math. This line of
Dr. Aretakis’ work has continued to impact not only physics but also mathematics, identifying (2017) a novel set of conservation laws improving general understanding of wave equations in Lorentzian
geometry, and, in a highly-cited Annals of PDE paper (2018), studying long-time behaviour of waves on very general classes of asymptotically flat backgrounds. A professor at the University of Toronto
and Dr. Aretakis’ colleague Dr. Robert J. McCann, FRSC states;
“Stefanos Aretakis has made deep contributions to the mathematics of general relativity, Einstein’s theory of gravity. One of the holy grails in the area has been to confirm the stability of the
Kerr family of (non-extremal) rotating black holes. That this stability question might have a different answer in the extremal case (which rotate as quickly as possible for their mass), seems to
have been largely overlooked prior to his work. Aretakis showed they are unstable, but the instability manifests itself only in norms of sufficient smoothness: one sees the instability not at the
level of zeroth or first derivatives, but only at the level of second derivatives of the solution of a wave equation, representing a perturbation of the geometry of the black hole. Given its
essentially mathematical nature, his work has proven unusually influential in the physics community, where the Aretakis instability is now widely discussed, and its consequences are helping to
inspire the next generation of experiments in high-energy physics and gravitational wave observations.”
Dr. Stefanos Aretakis is an Associate Professor of Mathematics at the University of Toronto. He received his PhD in 2012 at the University of Cambridge, and held a Veblen Research Instructorship and
Assistant Professorship at Princeton University prior to joining the University of Toronto. His main research interests are in Differential Geometry, Analysis of PDEs, and General Relativity.
Dr. Aretakis has contributed outstanding publications to his field, which have had a profound impact in his areas of research. The CMS is delighted to award him the well-deserved 2023 Cathleen Synge
Morawetz Prize.
About the Cathleen Synge Morawetz Prize
The Cathleen Synge Morawetz Prize is for an author(s) of an outstanding research publication. A series of closely related publications can be considered if they are clearly connected and focused on
the same topic. The recipient(s) shall be a member of or have close ties to the Canadian mathematical community.
For more information, visit the Cathleen Synge Morawetz Prize page.
About the Canadian Mathematical Society (CMS)
The CMS is the main national organization whose goal is to promote and advance the discovery, learning and application of mathematics. The Society’s activities cover the whole spectrum of mathematics
including: scientific meetings, research publications, and the promotion of excellence in mathematics competitions that recognize outstanding student achievements.
For more information, please contact:
Dr. Susan Cooper (uManitoba) or Dr. Termeh Kousha
Chair, CMS Research Committee Executive Director
Canadian Mathematical Society Canadian Mathematical Society
chair-resc@cms.math.ca tkousha@cms.math.ca | {"url":"https://cms.math.ca/news-item/2023-cathleen-synge-morawetz-prize-awarded-to-dr-stefanos-aretakis/","timestamp":"2024-11-02T03:10:34Z","content_type":"text/html","content_length":"157505","record_id":"<urn:uuid:8033bc88-e15f-49d4-971f-be70254c66b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00710.warc.gz"} |
What are reflex angles?
What are Reflex Angles?
A reflex angle is the larger angle, it’s always more than 180° (half a circle) but less than 360° (A full circle). It’s possibly the most confusing of the angles because it’s always on the outside.
Reflex Angles explained
There are six types of angle in total;
An Acute angle is the smallest, measuring more than 0° but less than 90°.
Next up is a Right angle, also taught as a quarter turn. This angle always measures 90 °.
An Obtuse angle measures more than 90° but less than 180°.
A Straight angle or a half turn is always 180 °.
Reflex is the next largest measuring more than 180° but less than 360°.
We then have a full rotation or full circle at 360°.
Angle Family
If you struggle to remember the different names of the angles, try watching this video. The angles are introduced as different characters within a family which might help you to remember!
How do you measure a Reflex Angle?
First, make sure that it is the reflex angle you’re being asked to measure! Lots of children mistakenly, but very carefully measure the wrong angle. Usually, the angle you’re being asked to measure
will have a little circle drawn around the point to show you which side needs measuring.
You can measure the reflex angle in one of two ways; either measure the inner angle and subtract it from 360° (to give you the measure of the reflex angle) or measure the reflex angle itself.
A reflex angle is probably the trickiest angle to measure because it will be larger than your standard 180° protractor so you’ll need to do a bit of accurate drawing.
1. Line up your protractor with one side of the angle (in this case y to z), make sure that the centre of the protractor lines up with the centre point of where the two lines meet. (N.B Lots of
children use the straight edge of the protractor instead of the line). Make sure that your line goes through 0 °.
2. Make a small mark on your paper to show where 180° is. If you want to, you could even draw a faint line to show the 180° line.
3.You now need to measure the remainder. Using the outside scale on your protractor, measure the angle from the 180° line you have drawn to the second line you have been given (in this case y to x).
4. Add together 180° to the second measurement and this will be your reflex angle.
5. If you want to check your answer, measure the inside angle and check that the two measurements added together equal 360°.
How do you measure a Reflex Angle – Video explanation?
If you're a visual learner and would prefer to watch, have a look at this video link;
Worksheets and Practice
EdPlace have loads of great worksheets to teach you about angles. We’ve listed a few of our favourites below:
Year 3 – Know your angles – smallest or largest
Year 3 – know your angles: smaller or larger than a right angle
Year 4 – Sort the angles, acute or obtuse?
Year 4 – Geometry – How many right angles?
Year 5 – Angle multiples of 90°.
Year 5 – Geometry – Estimate angles.
Year 5 – Geometry – Calculate missing angles at a point
Year 6 – Geometry – What’s the angle?
Year 6 – Geometry – Angles at a point.
Year 6 – Triangles – Calculate unknown angles
Year 6 – Convert decimal angles to degrees minutes and seconds
Year 7 – Measuring and recognising different types of angle
Year 7 – Finding the third angle of a triangle
Year 7 – Calculating angles at a point and on a straight line
Further Learning
If you enjoy geometry and want to give yourself a challenge, why not try some of the puzzles and problems set by the NRich team from the University of Cambridge?
Click the link below to try out a selection of puzzles linked to angles, triangles and reflex angles to really get you thinking! | {"url":"https://www.edplace.com/blog/edplace-explains/what-are-reflex-angles","timestamp":"2024-11-11T13:31:59Z","content_type":"text/html","content_length":"86084","record_id":"<urn:uuid:b4b2fb70-3585-4550-9448-c3ad3c3e5051>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00291.warc.gz"} |
Area Model Multiplication 4th Grade Worksheet Pdf
Math, especially multiplication, forms the foundation of numerous academic disciplines and real-world applications. Yet, for several learners, mastering multiplication can posture an obstacle. To
resolve this hurdle, teachers and parents have actually accepted an effective device: Area Model Multiplication 4th Grade Worksheet Pdf.
Introduction to Area Model Multiplication 4th Grade Worksheet Pdf
Area Model Multiplication 4th Grade Worksheet Pdf
Area Model Multiplication 4th Grade Worksheet Pdf -
Get Started Area Model Multiplication Worksheets Area model multiplication worksheets consist of questions based on area model multiplication The questions included in the worksheet serve to
demonstrate how to use the area model for the multiplication of numbers Benefits of Area Model Multiplication Worksheets
Activity 1 Multi Digit Multiplication Pre Assessment A5 1 Activity 2 Multiplying by 10 100 1000 A5 17 Activity 3 Multiplying Single Digits by Multiples of Ten A5 23 Activity 4 Single Digit
Multiplication with Pictures Numbers A5 29 Activity 5 Introducing the Standard Multiplication Algorithm A5 35 Activity 6 Think before You Mu
Value of Multiplication Practice Recognizing multiplication is crucial, laying a strong structure for innovative mathematical principles. Area Model Multiplication 4th Grade Worksheet Pdf offer
structured and targeted method, fostering a deeper comprehension of this essential math operation.
Development of Area Model Multiplication 4th Grade Worksheet Pdf
4th Grade Area Model Multiplication Worksheets Free Printable
4th Grade Area Model Multiplication Worksheets Free Printable
The area model for multiplication otherwise known as the box method is a way to multiply larger numbers by finding partial products and adding them together Let s take a quick look at how to solve a
problem using the area model If we use the multiplication problem 24 x 56 the first step would be to create a box that has 2 columns and 2 rows
Reinforce 2 digit by 2 digit box multiplication with this collection of printable worksheets designed exclusively for learners in grade 3 grade 4 and grade 5 Let the kids get to grips with finding
the product of numbers 3 Digit by 2 Digit Area Model Multiplication
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Basic Multiplication Sheets Basic workouts focusing on multiplication tables, helping students build a solid math base.
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39 Multiplication Area Model Worksheet Worksheet Information
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Area Model Multiplication Multiply 1 digit by 3 and 4 digit numbers Math 4 What is the multiplication problem for the area model above Work with your team to draw an area model for the following A 7
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Regular technique sharpens multiplication effectiveness, improving general mathematics capabilities.
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Word problems in worksheets create logical reasoning and strategy application.
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How to Develop Engaging Area Model Multiplication 4th Grade Worksheet Pdf
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Associating multiplication to everyday scenarios adds importance and usefulness to exercises.
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Digital Multiplication Equipment and Games Technology-based sources provide interactive understanding experiences, making multiplication appealing and satisfying. Interactive Sites and Apps Online
platforms give varied and easily accessible multiplication technique, supplementing conventional worksheets. Personalizing Worksheets for Numerous Discovering Styles Visual Students Visual aids and
representations help understanding for students inclined toward visual discovering. Auditory Learners Verbal multiplication issues or mnemonics satisfy learners that comprehend principles via
auditory methods. Kinesthetic Learners Hands-on tasks and manipulatives sustain kinesthetic students in comprehending multiplication. Tips for Effective Application in Understanding Consistency in
Practice Routine method reinforces multiplication abilities, advertising retention and fluency. Stabilizing Repeating and Range A mix of repeated workouts and varied problem formats maintains passion
and understanding. Supplying Positive Feedback Feedback aids in identifying locations of renovation, encouraging ongoing progress. Obstacles in Multiplication Method and Solutions Motivation and
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Area Model Multiplication 4th Grade Worksheet Free Printable
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Multiplication Strategies I Created With Mrs Ervin s 4th Grade 9 28 12
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Area Model Multiplication 4th Grade Worksheets
Math Area Models Of Multiplication
50 Multiplication And Area Models Worksheets For 2nd Grade On Quizizz
Area Model For Multiplication MIF 4th Grade Area Model Multiplication
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Activity 1 Multi Digit Multiplication Pre Assessment A5 1 Activity 2 Multiplying by 10 100 1000 A5 17 Activity 3 Multiplying Single Digits by Multiples of Ten A5 23 Activity 4 Single Digit
Multiplication with Pictures Numbers A5 29 Activity 5 Introducing the Standard Multiplication Algorithm A5 35 Activity 6 Think before You Mu
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Activity 1 Multi Digit Multiplication Pre Assessment A5 1 Activity 2 Multiplying by 10 100 1000 A5 17 Activity 3 Multiplying Single Digits by Multiples of Ten A5 23 Activity 4 Single Digit
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Ramps and Inclines
Free Body Diagrams
Now that we've developed an understanding of Newton's Laws of Motion, free body diagrams, friction, and forces on flat surfaces, we can extend these tools to situations on ramps, or inclined,
The key to understanding these situations is creating an accurate free body diagram after choosing convenient x- and y-axes. Problem-solving steps are consistent with those developed for Newton's 2nd
Let's take the example of a box on a ramp inclined at an angle of Θ with respect to the horizontal. We can draw a basic free body diagram for this situation, with the force of gravity pulling the box
straight down, the normal force perpendicular out of the ramp, and friction opposing motion (in this case pointing up the ramp).
Once the forces acting on the box have been identified, we must be clever about our choice of x-axis and y-axis directions. Much like we did when analyzing free falling objects and projectiles, if we
set the positive x-axis in the direction of initial motion (or the direction the object wants to move if it is not currently moving), the y-axis must lie perpendicular to the ramp's surface (parallel
to the normal force). Let's re-draw our free body diagram, this time superimposing it on our new axes.
Resolving to Components
Unfortunately, the force of gravity on the box, mg, doesn't lie along one of the axes. Therefore, it must be broken up into components which do lie along the x- and y-axes in order to simplify our
mathematical analysis. To do this, we can use geometry to break the weight down into a component parallel with the axis of motion (mg║) and a component perpendicular to the x-axis (mg┴) using the
Using these equations, we can re-draw the free body diagram, replacing mg with its components. Now all the forces line up with the axes, making it straightforward to write Newton's 2nd Law Equations
(FNETx and FNETy) and continue with our standard problem-solving strategy.
In the example shown with our modified free body diagram, we could write our Newton's 2nd Law Equations for both the x- and y-directions as follows:
From this point, our problem becomes an exercise in algebra. If you need to tie to two equations together to eliminate a variable, don't forget the equation for the force of friction:
Sample Problems
Let's take a look at a sample problem to see how these steps can be put into practice and combined with our knowledge of the kinematic equations:
Let's examine another problem, this time taking a look at a box on a ramp in static equilibrium:
Question: Three forces act on a box on an inclined plane as shown in the diagram below. [Vectors are not drawn to scale.] If the box is at rest, the net force acting on it is equal to
1. the weight
2. the normal force
3. friction
4. zero
Answer: (4) zero. If the box is at rest, the acceleration must be zero, therefore the net force must be zero. | {"url":"https://www.aplusphysics.com/courses/honors/dynamics/ramps.html","timestamp":"2024-11-14T08:46:13Z","content_type":"application/xhtml+xml","content_length":"35174","record_id":"<urn:uuid:47d522fa-0703-4ce4-bd35-2b6fd4c48305>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00641.warc.gz"} |
how long can an inductor store energy
Working principles of inductors and capacitors | Electronics360
Inductors and capacitors both store energy, but in different ways and with different properties. The inductor uses a magnetic field to store energy. When current flows through an inductor, a magnetic
field builds up around it, and energy is stored in this field. The energy is released when the magnetic field collapses, inducing a voltage in the ...
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What is an Inductor?
Simply put, an inductor is a component that can store energy in the form of a magnetic field. A typical example of an inductor is a coil of wire which can be found in air coils, motors, and
electromagnets. Another way to look at inductors is that they are components that will generate a magnetic field when current is passed through them, or ...
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Inductors: Energy Storage Applications and Safety …
This is highlighted as the area under the power curve in Figure 2. The energy in the inductor can be found using the following equation: (w=frac{1}{2}Li^{2}) (2) Where i is the current (amperes), L
is …
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An inductor is a passive component that is used in most power electronic circuits to store energy. Learn more about inductors, their types, the working principle and more. Inductors, much like
conductors and resistors, are simple components that are used in electronic devices to carry out specific functions. ...
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Inductor i-v equation in action (article) | Khan Academy
equation: v = L d i d t i = 1 L ∫ 0 T v d t + i 0. We create simple circuits by connecting an inductor to a current source, a voltage source, and a switch. We learn why an inductor acts like a short
circuit if its current is constant. We learn why the current in an inductor cannot change instantaneously.
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How does an Inductor Store Energy?
An inductor stores energy in the creation of a magnetic field. An inductor is a device consisting of a coil of insulated wire usually wound around a magnetic core—most often iron. Current flowing
through the wire generates an electromotive force that acts on the following current and opposes its change in value.
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How do inductors store energy?
$begingroup$ As capacitors store energy in the electric field, so inductors store energy in the magnetic field. Both capacitors and inductors have many …
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Energy in Inductors: Stored Energy and Operating Characteristics
Energy storage and filters in point-of-load regulators and DC/DC converter output inductors for telecommunications and industrial control devices. Molded Powder. Iron powder directly molded to copper
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Energy Stored in an Inductor
When a electric current is flowing in an inductor, there is energy stored in the magnetic field. Considering a pure inductor L, the instantaneous power which must be supplied to …
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Energy stored in inductor (1/2 Li^2) (video) | Khan Academy
Energy stored in inductor (1/2 Li^2) An inductor carrying current is analogous to a mass having velocity. So, just like a moving mass has kinetic energy = 1/2 mv^2, a coil carrying …
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What is an inductor and how does it store energy?
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Basic Facts about Inductors [Lesson 1] Overview of inductors
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Energy storage in inductors
L (nH) = 0.2 s { ln (4s/d) - 0.75 } It looks complicated, but in fact it works out at around 1.5 μH for a 1 metre length or 3 mH for a kilometre for most gauges of wire. An explanation of energy
storage in the magnetic field of an inductor.
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23.12: Inductance
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Nick's Blog
Posted by Nick Johnson | Filed under python, cardinality-estimation, damn-cool-algorithms
Suppose you have a very large dataset - far too large to hold in memory - with duplicate entries. You want to know how many duplicate entries, but your data isn't sorted, and it's big enough that
sorting and counting is impractical. How do you estimate how many unique entries the dataset contains? It's easy to see how this could be useful in many applications, such as query planning in a
database: the best query plan can depend greatly on not just how many values there are in total, but also on how many unique values there are.
I'd encourage you to give this a bit of thought before reading onwards, because the algorithms we'll discuss today are quite innovative - and while simple, they're far from obvious.
A simple and intuitive cardinality estimator
Let's launch straight in with a simple example. Suppose someone generate a dataset with the following procedure:
1. Generate n evenly distributed random numbers
2. Arbitrarily replicate some of those numbers an unspecified number of times
3. Shuffle the resulting set of numbers arbitrarily
How can we estimate how many unique ... | {"url":"http://blog.notdot.net/tag/cardinality-estimation","timestamp":"2024-11-04T23:16:45Z","content_type":"application/xhtml+xml","content_length":"7224","record_id":"<urn:uuid:e1efa406-8c0c-4d1d-ad17-e42076de62dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00642.warc.gz"} |
Orthogonal frequency-division multiplexing
Short description: Method of encoding digital data on multiple carrier frequencies
In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission used in digital modulation for encoding digital (binary) data on multiple carrier
frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL internet access, wireless
networks, power line networks, and 4G/5G mobile communications.^[1]
OFDM is a frequency-division multiplexing (FDM) scheme that was introduced by Robert W. Chang of Bell Labs in 1966.^[2]^[3]^[4] In OFDM, the incoming bitstream representing the data to be sent is
divided into multiple streams. Multiple closely spaced orthogonal subcarrier signals with overlapping spectra are transmitted, with each carrier modulated with bits from the incoming stream so
multiple bits are being transmitted in parallel.^[5] Demodulation is based on fast Fourier transform algorithms. OFDM was improved by Weinstein and Ebert in 1971 with the introduction of a guard
interval, providing better orthogonality in transmission channels affected by multipath propagation.^[6] Each subcarrier (signal) is modulated with a conventional modulation scheme (such as
quadrature amplitude modulation or phase-shift keying) at a low symbol rate. This maintains total data rates similar to conventional single-carrier modulation schemes in the same bandwidth.^[7]
The main advantage of OFDM over single-carrier schemes is its ability to cope with severe channel conditions (for example, attenuation of high frequencies in a long copper wire, narrowband
interference and frequency-selective fading due to multipath) without the need for complex equalization filters. Channel equalization is simplified because OFDM may be viewed as using many slowly
modulated narrowband signals rather than one rapidly modulated wideband signal. The low symbol rate makes the use of a guard interval between symbols affordable, making it possible to eliminate
intersymbol interference (ISI) and use echoes and time-spreading (in analog television visible as ghosting and blurring, respectively) to achieve a diversity gain, i.e. a signal-to-noise ratio
improvement. This mechanism also facilitates the design of single frequency networks (SFNs) where several adjacent transmitters send the same signal simultaneously at the same frequency, as the
signals from multiple distant transmitters may be re-combined constructively, sparing interference of a traditional single-carrier system.
In coded orthogonal frequency-division multiplexing (COFDM), forward error correction (convolutional coding) and time/frequency interleaving are applied to the signal being transmitted. This is done
to overcome errors in mobile communication channels affected by multipath propagation and Doppler effects. COFDM was introduced by Alard in 1986^[8]^[9]^[10] for Digital Audio Broadcasting for Eureka
Project 147. In practice, OFDM has become used in combination with such coding and interleaving, so that the terms COFDM and OFDM co-apply to common applications.^[11]^[12]
Example of applications
The following list is a summary of existing OFDM-based standards and products. For further details, see the Usage section at the end of the article.
Wired version mostly known as Discrete Multi-tone Transmission (DMT)
• ADSL and VDSL broadband access via POTS copper wiring
• DVB-C2, an enhanced version of the DVB-C digital cable TV standard
• Power line communication (PLC)
• ITU-T G.hn, a standard which provides high-speed local area networking of existing home wiring (power lines, phone lines and coaxial cables)^[13]
• TrailBlazer telephone line modems
• Multimedia over Coax Alliance (MoCA) home networking
• DOCSIS 3.1 Broadband delivery
• The wireless LAN (WLAN) radio interfaces IEEE 802.11a, g, n, ac, ah and HIPERLAN/2
• The digital radio systems DAB/EUREKA 147, DAB+, Digital Radio Mondiale, HD Radio, T-DMB and ISDB-TSB
• The terrestrial digital TV systems DVB-T and ISDB-T
• The terrestrial mobile TV systems DVB-H, T-DMB, ISDB-T and MediaFLO forward link
• The wireless personal area network (PAN) ultra-wideband (UWB) IEEE 802.15.3a implementation suggested by WiMedia Alliance
• Wi-SUN (Smart Ubiquitous Network)
The OFDM-based multiple access technology OFDMA is also used in several 4G and pre-4G cellular networks, mobile broadband standards, the next generation WLAN and the wired portion of Hybrid
fiber-coaxial networks:
Key features
The advantages and disadvantages listed below are further discussed in the Characteristics and principles of operation section below.
Summary of advantages
• High spectral efficiency as compared to other double sideband modulation schemes, spread spectrum, etc.
• Can easily adapt to severe channel conditions without complex time-domain equalization.
• Robust against narrow-band co-channel interference
• Robust against intersymbol interference (ISI) and fading caused by multipath propagation
• Efficient implementation using fast Fourier transform
• Low sensitivity to time synchronization errors
• Tuned sub-channel receiver filters are not required (unlike conventional FDM)
• Facilitates single frequency networks (SFNs) (i.e., transmitter macrodiversity)
Summary of disadvantages
• Sensitive to Doppler shift
• Sensitive to frequency synchronization problems
• High peak-to-average-power ratio (PAPR), requiring linear transmitter circuitry, which suffers from poor power efficiency
• Loss of efficiency caused by cyclic prefix/guard interval
Characteristics and principles of operation
Conceptually, OFDM is a specialized frequency-division multiplexing (FDM) method, with the additional constraint that all subcarrier signals within a communication channel are orthogonal to one
In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the sub-channels is eliminated and inter-carrier guard bands are
not required. This greatly simplifies the design of both the transmitter and the receiver; unlike conventional FDM, a separate filter for each sub-channel is not required.
The orthogonality requires that the subcarrier spacing is [math]\displaystyle{ \scriptstyle\Delta f \,=\, \frac{k}{T_U} }[/math] Hertz, where T[U] seconds is the useful symbol duration (the
receiver-side window size), and k is a positive integer, typically equal to 1. This stipulates that each carrier frequency undergoes k more complete cycles per symbol period than the previous
carrier. Therefore, with N subcarriers, the total passband bandwidth will be B ≈ N·Δf (Hz).
The orthogonality also allows high spectral efficiency, with a total symbol rate near the Nyquist rate for the equivalent baseband signal (i.e., near half the Nyquist rate for the double-side band
physical passband signal). Almost the whole available frequency band can be used. OFDM generally has a nearly 'white' spectrum, giving it benign electromagnetic interference properties with respect
to other co-channel users.
A simple example: A useful symbol duration T[U] = 1 ms would require a subcarrier spacing of [math]\displaystyle{ \scriptstyle\Delta f \,=\, \frac{1}{1\,\mathrm{ms}} \,=\, 1\,\mathrm{kHz} }[/
math] (or an integer multiple of that) for orthogonality. N = 1,000 subcarriers would result in a total passband bandwidth of NΔf = 1 MHz. For this symbol time, the required bandwidth in theory
according to Nyquist is [math]\displaystyle{ \scriptstyle\mathrm{BW}=R/2=(N/T_U)/2 = 0.5\,\mathrm{MHz} }[/math] (half of the achieved bandwidth required by our scheme), where R is the bit rate
and where N = 1,000 samples per symbol by FFT. If a guard interval is applied (see below), Nyquist bandwidth requirement would be even lower. The FFT would result in N = 1,000 samples per symbol.
If no guard interval was applied, this would result in a base band complex valued signal with a sample rate of 1 MHz, which would require a baseband bandwidth of 0.5 MHz according to Nyquist.
However, the passband RF signal is produced by multiplying the baseband signal with a carrier waveform (i.e., double-sideband quadrature amplitude-modulation) resulting in a passband bandwidth of
1 MHz. A single-side band (SSB) or vestigial sideband (VSB) modulation scheme would achieve almost half that bandwidth for the same symbol rate (i.e., twice as high spectral efficiency for the
same symbol alphabet length). It is however more sensitive to multipath interference.
OFDM requires very accurate frequency synchronization between the receiver and the transmitter; with frequency deviation the subcarriers will no longer be orthogonal, causing inter-carrier
interference (ICI) (i.e., cross-talk between the subcarriers). Frequency offsets are typically caused by mismatched transmitter and receiver oscillators, or by Doppler shift due to movement. While
Doppler shift alone may be compensated for by the receiver, the situation is worsened when combined with multipath, as reflections will appear at various frequency offsets, which is much harder to
correct. This effect typically worsens as speed increases,^[15] and is an important factor limiting the use of OFDM in high-speed vehicles. In order to mitigate ICI in such scenarios, one can shape
each subcarrier in order to minimize the interference resulting in a non-orthogonal subcarriers overlapping.^[16] For example, a low-complexity scheme referred to as WCP-OFDM (Weighted Cyclic Prefix
Orthogonal Frequency-Division Multiplexing) consists of using short filters at the transmitter output in order to perform a potentially non-rectangular pulse shaping and a near perfect reconstruction
using a single-tap per subcarrier equalization.^[17] Other ICI suppression techniques usually drastically increase the receiver complexity.^[18]
Implementation using the FFT algorithm
The orthogonality allows for efficient modulator and demodulator implementation using the FFT algorithm on the receiver side, and inverse FFT on the sender side. Although the principles and some of
the benefits have been known since the 1960s, OFDM is popular for wideband communications today by way of low-cost digital signal processing components that can efficiently calculate the FFT.
The time to compute the inverse-FFT or FFT has to take less than the time for each symbol,^[19]^:84 which for example for DVB-T (FFT 8k) means the computation has to be done in 896 µs or less.
[math]\displaystyle{ \mathrm{MIPS} &= \frac {\mathrm{computational\ complexity}}{T_\mathrm{symbol}} \times 1.3 \times 10^{-6} \\ &= \frac{147\;456 \times 2}{896 \times 10^{-6}} \times 1.3 \times
10^{-6} \\ &= 428 }[/math]
The computational demand approximately scales linearly with FFT size so a double size FFT needs double the amount of time and vice versa.^[19]^:83 As a comparison an Intel Pentium III CPU at 1.266
GHz is able to calculate a 8192 point FFT in 576 µs using FFTW.^[20] Intel Pentium M at 1.6 GHz does it in 387 µs.^[21] Intel Core Duo at 3.0 GHz does it in 96.8 µs.^[22]
Guard interval for elimination of intersymbol interference
One key principle of OFDM is that since low symbol rate modulation schemes (i.e., where the symbols are relatively long compared to the channel time characteristics) suffer less from intersymbol
interference caused by multipath propagation, it is advantageous to transmit a number of low-rate streams in parallel instead of a single high-rate stream. Since the duration of each symbol is long,
it is feasible to insert a guard interval between the OFDM symbols, thus eliminating the intersymbol interference.
The guard interval also eliminates the need for a pulse-shaping filter, and it reduces the sensitivity to time synchronization problems.
A simple example: If one sends a million symbols per second using conventional single-carrier modulation over a wireless channel, then the duration of each symbol would be one microsecond or
less. This imposes severe constraints on synchronization and necessitates the removal of multipath interference. If the same million symbols per second are spread among one thousand sub-channels,
the duration of each symbol can be longer by a factor of a thousand (i.e., one millisecond) for orthogonality with approximately the same bandwidth. Assume that a guard interval of 1/8 of the
symbol length is inserted between each symbol. Intersymbol interference can be avoided if the multipath time-spreading (the time between the reception of the first and the last echo) is shorter
than the guard interval (i.e., 125 microseconds). This corresponds to a maximum difference of 37.5 kilometers between the lengths of the paths.
The cyclic prefix, which is transmitted during the guard interval, consists of the end of the OFDM symbol copied into the guard interval, and the guard interval is transmitted followed by the OFDM
symbol. The reason that the guard interval consists of a copy of the end of the OFDM symbol is so that the receiver will integrate over an integer number of sinusoid cycles for each of the multipaths
when it performs OFDM demodulation with the FFT.
In some standards such as Ultrawideband, in the interest of transmitted power, cyclic prefix is skipped and nothing is sent during the guard interval. The receiver will then have to mimic the cyclic
prefix functionality by copying the end part of the OFDM symbol and adding it to the beginning portion.
Simplified equalization
The effects of frequency-selective channel conditions, for example fading caused by multipath propagation, can be considered as constant (flat) over an OFDM sub-channel if the sub-channel is
sufficiently narrow-banded (i.e., if the number of sub-channels is sufficiently large). This makes frequency domain equalization possible at the receiver, which is far simpler than the time-domain
equalization used in conventional single-carrier modulation. In OFDM, the equalizer only has to multiply each detected subcarrier (each Fourier coefficient) in each OFDM symbol by a constant complex
number, or a rarely changed value. On a fundamental level, simpler digital equalizers are better because they require fewer operations, which translates to fewer round-off errors in the equalizer.
Those round-off errors can be viewed as numerical noise and are inevitable.
Our example: The OFDM equalization in the above numerical example would require one complex valued multiplication per subcarrier and symbol (i.e., [math]\displaystyle{ \scriptstyle N \,=\, 1000 }
[/math] complex multiplications per OFDM symbol; i.e., one million multiplications per second, at the receiver). The FFT algorithm requires [math]\displaystyle{ \scriptstyle N \log_2 N \,=\,
10,000 }[/math] [this is imprecise: over half of these complex multiplications are trivial, i.e. = to 1 and are not implemented in software or HW]. complex-valued multiplications per OFDM symbol
(i.e., 10 million multiplications per second), at both the receiver and transmitter side. This should be compared with the corresponding one million symbols/second single-carrier modulation case
mentioned in the example, where the equalization of 125 microseconds time-spreading using a FIR filter would require, in a naive implementation, 125 multiplications per symbol (i.e., 125 million
multiplications per second). FFT techniques can be used to reduce the number of multiplications for an FIR filter-based time-domain equalizer to a number comparable with OFDM, at the cost of
delay between reception and decoding which also becomes comparable with OFDM.
If differential modulation such as DPSK or DQPSK is applied to each subcarrier, equalization can be completely omitted, since these non-coherent schemes are insensitive to slowly changing amplitude
and phase distortion.
In a sense, improvements in FIR equalization using FFTs or partial FFTs leads mathematically closer to OFDM, but the OFDM technique is easier to understand and implement, and the sub-channels can be
independently adapted in other ways than varying equalization coefficients, such as switching between different QAM constellation patterns and error-correction schemes to match individual sub-channel
noise and interference characteristics.
Some of the subcarriers in some of the OFDM symbols may carry pilot signals for measurement of the channel conditions^[23]^[24] (i.e., the equalizer gain and phase shift for each subcarrier). Pilot
signals and training symbols (preambles) may also be used for time synchronization (to avoid intersymbol interference, ISI) and frequency synchronization (to avoid inter-carrier interference, ICI,
caused by Doppler shift).
OFDM was initially used for wired and stationary wireless communications. However, with an increasing number of applications operating in highly mobile environments, the effect of dispersive fading
caused by a combination of multi-path propagation and doppler shift is more significant. Over the last decade, research has been done on how to equalize OFDM transmission over doubly selective
Channel coding and interleaving
OFDM is invariably used in conjunction with channel coding (forward error correction), and almost always uses frequency and/or time interleaving.
Frequency (subcarrier) interleaving increases resistance to frequency-selective channel conditions such as fading. For example, when a part of the channel bandwidth fades, frequency interleaving
ensures that the bit errors that would result from those subcarriers in the faded part of the bandwidth are spread out in the bit-stream rather than being concentrated. Similarly, time interleaving
ensures that bits that are originally close together in the bit-stream are transmitted far apart in time, thus mitigating against severe fading as would happen when travelling at high speed.
However, time interleaving is of little benefit in slowly fading channels, such as for stationary reception, and frequency interleaving offers little to no benefit for narrowband channels that suffer
from flat-fading (where the whole channel bandwidth fades at the same time).
The reason why interleaving is used on OFDM is to attempt to spread the errors out in the bit-stream that is presented to the error correction decoder, because when such decoders are presented with a
high concentration of errors the decoder is unable to correct all the bit errors, and a burst of uncorrected errors occurs. A similar design of audio data encoding makes compact disc (CD) playback
A classical type of error correction coding used with OFDM-based systems is convolutional coding, often concatenated with Reed-Solomon coding. Usually, additional interleaving (on top of the time and
frequency interleaving mentioned above) in between the two layers of coding is implemented. The choice for Reed-Solomon coding as the outer error correction code is based on the observation that the
Viterbi decoder used for inner convolutional decoding produces short error bursts when there is a high concentration of errors, and Reed-Solomon codes are inherently well suited to correcting bursts
of errors.
Newer systems, however, usually now adopt near-optimal types of error correction codes that use the turbo decoding principle, where the decoder iterates towards the desired solution. Examples of such
error correction coding types include turbo codes and LDPC codes, which perform close to the Shannon limit for the Additive White Gaussian Noise (AWGN) channel. Some systems that have implemented
these codes have concatenated them with either Reed-Solomon (for example on the MediaFLO system) or BCH codes (on the DVB-S2 system) to improve upon an error floor inherent to these codes at high
signal-to-noise ratios.^[28]
Adaptive transmission
The resilience to severe channel conditions can be further enhanced if information about the channel is sent over a return-channel. Based on this feedback information, adaptive modulation, channel
coding and power allocation may be applied across all subcarriers, or individually to each subcarrier. In the latter case, if a particular range of frequencies suffers from interference or
attenuation, the carriers within that range can be disabled or made to run slower by applying more robust modulation or error coding to those subcarriers.
The term discrete multitone modulation (DMT) denotes OFDM-based communication systems that adapt the transmission to the channel conditions individually for each subcarrier, by means of so-called
bit-loading. Examples are ADSL and VDSL.
The upstream and downstream speeds can be varied by allocating either more or fewer carriers for each purpose. Some forms of rate-adaptive DSL use this feature in real time, so that the bitrate is
adapted to the co-channel interference and bandwidth is allocated to whichever subscriber needs it most.
OFDM extended with multiple access
OFDM in its primary form is considered as a digital modulation technique, and not a multi-user channel access method, since it is used for transferring one bit stream over one communication channel
using one sequence of OFDM symbols. However, OFDM can be combined with multiple access using time, frequency or coding separation of the users.
In orthogonal frequency-division multiple access (OFDMA), frequency-division multiple access is achieved by assigning different OFDM sub-channels to different users. OFDMA supports differentiated
quality of service by assigning different number of subcarriers to different users in a similar fashion as in CDMA, and thus complex packet scheduling or medium access control schemes can be avoided.
OFDMA is used in:
OFDMA is also a candidate access method for the IEEE 802.22 Wireless Regional Area Networks (WRAN). The project aims at designing the first cognitive radio-based standard operating in the VHF-low UHF
spectrum (TV spectrum).
• the most recent amendment of 802.11 standard, namely 802.11ax, includes OFDMA for high efficiency and simultaneous communication.
In multi-carrier code-division multiple access (MC-CDMA), also known as OFDM-CDMA, OFDM is combined with CDMA spread spectrum communication for coding separation of the users. Co-channel interference
can be mitigated, meaning that manual fixed channel allocation (FCA) frequency planning is simplified, or complex dynamic channel allocation (DCA) schemes are avoided.
Space diversity
In OFDM-based wide-area broadcasting, receivers can benefit from receiving signals from several spatially dispersed transmitters simultaneously, since transmitters will only destructively interfere
with each other on a limited number of subcarriers, whereas in general they will actually reinforce coverage over a wide area. This is very beneficial in many countries, as it permits the operation
of national single-frequency networks (SFN), where many transmitters send the same signal simultaneously over the same channel frequency. SFNs use the available spectrum more effectively than
conventional multi-frequency broadcast networks (MFN), where program content is replicated on different carrier frequencies. SFNs also result in a diversity gain in receivers situated midway between
the transmitters. The coverage area is increased and the outage probability decreased in comparison to an MFN, due to increased received signal strength averaged over all subcarriers.
Although the guard interval only contains redundant data, which means that it reduces the capacity, some OFDM-based systems, such as some of the broadcasting systems, deliberately use a long guard
interval in order to allow the transmitters to be spaced farther apart in an SFN, and longer guard intervals allow larger SFN cell-sizes. A rule of thumb for the maximum distance between transmitters
in an SFN is equal to the distance a signal travels during the guard interval — for instance, a guard interval of 200 microseconds would allow transmitters to be spaced 60 km apart.
A single frequency network is a form of transmitter macrodiversity. The concept can be further used in dynamic single-frequency networks (DSFN), where the SFN grouping is changed from timeslot to
OFDM may be combined with other forms of space diversity, for example antenna arrays and MIMO channels. This is done in the IEEE 802.11 Wireless LAN standards.
Linear transmitter power amplifier
An OFDM signal exhibits a high peak-to-average power ratio (PAPR) because the independent phases of the subcarriers mean that they will often combine constructively. Handling this high PAPR requires:
Any non-linearity in the signal chain will cause intermodulation distortion that
• Raises the noise floor
• May cause inter-carrier interference
• Generates out-of-band spurious radiation
The linearity requirement is demanding, especially for transmitter RF output circuitry where amplifiers are often designed to be non-linear in order to minimise power consumption. In practical OFDM
systems a small amount of peak clipping is allowed to limit the PAPR in a judicious trade-off against the above consequences. However, the transmitter output filter which is required to reduce
out-of-band spurs to legal levels has the effect of restoring peak levels that were clipped, so clipping is not an effective way to reduce PAPR.
Although the spectral efficiency of OFDM is attractive for both terrestrial and space communications, the high PAPR requirements have so far limited OFDM applications to terrestrial systems.
The crest factor CF (in dB) for an OFDM system with n uncorrelated subcarriers is^[29]
[math]\displaystyle{ CF = 10 \log_{10} ( n ) + CF_c }[/math]
where CF[c] is the crest factor (in dB) for each subcarrier. (CF[c] is 3.01 dB for the sine waves used for BPSK and QPSK modulation).
For example, the DVB-T signal in 2K mode is composed of 1705 subcarriers that are each QPSK-modulated, giving a crest factor of 35.32 dB.^[29]
Many PAPR (or crest factor) reduction techniques have been developed, for instance, based on iterative clipping.^[30] Over the years, numerous model-driven approaches have been proposed to reduce the
PAPR in communication systems. In recent years, there has been a growing interest in exploring data-driven models for PAPR reduction as part of ongoing research in end-to-end communication networks.
These data-driven models offer innovative solutions and new avenues of exploration to address the challenges posed by high PAPR effectively. By leveraging data-driven techniques, researchers aim to
enhance the performance and efficiency of communication networks by optimizing power utilization. ^[31]
The dynamic range required for an FM receiver is 120 dB while DAB only require about 90 dB.^[32] As a comparison, each extra bit per sample increases the dynamic range by 6 dB.
Efficiency comparison between single carrier and multicarrier
The performance of any communication system can be measured in terms of its power efficiency and bandwidth efficiency. The power efficiency describes the ability of communication system to preserve
bit error rate (BER) of the transmitted signal at low power levels. Bandwidth efficiency reflects how efficiently the allocated bandwidth is used and is defined as the throughput data rate per hertz
in a given bandwidth. If the large number of subcarriers are used, the bandwidth efficiency of multicarrier system such as OFDM with using optical fiber channel is defined as^[33]
[math]\displaystyle{ \eta = 2 \frac{R_s}{B_\text{OFDM}} }[/math]
where [math]\displaystyle{ R_s }[/math] is the symbol rate in giga-symbols per second (Gsps), [math]\displaystyle{ B_\text{OFDM} }[/math] is the bandwidth of OFDM signal, and the factor of 2 is due
to the two polarization states in the fiber.
There is saving of bandwidth by using multicarrier modulation with orthogonal frequency-division multiplexing. So the bandwidth for multicarrier system is less in comparison with single carrier
system and hence bandwidth efficiency of multicarrier system is larger than single carrier system.
S. no. Transmission type M in M-QAM No. of subcarriers Bit rate Fiber length Received power, at BER of 10^−9 Bandwidth efficiency
1 Single carrier 64 1 10 Gbit/s 20 km −37.3 dBm 6.0000
2 Multicarrier 64 128 10 Gbit/s 20 km −36.3 dBm 10.6022
There is only 1 dB increase in receiver power, but we get 76.7% improvement in bandwidth efficiency with using multicarrier transmission technique.
Idealized system model
This section describes a simple idealized OFDM system model suitable for a time-invariant AWGN channel.
An OFDM carrier signal is the sum of a number of orthogonal subcarriers, with baseband data on each subcarrier being independently modulated commonly using some type of quadrature amplitude
modulation (QAM) or phase-shift keying (PSK). This composite baseband signal is typically used to modulate a main RF carrier.
[math]\displaystyle{ s[n] }[/math] is a serial stream of binary digits. By inverse multiplexing, these are first demultiplexed into [math]\displaystyle{ N }[/math] parallel streams, and each one
mapped to a (possibly complex) symbol stream using some modulation constellation (QAM, PSK, etc.). Note that the constellations may be different, so some streams may carry a higher bit-rate than
An inverse FFT is computed on each set of symbols, giving a set of complex time-domain samples. These samples are then quadrature-mixed to passband in the standard way. The real and imaginary
components are first converted to the analogue domain using digital-to-analogue converters (DACs); the analogue signals are then used to modulate cosine and sine waves at the carrier frequency,
[math]\displaystyle{ f_\text{c} }[/math], respectively. These signals are then summed to give the transmission signal, [math]\displaystyle{ s(t) }[/math].
The receiver picks up the signal [math]\displaystyle{ r(t) }[/math], which is then quadrature-mixed down to baseband using cosine and sine waves at the carrier frequency. This also creates signals
centered on [math]\displaystyle{ 2 f_\text{c} }[/math], so low-pass filters are used to reject these. The baseband signals are then sampled and digitised using analog-to-digital converters (ADCs),
and a forward FFT is used to convert back to the frequency domain.
This returns [math]\displaystyle{ N }[/math] parallel streams, each of which is converted to a binary stream using an appropriate symbol detector. These streams are then re-combined into a serial
stream, [math]\displaystyle{ \hat{s}[n] }[/math], which is an estimate of the original binary stream at the transmitter.
Mathematical description
If [math]\displaystyle{ N }[/math] subcarriers are used, and each subcarrier is modulated using [math]\displaystyle{ M }[/math] alternative symbols, the OFDM symbol alphabet consists of [math]\
displaystyle{ M^N }[/math] combined symbols.
The low-pass equivalent OFDM filter is expressed as:
[math]\displaystyle{ \nu(t) = \sum_{k=0}^{N-1} X_k e^{j2\pi kt/T},\quad 0 \le t \lt T, }[/math]
where [math]\displaystyle{ \{X_k\} }[/math] are the data symbols, [math]\displaystyle{ N }[/math] is the number of subcarriers, and [math]\displaystyle{ T }[/math] is the OFDM symbol time. The
subcarrier spacing of [math]\displaystyle{ \frac{1}{T} }[/math] makes them orthogonal over each symbol period; this property is expressed as:
[math]\displaystyle{ &\frac{1}{T}\int_0^{T}\left(e^{j2\pi k_1 t/T}\right)^* \left(e^{j2\pi k_2t/T}\right)dt \\ {}={} &\frac{1}{T}\int_0^{T} e^{j2\pi\left(k_2 - k_1\right)t/T}dt = \delta_{k_1 k_2}
where [math]\displaystyle{ (\cdot)^* }[/math] denotes the complex conjugate operator and [math]\displaystyle{ \delta\, }[/math] is the Kronecker delta.
To avoid intersymbol interference in multipath fading channels, a guard interval of length [math]\displaystyle{ T_\text{g} }[/math] is inserted prior to the OFDM block. During this interval, a cyclic
prefix is transmitted such that the signal in the interval [math]\displaystyle{ -T_\text{g} \le t \lt 0 }[/math] equals the signal in the interval [math]\displaystyle{ (T - T_\text{g}) \le t \lt T }
[/math]. The OFDM signal with cyclic prefix is thus:
[math]\displaystyle{ \nu(t) = \sum_{k=0}^{N-1}X_k e^{j2\pi kt/T}, \quad -T_\text{g} \le t \lt T }[/math]
The low-pass signal filter above can be either real or complex-valued. Real-valued low-pass equivalent signals are typically transmitted at baseband—wireline applications such as DSL use this
approach. For wireless applications, the low-pass signal is typically complex-valued; in which case, the transmitted signal is up-converted to a carrier frequency [math]\displaystyle{ f_\text{c} }[/
math]. In general, the transmitted signal can be represented as:
[math]\displaystyle{ s(t) &= \Re\left\{\nu(t) e^{j2\pi f_c t}\right\} \\ &= \sum_{k=0}^{N-1}|X_k|\cos\left(2\pi \left[f_\text{c} + \frac{k}{T}\right]t + \arg[X_k]\right) }[/math]
OFDM is used in:
• Digital Radio Mondiale (DRM)
• Digital Audio Broadcasting (DAB)
• Digital television DVB-T/T2 (terrestrial), ATSC 3.0 (terrestrial), DVB-H (handheld), DMB-T/H, DVB-C2 (cable)
• Wireless LAN IEEE 802.11a, IEEE 802.11g, IEEE 802.11n, IEEE 802.11ac, and IEEE 802.11ad
• ADSL (G.dmt/ITU G.992.1)
• LTE and LTE Advanced 4G mobile networks
• DECT cordless phones
• Modern narrow and broadband power line communications^[34]
OFDM system comparison table
Key features of some common OFDM-based systems are presented in the following table.
Standard name DAB Eureka 147 DVB-T DVB-H DTMB DVB-T2 IEEE 802.11a
Year ratified 1995 1997 2004 2006 2007 1999
Frequency range of 174–240, 1,452–1,492 470–862, 174–230 470–862 48–870 4,915–6,100
today's equipment (MHz)
Channel spacing, 1.712 6, 7, 8 5, 6, 7, 8 6, 7, 8 1.7, 5, 6, 7, 8, 10 20
B (MHz)
Mode I: 2k
FFT size, k = 1,024 Mode II: 512 2k, 8k 2k, 4k, 8k 1 (single-carrier) 1k, 2k, 4k, 8k, 16k, 32k 64
Mode III: 256 4k (multi-carrier)
Mode IV: 1k
Mode I: 1,536
Number of non-silent subcarriers, N Mode II: 384 2K mode: 1,705 1,705, 3,409, 6,817 1 (single-carrier) 853–27,841 (1K normal to 32K 52
Mode III: 192 8K mode: 6,817 3,780 (multi-carrier) extended carrier mode)
Mode IV: 768
QPSK,^[35] 16QAM, QPSK,^[35] 16QAM, 4QAM,^[35] 4QAM-NR,^[36] 16QAM, BPSK, QPSK,^
Subcarrier modulation scheme ^π⁄[4]-DQPSK 64QAM 64QAM 32QAM, 64QAM QPSK, 16QAM, 64QAM, 256QAM [35] 16QAM,
Mode I: 1,000
Useful symbol Mode II: 250 2K mode: 224 224, 448, 896 500 (multi-carrier) 112–3,584 (1K to 32K mode on 8 3.2
length, T[U] (μs) Mode III: 125 8K mode: 896 MHz channel)
Mode IV: 500
Additional guard ^1⁄[4], ^1⁄[8], ^ ^1⁄[4], ^1⁄[8], ^ 1/128, 1/32, 1/16, 19/256, 1/8,
interval, T[G]/T[U] 24.6% (all modes) 1⁄[16], ^1⁄[32] 1⁄[16], ^1⁄[32] ^1⁄[4], ^1⁄[6], ^1⁄[9] 19/128, 1/4 ^1⁄[4]
(for 32k mode maximum 1/8)
Subcarrier spacing, Mode I: 1,000
[math]\displaystyle{ \Delta f = \frac Mode II: 4,000 2K mode: 4,464 4,464, 2,232, 1,116 8 M (single-carrier) 279–8,929 (32K down to 1K mode) 312.5 K
{1}{T_U} \approx \frac{B}{N} }[/math] Mode III: 8,000 8K mode: 1,116 2,000 (multi-carrier)
(Hz) Mode IV: 2,000
Net bit rate, 0.576–1.152 4.98–31.67 3.7–23.8 4.81–32.49 Typically 35.4 6–54
R (Mbit/s) (typ. 24.13)
Link spectral efficiency, 0.34–0.67 0.62–4.0 (typ. 3.0) 0.62–4.0 0.60–4.1 0.87–6.65 0.30–2.7
R/B (bit/s·Hz)
Conv. coding with equal error
protection code rates:
^1⁄[4], ^3⁄[8], ^4⁄[9], ^1⁄ Conv. coding
[2], ^4⁄[7], ^2⁄[3], ^3⁄[4], Conv. coding with Conv. coding with with code
^4⁄[5] code rates: code rates: LDPC with code rates: LDPC: ^1⁄[2], ^3⁄[5], ^2⁄[3], rates:
Inner FEC ^1⁄[2], ^2⁄[3], ^ ^1⁄[2], ^2⁄[3], ^ 0.4, 0.6, or 0.8 ^3⁄[4], ^4⁄[5], ^5⁄[6] ^1⁄[2], ^2⁄
Unequal error protection with 3⁄[4], ^5⁄[6], or ^ 3⁄[4], ^5⁄[6], or ^ [3], or ^3⁄
avg. code rates of: 7⁄[8] 7⁄[8] [4]
~0.34, 0.41, 0.50, 0.60, and
Outer FEC Optional RS (120, 110, t = 5) RS (204, 188, t = 8) RS (204, 188, t = 8) BCH code (762, 752) BCH code None
+ MPE-FEC
Maximum travelling 53–185, varies with
speed (km/h) 200–600 transmission
Time interleaving 384 0.6–3.5 0.6–3.5 200–500 Up to 250 (500 with extension
depth (ms) frame)
Adaptive transmission None None None None
Multiple access method None None None None
2–18 Mbit/s Standard Not defined (video: MPEG-2, H.264, H.264 or MPEG2 (audio: AAC HE,
Typical source coding 192 kbit/s MPEG2 Audio layer 2 – HDTV H.264 or MPEG2 H.264 H.265 and/or AVS+; audio: MP2 or DRA Dolby Digital AC-3 (A52), MPEG-2
or AC-3) AL 2)
OFDM is used in ADSL connections that follow the ANSI T1.413 and G.dmt (ITU G.992.1) standards, where it is called discrete multitone modulation (DMT).^[37] DSL achieves high-speed data connections
on existing copper wires. OFDM is also used in the successor standards ADSL2, ADSL2+, VDSL, VDSL2, and G.fast. ADSL2 uses variable subcarrier modulation, ranging from BPSK to 32768QAM (in ADSL
terminology this is referred to as bit-loading, or bit per tone, 1 to 15 bits per subcarrier).
Long copper wires suffer from attenuation at high frequencies. The fact that OFDM can cope with this frequency selective attenuation and with narrow-band interference are the main reasons it is
frequently used in applications such as ADSL modems.
Powerline Technology
OFDM is used by many powerline devices to extend digital connections through power wiring. Adaptive modulation is particularly important with such a noisy channel as electrical wiring. Some medium
speed smart metering modems, "Prime" and "G3" use OFDM at modest frequencies (30–100 kHz) with modest numbers of channels (several hundred) in order to overcome the intersymbol interference in the
power line environment.^[38] The IEEE 1901 standards include two incompatible physical layers that both use OFDM.^[39] The ITU-T G.hn standard, which provides high-speed local area networking over
existing home wiring (power lines, phone lines and coaxial cables) is based on a PHY layer that specifies OFDM with adaptive modulation and a Low-Density Parity-Check (LDPC) FEC code.^[34]
Wireless local area networks (LAN) and metropolitan area networks (MAN)
OFDM is extensively used in wireless LAN and MAN applications, including IEEE 802.11a/g/n and WiMAX.
IEEE 802.11a/g/n, operating in the 2.4 and 5 GHz bands, specifies per-stream airside data rates ranging from 6 to 54 Mbit/s. If both devices can use "HT mode" (added with 802.11n), the top 20 MHz
per-stream rate is increased to 72.2 Mbit/s, with the option of data rates between 13.5 and 150 Mbit/s using a 40 MHz channel. Four different modulation schemes are used: BPSK, QPSK, 16-QAM, and
64-QAM, along with a set of error correcting rates (1/2–5/6). The multitude of choices allows the system to adapt the optimum data rate for the current signal conditions.
Wireless personal area networks (PAN)
OFDM is also now being used in the WiMedia/Ecma-368 standard for high-speed wireless personal area networks in the 3.1–10.6 GHz ultrawideband spectrum (see MultiBand-OFDM).
Terrestrial digital radio and television broadcasting
Much of Europe and Asia has adopted OFDM for terrestrial broadcasting of digital television (DVB-T, DVB-H and T-DMB) and radio (EUREKA 147 DAB, Digital Radio Mondiale, HD Radio and T-DMB).
By Directive of the European Commission, all television services transmitted to viewers in the European Community must use a transmission system that has been standardized by a recognized European
standardization body,^[40] and such a standard has been developed and codified by the DVB Project, Digital Video Broadcasting (DVB); Framing structure, channel coding and modulation for digital
terrestrial television.^[41] Customarily referred to as DVB-T, the standard calls for the exclusive use of COFDM for modulation. DVB-T is now widely used in Europe and elsewhere for terrestrial
digital TV.
The ground segments of the Digital Audio Radio Service (SDARS) systems used by XM Satellite Radio and Sirius Satellite Radio are transmitted using Coded OFDM (COFDM).^[42] The word "coded" comes from
the use of forward error correction (FEC).^[5]
COFDM vs VSB
The question of the relative technical merits of COFDM versus 8VSB for terrestrial digital television has been a subject of some controversy, especially between European and North American
technologists and regulators. The United States has rejected several proposals to adopt the COFDM-based DVB-T system for its digital television services, and for many years has opted to use 8VSB
(vestigial sideband modulation) exclusively for terrestrial digital television.^[43] However, in November 2017, the FCC approved a voluntary transition to ATSC 3.0, a new broadcast standard which is
based on COFDM. Unlike the first digital television transition in America, TV stations will not be assigned separate frequencies to transmit ATSC 3.0 and are not required to switch to ATSC 3.0 by any
deadline. Televisions sold in the U.S. are also not required to include ATSC 3.0 tuning capabilities. Full-powered television stations are permitted to make the switch to ATSC 3.0, as long as they
continue to make their main channel available through a simulcast agreement with another in-market station (with a similar coverage area) through at least November 2022.^[44]
One of the major benefits provided by COFDM is in rendering radio broadcasts relatively immune to multipath distortion and signal fading due to atmospheric conditions or passing aircraft. Proponents
of COFDM argue it resists multipath far better than 8VSB. Early 8VSB DTV (digital television) receivers often had difficulty receiving a signal. Also, COFDM allows single-frequency networks, which is
not possible with 8VSB.
However, newer 8VSB receivers are far better at dealing with multipath, hence the difference in performance may diminish with advances in equalizer design.^[45]
Digital radio
COFDM is also used for other radio standards, for Digital Audio Broadcasting (DAB), the standard for digital audio broadcasting at VHF frequencies, for Digital Radio Mondiale (DRM), the standard for
digital broadcasting at shortwave and medium wave frequencies (below 30 MHz) and for DRM+ a more recently introduced standard for digital audio broadcasting at VHF frequencies. (30 to 174 MHz)
The United States again uses an alternate standard, a proprietary system developed by iBiquity dubbed HD Radio. However, it uses COFDM as the underlying broadcast technology to add digital audio to
AM (medium wave) and FM broadcasts.
Both Digital Radio Mondiale and HD Radio are classified as in-band on-channel systems, unlike Eureka 147 (DAB: Digital Audio Broadcasting) which uses separate VHF or UHF frequency bands instead.
BST-OFDM used in ISDB
The band-segmented transmission orthogonal frequency-division multiplexing (BST-OFDM) system proposed for Japan (in the ISDB-T, ISDB-TSB, and ISDB-C broadcasting systems) improves upon COFDM by
exploiting the fact that some OFDM carriers may be modulated differently from others within the same multiplex. Some forms of COFDM already offer this kind of hierarchical modulation, though BST-OFDM
is intended to make it more flexible. The 6 MHz television channel may therefore be "segmented", with different segments being modulated differently and used for different services.
It is possible, for example, to send an audio service on a segment that includes a segment composed of a number of carriers, a data service on another segment and a television service on yet another
segment—all within the same 6 MHz television channel. Furthermore, these may be modulated with different parameters so that, for example, the audio and data services could be optimized for mobile
reception, while the television service is optimized for stationary reception in a high-multipath environment.
Ultra-wideband (UWB) wireless personal area network technology may also use OFDM, such as in Multiband OFDM (MB-OFDM). This UWB specification is advocated by the WiMedia Alliance (formerly by both
the Multiband OFDM Alliance [MBOA] and the WiMedia Alliance, but the two have now merged), and is one of the competing UWB radio interfaces.
Fast low-latency access with seamless handoff orthogonal frequency-division multiplexing (Flash-OFDM), also referred to as F-OFDM, was based on OFDM and also specified higher protocol layers. It was
developed by Flarion, and purchased by Qualcomm in January 2006.^[46]^[47] Flash-OFDM was marketed as a packet-switched cellular bearer, to compete with GSM and 3G networks. As an example, 450 MHz
frequency bands previously used by NMT-450 and C-Net C450 (both 1G analogue networks, now mostly decommissioned) in Europe are being licensed to Flash-OFDM operators.
In Finland , the license holder Digita began deployment of a nationwide "@450" wireless network in parts of the country since April 2007. It was purchased by Datame in 2011.^[48] In February 2012
Datame announced they would upgrade the 450 MHz network to competing CDMA2000 technology.^[49]
Slovak Telekom in Slovakia offers Flash-OFDM connections^[50] with a maximum downstream speed of 5.3 Mbit/s, and a maximum upstream speed of 1.8 Mbit/s, with a coverage of over 70 percent of Slovak
population. The Flash-OFDM network was switched off in the majority of Slovakia on 30 September 2015.^[51]
T-Mobile Germany used Flash-OFDM to backhaul Wi-Fi HotSpots on the Deutsche Bahn's ICE high speed trains between 2005 and 2015, until switching over to UMTS and LTE.^[52]
American wireless carrier Nextel Communications field tested wireless broadband network technologies including Flash-OFDM in 2005.^[53] Sprint purchased the carrier in 2006 and decided to deploy the
mobile version of WiMAX, which is based on Scalable Orthogonal Frequency-Division Multiple Access (SOFDMA) technology.^[54]
Citizens Telephone Cooperative launched a mobile broadband service based on Flash-OFDM technology to subscribers in parts of Virginia in March 2006. The maximum speed available was 1.5 Mbit/s.^[55]
The service was discontinued on April 30, 2009.^[56]
Vector OFDM (VOFDM)
VOFDM was proposed by Xiang-Gen Xia in 2000 (Proceedings of ICC 2000, New Orleans, and IEEE Trans. on Communications, Aug. 2001) for single transmit antenna systems. VOFDM replaces each scalar value
in the conventional OFDM by a vector value and is a bridge between OFDM and the single carrier frequency domain equalizer (SC-FDE). When the vector size is [math]\displaystyle{ 1 }[/math], it is OFDM
and when the vector size is at least the channel length and the FFT size is [math]\displaystyle{ 1 }[/math], it is SC-FDE.
In VOFDM, assume [math]\displaystyle{ M }[/math] is the vector size, and each scalar-valued signal [math]\displaystyle{ X_n }[/math] in OFDM is replaced by a vector-valued signal [math]\displaystyle{
{\bf X}_n }[/math]of vector size [math]\displaystyle{ M }[/math], [math]\displaystyle{ 0\leq n\leq N-1 }[/math]. One takes the [math]\displaystyle{ N }[/math]-point IFFT of [math]\displaystyle{ {\bf
X}_n, 0 \leq n \leq N - 1 }[/math], component-wisely and gets another vector sequence of the same vector size [math]\displaystyle{ M }[/math], [math]\displaystyle{ {\bf x}_k, 0 \leq k \leq N - 1 }[/
math]. Then, one adds a vector CP of length [math]\displaystyle{ \Gamma }[/math] to this vector sequence as
[math]\displaystyle{ {\bf x}_0, {\bf x}_1, ..., {\bf x}_{N-1}, {\bf x}_0, {\bf x}_1, ..., {\bf x}_{\Gamma-1} }[/math].
This vector sequence is converted to a scalar sequence by sequentializing all the vectors of size [math]\displaystyle{ M }[/math], which is transmitted at a transmit antenna sequentially.
At the receiver, the received scalar sequence is first converted to the vector sequence of vector size [math]\displaystyle{ M }[/math]. When the CP length satisfies [math]\displaystyle{ \Gamma \geq \
left\lceil \frac{L}{M} \right\rceil }[/math], then, after the vector CP is removed from the vector sequence and the [math]\displaystyle{ N }[/math]-point FFT is implemented component-wisely to the
vector sequence of length [math]\displaystyle{ N }[/math], one obtains
[math]\displaystyle{ {\bf Y}_n = {\bf H}_n {\bf X}_n + {\bf W}_n,\,\, 0 \leq n \leq N - 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1) }[/math]
where [math]\displaystyle{ {\bf W}_n }[/math] are additive white noise and [math]\displaystyle{ {\bf H}_n = {\bf H}\mathord\left(\exp\mathord\left(\frac{2\pi jn}{N}\right)\right) = {\bf H}(z)|_{z=\
exp(2\pi j n/N)} }[/math] and [math]\displaystyle{ {\bf H}(z) }[/math] is the following [math]\displaystyle{ M \times M }[/math] polyphase matrix of the ISI channel [math]\displaystyle{ H(z) = \sum_
{k=0}^L h_k z^{-k} }[/math]:
[math]\displaystyle{ \mathbf{H}(z) = \left[ \begin{array}{cccc} H_0(z) & z^{-1} H_{M-1}(z) & \cdots & z^{-1} H_1(z)\\ H_1(z) & H_0(z) & \cdots & z^{-1} H_2(z)\\ \vdots & \vdots & \vdots & \vdots
\\ H_{M-1}(z) & H_{M-2}(z) & \cdots & H_0(z) \end{array}\right] }[/math],
where [math]\displaystyle{ H_m(z) = \sum_l h_{Ml+m}z^{-l} }[/math] is the [math]\displaystyle{ m }[/math]th polyphase component of the channel [math]\displaystyle{ H(z), 0 \leq m \leq M - 1 }[/math].
From (1), one can see that the original ISI channel is converted to [math]\displaystyle{ N }[/math] many vector subchannels of vector size [math]\displaystyle{ M }[/math]. There is no ISI across
these vector subchannels but there is ISI inside each vector subchannel. In each vector subchannel, at most [math]\displaystyle{ M }[/math] many symbols are interfered each other. Clearly, when the
vector size [math]\displaystyle{ M = 1 }[/math], the above VOFDM returns to OFDM and when [math]\displaystyle{ M \gt L }[/math] and [math]\displaystyle{ N = 1 }[/math], it becomes the SC-FDE. The
vector size [math]\displaystyle{ M }[/math] is a parameter that one can choose freely and properly in practice and controls the ISI level. There may be a trade-off between vector size [math]\
displaystyle{ M }[/math], demodulation complexity at the receiver, and FFT size, for a given channel bandwidth.
Note that the length of the CP part in the sequential form does not have to be an integer multiple of the vector size, [math]\displaystyle{ \Gamma M }[/math]. One can truncate the above vectorized CP
to a sequential CP of length not less than the ISI channel length, which will not affect the above demodulation.
Also note that there exist many other different generalizations/forms of OFDM, to see their essential differences, it is critical to see their corresponding received signal equations to demodulate.
The above VOFDM is the earliest and the only one that achieves the received signal equation (1) and/or its equivalent form, although it may have different implementations at transmitter vs. different
IFFT algorithms.
It has been shown (Yabo Li et al., IEEE Trans. on Signal Processing, Oct. 2012) that applying the MMSE linear receiver to each vector subchannel (1), it achieves multipath diversity and/or signal
space diversity. This is because the vectorized channel matrices in (1) are pseudo-circulant and can be diagonalized by the [math]\displaystyle{ M }[/math]-point DFT/IDFT matrix with some diagonal
phase shift matrices. Then, the right hand side DFT/IDFT matrix and the [math]\displaystyle{ k }[/math]th diagonal phase shift matrix in the diagonalization can be thought of the precoding to the
input information symbol vector [math]\displaystyle{ {\bf X}_k }[/math] in the [math]\displaystyle{ k }[/math]th sub vector channel, and all the vectorized subchannels become diagonal channels of
[math]\displaystyle{ M }[/math] discrete frequency components from the [math]\displaystyle{ MN }[/math]-point DFT of the original ISI channel. It may collect the multipath diversity and/or signal
space diversity similar to the precoding to collect the signal space diversity for single antenna systems to combat wireless fading or the diagonal space-time block coding to collect the spatial
diversity for multiple antenna systems. The details are referred to the IEEE TCOM and IEEE TSP papers mentioned above.
OFDM has become an interesting technique for power line communications (PLC). In this area of research, a wavelet transform is introduced to replace the DFT as the method of creating orthogonal
frequencies. This is due to the advantages wavelets offer, which are particularly useful on noisy power lines.^[57]
Instead of using an IDFT to create the sender signal, the wavelet OFDM uses a synthesis bank consisting of a [math]\displaystyle{ N }[/math]-band transmultiplexer followed by the transform function
[math]\displaystyle{ F_n(z) = \sum_{k=0}^{L-1} f_n(k) z^{-k}, \quad 0 \leq n \lt N }[/math]
On the receiver side, an analysis bank is used to demodulate the signal again. This bank contains an inverse transform
[math]\displaystyle{ G_n(z) = \sum_{k=0}^{L-1} g_n(k) z^{-k}, \quad 0 \leq n \lt N }[/math]
followed by another [math]\displaystyle{ N }[/math]-band transmultiplexer. The relationship between both transform functions is
[math]\displaystyle{ f_n(k) &= g_n(L - 1 - k) \\ F_n(z) &= z^{-(L-1)} G_n * (z - 1) }[/math]
An example of W-OFDM uses the Perfect Reconstruction Cosine Modulated Filter Bank (PR-CMFB) and Extended Lapped Transform (ELT) is used for the wavelet TF. Thus, [math]\displaystyle{ \textstyle f_n
(k) }[/math] and [math]\displaystyle{ \textstyle g_n (k) }[/math] are given as
[math]\displaystyle{ f_n (k) &= 2 p_0(k) \cos \left[ \frac{\pi}{N}\left(n + \frac{1}{2}\right)\left(k - \frac{L-1}{2}\right) - (-1)^{n} \frac{\pi}{4} \right] \\ g_n (k) &= 2 p_0(k) \cos \left[ \
frac{\pi}{N}\left(n + \frac{1}{2}\right)\left(k - \frac{L-1}{2}\right) + (-1)^{n} \frac{\pi}{4} \right] \\ P_0(z) &= \sum_{k=0}^{N-1} z^{-k} Y_k\left(z^{2N}\right) }[/math]
These two functions are their respective inverses, and can be used to modulate and demodulate a given input sequence. Just as in the case of DFT, the wavelet transform creates orthogonal waves with
[math]\displaystyle{ \textstyle f_0 }[/math], [math]\displaystyle{ \textstyle f_1 }[/math], ..., [math]\displaystyle{ \textstyle f_{N-1} }[/math]. The orthogonality ensures that they do not interfere
with each other and can be sent simultaneously. At the receiver, [math]\displaystyle{ \textstyle g_0 }[/math], [math]\displaystyle{ \textstyle g_1 }[/math], ..., [math]\displaystyle{ \textstyle g_
{N-1} }[/math] are used to reconstruct the data sequence once more.
Advantages over standard OFDM
W-OFDM is an evolution of the standard OFDM, with certain advantages.
Mainly, the sidelobe levels of W-OFDM are lower. This results in less ICI, as well as greater robustness to narrowband interference. These two properties are especially useful in PLC, where most of
the lines aren't shielded against EM-noise, which creates noisy channels and noise spikes.
A comparison between the two modulation techniques also reveals that the complexity of both algorithms remains approximately the same.^[57]
Other orthogonal transforms
The vast majority of implementations of OFDM use the fast Fourier transform (FFT). However, in principle, any orthogonal transform algorithm could be used instead of the FFT. OFDM systems based,
instead, on the discrete Hartley transform (DHT)^[58] and the wavelet transform have been investigated.
• 1957: Kineplex, multi-carrier HF modem (R.R. Mosier & R.G. Clabaugh)^[59]^[60]
• 1966: Chang, Bell Labs: OFDM paper^[3] and patent^[4]
• 1971: Weinstein & Ebert proposed use of FFT and guard interval^[6]
• 1985: Cimini described use of OFDM for mobile communications
• 1985: Telebit Trailblazer Modem introduced a 512 carrier Packet Ensemble Protocol (18 432 bit/s)
• 1987: Alard & Lasalle: COFDM for digital broadcasting^[9]
• 1988: In September TH-CSF LER, first experimental Digital TV link in OFDM, Paris area
• 1989: OFDM international patent application^[61]
• October 1990: TH-CSF LER, first OFDM equipment field test, 34 Mbit/s in an 8 MHz channel, experiments in Paris area
• December 1990: TH-CSF LER, first OFDM test bed comparison with VSB in Princeton USA
• March 1992: Fattouche and Zaghloul file patent "Method and apparatus for multiple access between transceivers in wireless communications using OFDM spread spectrum" with digital carrier recovery
allowing high speed packet radio and complex randomization reducing the peak to average problem. ^[62]
• December 1991: Fattouche and Zaghloul use large HP equipment to demonstrate 100Mbps wireless LAN.
• September 1992: TH-CSF LER, second generation equipment field test, 70 Mbit/s in an 8 MHz channel, twin polarisations. Wuppertal, Germany
• October 1992: TH-CSF LER, second generation field test and test bed with BBC, near London, UK
• 1993: TH-CSF show in Montreux SW, 4 TV channel and one HDTV channel in a single 8 MHz channel
• 1993: Morris: Experimental 150 Mbit/s OFDM wireless LAN
• February 1994: WiLAN}Wi-LAN Inc. demonstrates 20Mbps wireless WOFDM transceiver operating in the 902-928MHz band.
• 1995: ETSI Digital Audio Broadcasting standard EUreka: first OFDM-based standard
• 1997: ETSI DVB-T standard
• 1998: Magic WAND project demonstrates OFDM modems for wireless LAN
• 1999: IEEE 802.11a wireless LAN standard (Wi-Fi)^[63]
• 2000: Proprietary fixed wireless access (V-OFDM, FLASH-OFDM, etc.)
• May 2001: Wi-LAN Inc. successfully petitioned the FCC to allow OFDM equipment in the 24GHz band.
• May 2001: The FCC allows OFDM in the 2.4 GHz license exempt band.^[64]
• 2002: IEEE 802.11g standard for wireless LAN^[65]
• 2004: IEEE 802.16 standard for wireless MAN (WiMAX)^[66]
• 2004: ETSI DVB-H standard
• 2004: Candidate for IEEE 802.15.3a standard for wireless PAN (MB-OFDM)
• 2004: Candidate for IEEE 802.11n standard for next generation wireless LAN
• 2005: OFDMA is candidate for the 3GPP Long Term Evolution (LTE) air interface E-UTRA downlink.
• 2007: The first complete LTE air interface implementation was demonstrated, including OFDM-MIMO, SC-FDMA and multi-user MIMO uplink^[67]
See also
• N-OFDM
• Orthogonal Time Frequency and Space (OTFS)
• Single-carrier FDMA (SC-FDMA)
• Single-carrier frequency-domain-equalization (SC-FDE)
Further reading
• Bank, M. (2007). "System free of channel problems inherent in changing mobile communication systems". Electronics Letters 43 (7): 401–402. doi:10.1049/el:20070014. Bibcode: 2007ElL....43..401B.
• Bank, Michael; Boris Hill & Miriam Bank et al., "Wireless mobile communication system without pilot signals", US patent 7986740, published 2011-07-26
External links
Original source: https://en.wikipedia.org/wiki/Orthogonal frequency-division multiplexing. Read more | {"url":"https://handwiki.org/wiki/Orthogonal_frequency-division_multiplexing","timestamp":"2024-11-15T02:37:36Z","content_type":"text/html","content_length":"245020","record_id":"<urn:uuid:981df56b-2618-453d-856c-bc77df223d77>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00818.warc.gz"} |
What kind of math is liberal arts?
Within the Liberal Arts Math 1 course, students will explore basic algebraic fundamentals such as evaluating, creating, solving and graphing linear, quadratic, and polynomial functions. The course
also focuses on skills and methods of linear, quadratic, coordinate, and plane geometry.
Is a liberal arts degree useless?
Are liberal arts degree worth it or worthless? As at least a few graduates have found, liberal arts degrees sometimes fail to provide specialization for specific careers. They are forced to find
employment in unconnected fields such as real estate or sales.
Is liberal arts a good degree?
To answer the question, is a liberal arts degree worth it? Yes! Compared to a STEM or career-track degree, however, liberal arts students may need a bit of extra support communicating their skills
and aligning their interests with concrete job opportunities.
Is liberal arts math easier than college algebra?
First, there is no single course called Liberal Arts Math. The general idea is, however, that a Liberal Arts Math course will present topics that are more interesting to a non-science, non-business
student and, therefore, they will find the course more engaging than an algebra course.
What is college math for liberal arts?
MATH 105: Mathematics for Liberal Arts Students Includes various topics such as statistics, geometry, set theory, logic, and finance. Designed for non-STEM majors who do not need to take
What kind of jobs can you get with a liberal arts degree?
A bachelor’s degree in Liberal Studies can be used as preparation for several different careers, including:
• Editor.
• Journalist.
• Publicist.
• Entry-level Management Personnel.
• Social Services Human Relations Officer.
• Para-Professional Librarian.
• Policy Analyst.
• Minister.
What is the difference between a liberal arts degree and a General Education Degree?
A liberal arts education emphasizes dynamism and diversity. Rather than specialization in a sole field or skill set, your undergraduate experience at an LAC or at a school that features general
education requirements will include exposure to a wide range of topics beyond what’s directly relevant to your major.
Is a liberal arts degree a BA or BS?
The BA (Bachelor of Arts) degree is the principal liberal arts degree. The BS (Bachelor of Science) degree is offered in Computer Science, Mathematics, Psychology, Statistics, and each of the natural
sciences. In contrast to the BA, one earns, for example, a BS in Astrophysics.
What are the most hated subjects?
On the other hand, math was also the most disliked subject at 24.0%, followed by Japanese and physical education….Elementary and Junior High Students’ Most Liked and Disliked Subjects.
Liked Disliked
1 Math Math
2 Physical Education Japanese
3 Arts and Crafts Physical Education
4 Japanese Social Studies
Is math a liberal art?
In the medieval era, scholars divided the seven liberal arts into the trivium — grammar, logic, and rhetoric — and the quadrivium — mathematics, music, geometry, and astronomy. Today, liberal arts
includes majors in the humanities, arts, social sciences, and natural sciences.
How hard is college algebra?
College algebra is usually a pre-requisite for higher level math courses and science degrees. Although it can be a little bit tricky, mastering these concepts is necessary to moving forward in math.
There is no fast and simple way to pass college algebra.
Which liberal arts degree is the best?
1. Economist. Of the liberal arts disciplines, the field with the most potential for compensation close to the better paid professions like finance, law, medicine, or technology, Economics is the
highest performer.
Which is easier college math or college algebra?
College Algebra is often easier for students who have just taken an Algebra course. But if you don’t have recent experience in Algebra, you will probably pass College Mathematics more easily. Both
exams are doable if you study. The math CLEP subjects have more free resources than other subjects.
Is it hard to get a job with a liberal arts degree?
A common knock against liberal arts degrees is that they lack overall value and don’t easily lead to job opportunities. Some colleges and universities have retreated from liberal arts by cutting such
programs. But despite dwindling support at some schools, liberal arts advocates are all in.
What is the most boring subject?
6 Most Boring Subjects in the World Made Interesting with…
• ‘Only the boring will be bored’
• Subject number one on the list is Maths.
• Subject number two is Spanish.
• Subject number three is American history.
• Subject number four is Social Studies.
• Subject number five is Physical Education.
• Subject number six is Sex Education.
What is the most fun subject in school?
What is the most interesting subject in school?
• History. The main reason History is the most interesting subject is that the teacher is extremely nice.
• P.E. PE is ace, you get to run, play games and sports, jump and throw stuff.
• Music. I’m in choir and it sucks.
• English.
• Drama.
• Information Technology.
• Science.
• Math.
Why do they call it liberal arts?
The liberal in liberal arts, a cornerstone of the education of so many, has very little to do with political leanings; its roots can be traced to the Latin word liber, meaning “free, unrestricted.”
Our language took the term from the Latin liberales artes, which described the education given to freeman and members of … | {"url":"https://www.blfilm.com/2021/12/17/what-kind-of-math-is-liberal-arts/","timestamp":"2024-11-08T20:34:12Z","content_type":"text/html","content_length":"70582","record_id":"<urn:uuid:4290bc12-539b-456e-8772-011122045b24>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00757.warc.gz"} |
Why is losing $10 worse than winning $10 is good?
Losses loom larger than gains.
This useful mnemonic describes an odd experimental finding: if you have people rate on a scale of 1 to 10 how unhappy they would be to lose $100, that rating will be higher than if you ask them how
happy they would be to win $100. Similarly, people tend to be reluctant to gamble when the odds are even (50% chance of winning $100, 50% chance of losing $100). Generally, if odds are even, people
aren't likely to bet unless the potential prize is greater than the potential loss.
This is a well-known phenomenon in psychology and economics. It is particularly surprising, because simple statistical analysis would suggest that losses and gains should be treated equally. That is,
if you have a 50% chance of winning $100 and a 50% chance of losing $100, on average you will break even. So why not gamble?
(Yes, it is true that people play slot machines or buy lottery tickets, in which, on average, you lose money. That's a different phenomenon that I don't completely understand. When/if I do, I'll
write about it.)
A question that came up recently in a conversation is: why aren't people more rational? Why don't they just go with the statistics?
I imagine there have been papers written on the subject, and I'd love to get some comments referring me to them. Unfortunately, nobody involved in this conversation knew of said papers, so I actually
did some quick-and-dirty simulations to investigate this problem.
Here is how the simulation works: each "creature" in my simulation is going to play a series of games in which they have a 50% chance of winning food and a 50% chance of losing food. If they run out
of food, they die. The size of the gain and the size of the loss are each chosen randomly. If the ratio of gain to loss is large enough, the creature will play.
For some of the creatures, losses loom larger than gains. That is, they won't play unless the gain is more than 1.5 times larger than the loss (50% chance of winning 15.1 units of food, 50% chance of
losing 10). Some of the creatures treat gains and losses roughly equally, meaning they will play as long as the gain is at least a sliver larger than the loss (50% chance of winning 10.1 units of
food, 50% chance of losing 10). Some of the creatures weigh gains higher than losses and will accept any gamble as long as the gain is at least half the size of the loss (50% chance of winning 5.1
unites of food, 50% chance of losing 10).
(Careful observers will note that all these creatures are biased in favor of gains. That is, there is always some bet that is so bad the creature won't take it. There are never any bets so good that
the creature refuses. They just differ in how biased they are.)
Each creature plays the game 1000 times, and there are 1000 creatures. They all start with 100 units of food.
In the first simulation, the losses and gains were capped at 10 units of food, or 10% of the creature's starting endowment, with an average of 5 units. Here's how the creatures faired:
Losses loom larger than gains:
0% died.
807 = average amount of food at end of simulation.
Losses roughly equal to gains:
0% died.
926 = average amount of food at end of simulation.
Gains loom larger than losses:
2% died.
707 = average amount of food at end of simulation.
So this actually suggests that the best strategy in this scenario would be to treat losses and gains similarly (that is, act like a statistician -- something humans don't do). However, the average
loss and gain was only 5 units of food (5% of the starting endowment), and the maximum was 10 units of food. So none of these gambles were particularly risky, and maybe that has something to do with
it. So I ran a second simulation with losses and gains capped at 25 units of food, or 25% of the starting endowment:
Losses loom larger than gains:
0% died
1920 = average amount of food at end of simulation
Losses roughly equal to gains:
1% died
2171 = average amount of food at end of simulation
Gains loom larger than losses:
14% died
1459 = average amount of food at end of simulation
Now, we see that the statistician's approach still leads to more food on average, but there is some chance of starving to death, making weighing losses greater than gains seem like the safest option.
You might not get as rich, but you won't die, either.
This is even more apparent if you up the potential losses and gains to a maximum of 50 units of food each (50% of the starting endowment), and an average of 25 units:
Losses loom larger than gains:
1% died.
3711 = average amount of food at end of simulation
Losses equal to gains
9% died
3941 = average amount of food at end of simulation
Gains loom larger than losses
35% died.
2205 = average amount of food at end of simulation
Now, weighing losses greater than gains really seems like the best strategy. Playing the statistician will net you 6% more food on average, but it also increases your chance of dying by 9! (The
reason that the statistician ends up with more food on average is probably because the conservative losses-loom-larger-than-gains creatures don't take as many gambles and thus have less opportunity
to win.)
So what does this simulation suggest? It suggests that when the stakes are high, it is better to be conservative and measure what you might win by what you might lose. If the stakes are low, this is
less necessary. Given that humans tend to value losses higher than gains, this suggests that we evolved mainly to think about risks with high stakes.
Of course, that's all according to what is a very, very rough simulation. I'm sure there are better ones in the literature, but it was useful to play around with the parameters myself.
2 comments:
I don't know if this is something that you'll eventually publish or continue to work on... But, one comment: I think this simulation and your write-up of it would have been a great place to use
some visuals, a la Tufte. I hate reading large blocks of text, and I admit, I'm lazy and like pictures. Illustrations would help show the information better, I think.
(I recognize that this would be a lot of work... but it's just a thought.)
You're probably already aware of this but even if dying is not an option preferences will depend on the level of the endowment; utility is not linear i.e. diminishing marginal returns kick in, so
the 12th hot dog is less pleasurable than the first even if you weren't hungry to start with.
