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An Eventful Horizon
Sep 26, 2023
Scientists utilize elements of the Haramein Quantum Gravity Holographic Solution to solve the Black Hole Information Loss Paradox
By: William Brown, scientist at the International Space Federation
Clinging to Locality
In our quotidian experience the feature of spacetime locality seems to be an indelible feature of a rational reality; the idea that effects follow their causes gives us a sense (however illusionary)
that there is a natural chronology to our reality. From the theory of relativity, we know that the relativity of simultaneity requires that no signal or information travel faster than the speed of
light. Faster-than-light, or superluminal signals results in closed timelike curves, and in general relativity closed timelike curves can break causality with remarkable and unsettling consequences.
At the classical level, they induce causal paradoxes, which some find disturbing enough to motivate conjectures that explicitly prevent their existence (Hawking's chronology protection conjecture).
If a signal were to travel faster than the speed of light, an effect might precede its cause—so for instance a superluminal spaceship could make a roundtrip voyage and return to a frame-of-reference
where it had not yet departed. However, such seeming faster-than-light causal paradoxes, like our FTL spaceship arriving before it departed, may be misinterpretations of time-like closed loops in
relativity, which do not necessarily involve violations of causality or chronology even in instances of faster-than-light travel (see, for example, Hossenfelder's explanation "I Think Faster Than
Light Travel is Possible, Here's Why).
The conceptual challenge of nonlocality extends to quantum mechanics as well, in which ontic theories that posit nonlocal realism—where particles really exist prior to state reduction or "collapse of
the wavefunction"—are considered to be incompatible with relativity because of the seeming superluminal update of state, and hence interpretations that are local but deny real particles prior to
reduction of the state vector are preferred, like the Bohr-Heisenberg model of the Copenhagen interpretation. Nevertheless, non-locality is consistent with current scientific knowledge. Closed
timelike curves are valid solutions of Einstein’s equations in general relativity. Thus, the originators of the AMPS firewall paradox, which was posited in an attempt to preserve locality while
simultaneously providing a mechanism to preserve quantum information, have forgone this conjecture and solved the AMPS paradox by permitting nonlocal action via Einstein-Rosen bridges. No matter how
indelible locality and classical causal structure seems to our rationale, it seems we may need to accept that both quantum physics and relativity theory have properties that permit non-local
interactions: in the former there are Einstein Podolsky Rosen (EPR) correlations, and in the latter, there are Einstein Rosen bridges (ERb), more popularly known as wormholes.
Unification of Quantum Mechanics and Relativity Found in Non-Locality
As the monikers for each non-local behavior would imply, they were first proposed and described by Einstein and his colleagues. Einstein, Podolsky, and Rosen brought to light EPR correlations, or
quantum entanglement as it is more popularly known, to show how certain solutions in quantum mechanics—like the superposition of the wavefunction—must be erroneous, as they seemingly permit
instantaneous interconnection would seem to require faster-than light signaling [1]. Interestingly, Einstein and Rosen also subsequently proposed and described the physical process by which quantum
entanglement may occur without violation of the light-speed limit, and that is via ERbs, a bridge geometry in spacetime between two universes which they originally discovered as a solution to the
particle problem in the general theory of relativity, which results from the singularities of point particles like the electron— they were describing fundamental particles as wormholes [2].
If fundamental particles are quantum wormholes, then the "spooky" nonlocal connection between them (entanglement) may be the result of their ERb spacetime geometry, what physicists Juan Maldacena and
Leonard Susskind have summarized as ERb = EPR [3]; which means quantum entanglement is the result of wormhole connections in spacetime, or correspondingly space is multiply-connected via quantum
entangled spacetime frames. While both ERb and EPR appear to permit non-locality, it is not clear if wormholes are indeed traversable (although there have been many studies showing how they can
potentially be traversable), so that any real detectable signal could be transferred via these spacetime shortcuts, and no viable mechanism has been thought of to utilize quantum entanglement for
superluminal signal transmission:
In the present work we describe a new quantum-mechanical paradox in which the presence or absence of an interference pattern in a path-entangled two photon system with variable entanglement,
controlled by measurement choice, would seem to permit retrocausal signaling from one observer to another. We also present an analysis of this scheme, showing how the subtleties of the quantum
formalism block the potential signal. In particular, even when interference patterns can be switched off and on, there is always a “signal” interference pattern and an “anti-signal” interference
pattern that mask any observable interference when they are added, even when entanglement and coherence are simultaneously present. [4] "An Inquiry into the Possibility of Nonlocal Quantum
Communication", John G. Cramer and Nick Herbert; arXiv:1409.5098 (2014).
Indeed, it is largely accepted that although quantum entanglement maintains a highly correlated state transcending locality limitations, the outcome of any detection or measurement is always random,
so there can be no useful information transmitted in the process.
A Return to Cool Horizons
This leads us to another seeming conundrum that has existed at the crossroads of quantum theory and general relativity for over 45 years, ever since Stephen Hawking’s analysis of the quantum vacuum
around event horizons showed that black holes may evaporate, leading to the so-called information loss paradox. The problem arises because particle-pairs of the quantum vacuum—readers of our ISF
articles will be well aware that the vacuum of space is not empty, but instead has an immensely large mass-energy density—that get separated by the event horizon of a black hole are entangled, but no
information can be derived from the escaped entanglement particle pair (remember nonlocal quantum communication is a no-go), so as energy is slowly radiated away from the black hole the entangled
Hawking particles carry no information about the inner microstates (volume entropy) of the evaporating black hole. The information seems to be lost, which would be a direct violation of physical laws
akin to “energy is never created nor destroyed”. In present-day (accepted) theory, as the black hole evaporates its entanglement entropy inexorably continually increases, which is to say the
“randomness” of the radiated Hawking particles continually increases, so that whatever ordered information was contained within the event horizon (its volume entropy) is converted into random
radiation and seemingly lost forever.
Yet, there are solutions to this paradox that rely on the ERb = EPR equivalence conjecture, in which there is a relationship and connection between what is inside the event horizon of the black hole
with what is on the outside, such that information need not be irretrievably lost or destroyed but may be accessible via the quantum wiring of spacetime. In an article in Scientific American,
physicist Ahmed Almheiri—one of the originators of the AMPS firewall paradox (see my 2016 article “Firewalls or Cool Horizons?”)—describes recent work with Juan Maldacena and others in which they
resolve the AMPS firewall and information loss paradox by showing how, via ERb = EPR, the information within the event horizon of a black hole is “secretly on the outside” [5, How the Inside of a
Black Hole Is Secretly on the Outside, Scientific American, 2022]. Essentially, wormhole connections between maximally entangled black holes allow them to swap their volume entropy, which creates
“black hole islands” within their interiors. The black hole island is the event horizon volume from another distant black hole entangled via a wormhole connection.
What Almheiri and Maldacena discovered is that when these black hole islands form, because of the no-cloning theorem in quantum mechanics, their entropy counts to the surface entropy of the black
hole and is no longer lost within the interior (the inaccessible region forever concealed within the event horizon). So, for information to be liberated, all a particle need do is travel deeper into
the black hole interior into the island, and the information will be accessible on the surface horizon.
The Haramein Generalized Holographic Solution
This conclusion is remarkably similar to the generalized holographic solution discovered by physicist Nassim Haramein in 2012. In his seminal work “quantum gravity and the holographic mass” Haramein
demonstrated that there is a simple and fundamental relationship between the information content, in terms of PSUs inside the volume of a black hole, and the PSU information on the surface event
horizon, giving a ratio that when multiplied by the Planck mass (the energy of that ratio), generates the exact mass-energy of the black hole [6]. The Schwarzschild solution to Einstein’s field
equations gives the exact same answer for the mass of a black hole, however, in the case of Haramein’s generalized holographic solution using quantum voxels, we have a quantized gravitational
solution (in fine-grained or discrete quantities). For an in-depth exploration of the Haramein quantum gravity holographic solution see RSF physicist Dr. Inés Urdaneta's mutli-part series "The
Generalized Holographic Model" (available for free in the Resonance Science Foundation science news & articles section).
Haramein’s generalized holographic solution predicted the ERb = EPR correspondence before it was popularized by Susskind, as Haramein described the apparent rest mass of the fundamental hadron
particle the proton as being the result of the bandwidth of information transfer from the interior entanglement entropy with the surface via Planck particle wormholes comprising the quantum black
To understand this, imagine the surface horizon of the proton treated with Haramein’s holographic solution, in which each Planck unit on the surface (there are ~10^40 Planck on the surface of one
proton) is the termination of a tiny Planck-scale vortex wormhole that is connected to (and thus entangled with) another Planck on another proton’s surface. Then imagine that each of the ~10^40
wormhole Planck terminations of one proton is connected to a different proton, like network cables connecting one proton with ~10^40 others in the rest of the universe. Of course, each of these ~10^
40 protons are themselves connected to another ~10^40 protons, which results in ~10^80 connected protons, which is the estimated number of protons in the universe today. A new picture emerges wherein
the Planck vacuum structure generates a fractal network of wormholes where the proton volume is an information hub, and the surface is the through-put capacity of the hub to communicate with other
Haramein’s generalized holographic solution was equivalently applied to astronomical scale black holes (the proton being a quantum or microscopic scale black hole), so he had predicted the
information, mass, and energy of all black holes was the result of the interiors being “information hubs” via a fractalized spacetime wormhole network—what we described in later publications as the
unified spacememory network, and subsequently utilized this connected universe perspective to describe the origins of consciousness and the evolution and development of all organized matter to higher
orders of organization and synergetic order [7].
So from Haramein’s discovery, we see that the Planck information within the volume of all protons in the universe is unified and shared across ~10^40 connections on each proton. The result is that
all the information of all protons is present in the volume Planck information of one proton. From what we learned from Haramein’s holographic solution, the mass of the object is the result of the
information within the volume (~10^60 Plancks in each proton) communicating across the boundary through ~10^40 connections to all other protons. The difference between the two, what Haramein
describes as a fundamental universal ratio (which he defines with the greek-letter most commonly used for physical ratios, phi φ) is the mass-energy-information of the Planck Spherical Units inside
that do not have access to a Planck wormhole termination on the surface, adding up to the rest mass of the proton (~10^-20):
In simple terms, there is a larger number of Plancks in the volume than the number of Planck wormhole terminations on the surface, thus only a certain amount of information-energy remains expressed
locally and that amount of information-energy happens to equal the mass of the proton.
One could think of it as all the outgoing information inside trying to transmit through the event horizon, and encountering a resistance or entropy since there is more information within the volume
than the surface can transmit, which leaves a local mass-energy equivalent to the mass of the proton. Yet, when we examine all the other protons acting on one, or the incoming information, then the
strong confinement mass-energy value is found for the proton-to-proton interaction, almost as a pressure exerted by the information of all other protons communicating with the one:
Since this is true for cosmological objects as well, then the mass and the confining force of gravity from the nuclei of atoms to universal structures, such as galaxies and stars, are the result of
the “impedance” of the universal information network across event horizons — the network singing across scales… “the music of the spheres.”
Unified Science- In Perspective
While the approach taken by Almheiri has some key differences, which can only be expected since most physicists are coming from a model that views the world in terms of randomness and “isolated
systems”, and only now discovering in their own equations that everything in the universe is fundamentally and inextricably connected, it is interesting to see such disparate approaches converging on
the same conclusions… a strong indication that the approach is the correct one to realize a fully unified theory of quantum gravity and unified physics!
As stated in the Scientific American article: the origins of the information paradox can be traced back to the incompatibility between the sequestering of information by the event horizon and the
quantum-mechanical requirement of information flow outside the black hole. Naive resolutions of this tension lead to drastic modifications of the structure of black holes; however, subtle yet
dramatic effects from fluctuating wormholes change everything. What emerges is a self-consistent picture that lets a black hole retain its regular structure as predicted by general relativity, albeit
with the presence of an implicit though powerful nonlocality. This nonlocality is screaming that we should consider a portion of the black hole's interior—the island—as part of the exterior, as a
single unit with the outside radiation. Thus, information can escape a black hole not by surmounting the insurmountable event horizon but by simply falling deeper into the island [5].
It is interesting to see that the originators of the AMPS firewall paradox have abandoned the theory in favor of nonlocality. Realizing that the equations of physics from general relativity to
Feynman's path integral are all permitting Einstein-Rosen bridge shortcuts across multiply-connected spacetime, it is becoming all-too apparent that wormholes —no matter how exotic they may seem to
our locality-centered worldview— must be considered as a foundational property of the structure of spacetime and Nature seems to have a way of permitting subtle nonlocality without generating
blatantly apparent cause-and-effect violations. Yet nevertheless, retrocausal signaling (i.e. nonlocal quantum communication) is occurring in a seemingly nuanced way, and no information is ever lost
to the universe. As we see from Haramein's generalized holographic solution, the universe is an immensely connected network (where the information of any one particle is shared across all particles
in a truly holographic manner via the planckian micro-wormhole connectivity architecture of the proton surface horizon), and since fundamental particles are micro-black holes, this extends equally to
astronomical scale black holes, the interiors of which are information hubs linking together spacetime across the universe.
[1] A. Einstein, B. Podolsky and N. Rosen, "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?" Phys. Rev. 47, 777-80 (1935).
[2] A. Einstein and N. Rosen, “The Particle Problem in the General Theory of Relativity,” Phys. Rev., vol. 48, no. 1, pp. 73–77, Jul. 1935, doi: 10.1103/PhysRev.48.73
[3] J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortschr. Phys., vol. 61, no. 9, pp. 781–811, Sep. 2013, doi: 10.1002/prop.201300020
[4] "An Inquiry into the Possibility of Nonlocal Quantum Communication", John G. Cramer and Nick Herbert; arXiv:1409.5098 (2014)
[5] Ahmed Almheiri, How the Inside of a Black Hole Is Secretly on the Outside, Scientific American, 2022.
[6] Haramein, N. (2012). Quantum Gravity and the Holographic Mass, Physical Review & Research International, ISSN: 2231-1815, Page 270-292
[7] Haramein, N., Brown W., & Val Baker, A. K. F. (2016). The Unified Spacememory Network: from cosmogenesis to consciousness, Journal of Neuroquantology.
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Insulation Resistance Test and Polarization Index - Electrical Concepts
Insulation Resistance Test and Polarization Index
Megger is an electrical device which is used to measure the Insulation Resistance by performing Insulation Resistance Test (IR Test). Basically Megger injects High DC Voltage across the Insulator and
ground due to which leakage current flows through the Insulator to the ground. By measuring this Leakage Current, Megger calculates the Insulation Resistance.
Suppose the DC Voltage applied by the Megger = V
And Leakage Current through the Insulator = I
Therefore from Ohm’s Law,
Insulation Resistance = V/I ohm
A typical Megger is shown in figure below.
Meggers are generally of Rating 500 V, 2.5 kV and 5 kV. In modern Megger as shown in figure above these range can be selected. 500 V is used to measure the Insulation resistance of control cable for
Insulator which can withstand up to 1.1 kV voltage. For High Voltage Transformer / High Voltage Machine or Equipment, 5 kV range is selected to perform Insulation Resistance Test. Megger has the
provision to directly read the Insulation Resistance during Insulation Resistance Test.
All Insulators are supposed to be pure capacitor having very few capacitance so as to take minimum charging current. When an Insulator is connected across a DC voltage it takes leakage current.
Leakage current can be divided into following types:
· Capacitive Charging Current
· Resistive or Conductive Current
· Surface Leakage Current
· Polarization Current.
Capacitive Charging Current:
When a DC voltage is applied across in insulator, because of its dielectric nature there will be an initial high charging current through the insulator from line to ground. Although
this current decays exponentially and becomes zero. Generally this current exists for initial 10 seconds of the test. But it takes nearly 60 seconds to decay totally. That is why it is always
recommended to do Megger or Insulation Resistance Test at least for 1 minute as it is proved that charging current totally becomes zero after 1 minute. Thus after 1 minute leakage current measured by
Megger won’t include the charging current due to Capacitance of Insulator.
Resistive or Conductive Current:
This current is purely conductive in nature, flows through the insulator as if the insulator is purely resistive. This is direct flow of electrons. Every insulator should have this component of
electric current. Resistive or Conductive leakage current through the Insulator will be more if moisture and contamination in Insulator is high.
The resistive or conductive component of insulator leakage current remains constant throughout the test.
Surface Leakage Current:
Due to dust, moisture and other contaminants on the surface of the insulator, there is one small component of leakage current through the outer surface of the insulator. Therefore before conducting
Megger or Insulation Resistance Test, Insulator should be cleaned properly so as to eliminate the surface leakage component of Leakage Current.
Polarization Current:
Because of presence of impurities and moisture in Insulator, insulator becomes polar in nature. Therefore when we apply high DC Voltage across the Insulator, the polar molecules i.e. dipole try to
align themselves in the direction of Electric Field. During the period the molecules try to align along the Electric Field, a current will flow through the Insulator because of movement of dipole
along their axis. This current is called Polarization Current and it lasts for short time and as soon as the polar molecules align themselves along the Electric Field, there will be no further
movement of dipole and hence polarization current will stop. It normally takes 10 minutes for Polarization Current to become Zero.
Thus if we take the Megger reading after 10 minute then Megger will not consider the Polarization Current for the calculation of Insulation Resistance.
So when we take Megger value of an insulator for 1 minute, the Insulation Resistance value will be free from the effect of capacitive component of leakage current. Again if we take Megger value of an
insulator for 10 minutes, the Megger result shows the value, free from effects of both capacitive component and polarization component of leakage current.
Polarization Index (PI):
Polarization index is the ratio of Megger value taken for 10 minutes to the Megger value taken for 1 minute.
Therefore, PI = Megger value after 10 minutes / Megger value after 1 minute
Significance of PI Test:
Let I = Total initial current during Polarization Index Test or PI test.
I[C] = Charging current due to Capacitance of Insulator.
I[R] = Resistive or Conductive Current.
I[S] = Surface leakage current.
I[P] = Polarization Current of the Insulator.
The value of Megger or Insulation resistance Test / IR Test after 1 minute,
R[1minute] = V/( I[R]+ I[S] + I[P]) …….I[C]= 0 after 1 minute
Similarly, Megger Value after 10 minutes,
R[10minute] = V/( I[R]+ I[S]) ………….I[P] = 0 after 10 minutes
So, from Polarization Index Test or PI Test,
PI Value = R[10minute ]/ R[1minute]
[ ]= (I[R]+ I[S] + I[P])/( I[R]+ I[S])
= [1+ I[P]/( I[R]+ I[S])]
│So, │
│ │
│PI Value = [1+ I[P]/( I[R]+ I[S])]│
From the above it is clear that, if the value of (I[R] + I[S]) >> I[P], the PI of insulator approaches to 1. Large value of I[R] or I[S] or both indicate unhealthiness of the insulation.
The value of PI becomes high if (I[R] + IS) is very small compared to I[P]. This equation indicates that high Polarization Index of an insulator implies healthiness of insulator. For good insulator
resistive leakage current is very very small.
The value of Polarization Index of an insulator should be more than 2. If the value of Polarization Index is less than1.5 then it means Insulator is unhealthy and shall not be used.
2 thoughts on “Insulation Resistance Test and Polarization Index”
1. The value of Ro (polarization resistance) is better to be large or small? and what does that have to do with the polarization index?
2. polarization resistance better to be small because polarization index depends on the value of the current Ip and when we do the test and wait for 10m the current value that appear on the ammeter
should be the current value that represent the insulation resistance not the sum of all resistance, so we need to clean the cable from dust every time we get a chance, since the polarization
resistance is very small this will be reflected on the value of PI and the values are as follows: less than 1 the cable is dangerous, between 1-2 not enough, between 2-4 good, larger then 4
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How to calculate rate of economic growth
The GDP growth rate indicates the current growth trend of the economy. When calculating GDP growth rates, the U.S. Bureau of Economic Analysis uses real GDP, which equalizes the actual figures to
filter out the effects of inflation. Using real GDP allows you to compare previous years without inflation affecting the results.
Here we examine the various ways to measure the total output of an economy, and changes in income and output; i.e., economic growth. The expenditure 7 May 2010 Simple Formula Net GDP Growth Rate =
(GDP of This Year – GDP of Last Year) 100 _ Average Inflation Rate During This Year GDP of Last Year Definition of rate of economic growth: This is calculated to determine the economic status
throughout the year. By contrast, the economic growth rate of India fell to 5.8% In the first quarter of 2019, the lowest growth rate in five years. Given the nation's rapid growth in recent years,
there was much hand-wringing over a severe slump in industrial output and a fall-off in car sales, both factors in the lower rate. The annual rate is equivalent to the growth rate over a year if GDP
kept growing at the same quarterly rate for three more quarters (or the same average rate). Calculating the real GDP growth rate Here's a step-by-step example for the Second Quarter. Go to Table
1.1.6, Real Gross Domestic Product, Chained Dollars, at the BEA website. Divide the annualized rate for Q2 2019 ($19.024 trillion) by the Q1 2019 annualized rate ($18.927 trillion). You should get
1.0051. Raise this to the power
Measure an economy's rate of productivity growth; Evaluate the power of sustained growth. Sustained long-term economic growth comes from increases in worker
3 Jan 2018 It is expressed in percentage. Formula: Growth rate = Real GDP in _ Real GDP in of real GDP current year previous year x 100 Real GDP in Here we examine the various ways to measure the
total output of an economy, and changes in income and output; i.e., economic growth. The expenditure 7 May 2010 Simple Formula Net GDP Growth Rate = (GDP of This Year – GDP of Last Year) 100 _
Average Inflation Rate During This Year GDP of Last Year Definition of rate of economic growth: This is calculated to determine the economic status throughout the year. By contrast, the economic
growth rate of India fell to 5.8% In the first quarter of 2019, the lowest growth rate in five years. Given the nation's rapid growth in recent years, there was much hand-wringing over a severe slump
in industrial output and a fall-off in car sales, both factors in the lower rate. The annual rate is equivalent to the growth rate over a year if GDP kept growing at the same quarterly rate for three
more quarters (or the same average rate). Calculating the real GDP growth rate Here's a step-by-step example for the Second Quarter. Go to Table 1.1.6, Real Gross Domestic Product, Chained Dollars,
at the BEA website. Divide the annualized rate for Q2 2019 ($19.024 trillion) by the Q1 2019 annualized rate ($18.927 trillion). You should get 1.0051. Raise this to the power
In this section we specify the estimating equation to relate lights to GDP growth, specify our
Part 1 Calculating an Annual Growth RateDetermine the time period you want to calculate. The annualized GDP growth rate is a measure of the increase or decrease of the GDP from one year to the next.
Find the GDP for two consecutive years. Use the formula for growth rate. Interpret your result as a percentage. It's what helps fiscal policy leaders and monetary policy leaders interpret the trends
in the economy so they can create policies that will promote growth. In order to calculate growth rates, we The focus of this video is how to calculate the economic growth rate. The topics covered in
the Economic Growth series: - calculating growth rates - economic growth vs. business cycle expansions Calculate the annual growth rate. The formula for calculating the annual growth rate is Growth
Percentage Over One Year = (() −) ∗ where f is the final value, s is the starting value, and y is the number of years. Example Problem: A company earned $10,000 in 2011.
Economic growth is measured by using data on GDP, which is a measure of the total During the 20th century the rate of economic growth in the developed
Real Economic Growth Rate: The real economic growth rate measures economic growth, in relation to gross domestic product (GDP), from one period to another, adjusted for inflation - in other words
Part 1 Calculating an Annual Growth RateDetermine the time period you want to calculate. The annualized GDP growth rate is a measure of the increase or decrease of the GDP from one year to the next.
Find the GDP for two consecutive years. Use the formula for growth rate. Interpret your result as a percentage.
To determine economic growth, the GDP is compared to the population, also know as the per capita income. When the per capita income increases it is called
The government's calculation of real GDP growth begins with the estimation of nominal GDP, which is the market value of the millions of goods and services. 28 Oct 2019 The Way We Measure the Economy
Obscures What Is Really Going we are missing the reality of inequality — and a chance to level the playing field. president and C.E.O. of the Washington Center for Equitable Growth. 11 Jun 2019
India's gross domestic product product (GDP) growth rate between this period should be about 4.5 per cent instead of the official estimate of
3 Jan 2018 It is expressed in percentage. Formula: Growth rate = Real GDP in _ Real GDP in of real GDP current year previous year x 100 Real GDP in Here we examine the various ways to measure the
total output of an economy, and changes in income and output; i.e., economic growth. The expenditure
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Rent Split Calculator: Easily Divide Your Rent Fairly
This rent split calculator helps you fairly divide your living expenses among roommates based on each person’s space and usage.
Rent Split Calculator
How to Use This Calculator:
Enter the total rent amount in the first field and the number of people the rent will be split between in the second field. After filling in both fields, click the “Calculate” button to see the
amount each person should pay.
How It Calculates the Results:
The calculator takes the total rent amount and divides it by the number of people to determine the amount of rent each person is responsible for. It rounds the result to two decimal places to ensure
the amount is in a valid currency format.
The calculator assumes that the rent is split evenly among all residents and does not account for different room sizes or individual agreements that would adjust individual contributions. It only
calculates with numerical values and requires both fields to be filled with valid numbers.
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Perimeter Institute Quantum Discussions
This series consists of weekly discussion sessions on foundations of quantum theory and quantum information theory. The sessions start with an informal exposition of an interesting topic, research
result or important question in the field. Everyone is strongly encouraged to participate with questions and comments.
Displaying 85 - 96 of 428
Massachusetts Institute of Technology (MIT)
Max Planck Institute of Quantum Optics
California Institute of Technology (Caltech)
Ludwig-Maximilians-Universitiät München (LMU)
Massachusetts Institute of Technology (MIT) - Department of Physics
University of KwaZulu-Natal
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Soon purchased an office chair for $167.87. She received a $19.95 rebate from the manufacturer an $8.95 rebate from the store. The sales tax in her state is 55%. What is the final price? | HIX Tutor
Soon purchased an office chair for $167.87. She received a $19.95 rebate from the manufacturer an $8.95 rebate from the store. The sales tax in her state is 55%. What is the final price?
Answer 1
The final price is #$215.40#.
A rebate is a partial refund, so start by subtracting the two rebates from the original price.
#$167.87 - $19.95 = $147.92#
#$147.92 - $8.95 = $138.97#
Next, the sales tax is 55%. In decimal terms, this is 0.55. However, she still has to pay full price for the chair AND the sales tax, so we add 1. Now multiply the decimal by the current chair price,
and find your final answer.
#$138.97 * 1.55 = $215.4035#
You end up with a longer decimal, but since we only go to the hundreds place (pennies) when paying for things, you have to round to the hundreds place. So your final answer is #$215.40#.
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Answer 2
To find the final price, subtract the rebates from the original price, then add the sales tax. So, $167.87 -$19.95 - $8.95 =$138.97. Then, calculate the sales tax: $138.97 * 0.55 =$76.43. Finally,
add the sales tax to the discounted price: $138.97 +$76.43 = $215.40. Therefore, the final price is$215.40.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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24/25 EHE - Further percentages
Want to make creations as awesome as this one?
24/25 EHE - Further percentages
Created on September 27, 2023
More creations to inspire you
Decide whether each of the units are Imperial or Metric
9) inch10) mole11) link12) lux13) candela14) slug15) kilogram16) gallon
1) mile2) ampere3) kelvin4) acre5) ounce6) fluid ounce7) watt8) metre
Further percentages
y dπ ₯
A = lw
Quizizz - 8Qs8
1. Convert 250g to kg2. All probabilities must add up to?3. β 274. β 10β 195. Write 24 as a product of its prime factors6. Estimate the length of this plane 7. Calculate the area of the triangle8.
Write 2% as a decimal
Pin: 7244 8989
y dπ ₯
A = lw
percentage change
y dπ ₯
A = lw
percentage change
y dπ ₯
A = lw
1). In a storm 144 fruit trees were left standing out of 180 fruit trees in an orchard. What is the percentage decrease in the number of trees? 2). A javelin thrower has best throw of 60m. In the
next competition he throws 72m. What is the percentage increase of his personal best? 3). A wine manufacturer puts down 250 bottles for 3 years. After 3 years only 220 bottles are in tact. What is
the percentage decrease in the number of bottles? 4). A man weighs 65Kg. After two weeks on a diet he weighs 58Kg. What is his percentage decrease in weight? 5). A board 130 cm long is trimmed to 104
cm. What percentage has been removed?
percentage change
y dπ ₯
A = lw
Repeated percentage change
y dπ ₯
A = lw
Repeated percentage change
y dπ ₯
A = lw
Repeated percentage change
y dπ ₯
A = lw
The price of a jacket was reduced in a sale by 20% to Β£368. How much did the jacket cost before the sale?
Reverse percentages
y dπ ₯
A = lw
Penny is given a 13% pay rise.She now earns Β£16,950 per year.How much did she earn before the pay rise?
Reverse percentages
y dπ ₯
A = lw
a) After a 10% increase, a laptop is priced at Β£550. What was its original price? b) A watermelon's weight increases by 20% after absorbing water. If it now weighs 4.8 kilograms, what was its
original weight? c) A shirt is sold for Β£18 after a 25% discount. What was its original price?
Reverse percentages
y dπ ₯
A = lw
Reverse Percentage [MF11.11]
Percentage Increase and Decrease [MF10.03]
Percentage Increase [MF10.01]
Repeated Percentage Increase and Decrease (Calculator) [MF11.05]
Percentage Change [MF11.04]
Century Nuggets
Then search for the nugget.
Click on this one
Further percentages
Head to 'my courses'
y dπ ₯
A = lw
y dπ ₯
A = lw
y dπ ₯
A = lw
y dπ ₯
A = lw
y dπ ₯
A = lw
1). In a storm 144 fruit trees were left standing out of 180 fruit trees in an orchard. What is the percentage decrease in the number of trees? 2). A javelin thrower has best throw of 60m. In the
next competition he throws 72m. What is the percentage increase of his personal best? 3). A wine manufacturer puts down 250 bottles for 3 years. After 3 years only 220 bottles are in tact. What is
the percentage decrease in the number of bottles? 4). A man weighs 65Kg. After two weeks on a diet he weighs 58Kg. What is his percentage decrease in weight? 5). A board 130 cm long is trimmed to 104
cm. What percentage has been removed?
y dπ ₯
f(π ₯+a)
A Level
A Level
A Level
A Level
y dπ ₯
f(π ₯+a)
A Level
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
3Β³β β + 5β
Without a calculator, work out
Challenge if finished early:
A Level
A Level
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
A Level
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
A Level
A Level
y dπ ₯
f(π ₯+a)
A Level
y dπ ₯
f(π ₯+a)
A Level
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
y dπ ₯
f(π ₯+a)
A Level
A Level
A Level
y dπ ₯
f(π ₯+a)
A Level
y dπ ₯
f(π ₯+a)
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How do I ...? > Example plots > EX09.GRF - Surface plot from an equation
This topic illustrates generating a 3D surface plot from an equation. First, open a new document window (File>New>3D surface). Now select Generate>Z=f(X,Y), which allows us to generate a 3D surface
with Z (height) values a function of X and Y. You are of course welcome to use any equation you like; this example uses Z=sin(x)*sin(x)*cos(y) with X and Y extents = + pi radians. Note that you can
enter the value “PI” for 3.14159… in the extents boxes in addition to numbers and simple equations.
We've used an interval on both the X and Y axes of PI/30, which should be more than sufficient (in this case) to capture all of the details of the equation without trying our patience: That interval
gives us 61 columns by 61 rows of data values for a total of 3721 points. On reasonably fast computers DPlot will take no more than a few seconds to plot 3D surfaces with 100,000 or so points.
This initially produces a two-dimensional plot with contour lines:
To switch to a 3D view, switch to shaded bands rather than contour lines, change the number of contour intervals or the upper and lower limits of those contours, and/or change the color scheme used
for the contours, select Options>Contour Options. The settings used for the example plot are shown here:
For a description of all of these options see the Contour Options topic.
To show the X and Y axis values as fractional values of pi rather than the default number format, right-click on any number on the respective axis and select “Pi Fractions”.
To change the extents of the axes and/or the tick mark interval used on each axis, select Options>Extents/Intervals/Size.
Page url: https://www.dplot.com/help/index.htm?ex09_grf.htm
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Pro Problems
Missile Launch
Pro Problems
Motion and Forces
Linear Motion
Missile Launch
A 5.0 kg missle is projected from rest to a speed of 4.5 x 10^3 m/s. How much time is required to do this by a net force of 5.3 x 10^5 N?
Presentation mode
Problem by Mr. Twitchell
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Assign this problem
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Similar Problems
A 1000-kg car moving north at 100km/h brakes to a stop in 50m. What are the magnitude and direction of the force?
Jamal is pulling a 4,900 N box with a force of 500 N. Jakob is pulling in the opposite direction with a force of 350 N. How fast will the box be moving after 5 seconds?
If a force accelerates 4.5 kg at 40 m/s^2, that same force would accelerate 18 kg by how much?
Howard is standing in the middle of a skating rink (we'll assume no friction), and his four friends Alice, Bernie, Christopher, and David approach him from each of the four compass directions -
north, east, south, and west, respectively.
Alice pushes on Howard with a force of x + 20 Newtons. Bernie pushes on him with a force of 2x + y Newtons. Christopher pushes on him with a force of 2x - y Newtons, and David pushes with a force of
5y Newtons.
Howard, though being pushed from all directions, does not move. Calculate the forces being applied in all four directions.
If the coefficient of kinetic friction between a 35 kg crate and the floor is .30, what horizontal force is required to move the crate at a steady speed across the floor? What horizontal force is
required if µ[k] is zero?
A 50 kg sled starts at rest, and is being pushed across a horizontal frictionless surface with a force of 30 N. The wind is pushing against the sled. The sled travels 100 meters while accelerating to
10 m/s. What is the force of the wind on the sled?
A 6000 kg tractor rests on a flatbed, held in place by chains. The chains provide a maximum horizontal force of 8000 N. When the flatbed is traveling 20 m/s, what is the minimum stopping distance if
the chains are not to break?
A mass, m[1], accelerates at 3 m/s^2 when a force, F, is applied. A second mass, m[2], accelerates at 1.0 m/s^2 when F is applied to it.
1. Find the value of the ratio m[1]/m[2].
2. Find the acceleration of the combined mass ( m[1] + m[2]) under the action of the force, F.
A 50 slug car can accelerate from 0 to 50 ft/s while traveling 1000 feet. The car then stops, and a passenger gets in. Now the car can accelerate from 0 to 50 ft/s while traveling 1050 feet. What is
the weight of the passenger?
An elevator has a mass of 2,000 kg. An upward force of 21,000 Newtons is applied to the elevator. The elevator travels up 50 floors, and each floor is 2.5 meters tall, with a 20 centimeter gap
between floors. How long does it take the elevator to arrive, and at what speed will it be traveling?
Accelerating Car
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Car and Motorcycle
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Formula using Sum and Countif not returning expected outcomes
Here's a version of the formula that works. However I need to include a second value and get an error when I try. I'm sure it's the formula syntax. Not sure if there is a better way to get the right
outcome. I basically am trying to determine % Complete OR Not Applicable
Working with just Pass as a value
=SUM((COUNTIF(CHILDREN(Status22), ="Pass")) / SUM(COUNT(CHILDREN(Status22))))
When a Not Applicable is selected the above formula results in a reduction to the % Complete and the desired outcome would be for Pass and Not Applicable to both count so if a pass or not applicable
are used the % complete would be the same.
=SUM((COUNTIF(CHILDREN(Status22), ="Pass") + (COUNTIF(CHILDREN(Status22), ="Not Applicable")) / SUM(COUNT(CHILDREN(Status22))))
When I use the formula above I get a number much higher than 100% which is the max I should be getting
Not sure the right way to structure the formula to get a % complete based on the sum of both Pass and Not Applicable versus overall number of children. Which is how I think the formula should be
structured. Any suggestions?
• Without taking a hard look at it... try this revision?
=COUNTIF(CHILDREN(Status22), ="Pass") + COUNTIF(CHILDREN(Status22), ="Not Applicable") / COUNT(CHILDREN(Status22))
You don't need all the SUM's cause the countif will automatically sum it for you and you're doing simple math to add and then divide each part.
• Th formula you gave me did not work. It returned 700%.
However the following formula did work
=SUM(COUNTIF(CHILDREN(Status22), ="Pass") + COUNTIF(CHILDREN(Status22), ="Not Applicable")) / COUNT(CHILDREN(Status22))
Thank you!
• Hmmm. Cool. Glad I helped you get on the right track!
Help Article Resources
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How to Combine Two Column Matrices in Python [Guide]
How to Combine Two Column Matrices in Python
In the field of linear algebra, matrices play a crucial role in solving various mathematical problems. One such problem is combining two matrices, which can be useful in a variety of applications
such as image processing, machine learning, and more. In this article, we will be discussing how to combine two column matrices in Python using the NumPy library.
Understanding Column Matrices
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. A column matrix, also known as a column vector… Is a matrix where the number of rows is greater than
the number of columns. In other words, it is a matrix that has a single column. An example of a column matrix is:
[[1], [2], [3]]
The Need for Combining Column Matrices
In various applications such as image processing, machine learning, and more. It may be necessary to combine two or more column matrices to form a single matrix. This can be useful for solving
problems that involve multiple sets of data.
For example, in image processing, it may be necessary to combine multiple columns of image data to form a single image. Similarly, in machine learning, it may be necessary to combine multiple columns
of feature data to form a single feature set.
How to Combine Two Column Matrices in Python Using NumPy
The NumPy library in Python provides a wide range of mathematical functions and tools to work with matrices. One of these tools is the hstack() function, which is used to stack matrices horizontally
(i.e., side by side) to form a single matrix.
Here is an example of how to combine two column matrices a and b using the hstack() function:
import numpy as np
a = np.array([[1], [2], [3]])
b = np.array([[4], [5], [6]])
c = np.hstack((a, b))
The output of this code will be:
[[1 4]
[2 5]
[3 6]]
As you can see, the hstack() function takes the two column matrices a and b and combines them horizontally to form a single matrix c. The resulting matrix c has the same number of rows as the
original matrices a and b but now has two columns.
Visualizing the Combination
graph LR
A[Matrix A] -- Horizontally--> C[Matrix C]
B[Matrix B] -- Horizontally--> C
It is important to note that the matrices being combined must have the same number of rows. If this is not the case, a ValueError will be raised.
In addition to the hstack() function, NumPy also provides the vstack() function, which is used to stack matrices vertically (i.e., one on top of the other) to form a single matrix.
Frequently Asked Questions
What is a column matrix?
A column matrix, also known as a column vector, is a matrix where the number of rows is greater than the number of columns. In other words, it is a matrix that has a single column.
What is the hstack() function in NumPy?
The hstack() function in NumPy is used to stack matrices horizontally (i.e., side by side) to form a single matrix.
What is the vstack() function in NumPy?
The vstack() function in NumPy is used to stack matrices vertically (i.e., one on top of the other) to form a single matrix.
Why is it important that the matrices being combined have the same number of rows?
It is important that the matrices being combined have the same number of rows because if this is not the case, a ValueError will be raised. The matrices being combined must have the same number of
rows in order for the hstack() function to work properly.
What are the benefits of combining column matrices?
Combining column matrices can be useful for solving problems that involve multiple sets of data, and can make data analysis and processing more efficient.
Combining column matrices in Python is a useful technique… That can be applied in a variety of fields such as image processing, machine learning, statistics, and engineering. The hstack() function
provided by the NumPy library is a simple and efficient way to combine column matrices, and it is important to ensure that the matrices being combined have the same number of rows. With the ability
to combine multiple columns of data. This technique can be used to solve a wide range of problems and make data analysis and processing more efficient.
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Prop/Speed calculator
Mar 19, 2004
Cruising area
Mobjack Bay (where?)
Boat name
A Little Noisy
Boat make
Excalibur 24
Express Racing 525
May 9, 2004
Cruising area
Malahide, Dublin
Boat make
Hydrostream V-king, 650SS OCR ,Ring 21, Ring 18, Phantom 18.
300Hp Mercury 2.4, 130 Yamaha, Bridgeport EFI, XR6, Merc 200.
I like this one,
This is a downloadable prop slippage calculator that is designed to easily calculate changes to speed and slip by changing variables such as engine rpm, horsepower, and prop pitch.
Mar 12, 2004
I tried the prop calc and it told me that it "thought I was a lying bastard and to get me GPS checked"....
Mar 13, 2004
Cruising area
In the bath
Boat make
Spectre 30
2 x Promax 225
Jono said:
I tried the prop calc and it told me that it "thought I was a lying bastard and to get me GPS checked"....
How long have you owned a Falcon?
dont understand that entered all the info and with my laser it said i had 0.08 slip. but with the chopper it said it was impossible as the slip was a minus
explain that please boffins
speed was gps'd over quite a distance
pitch of props measured by prop reveloutions
Mar 12, 2004
Cruising area
South Coast
Boat name
Boat make
Phantom 28
was the gps speed taken in one direction only?
was in boat on my own, did 2 or 3 runs then looked at max speed.
does that mean that i havent actually done 76mph then
My boat had negative prop slip too. Cye must be one amazing genius boat rigging dude
Your GPS seemed pretty much bang on next to your mondeo (shiney and new, but it is a ford) speedo when we tried it out if you remember. And that was in all sorts of dierctions and at lots of speeds
ranging from bugger all to over a ton.
hmmm go figure
The tide moves pretty fast at weston
Mar 12, 2004
Cruising area
South Coast
Boat name
Boat make
Phantom 28
I'm not trying to belittle your speed, Just pointing out that your 'max' speed would have been the highest 'speed over ground' that you attained, this would have been your actual water speed, plus
the speed of the tidal flow you were traveling in, which could be 4 or 5 mph, depending on where you are and point in the lunar cycle it was (tide height etc)
However, a P20 with a 2.5 efi ought to do circa 75-80, mebbe more, you should'a gone to windermere, you might be going even quicker than 75!
Mar 12, 2004
Cruising area
South Coast
Boat name
Boat make
Phantom 28
Johnny Boat Dude said:
The tide moves pretty fast at weston
but that's unlikely to affect the speedo in your mundane-o
that was my old sharpe havent tried my P20 yet
it affected it for a while didnt it johnny
Mar 12, 2004
Cruising area
South Coast
Boat name
Boat make
Phantom 28
even if i take 7mph off speed it still only says 0.8 whats that 8%?
what would you expect prop slip to be Mr F?
6" setback
20" mid
That 76 was in his sharp 19 with a mildly tuned 200 merc probably about 210hp at the prop. Enigine was on a 5.5" setback plate running a worked 26" chopper (mine but I'm to scared to use it) which we
were told is now 24" by prop revolutions. It was a lovely flat day and he came past me on one of his runs. It looked awesome that chopper picked the whole boat up, it was running with only a palm
sized contact point between the hull and the water, big rooster tail too. Looked well twitchy though, most excelent
Last edited:
May 9, 2004
Cruising area
Malahide, Dublin
Boat make
Hydrostream V-king, 650SS OCR ,Ring 21, Ring 18, Phantom 18.
300Hp Mercury 2.4, 130 Yamaha, Bridgeport EFI, XR6, Merc 200.
Burty said:
dont understand that entered all the info and with my laser it said i had 0.08 slip. but with the chopper it said it was impossible as the slip was a minus
explain that please boffins
speed was gps'd over quite a distance
pitch of props measured by prop reveloutions
Thats good slip no's, should be around 8-12% if set up right, you need slip, without it you have no thrust. Without thrust youre stopped!
Is your chopper a diferent pitch to the laser? Whats the pitch of the two props? Did you change the speed when you put in the pitch of the chopper?
Whats the pitch of both props and the speed with each?
Re: slip
Is your chopper a diferent pitch to the laser?
Whats the pitch of the two props?
chopper is a 24 laser is a 22.75
Did you change the speed when you put in the pitch of the chopper?
Whats the pitch of both props and the speed with each?
chopper 76mph @ 6200rpm
laser 72mph @ 6800rpm
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On Shortest Disjoint Paths in Planar Graphs
Yusuke Kobayashi and Christian Sommer
Discrete Optimization, Volume 7, Issue 4 (pp. 234-245), 2010
For a graph and a collection of vertex pairs, the disjoint paths problem is to find vertex-disjoint paths for each pair of nodes. In the corresponding optimization problem, the shortest disjoint path
problem, the vertex-disjoint paths have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or
short: min-sum), and minimizing the length of the longest path (min-max).
min-sum: We extend recent results by Colin de Verdière and Schrijver to prove that for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be
solved in polynomial time. We also prove that for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.
min-max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We also close
the gap between easy and hard instances by proving that the problem is weakly NP-hard for graphs with tree-width at least 3.
author = {Yusuke Kobayashi and Christian Sommer},
title = {On Shortest Disjoint Paths in Planar Graphs},
journal = {Discrete Optimization},
volume = {7},
number = {4},
year = {2010},
pages = {234--245},
url = {http://dx.doi.org/10.1016/j.disopt.2010.05.002},
doi = {10.1016/j.disopt.2010.05.002},
note = {Announced at ISAAC 2009}
Official version
Local version (184.2 KB)
Slides (269.9 KB)
Different from the single source shortest path problem, the single pair shortest path problem (find the shortest path between two nodes), and the multiple pairs shortest path problem, the shortest
disjoint paths problem cannot be solved by just running Dijkstra's algorithm or its bidirectional version. A k shortest path algorithm is not sufficient either. The difficulty of the shortest
disjoint paths problem comes from the difficulty of finding node disjoint paths (even without length restrictions). The disjoint paths problem for a fixed number of terminals in an undirected graph
can be solved using a polynomial-time algorithm that relies on graph minor theory. In each step, the algorithm finds a node that can be removed without altering the solution, for which crude
connectivity is sufficient. A straightforward generalization of this algorithm to the weighted version would mean to find a vertex whose removal changes neither the connectivity of the terminals nor
the optimality of the solution. Consequently, much less is known for the weighted version of the disjoint paths problem.
Home → Publications → On Shortest Disjoint Paths in Planar Graphs
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Units in SimWeights
Units in SimWeights
Units can be a big source of confusion when weighting simulation because the units most commonly used for flux are in a different system than those used by offline-software. IceTray uses a unit
system which is convenient for IceCube reconstructions but not for weighting. The units used in SimWeights were not chosen to be based on a consistent system, but were instead chosen to be the units
most often used for each quantity. For particle flux per centimeter square because even though for cosmic rays it can be given in either per centimeter squared or per meter squared, for neutrinos,
flux is almost always given per square centimeter. A table of the units used for each quantity is shown below.
Length \(\mathrm{m}\)
Effective Area \(\mathrm{m}^2\)
Solid angle \(\mathrm{sr}\)
Etendue \(\mathrm{cm}^2\cdot\mathrm{sr}\)
Energy \(\mathrm{GeV}\)
Generation Surface \(\mathrm{GeV}\cdot\mathrm{cm}^2\cdot\mathrm{sr}\)
Particle Flux \(\mathrm{GeV}^{-1}\cdot\mathrm{cm}^{-2} \cdot\mathrm{sr}^{-1}\cdot\mathrm{s}^{-1}\)
Weights \(\mathrm{s}^{-1}\)
The units listed here are for the most common case weighing case. The value returned by get_weights() will be whatever the units passed as the flux times the generation surface. For example, if you
were to pass a quantity that represented fluence in units of \(\mathrm{GeV}^{-1}\cdot\mathrm{cm}^{-2}\cdot\mathrm{sr}^{-1}\), then the result would be a unitless weight.
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An Online Math Program Helps Simplify Your Homeschool Day
Your homeschool day is chaotic enough without having to deal with tears over teaching math. In this post, find out how an online math program like CTC Math helps to simplify your homeschool day.
So often as a homeschool mom of many, it feels like I’m being pulled in several directions all day long. As soon as I finish helping one child, I’m able to sit down for about 3.2 seconds before
another child is calling for my help.
This can quickly lead to frustration for me and for my kids as they have to wait for my help. Because of that, I’m always on the lookout to simplify our homeschool days so we’re all less stressed.
Luckily, I’ve found several ways to have our curriculum work for us to make our days less hectic. One of those ways is to get a helping hand with teaching through online classes. Continue reading to
learn why online math learning is a necessity in our homeschool.
Disclosure: This is a sponsored post for CTCMath. I received free access to the product and was compensated for my time. I am not required to give a positive review and all opinions are my own. This
post contains affiliate links.
Through the many years we’ve been homeschooling, I’ve tried several different math programs. Some of the homeschool math curriculum used a mastery approach, while others used a spiral approach. We’ve
used math textbooks, living books, online math programs, and games to teach math.
All of these approaches to teaching math worked well, especially our first year or two when I was only homeschooling fourth grade and kindergarten. Then my children got older and started learning
more complex math concepts. I also started homeschooling my third child and added a baby into the mix as well. Math became a time-consuming, tear-inducing part of our homeschool day.
I quickly realized that having my children learn math online makes for much less chaos in our day. My children still learn the necessary math concepts for their levels, but I don’t have to be the one
teaching them. This frees up so much of my time, which also relieves much of the daily stress homeschool moms experience.
Why an Online Math Program Makes Your Day Easier
So how can an online math program simplify your homeschool day? Why can it be a more efficient method of teaching math than using a textbook when you’re homeschooling multiple children?
Imagine all of your children from K-12 doing their math lessons at the same time on separate devices. Instead of spending at least half an hour per child teaching math, you simply walk around and
help as needed. Your children can use headphones to listen to their lessons so that everyone can work at once.
What if you don’t have enough computers for each child to work at the same time? Then you can simply have your children work on math one at a time. Since they can listen to the lessons using
headphones, they won’t disrupt siblings doing something else. During this time, you can work with another child on a different subject, or have the others work independently on other assignments
while you tend to younger children or get lunch started.
How an Online Math Program Helps Us Simplify Our Homeschool Day
As I mentioned above, once my children got into more time-consuming math concepts, math became so stressful that it quickly became my least favorite part of our homeschool day. Not only was I trying
to teach concepts to three children in three different grades, I also had a young baby to tend to. I was stretched so thin!
As soon as I finished one lesson, I set that child off to work on math problems while I started a lesson with child two. Then, of course, as I was teaching that lesson, child one had a question. It
was just too much. With lessons for all three of my homeschool-age children, math was easily taking up half of our day, especially once that baby became a toddler to chase after. I knew using
textbooks with lessons I had to teach wasn’t working anymore.
Enter online math. Once we started using online math curriculum, I noticed we had more time in our homeschool day. Everyone’s math lessons can happen at the same time instead of taking half of our
school day.
How I Use CTC Math
Each weekend I write out plans for the next homeschool week. Thanks to CTC Math, preparing for math only takes a few minutes. Through my parent account, I create a weekly task for each boy. I can
choose which lessons I want the boys to work on, including the grade they need to receive to pass the lesson. They can be new concepts or a review of previous lessons in which they need more
Along with choosing four lessons for each boy to work on per week, I also create a quiz for Fridays. To do this, I simply add a question bank to the weekly task. I choose how many questions will be
on the quiz, which lessons it will cover, and the difficulty of the problems. I appreciate seeing how they do on this so I know if they should review any information.
My boys each have a Chromebook so that they can both work on math at the same time. They each use headphones as well for less noise and disruption. When it’s time for math, they can easily put in
their username and password to begin the day’s lesson. The lessons are short, which works well for attention spans. I receive an email report of their progress, so I can easily make sure they are
doing, and understanding, the work.
You can see how quick and easy it is to use CTCMath in your homeschool in this video.
When you’re homeschooling multiple children, teaching math can take up way too much of your day. You want to find a math curriculum that will work with your schedule and keep you from getting pulled
in too many directions.
Online math programs like CTC Math help ease the chaos in your day by teaching the math lessons for you. Your children can work on math at the same time while you walk around and help as needed or
tend to little ones. Or one child can focus on their online math learning while you work with a sibling in a different subject. Either way, you aren’t spending your entire day teaching math! Try out
the CTC Math free trial to learn how to simplify your chaotic homeschool days.
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[In Depth] Random Forests & Ensemble Learning: Concept And Application | Neuraldemy
In this tutorial, we are going to learn an advanced concept related to decision trees called random forests. The idea behind random forests is the concept of ensembling. Sir Francis Galton
(1822–1911), an English philosopher and statistician, was the brain behind the basic ideas of standard deviation and correlation. Once, during a visit to a livestock fair, Galton got interested in a
simple game where people tried to guess the weight of an ox. Lots of folks joined in, but no one hit the exact weight: 1,198 pounds.
But guess what? Galton discovered something cool – when he averaged out all the guesses, it was super close to the real weight: 1,197 pounds. This reminded him of the Condorcet jury theorem, showing
that combining many simple guesses can give a really good result. Fast forward to 2004, an American financial journalist named James Michael Surowiecki wrote a book called “The Wisdom of Crowds: Why
the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies, and Nations.” Surowiecki’s idea is pretty neat – when you gather info from different people, you can
make better decisions, sometimes even better than what a super-smart person might decide. This is the basic idea of ensemble learning in machine learning.
An ensemble in the context of machine learning refers to the technique of combining multiple individual models to create a stronger, more robust predictive model. The idea behind ensemble methods is
to leverage the diversity of multiple models to improve overall performance, generalization, and accuracy. A group of predictors is called an ensemble and an ensemble Learning algorithm is called an
Ensemble method.
Now the question is how do we ensemble models? What are the combination methods? Where does random forest fit in this whole thing? That’s what we are going to learn in this tutorial.
Table of Contents
1. Linear Algebra And Calculus For Machine Learning
2. Probability And Statistics For Machine Learning
3. Python, Numpy, Matplotlib And Pandas
4. Decision Trees – Learn here.
What You Will Learn
1. Bias-Variance Tradeoff
2. Bagging
3. Pasting
4. Ensemble Learning
5. Random Forests
6. Stacking
7. Application
Bias-Variance Tradeoff
To understand random forests, we need to understand the bias and variance tradeoffs in machine learning. Imagine you are building multiple models on different samples of a dataset. The bias in data
is also reflected in the predictions of these models. This means since we are training multiple models on different samples each will have its own bias output as each sample has its own bias. Meaning
our models will produce a range of predictions. Bias measures how far off in general these model’s predictions are from the correct value. When training multiple models on different samples, variance
can be observed in the variability of predictions across these models for the same input. The variance here is how much predictions for a given point vary between different realizations of the data
When dealing with predictive models, the prediction errors are mainly because of the errors due to bias and errors due to variance. Our goal in machine learning is to reduce bias and variance so that
the model can make fewer errors. But in the real world due to finite data, we have to settle with a tradeoff between bias-variance.
At its core, reducing bias and variance is nothing but dealing with model complexity or dealing with underfitting or overfitting. As our model becomes more complex, the bias reduces and variance
increases. So, we try to find an optimum model complexity. A simple model may not have enough capacity to capture the underlying patterns in the data, resulting in high bias. It makes strong
assumptions and may oversimplify the true relationships. As the model complexity increases, it becomes more flexible and capable of fitting complex patterns in the data. It can better approximate the
underlying relationships, leading to a reduction in bias.
Simple models are less sensitive to variations in the training data, resulting in lower variance. They are more stable and consistent but may miss intricate patterns. More complex models, with
greater flexibility, can fit the training data more closely. However, they become sensitive to noise and fluctuations, leading to higher variance. The model might start capturing random patterns
present in the training data that don’t generalize well.
Overfitting occurs when a highly complex model fits the training data too closely, capturing noise and leading to high variance. It performs well on the training data but poorly on new data.
Underfitting occurs when a model is too simple to capture the underlying patterns, resulting in high bias. It performs poorly on both training and new data.
A detailed discussion on this topic can be found in this tutorial here.
Our goal is to find the optimum balance between bias and variance. Going beyond optimum may result in overfitting and before optimum may result in underfitting. Regularization and Cross-Validation
techniques are often used to control model complexity and prevent overfitting by penalizing overly complex models.^1
Bagging (Bootstrap Aggregating)
A unique way to reduce the variance is by a method called bagging. It is an ensemble learning technique used to improve the stability and accuracy of machine learning models. The basic idea behind
bagging is to train multiple instances of the same learning algorithm on different subsets of the training data, and then combine their predictions to achieve a more robust and generalizable model.
What this is doing basically is creating a bigger population for us which will allow us to reduce the variance.
Here’s how bagging works:
1. Bootstrap Sampling:
□ Random subsets of the training data are created by sampling with replacement (bootstrap sampling). This means that some instances may be repeated in a subset while others may not be included
at all.
2. Model Training:
□ A base learning algorithm (e.g., decision tree) is trained independently on each of these bootstrap samples. As a result, multiple models are created, each exposed to a slightly different
perspective of the data.
3. Prediction Combination:
□ When making predictions, the individual models’ outputs are combined through a voting or averaging process. For classification tasks, the mode (most frequent class) is often used, while for
regression tasks, the average of predictions is taken.
Notes On Ensemble:
• Each model in the bagging ensemble is trained independently of the others. This independence means that if one model overfits a particular pattern or noise in the data, other models may not
necessarily follow the same pattern. As a result, the overall ensemble is less likely to be sensitive to specific instances in the training data, reducing the risk of overfitting. The diversity
introduced by bagging improves the stability and generalization performance of the ensemble. The combined knowledge of multiple models tends to be more reliable and less sensitive to variations
in the training data.
• If individual models make errors in certain instances, the ensemble has the potential to compensate for these errors. Some models may correctly predict instances where others fail, leading to a
more accurate and robust overall prediction.
• It helps reduce the variance of the model by combining predictions from multiple models trained on different subsets of the data. This is particularly beneficial when dealing with complex models
prone to overfitting such as decision trees. By training on diverse subsets of the data, bagging makes the model less sensitive to variations in the training data. It improves stability and
generalization performance. Bagging can improve a model’s robustness by reducing the impact of outliers or noisy instances in the training data. The independent training of models allows for
parallelization, making bagging suitable for distributed computing environments.
But what about bias?
It tends to reduce both bias and variance but not on individual models. The net result is that the ensemble, formed by combining predictions from multiple predictors, tends to have a similar bias to
that of individual predictors trained on the original training set. However, the variance of the ensemble is lower compared to a single predictor trained on the complete dataset.
Just like bagging, there is another sampling method called pasting.
In pasting, each base model is trained on a random subset of the training data, sampled without replacement. This means that once an instance is selected for a particular subset, it cannot be
selected again for that subset. Multiple base models (predictors) are trained independently on different subsets of the data so that each model sees a distinct subset of the training instances.
Advantages of Pasting:
1. Reduction in Variance:
□ Similar to bagging, pasting primarily aims to reduce variance. By training models on different subsets of the data, it helps create an ensemble that is less sensitive to the noise or
fluctuations present in any single training set.
2. Improved Generalization:
□ The diversity introduced by pasting contributes to improved generalization to unseen data. The ensemble is more likely to capture the underlying patterns in the data rather than memorizing
specific instances.
3. Parallelization:
□ Pasting allows for parallelization during training because each model is trained independently. This makes it computationally efficient and suitable for distributed computing environments.
Comparison with Bagging:
• Sampling Approach:
□ In bagging, sampling is done with replacement, allowing instances to be selected multiple times for a particular subset. In pasting, sampling is done without replacement, ensuring that each
instance is selected only once for a particular subset.
• Use Cases:
□ Bagging is often used with high-variance models, such as decision trees. Pasting can be suitable for situations where the base models are sensitive to the training instances and can
potentially overfit.
Now the question is how do we combine these model’s output to get the final output? Various methods are available but I am mentioning only the ones that are mostly used.
Combination Methods
There are various combination methods available. You don’t need to know all of them as most of them are not even implemented in Sklearn. Here are the important ones:
1. Voting (Classification):
□ In binary classification, each model in the ensemble predicts the class label for a given instance. The final prediction is determined by a majority vote. The class that receives the most
votes is chosen as the ensemble’s prediction.
□ For multiclass classification, a similar voting process is applied. Each model predicts a class, and the class with the highest number of votes is selected as the final prediction.
□ The voting process can be of two types: Soft Voting And Hard Voting.
□ In hard voting, each base model in the ensemble makes a classification prediction, and the final ensemble prediction is determined by a simple majority vote. The class that receives the most
votes is selected as the final prediction. For example, if there are three base models, and they predict classes A, B, and A, then the majority class is A, and the final prediction is A.
□ In soft voting, each base model provides a probability estimate for each class, and the final ensemble prediction is based on the average or weighted average of these probabilities. For
example, if there are three base models, and they provide probability estimates for classes A, B, and A as (0.8, 0.6, 0.7), then the final prediction is the class with the highest average
probability – A. It works well when individual models provide meaningful probability estimates. This is not for all models.
2. Averaging (Regression):
□ In regression tasks, where the goal is to predict a continuous numerical value, each model produces a numeric prediction. The final prediction is obtained by averaging these numeric
predictions. This can be a simple arithmetic mean or a weighted mean, where the weights are determined based on the models’ performance or other criteria.
3. Weighted Aggregation:
□ Each model’s prediction may be given a specific weight based on its performance or other considerations. The final prediction is then a weighted combination of individual predictions. This
approach allows for giving more influence to models that have demonstrated better accuracy on the validation set or during training.
Now that we have these basics in mind. Let’s come to our topic of discussion random forests.
What are Random Forests?
In machine learning, decision trees are like individual experts. They’re simple and intuitive, but they can sometimes make errors or be overly sensitive to the data they were trained on. Random
Forests, on the other hand, act as a group of decision trees, offering a more robust and accurate prediction by combining the strengths of multiple trees.
Example: Decision Trees vs. Random Forests
Decision Tree Scenario: Suppose you’re predicting whether someone will enjoy outdoor activities based on weather conditions. A decision tree might say, “If it’s sunny, they’ll likely enjoy it; if
it’s rainy, they won’t.” However, this decision tree might become too specific and make errors if it encounters a rare rainy day that people actually enjoy.
Random Forest Scenario: Now, consider a Random Forest consisting of several decision trees. Each tree might look at different aspects of the weather, like temperature, humidity, or wind speed. When
it’s time to make a prediction, each tree casts its vote on whether the person will enjoy the activity. The final decision is determined by the majority vote among all the trees. This way, even if
one tree makes a mistake due to unusual circumstances, the overall prediction remains reliable.
The random forest is based on applying bagging to decision trees, with one important extension: in addition to sampling the records, the algorithm also samples the variables. During the training
process, each decision tree in the Random Forest is exposed to a random subset of the training data and a random subset of the features. This introduces diversity among the trees, preventing them
from becoming too similar. When it’s time to make a prediction, each tree in the Random Forest “votes” on the outcome. For classification, the majority vote determines the predicted class; the
average of the individual predictions is taken for regression.
Advantages of Random Forests:
1. Reduced Overfitting: Decision trees can be prone to overfitting, meaning they memorize the training data instead of learning general patterns. Random Forests mitigate this by averaging over
multiple trees, which tends to smooth out individual peculiarities and provide a more generalized prediction.
2. Improved Accuracy: The collective wisdom of multiple trees often leads to more accurate predictions compared to a single decision tree.
3. Robustness: Random Forests are less sensitive to outliers or noisy data because they consider a broader range of perspectives.
4. Feature Importance: Random Forests can also provide insights into feature importance. By observing how much each feature contributes to the accuracy of predictions across the ensemble, one can
gain a better understanding of which features are more influential. We will learn about feature importance in the notebook in further detail.
Comparison with Decision Trees:
• Decision Trees:
□ Single, straightforward model.
□ Prone to overfitting.
□ Might be sensitive to the specific training data.
• Random Forests:
□ Ensemble of decision trees.
□ Reduces overfitting through averaging.
□ It is more robust due to the diversity of trees.
We will see the implementation of random forests in the notebook.
Please Note Random Forests are often considered “black box” models, meaning that their internal workings can be complex and challenging to interpret directly. Decision trees, in general, are capable
of capturing complex non-linear relationships in the data. When combined in an ensemble, the overall model’s complexity can increase, making it harder to interpret.
Random Forests also give a measure of feature importance, it’s often relative within the model but you should note that the importance values do not necessarily reflect real-world meanings or causal
relationships. The importance values are specific to the predictions made by the model.
“Random Patches” and “Random Subspaces”
Bagging methods come in many flavours but mostly differ from each other by the way they draw random subsets of the training set:
Random Patches:
• Especially with high-dimensional data, Random Patches involve creating multiple subsets of the training dataset by randomly sampling both instances (data points) and features (attributes or
• Each subset (patch) is used to train a separate model. These models are then aggregated to make predictions.
• The randomness in selecting both instances and features helps create diverse models, enhancing the overall model’s robustness and generalization.
• The Random Patches method is commonly used with algorithms like the Random Forest, where each tree is trained on a different subset of both instances and features.
Random Subspaces:
• Random Subspaces is a similar concept but focuses on randomizing only the features (attributes or variables) while keeping all instances in each subset.
• In this approach, each subset (subspace) is created by randomly selecting a subset of features for training a model.
• Again, the goal is to introduce diversity among the models by training them on different sets of features.
• Often used with algorithms like Bagged Decision Trees or other base learners.
• The key difference lies in what is randomized—Random Patches randomize both instances and features, while Random Subspaces randomize only features.
• Both methods contribute to reducing overfitting and improving the generalization performance of ensemble models.
• These techniques are particularly useful when dealing with high-dimensional data, as they introduce variability in the training process.
When training each decision tree in the ensemble, a random subset of the original training dataset is created by sampling with replacement. This means that some instances from the original dataset
may be included multiple times, while others may not be included at all. On average, each decision tree in the Random Forest is trained on about 63% of the original training instances. The reason for
this is the nature of bootstrapping, where, on average, about 63% of the instances are selected in each bootstrap sample. The remaining 37% of the instances that are not included in the bootstrap
sample for a particular tree are referred to as out-of-bag (OOB) instances. Since these instances are not used in training a specific tree, they can be considered as a validation set for that
particular tree. We use these OOB instances for evaluation. We will see its implementation in the notebook.
If you are curious about why 63% then please visit this post here. It has something to do with the probability of instance sampling. And do share your discovery in the forum.
To understand boosting, we need to know the concept of weak learners and strong learners.
A decision stump is a simple decision tree with only one level (depth of one). It makes a decision based on a single feature and a threshold. Now, consider a decision stump that predicts whether a
person will buy a product based on their age. The decision stump might say “yes” if the person is older than 30 and “no” otherwise. It is an example of a weak learner. A weak learner is a model that
performs slightly better than random chance or is just slightly better than random guessing. They are often simple models, such as decision stumps or shallow decision trees, which have limited
expressive power. These models may not perform well on their own, but they can be combined or boosted to create a strong learner.
Now consider a random forest, it is a strong leaner because it can achieve high predictive performance on its own. Strong learners are usually complex models with high expressive power and the
ability to capture intricate relationships in the data. They can be more resource-intensive and prone to overfitting, as they may learn the training data too well.
Boosting is an ensemble learning technique that combines multiple weak learners to create a strong learner. The goal is to improve overall predictive performance by sequentially training weak models
on the dataset, with each subsequent model giving more emphasis to the examples that the previous models misclassified. The final prediction is often made by combining the predictions of all the weak
learners, typically through a weighted sum or a voting mechanism. Bagging requires little tuning but boosting requires much greater care in its application.
There are various boosting algorithms available but the most popular ones are AdaBoost, Gradient boosting, and Stochastic Gradient Boosting.
1. AdaBoost (Adaptive Boosting)
AdaBoost is the most popular boosting algorithm used. It is called adaptive because, it does not need to know error bounds on the weak classifiers, nor does it need to know the number of classifiers
in advance. The algorithm assigns different weights to training examples based on their classification errors, allowing subsequent weak learners to focus on the instances that were misclassified by
previous models. AdaBoost is a greedy algorithm.
How AdaBoost Works:
• Weighted Instances: AdaBoost assigns weights to each training instance. Initially, all weights are equal.
• Sequential Training: Weak learners are trained iteratively, and the algorithm pays more attention to instances that are misclassified by giving them higher weights.
• Weighted Voting: The final prediction is a weighted combination of the weak learners. Each weak learner’s weight is determined based on its accuracy, with more accurate learners having a higher
say in the final prediction.
# AdaBoost Algorithm
# Input: Training dataset D, Number of weak learners T
# Initialize instance weights
# For each iteration t from 1 to T
for t in range(T):
# Train weak learner
weak_learner = train_weak_learner(D, weights)
# Calculate weighted error of weak learner
epsilon_t = calculate_weighted_error(weak_learner, D, weights)
# Calculate weak learner weight
alpha_t = 0.5 * log((1 - epsilon_t) / epsilon_t)
# Update instance weights
update_weights(weights, alpha_t, weak_learner, D)
# Combine weak learners into a strong learner
def final_strong_learner(x):
return sign(sum(alpha_t * weak_learner(x) for alpha_t, weak_learner in zip(alphas, weak_learners)))
# Output: Final strong learner is a linear combination of weak learners.
Code language: PHP (php)
Though not necessary If you are curious about learning its mathematical implementation I would recommend reading the first 20 pages of this paper.
AdaBoost uses exponential loss for updating instance weights, emphasizing instances that are misclassified more often. AdaBoost is sensitive to outliers and noise. Techniques such as limiting tree
depth or using more robust weak learners can mitigate these issues.
When To Use:
1. You Have Weak Learners:
□ AdaBoost works well when you have weak learners, which are models that perform slightly better than random chance. Common weak learners include shallow decision trees (stumps).
2. High-Dimensional Data:
□ AdaBoost can be effective in high-dimensional datasets where features might not be informative on their own, but their combinations contribute to better predictive performance.
3. Classification Tasks:
□ AdaBoost is primarily designed for classification tasks. It can be used when you have a binary or multiclass classification problem. However, Sklearn also allows you to use it for regression.
4. You Want to Combine Multiple Models:
□ If you want to combine the predictions of multiple models to create a strong ensemble classifier, AdaBoost is a suitable choice.
AdaBoost is a powerful ensemble learning algorithm, several variants and extensions have been developed to address specific challenges or improve its performance in certain scenarios. Here are some
notable variants of AdaBoost:
1. Real AdaBoost:
□ Idea: It extends AdaBoost to handle real-valued predictions rather than just binary classification.
□ Application: Real AdaBoost is suitable for problems where the target variable has multiple classes or when you need probabilistic predictions.
2. SAMME (Stagewise Additive Modeling using a Multiclass Exponential loss):
□ Idea: SAMME is an extension of AdaBoost designed for multi-class classification problems.
□ Application: It is commonly used when dealing with more than two classes, and it generalizes AdaBoost to work in the multi-class setting.
3. SAMME.R (Real SAMME):
□ Idea: SAMME.R is an improvement over SAMME, which handles real-valued class probabilities rather than discrete class labels.
□ Application: SAMME.R is beneficial when the base learner can provide class probabilities, such as in the case of decision trees with probability estimates.