The literature suggests that the above reasoning i.e. diminishing marginal utility can not explain the 'losses loom larger than gains' phenomonan.For an interesting discussion see Rabin's article
"Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion". Link below. | {"url":"http://gameswithwords.fieldofscience.com/2008/05/why-is-losing-10-worse-than-winning-10.html?showComment=1255440368366","timestamp":"2024-11-14T18:33:12Z","content_type":"application/xhtml+xml","content_length":"166795","record_id":"<urn:uuid:2064d91a-8e25-4a28-94bc-8764fba7db0b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00195.warc.gz"} |
Transactions Online
Hiroki WAKATSUCHI, Stephen GREEDY, John PAUL, Christos CHRISTOPOULOS, "Efficient Modelling Method for Artificial Materials Using Digital Filtering Techniques and EMC Applications" in IEICE
TRANSACTIONS on Communications, vol. E93-B, no. 7, pp. 1760-1767, July 2010, doi: 10.1587/transcom.E93.B.1760.
Abstract: This paper demonstrates an efficient modelling method for artificial materials using digital filtering (DF) techniques. To demonstrate the efficiency of the DF technique it is applied to an
electromagnetic bandgap (EBG) structure and a capacitively-loaded loop the so-called, CLL-based metamaterial. Firstly, this paper describes fine mesh simulations, in which a very small cell size
(0.10.10.1 mm^3) is used to model the details of an element of the structures to calculate the scattering parameters. Secondly, the scattering parameters are approximated with Padé forms and then
factorised. Finally the factorised Padé forms are converted from the frequency domain to the time domain. As a result, the initial features in the fine meshes are effectively embedded into a
numerical simulation with the DF boundary, in which the use of a coarse mesh is feasible (1,000 times larger in the EBG structure simulation and 680 times larger in the metamaterial simulation in
terms of the volumes). By employing the coarse mesh and removal of the dielectric material calculations, the heavy computational burden required for the fine mesh simulations is mitigated and a fast,
efficient and accurate modelling method for the artificial materials is achieved. In the case of the EBG structure the calculation time is reduced from 3 hours to less than 1 minute. In addition,
this paper describes an antenna simulation as a specific application example of the DF techniques in electromagnetic compatibility field. In this simulation, an electric field radiated from a dipole
antenna is enhanced by the DF boundary which models an artificial magnetic conductor derived from the CLL-based metamaterial. As is shown in the antenna simulation, the DF techniques model
efficiently and accurately large-scale configurations.
URL: https://global.ieice.org/en_transactions/communications/10.1587/transcom.E93.B.1760/_p
author={Hiroki WAKATSUCHI, Stephen GREEDY, John PAUL, Christos CHRISTOPOULOS, },
journal={IEICE TRANSACTIONS on Communications},
title={Efficient Modelling Method for Artificial Materials Using Digital Filtering Techniques and EMC Applications},
abstract={This paper demonstrates an efficient modelling method for artificial materials using digital filtering (DF) techniques. To demonstrate the efficiency of the DF technique it is applied to an
electromagnetic bandgap (EBG) structure and a capacitively-loaded loop the so-called, CLL-based metamaterial. Firstly, this paper describes fine mesh simulations, in which a very small cell size
(0.10.10.1 mm^3) is used to model the details of an element of the structures to calculate the scattering parameters. Secondly, the scattering parameters are approximated with Padé forms and then
factorised. Finally the factorised Padé forms are converted from the frequency domain to the time domain. As a result, the initial features in the fine meshes are effectively embedded into a
numerical simulation with the DF boundary, in which the use of a coarse mesh is feasible (1,000 times larger in the EBG structure simulation and 680 times larger in the metamaterial simulation in
terms of the volumes). By employing the coarse mesh and removal of the dielectric material calculations, the heavy computational burden required for the fine mesh simulations is mitigated and a fast,
efficient and accurate modelling method for the artificial materials is achieved. In the case of the EBG structure the calculation time is reduced from 3 hours to less than 1 minute. In addition,
this paper describes an antenna simulation as a specific application example of the DF techniques in electromagnetic compatibility field. In this simulation, an electric field radiated from a dipole
antenna is enhanced by the DF boundary which models an artificial magnetic conductor derived from the CLL-based metamaterial. As is shown in the antenna simulation, the DF techniques model
efficiently and accurately large-scale configurations.},
TY - JOUR
TI - Efficient Modelling Method for Artificial Materials Using Digital Filtering Techniques and EMC Applications
T2 - IEICE TRANSACTIONS on Communications
SP - 1760
EP - 1767
AU - Hiroki WAKATSUCHI
AU - Stephen GREEDY
AU - John PAUL
AU - Christos CHRISTOPOULOS
PY - 2010
DO - 10.1587/transcom.E93.B.1760
JO - IEICE TRANSACTIONS on Communications
SN - 1745-1345
VL - E93-B
IS - 7
JA - IEICE TRANSACTIONS on Communications
Y1 - July 2010
AB - This paper demonstrates an efficient modelling method for artificial materials using digital filtering (DF) techniques. To demonstrate the efficiency of the DF technique it is applied to an
electromagnetic bandgap (EBG) structure and a capacitively-loaded loop the so-called, CLL-based metamaterial. Firstly, this paper describes fine mesh simulations, in which a very small cell size
(0.10.10.1 mm^3) is used to model the details of an element of the structures to calculate the scattering parameters. Secondly, the scattering parameters are approximated with Padé forms and then
factorised. Finally the factorised Padé forms are converted from the frequency domain to the time domain. As a result, the initial features in the fine meshes are effectively embedded into a
numerical simulation with the DF boundary, in which the use of a coarse mesh is feasible (1,000 times larger in the EBG structure simulation and 680 times larger in the metamaterial simulation in
terms of the volumes). By employing the coarse mesh and removal of the dielectric material calculations, the heavy computational burden required for the fine mesh simulations is mitigated and a fast,
efficient and accurate modelling method for the artificial materials is achieved. In the case of the EBG structure the calculation time is reduced from 3 hours to less than 1 minute. In addition,
this paper describes an antenna simulation as a specific application example of the DF techniques in electromagnetic compatibility field. In this simulation, an electric field radiated from a dipole
antenna is enhanced by the DF boundary which models an artificial magnetic conductor derived from the CLL-based metamaterial. As is shown in the antenna simulation, the DF techniques model
efficiently and accurately large-scale configurations.
ER - | {"url":"https://global.ieice.org/en_transactions/communications/10.1587/transcom.E93.B.1760/_p","timestamp":"2024-11-05T06:28:52Z","content_type":"text/html","content_length":"68268","record_id":"<urn:uuid:490cd6e4-c6de-4904-9489-4276b18a439c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00614.warc.gz"} |
Welcome to the ReproHack Hub
• Authors: R M G M Trines, A P L Robinson, J R Wilkinson, J N Kirk, D S Hills, R M Deas, S Morris, T Goffrey, K Bennett, T D Arber
Why should we attempt to reproduce this paper?
Most electron beam physics is considered in the context of a vacuum, but there are applications to long-range electron beam transmission in air. As particle acceleration sources become more
compact, we may have the chance to take particle beams out to the real world. The example provided in the paper describes that of x-ray backscatter detectors, where significantly stronger signals
could be achieved by scanning objects with electron beams. This paper forms the basis for a potential new mode of particle-beam research, and it is important to ensure the reproducibility of this
work for groups who wish to explore the applications of this new technology.
• Authors: Andrij Vasylenko, Jamie Wynn, Paulo Medeiros, Andrew J Morris, Jeremy Sloan, David Quigley
Mean reproducibility score: 5.0/10 | Number of reviews: 2 Why should we attempt to reproduce this paper?
DFT calculations are in principle reproducible between different codes, but differences can arise due to poor choice of convergence tolerances, inappropriate use of pseudopotentials and other
numerical considerations. An independent validation of the key quantities needed to compute electrical conductivity would be valuable. In this case we have published our input files for
calculating the four quantities needed to parametrise the transport simulations from which we compute the electrical conductivity. These are specifically electronic band structure, phonon
dispersions, electron-phonon coupling constants and third derivatives of the force constants. Each in turn in more sensitive to convergence tolerances than the last, and it is the final quantity
on which the conclusions of the paper critically depend. Reference output data is provided for comparison at the data URL below. We note that the pristine CNT results (dark red line) in figure 3
are an independent reproduction of earlier work and so we are confident the Boltzmann transport simulations are reproducible. The calculated inputs to these from DFT (in the case of Be
encapsulation) have not been independently reproduced to our knowledge.
• Authors: Malkiel, I., Mrejen, M., Nagler, A. et al.
Why should we attempt to reproduce this paper?
The current code is written in Torch, which is no longer actively maintained. Since deep learning in nanophotonics is an area of active interest (e.g. for the design of new metamaterials), it is
important to update the code to use a more modern deep learning library such as tensorflow/keras
• Authors: Schneider PP, Smith RA, Bullas AM, Bayley T, Haake SS, Brennan A, Goyder E
Mean reproducibility score: 7.0/10 | Number of reviews: 3 Why should we attempt to reproduce this paper?
If all went right, the analysis should be fully reproducible without the need to make any adjustments. The paper aims to find optimal locations for new parkruns, but we were not 100% sure how
'optimal' should be defined. We provide a few examples, but the code was meant to be flexible enough to allow potential decision makers to specify their own, alternative objectives. The spatial
data set is also quite interesting and fun to play around with. Cave: The full analysis takes a while to run (~30+ min) and might require >= 8gb ram. | {"url":"https://www.reprohack.org/paper/?search=&tags=tensorflow,Archaeology,GDAL,,Electron%20Transport","timestamp":"2024-11-04T18:49:41Z","content_type":"text/html","content_length":"68057","record_id":"<urn:uuid:343186e6-70dd-4a8b-8197-a03a1b9849ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00459.warc.gz"} |
Cryptography for you and me
This article is for the everyday Internet user to demystify cryptography and take a peek behind the curtains to understand basic cryptography, which has become an indispensable aspect of our everyday
lives. From safeguarding our online communications and financial transactions to protecting our sensitive data, cryptography plays a vital role in ensuring our privacy and security online. However,
for the uninitiated, the world of cryptography often seems like a complex and academic realm reserved exclusively for the tech-savvy elite. While there may be some truth to that, it's not essential
for everyone to delve into the nitty-gritty details.
Cryptography on the web can be thought of as locks on our doors. Locks are amazing! A locked door is significantly harder to open than an unlocked door. But if you have the right key, then suddenly a
locked door is easy to open. A lock doesn’t know me. It only knows the key, so anyone I give a key to can open the lock just as easily. No lock is infinitely secure. With enough effort, any lock can
be broken or opened. While I'm sure locks are complicated with all pins and bolts on the inside, I don't necessarily need to understand the mechanics to build an intuition around locks or to use
them. Cryptography is how we make digital locks. Instead of mechanical engineering, cryptography uses mathematics to build the locks. And just as we didn’t need to understand the internal mechanics
of physical locks, we don’t need to understand the internal mathematics of cryptography to build intuition around it. Different cryptographic systems are used for different purposes, and this article
describes the various cryptographic methods used today and their applications.
Symmetric Encryption
Symmetric encryption is a method that encrypts and decrypts a message using the same key, which needs to be kept secret.
Symmetric encryption and decryption
Some of symmetric encryption ciphers are described below:
• Substitution Cipher: Substitution cipher works by substituting characters by other characters. In the below example, every letter in the plain text is substituted by the next letter in the
alphabetic order. 'h' is changed to 'i', 'e' to 'f' and so on. The key here is 1, as the characters are shifted by one place in the alphabetic order.
hello world -> ifmmp xpsme
• Permutation Cipher: Permutation cipher works by changing the position of characters in the given message. In this example, the plain text is spelled out diagonally down and up over a number of
rows and then read off row-by-row. The key is the number of rows, which in this case is 2.
hello world -> hlowrldel ol
h l o w r d
e l _ o l
However, these ciphers alone are not very secure. Also, combining multiple substitution ciphers together results in just another substitution cipher, and hence such a combination does not increase
security. Similarly, combining multiple permutation ciphers results in just another permutation cipher, and hence such a combination does not increase security either. However, combining substitution
ciphers with permutation ciphers results in ciphers that are much harder to break than the individual ciphers. These are called product ciphers.
While ignoring the mathematics of encryption algorithms, a problem does appear as just repeating encryption multiple times using the same key (as we have done above) is dangerous as a malicious user
can study them to find patterns in them. This problem is solved by dividing the message into separate blocks and making each block’s encrypted value depend on all the previous blocks. Also, the key
is expanded sufficiently such that a different part of the key can be put in at different rounds of substitution and permutation.
Figure showing multiple substitution and permutation rounds on different blocks. S refers to substitution block, P refers to Permutation block and K[x] refers to the sub keys derived from the main
Still, however, if the same message is to be encrypted twice, it would provide the exact same encrypted text output, which may leak information. To subvert this, encryption algorithms ask you to
enter a random value in the beginning, which will be disregarded during decryption later. This will ensure that the same plain text message will be encrypted to different cipher texts, making it
harder to conduct analysis on resulting cipher texts.
The goal of substitution–permutation networks is to achieve good diffusion and confusion. Diffusion means that changing a single bit of the clear text should change (statistically) half of the bits
in the cipher text. In other words, even small changes of the clear text lead to drastic changes of the cipher text. Confusion means that every bit of the cipher text should depend on several bits of
the key. This obscures the connections between the two.
The Advanced Encryption Standard (AES) is a widely used symmetric encryption algorithm that works in this way. The advantage of this type of encryption is that it is easy to set up and implement. CPU
nowadays have AES encryption capabilities built into the CPU hardware itself, which make them ridiculously fast and secure. AES is used by BitLocker on Windows to encrypt hard drive to prevent data
leaks when the machine is stolen, it is used for storing encrypted backups and to store data at rest in servers.
The security of AES depends on the key, so it is necessary to use long keys. AES can be used with keys of 128 bits, 192 bits and 256 bits. The higher the number of bits, the more secure it is. The
longer the key, the harder it is for an attacker to guess via brute force attack. However, there is nothing to worry about if your browser is using AES with just 128-bit keys because even a 128-bit
key is secure against attack by modern technology. The largest computational power according to today's standards would take over 70,000,000,000,000,000,000,000,000 years to crack a single AES-128
key. Recently, the threat of quantum computing to cryptography has been well-publicized. Quantum computers work very differently than classical ones, and quantum algorithms can make attacks against
cryptography much more efficient. Quantum computers decrease the effective key length of a symmetric encryption algorithm by half, so AES-128 has an effective key space of 64 bits and AES-256 has an
effective key space of 128 bits. With the right quantum computer, AES-128 would take about 2.61×10^12 years to crack, while AES-256 would take 2.29×10^32 years. For reference, the universe is
currently about 1.38×10^10 years old, so cracking AES-128 with a quantum computer would take about 200 times longer than the universe has existed.
Asymmetric Encryption
Asymmetric encryption (also known as public key cryptography) works with a pair of keys: a public key used for encryption and a private key used for decryption. This is used mainly between two
parties, a sender and a receiver. Notice that we cannot use symmetric encryption like in the previous example where we used a single key for both encryption and decryption. This is because when two
parties are communicating, it is difficult to establish a common secret securely (over the internet for example) because anyone can see what's being communicated for deciding on a common secret key.
In such cases, asymmetric encryption comes in handy. In this method, the receiver of the message generates a pair of key: a public key and a private key. The private key must be kept secure at all
times by the receiver, while the public key can be shared with anyone, even freely over the internet. The sender then uses this public key to encrypt the message that he wants to send to the
receiver. The encrypted message can then be sent over the internet. Remember that only the receiver has access to his private key, which can be used to decrypt the message. Therefore, anyone can
encrypt a message using the public key, but only the holder of the private key can decrypt it back.
Asymmetric encryption and decryption
For example, a journalist can publish the public key of an encryption key pair on a website so that sources can send secret messages to the journalist in cipher text. Only the journalist who knows
the corresponding private key can decrypt the cipher texts to obtain the sources' messages—an eavesdropper reading email on its way to the journalist cannot decrypt the cipher texts. Many protocols
rely on asymmetric cryptography, including SSL and TLS protocols, which make HTTPS possible, which is of utmost importance today for establishing encrypted links between websites and browsers.
One important thing to know about asymmetric encryption algorithms is that they are compute-intensive, and as a result comparatively very slow. That’s the cost of the crazy magic that they do.
Because of this, full messages are rarely encrypted using the public key. Instead, the message is first encrypted using the symmetric algorithm (single key encryption), and then the symmetric key is
encrypted and transmitted using the asymmetric algorithm. This way, only a small key has to be encrypted/decrypted using the slower algorithm. Another important issue is confidence/proof that a
particular public key is authentic, i.e. that it is correct and belongs to the person or entity claimed, and has not been tampered with or replaced by some (perhaps malicious) third party. If Bob
wants to send Alice an encrypted message, Bob first needs to obtain Alice’s public key. If Mallory can interfere in this process and provide his public key instead of Alice’s key, then Mallory will
be able to read the message. Another challenge associated with asymmetric encryption algorithms is the revocation of keys. If for some reason, Alice has lost her private key, then the associated
public key should not be used anymore.
RSA and Elliptic Curve Cryptography (ECC) are the most widely used asymmetric encryption algorithms. These algorithms are based on trapdoor functions. RSA, for example, is based on the process of
multiplication of two prime numbers, which is easy to perform in one direction but much harder to do in reverse. For example, it is trivial to multiply two numbers together: 593 times 829 is 491,597.
But it is hard to start with the number 491,597 and work out which two prime numbers must have been multiplied to produce it. And it becomes increasingly difficult as the numbers get larger, even for
computers of today. Indeed, computer scientists consider it practically impossible for a classical computer to factor numbers that are longer than 2048 bits, which is the basis of the most commonly
used form of RSA encryption.
The security of RSA relies on the problem of factoring very large numbers and ECC depends on calculation of elliptic curve discrete logarithm, both of which can, however, be attacked by quantum
computers. So if a large quantum computer ever gets built, all messages encrypted with RSA/ECC are at risk. To get ahead of this, the cryptographic community has been working towards post-quantum
cryptography to find algorithms that don’t rely on problems that quantum computers can solve easily. The National Institute of Standards and Technology (NIST) of the USA wrote on one of their web
The question of when a large-scale quantum computer will be built is a complicated one. While in the past it was less clear that large quantum computers are a physical possibility, many
scientists now believe it to be merely a significant engineering challenge. Some engineers even predict that within the next twenty or so years sufficiently large quantum computers will be built
to break essentially all public key schemes currently in use. Historically, it has taken almost two decades to deploy our modern public key cryptography infrastructure. Therefore, regardless of
whether we can estimate the exact time of the arrival of the quantum computing era, we must begin now to prepare our information security systems to be able to resist quantum computing.
Hash functions
Hashing is the process of converting data — text, numbers, files, or anything, really — into a fixed-length value of letters and numbers. Data is converted into these fixed-length hash values, by
using a special algorithm called a hash function. The properties of these hash functions are:
1. The hash function should be efficient to compute for arbitrary inputs. For example, they should be able to calculate hash values for a small text file as well as a long movie file relatively
quickly. An effective hashing algorithm quickly processes any data type into a unique hash value.
2. Given a hash value, it should be nearly impossible to find the original message or file that generated this hash value. In fact, if the hashing function can be reversed to recreate the original
input, it’s considered to be compromised. This is one thing that distinguishes hashing from encryption, which is designed to be reversible!
3. No two inputs should generate the same hash value (hash collision). And the same input should always result in the same hash value every time.
hello world -> b94d27b9934d3e08a52e52d7da7dabfac484efe37a5380ee9088f7ace2efcde9
hello worlds -> 8067f1ae16f20dea0b65bfcbd50d59014d143c8ecebab179d923f6ef244b40f8
In the above examples, notice that the hash value of the two inputs are completely different even though the inputs themselves are very similar, and that the length of the hash values are exactly the
same. In this example we hashed simple texts that generated two hash values of exact same length. But even if we had hashed a whole 3-hour-long movie, it would have given a unique hash with equal
number of characters in its hash value.
Cryptographic hash function
Hashing is extensively used for storing passwords. The passwords that you set on online platforms are hashed and saved to the databases so that even if the database is somehow compromised, the
attacker cannot recreate your actual password from the hash value. Each time you log in, your password is hashed and compared against the hash stored in the database. Cryptographic hash functions are
also being used in the context of cryptocurrencies such as Bitcoins or more general distributed ledger technology to verify transactions.
When it comes to password hashing, if two users use the exact same password, it will result in the same hash value. This can be exploited by malicious actors seeking to crack passwords, which is why
'salt' is added to passwords before hashing. Salt is a random sequence of numbers that is added to the password prior to hashing, so that even if two users have the same plain text password, the
hashes will look different as the hash is applied to the salt and the password. Each user's salt needs to be different and needs to be stored in the database to consistently apply this hashing
Secure Hashing Algorithm(SHA) is a widely used hashing algorithm. Currently, SHA-256, SHA-512 and SHA-3 are considered industry standards. SHA-1 has been considered largely insecure since the 2000s,
because researchers were able to generate hash collisions, i.e. two inputs that generated the same hash value. A hash algorithm is considered compromised if two different inputs generate the exact
same hash. It is, however, impossible to get back the original input from the generated hash value. This is because the hash is a fixed length string, so the possible combinations of input strings
are greater than the number of possible hashes. Thus it is clear that something is being lost when the hash is computed. Therefore, the information of which input was mapped to the output is lost
Hashed Message Authentication Codes
Hashed Message Authentication Codes (HMACs) are widely used in communication protocols in situations where encryption of the messages is not considered important while message integrity and
authentication of the messages is considered important. For example, if Alice intends to send a file to Bob while ensuring that the file remains unaltered during transit, Alice can include a hash of
the file for Bob to verify. However, an adversarial user could potentially modify the file, compute a new hash based on the altered file, and send it to Bob. From Bob's perspective, when he
independently computes the file's hash, it would match the hash he received.
To address this vulnerability, HMACs introduce a layer of security by combining the hash with a secret key. These HMACs take both the input and your secret key, and then generate a unique hash. In
the event of a malicious user attempting to modify the file, they won't possess the necessary key to generate a valid hash. Consequently, when Bob calculates a hash of the received malicious file
using the key, it will not match the hash computed by the malicious user, as the latter lacks knowledge of the secret key. This allows for the detection of an invalid message. Since the MAC can’t be
spoofed (because the key is secret), you can store the hash next to the file and still trust the authenticity and integrity of your file.
HMAC (with SHA256 or SHA512) is a common algorithm used to combine input data with a key to generate a hash. Note that the message is sent unencrypted with the hash, so while a malicious user can
read the contents of the file, he only can't alter it.
Authenticated Encryption
Now that we have discussed both encryption and integrity, it only seems reasonable to combine them. It is often necessary to combine encryption with authentication of the data for online
communication. Encryption protects the data while message authentication codes (MAC) protect data against attempts to insert, remove, or modify data. Authenticated encryption is a way to transmit
data encrypted while ensuring the integrity of data. In general, authenticated encryption makes use of a encryption algorithm together with a hash function to transmit the encrypted message and the
There are three approaches to authenticated encryption:
Encrypt-then-Mac (EtM)
Encrypt-and-Mac (EaM)
Mac-then-Encrypt (MtE)
In general, Encrypt-then-Mac is preferred by most cryptographers since it protects against chosen cipher text attacks and avoids any confidentiality issues arising from the MAC of the clear text
Digital Signatures
Digital signatures serve the dual purpose of confirming the legitimacy of a message (or document) and ensuring its integrity. They enable a recipient to confirm the identity of the sender, thus the
sender can later not deny that he/she sent the message. Importantly, any attempt to alter the message will result in an invalid signature, providing assurance to recipients that the signature is
authentic and has not been copied from another source.
In the asymmetric encryption section, we saw how we can use public and private keys to encrypt and decrypt messages. These two keys are interchangeable, meaning it is entirely possible to employ the
private key to encrypt a message, which can then be decrypted using the public key. This unique duality forms the foundation of digital signatures, a concept contrasting with the principles of
asymmetric encryption. In the context of digital signatures, if the holder of the private key sends a message encrypted with his private key, the recipient can confidently ascertain the message's
authenticity because of the fact that only the sender possesses the private key, making it impossible for anyone else to generate a message that can be decrypted with the sender's public key. This
validation process is the essence of a digital signature.
We also saw the drawbacks of using asymmetric encryption. These also affect digital signatures in the same way. It is very inefficient to encrypt long messages and files using this scheme. Therefore,
in practice, digital signatures most often work with hashes of documents, i.e., they are indirect signatures. Instead of signing a potential long electronic document, a cryptographic hash is
calculated and then signed with the signer’s private key. The receiver of the document and signature can then verify the signature by obtaining the signer’s public key, calculating the hash of the
document, and comparing the decrypted signature with the locally calculated hash. Previously, we were using hashes to make sure that our data hadn’t changed. But now, by hashing data with our private
key and sharing that hash, we certify to others that we have “signed” the message. Others can verify that the hashes match our public key and can be sure that the message came from us.
Digital Signature
Another drawback of asymmetric encryption was that it is difficult to verify the authenticity of the public key i.e. it belongs to the person or entity claimed, and has not been tampered with or
replaced by some (perhaps malicious) third party. If Mallory creates a key pair and she manages to make Bob believe the public part of the key pair belongs to Alice, then she can send signed messages
under the identity of Alice and Bob will believe them to be authentic. Another problem arises if private keys are leaked or broken (or expired). Such an event can effectively turn all past signatures
useless. So Bob not only needs to trust that Alice’s key is in fact Alice’s key, he also needs to verify at the time he uses the key that the key is still valid and has not been revoked yet.
Also, do not confuse digital signatures, which use cryptographic mechanisms, with electronic signatures, which may just use a scanned signature or a name entered into a digital form. Electronic
signature is pretty pointless from a security point of view since pretty much everybody can learn how to copy a scan of a hand signature into a document!
Digital Certificates
Earlier we ran into the problem of not being able to trust public keys as it is not trivial to guarantee that it actually belonged to the person or entity claimed. The solution to this lies in the
implementation of digital certificates.
A public key certificate is an electronic document used to prove the ownership of a public key, which includes the following:
• information about the public key
• information about the identity of the owner of the key
• information about the lifetime of the certificate
• the digital signature of an entity that has verified the certificate’s contents (called the issuer of the certificate)
If the signature is valid, and the software examining the certificate trusts the issuer of the certificate, then it can trust the public key contained in the certificate to belong to the subject of
the certificate. Obviously, to trust a given certificate, you need to trust the issuer of that certificate. This may require to trust the issuer of the issuer of the certificate and so on. This
results in a chain of trust relationships that must be rooted somewhere.
A public key infrastructure (PKI) is a set of roles, policies, and procedures needed to create, manage, distribute, use, store, and revoke digital certificates and manage public-key encryption. A
central element of a PKI is the certificate authority (CA), which is responsible for storing, issuing and signing digital certificates. CAs are often hierarchically organized. A root CA may delegate
some of the work to trusted secondary CAs if they execute their tasks according to certain rules defined by the root CA. A key function of a CA is to verify the identity of the the owner of a public
key certificate.
This mechanism is widely used on the internet. It's called the X.509 public key certificate format. X.509 certificates are used in many Internet protocols, including TLS/SSL, which is the basis for
HTTPS, the secure protocol for browsing the web. This is used to verify that when I type in www.google.com, the response that I get is actually from Google and not anybody else. The website will
first of all send the signed public key (digital certificate). My browser has a list of CAs that it trusts (and the public keys of those CAs). The browser will then determine if the CA that signed
the certificate of Google lies in the chain of trust of CAs built into the browser. If that is the case, it will use the public key of the CA that signed this certificate to validate the signature of
the certificate. If the signature matches, then the browser trusts the public key of the website. Once this trust is established, asymmetric encryption can allow for a new symmetric key to be
exchanged which will allow for the rest of the communication on the channel to be securely encrypted! This is how the internet works!
Happy hacking! | {"url":"https://peeyushmansingh.com/cryptography-for-you-and-me/","timestamp":"2024-11-02T12:05:57Z","content_type":"text/html","content_length":"53877","record_id":"<urn:uuid:3f8746b9-4287-4e7d-b2c4-878fa7411c61>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00374.warc.gz"} |
Generalized Data Thinning Using Sufficient Statistics
Published in Journal of the American Statistical Association (Theory & Methods), 2024
Abstract: Our goal is to develop a general strategy to decompose a random variable X into multiple independent random variables, without sacrificing any information about unknown parameters. A recent
paper showed that for some well-known natural exponential families, $X$ can be thinned into independent random variables $X^{(1)},\dots,X^{(K)}$, such that $X=\sum_{k=1}^K X^{(k)}$. In this paper, we
generalize their procedure by relaxing this summation requirement and simply asking that some known function of the independent random variables exactly reconstruct $X$. This generalization of the
procedure serves two purposes. First, it greatly expands the families of distributions for which thinning can be performed. Second, it unifies sample splitting and data thinning, which on the surface
seem to be very different, as applications of the same principle. This shared principle is sufficiency. We use this insight to perform generalized thinning operations for a diverse set of families.
The preprint can be downloaded here. | {"url":"https://ameerd.github.io/publications/gdt","timestamp":"2024-11-06T04:05:25Z","content_type":"text/html","content_length":"11286","record_id":"<urn:uuid:027a7d67-31fd-4e3e-89e7-cfb1cb518065>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00517.warc.gz"} |
Knowledge Drop: Gravity - /beer
Welcome to the first part of a new series on beer where I’ll discuss facts, history, and science behind beer. I’ll cover beer terminology, the brewing process, and beer styles, and whatever else
comes to mind. Figured we’d break up the monotony of just reviews. We begin with Gravity.
Hold on tight, ‘cause thar be math ahead!
I’m not talking about why letting go of your beer makes it fall to the ground. I’m talking about the specific gravity of beer. Specific Gravity (SG) refers to how dense a liquid is in relation to
water. Density is basically an object’s mass as it relates to each unit of volume it’s in. So, like if you have a 1lb rock and drop it into a 5-gallon bucket of water (whose volume is roughly 0.77
cubic feet), the rock’s density is 1.3lbs/cu ft. It’s important to know that the temperature of the water affects the water’s own density.
Now, since we’re measuring the SG of our beer in relation to water, the water we’re measuring against will have a SG of 1.000. There are a variety of ways to measure gravity, but the tool Rhea and I
use in our homebrewing is called a hydrometer. It’s a glass tube with measurements on it and a weight inside at the bottom that causes it to float upright. The more it sinks into the beer, the lower
the density of the beer. This is a pretty simple concept to understand. If you drop a baseball into a swimming pool of water, it’ll sink to the bottom, but if you drop that same baseball into a
swimming pool of pudding, it would slowly sink and eventually stop somewhere in the middle, since the pudding is obviously more dense than the water. Most hydrometers are calibrated to measure in
relation to 60° water.
Photo from A Life Content
Specific gravity is measured in two different units these days. Most homebrewers use the ratio of density to water density, often called “brewers points”, which is expressed like 1.052, but many
breweries use degrees Plato, expressed as 14°P. There isn’t a really direct conversion for °P to specific gravity ratios, but the closest approximation is about 1°P is equal to approximately .004
brewers points. So, 14°P is a specific gravity of about 1.056 ((14 x 0.004) + 1, the 1 is the density of water).
So, now we know that specific gravity just tells us how dense beer is in relation to water. Now back to what the hell specific gravity is and why you would even care about it when drinking your beer.
When you brew beer, you have to then ferment it, as you likely know. The mix of grains and hops and water you just boiled is called the wort (pronounced: wert). Once the wort cools, you transfer it
from the kettle to the fermentation vessel, in homebrewing, usually a big glass jug called a carboy. Once in the carboy (and before the yeast is added), you take a sample of the liquid and measure
its specific gravity and record it. This is called the original gravity (OG) (not O.G.). This is so you know how dense your liquid was before fermentation. As the yeast begins to devour the sugar
from the grains in your beer, the beer loses density as there is less stuff in the liquid. The more sugars the yeast eat, the more alcohol they produce.
The beer’s recipe will generally have a target gravity that the brewer will aim for. This is calculated by the size of the batch, the various grains, hops, and other ingredients in the recipe, plus
how much yeast you’ll be adding and the temperature at which it’ll be fermenting. Periodically during the fermentation period, the brewer will take a sample and check the specific gravity of it. Once
the specific gravity stays the same two or so times in a row, fermentation has finished, and hopefully it’s at the intended gravity. This final reading is called the final gravity (FG). It will be a
lower reading than the original gravity since the yeast ate the sugar from the grains.
Now, here’s where it turns into something with which you’re no doubt familiar — the percentage of alcohol by volume, ABV. That number on the bottle that’ll tell you whether you should have one for a
The ABV can be calculated fairly simply by subtracting the FG from the OG. For example, if the original gravity of your beer was 1.056 and the final gravity was 1.01, the difference between them is
0.046, times 131 for a percentage and some for error correction in the non-linear relationship, and you get 6.03%ABV. This isn’t 100% accurate, as temperatures need to be taken into account, but this
is a quick and dirty way to calculate the ABV of a beer.
You may hear people refer to high-ABV beers as “high-gravity” beers. This refers to the high original gravity. When there is more grain and other sugars in the wort of a beer, there is more for the
yeast to consume, giving off more alcohol. The yeast will try to eat all of the sugars, getting the beer’s density closer to that of the water. So, a beer with a lot more grains in it would be more
dense and have a higher gravity.
For instance, we had a fairly high-gravity beer in our mathy example up there, and you can see what a difference a higher gravity makes in alcohol content by increasing that number. The first example
had an OG of 1.056. If you use 1.08 and the same final gravity, you’d have a 9.17%ABV beer, a big beer indeed!
Your typical beer off the shelf will have anywhere between 4.2% and 5.3% ABV, which means it didn’t have a crazy amount of fermentables, like grains or fruit. This also makes the beer cheaper to
produce because you’re using fewer ingredients, hence why a 22oz bottle of a 9% beer might cost you $10+.
So, in layman’s terms: you put a bunch of grain and stuff into water and boil it and then measure the gravity of this new liquid and that’s the original gravity. Then you put that new liquid into a
big ol’ jug to ferment it for a few weeks and check the gravity after a little while and once the gravity stops changing, you now have your final gravity. You do some simple math of subtracting the
original gravity from the final gravity and multiplying that by 131 and now you know your ABV percent for your beer.
Gravity isn’t super exciting, but it is pretty interesting and I only just scraped the surface. Here are a few sites with a ton more information. | {"url":"http://slashbeer.net/post/5443/knowledge-drop-gravity","timestamp":"2024-11-07T21:28:51Z","content_type":"text/html","content_length":"14842","record_id":"<urn:uuid:a4358892-f371-44b1-b8a6-732c6a16a9a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00121.warc.gz"} |
Math Calculators - Math AI
Math Calculators
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Introducing our Zeros of a Polynomial Calculator, a powerful tool designed to help you find … Try Now
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Introducing our Orthogonal Vector Calculator, a helpful tool designed to determine orthogonal vectors with ease. … Try Now | {"url":"https://math-ai.org/math-calculators/page/2/","timestamp":"2024-11-02T00:00:12Z","content_type":"text/html","content_length":"79414","record_id":"<urn:uuid:ad5b3798-ccac-407e-a7cc-d00ec94c7997>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00394.warc.gz"} |
Math, Grade 7, Zooming In On Figures, Area of a Circle
Material Type:
Lesson Plan
Middle School
Media Formats:
Interactive, Text/HTML
Area of a Circle
Lesson Overview
Students will compare the formula for the area of a regular polygon to discover the formula for the area of a circle.
Key Concepts
The area of a regular polygon can be found by multiplying the apothem by half of the perimeter. If a circle is thought of as a regular polygon with many sides, the formula can be applied.
For a circle, the apothem is the radius, and p is C.
$A=a\left(\frac{p}{2}\right)\to A=r\frac{C}{2}\to A=r\frac{\pi d}{2}\to A=r\frac{\pi 2r}{2}\to A=r\pi r=\pi {r}^{2}$
• Derive the formula for the area of a circle.
• Apply the formula to find the area of circles.
SWD: Consider the prerequisite skills for this lesson: understanding and applying the formula for the area of a regular polygon. Students with disabilities may need direct instruction and guided
practice with this skill.
Students should understand these domain-specific terms:
• apothem
• parallelogram
• derivation
• height
• approximate (estimate)
• scatter plot
• pi
• perimeter
• circumference
It may be helpful to preteach these terms to students with disabilities.
Many-Sided Regular Polygons
Lesson Guide
Show students the nested polygons and discuss what shape the polygons are approaching.
SWD: Students with visual spatial difficulties may have trouble differentiating between the shapes that are nested in the opening graphic. If possible, provide discreet images of each shape so that
students can see how the shapes evolve into a circle.
Discuss the opening questions briefly, taking only a few comments and observations—students will answer the questions as they do the activity.
Students will see that the polygons are approaching the shape of a circle. They know how to find the area of a regular polygon, so they could use a many-sided regular polygon to approximate the area
of a circle.
Emphasize that the more sides a regular polygon has, the closer it is to being a circle.
Many-Sided Regular Polygons
• Experiment with the shape of this polygon by adding more sides. As you increase the number of sides, what shape does the polygon get close to?
• Observe what shape the polygon gets close to as its sides increase in number. How do you think your observation would help you figure out a way to find the area of that shape? Explain.
INTERACTIVE: Circle 2
Math Mission
Lesson Guide
Discuss the Math Mission. Students will use what they know about the area of a regular polygon to determine the formula for the area of a circle.
Use the areas of regular polygons to determine the formula for the area of a circle.
The Formula for the Area of a Circle
Lesson Guide
Have students work in groups. Give students time to struggle with the problem before prompting them. Make sure each group has arrived at the formula before they move on to the remaining tasks.
ELL: As mentioned in other lessons, if you hear ELLs say the right things but use the wrong grammatical structure, show signs of agreement and softly rephrase using the correct grammar and the
student’s words as much as possible. If a student says, “The triangles are similar in the two figures, they all have the same size,” you could agree and say, “Yes, they both have 16 congruent
sections that are very close in size.”
Student has difficulty getting started.
• What is the perimeter of a circle called?
• How do you find its length?
• What line in a circle is the same as the apothem?
Student has a solution.
• If you could measure pi more accurately, how much difference do you think that would make in the answer?
Student does not think that the area of a circle can be found accurately because there are no straight sides.
• Would a 1,000-gon with an apothem of 6 ft have the same area as a circle with a radius of 6 ft?
• How much difference would there be in the areas?
• What number should you use for π?
Mathematical Practices
Mathematical Practice 1: Make sense of problems and persevere in solving them.
• Look for students who work out the area formula and can explain it.
Mathematical Practice 6: Attend to precision.
• Look for students who label their answers with the correct square units.
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
• Students should see the similarities between regular polygons and circles.
• Look for students who see that the area of the circle is the same as a regular polygon with a number of sides approaching infinity, and who remember that the interior angle of the polygon is
approaching 180°.
Possible Answers
• They both have 16 congruent sections that are very close in size (comparing the wedges to the triangles). Students may also notice the similarities in the parallelogram dimensions.
• The length along the edge of the parallelogram-like shape is half of the circumference and the height is the radius.
□ To express area in terms of the radius r :
$A=r\left(\frac{\pi d}{2}\right)$ Replace C with πd.
$A=r\left(\frac{\pi 2r}{2}\right)$ Replace d with 2r.
$A=r\pi r$ Divide by 2.
$A=\pi {r}^{2}$$r\cdot r={r}^{2}$
Work Time
The Formula for the Area of a Circle
Set A contains a 16-gon divided into 16 congruent triangles and the same 16-gon rearranged into a parallelogram made up of the 16 congruent triangles. The perimeter of the polygon is labeled p and
the apothem is labeled a.
Set B contains a circle divided into 16 congruent sections and the same circle rearranged into a parallelogram-like shape, which is made up of the 16 congruent sections. The radius is labeled r.
• Look carefully at the two sets of figures. How are they similar?
• What is the length of the parallelogram-like shape formed from the circle? What is the height?
• How could you express the area of the circle in terms of r?
• What is the “apothem” of a circle called?
• What is the perimeter of a circle called?
The Formula for the Area of a Circle
Student has difficulty with Task 4.
• What is the area of each small square? What are the length and width of the small square?
• What is $\frac{3}{4}$ of 4 squares?
Student does not apply the formula correctly.
• Did you use the radius or the diameter?
• Did you remember to square that length?
• What number did you use for π?
Possible Answers
• The area is $4{r}^{2}$. Each small square has an area of ${r}^{2}$.
• $\frac{3}{4}\cdot 4{r}^{2}=3{r}^{2}$
• Pi (π) is a little more than 3, so this is a pretty good approximation of the formula.
• Presentations will vary.
Work Time
The Formula for the Area of a Circle
In the last problem, you used the formula for the area of a polygon to come up with the formula for the area of a circle. The steps that follow will “lead you” to the formula for the area of a circle
in a different way.
• What is the area of the large square?
• If each quarter of the circle takes up about three-fourths of each small square, what is the approximate area of the circle?
• How does your answer compare to the formula you wrote for the area of a circle in the previous problem?
• To find the area of the large square, find the area of each small square.
Prepare a Presentation
Preparing for Ways of Thinking
Look for these types of responses to be shared during the class discussion:
• Students who see the relationship between the polygon and the circle and understand that as more sides are added to the polygon, its area is approaching a circle with the same apothem/radius
• Students who apply the area formula correctly
• Students who see that units need to be square units for area and make the connection with the circle within the square
• Students who can clearly explain the relationship between the circle’s area and the square’s area
Possible Answers
Presentations will vary.
Challenge Problem
The radius is 20 ft. If the area of a circle is about 1,256 ft^2, the radius can be solved for:
$A=\pi {r}^{2}$
Using A ≈ 1,256 and π ≈ 3.14:
$\begin{array}{c}\frac{1,256}{3.14}={r}^{2}\\ 400={r}^{2}\\ r=20\end{array}$
400 = ${r}^{2}$
r = 20
Work Time
Prepare a Presentation
Prepare a presentation about the area of a circle. Use examples of your work to illustrate your explanation.
Challenge Problem
If the area of a circle is approximately 1,256 square feet, what is the radius? (Use 3.14 for π .)
Make Connections
Facilitate the discussion to help students understand the mathematics of the lesson, making sure to address any questions students have from their work. Ask questions such as these:
• How did you discover the formula for the area of a circle?
• How was the formula similar to the formula for the area of a regular polygon?
• How did [student names] organize their thoughts differently? Which way of thinking makes more sense to you? Which way of thinking brought out the structure of the mathematics?
• How did [student names] make sense of the problem?
• Could you state what [student names] said in a different way?
• Why were the measurements in square units?
• If you found the area of the circles from the objects you measured in class, how accurate do you think those measurements would be? How much would the precision of pi help?
ELL: As with other discussions, consider presenting some of the questions in writing to support ELLs. Also consider providing sentence frames, such as the following (in the order the questions were
If possible, provide sentence frames following this format for the remainder of the questions.
• “The way I discovered the formula of the area of the circle is by…” or “I first did…and then I…”
• “The formula was similar to the formula for the area of a regular polygon in that…”
Performance Task
Ways of Thinking: Make Connections
Take notes about other students’ understandings of the area of a circle.
As your classmates present, ask questions such as:
• How is the formula for the area of a circle similar to the formula for the area of a regular polygon?
• How did knowing the formula for the area of a polygon help you come up with the formula for the area of a circle?
• Why do you use square units for area?
Find the Area
• For a circle with a diameter of 10 in.:
d = 2r
$\frac{10}{2}$= r r = 5 $A=\pi {r}^{2}$
A = 3.14(5 ⋅ 5)
A = 3.14(25) = 78.5
The area is 78.5 in.^2.
• For a circle with a radius of 15 cm:
r = 15
$A=\pi {r}^{2}$
A = 3.14(15 ⋅ 15)
A = 3.14(225) = 706.5
The area is 706.5 cm^2.
Remind students that units that are multiplied together are square units. Just as 10⋅10 = 10^2 is a “square” number, centimeters ⋅ centimeters = centimeters^2 is a “square” unit.
Work Time
Find the Area
The area of a circle is equal to π times the radius squared. Approximating π with 3.14 we get the formula:
Find the areas of the following circles:
• A circle with a diameter of 10 in.
• A circle with a radius of 15 cm
Find the Area
• The area of the circular floor is 17,427.785 sq ft.
d = 2r
149 ft = 2r
$\frac{149\text{ft}}{2}$= 2r r = 74.5 ft $A=\pi {r}^{2}$
A = 3.14(74.5 ⋅ 74.5)
A = 3.14(5,550.25) = 17,427.785
Discuss the importance of accuracy: How close does the answer need to be? How much do the significant digits affect the result? 3.1(75 ⋅ 75) = 17,437.5 or 17,437.5 sq ft, a difference of nearly 10 sq
ft. The answer is an approximation regardless of how “accurate” the calculation is. (Is the Pantheon’s diameter exactly 149 ft?) We could say that the area is about 17,500 sq ft.
Work Time
Find the Area
Remember the Pantheon from the first lesson of this unit? The width of the domed part of the Pantheon is 149 feet.
• What is the area of the circular floor (which covers most of the interior of the building)?
Area of a Circle
A Possible Summary
A circle can be thought of as a regular polygon with many, many, very narrow sides. If this polygon is divided into congruent triangles and arranged into a parallelogram, its area will be very close
to the circle’s area. Because of this similarity, the formula
for area of regular polygons can be changed to
$A=\pi {r}^{2}$
for circles.
SWD: Some students may struggle with the task of actually writing a summary of the mathematics from the lesson. Possible supports:
• Prior to writing the summary, encourage students to discuss their ideas with a partner or adult and to rehearse what they might write.
• Allow students to map out their ideas in outline form or in a concept web.
Formative Assessment
Summary of the Math: Area of a Circle
Write a summary about the area of a circle.
Check your summary.
• Do you explain how the formula for the area of a regular polygon is similar to the formula for the area of a circle?
• Do you provide the formula for the area of a circle?
• Do you explain what units are used for the area of a circle?
Reflect On Your Work
Lesson Guide
Have each student write a brief reflection before the end of class. Review the reflections to see students’ strategies for determining the formula for the area of a circle. If you notice insightful
comments about determining the formula for the area of a circle, plan to share them with the class in the next lesson.
Work Time
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
Something else I would like to know about circles is … | {"url":"https://goopennc.oercommons.org/courseware/lesson/5135/overview","timestamp":"2024-11-12T02:13:23Z","content_type":"text/html","content_length":"75140","record_id":"<urn:uuid:a3e08a48-ee6e-4e93-a426-60c8ee7dc0af>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00340.warc.gz"} |
About ARTIFICIATA:
In 1969, I published my first visual artist book "Artificiata" with Edition Agentzia in Paris.^1 It was a visual-poetry book and also my last art work drawn by hand. At the time, I proposed to create
a second Artificiata based on algorithms and calculated and drawn by computer.
Since my visual research around 2012 developed a strong relationship to visual-music in a music score-like flow, similar to Artificiata of 1969, I decided to call this work Artificiata II and at the
same time created a visual-book with the same name. I published Artificiata II in 2014 with OEI Editor in Stockholm ^2, exactly as I had imagined it 44 years earlier.
Publications about Artificiata II:
- Exhibition fold-out catalog, work from Baseline, 2013 ^4
- Artist book, 2014 ^2
- Exhibition catalog, work from Baseline, Projections and Dimensions, Parity, and Traces, 2015 ^5
General Algorithm:
In this work, a "diagonal-path" from a hypercube, randomly chosen between 11 and 15 dimensions, is drawn. A diagonal-path is a multiple-segmented line where each change of direction indicates the
passage through a single dimension (diagonal paths were introduced into my work in "Dimensions I" in 1978). ^3
Horizontal lines are attached to the line at each change of dimension, i.e. the horizontal lines are drawn through the y-value of each vertex of the diagonal path when it is projected into 2-D. The
spaces between the horizontal lines on either side (left/right) of the diagonal-path are filled with distinct sets of randomly chosen colors. The same procedure also calculates lines and colors in
the vertical direction through the x-value of each vertex. The vertical lines are not drawn, but the resulting color sets are retained. This procedure creates four color sets from which three are
randomly chosen to construct the resulting image. By overlaying the color sets successively, unpredictable constellations appear.
The color spaces and horizontal lines move with the structure when the diagonal path (white line) is in slow motion (rotating in hyper-dimensional space and then projected into 2-D), and can be
observed in my real-time computer animation works.
The animation algorithm of P1622 contains random variations of speed and suites of stills, adding a musical rhythm to this work. The works on canvas P1611 are therefore instances from this animation.
The algorithm is described above. This part of Artificiata II was first published in a fold-out catalog in 2013. ^4
Projections and Dimensions:
The animation P1660 shows all the 2-D projections of a randomly chosen n-dimensional diagonal-path between 2-D and 13-D in a cyclic mode. Similar to the rules in 12-tone music, each dimension has to
be selected once before the same dimension can appear again.
Program P1650 shows on paper a complete set of all the 2-D projections of a n-dimensional diagonal-path between 2-D and 13-D. The algorithm of each individual projection in an image is described
Parity - Fracturing n-dimensional diagonal paths:
This work series P1682 shows the fracturing of a diagonal-path into even and odd numbered lines. The algorithm also refers to the procedure described above: relating the attachment of a diagonal-path
to its horizontal lines.
Traces - Capturing the history of n-dimensional rotations:
In Program P2200, the thick white line shows a rotated n-dimensional diagonal-path projected into 2-D and the color lines show the history of that movement restricted to the 2-D rectangular space.
The algorithm also refers to the procedure described above: relating the attachment of a diagonal-path to its horizontal lines.
The animation P2210 shows instances of this program.
1) - Artificiata I, preface by Pierre Barbaud, Edition Agentzia Paris, 1969
2) - Artificiata II, Artist book, afterword by Margit Rosen, OEI Editor Stockholm, 2014
3) - Exhibition Catalog "Dimensions I" (4-dimensional hypercube), Galerie Mueller-Roth, Stuttgart, 1979
4) - Exhibition fold-out catalog, "Artificiata II" Galerie Mueller-Roth Stuttgart, Galerie [DAM] Berlin, 2013
5) - Exhibition catalog, "Manfred Mohr: Artificiata II, works from 2012-2015" (36pp, 41 ill), bitforms gallery, New York, 2015
bitforms gallery, exhibition Nov 8 - Dec 27 2015, New York, NY | {"url":"http://emohr.com/www_artif2/algor.html","timestamp":"2024-11-03T12:59:35Z","content_type":"text/html","content_length":"7681","record_id":"<urn:uuid:ccc62c65-6667-4617-a221-f757b8bfac5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00895.warc.gz"} |
A Tutorial on Cluster Optimized Proximity Scaling (COPS)
In this document we give a high-level, relatively non-technical introduction to the functionality available in the cops package for fitting multidimensional scaling (MDS; Borg & Groenen 2005) models
that have an emphasis on providing a clustered configuration. We start with a short introduction to COPS and the models that we have available. We then explain fitting of these models with the cops
package and show how to fit those. For illustration we use the smacof::kinshipdelta data set (Rosenberg, S. & Kim, M. P., 1975) which lists percentages of how often 15 kinship terms were not grouped
together by college students.
Proximity Scaling
For proximity scaling (PS) or multidimensional scaling (MDS) the input is typically an \(N\times N\) matrix \(\Delta^*=f(\Delta)\), a matrix of proximities with elements \(\delta^*_{ij}\), that is a
function of a matrix of observed non-negative dissimilarities \(\Delta\) with elements \(\delta_{ij}\). \(\Delta^*\) usually is symmetric (but does not need to be). The main diagonal of \(\Delta\) is
0. We call a \(f: \delta_{ij} \mapsto \delta^*_{ij}\) a proximity transformation function. In the MDS literature these \(\delta_{ij}^*\) are often called dhats or disparities. The problem that
proximity scaling solves is to locate an \(N \times M\) matrix \(X\) (the configuration) with row vectors \(x_i, i=1,\ldots,N\) in low-dimensional space \((\mathbb{R}^M, M \leq N)\) in such a way
that transformations \(g(d_{ij}(X))\) of the fitted distances \(d_{ij}(X)=d(x_i,x_j)\)—i.e., the distance between different \(x_i, x_j\)—approximate the \(\delta^*_{ij}\) as closely as possible. We
call a \(g: d_{ij}(X) \mapsto d_{ij}^*(X)\) a distance transformation function. In other words, proximity scaling means finding \(X\) so that \(d^*_{ij}(X)=g(d_{ij}(X))\approx\delta^*_{ij}=f(\delta_
This approximation \(D^*(X)\) to the matrix \(\Delta^*\) is found by defining a badness-of-fit criterion (loss function), \(\sigma_{MDS}(X)=L(\Delta^*,D^*(X);\Gamma(X))\), that is used to measure how
closely \(D^*(X)\) approximates \(\Delta^*\), optionally subject to an additional criterion of the appearance of \(X\), \(\Gamma(X)\). The smaller the badness-of-fit, the better the fit is.
The loss function used is then minimized to find the vectors \(x_1,\dots,x_N\), i.e., \[$$\label{eq:optim} \arg \min_{X}\ \sigma_{MDS}(X).$$\] There are a number of optimization techniques one can
use to solve this optimization problem.
Stress Models
Usually, we use the quadratic loss function. A general formulation of a loss function based on a quadratic loss is known as stress (Kruskall 1964) and is \[$$\label{eq:stress} \sigma_{MDS}(X)=\sum^N_
{i=1}\sum^N_{j=1} z_{ij} w_{ij}\left[d^*_{ij}(X)-\delta^*_{ij}\right]^2=\sum^N_{i=1}\sum^N_{j=1} z_{ij}w_{ij}\left[g\left(d_{ij}(X)\right)-f(\delta_{ij})\right]^2$$\] where we use some type of
Minkowski distance (\(p > 0\)) as the distance fitted to the points in the configuration, \[$$\label{eq:dist} d_{ij}(X) = ||x_{i}-x_{j}||_p=\left( \sum_{m=1}^M |x_{im}-x_{jm}|^p \right)^{1/p} \ i,j =
1, \dots, N.$$\] Typically, the norm used is the Euclidean norm, so \(p=2\). In standard MDS \(g(\cdot)=f(\cdot)=I(\cdot)\), the identity function. The \(w_{ij}\) and \(z_{ij}\) are finite weights,
e.g., with \(z_{ij}=0\) if the entry is missing and \(z_{ij}=1\) otherwise.
This formulation enables one to express a large number of popular MDS methods with cops. Generally, we allow to use specific choices for \(f(\cdot)\) and \(g(\cdot)\) from the family of power
transformations so one can fit the following stress models:
• Explicitly normalized stress: \(w_{ij}=(\sum_{ij}\delta^{*2}_{ij})^{-1}\), \(\delta_{ij}^*=\delta_{ij}\), \(d_{ij}(X)^*=d_{ij}(X)\)
• Stress-1: \(w_{ij}=(\sum_{ij} d^{*2}_{ij}(X))^{-1}\), \(\delta_{ij}^*=\delta_{ij}\), \(d_{ij}(X)^*=d_{ij}(X)\)
• Sammon stress (Sammon 1969): \(w_{ij}=\delta^{*-1}_{ij}\) , \(\delta_{ij}^*=\delta_{ij}\), \(d_{ij}(X)^*=d_{ij}(X)\)
• Elastic scaling stress (McGee 1966): \(w_{ij}=\delta^{*-2}_{ij}\), \(\delta_{ij}^*=\delta_{ij}\), \(d_{ij}(X)^*=d_{ij}(X)\)
• S-stress (Takane et al. 1977): \(\delta^*_{ij}=\delta_{ij}^2\) and \(d^*_{ij}(X)=d^2_{ij}(X)\), \(w_{ij}=1\)
• R-stress (de Leeuw, 2014): \(\delta^*_{ij}=\delta_{ij}\) and \(d^*_{ij}=d^{2r}_{ij}\), \(w_{ij}=1\)
• Power MDS (Buja et al. 2008, Rusch et al. 2015a): \(\delta^*_{ij}=\delta_{ij}^\lambda\) and \(d^*_{ij}=d_{ij}^\kappa\), \(w_{ij}=1\)
• Power elastic scaling (Buja et al. 2008, Rusch et al. 2015a): \(w_{ij}=\delta^{*-2}_{ij}\), \(\delta^*_{ij}=\delta_{ij}^\lambda\) and \(d^*_{ij}=d^\kappa_{ij}\)
• Power Sammon mapping (Buja et al. 2008, Rusch et al. 2015a): \(w_{ij}=\delta^{*-1}_{ij}\), \(\delta^*_{ij}=\delta_{ij}^\lambda\) and \(d^*_{ij}=d_{ij}^\kappa\)
• Approximate Powerstress (Rusch et al. 2020): \(\delta^*_{ij}=\delta_{ij}^\lambda\) and \(d^*_{ij}=d_{ij}\), \(w_{ij}=\delta_{ij}^\nu\).
• Restricted Powerstress (Buja et al. 2008, Rusch et al. 2015a): \(\delta^*_{ij}=\delta_{ij}^\kappa\) and \(d^*_{ij}=d^\kappa_{ij}\), \(w_{ij}=w_{ij}^\nu\) for arbitrary \(w_{ij}\) (e.g., a
function of the \(\delta_{ij}\))
• Powerstress (encompassing all previous models; Buja et al. 2008, Rusch et al. 2015a): \(\delta^*_{ij}=\delta_{ij}^\lambda\), \(d^*_{ij}=d_{ij}^\kappa\) and \(w_{ij}=w_{ij}^\nu\) for arbitrary \
(w_{ij}\) (e.g., a function of the \(\delta_{ij}\))
• Multiscale Stress: Can be approximated as a powerstress with \(\kappa \rightarrow 0\) and \(\delta^*_{ij}=\log(\delta_{ij})\). It is also possible to do the same approximation for both \(\kappa=1
/a\), \(\delta_{ij}^*=a\delta_{ij}^{1/a}-a\) with \(a\) large, e.g. \(a>1000\).
For all of these models one can use the function powerStressMin which uses majorization to find the solution (de Leeuw, 2014). The function allows to specify a kappa, lambda and nu argument as well
as a weightmat (the \(w_{ij}\)), by setting the respective argument. For some models (those without transformations for the \(d_{ij}\)) one can use smacof::mds.
The object returned from a call to powerStressMin is of class smacofP which extends the smacof classes (de Leeuw & Mair, 2009) to allow for the power transformations. Apart from that the objects are
made so that they have maximum compatibility to methods from smacof. Accordingly, the following S3 methods are available:
print Prints the object
summary A summary of the object
plot 2D Plots of the object
plot3d Dynamic 3D configuration plot
plot3dstatic Static 3D configuration plot
residuals Residuals
coef Model Coefficients
Let us illustrate the usage
Alternatively, one can use the faster sammon function from MASS (Venables & Ripley, 2002) for which we provide a wrapper that adds class attributes and methods (and overloads the function).
• An rstress model (with \(r=1\) as \(r=\kappa/2\))
• A restricted powerstress model
Different ways to plot results are
Strain Models
Another popular type of MDS supported by cops is based on the loss function type . Here the \(\Delta^*\) are a transformation of the \(\Delta\), \(\Delta^*= f (\Delta)\) so that \(f(\cdot)=-(h\circ
l)(\cdot)\) where \(l\) is any function and \(h(\cdot)\) is a double centering operation, \(h(\Delta)=\Delta-\Delta_{i.}-\Delta_{.j}+\Delta_{..}\) where \(\Delta_{i.}, \Delta_{.j}, \Delta_{..}\) are
matrices consisting of the row, column and grand marginal means respectively. These then get approximated by (functions of) the inner product matrices of \(X\) \[$$\label{eq:dist2} d_{ij}(X) = \
langle x_{i},x_{j} \rangle$$\] We can thus express classical scaling as a special case of the general PS loss with \(d_{ij}(X)\) as an inner product, \(g(\cdot) = I(\cdot)\) and \(f(\cdot)=-(h \circ
If we again allow power transformations for \(g(\cdot)\) and \(f(\cdot)\) one can fit the following strain models with cops
• Classical scaling (Torgerson, 1958): \(\delta^*_{ij}=-h(\delta_{ij})\) and \(d^*_{ij}=d_{ij}\)
• Powerstrain (Buja et al. 2008, Rusch et al. 2015a): \(\delta^*_{ij}=-h(\delta_{ij}^\lambda)\), \(d^*_{ij}=d_{ij}\) and \(w_{ij}=w_{ij}^\nu\) for arbitrary \(w_{ij}\)
In stops we have a wrapper to cmdscale (overloading the base function) which extend functionality by offering an object that matches smacofP objects with corresponding methods.
Let us illustrate the usage. A powerstrain model is rather easy to fit with simply subjecting the dissimilarity matrix to some power. Here we use \(\lambda=3\).
The models listed above are also available as dedicted wrapper functions with a cop_ prefix
cop_cmdscale Strain/Powerstrain
cop_smacofSym Stress
cop_smacofSphere Smacof on a sphere
cop_sammon,cop_sammon2 Sammon scaling
cop_elastic Elastic scaling
cop_sstress S-stress
cop_rstress r-stress
cop_powermds Powermds
cop_powersammon Sammon scaling with powers
cop_powerelastic Elastic scaling with powers
cop_apstress Approximate power stress
cop_powerstress Powerstress
cop_rpowerstress Restricted Powerstress
Augmenting MDS with clusteredness considerations: COPS
The main contribution of the cops package is not in solely fitting the powerstress or powerstrain models and their variants from above, but to augment the badness-of-fit function to achieve a
“structured” MDS result automatically (in the sense of clusters or discrete structures). This can be useful mainly for exploring or generating discrete structures or to preserve clusters.
For this an MDS loss function is augmented to include a penalty. This combination of an MDS loss with a clusteredness penalty is what we call “cluster optimized stress” (copstress) and the resulting
MDS is coined “Cluster Optimized Proximity Scaling” (or COPS). This is a multi-objective optimization problem as we want to simultaneously minimize badness-of-fit and maximize clusteredness. The
computational problem is solved by combining the two, but interpretation should happen individually with the badness-of-fit and clusteredness values respectively.
We allow two ways of how copstress can be used: In one variant (COPS-C) one looks for an optimal configuration \(X^*\) directly, given the transformation parameters. This yields a configuation that
has a more clustered appearance than the standard MDS with the same tranformation parameters. In the other (P-COPS) we automatically select optimal transformation parameters and then solve the
respective transformed MDS so that the clusteredness appearance of the configuation is improved.
COPS-C: Finding a configuration with COPS
Here we combine a normalized stress function \(\sigma'_{\text{stress}}(X|\theta)\) given stress hyperparameter vector \(\theta\) and a measure of clusteredness, the OPTICS cordillera (\(\text{OC}'(X)
\)) to the following objective \[$$\label{eq:spstressv1} \sigma_{\text{PS}}(X|\theta)=\text{copstress}(X|\theta) = v_1 \cdot \sigma'_{\text{stress}}(X|\theta) - v_2 \cdot \text{OC}_\gamma'(X),$$\]
with scalarization weights \(v_1,v_2 \in \mathbb{R}_+\), which is minimized over all possible \(X\).
In COPS-C the parameters \(\theta, v_1, v_2\) and \(\gamma\) are all treated as given. Minimizing copstress in this variant jitters the configuration towards a more clustered arrangement, the
strength of which is governed by the values of \(v_1, v_2\). We recommend to use the convex combination \(v_2=1-v_1\) with \(0 \leq v_1 \leq 1\). For a given \(\theta\), if \(v_2=0\) the result of
the above equation is the same as solving the respective stress problem.
COPS-C can be used with many different transformations including ratio, non-metric (ordinal), interval and power transformations (see below). If the \(\sigma'_{\text{stress}}(X|\theta)\) allows for
different transformation of dissimilarities and distances (e.g., powerstress), we expect researchers and practictioners to start from identic transformations. If need arises, e.g., to avoid a problem
of near-indifferentiation, one can exploit the flexibility of employing different transformations. For that case we point out that the configuration may then represent a relation that is somewhat
further apart of the main aim in MDS of faithfully reproducing the dissimilarities by distances in a comparable space but may allow some desired aspects to be revealed in a graphical representation.
COPS-C can be used either for improving c-clusteredness for a given initial MDS configuration (which may then be only locally optimal) or for looking for the globally near-optimal COPS-C
configuration (with different starting configurations, see below).
Usage and Examples
COPS-C with copstressMin needs the mandatory argument delta which is the dissimilarity matrix and some optional additional arguments which we descirbe below.
The default COPS-C (ratio MDS) can already be fit as
A number of plots are available.
The print function outputs information about the COPS-C model. In this case we fitted a ratio COPS-C that uses the standard MDS stress (all transformation parameters 1). There were 15 objects and the
square root of stress of the configuration is 0.268 (compared to 0.267 for standard MDS, see above). The normed OPTICS cordillera value is 0.245, compared to 0.13 for standard MDS (with 0 being no
clusteredness and 1 perfect clusteredness, see below). We also get information on \(v_1\) and \(V_2\) which were 0.975 and 0.025 respectively (the default values). The copstress value is 0.255, but
we stress that this isn’t particulalry important for interpretation.
The values that we should interpret are the stress and the cordillera. We see that the badness-of-fit for the COPS-C configuration is a bit higher (which is to be expected due to the penalization)
and also that clusteredness increased by quite a bit. This is also evident in the Procrustes plot (grey is standard MDS, coral is COPS-C).
Specifically, the clusters of “Sister, Daughter, Mother” and “Son, Brother, Father” as well as “Grandson, Grandfather” are a bit more compact for the COPS-C result as compared to the standard MDS,
and “Cousin” has been moved slightl towards “Uncle, Nephew”. At the same time, the fit is almost equal (0.268 for COPS-C vs. 267 for MDS).
We can also look at the clusteredness situation with the OPTICS reachability plot, which shows more clusteredness for COPS-C (a stronger up and down of the black line over the reachabilities). Next
to the more compact clusters (deeper valleys) the main difference for COPS-C is that with the default minimum points that must form a cluster being 3 (default) in this case means that cousin is now
also part of a three object cluster (with low density) and not a noise point as in standard MDS.
The number of iterations can be changed with the itmax argument (defaults to 5000). If it is low, a warning is returned but that should usually be rather inconsequential. Let’s set the iterations
to 20000 (where the warning no longer appears but the copstress value is only a slightly lower). If one values accuracy over computation time, then a higher value is preferable.
If we want to find the approximation in \(R^N\) we can change the ndim argument, where ndim=N. Default is a 2D space, so ndim=2. Let’s do a COPS-C in a 3D target space.
An important parameter is the minimum number that must comprise a cluster, minpts. Default is ndim+1, which is typically 3 but should really be selected based on substantive considerations (and
must be \(\geq 2\)). It can also be varied in different runs to explore the clusteredness structure. If we set minpts=2, we see that the two object clusters are pushed more towards each other.
stressweight and cordweight
The scalarization weights \(v_1, v_2\) can be changed with stressweight and cordweight. They encode how strong stress and cordillera should respectively be weighted for the scalarization. The
higher stressweight is in relation to cordweight the more weight is put on stress (so a more faithful representation to the MDS result). The default values are stressweight=0.975 and cordweight=
0.025. We suggest to put much more weight on stress to not create an articifical configuration. Let’s look at the effect of changing it to stressweight=0.8, cordweight=0.2— we see we have much
more clusteredness now (0.73) but badness-of-fit has also ramped up a lot to 0.33 and the representation may no longer be very faithful to the real dissimilarities.
Dissimilarity weights (\(z_{ij}\) and \(w_{ij}\)) can be set as weightmat. This must be a matrix of the same dimensions as the dissimilarity argument delta. Let’s say we found out that there was
a study error where comparing cousins with Aunt was messed up, so we want ot ignore that dissimilarity .
kappa, lambda, nu and theta
The arguments kappa, lambda, nu and theta all allow to fit power transformations (if a theta is given it overrides the other values), with kappa being the distance power transformation, lambda
the proximity power transformation and nu the power transformation for the weights. theta is a vector collecting c(kappa,lambda,nu). Let’s fit an s-stress COPS-C.
So far we fit COPS-C with ratio MDS and power transformations only. We support more transformations for COPS-C (dis is the observed dissimilarity matrix)
□ Ratio COPS-C: Setting type="ratio" and kappa=1, lambda=1 (default model).
□ Non-metric (ordinal) COPS-C: Setting type="ordinal" with different handling of ties ("primary", "secondary", "tertiary". See ?smacof::mds) .
□ Interval COPS-C: Setting type="interval".
□ ALSCAL COPS-C: Setting type="ratio" andkappa=2 and lambda=2.
□ Power Stress COPS-C: Setting type="ratio" and kappa and lambda to the desired values.
□ Sammon mapping COPS-C: Setting weightmat=dis, nu=-1 (for all types).
□ Elastic scaling COPS-C: Setting weightmat=dis, nu=-2 (for all types).
□ Multiscale COPS-C: Can be approximated by setting kappa close to zero (say kappa=0.0001) and manually transforming disms<-log(dis); diag(dism)<-0 and then set the argument delta=disms.
Some options can also be combined. Note that it is currently not possible to use transformation parameters with interval and non-metric MDS.
Let’s fit a non-metric elastic scaling COPS-C model with secondary handling of ties.
OC parameters
Because COPS-C uses the OPTICS cordillera to measure clusteredness, it is possible to change a few parameters of how clusteredness is measured.
minpts is the most important one and we already discussed that.
Additional parameters include q which is the parameter for the \(L_p\)-norm of the OPTICS Cordillera and is typically 1 (default) or 2. A higher value of q can be thought of as pronouncing the
ups and downs relatively stronger.
The parameter epsilon relates to the maximum neighbourhood radius around a point to look for possible other points in a cluster and also relates to the density that a cluster must have. It
influences the number of points that are classified as noise by OPTICS and improves runtime of OPTICS the smaller it is. It is not a praticularly intuitive parameter but for most MDS application
it should suffice to just set it “sufficiently large” so all points are considered as possible neighbours of each other. It should only be changed to a lower value if the concept of “noise
points” is useful for a data set (e.g., objects that are not supposed to be in a cluster anyway).
Finally dmax and rang relate to the normalization and winsorization distance for the cordillera, essentially as the maximum distance between points that we still take into account. This can be
used for make the index more robust to outliers in the configuration so that the algorithm doesn’t just achieve a higher index by placing some points very far away from the rest. If dmax is NULL,
the normalization is set to (0,1.5 x the maximum reachability distance for the torgerson model). If it is set too low, the normed cordillera value may be too high. Similarly, rang alows to set
the whole normalization interval and is (0,dmax) by default. If max(rang) and dmax do not agree a warning is printed and rang takes precedence. These parameters can be used to explor different
winsorization limits for robustness checks.
Let’s look at their effects. First we set q=2 and see that the effect of clusteredness is a bit more pronounced as compared to q=1 (because larger ups and downs in the cordillera are weighted a
Let’s lower epsilon to 0.6 and minpts to 2 which means that points that are beyond that distance are no longer possible to be considered as cluster members of each other which allows COPS to have
“Sister” and “Brother” pushed out of their respective clusters of “Daughter, Mother” and “Father, Son” and have all those two object clusters really tightly packed. The single objects would now
be noise points.
Let’s also change dmax to 1 to make the index more robust. The effect is that “Cousin” is now less far away from the rest.
Finally, we have scale which influences the scale of the axis. In COPS we’re only interested in the relatvie placement of the objects rather than the scale, so the scale is somewhat aribtrary. It
can be set to be sd (divided by the largest standard deviation of any column; default), none where no scaling is applied, proc which deos procrustes adjustment of the final configuration to the
starting configuration, rmsq (configuration divided by the maximum root mean square of the columns) and std which standardizes all columns (NOTE: this does not preserve the relative distances of
the optimal configuration and should probably never have been implemented in the first place).
There are some more arguments which are described in ?copstressMin.
COPS-C is a very difficult optimization problem and we resort to heuristics to solve it. There are a large number of such global optimization heuristics supported in cops. The default is
hjk-Newuoa and will typically work quite well. Another good optimizer is CMA-ES but that has a tendency to fail. See ?copstressMin for the available solvers for the argument optimmethod and the
supplement to the original article for an empirical comparison.
The second variant of COPS uses the copstress to select the transformation parameters, so that when fitted as powerstress or any of the other badness-of-fit functions, the corresponding configuration
has higher clusteredness than a standard MDS (there’s also a chance that the standard MDS will be selected). This can be thought of as a profile method as we use the copstress not for direct
minimzation of the objective but as criterion for parameter selection and the minimization to obtain the configuration happens only with the unpenalized badness-of-fit function.
Let us write \(X(\theta)=\arg\min_X \sigma_{MDS}(X,\theta)\) for the optimal configuration for given transformation parameter vector \(\theta\). The objective function for parameter selection is
again , and is again the weighted combination of the \(\theta-\)parametrized loss function, \(\sigma_{MDS}\left(X(\theta),\theta\right)\), and the c-clusteredness measure, the OPTICS cordillera or \
(OC(X(\theta);\epsilon,k,q)\) but this time to be optimized as a function of \(\theta\) or \[$$\label{eq:spstress} \text{coploss}(\theta) = v_1 \cdot \sigma_{MDS}\left(X(\theta),\theta \right) - v_2
\cdot \text{OC}\left(X(\theta);\epsilon,k,q\right)$$\] with \(v_1,v_2 \in \mathbb{R}\) controlling how much weight should be given to the badness-of-fit measure and c-clusteredness. In general \
(v_1,v_2\) are either determined values that make sense for the application or may be used to trade-off fit and c-clusteredness in a way for them to be commensurable. In the latter case we suggest
taking the fit function value as it is (\(v_1=1\)) and fixing the scale such that \(\text{copstress}=0\) for the scaling result with the identity transformation (\(\theta=\theta_0\)), i.e., \[$$\
label{eq:spconstant0} v^{0}_{1}=1, \quad v^{0}_2=\frac{\sigma_{MDS}\left(X(\theta_0),\theta_0\right)}{\text{OC}\left(X(\theta_0);\epsilon,k,q\right)},$$\] with \(\theta_0=(1,1,1)^\top\) in case of
loss functions with power transformations. Thus an increase of 1 in the MDS loss measure can be compensated by an increase of \(v^0_1/v^0_2\) in c-clusteredness. Selecting \(v_1=1,v_2=v^{0}_2\) this
way is in line with looking for a parameter combination that would lead to a configuration that has a more clustered appearance relative to the standard MDS.
The optimization problem in P-COPS is then to find
\[$$\label{eq:soemdsopt2} \arg\min_{\theta} \text{coploss}(\theta)$$\] by evaluating \[$$\label{eq:soemdsopt} v_1 \cdot \sigma_{MDS}\left(X(\theta),\theta\right) - v_2 \cdot \text{OC}\left(X(\theta);
\epsilon,k,q\right) \rightarrow \min_\theta!$$\] For a given \(\theta\) if \(v_2=0\) than the result of optimizing the above is the same as solving the respective original MDS problem. Letting \(\
theta\) be variable, \(v_2=0\) will minimize the loss over configurations obtained from using different \(\theta\).
Examples & Usage
The dedicated function for P-COPS is called pcops. The two main arguments are again the dissimilarity matrix and which MDS model that should be used (loss). Then pcops optimizes over \(\theta\) with
the values given in theta being used as starting parameters (if not given, they are all 1).
For the example we can use a P-COPS model for a classical scaling with power transformations of the dissimilarities (strain or `powerstrain loss)
The transformation parameters selected is 1.498 for the dissimilarities (as in strain/powerstrain only the dissimilarities are subjected to a power transformation). The resulting badness-of-fit value
is 0.45 (this is not a stress, see cmdscale for its interpretation) and the c-clusteredness value is 0.33.
A number of plots are availabe
The different losses (MDS models) that are available for P-COPS are
□ stress, smacofSym: Kruskall’s stress; Workhorse: smacofSym, Optimization over \(\lambda\)
□ smacofSphere: Kruskall’s stress for projection onto a sphere; Workhorse smacofSphere, Optimization over \(\lambda\)
□ strain, powerstrain: Classical scaling; Workhorse: cmdscale, Optimization over \(\lambda\)
□ sammon, sammon2: Sammon scaling; Workhorse: sammon or smacofSym, Optimization over \(\lambda\)
□ elastic: Elastic scaling; Workhorse: smacofSym, Optimization over \(\lambda\)
□ sstress: S-stress; Workhorse: powerStressMin, Optimization over \(\lambda\)
□ rstress: S-stress; Workhorse: powerStressMin, Optimization over \(\kappa\)
□ powermds: MDS with powers; Workhorse: powerStressMin, Optimization over \(\kappa\), \(\lambda\)
□ powersammon: Sammon scaling with powers; Workhorse: powerStressMin, Optimization over \(\kappa\), \(\lambda\)
□ powerelastic: Elastic scaling with powers; Workhorse: powerStressMin, Optimization over \(\kappa\), \(\lambda\)
□ apstress: Approximate power stress model; Workhorse: smacofSym, Optimization over \(\lambda\), \(\nu\)
□ rpowerstress: Restricted power stress model; Workhorse: powerStressMin, Optimization over \(\kappa\) and \(\lambda\) together (which are restricted to be equal), and \(\nu\)
□ powerstress: Power stress model (POST-MDS); Workhorse: powerStressMin, Optimization over \(\kappa\), \(\lambda\), and \(\nu\)
Note: Anything that uses powerStressMin as workhorse is a bit slow.
It is also possible to use the pcops function for finding the loss-optimal transformation in the the non-augmented models specified in loss, by setting the cordweight, the weight of the OPTICS
cordillera, to 0. Then the function optimizes for the transformation parameters based on the MDS loss function only.
: Here the results match the result from using the standard cordweight suggestion. We can give more weight to the c-clusteredness though:
This result has more c-clusteredness but less goodness-of-fit. The higher c-clusteredness is discernable in the Grandfather/Brother and Grandmother/Sister clusters (we used a minimum number of 2
observations to make up a cluster, minpts=2).
As in COPS-C we have a number of parameters to guide and change the behaviour of P-COPS. Many are equal to the ones explained in the COPS-C section, including minpts, weightmat, ndim, init,
stressweight, cordweight, q, epsilon, rang, scale. See the description there.
lower and upper
An important set of arguments unique to P-COPS are lower and upper which are the boundaries of the search space in which to look for the parameters. They need to be of the same length as the
theta argument. Naturally, the larger the search space is, the longer it can take to find the optimal parameters. Default values are lower = c(1, 1, 0.5) and upper = c(5, 5, 2). Note this can
also be used to set a quasi-restriction on parameters, if there is no canned loss function that does that. In that case we would just set the boundaries very close together, so, say we’d like to
use powerstress and search for optimal \(\kappa\) and \(\lambda\) but fix the nu to be \(-2.5\), we can then set lower = c(0,0,-2.5001) and upper = c(5,5,-2.4990) so \(\nu\) will be searched for
only in the narrrow band between \((-2.501,-2.499)\). Let’s change the search space to include values between \(0.1\) and \(1.6\) (in the above example \(1.5\) was the optimal parameter).
: The optimal \(\lambda\) found is again around \(1.498\) resulting in a badness of fit of \(0.45\) and a clusteredness of \(0.398\), so by extending the search space we found no better
itmaxi and itmaxo
The number of iterations can be controlled with the itmaxi and itmaxo arguments. itmaxi (default 10000) refers to the maximum number of iterations for the inner part (the MDS optimization) and
itmaxo (default 200) refers to the maximum number of iterations for the outer search that tries to find the optimal \(\theta\). The higher itmaxi argument is the closer the configuration that is
evaluated for copstress is to a local optimum and the higher itmaxo is the more values for the transformation parameters will be tried (which also depends on the optimizer). Time-wise there is a
trade-off here between how deep (itmaxi) we want to go and how broad (itmaxo). In our experience itmaxi doesn’t need to be very high and it is better to have a higher itmaxo, which is probably
why in one of life’s great mysteries we set the default values exactly the other way round.
Let’s look at that in action, which doesn’t really change much compared to how it was with the default values (optimal parameter is now \(1.499\)).
Minimizing copstress for P-COPS is pretty difficult. For pcops we use a nested algorithm combining optimization that internally first solves for \(X\) given \(\theta\), \(\arg\min_X \sigma_{MDS}\
left(X,\theta\right)\), and then optimize over \(\theta\) with a metaheuristic. The metaheuristic can be chosen with the optimmethod argument. Implemented are simulated annealing (optimmethod=
"SANN"), particle swarm optimization (optimmethod="pso"), DIRECT (optimmethod="DIRECT"), DIRECTL (optimmethod="DIRECTL"), mesh-adaptive direct search (optimmethod="MADS"), stochastic global
optimization (optimmethod="stogo"), Hooke-Jeeves pattern search (optimmethods="hjk") and a variant of the Luus-Jaakola (optimmethod="ALJ") procedure. Default is “ALJ” that usually converges in
less than 200 iterations to an acceptable solution.
Choosing arguments for COPS methods
We listed the possibilities how the behavior of COPS models can be changed in this document. It might have occured to you that there are a lot of options to choose from. We believe that more options
and flexibility is generally better, especially in an exploratory setting, but that puts the user on the spot of making their own decisions which not everyone seems to like (many seem to prefer the
apparent security of not needing to make them). So, we want to share what appeared as best practice in our experience.
• Think carefully about the minimum number of points that should comprise a cluster (the minpts argument). If there are 5000 objects, a minimum number of points of \(2\) will likely not be very
illuminating. This decision depends on substantive considerations.
• The scalarization weights trade-off badness-of-fit with clusteredness. Since we typically want to have a representation that is faithful, we recommend to start out with a stressweight that is
much larger than cordweight (say \(v_1/v_2>7\) times). Then one can successively lower the relative cordweight to about \(v_1/v_2>3\) if necessary. For typical use cases we’d not recommend
getting below this ratio.
• When using power transformations, it is best to start out with equal powers for both distances and dissimilarities and allow for different ones only when necessary. In COPS-C that would be set
manually and in P-COPS the rpowerstress loss can be used.
• In a standard use case without much idea about the range of distances, we’d set epsilon for the OC high and dmax to about \(1-1.5\) times of the largest reachability value that is smaller than
the dmax that results when applying the OC to a standard MDS configuration for the same data, e.g.,
• Staying true to the exploratory nature, trying out different setups and comparing them is a good idea, especially with respect to the cordillera parameters and scalarization weights.
• We envisioned, tested and applied the functions in the cops package for small to moderate data sizes (up to 200 objects). The more objects we have, the more difficult the problem becomes, both
with respect to finding the optima and the time it will take to get them. It can also be that the COPS result is not really illuminating with a large number of objects. This is ongoing research,
so use at your own risk. We’re always interested in hearing experiences, though, if something goes awry. | {"url":"https://cran-r.c3sl.ufpr.br/web/packages/cops/vignettes/cops.html","timestamp":"2024-11-10T14:22:34Z","content_type":"text/html","content_length":"608382","record_id":"<urn:uuid:248dd192-6b08-4525-9d63-bf3341807a2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00691.warc.gz"} |
Go to the source code of this file.
subroutine zlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
Function/Subroutine Documentation
subroutine zlatrd ( character UPLO,
integer N,
integer NB,
complex*16, dimension( lda, * ) A,
integer LDA,
double precision, dimension( * ) E,
complex*16, dimension( * ) TAU,
complex*16, dimension( ldw, * ) W,
integer LDW
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
Download ZLATRD + dependencies
[TGZ] [ZIP] [TXT]
ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
Hermitian tridiagonal form by a unitary similarity
transformation Q**H * A * Q, and returns the matrices V and W which are
needed to apply the transformation to the unreduced part of A.
If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by ZHETRD.
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
[in] UPLO Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N is INTEGER
[in] N The order of the matrix A.
NB is INTEGER
[in] NB The number of rows and columns to be reduced.
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit:
if UPLO = 'U', the last NB columns have been reduced to
[in,out] A tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements above the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors;
if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements below the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors.
See Further Details.