You don’t need to remember each one of them. In scikit-learn, the default AdaBoost algorithm used for classification is a variant of SAMME, specifically SAMME.R (Real SAMME). This variant allows
handling real-valued class probabilities. For regression tasks, scikit-learn’s AdaBoost uses the exponential loss function and AdaBoost.R2 which is designed specifically for regression problems,
where the goal is to predict a continuous target variable. We will see what to use in practice.
2. Gradient Boosting
Our next boosting algorithm is gradient boosting which trains weak learners sequentially, with each one addressing the errors of the previous ones. Gradient Boosting aims to improve the accuracy of
predictions by optimizing a differentiable loss function through the iterative addition of weak learners. Gradient Boosting optimizes the loss function by minimizing gradients, whereas AdaBoost
focuses on adjusting instance weights to correct misclassification.
Components of Gradient Boosting:
a. Weak Learners (Base Models):
• Typically Decision Trees: Decision trees are commonly used as weak learners, often shallow trees to avoid overfitting.
• Can Be Other Models: While decision trees are common, Gradient Boosting can use other types of models as well.
b. Loss Function:
• Defines the Objective: A differentiable loss function is chosen based on the nature of the problem (regression or classification).
• Measures Prediction Error: The goal is to minimize the loss, which represents the discrepancy between predictions and true values.
c. Gradient Descent:
• Optimization Technique: Gradient Descent is used to minimize the loss function.
• Adjusts Predictions: At each stage, the new weak learner is trained to correct the errors made by the existing ensemble.
d. Shrinkage (Learning Rate):
• Controls Contribution of Each Weak Learner: A shrinkage parameter (learning rate) is introduced to control the contribution of each weak learner.
• Prevents Overfitting: Shrinkage serves as a regularization technique by penalizing the impact of each weak learner, preventing the model from fitting the training data too closely.
Gradient Boosting Algorithm:
1. Initialize the Model:
□ Set the initial prediction as the average (for regression) or log odds (for classification) of the target variable.
2. For each iteration (t = 1 to T):
□ Calculate the negative gradient of the loss function with respect to the current predictions.
□ Train a weak learner (e.g., decision tree) to predict the negative gradient.
□ Determine the step size (learning rate) to update the predictions.
□ Update the predictions by adding the product of the step size and the predictions of the weak learner.
3. Combine Weak Learners:
□ The final prediction is the sum of the initial prediction and the weighted sum of the weak learners’ predictions.
Challenges and Considerations:
• Computational Complexity: Training can be computationally expensive, especially with large datasets.
• Sensitivity to Noisy Data: Gradient Boosting can be sensitive to noisy data and outliers.
• Tuning Hyperparameters: Requires careful tuning of hyperparameters such as learning rate, tree depth, and number of iterations.
If you are looking for mathematical intuition please check this paper. The concepts explained above will be clear in the notebook.
Gradient Boosting has evolved over time, leading to various variants and extensions that address specific challenges or aim to improve certain aspects of the algorithm. Some notable variants of
Gradient Boosting include:
1. XGBoost (eXtreme Gradient Boosting):
□ Key Features:
☆ Regularization: Incorporates L1 and L2 regularization terms in the objective function to control model complexity.
☆ Parallelization: Enables parallel and distributed computing for faster training.
☆ Handling Missing Values: Can handle missing values in the dataset during training.
□ Advantages:
☆ Often achieves better performance and faster training compared to traditional Gradient Boosting.
☆ Widely used in various machine learning competitions.
2. LightGBM (Light Gradient Boosting Machine):
□ Key Features:
☆ Gradient-Based One-Side Sampling (GOSS): Efficiently selects instances with large gradients for training, reducing the number of instances used.
☆ Exclusive Feature Bundling: Optimizes the use of memory by bundling exclusive features together.
☆ Support for Categorical Features: Can handle categorical features directly.
□ Advantages:
☆ Designed for distributed and efficient training, particularly on large datasets.
☆ Efficiently handles large categorical feature spaces.
Now, let’s move to our next concept stacking. Imagine you’re tasked with predicting whether a student passes or fails an exam based on various features like study hours, attendance, and previous
grades. You decide to use these models: a Decision Tree, and a Logistic Regression. In stacking, instead of relying on just one of these models, you combine their strengths to create a more robust
Stacked generalization is a method for combining estimators to reduce their biases. More precisely, the predictions of each individual estimator are stacked together and used as input to a final
estimator to compute the prediction. This final estimator is trained through cross-validation. Stacking goes beyond simple averaging or voting by training a meta-model to make predictions based on
the outputs of diverse base models.
Imagine we have three types of predictors (models) – a Decision Tree, a Linear Regression, and a k-nearest Neighbors (KNN) model. We want to create a super predictor, a Blender, that combines their
1. Training the First Layer:
We take our dataset and split it into two parts – Training Set A and Hold-Out Set B.
Training Set A:
• Train our three models (Decision Tree, Linear Regression, KNN) on Training Set A. Each model learns to make predictions based on the features in this set.
Hold-Out Set B:
• We keep Hold-Out Set B untouched for now. These are instances our models have never seen during their training.
2. Making Predictions with First Layer:
Now, each model makes predictions on Hold-Out Set B. We get three sets of predictions – one from each model.
3. Creating a 3D Training Set for the Blender:
We create a new 3D training set using these three sets of predictions as features. The target values (actual outcomes) from Hold-Out Set B are still there.
4. Training the Blender (Meta-Model):
Our Blender is like a smart friend that learns from these three sets of predictions. We train a meta-model (Blender) on this new 3D training set.
5. Predicting with the Stacked Model:
Now, when a new instance comes in:
• Each model (Decision Tree, Linear Regression, KNN) predicts the outcome.
• These predictions become features for our Blender.
• The Blender combines these features to make the final prediction.
6. Layered Blending (Optional):
If we want to go even deeper, we can repeat this process. We split our data into three subsets, train models on the first subset, use those models to create a 3D training set for the next layer, and
so on.
7. Sequential Prediction:
For a new instance, predictions travel through each layer sequentially – first from the base models, then through the Blender (meta-model), and so on if we have multiple layers.
This stacking approach allows us to harness the collective intelligence of diverse models, improving our overall predictive performance.
Why Stack?
• Diversity Benefits:
□ Each model brings a unique perspective. Decision Trees capture complexity, Logistic Regression simplifies, and KNN looks at neighbors.
• Reducing Overfitting:
□ While individual models may overfit, combining them with a meta-model helps smooth out their individual quirks.
• Improved Generalization:
□ The ensemble can often generalize better to new, unseen data.
Implementations of Concepts In Python
Footnotes And Sources:
1. Scott Fortmann Roe ↩︎
Sources And Further Reading:
1. Data Mining with Decision Trees – Book
2. Hands-on Machine Learning Book
3. Friedman, J.H. (2001). Greedy function approximation: A gradient boosting machine. Annals of Statistics
4. Tianqi Chen, Carlos Guestrin, “XGBoost: A Scalable Tree Boosting System”
5. T. Hastie, R. Tibshirani and J. Friedman, “Elements of Statistical Learning Ed. 2”, Springer, 2009.
6. Understanding Random Forests From Theory To Application by Gilles Louppe.
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Compiler Error Index
8. Compiler Error Index¶
Elaboration on type errors produced by the compiler. Many error messages contain links to the sections below.
8.1. Uniqueness errors¶
8.1.1. “Using x, but this was consumed at y.”¶
A core principle of uniqueness typing (see In-place Updates) is that after a variable is “consumed”, it must not be used again. For example, this is invalid, and will result in the error above:
let y = x with [0] = 0
in x
Several operations can consume a variable: array update expressions, calling a function with unique-typed parameters, or passing it as the initial value of a unique-typed loop parameter. When a
variable is consumed, its aliases are also considered consumed. Aliasing is the possibility of two variables occupying the same memory at run-time. For example, this will fail as above, because y and
x are aliased:
let y = x
let z = y with [0] = 0
in x
We can always break aliasing by using a copy expression:
let y = copy x
let z = y with [0] = 0
in x
8.1.2. “Would consume x, which is not consumable”¶
This error message occurs for programs that try to perform a consumption (such as an in-place update) on variables that are not consumable. For example, it would occur for the following program:
let f (a: []i32) =
let a[0] = a[0]+1
in a
Only arrays with a a unique array type can be consumed. Such a type is written by prefixing the array type with an asterisk. The program could be fixed by writing it like this:
let f (a: *[]i32) =
let a[0] = a[0]+1
in a
Note that this places extra obligations on the caller of the f function, since it now consumes its argument. See In-place Updates for the full details.
You can always obtain a unique copy of an array by using copy:
let f (a: []i32) =
let a = copy a
let a[0] = a[0]+1
in a
But note that in most cases (although not all), this subverts the purpose of using in-place updates in the first place.
8.1.3. “Unique-typed return value of x is aliased to y, which is not consumable”¶
This can be caused by a function like this:
let f (xs: []i32) : *[]i32 = xs
We are saying that f returns a unique array - meaning it has no aliases - but at the same time, it aliases the parameter xs, which is not marked as being unique (see In-place Updates). This violates
one of the core guarantees provided by uniqueness types, namely that a unique return value does not alias any value that might be used in the future. Imagine if this was permitted, and we had a
program that used f:
let b = f a
let b[0] = x
The update of b is fine, but if b was allowed to alias a (hence occupying the same memory), then we would be modifying a as well, which is a violation of referential transparency.
As with most uniqueness errors, it can be fixed by using copy xs to break the aliasing. We can also change the type of f to take a unique array as input:
let f (xs: *[]i32) : *[]i32 = xs
This makes xs “consumable”, in the sense used by the error message.
8.1.4. “A unique-typed component of the return value of x is aliased to some other component”¶
Caused by programs like the following:
let main (xs: *[]i32) : (*[]i32, *[]i32) = (xs, xs)
While we are allowed to “consume” xs, as it is a unique parameter, this function is trying to return two unique values that alias each other. This violates one of the core guarantees provided by
uniqueness types, namely that a unique return value does not alias any value that might be used in the future (see In-place Updates) - and in this case, the two values alias each other. We can fix
this by inserting copies to break the aliasing:
let main (xs: *[]i32) : (*[]i32, *[]i32) = (xs, copy xs)
8.1.5. “Consuming parameter passed non-unique argument”¶
Caused by programs like the following:
let update (xs: *[]i32) = xs with [0] = 0
let f (ys: []i32) = update ys
The update function consumes its xs argument to perform an in-place update, as denoted by the asterisk before the type. However, the f function tries to pass an array that it is not allowed to
consume (no asterisk before the type).
One solution is to change the type of f so that it also consumes its input, which allows it to pass it on to update:
let f (ys: *[]i32) = update ys
Another solution to copy the array that we pass to update:
let f (ys: []i32) = update (copy ys)
8.1.6. “Non-consuming higher-order parameter passed consuming argument.”¶
This error occurs when we have a higher-order function that expects a function that does not consume its arguments, and we pass it one that does:
let apply 'a 'b (f: a -> b) (x: a) = f x
let consume (xs: *[]i32) = xs with [0] = 0
let f (arr: *[]i32) = apply consume arr
We can fix this by changing consume so that it does not have to consume its argument, by adding a copy:
let consume (xs: []i32) = copy xs with [0] = 0
Or we can create a variant of apply that accepts a consuming function:
let apply 'a 'b (f: *a -> b) (x: *a) = f x
8.1.7. “Function result aliases the free variable x”¶
Caused by definitions such as the following:
let x = [1,2,3]
let f () = x
To simplify the tracking of aliases, the Futhark type system requires that the result of a function may only alias the function parameters, not any free variables. Use copy to fix this:
8.1.8. “Parameter x refers to size y which will not be accessible to the caller¶
This happens when the size of an array parameter depends on a name that cannot be expressed in the function type:
let f (x: i64, y: i64) (A: [x]bool) = true
Intuitively, this function might have the following type:
val f : (x: i64, y: i64) -> [x]bool -> bool
But this is not currently a valid Futhark type. In a function type, each parameter can be named as a whole, but it cannot be taken apart in a pattern. In this case, we could fix it by splitting the
tuple parameter into two separate parameters:
let f (x: i64) (y: i64) (A: [x]bool) = true
This gives the following type:
val f : (x: i64) -> (y: i64) -> [x]bool -> bool
Another workaround is to loosen the static safety, and use a size coercion to give A its expected size:
let f (x: i64, y: i64) (A_unsized: []bool) =
let A = A_unsized :> [x]bool
in true
This will produce a function with the following type:
val f [d] : (i64, i64) -> [d]bool -> bool
This does however lose the constraint that the size of the array must match one of the elements of the tuple, which means the program may fail at run-time.
The error is not always due to an explicit type annotation. It might also be due to size inference:
let f (x: i64, y: i64) (A: []bool) = zip A (iota x)
Here the type rules force A to have size x, leading to a problematic type. It can be fixed using the techniques above.
8.2. Size errors¶
8.2.1. “Size x unused in pattern.”¶
Caused by expressions like this:
And functions like this:
Since n is not the size of anything, it cannot be assigned a value at runtime. Hence this program is rejected.
8.2.2. “Causality check”¶
Causality check errors occur when the program is written in such a way that a size is needed before it is actually computed. See Causality restriction for the full rules. Contrived example:
let f (b: bool) (xs: []i32) =
let a = [] : [][]i32
let b = [filter (>0) xs]
in a[0] == b[0]
Here the inner size of the array a must be the same as the inner size of b, but the inner size of b depends on a filter operation that is executed after a is constructed.
There are various ways to fix causality errors. In the above case, we could merely change the order of statements, such that b is bound first, meaning that the size is available by the time a is
bound. In many other cases, we can lift out the “size-producing” expressions into a separate let-binding preceding the problematic expressions.
8.2.3. “Unknowable size x in parameter of y”¶
This error occurs when you define a function that can never be applied, as it requires an input of a specific size, and that size is not known. Somewhat contrived example:
let f (x: bool) =
let n = if x then 10 else 20
in \(y: [n]bool) -> ...
The above constructs a function that accepts an array of size 10 or 20, based on the value of x argument. But the type of f true by itself would be ?[n].[n]bool -> bool, where the n is unknown. There
is no way to construct an array of the right size, so the type checker rejects this program. (In a fully dependently typed language, the type would have been [10]bool -> bool, but Futhark does not do
any type-level computation.)
In most cases, this error means you have done something you didn’t actually mean to. However, in the case that that the above really is what you intend, the workaround is to make the function fully
polymorphic, and then perform a size coercion to the desired size inside the function body itself:
let f (x: bool) =
let n = if x then 10 else 20
in \(y_any: []bool) ->
let y = y_any :> [n]bool
in true
This requires a check at run-time, but it is the only way to accomplish this in Futhark.
8.2.4. “Existential size would appear in function parameter of return type”¶
This occurs most commonly when we use function composition with one or more functions that return an existential size. Example:
The filter function has this type:
val filter [n] 't : (t -> bool) -> [n]t -> ?[m].[m]t
That is, filter returns an array whose size is not known until the function actually returns. The length function has this type:
val length [n] 't : [n]t -> i64
Whenever length occurs (as in the composition above), the type checker must instantiate the [n] with the concrete symbolic size of its input array. But in the composition, that size does not actually
exist until filter has been run. For that matter, the type checker does not know what >-> does, and for all it knows it may actually apply filter many times to different arrays, yielding different
sizes. This makes it impossible to uniquely instantiate the type of length, and therefore the program is rejected.
The common workaround is to use pipelining instead of composition whenever we use functions with existential return types:
xs |> filter (>0) |> length
This works because |> is left-associative, and hence the xs |> filter (>0) part will be fully evaluated to a concrete array before length is reached.
We can of course also write it as length (filter (>0) xs), with no use of either pipelining or composition.
8.3. Module errors¶
8.3.1. “Entry points may not be declared inside modules.”¶
This occurs when the program uses the entry keyword inside a module:
module m = {
entry f x = x + 1
Entry points can only be declared at the top level of a file. When we wish to make a function from inside a module available as an entry point, we must define a wrapper function:
module m = {
let f x = x + 1
entry f = m.f
8.4. “Module x is a parametric module¶
A parametric module is a module-level function:
module PM (P: {val x : i64}) = {
let y = x + 2
If we directly try to access the component of PM, as PM.y, we will get an error. To use PM we must first apply it to a module of the expected type:
module M = PM { val x = 2 : i64 }
Now we can say M.y. See Modules for more.
8.5. Other errors¶
8.5.1. “Literal out of bounds”¶
This occurs for overloaded constants such as 1234 that are inferred by context to have a type that is too narrow for their value. Example:
It is not an error to have a non-overloaded numeric constant whose value is too large for its type. The following is perfectly cromulent:
In such cases, the behaviour is overflow (so this is equivalent to 1u8).
8.5.2. “Type is ambiguous”¶
There are various cases where the type checker is unable to infer the full type of something. For example:
We know that r must be a record with a field called x, but maybe the record could also have other fields as well. Instead of assuming a perhaps too narrow type, the type checker signals an error. The
solution is always to add a type annotation in one or more places to disambiguate the type:
let f (r: {x:bool, y:i32}) = r.x
Usually the best spot to add such an annotation is on a function parameter, as above. But for ambiguous sum types, we often have to put it on the return type. Consider:
let f (x: bool) = #some x
The type of this function is ambiguous, because the type checker must know what other possible contructors (apart from #some) are possible. We fix it with a type annotation on the return type:
let f (x: bool) : (#some bool | #none) = #just x
See Type Abbreviations for how to avoid typing long types in several places.
8.5.3. “The x operator may not be redefined”¶
The && and || operators have magical short-circuiting behaviour, and therefore may not be redefined. There is no way to define your own short-circuiting operators.
8.5.4. “Unmatched cases in match expression”¶
Futhark requires match expressions to be exhaustive - that is, cover all possible forms of the value being pattern-matches. Example:
let f (x: i32) =
match x case 0 -> false
case 1 -> true
Usually this is an actual bug, and you fix it by adding the missing cases. But sometimes you know that the missing cases will never actually occur at run-time. To satisfy the type checker, you can
turn the final case into a wildcard that matches anything:
let f (x: i32) =
match x case 0 -> false
case _ -> true
Alternatively, you can add a wildcard case that explicitly asserts that it should never happen:
let f (x: i32) =
match x case 0 -> false
case 1 -> true
case _ -> assert false false
See here for details on how to use assert.
8.5.5. “Full type of x is not known at this point”¶
When performing a record update, the type of the field we are updating must be known. This restriction is based on a limitation in the type type checker, so the notion of “known” is a bit subtle:
let f r : {x:i32} = r with x = 0
Even though the return type annotation disambiguates the type, this program still fails to type check. This is because the return type is not consulted until after the body of the function has been
checked. The solution is to put a type annotation on the parameter instead:
let f (r : {x:i32}) = r with x = 0
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WeBWorK Standalone Renderer
$f(x) = x^{6}+3x+1$
. In this problem, we will show that
has exactly one root (or zero) in the interval
$\lbrack -5, -1 \rbrack$
(a) First, we show that
has a root in the interval
$(-5, -1)$
. Since
is a function on the interval
$\lbrack -5, -1 \rbrack$
$f(-5) =$
$f(-1) =$
, the graph of
$y = f(x)$
must cross the
-axis at some point in the interval
$(-5, -1)$
by the . Thus,
has at least one root in the interval
$\lbrack -5, -1 \rbrack$
(b) Second, we show that
cannot have more than one root in the interval
$\lbrack -5, -1 \rbrack$
by a thought experiment. Suppose that there were two roots
$x = a$
$x = b$
in the interval
$\lbrack -5, -1 \rbrack$
$a < b$
. Then
$f(a) = f(b) =$
. Since
is on the interval
$\lbrack -5, -1 \rbrack$
and on the interval
$(-5, -1)$
, by there would exist a point
in interval
so that
$f'(c) = 0$
. However, the only solution to
$f'(x) = 0$
$x =$
, which is not in the interval
, since
$(a,b) \subseteq \lbrack -5, -1 \rbrack$
. Thus,
cannot have more than one root in
$\lbrack -5, -1 \rbrack$
(Note: where the problem asks you to make a choice select the weakest choice that works in the given context. For example "continuous" is a weaker condition than "polynomial" because every polynomial
is continuous but not vice-versa. Rolle's theorem is a weaker theorem than the mean value theorem because Rolle's theorem applies to fewer cases.)
You can earn partial credit on this problem.
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How to calculate how much stock you want to buy
To calculate first the percentage of your funds you want to spend on a given stock, take your amount multiplied by the percent (We have $90000 and want to spend 20% of that, so we type in 90000 x 0.2
= ). We now know we want to spend not more than $18000.
Now subtract the commission for the stock purchase ($9.99). Then divide the remaining number (17990) by the ask price of the stock ($32.77), like this: 17990 / 32.77 = . Round the number you get down
to a whole number (remove the numbers after the decimal). Now you know you want to buy 549 stocks at the price of $32.77 each, which won’t cost more than $17,990.
You can check your math by multiplying the number of stocks (549) by the price (32.77). 549 x 32.77 = 17990.
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Applications to Robotic Systems Bo Zhang and Jeffrey Uhlmann Dept. Book Condition: New. In this work we focus on improving some of the earliest methods used for the analysis of gray-level
texture-based on statistical approaches: gray-level cooccurrence matrix (GLCM), gray-level difference matrix (GLDM), gray-level run-length matrix (GLRLM). It is a basic linear algebra tool and has a
wide range of applications in several domains like physics, engineering, and economics. In this tutorial, we’ll discuss two popular matrix multiplication algorithms: the naive matrix multiplication
and the Solvay Strassen algorithm. 5, No. Each chapter covers material that can comfortably be taught in one or two lessons. Asked by Wiki User. International Journal of Computer and Information
Sciences, Vol. It has seen increasing interactions with other areas of Mathematics. This implies that the leading diagonal elements of a skew – Hermitian matrix are either all zeros or all purely
imaginary. Thank you of Electrical Engineering & Computer Science University of Missouri-Columbia Abstract It is well-understood that the robustness of mechanical and robotic control systems depends
critically on minimizing sensitivity to arbitrary application-specific details whenever possible. 24 Department of Computer Science, Department of Mathematics, Birla Institute of Technology and
Science, Goa, India 25 Indian Institute of Technology Bombay, Mumbai, Maharashtra, India 26 New Technologies Research Centre, University of West Bohemia, Plze‹, Czech Republic ABSTRACT SymPy is an
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How do you use the definition of continuity to determine if g(x) = x^3 / x is continuous at x=0? | Socratic
How do you use the definition of continuity to determine if #g(x) = x^3 / x# is continuous at x=0?
1 Answer
Is ${\lim}_{x \rightarrow 0} g \left(x\right) = g \left(0\right)$?
$g \left(0\right)$ is not defined, so the function is not continuous at $0$.
Impact of this question
2210 views around the world
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Multiplication Riddle Worksheets
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Post shut-in arrest and recession solutions for a deflating hydraulic fracture in a permeable elastic medium
Published: 31 May 2022| Version 1 | DOI: 10.17632/4nxghz7cmr.1
Anthony Peirce
This data set provides the numerical solutions generated using an Implicit Moving Mesh Algorithm (IMMA) that has been adapted to include the k-g, r-g multiscale and k and r-vertex asymptotes to model
the post-shut-in arrest and recession of a radial hydraulic fracture in a permeable elastic medium. The theory behind this work is described in the paper "The arrest and recession dynamics of a
deflating radial hydraulic fracture in a permeable elastic medium" published in the JMPS (https://doi.org/10.1016/j.jmps.2022.104926). For the simulations I set the following parameters to unity: Ep
= 1 mup = 1 Cp = 1 Q0 = 1 Since V0 = Q0*ts, from eq (39) in the paper you can vary omega = ts/tmm~ = ts (since all the material parameters in t_mm~ are 1) and choose the value of phi^V that you want
by setting Kp as follows: Kp = (TS'*(phi).^(-65/9)).^(1/26); Recall: tmmt = (mup^4*Q0^6/Cp^18/Ep^4)^(1/7); omega = Ts/tmmt; phiV = (Ep^21*mup^5*Cp^10*Q0*Ts/Kp^26)^(9/65); Issue the command load
(Extract_Radial_MKC1_Ts_1em3_phi_50); to access the structures Results, which contains the structure Input. The file names embed the two dimensionless parameters: omega=Ts/tmm~=Ts (since tmm~=1) and
phiV. To avoid decimal points in the file name I have multiplied the phiV value by 100. So the above data file is for the case omega=10^{-3} and phiV=0.5. Input = struct('Ep',Ep,... % Pa plane strain
modulus 'mup',mup,... % Pa*s, alternate fluid viscosity 'Cp',Cp,... % m/s^1/2, alternate Carter's leak-off coefficient 'Kp',Kp,... % Pa*m^1/2, alternate fracture toughness 'Q0',Q0,... % m/s,
injection rate 'Ts',Ts,... % shut-in time 'omega',omega,... % dimensionless shut-in time 'phiV',phiV,... % arrest regime parameter 'Nr',Nr,... % number of grid points in r direction 'Nt',itcol); %
number of time steps till collapse Results=struct('pt',P(1,1:itcol),... % wellbore pressure versus t 'Rt',R(1:itcol),... % fracture radius versus t 'wt',W(1,1:itcol),... % wellbore aperture versus t
'eta',eta(1:itcol),... % efficiency versus t 'pr',P(:,1:itcol),... % fluid pressure versus r at all times Nt 'wr',W(:,1:itcol),... % fracture width versus r at all times Nt (because of the moving
mesh to plot in real space use plot(rho*R(it),wr(:,it))) 'rho',rho,... % lateral spatial coordinate 't',time(1:itcol),... % time 'keyindx',[its ita itd itcol],... % key indices keyindx(1)=its
(shut-in index), keyindx(2)=ita (arrest), keyindx(3)=itr (recession), keyindx(4)=itc (collapse) 'Input',Input); % Input Structure
Steps to reproduce
See the following article in the JMPS for a detailed description: https://doi.org/10.1016/j.jmps.2022.104926
The University of British Columbia
Hydraulic Fracturing
Related Links
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ONE IS EQUAL TO ZERO ... HOW COME? - edsmathscholar.com
Some people may think it is a weird question and say “Nonsense!” But if you ask the question to someone who has studied abstract algebra, you’ll possibly get another answer. This article introduces
the concept of ring in mathematics.
A non-empty set R is called a ring if two binary operations, denoted by + and ⋅, are defined in R such that all the properties below are satisfied:
1. (a + b) ∈ R for every a, b ∈ R
2. a + b = b + a for every a, b ∈ R
3. (a + b) + c = a + (b + c) for every a, b, c ∈ R
4. There exists 0 ∈ R such that a + 0 = a for every a ∈ R.
5. For every a ∈ R there exists an element -a ∈ R such that a + (-a) = 0
6. a⋅b ∈ R for every a, b ∈ R
7. a⋅(b⋅c) = (a⋅b)⋅c for every a, b, c ∈ R
8. For every a, b, c ∈ R both a⋅(b+c) = a⋅b + a⋅c and (b+c)⋅a = b⋅a + c⋅a apply.
The operation + is usually called the addition operation (simply, addition), while the ⋅ operation is usually called the multiplication operation (simply, multiplication). 0 in Property 4 is often
referred to as additive identity element or zero element. A ring R with the operations + and ⋅ is usually denoted by (R,+,-). Also note that Properties 1 to 5 above are the properties of abelian
groups. As a consequence, we have an equivalent definition of a ring as follows.
A non-empty set R is called a ring if two binary operations, denoted by + and ⋅, are defined in R such that all the properties below are satisfied:
1. (R,+) is an abelian group
2. a⋅b ∈ R for every a, b ∈ R
3. a⋅(b⋅c) = (a⋅b)⋅c for every a, b, c ∈ R
4. For every a, b, c ∈ R both a⋅(b+c) = a⋅b + a⋅c and (b+c)⋅a = b⋅a + c⋅a hold.
Is the set of all natural numbers,
On the other hand, the set of all integers,
In ring with unit element or ring with unity.
In a ring, the addition has to be commutative (Property 2), but the multiplication does not. Consider the set M[2] which is defined as follows.
M[2] is the set of all real-valued square matrices of order 2. It can be shown that (M[2],+,∙) is a ring. If A, B ∈ M[2], in general A∙B B∙A, so the multiplication is not commutative. A ring (R,+,∙)
is said to be commutative if a∙b = b∙a for every a, b ∈ R. Such a ring is called a commutative ring. Therefore, (M[2],+,∙) is not a commutative ring.
The simplest ring is the trivial ring (R,+,∙) with R = {e}, which is a set containing one and only one member denoted by e. The addition in R is defined as e + e = e and the multiplication is defined
as e∙e = e. With this definition, it is easy to prove that R is a ring with unity. Since e + e = e, the only possibility is e = 0 (as a consequence of Property 4) and because e.e = e, the only
possibility is e = 1. So we have proved that 0 = 1!
0 = 1 only holds in the trivial ring. If (R,+,∙) is a ring containing more than one member, the ring is called a nontrivial ring. In a nontrivial ring, is it possible that 0 = 1? In a nontrivial
ring, 0 ≠ 1. As a proof, assume that (R,+,∙) a nontrivial ring. Let x be any member of R. Suppose that 0 = 1. This implies x = x∙1 = x∙0 = 0. [The theorem x⋅0 = 0 is proved below.] Since x ∈ R is
arbitrary and x = 0, it follows that all elements of R are 0. In other words, R is the trivial ring. This contradicts the assumption that (R,+,∙) is a nontrivial ring. So, in a nontrivial ring, 0 ≠
Proof that x⋅0 = 0 for every x ∈ (R,+,⋅)
Let x be any member of (R,+,⋅). By Property 8, x⋅0 = x⋅(0 + 0) = x⋅0 + x⋅0. Since R is a group under addition, this equation implies that x⋅0 = 0. So, for every x ∈ (R,+,⋅) x⋅0 = 0 (q.e.d.)
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The Ultimate 7th Grade MCA Math Course (+FREE Worksheets)
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Aerodynamics of Boomerangs. Chapter 4
A gyroscope is a spinning device demonstrating the principle of conservation of angular momentum, in physics.
The traditional mathematical definition of the angular momentum of a particle about some origin is:
where L is the angular momentum of the particle, r is the position of the particle expressed as a displacement vector from the origin.
p = m*v is the linear momentum of the particle (more correct expression is m - mass of the particle.
If a system consists of several particles, the total angular momentum can be obtained by adding (integrating) all the angular moments of the constituent particles. Angular momentum can also be
calculated by multiplying the square of the distance to the point of rotation, the mass of the particle and the angular velocity.
Precession of Gyroscope
The device, once spinning, tends to resist changes to its orientation. If external force F1 is applied to some point which is at radius r from rotation axis of gyroscope, the spinning device begins
to rotate. The motion "seems to be strange" as it does not follow the applied force direction, but moves in a perpendicular one. This rotation of spinning plane is called precession.
The simplest explanation of the phenomenon is shown below.
The gyroscopic precession is fundamental phenomenon which explains why boomerang returns. See boomerang model in next chapter.
Exercise: get precession with disk sander by moving your wrist up and down
The angular momentum of turning bicycle wheels makes them act like gyroscopes to help stabilize the bicycle. This gyroscopic action also helps to turn the bicycle.
Stability of free-hand biking: see by yourselves how front wheel turns to the left/right when you shift your center of mass (and bend the bike) to the left/right side of a bicycle.
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A “better” class of puzzle
Due to the last few weeks being the only time of the year when I actually do some real work, as opposed to pretending (hope my employers don’t read this blog!), I haven’t done many puzzles in the
last 2-3 weeks.
At first, I thought the lack of practice was going to handicap me with this puzzle, as I got down to 16ac without putting in an answer. No need to worry though, as the top half came very quickly
The bottom half didn’t take long either, and with the help of a fair smattering of general knowledge, I ended up solving the puzzle in about 8 minutes. Considering the grid was still blank after
about 90 seconds, I am actually quite proud of that time.
This was an excellent puzzle – I always enjoy crosswords that test your knowledge as much as your ability to work out wordplay. I also noticed a “themelet” around gambling
1 ZOOM-LENS – Lens being a city in France, of course, as any football fan would instantly know.
9 UNDERMILK-WOOD – brilliant wordplay leading to Dylan Thomas’s famous radio play
10 BET-RAY – as an ex-bookie, I am disappointed I didn’t get this clue quicker. A Yankee is a type of bet where you make four selections, and have a combination of 11 bets, guaranteeing a return if
you have at least two winners
11 O(<=tar)O-RIO – I guessed that Samson was the name of an oratorio, and upon checking, discovered that Handel wrote it.
13 STATE-CR.-A-F.T.
15 DIGS – good one
18 GO FOR-BROKER(r)
21 CONSPIRE – (prison)* in CE
22 PI-RATE – as in Captain Hook
23 STRAIGHT FLUSH
25 SE(X)T ON
2 OPULENT – (up l note)*
3 MODERN TIMES – fantastic Chaplin movie
4 (p)EARLY
5 (<=nips)-OZ.-A – Baruch Spinoza, a Dutch 17th century philosopher
7 (d)R(a)I(n)-O
8 (g)UNDO(IN)G
12 ORDER AROUND – my employees would never have got this clue!
14 COGNITION – O in (gin tonic)*
17 LO(OK)SE-(grime)E – “butcher’s” is Cockney rhyming slang for LOOK (butcher’s hook)
20 KITCHEN – (the nick)*
16 comments on “A “better” class of puzzle”
1. 6:39 for this, profiting from spotting 9A quickly.
1. Off hand I can’t think of any other radio play famous enough to make it to the Times crossword so, like you, PB, I wrote it in early.
It was useful as I had pencilled in DAN for 7D tentatively hoping that the old city of that name (in Caanan) may have been a port, so I was able to correct that to RIO before it messed up my
thinking in the NE corner.
1. War of the Worlds?
1. Yes, maybe, but I think UMW was originally devised as a radio play whereas WotW was a book first and later adapted for radio/film.
I agree with you about STRAIGHT FLUSH.
2. 6:07 for me, although I have to admit I put a couple in without worrying about working out the wordplay. There are a lot of FRESHERs here in Cambridge this week, which may have helped. I also
have a good friend doing a PhD on Spinoza. 26A I think falls into the “Old chestnut” category. Jason J
3. Took the plunge this morning and joined the Times club, thus had the luxury of solving while at home wrapped in five duvets. OK, slight exaggeration.
20 minutes or so, which I’m very happy with.
Very best of luck to all at Cheltenham this weekend. Have fun!
4. After a mediocre week I was hoping for a confidence boost today before Sunday. I didn’t get it, after an utter disaster in the top left, only resolved by an adjacent amateur photographer who
volunteered the ‘zoom’ bit of ZOOM LENS, whereafter I finally got OPULENT, MODERN (TIMES) and BETRAY. Maybe ‘Yankee’ = BET is questionable without a ‘perhaps’ or similar, but that’s really no
5. 9:36 for me, but I was flagging after doing Wednesday’s and Thursday’s. I enjoyed this puzzle very much, but, like Neil, I’d have liked some sort of indication that “Yankee” is an example of a
BET; similarly for “Good group of clubs” (STRAIGHT FLUSH), where “perhaps” would have been easy to slip in.
6. 15-16 minutes here. I too had an unsolved grid until 16ac. From there, the bottom half flew in, but some of the top half had me scratching my head. For the second day running, it took far too
long to spot a simple anagram indication (this time at 2dn). The ZOOM bit of 1ac also took far longer than it should. Perhaps its a good job I decided not to try to qualify for Cheltenham. Good
luck to all those who did qualify. On that note, what chance is there that the regional finals will be rekindled in the future?
1. I can’t see them doing regional finals unless a generous sponsor can be found. It’s a pity, as many solvers who would be quite happy to go to Glasgow or York are unlikely to trek to
Cheltenham. Now that the link with the literary festival is pretty nominal, the one-day final could quite reasonably be held in different places each year, like the Open golf – there must be
lots of other universities with similar facilities that could be used. I may try suggesting this, though the organizers have probably had enough interfering suggestions from me …
7. Am I the only one who would have liked a ‘maybe’ or similar for the clubs? Unless I’ve misunderstood this, a straight flush can be in any suit.
8. “Yankee” and “straight flush” have been mentioned as needing a “perhaps” or some such. So also, I think, does “deal” in 9A. Yet things are OK in 15A, 22A, 8D. Are rules about all this changing?
9. Three puzzles in a week where I couldn’t get a glue, gak! Had ?e?ray from the fish, couldn’t think of anything that meant either yankee or shop, so I took a misguided stab at DEFRAY. Rest of
crossword took very little time. Oh, well… you learn something every month.
10. Another one over an hour. I, too, missed the anagram in 2d and bogged down for easily 15 minutes!
Didn’t know ‘bet’ for yankee so had to go searching for proof!
11. I, who lag fainting behind all of you most of the time, found this maybe the easiest puzzle ever. Filled most of it in nonstop, then even after “stuckness” found that the rest filled in steadily.
Too bad I never time myself.
I was mystified by “Yankee,” even though I am one, but “ray” and “shop” worked so I wrote it in anyway. “Under Milk Wood” was my first entry, and “Zoom Lens” my last — I had no idea Lens was a
French city.
12. Magnificent 7 “easies”:
6a Show surprise (4-2)
16a Mollusc left in river (4)
C L AM. Detritus from a picnic basket in a punt?
26a Local leader ignored by head of state (8)
(P) RESIDENT
6d Look like the next scene in film (4,5)
TAKE AFTER
19d More familiar person up for the first time? (7)
22d Adverts for pants (5)
24d Grass, speed not ecstasy (3)
RAT (E)
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The electrocomp corporation manufactures two electrical products: air
Part 1 of 5
The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and
drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of
wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit.
Now, Electrocomp’s management also realizes that there should be a minimum number of air conditioners produced in order to fulfill a contract. Also, due to an oversupply of fans in the preceding
period, a limit should be placed on the total number of fans produced.
1. If Electrocomp decides that at least 20 air conditioners should be produced but no more than 80 fans should be produced, the optimal production mix of air conditioners and fans is _____________
(The first number is for air conditioners and the second is for fans
a. (50, 30)
b. (20, 80)
c. (40, 60)
d. (26.67, 80)
2. In the optimal solution for the above question (Question 1), the slack for the four constraints – Total hours of wiring, Total hours of drilling, Number of air conditioners, Number of fans are
_________, respectively. (Hint: A negative number shows that the constraint is binding downward, i.e. it sets a lower rather than an upper limit. Take the negative sign out when interpreting the
a. 0, 0, 20, 20
b. 0, 0, 10, 20
c. 0, 0, 20, 10
d. 10, 10, 10, 10
3. If Electrocomp decides that at least 30 air conditioners should be produced but no more than 50 fans should be produced, the optimal production of air conditioners is 45 and the optimal
productions of fans is 50 (Only enter an integer and include no units.)
4. In the optimal solution for the above question (Question 3), the slack for the four constraints – Total hours of wiring, Total hours of drilling, Number of air conditioners, Number of fans are
__________, respectively. (Hint: A negative number shows that the constraint is binding downward, i.e. it sets a lower rather than an upper limit. You may take the negative sign out when interpreting
the results)
a. 0, 0, 15, 10
b. 5, 0, 15, 0
c. 0, 5, 15, 0
d. 0, 0, 10, 15
Part 2 of 5
The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of
redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 feet of good-quality redwood. Each
bench that Outdoor Furniture produces requires 4 labor hours and 10 feet of redwood; each picnic table takes 6 labor hours and 35 feet of redwood. Completed benches will yield a profit of $9 each,
and tables will result in a profit of $20 each.
5. The optimal production of benches is262 and the optimal production of tables is 25.(Please only include the integer and no units. Hint: Because of the resource constraints, you may not be able to
produce 4 tables even if the solution is 3.8; you may have to just produce 3 tables)
Part 3 of 5
A winner of the Texas Lotto has decided to invest $50,000 per year in the stock market. Under consideration are stocks for a petrochemical firm and a public utility. Although a long-range goal is to
get the highest possible return, some consideration is given to the risk involved with the stocks. A risk index on a scale of 1–10 (with 10 being the most risky) is assigned to each of the two
stocks. The total risk of the portfolio is found by multiplying the risk of each stock by the dollars invested in that stock. The attached table provides a summary of the return and risk. The
investor would like to maximize the return on the investment, but the average risk index of the investment should not be higher than 6.
Estimated Return
Risk Index
6. The optimal dollar amount invested in Petrochemical stock is 20000 dollars, and the optimal dollaramount invested in Utility stock is 30000 dollars. (Only use an integer and include no units.
Hint: The total risk on each stock is the risk index times the investment in dollars. When you use Excel QM, you need to end the total risk in the right hand side of the constraint not simply the
risk index.)
a. 20,000, 30,000
b. 30,000, 20,000
c. 10,000, 15,000
d. 15,000, 10,000
7. The average risk index (not in dollar amount) for the optimal investment is 6, and the estimated best/maximum return for this investment is 4200. (Please only use an integer and include no
Part 4 of 5
The Heinlein and Krampf Brokerage firm has just been instructed by one of its clients to invest $250,000 of her money obtained recently through the sale of land holdings in Ohio. The client has a
good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and
bonds they believe are well rated, but within the following guidelines:
(a) Municipal bonds should constitute at least 20% of the investment.
(b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers.
(c)
No more than 50% of the amount invested in municipal bonds should be placed in a high-risk, high-yield nursing home stock.
Subject to these restraints, the client’s goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare the attached list of
high-quality stocks and bonds and their corresponding rates of return.
Hint: You will need to rearrange one of the constraints in this problem so that all variables will be at the left hand side of the constraint and the right hand side of the constraint will be zero.
You may adjust he sign of the constraint accordingly. For example, rearranging x < y + 2 to x – y < 2.
Projected Rate of Return (%)
Los Angeles municipal bonds
Thompson Electronics, Inc
United Aerospace Corp.
Palmer Drugs
Happy Days Nursing Homes
8. The optimal money invested in the five stocks and bonds – Municipal bonds, Electronics, Aerospace, Drugs, and Nursing homes are ________ dollars, respectively.
a. 50,000, 50,000, 50,000, 50,000, 50,000
b. 50,000, 0, 0, 175,000, 25,000
c. 25,000, 0, 0, 175,000, 50,000
d. 50,000, 25,000, 25,000, 125,000, 25,000
Part 5 of 5
Mt. Sinai Hospital in New Orleans is a large, private, 600-bed facility, complete with laboratories, operating rooms, and x-ray equipment. In seeking to increase revenues, Mt. Sinai’s administration
has decided to make a 90-bed addition on a portion of adjacent land currently used for staff parking. The administrators feel that the labs, operating rooms, and x-ray department are not being fully
utilized at present and do not need to be expanded to handle additional patients. The addition of 90 beds, however, involves deciding how many beds should be allocated to the medical staff for
medical patients and how many to the surgical staff for surgical patients.
The hospital’s accounting and medical records departments have provided the following pertinent information. The average hospital stay for a medical patient is 8 days, and the average medical patient
generates $2,280 in revenues. The average surgical patient is in the hospital 5 days and receives a $1,515 bill. The laboratory is capable of handling 15,000 tests per year more than it was handling.
The average medical patient requires 3.1 lab tests and the average surgical patient takes 2.6 lab tests. Furthermore, the average medical patient uses one x-ray, whereas the average surgical patient
requires two x-rays. If the hospital was expanded by 90 beds, the x-ray department could handle up to 7,000 x-rays without significant additional cost. Finally, the administration estimates that up
to 2,800 additional operations could be performed in existing operating room facilities. Medical patients, of course, do not require surgery, whereas each surgical patient generally has one surgery
Assume that the hospital is open 365 days a year. Round off your answers to the nearest integer.
9. The optimal number of medical patients per year is 2791, and the optimal number of surgical patients per year is 2104. The maximum annual profit is9551040 dollars. (Please round to the closest
integer and include no units)
10. Among the 90 additional beds, _____ beds should be used for medical patients and _________ beds used for surgical patients.
a. 45, 45
b. 60, 30
c. 61, 29
d. 29, 61
11. At the optimal, there are 2 empty beds, 877.5lab tests of unused capacity, 1x-rays of unused capacity, and 696unused operation rooms available. (Please round to the closest integer and include no
https://doneassignments.com/wp-content/uploads/2021/08/logo-300x75.png 0 0 admin https://doneassignments.com/wp-content/uploads/2021/08/logo-300x75.png admin2021-08-26 14:30:172021-08-26 14:30:17The
electrocomp corporation manufactures two electrical products: air
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PPT - Measurement Tools Period 1 PowerPoint Presentation, free download - ID:15196
1. DQ 9-14-10 • List the 7 steps of the scientific method in order • In a sentence, pick one step and explain what it is done for this step.
8. Measurement Tools KEY CONCEPTS The basic laboratory tools that you will learn to use are the metric ruler, triple-beam balance, graduated cylinder, and Celsius thermometer.
9. 1. Name:______________________________ • Measurement:_________________________ • Units:________________________________
10. 2. Name:______________________________ • Measurement:_________________________ • Units:________________________________
11. 3. Name:__________________ • Measurement:______________ • Units:_____________________
12. 4. Name:______________________________ • Measurement:_________________________ • Units:________________________________
13. Wieviel Grad ist es in Deutschland? • (What is the temperature in Germany?)
18. Reading a Graduated Cylinder A graduated cylinder is an instrument used to measure small amounts of liquid. The scientists use this cylinder to measure liquid volume. There are many sizes of
graduated cylinders. Some measure 1,000 milliliters (mL) or 1 liter (L). Some measure 500 milliliters (mL). Some measure only in milliliters (mL). The lines on the graduated cylinder are called
graduations. The liquid usually curves up the side of a graduated cylinder. To achieve an accurate reading, it is important to remember to read the measurement at the lowest point or the bottom
of the curve. This low point is called the meniscus.
22. Sometimes scientists need to find the volume of small, irregularly shaped solid objects. They use the graduated cylinder and the water displacement method to calculate the volume. The graduated
cylinder is filled to a specific height and recorded. The object is then placed inside the graduated cylinder. The water will rise above the object. Then a second reading of the cylinder is
recorded. The first reading is subtracted from the second or higher reading. The difference is the volume of the solid object in milliliters or cubic centimeters (**remember 1 mL = 1 cm3)
23. Calculate the volume of the marbles, using the water displacement method. • What is the volume of the water in the graduated cylinder?
24. What is the volume of the water in the graduated cylinder after five marbles were placed in it?
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Exclusive-OR (XOR) Digital Logic Gate
Exclusive-OR Gate – Digital Logic XOR Gate
What is Logic XOR or Exclusive-OR Gate?
XOR gate also known as Exclusive OR gate is “A logic gate which produces HIGH state ‘1’ only when there is an odd number of HIGH state ‘1’ inputs”.
For 2-input gate it can be interpreted as “when both of the inputs are different, then the output is HIGH state ‘1’ and when the inputs are same, then the output is LOW state ‘0’”.
XOR gate can have two or more than two inputs but it has only one output.
XOR Gate Logic Symbol, Boolean Expression & Truth Table
XOR Gate Symbol
There are 3 types of symbols used for XOR gate all over the world.
American National Standards Institute (ANSI)/ MILITARY
International Electrotechnical Commission (IEC)/EUROPEAN
Deutsches Institut für Normung (DIN)/GERMANY
Boolean Expression
OUT = ( I̅N̅1 & IN2) + (IN1 & I̅N̅2) or OUT = (I̅N̅1 + I̅N̅2) & (IN1 + IN2) ;
Truth Table
XOR Gate Logic flow Schematic Diagram
Construction and Working Mechanism of XOR Gate
XOR Gate Using BJT and Diodes
Schematic of XOR gate using Diodes and BJT (NPN transistor) is given below. In which we have used two NPN transistors, 4 diodes and the resistor between Diode Bridge and NPN is used in series
configuration because BJTs operates on input current, not input voltages.
The diodes are used in a bridge configuration (Rectifier) to sort out the input logic into the positive level, means if there is a High state input it will always flow to the base of NPN to switch it
on. And the Low state will always flow out to the emitter of NPN transistor.
The first transistor is used for switching upon the input logic given to it. And the second NPN transistor is nothing more than just an Inverter. It only inverts the output of the first NPN
When inputs are different, the high state flow to NPN transistor’s base and turn it ON. The LOW state “0” flow through the emitter to the base of the 2nd NPN transistor and inverts into Logic HIGH
state “1” as output.
When the inputs are same, if it’s LOW state “0” then NPN will never turn on because there be no HIGH state input at its base. So Vcc will flow out to the inverter phase and it will invert into Low
state “0” as output. If it’s HIGH state “1” then the NPN will turn on but there will be no Logic 0 to flow through the emitter. So again Vcc will flow through an inverter and invert into low state”0”
as output.
XOR Gate Using MOSFET and Diodes
Discrete XOR gate can be made with MOSFETs and diodes. XOR schematic using N-MOSFETs and diodes is given below.
In this schematic 4 diodes are used in a bridge configuration for sorting out input logic. 2 NMOSFETs are used; 1st one for switching upon input logic and the 2nd one for Inverting output of 1st
NMOSFET. The resistor between Diode Bridge and 1st MOSFET is used in parallel configuration because MOSFETs operates on Gate’s voltages, not current.
When both inputs are different, then the High state will flow to NMOSFET’s gate and the LOW state will flow to NMOSFET’s source to build up potential at its gate, which will switch it ON. And the LOW
state will flow out to the inverter and inverts into HIGH state “1”.
When both of the inputs are same i.e. if both are HIGH state “1” than the HIGH state will flow to the gate of NMOSFET but there will be no potential at its source so NMOSFET will switch OFF. Hence
Vdd (HIGH state “1”) will flow to the inverter and inverts into LOW state “0” as output. When both inputs are LOW state “0”, again there will be no potential (voltage) at the gate of NMOSFET so it
will never turn ON and Vdd will flow out to the inverter and inverts into LOW state “0” as output.
XOR Gate From other Logic Gates: (Combinational Logic)
XOR operation can be achieved with a combination of different logic gates.
Boolean expression of XOR is given below.
Sum of Product
In this expression, we use the sum of Min terms. Min terms are the product of inputs for which output is HIGH state “1”.
SOP expression can be easily implemented with NAND gates.
According to the truth table given above the SOP (sum of product) expression is:
OUT = (I̅N̅1 & IN2) + (IN1 & I̅N̅2)
This expression can be implemented with NOT, AND, OR gates as shown in the figure given below.
Product of Sum
In this expression, we use the product of Max terms. Max terms are the sum of inputs for which output is LOW state “0”.
POS expression can be easily implemented with NOR gates.
According to the truth table given above the POS (product of sum) expression is :
OUT = (I̅N̅1̅+I̅N̅2̅) & (IN1+IN2) Expression 1
OUT = (IN1 & IN2)’ & (IN1+IN2) Expression 2 DE MORGAN’S LAW
Expression 1 can be implemented with NOT, AND, OR gates as shown in the figure given below.
Expression 2 uses NAND, AND & OR gate to reduce the number of used gates as shown in the figure given below.
XOR Gate From Universal Gates
Universal gates are those gates which can be implemented into any logic gate or logic function.
XOR Gate From NAND Gate
NAND gate is a universal gate. It can be implemented into any Logic function.
As we have discussed before SOP (sum of product) expression can be easily Implemented with NAND gates, so SOP expression for XOR gate is
OUT = { (I̅N̅1 & IN2) + (IN1 & I̅N̅2) }
OUT’ = { (I̅N̅1 & IN2) + (IN1 & I̅N̅2) }’ Taking complement on both sides
OUT’ = { (I̅N̅1 & IN2)’ & (IN1 & I̅N̅2)’ } De Morgan’s Law
OUT’’ = { (I̅N̅1 & IN2)’ & (IN1 & I̅N̅2)’ }’ Taking complement on both sides
OUT = [ { (IN1 & IN1)’ & IN2}’ & { IN1 & (IN2 & IN2) }’ ]’ (IN1 & IN1)’ = I̅N̅1
Now, this expression is in NAND form. And can easily be implemented with NAND gates as shown in the figure below.
XOR Gate From NOR GATE
NOR gate is also a universal gate. It can be implemented into any Logic function.
As we have discussed before POS (product of sum) expression can be easily implemented with NOR gates, so POS expression for XOR gate is given below
OUT = { (I̅N̅1̅ + I̅N̅2̅) & (IN1 + IN2) }
OUT’ = { (I̅N̅1̅ + I̅N̅2̅) & (IN1 + IN2) }’ Taking complement on both sides
OUT’ = { (I̅N̅1̅ + I̅N̅2̅)’ + (IN1 + IN2)’ } De Morgan’s Law
OUT’’ = { (I̅N̅1̅ + I̅N̅2̅)’ + (IN1 + IN2)’ }’ Taking complement on both sides
OUT = [ { (IN1 + IN1)’ +(IN2 + IN2)’ }’ + (IN1 + IN2)’]’ (IN1 + IN1)’ = I̅N̅1
Now, this expression is in NOR form. And can easily be implemented with NOR gates as shown in the figure below.
Multi-Input Exclusive OR Gate
XOR gate gives HIGH state “1” only when there is an odd number of HIGH state “1” inputs.
XOR gate can have more than two inputs but it has only one output.
Truth Table
The truth table of “3” inputs XOR gate is given below.
Combinational Logic
Combinational logic is the logic of making a schematic with the help of basic logic gates.
Sum of products (SOP) and products of the sum (POS) are two methods in combinational logic.
Sum of Products
SOP uses the idea of summation of minterms (product of inputs for which output is high)
According to the truth table given above, Sum of product expression and schematic for 3-input XOR gate is given below.
OUT = (I̅N̅1̅ & I̅N̅2̅ & IN3) + (IN1 & I̅N̅2̅ & I̅N̅3̅) + (I̅N̅1̅ & IN2 & I̅N̅3̅) + (IN1 & IN2 & IN3)
Product of Sum
POS uses the idea of the product of max terms (Sum of inputs for which output is LOW.)
According to the truth table given above, Product of sum expression and schematic for 3-input XOR gate is given below.
OUT = (I̅N̅1 + IN2 + IN3) & (IN1 + I̅N̅2 + IN3) & (IN1 + IN2 + I̅N̅3) & (I̅N̅1 + I̅N̅2 + I̅N̅3̅)
TTL and CMOS Logic XOR Gate IC’s
Some of XOR IC with pin configurations is given below.
TTL Logic XOR Gate
• 74136 Quad 2-input (with open collector outputs)
• 7486 Quad 2-input
CMOS Logic XOR Gate
Pinout for 7486 TTL XOR Gate IC
PIN Number Description
1 Input Gate 1
2 Input Gate 1
3 Output Gate 1
4 Input Gate 2
5 Input Gate 2
6 Output Gate 2
7 Ground
8 Output Gate 3
9 Input Gate 3
10 Input Gate 3
11 Output Gate 4
12 Input Gate 4
13 Input Gate 4
14 Positive Supply Voltage
Exclusive OR Gate Applications
Some common application and uses of XOR or Exclusive-OR gate are as follow:
• XOR is used as a comparator to know if the input signals are equal or not.
• XOR Gate is the crucial part of a Half Adder. it can produce the sum of two single bit numbers. Half Adder is the building block of the ALU (Arithmetic Logic Unit) which is used in every digital
• It can be used as parity checker to check if the data stream received is corrupted or not.
You may also read more about Digital logic gates.
Logic NOT Gate – Digital Inverter Logic Gate Digital Logic OR Gate Digital Logic AND Gate Digital Logic NOR Gate
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Solving Linear Equations Examples
This resource is a PDF document packed with examples of how to solve linear equations. It's aimed at secondary school students in Years 7, 8 and 9.
Why is solving linear equations important?
Linear equations crop up everywhere in real life! From understanding recipes to working out exchange rates, these equations help us model situations and solve problems. Mastering them unlocks a whole
world of applications.
Why is this resource helpful?
This resource breaks down linear equations into bite-sized chunks, using clear explanations and plenty of worked examples.
• It covers two key methods: balancing equations and function machine methods.
• Balancing equations involves adding or subtracting the same number to both sides of an equation to isolate the variable. It's like balancing a set of scales!
• Function machine methods involve visualising the equation as a machine that takes in a number, performs an operation, and outputs an answer. It's a fun and interactive way to grasp the concepts.
• Plus, there's a free printable PDF version you can download and use in class or for home learning.
Remember, practice is key to mastering linear equations. This resource provides a wealth of examples and clear guidance to help your students conquer them!
Also, have a look at our wide range of worksheets that are specifically curated to help your students practice their skills related to linear equations. These teaching resources and worksheets are in
PDF format and can be downloaded easily.
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Teaser - Introduction to Graph Theory
On my upcoming 5-part series on the fundamental ideas of graph theory, the most important concept in discrete math.
Hey folks!
This post is a teaser for an upcoming series on graph theory I’ll be writing in collaboration with
from . The series aims to introduce readers to the fundamentals of graph theory, and the articles will prioritize intuition over formalism without losing rigor, presenting the key ideas —as it is a
staple in both Tivadar’s and my own articles.
Why should you care?
Graph theory is essential for many practical reasons, whatever your field of expertise. Graphs are the quintessential abstract representation for a vast range of domains, from computer networks to
social networks, DNA, city planning, game theory… I could go on and on.
But, if you’re just a bit like Tivadar and me, and you love math for its beauty, then graph theory is one of the most fascinating subjects for the many unintuitive and significant theorems and
Am I the right person to teach you this?
If you’re a long-time reader of Mostly Harmless Ideas, you already know that I love writing about the most fascinating topics in Computer Science, always from an intuitive and pragmatic perspective.
You might also know that I have a Ph.D. in machine learning and do research on practical applications of language models.
What you might not know is that I’m also a full-time lecturer in a CS major, and I’m currently teaching an undergrad Discrete Math course, which is more than 60% graph theory.
As far as credentials matter —which is fairly little, to be honest— I think that covers it. What’s more important is that I’ve spent my entire career thinking about how to make CS education more
approachable. So whether you’re new to all of this or already know a lot about graphs, I believe I can give you a couple novel points of view that I’m sure will surprise you.
So, now that you’re convinced, here’s the deal.
The posts will be published first in The Palindrome and then cross-posted here after the series is complete. We plan to publish one article each week starting next week and should finish around the
end of the year. If you want to read them as they come out, subscribe to Tivadar!
The series is not algorithmic but theory-oriented. We will dive deep into why graphs have some of the intriguing properties they have. It’s all about the beauty of graphs rather than the pragmatics.
However, in that exploration, we will build some strong intuitions that will later —in a future series on graph algorithms— help us design clever computational strategies.
Since this is a series on theory, we will do lots of proofs! However, to keep articles useful for all readers, we will do them in two layers. First, we will see all the intuitions behind each
theorem, understand what it says, and why it must be true. Then, we will have an optional section for the math nerdy who wants to see the technicalities.
Ok, but what’s inside?
I'm glad you asked! Here’s a short table of the contents we have planned. We may adjust it a bit as we see how complex or long each topic gets.
1 - The Basics
• Why graphs matter, a bit of history, and some examples of applications of graphs.
• Basic concepts: vertices, edges, paths, and cycles.
• Undirected, directed, and weighted graphs.
• Connected graphs and connected components, cuts.
• Special types of graphs: complete and bipartite.
• Subgraphs, induced and spanning.
2 - Trees
• What is a tree, examples of applications of trees.
• Basic properties of trees.
• Spanning trees, what are they, and why are they useful.
• Properties of spanning trees.
• Some intuitions about finding minimum spanning trees.
3 - Matching and Bipartite Graphs
• What is matching in general and in bipartite graphs.
• Examples of matching problems.
• Properties of matching in general graphs.
• Finding a matching in general graphs (Berge’s theorem).
• Finding a matching in bipartite graphs (Hall’s theorem).
4 - Tours: Eulerian and Hamiltonian
• Why tours are important.
• Examples of problems that involve tours.
• Eulerian tours, when they exist, and how to find them.
• Hamiltonian tours, why are they hard to find.
• Sufficient conditions for Hamiltonian tours (Dirac’s theorems)
• Closures and closed characterizations of Hamiltonian tours.
5 - Drawing Graphs: Planarity and Coloring
• Examples of applications of planarity and graph coloring.
• Basic properties of planar graphs.
• When is a graph planar? (Kuratoski’s theorem)
• Why graph coloring is hard.
• Some unintuitive results about graph coloring.
• Coloring in planar graphs (the 4 colors theorem).
Are you ready for a deep dive into the fascinating world of graph theory? Hit that subscribe button, and I’ll be back in your inbox with the first article before you can say “induction”!
Mostly Harmless Ideas is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber.
Looking forward to it. Discrete Math was one of my favorite classes in college.
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Looking forward to this...
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8 Predictive trigger functions
Sometimes there are signs of the upcoming problem. These signs can be spotted so that actions may be taken in advance to prevent or at least minimize the impact of the problem.
Zabbix has tools to predict the future behaviour of the monitored system based on historic data. These tools are realized through predictive trigger functions.
Two things one needs to know is how to define a problem state and how much time is needed to take action. Then there are two ways to set up a trigger signalling about a potential unwanted situation.
First: trigger must fire when the system after "time to act" is expected to be in problem state. Second: trigger must fire when the system is going to reach the problem state in less than "time to
act". Corresponding trigger functions to use are forecast and timeleft. Note that underlying statistical analysis is basically identical for both functions. You may set up a trigger whichever way you
prefer with similar results.
Both functions use almost the same set of parameters. Use the list of supported functions for reference.
First of all you should specify the historic period Zabbix should analyse to come up with prediction. You do it in a familiar way by means of sec or #num parameter and optional time_shift like you do
it with avg, count, delta, max, min and sum functions.
(forecast only) Parameter time specifies how far in the future Zabbix should extrapolate dependencies it finds in historic data. No matter if you use time_shift or not, time is always counted
starting from the current moment.
(timeleft only) Parameter threshold specifies a value the analysed item has to reach, no difference if from above or from below. Once we have determined f(t) (see below) we should solve equation f(t)
= threshold and return the root which is closer to now and to the right from now or 999999999999.9999 if there is no such root.
When item values approach the threshold and then cross it, timeleft assumes that intersection is already in the past and therefore switches to the next intersection with threshold level, if any. Best
practice should be to use predictions as a complement to ordinary problem diagnostics, not as a substitution.1
Default fit is the linear function. But if your monitored system is more complicated you have more options to choose from.
fit x = f(t)
linear x = a + b*t
polynomialN^2 x = a[0] + a[1]*t + a[2]*t^2 + ... + a[n]*t^n
exponential x = a*exp(b*t)
logarithmic x = a + b*log(t)
power x = a*t^b
(forecast only) Every time a trigger function is evaluated it gets data from the specified history period and fits a specified function to the data. So, if the data is slightly different the fitted
function will be slightly different. If we simply calculate the value of the fitted function at a specified time in the future you will know nothing about how the analysed item is expected to behave
between now and that moment in the future. For some fit options (like polynomial) a simple value from the future may be misleading.
mode forecast result
value f(now + time)
max max[now <= t <= now + time] f(t)
min min[now <= t <= now + time] f(t)
delta max - min
avg average of f(t) (now <= t <= now + time) according to definition
To avoid calculations with huge numbers we consider the timestamp of the first value in specified period plus 1 ns as a new zero-time (current epoch time is of order 109, epoch squared is 1018,
double precision is about 10-16). 1 ns is added to provide all positive time values for logarithmic and power fits which involve calculating log(t). Time shift does not affect linear, polynomial,
exponential (apart from easier and more precise calculations) but changes the shape of logarithmic and power functions.
No warnings or errors are flagged if chosen fit poorly describes provided data or there is just too few data for accurate prediction.
To get a warning when you are about to run out of free disk space on your host you may use a trigger expression like this:
However, error code -1 may come into play and put your trigger in a problem state. Generally it's good because you get a warning that your predictions don't work correctly and you should look at them
more thoroughly to find out why. But sometimes it's bad because -1 can simply mean that there was no data about the host free disk space obtained in the last hour. If you are getting too many false
positive alerts consider using more complicated trigger expression5:
Situation is a bit more difficult with forecast. First of all, -1 may or may not put the trigger in a problem state depending on whether you have expression like {host:item.forecast(...)}<... or like
Furthermore, -1 may be a valid forecast if it's normal for the item value to be negative. But probability of this situation in the real world situation is negligible (see how operator = works). So
add ... or {host:item.forecast(...)}=-1 or ... and {host:item.forecast(...)}<>-1 if you want or don't want to treat -1 as a problem respectively.
For example, a simple trigger like {host:item.timeleft(1h,,X)} < 1h may go into problem state when the item value approaches X and then suddenly recover once value X is reached. If the problem is
item value being below X use: {host:item.last()} < X or {host:item.timeleft(1h,,X)} < 1h If the problem is item value being above X use: {host:item.last()} > X or {host:item.timeleft(1h,,X)} < 1h↩︎
Polynomial degree can be from 1 to 6, polynomial1 is equivalent to linear. However, use higher degree polynomials with caution. If the evaluation period contains less points than needed to determine
polynomial coefficients, polynomial degree will be lowered (e.g polynomial5 is requested, but there are only 4 points, therefore polynomial3 will be fitted).↩︎
For example fitting exponential or power functions involves calculating log() of item values. If data contains zeros or negative numbers you will get an error since log() is defined for positive
values only.↩︎
For linear, exponential, logarithmic and power fits all necessary calculations can be written explicitly. For polynomial only value can be calculated without any additional steps. Calculating avg
involves computing polynomial antiderivative (analytically). Computing max, min and delta involves computing polynomial derivative (analytically) and finding its roots (numerically). Solving f(t) = 0
involves finding polynomial roots (numerically).↩︎
But in this case -1 can cause your trigger to recover from the problem state. To be fully protected use: {host:vfs.fs.size[/,free].timeleft(1h,,0)}<1h and ({TRIGGER.VALUE}=0 and {host:vfs.fs.size
[/,free].timeleft(1h,,0)}<>-1 or {TRIGGER.VALUE}=1)↩︎
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WLL / (2 * cos(A/2))
30 Aug 2024
WLL / (2 * cos(A/2)) & Analysis of variables
Equation: WLL / (2 * cos(A/2))
Variable: WLL
Impact of Working Load Limit on SWL Function
X-Axis: 100.0to100.0
Y-Axis: SWL Function
The Impact of Working Load Limit on SWL Function: An Equation-Based Analysis
In the realm of engineering, particularly in the field of lifting and rigging, the Working Load Limit (WLL) plays a crucial role in ensuring the safe operation of equipment. In this article, we will
delve into the equation WLL / (2 * cos(A/2)), which is essential for understanding the relationship between WLL and the Safe Working Load (SWL) function. We will explore the context in which this
equation is used and its significance in maintaining optimal performance.
The Safe Working Load (SWL) function is a critical aspect of lifting equipment design, ensuring that loads are handled within safe limits to prevent accidents and injuries. The Working Load Limit
(WLL), on the other hand, represents the maximum weight capacity of a lift line or wire rope, as specified by manufacturers.
Equation: WLL / (2 * cos(A/2))
The equation WLL / (2 * cos(A/2)) is fundamental to calculating the effective SWL of lifting equipment. Here’s a breakdown of the variables and their significance:
• WLL: Working Load Limit, representing the maximum weight capacity of a lift line or wire rope.
• A: The angle of inclination, measured in radians (or degrees), at which the load is lifted.
The equation itself represents the relationship between WLL and the SWL function. In essence, it shows that the effective SWL decreases as the angle of inclination increases. This is because the
weight of the load remains constant, but the force required to lift it increases with the cosine of half the angle of inclination.
Mathematical Derivation
To understand the significance of this equation, let’s derive it mathematically:
1. The weight (W) of the load can be calculated as W = mg, where m is the mass and g is the acceleration due to gravity.
2. The force (F) required to lift the load at an angle A is given by F = W * sin(A).
3. Since we are interested in the effective SWL function, which represents the maximum weight capacity of the equipment, we can divide both sides of the equation by 2 * cos(A/2). This yields: SWL =
(WLL / (2 * cos(A/2))).
Implications and Consequences
The implications of this equation are far-reaching:
1. Reduced Effective SWL: As the angle of inclination increases, the effective SWL decreases proportionally.
2. Increased Risk: With reduced effective SWL, there is a higher risk of overloading the equipment, potentially leading to accidents or injuries.
In conclusion, the equation WLL / (2 * cos(A/2)) highlights the critical relationship between WLL and the SWL function in lifting and rigging applications. By understanding this equation and its
implications, engineers can ensure that equipment is operated within safe limits, minimizing risks associated with overloading or improper use.
To maintain optimal performance and prevent accidents:
1. Regularly inspect equipment: Verify that lift lines and wire ropes are in good condition and meet the manufacturer’s specifications.
2. Calculate effective SWL: Use the equation WLL / (2 * cos(A/2)) to determine the effective SWL for each lifting operation, taking into account the angle of inclination.
3. Follow manufacturer guidelines: Adhere to manufacturer recommendations regarding maximum load capacities, angles of inclination, and other safety parameters.
By embracing a proactive approach to equipment maintenance and operation, engineers can minimize risks associated with overloading or improper use, ultimately ensuring a safer working environment for
all personnel involved in lifting and rigging operations.
Related topics
Academic Chapters on the topic
Information on this page is moderated by llama3.1
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Conventionality of Simultaneity
First published Mon Aug 31, 1998; substantive revision Sat Jul 21, 2018
In his first paper on the special theory of relativity, Einstein indicated that the question of whether or not two spatially separated events were simultaneous did not necessarily have a definite
answer, but instead depended on the adoption of a convention for its resolution. Some later writers have argued that Einstein’s choice of a convention is, in fact, the only possible choice within the
framework of special relativistic physics, while others have maintained that alternative choices, although perhaps less convenient, are indeed possible.
1. The Conventionality Thesis
The debate about the conventionality of simultaneity is usually carried on within the framework of the special theory of relativity. Even prior to the advent of that theory, however, questions had
been raised (see, e.g., Poincaré 1898) as to whether simultaneity was absolute; i.e., whether there was a unique event at location A that was simultaneous with a given event at location B. In his
first paper on relativity, Einstein (1905) asserted that it was necessary to make an assumption in order to be able to compare the times of occurrence of events at spatially separated locations
(Einstein 1905, 38–40 of the Dover translation or 125–127 of the Princeton translation; but note Scribner 1963, for correction of an error in the Dover translation). His assumption, which defined
what is usually called standard synchrony, can be described in terms of the following idealized thought experiment, where the spatial locations A and B are fixed locations in some particular, but
arbitrary, inertial (i.e., unaccelerated) frame of reference: Let a light ray, traveling in vacuum, leave A at time t[1] (as measured by a clock at rest there), and arrive at B coincident with the
event E at B. Let the ray be instantaneously reflected back to A, arriving at time t[2]. Then standard synchrony is defined by saying that E is simultaneous with the event at A that occurred at time
(t[1] + t[2])/2. This definition is equivalent to the requirement that the one-way speeds of the ray be the same on the two segments of its round-trip journey between A and B.