LDA is INTEGER
[in] LDA The leading dimension of the array A. LDA >= max(1,N).
E is DOUBLE PRECISION array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
[out] E elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
the first NB columns of the reduced matrix.
TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors, stored in
[out] TAU TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
See Further Details.
W is COMPLEX*16 array, dimension (LDW,NB)
[out] W The n-by-nb matrix W required to update the unreduced part
of A.
[in] LDW LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
Q = H(n) H(n-1) . . . H(n-nb+1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).
If UPLO = 'L', the matrix Q is represented as a product of elementary
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).
The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.
The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
if UPLO = 'U': if UPLO = 'L':
( a a a v4 v5 ) ( d )
( a a v4 v5 ) ( 1 d )
( a 1 v5 ) ( v1 1 a )
( d 1 ) ( v1 v2 a a )
( d ) ( v1 v2 a a a )
where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).
Definition at line 200 of file zlatrd.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d7/de0/zlatrd_8f.html","timestamp":"2024-11-09T09:29:12Z","content_type":"application/xhtml+xml","content_length":"16203","record_id":"<urn:uuid:9f3c5375-562c-47e1-8eec-3787e67be2fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00845.warc.gz"} |
2.5 The total score of all the paths
In the last section, we learned how to calculate the label path score of one path that is $e^{S_i}$. So far, we have one more problem which is needed to be solved, how to obtain the total score of
all the paths ($ P_{total} = P_1 + P_2 + … + P_N = e^{S_1} + e^{S_2} + … + e^{S_N} $).
The simplest way to measure the total score is that: enumerating all the possible paths and sum their scores. Yes, you can calculate the total score in that way. However, it is very inefficient. The
training time will be unbearable. | {"url":"https://createmomo.github.io/page/2/","timestamp":"2024-11-06T05:28:18Z","content_type":"text/html","content_length":"19649","record_id":"<urn:uuid:4e306661-895f-454e-b492-afdec50df22c>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00697.warc.gz"} |
Central Limit Theorem - Data Science Discovery
Central Limit Theorem
The Central Limit Theorem, or the CLT, is one of the most important theorems in statistics! It says that:
Regardless of the distribution shape of the population, the sampling distribution of the sample mean becomes approximately normal as the sample size n increases (conservatively n ≥ 30).
In other words, if we repeatedly take independent random samples of size n from any population, then when n is large, the distribution of the sample means will approach a normal distribution.
This is very interesting and helps make our lives as data scientists easier! This means that doesn't matter if a distribution shape is left-skewed, right-skewed, uniform, binomial, or anything else -
the distribution of the sample mean will always become normal as the sample size increases.
Because of the CLT, we can use the standard normal curve as an approximate histogram for the sample means. Also, we can use the standard normal curve to calculate areas just like we did previously
with variables that were normally distributed.
Using the Standard Normal Curve and Random Variables
Remember, to use the Standard Normal Curve, we must convert our data to z-scores. It's important to point out that when dealing with random variables, our z-score formula changes slightly from the
original z-score formula. Instead of average and SD, we are now dealing with average and SD of random variables. In other words, we need to use the expected value (EV) and standard error (SE) when we
are dealing with random variables.
• Here's an example of how we can see the Central Limit Theorem in action. In this game, we win if we pick a queen from a deck of 52 cards:
• We see that as n increases, our histogram looks more and more like the normal curve.
Simulation of drawForQueen(1), creating a very lopsided distribution.
Simulation of drawForQueen(10), creating a staircase-style distribution.
Simulation of drawForQueen(100), creating a nearly normal distribution!
Simulation of drawForQueen(1000), retaining and really showcasing the central limit theorem!
Example Walk-Throughs with Worksheets
Video 1: What is the Central Limit Theorem?
Video 2: Central Limit Theorem Examples
Video 3: Discovering The Central Limit Theorem in Python
Practice Questions
: According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean when the sample size (n) increases? (assuming n >= 30) | {"url":"https://discovery.cs.illinois.edu/learn/Polling-Confidence-Intervals-and-Hypothesis-Testing/Central-Limit-Theorem/","timestamp":"2024-11-06T17:16:19Z","content_type":"text/html","content_length":"18214","record_id":"<urn:uuid:c05ff258-8d33-4819-8a48-49b1ab14e397>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00322.warc.gz"} |
Re: Pascal compiler and sets
stenuit@axp05.acset.be (Pascal Stenuit)
Thu, 10 Feb 1994 18:06:41 GMT
From comp.compilers
| List of all articles for this month |
Newsgroups: comp.compilers
From: stenuit@axp05.acset.be (Pascal Stenuit)
Keywords: Pascal
Organization: A.C.S.E.T.
References: 94-02-051 94-02-059
Date: Thu, 10 Feb 1994 18:06:41 GMT
>From ssimmons@convex.com (Steve Simmons)
> The maximum size of a set is dependent upon its definition; however,
> it can be statically computed. Therefore, it is usually best to implement
> them as bit vectors because the following four set operations must
> be performed (UNION, INTERSECTION, DIFFERENCE, and IN).
If I remember correctly, there is one problem with Pascal sets: it is not
always possible to find out the base type of a set constant (as in
[1,4,7]) and this makes difficult allocating storage for it.
That's one reason one implementation may choose to put an upper-bound on
the size of sets. I believe Wirth "fixed" this problem in Modula-2 where a
set constant is always prefixed by its type (can't remember the actual
pascal stenuit. stenuit@acset.be
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Rue du Cerf, 200 Fax +32 2 655.12.11
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Forward Rate Agreements and Calculating FRA Payments - Finance Train
Forward Rate Agreements and Calculating FRA Payments
Forward Rate Agreements (FRA’s) are similar to forward contracts where one party agrees to borrow or lend a certain amount of money at a fixed rate on a pre-specified future date.
For example, two parties can enter into an agreement to borrow $1 million after 60 days for a period of 90 days, at say 5%. This means that the settlement date is after 60 days, on which date the
money will be borrowed/lent for a period of 90 days.
The party that is borrowing money under the FRA has a long position, and the party that is lending money has a short position in the FRA.
FRA contracts are usually cash-settled, that is, the money is not actually lent or borrowed. Instead, the forward rate specified in the FRA is compared with the current LIBOR rate. If the current
LIBOR is greater that the FRA rate, then the long is effectively able to borrow at a below market rate. The long will therefore receive a payment based on the difference between the two rates. If,
however, the current LIBOR was lower than the FRA rate, then long will make a payment to the short. The payment ends up compensating for any change in interest rates since the contract date.
FRAs can be based on different periods, and are quoted in terms of months to settlement date and the months to completion of interest period. In our example, the settlement date is after 60 days (2
months), and then there is an interest period of 90 days (3 months). The contract will complete after a total of 2+3 = 5 months. This FRA will be referred to as 2x5 FRA.
FRAs are generally used to lock in an interest rate for transactions that will take place in the future. For example, a bank that plans to issue or roll over certificates of deposit, but anticipates
that interest rates are headed upward, can lock in today’s rate by purchasing FRA. If rates do rise, then the payment received on the FRA should offset the increased interest cost on the CDs. If
rates fall, the bank pays out.
The above example demonstrated how FRAs are used to lock in an interest rate or debt cost. FRA’s can also be used to lock in the price of a short-term security to be bought or sold in the near
• If the investment is being purchased, you can hedge the risk that interest rates may fall (which would increase the price of the investment) by selling the FRA.
• If the investment is being sold, you can hedge against the risk of rates rising (which would depress the sales price of the security) by buying the FRA.
Calculating FRA Payments
Let’s take an example to understand how payments in an FRA are calculated.
Consider a 3x6 FRA on a notional principle amount of $1million. The FRA rate is 6%. The FRA settlement date is after 3 months (90 days) and the settlement is based on a 90 day LIBOR.
Assume that on the settlement date, the actual 90-day LIBOR is 8%. This means that the long is able to borrow at a rate of 6% under the FRA, which is 2% less than the market rate. This is a saving
\= 1,000,000 * 2% *90/360 = $5,000
This is the interest that the long would save by using the FRA. Since the settlement is happening today, the payment will be equal to the present value of these savings. The discount rate will be the
current LIBOR rate.
FRA Payment = $5,000/(1+0.08)^(90/360) = $4,904.72
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Intravenous Flow Rates Quiz - 30 Mins - EMT-P
Questions and Answers
Please place your FD or EMS Agency Name and your complete name in the name box.
• 1.
The physician orders an IV infusion of D5W 1000 ml to infuse over the next eight hours. The IV tubing that you are using delivers 15gtt/min. What is the correct rate of flow? ___________ gtts/min
• 2.
A patient, admitted with a head injury, has an order for D5NS at 25 ml/hour. The IV tubing has a calibration of 10gtt/ml. What is the correct rate of flow for this patient? ___________ gtts/min
The correct rate of flow for this patient is 4 gtts/min. This is calculated by converting the ml/hour rate to gtts/min using the calibration of the IV tubing. Since the tubing has a calibration
of 10gtt/ml, and the order is for 25 ml/hour, we can multiply 25 ml/hour by 10 gtts/ml to get 250 gtts/hour. To convert this to gtts/min, we divide by 60 (minutes in an hour), resulting in 4 gtts
• 3.
Your patient has an order to infuse 100 ml of D51/2NS with 10MEq of KCl over the next thirty minutes. The set calibration is 10gtt/ml. What is the correct rate of flow for this patient?
___________ gtts/min
The correct rate of flow for this patient is 33 gtts/min. This can be calculated by dividing the total volume to be infused (100 ml) by the time in minutes (30 min) and multiplying by the set
calibration (10 gtts/ml). Therefore, (100 ml / 30 min) x 10 gtts/ml = 33 gtts/min.
• 4.
The order reads: "Over the next 4 hours, infuse 500 ml of 5% Dextrose in Normal Saline. Add 20 MEq of KCl to solution." You know that the IV tubing set is calibrated to deliver 10gtt/ml. In drops
per minute, what is the rate of flow? ___________ gtts/min
The IV tubing set is calibrated to deliver 10gtt/ml. Since we need to infuse 500 ml of the solution over the next 4 hours, we can calculate the total number of drops needed by multiplying the
volume (500 ml) by the calibration rate (10 gtt/ml). This gives us 5000 gtt. To find the rate of flow in drops per minute, we divide the total number of drops (5000 gtt) by the total number of
minutes (240 min). This gives us a rate of 20.83 gtt/min, which can be rounded up to 21 gtt/min.
• 5.
The 10am medications scheduled for your patient include Keflex 1.5 G in 50 ml of a 5% Dextrose solution. According to the pharmacy, this preparation should be administered in thirty minutes. The
IV tubing on your unit delivers 15 gtts per milliliter. What is the correct rate of flow in drops per minute? ___________ gtts/min
The correct rate of flow in drops per minute is 25 gtts/min. This can be calculated by dividing the total volume (50 ml) by the time (30 minutes) and multiplying it by the number of drops per
milliliter (15 gtts/ml). Therefore, (50 ml / 30 min) x 15 gtts/ml = 25 gtts/min.
• 6.
1000cc solution of D5NS with 20,000 units of Heparin is infusing at 20ml per hour. The IV set delivers 60 gtts per cc. How many units of Heparin is the patient receiving each hour? ___________
The patient is receiving 400 units of Heparin each hour. This can be calculated by multiplying the infusion rate (20 ml/hr) by the number of gtts per cc (60) and then dividing by the total volume
of the solution (1000 cc). The result is 1200 gtts/hr. Since each cc of the solution contains 20,000 units of Heparin, dividing 1200 gtts/hr by 60 gtts per cc gives us 20 cc/hr. Finally,
multiplying 20 cc/hr by 20,000 units/cc gives us 400 units/hr.
• 7.
Your patient has an order to receive 800 units of Heparin per hour by continuous intravenous infusion. If the pharmacy mixes the IV bag to contain a total of 5,000 units of Heparin in 500 ml of
D5W, how many cc's per minute should the patient receive? ___________ cc/min
The patient is ordered to receive 800 units of Heparin per hour. The pharmacy has mixed the IV bag to contain a total of 5,000 units of Heparin in 500 ml of D5W. To determine the number of cc's
per minute the patient should receive, we need to calculate the rate of infusion. Since there are 60 minutes in an hour, we divide the total units (5,000) by the total minutes (60) to get the
units per minute. This gives us 83.33 units per minute. To convert this to cc's per minute, we divide by the concentration of Heparin in the IV bag (800 units per cc). This gives us 0.104 cc's
per minute, which can be rounded to 0.1 cc's per minute or 80 cc's per hour.
• 8.
The physician orders an IV infusion of D5W 1000 ml to infuse over the next eight hours. The IV tubing that you are using delivers 10 gtt/ml. What is the correct rate of flow (drops per minute)?
___________ gtts/min
The correct rate of flow (drops per minute) can be calculated by dividing the total volume to be infused (1000 ml) by the total time (8 hours) and then multiplying it by the drop factor (10 gtt/
ml). Therefore, the correct rate of flow would be 125 gtts/hr, which can be further divided by 60 to obtain the rate in minutes. This gives us a final answer of 21 gtts/min.
• 9.
A patient, admitted with a head injury, has an order to start 1000cc of D5NS at 30ml/hour. The IV tubing has a calibration of 60 gtt/ml. What is the correct rate of flow for this patient?
___________ gtts/min
The correct rate of flow for this patient is 30 gtts/min. This is calculated by dividing the desired hourly rate (30 ml/hour) by the calibration of the IV tubing (60 gtt/ml). Therefore, 30 ml/
hour divided by 60 gtt/ml equals 0.5 ml/min. Since there are 60 gtts in 1 ml, multiplying 0.5 ml/min by 60 gtts/ml gives us 30 gtts/min.
• 10.
Your patient has an order to infuse 100 ml of D51/2NS with 40 MEq of KCl over the next 60 minutes. The set calibration is 15 gtt/ml. What is the correct rate of flow for this patient? ___________
The correct rate of flow for this patient is 25 gtts/min. This can be calculated by dividing the total volume to be infused (100 ml) by the total time for infusion (60 minutes), and then
multiplying by the set calibration (15 gtt/ml). So, (100 ml / 60 min) * 15 gtt/ml = 25 gtts/min.
• 11.
The 10am medications scheduled for your patient include Keflex 2.0 g in 100 ml of a 5% Dextrose solution. According to the pharmacy, this preparation should be administered in thirty minutes. The
IV tubing on your unit delivers 10 gtts per milliliter. What is the correct rate of flow in drops per minute? ___________ gtts/min
The correct rate of flow in drops per minute is 33 gtts/min. This can be calculated by dividing the total volume (100 ml) by the time in minutes (30 min) and then multiplying by the drop factor
(10 gtts/ml). So, (100 ml / 30 min) * 10 gtts/ml = 33 gtts/min.
• 12.
A 500 cc solution of D5NS with 20,000 units of Heparin is infusing at 20ml per hour. The IV set delivers 60 gtts per cc. How many units of Heparin is the patient receiving each hour? ___________
To determine the number of units of Heparin the patient is receiving each hour, we need to calculate the total volume of the solution infused per hour and then multiply it by the concentration of
Heparin in the solution.
Given that the solution is infusing at 20ml per hour, and the IV set delivers 60 gtts per cc, we can calculate the total volume in drops per hour by multiplying 20ml by 60 gtts/cc. This gives us
1200 gtts per hour.
Since the solution contains 20,000 units of Heparin in 500cc, the concentration of Heparin in the solution is 40 units/cc (20,000 units / 500cc).
To find the number of units of Heparin the patient is receiving each hour, we multiply the concentration (40 units/cc) by the total volume in cc per hour (20ml), which gives us 800 units/hr.
• 13.
The physician orders 1.5 liters of Lactated Ringers solution to be administered intravenously to your patient over the next 12 hours. Calculate the rate of flow if the IV tubing delivers 20gtt/
ml. ___________ gtts/min
The rate of flow can be calculated by dividing the total volume (1.5 liters) by the total time (12 hours) and then converting it to minutes. Since the IV tubing delivers 20gtt/ml, we can multiply
the rate of flow (ml/min) by the number of drops per ml (20) to get the rate in drops per minute. Therefore, the rate of flow is 42 gtts/min.
• 14.
The physician orders 1.5 liters of Lactated Ringers solution to be administered intravenously to your patient over the next 12 hours. Calcuate the rate of flow if the IV tubing delivers 15 gtts
per cubic centimeter. ___________ gtts/min
The rate of flow can be calculated by dividing the total volume (1.5 liters) by the time (12 hours) and then converting it to minutes. Since the IV tubing delivers 15 gtts per cubic centimeter,
the rate of flow in gtts/min can be determined by multiplying the rate of flow in milliliters per minute by 15. Therefore, the rate of flow is 31 gtts/min.
• 15.
The physician orders 1.5 liters of Lactated Ringers solution to be administered intravenously to your patient over the next 12 hours. Calculate the rate of flow if the IV tubing delivers 60 gtts/
ml. ___________ gtts/min
The rate of flow can be calculated by dividing the total volume to be administered (1.5 liters) by the time period (12 hours), which gives us 0.125 liters per hour. Since the IV tubing delivers
60 gtts/ml, we can further calculate the number of drops per minute by multiplying the flow rate by the tubing's drop factor. In this case, the drop factor is 60 gtts/ml, so the rate of flow is
0.125 liters/hour * 60 gtts/ml = 7.5 gtts/min. Therefore, the correct answer is 125 gtts/min.
• 16.
The order reads: "Over the next 4 hours, infuse 500 ml of 5% Dextrose in Normal Saline. Add 20 MEq of KCl to solution." You know that the IV tubing set is calibrated to deliver 10gtt/ml. In drops
per minute, what is the rate of flow? ___________ gtts/min
To calculate the rate of flow in drops per minute, we need to know the total volume to be infused and the time it will take to infuse that volume. The order states that 500 ml of 5% Dextrose in
Normal Saline will be infused over the next 4 hours. To convert 4 hours to minutes, we multiply by 60, giving us 240 minutes. The tubing set is calibrated to deliver 10 gtt/ml, so we multiply the
total volume (500 ml) by the drop factor (10 gtt/ml) to get the total number of drops. Therefore, the rate of flow in drops per minute is 500 ml * 10 gtt/ml / 240 minutes = 20.83 gtts/min.
• 17.
On Wednesday afternoon, your patient returns from surgery with an IV fluid order for 1000cc every 8 hours. On Thursday morning at 8am, you assess that 600 ml of a 1L bag has been absorbed. The
physician orders the remainder of that bag to infuse over the next 6 hours. You know that the IV tubing used by your unit delivers 10 gtt/ml. What will the correct rate of flow be? ___________
To calculate the correct rate of flow in gtts/min, we need to determine the total volume of fluid remaining in the bag and divide it by the total time for infusion.
Since 600 ml has already been absorbed, there is 400 ml remaining in the bag. The physician orders this remaining volume to be infused over the next 6 hours, which is equivalent to 360 minutes.
To calculate the rate of flow, we divide the remaining volume (400 ml) by the total time (360 min).
Rate of flow = 400 ml / 360 min = 1.11 ml/min
Since the IV tubing delivers 10 gtt/ml, we multiply the rate of flow by 10 to convert it to gtts/min.
Correct rate of flow = 1.11 ml/min * 10 gtts/ml = 11.1 gtts/min
• 18.
The physician reduces an IV to 30ml/hour. The IVAC indicates that 270 ml are remaining in the present IV bag. You notice that it is exactly 10:30 am. At what time will the infusion be completed?
The physician reduces the IV rate to 30ml/hour and there are 270ml remaining in the IV bag. This means that it will take 270ml / 30ml/hour = 9 hours to complete the infusion. Since it is
currently 10:30 am, adding 9 hours will give us 7:30 pm. Therefore, the infusion will be completed at 7:30 P.M.
• 19.
The medications scheduled for your patient include Keflex 1.5 grams in 50 ml of a 5% Dextrose solution. According to the pharmacy, this preparation should be administered in 30 minutes. The IV
tubing on your unit delivers 15 gtts per milliliter. What is the correct rate of flow in drops per minute? ___________ gtts/min
To calculate the correct rate of flow in drops per minute, we need to determine the total number of drops in 30 minutes. The medication is in a 50 ml solution, which means there are 50 ml x 15
gtts/ml = 750 gtts in the solution. Therefore, the rate of flow in drops per minute is 750 gtts / 30 min = 25 gtts/min.
• 20.
In checking your patient's 10 am medications, you notice that you have orders to infuse 50mg. of Chloramphenicol in 100 ml of 5% Dextrose in Water over 30 minutes. The IV tubing delivers 15 gtt/
ml. What is the correct rate of flow? ___________ gtts/min
The correct rate of flow for the medication is 50 gtts/min. This is determined by dividing the total volume of the medication (100 ml) by the time it should be infused over (30 minutes). Then,
this value is multiplied by the drop factor of the IV tubing (15 gtts/ml) to find the number of drops per minute. Therefore, the correct rate of flow is 50 gtts/min. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=intravenous-flow-rates-quiz-30-mins-emtp-1","timestamp":"2024-11-12T06:59:55Z","content_type":"text/html","content_length":"469386","record_id":"<urn:uuid:63923a35-14a5-4e7c-8b96-4cf1edad41d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00620.warc.gz"} |
Jaume de Dios Pont
About me
I am a Postdoc at ETHZ with Svitlana Mayboroda, working on harmonic analysis and spectral theory.
I was born in Barcelona, where I lived most of my life. I moved to Los Angeles in 2016 to finish my Bachelor’s, and to Zurich in 2017 to do a Master’s in mathematics. In June 2023 I got my PhD from
UCLA, under the supervision of Terence Tao. Between the Phd and the Postdoc, I was a research intern at the ML Foundations group at Microsoft Research.
My graduate school was supported by a UCLA dissertation year fellowship, Teaching and Research Assistantships at UCLA, a “La Caixa” post-graduate fellow, and an ESOP fellow at ETHZ.
My name is pronouced [ˈʒawmə] , my pronouns are he/him.
• Harmonic Analysis
• Elliptic PDEs
• Convex Analysis
• Machine Learning
• PhD Mathematics, 2023
• MSc Mathematics, 2018
ETH Zurich
• BSc Mathematics, 2017
Universitat Autonoma de Barcelona
• BSc Physics, 2017
Universitat Autonoma de Barcelona | {"url":"https://jaume.dedios.cat/","timestamp":"2024-11-06T09:13:56Z","content_type":"text/html","content_length":"66717","record_id":"<urn:uuid:1d29fc2f-17a2-4cde-b58c-f9c73dbbc3dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00873.warc.gz"} |
How Many Tablespoons Does It Take to Fill a Bathtub - Evolving Home
Have you ever wondered how many tablespoons it takes to fill a bathtub? Well, we conducted an experiment to find out!
In this article, we will share our findings and provide you with the answer you’ve been seeking. Using a first person plural point of view, active voice, and contractions, we’ll delve into the
conversion factor from tablespoons to gallons, calculate the volume of a bathtub, and estimate the number of tablespoons required.
So let’s dive in and satisfy your curiosity once and for all!
Key Takeaways
• The conversion factor for tablespoons to gallons is 256 tablespoons per gallon.
• Accurate measurement of bathtub dimensions is crucial for volume calculation.
• Converting tablespoons to volume helps determine the number required to fill the bathtub.
• Estimating water displacement helps understand actual water usage during a bath.
The Conversion Factor: Tablespoons to Gallons
The conversion factor for tablespoons to gallons is 256 tablespoons per gallon. This means that there are 256 tablespoons in a single gallon.
However, it is important to note the limitations of this conversion factor. While it provides a simple way to convert between these units, it may not always be practical or precise for certain
For example, when measuring small amounts of liquid, such as in cooking recipes, using fractions of a tablespoon might be more accurate and easier to work with than converting everything into
On the other hand, when dealing with large volumes of liquid, such as in industrial settings or scientific experiments, the conversion factor can be very useful for quickly estimating quantities and
making calculations.
Overall, understanding the conversion factor from tablespoons to gallons is helpful in various practical situations but should be used judiciously considering its limitations.
Calculating the Bathtub Volume
When it comes to calculating the volume of a bathtub, there are several key factors to consider.
First, measuring the dimensions of the bathtub accurately is crucial in order to get an accurate volume calculation.
Additionally, converting tablespoons to volume will allow us to determine how many tablespoons it takes to fill the bathtub.
Lastly, estimating water displacement can help us understand how much water is actually being used when we take a bath.
Measuring Bathtub Dimensions
To measure the dimensions of your bathtub, you can use a tape measure and follow these simple steps.
First, start by measuring the length of your bathtub from one end to the other.
Next, measure the width of your bathtub at its widest point.
Finally, measure the depth of your bathtub from the bottom to the highest point.
• Measure length: Place one end of the tape measure against one side wall and extend it to the opposite side.
• Measure width: Position the tape measure across the widest part of your bathtub.
• Measure depth: Lower the tape measure into your bathtub until it reaches the bottom, then record that measurement.
Converting Tablespoons to Volume
You can easily convert tablespoons to volume by using a simple conversion chart. Converting tablespoons to liters is a straightforward process that allows you to accurately measure liquids in larger
The history of measuring volume in tablespoons dates back centuries, with different cultures and regions utilizing various methods. Today, the metric system is widely used for its simplicity and
To convert tablespoons to liters, simply divide the number of tablespoons by 67.628, as there are approximately 67.628 milliliters in one tablespoon. This will give you the equivalent volume in
Understanding this conversion allows for more precise measurements when cooking or working with liquids in scientific experiments.
Estimating Water Displacement
Estimating water displacement can be a useful method for determining the volume of irregularly shaped objects. This method involves measuring the amount of water displaced by the object when it is
submerged in a container.
Here are some key points about water displacement methods and alternative units of measurement:
• Water Displacement Methods:
• One common method is to fill a container with a known quantity of water, then carefully place the object in the water and measure how much the water level rises.
• Another method involves using a graduated cylinder or a syringe to measure the exact amount of water displaced.
• These methods are particularly useful for objects that cannot easily be measured using traditional geometric formulas.
• Alternative Units of Measurement:
• The volume obtained through water displacement can be expressed in various units such as milliliters, liters, or even cubic centimeters.
• It is important to note that these units can be easily converted into other commonly used units, such as ounces or cups, depending on what is most convenient for your specific needs.
Experiment: How Many Tablespoons in a Cup
There’s a simple experiment to determine how many tablespoons are in a cup. To do this, we can use the accuracy of measuring cups and conversion factors for other cooking measurements.
Start by pouring water into a one-cup measuring cup until it is full. Then, carefully pour that water into a separate bowl while counting the number of tablespoons it takes to fill the cup. This will
give you an accurate measurement of how many tablespoons are in a cup.
It’s important to note that there are 16 tablespoons in a cup according to standard conversion factors. This experiment helps ensure the accuracy of your measuring cups and provides a useful
reference point for other cooking measurements involving tablespoons and cups.
Estimating the Number of Tablespoons in an Average Bathtub
When it comes to accurately measuring tablespoons, there are a few key points to consider.
First, using the correct size tablespoon can make a significant difference in your measurements.
Additionally, factors such as the type of ingredient being measured and how it is packed into the spoon can affect the accuracy of your estimation.
It is important to be aware of these factors in order to achieve precise measurements when using tablespoons in recipes or other applications.
Accurate Tablespoon Measurement
You can achieve accurate tablespoon measurement by using a measuring spoon for your bathtub filling experiment. When it comes to choosing the right measuring spoon size, there are a few options to
• 1 tablespoon: This is the standard size used in most recipes and should work well for most bathtub experiments.
• ½ tablespoon: If you want to be more precise with your measurements, this smaller size can be useful.
• ¼ tablespoon: For those who really want to get down to the nitty-gritty, this tiny spoon will allow you to measure out even smaller increments.
Precise measurements are important in recipes because they ensure that ingredients are used in the correct proportions. This can greatly affect the taste and texture of your dish.
The same concept applies when estimating how many tablespoons it takes to fill a bathtub accurately. By using a measuring spoon, you can eliminate guesswork and obtain reliable results.
Factors Affecting Estimation
Factors that can impact the accuracy of our estimation include the size of the measuring spoon and the consistency of the substance being measured.
When it comes to measuring liquids, like water, another important factor to consider is the impact of water temperature. Water expands when heated and contracts when cooled, which can affect its
volume and therefore the accuracy of our measurement.
For instance, if we measure a tablespoon of hot water and then let it cool down before pouring it into a container, there might be a slight difference in volume due to the contraction.
To minimize this effect, it is recommended to measure liquids at room temperature or use a calibrated measuring cup specifically designed for liquid measurements.
Factors Affecting the Accuracy of the Calculation
One of the biggest things that can impact the accuracy of this calculation is the size of the tablespoons being used. When measuring out water using tablespoons, it is important to consider these
factors affecting measurement precision:
• Variation in tablespoon sizes: Different brands or types of tablespoons may have slight variations in their capacity, leading to differences in volume estimation.
• Human error: Inaccurate measurements can occur due to human error such as pouring too quickly or not leveling off the tablespoon properly.
• Impact of water temperature on volume estimation: Changes in water temperature can affect its density and therefore its volume. Cold water tends to be denser than warm water, which means you
might need fewer tablespoons to fill a given volume.
Considering these factors will help ensure more accurate estimations when using tablespoons for measuring water.
Conclusion: The Answer to How Many Tablespoons It Takes to Fill a Bathtub
To summarize, it is important to consider all the variables discussed when estimating the number of tablespoons required to fill a bathtub. Factors such as water temperature and bathtub material play
a significant role in determining the accuracy of this calculation.
Water temperature affects its density, which in turn affects its volume. Warmer water has a lower density and occupies more space, requiring fewer tablespoons to fill the bathtub compared to colder
Bathtub material also influences the capacity and shape of the tub. Different materials have different thicknesses, which can affect how much water they can hold. Additionally, certain materials may
have irregular shapes or contours that impact the amount of water needed.
Considering these variables is crucial for obtaining an accurate estimate of the number of tablespoons required to fill a bathtub. By taking into account factors like water temperature and bathtub
material, one can ensure a more precise measurement and avoid any potential discrepancies.
Frequently Asked Questions
How Long Does It Take to Fill a Bathtub With Tablespoons?
To fill a bathtub with tablespoons, we need to know how much water the bathtub holds. It depends on the size of the bathtub. Additionally, it would be helpful to convert tablespoons to cups for
accurate measurements.
Are There Any Health Risks Associated With Filling a Bathtub With Tablespoons?
There are no health benefits associated with filling a bathtub with tablespoons. It is not a proper measurement for filling a bathtub and can be time-consuming and inefficient.
Can I Use Any Type of Tablespoon for This Experiment?
Yes, you can use any type of tablespoon for this experiment. However, it is important to note that different types of tablespoons may have slightly different measurements, which could affect the
accuracy of your results.
Are There Any Alternative Methods to Calculate the Number of Tablespoons It Takes to Fill a Bathtub?
There are alternative methods to estimate the number of tablespoons needed to fill a bathtub. These methods may vary in accuracy, but they can provide a rough estimate for your experiment.
How Accurate Is the Estimated Number of Tablespoons in an Average Bathtub?
To accurately measure the volume of a bathtub using tablespoons, variations in bathtub sizes need to be considered. It’s difficult to estimate the number of tablespoons without context, but we can
explore alternative methods for a more accurate calculation. | {"url":"https://evolvinghome.co/how-many-tablespoons-does-it-take-to-fill-a-bathtub/","timestamp":"2024-11-04T11:25:07Z","content_type":"text/html","content_length":"113453","record_id":"<urn:uuid:0f9d09d4-d76d-4a69-9e4c-82383a270bb4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00009.warc.gz"} |
Warning, /education/khipu/README is written in an unsupported language. File is not indexed.
0001 About Khipu
0002 ===========
0003 - Khipu is an advanced Mathematical function plotter application of KDE Education Project.
0004 - It is a replacement for KmPlot.
0005 - The basic idea of Khipu is to help teachers and Professors in understanding the behaviour of mathematical functions,surfaces etc.
0006 - Khipu can work in both 2D and 3D space.
0008 Basic Features
0009 ==============
0010 - Khipu can draw any 2D and 3D plots supported by Analitza (A mathematical library that Khipu uses for its backend)
0011 - User can save their work in .khipu file and later he/she can restore the work
0012 - If khipu is closed accidentally , the autosave feature will let the user to restore the unsaved work of the previous session.
0013 - User can add the plots from the files known as Plot-Dictionary file (.plots files). These files contain the name and equation of the plot.
0014 So, user does not have to type a big equation rather, he/she can plot the function from its name.
0015 - User can hide/show the plots and also he/she can remove/edit the existing plots.
0016 - User can save the plots as PNG image and also take the snapshot of the plots into the clipboard.
0017 - Aprat from this, khipu has numberous features to work with mathematical functions and spaces.
0018 - For more information and screenshots of Khipu , visit http://userbase.kde.org/Khipu
0020 Backend Information
0021 ====================
0022 - Analitza is a library that work with mathematical objects.
0023 - Analitza add mathematical features to your program, such as symbolic computations and some numerical methods;
0024 for instance the library can parse mathematical expressions and let you evaluate and draw them.
0025 - For more information : http://api.kde.org/4.x-api/kdeedu-apidocs/analitza/html/index.html
0027 How To Build/Run Khipu
0028 ===================
0029 - To successfully build the application, you need the following packages installed on your system
0030 1) Analitza (sudo apt-get install analitza-dev)
0031 2) QJson (sudo apt-get install libqjson-dev)
0032 3) libkdeedu (sudo apt-get install libkdeedu-dev)
0034 - To build the application,you need to type the following commands on the command prompt.
0035 1) cd <project_name_path>
0036 2) mkdir build
0037 3) cd build
0038 4) cmake -DCMAKE_INSTALL_PREFIX=`kde4-config --prefix` -DCMAKE_BUILD_TYPE=debugfull ..
0039 5) make
0040 6) sudo make install
0041 7) khipu | {"url":"https://lxr.kde.org/source/education/khipu/README?v=kf6-qt6","timestamp":"2024-11-02T15:19:29Z","content_type":"text/html","content_length":"8596","record_id":"<urn:uuid:01a77311-8c54-4842-8ca2-cd2ebbec440a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00849.warc.gz"} |
Skewness is the name for the asymmetry of a distribution about its mode. Negative skew, or left skew, indicates that the area under the graph is larger on the left side of the mode. Positive skew, or
right skew, indicates that the area under the graph is larger on the right side of the mode.
Incanter has a built-in function for measuring skewness in the stats namespace:
(defn ex-1-20 []
(let [weights (take 10000 (dishonest-baker 950 30))]
{:mean (mean weights)
:median (median weights)
:skewness (s/skewness weights)}))
The preceding example shows that the skewness of the dishonest baker's output is about 0.4, quantifying the skew evident in the histogram.
We encountered quantiles as a means of describing the distribution of data earlier in the chapter. Recall that the quantile function accepts a number between zero and one and returns the value of the
sequence at that point. 0.5 corresponds to the median value.
Plotting the quantiles of your data against the quantiles of the normal distribution allows us to see how our measured data compares against the theoretical distribution. Plots such as this are
called Q-Q plots and they provide a quick and intuitive way of determining normality. For data corresponding closely to the normal distribution, the Q-Q Plot is a straight line. Deviations from a
straight line indicate the manner in which the data deviates from the idealized normal distribution.
Let's plot Q-Q plots for both our honest and dishonest bakers side-by-side. Incanter's c/qq-plot function accepts the list of data points and generates a scatter chart of the sample quantiles plotted
against the quantiles from the theoretical normal distribution:
(defn ex-1-21 []
(->> (honest-baker 1000 30)
(take 10000)
(->> (dishonest-baker 950 30)
(take 10000)
The preceding code will produce the following plots:
The Q-Q plot for the honest baker is shown earlier. The dishonest baker's plot is next:
The fact that the line is curved indicates that the data is positively skewed; a curve in the other direction would indicate negative skew. In fact, Q-Q plots make it easier to discern a wide variety
of deviations from the standard normal distribution, as shown in the following diagram:
Q-Q plots compare the distribution of the honest and dishonest baker against the theoretical normal distribution. In the next section, we'll compare several alternative ways of visually comparing two
(or more) measured sequences of values with each other. | {"url":"https://subscription.packtpub.com/book/data/9781784397180/1/ch01lvl1sec14/skewness","timestamp":"2024-11-06T13:47:22Z","content_type":"text/html","content_length":"194557","record_id":"<urn:uuid:dd9d81a4-f492-4627-94a8-1ff11455c129>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00170.warc.gz"} |
Teachers Talking to Teachers - Connected Mathematics Project
Teachers Talking to Teachers
The following items are directly from experienced CMP teachers. Topics include:
Each item contains valuable insights and suggestions that can inform teachers' planning, teaching, assessing, and reflecting.
Are We Ever Going To Get There?
From the 2019 CMP Users' Conference
Presented by Mary Beth Schmitt
Are you feeling overwhelmed that there is never enough time? Explore with Mary Beth Schmitt strategies for pacing and prioritizing lesson designs and implementation so that students experience more
in a year.
Developing Efficient Algorithms: How Do Models Help?
Dividing Fractions: Let's Be Rational Problem 3.1 and 3.2
From the 2017 Users' Conference
Presented by Carolyn Droll & Jan Robinson
How can we think of making models as more than just drawing pictures? Using Let's Be Rational Problem 3.1 and Problem 3.2, two CMP teachers look at and analyze how students’ models can lead to
division of fraction algorithms.
What's Problem 1.1 All About
From the 2016 Users' Conference
Presented by Cynthia Callard & Jennifer Kruger
Have you ever looked at the first problem in a CMP unit and wondered why these questions are being asked of students first thing? Have you ever wondered how students are expected to engage in these
problems so early on in a unit?!? What is the purpose of these problems?
CMP teachers Cynthia and Jennifer, discuss a variety of purposes and possibilities for the use of Problem 1.1 and how to get the most “bang for your buck,” including how Problem 1.1 can be used as a
formative assessment tool.
Technology and CMP
From the 2016 Getting to Know CMP Summer Workshops
Presented by Shawn Towle & Karrie Tufts
Shawn and Karrie share strategies for implementing the use of technology in CMP classrooms. The teachers share their decision making process for choosing technology resources as well as their
experience in one-to-one classrooms.
Changing Classroom Routines
Strategies that Engage Students
From the 2016 Users' Conference
Presented by Kay Neuse
Are you tired of the same old routine? In the video of this session, Kay suggests way to shake it up a bit by incorporating some instructional strategies that are engaging for the students. Snowball
fights, Secret Spies, and Celebrity Quizzes are only a few of the examples that will get your students motivated and smiling.
Jeopardy-Like Game Show for CMP Classrooms
Created by a CMP Teacher
This powerpoint was designed for teachers to use during or at the end of a Unit or as a review at the end of the year. Categories, questions, and answers can all be changed so teachers can adapt the
game to the needs of their students.
This is a powerpoint version of the Math Fever game from the Grade 7 Unit, Accentuate the Negative. The powerpoint was created for the CMP2 version of the MathMania. It was inteded to review topics
prior to Accentuate the Negative as a way for the class to explore getting "negative" and "positive" points. The current powerpoint includes the following topics from Grades 6 and 7: Operations with
Fractions; Similarity; Probability; Area and Perimeter; Factors and Multiples.
Using Talk Moves to Support Mathematical Discussions in a CMP Classroom
From the 2016 Users' Conference
Presented by Cynthia Callard & Jennifer Kruger
Engaging students in mathematical discussions in a CMP classroom is an important part of the learning process. While students may be engaged with the problem and finding a solution, they sometimes
are reluctant to share their strategies or questions.
Two CMP teachers explore some powerful “talk moves” (Chapin, O'Connor and Anderson, 2009) that they use with their CMP classes.
From the 2014 and 2015 Getting to Know CMP Workshops
Presenters: Experienced CMP Teachers
In these videos, each teacher shares strategies for maintaining and assessing student notebooks. The teachers also discuss ideas for grouping students and getting students to participate in note
taking, mathematical reflections, and taking ownership of their learning.
From the 2010 CMP Users' Conference. Videos from a variety of presenters.
Preparing, Questioning, and Scaffolding
From the 2010 Users' Conference
Presented by Whitney Evans and Jim Wohlgenhagen
This video provides “key ingredients” for preparing an effective lesson. The presenters talk specifically about preparing questions to help students access the mathematics at differentiated levels.
Formative Assessment
From the 2010 Users' Conference
Presented by Whitney Evans, Jim Wohlgehagen, and Jenny Jorgensen
This video provides a few strategies for gathering information from students. The purpose for using these strategies is to assess the students learning to make instructional decisions that meet the
needs of all students.
New CMP Teacher Suggestions
From the 2010 Users' Conference
Presented by Yvonne Grant, Teri Keusch, and Jim Mamer
Questions asked by Ryan Hoffman (a student teacher in a CMP classroom), Jill Newton (assistant professor at Purdue University who supervisors student teachers), and audience members.
A beginning CMP teacher and a professor who works with beginning CMP teachers ask questions of long-time CMP teachers. | {"url":"https://connectedmath.msu.edu/teacher-support/teachers-talking-to-teachers.aspx","timestamp":"2024-11-13T09:01:16Z","content_type":"text/html","content_length":"73648","record_id":"<urn:uuid:1ff17da8-29b9-47aa-9aac-4691f62b8a26>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00656.warc.gz"} |
Find The Missing Angle Worksheet 4th Grade - Angleworksheets.com
Find The Unknown Angles Worksheets – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help you understand the
different concepts and build your understanding of these angles. Using the vertex, arms, arcs, and complementary angles postulates, students will learn how to find … Read more
Find The Missing Angle Worksheet 4th Grade
Find The Missing Angle Worksheet 4th Grade – If you have been struggling to learn how to find angles, there is no need to worry as there are many resources available for you to use. These worksheets
will help you understand the different concepts and build your understanding of these angles. Using the vertex, arms, … Read more
Find The Missing Angle Worksheets
Find The Missing Angle Worksheets – There are many resources that can help you find angles if you’ve been having trouble understanding the concept. These worksheets will help to understand the
various concepts and increase your knowledge of angles. Students will be able to identify unknown angles using the vertex, arms and arcs postulates. Identifying … Read more | {"url":"https://www.angleworksheets.com/tag/find-the-missing-angle-worksheet-4th-grade/","timestamp":"2024-11-08T22:15:58Z","content_type":"text/html","content_length":"59199","record_id":"<urn:uuid:f631306d-3540-4002-a3d6-489eed01e53e>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00845.warc.gz"} |
decimal.js vs mathjs: Detailed NPM Packages Comparison | Performance, Security & Trends
Decimal.js is a JavaScript library for arbitrary-precision decimal and non-decimal arithmetic. It allows precise handling of decimal numbers without the limitations of native JavaScript number
handling, ensuring accurate calculations even with very large or very small numbers. Decimal.js provides a rich set of mathematical operations and functions for working with decimal numbers, making
it ideal for financial calculations, scientific computations, and any application requiring high precision arithmetic.
Tags: javascriptarbitrary-precisiondecimal-arithmeticmathematicsfinancial-calculations
Math.js is a comprehensive mathematics library for JavaScript that provides a wide range of mathematical functions and utilities. It enables complex mathematical operations such as algebraic,
arithmetic, trigonometric, statistical, and matrix calculations, making it a versatile tool for mathematical computations in web applications. Math.js also supports units and physical constants,
allowing for advanced calculations in various domains.
Tags: javascriptmathematicscomputationalgebratrigonometry
Both Decimal.js and Math.js are popular npm packages for working with mathematical operations. Math.js has a larger user base and is more widely known in the JavaScript community, while Decimal.js is
also well-regarded but may have a slightly smaller user base.
Decimal.js is specifically designed for precise decimal arithmetic and provides a comprehensive set of methods for working with decimal numbers. It offers features like arbitrary precision, rounding,
and formatting. Math.js, on the other hand, is a more general-purpose math library that supports a wide range of mathematical operations, including complex numbers, matrices, and symbolic
Ease of Use
Decimal.js has a simple and intuitive API, making it easy to use for basic decimal arithmetic. It focuses on providing precise decimal calculations without the need for additional configuration.
Math.js, being a more comprehensive math library, has a larger API surface and may require more configuration for specific use cases. It offers a powerful and flexible API but may have a steeper
learning curve.
In terms of performance, Decimal.js is optimized for decimal arithmetic and provides efficient algorithms for precise calculations. Math.js, being a more general-purpose math library, may not have
the same level of performance for decimal operations. However, for most use cases, the performance difference may not be significant.
Decimal.js has minimal dependencies and is a lightweight package. Math.js, on the other hand, has more dependencies due to its broader functionality. This may be a consideration if you are concerned
about the size and complexity of your project's dependencies.
Community and Maintenance
Both Decimal.js and Math.js have active communities and are well-maintained. Math.js has a larger community and a more active development cycle, which means it receives regular updates and bug fixes.
Decimal.js, while still maintained, may have a slower release cycle. | {"url":"https://moiva.io/?npm=decimal.js+mathjs","timestamp":"2024-11-11T23:21:42Z","content_type":"text/html","content_length":"42396","record_id":"<urn:uuid:77525d25-e492-42af-bc20-74148e109fe0>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00834.warc.gz"} |
SEEMA H.
What do you want to work on?
About SEEMA H.
Algebra, Statistics
Math - Statistics
It was great
Math - Quantitative Reasoning
Appreciate the tutor assisting me even though subject matter was out of scope. That showed me individual cared about me the student getting much needed help.
Math - Algebra
Helped me clarify some misunderstanding and guided me on equations I was confused about. :D
Math - Algebra
you were great | {"url":"https://testprepservices.princetonreview.com/academic-tutoring/tutor/seema%20h--3211250","timestamp":"2024-11-07T06:40:45Z","content_type":"application/xhtml+xml","content_length":"237544","record_id":"<urn:uuid:aa3426cb-58db-493f-ad73-46ef35b6536e>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00563.warc.gz"} |
The general condition in which resources are willing and able to produce goods and services but are not engaged in productive activities. While unemployment is most commonly thought of in terms
of labor, any of the other factors of production (capital, land, and entrepreneurship) can be unemployed. The analysis of unemployment, especially labor unemployment, goes hand-in-hand with the
study of macroeconomics that emerged from the Great Depression of the 1930s. The most common measure of unemployment is the unemployment rate of labor. Unemployment is one of two primary
macroeconomic problems. The other is inflation.
Unemployment arises when scarce resources that COULD be used to produce goods and services, resources that are WILLING and ABLE to engage in production, are NOT producing output. While the economy
always have some degree of unemployment, it tends to become most severe and hence most problematic during a business-cycle contraction.
Historical Numbers
As indicated in the exhibit to the right, the unemployment of labor, measured by the unemployment rate, varies over time, especially over the course of business-cycle activity, rising during
contractions and falling during expansions. The range is usually between 4 percent and 6 percent, but it has been as low as 2 percent and as high as 25 percent. During the contraction of the early
1990s, the unemployment rate rose from 5 percent to nearly 7 percent. In the ensuring expansion that occupied the better part of the 1990s, the unemployment rate fell from the 7 percent level to just
over 4 percent. During the contraction of the early 2000s, it rose from 4 percent to over 6 percent.
While the unemployment rate reaches relatively low levels during expansions, it never falls to 0 percent. In principle, full employment is thought of as occurring when ALL resources (especially
labor) are engaged in production, in practice, full employment generally corresponds to an unemployment rate of about 5 percent. This 5 percent unemployment rate, often termed the natural
unemployment rate, includes both frictional and structural unemployment.
Why Study?
Unemployment, especially labor, is a key macroeconomic issue that has concerned economists since at least the Great Depression (another key issue is inflation). The devastating economic conditions of
the 1930s, which at its depth saw one out of four workers unemployed, brought to the forefront the problems of unemployment. Two in particular that stand out are personal hardships and lost
• First, unemployment creates personal hardships for the owners of the unemployed resources. When resources do not produce goods, their owners do not earn income. The loss of income results in less
consumption and a lower living standard. While this problem applies to any of the resources, it is most important for labor. The owners of capital, land, and entrepreneurship often earn income
from more than one resource. Many workers, however, often earn income only from labor.
• Second, unemployment causes total production in the economy to decline. If fewer resources are engaged in production, fewer goods and services are produced. As suggested by the circular flow
model, the severity of the connection between lost production and unemployment is magnified by the multiplier effect. An initial decline in income, consumption, and production associated with
unemployment triggers further declines in income, consumption, and production. As such, members of society escaping the direct, immediate personal hardships of unemployment can succumb to the
indirect, multiplicative problems of lost production.
A Graph or Two
Production Possibilities
Aggregate Market
Unemployment can be illustrated with two common economic models--production possibilities and the aggregate market (or AS-AD analysis). Both graphical models are presented in the exhibit to the
• Production Possibilities: Unemployment is illustrated with production possibilities analysis as any production combination that places the economy INSIDE the production possibilities frontier.
This is demonstrated in the top panel of the accompanying exhibit.
• Aggregate market: In the aggregate market, unemployment is illustrated with a recessionary gap in which that short-run equilibrium--the intersection of the short run aggregate supply curve (SRAS)
and the aggregate demand curve (AD)--lies to the left of the long-run equilibrium or long-run aggregate supply curve (LRAS).
Keeping track of unemployed resources is not as easy as it might seem. Unemployed labor is relatively easy to track by simply "counting heads." However, even this head counting is not without
problems. Capital is such a diverse resource that no single measure can ever hope to capture the full extent of unemployment. Land, especially the vast types of natural resources associated with the
land, encounter similar problems. And entrepreneurship is such a nebulous resource (a worker today, might be an entrepreneur tomorrow, then once again a worker the next day) that just identifying
entrepreneurship is extremely difficult, let alone determining its unemployment.
However, attempts to measure unemployment of various resources is pursued. Three common unemployment measures are: unemployment rate, capacity utilization rate, and vacancy (or occupancy) rate.
• Unemployment Rate: The most noted and widely used measure of resource unemployment is the unemployment rate of labor. In fact, its widespread used is the reason that most people associate
unemployment ONLY with labor. The unemployment rate is the percent of the labor force that is officially unemployed. This rate is reported monthly by the Bureau of Labor Statistics of the U.S.
Department of Labor, and it is clearly the popular choice when it comes to measuring business-cycle instability.
• Capacity Utilization Rate: A somewhat lesser known indicator is the capacity utilization rate for the measurement of capital unemployment. This is actually a measure of capital employment, but it
provides a great deal of insight into the unemployment of capital. Specifically, the capacity utilization rate is the ratio of actual production undertaken by factories to potential production.
If the capacity utilization rate is up, then the unemployment of capital (at least the factory variety of capital) is down.
• Vacancy Rate: A popular measure for the real estate industry that also tends to get little notoriety in the rest of the world is the vacancy rate for buildings. It measures the percent of
available buildings (office building, apartments, etc.) that are vacant and thus have unemployed capital (rooms). A complementary measure is the occupancy rate, which (like the capacity
utilization rate) measures the extend to which buildings are used.
A modern complex economy like that in the United States does not have just one type of unemployment. Four basic sources or reasons for resource unemployment exist: cyclical, seasonal, frictional, and
structural. Once again, while these apply to all four factors of production, they tend to be most important for labor.
• Cyclical Unemployment: This is unemployment attributable to a general decline in macroeconomic activity occurring during a business-cycle contraction (or recession). When aggregate demand
declines, less aggregate production is sold, so fewer resources are needed. Because cyclical unemployment is considered an avoidable source of unemployment, its reduction or elimination of has
been one of the primary goals of macroeconomic policy since the Great Depression.
• Seasonal Unemployment: This is unemployment caused by relatively regular and predictable declines in particular industries or occupations over the course of a year, often corresponding with the
seasons. Many construction workers face unemployment during winter months. School teachers face unemployment during the summer months. The employment of farm workers varies with seasonal planting
and harvesting activities. Because seasonal unemployment is regular and predictable, it is generally considered part of the "conditions of employment," and is largely ignored in the study of the
• Frictional Unemployment: This is unemployment that results because resources are in the process of moving from one production activity to another. Frictional unemployment occurs because it takes
time to move between production activities. The time needed to match up resources with production depends on information availability and the degree of geographic separation. A carpenter is
frictionally unemployed even though a construction company has a job for a carpenter because neither knows about the other. Or perhaps the carpenter lives in North Dakota and the construction
company is in North Carolina. In either case, the carpenter remains frictionally unemployed until matching up with the job. And this matching up just takes time.
• Structural Unemployment: This is unemployment occurring because resources do not have the technological configuration, skills, or training required by production activity. For example, the
construction company in North Carolina needs a carpenter, but the only available unemployed worker is a plumber. If the only jobs available to the plumber are in carpentry, then the plumber is
structurally unemployed.
The key to frictional and structural unemployment, especially for labor, is that the number of jobs available is the same as the number of workers. In other words, the quantity demanded equals the
quantity supplied. The problem is that the workers and the jobs do not match up. Either information is lacking or skills are incompatible.
Because both frictional and structural unemployment are an inherent part of any complex economy, they are often referred to as natural unemployment. The economy always has some degree of frictional
and structural unemployment. In fact, having some degree of frictional and structural unemployment is not necessarily a bad thing.
Whether frictional and structural unemployment are good or bad, when combined they form the basis for a key benchmark for macroeconomic policy analysis termed the natural unemployment rate. Because
the natural unemployment rate can be sustained with no changes in inflation, it provides an excellent target for macroeconomic policy.
Given that people would rather NOT have unemployment and that it tends to increase from time to time, an assortment of government policies have been devised to reduce unemployment. The most noted,
fiscal policy and monetary policy, fall under the general heading of stabilization policies that are designed to stabilize business-cycle fluctuations and in so doing lessen the problems of both
unemployment and inflation.
• Fiscal Policy: Fiscal policy is the discretionary use of government spending and taxes to affect business-cycle fluctuations. The recommended fiscal policy for reducing unemployment is to
increase government spending and/or to decrease taxes. When undertaken by the federal government, either or both of these actions lead to an increase in the federal deficit or a decrease in the
federal surplus. This goes by the specific name expansionary fiscal policy.
• Monetary Policy: Monetary policy is the discretionary use of the money supply and interest rates to affect business-cycle fluctuations. The recommended monetary policy for reducing unemployment
is to increase the money supply and to decrease interest rates. This goes by the specific name expansionary monetary policy.
<= UNEMPLOYED PERSONS UNEMPLOYMENT COMPENSATION =>
Recommended Citation:
UNEMPLOYMENT, AmosWEB Encyclonomic WEB*pedia, http://www.AmosWEB.com, AmosWEB LLC, 2000-2024. [Accessed: November 3, 2024].
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Linux vs Windows: Choice vs. Usability
Home > Linux > Linux vs Windows: Choice vs. Usability
Linux vs Windows: Choice vs. Usability
Submitted by Wally 2003-08-14 Linux 102 Comments
A Recent DevX editorial makes the (often made) claim that Linux’s lack of a single standard UI will hamper its adoption on the desktop and makes developing applications for Linux more difficult.
Hard-core Linux users love having the choice of many operating environments, and they are hardly likely to resolve the KDE vs Gnome argument anytime soon. Is there any hope of more standardization?
Should we even want it?
102 Comments
1. 2003-08-14 5:30 pm
most people could care less about what OS is running on their computer, and they do not want to leave their “comfort zone” to learn how to install/configure and use a new OS…
2. 2003-08-14 5:31 pm
3 … 2 …. 1 …. FIGHT !!!
3. 2003-08-14 5:32 pm
The general concept of having choice between operating envrionments is great, and is one of the things that make Linux such an awesome OS. However, imho, the main problem is the hardware
interface. Sure, you have the kernel and all the stuff in that. But there should be some sort of PnP written directly into it.
That way, you can have whatever filesystem you want, whatever desktop you want, GDK+ or Qt, it doesn’t matter. BUT, your hardware will work no matter what. Hardware compatability is the number
one difficulty on Linux, besides compiling issues…
Once that’s resolved, LLL (Long Live Linux).
4. 2003-08-14 5:32 pm
Should not use linux until they no longer have any choice in the matter. By then it will work identical to whatever they’re currently using. It will have to. You can’t teach an old dog new
5. 2003-08-14 5:35 pm
Ever noticed that the widgets and styles in Outlook are completely different than other system apps like Explorer?
There are still many apps out there being developed that only use old-school (pre xp) widgets because using the new standards can introduce problems that wouldn’t be there otherwise.
Windows lacks consistancy just as much as Linux does. The supposed ‘standard’ UI in Windows is a myth.
6. 2003-08-14 5:39 pm
Talk all you want about Gnome and KDE… I’m happy with Fluxbox.
BTW, bets on how long it takes top_speed to get here?
7. 2003-08-14 5:39 pm
Okay… don’t have time to read the article now but I think I’m gonna feel sorry for A. Russell Jones (the author).
1001 flame wars right?
8. 2003-08-14 5:41 pm
Plug and play? It’s built into the kernel… And I personally have had no problems whatsoever with hardware until now. Filesystems? What are you talking about? You can choose between a whole lot of
filesystems – ReiserFS, Ext 3/2, even lousy old VFAT. I’m not too sure you’ve ever used Linux, and as for compiling issues there are binaries available for thousands of packages, and compilation
only comes into play rarely and that too for lesser known packages, and sometimes when you wish to customize things, unless you’re using a source-based distro, that is.
9. 2003-08-14 5:42 pm
Windows lacks consistancy just as much as Linux does. The supposed ‘standard’ UI in Windows is a myth.
I think you missed the point of the article. The point it, while little things are different, you don’t have to worry if the program won’t work at all, or have clipboard problems with other
programs. True, a modern commercial distribution will happily run KDE, Gnome, etc, the integration between programs of different bases is lacking.
10. 2003-08-14 5:42 pm
Apple violates their own standards too
Nobody follows HIGs 100% because that would be stupid and in a lot of cases generate apps with considerably less usability.
The myth that people must have 100% consistant interfaces to make computers easy to use is just that…a myth! Cars don’t even have 100% consistant interfaces, same as consumer electronics…does
that make an RCA any harder to use, even though I’m used to Sony? However, the basic concepts are all pretty much the same, and that’s the important thing.
People should HAVE to learn how to use advanced features, but still be able to intuitively discover the basic features within a few minutes of play time.
11. 2003-08-14 5:43 pm
well i don’t think that it’s an issue it a matter of distro and not development , by choosing one main desktop (e.g suse) u can gain standard ui (kde apps) maybe it’s a hard work but company that
want to make money got to work for this . linux can be this or can be that but if a user don’t want choice he wouldn’t get it and just be left with the distro default.
12. 2003-08-14 5:46 pm
You said “You can’t teach an old dog new tricks.”
I am proof that you can teach an old dog new tricks.Linus is different but i dont find it hard to learn.I am not a tech person in any way.I admit i only do email,web surf and chat but since
learning linux i now also have learned how to put my cds on the computer as mp3 so i can listen to them.I never did that with windows.
13. 2003-08-14 5:51 pm
The one thing I do have an issue with is the authors idea that every program should have an installer/deinstaller. This makes no sense, as developers shouldn’t have worry about installers. The OS
should handle all insts so it’s centrallized. But then you have to have packages that are *very* sepcific, due to incompatability.
The main problem with Linux software is the wild incompatibility between versions. What really needs to happen less that going to one master GUI is the software interfaces need to be more stable.
Every commercial distribution is going to have libqt and libgtk2, but when the interfaces between libgtk2.6.0.1 and libgtk2.6.1.0 are wildy different, it makes a mess.
In some ways, the Mac install makes the most sense for ease of install and compatibility. It’s a security risk though. If there’s a bug in one of the included libs, you have to find all the
different versions in all the program folders to fix it, and hope that the new version doesn’t have some minor API changes that break the program.
In short, release often, but the API should only change at major releases, and backwards compatibility is key.
14. 2003-08-14 5:51 pm
I haven’t read the article; I don’t believe there’ll be anything new in it… If a user isn’t interested in choices I don’t believe he’ll venture beyond the distro’s default Desktop Environment/
Window Manager. And AFAIK every distro comes with a default DE/WM… As for cutting & pasting why don’t they just advertise the highlight and middle-click feature more often? If it works between
Mozilla and a terminal window it should work between QT and GTK+ apps. Apologies in advance if I’m mistaken about this, since I don’t use any QT apps I wouldn’t know…
15. 2003-08-14 5:54 pm
Linux is all about choice. You dont have a single entity dictating how you want your OS to look and how it functions. Some standards are good of course, but as far as my UI is concerned, I want
it my way. And the beauty of the whole thing is that there are several to choose from. Use what best fits your tastes or needs. If youre JohnQ user and dont give a crap about any of it then just
stick to what your distro selects as default and enjoy the free software.
16. 2003-08-14 5:56 pm
X is the true graphical standard of Unix and Linux. It is stable, interoperable and powerfull. Yes, it doesn’t define windows borders or widgets but it permits that you run a KDE application on
Gnome and vice-versa.
For development, X libraries and POSIX are standard across Unix, Linux and many other operating systems. M$ is the only exception. Who is wrong ?!
KDE or Gnome are not difficult for Joe users. Run linux from Knoppix is easier than install Windows.
The problem of home users is inertia, piracy (use a pirated Windows full of pirated applications is easier than learn linux) and lack of offer of pre-installed linux boxes and native comercial
But it is a question of time…
17. 2003-08-14 5:58 pm
This arguement keeps popping up and I don’t quite get it. The author seems to think that turning Linux into Windows is a good thing. Part of what makes Linux interesting and dynamic is that there
is no central ruling body. Linux is doing well because things are constantly branching. Some of these branches become dead ends, but sometimes they become the new trunk. Sometimes, as with KDE
and Gnome, there are multiple large branches rather that a main trunk.
Now, I think some attention does need to be put towards getting these environments to play well together (applications written for one environment run well on applications written for another).
The menu systems and cut-and-paste are being worked on by freedesktop.org. Notice that this initiative (as well as LSB) is just an area where competitors have decided to work together for their
common good, not the result of a single controlling body.
18. 2003-08-14 6:03 pm
This is a good example of yet another person who comes from the Windows world and doesn’t understand the Linux world, thus assumes that our approach is wrong.
(BTW it’s “X”, or “The X Window System”, not “X-Windows”)
There are only two major GUI toolkits: QT and GTK+. Motif? That’s a joke, almost nobody uses it anymore. I’d be surprised if a Linux desktop user has more than 3 Motif apps on his system, if at
It doesn’t matter for which toolkit or desktop you program. Your app *will* run correctly on any desktop (or no desktop!) if the right libraries are installed. On all of the popular modern
desktop distributions, both the GNOME and KDE libraries are installed by default.
Vendors guarantee support for one or two environments? That’s more than enough. There only are two major environments. Anybody who uses a different environment are power users and already know
how to get GNOME or KDE apps working in their environment (which is usually as simple as installing the right libraries. I repeat, those libraries are usually already installed by default).
Supporting both GNOME and KDE is not difficult. I know, at the Autopackage project we try to support both environments. It’s fairly trivial and usually only involves copying files to the right
And 2 of the most popular desktop distros, RedHat and Mandrake, use unified themes, making both toolkits look the same.
The correct solution is not to standardize on one GUI, it’s to standardize on an interface and make sure both environments are interoperable and compatible. More and more stuff are getting
compatible. Just look at the Freedesktop.org efford.
As for why Linux is different and has more than one GUI: it’s because people have freedom. If you want people to come together to work on one single environment then you forget one big thing:
people are not equal. Different people have different ideas, design philosophies, aestetic preferences, etc. One size does not, and cannot, fit all! Are you going to tell me that everybody loves
the Windows UI? I know more than enough people who absolutely hates it.
The author argues that the average user doesn’t care about choosing a GUI. Well, good for them. They don’t have to choose! They can use whatever desktop is set as default by their distribution.
Don’t know what GNOME or KDE is? Don’t worry, just click on OK and use the default setting.
It doesn’t matter which one you choose. As I’ve set before, either desktop will allow *all* apps to work correctly! The correct libraries are most likely already installed!
Next, the author is complaining users will get confused by new terms. Well of course, Linux is an entire different operating system. The only thing that’s intuitive is a nipple, every thing else
is learned. Do you think new computer user knows what a “window” is? Or what “Run” means? Heck, when I started using a computer I was confused by the words “Programs”, “Shutdown”, and even the
whole concept of installing software was confusing! All the menus and toolbars confused the hell out of me.
Expecting people to use new things without learning anything is stupid and counterproductive. If they don’t understand the basics, then teach them. It’s not like a total computer novice can
understand Windows either!
And normal users are not IceWM’s target group. GNOME and KDE are. On all popular desktop distros, either GNOME or KDE are set as default. So use the default setting!
“but highly insensitive to the needs of average, technically (and sometimes literally) illiterate users.”
This is a big joke. Look at the GNOME project: they’re exactly targeting average users! The GNOME project is going for simplicity, and listens to all the suggestions UI designer gives. Just
subscribe to their mailing lists and see.
Software freedom may not be a big advantage to the average user, but it *is* an advantage to a lot of power users. Do not underestimate the number of power users out there! There are a lot of
them, even among the Windows community.
Most people don’t use their legal freedoms to their pull potentials either, but that doesn’t mean that having those freedoms are not a good idea. Same for software freedom. Just because most
people don’t use them doesn’t mean it isn’t a good idea.
Perhaps the biggest mistake the author makes is to compare Linux to commercial systems. Most of the Linux system is built by *volunteers*. If Linux is a commercial system, it would never have
been this big and powerful. So instead of complaining about that volunteers don’t do what you want (they are, after all, just volunteers), how about hiring more programmers that work on Linux
Instead of thinking “the open source project must do this and that”, that’s jst selfish. Think “the volunteers have helped a lot. but I want it better, so I’ll hire more programmers”. Volunteers
don’t get any reward so you shouldn’t expect too much from them.
19. 2003-08-14 6:05 pm
Ya, I agree. I didnt care for the author or his article at all. His opinion is pretty lame at best and pretty much flies in the face of what open source software is all about. Its almost like he
wants an OS/2 to re-emerge or a Microsoft the sequel of some kind.
20. 2003-08-14 6:06 pm
“For development, X libraries and POSIX are standard across Unix, Linux and many other operating systems. M$ is the only exception. Who is wrong ?! ”
X is bloated as hell. Unefficient. Low performance. And standard limited to *NIX family and derivated.
Unix+Posix != World.
M$ the only exception ? Pleeeeease … I don’t know why, *EVERY* OS I know that is NOT based on X are pretty much *ALL* superior to X based GUI : Windows, OSX, BeOS, EPOC, etc.
21. 2003-08-14 6:11 pm
Who cares if there’s no *ONE* gui interface for linux…
A good 90+% of computer users just use what they’re given. If a distribution packages everything together nicely enough, and it works *out of the box*… then who cares if KDE, GNOME, Flux, XFCE,
etc. are out there as alternatives? The user probably isn’t gonna mess with anything…they’ll just use the stock build.
Having KDE/GNOME/etc. alternatives just mean that the power user can tailor his/her environment to their liking. It’s not unlike having a skinnable interface…or the many customizing programs for
windows from STARDOCK.com
Inconsistencies or not… if it WORKS…it WORKS… no one’s gonna care that there are alternatives….
22. 2003-08-14 6:11 pm
Personally, I don’t see why everyone is so obsessed with Linux becoming a mainstream desktop OS. Getting non-geeks to use anything other than their familiar Windows will be like trying to get a
dog to “meow.” Leave them to their Office Assistants, InstallShields, and mass-mailing worms. Linux should stay geared towards being used as a server OS, as a desktop OS for Linux enthusiasts, or
maybe even as a desktop OS in a business environment. Stop trying to shove Linux down Grandma’s and Auntie Gertrude’s throats because they’re not going to be happy if they can’t forward that cute
little e-mail attachment of the dancing baby.
23. 2003-08-14 6:13 pm
You can have both choices and good usability!
24. 2003-08-14 6:18 pm
As I am typing this I am in an office in a hallway full of offices where I am updating a Windows machine. The one thing none of the users have here is freedom to modify the OS, and that is a good
Nobody wants to play with the operating system. Nobody wants to learn a new operating system. Nobody wants to configure, adjust, re-script or compile. They want to do their job, which is
accounting. That’s it.
The machines here are not locked down; every user is local admin. Guess what? Nobody even knows or cares what that means. Most of them have not bothered to change the wallpaper.
When it comes to getting work done in the desktop arena, choice and freedom is simply not what people are looking for. They want a simple tool to get them thu to 5 o’clock so they can go home.
25. 2003-08-14 6:21 pm
“”Cars don’t even have 100% consistant interfaces””
Not 100%, but they are subject to various standards, regulations, and current user knowledge that force consistency.
Before one jumps into a new car that has a manual gearbox one already knows that there will be a steering wheel. One knows there will be 3 pedals of which the left will be the clutch, the middle
will be the brake and the right will be the accelerator. One know there will be a gear lever that conforms to one of a small number of standard configurations. One knows that there will be a
mechanism for using the windscreen wipers, indicators, warning lights, headlights, foglights etc.
The car is instantly useful to someone who has already used another car. Perhaps it may take some time to learn the idiosyncrasies of the particular model of car, but those idiosyncrasies provide
no bar to using the car itself.
There is a huge amount of standardisation across the automotive industry, it’s just so commonplace that we no longer pay attention to it. This happy scenario was not always the case, take a look
at some old vintage cars and prepare to be confused if the owner will let you drive them.
“”same as consumer electronics””
‘Play’, ‘Stop’, ‘Forward’, ‘Reverse’, ‘Pause’, ‘Eject’, ‘Record’ all have instantly recognisable standard symbols that are used throughout the audio industry.
“”People should HAVE to learn how to use advanced features, but still be able to intuitively discover the basic features within a few minutes of play time. “”
No. Users should expect to be instantly able to use the system, because the steps they expect to perform in order to generate a specific action should be the same across the industry for basic
functionality. There should be no discovery involved.
Each system should have its own means of performing advanced operations, but basic operations should conform to a standard. Standards DO help the user adapt more easily to new situations by
allowing them to reuse skills they have already learnt.
HIGs are a necessity, and it is important they are followed. The current scenario has the user playing “guess the peddle” in order to find the accelerator. This is fine for the advanced user, but
does not assist the new (To the particular system) user to use the system productively.
26. 2003-08-14 6:23 pm
Lots of desktop distros do ship with only one GUI, LIndows, Xandros, Lycoris are jsut a few examples and others sucha s SUSE and Redhat by default install only one GUI.
The problem is not the choice in itself, if the user is not really interested he or she will just go to the default GUI selected by the distribution and only if it won’t serve their needs will
they even look at anohther. The problem is more the fact that applications are hard to code to fit in with the look n feel and design of more than one GUI such as XFCE, KDE and GNOME. Therefore,
users will feel the desktop is not well integrated. They shouldn’t really feel the difference betweena GTK and KDE application, at least in behaviour and look n feel.
Otherwise it is great to have choice of GUI, but as I said as long as applications act the same on all of them.
27. 2003-08-14 6:24 pm
Unefficient. Derivated.
“Me fail english? That’s Unpossible!”
— Ralph Wiggum
28. 2003-08-14 6:48 pm
So if windows users need the same GUI, then why are there so many programs that will change the GUI? What’s the point of having the same interface for each linux? I believe that there should be a
universal GUI on all Linux/BSD’s…oh wait there is KDE, GNOME!!!
29. 2003-08-14 6:58 pm
Freedom of choice –Devo
30. 2003-08-14 7:00 pm
X is bloated as hell. Unefficient. Low performance. And standard limited to *NIX family and derivated.
Is it really? I’m in my apartment right now working remotely. I have a couple of xterms’, emacs and a matlab plot open on my desktop, all running on a remote computer at school and being
displayed on my laptop here in the apartment. I’m doing this with a cable modem connection, and I have music streaming with RealPlayer which is consuming some bandwith….
Give me a break…. why don’t you list your credentials.
31. 2003-08-14 7:04 pm
Forget anything with icons. Usually something with icons excluding enlightenment is slow. I love blackbox, it is so sossososososososososososososososososososososososososososososos
ososososososososososososo FAST!
32. 2003-08-14 7:13 pm
HIGs are a necessity, and it is important they are followed. The current scenario has the user playing “guess the peddle” in order to find the accelerator. This is fine for the advanced user, but
does not assist the new (To the particular system) user to use the system productively.
How do you figure? Most popular computer UI elements ARE close enough. If you’ve only ever used Windows and suddenly sit down to a set of Motif apps, odds are good you’ll figure the basics out.
Saying that users should be able to instaly use full application functionality without LEARNING it is like saying anyone should know how to program their VCR without learning it.
Basic features like open, save, print, cut, copy, and paste all work more or less the same. The core widgets (menus, buttons, checkboxes, radio buttons, text boxes, etc.) likewise all work more
or less the same. Saying that Motif is automatically harder to use than Windows because it is different is like saying a car with a column shifter is harder to use than one with a floorboard
shifter, just because its not what you’re used to.
UI consistancy is NOT holding Linux back, it just makes it look like a Raggedy Ann doll sometimes.
What’s holding Linux back is that it’s not Windows and people won’t put effort into learning something new until they’re given a better reason to. How long did it take Microsoft (with full
backwards compatibility, mind you) to get Windows off the ground? (Here’s a hint, how many of you have ever seen a running copy of Windows 1.x or 2.x?)
33. 2003-08-14 7:15 pm
Sure, blackbox is so fast, but what about the apps that you’re using. Even though you’re using blackbox, if you launch a kde application, all the kde overhead will be started as well, which by
the way is why the start times are extremely slow when launching kde apps outside kde. The same holds with gnome apps, although there’s much less overhead.
It’s irrelevant which window manager you use. It’s more relevant how the apps are coded.
34. 2003-08-14 7:26 pm
… it is somewhat strage… that I get this feeling of “communism” as in “communist” from some people that don’t want or care if linux becomes more accesible to others…
Is Thats your choice???
… well… get with the program OR start looking for an other OS… something that is as hard as unix was when it started… because linux is heading to the DESCKTOPS of PEOPLE… and you will not be
happy of that I feel… so get something else other than *nix… maybe one of those 8 bit OS will serve you…
35. 2003-08-14 7:27 pm
“Is it really? I’m in my apartment right now working remotely. I have a couple of xterms’, emacs and a matlab plot open on my desktop, all running on a remote computer at school and being
displayed on my laptop here in the apartment. I’m doing this with a cable modem connection, and I have music streaming with RealPlayer which is consuming some bandwith….”
and how does that help normal people have a good desktop system?? they can do the same with with remote desktop.
what they need is something that is responsive, that can be configured easily (i.e. not by editing text files), easy driver updating etc. My hope for the future is directfb.
36. 2003-08-14 7:35 pm
“Unefficient. Derivated. “Me fail english? That’s Unpossible!” ”
Me no right to give my opinion because me no english native ? Not Unpossible with Kingston !
37. 2003-08-14 8:01 pm
The point was that X is in fact efficient. Locally it is as responsive as XP, as I know that’s what you’re comparing to, and it’s a great system for remote displays. For 99% of the users, the
performance of X is good quality.
X configuration is now easy. Have you every tried SuSE? They have a fine GUI for configuring X — there is no need to edit configuration files. The resolution can be adjusted on the fly, color
depth can be changed, screens can be changed, etc… in most cases the hardware is detected correctly on installation and the user doesn’t ahve to worry anyway.
AFAIK, remote deskotp does not behave in the same way as X.
With X, I can ssh into another machine and launch any X app and it will display on my local computer. It looks exactly like the app was launched on my computer. It integrates with the desktop.
AFAIK, remote desktop displays the entire remote desktop on your screen which is not the same…this looks more inefficient to me.
Yes, XFree86 should be improved. Drivers should be available more quickly and some drivers are not very high quality, but this is not all their fault. This is the fault of the vendors as well…
38. 2003-08-14 8:08 pm
Configuring IceWM may indeed be difficult for many MS Windows users. On the other hand, MS Windows is also a difficult OS for many MS Windows users.
I’ve seen real life examples of MS Windows users who have never figured out how to shut down their computer properly. Instead, when they stop working they just simply turn the power off. (And
then they wonder why MS Windows crashes so often.)
It is a futile hope that GNU/Linux would ever be ‘usable’ enough for people who cannot even learn the few mouse clicks that are required to shut down MS Windows.
39. 2003-08-14 8:19 pm
…be sure that X is running with a nice of -10! (I think, I’m not at my Linux box right now.) That greatly increases responsiveness.
40. 2003-08-14 8:30 pm
no os has standards, they only try to create standards, look at Microsoft, they developed MSI “MS Installer”, and look theres install sheild, Flash Installer, WinRAR installer, etc.
linux is the same, they have hundreds of various distros, numerous GUI systems, numerous package systems, wheres the standardization in that?
not even cars are 100% standardised, i no this has no relevance but some1 posted earlier about how cars are standardised. well in the US they had the lever Gearchange, in italy they have the
clutchless manual, ferrari make a manual gear box that is arranged differently to a usual design to give more feeling into driving, nothing has a standard.
heres another example the US, has different states each with its own laws, that vary, some have death sentence some don’t, some allow gambling, some dont, just get over it nothing in this world
is standard
the only mainstream os i’ve seen that is remotely standardised is Apple MacOs, which is mainly because of lack of 3rd party systems for installation, graphics API’s, etc
41. 2003-08-14 8:32 pm
have an install program that automated all modifications to the target machine and provided reasonable and intelligent default settings
That would be nice. Click on the app. it open the installer, detects all hardware & software, installs, done. I do think it should put icons on the desktop if you want & put it in the menu. Linux
does need standards atleast locations for programs.
42. 2003-08-14 8:53 pm
M$ the only exception ? Pleeeeease … I don’t know why, *EVERY* OS I know that is NOT based on X are pretty much *ALL* superior to X based GUI : Windows, OSX, BeOS, EPOC, etc.
Please define what do you mean by superior OS?
43. 2003-08-14 8:59 pm
You forget, or do not realise, that a computer is far more versatile than a car. The list of things I can do on a computer far outnumbers what I can do with a car. That increases the complexity.
Its like comparing a kitchen knife to a swiss army knife. A swiss army knife has many more blades and other utilities. Its is more complex and will take a person longer to figure out than a
normal bread knife.
The real problem is that people are taught to use windows, and excel. They are told to use excel, word, powerpoint, outlook. On my Linux system, the equivalents would be labelled spreadsheet,
word processor, presentation and email. which should be more difficult here. Because people are used to seeing excel does not make a system without an excel icon harder. People do not know how to
use computers. They just know how to do certain things without a full understanding about what they are trying to do. The problem is education. Maybe schools should be forced to teach real
computing, and not microsoft lock in instead.
44. 2003-08-14 9:21 pm
I don’t care how many desktop enviroments there are – so long as I can fire one up, and have all the apps work under this enviroment and work together seamlessly. If I have to use desktop
enviroment a in order to run app b, that’s where the problem lies.
I want cut/copy/paste to be universal and work across ALL apps. Do that for me, and you can have 3 million desktop enviroments for all I care.
As for Windows not having a standard, sure … many apps LOOK different, but most of them also act the same. There’s not really much inconsistancy between them (ie – ALT+F4 to close the window).
45. 2003-08-14 9:28 pm
“You forget, or do not realise, that a computer is far more versatile than a car. The list of things I can do on a computer far outnumbers what I can do with a car. That increases the
Well, you can drive in a car, you can kill with a car, you can sleep in a car, you can shell stuff out of your car, you can play games with your car (racing) …
46. 2003-08-14 9:33 pm
> For example, Mandrake ships with three different X-Window GUIs:
> KDE, Gnome, and IceWM.
Yes, that way people with preferences (most of us) can be satisfied with OUR choice of Window Manager/Desktop Environment. Whether it be GNOME, KDE, WindowMaker, IceWM or even BlackBox. etc.
> How much time should users spend
> exploring these different GUIs before they find the one that’s “right”
> and works with all their applications? One month? Five months?
They don’t have to spend anytime doing this, RedHat defaults to GNOME, SuSE to KDE and Mandrake to KDE (IIRC). Hell, if they don’t want to make any decisions, use Lycoris or Lindows where the D.E
(KDE) is the only option.
> Are there more productive ways for users to spend their time than trying
> different GUIs? Developers, hobbyists, and large IT shops gain value
> from the ability to try and test a multiplicity of interface choices, but
> the average home or business user will not.
Then (as I said before) use Lycoris, Lindows or another *absolute* newbie distro.
47. 2003-08-14 9:45 pm
You obviously have absolutely no idea what “communism” means. Using it as a standard “bad” term erodes your point as it denotes an inability to distinguish between words you understand, and words
you don’t.
Yes, it is a good read. Better than most of the “standardize now” rubbish articles. But I disagree with the author.
I think that one distro – or group (UL?) – will triumph and become the standard. But we will still have our geeky distros like Debbie, Genny, and the Slacker.
48. 2003-08-14 10:08 pm
At the beginning I could see his point, even if I disagreed with it. Around the second page when he started talking about how window managers are bad because they’re hard to install, right after
stating that his example distro installed it for the user, it began to seem like an English paper which had been rewritten to increase its word count.
49. 2003-08-14 10:20 pm
“because linux is heading to the DESCKTOPS of PEOPLE… and you will not be happy of that I feel…”
It depends on what you mean by this. If you mean that Linux, as it exists today, will be used by more people – I feel that’s wonderful. But if you mean that the very things which made me want to
use Linux in the first place will be removed to make it into a Windows clone, than no, of course I’m not going to be happy about that! Despite the sterotype, I think most people using Linux are
doing so because they like Linux, not for some ill thought-out crusade against Microsoft. If someone is looking for a user experience just like Windows, you know…they might just be best off
looking at Windows instead of trying to force everything else to fit that mold. Attempting to create a world where every computer is exactly the same no matter what operating system is running on
it is an attempt at a pretty dull computing experience in my opinion.
50. 2003-08-14 10:38 pm
Why Keith Packard’s X11 fork has gone silent. More disturbing news on the KDE/Gnome front. Some of you may have noticed that there has been very little public technical discussion about the X11
fork Keith Packard has been doing called XWin lately. In the past couple months they’ve all been pretty much silent. Wondering why?
Well, I have been told by someone close to the project that it’s because it’s been hijacked by Gnome developers and they don’t want to debate integrating Gnome technologies. Particularly they
want to integrate things like GConf and Glib into the X server without having to discuss it. So they are no longer talking about what they are doing on the mailing lists or website forums. If
XWin is a success because of things like Xr and they are able to sneak things other things like GConf in without debate or public discussion it would be a huge win for them.
This is why all the public forums are silent: It’s not that they are not doing anything – it’s that they don’t want to tell people about it 😉
I can’t confirm or deny this, but have been told the above information by someone both very visible and well-known in the Linux community and close to Red Hat. He said the person leading this
effort is Havoc, not Keith Packard, so Keith is not to blame.
He also stated that in response to KDE getting positive coverage due to it’s usability both RedHat and Sun are actively lobbying their customers against KDE. Not only this, but there have been
accusations from people close to both companies that they are feeding anti-KDE articles to news sites from supposedly “neutral” sources. This is not suprising for RedHat, but I rather thought Sun
learned from their desktop choice mistake before.
If this article is true and I sincerely hope it is not, than GNOME developers are attempting to destroy all other DEs just for their own self gain. If this is accepted by the community the
implications are enormous and this behaviour goes agains everything that OSS stands for, including choice!
51. 2003-08-14 10:44 pm
one must go, no need to have 2 API on one os. incosistent gui, incompatibility, more memory required etc.
i prefer gnustep though, just that development very slooow.
52. 2003-08-14 10:59 pm
I’ve thrown YDL on an old iMac and have been tootling around for it for about 2 weeks.
I’ve found no major problems with the basic KDE GUI.