It is interesting to note (as pointed out by Jammer (2006, 49), in his comprehensive survey of virtually all aspects of simultaneity) that something closely analogous to Einstein’s definition of
standard simultaneity was used more than 1500 years earlier by St. Augustine in his Confessions (written in 397 CE). He was arguing against astrology by telling a story of two women, one rich and one
poor, who gave birth simultaneously but whose children had quite different lives in spite of having identical horoscopes. His method of determining that the births, at different locations, were
simultaneous was to have a messenger leave each birth site at the moment of birth and travel to the other, presumably with equal speeds. Since the messengers met at the midpoint, the births must have
been simultaneous. Jammer comments that this “may well be regarded as probably the earliest recorded example of an operational definition of distant simultaneity.”
The thesis that the choice of standard synchrony is a convention, rather than one necessitated by facts about the physical universe (within the framework of the special theory of relativity), has
been argued particularly by Reichenbach (see, for example, Reichenbach 1958, 123–135) and Grünbaum (see, for example, Grünbaum 1973, 342–368). They argue that the only nonconventional basis for
claiming that two distinct events are not simultaneous would be the possibility of a causal influence connecting the events. In the pre-Einsteinian view of the universe, there was no reason to rule
out the possibility of arbitrarily fast causal influences, which would then be able to single out a unique event at A that would be simultaneous with E. In an Einsteinian universe, however, no causal
influence can travel faster than the speed of light in vacuum, so from the point of view of Reichenbach and Grünbaum, any event at A whose time of occurrence is in the open interval between t[1] and
t[2] could be defined to be simultaneous with E. In terms of the ε-notation introduced by Reichenbach, any event at A occurring at a time t[1] + ε(t[2] − t[1]), where 0 < ε < 1, could be simultaneous
with E. That is, the conventionality thesis asserts that any particular choice of ε within its stated range is a matter of convention, including the choice ε=1/2 (which corresponds to standard
synchrony). If ε differs from 1/2, the one-way speeds of a light ray would differ (in an ε-dependent fashion) on the two segments of its round-trip journey between A and B. If, more generally, we
consider light traveling on an arbitrary closed path in three-dimensional space, then (as shown by Minguzzi 2002, 155–156) the freedom of choice in the one-way speeds of light amounts to the choice
of an arbitrary scalar field (although two scalar fields that differ only by an additive constant would give the same assignment of one-way speeds).
It might be argued that the definition of standard synchrony makes use only of the relation of equality (of the one-way speeds of light in different directions), so that simplicity dictates its
choice rather than a choice that requires the specification of a particular value for a parameter. Grünbaum (1973, 356) rejects this argument on the grounds that, since the equality of the one-way
speeds of light is a convention, this choice does not simplify the postulational basis of the theory but only gives a symbolically simpler representation.
2. Phenomenological Counterarguments
Many of the arguments against the conventionality thesis make use of particular physical phenomena, together with the laws of physics, to establish simultaneity (or, equivalently, to measure the
one-way speed of light). Salmon (1977), for example, discusses a number of such schemes and argues that each makes use of a nontrivial convention. For instance, one such scheme uses the law of
conservation of momentum to conclude that two particles of equal mass, initially located halfway between A and B and then separated by an explosion, must arrive at A and B simultaneously. Salmon
(1977, 273) argues, however, that the standard formulation of the law of conservation of momentum makes use of the concept of one-way velocities, which cannot be measured without the use of
(something equivalent to) synchronized clocks at the two ends of the spatial interval that is traversed; thus, it is a circular argument to use conservation of momentum to define simultaneity.
It has been argued (see, for example, Janis 1983, 103–105, and Norton 1986, 119) that all such schemes for establishing convention-free synchrony must fail. The argument can be summarized as follows:
Suppose that clocks are set in standard synchrony, and consider the detailed space-time description of the proposed synchronization procedure that would be obtained with the use of such clocks. Next
suppose that the clocks are reset in some nonstandard fashion (consistent with the causal order of events), and consider the description of the same sequence of events that would be obtained with the
use of the reset clocks. In such a description, familiar laws may take unfamiliar forms, as in the case of the law of conservation of momentum in the example mentioned above. Indeed, all of special
relativity has been reformulated (in an unfamiliar form) in terms of nonstandard synchronies (Winnie 1970a and 1970b). Since the proposed synchronization procedure can itself be described in terms of
a nonstandard synchrony, the scheme cannot describe a sequence of events that is incompatible with nonstandard synchrony. A comparison of the two descriptions makes clear what hidden assumptions in
the scheme are equivalent to standard synchrony. Nevertheless, editors of respected journals continue to accept, from time to time, papers purporting to measure one-way light speeds; see, for
example, Greaves et al. (2009). Application of the procedure just described shows where their errors lie.
3. Malament’s Theorem
For a discussion of various proposals to establish synchrony, see the supplementary document:
Transport of Clocks
The only currently discussed proposal is based on a theorem of Malament (1977), who argues that standard synchrony is the only simultaneity relation that can be defined, relative to a given inertial
frame, from the relation of (symmetric) causal connectibility. Let this relation be represented by κ, let the statement that events p and q are simultaneous be represented by S(p,q), and let the
given inertial frame be specified by the world line, O, of some inertial observer. Then Malament’s uniqueness theorem shows that if S is definable from κ and O, if it is an equivalence relation, if
points p on O and q not on O exist such that S(p,q) holds, and if S is not the universal relation (which holds for all points), then S is the relation of standard synchrony.
Some commentators have taken Malament’s theorem to have settled the debate on the side of nonconventionality. For example, Torretti (1983, 229) says, “Malament proved that simultaneity by standard
synchronism in an inertial frame F is the only non-universal equivalence between events at different points of F that is definable (‘in any sense of “definable” no matter how weak’) in terms of
causal connectibility alone, for a given F”; and Norton (Salmon et al. 1992, 222) says, “Contrary to most expectations, [Malament] was able to prove that the central claim about simultaneity of the
causal theorists of time was false. He showed that the standard simultaneity relation was the only nontrivial simultaneity relation definable in terms of the causal structure of a Minkowski spacetime
of special relativity.”
Other commentators disagree with such arguments, however. Grünbaum (2010) has written a detailed critique of Malament’s paper. He first cites Malament’s need to postulate that S is an equivalence
relation as a weakness in the argument, a view also endorsed by Redhead (1993, 114). Grünbaum’s main argument, however, is based on an earlier argument by Janis (1983, 107–109) that Malament’s
theorem leads to a unique (but different) synchrony relative to any inertial observer, that this latitude is the same as that in introducing Reichenbach’s ε, and thus Malament’s theorem should carry
neither more nor less weight against the conventionality thesis than the argument (mentioned above in the last paragraph of the first section of this article) that standard synchrony is the simplest
choice. Grünbaum concludes “that Malament’s remarkable proof has not undermined my thesis that, in the STR, relative simultaneity is conventional, as contrasted with its non-conventionality in the
Newtonian world, which I have articulated! Thus, I do not need to retract the actual claim I made in 1963…” Somewhat similar arguments are given by Redhead (1993, 114) and by Debs and Redhead (2007,
For further discussion, see the supplement document:
Further Discussion of Malament’s Theorem
4. Other Considerations
Since the conventionality thesis rests upon the existence of a fastest causal signal, the existence of arbitrarily fast causal signals would undermine the thesis. If we leave aside the question of
causality, for the moment, the possibility of particles (called tachyons) moving with arbitrarily high velocities is consistent with the mathematical formalism of special relativity (see, for
example, Feinberg 1967). Just as the speed of light in vacuum is an upper limit to the possible speeds of ordinary particles (sometimes called bradyons), it would be a lower limit to the speeds of
tachyons. When a transformation is made to a different inertial frame of reference, the speeds of both bradyons and tachyons change (the speed of light in vacuum being the only invariant speed). At
any instant, the speed of a bradyon can be transformed to zero and the speed of a tachyon can be transformed to an infinite value. The statement that a bradyon is moving forward in time remains true
in every inertial frame (if it is true in one), but this is not so for tachyons. Feinberg (1967) argues that this does not lead to violations of causality through the exchange of tachyons between two
uniformly moving observers because of ambiguities in the interpretation of the behavior of tachyon emitters and absorbers, whose roles can change from one to the other under the transformation
between inertial frames. He claims to resolve putative causal anomalies by adopting the convention that each observer describes the motion of each tachyon interacting with that observer’s apparatus
in such a way as to make the tachyon move forward in time. However, all of Feinberg’s examples involve motion in only one spatial dimension. Pirani (1970) has given an explicit two-dimensional
example in which Feinberg’s convention is satisfied but a tachyon signal is emitted by an observer and returned to that observer at an earlier time, thus leading to possible causal anomalies.
A claim that no value of ε other than 1/2 is mathematically possible has been put forward by Zangari (1994). He argues that spin-1/2 particles (e.g., electrons) must be represented mathematically by
what are known as complex spinors, and that the transformation properties of these spinors are not consistent with the introduction of nonstandard coordinates (corresponding to values of ε other than
1/2). Gunn and Vetharaniam (1995), however, present a derivation of the Dirac equation (the fundamental equation describing spin-1/2 particles) using coordinates that are consistent with arbitrary
synchrony. They argue that Zangari mistakenly required a particular representation of space-time points as the only one consistent with the spinorial description of spin-1/2 particles.
Another argument for standard synchrony has been given by Ohanian (2004), who bases his considerations on the laws of dynamics. He argues that a nonstandard choice of synchrony introduces
pseudoforces into Newton’s second law, which must hold in the low-velocity limit of special relativity; that is, it is only with standard synchrony that net force and acceleration will be
proportional. Macdonald (2005) defends the conventionality thesis against this argument in a fashion analagous to the argument used by Salmon (mentioned above in the first paragraph of the second
section of this article) against the use of the law of conservation of momentum to define simultaneity: Macdonald says, in effect, that it is a convention to require Newton’s laws to take their
standard form.
Many of the arguments against conventionality involve viewing the preferred simultaneity relation as an equivalence relation that is invariant under an appropriate transformation group. Mamone Capria
(2012) has examined the interpretation of simultaneity as an invariant equivalence relation in great detail, and argues that it does not have any bearing on the question of whether or not
simultaneity is conventional in special relativity.
A vigorous defense of conventionality has been offered by Rynasiewicz (2012). He argues that his approach “has the merit of nailing the exact sense in which simultaneity is conventional. It is
conventional in precisely the same sense in which the gauge freedom that arises in the general theory of relativity makes the choice between diffeomorphically related models conventional.” He begins
by showing that any choice of a simultaneity relation is equivalent to a choice of a velocity in the equation for local time in H.A. Lorentz’s Versuch theory (Lorentz 1895). Then, beginning with
Minkowski space with the standard Minkowski metric, he introduces a diffeomorphism in which each point is mapped to a point with the same spatial coordinates, but the temporal coordinate is that of a
Lorentzian local time expressed in terms of the velocity as a parameter. This mapping is not an isometry, for the light cones are tilted, which corresponds to anisotropic light propagation. He
proceeds to argue, using the hole argument (see, for example, Earman and Norton 1987) as an analogy, that this parametric freedom is just like the gauge freedom of general relativity. As the tilting
of the light cones, if projected into a single spatial dimension, would be equivalent to a choice of Reichenbach’s ε, it seems that Rynasiewicz’s argument is a generalization and more completely
argued version of the argument given by Janis that is mentioned above in the third paragraph of Section 3.
The debate about conventionality of simultaneity seems far from settled, although some proponents on both sides of the argument might disagree with that statement. The reader wishing to pursue the
matter further should consult the sources listed below as well as additional references cited in those sources.
• Anderson, R., I. Vetharaniam, and G. Stedman, 1998. “Conventionality of Synchronisation, Gauge Dependence and Test Theories of Relativity,” Physics Reports, 295: 93–180.
• Augustine, St., Confessions, translated by E.J. Sheed, Indianapolis: Hackett Publishing Co., 2nd edition, 2006.
• Ben-Yami, H., 2006. “Causality and Temporal Order in Special Relativity,” British Journal for the Philosophy of Science, 57: 459–479.
• Brehme, R., 1985. “Response to ‘The Conventionality of Synchronization’,” American Journal of Physics, 53: 56–59.
• Brehme, R., 1988. “On the Physical Reality of the Isotropic Speed of Light,” American Journal of Physics, 56: 811–813.
• Bridgman, P., 1962. A Sophisticate’s Primer of Relativity. Middletown: Wesleyan University Press.
• Debs, T. and M. Redhead, 2007. Objectivity, Invariance, and Convention: Symmetry in Physical Science, Cambridge, MA and London: Harvard University Press.
• Earman, J. and J. Norton, 1987. “What Price Spacetime Substantivalism? The Hole Story,” British Journal for the Philosophy of Science, 38: 515–525.
• Eddington, A., 1924. The Mathematical Theory of Relativity, 2nd edition, Cambridge: Cambridge University Press.
• Einstein, A., 1905. “Zur Elektrodynamik bewegter Körper,” Annalen der Physik, 17: 891–921. English translations in The Principle of Relativity, New York: Dover, 1952, pp. 35–65; and in J. Stachel
(ed.), Einstein’s Miraculous Year, Princeton: Princeton University Press, 1998, pp. 123–160.
• Ellis, B. and P. Bowman, 1967. “Conventionality in Distant Simultaneity,” Philosophy of Science, 34: 116–136.
• Feinberg, G., 1967. “Possibility of Faster-Than-Light Particles,” Physical Review, 159: 1089–1105.
• Giulini, D., 2001. “Uniqueness of Simultaneity,” British Journal for the Philosophy of Science, 52: 651–670.
• Greaves, E., A. Rodriguez, and J. Ruiz-Camaro, 2009. “A One-Way Speed of Light Experiment,” American Journal of Physics, 77: 894–896.
• Grünbaum, A., 1973. Philosophical Problems of Space and Time (Boston Studies in the Philosophy of Science, Volume 12), 2nd enlarged edition, Dordrecht/Boston: D. Reidel.
• Grünbaum, A., 2010. “David Malament and the Conventionality of Simultaneity: A Reply,” Foundations of Physics, 40: 1285–1297.
• Grünbaum, A., W. Salmon, B. van Fraassen, and A. Janis, 1969. “A Panel Discussion of Simultaneity by Slow Clock Transport in the Special and General Theories of Relativity,” Philosophy of Science
, 36: 1–81.
• Gunn, D. and I. Vetharaniam, 1995. “Relativistic Quantum Mechanics and the Conventionality of Simultaneity,” Philosophy of Science, 62: 599–608.
• Havas, P., 1987. “Simultaneity, Conventionalism, General Covariance, and the Special Theory of Relativity,” General Relativity and Gravitation, 19: 435–453.
• Jammer, M., 2006. Concepts of Simultaneity: From Antiquity to Einstein and Beyond, Baltimore: Johns Hopkins University Press.
• Janis, A., 1983. “Simultaneity and Conventionality,” in R. Cohen and L. Laudan (eds.), Physics, Philosophy and Psychoanalysis (Boston Studies in the Philosophy of Science, Volume 76), Dordrecht/
Boston: D. Reidel, pp. 101–110.
• Lorentz, H., 1895. Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegter Körpern, Leiden: E.J. Brill.
• Macdonald, A., 2005. “Comment on ‘The Role of Dynamics in the Synchronization Problem,’ by Hans C. Ohanion,” American Journal of Physics, 73: 454–455.
• Malament, D., 1977. “Causal Theories of Time and the Conventionality of Simultaniety,” Noûs, 11: 293–300.
• Mamone Capria, M., 2001. “On the Conventionality of Simultaneity in Special Relativity,” Foundations of Physics, 31: 775–818.
• Mamone Capria, M., 2012. “Simultaneity as an Invariant Equivalence Relation,” Foundations of Physics, 42: 1365–1383.
• Minguzzi, E., 2002. “On the Conventionality of Simultaneity,” Foundations of Physics Letters, 15: 153–169.
• Norton, J., 1986. “The Quest for the One Way Velocity of Light,” British Journal for the Philosophy of Science, 37: 118–120.
• Ohanian, H., 2004. “The Role of Dynamics in the Synchronization Problem,” American Journal of Physics, 72: 141–148.
• Pirani, F., 1970. “Noncausal Behavior of Classical Tachyons,” Physical Review, D1: 3224–3225.
• Poincaré, H., 1898. “La Mesure du Temps,” Revue de Métaphysique et de Morale, 6: 1–13. English translation in The Foundations of Science, New York: Science Press, 1913, pp. 223–234.
• Redhead, M., 1993. “The Conventionality of Simultaneity,” in J. Earman, A. Janis, G. Massey, and N. Rescher (eds.), Philosophical Problems of the Internal and External Worlds, Pittsburgh:
University of Pittsburgh Press, pp. 103–128.
• Reichenbach H., 1958. The Philosophy of Space & Time, New York: Dover.
• Rynasiewicz, R., 2012. “Simultaneity, Convention, and Gauge Freedom,” Studies in History and Philosophy of Modern Physics, 43: 90–94.
• Salmon, M., J. Earman, C. Glymour, J. Lennox, P. Machamer, J. McGuire, J. Norton, W. Salmon, and K. Schaffner, 1992. Introduction to the Philosophy of Science, Englewood Cliffs: Prentice Hall.
• Salmon, W., 1977. “The Philosophical Significance of the One-Way Speed of Light,” Noûs, 11: 253–292.
• Sarkar, S. and J. Stachel, 1999. “Did Malament Prove the Non-Conventionality of Simultaneity in the Special Theory of Relativity?” Philosophy of Science, 66: 208–220.
• Scribner, C., 1963. “Mistranslation of a Passage in Einstein’s Original Paper on Relativity,” American Journal of Physics, 31: 398.
• Spirtes, P., 1981. Conventionalism and the Philosophy of Henri Poincaré, Ph.D. Dissertation, University of Pittsburgh.
• Stein, H., 1991. “On Relativity Theory and Openness of the Future,” Philosophy of Science, 58: 147–167.
• Torretti, R., 1983. Relativity and Geometry, Oxford, New York: Pergamon.
• Winnie, J., 1970a. “Special Relativity Without One-Way Velocity Assumptions: Part I,” Philosophy of Science, 37: 81–99.
• Winnie, J., 1970b. “Special Relativity Without One-Way Velocity Assumptions: Part II,” Philosophy of Science, 37: 223–238.
• Zangari, M., 1994. “A New Twist in the Conventionality of Simultaneity Debate,” Philosophy of Science, 61: 267–275.
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A selection of mathematical poems, written by some of my former students.
20 comments to Poems
• Math Poem
My mind is turning into scrambled eggs!
What is x and y to the power of three?
Whole numbers, mixed numbers and absolute,
exponents, integers and factoring trees.
Terms, expressions and what is the root?
It all sounds like Greek to me.
I must write a poem for another class.
I’m running out of time much too fast.
So I’ll talk about digits and my reaction,
while I attempt to work these equations.
Mixed operations in an expression
must be done in the following manner.
Please Excuse My Dear Aunt Sally,
to help me remember the proper order.
Parentheses, exponents, multiplication,
addition or subtraction,
that is the order of proper action
for solving math, numbers and fractions.
With fractions when I multiply
it is best to quickly simplify,
the denominators remain the same
these do not need to change.
If fractions I add or subtract
I must remember it works like this,
each denominator must be the same
then add the top like a list.
Multiply two positives they remain just that.
Two negatives will spoil the batch.
Mixed signs will keep the minus too,
keep this in mind or stay confused.
I wish that I could remember more
But that is as far as I have gotten.
Off to bed I must go,
or tomorrow I will feel rotten
• \
VERY NICE POEM
• omg this is a very well written poem, i love it , well done joana ..!
• this poem is fantastic
• thanx i am going to say this poem on teachers day for my favorite maths teacher
• thnks i have got a nice poem for tell to my math teacher on the end of academic year
• It would be best for older kids not so much younger kids.
• teachers day poem 4 a maths teacher
• A long and sweet one. I loved it. thnks
• really well done. better than i’ve ever seen before
• its really interesting but it could hv been a bit shorter last 2 paragraphs are really nice
• this poem is better for younger kids
□ Not really, we don’t study those stuff, tho we are in grade 6, and we never learned it before, but it’s an amazing poem👏👏👏
• THE WORLD OF MATH
Roses are red violets are
blue.Multiplication, division is what you have to do. Addition, subtraction. Next is fractions.If you know how to multiply, then you can simplify. Please, excuse, my, dear, aunt, sally. Is what
we use for expressions.Like terms, coefficients don’t sound fun. This is the end now I must run
• Is a excellent poem I love this
• WOW WHAT A FANTASTIC POEM.!!!!!!!!!!!!!!1
• Orayt
• Very good poem joana
• Speechless…….. Math is so magically mathematical…….KUDOS
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Product forecasting of Economics Topics | Question AI
<div id="mw-content-wrapper"><div id="mw-content"><div id="content" class="mw-body" role="main"><div id="bodyContentOuter"><div class="mw-body-content" id="bodyContent"><div id="mw-content-text"
class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-parser-output"><p><b>Product forecasting</b> is the science of predicting the degree of success a new product will enjoy in
the <span title="Finance:Marketplace">marketplace</span>. To do this, the forecasting model must take into account such things as product awareness, distribution, <span title="Finance:Price">price</
span>, fulfilling unmet needs and competitive alternatives. </p><h2 id="qai_title_1"><span class="mw-headline" id="Bass_model">Bass model</span></h2><p>Bass model is one type of forecasting method
primarily used in new product forecasting. In general, there will be no historical demand for new product. Then, Bass model tries to capture shape of demand of existing product and apply new product.
</p><dl><dd><span class="noprint"><i>Main page: <span title="Bass diffusion model">Bass diffusion model</span></i></span></dd></dl><p><span style="opacity:.5">$\displaystyle{ \frac{f(t)}{1-F(t)} = p
+ \frac{q}{m} N(t) }$</span> </p><p>where, </p><ul><li>F(t) is the probability of adoption at time t</li><li>f(t) is the rate at which adoption is changing with respect to t</li><li>N(t) is the
number of adopters at time t</li><li>m is the total number of consumers who will eventually adopt</li><li>p is the coefficient of innovation</li><li>q is the coefficient of imitation</li></ul><p>
Multivariate techniques such as regression can be used to determine the values of p, q and N if historical sales data is available. </p><h2 id="qai_title_2"><span class="mw-headline" id=
"Fourt-Woodlock_model">Fourt-Woodlock model</span></h2><p>The Fourt-Woodlock model is another method used to estimate product sales. </p><p><span style="opacity:.5">$\displaystyle{ V = (HH \cdot TR \
cdot TU) + (HH\cdot TR\cdot MR \cdot RR \cdot RU) }$</span> </p><p>The left-hand-side of the equation is the volume of purchases per unit time (usually taken to be one year). On the right-hand-side,
the first parentheses describes trial volume, and the second describes repeat volume. </p><p>HH is the total number of households in the geographic area of projection, and TR ("trial rate")
is the percentage of those households which will purchase the product for the first time in a given time period. TU ("trial units") is the number of units purchased on this first purchase
occasion. MR is "measured repeat," or the percentage of those who tried the product who will purchase it at least one more time within the first year of the product's launch. RR is the
repeats per repeater: the number of repeat purchases within that same year. RU is the number of repeat units purchased on each repeat event. </p></div></div></div></div></div></div></div>
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...and the Worst Idea
Creating a Culture of Questions
was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules.
I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted.
The First Idea is the Best Idea and the Worst Idea
The first time students hear this, I usually get, "Gosh, that's mean."
But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it. But using the same logic, this idea should be the worst. This assumes
the flow of ideas that should follow.
I think this encourages two important things:
1. "If I go first, it doesn't matter that my idea isn't fully formed." This student has established a floor on which each other student can stand and/or build.
2. "I can take someone's idea and help them make it better." The real work is done by the first follower. This student chips away at any imperfections and helps the first student refine her idea.
Subsequent students then follow suit.
What's this look like?
Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.
Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept. Once we were finished, I
asked students to write down any questions they had.
Student 1: "I have a comment."
"Ok, what is it?"
Student 1: "No matter which points we choose, the slope simplifies to the same thing."
"Can you turn your observation into a question?"
Student 1: "Will that happen all the time?"
Now here is where it happens.
"I can misunderstand [Student 1]'s question, can we make this more precise?"
Student 2: "Will the slopes always simplify to the same thing?"
Student 3: "Will the slopes between two points always simplify to the same thing?"
"Are we only using two points?"
Student 4: "Will the slopes between three points always simplify to the same thing?"
Student 5: "Will the slopes between any two pairs of points always simplify to the same thing?"
Student 6: "Are the slopes between any two pairs of points always equal?"
"Are we really talking about any 4 points here?"
Student 7: "Are the slopes between any two pairs of points on a line always equal?"
I called this a Project. It's not. It's more of a problem-y kind of performance task learning opportunity assessment [S:of:S] [S: for:S] [S:of:S] for? learning that hits close to home. Literally. We
live in a huge agricultural area and kids don't know what an acre is. Anything that gives students a chance to wrestle with the fact that a piece of land can't have dimensions of 20 acres x 20 acres,
is a win. Anything that allows me to answer the question "What's an acre-foot?" by doing this, is a win.
In this [S:project :S] problem's first iteration, I was focused on the skills of equation writing, line graphing and solving mixture and work problems.
In the second iteration, I was less focused on the skills and more interested in having students explain what each component of an equation represented, why we'd want that equation and how graphing
inequalities made sense. We got to discuss why understanding the problem makes sense--kids tried to hire crews to prune cotton. For you city-slickers out there--you don't prune cotton. It doesn't
grow on trees. Students had to sign up via Google form to interview with me as they finished a task. I did something north of 175 interviews for one class that year.
This year, I've changed it a bit more. They are no longer tasks, they're constraints. There are fewer of them and they don't specifically tell kids what to do. Before, I told them to create
inequalities and graph them. Now, I'm removing some of the scaffold. They get to decide what tools they want to use. Before, I did this project after we had done systems, mixture and work problems.
This time, we have only done systems. They're going to have to work through the mixture/work stuff.
That's been the highlight--the mixture problems. I have a few students who went straight for that constraint and have been on a mission to figure out how to make sense of it.
Today, one boy asked, "Mr. Cox, how accurate to I need to be? I'm accurate to the trillionth, but I can't get it to be exactly 36%."
I said, "How accurate do you think you need to be? We're killing weeds, not sending someone to space."
So, with all that, here's the updated version complete with dynamic answer key.
If we can get students to flip their thinking from this:
If I know the rules, then I can do the math.
to this:
If I do the math, I can know the rules.
Then we've won.
So here's the idea:
One problem with multiple paths to solution. Students connect as many skills as they can to the problem. I listed eight possible skills two of which wouldn't necessarily apply to the problem.
Students had to assess themselves on the skills they demonstrated.
Question of the day: "Mr. Cox, is it possible to use all of these skills?"
Answer to Question of the day: "It's possible that some of the skills don't apply."
For this first iteration, I used the standard Ticket Problem.
Below are samples of student work.
As an exercise for the reader:
1) What are your thoughts on this process?
2) How did each student do?
Let me know in the comments.
Student A
Student B
Student C
Student D
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On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender-Wu formula for x<sup>2N</sup>-anharmonic oscillators
Rigorous results on the Laplace expansions of conditional Wiener integrals with functional integrands having a finite number of global maxima are established. Applications are given to the Bender-Wu
formula for the x ^2N-anharmonic oscillator. © 1983 American Institute of Physics.
Original language English
Pages (from-to) 255-266
Number of pages 12
Journal Journal of Mathematical Physics
Volume 24
Issue number 2
Publication status Published - 1982
Dive into the research topics of 'On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender-Wu formula for x^2N-anharmonic oscillators'. Together they form a unique
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Trading Quiz
Some of information in this article has been updated by referencing new statistics on 9/21/2020.
This page presents a quiz on technical analysis, mostly covering how chart patterns behave. See how well you do. Answers are below the following links.
1. True or False: Chart patterns in small cap stocks outperform mid and large caps.
Market capitalization is the stock's price multiplied by the number of shares outstanding. I consider a small cap stock as having a value up to $1 billion. Large caps are over $5 billion with mid
caps between those two. Various rating services have their own boundaries which seem to grow as a bull market progresses.
I computed the rise or decline after the breakout from various chart patterns with the same result. Small cap stocks outperformed their mid and large cap brothers (or sisters). So, the statement
above is true.
2. True or false: Bullish chart patterns perform best within a third of the yearly low.
I separated chart patterns by where the breakout price occurs in the prior 12-month trading range, just to see if I could determine a performance difference. For bearish patterns, the answer is
false. They don't show much performance difference, but that's also not the question I asked. I mentioned bullish chart patterns. For those guys, the answer is true. Bullish chart pattern tend to
perform better if the breakout price is within a third of the yearly low.
3. True or False: Chart patterns have low failure rates.
Statistics updated 9/21/2020.
How do you measure failure? I chose to count the number of patterns which failed to see price move more than 5% away from the breakout price as a failure.
Do chart patterns have low failure rates? The answer depends on what is meant by low. A Eve & Eve double bottom has a 12% failure rate. That means 12% of chart patterns fail to see price rise at
least 5% after the breakout. So, on the surface, the statement is true (if you consider 12% as being 'low'). However, 23% of Eve & Eve double bottoms fail to rise at least 10%; 32% can't reach
gains of 15%. Half of all EEDBs can't make 30%. The failure rate rises by leaps and bounds once you plug in more realistic numbers for profit opportunity.
For example, if you look at your old trades and find that you need to make at least 15% to cover your costs and make up for losses that you keep small, how often will an ascending triangle fail
to make at least 15%? Answer: 40% of the time. Wow.
4. True or False: Breakout day gaps suggest better performance from a chart pattern.
Statistics updated 9/21/2020.
This one is easy. A breakout gap occurs on the day when price closes beyond a trendline boundary or above/below the chart pattern's top/bottom. Price forms a gap, a hole where today's low is
above yesterday's high (for bullish gaps).
The answer is true. In a poll of chart patterns, I found that 68% of them have gaps where performance is better if a gap appears on the day of breakout. However, the performance boost might not
be as big as you expect.
For example, symmetrical triangles with breakout day gaps showed price rising 35%. Without a gap, price climbed an average of 34% (both are from bull markets).
5. True or False: Short (less than 3 months) price trends leading to the start of a chart pattern means below average performance.
I discovered the answer to this when I did research for my book, Trading Classic Chart Patterns
I determined where the trend started by the same method as I use to find the ultimate high or low, that is, a 20% trend change. I found that short-term price trends suggest, but do not guarantee,
a more powerful move. So, the answer to the quiz is false. A short term price trend leads to above average performance.
6. True or False: Support and resistance gets weaker over time.
This is an easy one. Of course support and resistance gets weaker over time according to the experts that haven't test it. Every time I tested this I found that time is not an important factor in
how powerful support or resistance is. In other words, it does not grow weaker over time, despite what everyone believes. If you think I'm kidding, test it yourself.
Here's a brief review. I found a bunch of horizontal consolidation regions (HCRs) and measured how often price stopped within them after a breakout. I found that if the HCR is close enough to the
chart pattern, price will fly through the HCR. However, the stopping power increases for HCRs up to a month away and then oscillates up and down in stopping power for at least 1.5 years. In other
words, a HCR 1.5 years old is just as powerful at stopping price as one that formed a month ago. The correct answer is false.
7. True or False: On a price basis (not time), support and resistance gets weaker the farther away it is from the current price.
I measured the vertical distance (price) from a chart pattern to the HCR for both upward and downward breakouts. The stopping power of HCRs increased in strength for HCRs up to 15% away (upward
breakouts) and then decreased after that. The same can be said for downward breakouts except that they weaken after 20% away. Thus, the answer is true, support and resistance tends to weaken the
further away it is from the top or bottom of a chart pattern.
8. True of False: Above average volume on the day of a chart pattern breakout means better performance.
Statistics updated 9/21/2020.
Let's use Eve & Eve double bottoms as a test case. In my book, Encyclopedia of Chart Patterns
In a poll of chart patterns, I found that 79% of the time, above average breakout volume helped performance.
While it's generally true that above average breakout volume means better performance, it also depends on the situation, like the one described for Eve & Eve double bottoms.
Here's what I wrote in my study of studies.
For both breakout directions, heavy breakout volume is very important to chart pattern performance after the breakout. Heavy breakout day volume means above the 30-day volume average (one month
of calendar days, not trading days) up to but not including the breakout day.
☆ Upward breakouts
Heavy: 79% (above average breakout volume helps)
Light: 21%
☆ Downward breakouts
Heavy: 67%
Light: 33%
9. True or False: Tall chart patterns outperform short ones.
Updated using new statistics 9/21/2020.
I computed the height of each pattern from highest high to lowest low and divided the result by the breakout price to standardize the numbers across all stocks.
In a study of various types of chart patterns, I found that tall ones outperform short ones 89% of the time. That's a bit misleading since it's a count the various types of patterns (double
bottoms, double tops, head-and-shoulders tops, and so on) versus a count of each tall pattern that beats a short one. Nevertheless, I found that tall patterns do better than short ones, so the
correct answer is true. The following shows the numbers (from study of studies).
☆ Upward breakouts
Tall: 89% (tall patterns outperform)
Short: 11%
☆ Downward breakouts
Tall: 100%
Short: 0%
10. True of False: Wide patterns outperform narrow ones.
Statistics updated 9/21/2020.
If you answered true, you'd be correct. In a study of the various chart pattern types, here's what I found.
☆ Upward breakouts
Narrow: 9%
Wide: 91% (wide patterns outperform)
☆ Downward breakouts
Narrow: 15%
Wide: 85%
11. True or False: An unconfirmed chart pattern is just squiggles on the page.
Confirmation often occurs when price closes beyond the chart pattern's boundary. For example, in a double bottom, a close above the peak between the two bottoms means the chart pattern becomes a
valid, significant chart pattern. A study I conducted of nearly 1,000 twin bottom patterns showed that 64% of them had price failing to confirm the pattern (closing above the middle peak). Thus,
it's true that if a chart pattern is not confirmed, it is just squiggles on the page. It has little significance.
12. True or False: Performance suffers after a throwback.
A throwback occurs within 30-days after the breakout from a chart pattern. You often see it as a looping price movement that returns price close to or at the breakout price. Often, a throwback
happens in about 6 days after price rises 8%, with price completing the journey back to the breakout price in about 12 days (round trip time). Once a throwback occurs, 65% of the time, price
resumes the upward price trend. A throwback applies only to upward breakouts. A study I conducted using double bottoms found that those chart patterns with throwbacks showed average gains of 35%.
Those without throwbacks climbed 45%, on average. Thus, it's true that if a throwback occurs, it hurts performance.
13. True or False: Performance suffers after a pullback.
Don't be fooled into thinking that I'm asking the same question twice. This one applies to pullbacks, not throwbacks. A pullback occurs after a downward breakout from a chart pattern. Price drops
for an average of 6 days, sinking 4% to 10% before pulling back to the breakout price. It completes the journey in about 11 days and 47% of the time price moves lower thereafter.
Does performance suffer after a pullback? Yes, it's true. Just like throwbacks, pullbacks seem to rob downward momentum, hurting the decline. I measured this in downward breakouts from chart
patterns and found it to be true. The performance results are not as startling as with throwbacks, but there is a clear performance degradation. A straw poll of chart pattern types found that 97%
of them have worse performance after a pullback than those without pullbacks.
How did you do? If you got all of them right, then run next door and tell your neighbor. And be sure to give them this tip: Don't eat yellow snow. That's not so much of a problem in the summer as it
is in the winter.
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Number Sense Archives - The Robertson Program for Inquiry-based Teaching in Mathematics and Science
I came to realize the power and presence of sound when I listened to an audio walk – a podcast that guides you through a specific place and tells you...
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In November 2023, Ontario’s top court upheld the validity of the mandatory Mathematics Proficiency Test (MPT) for new teachers entering the field. The MPT was first introduced in 2021 as... read more
• January 22, 2024
• Blog, Inquiry, Math Blog, Number Sense
It’s quite common at this time of year to find yourself around a table playing a game with friends and family. Other than joy and entertainment, we know there are... read more →
• December 13, 2023
• Blog, Inquiry, Math Blog, Number Sense
A parent recently told me about her child in Grade 3 who was struggling to identify how many hundreds, tens, and ones were in any given number. “Is this necessary?... read more →
• November 9, 2023
• Blog, Inquiry, Math Blog, Number Sense
Grade 4/5 Teacher Zoe Donoahue shares how she reinventeda primary daily math practice so it met the needs of her junior studentsWhen I moved to teaching grade 5, I challenged... read more →
• April 7, 2023
Dr. Andres Bustamante is on a mission to design and implement playful STEM learning activities in the places that children and families spend large amounts of time. The University of... read more →
• February 13, 2023
• Blog, Math Blog, Number Sense, OISE, Spatial Reasoning
Understanding place value is foundational to conceptualizing number It is imperative that children understand how one number relates to another when navigating our base-ten-dependent society. Yet, an
alarming number of... read more →
• August 29, 2022
Four quick and efficient assessments that help educators better understand the mathematical strengths and needs of students. Along with each assessment, we've suggested lessons available in our free
math lesson library... read more →
• August 20, 2022
Whole Number Bias and 3 Misconceptions about Fractions in Junior Math Whole Number Bias and 3 Misconceptions about Fractions in Junior Math Q&A: OISE Assistant Professor Dr. Zack Hawes explains...
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• May 26, 2022
Exploring Food Insecurity using Financial Literacy Using mathematics to understand food insecurity highlights factors that influence one's ability to access food. Due to the supply issues resulting
from the pandemic,... read more →
• February 17, 2022
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# BuckeyeCTF 2024 Fixpoint Writeup
In this chal, we're given a "fixed point" which is reached when you repeatedly apply a Base64 encoding function with a custom alphabet to any string, and told to figure out what that custom alphabet
was. To do this, we must first quickly revisit how Base64 works with a normal alphabet.
# Back to Base64-ics
Base64 is an encoding function which converts arbitrary binary data into printable characters. The standard alphabet is A-Z, a-z, 0-9, and + / for a total of 64 characters. The best way to see how
this works is to just look at it happen in practice. Suppose that the binary data we're encoding is "test12" in the UTF-8 format. This means that each of these characters is represented by 8 bits,
which is to say an 8-digit binary number. We need to convert this into Base64, where each character is 6 bits (2^6 = 64). To get everything to line up neatly, we will have to group the 8-bit
characters into groups of 3, giving us a chunk of 24 bits which we can then neatly encode with four 6-bit characters in Base64.
In the case of our test string, this means that we will first encode `tes` - we first convert it to binary using a tool like [CyberChef](https://gchq.github.io/CyberChef/#recipe=To_Binary('Space',8)&
input=dGVz), giving us 01110100 01100101 01110011. We then take that and group it up into chunks of 6 bits, giving 011101 000110 010101 110011. Finally, we look at what character each of these chunks
corresponds to in the [standard alphabet](https://en.wikipedia.org/wiki/Base64#Base64_table_from_RFC_4648), giving us `dGVz`. We can then repeat the process with the final 3 characters of the test
### Observation: In Base64, groups of 3 characters are converted into groups of 4.
# Fixed Points
Now that we've established how Base64 works normally, let's take a quick look at the concept of a fixed point. A "fixed point" refers to something which maps to itself when a function is applied to
it. For example, 1 is a fixed point of f(x)=x^2, because f(1)=1. The challenge helpfully provides us a fixed point for ordinary base64.[^1] The first 12 characters of it are `Vm0wd2QyUXlV`. Let's
look at what happens when we base64-encode the start here, once again in chunks of 3:
`Vm0` -> `Vm0w`
`wd2` -> `d2Qy`
`QyU` -> `UXlV`
### Observation: The first 3 characters of the fixed point encode into its first 4 characters, the second group of 3 characters encode into the second group of 4, etc.
# Custom Alphabet
This finally brings us to the actual challenge itself. We're given the fixed point and need to figure out the alphabet that corresponds to it. Let's look at the start of this fixed point as well:
The fact that this is a fixed point implies the following mappings:
`Nsl` -> `NslS`
`SBw` -> `Bwm6`
`m6y` -> `YNHH`
To get a sense for exactly what this means, we can walk through the whole encoding process for the first chunk. `Nsl` in UTF-8 corresponds to `01001110 01110011 01101100` in binary (we can use
CyberChef for this as before). Splitting this into chunks of 6 bits gives `010011 100111 001101 101100`. This means that in our Base64 alphabet, `010011` (in decimal, 19) corresponds to `N`, `100111`
(in decimal, 39) to `s`, etc.
We now have everything we need to solve this challenge. We just need to iterate through the characters of the fixed point provided by the author, run this calculation for each chunk of 3 characters,
and every time we discover a correspondence (e.g. 19 corresponds to `N`), we save that into our alphabet string at the relevant index (in this case, this means that we set the character at index 19
to `N`). After we have discovered every correspondence, we will have the flag.
Here's a Python script which does exactly this:
# Given in chal description
alphabet = "bctf{?????????????????????????FiXed???????????????????????p01nT}"
fixed = "NslSBwm6YNHHNreCNsmojw8zY9nGVzep9NoJ5LHpH3b8NKnQlB2Ca{XzIxeUyR85Y{COjRD09P4mFEAAFACZlAo0jwnGBrj7UAbwYBHDjBDEjBlMY{DWkE46YrVtaKh6ABDdVLoty
# Iterate through fixed point chars in groups of 3. We do len(fixed)//4 instead of 3 because otherwise we get an out of bounds error later.
for i in range(len(fixed)//4):
# Grab the 3 characters which will be converted into 4 by the b64 process (3*8 bits = 24 bits = 4*6 bits)
# Basically, Nsl gets converted to NslS, then SBw -> BWm6, etc.
triplet = fixed[3*i:3*i+3]
# Convert each char to 0-extended binary and join them
binary = ''.join(format(ord(x), 'b').zfill(8) for x in triplet)
# Convert each group of 6 bits to an int and update alphabet at that index
alphabet = alphabet[0:int(binary[0:6],2)] + fixed[4*i] + alphabet[int(binary[0:6],2)+1:]
alphabet = alphabet[0:int(binary[6:12],2)] + fixed[4*i + 1] + alphabet[int(binary[6:12],2)+1:]
alphabet = alphabet[0:int(binary[12:18],2)] + fixed[4*i + 2] + alphabet[int(binary[12:18],2)+1:]
alphabet = alphabet[0:int(binary[18:],2)] + fixed[4*i + 3] + alphabet[int(binary[18:],2)+1:]
# We did fixed[4*i] rather than 3*i because we're converting into groups of 4 chars at a time. That's why SBw -> BWm6, not SBwm
This outputs `bctf{DEPCmQqklUgj5yNBA93IHMYaVFiXedxroKsh4GuSvJW72OzwLR6Z8p01nT}`, which is the flag :)
[^1]: Eagle-eyed readers might notice that this technically isn't a fixed point. Base64-encoding any text will always produce text which is slightly longer, because each group of 3 characters is
converted into 4. If we keep applying the base64 function infinitely many times, the string will grow infinitely long. What the challenge author means is that at every step in this process, `base64
(string)` will begin with that same string. Mathematically keen readers are challenged to prove that any starting string will always converge in the limit to the same string upon repeated
applications of `base64()` (note that, despite what the challenge implies, two starting strings might not necessarily converge to the exact same result in a finite number of steps because base64 is
bijective, and the string length increases each time it's applied).
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Using Regression to model race performance in Python - Michael Hainke
Using Regression to model race performance in Python
Reading Time: 5 minutes
In this post I’ll cover how to do the following in Python:
• Use the Seaborn library to plot data and trendlines
• Generate a regression equation using the polyfit function
• Use the regression model to predict future race times
• Review how to improve model performance
This is the third and final post in a series on how to visualize and analyze race and personal running data with the goal of estimating future performance. In the first part I did a bit of
exploratory analysis of Whistler Alpine Meadows 25km distance race data to help set an overall goal for finishing time and required pace to achieve that goal. In the second post I dived into how to
use the Strava API to retrieve my activity data that we will use in this final post to build a simple model that can estimate race finishing time.
Using Seaborn to plot polynomial regression line
First let’s load in our data from the .csv file we saved in our last post, so we don’t need to reload the data from the API. Reading a .csv file is easy using the pandas function
splits = pd.read_csv('18-08-25 New Activity Splits.csv')
Before we return to plotting the data, let’s take another quick look at the data. Last time we plotted the ‘moving time’ vs. the elevation change, but there is also an ‘elapsed time’ in the data.
Let’s investigate further by creating and plotting a new variable which is the difference between these two times.
splits['time_diff'] = splits['elapsed_time'] - splits['moving_time']
plt.plot( 'elevation_difference', 'time_diff', data=splits, linestyle='', marker='o', markersize=3, alpha=0.1, color="blue")
In most cases the elapsed time and moving time are close, but there are a significant number of points where they are different. What causes this? Time spent stationary or with little movement is
captured in elapsed time but not moving time. This confirms what I’ve noticed when logging an activity through Strava, especially on steep or twisty trails where Strava is fooled into thinking
you’ve stopped. For this analysis, I’m going to use elapsed time, even if it means that the few cases where I actually ‘stopped’ for an extended period of time will be included in data. Using
elapsed time will provide a more conservative and realistic estimate of my pace.
Last time we plotted the data using the matplotlib plot function. This time let’s use the awesome Seaborn library to produce some nicer plots and include some trendlines and confidence intervals,
using the function regplot.
sns.regplot(x = 'elevation_difference', y = 'elapsed_time', data = splits ,order = 2)
plt.title('Running Pace vs. Elevation Change', fontsize=18, fontweight="bold")
plt.xlabel('Elevation Change (m)', fontsize=18)
plt.ylabel('1km Pace (sec)', fontsize=18)
Notice we used the parameter order to specify which order polynomial to try and fit to the data. I used 2 in this case, which produces a nice parabola which approximates the data pretty well. As a
stats refresher, the equation for a second degree polynomial (also known as a quadratic) is y = ax² + bx + c. The light blue cone represents the 95% confidence interval, which is calculated using
the bootstrap method. One drawback of this plot is it doesn’t allow us the flexibility of setting the various visual parameters that the matplotlib plot functions does. Specifically, I’d like to
make the individual points look like those in the first plot by changing the alpha level to better show the point density. Luckily, Python makes this easy by allowing us to combine 2 plot functions
onto one plot. I use the plot function to plot the individual points, and the regplot function to plot the trendline and confidence interval. Use ‘scatter=None’ to suppress plotting the individual
points in the regplot.
plt.plot( 'elevation_difference', 'elapsed_time', data=splits, linestyle='', marker='o', markersize=5, alpha=0.1, color="blue")
sns.regplot(x = 'elevation_difference', y = 'elapsed_time', scatter=None, data = splits ,order = 2)
plt.title('Running Pace vs. Elevation Change', fontsize=18, fontweight="bold")
plt.xlabel('Elevation Change (m)', fontsize=18)
plt.ylabel('1km Pace (sec)', fontsize=18)
Using Polyfit to generate the equation for the fitted model
So here’s the main drawback of using regplot, there’s no ability to have it provide the coefficients for the fitted lines and confidence intervals. If anyone knows how to do this, I would love to
hear about it in the comments! So let’s rely on a Numpy function, polyfit, to give the equation
coeff = np.polyfit(splits['elevation_difference'], splits['elapsed_time'], 2)
That will produce the following coefficient array (in decreasing order):
array([7.40646826e-03, 6.30941912e-01, 3.74015634e+02])
So our complete equation is: y = 0.0074*x² + 0.6310*x + 374
Apply equation to WAM course profile to estimate total time
Finally, let’s apply our model to the WAM course profile, which I manually created as a .csv file. Then we calculate the time using the coefficients from the polyfit function above.
# Load WAM course data
WAM = pd.read_csv('WAM_25k_course.csv')
# Calculate estimated time for each km based on elevation change
WAM['estimated_time'] = coeff[0]*WAM['elevation']**2 + coeff[1]*WAM['elevation'] + coeff[2]
This is what the overall data looks like:
km elevation estimated_time
0 1 0 374.015634
1 2 0 374.015634
2 3 13 383.469572
3 4 18 387.772284
4 5 68 451.167193
5 6 203 807.309992
6 7 158 658.599529
7 8 32 401.789998
8 9 27 396.450381
9 10 141 610.226439
10 11 190 761.268101
11 12 310 1281.369227
12 13 -120 404.955747
13 14 -23 363.421991
14 15 -78 369.863117
15 16 24 393.424365
16 17 -43 360.579691
17 18 -60 362.822405
18 19 -16 365.816619
19 20 -93 379.396580
20 21 -167 475.207328
21 22 -181 502.458454
22 23 -165 471.551317
23 24 -128 414.602645
24 25 -79 370.394991
Adding up the all the times and converting to minutes
WAM['estimated_time'].sum() / 60
This gives an estimated time of 202 minutes (3 hrs and 22 minutes). That would be an amazing time! But I suspect that it’s a bit optimistic as it uses a number of runs done on smooth road or track
which will be much faster than a trail run. To try and get a more accurate estimate, I went and manually classified my runs over the last year as either ‘trail’, ‘road’, or ‘track’ and entered the
information in the description field of the activity on Strava. After retrieving only the classified data again using the Strava API, I use the code below to recalculate my estimated finishing time
splits_trail = splits[splits['description'] == 'Trail']
coeff_trail = np.polyfit(splits_trail['elevation_difference'], splits_trail['elapsed_time'], 2)
WAM['estimated_time_trail'] = coeff_trail[0]*WAM['elevation']**2 + coeff_trail[1]*WAM['elevation'] + coeff_trail[2]
WAM['estimated_time_trail'].sum() / 60
This time I get an estimated finish time of 242 minutes (4 hrs and 2 minutes), which is almost exactly my goal of finishing in the middle of the pack!
Final Thoughts
This has been an interesting exercise and provided quite a bit of insight through some exploratory data analysis and some simple modelling that was relatively quick and easy to do. This is always a
good approach, as it allows you to iterate quickly to understand the process and data more fully, before diving into more complicated and time consuming modelling techniques. Our next step would
likely be to build a more complex regression model and/or another popular machine learning algorithm like Random Forest which can utilize other potential factors in estimating pace. We already
identified that the type of surface is almost certainly a factor in estimating performance. There are some other hypothesized factors that we could add to train our model to see if it improves
• Fatigue estimate (split completed at beginning, middle or end of activity)
• Temperature (hot day vs cold day)
• More granular terrain classifications (ie smooth trail vs. technical trail)
Perhaps I will tackle this in a future post, but for now you have a solid set of tools to do some pretty cool analysis of your own activities. We learned how to scrape race data from the web and
retrieve data using an API, some creative ways to to visualize that data, and finally how to build a simple regression model to predict future performance. Pretty cool!
2 thoughts on “Using Regression to model race performance in Python”
1. Nice job using analytics on race data and well explained.
2. Thanks for sharing information.
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What Are The Factors Of 69? - Tech Stray
What Are The Factors Of 69?
So do you have to discover the elements of sixty nine? In this brief guide, we’re going to describe what the elements of sixty nine are, the way you discover them, and list the detail pairs of sixty
nine to make the calculation art work. Let’s dive!
Sixty nine Definition Factors
When we talk approximately the factors of sixty nine, we really endorse all the fantastic and horrible integers (complete numbers) that may be divided further into sixty nine. If you’re taking 69 and
divide it by way of way of actually without a doubt certainly one of its factors, the answer can be some one of a kind issue of 69.
Sixty nine. How To Discover The Elements Of
We have honestly said that a hassle is a spread of that can be divided lightly into 69. So the way you find and list all the factors of sixty nine is to undergo each quantity as plenty as and along
side 69 and take a look at which numbers bring about a very good quotient (this means that that that no decimal locations).
Doing this via hand in big numbers can be time-eating, but it is as an alternative easy for a pc software to do it. Our calculator has completed it for you. Here are all the factors of sixty nine:
sixty nine 1 = 69
sixty nine three = 23
69 23 = 3
sixty nine 69 = 1
All those elements can be used to divide 69 and get a whole range. The complete listing of terrific factors for sixty nine are:
1, 3, 23, and 69
sixty nine. Negative Elements Of
Technically, you can also have horrible elements of sixty nine in math. If you need to calculate the elements of numerous for homework or a check, often the instructor or take a look at can be mainly
looking for exceptional numbers.
However, we are able to great convert wonderful numbers to terrible and those terrible numbers additionally may be factors of sixty nine:
-1, -three, -23, and -69
How Many Elements Does sixty nine Have?
As we are capable of see from the above calculation that there are a entire of 4 high pleasant elements for sixty nine and 4 horrible factors for 69 for an entire of 8 elements for the quantity 69.
Sixty nine has 4 awesome factors and sixty nine has 4 terrible factors. What are the negative numbers that may be a factor of 69?
Sixty 9. Upload The Factors Of
A issue pair is a aggregate of factors that can be improved to equal sixty nine. For 69, all feasible component pairs are listed underneath:
1 x sixty nine = sixty nine
3 x 23 = 69
We’ve moreover written a guide that is going into a bit greater element about trouble pairs for 69 in case you’re concerned!
As earlier than, we can also listing all horrible problem pairs for 69:
-1 x -sixty nine = 69
-three x -23 = sixty nine
Note in terrible problem pairs that due to the reality we’re multiplying minus with the resource of minus, the quit result is a fantastic quantity.
So there you’ve got it. A Complete Guide to the Factors of sixty nine. You have to now have the information and talents to calculate your own factors and factors for any type of your choice.
Feel free to attempt the calculator beneath to test each one-of-a-kind variety or, if you’re feeling fancy, take a pencil and paper and try to do it via using way of hand. Just make certain to pick
out the smallest variety!
Cite, Hyperlink Or Reference This Web Page
If you discover this cloth beneficial to your research, please do us a tremendous pick out and use the device under to ensure you speak with us nicely anywhere you use it. We simply recognize your
Sixty nine. Factors Of
The factors of sixty nine and the excessive factors of 69 are one-of-a-kind because of the reality seventy-9 is a composite quantity. Also, no matter being carefully related, the high elements of
sixty nine and the pinnacle factors of sixty nine are not precisely the identical.
In any case, by way of studying you may discover the solution to the question What are the factors of 69? And the entirety else you need to understand approximately the problem.
What Are The Elements Of sixty nine?
They are: sixty nine, 23, 3, 1. These are all factors of sixty nine, and each get right of entry to in the list can divide sixty nine without punctuation (module 0). Therefore the issue and
denominator of 69 may be used interchangeably.
As in the case of any natural massive variety extra than 0, the giant range itself, right right right here sixty nine, further to at least one are the factors and divisors of sixty nine.
Sixty 9. Top Factors Of
The excessive factors of 69 are the ones excessive numbers that exactly divide 69 without a the relaxation described thru Euclidean branch. In different terms, a top factorization of sixty nine
divides sixty nine with none break, modulo 0.
For sixty nine, the pinnacle elements are: 3, 23. By definition, 1 is not a top variety.
In addition to 1, what separates the top elements and the immoderate factors of the variety 69 is the phrase “high”. The first listing consists of both composite and top numbers, while the latter
consists of fine excessive numbers.
Sixty 9. Prime Factorization Of
The pinnacle factorization of sixty nine is three × 23. This is a very particular listing of pinnacle factors with their multiples. Note that the excessive factorization of 69 does no longer
encompass the no 1, but it consists of each instance of a remarkable high factorization.
69 is a composite massive range. Unlike pinnacle numbers, which have handiest one aspect, mixed numbers like 69 have at the least factors.
To make clear the which means that that of 69, 23, 3, 1 the rightmost and rightmostose the second rightmost and the second leftmost get right of entry to to achieve the 2nd factorization which
moreover produces 69.
The top factorization or integer factorization of 69 way identifying the set of high numbers which, at the same time as extended collectively, produce the perfect quantity 69. This is likewise
referred to as pinnacle decomposition of 69.
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• BPC •
Autor: BLINDER, S. M. & FANO, GUIDO
Año: 2017
Género: FÍSICA, MATEMÁTICAS, CIENCIA Y TECNOLOGÍA
Formato: PDF
This book is designed to make accessible to nonspecialists the still evolving concepts of quantum mechanics and the terminology in which these are expressed. The opening chapters summarize elementary
concepts of twentieth century quantum mechanics and describe the mathematical methods employed in the field, with clear explanation of, for example, Hilbert space, complex variables, complex vector
spaces and Dirac notation, and the Heisenberg uncertainty principle. After detailed discussion of the Schrödinger equation, subsequent chapters focus on isotropic vectors, used to construct spinors,
and on conceptual problems associated with measurement, superposition, and decoherence in quantum systems. Here, due attention is paid to Bell's inequality and the possible existence of hidden
variables. Finally, progression toward quantum computation is examined in detail: if quantum computers can be made practicable, enormous enhancements in computing power, artificial intelligence, and
secure communication will result. This book will be of interest to a wide readership seeking to understand modern quantum mechanics and its potential applications.
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The Magic Cafe Forums - Seven Queens "number to card" calculation (difficult)
Special user
Nashville TN
960 Posts
Seven Queens "number to card" calculation (difficult).
Note that this thread is for other magician articles/threads to reference.
The easy "card to number" calculation is documented elsewhere on another thread. This thread is specifically on how to calculate "number to card" for the Seven Queens stack (this will also work on
the Jackknife stack and the King Deuce stack).
Below is the Seven Queens stack:
7S, QD, 8D, KC, 10H, 9C, AC, 3S, 6H, 5C, JS, 2H, 4D
7H, QS, 8S, KD, 10C, 9D, AD, 3H, 6C, 5D, JH, 2C, 4S
7C, QH, 8H, KS, 10D, 9S, AS, 3C, 6D, 5S, JC, 2D, 4H
7D, QC, 8C, KH, 10S, 9H, AH, 3D, 6S, 5H, JD, 2S, 4C
Note that the possible offsets that may be used are: 0 or 13 or 26 or 39.
The Suits associated number: 1=spades, 2=hearts, 3=clubs, 4=diamonds. Note that 0 is also equal to diamonds.
Below is the Harry Riser type of twin pairs:
Ace and 7 (have a sharp angle at top)
2 and Queen (2 headed Queen)
3 and 8 (the 3 looks like half of an eight)
4 and King (the K and 4 have 4 corners)
5 and 10 (five and dime store)
6 and 9 (same symbol upside down)
Jack stands alone.
Also note that we have an imaginary 8 rung ladder with rungs numbered 0, 1, 2, 3, 0, 1, 2, 3.
This ladder will come in handy below. These rung numbers do NOT directly relate to the suits, it's the number of steps that are taken between the rungs that relate to the suits (this will be
explained a little further below).
Note that the term "mod 4" just means the REMAINDER number when dividing by 4.
A spectator names any number from one to 52.
The magician (or shill) mentally calculates:
1. Subtracts the nearest OFFSET that is lower than the named number (this gives a number from one to 13).
2. The Harry Riser type pair is the value of the card.
3. To determine the suit of the card:
a. Mod 4 the Harry Riser pair (result will be 0 or 1 or 2 or 3). From the bottom of the ladder this is the STARTING RUNG.
b. The TARGET RUNG is the first rung above that is equal to THE FIRST DIGIT OF THE OFFSET.
c. The NUMBER OF STEPS to get from the starting rung to the target rung equates to the suit value ie: 1 step=spades, 2 steps=hearts, 3 steps=clubs, 4 steps=diamonds (or zero steps equals diamonds).
The above rules work on the Seven Queens stack, Jackknife stack, and King Deuce stack.
The below examples are only for the Seven Queens stack.
Example: spectator names the number 28.
Magician or shill calculates:
Nearest lower possible offset is 26.
28 - 26 is 2
The 2's twin is the Queen (12).
Mod 4 the 12 gives 0.
How many steps to get from the 0 rung to the first digit of 26 rung (rung 2)?
We must take two steps up:
(rung 1, then rung 2).
The two steps equates to hearts.
Thus the answer is the Queen of Hearts.
Example: spectator names the number 18.
Magician or shill calculates:
Nearest lower possible offset is 13.
18 - 13 is 5
The 5's twin is the 10 (the card value).
Mod 4 the 10 gives 2.
How many steps to get from the '2" rung to the first digit of 13 rung (rung 1)?
We must take three steps up:
(rung 3, then rung 0, then rung 1).
The three steps equates to clubs.
Thus the answer is the Ten of Clubs.
Example: spectator names the number 39.
Magician or shill calculates:
Nearest lower possible offset is 26.
39 - 26 is 13
The King (13) twin is the 4 (the card value).
Mod 4 the 4 gives 0.
How many steps to get from the "0" rung to the first digit of 26 rung (rung 2)?
We must take two steps up:
(rung 1, then rung 2).
The two steps equates to hearts.
Thus the answer is the Four of Hearts.
Example: spectator names the number 6.
Magician or shill calculates:
Nearest lower possible offset is 0.
6 - 0 is 6
The 6 twin is the 9 (the card value).
Mod 4 the 9 gives 1.
How many steps to get from the "1" rung to the first digit of 0 rung (next rung 0)?
We must take three steps up:
(rung 2, then rung 3, then rung 0).
The three steps equates to clubs.
Thus the answer is the Nine of Clubs.
Example: spectator names the number 20.
Magician or shill calculates:
Nearest lower possible offset is 13.
20 - 13 is 7.
The 7's twin is the Ace (1). The card value.
Mod 4 the 1 gives 1 .
How many steps to get from the "1" rung to the first digit of 13 rung (rung 1)?
We must take zero or 4 steps up:
(rung 2, then rung 3, then rung 0, then rung 1).
The 0 or 4 steps equates to diamonds.
Thus the answer is the Ace of Diamonds.
Note that "zero steps up" or "four steps up" will get to the same target rung number thus zero or four steps up means diamonds.
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Pictures of Other Prisms
Football Star
Instructions page
Video tutorials
Pictures of Other Prisms
Other Prisms:
<< previous
>> next
Paper Models of Polyhedra
Platonic Solids
Archimedean Solids
Kepler-Poinsot Polyhedra
Other Uniform Polyhedra
Concave Pyramids
Truncated Pyramids
Other Pyramids
Concave Prisms
Concave Antiprisms
Twisted Prisms
Other Prisms
Other Polyhedra
Other Paper Models
Polyhedra Collections
Football (ball)
Selection 1
Selection of Pyramids
Selection of Prisms
Isosceles Tetracontahedra
New Paper Models
Complex Paper Models
Pictures of decorated polyhedra models
Download Page (PDF-files)
Simple Paper Models
Oblique Paper Models
Egyptian Pyramids
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Find Angles Of Triangle Worksheet - Angleworksheets.com
Angles Of Triangle Worksheet – This article will discuss Angle Triangle Worksheets as well as the Angle Bisector Theorem. In addition, we’ll talk about Isosceles and Equilateral triangles. You can
use the search bar to locate the worksheet you are looking for if you aren’t sure. Angle Triangle Worksheet This Angle Triangle Worksheet teaches students … Read more
Angle Of Triangle Worksheet
Angle Of Triangle Worksheet – In this article, we’ll talk about Angle Triangle Worksheets and the Angle Bisector Theorem. We’ll also discuss Equilateral triangles and Isosceles. If you’re unsure of
which worksheet you need, you can always use the search bar to find the exact worksheet you’re looking for. Angle Triangle Worksheet This Angle Triangle … Read more
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A Note on Various Forking Lemmas
First introduced by Pointcheval and Stern, the forking lemma is commonly used in proofs of security to demonstrate a reduction to breaking some known-to-be-hard mathematical problem. While the
original forking lemma is straightforward, keeping up with the number of variations of forking lemmas in the literature (and why each variant exists) is... not.
I wrote a short (informal) note that reviews the original forking lemma and several variations thereof, and discusses the differences among each variant. You can find the note here.
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Codac: constraint-programming for robotics
Codac (Catalog Of Domains And Contractors) is a C++/Python library providing tools for constraint programming over reals, trajectories and sets. It has many applications in state estimation or robot
What is constraint programming?
In this paradigm, users concentrate on the properties of a solution to be found (e.g. the pose of a robot, the location of a landmark) by stating constraints on the variables. Then, a solver performs
constraint propagation on the variables and provides a reliable set of feasible solutions corresponding to the problem. In this approach, the user concentrates on what is the problem instead of how
to solve it, thus leaving the computer dealing with the how.
What about mobile robotics?
In the field of robotics, complex problems such as non-linear state estimation, parameter estimation, delays, SLAM or kidnapped robot problems can be solved in a very few steps by using constraint
programming. Even though the Codac library is not meant to target only robotics problems, the design of its interface has been largely influenced by the needs of the above class of applications.
Codac provides solutions to deal with these problems, that are usually hardly solvable by conventional methods such as particle approaches or Kalman filters.
In a nutshell, Codac is a constraint programming framework providing tools to easily solve a wide range of problems.
• constraint-programming • dynamical-systems • state-estimation
• mobile robotics • tubes • SLAM
• interval-analysis • localization • solver
We only have to define domains for our variables and a set of contractors to implement our constraints. The core of Codac stands on a Contractor Network representing a solver.
In a few steps, a problem is solved by
1. Defining the initial domains (boxes, tubes) of our variables (vectors, trajectories)
2. Take contractors from a catalog of already existing operators, provided in the library
3. Add the contractors and domains to a Contractor Network
4. Let the Contractor Network solve the problem
5. Obtain a reliable set of feasible variables
For instance.
Let us consider the robotic problem of localization with range-only measurements. A robot is described by the state vector \(\mathbf{x}=\{x_1,x_2,\psi,\vartheta\}^\intercal\) depicting its position,
its heading and its speed. It evolves between three landmarks \(\mathbf{b}_1\), \(\mathbf{b}_2\), \(\mathbf{b}_3\) and measures distances \(y_i\) from these points.
The problem is defined by classical state equations:
\[\begin{split}\left\{ \begin{array}{l} \dot{\mathbf{x}}(t)=\mathbf{f}\big(\mathbf{x}(t),\mathbf{u}(t)\big)\\ y_i=g\big(\mathbf{x}(t_i),\mathbf{b}_i\big) \end{array}\right.\end{split}\]
where \(\mathbf{u}(t)\) is the input of the system, known with some uncertainties. \(\mathbf{f}\) and \(g\) are non-linear functions.
First step.
Defining domains for our variables.
We have three variables evolving with time: the trajectories \(\mathbf{x}(t)\), \(\mathbf{v}(t)=\dot{\mathbf{x}}(t)\), \(\mathbf{u}(t)\). We define three tubes to enclose them:
dt = 0.01 # timestep for tubes accuracy
tdomain = Interval(0, 3) # temporal limits [t_0,t_f]=[0,3]
x = TubeVector(tdomain, dt, 4) # 4d tube for state vectors
v = TubeVector(tdomain, dt, 4) # 4d tube for derivatives of the states
u = TubeVector(tdomain, dt, 2) # 2d tube for inputs of the system
float dt = 0.01; // timestep for tubes accuracy
Interval tdomain(0, 3); // temporal limits [t_0,t_f]=[0,3]
TubeVector x(tdomain, dt, 4); // 4d tube for state vectors
TubeVector v(tdomain, dt, 4); // 4d tube for derivatives of the states
TubeVector u(tdomain, dt, 2); // 2d tube for inputs of the system
We assume that we have measurements on the headings \(\psi(t)\) and the speeds \(\vartheta(t)\), with some bounded uncertainties defined by intervals \([e_\psi]=[-0.01,0.01]\), \([e_\vartheta]=
x[2] = Tube(measured_psi, dt).inflate(0.01) # measured_psi is a set of measurements
x[3] = Tube(measured_speed, dt).inflate(0.01)
x[2] = Tube(measured_psi, dt).inflate(0.01); // measured_psi is a set of measurements
x[3] = Tube(measured_speed, dt).inflate(0.01);
Finally, we define the domains for the three range-only observations \((t_i,y_i)\) and the position of the landmarks. The distances \(y_i\) are bounded by the interval \([e_y]=[-0.1,0.1]\).
e_y = Interval(-0.1,0.1)
y = [Interval(1.9+e_y), Interval(3.6+e_y), \ # set of range-only observations
b = [[8,3],[0,5],[-2,1]] # positions of the three 2d landmarks
t = [0.3, 1.5, 2.0] # times of measurements
Interval e_y(-0.1,0.1);
vector<Interval> y = {1.9+e_y, 3.6+e_y, 2.8+e_y}; // set of range-only observations
vector<Vector> b = {{8,3}, {0,5}, {-2,1}}; // positions of the three 2d landmarks
vector<double> t = {0.3, 1.5, 2.0}; // times of measurements
Second step.
Defining contractors to deal with the state equations.
The distance function \(g(\mathbf{x},\mathbf{b})\) between the robot and a landmark corresponds to the CtcDist contractor provided in the library. The evolution function \(\mathbf{f}(\mathbf{x},\
mathbf{u})=\big(x_4\cos(x_3),x_4\sin(x_3),u_1,u_2\big)\) can be handled by a custom-built contractor:
ctc_f = CtcFunction(
Function("v[4]", "x[4]", "u[2]",
"(v[0]-x[3]*cos(x[2]) ; v[1]-x[3]*sin(x[2]) ; v[2]-u[0] ; v[3]-u[1])"))
CtcFunction ctc_f(
Function("v[4]", "x[4]", "u[2]",
"(v[0]-x[3]*cos(x[2]) ; v[1]-x[3]*sin(x[2]) ; v[2]-u[0] ; v[3]-u[1])"));
Third step.
Adding the contractors to a network, together with there related domains, is as easy as:
cn = ContractorNetwork() # creating a network
cn.add(ctc_f, [v, x, u]) # adding the f constraint
for i in range (0,len(y)): # we add the observ. constraint for each range-only measurement
p = cn.create_interm_var(IntervalVector(4)) # intermed. variable (state at t_i)
# Distance constraint: relation between the state at t_i and the ith beacon position
cn.add(ctc.dist, [cn.subvector(p,0,1), b[i], y[i]])
# Eval constraint: relation between the state at t_i and all the states over [t_0,t_f]
cn.add(ctc.eval, [t[i], p, x, v])
ContractorNetwork cn; // creating a network
cn.add(ctc_f, {v, x, u}); // adding the f constraint
for(int i = 0 ; i < 3 ; i++) // we add the observ. constraint for each range-only measurement
IntervalVector& p = cn.create_interm_var(IntervalVector(4)); // intermed. variable (state at t_i)
// Distance constraint: relation between the state at t_i and the ith beacon position
cn.add(ctc::dist, {cn.subvector(p,0,1), b[i], y[i]});
// Eval constraint: relation between the state at t_i and all the states over [t_0,t_f]
cn.add(ctc::eval, {t[i], p, x, v});
Fourth step.
Solving the problem.
Fifth step.
Obtain a reliable set of feasible positions: a tube, depicted in blue. The three yellow robots illustrate the three instants of observation. The white line is the unknown truth.
You just solved a non-linear state-estimation without knowledge about initial condition.
In the tutorial and in the examples folder of this library, you will find more advanced problems such as Simultaneous Localization And Mapping (SLAM), data association problems or delayed systems.
Want to use Codac? The first thing to do is to install the library, or try it online:
Then you have two options: read the details about the features of Codac (domains, tubes, contractors, slices, and so on) or jump to the standalone tutorial about how to use Codac for mobile robotics,
with telling examples.
The following tutorial is standalone and tells about how to use Codac for mobile robotic applications, with telling examples:
We suggest the following BibTeX template to cite Codac in scientific discourse:
title={The {C}odac Library},
journal={Acta Cybernetica},
series = {Special Issue of {SWIM} 2022},
author={Rohou, Simon and Desrochers, Benoit and {Le Bars}, Fabrice},
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Algebra 1
absolute value
The absolute value of a number is its distance from 0 on the number line.
In statistics we say that there is an association between two variables if the two variables are statistically related to each other; if the value of one of the variables can be used to estimate the
value of the other.
average rate of change
The average rate of change of a function \(f\) between inputs \(a\) and \(b\) is the change in the outputs divided by the change in the inputs: \(\frac{f(b)-f(a)}{b-a}\). It is the slope of the line
joining \((a,f(a))\) and \((b, f(b))\) on the graph.
bell-shaped distribution
A distribution whose dot plot or histogram takes the form of a bell with most of the data clustered near the center and fewer points farther from the center.
bimodal distribution
A distribution with two very common data values seen in a dot plot or histogram as distinct peaks. In the dot plot shown, the two common data values are 2 and 7.
categorical data
Categorical data are data where the values are categories. For example, the breeds of 10 different dogs are categorical data. Another example is the colors of 100 different flowers.
categorical variable
A variable that takes on values which can be divided into groups or categories. For example, color is a categorical variable which can take on the values, red, blue, green, etc.
causal relationship
A causal relationship is one in which a change in one of the variables causes a change in the other variable.
In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the
coefficient is 1.
The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.
completing the square
Completing the square in a quadratic expression means transforming it into the form \(a(x+p)^2-q\), where \(a\), \(p\), and \(q\) are constants.
Completing the square in a quadratic equation means transforming into the form \(a(x+p)^2=q\).
constant term
In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes.
In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (\text-8)\). In the expression \(12x-4\) the constant term is -4.
A limitation on the possible values of variables in a model, often expressed by an equation or inequality or by specifying that the value must be an integer. For example, distance above the ground \
(d\), in meters, might be constrained to be non-negative, expressed by \(d \ge 0\).
correlation coefficient
A number between -1 and 1 that describes the strength and direction of a linear association between two numerical variables. The sign of the correlation coefficient is the same as the sign of the
slope of the best fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. When the correlation coefficient is closer to 1 or -1, the linear model fits the data
The first figure shows a correlation coefficient which is close to 1, the second a correlation coefficient which is positive but closer to 0, and the third a correlation coefficient which is close to
decreasing (function)
A function is decreasing if its outputs get smaller as the inputs get larger, resulting in a downward sloping graph as you move from left to right.
A function can also be decreasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is decreasing for \(x \ge 0\) because the
graph slopes downward to the right of the vertical axis.
dependent variable
A variable representing the output of a function.
The equation \(y = 6-x\) defines \(y\) as a function of \(x\). The variable \(x\) is the independent variable, because you can choose any value for it. The variable \(y\) is called the dependent
variable, because it depends on \(x\). Once you have chosen a value for \(x\), the value of \(y\) is determined.
For a numerical or categorical data set, the distribution tells you how many of each value or each category there are in the data set.
The domain of a function is the set of all of its possible input values.
A method of solving a system of two equations in two variables where you add or subtract a multiple of one equation to another in order to get an equation with only one of the variables (thus
eliminating the other variable).
equivalent equations
Equations that have the exact same solutions are equivalent equations.
equivalent systems
Two systems are equivalent if they share the exact same solution set.
exponential function
An exponential function is a function that has a constant growth factor. Another way to say this is that it grows by equal factors over equal intervals. For example, \(f(x)=2 \boldcdot 3^x\) defines
an exponential function. Any time \(x\) increases by 1, \(f(x)\) increases by a factor of 3.
factored form (of a quadratic expression)
A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form. For example, \(2(x-1)(x+3)\) and \((5x + 2)(3x-1)\) are both in factored
five-number summary
The five-number summary of a data set consists of the minimum, the three quartiles, and the maximum. It is often indicated by a box plot like the one shown, where the minimum is 2, the three
quartiles are 4, 4.5, and 6.5, and the maximum is 9.
A function takes inputs from one set and assigns them to outputs from another set, assigning exactly one output to each input.
function notation
Function notation is a way of writing the outputs of a function that you have given a name to. If the function is named \(f\) and \(x\) is an input, then \(f(x)\) denotes the corresponding output.
growth factor
In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.
growth rate
In an exponential function, the growth rate is the fraction or percentage of the output that gets added every time the input is increased by one. If the growth rate is 20% or 0.2, then the growth
factor is 1.2.
horizontal intercept
The horizontal intercept of a graph is the point where the graph crosses the horizontal axis. If the axis is labeled with the variable \(x\), the horizontal intercept is also called the \(x\)
-intercept. The horizontal intercept of the graph of \(2x + 4y = 12\) is \((6,0)\).
The term is sometimes used to refer only to the \(x\)-coordinate of the point where the graph crosses the horizontal axis.
increasing (function)
A function is increasing if its outputs get larger as the inputs get larger, resulting in an upward sloping graph as you move from left to right.
A function can also be increasing just for a restricted range of inputs. For example the function \(f\) given by \(f(x) = 3 - x^2\), whose graph is shown, is increasing for \(x \le 0\) because the
graph slopes upward to the left of the vertical axis.
independent variable
A variable representing the input of a function.
The equation \(y = 6-x\) defines \(y\) as a function of \(x\). The variable \(x\) is the independent variable, because you can choose any value for it. The variable \(y\) is called the dependent
variable, because it depends on \(x\). Once you have chosen a value for \(x\), the value of \(y\) is determined.
inverse (function)
Two functions are inverses to each other if their input-output pairs are reversed, so that if one function takes \(a\) as input and gives \(b\) as an output, then the other function takes \(b\) as an
input and gives \(a\) as an output.
You can sometimes find an inverse function by reversing the processes that define the first function in order to define the second function.
irrational number
An irrational number is a number that is not rational. That is, it cannot be expressed as a positive or negative fraction, or zero.
linear function
A linear function is a function that has a constant rate of change. Another way to say this is that it grows by equal differences over equal intervals. For example, \(f(x)=4x-3\) defines a linear
function. Any time \(x\) increases by 1, \(f(x)\) increases by 4.
linear term
The linear term in a quadratic expression (in standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may
need to be rewritten in standard form first.)
A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.
A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.
A mathematical or statistical representation of a problem from science, technology, engineering, work, or everyday life, used to solve problems and make decisions.
negative relationship
A relationship between two numerical variables is negative if an increase in the data for one variable tends to be paired with a decrease in the data for the other variable.
non-statistical question
A non-statistical question is a question which can be answered by a specific measurement or procedure where no variability is anticipated, for example:
• How high is that building?
• If I run at 2 meters per second, how long will it take me to run 100 meters?
numerical data
Numerical data, also called measurement or quantitative data, are data where the values are numbers, measurements, or quantities. For example, the weights of 10 different dogs are numerical data.
A data value that is unusual in that it differs quite a bit from the other values in the data set. In the box plot shown, the minimum, 0, and the maximum, 44, are both outliers.
perfect square
A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.
piecewise function
A piecewise function is a function defined using different expressions for different intervals in its domain.
positive relationship
A relationship between two numerical variables is positive if an increase in the data for one variable tends to be paired with an increase in the data for the other variable.
quadratic equation
An equation that is equivalent to one of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
quadratic expression
A quadratic expression in \(x\) is one that is equivalent to an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
quadratic formula
The formula \(x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\) that gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\) is not 0.
quadratic function
A function where the output is given by a quadratic expression in the input.
The range of a function is the set of all of its possible output values.
rational number
A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into \(b\) equal parts and finding the
point that is \(a\) of them from 0. We can always write a fraction in the form \(\frac{a}{b}\) where \(a\) and \(b\) are whole numbers, with \(b\) not equal to 0, but there are other ways to write
them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0.
We can also write this number as \(\frac{7}{10}\).
The numbers \(3\), \(\text-\frac34\), and \(6.7\) are all rational numbers. The numbers \(\pi\) and \(\text-\sqrt{2}\) are not rational numbers, because they cannot be written as fractions or their
relative frequency table
A version of a two-way table in which the value in each cell is divided by the total number of responses in the entire table or by the total number of responses in a row or a column.
The table illustrates the first type for the relationship between the condition of a textbook and its price for 120 of the books at a college bookstore.
│ │$10 or less│more than $10 but less than $30 │$30 or more│
│new │0.025 │0.075 │0.225 │
│used│0.275 │0.300 │0.100 │
The difference between the \(y\)-value for a point in a scatter plot and the value predicted by a linear model. The lengths of the dashed lines in the figure are the residuals for each data point.
skewed distribution
A distribution where one side of the distribution has more values farther from the bulk of the data than the other side, so that the mean is not equal to the median. In the dot plot shown, the data
values on the left, such as 1, 2, and 3, are further from the bulk of the data than the data values on the right.
solutions to a system of inequalities
All pairs of values that make the inequalities in a system true are solutions to the system. The solutions to a system of inequalities can be represented by the points in the region where the graphs
of the two inequalities overlap.
solution to a system of equations
A coordinate pair that makes both equations in the system true.
On the graph shown of the equations in a system, the solution is the point where the graphs intersect.
standard deviation
A measure of the variability, or spread, of a distribution, calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more
advanced courses.
standard form (of a quadratic expression)
The standard form of a quadratic expression in \(x\) is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not 0.
A quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation).
statistical question
A statistical question is a question that can only be answered by using data and where we expect the data to have variability, for example:
• Who is the most popular musical artist at your school?
• When do students in your class typically eat dinner?
• Which classroom in your school has the most books?
strong relationship
A relationship between two numerical variables is strong if the data is tightly clustered around the best fit line.
Substitution is replacing a variable with an expression it is equal to.
symmetric distribution
A distribution with a vertical line of symmetry in the center of the graphical representation, so that the mean is equal to the median. In the dot plot shown, the distribution is symmetric about the
data value 5.
system of equations
Two or more equations that represent the constraints in the same situation form a system of equations.
system of inequalities
Two or more inequalities that represent the constraints in the same situation form a system of inequalities.
two-way table
A way of organizing data from two categorical variables in order to investigate the association between them.
│ │has a cell phone│does not have a cell phone │
│10–12 years old│25 │35 │
│13–15 years old│38 │12 │
│16–18 years old│52 │8 │
uniform distribution
A distribution which has the data values evenly distributed throughout the range of the data.
variable (statistics)
A characteristic of individuals in a population that can take on different values.
vertex form (of a quadratic expression)
The vertex form of a quadratic expression in \(x\) is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a\) is not 0.
vertex (of a graph)
The vertex of the graph of a quadratic function or of an absolute value function is the point where the graph changes from increasing to decreasing or vice versa. It is the highest or lowest point on
the graph.
vertical intercept
The vertical intercept of a graph is the point where the graph crosses the vertical axis. If the axis is labeled with the variable \(y\), the vertical intercept is also called the \(y\)-intercept.
Also, the term is sometimes used to mean just the \(y\)-coordinate of the point where the graph crosses the vertical axis. The vertical intercept of the graph of \(y = 3x - 5\) is \((0,\text-5)\), or
just -5.
weak relationship
A relationship between two numerical variables is weak if the data is loosely spread around the best fit line.
zero (of a function)
A zero of a function is an input that yields an output of zero. If other words, if \(f(a) = 0\) then \(a\) is a zero of \(f\).
zero product property
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.
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Applied Mathematics Colloquium - Gadi Fibich (School of Mathematical Sciences, Tel Aviv University) - Department of Mathematics
Applied Mathematics Colloquium – Gadi Fibich (School of Mathematical Sciences, Tel Aviv University)
March 19 @ 3:30 pm - 4:30 pm
Spreading of Innovations on Networks
Abstract. Spreading (diffusion) of new products is a classical problem. Traditionally, it has been analyzed using the compartmental Bass model, which implicitly assumes that all individuals are
homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamental Bass model on networks, which is a particle model for the stochastic
adoption by each individual.
In this talk I will review the emerging mathematical theory for the Bass model on networks. I will present analytic tools that enable us to obtain explicit expressions for the expected adoption level
on various networks (complete, circular, d-regular, Erdos-Renyi, …), without employing mean-field type approximations. The main focus of the talk will be on the effect of network structure. For
example, which networks yield the slowest and fastest adoption? What is the effect of boundaries? Of heterogeneity among individuals?
Related Events
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Investigating Trends in Streamflow and Precipitation in Huangfuchuan Basin with Wavelet Analysis and the Mann-Kendall Test
College of Water Resources and Hydrology, Hohai University, Xikang Road 1, Nanjing 210098, China
Authors to whom correspondence should be addressed.
Submission received: 8 December 2015 / Revised: 16 February 2016 / Accepted: 18 February 2016 / Published: 2 March 2016
This study aims to investigate trends in streamflow and precipitation in the period 1954–2010 in a semiarid region of the Yellow River watershed, Huangfuchuan basin, China. The combination of the
wavelet transform and different Mann-Kendall (MK) tests were employed to figure out the basic trends structure in streamflow and precipitation and what time scales are affecting the observed trends.
The comparative analysis with five MK test methods showed that the modified MK tests with full serial correlation structure performed better when significant autocorrelations exhibited for more than
one lag. Three criteria were used to determine the optimal smooth mother wavelet, the decomposition level and the extension mode used in the discrete wavelet transform (DWT) procedure. The first
criteria referred to the relative error of the wavelet approximated component and the original series. The second one was the relative error of MK Z-values of approximation component and the original
series. Additionally, a new criterion (Er), based on the relative error of energy between the approximate component and the original series, was proposed in this study, with better performance than
the previous two criteria. Further, a new powerful index, the energy of the hydrological time series, was proposed to verify the dominant periodic components for the observed trends. The analysis
indicated that all monthly, seasonal and annual streamflow showed significant decreasing trends, while no significant trends were found in precipitation. Results from the DWT and MK tests revealed
that the main factors influencing the trends in the monthly and seasonal series in Huangfuchuan watershed are intra-annual cycles, while the leading factors affecting the trends in the annual series
are decadal events. Different driving factors (e.g., seasonal cycles, solar activities, etc.) related to the periodicities identified in these data types resulted in this discrepancy.
1. Introduction
The processes that occur in the atmosphere and the earth’s surface, such as precipitation and streamflow, are mainly driven by the energy exchange between the sun, the earth and the atmosphere [
]. The fifth IPCC assessment report [
] indicated the mean annual global air temperature exhibited a significant upward trend during the period from the 1880–2012, and the greatest increase was noted from 1979 to 2012 with a 0.25–0.27 °C
increment per decade [
]. One of the most obvious effects aroused by climate warming and change is to intensify the hydrological cycle [
]. Changes in hydrological cycle may in turn affect the availability and quality of water resources, and the sustainability of water management, particularly in dry regions [
The Yellow River is considered to be China’s mother river and the cradle of Chinese civilization, and it is a vital water source for hundreds of millions of people in the northern and north-western
parts of China [
]. The Yellow River is 5464 km long with a basin area of 0.8 million km
, which is mainly comprised of arid and semi-arid environments [
]. The Huangfuchuan basin, an important semiarid watershed in the middle reaches of the Yellow River, was selected as a meso-scale catchment representative of the semiarid climates that predominate
across the Yellow River watershed, in order to detect the effects of climate variability and change. A better understanding of climate variability and change on both a basin and regional scale is
obviously critical to water management and sustainable ecological conservation of arid and semiarid regions. Many studies which consider both climate variability and change have centered on the
assessments in hydro-climate parameters such as temperature, precipitation and streamflow [
]. Hydrological variables have been considered as useful indicators of how the climate has changed and varied over time, therefore, it is needful to research trends associated with hydrological
events [
The Mann-Kendall (MK) trend test has been widely used to trend detection in hydrology and climatology [
], due to its rank-based procedure with resistance to the influence of extreme values that facilitate the skewness variables [
]. But an obvious weakness of the MK test is that it is not accounting for the serial correlation which is very often seen in the hydro-climate data [
]. Studies have demonstrated that the presence of positive autocorrelation overestimated the significance of (both positive and negative) trends, while negative autocorrelation underestimated the
significance of (both positive and negative) trends, if the autocorrelation is not considered [
]. In an effort to remove the influence of serial correlation on the MK test, the modified MK test with lag one and trend-free pre-whitening [
] and the modified MK test with variance correction were proposed and applied [
]. Gautam and Acharya (2012) [
] used trend-free pre-whitening to deal with serial and cross-correlation in detecting trends of streamflow in Nepal. However, Kumar
et al.
(2009) [
] found that consideration of only lag-1 autocorrelation is not sufficient to remove all significant serial correlation in the hydrological time series. Khaliq
et al.
(2009) [
] recommended the variance correction approach, because not only lag-1 but also higher lags are considered for serial correlation. However, Yue and Wang (2004) pointed out that the modified MK test
with full autocorrelation structure is not powerful in the case that trend cannot be approximated by a linear trend [
Wavelet analysis is another effective way to analyze the time series owing to its capability of illustrating the localized characteristics of a series both in temporal and frequency domains [
]. Wavelet analysis has been extensively employed to determine the non-stationary trends and periodicities in the analysis of various hydrological and meteorological variables [
]. In order to better analyze the trends and the fluctuating patterns of the hydrological variables, the wavelet transform has been recently used in conjunction with the MK test [
]. Partal and Küçük (2006) [
] firstly co-utilized the wavelet transform and the original MK test to find which periodicities are mainly responsible for the trends of the annual total precipitation series and found that the
trend analysis on detailed components of the precipitation time series resulting from the discrete wavelet transform (DWT) can clearly explain the trend structure of data. Nalley
et al.
(2012) [
] used DWT to analyze trends in streamflow and precipitation in Quebec and Ontario with the modified MK test proposed by Hirsch and Slack (1984), which accounts for seasonality and serial dependence.
However, this modified MK test is not powerful when there is long-term persistence (with autoregressive parameter >0.6) or when there are less than 5 years’ worth of monthly data [
When applying DWT to the decomposition of hydrologic time series, two critical issues, wavelet choice and decomposition level, should be confirmed firstly. When choosing wavelet, both the wavelet’s
properties and the hydrologic series’ composition should be considered [
]. It is suggested that two conditions be followed, one is that the chosen wavelet must meet the regularity condition which is required for DWT [
]; another is that the appropriate wavelet should satisfy the need of accurately separating deterministic series from original series [
]. The Daubechies (db) wavelets not only meet these two conditions, but also are commonly used in hydro-meteorological wavelet-based studies [
]. The number of decomposition levels needs to be confirmed in order to avoid unnecessary levels of data decomposition. This number is affected by the number of data points and the mother wavelet
used. The highest decomposition level should be in agreement with the data point where the last subsampling becomes smaller than the filter length [
]. The reason is that for signals with a finite length, convolution processes cannot perform at both ends of the signal since there is no information available outside these boundaries [
]. As a consequence, we need to make an extension at both edges. Border extensions that are frequently used are symmetrization, periodic extension and zero-padding—each of them has its defect,
because of the discontinuities introduced at both ends of the signals [
]. In order to determine the appropriate mother wavelet, the decomposition levels and extension mode, two criteria have been proposed by de Artigas
et al.
(2006) and Nalley
et al.
(2012) [
], and which will be discussed in detail in
Section 3.1.4
The main purpose of this study is to investigate the possible trends and the basic structure of the trends in the mean streamflow and the total precipitation in Huangfuchuan watershed by analyzing
its monthly, seasonal and annual time series through the wavelet transform and different MK tests. The trend analysis through different MK tests was examined at first for the selection of appropriate
trend tests. Then the powerful trend tests were applied to determine the trends in the original data and the ones resulting from DWT. In the process of applying DWT, three criteria were used to
determine the smooth mother wavelet, the decomposition levels, and the extension mode. In addition, a new criterion based on the relative error of energy between the original data and the
approximation component decomposed from DWT was proposed and successfully employed in this study, and the usage of it was discussed in detail in
Section 3.1.4
. Finally, the trend structure in precipitation and streamflow in Huangfuchuan watershed was identified by the wavelet transform and powerful MK tests. Additionally, a new powerful index, the energy
of the hydrological time series, was proposed in this study and used to confirm the dominant periodic components for the observed trends, and is discussed in detail in
Section 4.3
Section 4.4
Section 4.5
2. Study Area and Data
The Yellow River consists of three major reaches, the upper, the middle and the lower reaches [
], and the middle reach of the Yellow River watershed contributes significantly to the total streamflow and sediment discharge of the Yellow River [
] The Huangfuchuan is a primary tributary of the right middle reach of the Yellow River with the length of the main channel 137 km and an average channel slope of 2.7% [
]. The Huangfuchuan watershed (as shown in
Figure 1
) is located at 110.3°~111.2°E and 39.2°~39.9°N, with a catchment area of 3246 km
that is characterized by a semi-arid continental climate. The basin’s average precipitation and mean temperature from 1961 to 2000 were 388 mm and 7.5 °C, respectively [
]. The Huangfu gauging station started in 1954 with 3175 km
of control area, which accounts for 98% of the area of the whole watershed. This area has complex geomorphological types including a feldspathic sandstone hilly-gully region, the loess hilly-gully
region and the sanded loess hilly-gully region [
]. The Huangfuchuan basin is considered to be fairly vulnerable to climate change due to vegetation deterioration, soil erosion and land desertification [
Daily streamflow and precipitation observations from 1954 to 2010 for the Huangfu gauging station of the Huangfuchuan basin were provided by the Yellow River Conservancy Commission (YRCC). A large
amount of work has been conducted by the YRCC to make sure of the quality of the data before they were released [
]. Stations with missing data of less than 3% can be considered as acceptable for hydrological research [
]. However, the Huangfu gauging station has fully complete observations over the chosen period. Therefore, the data used in this study is considered to be good quality. Monthly, seasonal and annual
mean streamflow and total precipitation data (see
Figure 2
) were then collected and investigated in order to research short-term monthly variations (e.g., intra-annual and inter-annual cycles), seasonal cycles and long-term fluctuations such as multi-year,
decadal and multi-decadal events [
Burn and Elnur (2002) [
] considered that the length of the time series of at least 25 years was required in order to obtain an average statistic in assessing the trends of the streamflow. In addition, Partal (2010) [
] thought the length of the time series of 40 years was adequate for trend analysis studies. Kumar
et al.
(2009) [
] suggested that the same length of records should be used when analyzing trends of different variables to avoid misleading conclusions. Therefore, both the streamflow and precipitation of the same
record length of 57 years for the period 1954–2010 would be adequate for the trend analysis in this study.
3. Methodology
3.1. Wavelet Transforms (WTs)
3.1.1. Continuous Wavelet Transform (CWT)
A discrete recording sequence existed in a continuous time series
), and the wavelet function Ψ(η) with a time variable (η) is defined according to the reference [
$Ψ ( η ) = Ψ ( s , γ ) = 1 s Ψ ( t − γ s )$
where η is the non-dimensional parameter,
stands time, γ is the translation factor (time shift) of the wavelet over the time series, and s represents the wavelet scale which ranges from 0 to +∞. When γ = 0 and
= 1, Ψ(
) represents the mother wavelet—all wavelets following this computation are the rescaled versions of the mother wavelet. In order to be acceptable as a wavelet, the function Ψ(η) has to meet the
condition of having a zero mean and be localized in time-frequency space [
]. As can be seen in Equation (1), when
is less than 1, Ψ(η) corresponds to a high-frequency function; when
is greater than 1, Ψ(η) corresponds to a low-frequency function.
The wavelet coefficients
of CWT for the time series
) are computed by using the convolution of
) with the scaled and translated versions of the wavelet, Ψ(η) [
$W Ψ ( s , γ ) = 1 s ∫ − ∞ ∞ x ( t ) Ψ * ( t − γ s ) d t$
where the asterisk symbol stands for the complex conjugate function. If the scale (
) and translation (γ) functions are smoothly changed along with extended time
, a scalogram can be produced from the calculation that indicates the amplitude of a specific scale and how it fluctuates over time [
The Morlet wavelet is widely used in natural time series applications as the basis wavelet function, which is defined as
$Ψ 0 ( η ) = π - 0.25 e i ω 0 η e − 0.5 η 2$
where ω
is non-dimensional frequency and ω
= 6 is used here to satisfy the admissibility condition. The advantage of the Morlet wavelet function provides a conductive definition of the signal in the spectral-space [
3.1.2. Global Wavelet Spectrum
Local spectrum can be measured by a vertical slice through a wavelet plot, the time-averaged wavelet spectrum, also called the global wavelet spectrum (GWS) [
], which is the average of this slice along the time-axis and can be expressed as
$W ¯ 2 ( s ) = 1 T ∑ t = 0 T − 1 | W ( t , s ) | 2$
is the number of points in the time series. The smoothed Fourier spectrum approaches the GWS when the amount of necessary smoothing decreases with the increasing scale. Therefore, an unbiased and
consistent estimation of the true power spectrum are provided by the GWS.
3.1.3. Discrete Wavelet Transform (DWT)
The DWT is normally based on the dyadic calculation of position and scale of a signal [
] and the form of DWT can be written as
$Ψ ( a , b ) ( t − γ s ) = s 0 − a / 2 Ψ ( t − b γ 0 s 0 a 2 a )$
where Ψ represents the mother wavelet;
are integers that control the wavelet dilation (scale) and translation (time), respectively;
is a fixed dilation step whose value is greater than 1; and γ
is the location parameter whose value is greater than zero. In general, for practical reasons, the parameters
and γ
are 2 and 1, respectively [
]. This is the DWT dyadic grid arrangement. Supposing a discrete time series
, where
occurs at a discrete time
, the wavelet coefficient for the DWT becomes
$W Ψ ( a , b ) = 2 − a / 2 ∑ t = 0 N − 1 x t Ψ ( t 2 a − b )$
where the wavelet coefficients
) are computed at scale
= 2
and location γ = 2
which reveal the variation of signals at different scales and locations.
3.1.4. Time Series Decomposition via DWT
The multilevel 1-D wavelet decomposition function in MATLAB was used to accomplish the conventional discrete wavelet analysis of signals on each streamflow and precipitation time series. Since the
trends of hydrologic time series are supposed to be gradual and represent slowly-changing processes, smoother wavelets should be better at detecting long-term time-varying behavior [
]. Therefore, smoother db wavelets (db5–db10) were then tried for each monthly, seasonal and annual dataset.
Based on de Artigas
et al.
(2006) [
], who analyzed monthly geomagnetic activity indices, the maximum decomposition level L is computed as
$L = L o g ( n 2 ν − 1 ) L o g ( 2 )$
where υ represents the number of vanishing moments of a db wavelet and n denotes the number of data points in a monthly time series. In MATLAB, the number of vanishing moments for a db wavelet equals
half of its starting filter length. For instance, db6 in MATLAB represents the Daubechies6 wavelet, which has a 12-point filter length. If someone uses db6 to analyze the signals, the value of υ is
For annual datasets with the length of 57 years, there were 684 data points for the monthly datasets and 228 for the seasonal datasets. Symmetrization, periodic extension and zero-padding were
performed with signal extension in MATLAB, extending the monthly, seasonal and annual data points to 1024, 256, and 256, separately. A calculation based on Equation (7) indicates that the maximum
monthly decomposition level values range from 5.8–6.8, 3.8–6.8 for seasonal level, and 3.8–4.8 for annual level. Since the decomposition levels should be a positive integer, the 6th and 7th
decomposition levels were used for each smooth db wavelet, 4th and 5th decomposition levels for seasonal data and 4th and 5th levels for annual data.
Three criteria were applied to determine the smooth mother wavelet, decomposition levels and the extension mode in the data analysis of each data type and dataset. The first one is the mean relative
error (MRE) that was proposed by de Artigas
et al.
(2006) [
], which was written as
$M R E = 1 n ∑ i = 1 n | a i − x i | | x i |$
represents the original data value of an signal with a record length of
, and
is the approximation value of
. However, this criteria cannot be used in the series that contains zero value, and there are zero values in the datasets that are used in this study. Therefore, the smallest mean absolute error (
) was applied in this study as the first criteria to give a similar evaluation, which was computed as
$M A E = 1 n ∑ j = 1 n | a j − x j |$
The second one is the lowest approximation MK
-value relative error (
) suggested by Nalley
et al.
(2012) [
], which was given as
$e r = | Z a − Z o | | Z o |$
represent the MK
-value of the last approximation for the decomposition level used and the original data, respectively.
The third criterion is the one proposed in this paper which is based on the energy of the series. Supposing a time series
is the length of data record, the total energy of
can be computed as below [
$E = ∑ i = 1 n ( x i ) 2$
Different combinations (of decomposition level, extension mode and mother wavelet) were examined that would generate the lowest approximation energy relative error (
). The computation of the relative error was done using the following equation:
$E r = | E a − E o | E o$
is the total energy of the original series and
is the total energy of the last approximation for the decomposition level used.
3.2. Trend Analysis
3.2.1. The Mann-Kendall (MK) Trend Test
In the test, the null hypothesis
assumes that the deseasonalized data (
) denotes a sample of
independent and randomly ordered variables. The alternative hypothesis
of a two-sided test states that the distribution of
is not identical for all
The Mann-Kendall test statistic
is calculated as:
$S = ∑ i = 1 n − 1 ∑ j = i + 1 n s g n ( x j − x i )$
$sgn ( θ ) = { 1 i f θ > 0 0 i f θ = 0 − 1 i f θ < 0$
For an independent data sample without tied values, the mean and variance of
are given by:
$E [ S ] = 0 V a r ( S ) = n ( n − 1 ) ( 2 n + 5 ) 18$
If tied values are present in the sample,
) is computed by:
$V a r ( S ) = [ n ( n − 1 ) ( 2 n + 5 ) − ∑ i = 1 n t i ( i − 1 ) ( 2 i + 5 ) ] / 18$
Then, the MK test statistic
for all those cases where
is larger than 10 is given by [
$Z = { S − 1 V a r ( S ) S > 0 0 S = 0 S + 1 V a r ( S ) S < 0$
Therefore, the H[0] should be accepted if $| Z | ≤ Z α 2$ at the α level of significance in a two-sided test for trend. If Z > 0, the time series has an upward trend and if Z < 0, then there is a
downward trend. Critical value Z[α/2] at α = 5% significance level of trend test equals ±1.96.
3.2.2. Mann-Kendall Test with Trend-free Pre-whitening (TFPW)
The TFPW procedure is proposed by Yue
et al.
(2002) [
] to detect a significant trend in a time series with significant serial correlation, including the following steps:
• Calculate the lag-1 (
= 1) autocorrelation coefficient (
) by:
$r k = ∑ t = 1 n − k ( x t − x ¯ ) ( x t + k − x ¯ ) ∑ t = 1 n ( x t − x ¯ ) 2$
If ${ − 1 − 1.96 n − 2 } n − 1 ≤ r 1 ≤ { − 1 + 1.96 n − 2 } n − 1$, then the data are considered to be serial-independent at the 5% significant level and it is not necessary to conduct TFPW.
Elsewhere data are assumed to be serial-correlated and TFPW is required.
• The magnitude of trend in sample data is estimated by the Theil-Sen approach (TSA) [
], the TSA slope β is computed as:
$β = m e d i a n [ x j − x i j − i ] for all i < j$
Then the series are detrened by using the following equation:
• Calculate the
of the detrended series by using the Equation (18) and the AR(1) is removed from the detrended series to get a residual series by:
$x i r = x i d − r 1 × x i d$
• The identified trend (β ×
) is added back to the residual series to get a blended series by using the following
Finally, the original MK test is applied to the blended series to assess the significance of the trend.
3.2.3. Modified Mann-Kendall Test by Variance Correction
The variance correction approach supposes that a correlated series is composed of
data in which
* of them are uncorrelated. Hamed and Rao (1998) [
] and Yue and Wang (2004) [
] proposed modified variance
)* for computing the MK statistic
$V ( S ) * = V ( S ) ⋅ n n * = c f ⋅ V ( S )$
$c f 1 = 1 + 2 n ( n − 1 ) ( n − 2 ) ∑ i = 1 n − 1 ( n − i ) ( n − i − 1 ) ( n − i − 2 ) ρ s ( i )$
$c f 2 = 1 + 2 ∑ i = 1 n − i ( 1 − i n ) ρ ( i )$
is variance correction factor according to Hamed and Rao (1998) [
according to Yue and Wang (2004) [
], ρ
) is the lag-
significant autocorrelation coefficient of time series ranks; ρ(
) is the lag-
significant autocorrelation coefficient of time series. The values of ρ
) and ρ(
) must be estimated from the detrended sample data and only significant values are used in Equations (24) and (25) since the insignificant values will have an adverse effect on the accuracy of the
estimated variance of
]. The modified MK test calculated by using the
are referred as to MK1998 and MKDD, respectively.
The expression given by Kendall (1955) [
] in Equation (26) was used to transform rank autocorrelation to normalized data autocorrelation:
$ρ ( i ) = 2 sin ( π 6 ρ s ( i ) )$
Matalas and Langbein (1962) [
] provided a formula for calculating
* for the lag-1 autoregressive process:
$n * = n 1 + 2 ρ 1 n + 1 − n ρ 1 2 + ( n − 1 ) ρ 1 n ( ρ 1 − 1 ) 2$
When there is no trend exists,
= ρ
(Yue and Wang, 2004).
$c f 3 = 1 + 2 ρ 1 n + 1 − n ρ 1 2 + ( n − 1 ) ρ 1 n ( ρ 1 − 1 ) 2$
The value of ρ[1] that is estimated form the detrended sample data and the modified MK test calculated by using the cf[3] is referred as to MKDD1.
3.2.4. Sequential Mann-Kendall Test
The progressive values
) of the Mann-Kendall test were determined in order to see the change of trend with time. Similar to
) is a standardized variable with zero mean, unit standard deviation, and sequentially fluctuating behavior around the zero level [
3.2.5. Determining the Dominant Periodic Components for the Observed Trends
Two indexes, the closeness of the MK
values and the sequential MK graphs (between the individual periodic component and original time series), were employed to determine the most influential periodic component(s) for trends observed in
a hydrological time series [
]. Since the results of the MK trend analysis on periodic (or detailed) component(s) can be better interpreted with their respective approximation added [
], therefore, in this study: (1) the MK value for each of the detailed components with respective approximation was compared to the MK
value of the original time series to see if they are close (even if the values were not significant); (2) the sequential MK graphs of each detailed component with the respective approximation were
examined in comparison with the original time series to see their proximity. If a detailed component (with the approximation added) meets the two requests, this detailed component could be considered
as the most dominant periodic component for the observed trends. In addition to the MK
value and the sequential MK graphs, a new index based on the energy of the hydrological time series was proposed in this study and also used to verify the dominant periodic components for the
observed trends.
4. Results and Discussion
4.1. Mann-Kendall Analysis
Streamflow and precipitation time series (from the beginning of 1954 to the end of 2010) in Huangfuchuan hydrologic station were collected for the analysis of trends.
Table 1
summarized the lag-1 autocorrelation coefficients or the autocorrelation functions (ACFs) and MK
values (using the five methods mentioned earlier) of the flow and precipitation with regards to the monthly, seasonal and annual series. All those three series showed significant downward trends
using different MK trend test methods. Monthly and seasonal total precipitation series exhibited weak upward trends; while annual total precipitation series showed poor downward trends.
It is documented that the presence of positive autocorrelation overestimated the significance of (both positive and negative) trends, while negative autocorrelation underestimated the significance of
(both positive and negative) trends, if the autocorrelation is not considered [
]. As can be seen, all monthly series for both streamflow and precipitation exhibited significant lag-1 positive autocorrelation. The absolute MK
values of monthly mean streamflow based on the modified MK test methods decreased to some degree (6% or more) compared to the original MK, in which TFPW and MKDD1 exhibited less decrease than MK1988
and MKDD. For monthly total precipitation, there is a decrease in the MK Z values of MKDD and MKDD1 compared to the original MK, while there is an increase in TFPW and MK1998 compared to the original
MK. The differences are considered as very small since there are no significant trends found in all these methods. Annual total precipitation showed lag-1 negative autocorrelation. Comparing to the
original MK, the absolute MK
values based on the modified MK test methods increased to some degree, where TFPW and MKDD1 had a relatively smaller increase compared to MK1988 and MKDD. Seasonal and annual mean streamflow series
and seasonal total precipitation series exhibited no significant autocorrelation. The results of different types of data from TFPW and MKDD1 were very similar to the original MK. The results of
annual mean streamflow from MK1998 and MKDD were almost the same as the original MK. However, within seasonal mean streamflow and seasonal total precipitation series, the MK
values from MK1998 and MKDD were significantly different from the original MK.
In general, the four modified Mann-Kendall tests limited the impacts of the autocorrelation on the trend assessment of the time series to a certain extent. However, MK1998 and MKDD had significant
different test results compared to TFPW and MKDD1. This is mainly attributed to the fact that TFPW and MKDD1 only account for lag-1 autocorrelation while MK1988 and MKDD take all significant ρ values
into consideration. The correlograms of monthly, seasonal and annual can be seen in
Figure 3
. As it is shown, those data have significant autocorrelations extending beyond the first lag. Obviously, consideration of only lag-1 autocorrelation (for TFPW and MKDD1) is not sufficient to remove
all significant serial correlation in the data series. It is also found that there is a presence of annual cycles with repeated ACF values at about every 12th lag for monthly series and every 4th lag
for seasonal series. This may explain why the MK Z values of seasonal mean streamflow and seasonal total precipitation series based on MK1998 and MKDD differed significantly from the original MK.
Although a time series exhibited no significant lag-1 autocorrelation, applying the original MK may obtain the wrong results because of the presence of annual cycles. Thus, the modified MK tests
(MK1998 and MKDD) that consider full serial correlation structure should be used even though there is no significant lag-1 autocorrelation due to the annual cycles.
In this study, significant autocorrelations could exhibit for more than just one lag in a lot of time series (see
Section 4.3.1
), therefore, the modified MK tests (MK1998 and MKDD) that consider full serial correlation structure were recommended here. However, based on the later analysis on monthly and annual total
precipitation, the original MK test was also applied due to the limitation of the modified MK test for full serial correlation structure. Therefore, these three MK tests were employed to examine the
trends in the original time series and those resulting from the wavelet decomposition.
4.2. Decomposition via DWT
Three criteria were applied to determine the smooth mother wavelet, the decomposition levels and the extension mode, which were used in the data analysis for each data type and dataset. The results
of monthly streamflow are (see
Table 2
) showing significant differences among the three criteria. The minimum
value (4.577) corresponds to the six decomposition levels, zero-padding extension and db5 wavelet. The minimum
value (0.516) responds to the seven decomposition levels, zero-padding extension and db7 wavelet. The minimum
value (0.792) corresponds to the seven decomposition levels, symmetrization extension and db5 wavelet. It is interesting to find that at the same level of decomposition, the extension modes were
symmetrical extension, periodic extension and zero-padding extension, accordingly, at the same time, the values of
were increasing (
Figure 4
). In other words, the symmetrical extension mode is more suitable than the other two extension modes to expand the hydrological time series according to the
criterion. Further analysis of other types of data also indicated that the symmetrical extension mode is most applicable to broaden hydrological time series, followed by the periodic extension mode
and the zero-padding extension mode, which performs worst based on the
criterion. Similarly, Kharitonenko also argued that the point-symmetric extension method performs better than other methods [
]. In addition, according to the comparison of linear fit of the original series and the approximate component decomposing from different db types in the three extension modes, we found that the ways
of boundary extension were symmetric extension, period extension and zero-padding extension in sequence, and the degree of proximity was overall descending, for example, total annual precipitation (
Figure 5
); similar results were also found in other types of data which are mainly consistent with the conclusions obtained from the criterion of
Further, the selected results by three criteria are given in
Table 3
directly, instead of listing all calculated values of other types of data, due to limited space. As shown, results vary from one criterion to another. Determining the best criterion with higher
accuracy and precision will facilitate the utilization of DWT for the decomposition results which depend on extension mode, decomposition level, and selected wavelet function [
]. All three criteria described the proximity between original time series and the approximate components that were obtained through the decomposition of the original series, therefore this study
made the linear fit of original series and approximate components in order to compare the degree of proximity and evaluate the best optional criterion, as shown in
Figure 6
. Linear-fitting trend lines of approximate components obtained from the criterion that is proposed in this study matched the trend line of the original series to the largest extent, except for the
annual total precipitation.
Overall, the criterion
in this study performed with higher accuracy and precision in comparison to the other two criteria, therefore, the selected results were used in DWT (see
Table 3
4.3. Decomposition and Analysis of Monthly Data
The monthly mean flow and total precipitation time series were decomposed into seven and six lower resolution levels via the DWT approach, respectively. The detail components represent the 2-month
periodicity (D1), 4-month periodicity (D2), 8-month periodicity (D3), 16-month periodicity (D4), 16-month periodicity (D4), 32-month periodicity (D5), 64-month periodicity (D6) and 128-month
periodicity (D7). A6 and A7 represent the approximate components at the sixth and seventh level of decomposition, respectively. The application of the discrete wavelet transform on the monthly flow
and precipitation are shown in
Figure 7
Figure 8
, respectively. The lower detailed levels with higher frequencies represent the rapidly changing component of the series; on the contrary, higher detailed levels with lower frequencies represent the
slowly changing component, such as approximation component A7 in
Figure 7
and A6 in
Figure 8
4.3.1. Monthly Mean Streamflow Series
The results of the trend analysis in the original and the wavelet components of the monthly mean flow series using three MK trend tests are shown in
Table 4
. As can be seen, the correlation coefficients of the detail components, approximations and different combinations are very high, with most of the values greater than 0.6. The modified MK test
proposed by Hirsch and Slack (1984), which accounts for seasonality and serial dependence, was not applied in this study because this method is not powerful when there is long-term persistence (with
autoregressive parameter >0.6) [
]. In addition, significant autocorrelation exhibits more than for one lag in most of the wavelet decomposition components as well as for different combinations, for example, the A7 and each detail
component with A7 decomposing from the monthly streamflow series via DWT (see
Figure 9
). This is why the two modified MK tests (MK1998 and MKDD) that consider full serial correlation structure were employed in this study. As shown in
Table 4
, the MK1998 and MKDD for autocorrelation performed well in most series with high ACFs. However, given the situation that the slope β of the trend approaches zero, such as for D2, D3 and D4, the two
modified MK tests are not robust even though the existing trend can be approximated by linear trend. Since this situation mainly occurs in the individual detail component and it has little influence
on each detail component with the approximation, it is not considered and explored in later trend analysis. In addition, the original MK test performs worse when the ACF is high, as expected.
However, the original MK test is still powerful in the case that the slope β of the trend approaches zero even though the ACFs of the series are extremely high (
> 0.9). This could explain why the original MK test was applied in this study.
Table 4
, the most effective periodic components vary from one MK test to another. The original MK, MK1998 and MKDD tests indicate that D2 and D3; D7; D3 are responsible for the real trend in the monthly
mean streamflow, respectively. Based on further analysis of different details of D1–D6 and the combination of D7 + A7, all three MK tests indicated that the MK
value approached the MK
value of the original series only under the combination of D7 + A7 and D3, which means the D7 is not the dominant periodic component for the trend. It is interesting to find that the total energy of
the detail components (D1–D7) and the approximation (A7) approach the energy of the original series; and the highest energy of the periodic components is D3, as presented in
Table 4
. It also can be been that the change of
was basically the same with the change of energy in detail component combinations. D3 with A5, which has the highest energy, also has the highest
(0.645). In addition, two sequential MK (the original MK and MKDD) graphs of the different periodic components with approximations corresponding to the original series of the monthly mean streamflow
are shown in
Figure 10
, in which the trend line of D3 with A7 is most similar to the trend line of the original series. This evidence proves that D3 is the dominant periodic component for the observed trend in the monthly
mean streamflow.
4.3.2. Monthly Total Precipitation
As shown in
Table 5
, all of the original MK, MK1998 and MKDD tests suggest that D3 is the dominant periodic component in affecting the trend of the monthly total precipitation series in
Table 5
. The DW3 has the highest energy and
, which corresponds with the results of the monthly mean streamflow. Three sequential MK graphs of the different periodic components with respect to the original series of the monthly total
precipitation are shown in
Figure 11
, in which the trend line of D3 with A6 is most similar to the trend line of the original series. It makes sense that D3 is the most effective periodic component for the real trend seen in the
monthly total precipitation. It is clear that the dominant periodic components in the monthly precipitation and streamflow are consistent. In addition, MK1998 and MKDD tests were not applicable in
the trend examination of D5 due to the negative values in the calculations of the correction factor
which can result in incorrect results. However, detailed are not explored in this study due to the relatively lesser impact on our results.
4.4. Decomposition and Analysis of Seasonal Data
The seasonal mean streamflow and total precipitation series were both decomposed into four detail components and one approximation. D1, D2, D3 and D4 represent the 6-month, 12-month, 24-month and
48-month fluctuations, respectively. The A4 represents the approximation components at the fourth level of decomposition. The D2 component in the seasonal series decomposition represents the annual
(12-month) periodicity which is very useful in confirming whether or not the annual cycles can explain the trends found in the flow and the precipitation series.
4.4.1. Seasonal Mean Streamflow Series
Table 6
, the original MK, MK1998 and MKDD tests indicate that D1 and D2; D1; D1 are the dominant periodic components in influencing the real trend in the seasonal mean streamflow series, respectively. As
shown in
Table 6
, the DW1 has the highest energy and C
in the periodic components. Three sequential MK graphs of the seasonal mean streamflow are presented in
Figure 12
, and the trend line of D1 with A4 is most harmonious with the trend line of the original series. Obviously, D1 is the most effective periodic component for the real trend observed in the seasonal
streamflow series.
4.4.2. Seasonal Total Precipitation Series
As shown in
Table 7
, the original MK, MK1998 and MKDD tests indicate that D1 and D2; D1; D1 are the most influential components in affecting the real trend in the seasonal mean streamflow series, respectively. The
sequential MK analysis of the seasonal total precipitation is shown in
Figure 13
, the trend line of D1 with A4 is most similar to the trend line of the original series. The original sequential MK graph also argues that D2 is the most influential periodic component for the trend.
As can be seen in
Table 7
, the energy of D1 and D2 accounted for 17% and 33% of the total energy of the original sequence, respectively, which accounted for most of the total energy of details (92%), and the higher
and energy indicate that D1 and D2 are probably the dominant periodicities. Further analysis was conducted and results showed that the energy of D1 + D2 + A4 accounted for 96% of the energy of the
original series while the correlation coefficient
of D1 + D2 + A4 and the corresponding original series was up to 0.961 (
Table 7
). The results from all three MK tests show that the MK values of the D1 + D2 + A4 and the MK values of the original series are very close to each other. Apparently, D1 and D2 are the dominant
periodic component for the trend of the seasonal total precipitation. It is important to say that the energy of the detail component combinations is a very important index to indicate the most
effective components. In addition, D2, which represented the annual (12-month) periodicity in the seasonal time and was considered as the most dominant periodic component, t indicating that annual
cycles can explain the trends found in streamflow.
4.5. Decomposition and Analysis of Annual Aata
In order to obtain a more thorough trend analysis, the annual mean streamflow and total precipitation time series were decomposed into four and five levels, respectively, which correspond to 2-year,
4-year, 8-year and 16-year variations. The continuous wavelet transform (CWT) and the global wavelet spectra (GWS) were also employed to analyze the annual data so as to explain its time-frequency
characteristics, perfectly.
4.5.1. Periodicities of Annual Streamflow and Precipitation Data
The CWT scalograms and the global wavelet spectra of the CWTs are shown in
Figure 14
Figure 15
to illustrate the general periodic structure of the streamflow and precipitation time series. Light regions on the scalogram plot and peaks in the GWS figure indicate the effective periodic events.
Figure 14
indicates that an obvious periodic event occurred for the annual streamflow with decreased trends intensity as periods ranged from 7–14 years, 17–24 years and 4–5 years, separately. The major
periodicities can be cataloged to decadal events, including the intense 10- and 10-year scale events with peaks of GSW figures. Periodic events do not possess an overall property, with its continuity
ending in the late 1990s, which can be explained by intensive human activities that probably resulted in a significant decrease and cutoff of streamflow [
]. Moreover,
Figure 15
illustrated a decreased intensity of obvious periodic events for the annual precipitation as periods ranged from 25–30 years, 11–13 years, 4–5 years, 7–10 and 17–24 years, respectively. Similar to
annual mean flow, the major periodicities of total precipitation are defined as decadal events, including the intense 28-year scale oscillation that is continues over time.
4.5.2. Annual Mean Streamflow Series
As shown in
Table 8
, the D1; D1,D3 and D4; D1 and D4 are the dominant periodic components for the real trend in the annual mean streamflow suggested by the original MK, MK1998 and MKDD test, respectively. D1 had the
as well as the highest energy which implied that D1 might be the most influential periodic component for the trend. The sequential MK analysis of the annual mean streamflow is exhibited in
Figure 16
, the trend line of D1 with A4 approaches to the trend line of the original series, suggested by three MK tests. The sequential MK graphs tests also indicate that D3 (MK1998) and D4 (MK1998 and MKDD)
could be the most effective periodic component for the trend found in the annual mean streamflow series. The reason why the original sequential MK test does not suggest that D4 is the dominant
periodicity is that D4 with A4 has a very high autocorrelation coefficient (see
Table 8
) (the presence of the positive autocorrelation will overestimate the significance of trends). In addition, it is vital to note that decadal events are the major periodicities found in the CWT and
GWS, which makes sense that D3 and D4 represent the decadal events found in the CWT figure and are the most effective periodic components. Therefore, D1, D3 and D4 are the most dominant periodic
components for the trend in the annual mean streamflow.
4.5.3. Annual Total Precipitation Series
Based on previous trend analyses, MK1998 and MKDD effectively limited the influence of autocorrelation series trend analyses. However, in the analysis of D1, D2, D1 with A5, and D2 with A5, MK1998
and MKDD tests were still not applicable to trend analyses because there were negative values in the calculations of the correction factor
Table 9
). Further studies are needed to improve the modified MK for complete autocorrelation structure on this issue, but this is not further explored in this study.
To avoid the compatibility issue that results from MK1998 and MKDD in the trend analysis of D1 with A5 and D2 with D5, which may ignore that D1 and D2 are the possible dominant periodic component for
the trend found in the annual total precipitation, the MKDD1 and TFPW tests were used to conduct trend examinations. As displayed in
Table 9
, the results of the MKDD1 and TFPW indicate that D1 with A5 and D2 with A5 are not the dominant periodic components. As discussed above, the MKDD1 and TFPW tests are not overall powerful, therefore,
they are not applied, to make sure which detail component is the most effective periodic component for the observed trend.
The D4; D4 and D5; D4 and D5 are the dominant periodic components for the real trend of the annual mean streamflow suggested by the original MK, MK1998, and MKDD, separately. The sequential MK
analysis of the annual total precipitation is presented in
Figure 17
, the trend line of D4 with A5 does not match well with the trend line of the original series. D4 has the lowest
, which also indicates that the difference of D4 with A5 and the original series is high. Therefore, D4 is not considered to be the most effective component in influencing the trend. This also
illustrates that when analyzing the dominant periodic component for the trend, both MK values and their sequential MK graph should be taken into consideration. In addition, MK1998 and MKDD also
indicate that D5 is the most effective periodic component for the trend found in the annual total precipitation series. It is worth mentioning that the intense 28-year scale oscillation which is
defined as a decadal event is found in the CWT and GWS, which figures out that D5 represents the decadal events found in the CWT figure and is the most influential periodic component. According to
previous analysis, the combination of the periodic component and respective approximation with higher energy has more significant impacts on the trend of the observed series. From the results of
trend assessment, D5 is the most effective periodic component for the observed trend, but it has the lowest energy. This might be due to the fact that the energy of A5 accounts for 92% of the energy
of the annual total precipitation series while the energy of the detail components only account for about 8% of the total energy. There is no big energy difference among each detail component with
A5. In this case, energy is no longer an important index to evaluate the most effective periodic component for the observed trends. To sum up, D5 is the most dominant periodic component for the trend
in the total precipitation streamflow.
4.6. Factors Related to Precipitation and Streamflow Variations
Previous trend analysis indicated that there are no significant trends in precipitation, while strong decreasing trends were found in streamflow. Precipitation variations have been closely associated
with the increasing global temperature and El Niño/Southern Oscillation (ENSO) [
]. Baddoo
et al.
(2015) [
] proved that ENSO events influenced precipitation in Huangfuchuan basin, with El Niño corresponding to precipitation decline and La Niña to precipitation increment, with a semiannual to annual lag.
Global temperature has been increasing since the 1980s and the climate aridity in Huangfuchuan basin has increased in the recent three decades [
]. It is found that the combination of climate change and human activities has resulted in significant decreasing streamflow, and intensive anthropogenic activities in the upper and middle reaches of
the Yellow River basin, such as reservoirs, agricultural irrigation and soil and water conservation measures, account for more of this reduction [
]. Zhou
et al.
] found that the primary cause of streamflow reduction in Huangfuchuan was attributed to water diversion for irrigation in 1979–1998, and soil conversation measures in 1999–2006.
For the monthly, seasonal and annual mean streamflow series analysis, the common periodic components that were found to be the most influential for the observed trends are 8 months, 6 months and 2
years, as well as 8–16 years, separately. For the monthly, seasonal and annual total precipitation series analysis, the common periodic components seen as the most influential for the observed trends
are 8 months, 6–12 months and 32 years, respectively. As can be seen, the different data types produced pretty different conclusions in terms of the most dominant periodicities for observed trends.
The leading factors that impact the trends in the monthly and seasonal series in the Huangfuchuan watershed are intra-annual cycles (6–12 months), which may be associated with strong seasonal and
annual cycles in the data. Studies indicated that shorter and discontinuous periodicities found in streamflow are likely influenced by human activities [
]. Therefore, the intra-annual periodic modes are also linked with anthropogenic activities. Inter-annual (2 and 8 years) and decadal (16 and 32 years) periodicities can be seen as the most
influential components for the observed trends in annual series, and decadal events are the major periodicities. Baddoo
et al.
(2015) found that the correlation between precipitation in Huangfuchuan and ENSO events is at the 2–7 year periodicities. It is suggested that the effect of ENSO on precipitation will in turn affect
the streamflow activities [
]. Li and Yang (2005) [
] found a correlation between precipitation and solar activities in the Yellow River basin at 9 and 11 years. The combination effects of solar activities and ENSO are found at 18–32 years [
]. Here, inter-annual periodicities are likely related to 2–7 year ENSO events and decadal periodic modes may be correlated to the combined effects of solar activities and ENSO cycles. Overall,
multiple factors (e.g., ENSO, solar activities) are influencing the periodicities identified in precipitation and streamflow over the Huangfuchuan basin.
5. Conclusions and Recommendations
The wavelet transform and different MK tests were employed to investigate the possible trends and the basic structure of the trends in the mean streamflow and total precipitation in Huangfuchuan
watershed. A comparative analysis of five different MK methods, including the original MK test and the modified MK tests with lag-one and full serial correlation, showed that consideration of only
lag-1 autocorrelation (for TFPW and MKDD1) is not sufficient to remove all significant serial correlation in the data series. The results of the trend analysis also indicated the significant downward
trends in all monthly, seasonal and annual mean streamflow series, but weak upward trends in monthly and seasonal total precipitation series and poor negative trends in the annual total precipitation
series. Precipitation variation in Huangfuchuan basin has been closely linked to ENSO events [
]. The combined effects of human activities and climate change account for the significant decreasing streamflow, however, intensive anthropogenic activities are the major factors [
The modified MK tests (MK1998 and MKDD) that consider full serial correlation structure performed better than the original MK and the modified test for lag-1 autocorrelation (TFPW and MKDD1), because
significant autocorrelations exhibit for more than just one lag in most of the wavelet decomposition components as well as for different combinations. But in the case that the slope β of the trend
approaches zero, the two modified MK tests are not robust even though the existing trend can be approximated by linear trend. The original MK test performs worse when the ACF is high, as expected.
However, the original MK test is still powerful in the case that the slope β of the trend approaches zero even though the ACFs of the series are extremely high (r > 0.9). Additionally, MK1998 and
MKDD tests are not applicable to the analysis of monthly and annual total precipitation series, and this can be attributed to negative values of the correction factor cf. Further discussion is not
explored here, since it had little influence on this study. But future studies are needed to improve the modified MK, with complete autocorrelation structure on this issue. In summary, this study
suggests that the original MK test and the modified MK test for full serial correlation structure should be applied together to better analyze the trend in the hydrological time series and the
wavelet decomposition components obtained from DWT.
Three criteria were explored to determine the most appropriate smooth mother wavelet, the decomposition levels and the extension mode in the DWT procedure. The result revealed that the criterion Er
based on the relative error of energy that is proposed in this study performed better in comparison to the other two criteria, MAE and er. Additionally, the usage of Er is very convenient and is not
influenced by the method itself compared to er (the effect of er is influenced by the MK test).
In addition to the MK Z values and the sequential MK graphs, a new and powerful index, the energy of the hydrological time series, was proposed and successfully utilized in this study to confirm the
dominant periodic components for the trends. Furthermore, this index is easy to apply and has few limits. However, it is important to note that if there is no significant energy difference among
different detail components with respective approximation, energy is no longer the key index to indicate which detail components are the most effective periodic component for the observed trend.
Overall, from the energy of the hydrological time series point of view, this not only provides a robust index to determine which periodic component is the dominant periodic component for the trend,
but also explores a new way of analyzing the hydrologic time series.
Intra-annual periodicities (6–12 months) were found to be the most influential components in producing the trends in monthly and seasonal series, which may be related to strong seasonal and annual
cycles in monthly and seasonal data and human activities [
]. In the CWT and GWS of the annual series, the periodicities located between 2- and 4-year scales are seen, but the major periodicities are decadal events. Inter-annual (2, 4 and 8 years) and
decadal (16 and 32 years) periodicities found in annual series are likely associated with 2–7-year ENSO cycles and the combination effects of 11-year solar activities and 2–7-year ENSO events,
respectively [
]. Additionally, although streamflow in the Huangfuchuan basin has been greatly influenced by human activities, the long-term fluctuations (decadal events) exhibited in annual streamflow are still
evident. This indicates that the long-term fluctuations in streamflow are more influenced by climate variabilities (e.g., ENSO cycles, solar activities). However, the combined effects of ENSO events
and solar activities in streamflow have not been extensively explored to date, especially in the yellow River basin. Thus, further studies could incorporate some linkages between the streamflow and
combined climatic variabilities. The above findings will contribute to not only analyzing the trends in the hydrological time series, but also water resource planning and management in the semi-arid
or arid river watersheds of China.
The funding support for this research paper was provided by the National Natural Science Foundation of China (NSFC51209064 and NSFC51579067).
Author Contributions
Yiqing Guan and Danrong Zhang were primarily accountable for data collection and design and coordination of the study. Yuzhuang Chen and Guangwen Shao were responsible for data analysis,
interpretation and writing of the paper.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. Monthly, seasonal and annual streamflow and precipitation plots of the study used. (a) annual streamflow; (b) annual precipitation; (c) seasonal streamflow; (d) seasonal precipitation; (e)
monthly streamflow; (f) monthly precipitation.
Figure 3. The correlograms of monthly, seasonal and annual series. The upper and lower solid lines represents the confidence intervals (95% confidence level). (a) monthly streamflow; (b) monthly
precipitation; (c) seasonal streamflow; (d) seasonal precipitation; (e) annual streamflow; (f) annual precipitation.
Figure 4. Er values of monthly mean streamflow series with different extension mode in six and seven levels (sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension). (a)
L = 6; (b) L = 7.
Figure 5. Linear fit of original series and approximate components of annual total precipitation series in three extension modes (sym: symmetrization extension; per: periodic extension; zpa:
zero-padding extension) with the decomposition level L = 4. The dash lines are the linear fits of the approximate components decomposing from different db types. (a) sym; (b) per; (c) zpa.
Figure 6. Linear fit of original series and approximate components of monthly, seasonal and annual streamflow and precipitation series in the three criteria. (a) monthly streamflow; (b) monthly
precipitation; (c) seasonal streamflow; (d) seasonal precipitation; (e) annual streamflow; (f) annual precipitation.
Figure 7. Original monthly mean streamflow series and its approximation (A7) and detail components (D1–D7) decomposed via DWT. (a) original data; (b) A7; (c) D1; (d) D2; (e) D3; (f) D4; (g) D5; (h)
D6; (i) D7.
Figure 8. Original monthly precipitation series and its approximation (A6) and detail components (D1–D6) decomposed via DWT. (a) original data; (b) A6; (c) D1; (d) D2; (e) D3; (f) D4; (g) D5; (h) D6.
Figure 9. The correlograms of A7 and each detail component with A7 decomposing from the monthly streamflow series via DWT. The upper and lower solid lines represent the confidence intervals (95%
confidence level). (a) A7; (b) A7 + D1; (c) A7 + D2; (d) A7 + D3; (e) A7 + D4; (f) A7 + D5; (g) A7 + D6; (h) A7 + D7.
Figure 10. Two sequential Mann-Kendall (MK and MKDD) graphs of monthly streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with
respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MKDD.
Figure 11. Three sequential Mann-Kendall graphs of monthly total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with
respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MK1998; (c) MKDD.
Figure 12. Three sequential Mann-Kendall graphs of seasonal mean streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with
respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MK1998; (c) MKDD.
Figure 13. Three sequential Mann-Kendall graphs of seasonal total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with
respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MK1998; (c) MKDD.
Figure 14. Continuous wavelet spectrum (a) and global wavelet spectrum (b) of the annual mean streamflow series.
Figure 15. Continuous wavelet spectrum (a) and global wavelet spectrum (b) of the annual total precipitation series.
Figure 16. Three sequential Mann-Kendall graphs of annual mean streamflow series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with respect
to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MK1998; (c) MKDD.
Figure 17. Three sequential Mann-Kendall graphs of annual total precipitation series exhibiting the progressive trend lines of each detail component (with the addition of the approximation) with
respect to the original series. The upper and lower dashed lines represent the confidence limits (α = 5%). (a) MK; (b) MK1998; (c) MKDD.
Table 1. Lag-1 autocorrelation functions (ACFs) and MK values (using different MK methods) of the original monthly, seasonal, and annual streamflow and precipitation series (MS: monthly streamflow;
SS: seasonal streamflow; AS: annual streamflow; MP: monthly precipitation; SP: seasonal precipitation; AP: annual precipitation).
Table 1. Lag-1 autocorrelation functions (ACFs) and
MK values (using different MK methods) of the
original monthly, seasonal, and annual streamflow
and precipitation series (MS: monthly streamflow;
SS: seasonal streamflow; AS: annual streamflow; MP:
monthly precipitation; SP: seasonal precipitation;
AP: annual precipitation).
Type ACF MK TFPW MK1998 MKDD MKDD1
MS 0.399 * −13.179 −12.410 −5.503 −4.025 −8.779
SS 0.065 −7.624 −7.764 −4.055 −3.188 −7.578
AS 0.166 −5.239 −5.376 −5.239 −5.239 −6.312
MP 0.482 * 0.202 0.380 0.228 0.193 0.120
SP −0.051 0.168 0.107 0.086 0.067 0.176
AP −0.266 * −0.902 −0.984 −1.286 −1.344 −1.196
Notes: * indicates significant lag-1 serial correlation at α = 5%.; significant trend (95% confidence level) results are shown in bold letters.
Table 2. MAE, er, and Er of monthly mean flow series (sym: symmetrization extension; per: periodic extension; zpa: zero-padding extension).
Table 2. MAE, er, and Er of monthly mean flow series (sym: symmetrization
extension; per: periodic extension; zpa: zero-padding extension).
Extension Mode Decomposition Levels Criterion db5 db6 db7 db8 db9 db10
MAE 4.637 4.658 4.627 4.632 4.643 4.638
L = 6 er 1.772 1.613 1.723 2.005 1.643 1.783
sym Er 0.798 0.803 0.805 0.799 0.8 0.805
MAE 4.667 4.68 4.642 4.634 4.66 4.666
L = 7 er 1.425 0.863 1.295 1.687 0.990 0.616
Er 0.792 0.798 0.812 0.805 0.795 0.802
MAE 4.672 4.679 4.697 4.699 4.677 4.684
L = 6 er 2.287 2.732 3.252 2.628 2.297 1.759
per Er 0.811 0.817 0.813 0.808 0.812 0.816
MAE 4.799 4.814 4.736 4.736 4.792 4.806
L = 7 er 1.352 1.793 1.142 1.142 1.787 1.505
Er 0.82 0.811 0.829 0.829 0.82 0.813
MAE 4.577 4.6 4.606 4.591 4.587 4.604
L = 6 er 1.907 1.131 1.469 1.656 1.715 1.341
zpa Er 0.816 0.819 0.817 0.813 0.816 0.819
MAE 4.577 4.606 4.616 4.595 4.581 4.604
L = 7 er 0.922 0.747 0.516 0.583 1.066 0.624
Er 0.828 0.823 0.829 0.834 0.829 0.824
Notes: The minimum MAE, er, and Er are indicated in bold format.
Table 3. The results of extension mode, decomposition levels and db type that were used in DWT of monthly, seasonal and annual flow and precipitation series in the three criteria (MS: monthly
streamflow; SS: seasonal streamflow; AS: annual streamflow; MP: monthly precipitation; SP: seasonal precipitation; AP: annual precipitation; sym: symmetrization extension; per: periodic extension;
zpa: zero-padding extension).
Table 3. The results of extension mode, decomposition levels and
db type that were used in DWT of monthly, seasonal and annual
flow and precipitation series in the three criteria (MS: monthly
streamflow; SS: seasonal streamflow; AS: annual streamflow; MP:
monthly precipitation; SP: seasonal precipitation; AP: annual
precipitation; sym: symmetrization extension; per: periodic
extension; zpa: zero-padding extension).
Criterion Data Type Extension Mode Decomposition Levels db Type
MS zpa 6 db5
MP zpa 7 db5
MAE SS zpa 4 db7
SP zpa 5 db7
AS zpa 4 db6
AP sym 4 db5
MS zpa 7 db7
MP per 6 db6
er SS sym 5 db10
SP per 4 db10
AS zpa 4 db7
AP per 4 db10
MS sym 7 db5
MP per 6 db8
Er SS sym 4 db5
SP sym 4 db9
AS sym 4 db9
AP per 5 db8
Table 4. Slope β (computed by TSA), Lag-1 ACFs, Mann-Kendall values (three MK tests) and energy of monthly mean streamflow series: original data, detail components (D1–D7), approximations (A7) and a
set of combinations of the details and their respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 4. Slope β (computed by TSA), Lag-1 ACFs, Mann-Kendall values
(three MK tests) and energy of monthly mean streamflow series:
original data, detail components (D1–D7), approximations (A7) and a
set of combinations of the details and their respective
approximations. C[0] is the correlation coefficients between the
decomposition combinations and the original series.
Series Slope (β) ACF MK MK1998 MKDD C[0] Energy
Original −0.0027 0.399 −13.179 * −5.503 * −4.025 * – 69,705
A7 −0.0103 0.995 −34.542 * −5.855 * −9.761 * 0.212 14,476
D1 −0.0003 −0.437 −0.647 −0.849 −0.942 0.556 18,159
D2 0 0.338 −0.066 −0.171 −0.122 0.385 8783
D3 0.0003 0.83 0.366 0.444 0.337 0.611 22,011
D4 0.0003 0.932 0.614 1.485 3.356 * 0.252 4569
D5 −0.0003 0.98 −1.287 −1.016 −0.806 0.171 1895
D6 −0.0002 0.995 −0.797 −0.579 −0.911 0.126 985
D7 0.0006 0.999 3.682 * 1.307 2.145 * 0.023 348
D1 + A7 −0.0102 −0.227 −16.555 * −15.260 * −20.989 * 0.595 32,641
D2 + A7 −0.0103 0.512 −15.830 * −12.121 * −15.987 * 0.438 23,381
D3 + A7 −0.0098 0.852 −10.262 * −9.374 * −7.963 * 0.645 36,696
D4 + A7 −0.0098 0.964 −18.084 * −12.563 * −16.646 * 0.333 18,692
D5 + A7 −0.0101 0.984 −23.363 * −10.236 * −15.233 * 0.275 16,204
D6 + A7 −0.0098 0.993 −25.500 * −8.628 * −11.843 * 0.244 15,522
Notes: * Indicates significant trend values at α = 5%; the most effective periodic components for trends are indicated in bold format.
Table 5. Lag-one ACFs, Mann-Kendall values (three MK tests) and energy of monthly total precipitation series: original data, details components (D1–D6), approximations (A6) and a set of combinations
of the details and their respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 5. Lag-one ACFs, Mann-Kendall values (three MK
tests) and energy of monthly total precipitation series:
original data, details components (D1–D6), approximations
(A6) and a set of combinations of the details and their
respective approximations. C[0] is the correlation
coefficients between the decomposition combinations and
the original series.
Series ACF MK MK1998 MKDD C[0] Energy
Original 0.482 0.202 0.228 0.193 - 2,438,503
A6 0.999 −9.241 * −1.510 −2.187 * 0.071 828,972
D1 −0.626 0.317 0.692 0.526 0.434 305,735
D2 0.369 −0.168 −0.270 −0.190 0.455 336,637
D3 0.852 0.325 0.385 0.313 0.716 840,561
D4 0.954 −0.047 −0.084 −0.269 0.25 98,838
D5 0.987 −1.789 enable enable 0.13 25,058
D6 0.997 0.682 0.465 0.696 0.08 12,201
D1 + A6 −0.581 −1.748 −1.841 −2.013 * 0.44 1,134,729
D2 + A6 0.385 −1.042 −0.938 −0.986 0.461 1,168,091
D3 + A6 0.853 −0.336 −0.337 −0.309 0.72 1,663,805
D4 + A6 0.958 −1.512 −0.899 −1.581 0.26 928,184
D5 + A6 0.989 −5.125 * −1.716 −2.319 * 0.148 854,880
D6 + A6 0.997 −4.703 * −1.178 −1.892 0.108 839,865
Notes: * Indicates significant trend values at α = 5%; the most effective periotic components for trends are indicated in bold format.