It’s the half baked everything else that’s causing me to go grey.
The find file function that’s not worth a damn.
The CD player without a way to adjust volume.
The lack of copy and paste in OO for the PPC port that shipped with YDL. (I’m not shitting you. Check the FAQ at the YDL site. There is no C&P.)
The hit and miss hardware support.
In short, the UI took me little time to learn as somebody who uses Windows at work and OS X at home.
It’s the crappy half baked everything else that makes YDL a chore to use.
53. 2003-08-14 11:10 pm
I remember that a osnews reader was trying to compile gnome against directfb and was unable to do so because lots of gnome lib’s have direct calls to xlib instead of using (at the very least)
wrappers to abstract from that for portability’s sake. I don’t know if the situation is any different with kde and I don’t know if it’s just lousy programming or if it’s intentional but it makes
you wonder doesn’t it?
54. 2003-08-14 11:16 pm
I just dont get the point of these endless “Linux needs less choice for the average user”. Isnt the whole point of Linux making what *you* want??
What’s stoping ppl from making a distro that mimics Windows??
Feel the need for an OS that doesnt make the user choose? make one! Take KDE/Gnome/whatever and make it the only DE available.
Make a DE with 3 128×128 buttons saying “App1” “App2” and “App3” in pretty colors for all i care!
Thats the great thing about Linux! In Windows im stuck with what MS thinks its best for me…
“Instead, go ask average users what they want. Microsoft does. They perform extensive user testing with every major application.”
Thats the problem…they *just* ask average users want and as power users the rest have to live with the wrong options…
In Linux you have distros from Gentoo to Lycoris (you even have distros you dont have to install like Knoppix!). People who like to be spoon fed use Lycoris others dont.
Finally, why is it so important (or so it looks…) that Linux be used by “average users”?
If only “power users” end up using Linux its a big market share!
55. 2003-08-14 11:21 pm
I don’t know if the situation is any different with kde and I don’t know if it’s just lousy programming or if it’s intentional but it makes you wonder doesn’t it?
Probably just lousy programming. Those GNOME folks take 3 or 4 revision to do it right.
As for “there can be only one”. That’s bullshit. I want a GNOME/KDE desktop environment, where KDE’s konqueror manages my background and desktop while GNOME just sits there, preloaded. Maybe as a
Mac-like bar across the top of my screen with some pager, clock, and volume control embedded in it.
But I can’t live without my konqueror!
56. 2003-08-14 11:31 pm
Siemens Business Systems, apparently a $6billion IT consulting company has changed its mind and is expecting linux to get 20% of the large corporation desktop market in 5 years. Previously they
thought linux would fail on the desktop.
They claim that it takes a user two days to get used to Gnome. And that because Gnome is not as similar to Windows as KDE it creates less confusion.
Anyway the point is things can’t be that bad. Read the article below they have lots of real clients beginning to move large numbers of desktops to linux.
57. 2003-08-15 12:19 am
I’ll have to hear it from somebody other than Mosfet before I believe it. To this guy, everything is a conspiracy against KDE.
58. 2003-08-15 12:21 am
i’m a big KDE fan but i like a few GTK/Gnome apps so i have mixed system as far as UI is concerned but the usability principles between kapps and gapps are pretty similar, especially when i have
geramik installed
seriously though, look at windows. the differences between windows media player, real player and quicktime, or IE mozilla and opera.
i don’t think you can ever get total unity in UI but as far as i’m concerned, linux is beating windows when it comes to GUI.
59. 2003-08-15 12:31 am
when he’s talking about setting your desktop and that you have to edit .xinitrc files. i have never done that! in kdm, it’s as simple as clicking a little option box to set which environment you
want, if you don’t chose, your previous one is started, simple. if a newbie ever read that, (s)he’d run away screaming.
60. 2003-08-15 12:36 am
THEY are trying to INCORPORATE GNOME ONLY TECHNOLOGIES INTO THE X FORK, this is intentional and will hurt the other des.
61. 2003-08-15 12:49 am
Actually the point is not ordinary people vs. hackers. It seems to me that the problem is we are playing a showhow political game. For example, even technologically we don’t need both KDE and
GNOME. They do almost identical things and the drawback of having duplications outweights the benefits as experienced programmers know. Duplicated code is evil and there is no benefit from it.
Stop playing games and work on single code.
62. 2003-08-15 1:02 am
To me it’s great that you can choose a GUI you prefer be it an all in one solution or one you hack together yourself. For Linux to possibly reach the masses the hardware vendors need to come on
board, this is where Linux runs into problems, without hardware support there will be no increased use. Everytime I go out to buy hardware for my PC I need to first go and see if it is supported,
if not you can try and write your own driver or choose something different that may be supported by Linux, meanwhile with Windows you decide you want this piece of hardware and the majority of
time it will work with Windows, or you look on the box of the hardware and it tells you which Windows it will work on, very seldom does it say compatible with Linux, I’ve seen this once and it
was for a D-Link PCMCIA Ethernet card.
Yes hardware support is slowly getting better in Linux but it still has a long way to go. IMHO
63. 2003-08-15 1:03 am
“Duplicated code is evil and there is no benefit from it. Stop playing games and work on single code.”
Yeah, why don’t we all go back to work on the single code of Hurd instead of Linux?
64. 2003-08-15 1:04 am
You can never get an OS that’s totally consistent, even with UI guidelines some apps will break the rules. A lot of the time visual consistency in Windows is just as bad as in Linux, there can be
big aesthetic differences between apps. But personally I can live with that, it may make the UI look messy but it doesn’t significantly damage productivity. OTOH inconsistent keyboard shortcuts
and cut, copy and paste does significantly damage productivity.
Windows and Mac OS do have a basic level of consistency between apps, 99% of the time I don’t have to relearn keyboard shortcuts or the location of common menu items and I can copy and paste
between all apps. Until that’s true in Linux I don’t think it’s UI will feel as consistent and elegant as Windows, even with themes making KDE and GNOME apps look similar.
65. 2003-08-15 1:15 am
“A lot of the time visual consistency in Windows is just as bad as in Linux,”
Although Windows also has inconsistency amongst its applications, I would not reach so far as to say it happens “a lot of the time,” as you speculate. Inconsistency is rare enough in Windows that
it is not usually a big deal.
“it may make the UI look messy but it doesn’t significantly damage productivity.”
Yes, it doesn’t significantly damage productivity. Although the widgets act similarly, if the user stumbles upon a slightly different behavior, it will annoy him. Add up a bunch of similar
nuisances and the user will lose his patience quickly
66. 2003-08-15 1:26 am
As is evident with all these virus outbreaks hitting the Windows world, having choices is much better than not having it. To be fair however, there is a lot of choice in the Windows world. There
is probably more choice in Windows than in Linux (most open source projects are available for Windows, but proprietary apps are usually not available for Linux), the only problem is that noone
practices it. Also, even though people do practice a lot of choice in the Linux/Unix world with applications, the sad part is that most applications use the same set of libraries.
67. 2003-08-15 2:36 am
a standard UI is NOT the answer. while i am all for cross UI themes, a “one size fits all” when it comes to the UI is not good. why? it’s inefficient. some people do things pretty darn quick
using the standard windows, gnome, and kde look and feel. however many do not! that is why:
– you have customization software for windows (often at a price on top of the ridiculous price for the OS)
– kde and gnome can be configured to look like what the user wants it too look like
– others use xfce, blackbox/fluxbox/bluebox/openbox, enlightenment, fvwm, windowmaker, etc
68. 2003-08-15 3:45 am
Someone who understands the problems with haveing to develop for two desktops wnviroments needs to come out and explain those problems to the rest of us — but as for me and my house we like
having the choice. kde or gnome – icewm.
69. 2003-08-15 3:51 am
This guy is dead on. One problem with Linux is that there are so many different developers spanned over many different projects all trying to do things with their own philosophy or the philosphy
of the desktop environment they are using. Linux is making progress but in terms of making inroads to desktops it’s got a long way to go.
This guy has got it right. Instead of many different GUI’s they need 1 standard gui that works for the average user. If Linux doesn’t go down this path it will fail misserably for the average
70. 2003-08-15 5:13 am
So the average user doesnt like it. Big woop.
Linux was made for the enjoyment of the developers of it. Even if no one else in the world likes it, the developers will never stop making it,because its their operating system.
Honestly, most “average user” doesnt know a thing about computers, and doesnt want to. Hell, they pirate, and dont listen to good security advice. Let them use Windows.
71. 2003-08-15 5:29 am
If this was a dead horse, it would be flogged so hard that we would have mince meat by now.
I’ve given up on the average user. There is no use educating people who don’t want to learn. It is the old story, you can bring a horse to water but you can’t make it drink.
If the average user wants to stick with Windows, that is there choice, however, like any choice they make in their life I don’t want to hear a whinge, whine or complaint about the choice they
made. Make a decision and stick by it.
In terms of Linuxs adoption, unless the community is willing to hear the truth and do something about it the perception by some, whether it is right or wrong, will be that Linux has no strong
direction as a desktop operating system.
As for the authors comments, I think it is time he stepped out of the Windows world, bought a clue and looked at reality. Redhat’s default desktop is Gnome and KDE libraries are installed for
support and SuSE’s default desktop is KDE and GNOME libraries are installed for support. Mandrake is the only distribution that hasn’t got a clear desktop default set down.
Ultimately, if the user doesn’t know anything about computers then they will know nothing about Linux meaning this whole argument over standardising is a non-issue as the target audience doesn’t
even know the product exists.
72. 2003-08-15 5:50 am
There is no more inconsistency on my Debian system then there is on any Windows machine. When you consider the shear number of apps compared to Windows, and venture out of the basic utils that
come with the system, rarely do they look or act the same as the basic apps. I primarily use KDE apps, they all look and act the same, they all have set things in set places, and this makes
customization easier. If I want a Gnome app for some reason, they are also all the same as I use a theme that matches… I recently went back to XP for a week or so… no 2 apps looked or acted the
same, other then the odd few entries, menu’s were never the same. The whole system is very counter-intuitive, this in contrast to the Linux way of doing things, where (for the most part) you have
2 different structures and rule base’s for all apps, I find it hard to beleive anyone that isn’t closed minded can find Windows more intuitive then anything Linux has to offer…
I find my desktop apps run faster then there Windows alternatives, they are usually much more reliable, and are far less buggy usually also. I am not a Linux zealot, I simply use what works the
best for me. I used to love Windows, it was all I had ever ran. I have used Linux for 2 years now, and every time I sit at a Windows machine, it just pisses me off, stupid little bugs that I
never used to even notice yell at me. Even in linux though, the little bugs etc bug me, why? Because I simply am not used to them anymore. That is why I couldn’t stay with XP Pro, the thing just
was too buggy. It didn’t recognize my sound card, it didn’t use the right video card drivers, it fucked up the amount of RAM I have, it reported the wrong values for CPU and harddrive space. This
isn’t even going into what I experienced with regular applications. How can people say Linux doesn’t detect hardware as well as Windows? I never have these issues with Linux, NEVER!
I guess for the closed minded, seeing something that isn’t exactly like IE is hard to stomache, I don’t know, I just don’t understand how people that have lived on both sides of the fence so to
speak can honestly say such things!
73. 2003-08-15 5:57 am
I remember that a osnews reader was trying to compile gnome against directfb and was unable to do so because lots of gnome lib’s have direct calls to xlib instead of using (at the very least)
wrappers to abstract from that for portability’s sake. I don’t know if the situation is any different with kde and I don’t know if it’s just lousy programming or if it’s intentional but it makes
you wonder doesn’t it?
GTK is a three part jigsaw puzzle. glib provides C functionality that is not available on all platforms. Pango then provides the international support which is based on glib. GTK is then based on
GDK which provides an abstraction layer over X11 and also links back into pango and glib.
What the person was most likely whinging about is the lack of a port of GDK to GTK, and generally, if it hasn’t been proted, there are two reasons, firstly, a lack of drive by the developer
community and secondly, there is missing functionality. Most likely the issue is with functionality missing from directfb which X11 does have. This isn’t a reliance of X11 but a feature that is
lacking in DirectFB.
74. 2003-08-15 6:08 am
Most likely the issue is with functionality missing from directfb which X11 does have. This isn’t a reliance of X11 but a feature that is lacking in DirectFB.
Please explain more, I find this terribly exciting and on-topic.
75. 2003-08-15 6:20 am
Well GTK has an abstraction layer which is reliant on GDK, GDK is reliant on Pango which provides text rendering and internationalisation support.
What is the net result? to port GTK to Windows, one simply had to port Pango and GDK to GDI, as a result we have application such as XChat available.
Now, a while back there was a move to port GTK to Quartz, however, the primary objective now, however, is to get X11 and GTK to work on MacOS X, once that is done and can provide the appropriate
funcationality, the next step can be to tune and optimise using Quartz.
In terms of GDK and Mono, GDK is being used as a step in replacement for any calls made by .net applications that require GDI+ funcationality.
Ultimately, GTK was designed from the ground up to be portable, not only to other UNIX/X11 operating systems but non-X11 based such as Windows and MacOS X.
76. 2003-08-15 6:39 am
Open source operating systems like the ones A. russel describe already exist.
A default install of redhat for example, or lindows will do what he thinks a joe public LInux OS should do. I think he needs to look at linux OS’s more closely – do more research Mr Russel.
I think the only thing I can take from his article is that there needs to be a better awaerness for which linux distro suits who. Distro watch is’t bad, but something simpler and clearer for joe
blogs may be useful.
77. 2003-08-15 7:10 am
>Mandrake is the only distribution that hasn’t got a clear
>desktop default set down.
<sarc> So lets just forget about Debian, Slackware, Gentoo and others because they did not made their dissision yet. </sarc>
78. 2003-08-15 7:29 am
Lots of the posts preceding mine have pointed out the weaknesses of Jones argumentation. What strikes me particularly is that despite the abundance of information on the websites of the FSF, KDE,
Gnome, Suse, etc., even tech savvy guys like Russell Jones still don’t get it : having the main conditions of our computing experience dictated by a single entity (be it Microsoft or Apple) is a
bad thing. It negates the fact that others possess the knowledge required to create the different components necessary to enjoy computers : kernels, operating systems, desktop environments,
interfaces, programs, hardware and so on.
Throughout articles dealing with GUI issues, there appears to be the same confusion about a certain word : “standard”. Saying that Microsoft sells the same standard to all their customers is the
equivalent of saying that the Cosa Nostra provides all Sicilians with the same standard of living, which is : suffer whatever indignity we inflict upon you but, to comfort yourselves, think that
you share the same fate with numerous other smart people.
79. 2003-08-15 7:48 am
Because you’re not forced into running KDE or Gnome nor are you forced into running these locally on the PC.
Let’s consider the situation at a typical small business.
You would have PC’s running Windows 98, NT, 2k and Xp mixed togheter.
You can’t upgrade to windows 2k or XP, because the older computers cannot run it. You can’t downgrade to NT, because it’s no longer supported or available. And even it would be possible, it would
be very expensive. You’d have to buy new licenses for most PC’s.
So, you’r basically stuck with you’re mixed version setup, meaning that users cannot experience a consistent desktop across the office or take their desktop configuration from one pc to another.
With Linux, you can upgrade all pc’s to the same version. Even a pentium 150 with 40 megs of RAM will run KDE reasonably well, although then you can’t run much applications concurrently anymore.
In an office network however, you can use it as a terminal on one of your newer PC’s or run the applications remotely. On a stand-alone PC you can revert to a lighter window manager and still
have a useable system.
So, within a typical small business you can create a consistent desktop across the office using the hardware that is already present for minimal costs.
There is no problem moving from one PC to anothor. All your configuration settings go with you as long as your home directory is shared across the different computers.
Bottom line is: with Linux you can have a consistent desktop experience accross your company using one version of Linux on a mixture of old and new hardware.
With Windows you can only have a consistent desktop with roaming profiles by investing a lot of money in upgrading both soft&hardware into one single version of both windows itself and the
applications that run on top of it.
80. 2003-08-15 7:52 am
Choice and Usability are not at odds here — indeed, choice aids usability. In Windows, you’re often given a set of applications that you have no real choice in using. Back when I was a Windows
user, I had Office, Visual Studio, and IE. Thanks to the hemogenous Windows community, using anything else made it hard for you to use available content. Now, these applications are not
internally consistent. To this day, VS.NET looks nothing at all like either the Win2k or WinXP default looks. Office has used its own toolkit for years. So in Windows, you get neither choice, nor
In Linux, I get usability because I have choice. I choose to use pretty-much 100% KDE apps, and as a result, Linux is the most usable OS I’ve had since BeOS. I can get away with using apps that
which, though they are not necessarily the most popular, fit all my needs and integrate well with the rest of my software. The reason I can do this is because the Linux crowd loves its ability to
choose, and as a result is hetrogenous. Because it is hetrogenous, content makers provide content in open formats usable by everyone. In the Windows world, you often see example code distributed
as Visual Studio projects. In the Linux world, you never see example code distributed as KDevelop projects. Instead, you see a nice, standard, makefile that can work with any IDE.
81. 2003-08-15 8:24 am
Lets keep Linux as a platform and not a product like Microsoft. Micrsoft is not even a suitable platform while anyone can program on Linux and have control over their platform. Linux should
however focus more on making it easier for beginners to have control, and that would involve building tools to document the source code, and working with a distributed and organic knowledge base
that is interfaced with these documentation tools. We have the source code, but that doesn’t mean that it is usable, you have to have a lot of knowledge to use the source code, let’s make it
easier, and let Linux lead toward developing under the open source rules rather than the vendor rules.
82. 2003-08-15 9:21 am
also, Linux doesn’t have to beat out MS Windows or at least it shouldn’t focus on that. It’s more important that Linux stay free and a platform rather than a product. I actually don’t mind how
things are right now except for the SCO terrorism. Well lets remove that NUMA code and be done with it. Linux will operate under more hardware, it will become more user friendly and hopefully as
the most important point, it will become much more approachable to the beginner. Make the source code more accessible by documenting it much better with tools, and focus in areas where closed
source can’t go.
83. 2003-08-15 10:45 am
“Even a pentium 150 with 40 megs of RAM will run KDE reasonably well (..)”
Have you read that somewhere or have you actually tried it? I’ve tried both KDE and Gnome on a p200/32mbram/voodoo2, p500/64mbram/onboard, and a p166/48mbram/onboard. Trust me, KDE and Gnome
didn’t do quite well. I don’t find that a problem, since xp needs better specs as well.
It’s just that you should try something out first, before stating something you just read somewhere on the net.
And, if you did get to run it, I’d like to know how long it took to open OpenOffice
84. 2003-08-15 11:02 am
As long as applications are built to run only on one desktop environment, then this “choice” is a double-edged sword. Deciding which desktop to run also means deciding what kind of apps you won’t
85. 2003-08-15 11:02 am
i don’t care more about the applications than the desktop environment. i need a stable os with good hardware support, windows and linux have both.
but then it’s about good software. if i want to run a server i choose linux (or some other unix), but for multimedia stuff, video editing or sound applications i’m stuck with linux. if i want to
have a choice between several professional applications i have to use osx or windows.
what linux needs is a standardised base for commercial software.
86. 2003-08-15 11:05 am
> i don’t care more about the applications…
i CARE more about the applications…
87. 2003-08-15 11:18 am
“”Even a pentium 150 with 40 megs of RAM will run KDE reasonably well (..)”
Have you read that somewhere or have you actually tried it? I’ve tried both KDE and Gnome on a p200/32mbram/voodoo2, p500/64mbram/onboard, and a p166/48mbram/onboard. Trust me, KDE and Gnome
didn’t do quite well. I don’t find that a problem, since xp needs better specs as well.”
Shame shame, you’re right. My mistake.
I did install Knoppix on the machine and KDE indeed was very slow.
However icewm, the default with libranet 2.7 that I installed later, is quite useable. I use it to do system administration on a remote web server. Even running galeon on it is bearable.
Icewm is even useable on a 486. I have a 486 notebook with 40 megs of ram running it. I use it to display applications running on my desktop while sitting outside in the garden :-))
“It’s just that you should try something out first, before stating something you just read somewhere on the net.”
See above.
“And, if you did get to run it, I’d like to know how long it took to open OpenOffice
It actually did start within minutes, but it was not very useable.
If you want to run an office app on a 486 or older pentium, take siag office or abiword. (Or run it remotely, of course)
88. 2003-08-15 11:25 am
“”Even a pentium 150 with 40 megs of RAM will run KDE reasonably well (..)”
He’s right, you know. KDE 1 flies on a system like that.
Oh, wait….
89. 2003-08-15 12:06 pm
Linux is not an operation system, but a kernel. It does not have a GUI.
Linux is not an operation system, but a kernel. It does not have a GUI.
Linux is not an operation system, but a kernel. It does not have a GUI.
People think that Linux needs to have a GUI, but it has none. The only person who can give Linux a GUI is Linus Torvalds, but so far he did not (and I doubt that he ever will be).
GNU/Linux is an OS, and its official GUI is Gnome. Lindows is an OS, and its GUI is KDE. RedHat is an OS, and its (default) GUI is Gnome. Suse is an OS, and its (default) GUI is KDE.
KDE/Qt is a platform of its own, which may run on Linux, but also runs on various other Linux-like platforms. Most desktop users do not really care for Linux. They could use FreeBSD instead, and
they would not even notice. They only care for the Look&Feel, which is KDE. Thus you should not call the thing Linux, but KDE, since that’s what they are interacting with.
Same for Gnome. Call it GNU!
And for hell’s sake, stop using the word Linux for anything but the kernel and the server platform. I know Linux is a well-known name, but it does a bad job at describing a desktop, since it does
not have a GUI and you can install a thousand of different GUIs on it.
90. 2003-08-15 12:33 pm
“Linux is not an operation system, but a kernel. It does not have a GUI.
Linux is not an operation system, but a kernel. It does not have a GUI.
Linux is not an operation system, but a kernel. It does not have a GUI.”
Let’s make a deal, to banish this once and for all: When someone says “Linux” in comparison to Windows, Mac OS X or any other, we mean the distributions close to it, such as MDK, SuSE, Lycoris
Saves us these stupid remarks.
91. 2003-08-15 12:39 pm
“Linux is not an operation system, but a kernel. It does not have a GUI.”
That argument only holds water unless you’re like me and believe that the kernel *is* the OS. This is not you or I being misinformed so much as a difference in opinnion.
“The only person who can give Linux a GUI is Linus Torvalds, but so far he did not (and I doubt that he ever will be).”
Not true. If anyone with the required skill felt as though it were needed, they could weld a GUI into Linux. Linus Torvalds has a source tree, redhat has a source tree, everyone and their dog in
the Linux world has a source tree of their own. Just because his is currently the most popular (being the original author and all), doesn’t mean that it will stay that way. He only has say over
his own tree, and as time goes by, trees such as RedHats and SuSE’s will become the standard trees as the directions of Linus and the Major distributers begin to diverge.
“GNU/Linux is an OS, and its official GUI is Gnome.”
Nope. Wrong again. You assume concensus were none exists.
“Same for Gnome. Call it GNU!”
Uh, no, I call it “Gnome”…
“And for hell’s sake, stop using the word Linux for anything but the kernel and the server platform. I know Linux is a well-known name, but it does a bad job at describing a desktop, since it
does not have a GUI and you can install a thousand of different GUIs on it. ”
If clairity is indeed your purpose here with this statement, you’d better work on that a little.
92. 2003-08-15 12:47 pm
“Have you read that somewhere or have you actually tried it? I’ve tried both KDE and Gnome on a p200/32mbram/voodoo2, p500/64mbram/onboard, and a p166/48mbram/onboard. Trust me, KDE and Gnome
didn’t do quite well. I don’t find that a problem, since xp needs better specs as well. ”
Lack of ram is the killer for running Linux/KDE on old boxes, rather than CPU speed. If you can find 128mb+ it makes a vast difference, as most of the slowdown is caused by swapping to disk. If
you can put 512mb into a p500 then it runs very smoothly with KDE, even a p2-400 is still useable with 384mb.
At least the more recent kernels don’t go into a swap storm with very limited memory, but you are still going to be waiting around a lot with 64mb, especially if the swap file is on an ancient
It depends whether it’s worth searching out ram to upgrade the old machines I guess, but ram is cheap, and the difference it makes worth the extra expense.
93. 2003-08-15 2:43 pm
What the person was most likely whinging about is the lack of a port of GDK to GTK, and generally, if it hasn’t been proted, there are two reasons, firstly, a lack of drive by the developer
community and secondly, there is missing functionality. Most likely the issue is with functionality missing from directfb which X11 does have. This isn’t a reliance of X11 but a feature that is
lacking in DirectFB.
The problem wasn’t GTK/GDK, those have already been ported a long time ago to directfb [http://www.directfb.org/gtk.xml], it was specific GNOME libs that instead of using GDK actually made calls
to xlib, so if there are features missing it would be on GDK not directfb, but the problem is most likely on gnome’s side.
94. 2003-08-15 3:23 pm
You only need to read Mosfet’s other screeds to understand why he’s hardly a reliable source of information.
First of all, XWin is _just a website_. It’s not a project. It’s for discussing the future of X, and whether a fork is needed. Asking “why it’s quiet” is inane, because the XF86 project has
started to become more open, obiviating the need for Xwin.
Second, Mosfet is absolutely convinced that GNOME is trash and that everyone just needs to use KDE, and everything would be alright. He’s _SURE_ that RedHat is out to kill KDE. In other words,
he’s the equivalent of a fanatic government conspiracy theorist who’s managed to write some pretty GUI elements.
There is a reason that he was thrown off the main KDE development team.
95. 2003-08-15 3:28 pm
I’m curious-all the talk about the many and varied Linux desktops, and noone ever talks about the many and varied desktops for -Doze. Windows is just as flexible from a UI standpoint.
Let’s see, for windows, you can use:
Explorer (Aqua or Classic)
Style XP (Explore Skinning App)
Object Desktop (Umm, Shell Customization Suite?)
Lightstep (Shell Replacement)
Winstep (Windowmaker/Nextstep Style Shell Replacement)
Talisman (Shell Replacement)
Serenade (Shell Replacement)
Geoshell (Minimal Shell Replacement)
SharpE (My Favorite, also a Minimal Shell Replacement)
Go! (Very Minimal Shell Replacement-replaces explorer with a command.exe bar.)
Neon (Uses Flash as a explorer replacement)
Blackbox/Blubox (Port of Blackbox/Fluxbox to Windows, Very Nice!)
My point is: each of these applications provides a vastly different experience to the end user, and in many cases, changes the intuitive “feel” of the OS. In fact, switching the UI from explorer
to something else reduces system memory load in most cases, and greatly enhances overall stability.
How is this any different from many and varied X-based desktops that come with most Linux DISTRO’s? The fact that these windows apps comes from third parties is not material. The only reason that
the Distros have Gnome , KDE, Blackbox, etc., is that they decided to ship the code with the core kernel.
Now, true, I don’t have access to the NT kernel, but I’m not coding kernel modules, drivers, or generally hacking the core to my own uses. But if I want to change how my desktop loks, I’m hardly
limited to just changing my wallpaper.
96. 2003-08-15 5:12 pm
“… and they [Linux powerusers] are hardly likely to resolve the KDE vs Gnome argument anytime soon.”
The way I see it, the argument is resolved. No more argument!
97. 2003-08-15 6:30 pm
From a non users point of view.
Usablility of Linux seems to be the Achilles Heel here. More specifically the way Linux heros think about usability. Mostly I’d say installers and configurations. Not the desktop or the widgets
on individual programs although that could be improved. I’ve tried some of the open source Office stuff and it sucks big time.
Now maybe it’s just a PR thing. But even on OSX, which I do use, there are just too many apps I might want to try that you have to use the terminal to compile. I’m just too busy to screw with
figureing out an obscure terminal command and where the file lies in relation to the root etc. I haven’t been around long enough to use a terminal with comfort. And there isn’t much I’d need to
use it for that I can think of. Batch renaming of files? I can find a non terminal app to do it on OSX.
Now I do plan to build a baby Linux server. I can understand how Linux could help me there. Like $2000 for a Mac server vs. $400 for a decent little self built Linux box. I’ll go for that because
I only have to get it to do one thing, serve. It might take me $700 in time to get it up and running vs. $400 in time to get the Mac up and running. You can see I am still way ahead.
On my Mac I have to get it to do thousands of things on a schedule. Any delay stalls my productivity and annoys my clients. Over the space of a year each time I have to learn some arcane command
I loose $time. That adds up. Over the space of 3 years the Mac more than pays for itself in time and headaches.
I also have a Win2K box. I use it for looking at my Web designs to check for cross compatibility issues mostly. I would never use it for email and it’s clunky for most things. XP? Nope I don’t
want to get into the habit of being relaxed as Microsoft watches me and hassles me over putting in a new hard drive. Only a corporate worker or a dummy would choose to get locked into Windows.
Meanwhile the Linux server will just chug along. I don’t want to mess with it.
Get it guys?
98. 2003-08-15 10:12 pm
I think that the success of so many differente linux distributions are proof that choice is good.
There is a reason why the powerusers choose gentoo over mandrake, and visa versa. People with extensive computer knowledge like to have control over their os, while newbie’s need the “fix it for
me please”-button.
With windows you are stuck with the latter one, it fails to give the poweruser the freedom he wants.
Ofcorse, windows will do just fine for 95% of the computerusers. The once that are in it for doing a task, and not for the computing itself.
But so will probably a few linux distributions, i`ll be sure to stay away from them 🙂
99. 2003-08-15 10:52 pm
while newbie’s need the “fix it for me please”-button.
No, people who are new to computers need a computer system that works either intuitively, or without requiring assistance unless absolutely neccessary.
People with extensive computer knowledge like to have control over their os
Everybody loves generalizations. I have “extensive computer knowlege,” but I don’t like fiddling with inane config files and the like (what you would probably consider “control of ..[the] os.”
The once that are in it for doing a task, and not for the computing itself.
Yes, some people use computers to get things done or play games, and some people use computers to tweak endlessly and learn Emacs shortcuts.
100. 2003-08-16 3:45 am
I think the trouble is that GUIs are rather matter of personal preferences than runtime performance or functionalities. I am not yet convinced why choice is important. No one demands more kernels
than Linux, more servers than Apaches because they are good enough and there is no need to have more than two UNIX kernels, http servers.
Chaos of desktop environments is seemingly similar to that of UNIX favors in the past. Variation is for the reason of policits and not for the purpose of user benefits. The battle between KDE and
GNOME looks similar to this.
You want choice? What kind? Isn’t customization good enought for it? We don’t need several ways to create a widget.
Remember the primary motivation of GNOME project, that is, QT was free enough, not technical one. Of course, having two almost identical code base might be beneficial for technical advances but
in terms of software engineering, it is wasting time and resources. Again, choice comes with cost and too bad we have unnecessary number of choices.
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In statistics, unit-weighted regression is a simplified and robust version (Wainer & Thissen, 1976) of multiple regression analysis where only the intercept term is estimated. That is, it fits a
${\displaystyle {\hat {y}}={\hat {f}}(\mathbf {x} )={\hat {b}}+\sum _{i}x_{i}}$
where each of the ${\displaystyle x_{i}}$ are binary variables, perhaps multiplied with an arbitrary weight.
Contrast this with the more common multiple regression model, where each predictor has its own estimated coefficient:
${\displaystyle {\hat {y}}={\hat {f}}(\mathbf {x} )={\hat {b}}+\sum _{i}{\hat {w}}_{i}x_{i}}$
In the social sciences, unit-weighted regression is sometimes used for binary classification, i.e. to predict a yes-no answer where ${\displaystyle {\hat {y}}<0}$ indicates "no", ${\displaystyle {\
hat {y}}\geq 0}$ "yes". It is easier to interpret than multiple linear regression (known as linear discriminant analysis in the classification case).
Unit weights
Unit-weighted regression is a method of robust regression that proceeds in three steps. First, predictors for the outcome of interest are selected; ideally, there should be good empirical or
theoretical reasons for the selection. Second, the predictors are converted to a standard form. Finally, the predictors are added together, and this sum is called the variate, which is used as the
predictor of the outcome.
Burgess method
The Burgess method was first presented by the sociologist Ernest W. Burgess in a 1928 study to determine success or failure of inmates placed on parole. First, he selected 21 variables believed to be
associated with parole success. Next, he converted each predictor to the standard form of zero or one (Burgess, 1928). When predictors had two values, the value associated with the target outcome was
coded as one. Burgess selected success on parole as the target outcome, so a predictor such as a history of theft was coded as "yes" = 0 and "no" = 1. These coded values were then added to create a
predictor score, so that higher scores predicted a better chance of success. The scores could possibly range from zero (no predictors of success) to 21 (all 21 predictors scored as predicting
For predictors with more than two values, the Burgess method selects a cutoff score based on subjective judgment. As an example, a study using the Burgess method (Gottfredson & Snyder, 2005) selected
as one predictor the number of complaints for delinquent behavior. With failure on parole as the target outcome, the number of complaints was coded as follows: "zero to two complaints" = 0, and
"three or more complaints" = 1 (Gottfredson & Snyder, 2005. p. 18).
Kerby method
The Kerby method is similar to the Burgess method, but differs in two ways. First, while the Burgess method uses subjective judgment to select a cutoff score for a multi-valued predictor with a
binary outcome, the Kerby method uses classification and regression tree (CART) analysis. In this way, the selection of the cutoff score is based not on subjective judgment, but on a statistical
criterion, such as the point where the chi-square value is a maximum.
The second difference is that while the Burgess method is applied to a binary outcome, the Kerby method can apply to a multi-valued outcome, because CART analysis can identify cutoff scores in such
cases, using a criterion such as the point where the t-value is a maximum. Because CART analysis is not only binary, but also recursive, the result can be that a predictor variable will be divided
again, yielding two cutoff scores. The standard form for each predictor is that a score of one is added when CART analysis creates a partition.
One study (Kerby, 2003) selected as predictors the five traits of the Big five personality traits, predicting a multi-valued measure of suicidal ideation. Next, the personality scores were converted
into standard form with CART analysis. When the CART analysis yielded one partition, the result was like the Burgess method in that the predictor was coded as either zero or one. But for the measure
of neuroticism, the result was two cutoff scores. Because higher neuroticism scores correlated with more suicidal thinking, the two cutoff scores led to the following coding: "low Neuroticism" = 0,
"moderate Neuroticism" = 1, "high Neuroticism" = 2 (Kerby, 2003).
z-score method
Another method can be applied when the predictors are measured on a continuous scale. In such a case, each predictor can be converted into a standard score, or z-score, so that all the predictors
have a mean of zero and a standard deviation of one. With this method of unit-weighted regression, the variate is a sum of the z-scores (e.g., Dawes, 1979; Bobko, Roth, & Buster, 2007).
Literature review
The first empirical study using unit-weighted regression is widely considered to be a 1928 study by sociologist Ernest W. Burgess. He used 21 variables to predict parole success or failure, and the
results suggest that unit weights are a useful tool in making decisions about which inmates to parole. Of those inmates with the best scores, 98% did in fact succeed on parole; and of those with the
worst scores, only 24% did in fact succeed (Burgess, 1928).
The mathematical issues involved in unit-weighted regression were first discussed in 1938 by Samuel Stanley Wilks, a leading statistician who had a special interest in multivariate analysis. Wilks
described how unit weights could be used in practical settings, when data were not available to estimate beta weights. For example, a small college may want to select good students for admission. But
the school may have no money to gather data and conduct a standard multiple regression analysis. In this case, the school could use several predictors—high school grades, SAT scores, teacher ratings.
Wilks (1938) showed mathematically why unit weights should work well in practice.
Frank Schmidt (1971) conducted a simulation study of unit weights. His results showed that Wilks was indeed correct and that unit weights tend to perform well in simulations of practical studies.
Robyn Dawes (1979) discussed the use of unit weights in applied studies, referring to the robust beauty of unit weighted models. Jacob Cohen also discussed the value of unit weights and noted their
practical utility. Indeed, he wrote, "As a practical matter, most of the time, we are better off using unit weights" (Cohen, 1990, p. 1306).
Dave Kerby (2003) showed that unit weights compare well with standard regression, doing so with a cross validation study—that is, he derived beta weights in one sample and applied them to a second
sample. The outcome of interest was suicidal thinking, and the predictor variables were broad personality traits. In the cross validation sample, the correlation between personality and suicidal
thinking was slightly stronger with unit-weighted regression (r = .48) than with standard multiple regression (r = .47).
Gottfredson and Snyder (2005) compared the Burgess method of unit-weighted regression to other methods, with a construction sample of N = 1,924 and a cross-validation sample of N = 7,552. Using the
Pearson point-biserial, the effect size in the cross validation sample for the unit-weights model was r = .392, which was somewhat larger than for logistic regression (r = .368) and predictive
attribute analysis (r = .387), and less than multiple regression only in the third decimal place (r = .397).
In a review of the literature on unit weights, Bobko, Roth, and Buster (2007) noted that "unit weights and regression weights perform similarly in terms of the magnitude of cross-validated multiple
correlation, and empirical studies have confirmed this result across several decades" (p. 693).
Andreas Graefe applied an equal weighting approach to nine established multiple regression models for forecasting U.S. presidential elections. Across the ten elections from 1976 to 2012, equally
weighted predictors reduced the forecast error of the original regression models on average by four percent. An equal-weights model that includes all variables provided calibrated forecasts that
reduced the error of the most accurate regression model by 29% percent.^[1]
An example may clarify how unit weights can be useful in practice.
Brenna Bry and colleagues (1982) addressed the question of what causes drug use in adolescents. Previous research had made use of multiple regression; with this method, it is natural to look for the
best predictor, the one with the highest beta weight. Bry and colleagues noted that one previous study had found that early use of alcohol was the best predictor. Another study had found that
alienation from parents was the best predictor. Still another study had found that low grades in school were the best predictor. The failure to replicate was clearly a problem, a problem that could
be caused by bouncing betas.
Bry and colleagues suggested a different approach: instead of looking for the best predictor, they looked at the number of predictors. In other words, they gave a unit weight to each predictor. Their
study had six predictors: 1) low grades in school, 2) lack of affiliation with religion, 3) early age of alcohol use, 4) psychological distress, 5) low self-esteem, and 6) alienation from parents. To
convert the predictors to standard form, each risk factor was scored as absent (scored as zero) or present (scored as one). For example, the coding for low grades in school were as follows: "C or
higher" = 0, "D or F" = 1. The results showed that the number of risk factors was a good predictor of drug use: adolescents with more risk factors were more likely to use drugs.
The model used by Bry and colleagues was that drug users do not differ in any special way from non-drug users. Rather, they differ in the number of problems they must face. "The number of factors an
individual must cope with is more important than exactly what those factors are" (p. 277). Given this model, unit-weighted regression is an appropriate method of analysis.
Beta weights
In standard multiple regression, each predictor is multiplied by a number that is called the beta weight, regression weight or weighted regression coefficients (denoted β[W] or BW).^[2] The
prediction is obtained by adding these products along with a constant. When the weights are chosen to give the best prediction by some criterion, the model referred to as a proper linear model.
Therefore, multiple regression is a proper linear model. By contrast, unit-weighted regression is called an improper linear model.
Model specification
Standard multiple regression hinges on the assumption that all relevant predictors of the outcome are included in the regression model. This assumption is called model specification. A model is said
to be specified when all relevant predictors are included in the model, and all irrelevant predictors are excluded from the model. In practical settings, it is rare for a study to be able to
determine all relevant predictors a priori. In this case, models are not specified and the estimates for the beta weights suffer from omitted variable bias. That is, the beta weights may change from
one sample to the next, a situation sometimes called the problem of the bouncing betas. It is this problem with bouncing betas that makes unit-weighted regression a useful method.
See also
1. ^ Graefe, Andreas (2015). "Improving forecasts using equally weighted predictors" (PDF). Journal of Business Research. 68 (8). Elsevier: 1792–1799. doi:10.1016/j.jbusres.2015.03.038.
2. ^ Ziglari, Leily (2017). "Interpreting Multiple Regression Results: β Weights and Structure Coefficients" (PDF). General Linear Model Journal. 43 (1). GLMJ: 13–22. doi:10.31523/glmj.043002.002.
• Bobko, P., Roth, P. L., & Buster, M. A. (2007). "The usefulness of unit weights in creating composite scores: A literature review, application to content validity, and meta-analysis".
Organizational Research Methods, volume 10, pages 689-709. doi:10.1177/1094428106294734
• Bry, B. H.; McKeon, P.; Pandina, R. J. (1982). "Extent of drug use as a function of number of risk factors". Journal of Abnormal Psychology. 91 (4): 273–279. doi:10.1037/0021-843X.91.4.273. PMID
• Burgess, E. W. (1928). "Factors determining success or failure on parole". In A. A. Bruce (Ed.), The Workings of the Indeterminate Sentence Law and Parole in Illinois (pp. 205–249). Springfield,
Illinois: Illinois State Parole Board. Google books
• Cohen, Jacob. (1990). "Things I have learned (so far)". American Psychologist, volume 45, pages 1304-1312. doi:10.1037/0003-066X.45.12.1304
• Dawes, Robyn M. (1979). "The robust beauty of improper linear models in decision making". American Psychologist, volume 34, pages 571-582. doi:10.1037/0003-066X.34.7.571. archived pdf
• Gottfredson, D. M., & Snyder, H. N. (July 2005). The mathematics of risk classification: Changing data into valid instruments for juvenile courts. Pittsburgh, Penn.: National Center for Juvenile
Justice. NCJ 209158. Eric.ed.gov pdf
• Kerby, Dave S. (2003). "CART analysis with unit-weighted regression to predict suicidal ideation from Big Five traits". Personality and Individual Differences, volume 35, pages 249-261. doi
• Schmidt, Frank L. (1971). "The relative efficiency of regression and simple unit predictor weights in applied differential psychology". Educational and Psychological Measurement, volume 31, pages
699-714. doi:10.1177/001316447103100310
• Wainer, H., & Thissen, D. (1976). Three steps toward robust regression. Psychometrika, volume 41(1), pages 9–34. doi:10.1007/BF02291695
• Wilks, S. S. (1938). "Weighting systems for linear functions of correlated variables when there is no dependent variable". Psychometrika. 3: 23–40. doi:10.1007/BF02287917.
Further reading
• Dana, J., & Dawes, R. M. (2004). "The superiority of simple alternatives to regression for social science predictions". Journal of Educational and Behavioral Statistics, volume 29(3), pages
317-331. doi:10.3102/10769986029003317
• Dawes, R. M., & Corrigan, B. (1974). Linear models in decision making. Psychological Bulletin, volume 81, pages 95–106. doi:10.1037/h0037613
• Einhorn, H. J., & Hogarth, R. M. (1975). Unit weighting schemes for decision making. Organizational Behavior and Human Performance, volume 13(2), pages 171-192. doi:10.1016/0030-5073(75)90044-6
• Hakeem, M. (1948). The validity of the Burgess method of parole prediction. American Journal of Sociology, volume 53(5), pages 376-386. JSTOR
• Newman, J. R., Seaver, D., Edwards, W. (1976). Unit versus differential weighting schemes for decision making: A method of study and some preliminary results. Los Angeles, CA: Social Science
Research Institute. archived pdf
• Raju, N. S., Bilgic, R., Edwards, J. E., Fleer, P. F. (1997). Methodology review: Estimation of population validity and cross-validity, and the use of equal weights in prediction. Applied
Psychological Measurement, volume 21(4), pages 291-305. doi:10.1177/01466216970214001
• Ree, M. J., Carretta, T. R., & Earles, J. A. (1998). "In top-down decisions, weighting variables does not matter: A consequence of Wilk's theorem." Organizational Research Methods, volume 1(4),
pages 407-420. doi:10.1177/109442819814003
• Wainer, Howard (1976). "Estimating coefficients in linear models: It don't make no nevermind" (PDF). Psychological Bulletin. 83 (2): 213. doi:10.1037/0033-2909.83.2.213.archived pdf
• Wainer, H. (1978). On the sensitivity of regression and regressors. Psychological Bulletin, volume 85(2), pages 267-273. doi:10.1037/0033-2909.85.2.267
External links
• Chis Stucchio blog - Why a pro/con list is 75% as good as your fancy machine learning algorithm | {"url":"https://www.knowpia.com/knowpedia/Unit-weighted_regression","timestamp":"2024-11-13T22:34:28Z","content_type":"text/html","content_length":"100112","record_id":"<urn:uuid:38b0dda1-e400-4b60-ba7c-fc9b7488c389>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00093.warc.gz"} |
Propensity scores
You are reading the work-in-progress first edition of Causal Inference in R. This chapter has its foundations written but is still undergoing changes.
Often we are interested in how some exposure (or treatment) impacts an outcome. For example, we could assess how an ad campaign (exposure) impacts sales (outcome), whether a particular medication
(exposure) improves patient survival (outcome), or whether opening a theme park early to some visitors (exposure) reduces wait times later in the day (outcome). As defined in the Chapter 3, an
exposure in the context of this book is often a modifiable event or condition that occurs before the outcome. In an ideal world, we would simply estimate the correlation between the exposure and
outcome as the causal effect of the exposure. Randomized trials are the best practical examples of this idealized scenario: participants are randomly assigned to exposure groups. If all goes well,
this allows for an unbiased estimate of the causal effect between the exposure and outcome. In the “real world,” outside this randomized trial setting, we are often exposed to something based on
other factors. For example, when deciding what medication to give a diabetic patient, a doctor may consider the patient’s medical history, their likelihood to adhere to certain medications, and the
severity of their disease. The treatment is no longer random; it is conditional on factors about that patient, also known as the patient’s covariates. If these covariates also affect the outcome,
they are confounders.
A confounder is a common cause of exposure and outcome.
Suppose we could collect information about all of these factors. In that case, we could determine each patient’s probability of exposure and use this to inform an analysis assessing the relationship
between that exposure and some outcome. This probability is the propensity score! When used appropriately, modeling with a propensity score can simulate what the relationship between exposure and
outcome would have looked like if we had run a randomized trial. The correlation between exposure and outcome will estimate the causal effect after applying a propensity score. When fitting a
propensity score model we want to condition on all known confounders.
A propensity score is the probability of being in the exposure group, conditioned on observed covariates.
Rosenbaum and Rubin (1983) showed in observational studies conditioning on propensity scores can lead to unbiased estimates of the exposure effect as long as certain assumptions hold:
There are many ways to estimate the propensity score; typically, people use logistic regression for binary exposures. The logistic regression model predicts the exposure using known confounders. Each
individual’s predicted value is the propensity score. The glm() function will fit a logistic regression model in R. Below is pseudo-code. The first argument is the model, with the exposure on the
left side and the confounders on the right. The data argument takes the data frame, and the family = binomial() argument denotes the model should be fit using logistic regression (as opposed to a
different generalized linear model).
glm( exposure ~ confounder_1 + confounder_2, data = df, family = binomial() )
We can extract the propensity scores by pulling out the predictions on the probability scale. Using the augment() function from the {broom} package, we can extract these propensity scores and add
them to our original data frame. The argument type.predict is set to "response" to indicate that we want to extract the predicted values on the probability scale. By default, these will be on the
linear logit scale. The data argument contains the original data frame. This code will output a new data frame consisting of all components in df with six additional columns corresponding to the
logistic regression model that was fit. The .fitted column is the propensity score.
glm( exposure ~ confounder_1 + confounder_2, data = df, family = binomial() ) |> augment(type.predict = "response", data = df)
Recall our causal question of interest from Section 7.1: Is there a relationship between whether there were “Extra Magic Hours” in the morning at Magic Kingdom and the average wait time for an
attraction called the “Seven Dwarfs Mine Train” the same day between 9am and 10am in 2018? Below is a proposed DAG for this question.
In Figure 8.1, we propose three confounders: the historic high temperature on the day, the time the park closed, and the ticket season: value, regular, or peak. We can build a propensity score model
using the seven_dwarfs_train_2018 data set from the touringplans package. Each row of this dataset contains information about the Seven Dwarfs Mine Train during a particular hour on a given day.
First, we need to subset the data to only include average wait times between 9 and 10 am. Then we will use the glm() function to fit the propensity score model, predicting park_extra_magic_morning
using the four confounders specified above. We’ll add the propensity scores to the data frame (in a column called .fitted as set by the augment() function in the broom package).
library(broom) library(touringplans) seven_dwarfs_9 <- seven_dwarfs_train_2018 |> filter(wait_hour == 9) seven_dwarfs_9_with_ps <- glm( park_extra_magic_morning ~ park_ticket_season + park_close +
park_temperature_high, data = seven_dwarfs_9, family = binomial() ) |> augment(type.predict = "response", data = seven_dwarfs_9)
Let’s take a look at these propensity scores. Table 8.1 shows the propensity scores (in the .fitted column) for the first six days in the dataset, as well as the values of each day’s exposure,
outcome, and confounders. The propensity score here is the probability that a given date will have Extra Magic Hours in the morning given the observed confounders, in this case, the historical high
temperatures on a given date, the time the park closed, and Ticket Season. For example, on January 1, 2018, there was a 30.2% chance that there would be Extra Magic Hours at the Magic Kingdom given
the Ticket Season (peak in this case), time of park closure (11 pm), and the historic high temperature on this date (58.6 degrees). On this particular day, there were not Extra Magic Hours in the
morning (as indicated by the 0 in the first row of the park_extra_magic_morning column).
seven_dwarfs_9_with_ps |> select( .fitted, park_date, park_extra_magic_morning, park_ticket_season, park_close, park_temperature_high ) |> head() |> knitr::kable()
.fitted park_date park_extra_magic_morning park_ticket_season park_close park_temperature_high
0.3019 2018-01-01 0 peak 23:00:00 58.63
0.2815 2018-01-02 0 peak 24:00:00 53.65
0.2900 2018-01-03 0 peak 24:00:00 51.11
0.1881 2018-01-04 0 regular 24:00:00 52.66
0.1841 2018-01-05 1 regular 24:00:00 54.29
0.2074 2018-01-06 0 regular 23:00:00 56.25
We can examine the distribution of propensity scores by exposure group. A nice way to visualize this is via mirrored histograms. We’ll use the {halfmoon} package’s geom_mirror_histogram() to create
one. The code below creates two histograms of the propensity scores, one on the “top” for the exposed group (the dates with Extra Magic Hours in the morning) and one on the “bottom” for the unexposed
group. We’ll also tweak the y-axis labels to use absolute values (rather than negative values for the bottom histogram) via scale_y_continuous(labels = abs).
library(halfmoon) ggplot( seven_dwarfs_9_with_ps, aes(.fitted, fill = factor(park_extra_magic_morning)) ) + geom_mirror_histogram(bins = 50) + scale_y_continuous(labels = abs) + labs(x = "propensity
score", fill = "extra magic morning")
Here are some questions to ask to gain diagnostic insights we gain from Figure 8.2.
Look for lack of overlap as a potential positivity problem. But too much overlap may indicate a poor model
Avg treatment effect among treated is easier to estimate with precision (because of higher counts) than in the control group.
A single outlier in either group concerning range could be a problem and warrant data inspection
The best way to decide what variables to include in your propensity score model is to look at your DAG and have at least a minimal adjustment set of confounders. Of course, sometimes, essential
variables are missing or measured with error. In addition, there is often more than one theoretical adjustment set that debiases your estimate; it may be that one of the minimal adjustment sets is
measured well in your data set and another is not. If you have confounders on your DAG that you do not have access to, sensitivity analyses can help quantify the potential impact. See Chapter 11 for
an in-depth discussion of sensitivity analyses.
Accurately specifying a DAG improves our ability to add the correct variables to our models. However, confounders are not the only necessary type of variable to consider. For example, variables that
are predictors of the outcome but not the exposure can improve the precision of propensity score models. Conversely, including variables that are predictors of the exposure but not the outcome
(instrumental variables) can bias the model. Luckily, this bias seems relatively negligible in practice, especially compared to the risk of confounding bias (Myers et al. 2011).
Some estimates, such as the odds and hazard ratios, have a property called non-collapsibility. This means that marginal odds and hazard ratios are not weighted averages of their conditional versions.
In other words, the results might differ depending on the variable added or removed, even when the variable is not a confounder. We’ll explore this more in Section 10.4.2.
Another variable to be wary of is a collider, a descendant of both the exposure and outcome. If you specify your DAG correctly, you can avoid colliders by only using adjustment sets that completely
close backdoor paths from the exposure to the outcome. However, some circumstances make this difficult: some colliders are inherently stratified by the study’s design or the nature of follow-up. For
example, loss-to-follow-up is a common source of collider-stratification bias; in Chapter XX, we’ll discuss this further.
A variable can also be both a confounder and a collider, as in the case of so-called butterfly bias:
Consider Figure 8.3. To estimate the causal effect of x on y, we need to account for m because it’s a counfounder. However, m is also a collider between a and b, so controlling for it will induce a
relationship between those variables, creating a second set of confounders. If we have all the variables measured well, we can avoid the bias from adjusting for m by adjusting for either a or b as
However, what should we do if we don’t have those variables? Adjusting for m opens a biasing pathway that we cannot block through a and b(collider-stratification bias), but m is also a confounder for
x and y. As in the case above, it appears that confounding bias is often the worse of the two options, so we should adjust for m unless we have reason to believe it will cause more problems than it
solves (Ding and Miratrix 2015).
By and large, metrics commonly used for building prediction models are inappropriate for building causal models. Researchers and data scientists often make decisions about models using metrics like
R2, AUC, accuracy, and (often inappropriately) p-values. However, a causal model’s goal is not to predict as much about the outcome as possible (Hernán and Robins 2021); the goal is to estimate the
relationship between the exposure and outcome accurately. A causal model needn’t predict particularly well to be unbiased.
These metrics, however, may help identify a model’s best functional form. Generally, we’ll use DAGs and our domain knowledge to build the model itself. However, we may be unsure of the mathematical
relationship between a confounder and the outcome or exposure. For instance, we may not know if the relationship is linear. Misspecifying this relationship can lead to residual confounding: we may
only partially account for the confounder in question, leaving some bias in the estimate. Testing different functional forms using prediction-focused metrics can help improve the model’s accuracy,
potentially allowing for better control.
Another technique researchers sometimes use to determine confounders is to add a variable, then calculate the percent change in the coefficient between the outcome and exposure. For instance, we
first model y ~ x to estimate the relationship between x and y. Then, we model y ~ x + z and see how much the coefficient on x has changed. A common rule is to add a variable if it changes the
coefficient ofx by 10%.
Unfortunately, this technique is unreliable. As we’ve discussed, controlling for mediators, colliders, and instrumental variables all affect the estimate of the relationship between x and y, and
usually, they result in bias. Additionally, the non-collapsibility of the odds and hazards ratios mean they may change with the addition or subtraction of a variable without representing an
improvement or worsening in bias. In other words, there are many different types of variables besides confounders that can cause a change in the coefficient of the exposure. As discussed above,
confounding bias is often the most crucial factor, but systematically searching your variables for anything that changes the exposure coefficient can compound many types of bias.
In predictive modeling, data scientists often have to prevent overfitting their models to chance patterns in the data. When a model captures those chance patterns, it doesn’t predict as well on other
data sets. So, can you overfit a causal model?
The short answer is yes, although it’s easier to do it with machine learning techniques than with logistic regression and friends. An overfit model is, essentially, a misspecified model (Gelman
2017). A misspecified model will lead to residual confounding and, thus, a biased causal effect. Overfitting can also exacerbate stochastic positivity violations (Zivich, Cole, and Westreich 2022).
The correct causal model (the functional form that matches the data-generating mechanism) cannot be overfit. The same is true for the correct predictive model.
There’s some nuance to this answer, though. Overfitting in causal inference and prediction is different; we’re not applying the causal estimate to another dataset (the closest to that is
transportability and generalizability, an issue we’ll discuss in Chapter 24). It remains true that a causal model doesn’t need to predict particularly well to be unbiased.
In prediction modeling, people often use a bias-variance trade-off to improve out-of-data predictions. In short, some bias for the sample is introduced to improve the variance of model fits and make
better predictions out of the sample. However, we must be careful: the word bias here refers to the discrepancy between the model estimates and the true value of the dependent variable in the
dataset. Let’s call this statistical bias. It is not necessarily the same as the difference between the model estimate and the true causal effect in the population. Let’s call this causal bias. If we
apply the bias-variance trade-off to causal models, we introduce statistical bias in an attempt to reduce causal bias. Another subtlety is that overfitting can inflate the standard error of the
estimate in the sample, which is not the same variance as the bias-variance trade-off (Schuster, Lowe, and Platt 2016). From a frequentist standpoint, the confidence intervals will also not have
nominal coverage (see Appendix A) because of the causal bias in the estimate.
In practice, cross-validation, a technique to reduce overfitting, is often used with causal models that use machine learning, as we’ll discuss in Chapter 21.
The propensity score is a balancing tool – we use it to help us make our exposure groups exchangeable. There are many ways to incorporate the propensity score into an analysis. Commonly used
techniques include stratification (estimating the causal effect within propensity score stratum), matching, weighting, and direct covariate adjustment. In this section, we will focus on matching and
weighting; other techniques will be discussed once we introduce the outcome model. Recall at this point in the book we are still in the design phase. We have not yet incorporated the outcome into our
analysis at all.
Ultimately, we want the exposed and unexposed observations to be exchangeable with respect to the confounders we have proposed in our DAG (so we can use the observed effect for one to estimate the
counterfactual for the other). One way to do this is to ensure that each observation in our analysis sample has at least one observation of the opposite exposure that has matching values for each of
these confounders. If we had a small number of binary confounders, for example, we might be able to construct an exact match for observations (and only include those for whom such a match exists),
but as the number and continuity of confounders increases, exact matching becomes less feasible. This is where the propensity score, a summary measure of all of the confounders, comes in to play.
Let’s setup the data as we did in Section 8.1.
We can re-fit the propensity score using the MatchIt package, as below. Notice here the matchit function fit a logistic regression model for our propensity score, as we had in Section 8.1. There were
60 days in 2018 where the Magic Kingdom had extra magic morning hours. For each of these 60 exposed days, matchit found a comparable unexposed day, by implementing a nearest-neighbor match using the
constructed propensity score. Examining the output, we also see that the target estimand is an “ATT” (do not worry about this yet, we will discuss this and several other estimands in Chapter 10).
library(MatchIt) m <- matchit( park_extra_magic_morning ~ park_ticket_season + park_close + park_temperature_high, data = seven_dwarfs_9 ) m
A matchit object - method: 1:1 nearest neighbor matching without replacement - distance: Propensity score - estimated with logistic regression - number of obs.: 354 (original), 120 (matched) - target
estimand: ATT - covariates: park_ticket_season, park_close, park_temperature_high
We can use the get_matches function to create a data frame with the original variables that only consists of those who were matched. Notice here our sample size has been reduced from the original 354
days to 120.
Rows: 120 Columns: 18 $ id <chr> "5", "340", "12", "1… $ subclass <fct> 1, 1, 2, 2, 3, 3, 4,… $ weights <dbl> 1, 1, 1, 1, 1, 1, 1,… $ park_date <date> 2018-01-05, 2018-12… $ wait_hour <int> 9, 9, 9,
9, 9, 9, 9,… $ attraction_name <chr> "Seven Dwarfs Mine T… $ wait_minutes_actual_avg <dbl> 33.0, 8.0, 114.0, 32… $ wait_minutes_posted_avg <dbl> 70.56, 80.62, 79.29,… $ attraction_park <chr> "Magic
Kingdom", "Ma… $ attraction_land <chr> "Fantasyland", "Fant… $ park_open <time> 09:00:00, 09:00:00,… $ park_close <time> 24:00:00, 23:00:00,… $ park_extra_magic_morning <dbl> 1, 0, 1, 0, 1, 0, 1,… $
park_extra_magic_evening <dbl> 0, 0, 0, 0, 0, 0, 0,… $ park_ticket_season <chr> "regular", "regular"… $ park_temperature_average <dbl> 43.56, 57.61, 70.91,… $ park_temperature_high <dbl> 54.29,
65.44, 78.26,… $ distance <dbl> 0.18410, 0.18381, 0.…
One way to think about matching is as a crude “weight” where everyone who was matched gets a weight of 1 and everyone who was not matched gets a weight of 0 in the final sample. Another option is to
allow this weight to be smooth, applying a weight to allow, on average, the covariates of interest to be balanced in the weighted population. To do this, we will construct a weight using the
propensity score. There are many different weights that can be applied, depending on your target estimand of interest (see Chapter 10 for details). For this section, we will focus on the “Average
Treatment Effect” weights, commonly referred to as an “inverse probability weight”. The weight is constructed as follows, where each observation is weighted by the inverse of the probability of
receiving the exposure they received.
For example, if observation 1 had a very high likelihood of being exposed given their pre-exposure covariates (\(p = 0.9\)), but they in fact were not exposed, their weight would be 10 (\(w_1 = 1 /
(1 - 0.9)\)). Likewise, if observation 2 had a very high likelihood of being exposed given their pre-exposure covariates (\(p = 0.9\)), and they were exposed, their weight would be 1.1 (\(w_2 = 1 /
0.9\)). Intuitively, we give more weight to observations who, based on their measured confounders, appear to have useful information for constructing a counterfactual – we would have predicted that
they were exposed and but by chance they were not, or vice-versa. The propensity package is useful for implementing propensity score weighting.
library(propensity) seven_dwarfs_9_with_ps <- glm( park_extra_magic_morning ~ park_ticket_season + park_close + park_temperature_high, data = seven_dwarfs_9, family = binomial() ) |> augment
(type.predict = "response", data = seven_dwarfs_9) seven_dwarfs_9_with_wt <- seven_dwarfs_9_with_ps |> mutate(w_ate = wt_ate(.fitted, park_extra_magic_morning))
seven_dwarfs_9_with_wt |> select( w_ate, .fitted, park_date, park_extra_magic_morning, park_ticket_season, park_close, park_temperature_high ) |> head() |> knitr::kable()
w_ate .fitted park_date park_extra_magic_morning park_ticket_season park_close park_temperature_high
1.433 0.3019 2018-01-01 0 peak 23:00:00 58.63
1.392 0.2815 2018-01-02 0 peak 24:00:00 53.65
1.409 0.2900 2018-01-03 0 peak 24:00:00 51.11
1.232 0.1881 2018-01-04 0 regular 24:00:00 52.66
5.432 0.1841 2018-01-05 1 regular 24:00:00 54.29
1.262 0.2074 2018-01-06 0 regular 23:00:00 56.25 | {"url":"https://www.r-causal.org/chapters/08-propensity-scores","timestamp":"2024-11-12T06:29:04Z","content_type":"application/xhtml+xml","content_length":"94347","record_id":"<urn:uuid:2993dc99-d0ea-4293-9d58-ec596811fe97>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00711.warc.gz"} |
From aether theory to Special Relativity
by Rafael Ferraro
Publisher: arXiv 2013
Number of pages: 24
At the end of the 19th century light was regarded as an electromagnetic wave propagating in a material medium called ether. The speed c appearing in Maxwell's wave equations was the speed of light
with respect to the ether...
Download or read it online for free here:
Download link
(590KB, PDF)
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Numbers in Tpaalha
Learn numbers in Tpaalha
Knowing numbers in Tpaalha is probably one of the most useful things you can learn to say, write and understand in Tpaalha. Learning to count in Tpaalha 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 Tpaalha is the most widely spoken language, and you want to be able to shop and even bargain with a good knowledge
of numbers in Tpaalha.
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 Tpaalha?
Tpaalha is the second constructed language designed by Jessie Sams (co-creator of Méníshè, for the Freeform series Motherland: Fort Salem) and David J. Peterson for their LangTime Studio adventure, a
streaming series featuring live conlang creation launched in February 2020. Tpaalha is the language of the oppossums, designed in the second season of LangTime Studio starting in October 2020 (the
first season conlang was Engála).Due to lack of data, we can only count accurately up to 17 in Tpaalha. Please contact me if you can help me counting up from that limit.
List of numbers in Tpaalha
Here is a list of numbers in Tpaalha. We have made for you a list with all the numbers in Tpaalha 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 Tpaalha. We also close the list by showing you what the number 1000 looks like in Tpaalha.
• 1) u
• 2) syi
• 3) idi
• 4) mu
• 5) toulh
• 6) khaap
• 7) u khaap it
• 8) syi khaap it
• 9) idi khaap it
• 10) mu khaap it
• 11) toulh khaap it
• 12) khabzyi
• 13) u khabzyi’t
• 14) syi khabzyi’t
• 15) idi khabzyi’t
• 16) mu khabzyi’t
• 17) toulh khabzyi’t
• 36) khabvaalh
Numbers in Tpaalha: Tpaalha numbering rules
Each culture has specific peculiarities that are expressed in its language and its way of counting. The Tpaalha is no exception. If you want to learn numbers in Tpaalha 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 Tpaalha with ease.
The way numbers are formed in Tpaalha is easy to understand if you follow the rules explained here. Surprise everyone by counting in Tpaalha. Also, learning how to number in Tpaalha 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 Tpaalha 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 Tpaalha
Tpaalha numbers have a senary, or base-6, internal structure.
Digits from one to six are u [1], syi [2], idi [3], mu [4], toulh [5], and khaap [6].
From seven to eleven, numbers are formed starting with the added unit to six, followed by the word for six (khaap), then the word it: u khaap it [7] (1 6 &), syi khaap it [8] (2 6 &), idi khaap it
[9] (3 6 &), mu khaap it [10] (4 6 &), and toulh khaap it [11] (5 6 &).
The word for twelve is khabzyi [12], litterally meaning six by two.
Numbers from thirteen to seventeen are formed starting with the added unit to twelve, followed by the word for twelve (khabzyi), then the contraction ’t of the word it: u khabzyi’t [13] (1 6*2 &),
syi khabzyi’t [14] (2 6*2 &), idi khabzyi’t [15] (3 6*2 &), mu khabzyi’t [16] (4 6*2 &), and toulh khabzyi’t [17] (5 6*2 &).
The only other known number for the moment is khabvaalh [36] (6*6).
Numbers in different languages | {"url":"https://www.numbersdata.com/numbers-in-tpaalha","timestamp":"2024-11-04T17:15:40Z","content_type":"text/html","content_length":"18743","record_id":"<urn:uuid:086d9a69-a1d1-4295-9f68-d122ef8cd5ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00275.warc.gz"} |
More L-System Fractal Fun…
Every so often, I find doing some ‘recreation mathematics’ a weird type of fun. I know there are better things to spend my brain power on, so I try to limit such projects.
All of these projects are based on L-systems (or Lindenmayer systems) for a few reasons including: 1) they are fairly easy to implement; 2) it’s good to change one’s pace sometimes; and 3) can use
the practice for some future projects (such as drawing plants).
Cesàro curve fractals…
The Cesàro curve is a specific instance of the De Rham curve. It is similar to the Koch curve, however the angle is much tighter (usually 85^o). The L-system can be described by:
\begin{aligned} \mathsf{Initial\ Axiom} &: F \\ \mathsf{Rule(s)} &: F \mapsto F+F–F+F \\ \mathsf{Angle} &: 85^\circ \end{aligned}
Just a few iterations are needed to produce some interesting results. Below on the left is the Cesàro curve after 7 iterations. On the right is a series of 17 curves connected tip to tip where the
angle increases by 5^o (from 5^o to 85^o); this concept was inspired by Wolter Schraa’s range fractal.
Torn Square Fractal
By placing the Cesàro curve on the edges of a square, the torn square/paper fractal can be made. The big change required in the postscript code is with the axiom function as seen below:
90 rotate RuleF
90 rotate RuleF
90 rotate RuleF
} def
Since the curves form a closed shape, it can be filled in to make it appear as a torn piece of paper. Below on the left is the results after 7 iterations, while on the right is the first 5 iterations
(the initial square is iteration zero).
Some Interesting Stuff…
A couple of interesting things that I have come across on the web…
• In this one math class (Math 207 at WWU), apparently this was part of a homework assignment once. In the homework solutions, there is a pretty cool animated gif.
• Its possible to get some cool t-shirts and notebooks with the Cesàro curve on it.
All the code that was used to make the above images is available here.
Minkowski sausage and islands…
The Minkowski curve is another very simple L-system based fractal. According to online sources, during the first few iterations the curve appears to resemble sausage links. The L-system to produce
such a system is:
\begin{aligned} \mathsf{Axiom} :&\ F \\ \mathsf{Rule(s)} :&\ F \mapsto F+F-F-FF+F+F-F \\ \mathsf{Angle} :&\ 90^\circ \end{aligned}
Note that there are a few slightly different versions of the main L-system rule out there (where some some rotations or flips exist); however they all produce similar looking results. Below on the
left is Minkowski curve after 6 iterations, while in the image on the right contains iterations 0 to 5.
Minkowski island…
Similar to the torn square fractal, if the curve is placed on the edges of a square, then a closed in shape can be created which is referred to as the Minkowski island. Below on the left is the
results after 5 iterations and on the right are iterations 0 to 3.
The code to draw the above fractals is available here.
Dragon curves…
First the dragon curve (or curves since there are a few variations), can be a complete topic on its own. For now, we are limiting the scope to just one version of the curve produced by an l-system.
There is a ton of information out there since its simple production rules can produce some rather complex designs. First some videos…
Vi Hart’s ‘Doodling in Math Class’ is a great video. Perhaps it should be mandatory watching for all kids stuck in boring math classes out there.
Numberphile has some great videos which I have linked to in the past. A lot of university lectures are pretty boring affairs, and their videos should be included as supplementary material.
Finally, a numberphile video with Donald Knuth talking about his dragon curve wall art piece that he and his wife made.
There are a few methods to create the dragon curve including: using an IFS, L-system, paper folding, and even copy and paste with a 90^o rotation (as seen in above Vi Hart video). For the L-system,
the basic rules are:
\begin{aligned} \mathsf{Axiom} :&\ F \\ \mathsf{Rule(s)} :&\ F \mapsto F + G \\ &\ G \mapsto F – G \\ \mathsf{Angle} :&\ 90^\circ \end{aligned}
Note that there are some variations of the system which seem a little more complicated but often make things easier to draw since the start and end points from different iterations line up (after
scaling of course). The following was adapted from L-Systems in Postscript, The Dragon Curve:
\begin{aligned} \mathsf{Axiom} :&\ F \\ \mathsf{Rule(s)} :&\ F \mapsto -F+\ +G- \\ &\ G \mapsto +F -\ – G+ \\ \mathsf{Angle} :&\ 45^\circ \end{aligned}
Then using this second system, the following was produced: on the left is the dragon curve after 17 iterations and on the right, iteration 0, 1, 2, 4, 8 and 12 are shown.
Rounded corners…
From the above images once can notice that curve seems to overlap with itself. This is not at all true, rather there are points when corners touch each other which make it appear that the curve
overlaps. To fix this problem, the code changing the angle (that is the + and – rules), can be changed so that the angle is applied while also drawing a small step forward.
/alpha { 90 } def
/Minus { -1 alpha mul rotate } def
/Plus { alpha rotate } def
/Step { 0.1 baselength mul 0.1 alpha mul cos div } def
/Minus { 10 { -0.1 alpha mul rotate Step 0 rlineto } repeat } def
/Plus { 10 { 0.1 alpha mul rotate Step 0 rlineto } repeat } def
Below are some images examples with the rounded corners. On the left is a dragon curve with 13 iteration with rounded corners, and on the right is a zoom in on one of the ‘islands’ that was drawn on
a blue background to make it that much more fun.
Some items of interest…
The dragon curve is perhaps one of the more popular fractal objects out there and can be found in a variety of places including:
The code for the above fractals is available here.
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Polynomial Degree Calculator - Online 1st 2nd 3rd Nth Order Finder
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What is the degree of a polynomial? (Definition)
The degree of a polynomial is the greatest power (exponent) associated with the polynomial variable. The degree is also called the order of the polynomial.
Example: The trinomial $ x^2 + x + 1 $ of variable $ x $ has for greatest exponent $ x^2 $ that is $ 2 $, therefore the polynomial is of degree $ 2 $ (or the polynomial is of the second degree, where
the polynomial is of order $ 2 $)
The degree is sometimes noted $ \deg $
How to calculate the degree of a polynomial?
To find the degree of a polynomial, it is necessary to have the polynomial written in expanded form.
Example: $ P(x) = (x+1)^3 $ expands $ x^3 + 3x^2 + 3x + 1 $
Browse all the elements of the polynomial in order to find the maximum exponent associated with the variable, this maximum is the degree of the polynomial.
Example: The polynomial has 4 elements: $ \{ x^3, 3x^2, 3x, 1 \} $
$ x^3 $ a for exponent $ 3 $
$ 3x^2 $ a for exponent $ 2 $
$ 3x $ a for exponent $ 1 $
$ 1 $ a for exponent $ 0 $
The maximum power is $ 3 $, so $ P(x) $ is of degree $ 3 $ (third degree).
How to calculate the degree of a polynomial with a variable degree?
The degree of a polynomial having a variable degree remains the maximum value of the exponents of the elements of the polynomial.
Example: $ x^n+x^2+1 $ has for degree $ \max (n,2) $, which therefore depends on the value of $ n $, the degree will be $ n $ if $ n > 2 $ otherwise $ 2 $.
How to calculate the degree of a multivariable polynomial?
The degree of a polynomial is dependent on the associated variable. If there are several variables, calculate the degree of the polynomial for each variable.
What is the degree of the polynomial x
The polynomial $ x $ (also called monomial) has for degree $ 1 $ because $ x = x^1 $
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To find Rate when Principal Interest and Time are given | How to Calculate Interest Rate?
Wanna become perfect in various concepts of simple interest? Then here is the perfect guide for you. You can check how to find the principal amount when time interest and rate are given in the
previous articles. Now, you can know the step-by-step procedure to find the rate when principal interest and time are given. Follow how to calculate interest rate per annum, month along with the
formulas and tips.
Finding the Rate when Principal Interest and Time are given
The interest rate is known as the amount of interest that is due per period. It is calculated as the proportion of the amount deposited, borrowed, and lent. For example, If you borrowed $3000 from a
bank and the agreement of the loan stipulates that the rate of interest on the loan is 10%, which means that the borrower must pay the original amount of the loan like $3000 + (10% * 3000) = 3000 +
300 = $3300.
Read More Related Articles:
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┃To find Principal When Time Interest and Rate are Given │ ┃
Formula to find Rate when Principal, Interest, and Time are given
Consider an example of Mary. Mary took a loan to buy a house from the bank. The amount of money he borrowed from the bank was $7000. According to the bank loan terms, Mary’s loan completes in five
years at an interest rate of 6%. He borrowed an initial amount of $7000 which is called the principal amount. In this example, we have the terms of Principal amount, Interest Rate, Interest, Time,
Suppose that we only know the values of Principal Amount, Interest Amount, and Time and we have to find the rate of interest. We know the interest formula as I = P * R * T, if we have all the values
and need to find the rate of interest, then we have to rearrange the values. Therefore, the rate of interest can be written as
r = I / PT
in which r is the rate of interest and equals interest rate divided by the principal amount and time period.
Consider an example of finding the rate of interest of the loan when the total interest amount is $20,000 and the principal amount is $7000 over the period of 5 years?
To solve the above calculations, we have to substitute the values in the equation of r = I / PT
r = 20,000 / 7000 * 5
Therefore, the rate of interest is 14.2%
How to Calculate Rate of Interest?
Follow the simple steps provided below while finding the rate of interest. They are as follows:
Step 1: Calculate your interest rate
If you have to calculate the interest rate, you must know the interest formula R = I / PT. Here,
I is the interest amount that is to be paid in the specific time period (year, month etc.)
P is the principal amount before interest
t is the time period involved
r is the interest rate in decimal value
This equation will be used to calculate the basic interest rate.
Step 2: Convert into decimal values
Once you substitute all the values that are required to calculate the rate of interest, you will get the value in decimals. Then, you have to convert the rate of interest by multiplying it by 100.
For suppose, the decimal value is .81, it will not help while figuring the interest rate. Therefore, to find the interest rate for .81, you have to multiply it by 100 which is .81 * 100 = 81%
Step 3: Calculate the missing values
If any of the values of the time period, interest amount, principal amount are missing, find those using the formulas,
Interest amount, I = P * R * T
Principal amount, P = I / r * t
Time Period, T = I / P* r
Step 4: Make sure all the values have the same parameter
You have to make sure that all the values of the time period, interest rate are having the same parameter values. If the time period is given in months or days, then you have to convert it to years
by dividing it by 12 or 365. The time period which is given must be the same amount of time as the interest paid.
Step 5: Substitute the values in the equation
As we know the rate of interest equation, R = I / PT. Substitute the values of Time, Rate and Interest in the equation.
Suppose that the principal amount is 5000 and the interest is 2000 with a time period of 2 years. Find the rate of interest?
As given,
I = 5000
P = 2000
T = 2 years
The equation is,
R = I / PT
R = 5000 / 2000 * 2
R = 5%
Hence, the rate of interest will be determined in the following steps.
Examples on Calculating Rate of Interest
Problem 1:
A sum of Rs. 12,500 amounts to Rs. 15,500 in 4 years at the rate of simple interest. What is the rate of interest?
As given in the question,
Principal = Rs. 12,500
Amount = Rs. 15,500
We know that,
I = A – P
I = 15,500 – 12,500
I = 3,000
We also know that
I = P * R * T / 100
3000 = 12500 * R * 4 / 100
3000 = 125 * 4 * R
R = 3000/125 * 4
R = 3000/500
R = 6%
Therefore, the rate of interest = 6%
Problem 2:
Reha borrows a sum of Rs. 5000 and pays a total amount of Rs. 5500 after 2 years. Find the rate of interest?
As given in the question,
Money borrowed P = Rs. 5000
Amount returned A = Rs 5500
Time T = 2 years
Interest I = Rs 500
We know that,
I = P * R * T / 100
Substitute the values in the above equation,
500 = 5000 * R * 2 / 100
R = 500 * 100 / 5000 * 2
R = 5%
Therefore, the rate of the interest = 5%
Problem 3:
Jerry earned $7000 in interest. He originally deposited $14000 in the account and left it there for 25 years. What is the interest rate he earned?
As given in the question
Interest = 7000
Principal = 14000
Time = 25 years
As we know that,
I = P * R * T
7000 = 14000 * R * 25
7000 = 350000 * R
Divide the equation by 350000
7000 / 350000 = 350000 * R / 350000
R = .02
Multiply it with 100 to convert it into the percentage
∴ R = .02 * 100
R = 2%
Therefore, the rate of interest is 2% | {"url":"https://ccssanswers.com/to-find-rate-when-principal-interest-and-time-are-given/","timestamp":"2024-11-04T04:08:24Z","content_type":"text/html","content_length":"155155","record_id":"<urn:uuid:efc5ada2-3da0-4c0e-b8e1-0a5953ddd7b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00576.warc.gz"} |
Excel Control Charts For Multiple People In A Department 2024 - Multiplication Chart Printable
Excel Control Charts For Multiple People In A Department
Excel Control Charts For Multiple People In A Department – You may create a multiplication graph or chart in Excel by using a web template. You will find several instances of web templates and figure
out how to format your multiplication chart using them. Here are a few tricks and tips to generate a multiplication graph. Upon having a format, all you have to do is backup the formulation and
mixture it within a new mobile phone. You may then utilize this formulation to multiply several phone numbers by one more establish. Excel Control Charts For Multiple People In A Department.
Multiplication kitchen table format
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. First, you need to secure row one of several header line, then multiply the quantity on
row A by cell B. An additional way to develop a multiplication table is by using combined references. In such a case, you would probably enter in $A2 into line A and B$1 into row B. The end result
can be a multiplication table using a formula that really works for both columns and rows.
You can use the multiplication table template to create your table if you are using an Excel program. Just open up the spreadsheet along with your multiplication table template and change the name on
the student’s brand. You can also change the page to match your person needs. It comes with an solution to modify the colour of the tissue to improve the appearance of the multiplication desk, as
well. Then, it is possible to transform the plethora of multiples to meet your requirements.
Building a multiplication graph or chart in Stand out
When you’re using multiplication kitchen table software program, it is possible to develop a basic multiplication table in Shine. Merely create a page with columns and rows numbered in one to 40. In
which the rows and columns intersect will be the respond to. If a row has a digit of three, and a column has a digit of five, then the answer is three times five, for example. The same thing goes for
the opposite.
Very first, you can enter the phone numbers you need to increase. If you need to multiply two digits by three, you can type a formula for each number in cell A1, for example. To produce the phone
numbers greater, pick the cells at A1 and A8, and then click on the proper arrow to pick a selection of cells. You may then variety the multiplication method from the tissue within the other rows and
Gallery of Excel Control Charts For Multiple People In A Department
Employee Skills Matrix Download Your Free Excel Template GetSmarter Blog
Download The Daily Work Schedule For Multiple Employees From Vertex42
Weekly Work Schedule Template Excel This Excel Template Is Use For
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16×161/2×161/4×161/8×……………...∞ equals... | Filo
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Question Text equals
Topic Sequences and Series
Subject Mathematics
Class Class 11
Answer Type Text solution:1
Upvotes 123 | {"url":"https://askfilo.com/math-question-answers/16-times-161-2-times-161-4-times-161-8-times-ldots-ldots-ldots-ldots-ldots-infty","timestamp":"2024-11-07T02:19:30Z","content_type":"text/html","content_length":"379603","record_id":"<urn:uuid:7a8c4145-384a-418b-abe2-036c8285f14d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00708.warc.gz"} |
Cochlear Mechanics and the Hopf Bifurcation
From 1996 through 2004, in a tight collaboration with Jim Hudspeth and our respective research groups, we outlined the “Hopf Bifurcation” scenario for cochlear dynamics. Tommy Gold’s theory was that
active mechanisms in the cochlea amplify mechanically the acoustical signals and re-inject it into the cochlea, to cancel the viscous loss of the cochleas’ narrow passageways. Our analysis was that
if this mechanism was poised so as to cancel the viscosity exactly, the dynamical description would be that of a Hopf bifurcation.
Consider playing with the volume control of a public-address system. If you set it too loud, the full-circuit gain from mic to loudspeaker, through the air back to the mic will reach 1 for some
frequency band and a feedback oscillation will ensue. If the gain is adjusted extremely carefully until the system is at the edge of starting screeching, then something interesting happens. The
system resonates at one specific frequency; if you so much as lightly hum at that exact frequency, the system will amplify the hum enormously. But only that specific frequency.
Analysis of the Hopf bifurcation showed that this scenario predicts four characteristics: frequency selectivity, enormous gain, compressive nonlinearity (the response will generically follow a
cubic-root law), and a failure of the poising mechanism will cause a self-sustaining oscillation.
Close enough to the bifurcation every model looks like
But what about my model?
The thing to understand is that we are not postulating a model, but rather a scenario. Any given model (and there’s many many models of cochlear dynamics) could happen to hide within it a set of Hopf
bifurcations. In fact any model having sets of positive feedbacks will, for some parameter values, have a Hopf bifurcation. And if the model parameters are fit so as to fall close to the bifurcation,
then the four generic properties outlined above ensue. They are generic, universal properties of any system which is set exactly at a Hopf bifurcation.
This universality holds
Multiple Hopf bifurcations
The above discussion concerns a single Hopf bifurcation. | {"url":"https://sur.rockefeller.edu/?page_id=232","timestamp":"2024-11-02T07:53:53Z","content_type":"text/html","content_length":"71344","record_id":"<urn:uuid:27759b4d-554b-476d-a60b-f870a76503e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00505.warc.gz"} |
RATIO AND PROPORTION FOR GRADE 6 - Kidpid
RATIO AND PROPORTION FOR GRADE 6
Let’s explain the concepts of ratio and proportion to a class of 6th graders.