Table 6. Lag-1 ACFs, Mann-Kendall values and energy of seasonal mean streamflow series: original data, details components (D1–D4), approximations (A4) and a set of combinations of the details and
their respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 6. Lag-1 ACFs, Mann-Kendall values and energy of
seasonal mean streamflow series: original data, details
components (D1–D4), approximations (A4) and a set of
combinations of the details and their respective
approximations. C[0] is the correlation coefficients
between the decomposition combinations and the original
Data ACF MK MK1998 MKDD C[0] Energy
Original 0.065 −7.624 * −4.055 * −3.188 * - 14,100
A4 0.969 −17.946 * −8.343 * −16.597 * 0.313 4878
D1 −0.359 1.5 0.772 0.571 0.765 6040
D2 0.205 0.56 0.993 0.677 0.449 2497
D3 0.745 0.044 0.1 0.059 0.246 491
D4 0.952 1.557 1.57 1.886 0.191 477
D1 + A4 −0.131 −6.562 * −3.633 * −2.651 * 0.822 11,032
D2 + A4 0.464 −8.928 * −8.132 * −7.344 * 0.552 7293
D3 + A4 0.869 −13.844 * −7.722 * −8.074 * 0.39 5462
D4 + A4 0.947 −13.443 * −9.512 * −12.182 * 0.393 5036
Notes: * Indicates significant trend values at α = 5%; the most effective periotic components for trends are indicated in bold format.
Table 7. Lag-one ACFs, Mann-Kendall values and energy of seasonal total precipitation series: original data, details components (D1–D4), approximations (A4) and a set of combinations of the details
and their respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 7. Lag-one ACFs, Mann-Kendall values and energy of
seasonal total precipitation series: original data, details
components (D1–D4), approximations (A4) and a set of
combinations of the details and their respective
approximations. C[0] is the correlation coefficients between
the decomposition combinations and the original series.
Data ACF MK MK1998 MKDD C[0] Energy
Original −0.051 0.168 0.086 0.067 - 600,926
A4 0.988 −5.513 * −1.902 −3.115 * 0.11 277,514
D1 −0.507 −0.072 −0.050 −0.033 0.556 100,382
D2 0.055 −0.143 −1.107 −0.955 0.783 196,786
D3 0.791 −0.129 −0.139 −0.116 0.243 20,315
D4 0.948 −0.446 −0.479 −0.542 0.116 4836
D1 + A4 −0.445 −1.106 −0.727 −0.501 0.567 376,755
D2 + A4 0.075 −0.920 −2.745 * −4.076 * 0.79 475,719
D3 + A4 0.819 −1.865 −1.131 −1.171 0.268 297,078
D4 + A4 0.959 −3.744 * −1.779 −2.304 * 0.161 281,563
D1 + D2 + A4 −0.126 0.05 0.025 0.019 0.961 578,056
Notes: * Indicates significant trend values at α = 5%; the most effective periotic components for trends are indicated in bold format.
Table 8. Lag-one ACFs, Mann-Kendall values and energy of annual mean streamflow series: original data, detail components (D1–D4), approximations (A4) and a set of combinations of the details and
their respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 8. Lag-one ACFs, Mann-Kendall values and energy of
annual mean streamflow series: original data, detail
components (D1–D4), approximations (A4) and a set of
combinations of the details and their respective
approximations. C[0] is the correlation coefficients
between the decomposition combinations and the original
Data ACF MK MK1998 MKDD C[0] Energy
Original 0.166 −5.239 * −5.239 * −5.239* - 1528
A4 0.948 −9.823 * −5.058 * −8.497* 0.6 1184
D1 −0.618 0.145 0.208 0.151 0.562 220
D2 0.322 0.048 0.101 0.088 0.45 149
D3 0.818 −0.805 −0.882 −1.295 0.252 18
D4 0.903 0.296 0.212 0.215 0.049 18
D1 + A4 0.18 −5.831 * −5.609 * −4.645 * 0.827 1390
D2 + A4 0.653 −6.877 * −8.021 * −7.182 * 0.753 1328
D3 + A4 0.937 −8.791 * −5.595 * −10.864 * 0.629 1229
D4 + A4 0.915 −10.925 * −5.980 * −5.717 * 0.631 1156
Notes: * Indicates significant trend values at α = 5%; the most effective periotic components for trends are indicated in bold format.
Table 9. Lag-one ACFs, Mann-Kendall values and energy of annual total precipitation: original data, detail components (D1–D5), approximations (A5) and a set of combinations of the details and their
respective approximations. C[0] is the correlation coefficients between the decomposition combinations and the original series.
Table 9. Lag-one ACFs, Mann-Kendall values and energy of annual total
precipitation: original data, detail components (D1–D5), approximations
(A5) and a set of combinations of the details and their respective
approximations. C[0] is the correlation coefficients between the
decomposition combinations and the original series.
Data ACF MK MK1998 MKDD MKDD1 TFPW C[0] Energy
Original −0.266 −0.902 −1.286 −1.344 −1.196 −0.984 - 10,820,675
A5 0.921 −2.265 −0.817 −0.890 −0.542 −10.994 * −0.182 9,977,842
D1 −0.691 −0.186 enable enable −0.426 −0.434 0.807 616,764
D2 0.27 −0.103 enable enable −0.079 −0.379 0.493 208,741
D3 0.796 0.062 0.092 0.105 0.022 −0.213 0.292 87,441
D4 0.938 −0.778 −0.529 −0.836 −0.162 −2.072 0.201 17,038
D5 0.93 −4.619 −1.636 −1.530 −1.163 −10.994 * 0.228 38,918
D1+A5 −0.688 −0.227 enable enable −0.518 0.213 0.8 10,628,434
D2+A5 0.276 −0.172 enable enable −0.130 −0.516 0.479 10,148,639
D3+A5 0.793 −0.062 −0.089 −0.101 −0.022 −1.095 0.272 10,078,044
D4+A5 0.944 −0.750 −0.632 −1.450 −0.152 −2.141 * 0.174 10,095,310
D5+A5 0.938 −5.032 * −1.834 −1.723 −1.299 −10.994 * 0.222 9,718,913
Notes: * Indicates significant trend values at α = 5%; the most effective periotic components for trends are indicated in bold format.
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peak finder algorithm
– • … is always challenging – More than a dozen algorithms have been published, If it’s not, then you’re going the other direction. Consider mid column and find maximum element in it. •Total
time ? In cases wherein manual peak integration is required to distinguish and detect the shoul-der and main peaks using traditional peak integration methods, i-Peak-Finder can automatically detect
shoulder peaks while maintaining consistent peak detection sensitivity throughout the entire chromatogram. Article PDF Available. Given an input array nums, where nums[i] ≠ nums[i+1], find a peak
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a signal based on peak properties. There might be multiple peak element in a array, we need to find any peak element. The content that I am using here to write this series is from MIT 6.006
Introduction to Algorithms, Fall 2011. And the algorithm will return 14 as a peak of the matrix. For example, 50 is peak element in {10, 20, 30, 40, 50}. We will see the recursion techniques to solve
this problem. So we take the above equation and expand it eventually we will get to the best case which is T(1) = Θ(1). We also concern about Scalability because back in the day’s large input was in
thousands, today it is in trillions it’s just a matter of time we call 10 to the power 18 fairly a large input. When you have a single column, find global maximum and you‘re done, Images used in the
blog are the screenshots of the Notes from MIT 6.006. The function performs a quadratic curve fitting to find the peaks and valleys. For example: In Array [1,4,3,6,7,5] 4 and 7 are Peak Elements. Let
index of mid column be ‘mid’, value of maximum element in mid column be ‘max’ and maximum element be at ‘mat[max_index][mid]’. We use cookies to ensure you have the best browsing experience on our
website. Algorithm to find peaks in a std::vector
MIT License 32 stars 4 forks Star Watch Code; Issues 2; Pull requests 1; Actions; Projects 0; Security; Insights; Dismiss Join GitHub today. Form a recursion and the peak element can be found in log
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it has to look at all the elements in the array. We can easily solve this problem in O(log(n)) time by using an idea similar to … Don’t stop learning now. Peak Element: peak element is the element
which is greater than or equal to both of its neighbors. in "An Efficient Algorithm for Automatic Peak Detection in Noisy Periodic and Quasi-Periodic Signals", Algorithms 2012, 5, 588-603. Let us
consider a number of arrays, we are representing them in symbols ( a — i ), we also assume that all the numbers are positive numbers. We can easily solve this problem in O(log(n)) time by using an
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given set of data. For example - In Array {1,4,3,6,7,5}, 4 and 7 are peak elements. 6. The array may contain multiple peaks, in that case return the index to any one of the peaks is fine. Find a
maximum element of these 6n elements, g = M[i][j]. Let’s start with the one dimensional version of peak Finder. Palshikar's [63] peak detection algorithm (S1) and Lehmann et al. Please write comments
if you find anything incorrect, or you want to share more information about the topic discussed above. “It is better to have an algorithm that is inefficient but correct rather have efficient
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is a data mining... | Find, read and cite all the research you need on ResearchGate. Therefore, 24 and 26 are both peak elements. So the last algorithm that will solve this problem is: So the
recurrence relation in terms of T(n,m) to this recursive algorithm is. Take mid as the starting point, this is classic case of divide and conquer approach as we will discard half of the array based
on certain condition. In the case where n = m the worst case complexity would be Θ(n²). Here in 21st century definition of large input is in trillions. Before starting out let’s first define
Algorithmic Thinking, According to the professor of MIT 6.006 Introduction to Algorithms Srini Devadas and I quote “Algorithmic Thinking is all about efficient procedures for solving problems on
large inputs”. If [n/2] < [n/2–1] then only look at left half from 1 to [n/2–1] to look for a peak, Else if [n/2] < [n/2+1] then only look at right half from [n/2+1] to n. Given the problem, we agree
that this algorithm is correct and finds a peak. Keywords timeseries . It’s true that 14 is a peak in a 1D case but looking from the perspective of a 2D 14 is not a peak which means the algorithm is
incorrect. Use (i, j) as a start point on row i to find 1D-peak on row i. I am really happy that we reduced the complexity to Θ(log n) as the complexity to find a peak in the 1D array is Θ(log n).
Now question is how to select m? 1D Peak Finder Algorithm. If g is greater than or equal to its neighbors, then by definition, that element is a peak element. Return anyone of 24 and 26. Let us again
assume that the peak is all the way to the right, so you start searching peak from the left all the way to the right, you will be looking at n elements to find a peak. SSE loop to walk likely primes.
SSE loop to walk likely primes. Now the peaks are clear; the results are reasonable and verifiable. Otherwise, there is always a case that you didn’t search hard enough. Formal Problem Statement -
Find a peak in a 2D array, where a is a 2D-peak iff a ≥ b, a ≥ d, a ≥ c, a ≥ e. If there are more than one peaks, just return one of them. A local peak is a data sample that is either larger than its
two neighboring samples or is equal to Inf. findpeaks(x, nups = 1, ndowns = nups, zero = "0", peakpat = NULL, minpeakheight = -Inf, minpeakdistance = 1, threshold = 0, npeaks = 0, sortstr = FALSE)
Arguments x numerical vector taken as a time series Because the peak detection algorithm uses a quadratic fit to find the peaks, it actually interpolates between the data points. Hope you got what I
meant in this blog. This problem is mainly an extension of Find a peak element in 1D array. Usage. And in that case, you want to be able to give an argument that you searched hard but could not find
it. Consider the following modified definition of peak element. Non- Inf signal endpoints are excluded. This series is not about algorithmic design it’s about algorithmic analysis. S. V. Chekanov1
and M. Erickson1,2 1 HEP Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA 2 Physics Department, The College of New Jersey, 2000 Pennington Road, Ewing, NJ
08628-0718, USA Correspondence should be addressed to S. V. Chekanov; … indexes, _ = scipy.signal.find_peaks(vector, height=7, distance=2.1) print('Peaks are: … References: Because the peak detection
algorithm uses a quadratic fit to find the peaks, it actually interpolates between the data points. These peaks may be correct, but it is difficult to determine whether this peak information is
really useful. Hello, just started learning algorithms. Approach 2: Recursive Binary Search. I couldn't find a good answer to how this formula was derived for the divide and conquer algorithm in a 1D
Peak-Finding problem. • Find a 1D-peak at i, j. So, we use divide and conquer method to find peak in O(logn) time. So, in this case, we will go to 12, 13, 14, 15, 16, 17,19, and 20. First, let’s
define a recurrence relation in terms of T(n) to this recursive algorithm Divide and Conquer. Here the algorithm will have to look at n/2 elements to find a peak. Given an array of size n, find a
peak element in the array. Peak element is the element which is greater than or equal to its neighbors. Codility's count passing cars in opposite directions in C#. Many time you are asked to do
something, and you can’t answer the question or find something that satisfies all the constraints required. Input: Array, arrA[] . By making use of this, and the fact that we can return any peak as
the result, we can make use of Binary Search to find the required peak … http://www.youtube.com/watch?v=HtSuA80QTyo, Related Problem: Hence the algorithm we design should be scalable to the growth of
the input. About the problem Basically, there's an array of numbers and we want to find a peak in this array (a peak is a number higher than the two numbers to the left and right of it). Given an
array of integers. For corner elements, we need to consider only one neighbour. 6. update. Easy to use and great results, but miss filtering. Nonparametric Peak Finder Algorithm. Why is this the
equation because n is the number of rows and m is the number of columns, In one case we will be breaking things down into half number of columns which is m/2 and In order to find the global maximum
we will be doing Θ(n) work. scipy.signal.find_peaks_cwt ... , however with proper parameter selection it should function well for different peak shapes. Now let’s look at a Straightforward Algorithm.
This looks like an efficient algorithm but does not work. T(n) = Θ(1) + …… + Θ(1) [This is a expanded form of the above equation], We gonna expand it log n times. Figure 8c shows the signal, smoothed
by using the same method as the peak detection algorithm, and then passed to the peak detection function. 2. Required height of peaks. Given the fact that we agreed on the correctness of the
algorithm now let us talk about the complexity of the algorithm. Naive Approach: The array can be traversed and the element whose neighbours are less than that element can be returned. Because I've
picked a column, and I'm just finding a 1D peak. Given an array, find peak element in it. Viewed 3k times 6 \$\begingroup\$ I'm reviewing MIT Introduction to Algorithm lectures/exercises and am
trying to implement a one dimensional peak finder algorithm. Due to the reasons discussed above, the program called Nonparametric Peak Finder (NPFinder) was developed using a numerical, iterative
approach to detect statistically significant peaks in event-counting distributions. Objective : In this article we will discuss an algorithm to Find a peak element in a Given Array. edit The
algorithm captures the position and shape of the probability peaks, even those corresponding to very different densities (blue and light green points in Fig. PeakFinder shows from any location the
names of all mountains and peaks with a 360° panoramic mountain view. Step 3: Search in {Ti} to find shapes of class 1-5, and process all matched shapes until all shapes of class 1,2 are The problem
is 2D peak my not exist in row i. Let’s choose the 3rd column from left as a middle. def detect_peak (data): nonlocal last, ascent_dist, ascent_start if data > last: if ascent_start is None:
ascent_start = last ascent_dist += 1 else: if ascent_dist: peak = last ascent_dist = 0 if (peak-ascent_start) > thresh: last = data ascent_start = … Let’s pick middle column j = m/2 and find a 1D
peak at (i, j). MaxCounters solution in C# from Codility. Following corner cases give better idea about the problem. You searched hard and could not find the answer is the proof of concept that the
solution might not be available. So the complexity of the algorithm is Θ(n log m), Well, this was quite a long blog. Implements a function find_peaks based on the Automatic Multi-scale Peak Detection
algorithm proposed by Felix Scholkmann et al. Peak valley detection in python. Algorithm Given an nxn matrix M: Take the ”window frame” formed by the first, middle, and last row, and first, middle, and
last column. Attempt # 1: Extend 1D Divide and Conquer to 2D. 10. So we can conclude that it is always better to reduce complexity as the input gets large. The paper studies the peak searching
algorithms and suggests future peak searching research tasks. Similarly, the signal shown in the figure on the left below could be interpreted as either as two broad noisy peaks or as 25 little
narrow peaks on a two-humped background. Its core is the comparison of what you see with the 3D model of the terrain in your camera view. Here position 2 is a peak if and only if b >= a and b >=c.
Find a peak element in a 2D array Last Updated: 25-09-2019 An element is a peak element if it is greater than or equal to its four neighbors, left, right, top and bottom. If you want the reference
from where I took content to write this blog then the reference has been listed below, A Solution to the (so-called) Paradox of the Ravens. The problem with the strictly derivative based peak finding
algorithms is that if the signal is noisy many spurious peaks are found. This is a convolution of vector with wavelet (width) for each width in widths. Finding the Moment of Inertia from a Point to a
Ring to a Disk to a Sphere. Hot Network Questions If a square wave has infinite bandwidth, how can we see it on an oscilloscope? detect_peaks from Marcos Duarte Codility's count passing cars in
opposite directions in C#. If a peak is flat, the function returns only the point with the lowest index. Nonparametric Peak Finder Algorithm Due to the reasons discussed above, the program called
Non-parametric Peak Finder (NPFinder) was developed using a numerical, iterative approach to detect statistically significant peaks in event-counting distributions. code. Experience. And we will find
a peak. So in this series we mostly concern about. i = m 2 • Pick middle column j = m/2. As of old saying goes by. Active 1 year, 1 month ago. MaxCounters solution in C# from Codility. 2. 2C) and
nonspherical peaks. Let us assume that the peak is in the middle, the numbers start increasing from left up to the middle and start decreasing. But, it takes O(n) time. 10. Moreover, points assigned
to the halo correspond to regions that by visual inspection of the probability distribution in Fig. Nonparametric Peak Finder Algorithm. Please use ide.geeksforgeeks.org, generate link and share the
link here. Parameters x sequence. Algorithm: Create two variables, l and r, initilize l = 0 and r = n-1 Iterate the steps below till l <= r, lowerbound is less than the upperbound Check if the mid
value or index mid = (l+r)/2, is the peak element or not, if yes then print the element and terminate. For the above three algorithms to find negative peaks, the raw data signal was negated, then
passed into the peak‐finding algorithm (note that Ridger algorithm finds both positive and negative peaks in a single pass). http://courses.csail.mit.edu/6.006/spring11/lectures/lec02.pdf
scipy.signal.find_peaks searches for peaks (local maxima) based on simple value comparison of neighbouring samples and returns those peaks whose properties match optionally specified conditions
(minimum and / or maximum) for their height, prominence, width, threshold and distance to each other. Greedy Ascent Algorithm works on the principle, that it selects a particular element to start
with. We will see the recursion techniques to solve this problem. Peaks are defined as a local maximum where lower values are present on both sides of a peak. For example neighbors for A [i] [j] are
A [i-1] [j], A [i+1] [j], A [i] [j-1] and A [i] [j+1]. 100 is the peak element in {100, 80, 60, 50, 20}. We need to return any one peak element. A peak element is an element that is greater than its
neighbors. The algorithm uses divide and conquer approach to find a peak element in the array in O(log n) time. And if it’s greater than, we’re going to follow that direction. r/algorithms: Computer
Science for Computer Scientists. Exercise: GitHub is where the world builds software. Looking at the row the peak is at 14. If you are equal and greater than the elements on left and right side than
you are the peak. import numpy as np import scipy.signal vector = np.array([0, 6, 25, 20, 15, 8, 15, 6, 0, 6, 0, -5, -15, -3, 4, 10, 8, 13, 8, 10, 3, 1, 20, 7, 3, 0]) print('Detect peaks with minimum
height and distance filters.') Attention reader! update, Else if the element on the right side of the middle element is greater then check for peak element on the right side, i.e. So if we try to do
the worst case analysis of the algorithm we will find that it would be Θ(nm) where n is the number of rows and m be the number of columns. Sign up. Writing code in comment? Solve the new problem with
half the number of columns. The Peak Finder panel displays the maxima, showing the x-axis values at which they occur. So we take the above equation and expand it eventually we will get to the best
case which is, T(n, m) = Θ(n) + …… + Θ(n) [This is a expanded form of the above equation], We gonna expand it log m times. In Greedy Ascent Algorithm, we have to make a choice from where to start.
Hot Network Questions In other words, the peaks found are not necessarily actual points in the input data but may be at fractions of an index and at amplitudes not found in the input array. The World
is moving faster than ever, things are getting bigger, we have the computational power that could handle large data (trillions) this does not mean efficiency is the main concern.
scipy.signal.find_peaks(x, height=None, threshold=None, distance=None, prominence=None, width=None, wlen=None, rel_height=0.5, plateau_size=None) [source] ¶ Find peaks inside a signal based on peak
properties. Press J to jump to the feed. We are mostly going to look at the n/2 position. We want to minimize the worst case number of elements to check after splitting, which is possible by
splitting the array in middle. Find Peaks Find peaks (maxima) in a time series. Let us assume that the peak is in the middle, the numbers start increasing from left up to the middle and start
decreasing. What Did Newton Do with his Time During Quarantine? It is roughly 6x faster then the R version in simple tests. 6. In case of the edges, you only have to look at only one side. Objective
: In this article we will discuss an algorithm to Find a peak element in a Given Array. i-PeakFinder can accurately detect shoulder peaks. it has to be considered a peak. Endpoints are not considered
peaks. 2A would not be assigned to any peak. So what we are really saying here is that the asymptotic complexity of the algorithm is linear. Return its indices (i;j). Keywords timeseries . First we
need to define the requirements for it to ... this time we only have {4} left so this is our base case, we only have one item and such this is a peak. Efficient Approach: Divide and Conquer can be
used to find a peak in O(Logn) time. Items attracting abnormal interest were identified by using three peak detection algorithms to validate the results as per Healy et al. Peak Element: peak element
is the element which is greater than or equal to both of its neighbors. Else traverse the array from the second index to the second last index, Else if the element on the left side of the middle
element is greater then check for peak element on the left side, i.e. Input: Array, arrA[] . Step 2: Remove all coincident points in set {Ti}. Algorithm I’: use the 1D algorithm •Observation: 1D peak
finder uses only O(log m) entries of B •We can modify Algorithm I so that it only computes B[j] when needed! If the input array is sorted in strictly decreasing order, the first element is always a
peak element. Now let’s look at the two dimensional version of peak finder, As we can guess a is a 2D peak if and only if. Lightweight Python algorithm to find peaks in single point streaming data.
What we are trying to advocate for this problem is that the algorithms we design should be general. 1D Peak Finder Algorithm. If all elements of input array are same, every element is a peak element.
If input array is sorted in strictly increasing order, the last element is always a peak element. pks = findpeaks (data) returns a vector with the local maxima (peaks) of the input signal vector,
data. Figure 5: Circled value is peak. So the complexity of the algorithm is Θ(log n). PeakFinderSavitzkyGolay extends PeakFinderBase, the abstract base class for all peak finding algorithms, and an
enumerable collection of all found peaks. So what’s the problem with this algorithm? Optionally, a subset of these peaks can be selected by specifying conditions for a peak’s properties. Let us
assume that the peak is in the middle, the numbers start increasing from left up to the middle and start decreasing. Research Article A Nonparametric Peak Finder Algorithm and Its Application in
Searches for New Physics. I agree we can scan billions of element in a matter of second but if you had an algorithm that required cubit complexity suddenly we are not talking about 10 to the power 9
we are talking about 10 to the power 27 and even current computer can’t handle that kind of numbers. 5. In other words, the peaks found are not necessarily actual points in the input data but may be
at fractions of an index and at amplitudes not found in the input array. We are going to tackle above concern using the classic data structure like arrays, linked list, stack and queue along with
classic algorithms like Search Algorithms, Sort algorithms, and Tree Algorithms. This function takes a 1-D array and finds all local maxima by simple comparison of neighboring values. Press question
mark to learn the rest of the keyboard shortcuts. Due to the reasons discussed above, the program called Nonparametric Peak Finder (NPFinder) was developed using a numerical, iterative approach to
detect statistically significant peaks in event-counting distributions. This is a divide and conquer algorithm. [61], i.e., Du et al. The algorithm don’t find all peaks on low sampled signals or on
short samples, and don’t have either a support for minimum peak height filter. Brute force approach to find peak in an array of integers will be to scan through it and for each element, check if
greater than it’s greater than previous and next element. For example, position 9 is a peak if i >= h. So the problem we solve right now is represented as “Find a peak if exists”. I've got a working
copy but it's a bit messy and I've had to put some array size constraints to get it working properly. From the menu, select Tools > Measurements > Peak Finder. Pick the middle column j = m/2 Find the
largest value in the current column span (global max) Compare to neighbors if larger than all this is the 2D peak Jump to left or right depending on comparison (divide and conquer) run recursively If
you are at … Ask Question Asked 4 years ago. You can enter values numerically, use the auto peak finder, interactively draw or edit your peaks with the mouse or some combination of these methods.
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Dalton's law of partial Pressures
When two or more gases, which do not react chemically, are mixed together in a vessel, the total pressure of the mixture is given by Dalton's law of partial pressures which states that,
"At constant temperature, the total pressure exerted by the gaseous mixture is equal to the sum of the individual pressures which each gas would exert if it occupies the same volume of mixture fully
by itself. Partial pressure is the measure of the pressure of an individual gas in a mixture of same volume and temperature.
Thus, if p1, p2, p3 .... are the partial pressures of the various gases present in a mixture, then the total pressure P of the gaseous mixture is given by
P = p1 + p2 + p3 ..., provided the volume and temperature of mixture and that of the individual gases are the same.
Equation of state of a Gaseous mixture
Let a gaseous mixture consists of n[A], n[B] and n[C] moles of three ideal gases A, B and C respectively, subjected to constant T and V, then, according to ideal gas equation.
p[A] = n[A]RT / V
p[B] = n[B] RT / V
pc = n[c] RT / V
where pA, pB, pC are the partial pressures of A,B,C gases respectively. Hence the total pressure of the mixture is given as
P = p[A] + p[B] + pc
P = n[A]RT/ V + n[B]RT/V + n[C]RT/V
PV = (n[A] + n[B] + n[C]) RT
This equation is known as equation of state of gaseous mixture.
Calculation of Partial Pressure
In order to calculate the pressure (p[A]) of the individual component say A, in a mixture (A and B), which is equal to the partial pressure of A, according to the equation of state of gaseous mixture
it is seen that,
P = Total pressure of the mixture
P = (n[A] + n[B]) x (RT/V)
But p[A] = ( n[A] /V ) RT and pB = (n[B]/V) x RT
The ratio is given by
p[A] / P = n[A] / (n[A] + n[B]) . V/RT . RT/V
x[A] = mole fraction of A.
Or p[A] = X[A]P
i.e:- Partial pressure, p[A] = mole fraction of A x total pressure. Similarly;
p[B] = X[B] P
Thus, the partial pressure of the individual component in the mixture can be calculated by the product of its mole fraction and total pressure.
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Machine Capacity Calculation in Pharmaceutical Industry
To determine the capacity in kgs the volume of machine and density of a powder are to be measured and then multiplied by Liters. Capacity in term of liters remain fix but in kgs vary depending upon
density of powder. For maximum capacity usually 60-70% of machine is considered and results are to be validated. For minimum capacity usually 30-40% of machine is considered and results are to be
For Rapid Mixer Granulator
Maximum Machine Capacity Calculation
Determine capacity in term of liters, for this add measured volume water in RMG.
Say it is 500 liters,
If density of powder is 0.75; then the volume of RMG will be 500×0.75=375 kg,
Now, 70% of it is = 375×70/100 = 262.50 kg,
So, the maximum capacity of RMG is 262.50 kg.
Minimum Machine Capacity Calculation
Determine capacity in term of liters, for this add measured volume water in RMG.
Say it is 500 liters,
If density of powder is 0.75; then the volume of RMG will be 500×0.75=375 kg,
Now, 40% of it is = 375×40/100 = 150.00 kg,
So, the minimum capacity of RMG is 150.00 kg.
For Coating Pan Capacity Calculation
Maximum capacity of the pan is calculated by the brim volume and the bulk density of the tablets.
Thus, Maximum Capacity of Coating Pan = Brim volume of pan (ltrs) x Bulk density of tablets (g/ml)
Brim volume = 300 ltrs.
Bulk density of tablets = 0.72 g/ml
Maximum capacity of coating pan = 300 x 0.72 = 216.0 Kg
If 60% of the maximum capacity is the minimum capacity of coating pan.
Minimum capacity of coating pan = 216.0 × 60% = 129.6 kg
• The coating pan's brim volume determines the coating machine's operational capacity.
• Bulk density depends on the tablet shape and size.
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If the integers m and n are chosen at random from 1 to 100 , th... | Filo
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Question Text If the integers and are chosen at random from 1 to 100 , then the probability that a number of the form is divisible by 5 , equals to Correct
Updated On Nov 25, 2022
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 2
Upvotes 246
Avg. Video Duration 5 min
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Syllabus | Microelectronic Devices and Circuits | Electrical Engineering and Computer Science | MIT OpenCourseWare
Course Meeting Times
Lectures: 2 sessions / week, 1 hour / session
Recitations: 2 sessions / week, 1 hour / session
Tutorials: 1 session / week, 1 hour / session
Required Text
Howe, Roger, and Charles Sodini. Microelectronics: An Integrated Approach. Upper Saddle River, NJ: Prentice Hall, 1996. ISBN: 9780135885185.
Reference Texts
Fonstad, Clifton. Microelectronic Devices and Circuits. New York, NY: McGraw-Hill, 1994. ISBN: 9780070214965.
Sedra, Adel, and Kenneth Smith. Microelectronic Circuits. New York, NY: Oxford University Press, 2007. ISBN: 9780195338836.
Horenstein, Mark. Microelectronic Circuits and Devices. New York, NY: Pearson, 1996. ISBN: 9780536846761.
Modular Series on Solid State Devices
Pierret, Robert. Semiconductor Fundamentals. Vol. I. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1988. ISBN: 9780201122954.
Neudeck, George. The PN Junction Diode. Vol. II. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1988. ISBN: 9780201122961.
———. The Bipolar Junction Transistor. Vol. III. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1989. ISBN: 9780201122978.
Pierret, Robert. Field Effect Devices. Vol. IV. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1990. ISBN: 9780201122985.
All the items below will enter into the computation of the final grade.
Two two-hour evening quizzes, open book, calculator required.
Final Exam
Three hours, open book, calculator required.
A total of eight problem sets will be handed out. The homework must be turned in by 4pm sharp on the due date. A 50% penalty will be applied to homework turned in after 4pm. Homework turned in after
solutions are distributed (about 3 days later) will be graded, but no credit will be given. All exceptions to this policy have to be approved by the lecturer. All homework sets will weigh equally
towards the final grade.
Design Problem and Web Labs
The design problem will be handed out one day after L12 and will be due one day after L17. There will also be two Web labs that will introduce you to real live device characterization. If you turn in
the design problem or any of the Web labs after its due date, the same policy as late homework applies.
We expect students to attend lectures and recitations, and we will keep track of attendance. The attendance record will be counted for 10% of the final grade.
The course grade will be determined as follows:
ACTIVITIES PERCENTAGES
Quiz 1 15%
Quiz 2 15%
Final exam 30%
Homework 10%
Design problem and web labs 20%
Attendance 10%
The final letter grade will also take into consideration non-numerical assessments of your command of the subject matter as evaluated by the lecturer, instructors, and TAs.
Policy for Academic Conduct (PDF)
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Free Body Diagrams
A free-body diagram (abbreviated as FBD, also called force diagram) is a diagram used to show the magnitude and direction of all applied forces, moments, and reaction and constraint forces acting on
a body. They are important and necessary in solving complex problems in mechanics.
What is and is not included in a free-body diagram is important. Every free-body diagram should have the following:
Always assume the direction of forces/moments to be positive according to the appropriate coordinate system. The calculations from Newton/Euler equations will provide you with the correct direction
of those forces/moments. Things that should not follow this are:
If forces/moments are present, always begin with a free-body diagram. Do not write down equations before drawing the FBD as those are often simple kinematic equations, or Newton/Euler equations.
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The interplay of classes of algorithmically random objects
We study algorithmically random closed subsets of $\cs$\!, algorithmically random continuous functions from $\cs$ to $\cs$\!, and algorithmically random Borel probability measures on $\cs$,
especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given
an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-Löf random for the pushforward measure if and only if its
preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply.
Our main results are the following. First we answer an open question in \cite{Barmpalias:2008aa} by showing that $\X\subseteq\cs$\! is a random closed set if and only if it is the set of zeros of a
random continuous function on $\cs$\!. As a corollary we obtain the result that the collection of random continuous functions on $\cs$ is not closed under composition. Next, we construct a computable
measure $Q$ on the space of measures on $\cs$ such that $\X\subseteq\cs$ is a random closed set if and only if $\X$ is the support of a $Q$-random measure. We also establish a correspondence between
random closed sets and the random measures studied in \cite{Culver:2014aa}. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random
continuous function is always contained in $(0,1)$.
is licensed under a
Creative Commons Attribution 3.0 License
Journal of Logic and Analysis ISSN: 1759-9008
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Sovereign Bond Index Weighting Under the EU Short Selling Regulation (SSR)
For the purposes of the EU Short Selling Regulation (SSR), the calculation of a sovereign bond constituent's weighting in a bond index differs from that of equity or equity-like index weightings.
While equity indices typically use market value weighting, for the SSR, face value weighting is required for bond indices to reflect nominal exposure.
Face Value-Weighted Bond Index Calculation
For a face value-weighted bond index, as required by the SSR, the Weighting of an individual bond can be calculated in two steps:
1. Step 1: Calculate the total index sum
The total sum of the bond constituents is calculated as follows:
Total Sum (index) = Sum((FaceValue of Bond1 x WeightingQuantity of Bond1)+(Face Value of Bond2 x WeightingQuantity of Bond2)+...)
2. Step 2: Calculate the weight of each bond
Once the total sum is determined, the weight of a specific bond is calculated as follows:
Weighting = (FaceValue* WeightingQuantity) / Total Sum (index)
• Weighting is the weight of the bond in the index.
• FaceValue refers to the face value of a single bond.
• WeightingQuantity is the number of units of the bond in the index
• Total Sum represents the total of all constituent bond face values multiplied by their respective WeightingQuantity.
This method ensures that each bond’s weight is proportional to its total face value contribution to the index. Key points to note:
• This weighting is based solely on FaceValue and WeightingQuantity.
• It does not take into account the current market price of the bonds.
• It differs from market value-weighted calculations, which rely on current market prices.
• WeightingQuantity is not a required input property; it is only mentioned here to clarify how to calculate each constituent bond’s Weighting.
Example Scenario:
Let's say we have a sovereign bond index with three bonds:
• Bond A: FaceValue = €1,000, WeightingQuantity = 10,000
• Bond B: FaceValue = €5,000, WeightingQuantity = 5,000
• Bond C: FaceValue = €2,000, WeightingQuantity = 15,000
Step 1: Calculate the total index sum
• Total Sum (index) = (FaceValue of BondA × WeightingQuantity of BondA) + (FaceValue of BondB × WeightingQuantity of BondB) + (FaceValue of BondC × WeightingQuantity of BondC)
• Total Sum (index) = (€1,000 × 10,000) + (€5,000 × 5,000) + (€2,000 × 15,000) = €10,000,000 + €25,000,000 + €30,000,000 = €65,000,000
Step 2: Calculate the weight of each bond
• Bond A Weighting = (€1,000 × 10,000) / €65,000,000 = 0.1538 or 15.38%
• Bond B Weighting = (€5,000 × 5,000) / €65,000,000 = 0.3846 or 38.46%
• Bond C Weighting = (€2,000 × 15,000) / €65,000,000 = 0.4615 or 46.15%
Note: WeightingQuantity is simply how many (number of) that bond is in the index.
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Transport calculations
Transport calculations
The conductivity, the Seebeck coefficient and the electronic contribution to the thermal conductivity in direction \(\alpha\beta\) are defined as [1] [2]:
\[\sigma_{\alpha\beta} = \beta e^{2} A_{0,\alpha\beta}\]
\[S_{\alpha\beta} = -\frac{k_B}{|e|}\frac{A_{1,\alpha\beta}}{A_{0,\alpha\beta}},\]
\[\kappa^{\text{el}}_{\alpha\beta} = k_B \left(A_{2,\alpha\beta} - \frac{A_{1,\alpha\beta}^2}{A_{0,\alpha\beta}}\right),\]
in which the kinetic coefficients \(A_{n,\alpha\beta}\) are given by
\[A_{n,\alpha\beta} = N_{sp} \pi \hbar \int{d\omega \left(\beta\omega\right)^n f\left(\omega\right)f\left(-\omega\right)\Gamma_{\alpha\beta}\left(\omega,\omega\right)}.\]
Here \(N_{sp}\) is the spin factor and \(f(\omega)\) is the Fermi function. The transport distribution \(\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right)\) is defined as
\[\Gamma_{\alpha\beta}\left(\omega_1,\omega_2\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}(\omega_1)v_{k,\beta}A_{k}\left(\omega_2\right)\right),\]
where \(V\) is the unit cell volume. In multi-band systems the velocities \(v_{k}\) and the spectral function \(A(k,\omega)\) are matrices in the band indices \(i\) and \(j\). The frequency depended
optical conductivity is given by
\[\sigma(\Omega) = N_{sp} \pi e^2 \hbar \int{d\omega \Gamma_{\alpha\beta}(\omega+\Omega/2,\omega-\Omega/2)\frac{f(\omega-\Omega/2)-f(\omega+\Omega/2)}{\Omega}}.\]
First perform a standard DFT+DMFT calculation for your desired material and obtain the real-frequency self energy.
If you use a CT-QMC impurity solver you need to perform an analytic continuation of self energies and Green functions from Matsubara frequencies to the real-frequency axis! This packages does NOT
provide methods to do this, but a list of options available within the TRIQS framework is given here. Keep in mind that all these methods have to be used very carefully. Especially for optics
calculations it is crucial to perform the analytic continuation in such a way that the real-frequency self energy is accurate around the Fermi energy as low-energy features strongly influence the
final results.
Below will describe the prerequisites from the different DFT codes.
Prequisites from Wien2k
Besides the self energy the Wien2k files read by the transport converter (convert_transport_input) are:
□ .struct: The lattice constants specified in the struct file are used to calculate the unit cell volume.
□ .outputs: In this file the k-point symmetries are given.
□ .oubwin: Contains the indices of the bands within the projected subspace (written by dmftproj) for each k-point.
□ .pmat: This file is the output of the Wien2k optics package and contains the velocity (momentum) matrix elements between all bands in the desired energy window for each k-point. How to use
the optics package is described below.
□ .h5: The hdf5 archive has to be present and should contain the dft_input subgroup. Otherwise convert_dft_input needs to be called before convert_transport_input.
These Wien2k files are read and the relevant information is stored in the hdf5 archive by using the following:
from triqs_dft_tools.converters.wien2k import *
from triqs_dft_tools.sumk_dft_tools import *
Converter = Wien2kConverter(filename='case', repacking=True)
SK = SumkDFTTools(hdf_file='case.h5', use_dft_blocks=True)
The converter convert_transport_input reads the required data of the Wien2k output and stores it in the dft_transp_input subgroup of your hdf file.
Wien2k optics package
The basics steps to calculate the matrix elements of the momentum operator with the Wien2k optics package are:
1. Perform a standard Wien2k calculation for your material.
2. Run x kgen to generate a dense k-mesh.
3. Run x lapw1.
4. For metals change TETRA to 101.0 in case.in2.
5. Run x lapw2 -fermi.
6. Run x optic.
Additionally the input file case.inop is required. A detail description on how to setup this file can be found in the Wien2k user guide [3] on page 166. The optics energy window should be chosen
according to the window used for dmftproj. Note that the current version of the transport code uses only the smaller of those two windows. However, keep in mind that the optics energy window has to
be specified in absolute values and NOT relative to the Fermi energy! You can read off the Fermi energy from the case.scf2 file. Please do not set the optional parameter NBvalMAX in case.inop.
Furthermore it is necessary to set line 6 to “ON” and put a “1” in the following line to enable the printing of the matrix elements to case.pmat.
Prequisites from Elk
The Elk transport converter (convert_transport_input) reads in the following files:
□ LATTICE.OUT: Real and reciprocal lattice structure and cell volumes.
□ SYMCRYS.OUT: Crystal symmetries.
□ PMAT.OUT: Fortran binary containing the velocity matrix elements.
□ .h5: The hdf5 archive has to be present and should contain the dft_input subgroup. Otherwise convert_dft_input needs to be called before convert_transport_input. It is recommended to call
convert_dft_input before convert_transport_input.
Except for PMAT.OUT, the other files are standard outputs from Elk’s groundstate calculation and are used in convert_dft_input. The PMAT.OUT file on the otherhand is generated by Elk by running task
120, see [4]. Note that unlike in the Wien2k transport converter, the Elk transport converter uses the correlated band window stored in the dft_misc_input (which originates from running
These Elk files are then read and the relevant information is stored in the hdf5 archive by using the following:
from triqs_dft_tools.converters.elk import *
from triqs_dft_tools.sumk_dft_tools import *
Converter = ElkConverter(filename='case', repacking=True)
SK = SumkDFTTools(hdf_file='case.h5', use_dft_blocks=True)
The converter convert_transport_input reads the required data of the Elk output and stores it in the dft_transp_input subgroup of your hdf file.
Using the transport code
Once we have converted the transport data from the DFT codes (see above), we also need to read and set the self energy, the chemical potential and the double counting:
with HDFArchive('case.h5', 'r') as ar:
chemical_potential,dc_imp,dc_energ = SK.load(['chemical_potential','dc_imp','dc_energ'])
As next step we can calculate the transport distribution \(\Gamma_{\alpha\beta}(\omega)\):
SK.transport_distribution(directions=['xx'], Om_mesh=[0.0, 0.1], energy_window=[-0.3,0.3],
with_Sigma=True, broadening=0.0, beta=40)
Here the transport distribution is calculated in \(xx\) direction for the frequencies \(\Omega=0.0\) and \(0.1\). To use the previously obtained self energy we set with_Sigma to True and the
broadening to \(0.0\). As we also want to calculate the Seebeck coefficient and the thermal conductivity we have to include \(\Omega=0.0\) in the mesh. Note that the current version of the code
repines the \(\Omega\) values to the closest values on the self energy mesh. For complete description of the input parameters see the transport_distribution reference.
The resulting transport distribution is not automatically saved, but this can be easily achieved with:
You can retrieve it from the archive by:
SK.Gamma_w, SK.Om_mesh, SK.omega, SK.directions = SK.load(['Gamma_w','Om_mesh','omega','directions'])
Finally the optical conductivity \(\sigma(\Omega)\), the Seebeck coefficient \(S\) and the thermal conductivity \(\kappa^{\text{el}}\) can be obtained with:
It is strongly advised to check convergence in the number of k-points!
Here we present an example calculation of the DFT optical conductivity of SrVO3 comparing the results from the Elk and Wien2k inputs. The DFT codes used 4495 k-points in the irreducible Brillouin
zone with Wannier projectors generated within a correlated energy window of [-8, 7.5] eV. We assume that the required DFT files have been read and saved by the TRIQS interface routines as discussed
previously. Below is an example script to generate the conductivities:
from sumk_dft_tools import *
import numpy
SK = SumkDFTTools(hdf_file=filename+'.h5', use_dft_blocks=True)
#Generate numpy mesh for omega values
om_mesh = list(numpy.linspace(0.0,5.0,51))
#Generate and save the transport distribution
SK.transport_distribution(directions=['xx'], Om_mesh=om_mesh, energy_window=[-8.0, 7.5], with_Sigma=False, broadening=-0.05, beta=40, n_om=1000)
#Generate and save conductivities
The optic_cond variable can be loaded by using SK.load() and then plotted to generate the following figure.
Note that the differences between the conductivities arise from the differences in the velocities generated in the DFT codes. The DMFT optical conductivity can easily be calculated by adjusting the
above example script by setting with_Sigma to True. In this case however, the SK object will need the DMFT self-energy on the real frequency axis.
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Kathryn Z. Hadley PhD Research
Astronomy picture of the day. Jan 10 2013
The Orion Bullets Image Credit: GeMS/GSAOI Team, Gemini Observatory, AURA Processing: Rodrigo Carrasco (Gemini Obs.), Travis Rector (Univ. Alaska Anchorage). Shock waves stream back around objects in
a star-forming region.
Computational modeling of hydrodynamic astrophysical systems is a ground-up approach to understanding the fundamental nature of fluids and the inherent behavior of astrophysical systems such as
protostellar and protoplanetary disks. Theoretical research in this area began in the 1980’s with modeling of very simple systems. It was predicted that gravitational instabilities would produce
internal structure in star-disk systems. Today, we have very highly evolved computational systems which allow us to carry out high-powered simulations of hydrodynamic structures. It is only in the
last few years that observation of star-disk systems has borne out the work of the theorists in this regime. Recent advances in observational techniques have, for the first time, allowed us to see
that protostellar disks contain internal structure, reminiscent of spiral arms seen in spiral galaxies.
Our modeling method for star-disk systems involves using an equilibrium model as an initial condition for the time-evolving model. We calculate the equilibrium disk structure by solving conservation
equations on a computational grid, using an initial guess for the mass density and angular momentum structure. We use an independent relationship between enthalpy and density to compare the result to
a previously defined tolerance. If the tolerance is not met, we improve the guess and reiterate until the tolerance is met. We then linearly perturb the equations and solve them on a cylindrical
grid. Each solution of the equations serves as the initial condition for the next time step. We monitor the growth of the perturbed density until it has either settled into exponential growth
(unstable) or a sufficient amount of time has elapsed to declare the model stable. We have developed an array of analysis tools useful in determining the properties of the unstable models. We see
that the geometry of the disk and the relative mass of the central star to the disk produces various kinds of modes in the disk. For systems where the disk mass is much greater than the mass of the
star, and inner edge of the disk is far removed from the star, the self-gravity of the disk is the dominating mechanism and J modes arise in the disk (named after the Jeans instability). In systems
where the star mass dominated, we see P modes and edge modes, depending on how far the inner edge of the disk lies from the star. Intermediate I modes arise where the self-gravity of the disk and the
pressure support are comparable.
Our nonlinear approach uses the same equilibrium models as initial conditions, but solves the system of equations by fluxing mass and angular momentum in the grid. This approach is much more
computationally expensive than the linear modeling, and it gives more information about the development of the systems, as it includes higher order coupling between the modes and evolves all of the
modes simultaneously. The nonlinear code we employ is an adaptation of the Chymera code, which is fully parallelized and includes several subroutines devoted to realistic equations of state and
radiative cooling. Chymera is second-order in space and time and includes artificial viscosity, enabling the correct handling of hydrodynamic shocks. The richness of the dual approach of combining
linear and nonlinear calculations is that the linear models allow an extensive sampling of parameter space, indicating regions of interest in which to focus nonlinear investigation. Linear models
also give a check for early behavior of the nonlinear models. The linear results allow us to perform a quasi-linear analysis, which is predictive of the nonlinear development.
Understanding the underlying reasons for the behavior of these systems is fundamental to this research. One of the main issues we address is how the bulk of the available mass in the system ends up
in the star, while the early central object is relatively low in mass, compared to the disk. We focus on understanding how mechanisms of angular momentum transport arise, and how various kinds of
modes inherent in the disks are driven. We are working on advancing the physics of the models by including radiative cooling and complicated equations of state, including molecular hydrogen and dust.
Our recent progress has been very promising, seeing that the formation of clumps of material can arise as a result of radiative cooling. Clumps like these may precede the formation of Jovian planets.
Typical models in this field include the star as a point mass. Very little work has been done where the star is included as a resolved object in the grid. Our recent work has shown that inclusion of
the star in this manner is important. Modes of oscillation in the star itself can gravitationally couple to the disk, changing the evolution of the system. I am currently conducting simulations in
both the linear and nonlinear regimes of systems including a resolved star. Nonlinear models are very computationally expensive in that very high resolution of the grid must be maintained in order to
correctly model the modes internal to the star. My work is fundamental at the moment, excluding cooling. I intend to add more complicated equations of state and radiative cooling to my models. This
methodology will shed light on many systems, such as first stars (those formed early in the universe) and protostellar and protoplanetary systems. Another application of this method would be to model
star-disk systems where the star is a white dwarf or a neutron star. Equations of state for systems like this are readily available, and can be included in our intact code in tabular form.
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Half Of Numbers Worksheet
Half Of Numbers Worksheet function as foundational devices in the realm of maths, giving a structured yet versatile system for students to check out and master numerical concepts. These worksheets
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sophisticated calculations, Half Of Numbers Worksheet cater to learners of diverse ages and skill levels.
Introducing the Essence of Half Of Numbers Worksheet
Half Of Numbers Worksheet
Half Of Numbers Worksheet -
These printable activities can be used for teaching fractions with halves and quarters only These are the most basic level of fraction worksheets If you re looking for fractions with numerators and
denominators between 1 and 10 please jump over to our main fraction worksheets page
Doc 44 KB A simple worksheet where children find half of the given number There are a few more challenging 2 digit numbers at the end Children are encouraged to record whether they have used cubes to
help them find half of a number to let us know if it violates our terms and conditions
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Finding Half Of A Number Worksheet Find Half And Quarter Of Small Numbers Fraction
Finding Half Of A Number Worksheet Find Half And Quarter Of Small Numbers Fraction
Our teacher made Ladybird Fractions Activity is a fun and engaging introduction to finding half of a number up to 20 It s the perfect way to consolidate your Early Level children s knowledge of
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This brilliant find a half resource is a fantastic way to see how well your children are doing with this topic Ready for a fun game Try this Interactive Halving Quiz
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Fractions Finding Half Of A Number Teaching Resources
Fractions Finding Half Of A Number Teaching Resources
Separate worksheets focus on halves thirds quarters and other basic fractions Halves Worksheet 1 Thirds Worksheet 2 Quarters Worksheet 3 Halves thirds quarters Worksheet 4 Worksheet 5
Half of numbers Half of numbers HudaMT Member for 3 years 3 months Age 3 10 Level 2 Language English en ID 698533 06 02 2021 Country code AE Country United Arab Emirates School subject Math 1061955
Main content Half of numbers 1994012 Finding half of numbers Other contents having Share Print
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different elements of life. These worksheets equip students to browse the elaborate surface of numbers, supporting an extensive gratitude for the style and reasoning inherent in maths.
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Halving Finding Half Of Differentiated Numbers By Landoflearning Teaching Resources Tes
Fun Activity On Fractions Half 1 2 Worksheets For Children
Check more of Half Of Numbers Worksheet below
Fun Activity On Fractions Half 1 2 Worksheets For Children
Doubling And Halving Maths Worksheet Ks1 Maths Worksheets Free Printable Math Worksheets 2nd
19 Cool Fractions Worksheets Ks1 Halves
Numeracy Finding Half Worksheet PrimaryLeap co uk
A Simple Way To Find Half Of The Large Number In Year 2 The Mum Educates
Halve Small Numbers 3 Fraction Worksheets For Year 1 age 5 6 By URBrainy
Halving Numbers Up To 20 And Then Beyond Teaching
Doc 44 KB A simple worksheet where children find half of the given number There are a few more challenging 2 digit numbers at the end Children are encouraged to record whether they have used cubes to
help them find half of a number to let us know if it violates our terms and conditions
Fraction Math Worksheets Math Salamanders
Here you will find a selection of Fraction worksheets designed to help your child understand what a half is both as a number and as an operator The sheets are graded so that the easier ones are at
the top halve different shapes and numbers to 20
Doc 44 KB A simple worksheet where children find half of the given number There are a few more challenging 2 digit numbers at the end Children are encouraged to record whether they have used cubes to
help them find half of a number to let us know if it violates our terms and conditions
Here you will find a selection of Fraction worksheets designed to help your child understand what a half is both as a number and as an operator The sheets are graded so that the easier ones are at
the top halve different shapes and numbers to 20
Numeracy Finding Half Worksheet PrimaryLeap co uk
Doubling And Halving Maths Worksheet Ks1 Maths Worksheets Free Printable Math Worksheets 2nd
A Simple Way To Find Half Of The Large Number In Year 2 The Mum Educates
Halve Small Numbers 3 Fraction Worksheets For Year 1 age 5 6 By URBrainy
Finding Half Butterfly Numbers TMK Education
Fun Activity On Fractions Half 1 2 Worksheets For Children
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Knight’s Tours Using a Neural Network
There was a paper in an issue of Neurocomputing that got me intrigued: it spoke of a neural network solution to the knight’s tour problem. I decided to write a quick C++ implementation to see for
myself, and the results, although limited, were thoroughly fascinating.
The neural network is designed such that each legal knight’s move on the chessboard is represented by a neuron. Therefore, the network basically takes the shape of the knight’s graph over an $n \
times n$ chess board. (A knight’s graph is simply the set of all knight moves on the board)
Each neuron can be either “active” or “inactive” (output of 1 or 0). If a neuron is active, it is considered part of the solution to the knight’s tour. Once the network is started, each active neuron
is configured so that it reaches a “stable” state if and only if it has exactly two neighboring neurons that are also active (otherwise, the state of the neuron changes). When the entire network is
stable, a solution is obtained. The complete transition rules are as follows:
$$U_{t+1} (N_{i,j}) = U_t(N_{i,j}) + 2 – \sum_{N \in G(N_{i,j})} V_t(N)$$
$$V_{t+1} (N_{i,j}) = \left\{
1 & \mbox{if}\,\, U_{t+1}(N_{i,j}) > 3\\
0 & \mbox{if}\,\, U_{t+1}(N_{i,j}) < 0\\
V_t(N_{i,j}) & \mbox{otherwise},
\end{array} \right.$$
where $t$ represents time (incrementing in discrete intervals), $U(N_{i,j})$ is the state of the neuron connecting square $i$ to square $j$, $V(N_{i,j})$ is the output of the neuron from $i$ to $j$,
and $G(N_{i,j})$ is the set of “neighbors” of the neuron (all neurons that share a vertex with $N_{i,j}$).
Initially (at $t = 0$), the state of each neuron is set to 0, and the output of each neuron is set randomly to either 0 or 1. The neurons are then updated sequentially by counting squares on the
chess board in row-major order and enumerating the neurons that represent knight moves out of each square.
Essentially, the network is configured to generate subgraphs of degree 2 within the knight’s graph. The set of degree-2 subgraphs naturally includes Hamiltonian circuits (re-entrant Knight’s Tours).
However, there are many other solutions that would satisfy the network that are not knight’s tours. For example, the network could discover two or more small independent curcuits within the knight’s
graph. In addition, there are certain cases that will cause the network to diverge (never become stable).
Knight’s Tour on an $8 \times 8$ board:
Not a Knight’s Tour, but still a solution (can you spot the four independent circuits?):
In fact, the probability of obtaining a knight’s tour on an $n \times n$ board virtually vanishes as n grows larger. Takefuji, at the time of his publication, only obtained solutions for n < 20.
Parberry was able to obtain a single knight's tour out of 40,000 trials for n = 26. I obtained one knight's tour out of about 200,000 trials for n = 28 (three days' worth of calculation on my Pentium
IV). Parberry wisely asserts that attempting to find a knight's tour for n > 30 using this method would be futile.
My implementation of this algorithm takes the shape of an application for Windows (although it’s perfectly runnable under Linux using wine). Several key features of the program include support for
arbitrary rectangular chess boards, as well as
statistical records of trials performed.
As seen in the screenshot, the program allows you to change the width and height of the chess board. You can then adjust the number of trials to perform. Click the Start button to begin the
calculation. The program will then display its progress in the statistics window on the right, showing the number of knight’s tours found, number of non-knight’s tours found, and number of divergent
patterns. By default, the program draws the chess board only when a knight’s tour has been found. However, if you enable the Display All check box, the program will draw all solutions it finds.
Notable Finds
Symmetric 10 x 3 knight’s tour
Symmetric 14 x 3 knight’s tour
24 x 24 knight’s tour (2 hours cpu time)
26 x 26 knight’s tour (8 hours cpu time)
28 x 28 knight’s tour (50 hours cpu time)
Undoubtedly, knight’s tours for n > 28 can easily be found using simpler combinatorial algorithms, which seems to make this neural network solution for the knight’s tour problem less than practical.
However, one cannot deny the inherent elegance in this kind of solution, which is what made it so interesting to investigate.
• I. Parberry. Scalability of a neural network for the knight’s tour problem. Neurocomputing, 12:19-34, 1996.
• Y. Takefuji and K. C. Lee. Neural network computing for knight’s tour problems. Neurocomputing, 4(5):249-254, 1992.
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What is
What is a 5-sigma level?
So, what does five-sigma mean? In short, five-sigma corresponds to a p-value, or probability, of 3×10-7, or about 1 in 3.5 million. If the p-value is low, for example 0.01, this means that there is
only a small chance (one percent for p=0.01) that the data would have been observed by chance without the correlation.
What does 5 standard deviations mean?
In most cases, a five-sigma result is considered the gold standard for significance, corresponding to about a one-in-a-million chance that the findings are just a result of random variations; six
sigma translates to one chance in a half-billion that the result is a random fluke.
What is the value of 5-sigma?
about 1 in 3.5 million
Five-sigma corresponds to a p-value, or probability, of 3×10-7, or about 1 in 3.5 million. This is where you need to put your thinking caps on because 5-sigma doesn’t mean there’s a 1 in 3.5 million
chance that the Higgs boson is real or not.
What is the full form of sigma?
Sigma is the 18th letter of the Greek alphabet and is equivalent to our letter ‘S’. The lower case sigma stands for standard deviation. …
What percent is 3 sigma?
Three-sigma limits set a range for the process parameter at 0.27% control limits. Three-sigma control limits are used to check data from a process and if it is within statistical control. This is
done by checking if data points are within three standard deviations from the mean.
How do you determine sigma level?
Sigma score produces an accurate analysis of observations or data, indicating the number of short-term standard deviations that fall above or below a specified mean. Also called sigma, sigma level or
standard score. The formula to determine sigma score is: Z = X – M / D.
What is the probability of 3 sigma?
Three sigma limits indicate data chosen randomly from a set of normally distributed data and has a 99.73% of probability of being within the acceptable standard deviation, translating into a
possibility of 1,350 defects per million opportunities.
How do you calculate Sigma?
How to Calculate Process Sigma How to Calculate Process Sigma Step 1: Define Your Opportunities Step 2: Define Your Defects Step 3: Measure Your Opportunities and Defects Step 4: Calculate Your Yield
Step 5: Look Up Process Sigma Assumptions
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Convex Functions¶
A function $f: \R^d \rightarrow \R$ is convex if for all $x, y \in \R^d$, and all $\eta \in [0,1]$
$$f(\eta x + (1 - \eta) y) \le \eta f(x) + (1 - \eta) f(y).$$
What this means graphically is that if we draw a line segment between any two points in the graph of the function, that line segment will lie above the function. There are a bunch of equivalent
conditions that work if the function is differentiable. In terms of the gradient, for all $x, y \in \R^d$,
$$(x - y)^T \left( \nabla f(x) - \nabla f(y) \right) \ge 0,$$
and in terms of the Hessian, for all $x \in \R^d$ and all $u \in \R^d$
$$u^T \nabla^2 f(x) u \ge 0;$$
this is equivalent to saying that the Hessian is positive semidefinite.
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AbstractIntroductionDataMethodExtract series of dry and rainfall eventsExtreme dry event modelling with annual maximum series and Gumbel
distributionEstimation of extreme dry event using PDSCharacteristics of PDSProbability distributions used to adjust PDSComparison of the AMS–G and PDS–E approachesResultsSelection of the lower bound in using PDSComparison of maximum dry event estimations using the AMS–G and PDS–E
approaches with the observed maximum dry eventDiscussion and conclusionsData availabilityAuthor contributionsCompeting interestsSpecial issue statementAcknowledgementsReferences
A proper simulation of precipitation is important. Precipitation is a very important element of climate that affects both the natural environment and human society. Events ranging from prolonged
droughts to short-term, high intensity floods are often associated with devastating impacts both to society and the environment (Hui et al., 2005). An alternative to the Markov chain process which is
typically used to simulate the occurrence of precipitation is to use a wet-dry spell model or alternating renewal model, that is, to simulate wet and dry spells separately by fitting their durations
to an appropriate probability distribution. Among the study using the wet-dry spell approach one can cite, for example, Bogardi and Duckstein (1993); Wilks (1999); Mathlouthi (2009); Mathlouthi and
Lebdi (2008, 2009, 2017); Dunxian et al. (2015); Konjit et al., 2016.
It is well known that dry spells cause major economic and human losses, and numerous studies have highlighted the need for drought prevention and mitigation plans (Vicente-Serrano and Beguería,
2003). The spatial and temporal assessment of dry spells is necessary in order to protect agriculture, water resources and other socio economic concerns, and areas at risk from droughts of long
duration and great intensity need to be determined. Sivakumar (1992) point up the importance of admitting partial patterns of extreme drought, which can then be used in the management of cultivated
areas (crop selection, irrigation planning, etc.) and water resources management.
The analysis of extremes in dry-spell series has been examined classically using annual maximum series (AMS) adjusted to a Gumbel distribution (Gupta and Duckstein, 1975; Lana and Burgueño, 1998).
The AMS are constructed by determining the maximum dry spell for each year, so the series length equals the number of years for which records are available. However, the main drawback is the loss of
the second, third, etc. largest annual dry spells, which might exceed the maximum dry spells of other years. An option approach is the partial duration series (PDS), which is constructed using the
values above a selected threshold regardless of the year in which they occurred (Hershfield, 1973; Vicente-Serrano and Beguería, 2003). Typically, the generalized Pareto (GP) distribution has been
used to model PDS (Bobée and Rasmussen, 1995). Although the PDS approach has obvious advantages over the AMS approach (Cunnane, 1973), it has been used only infrequently in precipitation dry-spell
analysis (Vicente-Serrano and Beguería, 2003).
In virtue, this paper is focused on the modelling of rainfall occurrences under Mediterranean climate by wet-dry spell approach. The intended objective is to determine whether the use of AMS with the
Gumbel distribution (AMS–G approach) is suitable for modelling extreme dry-spell risk; to analyse if PDS with a probability distribution, that best fits the data set, is adequate for modelling
extreme dry-spell risk and finally to compare both approaches with the observed maximum dry spells to determine the most suitable estimation of drought risk.
The study area is the Ichkeul basin (Northern Tunisia) with several dams for irrigation, drinking water and water transfer to other regions of the country. The precipitation irregularity and the
frequent dry spells are major restrictive factors in crop growth and water demand satisfaction imposed on dams. For this reason, this area is particularly suitable for examining this approach.
In the wet-dry spell approach, the time-axis is split up into intervals called wet periods and dry periods. A rainfall event is an uninterrupted sequence of wets periods. The definition of event is
associated with a rainfall threshold value which defines wet (Fig. 2). As this limit 3.6mmd-1 has been selected. This amount of water corresponds to the expected daily evapotranspiration rate,
marking the lowest physical limit for considering rainfall that may produce utilizable surface water resources. In this approach, the process of rainfall occurrences is specified by the probability
laws of the length of the wet periods, and the length of the dry periods (time between storms or inter-event time).
Event representation of the climatic cycle.
The rainfall event r in a given rainy season n will be characterized by its duration Dn,r, the temporal position within the rainy season, the dry event or inter-event time Zn,r and by the cumulative
rainfall amounts of Hn,r of Dn,r rainy days (Fig. 2). r=f(Dn,r,Hn,r,Zn,r) Where f is the function defined on R+∗, which to each event r associates a value D, H and Z themselves, real discrete random
variables. 2Zn,r=xn,r-xn,r-13Hn,r=∑k=1Dn,rhk Where hk represents the total daily rainfall in mm. Let hk>0 and at least a value of hk>3.6mm.
The varying duration of the events requires that the cumulative rainfall amounts corresponding to each event should be conditioned by the duration of the event. The identification and fitting of
conditional probability distributions to rainfall amounts may be problem especially in the case of short records and for events with extreme (long) durations (Foufoula-Georgiou and Georgakakos,
1991). The number of rainfall events per rainy season n is Nn and the length Ln of this last, of random duration, is defined as the time span between the start of the first and the end of the last
rainfall event. Ln=∑r=1NnDn,r+∑r=1Nn-1Zn,r The length of the climatic cycle Cn is determined as the time lapsed between the onsets of two subsequent rainy seasons. Cn=xn+1,r1-xn,r1
The distribution introduced by Gumbel is very useful for extreme dry event frequency modelling using the AMS–G approach (Gumbel, 1958; Vicente-Serrano and Beguería-Portugués, 2003). The Gumbel
distribution is a two-parameter distribution with constant skewness. It is a particular case of the three-parameter generalized extreme value (GEV) distribution, i.e. the limit distribution for
maxima series. The Gumbel is usually preferred to the GEV because of its ease of calculation. Its probability density function is fx=1∝e-x-β/α-e-x-β/α and its cumulative distribution function is
expressed by Fx=e-e-x-βα where x is the value of the variable, and α and β are scale and location parameters of the distribution, respectively. The mean and the variance are μ=β+0.5772α and σ2=π2α26,
The prospective maximum dry event for a T year period XT can be calculated using XT=β-αln-ln1-1/T
Although the preceding method has been widely used in the study of extreme dry spells, in the analysis of other hydrological and climatic variables (e.g. extreme rainfall, floods) many studies prefer
to use PDS or series of peaks over an upper limit. Given the dry spell series a={a1,a2,…,an}, for the station a, where ai is the duration of a given dry spell, the PDS b={b1,b2,…,bj} consists of all
the values of the original series that exceed a predetermined upper limit a0: bj=ai-a0∀ai>a0 The size of the series obtained depends, therefore, on the upper limit a0. For this reason, PDS use the
information contained in the original sample more efficiently, and permit the inclusion of more than one event per year, if they satisfy the conditions established in defining an extreme event (Chow
et al., 1988; Vicente-Serrano and Beguería-Portugués, 2003).
Many probability distributions have been adjusted to PDS hydrological series, including lognormal, Pearson III, Gamma, GEV, Weibull, etc. (Bobée et al., 1993; Vicente-Serrano and Beguería-Portugués,
2003). In this study, we evaluated the continuous probability distributions given by Hyfran software, and we found that the Exponential law is the best fitting probability distribution to PDS.
The parameter estimates is performed by the method of moments. A Chi-Squared goodness-of-fit Test is used to determine how well the theoretical distribution fits the empirical distribution obtained
from the sample data. The exponential (E) distribution function is ft=b⋅e-b(t-m)m<t<+∞ where b parameter of the exponential distribution, can be estimated as the reciprocal mean t‾ of the sample of
times observed: b=1t‾ and its cumulative distribution function is expressed by G(t)=1-e-b(t-m)m<t<+∞ The event XT in a period of T years is obtained using XT=m-1bln1-1T A major problem in using PDS
is the selection of the lower bound a0. This value should be low enough to ensure the inclusion of as much relevant information as possible, without violating the assumption of independence of the
peaks. Various methods have been proposed to determine the most appropriate lower bound (Ashkar and Rouselle, 1987; Madsen et al., 1997). However, according to Vicente-Serrano and Beguería-Portugués
(2003) Beguería (2003) has shown that the parameters and quantile estimations vary randomly with the threshold value, and no single value is entirely adequate. For this reason, in this paper, the
maximum dry event in the 42 year period was calculated using different lower bounds in the PDS–E approach. These bounds were defined using the percentiles of the dry event series every 0.5 from
percentiles 90 to 99.5. Dry event were considered extreme above the 90th percentile.
The maximum dry event observed in each series in the period 1968–2010 was extracted. These were compared with the 42 year estimates using the AMS–G and PDS–E approaches. It is clear that the maximum
dry event observed in a 42 year period does not necessarily correspond to a return period of 42 years. This limitation was partially overcome by using several rain gauges in the same region. The
goodness of fit was tested by means of the root-mean-square error (RMSE) (Willmott, 1982), the lowest value indicating the best estimation: RMSE=1n∑i=1nzi-z^i2 Where zi is the observed value and z^i
the estimated value using annual maximum or partial duration series; n is the number of rain gauges.
The main problem in using PDS involves the selection of the lower bound. In theory, the method is invariant to the variation in the lower bound. In practice, however, the results may vary greatly,
especially with the sample sizes that are common in hydro climatic studies. This is exemplified in Fig. 3, in which the maximum dry spells expected in 42 years are shown for five rain gauges, in
relation to the lower bound used. Whereas this value was expected to be similar independently of the lower bound chosen, it showed great random variation, being as 21% compared to the average in
some cases. Here, we assumed that the average of the different values would provide a good estimate of the unknown true value, this being less uncertain than using a unique, arbitrary, threshold.
Oscillation of the maximum dry event (days) estimations as a function of the selected percentile in the creation of the dry event PDS. Five representative rain gauges are shown.
Figures 4 and 5 compare AMS–G and PDS–E estimates with the observed maximum dry events. The AMS–G method estimated adequately in the majority of cases the duration of the observed maximum dry events.
The underestimation did not exceed 9d, which prudent use of this method. The PDS–E clearly overestimated the maximum dry events duration. The difference between predicted and observed values varies
from -5.4% to 25.7%.
The RMSE between the observed and estimated values is also highly indicative of the better performance of the AMS–G distribution. There was a better adjustment for the dry event series (RMSE=4.7
versus 9.2).
Differences between maximum dry events observed and estimated using AMS modelled using Gumbel distribution.
Differences between maximum dry events observed and estimated using PDS modelled using the Exponential distribution.
Figure 6 shows the spatial distribution of the maximum dry events observed in the study area between 1968 and 2010, along with the estimations using the PDS–E and AMS–G approaches. The longest dry
events are located in the southern areas, with values over 81 consecutive days of precipitation below 3.6mm. A negative southwestern gradient of the maximum dry event duration is established. The
same pattern is revealed by both estimations. There were significant contrasts between the south and west, with differences about 40d. The AMS–G map shows a much closer match to the observed data.
The Exponential estimation is clearly little higher than the observed figure.
Maximum dry events: (a) observed in 42 years (1968–2010); (b) predicted using the E distribution; (c) predicted using Gumbel distribution.
The absolute errors of the estimations are shown in Fig. 7. The high magnitude of the errors resulting from the PDS–E approach is evident. Here, the positive errors indicate the underestimation
provided by this approach. By contrast, the errors of the AMS–G approach include low negative values and the estimation is, in general, better.
Differences between maximum dry spells observed and estimated: (a) observed-estimated using Gumbel distribution; (b) observed-estimated using E distribution.
In this paper, we have used a PDS sampling in conjunction with an Exponential distribution. The results obtained have been compared with those obtained when adopting the AMS–G approach for the
maximum dry event series observed in the study area.
Different probability distributions can be used to fit both AMS and PDS. The Gumbel distribution is a two parameter extreme values distribution widely used in modelling AMS. It has been compared with
the one parameter Exponential distribution fitted to PDS. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution. It is obvious that a two
parameter distribution would fit the observed data better than a one parameter one. Nevertheless, the need to estimate a greater number of parameters introduces an extra source of uncertainty that
can affect the final estimates. Here, we find that the use of AMS-G is more efficient than PDS-E, contrary to what has been reported in several other studies. In this sense Moreno and Roldán (1999),
Mkhandi et al. (2000) and Vicente-Serrano and Beguería (2003) indicated that the use of PDS for the stochastic modelling of extremes has yielded good results in the analysis of hydrological
variables, whereas numerous studies have pointed out that AMS produces a significant loss of data for extreme modelling (Cunnane, 1973; Madsen et al., 1997; Vicente-Serrano and Beguería, 2003).