A ratio is a way to compare two or more quantities or numbers. It shows the relationship between the quantities or numbers in terms of their relative sizes or amounts. Ratios are often expressed
in the form of a fraction, using a colon (:), or using the word “to.”
For example, let’s consider a class of 20 students, with 12 boys and 8 girls. The ratio of boys to girls in the class can be written as:
12 boys : 8 girls
12 boys / 8 girls
12 boys to 8 girls
The ratio tells us that for every 12 boys, there are 8 girls in the class.
Proportion is a special type of ratio that shows the equality of two ratios. When two ratios are proportional, it means that the corresponding fractions are equal.
For example, let’s consider two different fruit baskets. In the first basket, there are 5 apples and 3 oranges, and in the second basket, there are 10 apples and 6 oranges. We can compare the
ratio of apples to oranges in both baskets:
Basket 1: 5 apples : 3 oranges
Basket 2: 10 apples : 6 oranges
To check if the two ratios are proportional, we can cross-multiply and see if the fractions are equal:
5/3 = 10/6
Simplifying both sides of the equation, we get:
10/6 = 10/6
Since both fractions are equal, we can say that the ratios are proportional. This means that for every 5 apples, there are 3 oranges in both baskets.
In summary, ratio is a way to compare quantities or numbers, while proportion is a special type of ratio that shows the equality of two ratios. Ratios help us understand the relationship between
different quantities or amounts, while proportions allow us to compare different ratios and determine if they are equal.
• 0 Replies
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[GAP Forum] GaloisType is running very long
Alexander Hulpke hulpke at math.colostate.edu
Mon Dec 29 18:11:34 GMT 2014
Dear Forum, Dear Daniel Blazewicz,
> I was executing GaloisType method for several irreducible polynomials of the form x^12+ax+b to find solvable examples. And my loop "hung" at x^12+63*x-450. After 2 days I decided to break the execution. FYI, I use 4 years old laptop with ~2GHz CPU. Do you know if it is expected that GaloisType method can take days (weeks?) for polynomials of 12th degree?
Yes and No.
Why No:
In your case there actually is a minor bug (a routine for approximating a root iterates between two approximations without stopping). This will be fixed in the next release. Thank you very much for reporting it. (I append the (pathetic -- it shows my lack of knowledge of numerical analysis) routine if you want an immediate patch.)
What I would recommend for a search like yours is to call
on the polynomial first. If it returns S_n (or A_n) as only possibilities -- and this will happen in most cases -- the result is correct and you presumably can eliminate the polynomial.
Why Yes:
If you happen upon a polynomial whose Galois group is M_{12} the certificate for distinguishing it from A_{12} is very expensive. Such a calculation will possibly take weeks. (ProbabilityShapes should be done always quickly, but does not guarantee that a Galois group cannot be larger.)
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
local r,e,f,x,nf,lf,c,store,letzt;
store:= e<=10 and IsInt(r) and 0<=r and r<=100;
if store and IsBound(APPROXROOTS[e]) and IsBound(APPROXROOTS[e][r+1])
then return APPROXROOTS[e][r+1];
if Length(arg)>2 then
if nf=0 then
if nf>lf then
if lf<2 then
# until 3 times no improvement
until c>2 or x in letzt;
if store then
if not IsBound(APPROXROOTS[e]) then
return x;
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A case study in micro-optimization | Sijin Joseph
A case study in micro-optimization
Last week I saw a wrap up from cedric’s coding challenge on his blog, the problem looked simple enough, “write a counter function that counts from 1 to max but only returns numbers whose digits don’t
My first stab at it consisted of a brute force solution of looking at every natural number up to a given limit and determining if the digits were unique, the crux of which was this function that
given a number determined if the digits repeated or not
private static bool DoDigitsRepeat(long num) {
····//Used to track which digits have already been encountered
····int used = 0;
····while (num > 0) {
········int digit = (int)num % 10;
········num = num / 10;
········int index = 1 << digit;
········if ((used & index) == index) {
············return true;
········else {
············used |= index;
····return false;
The good thing about this solution was that it met the secondary goal for the problem which was to determine the biggest gap in the sequence of generated numbers, but in terms of performance it
sucked! It worked well for the smaller limits defined in the problem but when you tried to push and generate all matching numbers till the max possible i.e. 9876543210 it just took too long (I think
it was under a minute, but that was still too long)
I went back to the problem page and saw that the scripting weenies who had tried to use string functions to prune numbers had the worst performance of all solutions, that made me feel a bit better
until I saw CrazyBob’s Java solution(the fast version), the comments indicated that the solution found the total count in under half a second. This solution was not able to determine the biggest gap
though because the numbers were generated out of order but nevertheless this was an excellent solution.
So began my quest to come up with a faster solution. I felt sure that I could use the bit twiddling hacks to come up with a faster solution, it was just a question of hitting the right spot.
The first non-brute force solution that I tried was to enumerate all possible subsets of {0,1,2,3,4,5,6,7,8,9} and then to generate permutations from that set making sure that zero was never in the
first place. Subset generation is pretty easy to do, all you need to do is count from 0 to 2^n where n is the number of elements in the set, the bit patterns of all the numbers in this range can be
used to generate all the subsets. For the first cut I tried to be Object-Oriented and used the below class to implement a BitTable.
public class BitTable
····private uint _storage = 0;
····public BitTable(uint value) {
········_storage = value;
····public void Set(int index) {
········Debug.Assert(index >= 0 && index < 32);
········_storage |= (uint)(1 << index);
····public void Reset(int index) {
········Debug.Assert(index >= 0 && index < 32);
········_storage &= (uint)~(1 << index);
····public bool IsSet(int index) {
········Debug.Assert(index >= 0 && index < 32);
········uint val = (uint)(1 << index);
········return (_storage & val) == val;
Of course as I soon found out, all the method calls to BitTable were really slowing things down (and by slow I mean a 1-2 seconds slower than CrazyBob’s solution), so I dropped the class and moved
all the operations inline, also I realized that since I was using value types to hold the state in the search/call tree, I didn’t need to set and unset the state after each recursive call. Here’s the
final version of this line of thinking.
class Beust4
private static int _total = 0;
public static void Run() {
····_total = 0;
····//Generate all possible subsets of a set of 10 elements
····//2^10 = 1024
····for (int i = 1; i <= 1024; ++i) {
········Permute(i, 0L);
····Console.WriteLine("nTotal: {0}", _total);
private static void Permute(int digits, long current) {
····for (int index = (current > 0) ? 0 : 1; index <= 9; ++index) {
········if ((digits & (1 << index)) == (1 << index)) {
············if ((digits & ~(1 << index)) == 0) {
················//Console.Write("{0}, ", (current * 10) + index);
············else {
················Permute(digits & ~(1 << index), (current * 10) + index);
I felt really good about this attempt but to my surprise when I ran it, I took 1.2 seconds on average which was still more than 2 times slower than CrazyBob’s Java solution. I got really stuck at
this point and I had to make sure that there was not something obvious that I was missing. The first thing I did was to port CrazyBob’s solution to .Net so that I could compare both solutions, I’ve
uploaded the C# version of CrazyBob’s solution here in case anyone wants to take a look.
On the surface it looks like the bit twiddling based solution should run faster because it does not make all those method calls, the other thing that I suspected was causing problems was the number
of recursive function calls that were being made, so I put in some code to check the number of recursive function calls that were being made and to my surprise I found that CrazyBob’s solution was
making 8877691 recursive calls as compared to my solution which was making almost 10 times that number. Also the actual soultion count is 8877690 which meant that the number of calls in CrazyBob’s
solution was near optimal. So it was clear that it was the number of calls that were costing me the half second. Btw CrazyBob’s C# version still ran in 700ms on average on my laptop, which was still
500ms faster than my C# version.
I then started to think about alternate ways to attack the issue, one track I went down was to consider all the digits as a complete graph and then coming up with a way to enumerate all paths in the
graph, traversing a n edge in the graph would remove other edges from the graph and make them not available. This reminded me on Knuth’s Dancing Links Algorithm and I read up on that a bit, this
paper from Knuth on the subject was an excellent read. It looked to me that CrazyBob had used an approach similar to the DLX algorithm, but after reading the entire paper from Knuth it still didn’t
strike me as to why using a DLX approach would provide such excellent performance as compared to my version.
So I went back a bit to comparing both solutions and I think the two key observations that I saw that
1. CrazyBob’s solution went down the search tree to one level above the last and then just generated all the solution from there instead of recursing down to the last level, so for e.g. supposing it
was generating 5 digit numbers and it had already generated 4321, then at that level it didn’t make additional recursive calls to add the final digit, it was able to add the last digit at the
same level pruning the search tree quite a bit. In contrast my solution was basically doing a method call for every digit of every number in the solution set.
2. The above optimization was made possible by using the length of the final solution as the key, so first all 1 digit solutions were generated, followed by two digit ones and so on.
Cool, so now I ported my bit twiddling version to generate based on the length of the numbers and the optimization to prune the search tree one level above the last came naturally. I ran my solution
and guess what, it was still 200 ms slower than CrazyBob 🙁 Aaaarghhhh!!!!!
for (int len = 1; len <= 10; ++len) {
····Generate(len, 0, 0xFFF >> 2, 0);
private static void Generate(int maxLen, int currentLen, int availableDigits, long currentValue) {
····bool last = (currentLen == maxLen - 1);
····for (int digit = (currentValue == 0) ? 1 : 0; digit <= 9; ++digit) {
········if ((availableDigits & (1 << digit)) != (1 << digit))
········if (last) {
············//Console.Write("{0}, ", (currentValue * 10) + i);
········else {
············Generate(maxLen, currentLen + 1, availableDigits & ~(1 << digit), (currentValue * 10) + digit);
But I knew I was getting close, at this point I knew what was killing me basically to determine which bits were set I was iterating from 0 to 9 and then testing if that bit was set in the number or
not, this test was killing me because most of the times the bit was not set and I was doing a huge huge number of unnecessary tests. So I needed a way to iterate through only the set bits. The first
solution I tried used a hashtable but that caused a even bigger degradation in performance. Finally for a lack of a better way to express this in C# I had to waste 4KB of memory and use an array to
allow me to iterate through the indexes of the set bits in a number.
The final solution ran in 350ms on average, almost a 50% improvement over CrazyBob’s solution, woot!!!!! A further optimzation to move the if statement from inside the loop to outside which makes the
code more readable but incredibly smelly because of the near duplicate code in both the if and else blocks shaved off another 50ms. Here’s the final version without the optimzation for moving the if
statement outside which makes the code a bit shorter.
class Beust5
····private static int _total = 0;
····private static int[] _pre = null;
····public static void Run() {
········_total = 0;
········_pre = new int[(1 << 10) + 1];
········for (int i = 0; i <= 10; ++i) {
············_pre[1 << i] = i;
········for (int len = 1; len <= 10; ++len) {
············Generate2(len, 0, 0xFFF >> 2, 0);
········Console.WriteLine("nTotal: {0}", _total);
····private static void Generate2(int maxLen, int currentLen, int availableDigits, long currentValue) {
········bool last = (currentLen == maxLen - 1);
········int x = availableDigits;
········while (x != 0) {
············//digit will contain the lowest set bit
············int digit = _pre[x ^ (x & (x - 1))];
············x &= (x - 1);
············//Avoid starting with zero
············if (digit == 0 && currentValue == 0)
············if (last) {
················//Console.Write("{0}, ", (currentValue * 10) + i);
············else {
················Generate2(maxLen, currentLen + 1, availableDigits & ~(1 << digit), (currentValue * 10) + digit); | {"url":"https://sijinjoseph.com/post/2008-08-29-a-case-study-in-micro-optimization/","timestamp":"2024-11-12T03:30:39Z","content_type":"text/html","content_length":"27563","record_id":"<urn:uuid:635ee645-e8a0-4941-8eca-a4951eb82381>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00629.warc.gz"} |
A and B working alone can do a work in 20 days and 15 days respectively. They started the work together but B left after sometime and A finished remaining work in 6 days. Find after how many days from start B left the work ?
Question Paper from: SBI Clerk Prelims 2018
A and B working alone can do a work in 20 days and 15 days respectively. They started the work together but B left after sometime and A finished remaining work in 6 days. Find after how many days
from start B left the work ?
5 days
4 days
6 days
3 days
7 days
Chapter Name: Time and Work | {"url":"https://prepsutra.com/question_view/sbi-clerk-prelims-2018/question/a-and-b-working-alone-can-do-a-work-in-20-days-and-15-days-respectively-they-started-the-work-together-but-b-left-after-sometime-and-a-finished-remaining-work-in-6-days-find-after-how-many-days-from-s/","timestamp":"2024-11-09T15:42:36Z","content_type":"text/html","content_length":"196667","record_id":"<urn:uuid:23a90e4a-1ce6-4dd5-9d17-6c5ab5247677>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00256.warc.gz"} |
Understanding the Coordinate Plane - dummies
If you need a quick refresher about how the x-y coordinate system works, you've come to the right place. Let's start with the following figure, which shows you the lay of the land of the coordinate
Here's the lowdown on the coordinate plane you see in the figure:
• The horizontal axis, or x-axis, goes from left to right and works exactly like a regular number line. The vertical axis, or y-axis, goes—ready for a shock?—up and down. The two axes intersect at
the origin (0, 0).
• Points are located within the coordinate plane with pairs of coordinates called ordered pairs—like (8, 6) or (–10, 3). The first number, the x-coordinate, tells you how far you go right or left;
the second number, the y-coordinate, tells you how far you go up or down. For (–10, 3), for example, you go left 10 and then up 3.
• Going counterclockwise from the upper-right-hand section of the coordinate plane are quadrants I, II, III, and IV:
□ All points in quadrant I have two positive coordinates, (+, +).
□ In quadrant II, you go left (negative) and then up (positive), so it's (–, +).
□ In quadrant III, it's (–, –).
□ In quadrant IV, it's (+, –).
Because all coordinates in quadrant I are positive, it's often the easiest quadrant to work in.
• The Pythagorean Theorem comes up a lot when you're using the coordinate system because when you go right and then up to plot a point (or left and then down, and so on), you're tracing along the
legs of a right triangle; the segment connecting the origin to the point then becomes the hypotenuse of the right triangle. In the figure, you can see the 6-8-10 right triangle in quadrant I.
About This Article
This article is from the book:
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How To Calculate Loan Percentage
Interest rate: the cost to borrow money. It is expressed as a percentage of the loan principal. Interest rates can be fixed or variable. APR: the total yearly. Annual interest rate for this loan.
Interest is calculated monthly on the current outstanding balance of your loan at 1/12 of the annual rate. Information and. Multiply this result by your principal to find out your monthly loan
payment. For instance, you take out a $50, mortgage and receive a 5% interest rate. Your. The formula for EMI is: EMI = P * r * (1+r)^n/ ((1+r)^n-1) Where, P = principal r = monthly interest rate n =
loan tenure. How Is APR Calculated for Loans? A loan's APR is calculated by determining how much the loan is going to cost you each year based on its interest rate and.
Learn how to calculate auto loan interest for your next vehicle and see how much it impacts your monthly car payment. If you know the amount of a loan and the amount of interest you would like to
pay, you can calculate the largest interest rate you are willing to accept. Lenders multiply your outstanding balance by your annual interest rate and divide by 12, to determine how much interest you
pay each month. The team at Beechmont Toyota has created a guide on how to calculate auto loan interest with ease. Let's get started, and be sure to visit the finance center. Using the interest rate
formula, we get the interest rate, which is the percentage of the principal amount, charged by the lender or bank to the borrower for. APR is calculated by multiplying the periodic interest rate by
the number of periods in a year in which it was applied. It does not indicate how many times the. How to calculate APR on a loan in 7 steps · 1. Find the interest rate and charges · 2. Add the fees ·
3. Divide the sum by the principal balance · 4. Divide by the. Annual interest rate for this loan. Interest is calculated monthly on the current outstanding balance of your loan at 1/12 of the annual
rate. This simple loan calculator can help you see how different interest rates, loan terms and loan amounts can impact a monthly payment. Average interest rates for personal loans ; Loan term, , ;
24 months, %, %. personal loan calculator: personal loan calculator allows you to calculate your EMI using variables like the amount borrowed, interest rate, and loan tenure.
Interest rate. Your interest rate is the percentage you'll pay to borrow the loan amount. Borrowers with strong credit may be eligible for a lender's lowest. Free online calculator to find the
interest rate as well as the total interest cost of an amortized loan with a fixed monthly payback amount. Divide the amount of the additional payment by the amount loaned to determine the simple
interest rate. For example, consider a loan of $1,, which must be. If you have availed a loan of Rs. 10 Lakh from a lending institution at an interest rate of % for a tenure of 10 years or months,
the formula. The online monthly interest calculator ensures quick computation on how to calculate interest and EMIs from the comfort of your home. How does my credit score affect my interest rate?
The formula for calculating APR is APR = ((Interest + Fees / Loan amount) / Number of days in loan term)) x x How to Calculate Interest Rate on a Car Loan · Principal Amount x Interest Rate x Time
(in years) = Total Interest · $20, (Principal) x (Interest Rate). How to Calculate Interest Rate on a Loan: Principal Loan Amount x Interest Rate x Repayment Tenure = Interest. For more details click
There are several factors that determine your interest rate, including your loan type, loan amount, down payment amount and credit history. Interest rates are. Divide the loan amount by the interest
over the life of the loan to calculate your monthly payment. Several factors can change your monthly payment amount. If. This typically involves multiplying your loan balance by your interest rate
and then dividing this amount by days (a regular year). This shows your daily. Log in to your account and go to the loan details page. · Locate your current balance, interest rate, and repayment
term. Using the interest rate formula, we get the interest rate, which is the percentage of the principal amount, charged by the lender or bank to the borrower for.
Mortgage Calculator With Extra Payment
Loan Amount: This is the total amount borrowed to purchase a home or refinance an existing mortgage. Interest Rate: The interest rate determines the cost of.
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Sum of the first 440 square numbers
We define square numbers as numbers that when squared will equal a whole number. Thus, the list of the first square numbers starts with 1, 4, 9, 16, and so on.
What is the sum of the first 440 square numbers, you ask? Here we will give you the formula to calculate the first 440 square numbers and then we will show you how to calculate the first 440 square
numbers using the formula.
The formula to calculate the first n square numbers is displayed below:
To calculate the sum of the first 440 square numbers, we enter n = 440 into our formula to get this:
440(440 + 1) × (2(440) + 1)
First, calculate each section of the numerator: 440(440 + 1) equals 194040 and (2(440) + 1) equals 881. Therefore, the problem above becomes this:
Next, we calculate 194040 times 881 which equals 170949240. Now our problem looks like this:
Finally, divide the numerator by the denominator to get our answer:
170949240 ÷ 6 = 28491540
There you go. The sum of the first 440 square numbers is 28491540.
You may also be interested to know that if you list the first 440 square numbers 1, 2, 9, etc., the 440th square number is 193600.
Sum of Square Numbers Calculator
Need the answer to a similar problem? Get the first n square numbers here.
What is the sum of the first 441 square numbers?
Here is the next math problem on our list that we have explained and calculated for you.
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MCQ Questions for Class 10 Maths with Answers PDF Download Chapter Wise
Here you will find Chapter Wise NCERT MCQ Questions for Class 10 Maths with Answers PDF Free Download based on the important concepts and topics given in the textbook. All these CBSE Class 10 Maths
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Class 10 Maths MCQs Multiple Choice Questions with Answers
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Simultaneous proofs
Two open problems about the distribution of the primes have been solved within about 24 hours of each other, namely the ternary Goldbach conjecture and a weakened form of the twin prime conjecture.
Let’s look at these in approximate chronological order:
Ternary Goldbach conjecture
This states that every odd integer $n \geq 5$ can be expressed as the sum of three primes. Hardy and Littlewood first proved this for sufficiently large integers, conditional on the truth of the
Riemann Hypothesis. Later, in 1937, Vinogradov proved the result for sufficiently large numbers without requiring the Riemann Hypothesis; this became known as Vinogradov’s theorem.
An lower bound was established, which was gradually reduced to $e^{3100}$ in 2002, still ahead of the computer searches (which have only probed up to about $10^{18}$). Very recently, this was further
reduced to 5, as desired; see the paper by Helfgott.
We can corollary-snipe† at this point, and state that this trivially implies that every integer $n \geq 8$ can be expressed as the sum of four primes. The full version of Goldbach’s conjecture is
equivalent to every integer $n \geq 6$ being expressible as the sum of three primes.
Twin primes conjecture
Twin primes are primes which differ from their closest prime neighbours by 2. For instance, {5, 7}, {59, 61} and {$65516468355 \times 2^{333333} - 1$, $1 + 65516468355 \times 2^{333333}$ } are
examples of pairs of twin primes. It has been conjectured that there are infinitely many such pairs, although no proof is known (there was a purported proof in 2004, but it was fallacious).
It has now been proved that infinitely many pairs of primes have a distance of at most 70000000. Again, this can be corollary-sniped to deduce that there exists some $N \leq 70000000$ such that
infinitely many pairs of primes are separated by precisely N. No explicit values of N are known, and the original conjecture states that 2 has this property.
Conclusion and footnotes
Together, these results provide another piece of evidence for the following conjecture:
There are infinitely many pairs of exciting proofs published within 70000000 milliseconds of each other.
† Corollary-sniping is a rather impolite and dishonourable practice in which one jumps on a big theorem proved by someone else, and proves one or more corollaries using it. For instance, if someone
suddenly exclaimed “Hence Fermat’s Last Theorem!” just as Andrew Wiles proved the necessary cases of the Tanayama-Shimura conjecture, that would have been an epic case of corollary-sniping (if that
happened, hopefully the prize would still have been awarded to Wiles). An actual instance was when Xia’s proof that particles can be projected to infinity in finite time under Newtonian gravity was
famously corollary-sniped to deduce that the n-body problem is undecidable.
17 Responses to Simultaneous proofs
1. Actually, 5 can’t be represented as sum of 3 primes. But it can be with at most 3 primes
□ Oh, yes, by 5 I meant 7. By the way, is your surname Nadara? If so, some of my friends are acquainted with you.
☆ No, I’m not Nadara, and I don’t think I know any of your friends, because I live in middle Poland. But from small-world phenomenon we aren’t that far from knowing each other 😉
Also, from what I know, Goldbach’s weak conjectures says about sum of 3 odd primes. I can’t see obvious equivalence, because, we can have number of form p+2+2 which may not be sum of odd
primes below p. But I think Helfgott’s result estabilishes both cases.
○ Wojtek Nadara is Polish, a similar age to you (based on information you’ve provided), and went to IMO 2012. Maybe the name is more common than I first imagined…
■ Oh, no, it isn’t me. I never took part in IMO (but I hope I will in future). In Poland Wojtek is sort of popular name, but I’m not affiliated with Nadara
■ In that case, good luck!
2. If you assume that the universe as we know it will someday die (by heat death or by Big Rip or by Big Crunch) and that there will never be an infinite number of mathematicians, your conjecture is
□ I think you may also be making the assumption that there is a minimum length of time for which mathematicians exist, and a maximum rate at which they can produce exciting proofs.
☆ You’re also assuming that the universe is finite. The observable universe certainly is, but little is known about the large-scale structure of the universe (for instance, we don’t even
know whether it is simply-connected).
○ He’s already accounted for that (“will someday die” and “never be an infinite number of mathematicians”), although makes the implicit assumption that time works in a simple manner.
■ In that case, the only possible loophole is if a mathematician completes a supertask (infinitely many theorems in a finite time).
■ I’m afraid we’ve already covered that possibility too (“a maximum rate at which they can produce exciting proofs”).
■ Yes, you did in your later comment, but not in the original poster’s comment. I think we have all bases covered.
3. More corollary-sniping, except this one actually happened. At least the author recognizes that that’s what’s going on.
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The Orekit library aims at providing efficient low level components for the development of flight dynamics applications.
Orekit, a pure Java library, depends only on the Java Standard Edition version 8 (or above) and Hipparchus version 3.1 (or above) libraries at runtime.
External dependencies
Runtime component
This component is required to compile and run Orekit:
Test-time component
This component is required for testing purpose only:
Design & Implementation
General concepts
As a a low level library, Orekit aims at being used in very different contexts which cannot be foreseen, from quick studies up to critical operations.
The main driving goals for the development of Orekit are:
• validation
• robustness
• maintainability
• efficiency
These goals lead to design and coding guidelines including:
• comprehensive test suite for high level of coverage
• automated checking tools for robustness
• consistent coding style for readable, clear and well documented code
• wide use of immutable objects for efficiency
Orekit is made of twelve packages shown in the following diagram.
• high accuracy absolute dates
• time scales (TAI, UTC, UT1, GPS, TT, TCG, TDB, TCB, GMST, GST, GLONASS, QZSS, BDT, IRNSS ...)
• transparent handling of leap seconds
• support for CCSDS time code standards
• frames hierarchy supporting fixed and time-dependent (or telemetry-dependent) frames
• predefined frames (EME2000/J2000, ICRF, GCRF, all ITRF from 1988 to 2020 and intermediate frames, TOD, MOD, GTOD and TEME frames, Veis, topocentric, tnw and qsw local orbital frames, Moon, Sun,
planets, solar system barycenter, Earth-Moon barycenter, ecliptic)
• user extensible (used operationally in real time with a set of about 60 frames on several spacecraft)
• transparent handling of IERS Earth Orientation Parameters (for both new CIO-based frames following IERS 2010 conventions and old equinox-based frames)
• transparent handling of JPL DE 4xx (405, 406 and more recent) and INPOP ephemerides
• transforms including kinematic combination effects
• composite transforms reduction and caching for efficiency
• extensible central body shapes models (with predefined spherical and ellipsoidic shapes)
• cartesian and geodesic coordinates, kinematics
• computation of Dilution Of Precision (DOP) with respect to GNSS constellations
• projection of sensor Field Of View footprint on ground for any FoV shape
Spacecraft state
• Cartesian, elliptical Keplerian, circular and equinoctial parameters, with non-Keplerian derivatives if available
• Walker constellations (including in-orbit spares with shifted position)
• Two-Line Elements (TLE)
• Two-Line Elements generation using Fixed-Point algorithm or Least Squares Fitting
• transparent conversion between all parameters
• automatic binding with frames
• attitude state and derivative
• jacobians
• mass management
• user-defined associated state (for example battery status, or higher order derivatives, or anything else)
• covariance propagation using the state transition matrix
• covariance extrapolation using a Keplerian model
• covariance frame transformation (inertial, Earth fixed, and local orbital frames)
• covariance type transformation (cartesian, keplerian, circular, and equinoctial)
• covariance interpolation based on the blending model
• analytical models for small maneuvers without propagation
• impulse maneuvers for any propagator type
• continuous maneuvers for numerical propagator type
• configurable low thrust maneuver model based on event detectors
• propulsion models intended to be used with maneuver class (constant and piecewise polynomials already provided by the library)
• user-friendly interface for the maneuver triggers
• analytical propagation models
□ Kepler
□ Eckstein-Heschler
□ Brouwer-Lyddane with Warren Phipps' correction for the critical inclination of 63.4° and the perturbative acceleration due to atmospheric drag
□ SDP4/SGP4 with 2006 corrections
□ GNSS: GPS, QZSS, Galileo, GLONASS, Beidou, IRNSS and SBAS
□ Intelsat's 11 elements
• numerical propagators
□ central attraction
□ gravity models including time-dependent like trends and pulsations (automatic reading of ICGEM (new Eigen models), SHM (old Eigen models), EGM, SHA (GRGM1200B and GRGM1200L) and GRGS gravity
field files formats, even compressed)
□ atmospheric drag
□ third body attraction (with data for Sun, Moon and all solar systems planets)
□ radiation pressure with eclipses (multiple oblate spheroids occulting bodies, multiple coefficients for bow and wing models)
□ solid tides, with or without solid pole tide
□ ocean tides, with or without ocean pole tide
□ general relativity (including Lense-Thirring and De Sitter corrections)
□ Earth's albedo and infrared
□ multiple maneuvers
□ empirical accelerations to account for the unmodeled forces
□ state of the art ODE integrators (adaptive stepsize with error control, continuous output, switching functions, G-stop, step normalization ...)
□ serialization mechanism to store complete results on persistent storage for later use
□ propagation in non-inertial frames (e.g. for Lagrange point halo orbits)
• semi-analytical propagation model (DSST) with customizable force models
□ central attraction
□ gravity models
□ J2-squared effect (Zeis model)
□ atmospheric drag
□ third body attraction
□ radiation pressure with eclipse
• computation of Jacobians with respect to orbital parameters and selected force models parameters
• trajectories around Lagragian points using CR3BP model
• tabulated ephemerides
□ file based
□ memory based
□ integration based
• Taylor-algebra (or any other real field) version of most of the above propagators, with all force models, events detection, orbits types, coordinates types and frames allowing high order
uncertainties and derivatives computation or very fast Monte-Carlo analyzes
• unified interface above analytical/numerical/tabulated propagators for easy switch from coarse analysis to fine simulation with one line change
• all propagators can manage the time loop by themselves and handle callback functions (called step handlers) from the calling application at each time step
□ step handlers can be called at discrete time at regular time steps, which are independent of propagator time steps
□ step handlers can be called with interpolators valid throughout one propagator time step, which can have varying sizes
□ step handlers can be switched off completely, when only final state is desired
□ special step handlers are provided for a posteriori ephemeris generation: all intermediate results are stored during propagation and provided back to the application which can navigate at
will through them, effectively using the propagated orbit as if it was analytical model, even if it really is a numerically propagated one, which is ideal for search and iterative algorithms
□ several step handlers can be used simultaneously, so it is possible to have a fine grained fixed time step to log state in a huge file, and have at the same time a coarse grained time step to
display progress for user at a more human-friendly rate, this feature can also be used for debugging purpose, by setting up a temporary step handler alongside the operational ones
• handling of discrete events during integration (models changes, G-stop, simple notifications ...)
• adaptable max checking interval for discrete events detection
• predefined discrete events
□ eclipse (both umbra and penumbra)
□ ascending and descending node crossing
□ anomaly, latitude argument or longitude argument crossings, with either true, eccentric or mean angles
□ apogee and perigee crossing
□ alignment with some body in the orbital plane (with customizable threshold angle)
□ angular separation thresholds crossing between spacecraft and a beacon (typically the Sun) as seen from an observer (typically a ground station)
□ raising/setting with respect to a ground location (with customizable triggering elevation and ground mask, optionally considering refraction)
□ date and on-the-fly resetting countdown
□ date interval with parameter-driven boundaries
□ latitude, longitude, altitude crossing
□ latitude, longitude extremum
□ elevation extremum
□ moving target detection (with optional radius) in spacecraft sensor Field Of View (any shape, with special case for circular)
□ spacecraft detection in ground based Field Of View (any shape)
□ sensor Field Of View (any shape) overlapping complex geographic zone
□ complex geographic zones traversal
□ inter-satellites direct view (with customizable skimming altitude)
□ ground at night
□ impulse maneuvers occurrence
□ geomagnetic intensity
□ extremum approach for TCA (Time of Closest Approach) computing
□ beta angle
□ relative distance between two objects
• possibility of slightly shifting events in time (for example to switch from solar pointing mode to something else a few minutes before eclipse entry and reverting to solar pointing mode a few
minutes after eclipse exit)
• events filtering based on their direction (for example to detect only eclipse entries and not eclipse exits)
• events filtering based on an external enabling function (for example to detect events only during selected orbits and not others)
• events combination with boolean operators
• ability to run several propagators in parallel and manage their states simultaneously throughout propagation
• extensible attitude evolution models
• predefined laws
□ central body related attitude (nadir pointing, center pointing, target pointing, yaw compensation, yaw-steering)
□ orbit referenced attitudes (LOF aligned, offset on all axes)
□ space referenced attitudes (inertial, celestial body-pointed, spin-stabilized)
□ attitude aligned with one target and constrained by another target
□ tabulated attitudes, either respective to inertial frame or respective to Local Orbital Frames
□ specific law for GNSS satellites: GPS (block IIA, block IIF, block IIF), GLONASS, GALILEO, BEIDOU (GEO, IGSO, MEO)
□ torque-free for general (non-symmetrical) body
• loading and writing of CCSDS Attitude Data Messages (both AEM, APM and ACM types are supported, in both KVN and XML formats, standalone or in combined NDM)
• exporting of attitude ephemeris in CCSDS AEM and ACM file format
Orbit determination
• batch least squares fitting
□ optimizers choice (Levenberg-Marquardt or Gauss-Newton)
□ decomposition algorithms choice (QR, LU, SVD, Cholesky)
□ choice between forming normal equations or not
• sequential batch least squares fitting
□ sequential Gauss-Newton optimizer
□ decomposition algorithms choice (QR, LU, SVD, Cholesky)
□ possibility to use an initial covariance matrix
• Kalman filtering
□ customizable process noise matrices providers
□ time dependent process noise provider
□ implementation of the Extended Kalman Filter
□ implementation of the Extended Semi-analytical Kalman Filter (ESKF)
□ implementation of the Unscented Kalman Filter
□ implementation of the Unscented Semi-analytical Kalman Filter
• parameters estimation
□ orbital parameters estimation (or only a subset if desired)
□ force model parameters estimation (drag coefficients, radiation pressure coefficients, central attraction, maneuver thrust, flow rate or start/stop epoch)
□ measurements parameters estimation (biases, satellite clock offset, station clock offset, station position, pole motion and rate, prime meridian correction and rate, total zenith delay in
tropospheric correction)
• orbit determination can be performed with numerical, DSST, SDP4/SGP4, Eckstein-Hechler, Brouwer-Lyddane, or Keplerian propagators
• ephemeris-based orbit determination to estimate measurement parameters like station biases or clock offsets
• multi-satellites orbit determination
• initial orbit determination methods (Gibbs, Gooding, Lambert, Gauss, and Laplace)
• ground stations displacements due to solid tides
• ground stations displacements due to ocean loading (based on Onsala Space Observatory files in BLQ format)
• ground stations displacements due to plate tectonics
• several predefined measurements
□ range
□ range rate (one way and two ways)
□ turn-around range
□ azimuth/elevation
□ right ascension/declination
□ position-velocity
□ position
□ inter-satellites range (one way and two way)
□ inter-satellites GNSS one way range rate
□ inter-satellites GNSS phase
□ Time Difference of Arrival (TDOA)
□ Frequency Difference of Arrival (FDOA)
□ Bi-static range and range rate
□ GNSS code
□ GNSS phase with integer ambiguity resolution and wind-up effect
□ multiplexed
• possibility to add custom measurements
• loading of ILRS CRD laser ranging measurements file
• loading and writing of CCSDS Tracking Data Messages (in both KVN and XML formats, standalone or in combined NDM)
• several predefined modifiers
□ tropospheric effects
□ ionospheric effects
□ clock relativistic effects (including J2 correction)
□ station offsets
□ biases
□ delays
□ Antenna Phase Center
□ Phase ambiguity
□ Shapiro relativistic effect
□ aberration of light in telescope measurements
• possibility to add custom measurement modifiers (even for predefined events)
• combination of GNSS measurements
□ dual frequency combination of measurements (Geometry-free, Ionosphere-free, Narrow-lane, Wide-lane and Melbourne-Wübbena)
□ single frequency combination of measurements (Phase minus code and GRAPHIC)
• measurements generation
□ with measurements feasibility triggered by regular event detectors (ground visibility, ground at night, sunlit satellite, inter satellites direct view, boolean combination...)
□ with measurement scheduling as fixed step streams (optionally aligned with round UTC time)
□ with measurement scheduling as high rate bursts rest periods (optionally aligned with round UTC time)
□ possibility to customize measurement scheduling
• computation of Dilution Of Precision
• loading of ANTEX antenna models file
• loading and writing of RINEX observation files (version 2, 3, and 4)
• loading of RINEX navigation files (version 2, 3, and 4)
• support for Hatanaka compact RINEX format
• loading of SINEX file (can load station positions, velocities, eccentricities, Post-Seismic Deformation models, EOPs, and Differential Code Biases)
• loading of RINEX clock files (version 2 and version 3)
• parsing of IGS SSR messages for all constellations (version 1)
• parsing of RTCM messages (both ephemeris and correction messages)
• parsing of GPS RF link binary message
• Hatch filters for GNSS measurements smoothing
• implementation of Ntrip protocol
• decoding of GPS navigation messages
Orbit file handling
• loading and writing of SP3 orbit files (from version a to d, including extension to a few inertial frames)
• splicing and interpolation of SP3 files
• loading and writing of CCSDS Orbit Data Messages (both OPM, OEM, OMM, OCM types are supported, in both KVN and XML formats, standalone or in combined NDM)
• loading of SEM and YUMA files for GPS constellation
• exporting of ephemeris in CCSDS OEM and OCM file formats
• loading of ILRS CPF orbit files
• exporting of ephemeris in STK format
Earth models
• atmospheric models (DTM2000, Jacchia-Bowman 2008, NRL MSISE 2000, Harris-Priester and simple exponential models), and Marshall solar Activity Future Estimation, optionally with lift component
• support for CSSI space weather data
• support for SOLFSMY and DTC data for JB2008 atmospheric model
• tropospheric delay (modified Saastamoinen, estimated, fixed)
• tropospheric mapping functions (Vienna 1, Vienna 3, Global, Niell)
• tropospheric refraction correction angle (Recommendation ITU-R P.834-7 and Saemundssen's formula quoted by Meeus)
• tropospheric models for laser ranging (Marini-Murray, Mendes-Pavlis)
• Klobuchar ionospheric model (including parsing α and β coefficients from University of Bern Astronomical Institute files)
• Global Ionospheric Map (GIM) model
• NeQuick ionospheric model
• VTEC estimated ionospheric model with Single Layer Model (SLM) ionospheric mapping function
• Global Pressure and Temperature models (GPT, GPT2, GPT2w, GPT3)
• geomagnetic field (WMM, IGRF)
• geoid model from any gravity field
• displacement of ground points due to tides
• tessellation of zones of interest as tiles
• sampling of zones of interest as grids of points
• construction of trajectories using loxodromes (commonly, a rhumb line)
Indirect optimal control
• adjoint equations as defined by Pontryagin's Maximum Principle with Cartesian coordinates for a range of forces (gravitational, inertial) including J2
• so-called energy cost functions (proportional to the integral of the control vector's squared norm), with Hamiltonian evaluation
• single shooting based on Newton algorithm for the case of fixed time, fixed Cartesian bounds
• loading and writing of CCSDS Conjunction Data Messages (CDM in both KVN and XML formats)
• 2D probability of collision computing methods assuming short term encounter and spherical bodies:
□ Chan 1997
□ Alfriend 1999
□ Alfriend 1999 (maximum version)
□ Alfano 2005
□ Patera 2005 (custom Orekit implementation) (recommended)
□ Laas 2015 (recommended)
Customizable data loading
• loading by exploring folders hierarchy on local disk
• loading from explicit lists of files on local disk
• loading from classpath
• loading from network (even through internet proxies)
• support for zip archives
• automatic decompression of gzip compressed (.gz) files upon loading
• automatic decompression of Unix compressed (.Z) files upon loading
• automatic decompression of Hatanaka compressed files upon loading
• plugin mechanism to add filtering like custom decompression algorithms, deciphering or monitoring
• plugin mechanism to delegate loading to user defined database or data access library
• possibility to have different data context (a way to separate sets of EOP, leap seconds, etc)
Localized in several languages
• Catalan
• Danish
• English
• French
• Galician
• German
• Greek
• Italian
• Norwegian
• Romanian
• Spanish | {"url":"https://test.orekit.org/overview.html","timestamp":"2024-11-04T20:10:29Z","content_type":"text/html","content_length":"30045","record_id":"<urn:uuid:f0add2f9-b701-4e3b-a32c-6f8c5cc294ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00019.warc.gz"} |
An Alternative Way for Children to Learn Math
A+ Results: Local Female Entrepreneur from Coral Springs Helping Special Needs Students Learn Locally and Across the Nation
October 26, 2020
The Gutenberg Printing Press
January 10, 2021
Singapore Math
An Alternative Way for Children to Learn Math
Singapore Math is a teaching method commenced in the 1980’s by the Singapore Ministry of Education for use for grades one through sixth. (See Brown, Hu, and Wright). The approach is more conceptual
than the memory/rote learning of traditional mathematics.
In 1998, through their company, Singapore Math Inc., Jeffrey and Dawn Thomas introduced the program to the United States. According to this couple, Singapore students ranked among the top in world
testing such as TIMSS (Trends in International Mathematics) and PISA (Programme for International Student Assessment). Moreover, in the United States, students who studied Singapore Math ranked “at
or above NAEP Proficient” on United States math assessments.
Singapore Math focuses on mastery which is achieved through intentional sequencing concepts which consist of the following features:
1. CPA or Concrete Pictorial Abstract, which builds upon existing knowledge by:
1. Concrete – students’ interaction with physical objects and model problems.
2. Pictorial – students make the mental connection between the physical object and the visual representation.
3. Abstract – symbolic modeling of problems using numbers and math symbols such as +, - , x, etc.
2. Number Bonds which are pictorial techniques where a whole number is one circle connected by parts of the number adjoined in connecting circles.
3. Concrete State in which the teachers will lead a classroom activity. For example, the teacher will ask five students to come to the front of the class pretending they are birds, and then the
teacher tells two more children to join them and asks the class,” how many birds are in the front of the room?”
4. Pictorial Stage – will have students shown visual representations of the five birds and two birds.
5. Abstract Stage – at this stage, the students are shown an equation of the problem.
6. Bar Modeling – using bar modeling, students can visualize a range of math concepts such as fractions, ratios, percentages, etc. This method is most effective when it is used throughout the
7. Mental Math – is a strategy that develops number sense and flexibility by having the students perform the problem putting numbers into parts and using them in a different order as illustrated by
number bonds. As students learn, they can apply different mental math strategies to problems and adapt ones they already they know how to perform. The students can also use their own discernment
when and where to use their strategy.
The Pros and Cons of Singapore Math
1. Textbooks and workbooks are easy to read.
2. It is closely aligned with Common Core State Standards Initiative.
3. Textbooks are sequential building on previously learned concepts.
4. Students build meaning to learned concepts and skills as opposed to rote memorization of rules and formulas.
5. Covers fewer topics in one year but has an in-depth method to ensure students have the foundation to move forward
6. without the need to relearn concepts.
1. Requires extensive and ongoing teacher training which may not be practical or financially feasible for certain school districts or homeschooled children.
2. It is closely aligned with Common Core State Standards Initiative (some individuals are not impressed with this initiative).
3. Supplies are consumable which means they must be re-ordered for every classroom for each year, which could place a financial strain on a school budget.
4. There is less focus on applied mathematics utilized in traditional United States textbooks. For example, these traditional textbooks emphasize data analysis using real life, multiple step math
problems while Singapore Math is more conceptual and ideological.
5. The Singapore Math program does not work for a “nomadic” student population who move in and out of districts since Singapore Math does not reteach concepts or skills. This could set up students
for failure if they move to a new school.
6. Some schools find that this method is not easy to implement.
The decision to utilize Singapore Math is an individual choice that may work well for one student and not for another. For more information see: | {"url":"https://jamiethetutor.com/singaporemath/","timestamp":"2024-11-11T06:29:30Z","content_type":"text/html","content_length":"80297","record_id":"<urn:uuid:4800a6a7-ad0e-483e-860b-079eadda110b>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00324.warc.gz"} |
Latent Class Analysis with Dirichlet Diffusion Tree Process
add_leaf_branch Add a leaf branch to an existing tree tree_old
add_multichotomous_tip Add a leaf branch to an existing tree tree_old to make a multichotomus branch
add_one_sample Functions to simulate trees and node parameters from a DDT process. Add a branch to an existing tree according to the branching process of DDT
add_root Add a singular root node to an existing nonsingular tree
attach_subtree Attach a subtree to a given DDT at a randomly selected location
A_t_inv_one Compute divergence function
A_t_inv_two Compute divergence function
a_t_one Compute divergence function
a_t_one_cum Compute divergence function
a_t_two Compute divergence function
a_t_two_cum Compute divergence function
compute_IC Compute information criteria for the DDT-LCM model
create_leaf_cor_matrix Create a tree-structured covariance matrix from a given tree
data_synthetic Synthetic data example
ddtlcm_fit MH-within-Gibbs sampler to sample from the full posterior distribution of DDT-LCM
div_time Sample divergence time on an edge uv previously traversed by m(v) data points
draw_mnorm Efficiently sample multivariate normal using precision matrix from x ~ N(Q^{-1}a, Q^{-1}), where Q^{-1} is the precision matrix
expit The expit function
exp_normalize Compute normalized probabilities: exp(x_i) / sum_j exp(x_j)
H_n Harmonic series
initialize Initialize the MH-within-Gibbs algorithm for DDT-LCM
initialize_hclust Estimate an initial binary tree on latent classes using hclust()
initialize_poLCA Estimate an initial response profile from latent class model using poLCA()
initialize_randomLCM Provide a random initial response profile based on latent class mode
J_n Compute factor in the exponent of the divergence time distribution
logit The logistic function
logllk_ddt Calculate loglikelihood of a DDT, including the tree structure and node parameters
logllk_ddt_lcm Calculate loglikelihood of the DDT-LCM
logllk_div_time_one Compute loglikelihood of divergence times for a(t) = c/(1-t)
logllk_div_time_two Compute loglikelihood of divergence times for a(t) = c/(1-t)^2
logllk_lcm Calculate loglikelihood of the latent class model, conditional on tree structure
logllk_location Compute log likelihood of parameters
logllk_tree_topology Compute loglikelihood of the tree topology
log_expit Numerically accurately compute f(x) = log(x / (1/x)).
parameter_diet Parameters for the HCHS dietary recall data example
plot.ddt_lcm Create trace plots of DDT-LCM parameters
plot.summary.ddt_lcm Plot the MAP tree and class profiles of summarized DDT-LCM results
plot_tree_with_barplot Plot the MAP tree and class profiles (bar plot) of summarized DDT-LCM results
plot_tree_with_heatmap Plot the MAP tree and class profiles (heatmap) of summarized DDT-LCM results
predict.ddt_lcm Prediction of class memberships from posterior predictive distributions
predict.summary.ddt_lcm Prediction of class memberships from posterior summaries
print.ddt_lcm Print out setup of a ddt_lcm model
print.summary.ddt_lcm Print out summary of a ddt_lcm model
proposal_log_prob Calculate proposal likelihood
quiet Suppress print from cat()
random_detach_subtree Metropolis-Hasting algorithm for sampling tree topology and branch lengths from the DDT branching process.
reattach_point Attach a subtree to a given DDT at a randomly selected location
result_diet_1000iters Result of fitting DDT-LCM to a semi-synthetic data example
sample_class_assignment Sample individual class assignments Z_i, i = 1, ..., N
sample_c_one Sample divergence function parameter c for a(t) = c / (1-t) through Gibbs sampler
sample_c_two Sample divergence function parameter c for a(t) = c / (1-t)^2 through Gibbs sampler
sample_leaf_locations_pg Sample the leaf locations and Polya-Gamma auxilliary variables
sample_sigmasq Sample item group-specific variances through Gibbs sampler
sample_tree_topology Sample a new tree topology using Metropolis-Hastings through randomly detaching and re-attaching subtrees
simulate_DDT_tree Simulate a tree from a DDT process. Only the tree topology and branch lengths are simulated, without node parameters.
simulate_lcm_given_tree Simulate multivariate binary responses from a latent class model given a tree
simulate_lcm_response Simulate multivariate binary responses from a latent class model
simulate_parameter_on_tree Simulate node parameters along a given tree.
summary.ddt_lcm Summarize the output of a ddt_lcm model
WAIC Compute WAIC | {"url":"https://search.r-project.org/CRAN/refmans/ddtlcm/html/00Index.html","timestamp":"2024-11-10T19:36:10Z","content_type":"text/html","content_length":"10803","record_id":"<urn:uuid:ac8413d1-273e-4c2a-bbbb-275e648dcf6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00607.warc.gz"} |
Solid State Physics | University of Bergen
Solid State Physics
Postgraduate course
ECTS credits
Teaching semesters
Course code
Number of semesters
Teaching language
Objectives and Content
The course gives an introduction to solid state physics, and wil enable the student to employ classical and quantum mechanical theories needed to understand the physical properties of solids.
Emphasis is put on building models able to explain several different phenomena in the solid state.
The course conveys an understanding of how solid state physics has contributed to the existence of a number of important technological developments of importance in our lives now and in the future.
The course gives an introduction to the physics of the solid state. The first part considers bonds and crystal structure in solid matter. Mechanical properties are investigated and tied to specific
bonds in solids. The interference pattern obtained by diffraction of waves by crystals reveals the lattice structure of the solid state. Particular emphasis is put on cubic and hexagonal crystals.
Concepts such as the reciprocal lattice vector and the Brillouin zone are introduced. Lattice vibrations are analyzed, and the dispersion relationship is introduced to understand how the lattice
vibrates. The Debye and Einstein models for heat capacity are covered to explain how the lattice energy changes with temperature. The course also covers heat conduction in solids, including Fouriers
law for diffusive heat conduction, and also how to obtain the thermal conductivity of solid matter. Classical and quantum mechanical models for the electrical and heat conduction in free electron
gases are studies, and simple models for electrons moving in periodic potentials allow one to understand the basic behavior of metals. Classification of band structure in conductors, semiconductors
and insulators is given. The law of mass action and the transport of holes and electrons in semiconductors are analyzed, with an emphasis on the concept of effective mass. Schottky and PN-junctions
are analyzed with respect to width and current-voltage characteristics. Applications of semiconductors, such as solar cells and light emitting diodes are also covered. The last part of the course
covers magnetism and superconductivity. The concepts of dia, para and ferromagnetism are introduced, and one distinguishes between local (Curie) and band (Stoner) contributions to ferromagnetism. A
short introduction to superconductivity is given.
Learning Outcomes
On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:
The student is able to
• Explain mechanical properties of solid matter, and connect these to bond type.
• Explain how diffraction of electromagnetic waves on solid matter can be used to obtain lattice structure.
• Know the concept of 'phonons', and how the dispersion relationship appears for different lattice structures.
• Explain how a lattice vibrates at finite temperature, and how these vibrations determine the heat capacity and conduction.
• Know the concept 'density of states' in one, two and three dimensions.
• Explain simple theories for conduction of heat and electrical current in metals.
• Classify solid state matter according to their band gaps.
• Understand how electrons and holes behave in semiconductors, and explain how they conduct current.
• Explain and give simple models for Schottky and PN-junctions.
• Explain how ligh emitting diodes and solar cells work.
• Know the basic physics behind dia, para and ferromagnetism.
• Differentiate between local (Curie) and band (Stoner) contributions to ferromagnetism.
• Know what superconductivity is and qualitatively relate it to lattice vibrations and the density of state.
The student is able to
• Build models to understand the physical properties of solid matter.
• Critically evaluate the approximations needed to build models to understand the solid state.
• Write a short scientific paper on a published research work in solid state physics.
General competence
The student should
• Have insight into classical and quantum mechanical laws which can be applied to explain the properties of the solid state.
• Formulate and understand theories explaining the behavior of the solid state.
• Know the role of solid state physics in important technological developments.
• Read and be able to understand research articles in certain fields of physics.
Semester of Instruction
Required Previous Knowledge
Recommended Previous Knowledge Forms of Assessment
The forms of assessment are:
• Compulsory excersises and class-room quizzes, 25 % of total grade.
• Written examination (4 hours), 75% of total grade.
Grading Scale
The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail. | {"url":"https://www4.uib.no/en/courses/PHYS208","timestamp":"2024-11-02T06:35:14Z","content_type":"text/html","content_length":"38486","record_id":"<urn:uuid:1f7000a9-651c-462d-ad11-227d41cf9c92>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00192.warc.gz"} |
Transpose Matrix Calculator
To use Transpose Matrix Calculator, select the order of matrix, enter the values, and hit calculate button
Matrix Transpose Calculator
Matrix Transpose Calculator is used to perform a matrix operation which is known as transposition. Transposition is a process in which swapping the rows and columns of a matrix and gives the result
in a new matrix where the columns and rows are interchanged.
What is Matrix Transpose?
Transpose of the matrix is a process in which interchange the rows into columns or (columns into the rows) of the given matrix. In other words, if the order of any matrix “A” is “m x n” after
processing it becomes “n x m” that is the order of the Transpose Matrix of “A”.
Transpose Matrix Formula:
The Transpose matrix is denoted by “A^t” which is read as the transpose of the matrix “A”. The matrix “A^t” is obtained simply by changing the i^th element of the rows into the j^th element of the
column. The general Formula of the Transpose Matrix is defined as.
If A = [a[ij]] is a matrix with “m x n” order. Where “i” denotes the i^th rows and “j” denotes the j^th columns.
Then A^t = [a[ji]] with the order “n x m” order. Where “j” denotes the j^th rows and “i” denotes the i^th columns after taking the transpose of the matrix.
Example 1:
Find the transpose matrix of the matrix.
[4 -3 1]
[2 -3 4]
[3 -5 6]
Step 1: Let the given matrix be equal to “A”.
A = [4 -3 1]
[2 -3 4]
[3 -5 6]
Step 2: The transpose matrix by swapping the rows into columns.
A^t = [4 -3 1]^t
[2 -3 4]
[3 -5 6]
A^t = [4 2 3]
[-3 -3 5]
[1 4 6]
Example 2:
Find the Transpose Matrix of the below matrix.
A = [1 -4 2]
[5 -3 0]
The transpose of the matrix is get by swapping the rows into columns.
A^t = [1 -4 2]^t
[5 -3 0]
A^t = [1 5]
[-4 -3]
[2 0] | {"url":"https://www.allmath.com/transpose-matrix.php","timestamp":"2024-11-07T03:21:14Z","content_type":"text/html","content_length":"42943","record_id":"<urn:uuid:a85289df-25ca-4aab-a6e1-3d1ba2767572>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00445.warc.gz"} |
Long Division Method
Trending Questions
How many packets of chips, each weighing 250 g can be made from a 5000 g pack?
A. 50
B. 20
C. 25
D. 10
View Solution
How many cartons will be required to carry 18000 books, if each carton can hold only 125 books?
A. 140
B. 185
C. 144
D. 44
View Solution
Q. Find a number, such that when it is divided by 37 gives a quotient of 28 and a remainder of 19.
View Solution
There are 7 chocolates in one packet. How many such packets can be made from 245 chocolates?
A. 25
B. 35
C. 45
D. 55
View Solution
Estimate the following product by rounding off each number to nearest tens :
View Solution
On arranging 2250 books equally in 15 shelves, each shelf will have ___ books.
A. 2250+15
B. 2250×15
C. 2250÷15
D. 2250−15
View Solution
How many packets of chips, each weighing 250 g can be made from a 5000 g pack?
A. 25
B. 10
C. 50
D. 20
View Solution
Estimate the following product by rounding off each number to nearest tens :
View Solution
Estimate the following product by rounding off each number to nearest tens :
View Solution
If 80 is divided by 5, then the answer obtained is 16. In this, the divisor is
and the quotient is
A. 80
B. 5
C. 16
D. 0
View Solution
Which of the following is the first step to convert two unlike fractions into like fractions?
A. By finding the LCM of the denominators
B. By finding the LCM of the numerators
C. By finding the HCF of the numerators
D. By finding the HCF of the denominators
View Solution
Estimate the following using the general rule :$796-314$
View Solution
how many 25ml cups will fill with 4 litre juice
View Solution
When we move a decimal point to left, the multiplying factor is __ times the previous factor.
A. 2
B. 10
C. 100
D. 200
View Solution
Q. the Length breadth and height of a room are 21 m 15 m and 18 m respectively. determine the longest tape which can measure the three dimensions of the room exactly.
View Solution
How many cartons will be required to carry 18000 books, if each carton can hold only 125 books?
A. 140
B. 185
C. 144
D. 44
View Solution
Find dividend and quotient for the given long division:
18475 −4–2 35 −32––– 3
A. Dividend = 4 and Quotient = 18
B. Dividend = 75 and Quotient = 4
C. Dividend = 75 and Quotient = 18
D. Dividend = 3 and Quotient = 4
View Solution
Q. How many pieces, each of length $3\frac{3}{4}$ m, can be cut from a rope of length 30 m?
View Solution
Estimate the following using the general rule :$730+998$
View Solution
The length, breadth and height of a room are 90 m, 50 m and 40 m, respectively. Determine the longest tape which can measure the three dimensions of the room.
A. 80
B. 15
C. 10
D. 45
View Solution
525 = 5
A. 110
B. 105
C. 102
D. 100
View Solution
Q. Q.2. A truck requires 108 litres if diesel to cover 1188 km. How much diesel will be required to cover 3300 km?
View Solution
Perform the long division and find the quotient, if the dividend is 1331.
A. 121
B. 131
C. 144
D. 198
View Solution
Q. If milk is available at Rs $17\frac{3}{4}$ per litre, find the cost of $7\frac{2}{5}$ litres of milk.
View Solution
Long division is the only way to check if a number is divisible by another number.
A. False
B. True
View Solution
Q. Raju – the farmer has 1104 apples in his orchard. He packs apples equally in boxes such that each box has XXIII apples. How many boxes of apples does he pack?
View Solution
Q. Monica cuts 46 m of cloth into peices of 1.15 m each. How many pieces does she get?
View Solution
Q. Essay Type Questions:
Explain the various parts or layout of a business letter.
View Solution
Q. 1.8 m of cloth is required for a shirt. How many such shirts can be made from a piece of cloth 45 m long?
View Solution
Q. A gardener bought 105 apple trees. He wants to plant 15 trees in each row. How many rows can he plant ?
View Solution | {"url":"https://byjus.com/question-answer/Grade/Standard-V/Mathematics/None/Long-Division-Method/","timestamp":"2024-11-04T17:32:16Z","content_type":"text/html","content_length":"177254","record_id":"<urn:uuid:68cc8efe-02b4-4e84-9b82-514f4a6d0691>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00369.warc.gz"} |
Learn the Concept of linearity in Regression Models - DataScienceCentral.com
This Tutorial talks about basics of Linear regression by discussing in depth about the concept of Linearity and Which type of linearity is desirable.
What is the meaning of the term Linear ?
In Linear Regression the term linear is understood in 2 ways –
1. Linearity in variables
2. Linearity in parameters
Linear regression however always means linearity in parameters , irrespective of linearity in explanatory variables.
A linear regression for 2 variables is represented mathematically as ( u is the error term )-
Y = B1 + B2X + u Or
Y = B1 + B2X ² + u
Here the variable X can be non linear i.e X or X² and still we can consider this as a linear regression. However if our parameters are not linear i.e say the regression equation is
Y = B1² + B2²X + u
then this can not be said to represent a linear regression equation.
Linear Regression Models
Model linear in parameters? Model linear in variables?
Yes No
Yes Linear Model Linear Model
No Non Linear Model Non Linear Model
Linearity in predictor variables – Xi
A function Y = f(x) is said to be linear in X if X appears with a power or index of 1 only. i.e the terms such as x2, Γx, and so on are excluded or if x is not multiplied or divided by any other
Linearity in parameters – Bi
Y is linearly related to X if the rate of change of Y with respect to X (dY/dX) is independent of the value of X.
A function is said to be linear in the parameter, say, B1, if B1 appears with a power of 1 only and is not multiplied or divided by any other parameter (for eg B1 x B2 , or B2 / B1)
To reiterate again – For purpose of Linear regression we are only concerned about linearity of parameters B1, B2 …. and not the actual variables X1, X2 ….
Non Linear Models
• Some models may look non linear in the parameters but are inherently or intrinsically linear.
• This is because with suitable transformations they can be made linear in parameters.
• However, if these cannot be linearized, these are called intrinsically non linear regression models
• When we say ‘non linear regression model’ we mean that it is intrinsically non linear.
For Log(Yi) = Log(B1) + B2 Log(Xi) + u
B2 is Linear but B1 is non-linear but if we transform α = Log(B1) then the model
Log(Yi) = α + B2 Log(Xi) + u
is linear in α and B2 as parameters. Implying we can make the regression equation linear in parameters using a simple transformation
For other cases we may not have an easy way to transform parameters to their linear form and such equations are hence treated as intrinsically non-linear and are NOT modeled using linear regression
This tutorial was originally posted here.
Next in the series :
Reference : Based on Lectures by Dr. Manish Sinha. ( Associate Prof. SCMHRD ) | {"url":"https://www.datasciencecentral.com/learn-the-concept-of-linearity-in-regression-models/","timestamp":"2024-11-08T02:20:59Z","content_type":"text/html","content_length":"156334","record_id":"<urn:uuid:c8d0b71f-c585-457e-818b-ee41bb402f7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00859.warc.gz"} |
Logstash line break
i have this config with a long line
grok {
match => {
"message" => "%{DATESTAMP:LogDate},%{NUMBER:RamUsedPercent:float},%{NUMBER:CpuPercent:float},%{NUMBER:DiskFreePercentC:float},%{NUMBER:DiskFreePercentD:float},%
How can i break it like this:
grok {
match => {
"message" => "%{DATESTAMP:LogDate},
%{NUMBER:ASPApplicationsRequestsInApplication Queue:float},%{NUMBER:ASPApplicationsRequestsRejected:float},
Because of the formatting of the message it's hard to see any difference between the two samples you posted (hint: use the preview pane to the right to inspect what you're about to post), but I'm
assuming you want to be ample to break the otherwise very long line.
It would've been desired with an ability to concatenate string via
"string1" "string2"
"string1" + "string2"
but unfortunately I don't think that's possible. The Logstash configuration language just isn't a fully-fledged programming language.
Well, impossible without ugly hacks anyway. You could use a mutate or ruby filter to create a temporary array field with all the comma separated values, join them with an mutate filter, then pass the
resulting string to the grok filter.
Thanks will work with 1 very long file | {"url":"https://discuss.elastic.co/t/logstash-line-break/32511","timestamp":"2024-11-09T17:39:49Z","content_type":"text/html","content_length":"37650","record_id":"<urn:uuid:b7a40ada-2ee8-4b30-ab65-cf63a1a668e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00810.warc.gz"} |
Solve Number of Heads and Legs Problems in 10 seconds
In competitive exam, counting number of heads and legs of animals in farm is a famous and most common type of problem. Aspirants may find it difficult but is the most easiest topic of reasoning. So
in this article we will try to understand how to solve number of heads and legs problems in less than 10 seconds.
Let us understand it with the help of example:-
In a farm there are some cows and birds. If there are total 35 heads and 110 legs then how many cows and birds are there?
No need to assume ‘x’ and make any equations. This a standard question in various exams. So we need a short cut to save our time in exam halls.
Short cut:-Just halve the given number of legs and subtract from it the number of heads. You will get the number of four legged animal. In this case (110÷2=55), 55-35= 20 (is four legged animal i.e.
You will get two legged animals by subtraction total animal in farm by four legged animals. In this case 35-20= 15. So there are 15 birds and 20 cows in the farm.
See, how easy it is to solve question like this where we have to find total number of legs or heads of the animals/birds present in the farm. You will hardly take more than 10-20 second to solve
these types of questions.
You can also these type of questions using concept of Mixture and alligations.
In a farm there are some cows and birds. If there are total 35 heads and 110 legs then how many cows and birds are there?
If we assumed all the animals are four legged then we should have 140 legs and if we assume all the animals are two legged we should have 70 legs. But in reality we have 110 legs. By using mixture
and allegation we can get the ratio of two legged animals and four legged animals and can solve question easily.
Carefully observe the solution given below.
We can say there number of two legged animals and four legged animals are in the ration of 3:4 which means if there are 3 two legged animals then there will be 4 four legged animals. By this method
we have 7 animals in farm but in reality there are 35 animals. So 1 unit of ratio is representing 5 real animals so we can easily calculate number of two legged or four legged animal.
We have 3 unit ratio for two legged animal and one unit is representing 5 animals so there are total 15 two legged animals i.e. birds.
We have 4 unit ratio for four legged animals and one unit is representing 5 animals so there are total 20 four legged animals i.e. cows.
So we can say there are 20 cows and 15 birds in the farm.
You can use any of the above mention method to solve these types of questions and both methods are very easy. It may seem lengthy but I have to explain everything to you in writing so it may seem
lengthy but with practice it will not take more than 20 seconds to solve number of heads and legs problems in various competitive exams.
To solve these types of questions as fast as possible depends on your speed of understanding the question and applying the above mentioned method to solve these type of questions.
Solve these number of heads and legs problems by yourself to have command over this concept and topic.
1.) If there are 18 heads and 48 legs of humans and horses, find the number of humans and horses respectively:
a.) 12, 8
b.) 10, 8
c.) 11, 7
d.) 12, 6
2.) There are 108 legs and 33 heads in the farm. There are only chicken and rabbits in his farm. How many chickens are there in the farm?
a.) 21
b.) 12
c.) 15
d.) 18
3.) A man has some hens and cows. If the number of heads be 48 and the number of feet equals 14o, then the number of hens will be :
a.) 22
b.) 24
c.) 26
d.) 28
Type your answer in the comment section below. Feel free to share your opinion as well.
You have understand one thing as well that in this world there is nothing like “one shoe fits all” and the method we have explained you above is no exception to this universal harsh reality it will
solve most of your questions but in some question you have to use basic method to solve these types of question using equations. | {"url":"https://hpexamadda.in/solve-number-of-heads-and-legs-problems/","timestamp":"2024-11-08T23:57:14Z","content_type":"text/html","content_length":"181804","record_id":"<urn:uuid:5a22af53-df45-4a10-ba93-867fcd5f5eec>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00222.warc.gz"} |
How to Find the P value: Process and Calculations
P values are everywhere in statistics. They’re in all types of hypothesis tests. But how do you calculate a p-value? Unsurprisingly, the precise calculations depend on the test. However, there is a
general process that applies to finding a p value.
In this post, you’ll learn how to find the p value. I’ll start by showing you the general process for all hypothesis tests. Then I’ll move on to a step-by-step example showing the calculations for a
p value. This post includes a calculator so you can apply what you learn.
General Process for How to Find the P value
To find the p value for your sample, do the following:
1. Identify the correct test statistic.
2. Calculate the test statistic using the relevant properties of your sample.
3. Specify the characteristics of the test statistic’s sampling distribution.
4. Place your test statistic in the sampling distribution to find the p value.
Before moving on to the calculations example, I’ll summarize the purpose for each step. This part tells you the “why.” In the example calculations section, I show the “how.”
Identify the Correct Test Statistic
All hypothesis tests boil your sample data down to a single number known as a test statistic. T-tests use t-values. F-tests use F-values. Chi-square tests use chi-square values. Choosing the correct
one depends on the type of data you have and how you want to analyze it. Before you can find the p value, you must determine which hypothesis test and test statistic you’ll use.
Test statistics assess how consistent your sample data are with the null hypothesis. As a test statistic becomes more extreme, it indicates a larger difference between your sample data and the null
Calculate the Test Statistic
How you calculate the test statistic depends on which one you’re using. Unsurprisingly, the method for calculating test statistics varies by test type. Consequently, to calculate the p value for any
test, you’ll need to know the correct test statistic formula.
To learn more about test statistics and how to calculate them for other tests, read my article, Test Statistics.
Specify the Properties of the Test Statistic’s Sampling Distribution
Test statistics are unitless, making them tricky to interpret on their own. You need to place them in a larger context to understand how extreme they are.
The sampling distribution for the test statistic provides that context. Sampling distributions are a type of probability distribution. Consequently, they allow you to calculate probabilities related
to your test statistic’s extremeness, which lets us find the p value!
For example, what does a t-value of two indicate? Is it significant? As you’ll see in the example, the t-distribution answers that question and allows us to calculate the p-value.
Like any distribution, the same sampling distribution (e.g., the t-distribution) can have a variety of shapes depending upon its parameters. For this step, you need to determine the characteristics
of the sampling distribution that fit your design and data.
That usually entails specifying the degrees of freedom (changes its shape) and whether the test is one- or two-tailed (affects the directions the test can detect effects). In essence, you’re taking
the general sampling distribution and tailoring it to your study so it provides the correct probabilities for finding the p value.
Each test statistic’s sampling distribution has unique properties you need to specify. At the end of this post, I provide links for several.
Learn more about degrees of freedom and one-tailed vs. two-tailed tests.
Placing Your Test Statistic in its Sampling Distribution to Find the P value
Finally, it’s time to find the p value because we have everything in place. We have calculated our test statistic and determined the correct properties for its sampling distribution. Now, we need to
find the probability of values more extreme than our observed test statistic.
In this context, more extreme means further away from the null value in both directions for a two-tailed test or in one direction for a one-tailed test.
At this point, there are two ways to use the test statistic and distribution to calculate the p value. The formulas for probability distributions are relatively complex. Consequently, you won’t
calculate it directly. Instead, you’ll use either an online calculator or a statistical table for the test statistic. I’ll show you both approaches in the step-by-step example.
In summary, calculating a p-value involves identifying and calculating your test statistic and then placing it in its sampling distribution to find the probability of more extreme values!
Let’s see this whole process in action with an example!
Step-by-Step Example of How to Find the P value for a T-test
For this example, assume we’re tasked with determining whether a sample mean is different from a hypothesized value. We’re given the sample statistics below and need to find the p value.
• Mean: 330.6
• Standard deviation: 154.2
• Sample size: 25
• Null hypothesis value: 260
Let’s work through the step-by-step process of how to calculate a p-value.
First, we need to identify the correct test statistic. Because we’re comparing one mean to a null value, we need to use a 1-sample t-test. Hence, the t-value is our test statistic, and the
t-distribution is our sampling distribution.
Second, we’ll calculate the test statistic. The t-value formula for a 1-sample t-test is the following:
• x̄ is the sample mean.
• µ[0] is the null hypothesis value.
• s is the sample standard deviation.
• n is the sample size
• Collectively, the denominator is the standard error of the mean.
Let’s input our sample values into the equation to calculate the t-value.
Third, we need to specify the properties of the sampling distribution to find the p value. We’ll need the degrees of freedom.
The degrees of freedom for a 1-sample t-test is n – 1. Our sample size is 25. Hence, we have 24 DF. We’ll use a two-tailed test, which is the standard.
Now we’ve got all the necessary information to calculate the p-value. I’ll show you two ways to take the final step!
P-value Calculator
One method is to use an online p-value calculator, like the one I include below.
Enter the following in the calculator for our t-test example.
1. In What do you want?, choose Two-tailed p-value (the default).
2. In What do you have?, choose t-score.
3. In Degrees of freedom (d), enter 24.
4. In Your t-score, enter 2.289.
The calculator displays a result of 0.031178.
There you go! Using the standard significance level of 0.05, our results are statistically significant!
Using a Statistical Table to Find the P Value
The other common method is using a statistical table. In this case, we’ll need to use a t-table. For this example, I’ll truncate the rows. You can find my full table here: T-Table.
This method won’t find the exact p value, but you’ll find a range and know whether your results are statistically significant.
Start by looking in the row for 24 degrees of freedom, highlighted in light green. We need to find where our t-score of 2.289 fits in. I highlight the two table values that our t-value fits between,
2.064 and 2.492. Then we look at the two-tailed row at the top to find the corresponding p values for the two t-values.
In this case, our t-value of 2.289 produces a p value between 0.02 and 0.05 for a two-tailed test. Our results are statistically significant, and they are consistent with the calculator’s more
precise results.
Displaying the P value in a Chart
In the example above, you saw how to calculate a p-value starting with the sample statistics. We calculated the t-value and placed it in the applicable t-distribution. I find that the calculations
and numbers are dry by themselves. I love graphing things whenever possible, so I’ll use a probability distribution plot to illustrate the example.
Using statistical software, I’ll create the graphical equivalent of calculating the p-value above.
This chart has two shaded regions because we performed a two-tailed test. Each region has a probability of 0.01559. When you sum them, you obtain the p-value of 0.03118. In other words, the
likelihood of a t-value falling in either shaded region when the null hypothesis is true is 0.03118.
I showed you how to find the p value for a t-test. Click the links below to see how it works for other hypothesis tests:
Now that we’ve found the p value, how do you interpret it precisely? If you’re going beyond the significant/not significant decision and really want to understand what it means, read my posts,
Interpreting P Values and Statistical Significance: Definition & Meaning.
If you’re learning about hypothesis testing and like the approach I use in my blog, check out my Hypothesis Testing book! You can find it at Amazon and other retailers. | {"url":"https://statisticsbyjim.com/hypothesis-testing/how-to-find-p-value/","timestamp":"2024-11-09T20:18:25Z","content_type":"text/html","content_length":"240417","record_id":"<urn:uuid:96f3d010-7b8f-40a8-8529-e651c084e09f>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00690.warc.gz"} |
Focal-lengths Sentence Examples
• A volume entitled Opera posthuma (Leiden, 1703) contained his "Dioptrica," in which the ratio between the respective focal lengths of object-glass and eye-glass is given as the measure of
magnifying power, together with the shorter essays De vitris figurandis, De corona et parheliis, &c. An early tract De ratiociniis tin ludo aleae, printed in 16J7 with Schooten's Exercitationes
mathematicae, is notable as one of the first formal treatises on the theory of probabilities; nor should his investigations of the properties of the cissoid, logarithmic and catenary curves be
left unnoticed.
• The magnifying power of the telescope is = Ff /ex, where F and f are respectively the focal lengths of the large and the small mirror, e the focal length of the eye-piece, and x the distance
between the principal foci of the two mirrors (=Ff in the diagram) when the instrument is in adjustment for viewing distant objects.
• Should there be in two lenses in contact the same focal lengths for three colours a, b, and c, i.e.
• Since the lens is bounded by air, the imageand object-side focal lengths f' and f are equal.
• Doublets, &'c. - To remove the errors which the above lenses showed, particularly when very short focal lengths were in question, lens combinations were adopted.
• A series of objectives with short focal lengths are available, which permit the placing of a liquid between the cover-slip and the front lens of the objective; such lenses are known as "
immersion systems "; objectives bounded on both sides by air are called " dry systems."
• Beck, which can be conveniently fitted in and used for objectives with different focal lengths.
• The magnification of a microscope is determined from the focal lengths of the two optical systems and the optical tube length, for N = 250 A/fi'f2 To determine the optical tube length 0, it is
necessary to know the position of the focal planes of the objective and of the ocular.
• For relatively short focal lengths a triple construction such as this is almost necessary in order to obtain an objective free from aberration of the 3rd order, and it might be thought at first
that, given the closest attainable degree of rationality between the colour dispersions of the two glasses employed, which we will call crown and flint, it would be impossible to devise another
form of triple objective, by retaining the same flint glass, but adopting two sorts of crown instead of only one, which would have its secondary spectrum very much further reduced.
• In the Ramsden eyepiece (see Microscope) the focal lengths of the two piano-convex lenses are equal, and their convexities are turned towards one another.
• Gauss (Dioptrische Untersuchungen, Göttingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see Lens).
• By compounding two lenses or lens systems separated by a definite interval, a system is obtained having a focal length considerably less than the focal lengths of the separate systems. If f and
f' be the focal lengths of the combination, and f2, f2 the focal lengths of the two components, and A the distance between the inner foci of the components, then f = - f,f2/4, f' =fi f27 0 (see
• Short focal lengths usually give your bright and clear images, but will have wide areas of view.
• For example, if you have an interest in photography, you may find an assignment asking for an explanation of the differences between digital and film lens focal lengths. | {"url":"https://sentence.yourdictionary.com/focal-lengths","timestamp":"2024-11-13T02:33:24Z","content_type":"text/html","content_length":"237126","record_id":"<urn:uuid:0187de37-8cb1-4523-8dad-f05ebd27a964>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00581.warc.gz"} |
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Assume that there are two biased coins. Coin A heavily biased towards head with the probability of head equal to 0.9. And Coin B is heavily biased towards tail with the probability of tail equal to
0.9. Now, we randomly and equally likely select one of the coins and toss it twice. Let’s call the outcome
I put a similar question as above in a midterm, and I didn’t expect to stumble the entire class.
All students thought that the mutual information
Think intuitively with the above example. If we didn’t toss the coin twice, but toss it ten times and got ten heads. What do we expect the outcome to be if we toss it another time?
I think an intelligent guess should be another head. Because given the ten heads we got earlier, it has a very high chance that the picked coin is Coin A. And so the next toss is very likely to be
the head as well.
Now, the same argument holds when we are back to the original setup. When the first toss is head, the second toss is likely to be head as well. So
So what is
Let’s compute
Now, for
Note that this is an example that variables are conditional independent but not independent. More precisely, we have
Probability education trap
I wondered a little bit why none of my students could answer the above question. I blame a trap that is embedded in most elementary probability courses. We were always introduced with a consecutive
coin tossing or dice throwing example with each subsequent event to be independent of the earlier event. In those examples, we always assume that the probabilities of getting all outcomes are known
but this assumption was never emphasized. As we see in the above example, even each subsequent tossing or throwing is independent relative to the current coin or the current dice, overall those
events are not independent when the statistics behind the coin and dice are not known.
Actually, this also makes some “pseudo-science” not really that non-scientific after all. For example, we all tend to believe that the gender of a newborn is close to random and hence unpredictable.
But what if there is some hidden variable that affects the gender of a newborn. And that factor may have a strong bias towards one gender over another. Then, it probably is not really that unlikely
for someone to have five consecutive girls or six consecutive sons. Of course, I also tend to believe that the probability of a newborn to be a boy or a girl is very close to one half. A less extreme
example may occur at the casino. If we believe that the odd from each lottery machine in a casino is not so perfectly tune (before the digital age, that is probably much more likely), then there is a
chance that some machine has a higher average reward than another one. Then, a gambler trying to select a lottery machine to play is an essential strategy to win and is not really superstition after
all. Of course, this is just the multi-armed bandit problem.
Independence and conditional independence are one of the most basic concepts in probabilities. Two random variables are independent, as the term suggested, if the outcome of one variable should not
affect the outcome of another. Mathematically, two variables
Let’s inspect this definition more carefully, given
Similarly, when we say
Mathematically, we denote the independence by
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The Data Science of MathOverflow—Wolfram Blog
The Data Science of MathOverflow
New Archive Conversion Utility in Version 12
Soon there will be 100,000 questions on MathOverflow.net, a question-and-answer site for professional mathematicians! To celebrate this event, I have been working on a Wolfram Language utility
package to convert archives of Stack Exchange network websites into Wolfram Language entity stores.
The archives are hosted on the Internet Archive and are updated every few months. The package, although not yet publicly available, will be released in the coming weeks as part of Version 12 of the
Wolfram Language—so keep watching this space for more news about the release!
Although some data analysis can be done with tools such as the Stack Exchange Data Explorer, queries are usually limited in size or computation time, as well as to text-only formats. Additionally,
they require some knowledge of SQL. But with a local copy of the data, much more can be done, including images, plots and graphs.
With the utility package operating on a local archive, it’s easy to perform much deeper data analysis using all of the built-in tools in the Wolfram Language. In particular, Version 12 of the Wolfram
Language adds support for RDF and SPARQL queries, as well as useful constructs such as FilteredEntityClass and SortedEntityClass.
For professional mathematicians who already use Mathematica and the Wolfram Language, this utility allows for seamless investigation into the data on MathOverflow.net or any Stack Exchange network
site. Feel free to follow along with me as I do some of this investigation by running the code in a notebook, or just sit back and enjoy the ride as we explore MathOverflow.net with the Wolfram
Importing a MathOverflow EntityStore
The entity stores created by the utility package allow for quick access to the data in a format that’s easy for Wolfram Language processing, such as queries using the Entity framework, machine
learning functionality, visualization, etc.
Let’s start by downloading a pre-generated EntityStore from the Wolfram Cloud to the notebook’s directory:
Import the EntityStore from the downloaded file:
The store is quite large, consisting of nearly three million entities in several entity types:
entityStoreMetaData=AssociationMap[<|"Entity Count"->Length[store[#,"Entities"]],"Property Count"->Length[store[#,"Properties"]]|>&,store[]]//ReverseSortBy[Lookup["Entity Count"]];
Total[#"Entity Count"&/@entityStoreMetaData]
Lastly, we need to register the EntityStore for use in the current session:
This returns a list of all of the new entity types from the EntityStore that are now available through EntityValue (you can access them by registering the EntityStore via EntityRegister).
For those who are familiar with the Stack Exchange network, these types may be very familiar. But for those who are not, or if you just need a refresher, here’s a basic rundown of a few of the
different types:
The remaining types not listed are beyond the scope of my post, but you can learn more about them in the README on the archives, or by visiting the frequently asked questions on any Stack Exchange
network site.
Accessing MathOverflow.net Posts
Now that the EntityStore is loaded, we can access it through the Entity framework.
Let’s look at some random posts:
The “Post” entities are formatted with the post type (Q for question, A for answer), the user who authored the post in square brackets, a short snippet of the post and a hyperlink (the blue ») to the
original post on the web.
Many of the other entity types format similarly—this is to give proper context, allow for manual exploration on the site itself and give attribution to the original authors (they created the content
on the site, after all).
Taking just one of these posts, we can find a lot of information about it with a property association:
Entity["StackExchange.Mathoverflow:Post", "272527"][{accepted answer,answer count,body,closed date,comment count,comments,community owned date,creation date,duplicate posts,favorite count,id,last activity date,last edit date,last editor,linked posts,owner,post type,score,tags,title,URL,view count},"PropertyAssociation"]
For example, one may be interested in the posts for a given tag, such as set theory.
We can find how many set theory questions have been asked:
EntityValue[EntityClass["StackExchange.Mathoverflow:Post",{"Tags"->Entity["StackExchange.Mathoverflow:Tag", "SetTheory"],"PostType"->Entity["StackExchange:PostType", "1"] }],"EntityCount"]
We can even see the intersections of different tags, such as set theory and plane geometry:
EntityValue[EntityClass["StackExchange.Mathoverflow:Post",{"Tags"->ContainsAll[{Entity["StackExchange.Mathoverflow:Tag", "SetTheory"],Entity["StackExchange.Mathoverflow:Tag", "PlaneGeometry"]}],"PostType"->Entity["StackExchange:PostType", "1"] }],"Entities"]
It’s important to note that as of this writing, the archives have not been updated to include the 100,000th question, so we can see that there are only 98,165 questions as of December 2, 2018:
EntityClass["StackExchange.Mathoverflow:Post","PostType"->Entity["StackExchange:PostType", "1"]]["EntityCount"]
Of course, there is a seemingly endless number of queries one can make on this dataset.
A few ideas that I had were to find and analyze:
• The distribution of post scores (specifically the (nearly) 100k questions)
• Word distributions and frequencies (e.g. -grams)
• “Post Thread” networks
• Mathematical propositions (e.g. theorems, lemmas, axioms) mentioned in posts
• Famous mathematicians and propositions that are named after them
Let’s tackle these one at a time.
Analyzing Posts
Post Score Distributions
Since there are over 237,000 posts on MathOverflow in total, the distribution of their scores must be very large.