Accordingly, the RMSE obtained by the AMS–G is lower than that obtained by the PDS–E when analysing the empirical maximum dry events for a 42 year time series.
One shortcoming of the PDS method is the selection of the upper limit used to define the PDS. We found that the final quantile estimates vary significantly when only small changes are made in the
upper limit used. This result has been reported previously by the study of Vicente-Serrano and Beguería, 2003. To cope with this problem, as proposed by Vicente-Serrano and Beguería, (2003), the use
of different upper limits when constructing a set of PDS, and then taking the average quantile estimates obtained with them. A set of PDS with limits ranging from percentiles 90 to 99.5, rising by
0.5 steps, was used in this paper. This proved to stabilize the variability of the quantile estimates. However, if this methodology is used on a more general scale, the upper limit range should be
defined more precisely because it may differ for each set of data.
This paper has revealed that the widely used AMS–G approach estimates adequately the observed extreme dry-spell risk in the study area, by contrast with the PDS-E.
The results obtained here are of potential importance for agrarian planning. The method used is of potential importance for agrarian planning and of benefit in crop management. It facilitates the
drawing of risk maps and the drafting of preventive and palliative plans for the mitigation of the effects of drought.
The data are printed in paper documents stored (archived) in office of the General Directorate of Water Resources and the Division of Dam Operation of Extreme North and Ichkeul of the Ministry of
Agriculture of Tunisia, 2019a, b (http://www.agriculture.tn/). These data are the property of this organization, and are available in situ.
Discussed the results and contributed to the final version of the manuscript: FL; Developed and performed the design and implementation of the research, the computations, the analysis of the data and
the results and the writing of the manuscript: MM.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Hydrological processes and water security in a changing world”. It is a result of the 8th Global FRIEND-Water Conference: Hydrological Processes and Water
Security in a Changing World, Beijing, China, 6–9 November 2018.
The authors acknowledge the help of General Directorate of Water Resources and Division of Dam Operation of Extreme North and Ichkeul of the Ministry of Agriculture of Tunisia. The authors thank the
reviewers for their relevant remarks that contributed to the improvement of this article.
Ashkar, F. and Rouselle, J.: Partial duration series modeling under the assumption of a Poissonian flood count, J. Hydrol., 90, 135–144, 1987. Beguería, S.: Identificación y características de las
fuentes de sedimento en áreas de montaña: erosión y transferencia de sedimento en la cuenca alta del río Aragón, PhD thesis (unpublished), University of Zaragoza, 2003. Bogardi, J. J. and Duckstein,
L.: Evénements de période sèche en pays semi-aride, Revue des Sciences de l'Eau, 6, 23–44, 1993. Bobée, B., Cavadias, G., Ashkar, F., Bernier, J., and Rasmussen, P.: Towards asystematic approach to
comparing distributions used in flood frequency analysis, J. Hydrol., 142, 121–136, 1993. Chow, V. T., Maidment, D. R., and Mays, L. W.: Applied Hydrology, McGraw-Hill, New York, NY, 572 pp., 1988.
Cunnane, C.: A particular comparison of annual maxima and partial duration series methods of flood frequency prediction, J. Hydrol., 18, 257–271, 1973. Dunxian, S., Ashok, K. M., Jun, X., Liping, Z.,
and Xiang, Z.: Wet and dry spell analysis using copulas, Int. J. Clim., 36, 476–491, 10.1002/joc.4369, 2015. Foufoula-Georgiou, E. and Georgakakos, K. P.: Hydrologic advances in space-time
precipitation modeling and forecasting, in: Recent advances in the modeling of hydrologic systems, edited by: Bowles, D. S. and O'Connell, P. E., NATO ASI Series, Serie C: mathematical and physical
sciences, Kluwer Academic Publishers, Dordrecht, The Netherlands, 345, 47–65, 1991. Gumbel, E. J.: Statistics of Extremes, Columbia University Press, 1958. Gupta, V. K. and Duckstein, L.: A
stochastic analysis of extreme droughts, Water Resour. Res., 11, 221–228, 1975. Hershfield, D. M.: On the probability of extreme rainfall events, B. Am. Meteorol. Soc., 54, 1013–1018, 1973. Hui, W.,
Xuebin, Z., and Elaine, M. B.: Stochastic modelling of daily precipitation for Canada, Atmos. Ocean, 43, 23–32, 10.3137/ao.430102, 2005. Konjit, S., Fitsume, Y., Asfaw, K., and Shoeb, Q.: Wet and dry
spell analysis for decision making in agricultural water management in the eastern part of Ethiopia, West Haraghe, Int. J. Water Res. Environ., 8, 92–96, 10.5897/IJWREE2016.0650, 2016. Lana, X. and
Burgueño, A.: Spatial and temporal characterisation of annual extreme droughts in Catalonia (NE Spain), Int. J. Clim., 18, 93–110, 1998. Madsen, H., Pearson, C. P., and Rosbjerg, D.: Comparison of
annual maximum series and partial duration methods for modelling extreme hydrologic events, 1. At-site modelling, Water Resour. Res., 33, 759–769, 1997. Mathlouthi, M.: Optimisation des règles de
gestion des barrages réservoirs pour des évènements extrêmes de sècheresse, Thèse de Doctorat, Institut National Agronomique de Tunisie, Tunis, Tunisie, 162 pp., 2009. Mathlouthi, M. and Lebdi, F.:
Event in the case of a single reservoir: the Ghèzala dam in Northern Tunisia, Stochastic Environ. Res. Risk Assess., 22, 513–528, 10.1007/s00477-007-0169-3, 2008. Mathlouthi, M. and Lebdi, F.:
Analyse statistique des séquences sèches dans un bassin du nord de la Tunisie, Hydrol. Sci. J., 54, 442–455, 10.1623/hysj.54.3.442, 2009. Mathlouthi, M. and Lebdi, F.: Frequency and severity of dry
spell phenomenon in Ghezala Dam reservoir (Tunisia), European Water, 60, 255–261, 2017. Ministry of Agriculture of Tunisia: General Directorate of Water Resources, available at: http://
www.agriculture.tn/, Daily rain gauge observations, 2019a. Ministry of Agriculture of Tunisia: General Directorate of Dams and Major Hydraulic Works, available at: http://www.agriculture.tn/,
Division of Dam Operation of Extreme North and Ichkeul, Hydraulic database of the Ghézala dam, 2019b. Mkhandi, S. H., Kachroo, R. K., and Gunasekara, T. A. G.: Flood frequency analysis of southern
Africa: II. Identification of regional distributions, Hydrol. Sci. J., 45, 449–464, 2000. Moreno, F. and Roldán, J.: Regional daily precipitation stochastic model parameters. Application to the
Guadalquivir valley in southern Spain, Phys. Chem. Earth Pt. B, 24, 35–47, 1999. Sivakumar, M. V. K.: Empirical analysis of dry spells for agricultural applications in West Africa, J. Climate, 5,
532–539, 1992. Vicente-Serrano, S. M. and Beguería Portugues, S.: Estimating extreme dry-spell risk in the middle Ebro Valley (Northeastern Spain): a comparative analysis of partial duration series
with a General Pareto distribution and annual maxima series with a Gumbel distribution, Int. J. Clim., 23, 1103–1118, 10.1002/joc.934, 2003. Wilks, D. S.: Interannual variability and extreme value
characteristics of several stochastic daily precipitation modes, Agric. Meteorol., 93, 153–169, 1999. Willmott, C. J.: Some comments on the evaluation of model performance, B. Am. Meteorol. Soc., 63,
1309–1313, 1982.
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Solving Equations/Inequalities with Variables On One Side Graphic Organizers | Math = Love
Solving Equations/Inequalities with Variables On One Side Graphic Organizers
We did six practice problems in our notebook over solving equations/inequalities with variables on one side. Students stapled these problems together and glued them on a single page. Some students
even chose to staple their paper with the steps on top of the practice problems to condense things even further. I kinda like this approach!
At the end of our first unit, students were really struggling with translating between words and algebra. So, I decided to continue giving my students practice translating by giving them every single
equation and inequality in WORDS instead of ALGEBRA. Was this the right decision? I’m not really sure. But, I can say that half way through this unit on solving equations/inequalities, my students
started translating between words and algebra like bosses! The continued practice did end up paying off. What it prevented, however, was the chance to look at different types of application problems.
I would like to combine these two approaches when I teach this again.
Here are close-ups of each:
Next time, I do want to make a few changes to this template. I want to switch around steps 7 and 8. By having my students check their solutions before they graphed them, some of my students became
confused and started putting the results they got from checking their work on the number line instead of the solution they found.
Free Download of Solving Equations/Inequalities with Variables On One Side of the Equal Sign Graphic Organizers
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2012 A-level H2 Mathematics (9740) Paper 2 Question 1 Suggested Solutions - The Culture SG2012 A-level H2 Mathematics (9740) Paper 2 Question 1 Suggested Solutions
All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.
KS Comments:
Students need to be reminded to put modulus due to the presence of ln. The integration formula in (ii) can be found in MF15 too.
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Game Theory, Cryptography, and Artificial Intelligence: A Comprehensive Guide
Navein Suresh & Keshav Shah
Machine Learning and all of its incredible capabilities stand stagnantly on their own, but what if there was a way to connect the way we secure personal data with machine learning architectures.
Through this article we want to explore the intersection between the fields of cryptography and game theory and how they can be used in the world of machine learning. We first introduce game theory
and cryptography and then delve into a summary of machine learning and how it can be used for predictions and labeling of complex data.
Brief Overview of Game Theory
Let’s begin by understanding what game theory is. Game theory is a branch of mathematics which is all centered around the study of strategic interactions during a “game”. We make a couple of key
assumptions about this “game”:
• There is a certain structure and established set of rules
• Individuals participating in the game make rational decisions (pursue the best possible strategy)
For the most part, at least for our purposes, games can either be categorized as sequential or simultaneous. Sequential games are when players make their decision at the same time and the results are
represented using a pay-off matrix. On the other hand, simultaneous games involve players making their decisions at the same time and possible results are portrayed using a decision tree. There are a
vast variety of executable strategies depending on the specific problem, prominent examples including Pure and Mixed Strategy Nash Equilibriums, Dominant Strategies, and more.
A dominant strategy in game theory is where regardless of what other players in the game choose, your decision will prevail and lead to a better outcome. There are not necessarily dominant strategies
for every game.
Nash Equilibrium is another particular area of interest when in Game Theory. Nash Equilibrium for a particular provides a set of strategies such that no player has any incentive to switch his/her
strategy. John Nash proved that every finite game (Finite number of strategies and finite number of players) has at least one Nash Equilibrium. Each outcome must be checked (in a pay-off matrix) to
ensure that each individual is satisfied with his/her choice. Nash Equilibriums come in two forms: Pure Strategy N.E. and Mixed Strategy N.E. One of these two strategies must exist.
Nash Equilibrium situation demonstrated using a table
Pure Strategy Nash Equilibrium is where an individual has a 100% chance (definite) of choosing a particular strategy such that the individual has no regrets after other players make a decision
Rather, in a Mixed Strategy Nash Equilibrium, an individual is presenting probability distributions of specific courses of actions, ensuring that the situation is stable and the other player can
choose a strategy without 100% certainty of loss.
One important term that’s introduced in game theory is the “zero-sum” game. Zero-sum games are defined in a way such that whenever an individual wins, the rest of the individuals (whether that be 1
more person or a 100 more) lose. The net change is 0 in this case. Nash Equilibrium Strategies are one way to solve “zero-sum” games. Real-life examples of “zero-sum” games include chess and
rock-paper-scissors-shoot. When one individual tries to maximize his or her reward, the other player’s reward is necessarily minimized.
The Minimax theorem is closely linked with zero-sum games. The idea behind the minimax theorem deals with maximizing the utility/gain that when dealt with the worst possible scenario. The formal
definition mathematically is stated below:
Minimax Formula
When applying to zero-sum games, it is the equivalent of establishing a Nash Equilibrium (in order to create a stable situation without any regret associated with any of the players).
Introduction to Cryptography
We can now move onto Cryptography. First, let us understand, what is cryptography in the broad sense? Well, cryptography is simply a method of protecting data and information by using specific codes
so that information can only be accessed by the person it is meant to be accessed by.
The process of using ciphertext indistinguishability is used in the backgrounds of many encryption systems in the real world to prevent cyberattacks from occurring. The idea behind these systems is
that an external source cannot encrypt ciphertexts even if they contain specific key words or text within a message. Now let us discuss what this truly means in the broad sense of encryption.
Encryption itself is a subsection of cryptography that involves the scrambling of data in order to give information in a safe manner where only authorized personnel can decrypt data and uncover the
message behind the encryption. Encryption itself is done with something called an encryption key: a mathematical relationship used between the sender and receiver of the data that is unusable by
external sources so that they can safely encrypt methods in a one-way path.
Encryption keys cannot be cracked unless a brute-force (guessing) method can calculate the encryption key using guess and check methods. This is very unlikely as there are millions of combinations of
encryption keys which would render it nearly impossible for brute-force methods to accurately guess.
Types of Encryption
Symmetric Encryption: Only one key is used for both encryption and decryption. This obviously depends on both parties having access to the key, and both parties must keep enough security so that
there cannot be a third party who can rendezvous with the encryption key and take a hold of the transfer of data between the intended 2 parties.
Asymmetric Encryption: This type of encryption involves distinct keys by both the sender and the receiver in order for data to both be encrypted and decrypted. This means the sender and receiver both
use 2 different types of keys in order for this process to occur. This type of encryption is also known as public key encryption as this involves the usage of public key private key pairings which
basically deliberates that data encrypted by private key can only be decrypted by a public key and vice versa.
Zero Knowledge Encryption
Zero knowledge encryption involves using mathematical techniques in order to verify attributes of data or knowledge without having to reveal the underlying data or sources used for this verification.
This encryption can be heavily implemented into the real world as monetary transactions commonly use this technique as they simply can’t access the balance of someone’s credit card and complete a
payment without having to access other related information connected to one’s credit card. Zero knowledge encryption is a completely probabilistic assessment so that means it does not have certainty,
but rather it analyzes small pieces of unlinked data in order to prove what can be probable.
A.I. In a Nutshell
Artificial Intelligence can be defined by the notion of giving computers characteristics of humans that cannot be replicated with normal computer science techniques. These include but are not limited
to, logical reasoning, creativity, decision-making, language, and social skills. These attributes are passed on to computers using machine learning.
Machine Learning
Machine Learning is a pipeline in which data (both inputs and outputs) is combined with a specific algorithm in order to create many inferences on the data that can be in the form of predictions
(supervised ML), patterns (unsupervised ML), or decisions (RL)
Supervised Learning: Given data that is labeled, the goal is to predict the labels of unlabeled data. In other words, it is machine learning with a known outcome, where the best algorithm to reach
that outcome is being determined. (Regressions)
Unsupervised Learning: Given a data set, find structures and patterns. The outcome or end goal is not known, and algorithms are created to summarize and group data.(Clustering)
Reinforcement Learning: Here the agent learns from its environment. Through trials it explores the full range of possibilities to determine the ideal behavior to maximize the reward and minimize
risk. Input state is observed by the agent.
1. Decision making function is used by an agent to perform an action.
2. Agent receives reward or reinforcement/feedback from the environment.
3. The action and state information regarding the reward is saved.
Leading from the types of ML, we get into features, labels, and models. Features are simply the input x variables in our data, labels are our output or y variables from the data, and models are
simply the relationship that is formed between both x and y. This leads to the difference between classification and regression, two basic types of relationships that can be made between x and y.
Classification vs Regression
Regression is used for supervised machine learning models where the data is prelabeled, so for instance predictions of future house prices based on how old a house is would be a proper application of
using regression. On the other hand, classification is putting data into a certain category based on probability. Classification uses logistic regression curves in order to classify data of 2 or more
classes so that the model can best match it to where it belongs on a sigmoid curve. Classification simply outputs a categorical style of output, an example being to check if an email is spam.
Obviously, every email on the internet cannot be labeled as spam or not, so emails can be classified using some of the words used within each email, and then the machine learning model can create a
probability curve based on the likelihood of a specific email being spam or not.
These basics of the machine learning pipeline express how data is not only used for input and output, but it can be used in the broad sense in order to make justifications based off of the patterns
that surround the data. This allows computers to make inferences that are at a much higher level then a simple computer program returning an output that is pre-programmed, hence the name artificial
intelligence is justified.
How to These Fields Intersect?
GANs (Generative Adversarial Networks), reinforcement learning, multi-agent AI systems, etc.… have vast applications of Game Theory. In fact, a lot of the online games played today are heavily
influenced by both these areas of study. Regulating Traffic with AI-powered self driving cars. When these cars are thought of and represented using game theory tools, the situation becomes
manageable. However, without game theory representation, each car makes a decision that is not necessarily beneficial for the entire traffic congestion, causing potential problems. Thus, when
employing tools such as Nash Equilibrium we are able to find a solution where each player (in this case cars) has a pathway out of the traffic jam where that car would not otherwise decide to change
its decision. In doing so, the optimal algorithm can be employed, eventually dispersing the traffic.
Fields that traditionally were considered highly abstract and isolated are now being brought together to solve modern-day problems. What’s even more inspiring and amazing is that there is active
research going on in all these areas, bringing us closer and closer to making our lives better and safer.
“The only thing greater than the power of the mind is the courage of the heart”
— John Nash
John Nash, American Mathematician and Nobel Prize Recipient
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Exciting 3rd Grade Math Activity for 100th Day of School: Solve 'The Case of the 100 Missing Treats
• >
• >
• 100th Day of School Math Mystery Activity - 3rd Grade Edition
100th Day of School Math Mystery Activity - 3rd Grade Edition
SKU: 100thDayofSchool_3rdGRADE
per item
100th Day of School Math Mystery Activity - 3rd Grade Math Worksheets Edition
The Case of the 100 Missing Treats
EASY PREP worksheets, print and solve! Useful for a fun way to practice and review math skills during the 100th Day of School.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
Suggestion: Pace the clues one by one to keep the class at the same point. If using the clues in a different order, keep the suspect list away from students until all five clues are completed and
Optional VIDEO HOOK available to use to introduce students to this math mystery activity. Check it out below:
>>NEW<< - Optional ENDING Video Clip: There is a short video that you can use at the end of the mystery to wrap up the activity.
- Print the student pages, staple or place in a folder, and your students are set to go.
- Have the video hook ready for viewing at the start of the lesson on an IWB, iPad or computer monitor screen.
Math Skills Required to Unlock Clues
There are five clues to crack to solve the mystery:
Clue 1: Make 100 (Missing Addend Equations)
Clue 2: Make 100 (Missing Subtrahend Equations)
Clue 3: Rounding to the nearest 10 or 100
Clue 4: Multiplication Facts Mix (Facts range from 2-12)
Clue 5: Number Patterns (Missing Number in a sequence, no rule given. Adding and subtracting, increasing or decreasing type of patterns.)
Students must use critical thinking skills to figure out what the clue is telling them to eliminate from the list of possibilities. Ensure students read and comprehend clues carefully!
Multiple Uses
- Suitable for independent, pairs, or group work.
- Use as part of your math centers, add to your sub tubs, make it part of your early finisher tasks, give for homework, or make it part of your classroom practice/review sessions.
I recommend pacing this activity by giving students one clue at a time. Once the whole class has completed a clue, then move on to the next clue either within the same lesson or the next math
session. New math content presented? Make a lesson out of it by modeling the math before diving into the clue. I like to say, "we must learn something new before attempting the next clue."
How long will this activity take?
Time to complete will vary anywhere between 30mins - 2 hours or more! It mainly depends on how familiar your students are with the math mystery format, as well as how difficult they find math skills
covered in the particular mystery. Please check the math skills outlined in the clues above to help determine suitability for your class.
MORE versions of this math mystery available in different grade editions. Check out the math skills involved in each packet to see what suits your class best.
100th Day of School Math Mystery Activity - 1st Grade Edition
100th Day of School Math Mystery Activity - 1st Grade Math Worksheets Edition
The Case of the 100 Missing Treats
• Useful for a fun way to practice and review math skills during the 100th Day of School.
• Students must use their math skills to unlock clues. Then, use their powers of deduction to narrow down the suspects to find who stole 100 treats from the school party hall!
• Easy prep! Just Print & Solve! Or go paperless with the new Google Slides option provided within your download.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
100th Day of School Math Mystery Activity - 2nd Grade Edition
100th Day of School Math Mystery Activity - 2nd Grade Math Worksheets Edition
The Case of the 100 Missing Treats
• Useful for a fun way to practice and review math skills during the 100th Day of School.
• Students must use their math skills to unlock clues. Then, use their powers of deduction to narrow down the suspects to find who stole 100 treats from the school party hall!
• Easy prep! Just Print & Solve! Or go paperless with the new Google Slides option provided within your download.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
100th Day of School Math Mystery Activity - 4th Grade Edition
100th Day of School Math Mystery Activity - 4th Grade Math Worksheets Edition
The Case of the 100 Missing Treats
• Useful for a fun way to practice and review math skills during the 100th Day of School.
• Students must use their math skills to unlock clues. Then, use their powers of deduction to narrow down the suspects to find who stole 100 treats from the school party hall!
• Easy prep! Just Print & Solve! Or go paperless with the new Google Slides option provided within your download.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
100th Day of School Math Mystery Activity - 5th Grade Edition
100th Day of School Math Mystery Activity - 2nd Grade Math Worksheets Edition
The Case of the 100 Missing Treats
• Useful for a fun way to practice and review math skills during the 100th Day of School.
• Students must use their math skills to unlock clues. Then, use their powers of deduction to narrow down the suspects to find who stole 100 treats from the school party hall!
• Easy prep! Just Print & Solve! Or go paperless with the new Google Slides option provided within your download.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
100th Day of School Math Mystery Activity - 6th Grade Edition
100th Day of School Math Mystery Activity - 6th Grade Math Worksheets Edition
The Case of the 100 Missing Treats
• Useful for a fun way to practice and review math skills during the 100th Day of School.
• Students must use their math skills to unlock clues. Then, use their powers of deduction to narrow down the suspects to find who stole 100 treats from the school party hall!
• Easy prep! Just Print & Solve! Or go paperless with the new Google Slides option provided within your download.
⭐A video hook is provided to set the stage to engage! Check out the video hook in the preview section.
⭐A bonus ENDING VIDEO clip is included to celebrate finishing the mystery.
(Please note: The videos now come with voiceovers to read the story).
__________________________________________________________________For more ideas, activities, and resources, subscribe to my newsletter to stay updated on new releases. Share how these activities
went by tagging @mrsjsresourcecreations on Instagram.InstagramPinterestFacebook___________________________________________________________________Please note: This is a digital download only. After
purchasing, you will receive an email with a link to download the resource packet. Please save the files for your future use or keep the email in your inbox to be able to re-access the download
again. Should you experience any difficulties, please contact me on [email protected]. Thank you!Mrs. J's Resource Creations 2020 (c)
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Expanded Form and Word Form Calculator
• Enter a number in the "Enter a number" field.
• Click "Calculate" to calculate the expanded and word form of the number.
• Your results will be displayed in the "Expanded Form" and "Word Form" fields.
• You can copy the results to the clipboard using the "Copy" button.
Understanding the Tool
1. Expanded Form in Mathematics:
• Definition: The expanded form of a number breaks it down into its individual place value components. For example, the number 1234 in expanded form is written as 1000 + 200 + 30 + 4.
• Purpose: This form helps understand the value of each digit in a number based on its position. It’s particularly useful in teaching place value concepts in early mathematics education.
2. Word Form of Numbers:
• Definition: The word form of a number is the way to write the number using words. For instance, 1234 is written as “one thousand two hundred thirty-four”.
• Significance: Converting numbers to word form aids in developing number sense and is a key skill in mathematical literacy.
Formulae and Process Used in the Tool
The Expanded Form and Word Form Calculator converts standard numerical representations into expanded and word forms. The process involves:
1. Decomposing Numbers for Expanded Form: Breaking down the number into its place values (units, tens, hundreds, etc.).
2. Converting to Word Form: Translating the numerical value into equivalent spoken or written words.
Step-by-Step Usage of the Tool
1. Input the Number:
2. Calculation:
□ The calculator processes the number to determine its expanded form and word form.
3. Results:
□ It displays the number in both expanded form and word form.
Benefits of Using the Expanded Form and Word Form Calculator
1. Educational Tool: Enhances understanding of place value and numerical representation in early math education.
2. Accuracy: Ensures precise conversion, important for educational accuracy.
3. Time-Saving: Offers quick transformations, useful for teachers and students.
4. Enhances Numerical Literacy: Helps in developing a deeper understanding of how numbers are constructed and communicated.
Practical Applications
1. Mathematics Education: Particularly useful in elementary and middle school education for teaching number representation.
2. Language Learning: Assists non-native speakers in learning to read and write numbers in English.
3. Financial Literacy: Useful in contexts where verbal or written communication of numerical values is required, such as in banking or commerce.
Facts and Additional Insights
1. Historical Perspective: Place value is fundamental in the decimal number system and has been a cornerstone of mathematics since ancient times.
2. Cultural Variations: Different languages and cultures may have unique ways of expressing numbers in word form.
3. Mathematics in Daily Life: The skills of converting numbers into expanded and word forms have practical applications in everyday tasks, such as writing checks or reading financial documents.
1. “Teaching Student-Centered Mathematics” by John A. Van de Walle – Offers insights into effective ways of teaching mathematical concepts, including number representation.
2. “Elementary and Middle School Mathematics: Teaching Developmentally” by Karen S. Karp – Discusses developmental approaches in teaching mathematics, with a focus on understanding numbers.
3. “Mathematics in the Early Years” by David C. Geary and Daniel B. Berch – Explores the foundations of mathematical understanding in early education.
The Expanded Form and Word Form Calculator is a valuable educational tool that is critical in developing foundational mathematical skills. By transforming numbers into expanded and word forms, it
provides a clear and tangible representation of place value and numerical structure, enhancing students’ understanding and fluency in working with numbers.
This tool not only simplifies the teaching and learning of key mathematical concepts but also supports the development of numerical literacy. This skill is essential in both academic and everyday
Last Updated : 03 October, 2024
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Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields,
including database systems, computer networks, and programming. You can read more about him on his bio page.
23 thoughts on “Expanded Form and Word Form Calculator”
1. Wmoore
The clarity and precision of the explanations provided make this article highly informative and beneficial for both educators and learners. It effectively communicates the value and relevance of
the expanded form and word form calculator.
2. Phoebe87
I couldn’t agree more, Wmoore. The article has skillfully articulated the significance and practical implications of place value and numerical representation.
3. Walker Stefan
The article’s insights into the historical perspective of place value and numerical representation provide a rich context for understanding the enduring importance of these concepts. It’s a
commendable aspect of this piece.
4. Shaw Ben
I find it interesting that the article emphasizes the practical applications of the expanded form and word form concepts in daily life. It truly underscores the relevance and utility of these
mathematical skills beyond the classroom setting.
5. Bennett Jack
I concur, Walker Stefan. The historical insights add depth and dimension to the significance of the expanded form and word form in mathematical understanding.
6. Khan Jake
The tool’s practical applications extend beyond mathematical education, making it a valuable resource for promoting both numerical and language literacy. The synergistic benefits it offers are
truly noteworthy.
7. Nikki96
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13 steps I took to prepare for my PhD viva - Dr Salma Patel
I submitted my thesis in Dec 2016. It took much longer than I had anticipated for the examiners and the chair to be approved due to administrative delays. Two months later (in late Feb) I received
confirmation that my exam would be at the end of April.
In terms of the arrangements of the day of the viva, I must say the graduate office in my department were great and said they would organise the day, and I should only worry about preparing for the
exam. I booked advance tickets so I could get to the University early and not have to pay a peak time train fare. I have a written about my viva experience here.
From Jan-March I was working with the public health team at Hackney City Council. I also had Open University marking to do in March. So I started preparing for my thesis approximately a month before
my viva, and this is how:
Step 1: The first thing I did was read this really useful three part blogpost by Fiona Noble. I decided I was going to tackle the preparation in a similar fashion.
Step 2: I knew I had to read the thesis cover to cover, but I was dreading it. I was overcome by fear that I may open the first page and spot mistakes. Or I may start reading it and realise it is a
complete rubbish. So I procrastinated for two days, looking for any excuse not to open the thesis, which is so unlike me. On the third day, I was forced to work in a cafe and lo and behold, I managed
to read almost half the thesis. Whilst reading the thesis, I placed post it notes where I thought I may be asked a question, where I thought something may not be clear and also the references that
were not fresh in my head. Because it had been three months since I had last read the thesis, it really felt like a fresh read, and when I completed reading it, thankfully I said to myself ‘it isn’t
bad at all’ (which means it is good!).
Step 3: I then sat at my desk with my computer, and started to go through all the post it notes. Where it was in relation to a reference, I re-read the abstract of the paper, just to refresh my
memory, and placed a short summary of the paper or the paper title on a larger post it note and stuck it inside the thesis, next to where it was mentioned. I also went through all the other post it
notes, answered the questions, and left small post it notes inside the thesis, in case I would be questioned in my viva and I’d forget the answer.
Step 4: I prepared questions that could come up in the thesis in a Q&A document. I looked online, and found many questions, and I had also bought PhD viva cards a year or so ago. I also went through
the archive of the PhD Viva website which I had set up in 2012 (but very sadly hacked and and deleted). So I sat with about 5 different long lists of questions and the cards in an attempt to
amalgamate them, so that I would end up with a very thorough list of questions. I did that, and towards the the end I realised that the list of questions from this list published online was actually
very comprehensive, and pretty much covered everything that was on other lists (I have copied the list of questions from that resource at the end of this post in case the online resource goes
haywire). I sat and typed up the answers to these questions in note form in a Word document. I realised quickly that the answer to most of these questions was narrated in my thesis, so there was much
copy pasting too. Rowen Murray’s book ‘How to Survive Your Viva’ is a book I dipped into regularly, during this stage and other stages too. It isn’t an essential read, but it is really detailed, and
if you are not feeling confident about answering the questions, it suggests really good ways to approach the answers.
(NB: After I completed this step I found this resource: list of 40 viva questions, which is shorter but looks almost as good too (and in hindsight post-viva, I think it is a better set of questions).
Step 5: I then received an email from my main supervisor asking me to prepare a 10 min presentation for the mock viva, an answer to the question ‘Tell me about your research’, as this is always the
first question. She suggested that I should structure it as follows:
1. about you – what disciplinary perspective are you approaching this from? Your motivation for doing the research topic.
2. research problem, aims, research questions
3. methodology
4. findings
5. conclusions and contributions
I was a bit taken aback to be honest at this point as I wasn’t expecting this type of open-ended question, and I hadn’t really come across it in the notes or books I had read, but once I had a think
about it, it made sense that the first question is likely to be quite open-ended, and as she advised, it made sense to prepare for the first question thoroughly, as it gives you a good strong start.
So I created a Powerpoint using a similar structure to the above (motivations for research, originality of the research, research question, findings, key contributions to knowledge, and key
contributions to practice). I had already prepared this in note format in Step 4 so it didn’t take long to put this presentation together. However, I did spend some time vocalizing the presentation
and practicing it in front of my husband. At this stage I was not sure whether I would use the Powerpoint in the viva, or just use the slides as a way to guide the answer to this question. My
supervisor suggested I should go with whichever I prefer, and I left that decision for closer to the time.
Step 6: I placed bookmarks using post it notes in my thesis, across the top went Chapter numbers and across the side were key areas of my thesis, that I was pretty certain I may need to look at
during my thesis (pages that had limitations of study for example, or why I had chosen one mode of survey over another for example).
Step 7: I had been compiling a list of new papers published since I had submitted that I thought I may be asked to comment about. I printed off the abstracts and read through them. I also went back
to the document prepared in Step 4, and read through it again critically and added more notes to it, and made some of the notes briefer. I also added a few notes to the thesis itself.
Step 8: I re-read the entire thesis for the second time before the mock viva (I suspect this probably wouldn’t be required if you do not have such a huge gap between submission and viva). I also
practiced the presentation and took all my notes along to the mock viva
Step 9: The mock viva ended up being more of a chat than a mock viva. I received some good feedback on the presentation for the viva, and we discussed a few questions that may come up. One of the
questions I hadn’t thought of was: Would your findings have implications on any other fields outside of healthcare? My supervisor also advised me that if both examiners asked me to make a change, or
argued a point strongly, I should accept their advice and say “I’d be happy to make that change” rather than arguing with them.
Step 10: I updated the PPT based on feedback from my supervisor, and I went through the Q&A document, and highlighted the key questions that I needed to take as notes in the interview. I copied those
questions and placed them on to 2 pages. I also placed some of the question and answers inside the thesis, and referenced it with page numbers on this 1 page (double page) notes document that I
decided I would take into the thesis (parts of this double page notes document can be seen in the image at the top of this post).
Step 11: I re-read the entire thesis (for the third time) two weeks before the viva and typed up any typos/errors I found. I also updated the Double page notes document during this time.
Step 12: In the final week, I practiced the presentation, practiced answering viva questions using the viva cards by myself and got my husband to ask me the questions too. I also read the thesis
again (but this time missing some chapters). I printed off the PPT two days before and got all the things ready for the viva using my last-minute checklist (see below). The day before I went through
the Thesis Defence Checklist and ensured it was all ticked off.
Step 13: On the day of the viva I got myself to the venue, had the viva and I passed with very few small minor corrections (8 to be precise). I have written about my viva experience in detail here.
Reflection post-viva:
In hindsight, I probably over prepared slightly, and the majority of the questions I had prepared for never came up. But the preparation gave me a huge amount of confidence, and it meant that I knew
my thesis inside out, so during the viva I easily navigated to certain pages. If I had to go back I would probably use the 40 questions listed below to prep rather than the longer list of questions.
I would definitely not skip the two mock vivas, and the PPT presentation I prepared, because presenting the PPT at very start of the viva meant that I started off very strong and was very well
prepared for their initial few questions. (BTW I used printouts of the PPT to talk over the slides (I gave the examiners a copy too), rather than formally presenting the PPT visually using a digital
Possible viva questions, a long list compiled by ddubdrahcir
1. Summarise your thesis in a sentence.
2. Does the title represent the content?
3. Describe your thesis in brief.
4. How did you decide to order your thesis?
5. What is your overall argument?
6. Summarise the context.
7. Why did you choose this topic?
8. Why is this topic important, and to whom is it relevant?
9. What are the key findings?
10. What is original here; what are your contributions to knowledge?
11. What justifies this thesis as a doctorate?
1. Where did you draw the line on what you included in your literature review?
2. Where did you draw the line on what you included in the theoretical literature?
3. How did the literature inform your choice of topic and the thesis overall?
4. What three publications would you say have been most influential in your work?
5. Where does your work fit into the literature?
6. Who are the key names in this area?
7. Who are the project’s key influences?
8. How does your work differ from theirs?
9. Do the findings confirm, extend, or challenge any of the literature?
10. How does your work connect to that of your reviewers?
Research Design and Methodology
1. Summarise your research design.
2. Did you think about applying a different design?
3. What are the limitations of this kind of study?
4. Is there anything novel in your method?
5. What problems did you have?
6. How did you develop your research questions?
7. Did the research questions change over the course of the project?
8. How did you translate the research questions into a data collection method?
9. What are the philosophical assumptions in your work?
10. Where are YOU in this study?
11. Describe your sample.
12. How did you recruit your sample?
13. What boundaries did you set on your sample?
14. What are the weaknesses of your sample?
15. What boundaries did you set on your data collection?
16. What are the strengths and weaknesses of your data?
17. What other data would you like (or have liked) to collect?
18. What is the theoretical framework in this study?
19. Why did you choose this conceptual framework?
20. Did you think about using any other theories, and if so, why did you reject them?
21. What ethical procedures did you follow?
22. What ethical issues arose in the course of your study and how did you address them?
1. Describe your frame of analysis.
2. How did you construct this framework?
3. What didn’t you include in the framework?
4. What problems did you have in the analysis?
5. Did you combine induction and deduction in your analysis? Can you share some examples?
6. Describe the findings in more detail.
7. Briefly summarise the findings as they relate to each of the research questions.
8. How do you think the theoretical framing was helpful? Can you share some examples?
9. What other data could you have included, and what might it have contributed?
10. Could the findings have been interpreted differently?
1. What are the strengths and weakness of your study?
2. What sense do you have of research being a somewhat untidy, or iterative and constantly shifting process?
3. How confident are you in your findings and conclusions?
4. What the implications of your findings?
5. How has the context changed since you conducted your research?
6. Where do your findings sit in the field in general?
7. How do you see this area developing over the next 5-10 years?
8. Where does your work fit within this?
9. To whom is your work relevant?
10. What haven’t you looked at, and why not?
11. What, if any, of your findings are generalisable?
12. How would you like to follow this project up with further research?
13. What would you publish from this research, and in which journals?
1. How did the project change as you went through?
2. How has your view of the area changed as you have progressed through your research?
3. How did your thinking change over the course of the project?
4. How have you changed as a result of undertaking this project?
5. What did you enjoy about your project?
6. What are you proudest of in the thesis?
7. What were the most difficult areas?
8. What surprised you the most?
9. If you started this study again, what would you do differently?
40 viva questions (a shorter list), compiled by Rebecca at OU Blog
1. Can you start by summarising your thesis?
2. Now, can you summarise it in one sentence?
3. What is the idea that binds your thesis together?
4. What motivated and inspired you to carry out this research?
5. What are the main issues and debates in this subject area?
6. Which of these does your research address?
7. Why is the problem you have tackled worth tackling?
8. Who has had the strongest influence in the development of your subject area in theory and practice?
9. Which are the three most important papers that relate to your thesis?
10. What published work is closest to yours? How is your work different?
11. What do you know about the history of [insert something relevant]?
12. How does your work relate to [insert something relevant]?
13. What are the most recent major developments in your area?
14. How did your research questions emerge?
15. What were the crucial research decisions you made?
16. Why did you use this research methodology? What did you gain from it?
17. What were the alternatives to this methodology?
18. What would you have gained by using another approach?
19. How did you deal with the ethical implications of your work?
20. How has your view of your research topic changed?
21. How have you evaluated your work?
22. How do you know that your findings are correct?
23. What are the strongest/weakest parts of your work?
24. What would have improved your work?
25. To what extent do your contributions generalise?
26. Who will be most interested in your work?
27. What is the relevance of your work to other researchers?
28. What is the relevance of your work to practitioners?
29. Which aspects of your work do you intend to publish – and where?
30. Summarise your key findings.
31. Which of these findings are the most interesting to you? Why?
32. How do your findings relate to literature in your field?
33. What are the contributions to knowledge of your thesis?
34. How long-term are these contributions?
35. What are the main achievements of your research?
36. What have you learned from the process of doing your PhD?
37. What advice would you give to a research student entering this area?
38. You propose future research. How would you start this?
39. What would be the difficulties?
40. And, finally… What have you done that merits a PhD?
Last minute checklist for the viva day:
Place on table:
• Thesis
• Blank paper and working Pen
• Presentation slides printed
• List of corrections
• Double sided notes
• Detailed Question notes (just incase?)
• Water?
Keep in bag:
• List of recent papers published
• All my published papers
• Spare pen and notebook
• Tissue pack
• Chewing gum
• Tickets
• Phone/charger?
• Cash: £20
Last minute generic advice:
• Useful phrases for the viva:
□ Can you rephrase the question?/Is that what you are asking?
□ I am aware …. However …
□ That’s an interesting point, but the way I was thinking about it was …
□ Is that answering your question?
□ I am happy to correct that.
8 Comments
1. […] preparing for the viva. I have written about the 13 steps I took to prepare for the viva in detail here, so I will not go into that […]
2. Thank you very much for this post. I am preparing for my viva voce which will be on 20 Jan. When i re-read my thesis i felt overwhelmed by some parts of my thesis. But i will do my best and be
prepared, so thank you for sharing this
3. it was useful
4. Thank you Salma for sharing your experiences in Viva Land. Much appreciated
5. Dear Dr.salma
Thank you for sharing your valuable viva experience , I will use your post as reference for preparing for my Viva next month ,
i have one question do think the 40 questions list is enough for viva preparation ( general questions)?
Thank you !
□ Hi Munannad,
Personally I think the 40 questions as preparation is most likely sufficient as it covers most areas, and there’s only so much you can prepare anyways, as some questions will come that you
did not anticipate. Do have a mock viva if you can. But don’t worry, you know your thesis inside out and can look into it too to find references or read passages.
All the best with your viva!
Best wishes,
6. Thank you Dr Salma,
Reading through your viva experience has given so much confidence and structure. I will be having my viva voce PhD assessment in Mach 2020 and I am vigorously preparing for it with all my might.
I have also just published an article based on my thesis. So, I thank you very much for sharing you experience, especially the 40 questions Best Wishes
7. Thank you. This is very useful post. Very thoughtful of you to have penned this so others find it helpful
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How to Manually Work with Entity Numbers in Selections
In Part 1 of this blog series, I introduced how you can export a model M-file from COMSOL Multiphysics® simulation software to learn about the structure of the COMSOL Application Programming
Interface (API). One important part of a model M-file is the selections that are made in order to set up properties for the domain, boundaries, etc. These selections are identified using numbers.
Here, we explain how you can automate the handling of the entity numbers using LiveLink™ for MATLAB®.
Handling Selections When Changing the Geometry
When large changes are introduced to a model’s geometry, keeping track of the numbers that domains, boundaries, edges, or vertices are assigned is a challenge. These numbers are used to specify where
certain settings should be applied:
The upper part of the above figure shows a heat sink with only one fin above the heat sink base. This is not a very efficient design, so I’ll add further fins in order to calculate the heat sink’s
performance for different designs.
As you can see in the lower part of the figure, the numbering of the boundaries changes when a second fin is introduced on top of the heat sink base. Naturally, the boundaries that are a part of the
newly introduced fin have to get numbers that are different from the previous design, since these boundaries have not been a part of the model so far. Introducing the new boundaries also means that
some of the old numbers will change.
Note that the numbering is usually very difficult to predict, even for a simple 2D model like this. For a more complicated model in 2D and 3D, it becomes even more complicated. The examples and
code used here could just as easily have been made with a 3D model, but I’ve chosen to go with a 2D model, as it becomes much easier to follow what’s going on.
When applying model settings to boundaries, the numbering becomes important. The figure below shows what the boundary condition for the cool side of the heat sink looks like:
The boundaries are easily identified in the Graphics window on the right. The corresponding numbers are seen in the Heat Flux settings window in the middle (circled in pink).
Selections in the Model M-file Code
Let’s have a look at what the corresponding selections look like in the model M-file code:
The first set is for the upper part, where the cooling takes place, and the second set represents the hot part of the heat sink. When the second fin (or more, depending on your modeling scenario) is
added to the model, we normally have little idea of which selection numbers to use.
There are two distinct ways of keeping track of the entity numbers:
1. Using LiveLink™ for MATLAB® functionality to track geometric entities based on their coordinates
2. Working with selections from geometric operations to track geometric entities
Method 1 is usually the easiest when you are introducing very large changes (such as new objects, for instance, like we will do here) into a geometry and the number of geometric entities change. This
method makes it very easy to work with a model on the command line when changing the model bit by bit.
Method 2 usually requires more steps to set up, but has the added benefit that once the model has been set up, you can avoid the use of entity numbers altogether. The model can be used either via
LiveLink™ for MATLAB® or directly in the COMSOL Multiphysics® user interface (UI), with ease.
Note that if you have a model where you don’t add or remove objects and you do not change its topology, then the numbering of the selections in the model will automatically be updated when you
change model settings, and, for example, move objects from one location to another.
Coordinate Based Selections Using LiveLink™ for MATLAB® Functionality
The model is available as both an M-file and an MPH-file. Typically, the best solution is to use the MPH-file as a base for the changes to the model. Loading an MPH-file is usually faster than
running the corresponding M-file. The MPH-file can, of course, contain meshes and solutions that can’t be saved as part of the M-file, which makes plotting model results a lot easier.
We load the model using this command: model = mphload('heat_param1')
A new fin is introduced into the model by these simple commands:
The design is verified by using this command:
which plots the geometry in a MATLAB® figure.
This produces the lower part of the heat sink figure . In this figure, the numbers can be identified visually, but we would like to automate the process such that we can vary both the number of fins
and their design. Getting the entity numbers of the boundaries that we need is easy if we use the mphselectbox command. This command selects entities using a box for which you have to specify the
coordinates of two opposite corners. There is a corresponding function called mphselectcoords that selects entities inside a circle (or a sphere in 3D), but we will not use that function in this
Below, are the mphselectbox commands that we have to use for this model. For the cooling fins, we supply one set of coordinates that cover the entire upper part of the heat sink. This means that when
I add more fins to the heat sink, these new fins will be covered by the rectangle and I can use the same code to get the selection numbers at that time. For the hot, lower part of the heat sink, I
have to use two calls to mphselectbox and take the union of the selection numbers. “Union” is a standard MATLAB® function that takes the union of a set. It is also possible to use the MATLAB®
functions, “setdiff” and “intersect”, in order to play around with the selection numbers to get the result you wish.
The output looks like this:
Now it is easy to update the model settings with the correct numbers:
Once the selections have been set up, it is easy to adjust the number of fins. The following script shows how to set up a simple loop that will add four fins (one by one) to a model and solve. The
goal is to find the effect of adding more fins to the heat sink.
The first line is used to load the model from an MPH-file. This way, we don’t have to run an M-file, which usually takes longer time than loading an MPH-file.
In the for loop, some more fins are added to the model with a proper size and location. The selection numbers are retrieved using mphselectbox and the model is solved. We generate a plot of each
result to be studied when the analysis is completed.
The method “uniquetag” is used to get a tag for the new fin (Rectangle). When we use uniquetag, we don’t have to guess what tags are already taken and which might be available. This use of uniquetag
with an ‘r’ as argument will return a tag that consists of ‘r’ and a number that is available.
What We Have Learned
When we change the geometry in a model (especially when we introduce new topology), it is important to keep track of the entity numbers for specifying model settings. An efficient way of doing this
in a model M-file, or at the MATLAB® command line, is by using the wrapper functions, mphselectbox and mphselectcoords, that return entity numbers based on entity coordinates.
Next Up
The next blog post in this Working with M-files series will discuss how we can set up a model using selections from geometric operations. This leads to a model that is easy to handle both with
LiveLink™ for MATLAB® and from within the COMSOL Multiphysics® user interface. Using this method means we are working directly with the selection objects and, in many cases, can avoid the use of
entity numbers altogether.
Other Posts in This Series
MATLAB is a registered trademark of The MathWorks, Inc.
Comments (9)
Alain Glière
January 21, 2016
Hi Lars,
I would like to use the parameters of my geometry to create the select box. Instead of :
coordBox = [-300e-6 300e-6 ; -300e-6 300e-6 ; 0.49e-6 0.51e-6];
idx_ftri1 = mphselectbox(model, ‘geom1’, coordBox, ‘boundary’);
I would like to something like :
coordBox = [‘-wSi/2’ ‘wSi/2′;’-wSi/2′ ‘wSi/2’; ‘0.99*hGe’ ‘1.01*hGe’];
but it does not work. I also tried with cells without success.
Thanks in advance,
Alain Glière
Lars Gregersen
January 21, 2016 COMSOL Employee
Hi Alain
Parameters defined in your Comsol model are not automatically known to Matlab.
If you wish to evaluate your expressions in Matlab you have to use the function mphevaluate that can calculate the value of your expressions (including unit if needed).
Georgios Yiannakou
February 9, 2016
Hello, I have a geometry with 720 boundary sections, and I want to specify a different value of Magnetic Flux Density to each one ( I use mfnc physics interface). Is there any way of doing it, or is
it possible to customise the name of the boundaries? Thanks.
Ehsan Aalaei
May 14, 2019
HI Lars Gregersen
In my case, This method apparently is useless when two lines ( or the couple points of them) are in the same box. Do you have any suggestion for this problem?
one solution which comes into my mind was measuring the length of lines and get the desired line with specific length ( It’s possible using “if-condition” and measure function), but those two lines
which are in the same box may be identical in length!
Lars Gregersen
May 14, 2019 COMSOL Employee
Hi Ehsan
I’m not really sure about what you have and what you seek. I interpret you post as: you have 2 vertices (in a box) and you wish to obtain the entity number of the edge between them. Perhaps you can
use this example:
mphopen busbar
idx1 = mphgetadj(model,”,’edge’,’point’,2)
idx2 =mphgetadj(model,”,’edge’,’point’,4)
If you don’t know the indices of your points, but instead know the coordinates then you should use mphselectcoords.
It should all be a matter of extracting the necessary indices and use Matlab’s set operations.
Ehsan Aalaei
May 14, 2019
Thanks Lars for your consideration
In fact, I have the coordinates (and definitely their entity numbers) of two specific points (for instance the point 1 and 2) then, I’m gonna find the entity number of that edge which is made by
those points (it might be a curve or even more complicated edge)
I don’t wanna use the “mphselectbox” or “mphselectcoords” because of this problem;
according to the schematic that I’ve created blew
Imagine in 2D, you have two lines and 3 points
(((( Line1 : between point1 and point2))) & (((Line2 : between point2 and point3)))
and you create a box using “mphselectbox” and the coordinates of the “point1” and “point2” ***to get the entity number of “line1″**. (actually,the entity number of “line1” is our goal)
as you see in this schematic, the result of “mphselectbox” function is the entity number of
“line1” and also “line2” !!!!!!!!!!!!
because “point3” is in this box (which is the smallest box contains “point1” and “point2”)
(point1) ———–(point3)
| \ |
| \ |
| \ | (Line2)
| (Line1) \ |
| \ |
| \ |
| \ |
|_ __ _ _ _ _ _ __(point2)
I have a complicated geometry with many points and edges in 3D and I’m gonna get the entity number of every edge using the start and end points of each edge.
Ehsan Aalaei
May 14, 2019
(point1) ———–(point3)
|~~~~~~\~~~~| (line2)
|_ __ _ _ _ _ _ __(point2)
Lars Gregersen
May 15, 2019 COMSOL Employee
Thanks for you nice drawings 🙂
You could use mphgetadj as shown in the example above. If you have a lot of edges you wish to get the indices for, you may be better off using the functions getStartEnd and getAdj from the COMSOL
Ehsan Aalaei
May 16, 2019
thank you, Lars
It works!!! 🙂
“getStartEnd” returns the start and end vertices (their IDs) of all edges in the first and second
row of the returned matrix. In addition, the number of columns indicate the entity number of related edge.
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FIR Nyquist (L-th band) Filter Design
This example shows how to design lowpass FIR Nyquist filters. It also compares these filters with raised cosine and square root raised cosine filters. These filters are widely used in pulse-shaping
for digital transmission systems. They also find application in interpolation/decimation and filter banks.
Magnitude Response Comparison
The plot shows the magnitude response of an equiripple Nyquist filter and a raised cosine filter. Both filters have an order of 60 and a rolloff-factor of 0.5. Because the equiripple filter has an
optimal equiripple stopband, it has a larger stopband attenuation for the same filter order and transition width. The raised-cosine filter is obtained by truncating the analytical impulse response
and it is not optimal in any sense.
NBand = 4;
N = 60; % Filter order
R = 0.5; % Rolloff factor
TW = R/(NBand/2); % Transition Bandwidth
f1 = fdesign.nyquist(NBand,'N,TW',N,TW);
hEq = design(f1,'equiripple',Zerophase=true,SystemObject=true);
coeffs = rcosdesign(R,N/NBand,NBand,'normal');
coeffs = coeffs/max(abs(coeffs))/NBand;
hRC = dsp.FIRFilter(Numerator=coeffs);
FA = filterAnalyzer(hEq,hRC);
setLegendStrings(FA, ["Equiripple NYQUIST design", "Raised Cosine design"]);
In fact, in this example it is necessary to increase the order of the raised-cosine design to about 1400 in order to attain similar attenuation.
Impulse Response Comparison
Here we compare the impulse responses. Notice that the impulse response in both cases is zero every 4th sample (except for the middle sample). Nyquist filters are also known as L-th band filters,
because the cutoff frequency is $\frac{\pi }{\mathit{L}}$ and the impulse response is zero every L-th sample. In this case, we have 4th band filters.
f1.FilterOrder = 38;
hEq1 = design(f1,'equiripple',Zerophase=true,SystemObject=true);
coeffs = rcosdesign(R,f1.FilterOrder/NBand,NBand,'normal');
coeffs = coeffs/max(abs(coeffs))/NBand;
hRC1 = dsp.FIRFilter(Numerator=coeffs);
FA = filterAnalyzer(hEq1,hRC1,Analysis='impulse');
setLegendStrings(FA,["Equiripple NYQUIST","Raised Cosine"]);
Nyquist Filters with a Sloped Stopband
Equiripple designs allow for control of the slope of the stopband of the filter. For example, the following designs have slopes of 0, 20, and 40 dB/(rad/sample)of attenuation:
f1.FilterOrder = 52;
f1.Band = 8;
f1.TransitionWidth = .05;
hEq1 = design(f1,'equiripple', SystemObject=true);
eq2 = design(f1,'equiripple', StopbandShape='linear', StopbandDecay=20,SystemObject=true);
eq3 = design(f1,'equiripple', StopbandShape='linear', StopbandDecay=40,SystemObject=true);
FA = filterAnalyzer(hEq1,eq2,eq3);
Minimum-Phase Design
We can design a minimum-phase spectral factor of the overall Nyquist filter (a square-root in the frequency domain). This spectral factor can be used in a similar manner to the square-root
raised-cosine filter in matched filtering applications. A square-root of the filter is placed on the transmitter's end and the other square root is placed at the receiver's end.
f1.FilterOrder = 30;
f1.Band = NBand;
f1.TransitionWidth = TW;
hEq1 = design(f1,'equiripple', Minphase=true, SystemObject=true);
coeffs = rcosdesign(R,N/NBand,NBand);
coeffs = coeffs / max(coeffs) * (-1/(pi*NBand) * (pi*(R-1) - 4*R));
srrc = dsp.FIRFilter(Numerator=coeffs);
FA = filterAnalyzer(hEq1,srrc);
setLegendStrings(FA,["Minimum-phase equiripple design","Square-root raised-cosine design"]);
Decreasing the Rolloff Factor
The response of the raised-cosine filter improves as the rolloff factor decreases (shown here for rolloff = 0.2). This is because of the narrow main lobe of the frequency response of a rectangular
window that is used in the truncation of the impulse response.
f1.FilterOrder = N;
f1.TransitionWidth = .1;
hEq1 = design(f1,'equiripple', Zerophase=true, SystemObject=true);
R = 0.2;
coeffs = rcosdesign(R,N/NBand,NBand,'normal');
coeffs = coeffs/max(abs(coeffs))/NBand;
hRC1 = dsp.FIRFilter(Numerator=coeffs);
FA = filterAnalyzer(hEq1,hRC1);
setLegendStrings(FA,["NYQUIST equiripple design","Raised Cosine design"]);
Windowed-Impulse-Response Nyquist Design
Nyquist filters can also be designed using the truncated-and-windowed impulse response method. This can be another alternative to the raised-cosine design. For example we can use the Kaiser window
method to design a filter that meets the initial specs:
f1.TransitionWidth = TW;
kaiserFilt = design(f1, 'kaiserwin', SystemObject=true);
The Kaiser window design requires the same order (60) as the equiripple design to meet the specs. In contrast, we required an extraordinary 1400th-order raised-cosine filter to meet the stopband
FA = filterAnalyzer(hEq,hRC,kaiserFilt);
setLegendStrings(FA,["Equiripple design", "Raised Cosine design", "Kaiser window design"]);
Nyquist Filters for Interpolation
Besides digital data transmission, Nyquist filters are attractive for interpolation purposes. The reason is that every L samples you have a zero sample (except for the middle sample) as mentioned
before. There are two advantages to this, both are obvious by looking at the polyphase representation.
fm = fdesign.interpolator(4,'nyquist');
kaiserFilt = design(fm,'kaiserwin',SystemObject=true);
FA = filterAnalyzer(kaiserFilt,PolyphaseDecomposition=true);
The polyphase subfilter #4 is an allpass filter, and in fact, it is a pure delay. To verify that, select impulse response or view the filter coefficients in filterAnalyzer. That pure delay branch of
the polyphase filter has the following characteristics:
• All of its coefficients are zero except for one, leading to an efficient implementation of that polyphase branch.
• The interpolation filter preserves the input samples values, i.e. $\mathit{y}\left(\mathrm{Lk}\right)=\mathit{u}\left(\mathit{k}\right)$, even though the filter is not ideal.
Related Topics
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Full Cost of Living comparison Dhaka vs Cairo.
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These prices were last updated on November 09, 2024. Exchange rate: 0.41263 EGP / BDT
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II. A grid of O-type stars in the Galaxy and the Magellanic Clouds
Issue A&A
Volume 648, April 2021
Article Number A36
Number of page(s) 16
Section Stellar atmospheres
DOI https://doi.org/10.1051/0004-6361/202038384
Published online 08 April 2021
A&A 648, A36 (2021)
New predictions for radiation-driven, steady-state mass-loss and wind-momentum from hot, massive stars
II. A grid of O-type stars in the Galaxy and the Magellanic Clouds
^1 KU Leuven, Instituut voor Sterrenkunde, Celestijnenlaan 200D, 3001 Leuven, Belgium
e-mail: robin.bjorklund@kuleuven.be
^2 LMU München, Universitätssternwarte, Scheinerstr. 1, 81679 München, Germany
^3 Centro de Astrobiologia, Instituto Nacional de Tecnica Aerospacial, 28850 Torrejon de Ardoz, Madrid, Spain
Received: 8 May 2020
Accepted: 12 August 2020
Context. Reliable predictions of mass-loss rates are important for massive-star evolution computations.
Aims. We aim to provide predictions for mass-loss rates and wind-momentum rates of O-type stars, while carefully studying the behaviour of these winds as functions of stellar parameters, such as
luminosity and metallicity.
Methods. We used newly developed steady-state models of radiation-driven winds to compute the global properties of a grid of O-stars. The self-consistent models were calculated by means of an
iterative solution to the equation of motion using full non-local thermodynamic equilibrium radiative transfer in the co-moving frame to compute the radiative acceleration. In order to study winds in
different galactic environments, the grid covers main-sequence stars, giants, and supergiants in the Galaxy and both Magellanic Clouds.
Results. We find a strong dependence of mass-loss on both luminosity and metallicity. Mean values across the grid are Ṁ~L[*]^2.2 and Ṁ~L[*]^0.95; however, we also find a somewhat stronger dependence
on metallicity for lower luminosities. Similarly, the mass loss-luminosity relation is somewhat steeper for the Small Magellanic Cloud (SMC) than for the Galaxy. In addition, the computed rates are
systematically lower (by a factor 2 and more) than those commonly used in stellar-evolution calculations. Overall, our results are in good agreement with observations in the Galaxy that properly
account for wind-clumping, with empirical Ṁ versus Z[*] scaling relations and with observations of O-dwarfs in the SMC.
Conclusions. Our results provide simple fit relations for mass-loss rates and wind momenta of massive O-stars stars as functions of luminosity and metallicity, which are valid in the range T[eff] =
28 000–45 000 K. Due to the systematically lower values for Ṁ, our new models suggest that new rates might be needed in evolution simulations of massive stars.
Key words: stars: atmospheres / stars: early-type / stars: massive / stars: mass-loss / stars: winds, outflows / Magellanic Clouds
© ESO 2021
1 Introduction
Hot, massive stars with masses ≿9 M[⊙] of spectral type O and B lose a significant amount of mass due to their radiation-driven stellar winds (Castor et al. 1975). This mass loss has a dominant
influence on the life cycles of massive stars as well as in determining the properties of the remnants left behind when these stars die (e.g. Smith 2014). The rates at which these stars lose mass,
which is of the order of Ṁ ~ 10^−5…−9 M[⊙] yr^−1, comprise a key uncertainty in current models of stellar evolution (even on the main sequence where the stars are typically “well behaved”, e.g.
Keszthelyi et al. 2017), simply because the mass of a star is the most important parameter when determining its evolution. In addition to the loss of mass, angular momentum is also lost through
stellar winds affecting the surface rotation speeds of these stars. Moreover, uncertainties related to mass loss have consequences on galactic scales beyond stellar physics, as massive stars provide
strong mechanical and radiative feedback to their host environment (Bresolin et al. 2008).
It is therefore important to have reliable quantitative predictions of mass-loss rates and wind-momenta of massive stars. In the first paper of this series (Sundqvist et al. 2019, from here on Paper
I), we developed a new method to provide mass-loss predictions based on steady-state wind models using radiative transfer in the co-moving frame (CMF) to compute the radiative acceleration g[rad].
Building on that, this paper now presents the results from a full grid of models computed for O-stars in the Galaxy, as well as the Small and the Large Magellanic Cloud, analysing the general
dependence on important stellar quantities, such as luminosity and metallicity. Paper I shows that simulations using CMF radiative transfer suggest reduced mass-loss rates as compared to the
predictions normally included in models of massive star evolution (Vink et al. 2000, 2001). Although this paper focuses on the presentation of our new rates for O-stars in different galactic
environments, a key aim for future publications within this series will be to directly implement the results of the new wind models into calculations of massive-star evolution.
Since most of the important spectral lines driving hot-star winds are metallic, a strong dependence on metallicity Z[*] is expected for the mass-loss rate (Kudritzki et al. 1987; Vink et al. 2001;
Mokiem et al. 2007). To investigate this, here, we compute models tailored to our local Galactic environment, assuming a metal content similar to that in the Sun, as well as for the Magellanic
Clouds. The metallicities of these external galaxies are about half the Galactic one for the Large Magellanic Cloud (LMC) and a fifth for the Small Magellanic Cloud (SMC). Using these three regimes,
we aim to study the mass-loss rate as a function of Z[*]. The Magellanic Clouds are interesting labs for stellar astrophysics because the distances to the stars are relatively well constrained,
providing values of their luminosities and radii. Another reason we are focusing on the Magellanic Clouds is that quantitative spectroscopy of individual stars there has been performed and compiled
into an observed set of scaling-relations for wind-momenta (Mokiem et al. 2007). While the quantitative spectroscopy of individual, hot, massive stars is also possible nowadays in galaxies further
away (e.g. Garcia et al. 2019), such studies are only in their infancy.
In order to derive global dependencies and relations for the mass-loss rates and wind-momenta, we perform a study using a grid of O-star models. Thanks to the fast performance of the method, as
explained in detail in Paper I, this is finally possible for the hydrodynamically consistent steady-state models with a non-paramterised CMF line-force computed without any assumptions about
underlying line-distribution functions. In Sect. 2, we briefly review our method for computing mass-loss rates, highlighting one representative model from the grid. In Sect. 3, the results of the
full grid of models are shown, first for the Galaxy and then including the Magellanic Clouds, in terms of computed wind-momenta and mass-loss rates. From these results we derive simple fit relations
for the dependence on luminosity and metallicity. Section 4 provides a discussion of the results, highlighting the general trends and comparing to other existing models and to observations.
Additionally we address the implication for stellar evolution and current issues such as the so-called weak wind problem. Section 5 contains the conclusions and future prospects.
2 Methods
A crucial part about the radiation-driven steady-state wind models used in our research is that they are hydrodynamically consistent. This means that the equation of motion (e.o.m.) in the
spherically symmetric, steady-state case is solved as described in Paper I. This e.o.m. reads $v(r)dv(r)dr(1−a2(r)v2(r)) = grad(r)−g(r)+2a2(r)r−da2(r)dr.$(1)
Here, v(r) is the velocity, a(r) the isothermal sound speed, g[rad](r) the radiative acceleration, g(r) = GM[*]∕r^2 the gravity, with gravitation constant G, M[*] the stellar mass used in the model,
and r the radius coordinate. The temperature structure T(r) enters the equation through the isothermal sound speed $a2(r) = kbT(r)μ(r)mH,$(2)
with k[B] Boltzmann’s constant, μ(r) the mean molecular weight, and m[H] the mass ofa hydrogen atom. Equation (1) has a sonic point where v(r) = a(r). Because in the above formulation g[rad] is only
an explicit function of radius, and not of the velocity gradient, the corresponding critical point in the e.o.m. is this sonic point (also see the discussion in Paper I). The radiative acceleration
also depends on velocity and mass loss, of course, but in our method these dependencies are implicit; they are accounted for through the iterative updates of the velocity and density structure and do
not affect the critical point condition in an explicit way.
For given stellar parameters luminosity L[*], mass M[*], radius R[*] (to be defined below), and metallicity Z[*], the e.o.m. (1) is solved to obtain v(r) for the subsonic photosphere and supersonic
radiation-driven wind. For a steady-state mass-loss rate Ṁ the mass conservation equation Ṁ = 4πr^2ρv gives the density structure ρ(r). The wind modelsfurther rely on the NLTE (non-local
thermodynamic equilibrium) radiative transfer inFASTWIND (see Paper I and Puls 2017) for the computation of g[rad], by means ofa co-moving frame (CMF) solution and without using any parametrised
distribution functions. The atomic data are taken from the WM-BASIC data base (Pauldrach et al. 2001). This compilation consists of more than a million spectral lines, including all major metallic
elements up to Zn and all ionisation stages relevant for O-stars. The data base is the same as the one utilised in Paper I, as well as in previous versions of theFASTWIND code (see, e.g. Puls et al.
2005). Also the hydrogen and helium model atoms are identical to those used in our previous work. We note that since WM-BASIC is calibrated for diagnostic usage in the UV regime, its principal
database should be ideally suited for the radiative force calculations in focus here. In the NLTE and g[rad] calculation, we account for pure Doppler-broadening alone^1 as depth and mass dependent
profiles, including also a fixed microturbulent velocity v[turb] (see further Paper I, and Sects. 2.3 and 4.5 of this paper). More specific details of the NLTE and radiative transfer in the new CMF
FASTWIND v11 are laid out in detail in Puls et al. (2020, see also Puls 2017 and Paper I).
The steady-state Ṁ and v(r) are converged in the model computation starting from a first guess. For all simulations presented in this paper, the start-value for Ṁ was taken as the mass-loss rate
predicted by the Vink et al. (2001) recipe using the stellar input parameters of the model. The initial velocity structure is obtained by assuming that a quasi-hydrostatic atmosphere connects at v
[tr] ≈ 0.1a(T = T[eff]) to a so-called β-velocity law $v(r) = v∞(1−bR∗r)β,$(3)
with v[∞] the terminal wind speed, R[*] the stellar radius, β a positive exponent, and b a constant derived from the transition velocity v[tr]. We further define the stellar radius $R∗≡r(τ˜F = 2/3),$
where $τ˜F$ is the spherically modified flux-weighted optical depth $τ˜F(r) = ∫r∞ρ (r^)κF(r^)(R∗r^ )2dr^,$(5)
for the flux-weighted opacity κ[F] (cm^2 g^−1). The flux-weighted opacity is related to the radiative acceleration as g[rad] = κ[F]L[*]∕(4πcr^2). After each update of the hydrodynamical structure
(see below), an NLTE/radiative transfer loop is carried out, to converge the occupation numbers and g[rad]. The velocity gradient at the wind critical point is then computed by applying l’Hôpital’s
rule, after which the momentum Eq. (1) is solved with a Runge–Kutta method to obtain the velocity structure v(r) above and below the critical point. This integration is performed by shooting both
outwards from the sonic point to a radius of about 100R[*] to 140R[*] and inwards towards the star, stopping at r = r[min] when a column mass $mc = ∫rmin∞ρ (r)dr = 80$ g cm^−2 is reached.
In addition to the velocity structure, the temperature structure is also updated every hydrodynamic iteration. We use a simplified method similar to Lucy (1971) to speed up the convergence; the
temperature throughout the radial grid is calculated as $T(r) = Teff(W(r)+34τ˜F(r))1/4,$(6)
where W(r) is the dilution factor given by $W(r) = {12(1−1−R∗2r2) if r>R∗,12 if r≤R∗.$(7)
The effective temperature T[eff] in Eq. (6) is defined as $σTeff4≡L∗/4πR∗2$, with σ the Stefan–Boltzman constant and R[*] as defined in Eq. (4). Additionally there is a floor temperature of T ≈ 0.4T
[eff] such as in previous versions ofFASTWIND. This temperature structure is held fixed during the following NLTE iteration, meaning that, formally, perfect radiative equilibrium is not achieved;
however, the effects from this on the wind dynamics are typically negligible (see Paper I).
As described in Paper I, the singularity and regularity conditions applied in CAK-theory cannot be used to update the mass-loss rate because in the approach considered here g[rad] does not explicitly
depend on density, velocity, or the velocity gradient. Instead, here at iteration i the mass-loss rate used in iteration i + 1 is updated to counter the current mismatch in the force balance at the
critical sonic point (see also Sander et al. 2017). For a hydrodynamically consistent solution, the quantity $frc = 1−2a2rg+da2dr1g$(8)
should be equal to Γ = g[rad]∕g at the sonic point. In order to fulfill the e.o.m. (1) the current mismatch is countered by updating the mass-loss rate according to $\Mdoti+1 = \Mdoti(Γ/frc)1/b$,
following the basic theory of line-driven winds where g[rad] ∝ 1∕Ṁ^b (Castor et al. 1975). Our models take a value of b = 1 in the iteration loop, providing a stable way to converge the steady-state
mass-loss rate. From this Ṁ and the computed velocity field, a new density is obtained from the mass conservation equation.
2.1 Convergence
The new wind structure (v(r), ρ(r), T(r), Ṁ) is used in the next NLTE iteration loop to converge the radiative acceleration once more in the radiative-transfer scheme. This method is then iterated
until the error in the momentum equation is small enough to consider the model as converged and thus hydrodynamically consistent. By rewriting the e.o.m. (1) the quantity describing the current error
is $ferr(r) = 1−ΛΓ,$(9)
with $Λ = 1g(vdvdr(1−a2v2)+g−2a2r+da2dr),$(10)
for each radial position r. For a hydrodynamically consistent model f[err] is zero everywhere and Γ should thus be equal to Λ. As such, in the models we need the maximum error in the radial grid
$ferrmax = max(abs(ferr))$(11)
to be close enough to zero; in this paper we require a threshold 0.01, meaning the converged model is dynamically consistent to within a percent. Additional convergence criteria apply to the
mass-loss rate and the velocity structure. These quantities are not allowed to vary by more than 2% for the former and 3% for the latter between the last two hydrodynamic iteration steps reaching
Table 1
Parameters of the characteristic model as described in Sect. 2.2.
2.2 Generic model outcome
As a first illustration, some generic outcomes of one characteristic simulation are now highlighted; the model parameters are listed in Table 1. The top panel of Fig. 1 shows the evolution of Γ for
several iterations in the scheme starting from an initial β-law structure. The characteristic pattern of a steep wind acceleration starting around the sonic point, where Γ ≈1, is clearly visible
throughout the iteration loop. The model starts off being far from consistency (yellow), but as the error in the hydrodynamical structure becomes smaller the solution eventually relaxes to a final
converged velocity structure (dark blue). The innermost points deepest in the photosphere remain quite constant, since the deep photospheric layers relax relatively quickly. In the bottom panel of
Fig. 1 a colour plot of the iterative evolution of the model error f[err] throughout the wind can be seen. The point of maximum error at each iteration is marked with a plus and the dash-dotted line
shows where the velocity equals the sound speed. The dashed lines further show the boundaries within which $ferrmax$ is computed, where we note that the part at very low velocity is excluded because
here the opacity is parametrised (see below). In addition a few of the outermost points are formally excluded in the calculation of $ferrmax$ (due to resolution considerations). Since the calculation
of $ferrmax$ excludes the innermost region and a few outermost points, these points do not contribute to the condition of convergence based on the error in f[err]. Nonetheless the models do provide
reliable terminal wind speeds, as they additionally require the complete velocity structure (including v[∞]) to be converged to better than 3% between the final two iteration steps (see above). The
figure illustrates explicitly how both the overall and maximum errors generally decrease throughout the iteration cycle of the simulation. We note that, after some initial relaxation, for this
particular model the position of maximum error always lies in the supersonic region, often quite close to the critical sonic point.
At the end of the sequence, the model is dynamically consistent and Γ matches the other terms in the equationof motion Λ. This is illustrated in Fig. 2. This figure compares Γ and Λ at each radial
point of the converged model, showing a clear match between the quantities. Only below a velocity v ≲ 0.1 km s^−1 there is some discrepancy; this arises because in these quasi-static layers the
flux-weighted opacity is approximated by a Kramer-like parametrisation (see Paper I), which is useful to stabilise the base in the deep subsonic atmosphere. It is important to point out that this
parametrisation is applied only at low velocities and high optical depths and so does not affect the structure of the wind or the derived global parameters.
The behaviour of the mass-loss rate Ṁ is important to understand for the purposes of this paper. The top panel of Fig. 3 shows $ferrmax$ of the model for all iterations versus the mass-loss rate
computed for that iteration. The general trend is that $ferrmax$ decreases quite consistently during the iteration cycle. The mass-loss rate can be seen to converge to one value as the structure gets
closer to dynamical consistence. Indeed, in the last couple of iterations the value of Ṁ only changes minimally from its former value. In the bottom panel of Fig. 3, the same plot is shown for the
iterative evolution of the terminal wind speed. Also this quantity displays a stable convergence behaviour towards one final value. The quantities Ṁ and v[∞] are from Fig. 3 seen to have an
anti-correlation, as for this model the total wind-momentum rate Ṁ v[∞] does not vary much after the first few iterations. Finally, Fig. 4 shows the converged velocity structure versus the modified
radius coordinate $rrmin−1$, where r[min] is the inner-most radial point of the grid. This figure illustrates the very steep acceleration around the sonic point that is characteristic for our O-star
models, followed by a supersonic β-law-like behaviour typical of a radiation-driven stellar wind. In the figure we add a fit to the velocity structure using a double β-law defined by Eq. (22) and
elaborate further on this comparison in Sect. 4.5.
Fig. 1
Top panel: value of Γ versus scaled radius-coordinate for 7 (non-consecutive) hydrodynamic iterations over the complete run. The starting structure (yellow) relaxes to the final converged structure
(dark blue). Bottom panel: colour map of log (f[err]) for all hydrodynamic iterations; on the abscissa is hydrodynamic iteration number and on the ordinate scaled wind velocity. The pluses indicate
the location of $ferrmax$ for each iteration, the dashed lines the limits between which $ferrmax$ is computedand the dash-dotted line the location of the sonic point.
Fig. 2
Top panel: final converged structure of the characteristic model showing Γ in black squares and Λ (see text for definition) as a green line. The black dashed lines show the location of the sonic
point approximately at Γ = 1 (but not exactly because of the additional pressure terms). Bottom panel: same as in the top panel, however versus velocity (which resolves the inner wind more).
2.3 Grid setup
In order to study the mass-loss rates of O stars in a quantitative way, a model-grid was constructed by varying the fundamental input stellarparameters. For the Galactic stars, we used the calibrated
stellar parameters obtained from a theoretical T[eff] scale of a set of O stars from Martins et al. (2005; adopting the values from their Table 1). The Martins et al. parameters are retrieved by
means of a grid of non-LTE, line-blanketed synthetic spectra models using the code CMFGEN (Hillier & Miller 1998). Here, a self-consistent wind model as described above was calculated for the stellar
parameters of each star in the grid, resulting in predictions of the wind structure, terminal velocity, and mass-loss rate. The wind models were computed with the microturbulent velocity kept at a
standard value for O-stars of v[turb] = 10 km s^−1 (see also Paper I) and the helium number abundance was kept fixed at Y[He] = n[He]∕n[H] = 0.1. Moreover, the simulations were performed without any
inclusion of clumping or high-energy X-rays. As discussed in detail in Paper I, while such wind clumping is both theoretically expected (Owocki et al. 1988; Sundqvist et al. 2018; Driessen et al.
2019)and observationally established (see Sect. 4.2), it is still uncertain what effect this might have on theoretically derived global mass-loss rates (indeed, in the simple tests performed in Paper
I the effect was only marginal). In addition, any inclusion of wind clumping into steady-state models will inevitably be of ad-hoc nature (see also discussion in Paper I). A study by Muijres et al.
(2011), for example, shows that introducing clumps can sometimes change their predicted mass-loss rate by as much as an order of magnitude. The models used in their study, however, use a global
energy constraint to derive the mass-loss rate for an assumed fixed β velocity law. As such, they are not locally consistent and thus might not necessarily fulfill the force requirements around the
sonic point. By contrast,the models presented here are (by design) both locally and globally consistent, with a mass-loss rate that is primarily sensitive to the conditions around the critical sonic
point. It might only be influenced if these corresponding regions where Ṁ is determined, are strongly clumped. This is a key difference between the mass-loss rates derived here and those in Muijres
et al. (2011) and a prime reason that, contrary to their findings, Ṁ in our models does not seem to change significantly when introducing clumping in the supersonic parts. However, as also discussed
in Paper I, the terminal velocities are typically affected by adding such clumping. Namely, since g[rad] is altered in the supersonic regions due to the presence of the clumps, this can lead to
modified values of v[∞]. So even though Ṁ is barely influenced when including typical wind-clumping, it remains an uncertainty of the current models and future work should aim for a more systematic
study of possible feedback effects from clumping upon also the steady-state e.o.m.
In any case, the models presented here contain 12 spectroscopic dwarfs, 12 giants and 12 supergiants for each value of metallicity Z[*]. For simplicity, the same stellar parameters (L[*], M[*], R[*])
as for the Galactic grid were assumed to create models for the LMC and SMC, changing only their metallicity. This set-up has the advantage of enabling a rather direct comparison for the
model-dependence on metallicity, independent of the other input parameters. The used metallicities in the grid are Z[Galaxy] = Z[⊙], Z[LMC] = 0.5 Z[⊙], and Z[SMC] = 0.2 Z[⊙], respectively. The value
of the Solar metallicity was here taken to be Z[⊙] = 0.013 (Asplund et al. 2009). In total this gives 108 models, with input stellar parameters as listed in Table A.1.
Fig. 3
Top panel: iterative behaviour of the mass-loss rate as $ferrmax$ decreases towards a value below 1%. The colour signifies the iteration number starting from Ṁ as predicted by the Vink et al recipe
in light green. Bottom panel: iterative behaviour of the terminal velocity v[∞] towards convergence.
Fig. 4
Converged velocity structure for the characteristic model of Sect. 2.2, showing velocity over terminal wind speed versus the scaled radius-coordinate in black. The green line shows a fit using a
double β-law following Eq. (22) (see text for details).
3 Results
The results for all 108 models are added to Table A.1, containing the derived values for Ṁ and v[∞]. The runs typically took about 50 iterative updates of the hydrodynamical structure to converge,
where for most parts the corresponding calculation of g[rad] (aside from the first ones) takes 10–15 NLTE radiative transfer steps per hydrodynamic update. Using the criteria as presented in Sect.
2.1, all models presented in this paper are formally converged. The following subsections highlight the results for the Galaxy, as well as the Small and the Large Magellanic Clouds.
3.1 The Galaxy
When studying the overall behaviour of line-driven winds, it is useful to look at a modified wind-momentum rate as function of the stellar luminosity: $M˙v∞R∗∝L∗x$(12)
where the left hand side is the so-called modified wind-momentum rate $Dmom≡M˙v∞R∗$ (Kudritzki et al. 1995; Puls et al. 1996), which is proportional to the luminosity to some power x.
The key advantage when using this modified wind-momentum is that basic line driven wind theory predicts the dependence on M[*] to scale out (or at least to become of only second order impact).
Namely, from (modified) CAK theory the following relations can be found (e.g. Puls et al. 2008): $M˙∝L∗1/αeffMeff1−1αeff,v∞∝vesc∝MeffR∗.$(13)
Here, the effective escape speed from the stellar surface is $vesc = 2GMeffR∗$ for an effective stellar mass M[eff] = M[*](1 − Γ[e]), reduced by electron scattering $Γe = κeL∗GM∗4πc$(14)
with an opacity κ[e] (cm^2 g^−1). Equation (13) above further introduces α[eff] = α − δ, where α, describing the power law distribution of the line strength of contributing spectral lines in CAK
theory, takes values between 0 and 1 and the parameter δ accounts for ionising effects in the wind. For a simple α[eff] = 2∕3 we thus have^2 $M˙v∞R∗∝L∗1/αeff,$(15)
as in the wind-momentum-luminosity relation (WLR) of Eq. (12) for x = 1∕α[eff]. However, even if α[eff] is not exactly 2/3, the dependence on M[*] will still be relatively weak. The validity of the
WLR relation is an overall key success of line-driven wind theory and the basic concept has been observationally confirmed by a multitude of studies (see Puls et al. 2008, for an overview).
The Galactic WLR for the radiation-hydrodynamic wind models presented here is shown in the top panel of Fig. 5. The figure indeed shows a quite tight relationship between the modified wind-momentum
rate and luminosity; fitting the models according to Eq. (12) above, a slope x = 2.07 ± 0.32 is derived (with the error mentioned being the 1σ standard deviation). Interpreted in terms of the
(modified) CAK theory above, this would imply a α[eff]≈ 0.5 for our models in the Galaxy, in rather good agreement with the typical O-star values α ≈0.6 and δ ≈ 0.1 (Puls et al. 2008).
Next, the mass-loss rate versus luminosity is plotted in the bottom panel of Fig. 5. From this figure, we infer a rather steep dependence of Ṁ on L[*]; fitting a simple power-law $M˙∝L∗y$ here gives
y = 2.16 ± 0.34. Within our grid, we further do not find any strong systematic trends of mass-loss rate (or wind-momentum rate) with respect to spectral type (in the considered temperature range) and
luminosity class.
Figure 6 shows the terminal wind speeds for the Galactic models. We obtain a mean value v[∞] ∕v[esc] = 3.7 for the complete Galactic sample, however with a relatively large scatter (1σ standard
deviation of 0.8). There is a systematic trend of increasing v[∞]∕v[esc] ratios for lower luminosities (see also Paper I), but also here the corresponding scatter is significant. Overall, although
these Galactic v[∞] ∕v[esc] values are quite high for O-stars, the significant scatter we find is generally consistent with observational studies. Section 4.3 further addresses this, including a
discussion about if a reduction in v[∞] might also affect the prediction of Ṁ.
Fig. 5
Top panel: modified wind-momentum rate versus luminosity for all Galactic models. The solid black line is a linear fit through the points and the dashed line shows the theoretical relation by Vink
et al. (2000). Bottom panel: mass-loss rates versusluminosity for all Galactic models with a linear fit as a solid black line. The dashed line is a fit through the mass-loss rates computed using
the Vink et al. recipe, the dash-dotted line is the relation derived by Krtička & Kubát (2017) and the dotted line is the relation computed from the results from Lucy (2010).
3.2 All models
In the top panel of Fig. 7 the modified wind-momenta for the Magellanic Cloud simulations are added to those of the Galaxy, alsoincluding power-law fits to the models in the LMC and SMC. The bottom
panel of Fig. 7 shows the same plot for Ṁ versus L[*]. On both figures, there is a clear systematic trend that the mass-loss and wind-momentum rates are always lower for the LMC than for the Galaxy
and lower still for the SMC. This is as expected for radiation-driven O-star winds since the majority of driving is done by metallic spectral lines. Inspection of the slope of the WLR at the LMC
reveals a similar trend as for the Galaxy and we derive x = 2.12 ± 0.34 from a fit accordingto Eq. (12). On the other hand, already from simple visual inspection it is clear that for low-luminosity
dwarfs in the SMC the overall slope changes significantly; indeed, an overall fit to Eq. (12) for all SMC stars here results in a higher x = 2.56 ± 0.44.
Another feature visible for the SMC supergiant models is a bump in D[mom] at $logL∗106 L⊙≈−0.3$. As discussed in Sect. 4.1, this feature most likely arises because of the different effective
temperatures of these models, which affects the ionisation balance of important driving elements in the wind.
The derived slopes of the mass-loss rate versus luminosity, $M˙∝L∗y$, for the LMC and SMC are y = 2.17 ± 0.34 and y = 2.37 ± 0.40, respectively.We note that while the dependence for the LMC is
virtually unchanged with respect to the Galaxy, the SMC models again display a steeper dependence, driven by the very low mass-loss rates found for the stars with the lowest luminosities. As
discussed further in Sect. 4, these findings are generally consistent with the line-statistics predictions by Puls et al. (2000) that α[eff] becomes lower both for lower density winds and for winds
of lower metallicity.
Fig. 6
Terminal wind speed over photospheric effective escape speed (corrected with 1-Γ[e], see text) versus luminosity. Values for Galactic metallicity for all three luminosity classes are shown.
Fig. 7
Top panel: modified wind-momentum rate of all models versus luminosity. The dashed lines show linear fits through each of the three sets of models. The markers show the different luminosity
classes, consistent with previous plots. Bottom panel: same as the top panel, but for the mass-loss rate.
3.3 Function of metallicity
Examining trends of the modified wind-momentum rate for the Galaxy, the Large, and the Small Magellanic Cloud, a dependence on metallicity is next derived. As discussed above, Fig. 7 shows that the
wind-momenta of all models with the same metallicity follow a quite tight correlation with L[*]. As such, to investigate the metallicity dependence we consider the three models with identical stellar
parameters, but with different Z[*], assuming for each triplet a simple dependence $M˙v∞R∗∝Z∗n.$(16)
The derived values of n are plotted in Fig. 8, where the distribution gives a mean value n = 0.85 with a 1σ standard deviation of 0.29; this significant scatter is not surprising since we consider
only three different metallicities in the fits. However, inspection of Fig. 8 also reveals a trend of decreasing n with increasing luminosity. Approximating it here by a simple linear fit with
respect to log (L[*]∕10^6 L[⊙]), we find n(L[*]) = −0.73log(L[*]∕10^6 L[⊙]) + 0.46, providing an analytic approximation for the dependence of D[mom] on Z[*] in function of L[*].
The same analysis is performed also for Ṁ, assuming a dependence $M˙∝Z∗m.$(17)
The distribution of the exponent m here gives amean value 0.95. The scatter around this mean is also significant, with a 1σ standard deviation of 0.21. If we again approximate the dependence of the
factor m with L[*], we find m(L[*]) = −0.32log(L[*]∕10^6 L[⊙]) + 0.79.
Building on the combined results above, we can now construct final relations for both the modified wind-momentum rate and mass-loss rate of the form: ${M˙v∞R∗ = A(L∗106 L⊙)y(Z∗Z⊙)n,M˙ = A(L∗106 L⊙)y
To obtain the fitting coefficients (which will be different for the wind-momenta and mass-loss relations), we simply combine the scalings found in Sect. 3.1 for the Galaxy ($M˙v∞R∗∝L∗2.1$ and
$M˙∝L∗2.2$) with those found above for the metallicity dependence ($M˙v∞R∗∝Z∗n(L∗)$ and $M˙∝Z∗m(L∗)$). Moreover, we also performed a multi-linear regression where log(Ṁ) depends on log(Z[*]), log (L
[*]), and log (Z[*]) ⋅ log(L[*]); the fitting coefficients found using these two alternative methods are indeed identical to the second digit. Thus the final relations are $log(M˙v∞R∗) =
−1.55+0.46log(Z∗Z⊙)+[2.07−−0.73log(Z∗Z⊙)]log(L∗106 L⊙)$(19)
and $log(M˙) = −5.55+0.79log(Z∗Z⊙)+[2.16−−0.32log(Z∗Z⊙)]log(L∗106 L⊙),$(20)
with the wind momentum rate in units of M[⊙] yr^−1 km s^−1 R[⊙]^0.5 and the mass − loss rate in units of M[⊙] yr^−1. When the complete model-grid sample is considered, these fitted relations give
mean values that agree with the original simulations to within 10%, with standard deviations 0.33 and 0.39 for the wind-momentum rate and the mass-loss rate, respectively. For none of the models is
the ratio between the actual simulation and the fit larger than a factor 2.1 or smaller than 0.36.
Finally, we can set up the same analysis to derive the dependence of the terminal wind speed on metallicity, assuming v[∞] ∝ Z^p(L[*]). From this we find that the exponent p also varies approximately
linearly with the log of luminosity as p(L[*]) = − 0.41 log(L[*]∕10^6 L[⊙]) − 0.32. This is consistent with the results above and indeed can be alternatively obtained by simply combining the
previously derived Ṁ and D[mom] relations (because p = n − m). The linear behaviour of p here causes low luminosity stars to have a positive exponent with Z, which flattens out and gets negative for
higher luminosities. So as a general trend, stars with log (L[*]∕10^6 L[⊙]) above roughly −0.78 tend to have slightly decreasing v[∞] with increasing metallicity, while for stars below −0.78 it is
the other way around.
Fig. 8
Value of the exponent n showing the metallicity dependence of D[mom] together with a linear fit, showing the linear dependence of this exponent with log (L[*] ∕10^6 L[⊙]). See text for details.
4 Discussion
4.1 General trends
The WLR results presented in the previous section show that our results would be overall consistent with standard CAK line-driven theory for α[eff] ≈ 0.5, at least for the Galactic and LMC cases. As
already mentioned, this is in reasonably good agreement with various CAK and line-statistics results (see overview in Puls et al. 2008). For the O-stars in the SMC, on the other hand, we find a
steeper relation with L[*], implying an overall α[eff] ≈ 0.42 if interpreted by means of such basic CAK theory. However, Fig. 7 shows that the WLR here exhibits significant curvature with a steeper
dependence for lower luminosities, making interpretation in terms of a single slope somewhat problematic. The model-grid indicates that α[eff] decreases both with decreasing metallicity and with
decreasing luminosity, suggesting that α[eff] generally becomes lower for lower wind densities. This is consistent with the line-statistics calculations by Puls et al. (2000) and may (at least
qualitatively) be understood via the physical interpretation of α as the ratio of the line-force due to optically thick lines to the total line-force (lower wind densities should generally mean a
lower proportion of optically thick contributing lines).
We further find a quite steep dependence of mass-loss and wind-momentum rate on metallicity. Power-law fits give average values $M˙v∞R∗∝Z∗0.85$ and $M˙∝Z∗0.95$, however the fitting also reveals a
somewhat steeper dependence for the lower-luminosity stars in our sample. The final fit relations presented in the previous section (Eqs. (19)–(20)) take this into account and give $M˙∝Z∗0.92$ for
the stars in our sample with luminosities above the mean (log(L[*]∕10^6 L[⊙]) > −0.7) and $M˙∝Z∗1.06$ for the ones below this mean (log(L[*]∕10^6 L[⊙]) < −0.7). Again this is generally consistent
with Puls et al. (2000), who derive a scaling relation $M˙∝Z∗(1−α)/αeff$. Inserting in this relation our Galactic value α[eff] ≈ 0.48 and a typical O-star δ ≈ 0.1 (see previous sections) gives
$M˙∝Z∗0.9$, whereas using the lower α[eff] ≈ 0.42 derived for the SMC yields $M˙∝Z∗1.1$. These values are in rather good agreement with the slopes we find from the scaling relations derived directly
from the model-grid results. This tentatively suggests that simplified line-statistics calculations such as those in Puls et al. (2000) might perhaps be used towards further calibration (and physical
understanding) of the scaling relations derived in this paper, provided accurate values of α[eff] (and $Q¯$, which is the line strength normalisation factor due to Gayley 1995) can be extracted from
full hydrodynamic models.
As for the dependence of the terminal wind speed on metallicity, the exponent seems to vary across the grid, being positive for low-luminosity stars and negative for higher luminosity stars. This
might be a manifestation of two (or more) competing processes. We already found that Ṁ always increases when increasing metallicity, which means that winds of high Z[*] tend to be denser and thus
harder to accelerate to high speeds. On the other hand, a higher metallicity also means higher abundances of important driving lines, providing a stronger acceleration that should increase the speed.
For the low luminosity dwarfs, the second effect might be more dominant because these stars already have a low mass-loss rate. As a mean value we find that v[∞] depends on the metallicity as
$v∞∝Z∗−0.10 ± 0.18$. Previous studies such as Leitherer et al. (1992) and Krtička (2006) find this dependence to be $v∞∝Z∗0.13$ and $v∞∝Z∗0.06$ respectively.While these exponents are positive, all
dependencies are very shallow.
As mentioned in previous sections, we do not find any strong trends with spectral type and luminosity class within our model-grid. The mass-loss rates and wind-momenta follow almost the same
relations for spectroscopic dwarfs, giants, and supergiants. The dwarfs do show some deviation from this general trend, in particular for the SMC models, but this is mainly due to the fact that they
span a much larger stellar-parameter range than the other spectral classes, thus reaching lower luminosities and so also the low mass-loss rates where the SMC WLR starts to exhibit significant
curvature (see above).
The dominant dependence of Ṁ on just L[*] and Z[*] within our O-star grid may seem a bit surprising, especially in view of CAK theory which predicts an additional dependence $M˙∝Meff−0.5…1$ (see Eq.
(13)). However, including an additional mass-dependence in our power-law fits to the grid does not significantly improve the fit-quality. Moreover, a multi-regression fit to $M˙∝L∗aM∗b$ for the
Galactic sample shows that if the mass-luminosity relation within our grid $L∗∝M∗c$ is also accounted for, the derived individual exponents return the previously found $M˙∝L∗a+b/c = L∗2.2$ (which
means that we get the same result as when previously neglecting any mass-dependence). Figure 9 shows this mass-luminosity relation, where a simple linear fit yields $L∗∝M∗2.3$ for our model grid. We
note that these results are generally consistent with the alternative CMF models by Krtička & Kubát (2018), who also find no clear dependence on mass or spectral type within their O-star grid.
On the other hand, when running some additional test-models outside the range of our O-star grid, by assuming a fixed luminosity but changing the mass, we do find that including an additional
mass-dependence sometimes can improve the fits. Specifically, it is clear that reducing the mass for such constant-luminosity models generally tends to increase Ṁ. This is in qualitative agreement
with the results expected from CAK theory (see above), however, we note that these test-models have stellar parameters that no longer fall within the O-star regime that is the focus of this paper. In
any case, when extending our grid towards a more full coverage of massive stars in various phases the question regarding an explicit mass-dependency will need to be revisited.
A notable feature for the SMC models is the bump that occurs at $logL∗106 L⊙≈−0.3$ for the supergiants. Figure 10 shows a zoom-in on the simulations that comprise this bump. For each of these models
we examined the dominant ionisation stage at the sonic point of a selected list of important wind-driving elements (Fe, C, N, O, and Ar). As we note above, the mass-loss rate in our simulations is
most sensitive to the conditions around the sonic point. If the ionisation stage of an important element changes there, the opacity and thus also the radiative acceleration may also be modified. In
Fig. 10 the models have increasing effective temperature and luminosity from left to right. The simple scaling relation of Eq. (20) would then predict increasing mass-loss rates from left to right,
in contrast to what is observed in some of these models. As illustrated in the figure, this irregularity coincides with the temperatures at which several key driving-elements change their ionisation
stages, which seem to produce a highly non-linear effect over a restricted range in the current model-grid. Although we have not attempted to explicitly account for such non-linear temperature/
ionisation-effects in the fitting-relations derived in this paper, this will certainly be important to consider when extending our grids towards lower temperatures (in particular the regime where Fe
IV recombines to Fe III, where this effect may become very important, e.g. Vink et al. 1999).
Fig. 9
Mass versus luminosity (from Martins et al. 2005) of all models, both on logarithmic scale to present a linear relation.
Fig. 10
Zoom in on the ‘bump’ that is notably present for the supergiants at SMC metallicity plotted as mass-loss rate versus luminosity on the lower axis and effective temperature on the upper axis. The
dashed line shows the reference slope of 2.37 derived from all SMC models. Each model shows the dominant ionisation stage at the critical point of five elements. They are connected when a
transition occurs between models.
4.2 Comparison to other models
A key result from the overall grid analysis is that the computed mass-loss rates are significantly and systematically lower than those predicted by Vink et al. (2000, 2001), which are the ones most
commonly used in applications such as stellar evolution and feedback. Figure 5 compares the Galactic modified wind-momenta and mass-loss rates of our models directly to these Vink et al. predictions.
Their theoretical WLR presented in the top panel is constructed from a fit to the objects of their sample of observed stars. This sample is used to ensure realistic parameters for v[∞] and R[*],
which together with the theoretical mass loss obtained from the recipe for Ṁ make up the modified wind-momentum rate (see Vink et al. 2000, their Sect. 6.2). The original predictions are calculated
with a higher value for metallicity of Z[⊙] = 0.019, but this is scaled down here to our value of Z[⊙] to remove the metallicity effect in this comparison. We note, however, that such a simple
scaling actually overestimates the effect somewhat, since the abundance of important driving elements such as iron has not changed much (see Paper I). Although both set of models follow rather tight
power-laws with similar exponents (x = 2.06 vs. x = 1.83), there is a very clear offset of about 0.5 dex between the two model-relations over the entire luminosity range. Indeed, every single one of
our models lies significantly below the corresponding one by Vink et al. This is further illustrated in Fig. 11, showing a direct comparison between the mass-loss rates from the full grid (that is to
say for all metallicities) and those computed by means of the Vink et al. recipe. In addition to a systematic offset, this figure also highlights the very low mass-loss rates we find for
low-luminosity stars, in particular in the SMC (reflected in the final fit-relations, Eqs. (19)–20, through the steeper metallicity dependence at low luminosities). Regarding the metallicity scaling,
the $M˙∝Z∗0.85$ obtained byVink et al. (2001) (neglecting then any additional dependence on v[∞] ∕v[esc]) is in quite good agreement with the overall values derived here. As discussed in Paper I, the
systematic discrepancies between our models and those by Vink et al. are most likely related to the fact that their mass-loss predictions are obtained on the basis of a global energy balance argument
using Sobolev theory (and a somewhat different NLTE solution technique) to compute the radiative acceleration for a pre-specified β velocity law. By contrast, the models presented here use CMF
transfer to solve the full equation of motion and to obtain a locally consistent hydrodynamic structure (which may strongly deviate from a β-law in those regions where Ṁ is initiated).
The bottom panel in Fig. 5 further compares our galactic results to those by Krtička & Kubát (2017, 2018). These authors also make use of CMF radiative transfer in the calculation of g[rad] and they
also find lower mass-loss rates as compared to Vink et al. However, the dependence on luminosity derived by Krtička & Kubát is weaker than that obtained here; this occurs primarily because they
predict higher mass-loss rates for stars with lower luminosities, which tends to flatten their overall slope (see Fig. 5 and also Paper I). Although there are some important differences between the
modelling techniques in Krtička & Kubát (2017) and those applied here^3, the overall systematic reductions of Ṁ found both here and by them may point towards the need for a re-consideration of the
mass-loss rates normally applied in simulations of the evolution of massive O-stars (as further discussed in Sect. 4.6 below).
A third comparison of our results, now with those from Lucy (2010), is shown in the bottom panel of Fig. 5. Using a theory of moving reverse layers (MRL, Lucy & Solomon 1970), they compute the mass
flux J = ρv for a grid of O-stars with different T[eff] and log g. The MRL method assumes a plane parallel atmosphere and therefore does not yield a spherical mass-loss rate Ṁ directly. As such, we
obtained Ṁ for the Lucy (2010) simulations by simply assuming the stellar parameters of the models in our own grid, extracting the mass-loss rate from $M˙ = 4πR∗2J$. The relation shown in the bottom
panel of Fig. 5 is then computed by performing a linear fit through the resulting mass-loss rates. This relation derived from the Lucy (2010) models also falls well below the Vink et al. curve, even
predicting somewhat lower rates than us at low luminosities. We note that in the MRL method again no Sobolev theory is used for the Monte-Carlo calculations determining g[rad] and the mass flux.
Fig. 11
Direct comparison of the mass-loss rates calculated in this work versus those predicted by the Vink et al. recipe. The dashed line denotes the one-to-one correspondence. The different markers show
the luminosity classes consistent with previous plots.
4.3 Comparison to observations
Observational studies aim to obtain empirical mass-loss rates based on spectral diagnostics in a variety of wavebands, ranging from the radio domain, over the IR, optical, UV, and all the way to
high-energy X-rays. As discussed in Paper I (see also Puls et al. 2008; Sundqvist et al. 2011, for reviews), a key uncertainty in such diagnostics regards the effects of a clumped stellar wind on the
inferred mass-loss rate. Indeed, if neglected in the analysis, such wind clumping may lead to empirical mass-loss rates that differ by very large factors for the same star, depending on which
diagnostic is used to estimate this Ṁ (Fullerton et al. 2006). More specifically, Ṁ inferred from diagnostics depending on the square of density (for instance radio emission and H[α]) are typically
overestimated if clumping is neglected in the analysis. On the other hand, if porosity in velocity space (see Sundqvist & Puls 2018) is neglected, this may cause underestimations of rates obtained
from UV line diagnostics (Oskinova et al. 2007; Sundqvist et al. 2010; Šurlan et al. 2013). Finally, regarding Ṁ determinations based on absorption of high-energy X-rays (Cohen et al. 2014), these
have been shown to be relatively insensitive to the effects of clumping and porosity for Galactic O-stars (Leutenegger et al. 2013; Hervé et al. 2013; Sundqvist & Puls 2018).
4.3.1 The Galaxy
Considering the above issues, Fig. 12 compares the predictions from this paper with a selected sample of observational studies of Galactic O-star winds. The selected studies are based on X-ray
diagnostics (Cohen et al. 2014), UV+optical analyses accounting for the effects of velocity-space porosity (Sundqvist et al. 2011; Šurlan et al. 2013; Shenar et al. 2015), and UV+optical+IR (Najarro
et al. 2011) and UV+optical (Bouret et al. 2012) studies accounting for optically thin clumping (but neglecting the effects of velocity-space porosity^4). The figure shows the modified wind-momentum
rate from the selected observations (using different markers) together with our relation Eq. (19) (the solid line). Simple visual inspection of this figure clearly shows that our newly calculated
models seem to match these observations quite well. More quantitatively, the dashed line is a fit through the observed wind-momenta, excluding the two stars with L[*] ∕L[⊙] < 10^5 but otherwise
placing equal weights for all observational data points (also for those stars that are included in several of the chosen studies). For the observed stars with luminosities L[*]∕L[⊙] > 10^5, there is
an excellent agreement with the theoretically derived relation Eq. (19), although there is also a significant scatter present in the data (quite naturally considering the difficulties in obtaining
empirical Ṁ, see above).
Regarding the two excluded low-luminosity stars, these seem to indicate a weak wind effect also at Galactic metallicity. Also observational studies of Galactic dwarfs (Martins et al. 2005; Marcolino
et al. 2009) and giants (de Almeida et al. 2019) find that the onset of the weak wind problem seems to lie around log (L[*]∕L[⊙]) ≈ 5.2, where such an onset was already indicated in the data provided
by Puls et al. (1996). Although our mass-loss rates in this regime are significantly lower than those of Vink et al. (2000), they are still not as low as those derived from these observational
studies. Further work would be required extending our current Milky Way grid to even lower luminosities, in order to examine if such an extension would yield a similar downward curvature in the WLR
(albeit at a lower onset luminosity) as suggested by the observational data. Based on our results for SMC stars, we do expect that such a grid-extension might eventually start to display a
significant curvature in the WLR also for Galactic objects. Nonetheless, the mismatch in the onset-luminosity between theory and observations would then still need to be explained. On the other hand,
further work is also needed to confirm the very low empirical mass-loss rates at these low luminosities, as none of the UV-based analyses cited above account for velocity-porosity.
Fig. 12
Observed wind-momenta for stars in the Galaxy from the studies discussed in the text are shown by different markers. The solid black line is our derived relation (19) and the dashed line is a fit
through the observations excluding the data points with L[*] ∕L[⊙] < 10^5 (see text).
Fig. 13
Observations of SMC stars from Bouret et al. (2013) are shown with black squares. Both the relation as predicted by Vink et al. (2001) and from Eq. (20) are shown as comparison together with a
linear fit through the observational data.
4.3.2 Low metallicity
The comparison above focused exclusively on Galactic O-stars, since similar analyses including adequate corrections for clumping are, to this date, scarce for Magellanic Cloud stars. However, a few
such studies do exist, for example the one performed by Bouret et al. (2013) for O dwarfs in the SMC. Again, porosity in velocity space has not been accounted for in deriving Ṁ from the
observationsin this study. Nonetheless, a comparison of our predicted rates to the empirical ones derived by Bouret et al. is shown in Fig. 13. The figure displays our fit relation found for the SMC
stars together with the results from Bouret et al. (2013). Additionally the figure shows a fit to this observed sample of stars (dash-dotted line) and again the relation for the samestars as would be
predicted by the Vink et al. recipe. It is directly clear from this figure that, again, our relation matches these observations of SMC O dwarfs significantly better than previous theoretical
predictions. Indeed, since no big systematic discrepancies are found here between our predictions and the observations, the previously discussed mismatch for low-luminosity Galactic O-dwarfs (“weak
wind problem”) does not seem to be present here for SMC conditions. This effect is also reflected through the significantly higher overall WLR slope we find for SMC models as compared to Galactic
ones, x[SMC] = 2.56 ± 0.44 versus x[Gal] = 2.07 ± 0.32. In terms of line-statistics, a quite natural explanation for this behaviour is that α[eff] decreases with wind density (see previous sections)
which produces a steeper slope both for decreasing metallicity and decreasing luminosity. Another consequence of the found agreement could be that velocity porosity effects are negligible in the
observations of SMC stars.
Moreover, we may also (in a relative manner) compare our predicted scaling relations to observations in different galactic environments, as long as we assume that clumping properties do not vary
significantly between the considered galaxies (and so does not significantly affect the relative observational results). In a large compilation-study of O-stars in the Galaxy, LMC, and SMC, Mokiem et
al. (2007) obtain observationally inferred values of D[mom]. Specifically, Mokiem et al. (2007) used the H[α] emission line to derive the mass-loss rate. This diagnostic is highly dependent on the
clumping factor, but comparison with our scaling results will still be reasonable if the clumping in the H[α] forming region does not vary too much between the different metallicity environments.
Neglecting such potential clumping effects, Mokiem et al. derived the empirical WLR slopes x[Gal] = 1.86 ± 0.2, x[LMC] = 1.87 ± 0.19, and x[SMC] = 2.00 ± 0.27. Within the 1σ errors, this is in
agreement with the theoretical values x[Gal] = 2.07 ± 0.32, x[LMC] = 2.12 ± 0.34, and x[SMC] = 2.56 ± 0.44 obtained here (although the general trend is that we find somewhat steeper relations). In
this respect, we note also that the tight 1σ limits in the empirical relations obtained by Mokiem et al. (2007) only reflect their fit-errors and not additional systematic errors due to uncertainties
in, such as stellar luminosities and adopted metallicities. As such, the overall agreement between their empirical relations and the theoretical WLR slopes presented in this paper is encouraging.
Moreover, by considering their results at a fixed luminosity log L[*]∕L[⊙] = 5.75, Mokiem et al. (2007) also derived an empirical mass-loss vs. metallicity relation $M˙∝Z∗0.83 ± 0.13$; again this is
in rather good agreement with the theoretical $M˙∝Z∗0.87$ obtained at such a logL[*]∕L[⊙] = 5.75 from Eq. (20) of this paper.
4.3.3 Terminal wind speed
As can be seen in Fig. 6, v[∞]∕v[esc] ratios range between approximately 2.5 to 5.5 for our Galactic models. Similar values are found for the Magellanic Cloud simulations, but the scatter is
significant and it is difficult to draw general conclusions from the model data. A significant scatter in v[∞]∕v[esc] is however quite consistent with observational compilations such as the one
presented by Garcia et al. (2014; see in particular their Fig. 9) and Lamers et al. (1995). Overall though, our mean values are somewhat higher than observations generally suggest (see also Kudritzki
& Puls 2000), in particular for low-luminosity objects in the Galaxy. This issue was also addressed in Paper I, where it was argued that the high v[∞] for such low-density winds might be naturally
reduced as a consequence of inefficient cooling of shocks in the supersonic wind (see also Lucy 2012). Namely, if a significant portion of the supersonic wind were shock-heated, this would lower g
[rad] in these layers and so also potentially reduce v[∞]. Moreover, if the wind is shock-heated, empirical estimates of v[∞] might be misinterpreted as it may no longer be possible to accurately
derive this quantity from observations of UV wind lines. Tailored radiation-hydrodynamical simulations of line-driven instability (LDI) induced shocks in low-density winds are underway in order to
examine this in detail and will be presented in an upcoming paper (Lagae et al., in prep.). For now we study how a reduced wind driving in supersonic layers may affect v[∞] and Ṁ in our steady-state
simulations by running a few additional models where g[rad] in layers with speeds above 1000 km s^−1 simply is reduced by an ad-hoc factor of two. While these models then indeed converged to a
significantly lower v[∞] than previously, Ṁ was only marginally affected. This suggests that (at least within the steady-state simulation set-up used here) a reduction of the outer wind driving does
not necessarily lead to an increased mass-loss rate, because in our simulations Ṁ is mostly sensitive to the conditions at the sonic point and is not as much affected by the reduction of g[rad] in
the highly supersonic regions. Although shock-heating in the outer winds in low-luminosity objects might explain the high predicted values of the low luminosity objects, it is unclear to what degree
the increasing trend in v[∞] /v[esc] for decreasing L[*] might still persist in such models. Similarly, it is also somewhat unclear how this matches with observations, as Lamers et al. (1995) do not
find such a trend while Garcia et al. (2014) do show a similar trend, at least for dwarf stars. All these observations show significant scatter though, such that a clear prediction is difficult to
obtain from the present data. To a major part, the scatter in v[∞] is understandable though, since in the outer wind only a few dozen strong resonance lines are responsible for the acceleration
(mostly from C, N, O, Ne, and Ar). Thus, with only a few lines, a certain difference in the abundances (for example because of different ages and mixing) can have a significant effect on v[∞],
leading to the observed scatter.
4.4 Influence of v[turb]
As discussed in Paper I, the turbulent velocity v[turb] can have a significant effect on g[rad] around the sonic point and so also affect the predicted Ṁ. In general, increasing v[turb] tends to
decrease Ṁ (Lucy 2010; Krtička & Kubát 2017; Paper I), mostly because then higher velocities are required to Doppler shift line profiles out of their own absorption shadows, reducing g[rad] in the
critical layers. To study the behaviour of our grid with v[turb], we ran a set of additional simulations where, for each of our previous models, we increased v[turb] to 12.5 km s^−1 and decreased it
to 7.5 km s^−1. This way a slope of logṀ with logv[turb] could be found for each model in the grid. The mean value obtained is $∂logM˙∂logvturb = −1.06 ± 0.40,$(21)
where the error is derived from the 1σ spread of the slope for all models. Also Lucy (2010) studied the behaviour of the mass flux J with v[turb]. Similar to this study, he finds an inverse
dependence, but with a somewhat steeper slope −1.46. We note, however, that this slope was derived from a single model at T[eff] = 40 000 K and log g = 3.75 whereas we here consider an average across
our full grid. Indeed, inspection of a single model similar to the one used in Lucy (2010), with T[eff] = 40 062 K and log g = 3.92, reveals a slope −1.49 that is in good agreement with their result.
Furthermore, considering the large scatter on the slope that we find, with values between a maximum of -0.07 and a minimum of −1.88, the result ofEq. (21) is also still in agreement with that of Lucy
(2010). But although the scatter around our derived mean-slope thus is significant, there are no clear trends within the grid. As such, to obtain a first-order approximation accounting for a v[turb]
that deviates from the standard value 10 km s^−1, the Ṁ relations in Sect. 3.3 may simply be scaled according to Eq. (21).
4.5 Comparison to β velocity law
Most wind models of hot, massive stars used for spectroscopic studies actually do not solve the e.o.m. for the outflow. Instead, these models assume an empirical ‘β-type’ wind velocity-law that
connects to a quasi-hydrostatic photosphere. This is the case also for the “standard” version ofFASTWIND, used here as a starting condition for our self-consistent simulations (see Sect. 2). For the
prototypical simulation presented in Sect. 2.2, Fig. 4 illustrates a fit to the self-consistent calculated velocity using such a β-law. More specifically, a “double” β-law similar to that presented
in Sect. 5.2 of Paper I is used, however in order to obtain a better fit to the very steep acceleration of the transition region we have also included a modification-term in dependence on the
photospheric scale-height. The full expression used to match the velocity structure is $v(r) = (v∞−vexp)(1−rtrr)β+vexp1+(vexpvtr−1)exp(rtr−rH),$(22)
where $β = β1+(β2−β1)(1−rtrr).$(23)
In this relation, the parameters v[∞], β[1], β[2], v[exp], and H are obtained by fitting to the numerically derived velocity structure above the transition radius r[tr], defined according to v(r[tr])
= v[tr]. Introduced here is the parameter v[exp], which is roughly the velocity at which v(r) has its biggest curvature and thus controls how far the exponential behaviour of the velocity holds in
the inner wind. Also introduced is H, which is thescale height setting the stratification in the photosphere, controlled by the density structure of the model.
With this, the model displayed in Fig. 4 can be fitted down to a transition velocity v[tr] ≈ 0.1a, for a primary beta-factor β[1] = 0.8. We note though, that in particular the more low-density winds
in our sample typically require much higher transition velocities in order to be well fit, often values well above v[tr] ≈ 0.5a are found (see also discussion in Paper I). Moreover, most of the
best-fit β[1] -values lie between 0.5–0.8, indicating a steep acceleration of the inner wind across our grid. A more detailed study aiming to develop an improved velocity-parametrisation for
spectroscopic studies is underway and will be presented in an upcoming paper. There, we also plan to examine in detail what effects the predicted steep acceleration in the transition region may have
upon the formation of various strategic spectral lines used for diagnostic work.
4.6 Implications for stellar evolution
The stellar mass is the most important parameter defining the evolution of a star and accurate mass-loss rates are crucial to determine the corresponding evolutionary pathways. Codes for stellar
structure and evolution use prescribed recipes calculating the change in stellar mass in between time steps. Many codes, such as MESA (Paxton et al. 2019, and references therein), offer the Vink et
al. recipe as an option to calculate Ṁ for hot, hydrogen-rich stars (excluding then the classical Wolf-Rayet (WR) stars which require different prescriptions for Ṁ). As clearly illustrated by Fig. 11
, the results presented here suggest that the O-star mass-loss rates should be significantly lower. In addition, this is also supported by the comparison to observations in Sect. 4.3. Even though the
calculations here are performed for massive stars in their early phases of evolution (mostly on the main-sequence), the lower rates will not only affect the stars there, but also impact the
properties of post-main-sequence stages. One consequence is that the luminosity at which the stars end their main-sequence evolution and cross the Hertzprung gap is changed, as seen directly in our
evolution calculations of a 60 M[⊙] star using MESA (where we have simply reduced the amount of mass loss on the main sequence to be in accordance with the models presented here). Another effect is
that a lower mass loss means that angular momentum is lost less rapidly so that the star keeps a higher surface rotation. Keszthelyi et al. (2017) computed models with reduced mass-loss rates (factor
2 to 3) and find the surface rotation speeds at the end of the main-sequence to remain rather high, possibly requiring an additional source of angular momentum loss to reconcile with observed values.
Moreover, Belczynski et al. (2010) studied the masses of compact objects originating from single stars. Here, the adopted mass loss during the life of the progenitor star is crucial in determining
the maximum black hole mass that can be achieved. The determinations of the black hole masses in Belczynski et al. (2010) were done assuming the Vink et al. rates; adopting lower rates would reduce
the amount of mass lost and thus possibly increase the resulting black hole mass. Getting direct measurements of these black hole masses is not straightforward. However, by taking advantage of
gravitational wave astronomy, detections such as GW150914 (Abbott et al. 2016) provide observational constraints. Derived black-hole masses turn out to be relatively high, ≿ 25 M[⊙], which, with
current wind prescriptions, can only be created in low metallicity environments. With lower values of the mass-loss rate during the evolution of the star, such as proposed here in this paper (or as
in magnetic massive stars, Petit et al. 2017), the ‘heavy’ black holes as detected by gravitational waves might in principle also be produced in high-metallicity environments (Belczynski et al. 2020
). So far though, in the Galaxy, no observationsof such heavy mass black holes (from single stars) exist as all of them have a mass less than 15 M[⊙] (as found in studies such as Shaposhnikov &
Titarchuk 2007; Torres et al. 2020). Depending on the initial mass of the progenitor stars, these values can be explained both by our proposed mass-loss rates as well as with those predicted by Vink
et al. (2001).
Significant mass loss is further necessary to create the naked Helium core that is a classical WR-star through wind-stripping. In low metallicity environments such as the SMC, this is generally
considered to be difficult because of the strong metallicity dependence resulting in low mass-loss rates. Considering our results here of lower O-star mass-loss rates, such (steady) wind-stripping
would be more difficult to obtain also in the Galaxy, potentially leading to an increase of the lower limit for the initial mass of WR-stars created by this channel. In order to explain the observed
number of WR-stars, alternative pathways might thus be necessary. In this respect, a straightforward option is binarity where the outer layers of stars can be removed through binary interaction such
as Roche-lobe overflow (e.g. Götberg et al. 2018). Recent studies have shown that a large majority of massive stars reside in such binary systems (Sana et al. 2014). On the other hand, a second
possible channel is eruptive mass loss in the luminous blue variable stage (LBV; Smith 2014). Indeed, significant fractions of the stellar mass can be removed in such eruptive events; the LBV η
-Carina, for example, lost 10 M[⊙] in just 10 yr in the 19th century. Also considering that the (so far confirmed) binary fraction of WR-stars in the SMC is lower than that of the Galaxy, or at least
similar (Foellmi et al. 2003), this latter pathway might prove to be of increased importance.
5 Summary and future prospects
We calculated a grid of steady-state wind models of O-stars by varying fundamental stellar parameters in three metallicity regimes corresponding to the Galaxy, the Small, and the Large Magellanic
Clouds. The models provide predictions of global wind parameters such as mass-loss rate and wind-momentum rate, allowing us to analyse how these quantities depend on fundamental stellar parameters
such as luminosity and metallicity.
From our grid, we find steep dependencies of the mass-loss rate with both luminosity and metallicity, with mean values $M˙~L∗2.2$ and $M˙~Z∗0.95$. The metallicity dependence is further found to vary
across the luminosity range. Accounting for this, results in the final fit relations for the wind-momentum rate and mass-loss rate presented in Sect. 3.3. Additionally, a clear change in the slope
for the predicted WLR for dwarfs in the SMC is found, pointing towards the occurrence of weak winds in the models.
Our computed mass-loss rates are significantly lower for all models than those predicted by Vink et al. (2000, 2001), which are the ones usually implemented in evolution calculations of massive
stars. Such lower O-star wind-momenta and mass-loss rates are also in general accordance with observational studies in the Galaxy that properly account for the effects of clumping upon the
diagnostics used to infer the empirical mass-loss rates. Regarding the metallicity dependence, our scaling predictions are (within the errors) in agreement with the larger empirical study by Mokiem
et al. (2007).
The systematically reduced mass-loss rates for all models strengthens the claim that new rates might be needed in evolution simulations of massive stars. Namely, adopting different rates can
significantly affect the evolution of the massive star, for example by changing its spin-down time and altering the initial mass needed in order to produce a wind-stripped Wolf-Rayet star. As such, a
key follow-up work to this study will be to now extend the grid presented here to include massive stars outside the O-star domain, to incorporate our new models into simulations of massive-star
evolution, and to analyse in detail the corresponding effects.
R.B. and J.O.S. acknowledge support from the Odysseus programme of the Belgian Research Foundation Flanders (FWO) under grant G0H9218N. J.O.S. also acknowledges support from the KU Leuven C1 grant
MAESTRO C16/17/007. F.N. acknowledges financial support through Spanish grants ESP2017-86582-C4-1-R and PID2019-105552RB-C41 (MINECO/MCIU/AEI/FEDER) and from the Spanish State Research Agency (AEI)
through the Unidad de Excelencia “María de Maeztu”-Centro de Astrobiología (CSIC-INTA) project No. MDM-2017-0737. We would also like to thank the referee for useful comments that led to additional
improvements of the paper.
Appendix A Model parameters
Table A.1
Input parameters of all models in the grid together with the resulting mass-loss rate and terminal wind speed.
Relaxing this assumption would have an only marginal effect on the occupation numbers (e.g. Hamann 1981; Lamers et al. 1987). Also for the calculation of g[rad], pure Doppler-broadening is
sufficient, since (i) only few strong lines have significant natural and – in the photosphere – collisionally line-broadened wings that could contibute to g[rad]; and (ii), because of line-overlap
effects, these wings are typically dominated by the (Doppler-core) line opacity from other transitions, which then dominate the acceleration.
Such a value results from considering the distribution of oscillator strengths for resonance lines within a hydrogenic ion and neglecting δ, see Puls et al. (2000).
In particular, since Krtička & Kubát (2017, 2018) scale their CMF line force to the corresponding Sobolev force, this means that their critical point is no longer the sonic point and so that the
nature of their basic hydrodynamic steady-state solutions may be quite different from those presented here. See also discussion in Paper I.
Although these two last studies indeed do not account for velocity-porosity, they do attempt to adjust their studies accordingly; Najarro et al. (2011) by analysis of IR lines that should be free of
such effects and Bouret et al. (2012) by scaling down the phosphorus abundance, thus mimicking the effect velocity–porosity would have on the formation of the unsaturated UV PV lines. As such, we opt
here to include also these two studies in our selected sample for observational comparisons.
All Tables
Table 1
Parameters of the characteristic model as described in Sect. 2.2.
Table A.1
Input parameters of all models in the grid together with the resulting mass-loss rate and terminal wind speed.
All Figures
Fig. 1
Top panel: value of Γ versus scaled radius-coordinate for 7 (non-consecutive) hydrodynamic iterations over the complete run. The starting structure (yellow) relaxes to the final converged structure
(dark blue). Bottom panel: colour map of log (f[err]) for all hydrodynamic iterations; on the abscissa is hydrodynamic iteration number and on the ordinate scaled wind velocity. The pluses indicate
the location of $ferrmax$ for each iteration, the dashed lines the limits between which $ferrmax$ is computedand the dash-dotted line the location of the sonic point.
In the text
Fig. 2
Top panel: final converged structure of the characteristic model showing Γ in black squares and Λ (see text for definition) as a green line. The black dashed lines show the location of the sonic
point approximately at Γ = 1 (but not exactly because of the additional pressure terms). Bottom panel: same as in the top panel, however versus velocity (which resolves the inner wind more).
In the text
Fig. 3
Top panel: iterative behaviour of the mass-loss rate as $ferrmax$ decreases towards a value below 1%. The colour signifies the iteration number starting from Ṁ as predicted by the Vink et al recipe
in light green. Bottom panel: iterative behaviour of the terminal velocity v[∞] towards convergence.
In the text
Fig. 4
Converged velocity structure for the characteristic model of Sect. 2.2, showing velocity over terminal wind speed versus the scaled radius-coordinate in black. The green line shows a fit using a
double β-law following Eq. (22) (see text for details).
In the text
Fig. 5
Top panel: modified wind-momentum rate versus luminosity for all Galactic models. The solid black line is a linear fit through the points and the dashed line shows the theoretical relation by Vink
et al. (2000). Bottom panel: mass-loss rates versusluminosity for all Galactic models with a linear fit as a solid black line. The dashed line is a fit through the mass-loss rates computed using
the Vink et al. recipe, the dash-dotted line is the relation derived by Krtička & Kubát (2017) and the dotted line is the relation computed from the results from Lucy (2010).
In the text
Fig. 6
Terminal wind speed over photospheric effective escape speed (corrected with 1-Γ[e], see text) versus luminosity. Values for Galactic metallicity for all three luminosity classes are shown.
In the text
Fig. 7
Top panel: modified wind-momentum rate of all models versus luminosity. The dashed lines show linear fits through each of the three sets of models. The markers show the different luminosity
classes, consistent with previous plots. Bottom panel: same as the top panel, but for the mass-loss rate.
In the text
Fig. 8
Value of the exponent n showing the metallicity dependence of D[mom] together with a linear fit, showing the linear dependence of this exponent with log (L[*] ∕10^6 L[⊙]). See text for details.
In the text
Fig. 9
Mass versus luminosity (from Martins et al. 2005) of all models, both on logarithmic scale to present a linear relation.
In the text
Fig. 10
Zoom in on the ‘bump’ that is notably present for the supergiants at SMC metallicity plotted as mass-loss rate versus luminosity on the lower axis and effective temperature on the upper axis. The
dashed line shows the reference slope of 2.37 derived from all SMC models. Each model shows the dominant ionisation stage at the critical point of five elements. They are connected when a
transition occurs between models.
In the text
Fig. 11
Direct comparison of the mass-loss rates calculated in this work versus those predicted by the Vink et al. recipe. The dashed line denotes the one-to-one correspondence. The different markers show
the luminosity classes consistent with previous plots.
In the text
Fig. 12
Observed wind-momenta for stars in the Galaxy from the studies discussed in the text are shown by different markers. The solid black line is our derived relation (19) and the dashed line is a fit
through the observations excluding the data points with L[*] ∕L[⊙] < 10^5 (see text).
In the text
Fig. 13
Observations of SMC stars from Bouret et al. (2013) are shown with black squares. Both the relation as predicted by Vink et al. (2001) and from Eq. (20) are shown as comparison together with a
linear fit through the observational data.
In the text
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Astronomical Units to Leagues Converter
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How to use this Astronomical Units to Leagues Converter π €
Follow these steps to convert given length from the units of Astronomical Units to the units of Leagues.
1. Enter the input Astronomical Units value in the text field.
2. The calculator converts the given Astronomical Units into Leagues in realtime β using the conversion formula, and displays under the Leagues label. You do not need to click any button. If the
input changes, Leagues value is re-calculated, just like that.
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5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Astronomical Units to Leagues?
The formula to convert given length from Astronomical Units to Leagues is:
Length[(Leagues)] = Length[(Astronomical Units)] / 3.2273405322519826e-8
Substitute the given value of length in astronomical units, i.e., Length[(Astronomical Units)] in the above formula and simplify the right-hand side value. The resulting value is the length in
leagues, i.e., Length[(Leagues)].
Calculation will be done after you enter a valid input.
Consider that the average distance from Earth to the Sun is 1 astronomical unit (AU).
Convert this distance from astronomical units to Leagues.
The length in astronomical units is:
Length[(Astronomical Units)] = 1
The formula to convert length from astronomical units to leagues is:
Length[(Leagues)] = Length[(Astronomical Units)] / 3.2273405322519826e-8
Substitute given weight Length[(Astronomical Units)] = 1 in the above formula.
Length[(Leagues)] = 1 / 3.2273405322519826e-8
Length[(Leagues)] = 30985264.4927
Final Answer:
Therefore, 1 AU is equal to 30985264.4927 lea.
The length is 30985264.4927 lea, in leagues.
Consider that the distance from Earth to Mars at its closest approach is approximately 0.5 astronomical units (AU).
Convert this distance from astronomical units to Leagues.
The length in astronomical units is:
Length[(Astronomical Units)] = 0.5
The formula to convert length from astronomical units to leagues is:
Length[(Leagues)] = Length[(Astronomical Units)] / 3.2273405322519826e-8
Substitute given weight Length[(Astronomical Units)] = 0.5 in the above formula.
Length[(Leagues)] = 0.5 / 3.2273405322519826e-8
Length[(Leagues)] = 15492632.2464
Final Answer:
Therefore, 0.5 AU is equal to 15492632.2464 lea.
The length is 15492632.2464 lea, in leagues.
Astronomical Units to Leagues Conversion Table
The following table gives some of the most used conversions from Astronomical Units to Leagues.
Astronomical Units (AU) Leagues (lea)
0 AU 0 lea
1 AU 30985264.4927 lea
2 AU 61970528.9855 lea
3 AU 92955793.4782 lea
4 AU 123941057.971 lea
5 AU 154926322.4637 lea
6 AU 185911586.9565 lea
7 AU 216896851.4492 lea
8 AU 247882115.942 lea
9 AU 278867380.4347 lea
10 AU 309852644.9275 lea
20 AU 619705289.855 lea
50 AU 1549263224.6375 lea
100 AU 3098526449.275 lea
1000 AU 30985264492.7499 lea
10000 AU 309852644927.4993 lea
100000 AU 3098526449274.9927 lea
Astronomical Units
An astronomical unit (AU) is a unit of length used in astronomy to measure distances within our solar system. One astronomical unit is equivalent to approximately 149,597,870.7 kilometers or about
92,955,807.3 miles.
The astronomical unit is defined as the mean distance between the Earth and the Sun.
Astronomical units are used to express distances between celestial bodies within the solar system, such as the distances between planets and their orbits. They provide a convenient scale for
describing and comparing distances in a way that is more manageable than using kilometers or miles.
A league is a unit of length that was traditionally used in Europe and Latin America. One league is typically defined as three miles or approximately 4.83 kilometers.
Historically, the league varied in length from one region to another. It was originally based on the distance a person could walk in an hour.
Today, the league is mostly obsolete and is no longer used in modern measurements. It remains as a reference in literature and historical texts.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Astronomical Units to Leagues in Length?
The formula to convert Astronomical Units to Leagues in Length is:
Astronomical Units / 3.2273405322519826e-8
2. Is this tool free or paid?
This Length conversion tool, which converts Astronomical Units to Leagues, is completely free to use.
3. How do I convert Length from Astronomical Units to Leagues?
To convert Length from Astronomical Units to Leagues, you can use the following formula:
Astronomical Units / 3.2273405322519826e-8
For example, if you have a value in Astronomical Units, you substitute that value in place of Astronomical Units in the above formula, and solve the mathematical expression to get the equivalent
value in Leagues.
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Long Multiplication with Google Sheets
My two youngest boys are still in elementary school, so the topic of long multiplication comes up frequently on homework.
Long multiplication is when you multiply two numbers, such as a 3-digit number times a 2-digit number, and you show all the steps by first multiplying by the ones digit, then by the tens digit, then
adding together those partial products to get the final answer.
This is a very structured step-by-step method, so it is important for students to be able to follow the process in the correct order. I wondered if Google Sheets could be used to guide a student
through the steps, help keep all the numbers organized properly, and provide feedback if the student does something wrong.
The answer is yes!
What I did was create some sheets that allow a student to type in the original multiplication problem, and then using hidden columns, custom formulas, and conditional formatting, it provides the
student with feedback each step along the way.
Below are links to access view-only copies of the sheets. You can simply click “File” then “Make a copy” to make your own copies to edit.
Once you have made a copy of the sheet you can:
• Type in your 2X2 or 3X2 multiplication problem in the gray boxes, one digit per box
• Multiply by the ones digit and put your results in the yellow boxes
• Multiply by the tens digit and put your results in the blue boxes
• Add those numbers and put your results in the green boxes
• Any wrong answers will be marked in red
• When done, delete all your numbers to start over with a new problem
Below is a short video showing you an example of using the sheet to work out a long multiplication problem.
Feel free to make copies of these Google Sheets templates to use with students learning long multiplication. I would love to hear any suggestions for improvement or feedback on results.
11 comments:
1. Very interesting.
2. That is awesome!! Thank you for sharing!
3. You are incredibly creative Eric! THANKS!! for sharing all your WONDERFUL tools and ideas with the rest of us!! ;-)
1. Thanks! You are so welcome!
4. Thanks for sharing. I just shared the link with the teachers in my building!!
5. Mike FeldmanJanuary 19, 2016 at 5:42AM
Is there anything like this for Long division?
1. I have not made one for long division, but will certainly take a look at that.
6. Mike FeldmanJanuary 19, 2016 at 2:10PM
Eric, I think I have one for long division. I am still checking it for errors. But I made one that is 2x1, 3x2 and 3x1. When I think I am done I can send it to you on twitter, if you wouldn't
mind checking it out, and of course suggestions would be welcome!
7. This is really great because it gives instant feedback to students. Can you make a version that requires a digit instead of a blank space on the second row for the product of the tens digit by
the ones digit? We teach it so that a 0 is placed in the ones column instead of leaving the space blank.
8. Texting Spy Apps as Your Valuable Asset. Click here for details.
9. You are crazy! I just love all the sheets templates...that's all I have explored yet. Thank you soooooooo much for sharing these. :)
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The Game Is In Theory
Game theory is promoted as a system where you can apply to any aspect of life. This area of economics has had excellent public relations for a long time. It traps people to take a simple idea and
apply it to complicated situations. Its call for imagination is muddling the gap between its usefulness and how much it is being used.
Game theory is predictive of the behavior of agents who abide by that theory, and are thus worth off due to this behavior. As how a set of fables can only give insights but never advice in real life,
game theory misses out on a lot of information and relevant details. This loss comes from formal language and abstractions that are far away from \(1 - \epsilon\) of the population.
The expected utility model proposed by von Neumann-Morgenstern was the basis for game theory. People still use this principle to model uncertainty even after Duncan and Howard^[1] criticized it for
decades and offered better alternatives. There are many ideas in game theory that have not been fully developed. And most of them may fall in the sphere of pure intellectual analysis and not always
hold practical relevance.
Game theory makes simple abstractions of strategic situations, and formalizes them into [often complex] models. It is only good at offering strategy-proof alternatives. Even so, one good real-life
example of its strategic employment is in security. Many research publications have promoted its broad usage in security. Contributions from Thomas Schelling, von Neumann and John Nash while he was
at RAND Corp., Milind Tambe^[2] and team towards better security are some examples. Discourse comprehension and formalizing argumentation is another unsolved interesting problem for game theory. The
best human argumentation model may only be achieved by a highly performant game theoretic model.
For any decision problem, the combined optimization + control theoretic approach often offers a better alternative. MIP with a class of nonlinear convex constraints using an FSM/MPC is an effective
template for this. Social and economic policies hedge on micro-economic theories with strong game theoretic considerations. Market design, however has aspects of efficiency, optimization and
non-strategic based dynamics. We should stop finding solutions in game theory to problems that theory has nothing to say about.
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A particle of mass 2 m is connected by an in extensible string of length 1.2m to a ring of mass m which is free to slide on a horizontal smooth rod. Initially the ring and the particle are at the same level with the string taut. Both are then released simultaneously. The distance in meters moved by the ring when the string becomes vertical is:
A particle of mass 2m is connected by an in extensible string of length 1.2m to a ring of mass m which is free to slide on a horizontal smooth rod. Initially the ring and the particle are at the same
level with the string taut. Both are then released simultaneously. The distance in meters moved by the ring when the string becomes vertical is:
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FAQ: How do I interpret odds ratios in logistic regression?
When a binary outcome variable is modeled using logistic regression, it is assumed that the logit transformation of the outcome variable has a linear relationship with the predictor variables. This
makes the interpretation of the regression coefficients somewhat tricky. In this page, we will walk through the concept of odds ratio and try to interpret the logistic regression results using the
concept of odds ratio in a couple of examples.
From probability to odds to log of odds
Everything starts with the concept of probability. Let’s say that the probability of success of some event is .8. Then the probability of failure is 1 – .8 = .2. The odds of success are defined as
the ratio of the probability of success over the probability of failure. In our example, the odds of success are .8/.2 = 4. That is to say that the odds of success are 4 to 1. If the probability
of success is .5, i.e., 50-50 percent chance, then the odds of success is 1 to 1.
The transformation from probability to odds is a monotonic transformation, meaning the odds increase as the probability increases or vice versa. Probability ranges from 0 and 1. Odds range from 0
and positive infinity. Below is a table of the transformation from probability to odds and we have also plotted for the range of p less than or equal to .9.
p odds
.001 .001001
.01 .010101
.15 .1764706
.2 .25
.25 .3333333
.3 .4285714
.35 .5384616
.4 .6666667
.45 .8181818
.5 1
.55 1.222222
.6 1.5
.65 1.857143
.7 2.333333
.75 3
.8 4
.85 5.666667
.9 9
.999 999
.9999 9999
The transformation from odds to log of odds is the log transformation (In statistics, in general, when we use log almost always it means natural logarithm). Again this is a monotonic
transformation. That is to say, the greater the odds, the greater the log of odds and vice versa. The table below shows the relationship among the probability, odds and log of odds. We have also
shown the plot of log odds against odds.
p odds logodds
.001 .001001 -6.906755
.01 .010101 -4.59512
.15 .1764706 -1.734601
.2 .25 -1.386294
.25 .3333333 -1.098612
.3 .4285714 -.8472978
.35 .5384616 -.6190392
.4 .6666667 -.4054651
.45 .8181818 -.2006707
.5 1 0
.55 1.222222 .2006707
.6 1.5 .4054651
.65 1.857143 .6190392
.7 2.333333 .8472978
.75 3 1.098612
.8 4 1.386294
.85 5.666667 1.734601
.9 9 2.197225
.999 999 6.906755
.9999 9999 9.21024
Why do we take all the trouble doing the transformation from probability to log odds? One reason is that it is usually difficult to model a variable which has restricted range, such as probability.
This transformation is an attempt to get around the restricted range problem. It maps probability ranging between 0 and 1 to log odds ranging from negative infinity to positive infinity. Another
reason is that among all of the infinitely many choices of transformation, the log of odds is one of the easiest to understand and interpret. This transformation is called logit transformation. The
other common choice is the probit transformation, which will not be covered here.
A logistic regression model allows us to establish a relationship between a binary outcome variable and a group of predictor variables. It models the logit-transformed probability as a linear
relationship with the predictor variables. More formally, let $Y$ be the binary outcome variable indicating failure/success with $\{0,1\}$ and $p$ be the probability of $y$ to be $1$, $p = P(Y=1)$.
Let $x_1, \cdots, x_k$ be a set of predictor variables. Then the logistic regression of $Y$ on $x_1, \cdots, x_k$ estimates parameter values for $\beta_0, \beta_1, \cdots, \beta_k$ via maximum
likelihood method of the following equation
$$logit(p) = log(\frac{p}{1-p}) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k.$$
Exponentiate and take the multiplicative inverse of both sides,
$$\frac{1-p}{p} = \frac{1}{exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}.$$
Partial out the fraction on the left-hand side of the equation and add one to both sides,
$$\frac{1}{p} = 1 + \frac{1}{exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}.$$
Change 1 to a common denominator,
$$\frac{1}{p} = \frac{exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)+1}{exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}.$$
Finally, take the multiplicative inverse again to obtain the formula for the probability $P(Y=1)$,
$${p} = \frac{exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}{1+exp(\beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k)}.$$
We are now ready for a few examples of logistic regressions. We will use a sample dataset, https://stats.idre.ucla.edu/wp-content/uploads/2016/02/sample.csv, for the purpose of illustration. The
data set has 200 observations and the outcome variable used will be hon, indicating if a student is in an honors class or not. So our p = prob(hon=1). We will purposely ignore all the significance
tests and focus on the meaning of the regression coefficients. The output on this page was created using Stata with some editing.
Logistic regression with no predictor variables
Let’s start with the simplest logistic regression, a model without any predictor variables. In an equation, we are modeling
logit(p)= β[0 ]
Logistic regression Number of obs = 200
LR chi2(0) = 0.00
Prob > chi2 = .
Log likelihood = -111.35502 Pseudo R2 = 0.0000
hon | Coef. Std. Err. z P>|z| [95% Conf. Interval]
intercept | -1.12546 .1644101 -6.85 0.000 -1.447697 -.8032217
This means log(p/(1-p)) = -1.12546. What is p here? It turns out that p is the overall probability of being in honors class ( hon = 1). Let’s take a look at the frequency table for hon.
hon | Freq. Percent Cum.
0 | 151 75.50 75.50
1 | 49 24.50 100.00
Total | 200 100.00
So p = 49/200 = .245. The odds are .245/(1-.245) = .3245 and the log of the odds (logit) is log(.3245) = -1.12546. In other words, the intercept from the model with no predictor variables is the
estimated log odds of being in honors class for the whole population of interest. We can also transform the log of the odds back to a probability: p = exp(-1.12546)/(1+exp(-1.12546)) = .245, if we
Logistic regression with a single dichotomous predictor variables
Now let’s go one step further by adding a binary predictor variable, female, to the model. Writing it in an equation, the model describes the following linear relationship.
logit(p) = β[0 ]+ β[1]*female
Logistic regression Number of obs = 200
LR chi2(1) = 3.10
Prob > chi2 = 0.0781
Log likelihood = -109.80312 Pseudo R2 = 0.0139
hon | Coef. Std. Err. z P>|z| [95% Conf. Interval]
female | .5927822 .3414294 1.74 0.083 -.0764072 1.261972
intercept | -1.470852 .2689555 -5.47 0.000 -1.997995 -.9437087
Before trying to interpret the two parameters estimated above, let’s take a look at the crosstab of the variable hon with female.
| female
hon | male female | Total
0 | 74 77 | 151
1 | 17 32 | 49
Total | 91 109 | 200
In our dataset, what are the odds of a male being in the honors class and what are the odds of a female being in the honors class? We can manually calculate these odds from the table: for males, the
odds of being in the honors class are (17/91)/(74/91) = 17/74 = .23; and for females, the odds of being in the honors class are (32/109)/(77/109) = 32/77 = .42. The ratio of the odds for female to
the odds for male is (32/77)/(17/74) = (32*74)/(77*17) = 1.809. So the odds for males are 17 to 74, the odds for females are 32 to 77, and the odds for female are about 81% higher than the odds for
Now we can relate the odds for males and females and the output from the logistic regression. The intercept of -1.471 is the log odds for males since male is the reference group (the variable female
= 0). Using the odds we calculated above for males, we can confirm this: log(.23) = -1.47. The coefficient for female is the log of odds ratio between the female group and male group: log(1.809) =
.593. So we can get the odds ratio by exponentiating the coefficient for female. Most statistical packages display both the raw regression coefficients and the exponentiated coefficients for
logistic regression models. The table below is created by Stata.