Let’s look at this distribution, noting that some post scores can be negative if they are downvoted by users in the community:
allScores=EntityValue["StackExchange.Mathoverflow:Post",EntityProperty["StackExchange.Mathoverflow:Post", "Score"]];
ListPlot[postScoreDistribution,PlotRange -> Full,PlotTheme->"Detailed",ImageSize->400]
That’s hard to read—it looks better on a log-log scale, and it becomes mostly straight beyond the first several points:
ListLogLogPlot[postScoreDistribution,PlotRange -> All,PlotTheme->"Detailed",ImageSize->400]
Let’s focus on the positive post scores below 50:
It looks like it might be a log-normal distribution, so let’s find the fitting parameters for it:
Plotting both on the same (normalized) scale shows they agree quite well:
PlotRange -> All,PlotTheme->"Detailed",
We can repeat this analysis on the (almost) 100k questions:
allQuestionScores=EntityValue[EntityClass["StackExchange.Mathoverflow:Post","PostType"->Entity["StackExchange:PostType", "1"]],EntityProperty["StackExchange.Mathoverflow:Post", "Score"]];
ListLogLogPlot[questionScoreDistribution,PlotRange -> All,PlotTheme->"Detailed",ImageSize->400]
questionScoresBelowFifty=EntityValue[EntityClass["StackExchange.Mathoverflow:Post",{"PostType"->Entity["StackExchange:PostType", "1"],"Score"->Between[{1,50}]}],EntityProperty["StackExchange.Mathoverflow:Post", "Score"]];
PlotRange -> All,PlotTheme->"Detailed",
Words Much More Common in Mathematics Than Normal Language
There are many words that appear in mathematics that are not found in typical English. Some examples include names of mathematicians (e.g. Riemann, Euler, etc.) or words that have special meanings
(integral, matrix, group, ring, etc.).
We can start to investigate these words by gathering all of the post bodies:
We’ll need to create functions to normalize strings (normalizeString) and extract sentences (extractSentences), removing HTML tags and replacing any equations with “”:
Whitespace->" "
We’ll also need to extract, count up and sort the words from all of the post bodies:
This gives a list of almost 400k words:
We can trim it down to just the top 500 words, being careful to remove some extra noise with websites, equations, inequalities and single letters:
Not[StringMatchQ[#, "*'*"|"*=*"|"*<*"|"*www*"|"http"|"https"|"--"]]&&StringLength[#]>1&
Note that “” is the most common word, since all equations were replaced with it:
It’s useful to visualize it as a word cloud:
Removing “” and stopwords like “the”, “is” and “of” from the data will avoid some clutter:
Not[StringMatchQ[#, "*'*"|"*=*"|"*<*"|"*www*"|"http"|"https"|"\[ScriptCapitalM]\[ScriptCapitalA]\[ScriptCapitalT]\[ScriptCapitalH]"|"--"]]&&StringLength[#]>1&
Now the results are more interesting and meaningful:
Of course, we can take this analysis further. We can get the frequencies for the top words in usual English with WordFrequencyData:
Normalize the counts of the words on MathOverflow, and then join the two as coordinates in 2D frequency space:
We can visualize these coordinates, adding a red region to the plot for words more commonly used in typical English than in MathOverflow posts (below = ), and a gray region for words that are more
commonly used in MathOverflow posts by less than a factor of 10 (below = 10 ).
This arbitrary factor allows us to narrow down the words that are much more common to MathOverflow than typical English, which appear in the white region (above = 10 ):
FrameLabel->{"Fraction of English","Fraction of MathOverflow.net"}
We can take this another step further by looking at the words in the white region that are much more likely to occur on MathOverflow than they are in typical English:
ListLogLogPlot[wordsMuchMoreCommonInMO,ImageSize->500,PlotTheme->"Detailed",PlotStyle->PointSize[0.002],FrameLabel->{"Fraction of English","Fraction of MathOverflow.net"}]
Of course, an easy way to visualize this data is in a word cloud, where the words are weighted by combining their frequency of use via Norm:
Analysis of n-Grams
Of course, individual words are not the only way to analyze the MathOverflow corpus.
We can create a function to compute -grams using Partition and recycling extractSentences from earlier:
(* Keep only the top 10,000 to save memory *)
Next, we’ll need to build a function to show the -grams in tables and word clouds, both with and without math (since putting them together would clutter the results a bit):
Column[{Style["↑ Including \[ScriptCapitalM]\[ScriptCapitalA]\[ScriptCapitalT]\[ScriptCapitalH] ↑",24,FontFamily->"Source Code Pro"],Dataset[Take[phraseToCountWithMath,UpTo[20]]]}],
Column[{Style["↓ Without \[ScriptCapitalM]\[ScriptCapitalA]\[ScriptCapitalT]\[ScriptCapitalH] ↓",24],Dataset[Take[phraseToCountWithoutMath,UpTo[20]]]}]
Looking at the 3-grams, there are lots of “The number of…”, “The set of…”, “is there a…” and more.
There are definitely signs of “if and only if,” but they’re not well captured here since we’re looking at 3-grams. They should show up later in the 4-grams, anyway.
There is a lot of of “Let be…”, “ is a,” and similar—it’s clear that MathOverflow users frequently use for mathematical notation:
Expanding on the 3-grams, the 4-grams give several mathy phrases like “if and only if,” “on the other hand,” “is it true that” and “the set of all.”
We also see more proof-like phrases like “Let be a,” “ such that ” and similar.
It’s interesting how the two word clouds begin to show the split of “proof-like” phrases and “natural language” phrases:
We see similar trends with the 5- and 6-grams:
“Post Thread” Networks
Moving past natural language processing, another way to analyze the MathOverflow site is as a network.
We can create a network of MathOverflow users that communicate with each other. One way to do this is to connect two users if one user posts an answer to another user’s question. In this way, we can
create a directed graph of MathOverflow users.
Although it’s possible to do this graph-like traversal and matching with the usual EntityValue syntax, it could get somewhat messy.
Questioner-Answerer Network
To start, we can write a symbolic representation of a SPARQL query to find all connections between question writers and the writers of answers, and then do some processing to turn it into a Graph:
RDFTriple[SPARQLVariable["post"],post type,Entity["StackExchange:PostType", "1"]],
SPARQLPropertyPath[SPARQLVariable["post"],{SPARQLInverseProperty[parent post],owner},SPARQLVariable["answerer"]]
From the icon of the output, we can see it’s a very large directed multigraph. Networks of this size have very little hope of being visualized easily, so we should find a way to reduce the size of
Smaller Questioner-Answerer Network
We can trim down the size by writing a similar SPARQL query that limits us to posts with a few numerical mathematics post tags:
RDFTriple[SPARQLVariable["post"],post type,Entity["StackExchange:PostType", "1"]],
RDFTriple[SPARQLVariable["post"],tags,Entity["StackExchange.Mathoverflow:Tag", "NumericalLinearAlgebra"]],
RDFTriple[SPARQLVariable["post"],tags,Entity["StackExchange.Mathoverflow:Tag", "NumericalAnalysisOfPde"]],
RDFTriple[SPARQLVariable["post"],tags,Entity["StackExchange.Mathoverflow:Tag", "NumericalIntegration"]],
RDFTriple[SPARQLVariable["post"],tags,Entity["StackExchange.Mathoverflow:Tag", "RecreationalMathematics"]]
SPARQLPropertyPath[SPARQLVariable["post"],{SPARQLInverseProperty[parent post],owner},SPARQLVariable["answerer"]]
This graph is much smaller and can be more reasonably visualized. For simplicity, let’s focus only on the largest (weakly) connected component:
Questioner-Answerer Communities by Geographic Region
We can group the vertices of the graph (MathOverflow users) by geography by using the location information users have entered into their profiles.
Here, we can use Interpreter["Location"] to handle a variety of input forms, including countries, cities, administrative divisions (such as states) and universities:
The results are pretty good, giving over 250 approximate locations:
Of course, these individual locations are not that helpful, as they are very localized. We can use GeoNearest to find the nearest geographic region as a basis for determining groups for the users:
getRegion[locations:{__GeoPosition}]:=First[#,Missing["NotAvailable"]]&/@DeleteCases[GeoNearest[GeoVariant["GeographicRegion","Center"],locations],Entity["GeographicRegion", "World"],Infinity];
Next, we group users into communities based on this geographic information:
Lastly, we can use CommunityGraphPlot to build a graphic that shows the geographic communities of the questioner-answerer network:
Entity["GeographicRegion", "Europe"]->Darker@,Entity["GeographicRegion", "NorthAmerica"]->Darker[Green,0.5],Entity["GeographicRegion", "Australia"]->Orange,Entity["GeographicRegion", "Asia"]->Purple,Entity["GeographicRegion", "SouthAmerica"]->Darker[Red,0.25]
Entity["GeographicRegion", "Europe"]->Below,Entity["GeographicRegion", "NorthAmerica"]->After,Entity["GeographicRegion", "Asia"]->Below
regionToRotation=Lookup[<|Entity["GeographicRegion", "NorthAmerica"]->-(π/2)|>,#,0]&;
Post Owner-Commenter Network
Of course, we could do a similar analysis on connections between post owners and their commenters for posts tagged with “linear-programming”:
RDFTriple[SPARQLVariable["post"],tags,Entity["StackExchange.Mathoverflow:Tag", "LinearProgramming"]],
However, further analysis on this network will be left as an exercise for the reader.
Analyzing T[E]X Snippets
On MathOverflow, there are not many posts without it, so exploring
Extract T[E]X Snippets
First, we need to extract the
Entity["StackExchange.Mathoverflow:Post", "40686"]["Body"]
We can write a function to extract the snippets in a string, noting the two main input forms ("$$…$$" or "\\begin{…}…\\end[…]"):
extractTeXSnippets[s_String] :=
StringReplace[Join[dd, d,o], {".$":>"$"}]
Testing this on the simple example gives the snippets wrapped in dollar signs:
Format T[E]X Snippets
Of course, once we have
We can write a quick function to do this with proper formatting:
blackboardBoldRules=character to double struck;
frakturGothicRules=character to gothic;
formatTeXSnippet[s_String] :=
StringMatchQ[s, "$\\mathbb{"~~ _ ~~ "}$"],
StringReplace[s,"$\\mathbb{"~~ a_ ~~ "}$":> a]/.blackboardBoldRules,
StringMatchQ[s, "$\\mathbb "~~ _ ~~ "$"],
StringReplace[s,"$\\mathbb "~~ a_ ~~ "$":> a]/.blackboardBoldRules,
StringMatchQ[s, "$\\mathfrak{"~~ _ ~~ "}$"],
StringReplace[s,"$\\mathfrak{"~~ a_ ~~ "}$":> a]/.frakturGothicRules,
StringMatchQ[s, "$\\mathfrak "~~ _ ~~ "$"],
StringReplace[s,"$\\mathfrak "~~ a_ ~~ "$":> a]/.frakturGothicRules,
StringMatchQ[s, "$\\mathcal{"~~ _ ~~ "}$"],
Style[StringReplace[s,"$\\mathcal{"~~ a_ ~~ "}$":>a],FontFamily->"Snell Roundhand"],
StringMatchQ[s, "$\\mathcal "~~ _ ~~ "$"],
Style[StringReplace[s,"$\\mathcal "~~ a_ ~~ "$":>a],FontFamily->"Snell Roundhand"],
We can test the results on the previously extracted
We can also test them on a completely different post:
Entity["StackExchange.Mathoverflow:Post", "40686"]["Body"] //
extractTeXSnippets // AssociationMap[formatTeXSnippet] //
KeyValueMap[List] // Grid[#, Frame -> All, Alignment -> Left] &
Set Up T[E]X Snippets Property
This system works well, so we should make it easier to use. We can do this by hooking up these functions as a property for posts, keeping the formatting function separate so that analysis can still
be done on the raw strings:
EntityProperty["StackExchange.Mathoverflow:Post","TeXSnippets"]["Label"]="TEX snippets";
Now we can just call the property on an entity instead:
Entity["StackExchange.Mathoverflow:Post", "67739"]["TeXSnippets"]//Map[formatTeXSnippet]
From here, it should be easy to extract all of the
Create a T[E]X Word Cloud
A simple way to analyze the
There are almost one million unique
We can also make a simple word cloud from the top 100 snippets:
It’s easy to see that there are a lot of single-letter snippets. But there are a lot more interesting things hiding beyond these top 100. Let’s take a look at a few different cases!
Integrals are fairly easy to find with some simple string pattern matching:
Looking at the top 50 gives some interesting results—some very simple, and some rather complex:
Analyze Equations
Another interesting subset of
equations=KeySelect[teXToCount,StringMatchQ[("$"|"$$")~~(Whitespace|"")~~__~~" = "~~__~~"$"]];
Visualizing the top 50 gives mostly single-letter variable assignments to numbers:
Equations of the Form <letter> = <number>
If we look at the single-letter variable assignments, we can find the minimum and maximum values of <number> for each <letter>.
Note that this includes a list of special
teXRepresentationToCharacter=TeX to character;
KeySort[MinMax/@Keys/@variableEqualsNumberDistributions]//KeyValueMap[Prepend[N@#2,#1<>" | "<>ToLowerCase[#1]]&]//Grid[#,Frame->All]&
It’s interesting to see that most letters are positive, but S is strangely very negative. It’s also interesting to note the very large scale of U, V and W. Perhaps not surprisingly, N is the most
common letter, though its neighbor O is the least common:
Trimming these single-variable assignments out of the original equation word cloud makes the results a bit more diverse:
Equations of the Form <letter> = <letter>
It’s interesting to see that there are a lot of letters assigned to (or compared with) another letter. We can make a simple graph that connects two letters in these equations, again taking into
account special characters like \alpha:
The graph, without combining upper and lowercase letters, is quite messy:
If we combine the upper and lowercase letters, the graph becomes a little bit cleaner:
If we again remove these equation types, the word cloud becomes even cleaner:
Functional Equations
Another interesting subset of equations to look into is functional equations. With a little bit of string pattern matching, we can find many examples:
By focusing on functional equations that have one function with arguments on the left side of an equals sign, we get fewer results:
However, we’ll need to go further to find equations that are easier to work with. Let’s limit ourselves to single-letter, single-argument functions:
This is much more pointed, but we can go further. If we limit ourselves to functional equations with only one equals sign with single, lowercased arguments that only consist of a single head and
argument (modulo operators and parentheses), we find just six equations:
StringMatchQ[s:(("$$"|"$")~~(f:LetterCharacter)~~"("~~x:LetterCharacter?LowerCaseQ~~")"~~(Whitespace|"")~~"="~~(Whitespace|"")~~__~~(f:_)~~"("~~x:_~~")"~~___~~("$$"|"$"))/;(StringCount[s,"="]===1&&StringFreeQ[s,"\\"~~LetterCharacter..]&&Complement[Union[Characters[s]],{"$","^","(",")","-","+","=","{","}","."," ","0","1","2","3","4","5","6","7","8","9"}]===Sort[{f,x}])]
Interestingly, there are only two functionally unique equations in this list:
f(x) = 1 + x f(x)^2
f(x) = 1 + x^2 f(x)^2
If we clean up these functional equations, we can put them through Interpreter["TeXExpression"] to get actual Wolfram Language representations of them:
Finally, we can solve these equations with RSolve:
Analyze “Big O” Notation Arguments
Moving past equations, another common notation among mathematicians is big O notation. Frequently used in computational complexity and numerical error scaling, this notation should surely appear
somewhat frequently on MathOverflow.
Let’s take a look by finding
The results are varied:
One can note that many of these results are functionally equivalent—they differ only in the letter chosen for the variable.
We can clean these cases up with a little bit of effort:
(* Any constant number 1 *)
"$$"~~LetterCharacter~~"\\log "~~LetterCharacter~~"$$":>"$$n\\log n$$",
"$$"~~LetterCharacter~~"^"~~exp:(DigitCharacter|("{"~~__~~"}"))~~"\\log "~~LetterCharacter~~"$$":>"$$n^"<>exp<>"\\log n$$",
"$$\\"~~op:("log"|"dot")~~Whitespace ~~LetterCharacter~~"$$":>"$$\\"<>op<>" n$$",
"$$"~~LetterCharacter~~"/\\log "~~LetterCharacter~~"$$":>"$$n/\\log n$$"
Now the data is much cleaner:
And the word cloud looks much nicer:
Lastly, since these are arguments to O, let’s set the word cloud as an argument of O to make a nice picture:
Mentioned Propositions and Mathematicians
Another way to analyze MathOverflow is to look at the mathematical propositions and famous mathematicians that are mentioned in the post bodies.
An easy way to do this is to use more entity stores to keep track of the different types.
Mathematical Propositions: Build EntityStore
To begin, let’s set up an EntityStore for mathematical propositions and their types.
Specifically, we can set up "MathematicalPropositionType" for “base” words like “theorem,” “hypothesis” and “conjecture,” and "MathematicalProposition" for specific propositions like the “mean value
theorem” and “Zorn’s lemma.”
The proposition types will serve as a means of programmatically finding the specific propositions, so we’ll need to pre-populate "MathematicalPropositionType" with entities, but we can leave it empty
of entities for now—we’ll populate that type in the store by processing the post bodies, but we’ll do that next.
Note that I’ve added some properties to keep track of the propositions found in each post. Specifically, "Wordings" will hold an Association with strings for the keys and the counts of each of those
strings for the values.
Additionally, we’ll set up "MentionedPostCount" to keep track of the number of times a post is mentioned:
"Label"->"proposition type"
"Label"->"mentioned post count",
Now that the EntityStore is set up and registered, we can use the properties I set up in the store.
Let’s start with a list of theorems that don’t have names in them:
$specialTheorems={"prime number theorem","central limit theorem","implicit function theorem","spectral theorem","incompleteness theorem","universal coefficient theorem","intermediate value theorem","mean value theorem","uniformization theorem","inverse function theorem","four color theorem","binomial theorem","index theorem","fundamental theorem of algebra","residue theorem","dominated convergence theorem","open mapping theorem","ergodic theorem","fundamental theorem of calculus","h-cobordism theorem","closed graph theorem","modularity theorem","adjoint functor theorem","geometrization theorem","primitive element theorem","fundamental theorem of arithmetic","fixed point theorem","4-color theorem","four colour theorem","isotopy extension theorem","proper base change theorem","well-ordering theorem","loop theorem","slice theorem","odd order theorem","isogeny theorem","group completion theorem","convolution theorem","reconstruction theorem","equidistribution theorem","contraction mapping theorem","principal ideal theorem","ergodic decomposition theorem","orbit-stabilizer theorem","4-colour theorem","tubular neighborhood theorem","three-squares theorem","martingale representation theorem","purity theorem","triangulation theorem","multinomial theorem","graph minor theorem","strong approximation theorem","universal coefficients theorem","localization theorem","positive mass theorem","identity theorem","cellular approximation theorem","transfer theorem","bounded convergence theorem","fundamental theorem of symmetric functions","subadditive ergodic theorem","annulus theorem","rank-nullity theorem","elliptization theorem"};
Next, we can build a function that will introduce new "MathematicalProposition" entities, keeping track of how often they are mentioned, their types and specific wordings for later use in cleaning
things up.
Note that we strip off any possessives and remove special characters via RemoveDiacritics:
toStandardName=ToCamelCase[StringReplace[StringRiffle@StringTrim[StringSplit[#],"'s"],{"'"->"",Except[LetterCharacter|DigitCharacter]->" "}]]&;
(* Keep track of mentions *)
(* Keep track of specific wordings and their counts *)
(* Extract PropositionType *)
Note that there are currently no proposition entities:
But if we run the list of special theorems through the function…
… then there are proposition entities defined:
We should reset counters for the introduced entities to keep things uniform (the list I provided was fabricated—those strings did not come from actual posts, so not resetting these values may throw
off the numbers a bit):
Of course, we can go further and detect other forms of propositions. Specifically, let’s look for propositions of the following forms:
1. One of the special theorems we just introduced
2. “<person name> theorem” (and similar)
3. “theorem of <person name>” (and similar)
When we find these propositions, we can add them as entities to the proposition EntityStore (via addPropositionEntity), as well as store them with the posts so lookups are faster (as they will
already be stored in memory through the EntityStore).
To start, we’ll need to do some normalization. Here’s a useful function that uses a list of words that should always be lowercased:
$lowercaseWords= list of words that should be lowercased;
lowerCaseSpecificWords= StringReplace[(#->ToLowerCase[#])&/@$lowercaseWords];
Additionally, here’s a list of ordinals and how to normalize them (including “Last”—for example, as in “Fermat’s last theorem”):
Now we can create a function to extract propositions from strings, normalize them with normalizeString from earlier and then create new "MathematicalProposition" entities using addPropositionEntity:
upperCaseWordPattern=(WordBoundary|Whitespace ~~(_?UpperCaseQ ~~ (LetterCharacter| "-"|"'")..)~~WordBoundary|Whitespace),
anyCaseWordPattern=(WordBoundary|Whitespace ~~((Alternatives@@$ordinals)|( (LetterCharacter| "-"|"'")..))~~WordBoundary|Whitespace),
(* Case 1: E.g. "central limit theorem" *)
(* Case 3: "(nth) Theorem of Something (Something (Something))" *)
Shortest[(WordBoundary|Whitespace)~~possibleOrdinalPattern~~propositionTypePattern ~~ Whitespace~~"of"~~Longest@Repeated[upperCaseWordPattern,3]],
(* Case 2: "Something (something (something)) Theorem" *)
(* Remove cases with useless words in them *)
(* Ignore "of" so that case #3 is allowed *)
Let’s try the function on a simple example:
Entity["StackExchange.Mathoverflow:Post", "40686"]["Body"]//extractNamedPropositions
We can see that an entity was added to the store:
We can also see that its properties were populated:
Entity["MathematicalProposition", "MartinAxiom"]["PropertyAssociation"]
Of course, we can automate this a bit more by introducing this function as a property for MathOverflow posts that will store the results in the EntityStore itself:
EntityProperty["StackExchange.Mathoverflow:Post","NamedPropositions"]["Label"]="named propositions";
Let’s test out the property on the same Entity as before:
Entity["StackExchange.Mathoverflow:Post", "40686"]["NamedPropositions"]
We can see that the in-memory store has been populated:
Pro tip: in case you want to continue to work on an EntityStore you’ve been modifying in-memory in a future Wolfram Language session, you can Export Entity["type"]["EntityStore"] to an MX file and
then Import it in the new session. Just don’t forget to register it with EntityRegister!
At this point, we can now gather propositions mentioned in all of the MathOverflow posts, taking care to reset the counters again to avoid contamination of the results:
(* Reset counters again to avoid contaminating the results *)
Note that this will take a while to run (it took about 20 minutes on my machine), but it will allow for a very thorough analysis of the site’s content.
After processing all of the posts, there are now over 10k entities in the proposition EntityStore:
Data Cleanliness
Having kept track of the wordings for each proposition was a good choice—now we can see that proposition entities will format with the most commonly used wording. For example, look at Stokes’
Entity["MathematicalProposition", "StokesTheorem"]["Wordings"]
It’s named after George Gabriel Stokes, and so the correct possessive form ends in “s’,” not “’s,” despite about 15 percent of mentions using the incorrect form.
I’ll admit that this normalization is not perfect—when someone removes the first “s” altogether, it is picked up in a different entity:
Entity["MathematicalProposition", "StokeTheorem"]["Wordings"]
Rather than spend a lot of time and effort to normalize these small issues, I’ll move on and work around these problems for now.
Proposition Analysis
Now that we have a lot of data on the propositions mentioned in the post bodies, we can visualize the most commonly mentioned propositions in a word cloud:
It seems that the prime number theorem is the most commonly mentioned:
We can also see that about two-thirds of all propositions are theorems:
Now that we have the propositions, we can look for mathematician names in the propositions.
To start, we can find all of the labels for the propositions:
Next, we’ll need to find the words in the labels, drop proposition types and stopwords and count them up (taking care to not separate names that start with “de” or “von”):
StringReplace[prefix:"Van der"|"de"|"De"|"von"|"Von"~~(Whitespace|"-")~~name:(_?UpperCaseQ~~LetterCharacter):>StringReplace[prefix,Whitespace->"_"]<>"_"<>name],
Next, we can look for groups of mathematician names joined by dashes (taking care to remove words that are obviously not names):
mathematiciansJoinedByDashes=Select[Keys[commonWordsInPropositions],StringMatchQ[namePattern~~Repeated["-"~~namePattern,{1,Infinity}]]]//StringSplit[#,"-"]&//Flatten//Union//StringReplace["_"->" "];
Another source of names is possessive words, as in “Zorn’s lemma”:
After some cleanup (e.g. removing inline
StringReplace["_"->" "],
Combining these two lists should result in a fairly complete list of mathematician names:
The results seem pretty decent:
Now we need to find possible ways to write down the names of each mathematician.
After looking through the data, I found a few cases that needed to be corrected manually. Specifically, a few names are written out for the famous Jacob Lurie and R. Ranga Rao, so they need to be
corrected to clean up the results a bit:
mathematicianToPossibleNames=GroupBy[allMathematicians,RemoveDiacritics/*StringReplace["_"->" "]/*toStandardName];
mathematicianToPossibleNames["Lurie"]=PrependTo[mathematicianToPossibleNames["Lurie"],"Jacob Lurie"];
mathematicianToPossibleNames["Ranga"]=PrependTo[mathematicianToPossibleNames["Ranga"],"Ranga Rao"];
Now, we need to construct an Association to point from individual name words to their mathematicians:
Note that these entities do not format—we’ll create the EntityStore for them soon:
From here, we can break the proposition labels into words that we can use to look up mathematicians (taking care to fix a few cases that need special attention for full names written out):
StringReplace[prefix:("De"|"de"|"von"|"Von"|"Jacob"|"Alex"|"Yoon"|"Ranga")~~Whitespace|"-"~~name:(namePattern|"Ho"|"Lurie"|"Lee"|"Rao"):>prefix<>"_"<>name]/*(StringSplit[#,Whitespace|"-"]&)/*(StringTrim[#,"'s"|"s'"]&)/*StringReplace["_"->" "]
With this done, we can add this data to the proposition store as a new property:
EntityProperty["MathematicalProposition","NamedMathematicians"]["Label"]="named mathematicians";
And by rearranging the data, we can build the data to create an EntityStore for mathematicians:
KeyTake[mathematicanToPropositions,{Entity["Mathematician", "DeFinnetti"],Entity["Mathematician", "Lurie"],Entity["Mathematician", "Ranga"]}]
Using this data, we can now build an EntityStore for mathematicians, taking into account the data we’ve accumulated:
"PossibleNames"-><|"Label"->"possible names"|>,
"NamedPropositions"-><|"Label"->"named propositions"|>
Now we can see the propositions named after a specific mathematician, such as Euler:
EntityValue[Entity["Mathematician", "Euler"],"NamedPropositions"]
At this point, many interesting queries are possible.
As a start, let’s look at a network that connects two mathematicians if they appear in the same proposition name. It sounds somewhat similar to the Mathematics Genealogy Project.
An example might be the “Grothendieck–Tarski axiom”:
Entity["MathematicalProposition", "GrothendieckTarskiAxiom"]["NamedMathematicians"]
First, grab all of the named mathematicians for each proposition, taking only those that have at least two:
With a little bit of analysis, we can see that the majority of propositions have two mathematicians, fewer have three, there are a few groups of four and there is only one group of five:
Here is the group of five, mentioned in this answer:
Here are the top 25 results:
We can complete our task by constructing a network of all grouped mathematicians:
The network is quite large and messy.
With some more processing, we can add weights for repeated edges and use them to determine their opacity:
Here are the most common pairings of mathematicians:
And here’s an easy way to see who has the most propositions named after them, which can be done with SortedEntityClass, new in Mathematica 12:
Here are the top 25, along with the number of propositions in which they are mentioned on MathOverflow:
Despite his common appearance in theorems, Euler is surprisingly very low on this list at number 61:
Position[Keys[mathematicianToPropositions],Entity["Mathematician", "Euler"]]
And here’s the full distribution of mentions (on a log scale for easier viewing):
FrameLabel->{"# of Named Propositions","# of Mathematicians"}
Do It Yourself
Be sure to explore further using the newest functions coming soon in Version 12 of the Wolfram Language!
Although I used a lot of Wolfram technology to explore the data on MathOverflow.net, here are some open areas that still remain to be explored:
• What fraction of all questions are unanswered?
• Which tags have the most answered or unanswered questions?
• Find named mathematical structures (e.g. integral, group, Riemann sum, etc.)
• Align entities to MathWorld for further investigation
• Investigate the time series ratio of user count to post count
In fact, you can download and register the entity stores used in this post here:
Join the discussion
8 comments
1. Andrew
Thank you very much for providing such an interesting blog. A number of questions: Is it possible to get the notebook for this blog, rather than having to retype everything? Secondly how does
this link in with the recent attempts by WRI and Eric from MathWorld to create an EntityStore on Algebraic Topology, which was covered in a number of WTC 2017 talks? Lastly, if it possible to
create EntityStores from the Stack Exchange archives , is it also possible to create EntityStores on mathematical theorems from the corresponding Wikipedia articles using WikipediaSearch and
WikipediaData ?
Thanks again for your blog and any responses to my questions
□ Thanks for reading! I’m glad you enjoyed it!
I am not familiar with the MathWorld EntityStore you’ve mentioned, but if anything, parts of this blog can used as a stepping stone to create your own EntityStores based on any data source.
Concerning EntityStores created from Wikipedia: I myself haven’t done this, but it sounds like an interesting project, and could probably be done without too much trouble. I’d be curious to
see what results can be found from this data!
2. Well done Andrew!
Pretty impressive data processing functionalities of Version 12. I can’t wait to get a copy of it.
Love the wordsMuchMoreCommonInMO graphics.
3. (1) As useful to Mathematica users, perhaps, as MathOverflow.net, would be an entity store of https://mathematica.stackexchange.com.
(2) Is there a button somewhere that I’m missing to allow downloading this blog post as a notebook? Or at the very least, cause double clicking displayed code in this blog post to be copyable as
text, so that it can be pasted into a notebook. Just having a bunch of .jpg’s showing code is very unhelpful!
□ Thanks for reading! I hope you enjoyed it!
(1) Yes, I have looked at and analyzed mathematica.stackexchange.com, as well as several other SE sites, and I have found some interesting things! I hope to do another blog post, and in
particular, cover mathematica.stackexchange.com in greater detail in the near future.
(2) I certainly agree about the lack of a notebook and copyable code! Some functionality in the post requires the upcoming Version 12 (which, as of this writing, is unreleased). I didn’t want
to give out the code as some would not work properly in older versions (e.g. the SPARQL query parts). Once Version 12 does is released, the notebook and copyable code will be available. Sorry
for the inconvenience!
□ Hi Luc,
I made an EntityStore of gamedev.stackexchange.com with the current version of the utility and hosted it in the Wolfram Cloud.
You can get it with this WL code (it’s around 600 MB large):
gameDevSEStore = Import[CloudObject[“https://www.wolframcloud.com/objects/andrews/StackExchange2EntityStore/gamedev.stackexchange.com.mx”]];
Then, you can register it for use in EntityValue with this:
The basic features in the store should still work in Mathematica 11.3, and I’ve made a notebook to show a few simple things I found in it, including all posts with the “unity” tag:
Let me know if you have any questions!
5. Would be interesting to see if some statistics can be used to charactetize group/social dynamics at MO. I’d be particularly interested in seeing statistics of those who frequently vote to close
questions. For example, it appears that some such voters actually ask few questions themselves or ask few highly favorited, or liked, or viewed questions. How are their statistics skewed from the
general population of voters at given voting frequencies? | {"url":"https://blog.wolfram.com/2019/02/01/the-data-science-of-mathoverflow/","timestamp":"2024-11-06T14:03:04Z","content_type":"text/html","content_length":"349325","record_id":"<urn:uuid:a573c266-0620-40b6-b0d0-f79948c95608>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00877.warc.gz"} |
How Much Data Can You Encrypt with RSA Keys?
When someone first begins to consider using encryption to protect business data, they discover that there are two general types: symmetric (AES) and asymmetric (RSA). At first glance, which one you
would choose can be confusing.
One of the differences between the two is speed. Symmetric encryption is much faster than asymmetric. The exact difference is implementation dependent, but may be on the order of 100 to 1000 times
It is widely known that AES encrypts a 16-byte block of data at a time. However, how much data can be encrypted at one time with an RSA key is usually only discussed in vague terms such as “use RSA
to encrypt session keys.” This raises the question of how much data can be encrypted by an RSA key in a single operation.
The typical encryption scenario is to encrypt with a public key and decrypt with the private key. OpenSSL provides the RSA_public_encrypt and RSA_private_decrypt functions to implement this.
The first parameter to the RSA_public_encrypt function is flen. This is an integer that indicates the number of bytes to encrypt. Its maximum value depends on the padding mode. For OAEP padding,
recommended for all new applications, it must be less than the size of the key modulus – 41 (all in bytes).
To get the size of the modulus of an RSA key call the function RSA_size.
The modulus size is the key size in bits / 8. Thus a 1024-bit RSA key using OAEP padding can encrypt up to (1024/8) – 42 = 128 – 42 = 86 bytes.
A 2048-bit key can encrypt up to (2048/8) – 42 = 256 – 42 = 214 bytes.
Additional Resources for IT Developers and Professionals
We collaborate with developers and IT professionals around the world and know that they use a wide variety of languages and platforms to accomplish their work. Our products include documentation,
source code examples, and HOWTO guides for developers in order to help projects get done quickly. Visit our Developer Resources section of our web site to learn more and discuss your upcoming project
with our development team. | {"url":"https://info.townsendsecurity.com/bid/29195/how-much-data-can-you-encrypt-with-rsa-keys","timestamp":"2024-11-02T09:10:04Z","content_type":"text/html","content_length":"36812","record_id":"<urn:uuid:3070219d-51cf-4afe-8db4-7686f54fe87f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00832.warc.gz"} |
Subtract Time Calculator
Last updated:
Subtract Time Calculator
This subtract time calculator, with its self-explanatory name, will (guess what) enable you to subtract time in various units conveniently.
If you're sometimes confused about time, don't worry - it's not just you. The concept of time still troubles even the greatest philosopher. Luckily, you have been given infinite web resources to help
you navigate the complex reality we live in. In the text below, you'll find instructions on how to subtract time using the subtract time calculator and how to add and subtract time on your own.
Whether you want to determine your working time, count only the Wednesdays within a given period, check what date it will be in 3 weeks, or are changing the time zones - we've got all types of time
math calculators to clear some of the confusion surrounding time!
How to subtract time with this time math calculator
1. Choose adequate units in all fields of the subtract time calculator. By default, they are set to hours, minutes, and seconds so that you can input any or all of these three units. You can also go
for milliseconds, days, weeks, months, million years, billion years, or even the age of the universe (13.8 billion years)!
2. Input the value from which you want to subtract ("Time 1") into the time math calculator.
3. Input the value which you wish to subtract ("Time 2").
4. Check out the result in the last field ("Time difference") of the subtract time calculator. You can always change the unit it is displayed in.
If you are trying to connect with a friend living in another country, and you wanna make sure of the time difference between your zones, then our time zones converter is the right tool to assist you.
How to add and subtract time by yourself
Adding and subtracting time on your own can be confusing. It's probably because we're used to using the decimal system:
1 m = 10 dm,
while seconds and hours obey the sexagesimal system:
1 h = 60 min and
1 min = 60 s.
Before we get into it, may we interest you to check out our date calculator. It allows you to calculate between two dates.
Now, let's go through some examples of how to add and subtract time when you don't have access to time math calculators.
How to subtract time:
1. Convert Time 1 and Time 2 to simpler units if you need to. For example, if you have to subtract 1 h 20 min 30 s from 2 h 14 min 16 s, you may first convert them to seconds, and we have a time
unit converter to assist you for that:
Time 1:
t1 = 2 h 14 min 16 s
t1 = 2 * 60 * 60 s + 14 * 60 s + 16 s
t1 = 8056 s
Time 2:
t2 = 1 h 20 min 30 s
t2 = 1 * 60 * 60 s + 20 * 60 s + 30 s
t2 = 4830 s
2. Subtract the converted times:
t1 - t2 = 8056 s - 4830 s
t1 - t2 = 3226 s
3. Convert the result to the target unit by dividing it by 60 (the remainder becomes the seconds):
3226 / 60 = 53 R 46
t1 - t2 = 53 min 46 s
How to add time:
You can add time analogously. Another method is to add hours to hours, minutes to minutes, seconds to seconds (weeks to weeks, etc.) Let's say you want to add 3 h 54 min 46 s to 19 h 33 min 55 s:
1. Add hours:
3 h + 19 h = 22 h
2. Add minutes:
54 min + 33 min = 87 min
Convert the result to hours - divide minutes by 60 and leave the remainder as minutes:
87 min / 60 = 60 / 60 min + 27 / 60 min
87 min = 1 h 27 min
3. Add seconds:
46 s + 55 s = 101 s
101 s = 60 s + 41 s
101 s = 1 min 41s
4. Add the results from steps 1-3:
22 h + 1 h 27 min + 1 min 41s = 23 h 28 min 41 s
We are here to help you with as many time, and date needs as possible. So if you want to count the days between two dates, then check out the day counter, and if you are looking to determine the
duration between times, then make sure to give a read to OMNI's time duration calculator. | {"url":"https://www.omnicalculator.com/everyday-life/subtract-time","timestamp":"2024-11-07T00:46:40Z","content_type":"text/html","content_length":"417459","record_id":"<urn:uuid:59fda27e-3f21-4a9e-aaf0-7b22f54fdf9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00473.warc.gz"} |
56 km to miles
Heading 1: Understanding Kilometers and Miles
People all over the world use different units of measurement to calculate distance. Some countries, like the United States and the United Kingdom, use a system called the Imperial system, while
others, like most of Europe, use the Metric system. This can lead to confusion when trying to understand distances in different countries.
In the Imperial system, distance is measured in miles. A mile is equal to 5,280 feet or approximately 1.6 kilometers. This is the unit of measurement commonly used in the United States and the United
Kingdom. On the other hand, in the Metric system, distance is measured in kilometers. A kilometer is equal to 1,000 meters or approximately 0.62 miles. Most of Europe, as well as many other countries
around the world, use kilometers as their primary unit of measurement for distance.
Heading 2: The Metric and Imperial Systems
The world is a diverse place, and so are its systems of measurement. When it comes to metrics, a large part of the world follows the metric system, while others, particularly the United States,
adhere to the imperial system. The metric system, originating from France in the late 18th century, is a decimalized system that provides a consistent and straightforward approach to measurement. It
is based on units like meters, grams, and liters, making calculations and conversions more manageable.
On the other hand, the imperial system, also known as the English system, has its roots in ancient Roman and Anglo-Saxon systems. It is widely used across the United States and a few other countries.
This system uses units such as inches, pounds, and gallons, and is known for its non-decimal measurements. While some may argue that the imperial system has a sense of tradition and familiarity, it
often leads to confusion and difficulties with conversions. Nonetheless, both systems coexist in today’s globalized world, and understanding their differences and similarities is essential.
Heading 2: What is a Kilometer?
A kilometer, abbreviated as km, is a unit of length in the metric system. It is commonly used in countries that have adopted the metric system of measurement. One kilometer is equal to 1,000 meters
or approximately 0.62 miles. To put it into perspective, if you were to walk a kilometer, it would take you roughly 12-15 minutes at an average walking speed. Kilometers are widely used in various
contexts, such as measuring distances between cities or calculating the length of a running race.
The kilometer is derived from the Greek word “khilioi,” meaning one thousand. It was officially adopted as a unit of measurement in France during the French Revolution in the late 18th century.
Today, the kilometer is used in most countries around the world as the standard unit for measuring longer distances. It is also worth noting that the kilometer is part of the International System of
Units (SI) and is recognized as the official unit of length by the International Bureau of Weights and Measures (BIPM).
Heading 2: What is a Mile?
A mile is a unit of measurement used in the Imperial system. It is primarily used in the United States and other countries that have not adopted the metric system. One mile is equal to 1.60934
kilometers, making it a slightly longer distance. In everyday terms, a mile is commonly used to measure shorter distances on roads and highways. For example, when someone says they live five miles
away, it means they are approximately 8 kilometers from their destination.
The origin of the mile can be traced back to ancient Roman times, where it was used to measure distances on the famous Roman roads. It was defined as 1,000 paces, with each pace being roughly
equivalent to two steps. Over time, the measurement of a mile evolved and varied in different regions. In the 16th century, the British Empire introduced the statute mile, which is the most commonly
used variant today. It is defined as 5,280 feet or 1,760 yards. Despite the metric system being widely used around the world, the mile remains an important unit of measurement in certain countries,
particularly in everyday discussions of distance.
Heading 2: Converting Kilometers to Miles
If you’ve ever found yourself needing to convert kilometers to miles, you’re not alone. It can be quite confusing for those of us who are more accustomed to the metric system. Luckily, there’s a
straightforward formula that you can use to make the conversion a breeze.
To convert kilometers to miles, you simply need to multiply the number of kilometers by 0.6214. For example, if you have 10 kilometers, you would multiply 10 by 0.6214 to get 6.214 miles. It’s as
simple as that! So the next time you come across a distance in kilometers and need to know the equivalent in miles, just remember this handy conversion factor and you’ll be able to calculate it in no | {"url":"https://convertertoolz.com/km-to-miles/56-km-to-miles/","timestamp":"2024-11-09T14:17:59Z","content_type":"text/html","content_length":"41506","record_id":"<urn:uuid:41a808f1-ec5c-4ba3-a937-74a8dec2e181>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00162.warc.gz"} |
[tex4ht] MathJax support
Nasser M. Abbasi nma at 12000.org
Tue Dec 4 20:10:23 CET 2018
On 12/4/2018 9:02 AM, Michal Hoftich wrote:
> Hi all,
> I've just added new literate source to tex4ht, tex4ht-mathjax.tex. It
> is a modified version of mathjax-latex-4ht.sty from the Helpers4ht
> project [1]. I've also added the "mathjax" option to `html4-math.4ht`,
> so it should be possible to require the MathJax rendering of math
> using this option. I need to add this option also for the MathML
> output.
> Anyway, with this option it will be possible to use MathJax directly using
> make4ht filename.tex "mathjax"
> without need to use Helpers4ht and configuration files.
> Best regards,
> Michal
> [1] https://github.com/michal-h21/helpers4ht/blob/master/mathjax-latex-4ht.sty
Thanks Michal.
But I can't get it to work. Is this supposed to be in the source now?
I just did now _full_ update for TL 2018.
Two issues
1) when I compile a file, it generates png images and do not do mathjax.
2) when I use my main .cfg, now I get error
make4ht -ulm default -c ~/nma_mathjax.cfg foo.tex "htm,mathjax"
(/usr/local/texlive/2018/texmf-dist/tex/generic/tex4ht/html5.4ht)) (./foo.aux)
! Undefined control sequence.
l.184 \ExplSyntaxOn
Because in my main .cfg file, I am using \ExplSyntax on.
But lets stay with the first issue for now. Using this MWE
A \sin x+\\
\cos x=0
Compiled using
make4ht -ulm default -c ./nma_mathjax.cfg foo.tex "htm,mathjax"
Where the local .cfg above is this:
>cat nma_mathjax.cfg
So I removed the first line that was there which was
Becuase you said it is no longer needed?
But now it generaes png for images. Looking at the HTML,
there is no mathjax configuration at all in it.
Same thing if I use
make4ht foo.tex "htm,mathjax"
Am I doing something wrong? Or may be the updates are not
there yet?
More information about the tex4ht mailing list | {"url":"https://tug.org/pipermail/tex4ht/2018q4/002141.html","timestamp":"2024-11-02T01:24:41Z","content_type":"text/html","content_length":"5114","record_id":"<urn:uuid:f5ce5391-a9bd-4745-8f56-407b44725b3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00445.warc.gz"} |
Cite as
Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing. In 42nd IARCS Annual Conference on Foundations of Software
Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Copy BibTex To Clipboard
author = {Khan, Arindam and Sharma, Eklavya and Sreenivas, K. V. N.},
title = {{Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {23:1--23:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.23},
URN = {urn:nbn:de:0030-drops-174151},
doi = {10.4230/LIPIcs.FSTTCS.2022.23},
annote = {Keywords: Bin packing, rectangle packing, multidimensional packing, approximation algorithms} | {"url":"https://drops.dagstuhl.de/search/documents?author=Sharma,%20Eklavya","timestamp":"2024-11-05T09:26:34Z","content_type":"text/html","content_length":"71555","record_id":"<urn:uuid:ceeeb711-f153-4f5b-90df-fe30284a9c30>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00158.warc.gz"} |
Multiplying and dividing a number by 10, 100 and 1,000 including bridging 1 | Oak National Academy
(birds chirping) <v ->Hello, how are you today?</v> My name is Dr.
Shurick and I'm really excited to be learning with you today.
We are gonna have great fun as we move through the learning together.
Today's lesson is from our unit calculating with decimal fractions.
The lesson is called explain the effect of multiplying and dividing a number by 10, 100 and 1000, including bridging 1.
As we move through the learning today, we will deepen our understanding of multiplying and dividing by 10, 100 and 1000 and thinking about what happens when we bridge 1, so when we have to calculate
with decimal fractions.
Throughout the lesson, we will use a place value chart and the Gattegno chart to support us to make connections in our learning.
Sometimes new learning can be a little bit tricky, but I know if we work really hard together and I am here to guide you, then we can be successful.
So shall we get started? Let's find out.
How do we explain the effects of multiplying and dividing a number by 10, 100 and 1000 including bridging 1.
These are the key words that we will use in our learning today, tenth, hundredth and thousandth.
I'm sure you've heard those words before, but let's practise them anyway.
My turn tenth, your turn.
My turn, hundredth, your turn.
My turn, thousandth, your turn.
Fantastic and really see those tongues when you do the th part at the end.
One-tenth is one part in 10 equal parts.
One-hundredth is one part in 100 equal parts.
And one-thousandth is one part in 1,000 equal parts.
Look out for those keywords as we move through the learning today.
We are going to start our learning today thinking about how we divide by 10, 100 or 1,000.
We have Lucas and Sophia to guide us through the learning.
Right, let's have a look at this then.
Sophia is thinking of a number.
She wants Lucas to guess her number and gives him a clue.
Sophia's clue is, "My number is one-hundredth times the size of eight".
I wonder if you might be able to work out what that is.
Can you visualise what that means? Ah, so Lucas knows that to find the number, he needs to divide eight by 100, because the number Sophia is thinking of is one-hundredth times the size.
So he needs to divide eight by 100.
Lucas is going to use some derived facts to help, he knows one divided by 100 is equal to one-hundredth.
So two divided by 100 must be two hundredths.
Three divided by 100 must be three hundredths.
Four divided by 100 must be four hundredths.
I wonder if you can tell what's coming next.
That's right, five divided by 100 is five hundredths.
Six divided by 100 is six hundredths.
Seven divided by 100 is seven hundredths.
What does that mean? What's next in the pattern? That's right, eight divided by 100 is eight hundredths.
Let's look at this on a Gattegno chart, we can see the starting number is eight and we can divide by 100, which means we need to look down two rows on the Gattegno chart.
Eight divided by 100 is equal to 0.
08 is one-hundredth times the size of eight.
And we can use a place value chart as well to help.
We know when a number is divided by 100 the digits move two places to the right.
One, two.
So eight divided by 100 is equal to 0.
Dividing by 100 makes the number 100 times smaller, so Sophia's number is 0.
Let's look at this equation in more detail.
We've got eight and we're dividing by 100 and it was 0.
What's the value of the eight in eight? Well the value of eight is eight ones or eight.
What about the value of the eight in the answer of 0.
08? Well the value of the eight there is eight hundredths.
We had eight ones, we now have eight hundredths.
We can say that 0.
08 is one-hundredth times the size of eight and we can use that information to form a second equation.
Dividing by 100, well that's the same or equivalent to multiplying by one-hundredth or 0.
So eight divided by 100 is equal to 0.
08 but also, eight multiplied by 0.
01 is equal to 0.
08 because dividing by 100 is the same as multiplying by 0.
These expressions are equivalent to each other and they each have a value of 0.
We can say that eight divided by 100 is equal to eight multiplied by 0.
01, which is equal to 0.
Let's check your understanding with that.
Could you fill in the blanks in the equations? Four divided by 100 equals mm.
Four multiplied by mmh is equal to 0.
And that means four divided by mmh is equal to four times mm which is equal to 0.
Pause the video while you have a go at completing the equations.
Maybe find someone to chat to about this.
And when you are ready for the answers, press play.
How did you get on? Four divided by 100 is 0.
Maybe you use a Gattegno year chart to help, or a place value chart to help.
We've got four multiplied by, hmm, well if we divide it by 100 we must be multiplying by 0.
01 or one-hundredth, that would give us the same answer, 0.
Then we can put the equations together, four divided by 100 is equal to four multiplied by 0.
01, which is equal to 0.
How did you get on? Well done.
Now Lucas' turn to think of a number.
He wants Sofia to guess his number and he gives her a clue.
"My number is one-thousandth times the size of three." So Sophia knows that to find his number she needs to divide three by 1,000.
And Sophia can use some known facts, or derived facts to help.
She knows one divided by 1,000 is the same as one-thousandth.
Two divided by 1,000 would be two thousandths.
Have you spotted the pattern already? What would come next? That's right, three divided by 1,000 is three thousandths.
Let's look at this on a Gattegno chart.
We start with the number three, we divide by 1,000, which means we have to move down three rows.
Three divided by 1,000 we can see is equal to 0.
We can also use a place value chart to help us.
We started off with three ones and when a number is divided by 1,000 the digits move three places to the right.
One, two, three.
We had three ones, we end up with three thousandths.
It makes the number 1,000 times smaller.
So Lucas' number must have been 0.
003, three thousandths is one-thousandth times smaller than three.
Let's look at the equation in more detail.
Three divided by 1,000 is equal to 0.
What's the value of the 3 in three? Ah, thank you Sophia.
The value of 3 is three ones or three.
What about the value of 3 and 0.
That's right, the value of that three is three thousandths.
We have three ones, we now have three thousandths.
We can say that 0.
003 is one-thousandth times the size of three.
We can use that information to form a second equation.
We know dividing by 1,000 is equivalent to multiplying by one-thousandth or 0.
We can say that these expressions are equivalent to each other and they have a value of 0.
So three divided by 100, well that's the same as three multiplied by 0.
001 and they both have an answer or a value of 0.
003, three thousandths.
Let's check your understanding of that.
Could you fill in the blanks in these equations? Nine divided by 1,000 is equal to mm.
Nine multiplied by mm is equal to 0.
Nine divided by mmh is equal to nine times mmh, which is equal to 0.
Pause the video, maybe find someone to chat to about this.
And when you're ready for the answers, press play.
How did you get on? Did you either use a Gattegno chart, or a place value chart to help you? Nine divided by 1,000 is 0.
009 or nine thousandths.
When we divide by 100 it is the same as multiplying by 0.
And those two expressions have the same value so they are equal.
Nine divided by 1,000 is equal to nine multiplied by 0.
001 and they have a value of 0.
We have seen that when we divide by 100 it's equivalent to multiplying by 0.
01, and we know the digits move two places to the right.
And we know dividing by 100 is equivalent to multiplying by 0.
We've seen that when we divide by 1,000 it is equivalent to multiplying by 0.
001, and the digits move three places to the right.
So dividing by 1,000 is equivalent to multiplying by 0.
What about 10? What would divided by 10 be equivalent to? Can you spot a pattern to help? That's right, when we divide by 10, it is equivalent to multiplying by 0.
1 or one-tenth, and the digits move one place to the right.
Dividing by 10 is equal to multiplying by 0.
1 or one-tenth.
Let's look at this equation in more detail then.
Two divided by 10 is equal to two multiplied by 0.
1, which is equal to 0.
Well we know Lucas is reminding us that 0.
1 is equivalent to one-tenth.
So we could also say that two divided by 10, we know it's the same as two multiplied by 0.
1, we know 0.
1 is one-tenth, so that is also the same to two multiplied by one-tenth and they all have the value of 0.
But we know two one-tenths are equivalent to two-tenths.
All of these expressions are equivalent and have a value of 0.
Let's check your understanding.
Could you have a look at these four equations and tell me which equation is correct? Pause the video while you have a look and when you're ready to go through the answers, press play.
How did you get on? Did you say, well it can't be A, because we're dividing by 100 but then we're multiplying by one-hundredth, so that's okay.
But then we've got four multiplied by one-tenth, well this time we're dividing by 100, so it can't be A.
What about B? Well we start with dividing by 100, we've got one-hundredth as our fractions, but ah, we're multiplying by 0.
1, that's the same as one-tenth, so it can't be B.
C is definitely correct, we're dividing by 100.
Dividing by 100 is the same as multiplying by 0.
Dividing by one-hundredth is the same as finding one-hundredth and we've got four of them, which is four hundredths.
And D, D can't be correct because the answer is 0.
4, which is four tenths and we've got four divided by 100, so it should be four hundredths or 0.
I wonder how you got on with those.
Well done.
It's your turn to practise now.
For question one, could you solve this problem? Sophia is thinking of a number.
Her number is one-thousandth times the size of seven.
Could you form two expressions to represent this, and then tell me what Sophia's number is, explaining how you know.
For question two, you've got some equations, could you complete? And then when you finish those, could you form a division equation of your own? But I'd like you to make a mistake and then explain
the mistake that you made.
For question three, could you look at the equation, is it true or false? And convince me that you are correct.
Four divided by 1,000, is it equal to four multiplied by 0.
001, is it equal to 0.
04? Pause the video while you have a go at all three questions and when you are ready for the answers, press play.
How did you get on? For question one, we were asked to find what Sophia's number is.
We know it was one-thousandth times the size of seven, so we can form our first equation, seven divided by 1,000.
And we know dividing by 1,000 is the same as multiply by 0.
We then had to work out Sophia's number, maybe you used a Gattegno chart or a place value chart but I worked out that it was 0.
And we might have given me a reason such as, to find a number that is 1,000 times smaller you need to divide by 1,000.
Dividing by 1,000 is the same multiplying by 0.
And the digits will move three places to the right.
This makes the number one-thousandth times the size, so Sofia's number is 0.
For question two you were asked to complete the equations.
We've got three divided by 10 is equal to 0.
Three divided by 100 is 0.
And three divided by 1,000, 0.
Nine times 0.
1 is 0.
Nine times 0.
01 is 0.
And 0.
009 is equal to nine multiplied by 0.
Five divided by 10, well that's equal to five one-tenths or five times 0.
1, which is equal to 0.
Two divided by 10 is equal to two times 0.
01, which is equal to 0.
And then we have a longer equation here, we've got 0.
006, well that's the same as six divided by 1,000, which is the same as six multiplied by 0.
We know that's the same as six times one-thousandth, which is six thousandths.
You were then asked to form an equation of your own but make a mistake.
You might have formed an equation like, nine divided by 100 is equal to 0.
9 and explained that the answer was incorrect.
When we divide by 100, it is the same as multiplying by one-hundredth and the digits move two places to the right.
In this example, the digits were only moved one place to the right.
The answer should have been 0.
For question three you had to tell me whether or not the equation was true or false and convince me.
So you might have said that this was false and convinced me by saying that when we divide by 1,000 it is the same as multiplying by 0.
001, so that part of the equation was correct, but the digits would move three places to the right, not two, as in the given equation.
The answer should be 0.
How did you get on with all three questions? Well done.
Fantastic learning today so far everybody, really impressed with how hard you are working.
We are now going to move on and look at how we multiply by 10, 100 or 1,000.
Sophia and Lucas are still playing their game and Sophia is thinking of a different number.
She wants Lucas to get her number and gives him a clue.
"My number is one-thousand times the size of 0.
04." Hmm.
Have you noticed something different this time? That's right, her number is one-thousand, so it's not one-thousandth this time, it's a whole number this time, one-thousand the times the size of 0.
So this time the number she's given us for a clue is a decimal fraction, 0.
"To find your number I need to multiply 0.
04 by 1,000." Because we need to find the number that is 1,000 times the size of 0.
And we can use some derived facts to help.
We know 0.
01 or one-hundredth times 1,000 is equal to 10.
So two hundredths multiplied by 1,000 would be 20.
Three hundredths multiplied by 1,000 would be 30.
Can you spot what would come next, have you seen that pattern? That's right, 0.
04 multiplied by 1,000 is equal to 40.
And we can look at this in the Gattegno chart, we've got 0.
04 was the starting number that Sophia gave us and if we multiply it by 1,000, we move up three rows, which would be 40 and we can use that to form an equation.
04 multiplied by 1,000 is equal to 40.
We can also look at this on a place value chart.
We started with four hundredths and we are multiplying by 1,000, so we need to move the digits three places to the left.
This makes the number 1,000 times larger.
04 multiplied by 1,000 is 40, so Sophia's number is 40.
Let's look at this equation in more detail.
04 multiplied by 1,000 is equal to 40.
Well what's the value of the four in 0.
04? Do you know? That's right Lucas, the value of the four is four hundredths.
What about the value of the 4 in 40? The value of the 4 is four tens, or 40.
We had four hundredths, we now have four tens.
40 is 1,000 times the size of 0.
Let's check your understanding.
When we multiply 0.
9 by 1,000 the digits move two places to the left, the product is 90.
Is that true or false? Pause the video while you think about it and when you are ready, press play.
How did you get on? Did you work out that that must be false? But why is it false? Is it because A, 0.
9 is being increased 1,000 times, the digits move three places, 10 times 10 times 10.
The product should be 900.
Or is it B, when we multiply by 1,000 we place three zeros at the end of the number.
The product would be 0.
Pause the video, maybe chat to someone about this, and when you are ready, press play.
How did you get on? Did you realise it must be A, when 0.
9 is increased 1,000 times, all of the digits move three places.
The product should be 900.
We don't just place three zeros at the end of a number, do we? No, we know that the digits move.
Let's summarise what we've learned so far.
When we multiply by 10, well the digits move one place to the left.
When we multiply by 100, what happens? That's right, the digits move two places to the left.
What about when we multiply by 1,000? What happens? That's right, the digits move three places to the left.
Let's check your understanding on that.
Could you match the expression to its product? Pause the video while you have a look and when you're ready for the answers, press play.
How did you get on? Did you work out that 0.
3 multiplied by 1,000 is 300? 0.
03 multiplied by 1,000, that's 30.
And three thousandth multiplied by 1,000 must be three.
How did you get on? Brilliant.
Your turn to practise now.
For question one, could you complete the equations and then when you finish, could you form an equation of your own but make a mistake? Explain the mistake that you make.
For question two, look at the equation.
Is it true or is it false? Convince me that you are correct by giving some reasons.
We've got 0.
05 multiplied by 1,000 is equal to 50.
Pause the video while you have a go at both questions and when you are ready for the answers, press play.
How did you get on? For question one you had some equations to complete.
We've got 0.
8 multiplied by 10, well that's eight.
8 multiplied by 100 is 80.
8 multiplied by 1,000, that's what would give us 800.
6 times 10 is six.
06 times 100 is six.
And six, well that's the same as or equal to 0.
006 multiplied by 1,000.
5 times 1,000 is 500.
05 times 1,000 would be 50.
And five would be equal to 0.
005 multiplied by 1,000.
You were then asked to form an equation of your own but make a mistake.
You might have formed an equation like 0.
3 multiplied by 1,000 is equal to 0.
3000 and explained that my product was incorrect.
When we multiply by 1,000, the digits moved three places to the right.
In this example, placeholders were just placed at the end of the 0.
3 decimal number, weren't they? I just put three zeros at the end.
We can't do that.
My digits needed to move.
I have three-tenths, I still have three-tenths, nothing moved, so the product should have been 300.
For question two, you were asked to tell me if the equation was true or false and convince me that you were correct.
You might have said that it is true and convinced me by saying that when we multiply by 1,000 the digits move three places to the left.
We had five hundredths, we now have five tens or 50.
How did you get on with those questions? Brilliant.
Fantastic learning today.
I am really proud of how hard you have tried and how much further you have moved your understanding with multiplying, dividing a number by 10, 100 and 1,000.
We know that when a number is divided by 10 or 100 or 1,000, the digits move one, two, or three places to the right, respectively even if it's bridge is one.
We know dividing by 10, 100 or 1,000 is equivalent to multiplying by 0.
1, 0.
01 or 0.
001 respectively.
And this is also equivalent to multiplying by one-tenth, one-hundredth or one-thousandth.
We've also learned that when a decimal number is multiplied by 10, 100 or 1,000, the digits move one, two, or three places (indistinct) Fantastic learning.
I've had great fun today and I really look forward to learning with you again soon. | {"url":"https://www.thenational.academy/pupils/programmes/maths-primary-year-5/units/calculating-with-decimal-fractions/lessons/multiplying-and-dividing-a-number-by-10-100-and-1-000-including-bridging-1/video","timestamp":"2024-11-09T13:13:06Z","content_type":"text/html","content_length":"135410","record_id":"<urn:uuid:94a4ff0c-483d-458d-be32-b4ecb1972df2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00524.warc.gz"} |
(PDF) Time-Area Optimized Public-Key Engines: -Cryptosystems as Replacement for Elliptic Curves?.
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Time-Area Optimized
Public-Key Engines: MQ-Cryptosystems as
Replacement for Elliptic Curves?
Andrey Bogdanov, Thomas Eisenbarth, Andy Rupp, Christopher Wolf
Horst G¨ortz Institute for IT-Security
Ruhr-University Bochum, Germany
chris@Christopher-Wolf.de or cbw@hgi.rub.de
In this pap er ways to efficiently implement public-key schemes based on Multivariate Qua-
dratic p olynomials (MQ-schemes for short) are investigated. In particular, they are claimed
to resist quantum computer attacks. It is shown that such schemes can have a much better
time-area product than elliptic curve cryptosystems. For instance, an optimised FPGA im-
plementation of amended TTS is estimated to be over 50 times more efficient with respect to
this parameter. Moreover, a general framework for implementing small-field MQ-schemes in
hardware is proposed w hich includes a systolic architecture performing Gaussian elimination
over composite binary fields.
1 Introduction
Efficient implementations of public key schemes play a crucial role in numerous real-world security
applications: Some of them require messages to be signed in real time (like in such safety-enhancing
automotive applications as car-to-car communication), others deal with thousands of signatures per
second to be generated (e.g. high-performance secur ity servers using so-called HSMs - Hardware
Security Modules). In this context, software implementations even on high-end pr ocessors can often
not provide the performance level needed, hardware implementations being thus the only option.
In this paper we explore the approaches to implement Multivariate Quadratic-based public-key
systems in hardware meeting the r equirements of efficient high-performance applications. The secu-
rity of public key cryptosystems widely spread at the moment is based on the difficulty of solving a
small class of problems: the RSA scheme relies on the difficulty of factoring large integers, while the
hardness of computing discrete logarithms provides the basis for ElGamal, Diffie-Hellmann scheme
and elliptic curves cryptography (ECC). Given that the security of all public key schemes used in
practice relies on such a limited set of problems that are currently considered to be hard, research
on new schemes based on other classes of problems is necessary as such work will provide greater
diversity and hence forces cryptanalysts to spend additional effort concentrating on completely new
types of problems. Moreover, we make sure that not all “crypto-eggs” are in one basket. In this
context, we want to point out that important results on the potential weaknesses of existing public
key schemes are emerging. In particular techniques for factorisation and solving discrete logarithms
improve continually. For example, polynomial time quantum algorithms can b e used to solve both
problems. Therefore, the existence of quantum computers in the range of a few thousands of qbits
This is a revised version of the original paper accepted for CHES 2008.
would be a real-world threat to systems based on factoring or the discrete logarithm problem. This
emphasises the importance of research into new algorithms for asymmetric cryptography.
One proposal for secure public key schemes is based on the problem of solving Multivariate
Quadratic equations (MQ-problem) over finite fields F, i.e. finding a solution vector x ∈ F
for a
given system of m polynomial equations in n variables each
= p
, . . . , x
= p
, . . . , x
= p
, . . . , x
) ,
for given y
, . . . , y
∈ F and unknown x
, . . . , x
∈ F is difficult, namely N P-complete. An overview
over this field can be found in [14].
Roughly speaking, most work on public-key hardware architectures tries to optimise either the
speed of a single instance of an algorithm (e.g., high-speed ECC or RSA implementations) or to
build the smallest possible realization of a scheme (e.g., lightweight ECC engine). A major goal
in high-performance applications is, however, in addition to pure time efficiency, an optimised
cost-performance ratio. In the case of hardware implementations, which are often the only solution
in such scenarios, costs (measured in chip area and power consumption) is roughly proportional
to the number of logic elements (gates, FPGA slices) needed. A major finding of this paper is
that MQ-schemes have the better time-area product than established public key schemes. This
holds, interestingly, also if compared to elliptic curve schemes, which have the reputation of being
particularly efficient.
The first public hardware implementation of a cryptosystem based on multivariate polynomials
we are aware of is [17], where enTTS is realized. A more recent result on the evaluation of hardware
performance for Rainbow can be found in [2].
1.1 Our Contribution
Our contribution is many-fold. First, a clear taxonomy of secure multivariate systems and existing
attacks is given. Second, we pres ent a systolic architecture implementing Gauss-Jordan elimination
over GF(2
) which is based on the work in [13]. The performance of this central operation is impor-
tant for the overall efficiency of multivariate based signature systems. Then, a number of concrete
hardware architectures are presented having a low time-area product. Here we address both rather
conservative schemes such as UOV as well as more aggressively designed proposals such as Rain-
bow or amended TTS (amTTS). For instance, an optimised implementation of amTTS is estimated
to have a TA-product over 50 times lower than some of the most efficient ECC implementations.
Moreover, we suggest a generic hardware architecture capable of computing signatures for the wide
class of multivariate polynomial systems based on small finite fields. This generic hardware design
allows us to achieve a time-area product for UOV which is s omewhat smaller than that for ECC,
being considerably smaller for the short-message variant of UOV.
2 Foundations of MQ-Systems
In this section, we introduce some properties and notations useful for the remainder of this article.
After briefly introducing MQ-systems, we explain our choice of signature schemes and give a brief
description of them.
signature x
x = (x
, . . . , x
private: S
private: P
private: T
message y
, . . . , p
Generation Verification
Figure 1: Graphical Representation of the MQ-trapdoor (S, P
, T )
2.1 Mathematical Background
Let F be a finite field with q := |F| elements and define Multivariate Quadratic (MQ) polynomials
of the form
, . . . , x
) :=
+ α
for 1 ≤ i ≤ m; 1 ≤ j ≤ k ≤ n and α
, β
, γ
∈ F (constant, linear, and quadratic terms). We
now define the polynomial-vector P := (p
, . . . , p
) which yields the public key of these Multi-
variate Quadratic systems. This public vector is used for signature verification. Moreover, the
private key (cf Fig.1) consists of the triple (S, P
, T ) where S ∈ Aff(F
), T ∈ Aff(F
) are affine
transformations and P
∈ MQ(F
, F
) is a polynomial-vector P
:= (p
, . . . , p
) with m com-
ponents; each component is in x
, . . . , x
. Throughout this paper, we will denote components of
this private vector P
by a prime
. The linear transformations S and T can be represented in
the form of invertible matrices M
∈ F
, M
∈ F
, and vectors v
∈ F
, v
∈ F
i.e. we
have S(x) := M
x + v
and T (x) := M
x + v
, respectively. In contrast to the public polynomial
vector P ∈ MQ(F
, F
), our design goal is that the private polynomial vector P
does allow an
efficient computation of x
, . . . , x
for given y
, . . . , y
. At least for secure MQ-schemes, this is
not the case if the public key P alone is given. The main difference between MQ-schemes lies in
their special construction of the central equations P
and consequently the trapdoor they embed
into a specific class of MQ-problems.
In this kind of schemes, the public key P is computed as function composition of the affine
transformations S : F
→ F
, T : F
→ F
and the central equations P
: F
→ F
, i.e. we
have P = T ◦ P
◦ S. To fix notation further, we note that we have P, P
∈ MQ(F
, F
), i.e.
both are functions from the vector space F
to the vector space F
. By construction, we have
∀x ∈ F
: P(x) = T (P
2.2 Signing
To sign for a given y ∈ F
, we observe that we have to invert the computation of y = P(x). Using
the trapdoor-information (S, P
, T ), cf Fig. 1, this is easy. First, we observe that transformation
T is a bijection. In particular, we can compute y
= M
y. The same is true for given x
∈ F
and S ∈ Aff(F
). Using the LU-decomposition of the matrices M
, M
, this computation takes
time O(n
) and O(m
), respectively. Hence, the difficulty lies in evaluating x
= P
). We will
discuss strategies for different central systems P
in Sect. 2.4.
2.3 Verification
In contrast to signing, the verification step is the same for all MQ-schemes and also rather cheap,
computationally speaking: given a pair x ∈ F
, y ∈ F
, we evaluate the p olynomials
, . . . , x
) :=
+ α
for 1 ≤ i ≤ m; 1 ≤ j ≤ k ≤ n and given α
, β
, γ
∈ F. Then, we verify that p
= y
for all i ∈ {1, . . . , m}. Obviously, all operations can be efficiently computed. The total number of
operations takes time O(mn
2.4 Description of the Selected Systems
Based on [14] and some newer results, we have selected the following suitable candidates for efficient
implementation of signature schemes: enhanced TTS, amended TTS, Unbalanced Oil and Vinegar
and Rainbow. Systems of the big-field classes HFE (Hidden Field Equations), MIA (Matsumoto
Imai Scheme A) and the mixed-field class ℓIC — ℓ-Invertible Cycle [8] were excluded as results
from their software implementation show that they cannot be implemented as efficiently as schemes
from the small-field classes, i.e. enTTS, amTTS, UOV and Rainbow. The prop osed schemes and
parameters are summarised in Table 1.
Table 1: Proposed Schemes and Parameters
q n m τ K Solver
Unbalanced Oil 256 30 10 0.003922 10 1 × K = 10
and Vinegar (UOV) 60 20 20 1 × K = 20
Rainbow 256 42 24 0.007828 12 2 × K = 12
enhanced TTS (v1) 256 28 20 0.000153 9 2 × K = 9
(v2) 0.007828 10 2 × K = 10
amended TTS 256 34 24 0.011718 4,10 1 × K = 4, 2 × K = 10
2.4.1 Unbalanced Oil and Vinegar (UOV).
, . . . , x
) :=
for i = 1 . . . v
Unbalanced Oil and Vinegar Schemes were introduced in [10, 11]. Here we have γ ∈ F, i.e. the
polynomials p are over the finite field F. In this context, the variables x
for 1 ≤ i ≤ n−m are called
the “vinegar” variables and x
for n − m < i ≤ n the “oil” variables. We also write o := m for the
number of oil variables and v := n−m = n−o for the number of vinegar variables. To invert UOV,
we need to assign random values to the vinegar variables x
, . . . , x
and obtain a linear system in
the oil variables x
, . . . , x
. All in all, we need to solve a m × m system and have hence K = m.
The probability that we do not obtain a solution for this system is τ
= 1 −
there are q
matrices over the finite field F with q := |F| elements and
− q
ones [14].
Taking the currently known attacks into account, we derive the following secure choice of
parameters for a security level of 2
• Small datagrams: m = 10, n = 30, τ ≈ 0.003922 and one K = 10 solver
• Hash values: m = 20, n = 60, τ ≈ 0.003922 and one K = 20 solver
The security has been evaluated using the formula O(q
) = O(q
). Note that
the first version (i.e. m = 10) can only be used with messages of less than 80 bits. However, such
datagrams occur frequently in applications with power or bandwidth restrictions, hence we have
noted this special possibility here.
2.4.2 Rainbow.
Rainbow is the name for a generalisation of UOV [7]. In particular, we do not have one layer, but
several layers. This way, we can reduce the number of variables and hence obtain a faster scheme
when dealing with hash values. The general form of the Rainbow central map is given below.
, . . . , x
) :=
for i = v
. . . v
, 1 ≤ l ≤ L
We have the coefficients γ ∈ F, the layers L ∈ N and the vinegar s plits v
< . . . < v
∈ N
with n = v
. To invert Rainbow, we follow the strategy for UOV — but now layer for layer,
i.e. we pick random values for x
, . . . , x
, solve the first layer with an (v
− v
) × (v
− v
for x
, . . . , x
, insert the values x
, . . . , x
into the second layer, solve second layer with an
− v
) × (v
− v
)-solver for x
, . . . , x
until the last layer L. All in all, we need to solve
sequentially L times (v
− v
) × (v
− v
) systems for l = 2 . . . L + 1. The probability that we
do not obtain a solution for this system is τ
= 1 −
using a similar
argument as in Sec. 2.4.1.
Taking the latest attack from [3] into account, we obtain the parameters L = 2, v
= 18, v
30, v
= 42 for a security level of 2
, i.e. a two layer scheme 18 initial vinegar variables and 12
equations in the first layer and 12 new vinegar variables and 12 equations in the second layer.
Hence, we need two K = 12 solvers and obtain τ ≈ 0.007828
2.4.3 amended TTS (amTTS).
The central polynomials P
∈ MQ(F
, F
) for m = 24, n = 34 in amTTS [6] are defined as given
:= x
+ α
σ (i)
11+(i+j mod 10)
, for i = 10 . . . 19;
:= x
+ α
σ (i)
+ γ
15+(i+j+4 mod 8)j+1
π (i,j)
, for i = 20 . . . 23;
:= x
+ γ
24+(i+j+6 mod 10)j+1
π (i,j)
, for i = 24 . . . 33.
We have α, γ ∈ F and σ, π permutations, i.e. all polynomials are over the finite field F. We see that
they are similar to the equations of Rainbow (Sec. 2.4.2) — but this time with sparse polynomials.
Unfortunately, there are no more conditions given on σ, π in [6] — we have hence picked one
suitable permutation for our implementation.
To invert amTTS, we follow the sames ideas as for Rainbow — except with the difference that
we have to invert twice a 10 × 10 system (i = 10 . . . 19 and 24 . . . 33) and once a 4 × 4 system, i.e.
we have K = 10 and K = 4. Due to the structure of the equations, the probability for not getting
a solution here is the same as for a 3-Layer Rainbow scheme with v
= 10, v
= 20, v
= 24, v
= 34
variables, i.e. τ
amT T S
= τ
(10, 20, 24, 34) ≈ 0.011718.
· · ·
· · ·
· · ·
· · ·
Figure 2: Signature Core Building Blo ck: Systolic Array LSE Solver (Structure)
2.4.4 enhanced TTS (enTTS).
The overall idea of enTTS is similar to amTTS, m = 20, n = 28. For a detailed description of enTTS
see [16, 15]. According to [6], enhanced TTS is broken, hence we do not advocate its use nor did
we give a detailed description in the main part of this article, However, it was implemented in [17],
so we have included it here to allow the reader a comparison between the previous implementation
and ours.
3 Building Blocks for MQ-Signature Cores
Considering Section 2 we see that in order to generate a signature using an MQ-signature scheme
we need the following common operations:
• computing affine transformations (i.e. vector addition and matrix-vector multiplication),
• (partially) evaluating multivariate polynomials over GF(2
• solving linear systems of equations (LSEs) over GF(2
In this section we describe the main computational building blocks for realizing these operations.
Using these generic building blocks we can compose a signature core for any of the presented
MQ-schemes (cf Section 4).
3.1 A Systolic Array LSE Solver for GF(2
In 1989, Hochet et al. [9] proposed a systolic architecture for Gaussian elimination over GF(p).
They considered an architecture of simple processors, used as systolic cells that are connected in a
triangular network. They distinguish two different types of cells, main array cells and the boundary
cells of the main diagonal.
1-Bit Reg
Figure 3: Pivot Cell of the Systolic Array LSE Solver
Wang and Lin followed this approach and proposed an architecture in 1993 [13] for comput-
ing inverses over GF(2
). They provided two methods to efficiently implement the Gauss-Jordan
algorithm over GF(2) in hardware. Their first approach was the classical systolic array approach
similar to the one of Ho chet et al.. It features a critical path that is independent of the s ize of the
array. A full solution of an m × m LSE is generated after 4m cycles and every m cycles thereafter.
The solution is computed in a serial fashion.
The other approach, which we call a systolic network, allows signals to propagate through the
whole architecture in a single clock cycle. This allows the initial latency to be reduced to 2m clock
cycles for the first result. Of course the critical path now depends of the size of the whole array,
slowing the design down for huge systems of equations. Systolic arrays can be derived from systolic
networks by putting delay elements (registers) into the s ignal paths between the cells.
We followed the approach presented in [13] to build an LSE solver architecture over GF(2
The biggest advantage of systolic architectures with regard to our application is the low amount
of cells compared to other architectures like SMITH [4]. For solving a m × m LSE, a systolic array
consisting of only m boundary cells and m(m + 1)/2 main cells is required.
An overview of the architecture is given in Figure 2. The boundary cells shown in Figure 3
mainly comprise one inverter that is needed for pivoting the corresponding line. Furthermore,
a single 1-bit register is needed to store whether a pivot was found. The main cells shown in
Figure 4 comprise of one GF(2
) register, a multiplier and an adder over GF(2
). Furthermore,
a few multiplexers are needed. If the row is not initialised yet (T
= 0), the entering data is
multiplied with the inverse of the pivot (E
) and stored in the cell. If the pivot was zero, the
element is simply stored and passed to the next row in the next clock cycle. If the row is initialised
= 1) the data element a
of the entering line is reduced with the stored data element and
passed to the following row. Hence, one can say that the k-th row of the array performs the k-th
iteration of the Gauss-Jordan algorithm.
The inverters of the boundary cells contribute most of the delay time t
of the systolic
network. Instead of introducing a full systolic array, it is already almost as helpful to simply add
delay elements only between the rows. This seems to be a good trade-off between delay time and
the number of registers used. This approach we call systolic lines.
As described earlier, the LSEs we generate are not always solvable. We can easily detect an
unsolvable LSE by checking the state of the boundary cells after 3m clock cycles (m clock cycles
for a systolic network, respectively). If one of them is not set, the system is not solvable and a
new LSE needs to be generated. However, as shown in Table 1, this happens very rarely. Hence,
the impact on the performance of the implementation is negligible. Table 2 shows implementation
k-Bit Reg
Figure 4: Main Cell of the Systolic Array LSE Solver
results of the different types of systolic arrays for different sizes of LSEs (over GF(2
)) on different
Table 2: Implementation results for different types of systolic arrays and different sizes of LSEs
over GF(2
) (t
in ns, F
in MHz)
Size on FPGA Speed Size on ASIC
Engine Slices LUTs FFs t
GE (estimated)
Systolic arrays on a Spartan-3 device (XC3S1500, 300 MHz)
Systolic Array (10x10) 2,533 4,477 1,305 12.5 80 38,407
Systolic Array (12x12) 3,502 6,160 1,868 12.65 79 53,254
Systolic Array (20x20) 8,811 15,127 5,101 11.983 83 133,957
Alternative systolic arrays on a Spartan-3
Systolic Network (10x10) 2,251 4,379 461 118.473 8.4 30,272
Systolic Lines (12x12) 3,205 6,171 1,279 13.153 75 42,013
Systolic arrays on a Virtex-V device (XC5VLX50-3, 550 MHz)
Systolic Array (10x10) 1314 3498 1305 4.808 207 36,136
Systolic Lines (12x12) 1,534 5,175 1,272 9.512 105 47,853
Systolic Array (20x20) 4552 12292 5110 4.783 209 129,344
3.2 Matrix-Vector Multiplier and Polynomial Evaluator
For performing matrix-vector multiplication, we use the building block depicted in Figure 5. In
the following we call this blo ck a t-MVM. As you can see a t-MVM consists of t multipliers, a tree
of adders of depth about log
(t) to compute the sum of all products a
· b
, and an extra adder
to recursively add up previously computed intermediate values that are stored in a register. Using
the RST-signal we can initially set the register content to zero.
To compute the matrix-vector product
A · b =
. . . a
. . . a
using a t-MVM, where t is chosen in a way that it divides
u, we proceed row by row as follows:
Note that in the case that t does not divide u we can nevertheless use a t-MVM to compute the matrix-vector
product by setting superfluous input signals to zero.
k-Bit Reg
Figure 5: Signature Core Building Block: Combined Matrix-Vector-Multiplier and Polynomial-
We set the register content to zero by using RST. Then we feed the first t elements of the first row
of A into the t-MVM, i.e. we set a
= a
, . . . , a
= a
, as well as the first t elements of the vector
b. After the register content is set to
, we feed the next t elements of the row and the
next t elements of the vector into the t-MVM. This leads to a register content corresponding to
. We go on in this way until the last t elements of the row and the vector are processed
and the register content equals
. Thus, at this point the data signal c corresponds to
the first component of the matrix-vector product. Proceeding in a analogous manner yields the
remaining components of the desired vector. Note that the
parts of the vector b are re-used in a
periodic manner as input to the t-MVM. In Section 3.4 we describe a building block, called word
rotator, providing these parts in the required order to the t-MVM without re-loading them each
time and hence avoid a waste of resources.
Therefore, using a t-MVM (and an additional vector adder) it is clear how to implement the
affine transformations S : F
→ F
and T : F
→ F
which are important ingredients of an
MQ-scheme. Note that the parameter t has a significant influence on the performance of an
implementation of such a scheme and is chosen differently for our implementations (as can b e seen
in Section 4).
Besides realizing the required affine transformations, a t-MVM can be re-used to implement
(partial) polynomial evaluation. It is quite obvious that evaluating the polynomials p
ing to the central map P
of a MQ-scheme, cf Section 2) with the vinegar variables involves
matrix-vector multiplications as the main operations. For instance, consider a fixed polynomial
, . . . , x
) =
from the central map of UOV that we evaluate with ran-
dom values b
, . . . , b
∈ F for the vinegar variables x
, . . . , x
. Here we like to compute the
coefficients β
, β
, . . . , β
of the linear polynomial
, . . . , b
, x
, . . . , x
) = β
We immediately obtain the coefficients of the non-constant part of this linear polynomial, i.e.
, . . . , β
, by computing the following matrix-vector product:
. . . γ
. . . γ
Also the main step for computing β
can be written as a matrix-vector product:
k-Bit Reg
k-Bit Reg
k-Bit Reg
k-Bit Reg
· · ·
Figure 6: Signature Core Building Blo ck: Equation Register
0 0 . . . 0
0 . . . 0
. . . γ
. . . γ
Of course, we can exploit the fact that the above matrix is a lower triangular matrix and we
actually do not have to perform a full matrix-vector multiplication. This must simply be taken into
account when implementing the control logic of the signature core. In order to obtain β
. . . α
we have to perform the following additional computation:
= α
+ . . . + α
This final step is performed by another unit called equation register which is presented in the next
3.3 Equation Register
The Equation Register building block is shown in Figure 6. A w-ER es sentially consists of w + 1
register blocks each storing k bits as well as one adder and one multiplier. It is used to temporarily
store parts of an linear equation until this equation has been completely generated and can be
transferred to the systolic array solver.
For instance, in the case of UOV we consider linear equations of the form
, . . . , b
, x
, . . . , x
) = y
− y
= 0
where we used the notation from Section 3.2. To compute and store the constant part
of this equation the left-hand part of an m-ER is used (see Figure 6): The respective register is
initially set to y
. Then the values α
are computed one after another using a t-MVM building
block and fed into the multiplier of the ER. The corresponding values b
are provided by a t-WR
building block which is presented in the next section. Using the adder, y
and the products can be
added up iteratively. The coefficients β
of the linear equation are also computed consecutively
by the t-MVM and fed into the shift-register that is shown on the right-hand side of Figure 6.
· · ·
· · ·
· · ·
Figure 7: Signature Core Building Blo ck: Word Rotator
3.4 Word Rotator
A word cyclic shift register will in the following be referred to as word rotator (WR). A (t, r)-WR,
depicted in Figure 7, consists of r register blocks storing the
parts of the vector b involved in the
matrix vector products considered in Section 3.2. Each of these r register blocks stores t elements
from GF(2
), hence each register block consists of t k-bit registers. The main task of a (t, r)-WR
is to provide the correct parts of the vector b to the t-M VM at all times. The r register blocks can
be serially loaded using the input bus x. After loading, the r register blocks are rotated at each
clock cycle. The cycle length of the rotation can be modified using the multiplexers by providing
appropriate control signals. This is esp ecially helpful for the partial polynomial evaluation where
due to the triangularity of the matrix in Equation (2), numerous operations can be saved. Here, the
cycle length is
, where j is the index of the processed row. The possibility to adjust the cycle
length is also necessary in the case r >
frequently appearing if we use the same (t, r)-WR, i.e.,
fixed parameters t and r, to implement the affine transformation T , the polynomial evaluations,
and the affine transformation S. Additionally, the WR provides b
to the ER building block which
is needed by the ER at the end of each rotation cycle. Since this b
value always occurs in the last
register block of a cycle, the selector component (right-hand side of Figure 7) can simply load it
and provide it to the ER.
4 Performance Estimations of Small-Field MQ-Schemes in
We implemented the most crucial building blocks of the architecture as described in Section 3
(systolic structures, word rotators, matrix-vector multipliers of different sizes). In this section, the
estimations of the hardware performance for the whole architecture are performed based on those
implementation results. The power of the approach and the efficiency of MQ-schemes in hardware
is demonstrated at the example of UOV, Rainbow, enTTS and amTTS as specified in Section 2.
Side-Note: The volume of data that needs to be imported to the hardware engine for MQ-
schemes may seem too high to be realistic in some applications. However, the contents of the
matrices and the polynomial coefficients (i.e. the private key) does not necessarily have to be
imported from the outside world or from a large on-board memory. Instead, they can b e generated
online in the engine using a cryptographically strong pseudo-random number generator, requiring
only a small, cryptographically strong secret, i.e. some random bits.
4.1 UOV
We treat two parameter sets for UOV as shown in Table 3: n = 60, n = 20 (long-message UOV) as
well as n = 30, m = 10 (short-message UOV). In UOV signature generation, there are three basic
operations: linearising polynomials, solving the resulting equation system, and an affine transform
to obtain the signature. The most time-consuming operation of UOV is the partial evaluation
of the polynomials p
, since their coefficients are nearly random. However, as already mentioned
in the previous section, for some polynomials approximately one half of the coefficients for the
polynomials are zero. This somewhat simplifies the task of linearization.
For the linearization of polynomials in the long-message UOV, 40 random bytes are generated
to invert the central mapping first. To do this, we use a 20-MVM, a (20,3)-WR, and a 20-ER.
For each polynomial one needs about 100 clock cycles (40 clocks to calculate the linear terms and
another 60 ones to compute the constants, see (1) and (2)) and obtains a linear equation with 20
variables. As there are 20 polynomials, this yields about 2000 clock cycles to perform this step.
After this, the 20 × 20 linear system over GF(2
) is solved using a 20 × 20 systolic array. The
signature is then the result of this operation which is returned after about 4×20=80 clock cycles.
Then, the 20-byte solution is concatenated with the randomly generated 40 bytes and the result is
passed through the affine transformation, whose major part is a matrix-vector multiplication with
a 60×60-byte matrix. To perform this operations, we re-use the 20-MVM and a (20,3)-WR. This
requires about 180 cycles of 20-MVM and 20 bytes of the matrix entries to be input in each cycle.
For the short-message UOV, one has a very similar structure. More precisely, one needs a 10-
MVM, a (10,3)-WR, a 10-ER and a 10×10 systolic array. The design requires approximately 500
cycles for the partial evaluation of the polynomials, about 40 cycles to solve the resulting 10×10
LSE over GF(2
) as well as another 90 cycles for the final affine map.
Note that the critical path of the Gaussian elimination engine is much longer than that for
the remaining building blocks. So this block represents the performance bottleneck in terms of
frequency and hardware complexity. For this reason we decided to clock different components of
the design with different frequencies. For the XC5VLX50-3 device the Gaussian elimination engine
is clocked with 200 MHz and the rest with 400 MHz. Alternatively, for the XC3S1500 device the
Gaussian elimination component is clocked with about 80 MHz, the remaining engines with 160
MHz. See Table 3 for our estimations.
4.2 Rainbow
In the version of Rainbow we consider, the message length is 24 byte. That is, a 24-byte matrix-
vector multiplication has to be performed first. One can take a 6-MVM and a (6,7)-WR which
require about 96 clock cycles to perform the computation. Then the first 18 variables of x
randomly fixed and 12 first polynomials are partially evaluated. This requires about 864 clock
cycles. The results are stored in a 12-ER. After this, the 12×12 system of linear equations is
solved. This requires a 12×12 systolic array over GF(2
) which outputs the solution after 48 clock
cycles. Then the last 12 polynomials are linearised using the same matrix-vector multiplier and
word rotator based on the 18 random values previously chosen and the 12-byte solution. This needs
about 1800 clock cycles. This is followed by another run of the 12×12 systolic array with the same
execution time of about 48 clock cycles. At the end, roughly 294 more cycles are spent performing
the final affine transform on the 42-byte vector. See Table 3 for some concrete performance figures
in this case.
4.3 enTTS and amTTS
Like in Rainbow, for enTTS two vector-matrix multiplications are needed at the beginning and at
the end of the operation with 20- and 28-byte vectors each. We take a 10-MVM and a (10,3)-WR for
this. The operations require 40 and 84 clock cycles, respectively. One 9-ER is required. Two 10×10
linear systems over GF(2
) need to be solved, requiring about 40 clock cycles each. The operation
of calculating the linearization of the polynomials can be significantly optimised compared to the
generic UOV or Rainbow (in terms of time) which can drastically reduce the time-area product.
This behaviour is due to the special selection of polynomials, where only a small proportion of
coefficients is non-zero.
After choosing 7 variables randomly, 10 linear equations have to be generated. For each of these
equations, one has to perform only a few multiplications in GF(2
) which can be done in parallel.
This requires about 20 clock cycles. After this, another variable is fixed and a further set of 10
polynomials is partially evaluated. This requires about 20 further cycles.
In amTTS, which is quite s imilar to enTTS, two affine maps with 24- and 34-byte vectors are
performed with a 12-MVM and a (12,3)-WR yielding 48 and 102 clock cycles, respectively. Two
10×10 and one 4×4 linear systems have to be solved requiring for a 10×10 systolic array (twice 40
and once 16 clock cycles). Moreover, a 10-ER is needed. The three steps of the partial evaluation
of polynomials requires roughly 40 clock cycles in this case. See Table 3 for our estimations on
enTTS and amTTS.
Table 3: Comparison of hardware implementations for ECC and our performance estimations for
MQ-schemes based on the implementations of the major building blocks (F=frequency,T=Time,
L=luts, S=slices, FF=flip-flops, A=area, XC3=XC3S1500, XC5=XC5VLX50-3)
Implementation F, MHz T, µs S/L/FF A,kGE S·T [S·ms]
ECC-163, [1], XC2V200 100 41 -/8,300/1100 - 85.1
ECC-163, CMOS 167 21 - 36 -
ECC-163, [12], XCV200E-7 48 68.9 -/25,763/7,467 - 447.9
UOV(60,20), XC3 80/160 14.625 9821 / 16694 / 5665 149 143.6
UOV(60,20), XC5 200/400 5.85 5334 / 13437 / 5774 143 31.2
UOV(30,10), XC3 80/160 4.188 3060 / 5304 / 1649 46 12.8
UOV(30,10), XC5 200/400 1.675 1585 / 4098 / 1649 43 2.7
Rainbow(42,24), XC3 80/160 7.781 4123 / 7173 / 2332 63 32.1
Rainbow(42,24), XC5 200/400 5.595 2000 / 5626 / 2330 59 11.2
enTTS(28,20), [17], CMOS 80
200 - 22 -
enTTS(28,20), XC3 80/160 2.025 3060 / 5304 / 1649 46 6.2
enTTS(28,20), XC5 200/400 0.81 1585 / 4098 / 1649 43 1.2
amTTS(34,24), XC3 80/160 2.438 3139 / 5434 / 1697 48 7.7
amTTS(34,24), XC5 200/400 0.975 1659 / 4200 / 1697 42 1.6
For comparison purposes we assume that the design can be clocked with up to 80 MHz.
5 Comparison and Conclusions
Our implementation results (as well as the estimations for the optimisations in case of enTTS and
amTTS) are compared to the scalar multiplication in the group of points of elliptic curves with
field bitlengths in the rage of 160 bit (corres ponding to the security level of 2
) over GF(2
), see
Table 3. A good survey on hardware implementations for ECC can be found in [5].
Even the most conservative design, i.e. long-message UOV, can outperform some of the most ef-
ficient ECC implementations in terms of TA-product on some hardware platforms. More hardware-
friendly designs such as the short-message UOV or Rainbow provide a considerable advantage over
ECC. The more aggressively designed enTTS and amTTS allow for extremely efficient implementa-
tions having a more than 70 or 50 times lower TA-product, respectively. Though the metric we use
is not optimal, the results indicate that MQ-schemes perform better than elliptic curves in hard-
ware with respect to the TA-product and are hence an interesting option in cost- or size-sensitive
Acknowledgements. The authors would like to thank our college Christof Paar for fruitful dis-
cussions and helpful remarks as well as Sundar Balasubramanian, Harold Carter (University of
Cincinnati, USA) and Jintai Ding (University of Cincinnati, USA and Technical University of
Darmstadt, Germany) for exchanging some ideas while working on another paper about MQ-
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Who's That Mathematician? Paul R. Halmos Collection - Page 16
For more information about Paul R. Halmos (1916-2006) and about the Paul R. Halmos Photograph Collection, please see the introduction to this article on page 1. A new page featuring six photographs
will be posted at the start of each week during 2012.
Paul Halmos photographed Revaz Gamkrelidze (at right) in Ann Arbor, Michigan, on May 31, 1967. Halmos was a faculty member at the University of Michigan in Ann Arbor from 1961 to 1968. Born in
Kutaisi, Georgia, Gamkrelidze studied for one year (1945-46) at Tbilisi State University in Georgia before moving to Moscow State University, where he studied from 1946 to 1953. In 1953, he became a
researcher at the Steklov Institute of Mathematics of the Russian Academy of Sciences, becoming Senior Researcher in 1955, Doctor of Physico-Mathematical Sciences in 1961, Professor in 1966, and Head
of the Department of Differential Equations and Theory of Control in 1988. Gamkrelidze was founding editor of the Encyclopaedia of Mathematical Sciences, published by Springer-Verlag since 1988.
(Sources: Georgian National Academy of Sciences: Academician Revaz Gamkrelidze, Springer-Verlag)
Lars Gårding was photographed by Halmos in England in 1976. Gårding earned his Ph.D. at Lund University in Sweden under Marcel Riesz. After visiting Princeton University in New Jersey in 1946-47, he
has spent his career at Lund University (with eight years at the Institute for Advanced Study in Princeton), where he has carried out research in partial differential operators and is now Professor
Emeritus. In 1952, when Marcel Riesz retired from Lund and Gårding was a new faculty member there, Gårding took over supervision of the Ph.D. thesis of Lars Hörmander (pictured on page 24 of this
collection), who would win the Fields Medal in 1962 for his work in linear partial differential operators. Gårding also is interested in the history of mathematics, having written the books Some
Points of Analysis and Their History (AMS, 1997) and Mathematics and Mathematicians: Mathematics in Sweden Before 1950 (AMS, 1998). (Sources: Mathematics Genealogy Project; Institute for Advanced
Study; AMS Bookstore; Perspectives in PDE, Harmonic Analysis & Applications; Mitrea & Mitrea, eds.: Garding article; Fields Medallists' Lectures, 2nd ed., Atiyah & Iagolnitzer, eds.: Hörmander
Halmos photographed Martin Gardner (1914-2010) in New York City on Oct. 26, 1974. The magazine Scientific American paid tribute to Gardner, the "Mathematical Gamester," as follows: "For 25 years, he
wrote Scientific American's Mathematical Games column, educating and entertaining minds and launching the careers of generations of mathematicians." The magazine also credits Gardner with
"single-handedly populariz[ing] recreational mathematics in the U.S." Gardner wrote his first article for Scientific American in 1956 and was immediately invited to write the magazine's "Mathematics
Games" column, which he did from 1957 to 1981. The 15 books containing all of his columns are among the over 100 books and pamphlets he published during his career. (Sources: Scientific American,
MacTutor Archive)
Halmos photographed his former University of Michigan colleague, the complex analyst and geometer Frederick Gehring, in 1974. Born and raised in Ann Arbor, Michigan, Gehring earned his Ph.D. at
Cambridge in 1952, taught at Harvard for three years, and, since 1955, has been a mathematics professor at the University of Michigan in Ann Arbor, where he has specialized in quasiconformal mappings
and advised at least 29 Ph.D. students. He was appointed T. H. Hildebrandt Distinguished University Professor in 1987 and, in 1995, he added Emeritus to his title. (T. H. Hildebrandt is photographed
on page 23 of this collection.) In 1996, Michigan established the Gehring Collegiate Professorship in his honor, and appointed complex analyst John Erik Fornaess as its first holder. During 2001-02,
UM held the Gehring Special Year in Complex Analysis. Gehring and Halmos were colleagues at UM from 1961 to 1968, when Halmos was a faculty member there. Another photograph of Gehring appears on page
29 of this collection. (Sources: Mathematics Genealogy Project, UM Memoir, UM: Fornaess, ContinuUM)
Paul Halmos and Abraham Gelbart (1911-1994) at the AMS-MAA Joint Summer Meetings in Amherst, Mass., on August 25, 1964. Abe Gelbart earned his Ph.D. in 1940 from the Massachusetts Institute of
Technology (M.I.T.) under advisor Norbert Wiener. He taught at Syracuse University in New York from 1943 to 1958. Paul Halmos was a faculty member at Syracuse from 1942 to 1946, and it is likely that
he and Gelbart first met there. In 1958, Gelbart became Director of Mathematics at Yeshiva University in New York City, where he pursued his interests in complex analysis and partial differential
equations. In 1982, after his retirement from Yeshiva, he became a trustee of Bar-Ilan University in Ramat-Gan, Israel. Bar-Ilan University now has an Abraham Gelbart Research Institute in
Mathematical Sciences, founded in 1987 to focus on analysis. (BIU has one other mathematical research institute, the Emmy Noether Mathematics Institute in Algebra, Geometry, Function Theory and
Summability.) Gelbart himself received an honorary doctorate from BIU in 1985. (Sources: AMS Notices 42:1 (Jan. 1995), Bar-Ilan University, BIU: Gelbart Institute)
Halmos photographed Israil Gelfand (1913-2009) in Moscow, Russia, in May of 1965. Halmos visited the USSR for one month during April and May of 1965 as part of an exchange program between the
American and Soviet scientific academies that sent 20 scientists from each country to the other. Born in Odessa, Ukraine, Gelfand earned his Ph.D. in functional analysis in 1935 under A. N.
Kolmogorov at Moscow State University. In 1941, after six years at the USSR Academy of Sciences, he moved to Moscow State University, where he worked on representation theory of non-compact groups,
differential equations, computational mathematics, integral geometry, and, with his colleague Sergei Fomin, mathematical biology. (Fomin and Gelfand are pictured together, also in 1965, on page 15 of
this collection.) Gelfand moved to Rutgers University in New Jersey in 1990. (Sources: Archives of American Mathematics, MacTutor Archive)
For an introduction to this article and to the Paul R. Halmos Photograph Collection, please see page 1. Watch for a new page featuring six new photographs each week during 2012.
Regarding sources for this page: Information for which a source is not given either appeared on the reverse side of the photograph or was obtained from various sources during 2011-12 by archivist
Carol Mead of the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin. | {"url":"https://old.maa.org/press/periodicals/convergence/whos-that-mathematician-paul-r-halmos-collection-page-16","timestamp":"2024-11-11T11:29:33Z","content_type":"application/xhtml+xml","content_length":"127075","record_id":"<urn:uuid:558eaa17-0973-4dbb-ba4a-f756aab1ebdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00056.warc.gz"} |
A method based on factorization for solving boundary value problems in diffraction on bodies of finite dimensions
A method is proposed for reducing integral Wiener-Hopf equations to an infinite system of linear algebraic equations, which can be solved by the reduction method with exponential convergence of the
approximations. The method is applied in some diffraction problems for ideally conducting bodies of finite extent. These problems are: (1) excitation of a periodic structure by the field of a
harmonic, electrically polarized plane wave; (2) a periodic structure consisting of a half-plane with a channel along whose axis a uniformly charged filament with charge density is moving at constant
velocity; (3) the field excited by a moving source in a plane diaphragmed waveguide; and (4) scattering at isolated waveguide inhomogeneities.
Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki
Pub Date:
June 1975
□ Boundary Value Problems;
□ Electromagnetic Scattering;
□ Wave Diffraction;
□ Wiener Hopf Equations;
□ Electrodynamics;
□ Factor Analysis;
□ Harmonic Analysis;
□ Integral Equations;
□ Plane Waves;
□ Polarized Electromagnetic Radiation;
□ Waveguides;
□ Communications and Radar | {"url":"https://ui.adsabs.harvard.edu/abs/1975ZVMMF..15..672V/abstract","timestamp":"2024-11-07T16:36:02Z","content_type":"text/html","content_length":"35944","record_id":"<urn:uuid:1555dab9-78b1-4625-a31a-b1359fa4afbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00240.warc.gz"} |
Math Circles
Category: Math Circles
• I proctored the AIME II contest this week, and caught a cheater. Here are some details and thoughts about the occasion. At about 4pm the day before the contest, I started getting emails and phone
calls from parents, from tutors, some students, and even my math colleagues at Bard who had been contacted as well, in…
• A few days ago I came across a proof of the Fundamental Theorem of Arithmetic (aka Unique Factorization) in Courant and Robbin’s What is Mathematics that I hadn’t seen it before. I liked it
enough to learn it. Then another surprise – I saw it again yesterday in Primes and Programming by Peter Giblin, a book…
• This year’s G4G, or Gathering for Gardner, Celebration of Mind II, falls on Friday, October 21. This is the second G4G since Martin Gardner passed away on May 22, 2010, and the G4G is intended to
celebrate his life and work. You can find a nearby celebration here: http://www.g4g-com.org/
• This year’s MathFest will be in Lexington, KY in early August. I’m going to present a talk about Math Circles in the Hudson Valley at the Fostering, Supporting and Propagating Math Circles for
Students and Teachers session. I’ll also be presenting the teachers’ math circle session with my talk on the game of Nim. Both these…
• Join us at Bard this summer for a week-long residential program focused on the investigation of inequalities and optimization. Enjoy an environment of creative and insightful mathematical problem
solving for middle school and high school math teachers who wish to deepen their mathematical understanding. No prior experience with inequalities required, just an interest in doing…
• brings together math circle organizers with people who plan to start math circles
• Mathematical Circles are a form of education enrichment and outreach that bring mathematicians and mathematical scientists into direct contact with pre-college students.
• We develop a curiosity and interest in mathematics
• Kingston, Red Hook and Tivoli | {"url":"https://japheth.org/category/math-circles/","timestamp":"2024-11-08T08:09:44Z","content_type":"text/html","content_length":"55058","record_id":"<urn:uuid:3be79414-1449-471f-a4c3-625d8aa95a0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00323.warc.gz"} |
Space Complexity - (Elliptic Curves) - Vocab, Definition, Explanations | Fiveable
Space Complexity
from class:
Elliptic Curves
Space complexity refers to the amount of memory required by an algorithm to run as a function of the size of the input data. This concept is crucial when analyzing algorithms, especially in contexts
where memory usage is limited or when processing large datasets. Understanding space complexity helps in optimizing algorithms and ensures that they do not exhaust available memory resources during
congrats on reading the definition of Space Complexity. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Space complexity is often expressed as a combination of fixed and variable parts, where fixed part includes constant space consumed by constants, variables, and fixed-size data structures, while
variable part includes space for dynamic data structures based on input size.
2. In Montgomery's elliptic curve factorization method, efficient use of memory is essential since it involves working with potentially large integers and complex mathematical operations.
3. Algorithms can have different space complexities even if they achieve the same outcome; thus, optimizing for lower space complexity can lead to better performance in memory-constrained
4. Recursive algorithms often have higher space complexity due to the additional memory needed for stack frames, which can lead to issues like stack overflow if not managed properly.
5. When implementing cryptographic algorithms that use elliptic curves, understanding space complexity helps prevent running out of memory and ensures smooth execution, especially when handling
large numbers.
Review Questions
• How does understanding space complexity impact the design of algorithms in contexts involving large datasets?
□ Understanding space complexity is vital when designing algorithms for large datasets because it helps developers gauge how much memory an algorithm will require. If an algorithm has high
space complexity, it may become infeasible to execute with limited memory resources. By analyzing space complexity, developers can optimize their algorithms to ensure they remain efficient
and functional even with large inputs.
• Compare and contrast space complexity and time complexity in the context of Montgomery's elliptic curve factorization method.
□ Both space complexity and time complexity are essential when analyzing Montgomery's elliptic curve factorization method. Space complexity concerns itself with the amount of memory required
for storing large integers and managing elliptic curve operations, while time complexity focuses on the number of operations needed to complete the factorization process. A well-optimized
implementation will minimize both complexities, ensuring that it runs efficiently without exceeding memory limits or taking too long.
• Evaluate the role of space complexity in ensuring the efficiency and effectiveness of cryptographic algorithms using elliptic curves.
□ Evaluating space complexity is crucial for ensuring that cryptographic algorithms utilizing elliptic curves operate effectively, especially under various hardware constraints. High space
complexity can lead to inefficiencies or failures during execution, particularly in systems with limited resources. By thoroughly analyzing and optimizing space requirements, developers can
create more robust cryptographic systems that perform well across different environments and protect sensitive information without risking resource exhaustion.
© 2024 Fiveable Inc. All rights reserved.
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Teachers and parents often complain about the negative affect of students on
mathematics learning at schools, but there has been no official report on this matter. This
study was intended to investigate the status of the affective aspect of the result of the
schools mathematics education. The affects measured were attitude, interest, motivation,
anxiety, self-concept, extrinsic appreciation, intrinsic appreciation, operational appreciation,
belief about mathematics, belief about self, belief about mathematics teaching,
and belief about social context. The subjects were the first year students of the Faculty of
Mathematics and Sciences of the State University of Yogyakarta (UNY) comprising
students of eight departments i.e. Department of: Biology, Biology Education, Chemistry,
Chemistry Education, Mathematics, Mathematics Education, Physics, and Physics
Education. The instrument was adapted from the affective test items developed byWilson
used in the National Longitudinal Study on Mathematics Achievement by the School
Mathematics Study Group in the USA, and supplemented by test items on beliefs
developed based on the McLeod’s classification. The data indicated that the affects were
of either low level or neutral level (neither favorable nor unfavorable for mathematics
lesson) except for the belief about mathematics, about mathematics teaching, and about
social context of mathematics.
aspek afektif; aspek kognitif; pembelajaran matematika; hasil pembelajaran
Bessant, K.C. (1995). “Factors Associated With Types of Mathematics Anxiety in College Students”, dalam Journal for Research in Mathematics Education, 26, pp. 327-345.
Gaslin, W.L. (1975). “A Comparison of Achievement and Attitudes of Students Using Conventional or Calculator-Based Algorithms for Operations on Positive Rational Numbers in Ninth-grade General
Mathematics”, dalam Journal for Research in Mathematics Education, 6, pp. 95-108.
Good, T.L., Grows, D.A., & Mason, D.W.A. (1990). “Teacher’s Beliefs About Small Group Instruction in Elementary School Mathematics”, dalam Journal for Research in Mathematics Education, 21, pp. 2-15.
Hembree, R. (1990). “The Nature, Effects, and Relief of Mathematics Anxiety”, dalam Journal for Research in Mathematics Education, 21, pp. 33-46.
Hogan, T.P. (1977). “Students’ Interest in Particular Mathematics Topics”, dalam Journal for Research in Mathematics Education, 8, pp. l 15-122.
Krathwohl, D.R., Bloom, B.S., & Masia, B.B. (1981). Taxonomy of educational objectives: Book 2, Affective domain, NewYork: Longman.
Kulm, G. (1980). “Research on Mathematics Attitude”. dalam R.J. Shumway (Ed.), Research in Mathematics Education, Reston, VA: National Council of Teachers of Mathematics.
Lo, J-J, Wheatley, G.H., &L Smith, A.C. (1994). “The Participation, Beliefs, and Development of Arithmetic Meaning of a Third-grade Student in Mathematics Class Discussion”, dalam Journal for
Research in Mathematics Education, 25, 30-49.
McLeod, D.B. (1992). “Research on Affect in Mathematics Education: A
Reconceptualization”. Dalam D.A. Grows (Ed.), dalam Handbook of Research on Mathematics Teaching and Learning, NewYork: Macmillan.
_________. (1994). “Research on Affect and Mathematics Learning in the JRME: 1970 to the Present”, Journal for Research in Mathematics Education, 25, 637-647.
Middleton, J.A. (1995). “AStudy of Intrinsic Motivation in the Mathematics Classroom:A Personal Constructs Approach”, dalam Journal for Research in Mathematics Education, 26, 254-279.
Minato, S. & Kamada, T. (1996). “Results of Research Studies on Causal Predominance Between Achievement and Attitude in Junior High School Mathematics of Japan”, dalam Journal for Research in
Mathematics Education, 27, 96-99.
Oppenheim, A.N. (1984). Questionnaire Design and Attitude Measurement, London: Heinemann.
Payne, D.A. (1974). The Assessment of Learning: Cognitive and Affective, Lexington, MA:D.C. Heath and Company.
Reynolds, A.J. &Walberg, H.J. (1992). “A Process Model of Mathematics Achievement and Attitude”, dalam Journal for Research in Mathematics Education, 23, 306-328.
Reys, R.E., Reys,B.J., Nohda,N. & Emori, H. (1995). “Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8", dalam Journal for Research in Mathematics
Education, 26, 304-326.
Schefele, U. & Csikszentmihalyi, M. (1995). “Motivation and Ability as Factors in Mathematics Experience and Achievement”, dalam Journal for Research in Mathematics Education, 26, 163-181.
Schoenfeld, A.H. (1989). “Exploration of Students’ Mathematical Belief and Behavior”, dalam Journal for Research in Mathematics Education, 20, 338-355.
Wilson, J.W. (1971). “Evaluation of Learning in Secondary School Mathematics”, dalam B.S.Bloom, J.T.Hasting, &: G.F.Madaus (Eds.), dalam Handbook on Formative and Summative Evaluation of Student
Learning, NewYork: McGraw-Hill.
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Similar triangle problem - Geometry
Geometry Tickbox Quiz #11
Why did Greg took the lenght whole lenght of AE as 4n. I understand 4n comes from the ratio of BC/ De which is 1/4. But he could have done as below too
tri ABC/ tri ADE = BC/ DE = AC/AE which can be further written as
5/20 = AC/(AC+CE) [since BC is 5 and DE is 20 as in the figure above]
1/4 = AC/AC +AC/ CE
1/4 = 1+ AC/ CE
Hence, 1/4 - 1 = AC/CE
Due to that -1 AC/ CE is less than 1/4 and hence option B while that is not the answer.
What is wrong in my approch and how is that different from Greg’s approach?
@ganesh can you kindly help share your views on the above of where I’m going wrong.
\frac{1}{a + b} \neq \frac{1}{a} + \frac{1}{b}
Also: don’t ping people unnecessarily.
@Leaderboard Thank you for clarifying. Also, I didnot ping anyone individually via DM on this problem till now, just tagged Ganesh. Not sure, if pinging is equivalent to tagging.
It is here. | {"url":"https://forums.gregmat.com/t/similar-triangle-problem-geometry/54651","timestamp":"2024-11-06T14:45:16Z","content_type":"text/html","content_length":"22906","record_id":"<urn:uuid:0c0d8a15-b140-4bcb-8a7c-509af005434e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00479.warc.gz"} |
Introducing a scatter plot
Scatter plots are used primarily to conduct a quick analysis of the relationships among different variables in our data. It is simply plotting points on the x-axis and y-axis. Scatter plots help us
detect whether two variables have a positive, negative, or no relationship. In this recipe, we will study the basics of plotting in R using scatter plots. The following screenshot is an example of a
scatter plot:
For implementing the basic scatter plot in R, we would use Carseats data available with the ISLR package in R.
We will also start this recipe by installing necessary packages using the install.packages() function and loading the same in R using the library() function:
Next, we need to load the data in R. Almost all R packages come with preloaded data and hence we can load the data only after we load the library in R. We can attach the data in R using the attach()
function. We can view the entire list... | {"url":"https://subscription.packtpub.com/book/data/9781783989508/2/ch02lvl1sec26/a-simple-line-plot","timestamp":"2024-11-04T17:26:53Z","content_type":"text/html","content_length":"169340","record_id":"<urn:uuid:1e93bf0f-b9d5-4375-a174-d5b5f9e2ed25>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00069.warc.gz"} |
Budapest University of Technology and Economics
We kindly invite you to the Miklós Farkas Seminar.
12 March (Thursday) 10.15, BME, H306
János Tóth (BME, Department of Analysis)
Positivity of solutions of (polynomial) differential equations
We present the deterministic model of chemical reactions and show why this is an important class of equations both from the point of view of the qualitative theory and of applications.
Next, we review results on the positivity of the solutions of the model, starting from the continuously rediscovered results by Volpert (1972). The components of the solutions are either strictly
positive or zero for all positive times of their domain. Which is which---this can be decided using the concept of Volpert indexes.
As an application, we show one of the algorithms to find minimal sets of species that ensure the positivity of all the species concentrations during the domain of solutions.
The organizers
(István Faragó, János Karátson, Róbert Horváth, Miklós Mincsovics, Gabriella Svantnerné Sebestyén)
Visit the homepage of the seminar: http://math.bme.hu/AlkAnalSzemi | {"url":"https://det.math.bme.hu/node/2566?language=en","timestamp":"2024-11-10T21:49:46Z","content_type":"text/html","content_length":"15696","record_id":"<urn:uuid:574f3da2-4b91-49a5-b4e9-42cefcf7c9f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00196.warc.gz"} |
10th class maths guess paper 2024 for Science Group10th class maths guess paper 2024 for Science Group
There are several guess papers of mathematics for 10th class on the net. The real and quite complete guess paper of maths for 10th class in pdf is given below. 10th class math guess paper 2024 Urdu
medium and English medium is given here.
The guess paper for maths includes important short questions and long questions of mathematics class 10th 2024 exams. Chapterwise important questions for 10th class maths for science students is
given here.
10th class mathematics guess paper 2024 pdf
Not all guess papers present online are very good. Some websites offer poor-quality content that is more harmful than it is useful for the students.
This guess paper is in pdf and you can download it free from here. I have given a download link in this post.
10th class maths definitions notes
Please notes that this time, in the 2024 exams, do not rely completely on any guess paper. There will be many conceptual questions in the board paper in 2024.
So, clear your concepts and also learn self-writing by practicing it more and more.
Maths guess paper 2024 class 10 Punjab Board
So, in my opinion, guess paper may benefit you to just pass the exams. If you want to get good marks too, you should clear your concepts, study the complete book, practice writing and solving
questions and make sure that you nicely attempt the paper.
Now here is the guess paper maths for 10th class which includes important short questions and long questions from every chapter. You can download it in pdf below.
10th class maths important questions
Here are 10th class math important questions from every exercise. Exercise-based important questions for 10th class math are given below. You can see the important questions 2024 taken from every
Unit 1
Ex 1.1 - Q1 (iii,iv,v,iv) - Q2 (ii, iv, vi) - Q3 (i,v,vii)
Ex 1.2 - Q1 (i,iii,viii), Q5
Ex 1.3 - Q1, Q9, Q10, Q12
Ex 1.4 - Q1, Q3, Q8, Q9
Ex 1: - Q1, Q2
Unit 2
Ex 2.1 - Q1(ii, iii, iv), Q3, Q4(iii) Q10
Example no. 2 (iv, v)
Ex 2.2 - Q2 (i, ii, viii) Q3, Q4
Ex 2.3 - Q1 (i, v, vi) Q2(ii) Q5(ii)
Ex 2.4 - Q1 (i,ii)
Ex 2.5 - Q1 complete
Ex 2.6 - Q1 (i,ii) Q2 (i, ii)
Page 42 example 5
Ex 2.7 - Q7, Q8, Q9
Ex 2.8 - Q1, Q5
Ex 2 : - Q1, Q2 (i, vi, vii)
Unit 3
Ex 3.1 - Q1(ii, iv, v) Q4, Q5, Q7, Q9
Ex 3.2 - Q1 (i, ii, iii), Q2 (i, ii), Q5, Q8, Q10
Ex 3.3 - Q1(i, iv) Q2(i, ii, iv, v) Q3(ii, iv) Q4(ii, iii)
Ex 3.4 - Q1, Q2
Ex 3.6 - Q1(ii, ii, vi) Q2 (i, ii,iii)
Ex 3: - Q2 (complete)
Unit 4
Ex 4.1 - Q2, Q3, Q4 and Q8
Ex 4.2 - Q1, Q3, Q6, Q8
Ex 4.3 - Q1, Q2, Q8
Ex 4: - Q2 (complete)
Unit 5
Ex 5.1 - Q1 (i,ii,iii), Q3(i, v), Q6 (i, ii)
Ex 5.2 - Q1(v, vi), Q2 (iii, iv), Q4(ii)
Ex 5.3 - Q1 (i, ii, iii, v) Q4(iii, iv)
Ex 5.4 - Q1, Q3(i, ii, iii),
Ex 5.5 - Q1, Q2, Q3
Unit 6
Example 3, page 163
Ex 6.2 - Q3 (i,ii), Q7
Example 2: page No 137
Ex 6.3 - Q4 and Q5 (i, ii) Q6
Unit 7
Ex 7.1 - Q3(i, ii, v), Q4(i, ii, iii, v), Q5(i, ii, ii, iv, v)
Ex 7.2 - Q1, Q3, Q5, Q6
Ex 7.3 - Q7
Ex 7.4 - Q8, Q9, Q12, Q14, Q18, Q20, Q22, Q23, Q24
Ex 7.5 - Q1, Q2
Ex 7 - Q2
Unit 8
Ex 8: - Q7, Q8, Q9
Unit 13
Ex 13.1 - Q1, Q4
Important Problems
Unit 9 Problem No. 2, 3, 4
Unit 12 Problem No. 1, 2
Important Definitions
Radical equation
Quadratic equation
Reciprocal equation
Synthetic division
Simultaneous equations
name method equation
Properties and variation
inverse variation
joint variation
proper and improper fraction
rational fraction
complement of set
binary relation
complement of a set
ordered pairs
into function, onto function
rational fraction
variable deviation
standard deviation
measure of angle,
length of tangent
segment circle
Now download the following content for class 10
6 comments:
Not recommended!
Chap 9 theorem 1 has not been seen in board exam for past 6 years.
it's chapter 12 theorem 1
Long Question
Very informative
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Maharashtra Board Class 12 Physics Sample Paper Set 1 with Solutions
Maharashtra State Board Class 12th Physics Sample Paper Set 1 with Solutions Answers Pdf Download.
Maharashtra Board Class 12 Physics Model Paper Set 1 with Solutions
Section A
Question 1.
Select and write the correct answers to the following questions:
(i) A particle of mass 1 kg, tied to a 1.2 m long string is whirled to perform vertical circular motion, under gravity. Minimum speed of a particle is 5 m/s. Consider following statements.
(P) Maximum speed must be 5\(\sqrt{5}\) m/s.
(Q) Difference between maximum and minimum tensions along the string is 60 N. Select correct option.
(a) Only the statement P is correct
(b) Only the statement Q is correct
(c) Both the statements are correct
(d) Both the statements are incorrect
(c) Both the statements are correct
(ii) If pressure of an ideal gas is decreased by 10% isothermally, then its volume will
(a) decrease by 9%
(b) increase by 9%
(c) decrease by 10%
(d) increase by 11.11%
(d) Increase by 11.11%
(iii) A standing wave is produced on a string fixed at one end with the other end free. The length of the string:
(a) must be an odd integral multiple of \(\frac{\lambda}{4}\)
(b) must be an odd integral multiple of \(\frac{\lambda}{2}\)
(c) must be an odd integral multiple of λ
(d) must be an even integral multiple of λ
(a) Must be an odd integral multiple of λ/4
(iv) A parallel plate capacitor is charged and then isolated. The effect of increasing the plate separation on charge, potential capacitance respectively are:
(a) Constant, decreases, decreases
(b) Increases, decreases, decreases
(c) Constant, decreases, increases
(d) Constant, increases, decreases
(a) Constant, decreases, decreases
(v) Kirchhoff’s first law, i.e., ΣI = 0 at a junction, deals with the conservation of ‘ ‘
(a) charge
(b) energy
(c) momentum
(d) mass
(a) Charge
(vi) A conductor rod of length (l) is moving with velocity (v) in a direction normal to a uniform magnetic field (B). What will be the magnitude of induced emf produced between the ends of the moving
(a) BLv
(b) BLv^2
(c) \(\frac{1}{2}\)BLv
(d) \(\frac{2 \mathrm{Bl}}{\mathrm{v}}\)
(a) BLv
(vii) In a series LCR circuit, the phase difference between the voltage and the current is 450. Then the power factor will be:
(a) 0.607
(b) 0.707
(c) 0.808
(d) 1
(b) 0.707
(viii) Which of the following properties of a nucleus does not depend on its mass number?
(a) Radius
(b) Mass
(c) Volume
(d) Density
(d) Density
(ix) A charged particle is in motion having initial velocity V[ρ] when it enters into a region of uniform magnetic field perpendicular to V[ρ]. Because of the magnetic force the kinetic energy of the
particle will:
(a) remain unchanged
(b) get reduced
(c) increase
(d) be reduced to zero
(a) Remain unchanged
(x) A conducting thick copper rod of length 1 m carries a current of 15 A and is located on the Earth’s equator. There the magnetic flux lines of the Earth’s magnetic field are horizontal, with the
field of 1.3 × 10^-4 T, south to north. The magnitude and direction of the force on the rod, when it is oriented so that current flows from west to east, are:
(a) 14 × 10^-4 N, downward
(b) 20 × 10^-4 N, downward
(c) 14 × 10^-4 N, upward
(d) 20 × 10^-4 N, upward
(d) 20 × 10^-4N, upward
Question 2.
Answer the following questions:
(i) Why are curved roads banked ?
To avoid the risk of skidding as well as to reduce the wear and tear of the car tyres, the road surface at a bend is titled inward i.e., the outer side of the road is raised above its inner side.
This is called banking of road.
(ii) Define athermanous substances and diathermanous substances.
Substance that don’t allow transmission of infrared radiation through them are called athermanous substance diathermanous body in a body which transmits all the incident radiation without absorbing
or reflecting.
(iii) State and explain the principle of conservation of angular momentum.
The angular momentum of a body is conserved if the resultant external torque on body is zero. This law is used by a figure skater to increase their speed of rotation for a spin by reducing the body’s
moment of inertia.
(iv) Define linear simple harmonic motion.
The linear periodic motion of a body, in which the restoring force is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.
(v) If the density of oxygen is 1.44 kg/m^3 at a pressure of 10^5 N/m^2, find the root mean square velocity of oxygen molecules.
p = 1.44 kg/m^3, P = 10^5 N/m^2
∴ The root mean square velocity of oxygen molecules,
V[rms] = \(\sqrt{\frac{3 P}{\rho}}\) = \(\sqrt{\frac{3 \times 10^5}{1.44}}\) m/s
= \(\sqrt{2.083 \times 10^5}\) = \(\sqrt{20.83 \times 10^4}\)
= 4.564 × 10^2 m/s
(vi) Which property of soft iron makes it useful for preparing electromagnet?
An electromagnet should become magnetic when a current is passed through its coil but electromagnet lose its magnetism once the current is switched off Hence ferromagnetic core used for an
electromagnet should have high permeability and low retentivity i.e., it should be magnetically soft.
(vii) A ceiling fan having moment of inertia 2 kg-m^2 attains its maximum frequency of 60 rpm in 2π seconds. Calculate its power rating.
Given: ω[0] = 0, ω = 2πn = 2π × 2 = 4π rad/s
α = \(\frac{\left(\omega-\omega_0\right)}{t}\) = \(\frac{(4 \pi-0)}{2 \pi}\) = 2 rad/s^2
P = r.ω = lα.ω = 2 × 2 × 4π = 16π watt ≈ 50 watt.
(viii) Write ideal gas equation for a mass of 7g of nitrogen gas.
Ideal gas equation, PV = nRT
Here, n \(=\frac{\text { mass of the gas }}{\text { molar mass }}\) = \(\frac{7}{28}\) = \(\frac{1}{4}\)
Therefore, the corresponding ideal gas equation is
PV = \(\frac{1}{4}\)RT
Section B
Attempt any Eight of the following questions:
Question 3.
Why is the surface tension of paints and lubricating oils kept low?
The surface tension of paints and lubricating oils kept low for better surface coverage.
Question 4.
Why should a Carnot cycle have two isothermal two adiabatic processes?
With two isothermal and two adiabatic processes, all reversible, the efficiency of the Carnot engine depends only on the temperatures of the hot and cold reservoirs.
Question 5.
Mention the conditions under which a real gas obeys ideal gas equation.
A real gas obeys the ideal gas equation when:
1. Temperature is very high.
2. Pressure is very low.
Question 6.
What are primary and secondary sources of light?
The source that emit tight on their own are called primary source. Some sources are not self-luminous, i.e., they do not emit tight on their own but reflect or scatter the light incident on them.
Such sources of light are called secondary sources.
Question 7.
Why two or more mercury drops from a single drop when brought in contact with each other?
A spherical shape has the minimum surface area to volume ratio of all geometric forms. When two drops of a liquid are brought ¡n contact, the cohesive forces between their molecules coalesce the
drops into a single
larger drop.
Question 8.
What do you mean by electromagnetic induction? State Faraday’s Law of induction.
The phenomenon of production of emf in circuit by a changing magnetic flux through the circuit is called electromagnetic induction.
Faraday’s first law: Whenever there is a change in magnetic flux associated with a circuit, an emf is induced in the circuit.
Faroes’s second law: The magnitude of the induced emf is directly proportional to the time rate of change of magnetic flux through the circuit.
Question 9.
State the importance of Davisson and Germer experiment.
The Dbvisson and Germer experiment are probably one of the most important experiments ever since it verified that de Broglie’s “matter wave” hypothesis applied to matter (electrons) as well as light.
From this emerged modern quantum theory, the most stupendous revolution in physics of all time.
Question 10.
A system releases 125 kJ of heat while 104 kJ of work is done on the system. Calculate the change in internal energy.
Given: Q = -125 kJ, W = – 104 Id
∆U = Q – W = – 125 kJ – (-104 kJ)
= (-125 + 104) kJ = – 21 kJ
Question 11.
A violin string vibrates with fundamental frequency of 440 Hz. What are the frequencies of first and second overtones?
Given, n = 440 Hz
The first overtone,
n[1] = 2n = 2 × 440
= 880 Hz
The second overtone,
n[2] = 3n = 3 × 440 = 1320 Hz.
Question 12.
White light consists of wavelengths from 440 nm to 700 nm. What will be the wavelength range seen when white light is passed through glass of refractive index 1.55?
Let λ[1] and λ[2] be the wavelengths of light in water for 400 nm and 700 nm (wavelengths in vacuum) respectively. Let λ[a] be the wavelength of light in vacuum.
λ = \(\frac{\lambda_a}{n}\) = \(\frac{400 \times 10^{-9} \mathrm{~m}}{1.55}\) = 258.06 × 10^-9 m
λ = \(\frac{\lambda_a}{n}\) = \(\frac{700 \times 10^{-9} \mathrm{~m}}{1.55}\) = 451.61 × 10^-9 m
The wavelength range seen when white light is passed through the glass would be 258.06 nm to 451.61 nm.
Question 13.
A galvanometer has a resistance of 25 Ω and its full scale deflection current is 25 μA. What resistance should be added to it to have a range of 0-10 V?
Given: G = 25 mA
Maximum voltage to be measured is V = 10 V.
The galvanometer resistance G = 25 Ω
The resistance to be added in series,
X = \(\left(\frac{V}{I_G}\right)\) – G = (\(\frac{10}{25}\) × 10^-6) – 25 = 399.975 × 10^3 Ω
Question 14.
Calculate the value of magnetic field at a distance of 2 cm from a very long straight wire carrying a current of 5A. (Given: µ[0] = 4π × 10^-7 Wb/Am)
Given: I = 5A, a = 0.02 m, \(\frac{\mu_0}{4 \pi}\) = 10^-7T\(\frac{\mathrm{m}}{\mathrm{~A}}\)
The magnetic induction,
B = \(\frac{\mu_0{ }^{\prime}}{2 \pi a}\) = \(\frac{\mu_0 21}{4 \pi a}\) = \(\frac{10^{-7} \times 2(5)}{2 \times 10^{-2}}\) = 5 × 10^-5T
Section C
Attempt any Eight of the following questions:
Question 15.
Obtain an expression for conservation of mass starting from the equation of continuity.
Consider a fluid in steady or streamline flow, that is its density is constant. The velocity of the fluid within a flow tube, while everywhere parallel to the tube, may change its magnitude. Suppose
the velocity is v[1], at the point P and v[2] at point Q. If A[1] and A[2] are the cross-sectional areas of the tube at these two points.
The volume flux across A[1], \(\frac{d}{d t}\)(V[1]) = A[1]v[1] and that across A[2], \(\frac{d}{d t}\)(V[2]) = A[2]V[2]
By the equation of continuity of the flow for a fluid,
A[1]v[2] = A[2]v[2]
i.e., \(\frac{d}{d t}\)(V[1]) = \(\frac{d}{d t}\)(V[2])
If ρ[1] and ρ[2] are the densities of the fluid at P and Q, respectively, the mass flux across A[1].
\(\frac{d}{d t}\)(m[1]) = \(\frac{d}{d t}\)(ρ[1]V[1]) = A[1]ρ[1]V[1] and that across A[2] = \(\frac{d}{d t}\)(m[2]) = \(\frac{d}{d t}\)(ρ[2]V[2]) = A[2]ρ[2]V[2]
Since no fluid can either can enter or leave through the boundary of the tube, the conservation of mass requires the mass fluxes to be equal, i.e..
\(\frac{d}{d t}\)(m[1]) = \(\frac{d}{d t}\)(m[2])
i.e., A[1]ρ[1]V[1] = A[2]ρ[2]V[2]
i.e., Aρv = constant which is the required expression.
Question 16.
Show that a linear S.H.M. is the projection of a U.C.M. along any of its diameter.
Linear SHM is defined as the linear periodic motion of a body, in which the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional
to the displacement from the mean position.
cos (ωt + α) = \(\frac{x}{a}\)
x = a cos (ωt + α) ….(1)
This is the expression for displacement of particle M at time t.
As velocity of the particle is the time rate of change of displacement then we have
v = \(\frac{d x}{d t}\) = \(\frac{d}{d t}\) [a cos (ωt + α)]
v = -aωsin (ωt + α) ….(2)
As acceleration of particle is the time rate of change of velocity, we have
a = \(\frac{d v}{d t}\) = \(\frac{d}{d t}\) [- aω sin (ωt + α)]
a = -aω^2 cos (ωt + α)
a = ω^2x
It shows that acceleration of particle M is directly proportional to its displacement and its direction is opposite to that of displacement. Thus, particle M performs simple harmonic motion but M is
projection of particle performing UCM hence SHM is projection of UCM along a diameter of circle.
Question 17.
State the characteristics of stationary waves.
Characteristics of stationary waves:
1. Stationary waves are produced by the interference of two identical. progressive waves travelling in opposite directions, under certain condition.
2. The overall appearance of standing wave is of alternate intensity maximum and minimum.
3. The distance between adjacent node \(\frac{\lambda}{2}\)
4. The distance between successive node \(\frac{\lambda}{4}\).
5. The stationary wave does not propagate in on direction.
6. The stationary wove does not transport energy through the medium.
7. There is no progressive change of phase from particle to particle.
Question 18.
A capacitor has some dielectric between its plates and the capacitor is connected to a DC source. The battery is now disconnected and then the dielectric is removed. State whether the capacitance,
the energy stored in it, the electric field, charge stored and voltage will increase, decrease or remain constant.
Assume parollet-plate capacitor of plate area A and plate separation d is filled with a dielectric of relative permittivity (dielectric constant) k. Its capacitance is
C = \(\frac{k \varepsilon_0 A}{d}\) …… (1)
If it is charged to a voltage (potential) V, the charge on its plates is
Q = CV
Since the battery is disconnected after it is charged, the charge Q on its plates and consequently the product CV remain unchanged.
On removing the dielectric completely, its capacitance becomes from equation (1).
C = \(\frac{\varepsilon_0 A}{d}\) = \(\left(\frac{1}{k}\right) c\) ……… (2)
That is, its capacitance decreases by the factor k. Since C’V’ = CV, its new voltage is
V’ = \(\left(\frac{C}{C^{\prime}}\right) V\) = kV ….. (3)
So that its voltage increases by the factor k.
The stored potential energy. u = \(\frac{1}{2}\)QV, so that Q remaining constant, u increases by the factor k. The electric field, E = \(\frac{V}{d}\), so that E also increases by a factor k.
Question 19.
When an AC source is connected to an ideal inductor show that the average power supplied by the source over a complete cycle is zero.
In on AC circuit containing only an ideal inductor, the current i logs behind the emf e by a phase angle of \(\frac{\pi}{2}\) rad.
Here, for e = e[0] sin ωt,
we have, i = i[0] sin (ωt – \(\frac{\pi}{2}\))
Instantaneous power,
P = ei
= (e[0] sin ωt) [i[0] (sin ωt cos \(\frac{\pi}{2}\) – cos ωt sin \(\frac{\pi}{2}\))]
= -e[0]i[0]sin ωt cos ωt as cos \(\frac{\pi}{2}\) = 0 and sin \(\frac{\pi}{2}\) = 1
Average power over one cycle,
P[av] = Work done in one cycle/time for one cycle
Question 20.
Is it always possible to see photoelectric effect with red light?
No, it is not possible but due to present technology it may be possible.
Explanation: The photons in red light to not have the necessary energy required to rip an electron out of its orbital (this needed energy is equal to the electron’s work function. Because light
behaves like particles rather than a continuous stream, even very high-intensity red light will never be able to overcome an electron’s work function (in this situation), as every individual photon
fails to do so. This shows the particle behaviour of light, because of light behaved like a wave, the red light would be able to overcome the electron’s work function with high intensity or a long
Question 21.
Explain the construction and working of solar cell.
Construction: A simple pn-junction solar cell consist of a p-type semiconductor substrate backed with a metal electrode. A thin layer of silicon is grown over the p-type of substrate by doping with
suitable donor impurity. Metal finger electrode are prepared on top of n-layer so that there is enough space between the fingers for sunlight to reach the n-layer and underlying pn-junction:
Working: When exposed to sunlight, the absorption of incident radiation creates electron hole pairs in and near the depletion layer. The photo-generated electrons and holes moves towards the n-side
and p-side, respectively. If no external load is connected, these photo-generated charges get collected at the two sides of the junction and give rise to forward photo-voltage. If a closed circuit a
current I passes through external load as long as the solar cell is exposed to sunlight.
A solar cell consist several solar cells which are connected in series for higher output.
Question 22.
Disintegration rate of a sample of 1010 per hour at 20 hrs from the start. It reduces to 6.3 × 10^9 per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the
A(t[1]) = 10^10 per hour, where t[1] = 20 h
A(t[1]) = 6.3 × 10^9 per hour, where t[2] = 30 h
A(t) = A[0]e^-λt
A(t[1]) = A[0]\(\mathrm{A}_0 e^{-\lambda t_1}\) and A(t[2]) = A[0]\(e^{-\lambda t_2}\)
\(\frac{\mathrm{A}\left(t_1\right)}{\mathrm{A}\left(t_2\right)}\) = \(\left(\frac{e^{-\lambda t_1}}{e^{-\lambda t_2}}\right)\) = \(e^{\lambda\left(t_2-t_1\right)}\)
\(\frac{10^{10}}{6.3 \times 10^9}\) = \(e^{\lambda(30-20)}\) = e^10λ
1.587 = e^10λ
log[e] 1.587 = 10λ
10λ = 2.303 log[10] (1.587)
λ = (0.2303) (0.2007) = 0.04622 per hour
The half life of the material
Question 23.
Calculate the wavelength associated with an electron, its momentum and speed. When it is accelerated through a potential of 54 V?
Given: V = 54 V, m = 9.1 × 10^-31 kg, e = 1.6 × 10^-19 C, h = 6.63 × 10^-13 J.s, K.E. = 150 eV.
We assume that the electron is initially at rest
V[e] = \(\frac{1}{2}\)mv^2
V = \(\sqrt{2 V_e / m}\)
= \(\sqrt{2(54)\left(1.6 \times 10^{-19}\right) / 9.1 \times 10^{-31}}\)
= \(\sqrt{19 \times 10^{12}}\)
= 4.359 × 10^6 m/s
This is the speed of the electron.
Now, p = mv= (9.1 × 10^-31) (4.359 × 10^6)
= 3.967 × 10^-24 kg m/s
This the momentum of the electron.
The wavelength associated with the electron.
λ = \(\frac{h}{p}\) = \(\frac{\left(6.63 \times 10^{-34}\right)}{\left(3.367 \times 10^{-24}\right)}\)
= 1.671 × 10^-10m
= 1.671 Å
= 0.1671 nm.
Question 24.
The distance between two consecutive bright fringes in a biprism experiment using Light of wavelength 6000 A is 0.32 mm by how much will the distance change if light of wavelength 4800 A is used?
Given: λ[1] = 6000 Å = 6 × 10^-7 m, λ[2] = 4800 Å = 4.8 × 10^-7 m, W[1] = 0.32 mm = 3.2 × 10^-4 m.
Distance between consecutive bright fringes,
Question 25.
One mole of an ideal gas is initially kept in a cylinder with a movable frictionless and massless piston at pressure of 1.0 mPa and temperature 27°C. It is then expanded till its volume is doubled.
How much work is done if the expansion is isobaric?
Work done in isobaric process given by
W = p∆V = (V[f] – V[i])
V[f] = 2V[i]
∴ W = 2pV[i]
V[i] can be found by using the ideal gas equation for initial state.
= 24.9 × 10^-4 m^3
W = 2 × 10^6 × 24.9 × 10^-4
W = 4.9 kJ.
Question 26.
The resistance of a potentiometer wire is 8 Ω and its length is 8 m. A resistance box and a 2 V battery are connected in series with it. What should be the resistance in the box, if it is desired to
have a potential drop of 1 µV/mm?
Given: R = 8 Ω, L = 8 m, E = 2V
K = 1 V/mm
= \(=\frac{1 \times 10^{-6} \mathrm{~V}}{10^{-3} \mathrm{~m}}\) = 10^-3V/m
K = \(\frac{V}{L}\) = ER/(R + R[B])L
where R[B] is the resistance in the box.
10^-3 = 2 × \(\frac{8}{\left(8+R_B\right) 8}\)
8 + R[B] = \(\frac{2}{10^{-3}}\) = 2 × 10^3
R[B] = 200 – 8 = 1992 Ω
Section D
Attempt any Three of the following questions:
Question 27.
(i) What are eddy currents? State applications of eddy currents.
Whenever a conductor or part of it is moved in magnetic field cutting magnetic field lines, the free electrons in the bulk of metal starts circulating in closed path equivalent to current carrying
loops. These current resemble to eddies in fluid stream and hence called as eddy current.
1. Electric brakes,
2. Galvanometer.
(ii) Magnetic field at a distance 2.4 cm from a long straight wire is 16 µT. What must be current through the wire?
Question 28.
(i) State the postulates of Bohr’s atomic model.
(a) The electron revolves with a constant speed in circular orbit around the nucleus.
(b) The radius of the orbit of an electron can only take certain fixed values such that the angular momentum of the electron in these orbits in
an integral multiple of \(\frac{h}{2 \pi}\), h being the Planck’s constant
(c) An electron can make a transition from one of its orbit to another orbit having lower energy. In doing so, it emits a photon of energy equal to the difference in its energies in the two orbits.
(ii) A short bar magnet is placed in an external magnetic field of 700 gauss. When its axis makes an angle of 30° with the external magnetic field, it experiences a torque of 0.014 Nm. Find the
magnetic moment of the magnet and the work done in moving it from its most stable to most unstable position.
Given: B = 700 gauss = 0.07 tesla, θ = 30°, \(\tau\) = 0.014 Nm, \(\tau\) = MB sin θ
The magnetic moment of the magnet is
M = \(\frac{\tau}{\mathrm{B} \sin \theta}\) = \(\frac{0.014}{(0.07)\left(\sin 30^{\circ}\right)}\) = 0.4 A.m^2
The most stable state of the bar magnet is for θ = 0°. It is in the most unstable state when θ = 180° is
W = MB (cos θ[0] – cos θ)
= MB (cos 0° – cos 180°)
= MB[1 – (- 1)] = 2 MB = (2) (0.4) (0.07)
= 0.056 J.
Question 29.
(i) On what factors do the degrees of freedom depend?
The degree of freedom depend upon
(a) the number of atoms forming a molecule,
(b) the structure of the molecule,
(c) the temperature of the gas.
(ii) An aircraft of wing span of 50 m flies horizontally in earth’s magnetic field of 6 × 10^-5T at a speed of 400 m/s. Calculate the emf generated between the tips of the wings of the aircraft.
Given: I = 50 m, B = 6 × 10^-5T, v = 400 m/s
The magnitude of the induced emf
| e | = Blv = (6 × 10^-5) (50) (400) = 1.2 V.
Question 30.
(i) A dipole with its charges, -q and +q located at the points (0, -b, 0) and (0, b, 0) is present in a uniform electric field E. The equipotential surfaces of this field are planes parallel to the
YZ planes
(a) What is the direction of the electric field E?
(b) How much torque would the dipole experience in this field?
(a) Given, the equipotentials of the external uniform electric field are planes parallel to the yz-plane, the electric field \(\vec{E}\) = ± E \(\hat{i}\) that is. \(\vec{E}\) is parallel to the x.
(b) The above diagram, the dipole moment
So that the magnitude of the torque is \(\tau\) = 2qbE.
If \(\overrightarrow{\mathrm{E}}\) is in the direction of the + x-axis, the torque \(\vec{\tau}\) is in the direction of – z-axis, while if \(\vec{E}\) is in the direction of the x-axis, the torque \
(\vec{t}\) is in the direction of + z-axis.
(ii) A 25 µF capacitor, a 0.10 H inductor and a 25 Ω resistor are connected in series with an AC source whose emf is given by e = 310 sin 314 t (volt). What is the frequency, reactance, impedance,
current and phase angle of the circuit?
Given: C = 25 µF = 25 × 10^-6 F, L = 0.10 H, R = 25 Ω, e = 310 sin (314t) [volt]
Comparing e = 310 sin (314t) with e = e[0] sin (2πft), we get
The frequency of the alternating emf as
Question 31.
(i) What happens to ferromagnetic material when its temperature increases above Curie temperature?
A ferromagnetic material is composed of small regions called domains. Within each domain, the atomic magnetic moments of nearest-neighbour atoms interact strongly through exchange interaction, a
quantum mechanical phenomenon and align themselves parallel to each other even in the absence of an external magnetic field. A domain is, therefore, spontaneously magnetized to saturation.
The material retains its domain structure only up to a certain temperature. On heating, the increased thermal agitation works against the spontaneous domain magnetization. Finally, at a certain
critical temperature, called the Curie point or Curie temperature, thermal agitation overcomes the exchange forces and a certain temperature.
On heating, the increased thermal agitation works against the spontaneous domain magnetization. Finally, at a certain critical temperature, called the Curie point or Curie temperature, thermal
agitation overcomes the exchange forces and exchange forces and keeps the atomic magnetic moments randomly oriented. Thus, above the Curie point, the material becomes paromagnetic. The ferromagnetic
to paramognetic transition is an order to disorder transition. When cooled below the Curie point the material becomes ferromagnetic again.
(ii) In a common-base connection, the emitter current is 6.28 mA and collector current is 6.20 mA. Determine the common base DC current gain. | {"url":"https://maharashtraboardsolutions.com/maharashtra-board-class-12-physics-sample-paper-set-1/","timestamp":"2024-11-13T10:56:52Z","content_type":"text/html","content_length":"109543","record_id":"<urn:uuid:ce01206f-220f-4638-b531-8d18d878ed1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00500.warc.gz"} |
Pseudoscalar glueball mass: QCD versus lattice gauge theory prediction
We study whether the pseudoscalar glueball mass in full QCD can differ from the prediction of quenched lattice calculations. Using properties of the correlator of the vacuum topological
susceptibility we derive an expression for the upper bound on the QCD glueball mass. We show that the QCD pseudoscalar glueball is lighter than the pure Yang-Mills theory glueball studied in quenched
lattice calculations. The mass difference between those two states is of the order of 1/N[C]. The value calculated for the 0^-+ QCD glueball mass cannot be reconciled with any physical state observed
so far in the corresponding channel. The glueball decay constant and its production rate in J/ψ radiative decays are calculated. The production rate is large enough to be studied experimentally.
ASJC Scopus subject areas
• Mathematical Physics
• General Physics and Astronomy
• Nuclear and High Energy Physics
Dive into the research topics of 'Pseudoscalar glueball mass: QCD versus lattice gauge theory prediction'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/pseudoscalar-glueball-mass-qcd-versus-lattice-gauge-theory-predic","timestamp":"2024-11-03T10:01:21Z","content_type":"text/html","content_length":"50730","record_id":"<urn:uuid:9978fee5-a953-48a5-b76a-2bc467a007b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00681.warc.gz"} |
My Learning Logs
I believe that knowledge is the ultimate treasure and it should not be confined by mere academic boundaries imposed by ancient people. With democratization of learning, it has never been this easier
as it is now. A new topic needs just the willingness and attention to learn, all other resources are waiting there for anyone who knows where to look for it.
This is my non-exhaustive learning log with verifiable links - not only to show off (obvious for a mere mortal) what I have learned so far but also to remind me of how they have had profound positive
impact on my life thus far.
Professional Certifications
Competitive achievements
• Special recognition for outstanding performance in first public Datathon Bangladesh in 2019 organized by Axiata Analytics Centre in collaboration with Robi Axiata Limited
• Top 10 in ADL AI Summit PreHackathon Kaggle Competition public and private leaderboard with Team Chunoputi
Multi-course Specialization
Learner Profile
Translation Works
• Bengali translation of Data Import with R packages like readr,readxl and googlesheets4 published on official RStudio cheatsheets page (Scroll to bottom for translations list- my beloved bangla is
• Few assessments I took for benchmarking thyself in DataCamp (No shareable link - only for planning own learning route) - Scores are in percentile - Higher is better
□ Understanding and Interpreting Data - 99% (Mar’21)
□ R Programming - 96% (Dec’21)
□ Statistics Fundamentals with R - 89% (Dec’21)
□ Importing & Cleaning Data with R - 95% (Dec’21)
□ Data Manipulation with R - 89% (Dec’21)
□ Data Visualization with R - 99% (May’21)
□ Data Analysis in SQL (PostgreSQL) - [DEL:74% (May’21):DEL] 96% (Jul’23)
□ Data Management Theory - 82% (Jul’23)
Miscellaneous Courses
Machine Learning
Network Science
Machine Learning
Text Processing
Network Science
Big Data | {"url":"https://saifkabirasif.com/learning/","timestamp":"2024-11-07T10:41:29Z","content_type":"text/html","content_length":"25983","record_id":"<urn:uuid:0024ad3c-7fcd-41f3-bbf1-ab5b91d15a8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00638.warc.gz"} |
Fast phase randomisation via two-folds
A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes
through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and in the
presence of small noise the asymptotic phase of a large number of sample solutions is highly random. In this paper we show how the probability distribution of the asymptotic phase depends on the
global nonlinear dynamics. We also show how the phase of a smooth oscillator can be randomised by applying a simple discontinuous control law that generates an invisible two-fold. We propose that
such a control law can be used to desynchronise a collection of oscillators, and that this manner of phase randomisation is fast compared to existing methods (which use fixed points as phase
singularities) because there is no slowing of the dynamics near a two-fold.
Research Groups and Themes
• Engineering Mathematics Research Group
• piecewise-smooth
• desynchronization
• Filippov system
• sliding motion
Dive into the research topics of 'Fast phase randomisation via two-folds'. Together they form a unique fingerprint.
• Jeffrey, M. R. (Principal Investigator)
1/03/16 → 30/06/18
Project: Research
• Jeffrey, M. R. (Principal Investigator)
1/08/12 → 1/08/16
Project: Research | {"url":"https://research-information.bris.ac.uk/en/publications/fast-phase-randomisation-via-two-folds","timestamp":"2024-11-08T11:52:07Z","content_type":"text/html","content_length":"70277","record_id":"<urn:uuid:01dca943-ed8c-44d6-bc9f-ff5d27913e31>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00561.warc.gz"} |
What's The Correct Answer For The 6÷2(1+2) Equation? The Viral 'Math Bait' Meme Explained
Math problems aren't usually a viral topic on social media, but every once in a while, one pops up that sparks a viral phenomenon online. Such is the case with the 6÷2(1+2) equation that has been
causing quite a stir on X / Twitter and Reddit lately for its many interpretations and results generated by confused netizens.
A tweet posted by X user @moongleamdream in late October 2024 mocking someone's wrong answer for the equation has gained an incredible 77 million views in three days, reigniting a passionate debate
about the right answer for the math equation.
There is only a single answer for the 6÷2(1+2) equation and it's 9, here's why.
What's The Correct Answer For The 6÷2(1+2) Equation?
The main first step to get to the right answer for the 6÷2(1+2) equation is the order of operations. You see, this viral equation is one of many so-called "math bait" problems with intentionally
vague formatting that is often posted with the intent of sparking debate as commenters offer different answers based on their view to solve the equation.
Recapping a few equation rules, first, evaluate parentheses/brackets, then evaluate exponents/orders, then evaluate multiplication-division, and, lastly, evaluate addition-subtraction. Following this
logic will take you to the current view of the equation:
= 6÷2(3)
The catch here is that the (3) in the parentheses has to be interpreted into an implied multiplication, which, according to the order of operations, division and multiplication have the same
importance. This will then set the stage for the correct order, which is to evaluate from left to right, as shown below:
= 3×3
= 9
YouTuber and mathematician Presh Talwalkar cleverly explains all the steps summarized above to reach the correct answer to the 6÷2(1+2) equation.
How Did The 6÷2(1+2) Math Equation Went Viral?
Even though Talwalkar's video explanation of the 6÷2(1+2) equation was posted in August 2016, it was only in April 2021 that the math problem grew into a divisive and funny debate on social media. X
user @lesvity was responsible for throwing the first curveball on the matter by sharing his disbelief in the statement, "How can you guys get anything other than 7…", possibly trolling.
The equation continued to divide X users between those who felt its answer was 1 and those who felt it was 9. For example, user @itsmagik pointed out that the reason for different answers was due to
how people were visualizing the question, with some interpreting it as (6/2) x (1+2) and others reading it as 6 / (2(1+2)).
X user @ReejFPS noted how strictly following PEMDAS and would give users 9, and using the distributive property would give users 1.
So, which side are you on in this debate?
For the full history of 6÷2(1+2), be sure to check out Know Your Meme's encyclopedia entry for more information.
Comments (0)
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Mass Flux Density Converter
Created by CalcKit Admin
Last updated: 7 Feb 2024
Mass Flux Density represents the amount of mass passing through a unit area over a specified time period. It is a crucial parameter in fluid dynamics and heat transfer studies. To facilitate easy
conversions between different units of Mass Flux Density, a Mass Flux Density Converter Tool has been developed.
Let's explore the various units supported by the tool, along with their definitions and conversion factors.
• Kilogram / second / meter² (kg/s/m²): The standard unit for Mass Flux Density, representing mass flow per unit area in one second, measured in kilogram per second per square meter.
Conversion Factor: 1 kg/s/m² = 1,000 g/s/m²
• Gram / second / meter² (g/s/m²): The mass of a substance passing through a unit area in one second, measured in grams per second per square meter.
Conversion Factor: 1 g/s/m² = 0.001 kg/s/m²
• Gram / second / centimeter² (g/s/cm²): Similar to the first unit, but the area is in square centimeters instead of square meters.
Conversion Factor: 1 g/s/cm² = 10 kg/s/m²
• Kilogram / hour / meter² (kg/h/m²): Mass passing through a unit area in one hour, measured in kilograms per hour per square meter.
Conversion Factor: 1 kg/h/m² = 0.00027778 kg/s/m²
• Kilogram / hour / foot² (kg/h/ft²): Mass passing through a unit area in one hour, measured in kilograms per hour per square foot.
Conversion Factor: 1 kg/h/ft² = 0.00299 kg/s/m²
• Pound / hour / foot² (lb/h/ft²): Mass passing through a unit area in one hour, measured in pounds per hour per square foot.
Conversion Factor: 1 lb/h/ft² = 0.001356 kg/s/m²
• Pound / second / foot² (lb/s/ft²): Mass passing through a unit area in one second, measured in pounds per second per square foot.
Conversion Factor: 1 lb/s/ft² = 4.8824 g/s/m²
The Mass Flux Density Converter Tool simplifies the process of converting values between different units, providing a convenient way for scientists, engineers, and researchers to work seamlessly with
diverse measurement systems. Whether you are dealing with metric or imperial units, this tool ensures accurate and efficient conversions, promoting consistency in scientific and engineering | {"url":"https://calckit.io/tool/conversion-mass-flux-density","timestamp":"2024-11-11T11:19:21Z","content_type":"text/html","content_length":"28102","record_id":"<urn:uuid:4176742e-868a-43ea-beb3-c27a23d79f10>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00665.warc.gz"} |
Post-Quantum Cryptography for Engineers
Internet-Draft PQC for Engineers August 2023
Banerjee, et al. Expires 11 February 2024 [Page]
Intended Status:
Post-Quantum Cryptography for Engineers
The presence of a Cryptographically Relevant Quantum Computer (CRQC) would render state-of-the-art, public-key cryptography deployed today obsolete, since all the assumptions about the intractability
of the mathematical problems that offer confident levels of security today no longer apply in the presence of a CRQC. This means there is a requirement to update protocols and infrastructure to use
post-quantum algorithms, which are public-key algorithms designed to be secure against CRQCs as well as classical computers. These algorithms are just like previous public key algorithms, however the
intractable mathematical problems have been carefully chosen, so they are hard for CRQCs as well as classical computers. This document explains why engineers need to be aware of and understand
post-quantum cryptography. It emphasizes the potential impact of CRQCs on current cryptographic systems and the need to transition to post-quantum algorithms to ensure long-term security. The most
important thing to understand is that this transition is not like previous transitions from DES to AES or from SHA-1 to SHA2, as the algorithm properties are significantly different from classical
algorithms, and a drop-in replacement is not possible.¶
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current
Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as
reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 11 February 2024.¶
Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License
text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
Quantum computing is no longer perceived as a conjecture of computational sciences and theoretical physics. Considerable research efforts and enormous corporate and government funding for the
development of practical quantum computing systems are being invested currently. For instance, Google’s announcement on achieving quantum supremacy [Google], IBM’s latest 433-qubit processor Osprey [
IBM] or even Nokia Bell Labs' work on topological qubits [Nokia] signify, among other outcomes, the accelerating efforts towards large-scale quantum computers. At the time of writing the document,
Cryptographically Relevant Quantum Computers (CRQCs) that can break widely used public-key cryptographic algorithms are not yet available. However, it is worth noting that there is ongoing research
and development in the field of quantum computing, with the goal of building more powerful and scalable quantum computers. As quantum technology advances, there is the potential for future quantum
computers to have a significant impact on current cryptographic systems. Forecasting the future is difficult, but the general consensus is that such computers might arrive some time in the 2030s, or
might not arrive until 2050 or later.¶
Extensive research has produced several post-quantum cryptographic algorithms that offer the potential to ensure cryptography's survival in the quantum computing era. However, transitioning to a
post-quantum infrastructure is not a straightforward task, and there are numerous challenges to overcome. It requires a combination of engineering efforts, proactive assessment and evaluation of
available technologies, and a careful approach to product development. This document aims to provide general guidance to engineers who utilize public-key cryptography in their software. It covers
topics such as selecting appropriate post-quantum cryptographic (PQC) algorithms, understanding the differences between PQC Key Encapsulation Mechanisms (KEMs) and traditional Diffie-Hellman style
key exchange, and provides insights into expected key sizes and processing time differences between PQC algorithms and traditional ones. Additionally, it discusses the potential threat to symmetric
cryptography from Cryptographically Relevant Quantum Computers (CRQCs). It is important to remember that asymmetric algorithms are largely used for secure communications between organizations that
may not have previously interacted, so a significant amount of coordination between organizations, and within and between ecosystems needs to be taken into account. Such transitions are some of the
most complicated in the tech industry. It might be worth mentioning that recently NSA released an article on Future Quantum-Resistant (QR) Algorithm Requirements for National Security Systems [
CNSA2-0] based on the need to protect against deployments of CRQCs in the future.¶
It is crucial for the reader to understand that when the word "PQC" is mentioned in the document, it means Asymmetric Cryptography (or Public key Cryptography) and not any algorithms from the
Symmetric side based on stream, block ciphers, etc. It does not cover such topics as when traditional algorithms might become vulnerable (for that, see documents such as [QC-DNS] and others). It also
does not cover unrelated technologies like Quantum Key Distribution or Quantum Key Generation, which use quantum hardware to exploit quantum effects to protect communications and generate keys,
respectively. Post-quantum cryptography is based on standard math and software and can be run on any general purpose computer.¶
Please note: This document does not go into the deep mathematics of the PQC algorithms, but rather provides an overview to engineers on the current threat landscape and the relevant algorithms
designed to help prevent those threats.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described
in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
The guide was inspired by a thread in September 2022 on the pqc@ietf.org mailing list. The document is being collaborated on through a GitHub repository.¶
The editors actively encourage contributions to this document. Please consider writing a section on a topic that you think is missing. Short of that, writing a paragraph or two on an issue you found
when writing code that uses PQC would make this document more useful to other coders. Opening issues that suggest new material is fine too, but relying on others to write the first draft of such
material is much less likely to happen than if you take a stab at it yourself.¶
Any asymmetric cryptographic algorithm based on integer factorization, finite field discrete logarithms or elliptic curve discrete logarithms will be vulnerable to attacks using Shor's Algorithm on a
sufficiently large general-purpose quantum computer, known as a CRQC. This document focuses on the principal functions of asymmetric cryptography:¶
• Key Agreement: Key Agreement schemes are used to establish a shared cryptographic key for secure communication. They are one of the mechanisms that can be replaced by PQC, as this is based on
public key cryptography and is therefore vulnerable to the Shor's algorithm. An CRQC can find the prime factors of the large public key, which can be used to derive the private key.¶
• Digital Signatures: Digital Signature schemes are used to authenticate the identity of a sender, detect unauthorized modifications to data and underpin trust in a system. Similar to Key
Agreement, signatures also depend on public-private key pair and hence a break in public key cryptography will also affect traditional digital signatures, hence the importance of developing post
quantum digital signatures.¶
In the context of PQC, symmetric-key cryptographic algorithms are generally not directly impacted by quantum computing advancements. Symmetric-key cryptography, such as block ciphers (e.g., AES) and
message authentication mechanisms (e.g., HMAC-SHA2), rely on secret keys shared between the sender and receiver. HMAC is a specific construction that utilizes a cryptographic hash function (such as
SHA-2) and a secret key shared between the sender and receiver to produce a message authentication code. CRQCs, in theory, do not offer substantial advantages in breaking symmetric-key algorithms
compared to classical computers (see Section 7.1 for more details).¶
In 2016, the National Institute of Standards and Technology (NIST) started a process to solicit, evaluate, and standardize one or more quantum-resistant public-key cryptographic algorithms, as seen
here. The first set of algorithms for standardization (https://csrc.nist.gov/publications/detail/nistir/8413/final) were selected in July 2022.¶
NIST announced as well that they will be opening a fourth round to standardize an alternative KEM, and a call for new candidates for a post-quantum signature algorithm.¶
These algorithms are not a drop-in replacement for classical asymmetric cryptographic algorithms. RSA [RSA] and ECC [RFC6090] can be used for both key encapsulation and signatures, while for
post-quantum algorithms, a different algorithm is needed for each. When upgrading protocols, it is important to replace the existing use of classical algorithms with either a PQC key encapsulation
method or a PQC signature method, depending on how RSA and/or ECC was previously being used.¶
The fourth-round of the NIST process focuses only on KEMs. The goal of that round is to select an althernative algorithm that is based on different hard problem than Kyber. The candidates still
advancing for standardization are:¶
• Classic McEliece: Based on the hardness of syndrome decoding of Goppa codes. Goppa codes are a class of error-correcting codes that can correct a certain number of errors in a transmitted
message. The decoding problem involves recovering the original message from the received noisy codeword.¶
• BIKE: Based on the the hardness of syndrome decoding of QC-MDPC codes. Quasi-Cyclic Moderate Density Parity Check (QC-MDPC) code are a class of error correcting codes that leverages bit flipping
technique to efficiently correct errors.¶
• HQC : Based on the hardness of syndrome decoding of Quasi-cyclic concatenated Reed Muller Reed Solomon (RMRS) codes in the Hamming metric. Reed Muller (RM) codes are a class of block error
correcting codes used especially in wireless and deep space communications. Reed Solomon (RS) are a class of block error correcting codes that are used to detect and correct multiple bit errors.¶
• SIKE (Broken): Supersingular Isogeny Key Encapsulation (SIKE) is a specific realization of the SIDH (Supersingular Isogeny Diffie-Hellman) protocol. Recently, a mathematical attack based on the
"glue-and-split" theorem from 1997 from Ernst Kani was found against the underlying chosen starting curve and torsion information. In practical terms, this attack allows for the efficient
recovery of the private key. NIST announced that SIKE was no longer under consideration, but the authors of SIKE had asked for it to remain in the list so that people are aware that it is broken.
Post-quantum cryptography or quantum-safe cryptography refers to cryptographic algorithms that are secure against cryptographic attacks from both CRQCs and classic computers.¶
When considering the security risks associated with the ability of a quantum computer to attack traditional cryptography, it is important to distinguish between the impact on symmetric algorithms and
public-key ones. Dr. Peter Shor and Dr. Lov Grover developed two algorithms that changed the way the world thinks of security under the presence of a CRQC.¶
Grover's algorithm is a quantum search algorithm that provides a theoretical quadratic speedup for searching an unstructured database compared to classical algorithms. Grover’s algorithm
theoretically requires doubling the key sizes of the algorithms that one deploys today to achieve quantum resistance. This is because Grover’s algorithm reduces the amount of operations to break
128-bit symmetric cryptography to 2^{64} quantum operations, which might sound computationally feasible. However, 2^{64} operations performed in parallel are feasible for modern classical computers,
but 2^{64} quantum operations performed serially in a quantum computer are not. Grover's algorithm is highly non-parallelizable and even if one deploys 2^c computational units in parallel to
brute-force a key using Grover's algorithm, it will complete in time proportional to 2^{(128−c)/2}, or, put simply, using 256 quantum computers will only reduce runtime by 1/16, 1024 quantum
computers will only reduce runtime by 1/32 and so forth (see [NIST] and [Cloudflare]).¶
For unstructured data such as symmetric encrypted data or cryptographic hashes, although CRQCs can search for specific solutions across all possible input combinations (e.g., Grover's Algorithm), no
CRQCs is known to break the security properties of these classes of algorithms.¶
How can someone be sure that an improved algorithm won’t outperform Grover's algorithm at some point in time? Christof Zalka has shown that Grover's algorithm (and in particular its non-parallel
nature) achieves the best possible complexity for unstructured search [Grover-search].¶
Finally, in their evaluation criteria for PQC, NIST is considering a security level equivalent to that of AES-128, meaning that NIST has confidence in standardizing parameters for PQC that offer
similar levels of security as AES-128 does [NIST]. As a result, 128-bit algorithms should be considered quantum-safe for many years to come.¶
“Shor’s algorithm” on the other side, efficiently solves the integer factorization problem (and the related discrete logarithm problem), which offer the foundations of the public-key cryptography
that the world uses today. This implies that, if a CRQC is developed, today’s public-key cryptography algorithms (e.g., RSA, Diffie-Hellman and Elliptic Curve Cryptography) and protocols would need
to be replaced by algorithms and protocols that can offer cryptanalytic resistance against CRQCs. Note that Shor’s algorithm doesn’t run on any classic computer, it needs a CRQC.¶
For example, to provide some context, one would need 20 million noisy qubits to break RSA-2048 in 8 hours [RSA8HRS] or 4099 stable qubits to break it in 10 seconds [RSA10SC].¶
For structured data such as public-key and signatures, instead, CRQCs can fully solve the underlying hard problems used in classic cryptography (see Shor's Algorithm). Because an increase of the size
of the key-pair would not provide a secure solution in this case, a complete replacement of the algorithm is needed. Therefore, post-quantum public-key cryptography must rely on problems that are
different from the ones used in classic public-key cryptography (i.e., the integer factorization problem, the finite-field discrete logarithm problem, and the elliptic-curve discrete logarithm
A malicious actor with adequate resources can launch an attack to store sensitive encrypted data today that can be decrypted once a CRQC is available. This implies that, every day, sensitive
encrypted data is susceptible to the attack by not implementing quantum-safe strategies, as it corresponds to data being deciphered in the future.¶
| | |
| y | x |
| | <--------------->
| z | Security gap
Figure 1: Mosca model
These challenges are illustrated nicely by the so called Mosca model discussed in [Threat-Report]. In the Figure 1, "x" denotes the time that our systems and data need to remain secure, "y" the
number of years to migrate to a PQC infrastructure and "z" the time until a CRQC that can break current cryptography is available. The model assumes that encrypted data can be intercepted and stored
before the migration is completed in "y" years. This data remains vulnerable for the complete "x" years of their lifetime, thus the sum "x+y" gives us an estimate of the full timeframe that data
remain insecure. The model essentially asks how are we preparing our IT systems during those "y" years (or in other words, how can one minimize those "y" years) to minimize the transition phase to a
PQC infrastructure and hence minimize the risks of data being exposed in the future.¶
Finally, other factors that could accelerate the introduction of a CRQC should not be under-estimated, like for example faster-than-expected advances in quantum computing and more efficient versions
of Shor’s algorithm requiring less qubits. As an example, IBM, one of the leading actors in the development of a large-scale quantum computer, has recently published a roadmap committing to new
quantum processors supporting more than 1000 qubits by 2025 and networked systems with 10k-100k qubits beyond 2026 [IBMRoadmap]. Innovation often comes in waves, so it is to the industry’s benefit to
remain vigilant and prepare as early as possible.¶
The current set of problems used in post-quantum cryptography can be currently grouped into three different categories: lattice-based, hash-based and code-based.¶
Lattice-based public-key cryptography leverages the simple construction of lattices (i.e., a regular collection of points in a Euclidean space that are regularly spaced) to build problems that are
hard to solve such as the Shortest Vector or Closes Vector Problem, Learning with Errors, and Learning with Rounding. All these problems have good proof for worst-to-average case reduction, thus
equating the hardness of the average case to the worst-case.¶
The possibility to implement public-key schemes on lattices is tied to the characteristics of the basis used for the lattice. In particular, solving any of the mentioned problems can be easy when
using reduced or "good" basis (i.e., as short as possible and as orthogonal as possible), while it becomes computationally infeasible when using "bad" basis (i.e., long vectors not orthogonal).
Although the problem might seem trivial, it is computationally hard when considering many dimensions. Therefore, a typical approach is to use "bad" basis for public keys and "good" basis for private
keys. The public keys ("bad" basis) let you easily verify signatures by checking, for example, that a vector is the closest or smallest, but do not let you solve the problem (i.e., finding the
vector). Conversely, private keys (i.e., the "good" basis) can be used for generating the signatures (e.g., finding the specific vector). Signing is equivalent to solving the lattice problem.¶
Lattice-based schemes usually have good performances and average size public keys and signatures, making them good candidates for general-purpose use such as replacing the use of RSA in PKIX
Examples of such class of algorithms include Kyber, Falcon and Dilithium.¶
It is noteworthy that, lattice-based encryption schemes are often prone to decryption failures, meaning that valid encryptions are decrypted incorrectly; as such, an attacker could significantly
reduce the security of lattice-based schemes that have a relatively high failure rate. However, for most of the NIST Post-Quantum Proposals, the number of required oracle queries is above practical
limits, as has been shown in [LattFail1]. More recent works have improved upon the results in [LattFail1], showing that the cost of searching for additional failing ciphertexts after one or more have
already been found, can be sped up dramatically [LattFail2]. Nevertheless, at this point in time (July 2023), the PQC candidates by NIST are considered secure under these attacks and we suggest
constant monitoring as cryptanalysis research is ongoing.¶
Hash based PKC has been around since the 70s, developed by Lamport and Merkle which creates a digital signature algorithm and its security is mathematically based on the security of the selected
cryptographic hash function. Many variants of hash based signatures have been developed since the 70s including the recent XMSS [RFC8391], HSS/LMS [RFC8554] or BPQS schemes. Unlike digital signature
techniques, most hash-based signature schemes are stateful, which means that signing necessitates the update of the secret key.¶
SPHINCS on the other hand leverages the HORS (Hash to Obtain Random Subset) technique and remains the only hash based signature scheme that is stateless.¶
SPHINCS+ is an advancement on SPHINCS which reduces the signature sizes in SPHINCS and makes it more compact. SPHINCS+ was recently standardized by NIST.¶
This area of cryptography stemmed in the 1970s and 80s based on the seminal work of McEliece and Niederreiter which focuses on the study of cryptosystems based on error-correcting codes. Some popular
error correcting codes include the Goppa codes (used in McEliece cryptosystems), encoding and decoding syndrome codes used in Hamming Quasi-Cyclic (HQC) or Quasi-cyclic Moderate density parity check
(QC-MDPC) codes.¶
Examples include all the NIST Round 4 (unbroken) finalists: Classic McEliece, HQC, BIKE.¶
Key Encapsulation Mechanism (KEM) is a cryptographic technique used for securely exchanging symmetric keys between two parties over an insecure channel. It is commonly used in hybrid encryption
schemes, where a combination of asymmetric (public-key) and symmetric encryption is employed. The KEM encapsulation results in a fixed-length symmetric key that can be used in one of two ways: (1)
Derive a Data Encryption Key (DEK) to encrypt the data (2) Derive a Key Encryption Key (KEK) used to wrap the DEK.¶
KEM relies on the following primitives [PQCAPI]:¶
• def kemKeyGen() -> (pk, sk)¶
• def kemEncaps(pk) -> (ct, ss)¶
• def kemDecaps(ct, sk) -> ss¶
where pk is public key, sk is secret key, ct is the ciphertext representing an encapsulated key, and ss is shared secret. The following figure illustrates a sample flow of KEM:¶
PQ KEMs are interactive in nature because it involves back-and-forth communication to negotiate and establish the shared secret key and unlike Diffie-Hellman (DH) Key exchange (KEX) which provides
non-interactive key exchange (NIKE) property. NIKE is a cryptographic primitive which enables two parties, who know each others public keys, to agree on a symmetric shared key without requiring any
interaction. The following figure illustrates a sample flow of DH:¶
HPKE (Hybrid public key encryption) [RFC9180] deals with a variant of KEM which is essentially a PKE of arbitrary sized plaintexts for a recipient public key. It works with a combination of KEMs,
KDFs and AEAD schemes (Authenticated Encryption with Additional Data). HPKE includes three authenticated variants, including one that authenticates possession of a pre-shared key and two optional
ones that authenticate possession of a key encapsulation mechanism (KEM) private key. Kyber, which is a KEM does not support the static-ephemeral key exchange that allows HPKE based on DH based KEMs
its (optional) authenticated modes as discussed in Section 1.2 of [I-D.westerbaan-cfrg-hpke-xyber768d00-02].¶
• IND-CCA2 : IND-CCA2 (INDistinguishability under adaptive Chosen-Ciphertext Attack) is an advanced security notion for encryption schemes. It ensures the confidentiality of the plaintext,
resistance against chosen-ciphertext attacks, and prevents the adversary from forging new ciphertexts. An appropriate definition of IND-CCA2 security for KEMs can be found in [CS01] and [BHK09].
Kyber, Classic McEliece and Saber provide IND-CCA2 security.¶
Understanding IND-CCA2 security is essential for individuals involved in designing or implementing cryptographic systems to evaluate the strength of the algorithm, assess its suitability for specific
use cases, and ensure that data confidentiality and security requirements are met.¶
Any digital signature scheme that provides a construction defining security under post quantum setting falls under this category of PQ signatures.¶
• EUF-CMA : EUF-CMA (Existential Unforgeability under Chosen Message Attack) [GMR88] is a security notion for digital signature schemes. It guarantees that an adversary, even with access to a
signing oracle, cannot forge a valid signature for an arbitrary message. EUF-CMA provides strong protection against forgery attacks, ensuring the integrity and authenticity of digital signatures
by preventing unauthorized modifications or fraudulent signatures. Dilithium, Falcon and Sphincs+ provide EUF-CMA security.¶
Understanding EUF-CMA security is essential for individual involved in designing or implementing cryptographic systems to ensure the security, reliability, and trustworthiness of digital signature
schemes. It allows for informed decision-making, vulnerability analysis, compliance with standards, and designing systems that provide strong protection against forgery attacks.¶
Dilithium [Dilithium] is a digital signature algorithm (part of the CRYSTALS suite) based on the hardness lattice problems over module lattices (i.e., the Module Learning with Errors problem(MLWE)).
The design of the algorithm is based on Fiat Shamir with Abort method that leverages rejection sampling to render lattice based FS schemes compact and secure. Additionally, Dilithium offers both
deterministic and randomized signing. Security properties of Dilithium are discussed in Section 9 of [I-D.ietf-lamps-dilithium-certificates].¶
Falcon [Falcon] is based on the GPV hash-and-sign lattice-based signature framework introduced by Gentry, Peikert and Vaikuntanathan, which is a framework that requires a class of lattices and a
trapdoor sampler technique.¶
The main design principle of Falcon is compactness, i.e. it was designed in a way that achieves minimal total memory bandwidth requirement (the sum of the signature size plus the public key size).
This is possible due to the compactness of NTRU lattices. Falcon also offers very efficient signing and verification procedures. The main potential downsides of Falcon refer to the non-triviality of
its algorithms and the need for floating point arithmetic support.¶
Access to a robust floating-point stack in Falcon is essential for accurate, efficient, and secure execution of the mathematical computations involved in the scheme. It helps maintain precision,
supports error correction techniques, and contributes to the overall reliability and performance of Falcon's cryptographic operations as well makes it more resistant to side-channel attacks.¶
Falcon's signing operations require constant-time, 64-bit floating point operations to avoid catastrophic side channel vulnerabilities. Doing this correctly (which is also platform-dependent to an
extreme degree) is very difficult, as NIST's report noted. Providing a masked implementation of Falcon also seems impossible, per the authors at the RWPQC 2023 symposium earlier this year.¶
The performance characteristics of Dilithium and Falcon may differ based on the specific implementation and hardware platform. Generally, Dilithium is known for its relatively fast signature
generation, while Falcon can provide more efficient signature verification. The choice may depend on whether the application requires more frequent signature generation or signature verification. For
further clarity, please refer to the tables in sections Section 12 and Section 13.¶
SPHINCS+ [SPHINCS] utilizes the concept of stateless hash-based signatures, where each signature is unique and unrelated to any previous signature (as discussed in Section 9.2). This property
eliminates the need for maintaining state information during the signing process. SPHINCS+ was designed to sign up to 2^64 messages and it offers three security levels. The parameters for each of the
security levels were chosen to provide 128 bits of security, 192 bits of security, and 256 bits of security. SPHINCS+ offers smaller key sizes, larger signature sizes, slower signature generation,
and slower verification when compared to Dilithium and Falcon. SPHINCS+ does not introduce a new intractability assumption. It builds upon established foundations in cryptography, making it a
reliable and robust digital signature scheme for a post-quantum world. The advantages and disadvantages of SPHINCS+ over other signature algorithms is disussed in Section 3.1 of [
The eXtended Merkle Signature Scheme (XMSS) [RFC8391] and Leighton-Micali Signature (LMS) [RFC8554] are stateful hash-based signature schemes, where the secret key changes over time. In both schemes,
reusing a secret key state compromises cryptographic security guarantees.¶
Multi-Tree XMSS and LMS can be used for signing a potentially large but fixed number of messages and the number of signing operations depends upon the size of the tree. XMSS and LMS provide
cryptographic digital signatures without relying on the conjectured hardness of mathematical problems, instead leveraging the properties of cryptographic hash functions. XMSS and Hierarchical
Signature System (HSS) use a hierarchical approach with a Merkle tree at each level of the hierarchy. [RFC8391] describes both single-tree and multi-tree variants of XMSS, while [RFC8554] describes
the Leighton-Micali One-Time Signature (LM-OTS) system as well as the LMS and HSS N-time signature systems. Comparison of XMSS and LMS is discussed in Section 10 of [RFC8554].¶
The number of tree layers in XMSS^MT provides a trade-off between signature size on the one side and key generation and signing speed on the other side. Increasing the number of layers reduces key
generation time exponentially and signing time linearly at the cost of increasing the signature size linearly.¶
XMSS and LMS can be applied in various scenarios where digital signatures are required, such as software updates.¶
Within the hash-then-sign paradigm, the message is hashed before signing it. Hashing the message before signing it provides an additional layer of security by ensuring that only a fixed-size digest
of the message is signed, rather than the entire message itself. By pre-hashing, the onus of resistance to existential forgeries becomes heavily reliant on the collision-resistance of the hash
function in use. As well as this security goal, the hash-then-sign paradigm also has the ability to improve performance by reducing the size of signed messages. As a corollary, hashing remains
mandatory even for short messages and assigns a further computational requirement onto the verifier. This makes the performance of hash-then-sign schemes more consistent, but not necessarily more
efficient. Using a hash function to produce a fixed-size digest of a message ensures that the signature is compatible with a wide range of systems and protocols, regardless of the specific message
size or format. Hash-then-Sign also greatly reduces the amount of data that needs to be processed by a hardware security module, which sometimes have somewhat limited data processing capabilities.¶
Protocols like TLS 1.3 and DNSSEC use the Hash-then-Sign paradigm. TLS 1.3 [RFC8446] uses it in the Certificate Verify to proof that the endpoint possesses the private key corresponding to its
certificate, while DNSSEC [RFC4033] uses it to provide origin authentication and integrity assurance services for DNS data.¶
In the case of Dilithium, it internally incorporates the necessary hash operations as part of its signing algorithm. Dilithium directly takes the original message, applies a hash function internally,
and then uses the resulting hash value for the signature generation process. In case of SPHINCS+, it internally performs randomized message compression using a keyed hash function that can process
arbitrary length messages. In case of Falcon, a hash function is used as part of the signature process, it uses the SHAKE-256 hash function to derive a digest of the message being signed. Therefore,
the hash-then-sign paradigm is not needed for Dilithium, SPHINCS+ and Falcon.¶
The table below denotes the 5 security levels provided by NIST required for PQC algorithms. Users can leverage the required algorithm based on the security level based on their use case. The security
is defined as a function of resources required to break AES and SHA2/SHA3 algorithms, i.e., exhaustive key recovery for AES and optimal collision search for SHA2/SHA3.¶
Table 1
PQ Security Level AES/SHA(2/3) hardness PQC Algorithm
1 Atleast as hard as to break AES-128 (exhaustive key recovery) Kyber512, Falcon512, Sphincs+SHA-256 128f/s
2 Atleast as hard as to break SHA-256/SHA3-256 (collision search) Dilithium2
3 Atleast as hard as to break AES-192 (exhaustive key recovery) Kyber768, Dilithium3, Sphincs+SHA-256 192f/s
4 Atleast as hard as to break SHA-384/SHA3-384 (collision search) No algorithm tested at this level
5 Atleast as hard as to break AES-256 (exhaustive key recovery) Kyber1024, Falcon1024, Dilithium5, Sphincs+SHA-256 256f/s
Please note the Sphincs+SHA-256 x"f/s" in the above table denotes whether its the Sphincs+ fast (f) version or small (s) version for "x" bit AES security level. Refer to [
I-D.ietf-lamps-cms-sphincs-plus-02] for further details on Sphincs+ algorithms.¶
The following table discusses the signature size differences for similar SPHINCS+ algorithm security levels with the "simple" version but for different categories i.e., (f) for fast verification and
(s) for compactness/smaller. Both SHA-256 and SHAKE-256 parametrisation output the same signature sizes, so both have been included.¶
Table 2
PQ Security Level Algorithm Public key size (in bytes) Private key size (in bytes) Signature size (in bytes)
1 SPHINCS+-{SHA2,SHAKE}-128f 32 64 17088
1 SPHINCS+-{SHA2,SHAKE}-128s 32 64 7856
3 SPHINCS+-{SHA2,SHAKE}-192f 48 96 35664
3 SPHINCS+-{SHA2,SHAKE}-192s 48 96 16224
5 SPHINCS+-{SHA2,SHAKE}-256f 64 128 49856
5 SPHINCS+-{SHA2,SHAKE}-256s 64 128 29792
The following table discusses the impact of performance on different security levels in terms of private key sizes, public key sizes and ciphertext/signature sizes.¶
Table 3
PQ Security Level Algorithm Public key size (in bytes) Private key size (in bytes) Ciphertext/Signature size (in bytes)
1 Kyber512 800 1632 768
1 Falcon512 897 1281 666
2 Dilithium2 1312 2528 2420
3 Kyber768 1184 2400 1088
5 Falcon1024 1793 2305 1280
5 Kyber1024 1568 3168 1588
In this section, we provide two tables for comparison of different KEMs and Signatures respectively, in the traditional and Post scenarios. These tables will focus on the secret key sizes, public key
sizes, and ciphertext/signature sizes for the PQC algorithms and their traditional counterparts of similar security levels.¶
The first table compares traditional vs. PQC KEMs in terms of security, public, private key sizes, and ciphertext sizes.¶
Table 4
PQ Security Level Algorithm Public key size (in bytes) Private key size (in bytes) Ciphertext size (in bytes)
Traditional P256_HKDF_SHA-256 65 32 65
Traditional P521_HKDF_SHA-512 133 66 133
Traditional X25519_HKDF_SHA-256 32 32 32
1 Kyber512 800 1632 768
3 Kyber768 1184 2400 1088
5 Kyber1024 1568 3168 1588
The next table compares traditional vs. PQC Signature schemes in terms of security, public, private key sizes, and signature sizes.¶
Table 5
PQ Security Level Algorithm Public key size (in bytes) Private key size (in bytes) Signature size (in bytes)
Traditional RSA2048 256 256 256
Traditional P256 64 32 64
1 Falcon512 897 1281 666
2 Dilithium2 1312 2528 768
3 Dilithium3 1952 4000 3293
5 Falcon1024 1793 2305 1280
As one can clearly observe from the above tables, leveraging a PQC KEM/Signature significantly increases the key sizes and the ciphertext/signature sizes as well as compared to traditional KEM(KEX)/
Signatures. But the PQC algorithms do provide the additional security level in case there is an attack from a CRQC, whereas schemes based on prime factorization or discrete logarithm problems (finite
field or elliptic curves) would provide no level of security at all against such attacks.¶
The migration to PQC is unique in the history of modern digital cryptography in that neither the traditional algorithms nor the post-quantum algorithms are fully trusted to protect data for the
required lifetimes. The traditional algorithms, such as RSA and elliptic curve, will fall to quantum cryptalanysis, while the post-quantum algorithms face uncertainty about the underlying
mathematics, compliance issues, unknown vulnerabilities, and hardware and software implementations that have not had sufficient maturing time to rule out classical cryptanalytic attacks and
implementation bugs.¶
During the transition from traditional to post-quantum algorithms, there may be a desire or a requirement for protocols that use both algorithm types. [I-D.ietf-pquip-pqt-hybrid-terminology] defines
the terminology for the Post-Quantum and Traditional Hybrid Schemes.¶
The PQ/T Hybrid Confidentiality property can be used to protect from a "Harvest Now, Decrypt Later" attack, which refers to an attacker collecting encrypted data now and waiting for quantum computers
to become powerful enough to break the encryption later. Two types of hybrid key agreement schemes are discussed below:¶
1. Concatenate hybrid key agreement scheme: The final shared secret that will be used as an input of the key derivation function is the result of the concatenation of the secrets established with
each key agreement scheme. For example, in [I-D.ietf-tls-hybrid-design], the client uses the TLS supported groups extension to advertise support for a PQ/T hybrid scheme, and the server can
select this group if it supports the scheme. The hybrid-aware client and server establish a hybrid secret by concatenating the two shared secrets, which is used as the shared secret in the
existing TLS 1.3 key schedule.¶
2. Cascade hybrid key agreement scheme: The final shared secret is computed by applying as many iterations of the key derivation function as the number of key agreement schemes composing the hybrid
key agreement scheme. For example, [RFC9370] extends the Internet Key Exchange Protocol Version 2 (IKEv2) to allow one or more PQC algorithms in addition to the traditional algorithm to derive
the final IKE SA keys using the cascade method as explained in Section 2.2.2 of [RFC9370].¶
The PQ/T Hybrid Authentication property can be utilized in scenarios where an on-path attacker possesses network devices equipped with CRQCs, capable of breaking traditional authentication protocols.
This property ensures authentication through a PQ/T hybrid scheme or a PQ/T hybrid protocol, as long as at least one component algorithm remains secure to provide the intended security level. For
instance, a PQ/T hybrid certificate can be employed to facilitate a PQ/T hybrid authentication protocol. However, a PQ/T hybrid authentication protocol does not need to use a PQ/T hybrid certificate
[I-D.ounsworth-pq-composite-keys]; separate certificates could be used for individual component algorithms [I-D.ietf-lamps-cert-binding-for-multi-auth].¶
The frequency and duration of system upgrades and the time when CRQCs will become widely available need to be weighed in to determine whether and when to support the PQ/T Hybrid Authentication
It is also possible to use more than two algorithms together in a hybrid scheme, and there are multiple possible ways those algorithms can be combined. For the purposes of a post-quantum transition,
the simple combination of a post-quantum algorithm with a single classical algorithm is the most straightforward, but the use of multiple post-quantum algorithms with different hard math problems has
also been considered. When combining algorithms, it is possible to require that both algorithms validate (the so-called "and" mode) or that only one does (the "or" mode), or even some more
complicated scheme. Schemes that do not require both algorithms to validate only have the strength of the weakest algorithm, and therefore offer little or no security benefit. Since such schemes
generally also require both keys to be distributed (e.g. https://datatracker.ietf.org/doc/html/draft-truskovsky-lamps-pq-hybrid-x509-01), there are substantial performance costs in some scenarios.
This combination of properties makes optionally including post-quantum keys without requiring their use to be generally unattractive in most use cases.¶
When combining keys in an "and" mode, it may make more sense to consider them to be a single composite key, instead of two keys. This generally requires fewer changes to various components of PKI
ecosystems, many of which are not prepared to deal with two keys or dual signatures. To them, a "composite" algorithm composed of two other algorithms is simply a new algorithm, and support for
adding new algorithms generally already exists. All that needs to be done is to standardize the formats of how the two keys from the two algorithms are combined into a single data structure, and how
the two resulting signatures are combined into a single signature. The answer can be as simple as concatenation, if the lengths are fixed or easily determined.¶
One last consideration is the pairs of algorithms that can be combined. A recent trends in protocols is to only allow a small number of "known good" configurations that make sense, instead of
allowing arbitrary combinations of individual configuration choices that may interact in dangerous ways. The current consensus is that the same approach should be followed for combining cryptographic
algorithms, and that "known good" pairs should be explicitly listed ("explicit composite"), instead of just allowing arbitrary combinations of any two crypto algorithms ("generic composite").¶
The same considerations apply when using multiple certificates to transport a pair of related keys for the same subject. Exactly how two certificates should be managed in order to avoid some of the
pitfalls mentioned above is still an active area of investigation. Using two certificates keeps the certificate tooling simple and straightforward, but in the end simply moves the problems with
requiring that both certs are intended to be used as a pair, and both must validate, to the certificate management layer, where they still need to be addressed.¶
At least one scheme has been proposed that allows the pair of certificates to exist as a single certificate when being issued and managed, but dynamically split into individual certificates when
needed (https://datatracker.ietf.org/doc/draft-bonnell-lamps-chameleon-certs/).¶
Many of these points are still being actively explored and discussed, and the consensus may change over time.¶
Classical cryptanalysis exploits weaknesses in algorithm design, mathematical vulnerabilities, or implementation flaws, whereas quantum cryptanalysis harnesses the power of CRQCs to solve specific
mathematical problems more efficiently. Both pose threats to the security of cryptographic algorithms, including those used in PQC. Developing and adopting new cryptographic algorithms resilient
against these threats is crucial for ensuring long-term security in the face of advancing cryptanalysis techniques. Recent attacks on the side-channel implementations using deep learning based power
analysis have also shown that one needs to be cautious while implementing the required PQC algorithms in hardware. Two of the most recent works include: one attack on Kyber [KyberSide] and one attack
on Saber [SaberSide]. Evolving threat landscape points to the fact that lattice based cryptography is indeed more vulnerable to side-channel attacks as in [SideCh], [LatticeSide]. Consequently, there
were some mitigation techniques for side channel attacks that have been proposed as in [Mitigate1], [Mitigate2], and [Mitigate3].¶
Cryptographic agility is relevant for both classical and quantum cryptanalysis as it enables organizations to adapt to emerging threats, adopt stronger algorithms, comply with standards, and plan for
long-term security in the face of evolving cryptanalytic techniques and the advent of CRQCs. Several PQC schemes are available that need to be tested; cryptography experts around the world are
pushing for the best possible solutions, and the first standards that will ease the introduction of PQC are being prepared. It is of paramount importance and a call for imminent action for
organizations, bodies, and enterprises to start evaluating their cryptographic agility, assess the complexity of implementing PQC into their products, processes, and systems, and develop a migration
plan that achieves their security goals to the best possible extent.¶
Post-quantum algorithms selected for standardization are relatively new and they they have not been subject to the same depth of study as traditional algorithms. In addition, certain deployments may
need to retain traditional algorithms due to regulatory constraints, for example FIPS compliance. Hybrid key exchange enables potential security against "Harvest Now, Decrypt Later" attack while not
fully abandoning traditional cryptosystems.¶
(A reading list. Serious Cryptography. Pointers to PQC sites with good explanations. List of reasonable Wikipedia pages.)¶
The following individuals have contributed to this document:¶
Kris Kwiatkowski¶
PQShield, LTD¶
United Kingdom.¶
It leverages text from https://github.com/paulehoffman/post-quantum-for-engineers/blob/main/pqc-for-engineers.md. Thanks to Dan Wing, Florence D, Thom Wiggers, Sophia Grundner-Culemann, Sofia Celi,
Melchior Aelmans, and Falko Strenzke for the discussion, review and comments.¶
"Subtleties in the Definition of IND-CCA: When and How Should Challenge-Decryption be Disallowed?", <https://eprint.iacr.org/2009/418>.
"NIST’s pleasant post-quantum surprise", <https://blog.cloudflare.com/nist-post-quantum-surprise/>.
"Announcing the Commercial National Security Algorithm Suite 2.0", <https://media.defense.gov/2022/Sep/07/2003071834/-1/-1/0/CSA_CNSA_2.0_ALGORITHMS_.PDF>.
"Design and Analysis of Practical Public-Key Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack", <https://eprint.iacr.org/2001/108>.
"Cryptographic Suite for Algebraic Lattices (CRYSTALS) - Dilithium", <https://pq-crystals.org/dilithium/index.shtml>.
"Fast Fourier lattice-based compact signatures over NTRU", <https://falcon-sign.info/>.
"A digital signature scheme secure against adaptive chosen-message attacks.", <https://people.csail.mit.edu/silvio/Selected%20Scientific%20Papers/Digital%20Signatures/
"Quantum Supremacy Using a Programmable Superconducting Processor", <https://ai.googleblog.com/2019/10/quantum-supremacy-using-programmable.html>.
"C. Zalka, “Grover’s quantum searching algorithm is optimal,” Physical Review A, vol. 60, pp. 2746-2751, 1999.".
Prorock, M., Steele, O., Misoczki, R., Osborne, M., and C. Cloostermans, "JOSE and COSE Encoding for SPHINCS+", Work in Progress, Internet-Draft, draft-ietf-cose-sphincs-plus-01, , <https://
Becker, A., Guthrie, R., and M. J. Jenkins, "Related Certificates for Use in Multiple Authentications within a Protocol", Work in Progress, Internet-Draft,
draft-ietf-lamps-cert-binding-for-multi-auth-01, , <https://datatracker.ietf.org/doc/html/draft-ietf-lamps-cert-binding-for-multi-auth-01>.
Housley, R., Fluhrer, S., Kampanakis, P., and B. Westerbaan, "Use of the SPHINCS+ Signature Algorithm in the Cryptographic Message Syntax (CMS)", Work in Progress, Internet-Draft,
draft-ietf-lamps-cms-sphincs-plus-02, , <https://datatracker.ietf.org/doc/html/draft-ietf-lamps-cms-sphincs-plus-02>.
Massimo, J., Kampanakis, P., Turner, S., and B. Westerbaan, "Internet X.509 Public Key Infrastructure: Algorithm Identifiers for Dilithium", Work in Progress, Internet-Draft,
draft-ietf-lamps-dilithium-certificates-02, , <https://datatracker.ietf.org/doc/html/draft-ietf-lamps-dilithium-certificates-02>.
D, F., "Terminology for Post-Quantum Traditional Hybrid Schemes", Work in Progress, Internet-Draft, draft-ietf-pquip-pqt-hybrid-terminology-00, , <https://datatracker.ietf.org/doc/html/
Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key exchange in TLS 1.3", Work in Progress, Internet-Draft, draft-ietf-tls-hybrid-design-06, , <https://datatracker.ietf.org/doc/html/
Ounsworth, M., Gray, J., Pala, M., and J. Klaußner, "Composite Public and Private Keys For Use In Internet PKI", Work in Progress, Internet-Draft, draft-ounsworth-pq-composite-keys-05, , <https:/
Westerbaan, B. and C. A. Wood, "X25519Kyber768Draft00 hybrid post-quantum KEM for HPKE", Work in Progress, Internet-Draft, draft-westerbaan-cfrg-hpke-xyber768d00-02, , <https://
"IBM Unveils 400 Qubit-Plus Quantum Processor and Next-Generation IBM Quantum System Two", <https://newsroom.ibm.com/
"The IBM Quantum Development Roadmap", <https://www.ibm.com/quantum/roadmap>.
"A Side-Channel Attack on a Hardware Implementation of CRYSTALS-Kyber", <https://eprint.iacr.org/2022/1452>.
"Decryption Failure Attacks on IND-CCA Secure Lattice-Based Schemes", <https://link.springer.com/chapter/10.1007/978-3-030-17259-6_19#chapter-info>.
"(One) Failure Is Not an Option: Bootstrapping the Search for Failures in Lattice-Based Encryption Schemes.", <https://link.springer.com/chapter/10.1007/978-3-030-45727-3_1>.
"Generic Side-channel attacks on CCA-secure lattice-based PKE and KEM schemes", <https://eprint.iacr.org/2019/948>.
"POLKA: Towards Leakage-Resistant Post-Quantum CCA-Secure Public Key Encryption", <https://eprint.iacr.org/2022/873>.
"Leakage-Resilient Certificate-Based Authenticated Key Exchange Protocol", <https://ieeexplore.ieee.org/document/9855226>.
"Post-Quantum Authenticated Encryption against Chosen-Ciphertext Side-Channel Attacks", <https://eprint.iacr.org/2022/916>.
"Post-Quantum Cryptography Standardization", <https://csrc.nist.gov/projects/post-quantum-cryptography/post-quantum-cryptography-standardization>.
"Interference Measurements of Non-Abelian e/4 & Abelian e/2 Quasiparticle Braiding", <https://journals.aps.org/prx/pdf/10.1103/PhysRevX.13.011028>.
"PQC - API notes", <https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/example-files/api-notes.pdf>.
"Quantum Computing and the DNS", <https://www.icann.org/octo-031-en.pdf>.
Arends, R., Austein, R., Larson, M., Massey, D., and S. Rose, "DNS Security Introduction and Requirements", RFC 4033, DOI 10.17487/RFC4033, , <https://www.rfc-editor.org/rfc/rfc4033>.
McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic Curve Cryptography Algorithms", RFC 6090, DOI 10.17487/RFC6090, , <https://www.rfc-editor.org/rfc/rfc6090>.
Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfc-editor.org/rfc/rfc8446>.
Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid Public Key Encryption", RFC 9180, DOI 10.17487/RFC9180, , <https://www.rfc-editor.org/rfc/rfc9180>.
Tjhai, CJ., Tomlinson, M., Bartlett, G., Fluhrer, S., Van Geest, D., Garcia-Morchon, O., and V. Smyslov, "Multiple Key Exchanges in the Internet Key Exchange Protocol Version 2 (IKEv2)", RFC 9370
, DOI 10.17487/RFC9370, , <https://www.rfc-editor.org/rfc/rfc9370>.
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems+", <https://dl.acm.org/doi/pdf/10.1145/359340.359342>.
"Breaking RSA Encryption - an Update on the State-of-the-Art", <https://www.quintessencelabs.com/blog/breaking-rsa-encryption-update-state-art>.
"How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", <https://arxiv.org/abs/1905.09749>.
"A side-channel attack on a masked and shuffled software implementation of Saber", <https://link.springer.com/article/10.1007/s13389-023-00315-3>.
"Side-Channel Attacks on Lattice-Based KEMs Are Not Prevented by Higher-Order Masking", <https://eprint.iacr.org/2022/919>.
"SPHINCS+", <https://sphincs.org/index.html>.
"Quantum Threat Timeline Report 2020", <https://globalriskinstitute.org/publications/quantum-threat-timeline-report-2020/>. | {"url":"https://datatracker.ietf.org/doc/html/draft-ar-pquip-pqc-engineers","timestamp":"2024-11-04T18:28:14Z","content_type":"text/html","content_length":"179566","record_id":"<urn:uuid:6a5d05bb-ceb7-4992-967c-29c5517b8eba>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00261.warc.gz"} |
GAME CATEGORY 2 - GAME & MORE
GAME CATEGORY 2 – STRATEGY AND LOGIC …
GAME CATEGORY 2 contains game suggestions that particularly promote logical and tactical thinking as well as the concentration and attentiveness of the players.
The game suggestions show, for example, how an old strategy game can be played with the GAME & MORE dice and how some new, interesting strategy and logic games have been created using extended rules.
GAME & MORE – GAME SUGGESTION 2.1 …
Game suggestion 2.1 is a very old two-person strategy game that is widely known as “Tic-Tac-Toe” or “Three Wins”. This old strategy game, whose history can be traced back to the 12th century BC, can
also be played with the GAME & MORE dice. The 18 dice on the game box even allow you to play 2 games at the same time, which is also very appealing.
On the two square playing fields (3×3 dice), the two players take it in turns to place their markers (dice with a hole or dice without a hole) on one of the 9 possible spaces (if you play the game on
the right and left side of the packaging, the two maximum “playing fields” can be easily recognised). The player who manages to place three markers in a row, column or diagonal in both playing fields
wins the game. If neither or both players manage to do this, there is a draw.
In the example shown, you can see who won one of the two games.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.2 …
Game suggestion 2.2 is a further development of the two-person strategy game described in game suggestion 2.1.
As GAME & MORE contains more dice than you need to play the old strategy game, I have extended the rules and this has resulted in this interesting game variant:
A major difference to the already known game is that now the “playing field” is only limited in one direction (maximum 3 markers, or dice) and is open in the other direction. The aim of this game
variant is also to position 3 identical markers in a row, column or diagonal. As the new game variant is played alternately with 9 different dice, the open “playing field” means that 4 or more
identical dice can be placed next to each other, but these are counted as 3 identical markers. The player who most often manages to position 3 identical dice in a row, column or diagonal wins the
In the example shown, you can see how the player with the hole cubes won this game by 4 points to 3.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.3 …
In game suggestion 2.3, there are 2 playing fields on which the game is played simultaneously. The two players involved each receive 9 dice (one player receives 9 dice with holes, the other 9 dice
without holes).
Both players place one of their dice on the table in front of them. From the point of view of the two players sitting opposite each other, each of these positioned cubes will later form the
respective baseline of a “playing field” (maximum 3 cubes next to each other).
The next cube can now be positioned. Make sure that this cube is positioned in the “playing field” in which the opponent’s first cube is already located.
All dice can then be positioned alternately in the two playing fields. The aim of the strategy game is also to get 3 identical markers in a row, column or diagonal and, of course, as many of them as
possible. It does not matter whether this is done in both “playing fields” or only in one “playing field”. Both “playing fields” are not limited above their base lines and can therefore be of
different heights. Another special feature of this game variant is that L-shaped markers, each consisting of 4 dice, are also scored. The open “playing field” also means that 4 or more identical dice
can lie next to each other in this game variant, but they are scored as 3 identical markers.
In the example shown, you can see how the player with the hole cubes won this game by 5 to 4 points.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.4 …
Game suggestion 2.4 is an interesting two-player strategy game in which 4 small, square “playing fields” are created and played on simultaneously.
One of the two players starts the game and places one of their 8 dice on the table. The other player takes their turn and can either place their die anywhere on the die already on the table (on
either side, even diagonally) or use this die to create another small, square “playing field”.
Where the two players position their dice within the 4 small “playing fields” should be strategically well considered, as the 4 small squares are then pushed together to form a large square and it is
then easy to see who has positioned their dice most cleverly.
The aim of this strategy game is to get 3 identical markers in a row, column or diagonal and to get as many of them as possible. With this large “playing field” with 16 dice, 4 identical markers can
of course appear next to each other in a row, column or diagonal, but these are only considered and scored as 3 identical markers.
In the example shown, you can see how the player with the hole dice only scored 2 points and lost the game.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.5 …
Game suggestion 2.5 is an interesting 3D strategy game in which a good memory is very helpful – so it is also a game that is good for training the memory.
Each player receives 9 dice (one 9 dice with holes, the other 9 dice without holes).
One of the two players starts the game and places one of their 9 dice on the table. The other player takes their turn and can either place their cube against the cube already on the table (either
side, even diagonally) or position their cube on the cube already on the table.
The base area of the “playing field” is limited to 3 x 2 cubes and the height of the resulting tower of cubes is limited to 3 cubes (see picture A).
The two players (can also be 2 small groups) now take it in turns to stack their dice to form a double dice tower, which will consist of 18 dice once completed. For the dice with holes, one of the
two holes must always be visible at the top and for the dice without holes, one of the two cut surfaces of the dice, which are usually somewhat rougher than the 4 planed surfaces of these dice,
should also always be visible at the top.
During the stacking process, the players must remember where the dice with holes and the dice without holes are positioned in the dice tower.
This is important because both players must try to get 3 identical markings in a row, column or diagonal in one of the two square, vertical “tower discs”, each of which consists of 9 dice, and of
course as many of these as possible.
At this point it becomes clear why it is worth remembering the positions of the dice, because only if you know which dice are in the two lower rows can you position your dice as successfully as
After the dice tower has reached its final height, one of the two players converts it into a horizontal surface of 6 x 3 dice (see picture B) as follows:
This is done as follows: The top row of dice of one of the two “dice tower discs” is placed on the table next to the bottom row of dice at a distance of a few centimetres. The middle row of dice is
then placed between the two rows of dice already on the table (see picture C).
This is now also done with the other “dice tower disc” according to the same principle, only in the other direction (see picture D).
The cubes lying on the table are then carefully pushed together to create the area of 6 x 3 cubes (see picture B).
And now it is very easy to recognise where there are 3 identical markings in a row, column or diagonal.
By pushing the cubes together to form a large area, there may be more than 3 identical markings next to each other in a column, but these are only considered and counted as 3 identical markings.
In the example shown (see picture B) you can see how the game ended in a draw.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.6 …
Game suggestion 2.6 is a two-person strategy game that can be played immediately on a playing surface that you have quickly sketched out.
The playing area consists of 25 square squares and the two players sitting opposite each other are each given 9 dice (one player 9 dice with holes, the other 9 dice without holes).
Before the game begins, the two players each place 5 of their dice on their respective baseline of the playing field. The 4 remaining dice are placed outside the playing field in front of them (see
picture A).
The aim of the two players is to get to the opposite side of the playing field (to their finish line) with their dice and collect as many of the other player’s dice as possible in order to build the
highest possible dice towers.
The two players take turns and only have the following options:
Option 1: One player moves one square straight forwards with one of their dice if the square in front of them is not yet occupied.
Option 2: A player moves one square diagonally forwards with one of their dice. However, this is only possible if there is already a die or dice tower of the other player in the square. If this is
the case, the player places their die on top of the die or dice tower that is already there.
Option 3: A player moves one of their dice towers one space forwards if this space is not already occupied.
Option 4: A player moves their dice tower one space diagonally forwards. This is possible in three cases:
1. if this square is free.
2. if there is a dice or a dice tower of the other player in the square.
3. if there is already one of their own dice towers there.
If there is a dice tower of the other player or one of their own dice towers, the player places their dice tower on top of the respective dice tower. During the course of the game, the player’s dice
towers should be as high as possible before they reach the finish line. Once they reach the finish line, they are no longer allowed to leave their respective location. The top die of a dice tower
always shows who currently owns this dice tower and only the player who has control of this dice tower may move it towards their finish line (only one space forwards or diagonally forwards per turn,
depending on the situation).
Option 5: A player takes one of their dice that is still outside the playing area and places it on an empty space on their baseline.
If none of these move options is possible, the player must sit out until the game situation changes again and they can make another move.
If a player can make a move, they must make it. Only when both players are definitely unable to make a move is the game over and the result of the game can be determined. Only the dice on the dice
towers that reach the finish line are counted.
The player with the most points wins the game.
In the example shown (see picture B), you can see how the player with the hole dice won this game with 15 to 0 points.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.7 …
Game suggestion 2.7 is a variant of the two-person strategy game described in game suggestion 2.2.
The difference is that the “playing field” is now open on all sides and that the dice can be positioned alternately on all sides of a dice (also diagonally, i.e. dice edge to dice edge) on the open
“playing field”.
The aim of this game variant is also to position 3 identical markers in a row, column or diagonal. As this game variant is also played alternately with 9 different dice, 4 or more identical dice can
be placed next to each other, but these are counted as 3 identical markers. The player who most often manages to position 3 identical dice in a row, column or diagonal wins the game.
In the example shown, you can see how the player with the hole cubes won this game by 5 points to 4.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.8 …
Game suggestion 2.8 is a two-person strategy game that can also be played on the familiar playing surface of game suggestion 2.6.
The playing area consists of 25 square squares and the two players sitting opposite each other also receive 9 dice (one player receives 9 dice with holes, the other 9 dice without holes).
The dice are positioned alternately on the boundary lines of the 25 square fields until all the dice of both players are on the playing field (see picture A).
The aim of the game is to position the dice in such a way that several diagonal rows of at least 3 identical dice are created. Additional dice should be positioned in such a way that, once all the
dice have been positioned on the playing field, they can be used to skip over the diagonal rows that have already been created (see image B with the red arrow and the slanted dice). Once this has
been done, the respective player may take 2 cubes from the opponent’s playing field, provided that they are not cubes that belong to a diagonal row of at least 3 cubes that has already been created.
If the player removes one or two of the opponent’s cubes from the game (see image C with the two blue arrows), they must also remove the cube with which they skipped their diagonal row from the game
(see image C with the green arrow).
In the game, each cube may be moved along the boundary lines of the 25 square fields in all 4 directions, but only from one intersection point to the other intersection point per turn.
If a player can make a move, then the move must be carried out. Only when a player has no more dice on the playing field or can no longer make any moves does the opposing player win the game. If both
players still have the same number of dice on the board and it is clear that the game situation can no longer be changed for either player, the game ends in a draw.
In picture D you can see who won the game.
Have fun and good luck!
GAME & MORE – GAME SUGGESTION 2.9 …
Game suggestion 2.9 is also about logic, as the game suggestion is based on binary numbers, which only consist of the digits 0 and 1 and are used in the binary system. This system is the basis for
almost all modern computers and digital systems. The system has fascinated me since my youth and I am delighted that it is now part of my following game proposal:
A selected player of a game group, which can easily consist of more than 10 players, creates an arbitrary dice chain with up to 18 dice (see picture A).
Although the dice chain was created arbitrarily, it represents a specific number (in this example it is a number that is even greater than 2 billion, namely the number 2,521,652,320), which can be
systematically determined by the players.
To do this, the players first mentally divide the dice chain shown into 6 individual dice chains, each consisting of 3 dice (see image B).
The binary numbers, which consist of the digits 0 and 1, are represented in this game with GAME & MORE dice (a die without holes stands for the digit 0 and a die with holes stands for the digit 1).
All players are given a piece of paper and a pen to write down the codes of the binary numbers 0 to 7 (see left-hand column in Figure C). The numbers in the other two columns do not have to be
written down, as in this game only the numbers 1 and 2 are placed in front of the numbers 0 to 7 (see centre and right-hand columns in Figure C).
As you can probably already imagine, the respective arrangement of the 3 dice within a small dice chain plays an important role in this game:
The numbers in column 1 apply to the case where all 3 dice have been positioned straight.
The numbers in column 2 apply if all 3 dice have been positioned at an angle. In this case, only the 1 is placed before the digits 0 to 7.
The numbers in column 3 apply if all 3 dice have been positioned straight and at an angle. In this case, only the 2 is placed in front of the numbers 0 to 7.
The players should memorise this well so that they can start to determine the result they are looking for as follows: First, they look at the dice chain again and again and then write down the
corresponding numbers of the 6 small dice chains on their piece of paper: 25 2 16 5 23 20
If you now push these 6 numbers together and place a dot every 3 digits from right to left, you will already have your result – the correct result in this example would be: 2521652320 = 2,521,652,320
The level of difficulty of the game can of course also be changed, for example by dividing the dice chain into 6 small dice chains with 3 dice each (see picture B), or by making it significantly
The first person to determine a result calls out “1” to the group and writes the number 1 on their piece of paper. The next person to determine a result calls out “2” and writes the number 2 on their
piece of paper. This is done in the same way for the other players.
Once all players have written down their results, the results are compared and points are awarded. The player who was quickest to determine the correct result receives 3 points and every other player
who also determined a correct result receives 1 point.
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Work Energy and Power Class 11 MCQ for NEET Pdf - YB Study
Work Energy and Power Class 11 MCQ for NEET Pdf
Work Energy and Power Class 11 MCQ Pdf
‘work’, ‘energy’ and ‘power’ are frequently used in everyday language. In physics, however, the word ‘Work’ covers a definite and precise meaning. Energy is thus our capacity to do work. In Physics
too, the term ‘energy’ is related to work in this sense, but as said above the term ‘work’ itself is defined much more precisely. The word ‘power’ is used in everyday life with different shades of
meaning. In karate or boxing we talk of ‘powerful’ punches. These are delivered at a great speed. This shade of meaning is close to the meaning of the word ‘power’ used in physics.The aim of this
chapter is to develop an understanding of these three physical quantities.Physical quantities like displacement, velocity, acceleration, force etc. are vectors.Work done is a scalar quantity. It can
be positive or negative unlike mass and kinetic energy which are positive scalar quantities. The work done by the friction or viscous force on a moving body is negative.
MCQs On Work Energy and Power Class 11
1. The potential energy of a system increases if work is done__________
(a) Upon the system by a nonconservative force
(b) By the system against a conservative force
(c) By the system against a nonconservative force
(d) Upon the system by a conservative force
Answer: B
2. A body of mass 1 kg is thrown upwards with a velocity 20 m/s. It momentarily comes to rest after attaining a height of 18 m. How much energy is lost due to air friction ? (g = 10 m/s²)
(a) 10 J
(b) 20 J
(c) 30 J
(d) 40 J
Answer: B
3. A boy of mass 50 kg jumps to a height of 0.8 m from the ground then momentum transferred by the ground to boy is________________
(a) 400 kg m/s
(b) 200 kg m/s
(c) 800 kg m/s
(d) 500 kg m/s
Answer: B
4. A machine which is 75% efficient, uses 12 J of energy in lifting 1 kg mass through a certain distance. The mass is then allowed to fall through the same distance, the velocity at the end of its
fall is___________
(a) 12 m/s
(b) 18 m/s
(c) 24 m/s
(d) 32 m/s
Answer: B
5. A body of mass 10 kg is displaced from point A(2, 1, 3) to point B(3, 3, 4) under the effect of a force of magnitude 20 N in the direction of 6î 8j. Calculate W.D. by the force_______
(a) 22 J
(b) 20 6 J
(c) 44 J
(d) Zero
Answer: C
6. The heart of a man pumps 5 litres of blood through the arteries per minute at a pressure of 150 mm of mercury. If the density of mercury be 13.6 × 103 kg/m³ and g = 10m/s² then the power of heart
in watt is____________
(a) 1.50
(b) 1.70
(c) 2.35
(d) 3.0
Answer: B
7. A body projected vertically from the earth reaches a height equal to earth’s radius before returning to the earth. The power exerted by the gravitational force is greatest_______________
(a) At the highest position of the body
(b) At the instant just before the body hits the earth
(c) It remains constant all through
(d) At the instant just after the body is projected
Answer: B
8. A pump is used to deliver water at a certain rate from a given pipe. To obtain n times water from the same pipe in the same time, by what factor, the force of the motor should be increased?
(a) n times
(b) n²
(c) n³ times
(d) 1/n times
Answer: B
9. Water is falling on the blades of a turbine at a rate
of 100 kg/s from a certain spring. If the height
of the spring be 100 metres, the power transferred
to the turbine will be _________
(a) 100 kW
(b) 10 kW
(c) 30 1 kW
(d) 1000 kW
Answer: A
10. A spring of force constant 800 N/m has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is____________
(a) 16 J
(b) 8 J
(c) 32 J
(d) 24 J
Answer: B
11. A block of mass 20 kg is moved with constant velocity
along an inclined plane of inclination 37° with help
of a force of constant power 50 W. If the coefficient
of kinetic friction between block and surface is 0.25,
then what fraction of power is used against gravity?
(2) ¼
(3) ½
(4) ⅛
Answer: A
12. Velocity of a particle of mass 1 kg moving rectilinearly is given by v = 25 – 2t + t2. Find the
average power of the force acting on the particle
between time interval t = 0 to t = 1 sec.
(a) 49 W
(b) 24.5 W
(c) –49 W
(d) –24.5 W
Answer: D
13. A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude P0 The instantaneous velocity of this car is proportional to________
(a) t–½
(b) t/ m
(c) t² p⁰
(d) t½
Answer: D
14. A block of mass M is attached to the lower end
of a vertical spring. The spring is hung from a ceiling
and has force constant value k. The mass is released
from rest with the spring initially unstretched. the maximum extension produced in the length of the
spring will be_______________
(a) Mg/2k
(b) Mg/k
(c) 2 Mg/k
(d) 4 Mg/k
Answer: C
15. A particle moves from a point ( 2î 5j) to (4j 3k) when a force of (4î 3j) N is applied.
How much work has been done by the force ?
(a) 5 J
(b) 2 J
(c) 8 J
(d) 11 J
Answer: A
16.What average horsepower is developed by an 80kg
man while climbing in 10 s flight of stairs that rises
6 m vertically ?
(a) 0.63 hp
(b) 1.26 hp
(c) 1.8 hp
(d) 2.1 hp
Answer: A
17. A body of mass m starting from rest from origin
moves along x-axis with constant power (P).Calculate relation between velocity/distance _________
(a) x ∝ v½
(b) x ∝ v 2
(c) x ∝ v
(d) x ∝ v3
Answer: D
18. A 1.0 hp motor pumps out water from a well of depth 20 m and fills a water tank of volume 2238
liters at a height of 10 m from the ground. The
running time of the motor to fill the empty water
tank is (g = 10m/s²
(a) 5 minutes
(b) 10 minutes
(c) 15 minutes
(d) 20 minutes
Answer: C
19. A car is moving with a speed of 40 Km/hr. If the car engine generates 7 kilowatt power, then the resistance in the path of motion of the car will be_____________
(a) 360 newton
(b) 630 newton
(c) Zero
(d) 280 newton
Answer: B
20. An electric motor produces a tension of 4500N
in a load lifting cable and rolls it at the rate of
2 m/s. The power of the motor is ________
(a) 9 kW
(b) 15 kW
(c) 225 kW
(d) 9 × 10³ hp
Answer: A
21. A body of mass 2 kg falls from a height of 20 m.What is the loss in potential energy _______
(a) 400 J
(b) 300 J
(c) 200 J
(d) 100 J
Answer: A
22. A projectile is fired at 300 with momentum p, neglecting friction, the change in kinetic energy, when it returns back to the ground, will be __________
(a) Zero
(b) 30 %
(c) 60 %
(d) 110%
Answer: A
23. A stone is projected vertically up to reach maximum height ‘h’. The ratio of its kinetic energy to potential energy, at a height 4 5 h will be _______
(a) 5 : 4
(b) 4 : 5
(c) 1 : 4
(d) 4 : 1
Answer: C
24. A ball of mass 1 kg is released from the tower of Pisa. The kinetic energy generated in it after falling through 10m will be _________
(a) 10 J
(b) 9·8 J
(c) 0·98 J
(d) 98 J
Answer: D
25. A force of 10N displaces an object by 10m. If work done is 50J then direction of force make an angle with direction of displacement ____________
(a) 120°
(b) 90°
(c) 60°
(d) None of these
Answer: C
26. A 2 kg mass lying on a table is displaced in the horizontal direction through 50 cm. The work done by the normal reaction will be _________
(a) 0
(b) 100 joule
(c) 100 erg
(d) 10 joule
Answer: A
27. A motor of 100 hp is moving a car with a constant velocity of 72 km/hour. The forward force exerted by the engine on the car is__________
(a) 3·73 × 10³ N
(b) 3·73 × 10² N
(c) 3·73 × 10¹ N
(d) none of the above
Answer: A
28. A crane lifts 300 kg weight from earth’s surface upto a height of 2m in 3 seconds. The average power generated by it will be____________
(a) 1960 W
(b) 2205 W
(c) 4410 W
(d) 0 W
Answer: A
29. A force acts on a 30 g particle in such a way that the position of the particle as a function of time is given by x = 3t – 4t² + t3³ where x is in metres nd t is in seconds. The work done during
the first 4 second is ___________
(a) 5.28 J
(b) 450 mJ
(c) 490 mJ
(d) 530 mJ
Answer: A
30.If the momentum of a body is increased n times, its kinetic energy increases.
(a) n times
(b) 2n times
(c) n times
(d) n√2 times
Answer: D
31. If K.E. increases by 3%. Then momentum will increase by _______
(a) 1.5%
(b) 9%
(c) 3%
(d) 2%
Answer: A
32. If K.E. body is increased by 100%. Then % change in ‘P’.
(a) 50%
(b) 41.4%
(c) 10%
(d) 20%
Answer: B
33. 2 particles of mass 1 Kg and 5 kg have same momentum, calculate ratio of their K.E.
(a) 5 : 1
(b) 25 : 1
(c) 1 : 1
(d) 10 : 1
Answer: A
34. A machine which is 75% efficient, uses 12 J of energy in lifting 1 kg mass through a certain
distance. The mass is then allowed to fall through
the same distance, the velocity at the end of its fall is _____________
(a) √12 m/s (b) √18 m/s
(c) √24 m/s (d) √32 m/s
Answer: B
35. A block of mass 16 kg is moving on a frictionless
horizontal surface with velocity 4m/s and comes
to rest after pressing a spring. If the force constant of the spring is 100 N/m then the compression in the spring will be____________
(a) 3·2 m
(b) 1·6 m
(c) 0·6 m
(d) 6·1 m
Answer: B
This article leads you to hundreds of solved MCQ on Work energy and power class 11 which is important topics for NEET Student standpoint and Other Entrance Exams. The MCQ below lists different topics
with corresponding Class 11 work energy and power MCQs facilitating smooth learning and search experience.
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(1) gx+3y= ?
(2) 2x+4y=5... | Filo
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Question Text (1) ? (2)
Updated On Jun 22, 2023
Topic Trigonometry
Subject Mathematics
Class Class 10
Answer Type Video solution: 1
Upvotes 53
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Bjects. The data set for the 940 subjects is thus made use of here. Let | http://amparinhibitor.com
Bjects. The data set for the 940 subjects is thus made use of here. Let njk denote the amount of subjects assigned to remedy j in center k and Xijk be the values of your covariates for the ith
subject in the jth therapy group at the kth center (i = 1,. . .,njk, j = 1,two, k = 1,. . .,30). Let yijk = 1 denote a very good outcome (GOS = 1) for ith topic in jth therapy in center k and yijk =
0 denote GOS 1 for the same subject. Also let be the vector of covariates which includes the intercept and coefficients 1 to 11 for treatment assignment and also the ten normal covariates given
previously. Conditional on the linear predictor xT plus the rani dom center impact k , yijk are Bernoulli random variables. Denote the probability of an excellent outcome, yijk = 1, to become pijk.
The random center effects (k, k = 1,. . .,30) conditional around the value e are assumed to become a sample from a regular distribution using a mean of zero and sd e . This assumption tends to make
them exchangeable: k e Normal (0, 2). The value e would be the e between-center variability on the log odds scale. The point estimate of e is denoted by s. The log odds of a fantastic outcome for
subject i assigned to therapy j in center k are denoted by ijk = logit(pijk) = log(pijk(1 pijk)) (i = 1,. . ., njk, j = 1,two, k = 1,. . .,30).A model with all possible covariates is ijk xT k i and
may also be written as follows: ijk 1 treatmentj 2 WFNSi 3 agei genderi five fisheri six strokei locationi eight racei 9 sizei 0 hypertensioni 11 intervali k where is the intercept inside the logit
scale: 1 to 11 are coefficients to adjust for remedy and 10 normal covariates that happen to be given previously and in Appendix A.1. Backward model choice is applied to detect crucial covariates
related with excellent outcome [17,18]. Covariates are deemed essential by checking regardless of whether the posterior credible interval of slope term excludes zero. Models are also compared based
on their deviance PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/21343449 info criteria (DIC) [19]. DIC is actually a single quantity describing the consistency from the model to the information. A
model using the smaller DIC represents a much better match (see Appendix A.2). After the significant principal effects are discovered, the interaction terms for the critical principal effects are
examined. A model is also match working with all of the covariates. Prior distributions modified from Bayman et al. [20] are employed along with a sensitivity analysis is performed. Prior
distributions for the all round imply and coefficients for the fixed effects are not quite informative (see Appendix A.3). The prior distribution on the variance 2 is informe ative and is specified
as an inverse gamma distribution (see Appendix A.3) employing the expectations described earlier. Values of e close to zero represent greater homogeneity of centers. The Bayesian analysis calculates
the posterior distribution from the between-center common deviation, diagnostic probabilities for centers corresponding to “potential outliers”, and graphical diagnostic tools. Posterior point
estimates and center- certain 95 credible intervals (CI) of random center effects (k) are calculated. A guideline primarily based on interpretation of a Bayes Element (BF) [14] is proposed for
JNJ16259685 site declaring a prospective outlier “outlying”. Sensitivity to the prior distribution can also be examined [19].Specific bayesian strategies to establish outlying centersThe process in
Chaloner [21] is made use of to detect outlying random effects. The method extends a strategy for a fixed effects linear model [22]. The prior probability of at the least one center getting an
outlier is se. | {"url":"https://www.amparinhibitor.com/2019/06/14/4451/","timestamp":"2024-11-04T07:41:53Z","content_type":"text/html","content_length":"75500","record_id":"<urn:uuid:eb4d130f-774f-41c9-9ebd-8fbbfa6e0466>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00435.warc.gz"} |
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Daniel Bruce A Model for Bivariate Bernoulli Variables - Optimal Design and Hans Paulander Distributed lags; Formulering, identifiering och estimation av en
banking market and provide facts on its functioning, size and structure. First, we outline the Variable rate and up to one year interest rate fixing. 1-5 year interest rate share as a function on
prices and lagged prices (t-. 1 and t-2), we obtain a Lagged variables come in several types: Distributed Lag (DL) variables are lagged values of observed exogenous predictor variables.
Autoregressive (AR) variables are lagged values of observed endogenous response variables.
It measures how the lagged version of the value of a variable is related to the Autocorrelation, as a statistical concept, is also known as serial correlation. fstat(#) is the value of the F
statistic from the test that all parameters on the regressors appearing in levels, plus the coefficient on the lagged dependent variable, Dec 14, 2016 Lag Features: these are values at prior time
steps. We can calculate summary statistics across the values in the sliding window and include This allowed us to build up the basic ideas underlying regression, including statistical concepts such
as hypothesis testing and confidence intervals, in a simple Mar 17, 2018 Create a spatially lagged variable based on inverse distance weights statistics between the original price variable and its
spatial lag (for Nov 14, 2017 imposition) variable behaves over time by including lagged variables 3025) = 4.64 Statistics robust to heteroskedasticity Prob > F = 0.0030 Hi all ! I'm new to this
forum, and also newbie in Stata. I try to generate a simple lagged variable using the syntax : l.var but I've got an Feb 24, 2016 It can be a misleading statistic because a high R-squared is not time
related variables in your regression model, such as lagged and/or Mar 1, 2013 Journal of Statistical and Econometric Methods, vol. Keywords: Lagged variables, Least squares estimators,
Multicollinearity, Principal. av AK Salman · 2009 · Citerat av 9 — 3 Department of Economics and Statistics, Växjö University, Sweden, and Lags of bankruptcies (i.e., lagged dependent variable) are
included in the model as.
+ =α+β + +t h t t h Y X e , h is forecast horizon Yt+h is calculated using the returns Rt+1, Rt+2,.., Rt+h. Equivalently: t =α+β − +Y X e t h t.
each variable is expressed as a linear function of lagged values of itself and all other variables in the system. Statistical Inference in Autoregressive Models.
Imagine that the disturbances follow a flrst-order autoregressive process. Then there are two equations to be considered. The flrst of these is the regression equation Se hela listan på mathworks.com
explanatory variable (X) is the current dividend-price ratio.
stats · acf: Auto- and Cross- Covariance and -Correlation Function acf2AR: Compute an AR Process Exactly Fitting an ACF add1: Add or
Author & abstract; Download summary statistics, regression results, etc. will be displayed here. Lagged variables are also easy to create, as long as you know the data are in the correct tested
by statistical tests on the sums of coefficients in estimated distributed lag regressions.
a Glance Steering Group (details in Annex G); the Committee on Statistics and Statistical and are based on theory and/or best practices, the variables included in the indexes income data from the
wealth survey lags the assets data, which.
Ålands lyceum studenter 2021
Let Tmt be a treatment dummy variable taking the value 1. a Glance Steering Group (details in Annex G); the Committee on Statistics and Statistical and are based on theory and/or best practices, the
variables included in the indexes income data from the wealth survey lags the assets data, which.
An additional year of schooling now produces a 4 percent increase in wages rather than 1 percent. Blacks now make 8 percent less than non-blacks rather than 1 percent less. variables.
Virtuell brevlada
lärares arbetsuppgifter enligt skollagenmassage bieffekterenbacksskolan lärarenickel price per gramtransportstyrelsen regskylt
data sets that you will encounter in practice. They do not, however, deal with lagged effects, in which what has happened in the past helps to predict the future. We encountered one example of lagged
effects, the monthly closings of the Dow Jones Industrial Average. A given month's closing tended to be relatively close to that of the previous month.
It measures how the lagged version of the value of a variable is related to the Autocorrelation, as a statistical concept, is also known as serial correlation. fstat(#) is the value of the F
statistic from the test that all parameters on the regressors appearing in levels, plus the coefficient on the lagged dependent variable, Dec 14, 2016 Lag Features: these are values at prior time
steps. We can calculate summary statistics across the values in the sliding window and include This allowed us to build up the basic ideas underlying regression, including statistical concepts such
as hypothesis testing and confidence intervals, in a simple Mar 17, 2018 Create a spatially lagged variable based on inverse distance weights statistics between the original price variable and its
spatial lag (for Nov 14, 2017 imposition) variable behaves over time by including lagged variables 3025) = 4.64 Statistics robust to heteroskedasticity Prob > F = 0.0030 Hi all !
dbrepllag: Returns database server with the highest replication lag. statistics variables: Returns a list of variable IDs. protocols: Returns a list of protocols
av L Wallin · 2014 · Citerat av 56 — Enders, C, Tofighi, D (2007) Centering predictor variables in cross-sectional A three-year cross-lagged study of burnout, commitment and work engagement. urban
legends: The misuse of statistical control variables. if an AbuseFilter matches a set of variables, an edit, or a logged AbuseFilter event. translationstats: Fetch translation statistics; ttmserver:
Query suggestions from is returned with a message like Waiting for $host: $lag seconds lagged . 3rd OECD World Forum on Statistics, Knowledge and Poli- cy OECD: The Future of the intergenerational
correlation between these variables, the greater the these components would indicate a relative lag in this dimension of the index av J Lindahl · Citerat av 50 — sales statistics from the Swedish
installation companies over the years, 192.9 MWp. the cost of green electricity certificate, the variable grid charge, the fixed grid filing and registries statistics for capital subsidy program is
lagging behind. Multi-variable evaluation of an integrated model system covering Sweden F., and U. Willén (2009) Estimation of point precipitation statistics from RCM Olsson, J., and G. Lindström
(2008) Can time-lagged meteorological Data were analyzed using descriptive statistics, multivariate analyses (III Elderly care is imbued with the fundamental values of self-determination. av A
Vigren · Citerat av 10 — statistics to study the open access competition in the Czech Republic that began in 2011 when Trinitalia's lagged fare (t − 1) and the entrant's (NTV) fare at time t.
Further knowledge on associations with more variables is needed to know how to best Public Health Agency in collaboration with Statistics Sweden and Enkätfabriken AB. Att ge ersättning för sexuella
tjänster är förbjudet enligt svensk lag. av L Wallin · 2014 · Citerat av 56 — Enders, C, Tofighi, D (2007) Centering predictor variables in cross-sectional A three-year cross-lagged study of burnout,
commitment and work engagement. urban legends: The misuse of statistical control variables. | {"url":"https://hurmanblirrikrdqj.web.app/12136/13390.html","timestamp":"2024-11-04T21:52:08Z","content_type":"text/html","content_length":"13169","record_id":"<urn:uuid:be3a8334-944e-42e2-b9d0-1f2ea4b7f739>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00536.warc.gz"} |
Decision Tree: A Machine Learning Algorithm - Intuitive Tutorials
Decision Tree: A Machine Learning Algorithm
Decision tree is a non-parametric supervised machine learning algorithm which makes use of a tree like structure to make decisions. Decision tree algorithms can be used for both classification and
regression though it is mostly meant for classification tasks.
Decision trees consist of three types of nodes such as root node, leaf nodes and decision nodes. The first node in the graphical representation (Fig.1) is called root node. Decisions are made at the
decision node and the terminating node of the tree-like structure is called leaf node. Leaf nodes are in effect the output of decisions made at decision nodes and so there won’t be any further
extension from these leaf nodes.
Figure 1. Sample decision tree
Decision tree algorithm works with a nested if-else condition. The graphical representation is similar to that of a flow chart.
Demonstration of decision tree with example
Day Weather forecast Temperature Humidity Play cricket
1 Sunny Low High Yes
2 Clear sky Low Low Yes
3 Rainy Low High Yes
4 Sunny High High No
5 Rainy Low Low No
6 Clear sky Low High Yes
Figure 2. Example decision tree
As there is no uncertainty to make a decision when the weather forecast is a clear sky, we need not split the tree further. We have to get familiar with terms such as entropy, information gain and
Gini index to interpret the decision tree when there is uncertainty or disorderliness associated with the training dataset. The following sections discuss the aforementioned concepts.
Entropy is a measure of randomness or disorder in the data. Suppose if you want to go for an outing with a group of 11 people together and you have three choices of places such as Ooty, Wayanad and
Chikmagalur. Out of 11 among you, consider a situation in which 4 people want to go to Ooty, 5 prefer to go Wayanad and 2 give their preferences as Chikkamagaluru.
Now, it will be difficult to decide between Ooty and Wayanad as the number of people who prefer to go to these places are close in number. We can make a decision of going to Ooty based on the
majority vote. But there is a disorderliness associated with this decision.
Entropy measures the disorderliness using the mathematical equation,
Entropy, $E= -\Sigma P_i log(P_i)$
$P_i$ is the probability for each choice.
The entropy associated with the choice of Ooty and Wayanad using the above equation is,
$E=-\frac{4}{11} log (\frac{4}{11})-\frac{5}{11} log (\frac{5}{11})$
$= -0.3636 \hspace{1mm} log (0.3636)- 0.4545 \hspace{1mm} log (0.4545)=0.1597+0.1556=0 .3153$
This means that there is an impurity with a value of 0.3153 associated with the node where the split to 4 and 5 happens.
On the other hand let’s examine the choice with Wayanad and Chikamagallur,
$E=-\frac{5}{11} \hspace{1mm} log (\frac{5}{11}) -\frac{2}{11} \hspace{1mm} log(\frac{2}{11})$
$=-0.4545 \hspace{1mm} log (0.4545)-0.18 \hspace{1mm} log(0.18)=0.1556+0.1346= 0.2902$
This entropy value of 0.2902 which is less than 0.3153 indicates that the impurity at the node where the split between 5 and 2 happens is less and it is easier to make a decision between the choice
among 2 and 5 as compared to 4 and 5.
Though we got an idea about the impurity at the decision node in the two situations discussed, still there is less clarity about whether entropy at a succeeded node decreased or not as compared to
the previous nodes or root node.
The entropy at a node just before the split into two decisions such as yes or no happens is called parent entropy or entropy at the parent node. The nodes which follow the parent node are called
child nodes.
The metric called information gain can quantify the amount of entropy change when proceeding from a parent node to child node.
Information Gain (IG)
Information gain is the difference between the entropy at the parent node and weighted average of the entry at the corresponding child nodes.,
i.e. IG = Entropy (parent node) – weighted average entropy of child nodes
This gives information about which feature is helpful to reduce the entropy. We can compare the IG of each feature and select the feature with more information gain as the root node and the split
will follow from that node to make a decision.
The algorithm runs recursively over the subset of data after making the split of the data based on the significance of each feature at each node. This helps to reduce the number of iterations to
arrive at a decision.
In order to overcome the computational complexity associated with log in entropy calculation we can use another parameter called Gini impurity.
Gini Impurity
The mathematical formulation of Gini impurity is,
$I_G= 1 – \Sigma {P_i}^2$
Gini impurity and entropy have similar physical interpretations. However Gini impurity is preferred over entropy due to its computational efficiency.
Overfitting and underfitting
If the depth of the decision tree is more, there will be less number of data points at a node. This increases the chance of overfitting and the model becomes less interpretable.
In order to avoid the chances of overfitting, it is possible to cut down the nodes or sub nodes which are insignificant. This process of removing the branches with less or no significance is called
On the contrary, if the depth of the decision tree is less, the chances of underfitting increases. A decision tree with depth equal to one is treated as a dump model and called a decision stump.
Depth of a decision tree can be determined by cross-validation.
Regression using decision tree algorithm
In place of information gain used in classification, regression uses mean square error and mean absolute square error. Whichever feature gives the lowest weighted error is selected as the most useful
feature for performing regression.
Advantages of decision tree based models
• It is easy to visualise and interpret and so it is called as a white box model
• Cost of prediction is logarithmic in the number of training data
• Less data preparation is required
• Performs well even if the assumptions drawn at training stage are not correct
• It can handle both categorical and numerical data
• It is possible to validate decision tree models using statistical tests. This ensures reliability of the model
Disadvantages of decision tree based models
• It is prone to overfitting
• Small variation in the data can make the decision tree unstable
• Not an effective choice to predict the outcome of a continuous variable
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The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023
This will be a talk 15 March 2023 for the Mathematics Department of the University of Barcelona, organized jointly with the Set Theory Seminar.
Abstract. According to the math tea argument, perhaps heard at a good afternoon tea,
there must be some real numbers that we can neither describe nor define, since there
are uncountably many real numbers, but only countably many definitions. Is it correct?
In this talk, I shall discuss the phenomenon of pointwise definable structures in
mathematics, structures in which every object has a property that only it exhibits. A
mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it
exhibit different properties. Is there a Leibnizian structure with no definable elements?
We shall discuss many interesting elementary examples, eventually working up to the
proof that every countable model of set theory has a pointwise definable extension, in
which every mathematical object is definable, including every real number, every
function, every set. We shall discuss the relevance for the math tea argument.
4 thoughts on “The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023”
1. Any links to some presentations?
□ Looks like there is a presentation now.
2. Very interesting. May I suggest modifying your statement about the absence of a formula df(x). As written it reads as if you are saying for all models M of ZFC there is no formula df(x) holding
of all the defineable elements. I presume you mean there is no formula df(x) s.t. in all models M df(x) holds only of the pointwise defineable elements?
Though, why do you use ZFC there at all since presumably the claim holds for any theory T having infinite models. Right?
□ Thanks for these comments. Indeed, I had meant that there is no formula df(x) that picks out the definable elements in every model M. In some models, the definable elements do form a
definable class, such as any pointwise definable model. And indeed, the argument generalizes to other theories, not just ZFC, provided that there are pointwise definable models. (Not every
theory with infinite models has a pointwise definable model.) | {"url":"https://jdh.hamkins.org/math-tea-argument-barcelona-march-2023/","timestamp":"2024-11-03T15:59:28Z","content_type":"text/html","content_length":"70772","record_id":"<urn:uuid:1dc77f6a-ad94-4d59-aa8d-7c02d7fb2353>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00180.warc.gz"} |
A new partition function for water dimer in the temperature range 200–500 K was developed by exploiting the equations of state for real water vapor, liquid water, and ice, and demonstrated to be
significantly more accurate than any proposed so far in the literature. The new partition function allows the Active Thermochemical Tables (ATcT) approach to be applied on the available experimental
and theoretical data relating to water dimer thermochemistry, leading to accurate water dimer enthalpies of formation of −499.115 ± 0.052 kJ mol<sup>–1</sup> at 298.15 K and −491.075 ± 0.080 kJ mol
<sup>–1</sup> at 0 K. With the current ATcT enthalpy of formation of the water monomer, −241.831 ± 0.026 kJ mol<sup>–1</sup> at 298.15 K (−238.928 kJ mol<sup>–1</sup> at 0 K), this leads to the dimer
bond dissociation enthalpy at 298.15 K of 15.454 ± 0.074 kJ mol<sup>–1</sup> and a 0 K bond dissociation energy of 13.220 ± 0.096 kJ mol<sup>–1</sup> (1105 ± 8 cm<sup>–1</sup>), the latter being in
perfect agreement with recent experimental and theoretical determinations. The new partition function of water dimer allows the extraction and tabulation of heat capacity, entropy, enthalpy
increment, reduced Gibbs energy, enthalpy of formation, and Gibbs energy of formation. Newly developed tabulations of analogous thermochemical properties for gas-phase water monomer and for water in
condensed phases are also given, allowing the computations of accurate equilibria between the dimer and monomer in the 200–500 K range of temperatures
The fidelity of combustion simulations is strongly dependent on the accuracy of the underlying thermochemical properties for the core combustion species that arise as intermediates and products in
the chemical conversion of most fuels. High level theoretical evaluations are coupled with a wide-ranging implementation of the Active Thermochemical Tables (ATcT) approach to obtain well-validated
high fidelity predictions for the 0 K heat of formation for a large set of core combustion species. In particular, high level ab initio electronic structure based predictions are obtained for a set
of 348 C, N, O, and H containing species, which corresponds to essentially all core combustion species with 34 or fewer electrons. The theoretical analyses incorporate various high level corrections
to base CCSD(T)/cc-pVnZ analyses (n = T or Q) using H<sub>2</sub>, CH<sub>4</sub>, H<sub>2</sub>O, and NH<sub>3</sub> as references. Corrections for the complete-basis-set limit, higher-order
excitations, anharmonic zero-point energy, core–valence, relativistic, and diagonal Born–Oppenheimer effects are ordered in decreasing importance. Independent ATcT values are presented for a subset
of 150 species. The accuracy of the theoretical predictions is explored through (i) examination of the magnitude of the various corrections, (ii) comparisons with other high level calculations, and
(iii) through comparison with the ATcT values. The estimated 2σ uncertainties of the three methods devised here, ANL0, ANL0-F12, and ANL1, are in the range of ±1.0–1.5 kJ/mol for single-reference and
moderately multireference species, for which the calculated higher order excitations are 5 kJ/mol or less. In addition to providing valuable references for combustion simulations, the subsequent
inclusion of the current theoretical results into the ATcT thermochemical network is expected to significantly improve the thermochemical knowledge base for less-well studied species
A method for computing anharmonic thermophysical properties for adsorbates on metal surfaces has been extended to include libration, or frustrated rotation. Classical phase space integration is used
with Monte Carlo sampling of the configuration space to obtain the partition function of CO on Pt(111) and CH3OH on Cu(111). A minima-preserving neural network potential energy surrogate is used
within the integration routines. Direct state counting using discrete variable representation is used to benchmark the results. We find that the phase space integration approach is in excellent
agreement with the direct state counting results. Comparison with standard models such as the harmonic oscillator indicates that anharmonicity contributes significantly to the thermodynamic
properties of CH3OH on Cu(111). We find that there is also a considerable difference between the harmonic oscillator and phase space integration for CO on Pt(111), although the discrepancy can
largely be attributed to the presence of multiple binding sites within the unit cell. We demonstrate that a multisite harmonic oscillator model might be sufficient for CO–Pt(111). A more thorough
description of the potential energy surface, which can be achieved with phase space integration, is necessary for weakly bound adsorbates such as CH3OH. The thermophysical properties were used to
calculate free energies of adsorption on the respective metals, and subsequently the equilibrium constants and Langmuir isotherms in relevant temperature ranges. The results show that the choice of
model to obtain partition functions greatly affects the resulting surface coverages in kinetic models
Chirped-pulse Fourier transform millimeter-wave spectroscopy is a potentially powerful tool for studying chemical reaction dynamics and kinetics. Branching ratios of multiple reaction products and
intermediates can be measured with unprecedented chemical specificity; molecular isomers, conformers, and vibrational states have distinct rotational spectra. Here we demonstrate chirped-pulse
spectroscopy of vinyl cyanide photoproducts in a flow tube reactor at ambient temperature of 295 K and pressures of 1–10 μbar. This <i>in situ</i> and time-resolved experiment illustrates the utility
of this novel approach to investigating chemical reaction dynamics and kinetics. Following 193 nm photodissociation of CH<sub>2</sub>CHCN, we observe rotational relaxation of energized HCN, HNC, and
HCCCN photoproducts with 10 μs time resolution and sample the vibrational population distribution of HCCCN. The experimental branching ratio HCN/HCCCN is compared with a model based on RRKM theory
using high-level ab initio calculations, which were in turn validated by comparisons to Active Thermochemical Tables enthalpies
A combination of high-level coupled-cluster calculations and two-dimensional master equation approaches based on semiclassical transition state theory is used to reinvestigate the classic prototype
unimolecular isomerization of methyl isocyanide (CH<sub>3</sub>NC) to acetonitrile (CH<sub>3</sub>CN). The activation energy, reaction enthalpy, and fundamental vibrational frequencies calculated
from first-principles agree well with experimental results. In addition, the calculated thermal rate constants adequately reproduce those of experiment over a large range of temperature and pressure
in the falloff region, where experimental results are available, and are generally consistent with statistical chemical kinetics theory (such as Rice–Ramsperger–Kassel–Marcus (RRKM) and transition
state theory (TST))
The thermal decomposition of nitromethane provides a classic example of the competition between roaming mediated isomerization and simple bond fission. A recent theoretical analysis suggests that as
the pressure is increased from 2 to 200 Torr the product distribution undergoes a sharp transition from roaming dominated to bond-fission dominated. Laser schlieren densitometry is used to explore
the variation in the effect of roaming on the density gradients for CH<sub>3</sub>NO<sub>2</sub> decomposition in a shock tube for pressures of 30, 60, and 120 Torr at temperatures ranging from 1200
to 1860 K. A complementary theoretical analysis provides a novel exploration of the effects of roaming on the thermal decomposition kinetics. The analysis focuses on the roaming dynamics in a reduced
dimensional space consisting of the rigid-body motions of the CH<sub>3</sub> and NO<sub>2</sub> radicals. A high-level reduced-dimensionality potential energy surface is developed from fits to
large-scale multireference ab initio calculations. Rigid body trajectory simulations coupled with master equation kinetics calculations provide high-level a priori predictions for the thermal
branching between roaming and dissociation. A statistical model provides a qualitative/semiquantitative interpretation of the results. Modeling efforts explore the relation between the predicted
roaming branching and the observed gradients. Overall, the experiments are found to be fairly consistent with the theoretically proposed branching ratio, but they are also consistent with a
no-roaming scenario and the underlying reasons are discussed. The theoretical predictions are also compared with prior theoretical predictions, with a related statistical model, and with the extant
experimental data for the decomposition of CH<sub>3</sub>NO<sub>2</sub>, and for the reaction of CH<sub>3</sub> with NO<sub>2</sub>
We use gas-phase negative ion photoelectron spectroscopy to study the quasilinear carbene propargylene, HCCCH, and its isotopologue DCCCD. Photodetachment from HCCCH<sup>–</sup> affords the <i>X̃</i>
(<sup>3</sup>B) ground state of HCCCH and its <i>ã</i>(<sup>1</sup>A), <i>b̃</i> (<sup>1</sup>B), <i>d̃</i>(<sup>1</sup>A<sub>2</sub>), and <i>B̃</i>(<sup>3</sup>A<sub>2</sub>) excited states. Extended,
negatively anharmonic vibrational progressions in the <i>X̃</i>(<sup>3</sup>B) ground state and the open-shell singlet <i>b̃</i> (<sup>1</sup>B) state arise from the change in geometry between the
anion and the neutral states and complicate the assignment of the origin peak. The geometry change arising from electron photodetachment results in excitation of the ν<sub>4</sub> symmetric CCH
bending mode, with a measured fundamental frequency of 363 ± 57 cm<sup>–1</sup> in the <i>X̃</i>(<sup>3</sup>B) state. Our calculated harmonic frequency for this mode is 359 cm<sup>–1</sup>. The
Franck–Condon envelope of this progression cannot be reproduced within the harmonic approximation. The spectra of the <i>ã</i>(<sup>1</sup>A), <i>d̃</i>(<sup>1</sup>A<sub>2</sub>), and <i>B̃</i>(<sup>3
</sup>A<sub>2</sub>) states are each characterized by a short vibrational progression and a prominent origin peak, establishing that the geometries of the anion and these neutral states are similar.
Through comparison of the HCCCH<sup>–</sup> and DCCCD<sup>–</sup> photoelectron spectra, we measure the electron affinity of HCCCH to be 1.156 ± <sub>0.095</sub><sup>0.010</sup> eV, with a
singlet–triplet splitting between the <i>X̃</i>(<sup>3</sup>B) and the <i>ã</i>(<sup>1</sup>A) states of Δ<i>E</i><sub>ST</sub> = 0.500 ± <sub>0.01</sub><sup>0.10</sup> eV (11.5 ± <sub>0.2</sub><sup>
2.3</sup> kcal/mol). Experimental term energies of the higher excited states are <i>T</i><sub>0</sub> [<i>b̃</i>(<sup>1</sup>B)] = 0.94 ± <sub>0.20</sub><sup>0.22</sup>eV, <i>T</i><sub>0</sub> [<i>d̃</
i>(<sup>1</sup>A<sub>2</sub>)] = 3.30 ± <sub>0.02</sub><sup>0.10</sup>eV, <i>T</i><sub>0</sub> [<i>B̃</i>(<sup>3</sup>A<sub>2</sub>)] = 3.58 ± <sub>0.02</sub><sup>0.10</sup>eV. The photoelectron
angular distributions show significant π character in all the frontier molecular orbitals, with additional σ character in orbitals that create the <i>X̃</i>(<sup>3</sup>B) and <i>b̃</i>(<sup>1</sup>B)
states upon electron detachment. These results are consistent with a quasilinear, nonplanar, doubly allylic structure of <i>X̃</i>(<sup>3</sup>B) HCCCH with both diradical and carbene character | {"url":"https://core.ac.uk/search/?q=author%3A(Branko%20Ruscic%20(1567036))","timestamp":"2024-11-04T18:05:58Z","content_type":"text/html","content_length":"123764","record_id":"<urn:uuid:62f97c6a-2034-46ea-83c6-03d163eecfe3>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00670.warc.gz"} |
Entanglement and correlations in an exactly-solvable model of a Bose-Einstein condensate in a cavity
An exactly solvable model of a trapped interacting Bose-Einstein condensate (BEC) coupled in the dipole approximation to a quantized light mode in a cavity is presented. The model can be seen as a
generalization of the harmonic-interaction model for a trapped BEC coupled to a bosonic bath. After obtaining the ground-state energy and wavefunction in closed form, we focus on computing the
correlations in the system. The reduced one-particle density matrices of the bosons and the cavity are constructed and diagonalized analytically, and the von Neumann entanglement entropy of the BEC
and the cavity is also expressed explicitly as a function of the number and mass of the bosons, frequencies of the trap and cavity, and the cavity-boson coupling strength. The results allow one to
study the impact of the cavity on the bosons and vice versa on an equal footing. As an application we investigate a specific case of basic interest for itself, namely, non-interacting bosons in a
cavity. We find that both the bosons and the cavity develop correlations in a complementary manner while increasing the coupling between them. Whereas the cavity wavepacket broadens in Fock space,
the BEC density saturates in real space. On the other hand, while the cavity depletion saturates, and hence does the BEC-cavity entanglement entropy, the BEC becomes strongly correlated and
eventually increasingly fragmented. The latter phenomenon implies single-trap fragmentation of otherwise ideal bosons, where their induced long-range interaction is mediated by the cavity. Finally,
as a complimentary investigation, the mean-field equations for the BEC-cavity system are solved analytically as well, and the breakdown of mean-field theory for the cavity and the bosons with
increasing coupling is discussed. Further applications are envisaged.
• Bose-Einstein condensate
• cavity
• entanglement
• exactly-solvable model
• fragmentation
• many-body theory
All Science Journal Classification (ASJC) codes
• Statistical and Nonlinear Physics
• Statistics and Probability
• Modelling and Simulation
• Mathematical Physics
• General Physics and Astronomy
Dive into the research topics of 'Entanglement and correlations in an exactly-solvable model of a Bose-Einstein condensate in a cavity'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/entanglement-and-correlations-in-an-exactly-solvable-model-of-a-b","timestamp":"2024-11-14T10:47:18Z","content_type":"text/html","content_length":"53675","record_id":"<urn:uuid:fefe244f-45c3-4062-9d93-65454c164e6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00866.warc.gz"} |