Logistic regression Number of obs = 200
LR chi2(1) = 3.10
Prob > chi2 = 0.0781
Log likelihood = -109.80312 Pseudo R2 = 0.0139
hon | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
female | 1.809015 .6176508 1.74 0.083 .9264389 3.532379
Logistic regression with a single continuous predictor variable
Another simple example is a model with a single continuous predictor variable such as the model below. It describes the relationship between students’ math scores and the log odds of being in an
honors class.
logit(p) = β[0 ]+ β[1]*math
Logistic regression Number of obs = 200
LR chi2(1) = 55.64
Prob > chi2 = 0.0000
Log likelihood = -83.536619 Pseudo R2 = 0.2498
hon | Coef. Std. Err. z P>|z| [95% Conf. Interval]
math | .1563404 .0256095 6.10 0.000 .1061467 .206534
intercept | -9.793942 1.481745 -6.61 0.000 -12.69811 -6.889775
In this case, the estimated coefficient for the intercept is the log odds of a student with a math score of zero being in an honors class. In other words, the odds of being in an honors class when
the math score is zero is exp(-9.793942) = .00005579. These odds are very low, but if we look at the distribution of the variable math, we will see that no one in the sample has math score lower
than 30. In fact, all the test scores in the data set were standardized around mean of 50 and standard deviation of 10. So the intercept in this model corresponds to the log odds of being in an
honors class when math is at the hypothetical value of zero.
How do we interpret the coefficient for math? The coefficient and intercept estimates give us the following equation:
log(p/(1-p)) = logit(p) = – 9.793942 + .1563404*math
Let’s fix math at some value. We will use 54. Then the conditional logit of being in an honors class when the math score is held at 54 is
log(p/(1-p))(math=54) = – 9.793942 + .1563404 *54.
We can examine the effect of a one-unit increase in math score. When the math score is held at 55, the conditional logit of being in an honors class is
log(p/(1-p))(math=55) = – 9.793942 + .1563404*55.
Taking the difference of the two equations, we have the following:
log(p/(1-p))(math=55) – log(p/(1-p))(math = 54) = .1563404.
We can say now that the coefficient for math is the difference in the log odds. In other words, for a one-unit increase in the math score, the expected change in log odds is .1563404.
Can we translate this change in log odds to the change in odds? Indeed, we can. Recall that logarithm converts multiplication and division to addition and subtraction. Its inverse, the
exponentiation converts addition and subtraction back to multiplication and division. If we exponentiate both sides of our last equation, we have the following:
exp[log(p/(1-p))(math=55) – log(p/(1-p))(math = 54)] = exp(log(p/(1-p))(math=55)) / exp(log(p/(1-p))(math = 54)) = odds(math=55)/odds(math=54) = exp(.1563404) = 1.1692241.
So we can say for a one-unit increase in math score, we expect to see about 17% increase in the odds of being in an honors class. This 17% of increase does not depend on the value that math is held
Logistic regression with multiple predictor variables and no interaction terms
In general, we can have multiple predictor variables in a logistic regression model.
logit(p) = log(p/(1-p))= β[0 ] + β[1]*x1 + … + β[k]*xk
Applying such a model to our example dataset, each estimated coefficient is the expected change in the log odds of being in an honors class for a unit increase in the corresponding predictor variable
holding the other predictor variables constant at certain value. Each exponentiated coefficient is the ratio of two odds, or the change in odds in the multiplicative scale for a unit increase in the
corresponding predictor variable holding other variables at certain value. Here is an example.
logit(p) = log(p/(1-p))= β[0 ] + β[1]*math + β[2]*female + β[3]*read
Logistic regression Number of obs = 200
LR chi2(3) = 66.54
Prob > chi2 = 0.0000
Log likelihood = -78.084776 Pseudo R2 = 0.2988
hon | Coef. Std. Err. z P>|z| [95% Conf. Interval]
math | .1229589 .0312756 3.93 0.000 .0616599 .1842578
female | .979948 .4216264 2.32 0.020 .1535755 1.80632
read | .0590632 .0265528 2.22 0.026 .0070207 .1111058
intercept | -11.77025 1.710679 -6.88 0.000 -15.12311 -8.417376
This fitted model says that, holding math and reading at a fixed value, the odds of getting into an honors class for females (female = 1)over the odds of getting into an honors class for males (
female = 0) is exp(.979948) = 2.66. In terms of percent change, we can say that the odds for females are 166% higher than the odds for males. The coefficient for math says that, holding female and
reading at a fixed value, we will see 13% increase in the odds of getting into an honors class for a one-unit increase in math score since exp(.1229589) = 1.13.
Logistic regression with an interaction term of two predictor variables
In all the previous examples, we have said that the regression coefficient of a variable corresponds to the change in log odds and its exponentiated form corresponds to the odds ratio. This is only
true when our model does not have any interaction terms. When a model has interaction term(s) of two predictor variables, it attempts to describe how the effect of a predictor variable depends on
the level/value of another predictor variable. The interpretation of the regression coefficients become more involved.
Let’s take a simple example.
logit(p) = log(p/(1-p))= β[0] + β[1]*female + β[2]*math + β[3]*female*math
Logistic regression Number of obs = 200
LR chi2(3) = 62.94
Prob > chi2 = 0.0000
Log likelihood = -79.883301 Pseudo R2 = 0.2826
hon | Coef. Std. Err. z P>|z| [95% Conf. Interval]
female | -2.899863 3.094186 -0.94 0.349 -8.964357 3.164631
math | .1293781 .0358834 3.61 0.000 .0590479 .1997082
femalexmath | .0669951 .05346 1.25 0.210 -.0377846 .1717749
intercept | -8.745841 2.12913 -4.11 0.000 -12.91886 -4.572823
In the presence of interaction term of female by math, we can no longer talk about the effect of female, holding all other variables at certain value, since it does not make sense to fix math and
femalexmath at certain value and still allow female change from 0 to 1!
In this simple example where we examine the interaction of a binary variable and a continuous variable, we can think that we actually have two equations: one for males and one for females. For males
(female=0), the equation is simply
logit(p) = log(p/(1-p))= β[0 ]+ β[2]*math.
For females, the equation is
logit(p) = log(p/(1-p))= (β[0 ]+ β[1]) + (β[2] + β[3 ])*math.
Now we can map the logistic regression output to these two equations. So we can say that the coefficient for math is the effect of math when female = 0. More explicitly, we can say that for male
students, a one-unit increase in math score yields a change in log odds of 0.13. On the other hand, for the female students, a one-unit increase in math score yields a change in log odds of (.13 +
.067) = 0.197. In terms of odds ratios, we can say that for male students, the odds ratio is exp(.13) = 1.14 for a one-unit increase in math score and the odds ratio for female students is exp
(.197) = 1.22 for a one-unit increase in math score. The ratio of these two odds ratios (female over male) turns out to be the exponentiated coefficient for the interaction term of female by math:
1.22/1.14 = exp(.067) = 1.07.
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Problem D: Defensive Tower
On the mainland, there is a fire-breathing dragon called "lanran", who is always burning cities and attacking people. So Pisces decides to build some defensive towers in the kingdom to protect his
people. A defensive tower is able to protect the city where it is located and the cities adjacent to it. However, building a tower costs a lot, so Pisces could only build at most \(\lfloor n/2\rfloor
\) defensive towers (\(n\) is the total number of cities in the kingdom). Please find a way to build defensive towers in order to protect the whole kingdom. If there are multiple answers, print any.
By saying that "two cities are adjacent", it means that there is one undirected road connecting them.
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Year 10+ 3D Geometry
Many things, like party hats and ice cream cones are shaped like cones. Wizard's hats and dunce caps are also cone-shaped. A witch's hat is shaped a bit like a cone with a brim.
We see traffic cones on our roads. They have a shape like a cone, with a bit sawn off the top and a widened-out base so that they don't topple over. We call objects that are shaped like cones
conical. So, what exactly is a cone, and what are its properties?
The Structure of a Cone
Cones are solids with a flat base, and one curved side. The flat base may be shaped like a circle or an ellipse. Most of the time, it's a circle. Cones are not polyhedrons as they have a curved
surface, and no straight edges.
The flat bottom of a cone is called its base, and the pointy bit at the top of the cone is called its apex, as shown in the picture below.
How Do Mathematicians Build a (Circular) Cone?
You may have made a cone yourself at some stage, by taking a bit of cardboard shaped like the one shown below and bending it around until the edges meet.
I've made them as witches and wizards hats, and as lolly baskets for the Christmas tree.
Mathematicians have a slightly more abstract view of how to create a cone. They take a right-angled triangle and rotate it around one of its two shorter sides:
The side that the triangle is rotated around is called the
of the cone.
Right and Oblique Cones
Just as with pyramids, there are two types of cones: right cones are cones in which the apex lies directly about the centre of the base, and oblique cones are cones in which it doesn't:
In the above picture, the cone on the left is a right cone, while the cone on the right is an oblique cone.
Finding the Surface Area of a Cone
The surface area of a cone may be split up into two parts:
• The area of the base. For a circular cone, this is \(\pi r^2\), where \(r\) is the radius of the cone.
• The area of the curved surface. This is equal to \(\pi r s\), where \(r\) is the radius of the cone, and \(s\) is the slant height of the cone as shown in the picture on the left.
Adding these two quantities together gives the surface area of the cone:
\(SA =\pi r^2 + \pi rs = \pi r(r + s)\).
Note that we can use Pythagoras' theorem to find \(s = \sqrt{r^2 + h^2}\).
For example, if we were asked to find the surface area of a cone with height \(h = 3\) cm and radius \(r = 4\) cm, then we would calculate \(s = \sqrt{3^2 + 4^2} = 5\) cm, and
\(SA = \pi (4)(4 + 3) = 28 \pi \approx 88 \text{ cm}^2\).
Finding the Volume of a Cone
The formula for the volume of a right circular cone is
\(V = \dfrac{1}{3} \pi r^2 h\),
where \(h\) is the height of the cone, and \(r\) is the radius of its base.
For example, the volume of a cone with height \(h = 3\) cm and base radius \(r = 4\) cm is given by
\(V = \dfrac{1}{3} \pi (4)^2(3) = 16 \pi \approx 50.3 \text{ cm}^3\).
Volumes of Cones and Volumes of Cylinders
Did you notice something about the formula for the volume of a cone? It looks a lot like the formula for the volume of a cylinder.
In fact, we can draw a cylinder of the same base radius and height around a cone, as shown in the picture below:
The formula for the volume of the cylinder is \(V = \pi r^2 h\), so the volume of the cone is equal to \(\dfrac{1}{3}\) of the volume of the cylinder of the same base radius and height.
If \(\dfrac{1}{3}\) is sounding awfully familiar to you, that's because the volume of a right pyramid is also equal to \(\dfrac{1}{3}\) of the volume of the prism of the same base and height. This
isn't a coincidence: you can think of a cone as a "pyramid with a circular base".
There are several lessons related to 3D geometry such as
1. Euler's formula
2. Vertices, Edges and Faces
3. Volumes of 3D shapes
4. etc
Even though we've titled this lesson series to be more inclined for Year 10 or higher students, however, these lessons can be read and utilized by lower grades students.
Understanding of 3D shapes
Year 10 or higher, but suitable for Year 8+ students
Learning Objectives
Get to know 3D Geometry
Author: Subject Coach
Added on: 27th Sep 2018
You must be logged in as Student to ask a Question.
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Calculate Fortune's Formula with Perl
The Kelly criterion is an equation for deriving the optimal fraction of a bankroll to place on a bet, given the probability and the betting odds. I read about it a few years ago in William
Poundstone’s page turner, Fortune’s Formula. To use the Kelly criterion in Perl code, you can used Algorithm::Kelly, a module I released last week.
Using Algorithm::Kelly
Algorithm::Kelly exports the optimal_f sub, which takes two parameters: the probability of the event occurring (a value between 0.00 and 1.00) and the payoff (net betting odds). optimal_f returns the
optimal fraction of your betting bankroll to place on the bet.
For example if I want to find the optimal f of a bet which has a 50% chance of winning, and pays 3-to-1:
use Algorithm::Kelly;
my $optimal_f = optimal_f(0.5, 3);
Here optimal_f returns a value of 0.25, which means I should place 25% of my bankroll on this bet. Let’s look at another example: a bet which has 12% chance of occurring and pays 5-to-1. I can also
calculate optimal f at the command line:
$ perl -MAlgorithm::Kelly -E 'say optimal_f(0.12, 5)';
So this time, optimal f is -0.056, or negative 5.6%, which means I shouldn’t take this bet as the odds are not generous enough, given the probability of the bet winning. This is tremendously useful:
the optimal fraction can be used to eliminate bad bets, and also rank competing betting options, to find the best value bet.
Practical pitfalls
The Kelly criterion is only as accurate as its inputs, and whilst it’s easy to look up the odds offered for a particular bet, precisely calculating the probability of the bet winning is usually a far
more difficult task. It’s easy to calculate the probability for casino games like roulette, but they have negative optimal fs and are not worth pursuing. Some successful sport bettors use statistical
modeling techniques to estimate the probability of a bet winning - but this is only an estimate.
The second issue with the Kelly criterion is the size of optimal f. The Kelly criterion will always maximize return over the long term, but there is not an infinite market of bets available, and
regularly risking high percentages of your bankroll will mean a big short term losses. Further, even if you have a sizable bankroll, many markets are simply not liquid enough to accommodate the size
of bets recommended by optimal f. Bettors will often use a “half-Kelly” instead, which is the optimal f of a bet divided by 2.
This article was originally posted on PerlTricks.com.
Something wrong with this article? Help us out by opening an issue or pull request on GitHub
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Riemannian Manifold
Review of Short Phrases and Links
This Review contains major "Riemannian Manifold"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion.
2. A Riemannian manifold (M,g) is a smooth manifold M, together with a Riemannian metric on M, i.e. (Web site)
3. A (pseudo -) Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.
4. A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it. (Web site)
5. A Riemannian manifold is collapsed with a lower curvature bound if the sectional curvature is at least −1 and the volume of every unit ball is small.
1. In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.
2. In mathematics, a Hermitian manifold is the complex analog of a Riemannian manifold. (Web site)
3. In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. (Web site)
1. In general, geodesics can be defined for any Riemannian manifold.
1. More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form F(P) = 0 such that dF is nowhere zero.
2. Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω.
1. A manifold $M$ together with a Riemannian metric tensor $g$ is called a Riemannian manifold.
1. The Riemannian metrics of tangent and cotangent bundles of a Riemannian manifold were defined by S.Sasaki [1], I.Sato [2] and N.Bhatia, N.Prakash [3]. (Web site)
1. A manifold together with a Riemannian metric tensor is called a Riemannian manifold.
1. In general, a manifold is not a linear space, but the extension of concepts and techniques from linear spaces to Riemannian manifold are natural. (Web site)
2. In this setting, the linear space has been replaced by Riemannian manifold and the line segment by a geodesic. (Web site)
1. I will discuss the Dirichlet Problem for such harmonic functions on bounded domains in a riemannian manifold. (Web site)
1. As David Henderson, Taimina's husband, has explained, a hyperbolic plane "is a simply connected Riemannian manifold with negative Gaussian curvature".
1. The Riemann sphere is only a conformal manifold, not a Riemannian manifold.
1. The converse is also true: a Riemannian manifold is hyperkähler if and only if its holonomy is contained in Sp(n).
1. Abstract: Let M be a closed Riemannian manifold of dimension d.
2. Theorem: Let be a closed Riemannian manifold.
1. So let us construct a Riemannian manifold that has vanishing curvature outside of a compact set. (Web site)
2. Conversely, we can characterize Eucldiean space as a connected, complete Riemannian manifold with vanishing curvature and trivial fundamental group.
1. A Riemannian manifold M is geodesically complete if for all, the exponential map exp p is defined for all, i.e. (Web site)
1. That makes possible to define geodesic flow on unit tangent bundle U T(M) of the Riemannian manifold M when the geodesic γ V is of unit speed.
1. Abstract: Connes has demonstrated that for a compact spin Riemannian manifold, the geodesic metric is determined by the Dirac operator.
1. If is a biwave map from a compact domain into a Riemannian manifold such that (3.19) then is a wave map. (Web site)
1. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. (Web site)
2. Let us denote by a complete simply connected m - dimensional Riemannian manifold of constant sectional curvature k, i.e. (Web site)
1. It is known that a cut - The dimension of a cut locus on a smooth Riemannian manifold.
1. Zaved., Matematika (1987) no.5, 25-33) Grigor'yan, A., On the fundamental solution of the heat equation on an arbitrary Riemannian manifold, Math. (Web site)
1. Let (M, g) be a Riemannian manifold, and a Riemannian submanifold. (Web site)
1. Every compact, simply connected, conformally flat Riemannian manifold is conformally equivalent to the round sphere. (Web site)
2. Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat Riemannian manifold s. (Web site)
3. Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of isothermal coordinates. (Web site)
1. The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but contains less information. (Web site)
2. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. (Web site)
3. However, the Ricci curvature has no analogous topological interpretation on a generic Riemannian manifold. (Web site)
1. This work proposes a novel algorithm for clustering data sampled from multiple submanifolds of a Riemannian manifold. (Web site)
1. Let (N, h) be a Riemannian manifold with LeviCivita connection ∇ and (M, g) be a submanifold with the induced metric.
1. Parallel to this discussion, the notion of a Riemannian manifold will be introduced. (Web site)
1. Abstract: The loop space of a Riemannian manifold has a family of canonical Riemannian metrics indexed by a Sobolev space parameter.
1. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M.
2. It is known that the spin structure on Riemannian manifold can be extended to noncommutative geometry using the notion of spectral triple. (Web site)
1. This motivates the definition of geodesic normal coordinates on a Riemannian manifold. (Web site)
2. A space form is by definition a Riemannian manifold with constant sectional curvature.
1. Any two points of a complete simply connected Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
2. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold.
1. In this paper, we discuss various concepts, definitions and properties for the functions on Riemannian manifold. (Web site)
2. In this paper, we extend the Brézis-Wainger result onto a compact Riemannian manifold. (Web site)
1. The Riemannian curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.
1. The restriction of a Killing vector field to a geodesic is a Jacobi field in any Riemannian manifold. (Web site)
2. The restriction of a Killing field to a geodesic is a Jacobi field in any Riemannian manifold.
1. The Riemannian manifold of covariance matrices is transformed into the vector space of symmetric matrices under the matrix logarithm mapping.
2. Let denote the vector space of smooth vector fields on a smooth Riemannian manifold.
1. One can think of Ricci curvature on a Riemannian manifold, as being an operator on the tangent bundle.
2. One can think of Ricci curvature on a Riemannian manifold, as being an operator on the tangent space. (Web site)
1. On a (pseudo -) Riemannian manifold M a geodesic can be defined as a smooth curve γ(t) that parallel transports its own tangent vector.
1. We also give an example of a connection in the normal bundle of a submanifold of a Riemannian manifold and study its properties.
1. In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold.
2. Riemannian manifold s (but not pseudo-Riemannian manifold s) are special cases of Finsler manifolds.
3. Oriented Riemannian manifold s and pseudo-Riemannian manifold s have a canonical volume form associated with them.
1. General relativity is also a local theory, but it is used to constrain the local properties of a Riemannian manifold, which itself is global.
1. A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function f p(x) = d i s t 2(p, x) is 1- convex.
2. Let f be a smooth nondegenerate real valued function on a finite dimensional, compact and connected Riemannian manifold.
3. Distance function and cut loci on a complete Riemannian manifold. (Web site)
1. Let (M,g) be a Riemannian manifold, and S subset M a Riemannian submanifold.
1. A subset K of a Riemannian manifold M is called totally convex if for any two points in K any geodesic connecting them lies entirely in K, see also convex.
2. A function f on a Riemannian manifold is a convex if for any geodesic γ the function is convex.
3. A function f on a Riemannian manifold is a convex if for any geodesic γ the function is convex.
1. The field of velocities of a (local) one-parameter group of motions on a Riemannian manifold.
1. A case of particular interest is a metric linear connection: this is a metric connection on the tangent bundle, for a Riemannian manifold. (Web site)
1. An example of a Riemannian submersion arises when a Lie group G acts isometrically, freely and properly on a Riemannian manifold (M, g). (Web site)
1. On a Riemannian manifold one has notions of length, volume, and angle.
2. Informally, a Riemannian manifold is a manifold equipped with notions of length, angle, area, etc. (Web site)
Related Keywords
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Differentiable Manifold * Differential Forms * Differential Geometry * Dimension * Elliptic Operator * Euclidean Space * Finsler Manifold * Geodesic * Geodesics * Injectivity Radius * Inner
Product * Isometric * Isometries * Manifold * Manifolds * Metric * Metric Space * Metric Tensor * Notion * Point * Riemannian Geometry * Riemannian Manifolds * Riemannian Metric * Riemann
Curvature Tensor * Scalar Curvature * Smooth Manifold * Space * Structure * Tangent Bundle * Tangent Space * Tangent Spaces * Tensor * Theorem * Topological Dimension * Vector
1. Books about "Riemannian Manifold" in Amazon.com
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Supermarket displays
In this unit students explore the number patterns created when tins are stacked in different arrangements and keep track of the numbers involved by drawing up a table of values.
About this resource
Specific learning outcomes:
• Identify patterns in number sequences.
• Systematically “count” to establish rules for sequential patterns.
• Use rules to make predictions.
Achievement objectives
NA2-8: Find rules for the next member in a sequential pattern.
Description of mathematics
Patterns are an important part of mathematics. It is valuable to be able to recognise the relationships between things. This enhances our understanding of how things are interrelated and allows us to
make predictions.
Patterns also provide an introduction to algebra. The rules for simple patterns can be discovered in words and then written using more concise algebraic notation. There are two useful rules that we
concentrate on here.
• The recurrence rule explains how a pattern increases. It tells us the difference between two successive terms. A pattern of 5, 8, 11, 14, 17, … increases by 3 each time. Therefore, the recurrence
rule says that the number at any stage in the pattern is 3 more than the previous number.
• The general rule tells us about the value of any number in the pattern. For the pattern above, the general rule is that the number connected to any term of the sequence is 2 plus 3 times the
number of the term. For instance, the third number in the sequence above is 2 plus 3 x 3, which equals 11. The sixth number is 2 plus 3 x 6 = 20. To see why this general rule works, it is useful
to write the initial term (5) in terms of the increase (3). So 5 = 2 + 3.
It should be noted that there are many rules operating in these more complicated patterns. Encourage students to look for any relation between the numbers involved.
In this unit, we ask students to construct tables so that they can keep track of the numbers in the patterns. The tables will also make it easier for the students to look for patterns.
In addition to the algebraic focus of the unit, there are many opportunities to extend the students computational strategies. By encouraging the students to explain their calculating strategies, we
can see where the students are in terms of the Number Framework. As the numbers become larger, expect the students to use a range of part-whole strategies in combination with their knowledge of the
basic number facts.
Opportunities for adaptation and differentiation
This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
• providing students with additional time to explore the patterns by drawing and counting tins, before expecting them to continue the patterns using only numbers
• working in small groups with students who need additional support, solving problems together.
The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:
• growing number patterns could be explored using the context of tukutuku panels in the wharenui, or the layout of seedlings for a community garden
• te reo Māori vocabulary terms such as tauira tau (number patterns), raupapa tau (number sequence), tini (tin), hokomaha (supermarket), and kapa (row) as well as numbers in te reo Māori could be
introduced in this unit and used throughout other mathematical learning.
Required resource materials
See Materials that come with this resource to download:
• Tins
• Supermarket displays 1 (.pdf)
• Supermarket displays 2 (.pdf)
• Supermarket displays 3 (.pdf)
Today we look at the number patterns in a tower of tins (tini).
Tell the students that today we will stack tins for a supermarket (hokomaha) display.
Show the students the arrangement:
• How many tins are in this arrangement?
• How many tins will be in the next row (kapa)?
• Then how many tins will there be altogether?
• How did you work that out?
Encourage the students to share the strategy they used to work out the number of tins.
• I can see 4 tins and know that you need 5 more on the bottom. 4 + 5 = 9
• I know that 1 + 3 + 5 = 9 because 5+3= 8 and 1 more is 9.
These strategies illustrate the student’s knowledge of basic addition facts.
Show the students the next arrangement of tins. They can check that their predictions were correct.
• How many tins will be in the next row?
• Then how many tins will there be altogether?
• How did you work that out?
Add seven tins to the arrangement and ask the same questions. As the numbers increase, expect the range of strategies used to be more varied.
Encourage the students to share the strategies they used to work out the number of tins.
I know that we need to add 7 to 9, which is 16. (knowledge of basic facts) I know that 7+ 9 = 16 because 7 + 10 = 17, and this is one less. (early part-whole reasoning) I know that we are adding on
odd numbers each time. 1+3+5+7 = 16, because 7+3 is 10 + 5 + 1 = 16.
• 16 + 9 = 25. I counted on from 16. (advanced counting strategy)
• 16 + 10 = 26 so it is one less, which is 25. (part-whole strategy)
Encourage the students to explain their strategies for "counting" the number of tins.
As a class, share the patterns noted.
Gather the students back together as a class to share solutions.
Discuss the methods that the groups have used to keep track of the number of tins.
Work with students to make a table showing the number of rows and total number of tins. Complete the first couple of rows together.
Ask the small groups to complete their own copy of the table on Supermarket displays 1. As they complete the chart, ask:
Can you spot any patterns? Write down what you notice? Can you predict how many tins would be needed when there are 15 in the bottom row?
Ask the students to work in small groups to find out how many tins are needed. As the students work circulate asking:
How are you keeping track of the numbers? Do you know how many tins will be on the bottom row? How do you know?
Tell the students that the supermarket has asked for the display to be 10 rows high.
How many tins will you need altogether?
Over the next 2-3 sessions, the students work with a partner to investigate the patterns in other stacking problems. Consider pairing together students with mixed mathematical abilities (tuakana/
teina). We suggest the following introduction to each problem.
Pose the problem to the class and ask the students to think about how they might solve it. In particular, encourage them to think about the table of values that they would construct to keep track of
the numbers.
Share tables.
Ask the students to work with their partner to construct and complete their own table.
Write the following questions on the board for the students to consider as they solve the problem.
• How many tins are in the first row?
• How many are in the second row?
• By how much is the number of tins changing as the rows increase?
• What patterns do you notice?
• Can you predict how many tins would be needed for the bottom row if the stack was 15 rows high?
• Explain the strategy you are using to count the tins to your partner?
• Did you use the same strategy?
• Which strategy do you find the easiest?
As the students complete the tables and solve the problem, circulate and ask them to explain the strategies that they are using to "count" the numbers of tins in the design.
Share solutions as a class.
Problem one
Use Supermarket display 1 for this problem.
A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
• How many tins are needed altogether?
• What patterns do you notice?
Problem two
Use Supermarket display 2 for this problem.
A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
• How many tins are needed altogether?
• What patterns do you notice?
Problem three
Use Supermarket display 3 for this problem.
A food demonstrator likes her products displayed using a cross pattern. The display has to be 10 products wide.
How many products are needed altogether?
What patterns do you notice?
In this session, the students create their own "growth" pattern for others to solve.
Display the growth patterns investigated over the previous sessions.
Gather the students as a class and tell them that their task for the day is to invent a pattern for the supermarket to use to display objects.
Ask the students in small groups to decide on a pattern and the way that it will grow. (A supply of counters may be helpful for some students.)
Direct students to construct a table to keep track of their pattern (up to the 10^th model). Model how to construct and use this. Alternatively, you could provide a graphic organiser for students to
Once they have constructed the table, ask them to record the any patterns that they spot in the numbers. Ask them also to make predictions about the 15^th and 20^th model.
Direct students to swap problems with another group. When the problem has been solved, they should compare solutions with each other.
Home Link
Dear parents and whānau,
In math this week, we have been looking at patterns. Patterns are an important part of mathematics. It is always valuable to be able to recognise the relationships between things to help us see how
things are interrelated and allow us to make predictions.
The patterns below are to do with buildings. We have been learning about how patterns like these can be continued. An important part of this has been learning to use tables to keep track of the
pattern and relationships between terms.
Ask your child if they can continue the pattern below and say what patterns they notice in the numbers. Can they draw or fill out a table to show how the pattern would progress? Can you work out how
many crosses would be in the triangle with 15 crosses along the bottom?
│Number of crosses high│Number of crosses along bottom │Number altogether│
│1 │1 │1 │
│2 │2 │3 │
│3 │3 │6 │
The quality of the images on this page may vary depending on the device you are using.
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There are Different Ways to Measure Mass - Fact or Myth?
There are different ways to measure mass, but all of them are related to rest-mass (invariant mass) the “true” inertial mass of an object at rest. We can measure mass as inertial mass, or as rest
mass (invariant mass); OR we can measure mass as relativistic mass, active gravitational mass, passive gravitational mass, quantum mass, or mass-energy. [1][2][3]
Assuming Einstein’s logic that “the laws of physics are the same in all inertial frames of reference“, we need only use a little math to see all the above mass types through the lens of Newton’s F=ma
(force equals mass times acceleration). This is because they are all just equivalent affects of mass-energy.[1][2][3]
TIP: An Inertial Frame is a “frame of reference” that moves with constant velocity and direction with respect to the observer. An “accelerated frame” is a frame of reference accelerating away from
the object being observed. How mass is measured is dependent on frame of reference (you can blame relativity for that one). We have to “calibrate” our “mass-measuring” tools to account for relativity
and reference frame.
NOTE: All mass arises from the same set of phenomena (elementary particles, their interactions, and the fundamental forces) and can be measured with F=ma and a little math (as the laws of physics is
the same in all INERTIAL frames). So we can distinguish between aspects of mass, but they can be shown to have proportional (and in some cases equal) values.[1]
NOTE: Quantum level mass manifests itself as a difference between an object’s quantum frequency and its wave number (both types of frequency fluctuations, like an EKG or light wave). This explains
how angular momentum (spin) can add to mass at quantum levels by changing frequency (related to mass-energy equivalence). Changes in mass are very tiny, as one might imagine. The invariant mass of
elementary particles adds to the mass of a system, but most of a systems mass comes from fermions being bound in the nucleus and protons of atoms, and those atoms being bound to each other. In other
words, quantum level mass is very tiny, the main source of mass in objects comes from particles binding kinetic energy into potential energy in larger systems.
To show all mass types are analogous and have proportional (and sometimes equal value) let’s take a look at a simple example.
I have the worlds heaviest bowling ball and I’m at the top of an infinitely steep snowy hill. The ball represents rest-mass (bound potential energy of a system at rest), the hill represents
acceleration, and the snow represents kinetic energy (what most people think of as energy, unbound electromagnetic force).
That bowling ball is my “bound” system (meaning all particles in the system are bound, sort of like how a cell has a membrane, or we have skin). That can be a quark, quarks binding together in a
nucleus, atoms binding together as a person, or a planet, etc. As long as kinetic and potential energy is bound as mass in the system, the metaphor will work.
If I weigh the bowling ball at rest (analogous, but not the same as measuring mass), I get the invariant rest-mass of the object (the mass of an object at rest).
If I try to move the bowling ball from it’s state of rest, I can find it’s inertial mass which is the amount of acceleration needed to move the mass (F=ma).
If I push with enough force to roll the bowling ball down the snowy hill, it will pick up more and more snow (energy) as it rolls down the hill (accelerates).
If I freeze frame as the ball rolls down the hill (analogous to an inertial frame moving at a constant velocity and direction as an object) and measure the ball, I will measure a greater mass, and I
will measure that more acceleration is needed to move the mass as the snow (energy) has “weighed down” (added mass) the bowling ball.
The mass from the snow is relativistic mass (energy acting as mass at high speeds). The relativistic mass increases exponentially with acceleration, but despite this, we know that under that snow is
the same bowling ball. The mass of that bowling ball hasn’t varied (it’s invariant), even though the mass from energy is “relative” to acceleration.
Given this, we know we can just factor out the snow (kinetic energy added to the system due to acceleration) and find both the total mass from snow added to the already bound system, and the original
mass of the bowling ball.
If we take the bowling ball in it’s snowy state, or in it’s pure bowling ball state, and put it on a “spacetime scale” that measures gravitational mass, we will find that the bowling ball weighs down
both space and time resulting in a curvature of spacetime. The snowy ball weighs down the scale more than the bowling ball alone, because the snowy ball has more mass.
However, if we take just a single snow flake (massless energy particle) and put it on the scale there is no affect. The kinetic energy in the system only adds mass when it’s bound or being used for
the acceleration of an object with mass, a single snowflake on it’s own is massless. The parts of a system (almost) always have less mass than the system, because bound energy acts as mass.
The above said, some of the elementary particles that make up the bowling ball have mass (like quarks). On a very small scale, given the relationships above, it can be useful to think of all mass and
energy as simply mass-energy.
With a little math we can calculate the rest mass, relativistic mass, inertial mass, or any affects of gravity as proportional, as they are all arising from the same interactions of elementary
particles. The affects of all this can be measured as mass-energy and it’s affects on spacetime.
All systems (a particle, or interaction of particles) have the properties of mass and/or energy. Generally, when systems interact they gain mass, when systems speed up they gain energy. If we factor
out the motion of the system, we get rest-mass (the potential energy of the system). Einstein tells us that mass and energy are equivalent properties of a system, so we can consider rest mass
(equivalently, rest-energy) to be the true mass of the system. If a system has rest mass it’s matter, if it doesn’t it’s a mass-less light speed particle(s).
If we factor motion back in, we get relativistic mass-energy which changes with speed (as energy is needed to accelerate, and in a closed system that means potential energy is used as kinetic energy
for acceleration).
If we consider the fact that mass-energy curves spacetime, we get gravity (which also affects relativistic mass).
Simply put, each mass type is a different way to measure the same fundamental properties which arise from particle interactions and can all be measured with newton’s F=ma.
E=mc2, tells us mass is a property of a system that can be thought of as potential energy, and energy is a property of a system that can be thought of as kinetic energy. If we measure the total mass
and energy of a system at rest (by factoring out motion), we get a system’s true mass-energy and we can apply inertial F = ma, just like on earth.
This seemingly complex tidbit actually makes life simple. We really only need to understand two types of mass since it’s all a property of energy.
Inertial mass (resistance to acceleration): A measure of the inertia of an object is the amount of resistance an object offers to acceleration under a directly applied force. This is what we think of
as mass in Newtonian physics (from Newton’s second law). It represents the ‘m’ in this equation: F = ma (Force = Inertial Mass x Acceleration) or m=F/a (Inertial mass = force / by acceleration).
Rest mass (inertial mass measured for a special “resting frame”): If we want to find the “true” mass of an object, we can assume a special “resting frame” of reference. This is based on Einstein’s
second postulate “The laws of physics are the same in all inertial frames of reference”. It’s the ‘M’ in this equation: ‘Rest Energy‘ (E) = ‘Rest Mass’ (M) x ‘the Speed of Light‘ (C) squared (2) (Eo
= mo c2) or just E=MC2.[3]
Mass is a property of energy. To make the concept easier to understand, we assume a frame of rest and simply measure the inertial mass of an object, just as we would on earth. We take that rest mass,
plug it into E=MC2 and we have the concepts that we need to understand this manifestation of the universe.
The rest mass of the particles in a system is (almost) always less than the rest mass of the total system. This is because rest mass factors out the momentum of the system, but counts the kinetic
energy of the particles. At the core of all systems is massless energy particles. So at some level all mass is a product of energy (although this is somewhat shown by science, many elementary
particles have mass, so it’s not very useful to think of the universe as ONLY massless energy particles).
The types of mass below are all ways to measure mass and can all be converted back to E=MC2, which is just the inertial mass of an object measured from a special resting frame. Mass is a property of
energy, so no matter what shenanigans energy is up to we can measure it as mass. Ok, now things get strange.
Relativistic mass (inertial mass in motion): When inertial mass as measured by an observer, with respect to whom the object is at motion, we can calculate relativistic mass. Einstein, modern
physicists, and your neighbor down the street all agree that relativistic mass is a confusing and mostly unnecessary concept unless you are a physicist. This is because when you calculate
relativistic mass you have to account for changes in kinetic energy and potential energy. When speed increases relativistic mass appears to increase, but actually potential energy (measured as mass)
is being used as kinetic energy (and with a little math we can show invariant mass doesn’t actually change). It is much easier to just use “the Lorentz factor” to factor out the movement.
Using the Lorentz transformation to find rest mass: Relativistic mass is the ‘M’ in this equation, the little ‘m’ is rest mass: $M = m/\sqrt{1 - v^2/c^2}$ ( v is the speed of the object, c is the
speed of light, ‘M’ is the relativistic mass, and ‘m’ is the rest mass. The non-m part is “the Lorentz factor”). This is also the ‘M’ in this equation: ‘Kinetic Energy‘ (E) = ‘Relativistic Mass‘ (M)
x ‘the Speed of Light‘ (C) squared (2) (E=MC2)[3]
Gravitational mass (a way to describe inertial mass as gravity): Gravity is the result of energy not moving at light speed and its impact on spacetime, which affects other energy. When we measure
active gravitational mass we measure the gravitational force exerted by an object. When we measure passive gravitational mass we measure the gravitational force experienced by an object in a known
gravitational field. Simply put, when energy does anything other than travel light speed, it has mass-energy. That mass-energy can be measured as gravity. Gravity also affects relativistic mass.[1]
Invariant mass (a way to describe the fact that rest mass doesn’t vary with speed or gravity).
Mass-Energy. Mass-energy measures the total amount of energy contained within a body, using E=MC2.[1] We already talked about mass energy above, but keep in mind all the energy in a system can be
measured as mass. Energy doing anything other than traveling at light speed warps spacetime and can be measured as mass. We can simplify this and measure all potential and kinetic energy as
NOTE: Relativistic mass is different than relative mass. Relative mass is the mass of an object relative to another object.
In simple terms, objects have potential energy (M) and kinetic energy (E). Pure kinetic energy is light speed (C), potential energy is mass. We can measure this all as mass-energy, use the simple
rest mass which accounts for mass-energy, or measure it from relative frames of motion and make life complicated.
As noted above, when objects increase in speed they seem to gain mass, but this isn’t what is actually happening. Kinetic energy is being added to the system to accelerate the mass faster and faster.
The relativistic mass increases and more and more energy is needed to accelerate the body. According to mass-energy equivalence and infinite amount of energy would be needed to accelerate something
with non-zero rest mass to light speed due to the way relativistic mass works.
The true mass of an object never changes, but relativistic mass does. If we want to look at things from a relative non-resting frame, then we need to be able to prove that relativistic mass and
rest-mass are interchangeable.
Both mass types work with E=MC2 (with a little math involved) because rest mass accounts for the kinetic energy used for speed in relativistic mass (as rest mass is a special case where we assume a
relative resting frame of reference for the observer and the system and factor out the impacts of speed).
TIP: ‘Relativistic mass’ is the mass of a system measured by an observer traveling at a relative speed to the object. Rest mass (or mass at rest) is a measure of the inertia of the system (the
tendency of the system to resist changes in velocity). Rest mass is equal to inertial mass. Rest mass gives us a universal frame of reference for discussing E=MC2. Einstein thought rest mass was the
best starting point. Many people argue that mass should always be considered “rest mass” and that saying “rest mass” is redundant. Rest mass is “true mass”, relativistic mass only matters when one is
dealing with physics and math.
Einstein thought that understanding relativity was best done by looking at rest mass. We can use equations to account for relativistic mass, but it’s a little like trying to track the orbits of
planets with a telescope (they appear to move in random patterns).
It’s easier to just say ‘mass’ means “total energy of an object at rest” instead of trying to factor in the way in which velocity affects mass-energy and how relativistic mass works.
“It is not good to introduce the concept of the mass $M = m/\sqrt{1 - v^2/c^2}$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the
‘rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion.” — Albert Einstein in letter to Lincoln Barnett, 19 June 1948 (quote
from L. B. Okun (1989), p. 42).[4]
What Einstein is saying is that rest mass is ‘real’ and relativistic mass is relative to frames of reference. We can number crunch our way around frames of reference, but it’s much better to view
this all from a resting frame first and then discuss the moving part after.
FACT: Einstein’s version of E=MC2 can be read in full here. It’s less than 3 pages long. The paper doesn’t actually say ‘E=MC2‘ it says ‘m=L/c2‘ or rather “If a body gives off the energy L in the
form of radiation, its mass diminishes by L/c2… the mass of a body is a measure of its energy-content”. The L stands for “Lagrangian”, a type of energy. This was later translated to a simpler and
easier to meme E=MC2.
The above ideas may seem like it’s all Einstein and Newton, but it’s built on Galilean relativity, and work from many others. We love heroes, but only more reason to look at the other heroes of
relativity, mass-energy, and physics in general.
There are a few different types of mass, which are really just ways to measure the effects of mass-energy. How we measure has to do with our frame of reference, and the type of effect we are looking
at. All affects of mass have proportional value, because they all arise from the same phenomena.
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Harvard class studies the philosophy of physics
Science & Tech
Yes, cat is both dead, alive
New class combines philosophy, physics to look at quantum theory
Walk into Jacob Barandes’ new class and the topic of discussion might be a philosophical exploration of whether a cat could be simultaneously alive and dead. A visit on a different day may find the
lecturer filling the blackboard with a mathematical equation that stretches 20 feet before continuing on a new line, over and over. Or students might be dissecting an example of classical physics
such as Newton’s laws of motion.
So, exactly what kind of class is this?
It’s officially designated Physics 137, “Conceptual Foundations of Quantum Mechanics,” but it’s really part physics and part philosophy, with a hearty infusion of math and logic.
The subject sits within a much larger field called the philosophy of science, a branch of study that examines the theoretical foundations, methods, and implications of science in the real world. In
this case, Barandes is applying the class’ inquiry to quantum theory.
“This is physics by scrutinizing,” said Barandes, who is also co-director of graduate studies for the Department of Physics. “This is taking our best scientific theories, dissecting them,
disassembling them, looking at the pieces piece by piece, trying to understand them and how they fit together, and the larger wholes that they form.”
Quantum theory, which explains the nature and behavior of matter and energy on the atomic and subatomic levels, is often described as the best-tested and most predictive scientific theory out there,
one that makes possible precision technology such as atomic clocks and particle accelerators. Much of our modern technology — including smartphones, lasers, LEDs, and MRI machines — relies on it.
But when it comes to painting a picture of the real world, quantum theory can feel unwieldy and counterintuitive. Take, for instance, the notion of particles being in more than one place at a time.
The class aims to explore why quantum theory contains so many strange and exotic mathematical structures and seemingly illogical possibilities and to get a sense of the different ways the world would
appear depending on how aspects of the theory are interpreted.
Students and faculty from across Harvard explain the basics of quantum physics and why it’s so important.
It delves into the century-long effort to resolve these mysteries and hits on ideas from quantum theory like entanglement, superposition, and, of course, parallel universes and Schrödinger’s cat
(both alive and dead in a box).
“[One of the goals is] to reformulate the classical picture as closely as we can to quantum theory so that we can pinpoint as precisely as possible what it is that we’re changing when we go from
classical to the quantum case,” Barandes said.
What sets the class apart from many quantum physics courses is that this one is less about calculating numerical predictions and more about learning foundationally and logically how the theory works
and what it tells us about the world around us. Through it all, students are encouraged to ask the ultimate philosophical question: Why?
At one recent class, for example, as Barandes was working through a quantum equation, a student’s hand shot up and Barandes was asked why he chose one specific example over another to illustrate the
point of the lesson. The class then went into an extended debate over the logic behind that decision.
Lavanya Singh, a senior concentrating in computer science and philosophy, says this type of discussion usually wouldn’t happen in a more technically focused class.
“Why have we decided to model the system in this way? Why are these the operations we have chosen? What if we did it differently?” Singh said. “Those are usually questions that are not the point of a
technical class, but in this class the [instructor] was really happy to entertain those questions because that is the point. The point is to understand why we are making the decisions that we are.”
Students say this level of understanding, especially when it comes to a theory as counterintuitive as quantum can be, is one of the main reasons they took the course.
“I studied quantum mechanics last year and found it as bewildering a subject as anything in physics,” said Samuel Buckley-Bonanno ’22. “I was still wondering about things in it that didn’t make any
conceptual sense, so this seemed like the obvious class to take for me, and it’s proven to be really interesting. It’s changed many of the kinds of frameworks in which I’ve been thinking about these
sorts of ideas.”
After devoting the first half of the semester to a historical survey, a review of classical physics concepts, and the transition to quantum theory, the second half will examine the internal logic of
the theory. Students are eager to see what all that yields.
“Humanity is still confused about quantum theory,” Singh said. “It feels like the point of the class is helping me distinguish which are the questions I just don’t understand, and which questions are
the ones humanity doesn’t understand.”
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Existence of Mutual Stabilization in Chaotic Neural Models
Date of Award
Spring 2021
Project Type
Program or Major
Applied Mathematics
Degree Name
Doctor of Philosophy
First Advisor
Kevin M Short
Second Advisor
John Gibson
Recent work has demonstrated that interacting chaotic systems can establish persistent, periodic behavior, called mutual stabilization, when certain information is passed through interaction
functions. In particular, this was first shown with two interacting cupolets (Chaotic Unstable Periodic Orbit-lets) of the double scroll oscillator. Cupolets are highly accurate approximations of
unstable periodic orbits of a chaotic attractor that can be generated through a control scheme that repeatedly applies perturbations along Poincaré sections. The decision to perturb or not to perturb
the trajectory is determined by a bit in a binary control sequence. One interaction function used in the original cupolet research was based on integrate-and-fire dynamics that are often seen in
neural and laser systems and was used to demonstrate mutual stabilization between two double scroll oscillators. This result provided the motivation for this thesis where the stabilization of chaos
in mathematical models of communicating neurons is investigated.
This thesis begins by introducing mathematical models of neurons and discusses the biological realism of the models. Then, we consider the two-dimensional FitzHugh-Nagumo (FHN) neural model and we
show how two FHN neurons can exhibit chaotic behavior when communication is mediated by a coupling constant, g, representative of the synaptic strength between the neurons. Through a bifurcation
analysis, where the synaptic strength is the bifurcation parameter, we analyze the space of possible long-term behaviors of this model. After identifying regions of periodic and chaotic behavior, we
show how a synaptic sigmoidal learning rule transitions the chaotic dynamics of the system to periodic dynamics in the presence of an external signal. After the signal passes through the synapse,
synaptic learning alters the synaptic strength and the two neurons remain in a persistent, mutually stabilized periodic state even after the signal is removed. This result provides a proof-of-concept
for chaotic stabilization in communicating neurons.
Next, we focus on the 3-dimensional Hindmarsh-Rose (HR) neural model that is known to exhibit chaotic behavior and bursting neural firing. Using this model, we create a control scheme using two
Poincaré sections in a manner similar to the control scheme for the double scroll system. Using the control scheme we establish that it is possible to generate cupolets in the HR model. We use the HR
model to create neural networks where the communication between neurons is mediated by an integrate-and-fire interaction function. With this interaction, we show how a signal can propagate down a
unidirectional chain of chaotic neurons. We further show how mutual stabilization can occur if two neurons communicate through this interaction function. Lastly, we expand the investigation to more
complicated networks including a feedback network and a chain of neurons that ends in a feedback loop between the two terminal neurons. Mutual stabilization is found to exist in all cases. At each
stage, we comment on the potential biological implications and extensions of these results.
Recommended Citation
Parker, John, "Existence of Mutual Stabilization in Chaotic Neural Models" (2021). Doctoral Dissertations. 2589.
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Buckling of mono-axially compressed rectangular grids
A theoretical analysis of the buckling of mono-axially compressed rectangular grid is carried out. The grid is composed of two orthogonal orders of continuous beams, simply supported at the ends. The
critical load is determined by two different models: (i) the Kirchhoff’s homogeneous orthotropic plate, and (ii) a system of beams, solved by the Rayleigh-Ritz method. Parametric studies are carried
out and numerical examples developed. The analytical solutions are validated by comparison with numerical Finite Element analyses.
1. Introduction
Grid structures are structural systems extensively used in many engineering applications for their many advantages: (a) they are efficient in transferring concentrated loads, (b) participate as a
whole to the load carrying action, (c) are easy to inspect or repair because are open structures.
In the past, grid structures consisting of ribs and skins have been largely studied. Ref. [1] grouped analytical models for grid structures into two categories: exact models, which consider the grid
geometry simulating ribs and skin individually, and equivalent models, which smear the grid as a homogenized plate. Grid structures can have different configuration: with or without skins (at the
inner and outer surface of the shell), made of composite materials, multiple geometries, and shapes. In recent years, many studies have been carried out on the buckling behavior of reinforced
cylindrical shells with cross stiffeners [2]. However, rectangular beam grids are the most common in practice and their buckling load is an important parameter to know.
In this work, the buckling behavior of rectangular beam grids is studied analytically. The critical load of mono-axially compressed rectangular beam grids is derived by using two different models:
(i) the Kirchhoff's orthotropic plate, and (ii) a planar system of beams, both axially loaded in-plane. The two models are intrinsically different: the plate is meant as an equivalent model of
homogenized continuum, in which the beams “are smeared” on the plate; the frame, instead, preserves the discrete nature of the constituent members. To solve the plate, the classic separation of the
variables is adopted; to solve the frame, it is conjectured that the envelope of the buckled beams is a smooth function, so that a variational procedure can be applied.
Indeed, the Rayleigh-Ritz energy method had already been adopted long time ago [3], but it seems to have been forgotten nowadays. Here, the method of Ref. [3] is reproposed with slight changes. On
the other hand, several orthotropic plate theories are available in literature [4]-[7]. The Kirchhoff's orthotropic model is chosen here for its simplicity.
The paper is organized as follows. In Section 2 the problem is posed. In Section 3 the Kirchhoff’s orthotropic theory is used for an equivalent plate, and the bifurcation condition is derived. In
Section 4 the Rayleigh-Ritz energy method is worked out for the beam assembly, and the relevant critical load is computed. In Section 5 some parametric studies are carried out and the analytical
solutions are validated by comparison with numerical Finite Element results. Finally, in Section 6 some conclusions are drawn.
2. Problem position
A rectangular $\mathcal{l}×b$ beam grid with rectangular $\mathrm{\Delta }x×\mathrm{\Delta }y$ mesh is considered (see Fig. 1). The grid is composed of two orders of continuous beams, simply
supported at the ends. The $x$-direction and $y$-direction beams are prestressed by forces ${P}_{x}$ and ${P}_{y}$, respectively, positive when they generate compression. Therefore, in general, the
grid is biaxially prestressed. $p$ and $q$ are the numbers of beams in $x$-direction and in $y$-direction, respectively. It is consequently $\mathcal{l}/\mathrm{\Delta }x=q+1$ and $b/\mathrm{\Delta }
y=p+1$. Each order of beams consists of equal beams of uniform section. The flexural and torsional stiffness of the single beam in $i$-direction are denoted by $E{I}_{i}$ and $G{J}_{i}$,
3. Equivalent orthotropic plate model
A rectangular Kirchhoff’s orthotropic plate, simply supported on the whole contour is considered. The plate is monoaxially prestressed by forces ${p}_{x}$ uniformly distributed and normal to the
edges. Denoting by $w\left(x,y\right)$ the out of plane displacement of the point $\left(x,y\right)$, the indefinite equilibrium equation is written as:
where ${D}_{i}\left(i=x,y\right)$ and $H$ are the flexural and the torsional stiffnesses of the plate, respectively. They are evaluated by enforcing energy equivalences between the plate and the
grid, and turn out to be equal to:
${D}_{x}=\frac{E{I}_{x}}{\mathrm{\Delta }y},{D}_{y}=\frac{E{I}_{y}}{\mathrm{\Delta }x},H=\frac{1}{2}\left(\frac{G{J}_{x}}{\mathrm{\Delta }y}+\frac{G{J}_{y}}{\mathrm{\Delta }x}\right).$
A linear combination with unknown coefficients of products of sinusoidal functions is used as trial functions, namely:
$w\left(x,y\right)={\sum }_{n=1}^{N}{\sum }_{m=1}^{M}{A}_{nm}\mathrm{sin}\left(\frac{n\pi }{\mathcal{l}}x\right)\mathrm{sin}\left(\frac{m\pi }{b}y\right),$
where ${A}_{nm}$ are unknown coefficients and $n$, $m$ are integer numbers, even unknowns. Each term of the series describes a deformation in which the plate bends in $n$ half-waves (of length $\
mathcal{l}/n$) in the $x$-direction and $m$ half-waves (of length $b/m$) in the $y$-direction. Denoting by $\alpha ≔\mathcal{l}/b$ the aspect ratio and substituting Eq. (3) in Eq. (1), ${\infty }^{2}
$ critical loads are found, each associated with the pair $\left(n,m\right)$, i.e:
${p}_{xc}\left(n,m\right)={D}_{y}\frac{{\pi }^{2}}{{b}^{2}}\left(\frac{{D}_{x}}{{D}_{y}}\frac{{n}^{2}}{{\alpha }^{2}}+2\frac{H}{{D}_{y}}{m}^{2}+{m}^{4}\frac{{\alpha }^{2}}{{n}^{2}}\right).$
Since we are interested in the smallest of these loads, we need to minimize ${p}_{xc}\left(n,m\right)$ with respect to the two variables. Accordingly, $m=1$ must be taken, while $n$ must be
determined by trial and error. By letting $\delta ≔{D}_{x}/{D}_{y}$, $\gamma ≔H/{D}_{y}$ and ${p}_{xc}={\mu }_{xc}{\pi }^{2}{D}_{x}/{b}^{2}$, the nondimensional critical load ${\mu }_{xc}$ is
determined as:
${\mu }_{xc}=\left(\frac{{n}^{2}}{{\alpha }^{2}}+2\frac{\gamma }{\delta } +\frac{{\alpha }^{2}}{{\delta n}^{2}}\right).$
It is observed that, when $\alpha$ is an integer, the minimum critical load occurs at $n=\alpha$, and it is equal to ${\mu }_{xc,min}=1+2\frac{\gamma }{\delta }+\frac{1}{\delta }.$The same limit
value is found for any $\alpha$ sufficiently large. When the plate is isotropic (i.e., when $\delta =\gamma =1$), then, the well-known value ${\mu }_{xc,min}=4$ is recovered.
4. Beam assembly model
The grid is considered as an assembly of beams undergoing flexure and torsion. It is conjectured that, in the buckled configuration, the beams lie on a smooth surface $w=w\left(x,y\right)$, which is
still given by Eq. (3), so that, by using the Raileigh-Ritz model, the problem can be solved in closed-form.
The Total Potential Energy (TPE) of the grid is defined as sum of flexural ($f$), torsional ($t$) and geometric ($g$) energy of the two orders of beams, $\mathrm{\Pi }={\mathrm{\Pi }}_{x}^{f}+{\
mathrm{\Pi }}_{y}^{f}+{\mathrm{\Pi }}_{xy}^{t}+{\mathrm{\Pi }}_{x}^{g}+{\mathrm{\Pi }}_{y}^{g}$.
By accounting for ${\theta }_{x}={w}_{,y}$, ${\theta }_{y}={w}_{,x}$, the single energy terms read:
${\mathrm{\Pi }}_{x}^{f}=\sum _{i=1}^{p}\frac{1}{2}\underset{0}{\overset{\mathcal{l}}{\int }}E{I}_{x}{w}_{,xx}^{2}\left({x}_{i},\stackrel{-}{{y}_{i}}\right)dx,{\mathrm{\Pi }}_{y}^{f}=\sum _{j=1}^{q}\
frac{1}{2}\underset{0}{\overset{b}{\int }}E{I}_{y}{w}_{,yy}^{2}\left(\stackrel{-}{{x}_{j}},{y}_{j}\right)dy,$${\mathrm{\Pi }}_{xy}^{t}=\sum _{i=1}^{p}\frac{1}{2}\underset{0}{\overset{\mathcal{l}}{\
int }}G{J}_{x}{w}_{,yx}^{2}\left({x}_{i},\stackrel{-}{{y}_{i}}\right)dx+\sum _{j=1}^{q}\frac{1}{2}\underset{0}{\overset{b}{\int }}G{J}_{y}{w}_{,xy}^{2}\left(\stackrel{-}{{x}_{j}},{y}_{j}\right)dy,$$
{\mathrm{\Pi }}_{x}^{g}=\sum _{i=1}^{p}\frac{1}{2}\underset{0}{\overset{\mathcal{l}}{\int }}{P}_{x}{w}_{,x}^{2}\left({x}_{i},\stackrel{-}{{y}_{i}}\right)dx,{\mathrm{\Pi }}_{y}^{g}=\sum _{j=1}^{q}\
frac{1}{2}\underset{0}{\overset{b}{\int }}{P}_{y}{w}_{,y}^{2}\left(\stackrel{-}{{x}_{j}},{y}_{j}\right)dy.$
Assume, for simplicity's sake, that $E{I}_{x}=E{I}_{y}=EI$ and $G{J}_{x}=G{J}_{y}=GJ$. By imposing stationarity of the TPE, the equilibrium equation follows, from which the buckling critical load is
derived as:
${P}_{xc}=\frac{{\pi }^{2}}{{b}^{2}{\mathcal{l}}^{2}}\left(\frac{{m}^{2}{n}^{2}b\mathcal{l}GJ\left(b{\sum }_{j=1}^{q}{\mathrm{cos}}^{2}\left(\frac{jn\mathrm{\Delta }x}{\mathcal{l}}\pi \right)+\
mathcal{l}{\sum }_{i=1}^{p}{\mathrm{cos}}^{2}\left(\frac{im\mathrm{\Delta }y}{b}\pi \right)\right)}{\left({m}^{2}\mathcal{l}\mathcal{}\rho {\sum }_{j=1}^{q}{\mathrm{sin}}^{2}\left(\frac{jn\mathrm{\
Delta }x}{\mathcal{l}}\pi \right)+{n}^{2}b{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(\frac{im\mathrm{\Delta }y}{b}\pi \right)\right)}+\frac{EI\left({m}^{4}{\mathcal{l}}^{3}{\sum }_{j=1}^{q}{\mathrm
{sin}}^{2}\left(\frac{jn\mathrm{\Delta }x}{\mathcal{l}}\pi \right)+{n}^{4}{b}^{3}{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(\frac{im\mathrm{\Delta }y}{b}\pi \right)\right)}{\left({m}^{2}\mathcal{l}\rho
{\sum }_{j=1}^{q}{\mathrm{sin}}^{2}\left(\frac{jn\mathrm{\Delta }x}{\mathcal{l}}\pi \right)+{n}^{2}b{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(\frac{im\mathrm{\Delta }y}{b}\pi \right)\right)}\right),$
where, $\rho ≔{P}_{y}/{P}_{x}$ is the ratio between the magnitudes of the two loads and $n$, $m$ are the values which minimize ${P}_{xc}$. In the special case of $\rho =0$, by introducing the
dimensionless parameters $\mathrm{\Delta }\xi ≔\mathrm{\Delta }x/\mathcal{l}$, $\mathrm{\Delta }\eta ≔\mathrm{\Delta }y/b$ and by letting $\alpha ≔l/b$ the aspect ratio of the grid, $\beta =GJ/EI$
the torsional-to-flexural stiffness ratio, and ${P}_{xc}={\mu }_{xc}{\pi }^{2}EI/{b}^{2}$, the critical nondimensional load ${\mu }_{xc}$ is determinated:
${\mu }_{xc}=\frac{{m}^{2}\beta \left(\frac{1}{\alpha }{\sum }_{j=1}^{q}{\mathrm{cos}}^{2}\left(jn\mathrm{\Delta }\xi \pi \right)+{\sum }_{i=1}^{p}{\mathrm{cos}}^{2}\left(im\mathrm{\Delta }\eta \pi \
right)\right)}{{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(im\mathrm{\Delta }\eta \pi \right)}+\frac{\left(\frac{{m}^{4}}{{n}^{2}}\alpha {\sum }_{j=1}^{q}{\mathrm{sin}}^{2}\left(jn\mathrm{\Delta }\xi \
pi \right)+{n}^{2}\frac{1}{{\alpha }^{2}}{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(im\mathrm{\Delta }\eta \pi \right)\right)}{{\sum }_{i=1}^{p}{\mathrm{sin}}^{2}\left(im\mathrm{\Delta }\eta \pi \
It should be noticed that when $m=p+1$, Eq. (8) becomes indeterminate, because the flexural and prestress energies are zero (indeed, the beams are located at the nodal lines of the deformed
5. Numerical results
A parametric study is carried out by using the beam assembly model (Eq. (8)). Since ${\mu }_{xc}={\mu }_{xc}\left(∆\xi ,∆\eta ,\alpha ,\beta ,n,m\right)$, as a first analysis, $∆\xi$, $∆\eta$ are
fixed and ${\mu }_{xc}$ (minimized with respect to $n,m$) plotted vs the aspect ratio $\alpha$ for different stiffness ratios $\beta$ (Fig. 2(a)). The classic signature curves, typical of homogeneous
and isotropic plates, is obtained. However, differently from those, it is observed that the critical load slowly decreases with $\alpha$. This result can be interpreted by making use of the
equivalent plate model. Indeed, since the nondimensional steps $∆\xi$, $∆\eta$ are kept constants, also the number of beams parallel to the $x$ and $y$-direction are constant, so that, while the
aspect ratio increases, the equivalent stiffnesses per unit length of the plate decrease, entailing a decreasing of the critical load.
As a second analysis, only $∆\eta$ is kept constant, while the $\mathrm{\Delta }\xi = \mathrm{\Delta }\eta /\alpha$ is taken, thus obtaining the plot in Fig. 2(b). It is seen that the critical load
now tends to an asymptotic value for large $\alpha$. Indeed, the case analyzed is representative of a family of grids in which the width $b$ is kept constant together with the number of beams in the
$x$-direction, while the length $\mathcal{l}$ is increased, together with the number of beams in the $y$-direction, to keep their dimensional step $\mathrm{\Delta }x$ constant. Again, by resorting to
the equivalent plate model, since the plate stiffnesses per unit length are equal for any member of the family, the tendency to an asymptotic value is consistently observed.
Fig. 2Frame model nondimensional critical load of the rectangular grid, subject to mono-axial compression vs the aspect ratio α, for β=0,…,1: a) Δξ=Δη=0.09; b) Δη=0.01, Δξ=Δη/α
The analytical results are compared with a Finite Element analysis for validation. The FE model is made of an assembly of Euler-Bernoulli beams. Constraints at the ends do not permit $z$
-displacements for all the beams, and $x$- and $y$-displacement for beams in the $y$ and $x$-directions, respectively. The loads are applied as concentrated forces at the ends of the beams, directed
as the beam longitudinal axes (see Fig. 1). One of the (analytical) curves in Figs. 2(a) and 2(b) is compared in Figs. 3(a) and 3(b) with FE results, and a very good agreement is found. In the same
figures, the results provided by the Kirchhoff’s orthotropic plate theory (Eq. (5)) are also reported; they turn out to be close to those relevant to the assembly beam model.
Fig. 3Nondimensional critical load of the rectangular grid, subject to mono-axial compression vs the aspect ratio α, for β= 0.25: a) Δξ=Δη=0.09; b) Δη=0.01, Δξ=Δη/α. Rayleigh-Ritz energy method
(solid black line), Kirchhoff's orthotropic plate theory (dashed cyan line), FE analysis (dots)
As a third analysis, a study is performed, in which the dimension $b$, $\mathcal{l}$ of the grid are kept constant, while the number of beams is increased. Fig. 4 compares the two analytical model
with FE results. It refers to a square dominion ($b=\mathcal{}\mathcal{l}$) with an equal number of beams ($p=q$). It is seen that when the number of beams is sufficiently large (e.g., $q=\text{9}$,
corresponding to $\mathrm{\Delta }\eta =\text{0.1}$) the error is reasonable (about 6 %), and tends to zero with $q$. However, when $q$ is smaller, the grid model gives a result better than the
homogenized plate.
Fig. 4Comparison of results: a) nondimensional critical load of the rectangular grid, subject to mono-axial compression vs the number of beams q=p, for β=0.25; b) percentage error (PE) from FE
results. Rayleigh-Ritz energy method (blue), Kirchhoff's orthotropic plate theory (magenta), FE analysis (dots)
6. Conclusions
A buckling analysis of a simply supported rectangular grid, subject to mono-axial compression, was carried out. Two models were used: (a) a homogenized Kirchhoff’s orthotropic plate, and (b) an
assembly of beams tackled via an energy method. Analytical results were compared with Finite Element analyses. The following conclusions were drawn.
1) The Rayleigh-Ritz energy method supplies results in good agreement with FE models for all the cases analyzed.
2) The Kirchhoff's orthotropic plate model is also in good agreement with FE results only when the number of beams is sufficiently large.
The results should be important to the research, since based on analytical solutions that, unlike the numerical approaches, allow to study the influence of the parameters without redoing the model.
The main contribution of this study is the interpretation of the grid mechanical behavior. Indeed, it has been proven here, that the surface enveloping the buckled grid is close to that describing
the buckling of a homogeneous plate, while giving better results.
• H.-J. Chen and S. W. Tsai, “Analysis and optimum design of composite grid structures,” Journal of Composite Materials, Vol. 30, No. 4, pp. 503–534, Mar. 1996, https://doi.org/10.1177/
• M. Zarei, G. H. Rahimi, M. Hemmatnezhad, and F. Pellicano, “On the buckling load estimation of grid-stiffened composite conical shells using vibration correlation technique,” European Journal of
Mechanics – A/Solids, Vol. 96, p. 104667, Nov. 2022, https://doi.org/10.1016/j.euromechsol.2022.104667
• H. L. Cox and H. E. Smith, “The buckling of grids of stringers and ribs,” Proceedings of the London Mathematical Society, Vol. s2-48, No. 1, pp. 1–26, 1945, https://doi.org/10.1112/plms/s2-48.1.1
• G. Z. Harris, “Buckling and Postbuckling of Orthotropic Plates,” AIAA Journal, Vol. 14, No. 11, pp. 1505–1506, Nov. 1976, https://doi.org/10.2514/3.61487
• G. S. Johnston, “Buckling of orthotropic plates due to biaxial in-plane loads taking rotational restraints into account,” Fibre Science and Technology, Vol. 12, No. 6, pp. 435–443, Nov. 1979,
• L. P. Kollar and I. A. Veres, “Buckling of rectangular orthotropic plates subjected to biaxial normal forces,” Journal of Composite Materials, Vol. 35, No. 7, pp. 625–635, 2001.
• I. Hwang and J. S. Lee, “Buckling of orthotropic plates under various inplane loads,” KSCE Journal of Civil Engineering, Vol. 10, No. 5, pp. 349–356, Sep. 2006, https://doi.org/10.1007/bf02830088
About this article
Mathematical models in engineering
beam grid systems
buckling analysis
Rayleigh-Ritz method
Kirchhoff’s orthotropic plate
The authors have not disclosed any funding.
Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Conflict of interest
The authors declare that they have no conflict of interest.
Copyright © 2023 Francesca Pancella, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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14th International Young Researchers Workshop on Geometry, Mechanics and Control - Georg-August-University Göttingen
14th International Young Researchers Workshop on Geometry, Mechanics and Control
Göttingen, December 16 - 18, 2019
Daniele Angella (Firenze): Cohomological and metric aspects in complex non-Kähler geometry
Gabriella Pinzari (Napoli): Some application of perturbative theory to the N-body problem
Dimitra Panagou (Michigan): Multi-Agent Control for Safety-Critical Systems
Location: Sitzungszimmer, Mathematisches Institut, Bunsenstr. 3-5, D-37073 Göttingen
Contributed talks
Martin Kohlmann The Cauchy problem for a weakly dissipative μ-DP equation
V. I. Arnold found out in the 1960s that the motion of inertia rigid objects in Classical Mechanics and the incompressible flow of some ideal fluid can be described by the same mathematical approach:
both can be recast as a geodesic flow on a suitable Lie group. While the configuration space for a rigid three-dimensional body is the Lie group SO(3), fluid flow can be modeled on suitable
diffeomorphism groups. In this talk, we explain how the geometric reformulation of a variant of the Degasparis-Procesi equation occurring in shallow water theory can be used to obtain results on
well-posedness, conservation laws and blow up.
Alexandre Simoes Exact discrete nonholonomic mechanics: an open problem
The existence of an exact discrete lagrangian function for nonholonomic systems is still an open problem in the field of geometric integration. In the last few decades, an effort has been made to
introduce geometric numerical methods, such as variational integrators, which preserve geometric structure. In the case of variational integrators, we discretize the principal action of the system
and then we apply a discrete variational principle to obtain the discrete-time equations of motion, whose solutions are sequences of points which approximate the solution for the continuous-time
problem. In this talk we will restrict to nonholonomic mechanics. Although there are plenty of discrete descriptions at our disposable in the existing literature, the problem of finding an exact
discrete lagrangian function for onholonomic mechanics or, more generally, the problem of integrating exactly the continuous-time nonholonomic problem was not yet solved. We will show how to produce
a family of nonholonomic integrators with the special property that we can distinguish one of them that exactly integrates the continuous-time non-holonomic problem. This discovery could make an
advance to the study of error analysis of numerical methods and would have many applications on subjects such as optimal control.
Antonio Bueno The structure of complete surfaces in R^3 with prescribed mean curvature
Let H be a C^1 function defined in the 2-sphere of the Euclidean space R^3. An oriented surface M is said to have prescribed mean curvature equal to H if its mean curvature function at every point is
given by the value of the function H at the image of the Gauss map at p. In this talk we give a structure result concerning the existence and uniqueness of properly embedded surfaces with prescribed
mean curvature. This is a joint work with José Antonio Gálvez and Pablo Mira.
Layth L. Alabdulsada On the Connections of Sub-Finslerian Geometry
A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a k-dimensional smooth distribution only, not on the whole tangent manifold. Our
purpose is to construct a generalized non-linear connection for a sub-Finslerian manifold, called L-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler
structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.
Theocharis Papantonis Representation up to homotopy of Lie n-algebroids
In this talk, we will discuss about 'representations up to homotopy' which are considered to be the 'correct' notion of representations of higher Lie algebroids. We will construct an important class
of representations called 'adjoint representation' and see an interesting connection between this class and higher Poisson and symplectic structures.
Veronica Arroyo Control-oriented learning for formation control of Lagrangian systems
Methods such as deep learning and reinforced learning have been successful in modeling and controlling many dynamical systems. For such methods to achieve the desired performance which appears in the
real world, but we don't know which dynamics follow, we often require multiple systems runs over a large sum of data that sometimes may not be available in several scenarios. In this poster, we
present a method based on quadratic programming to approximate the Lagrangian associated with the distance-based formation problem from limited data from an observation of the trajectory. We further
show how to obtain bounds for the approximation errors. This is joint work together with Leonardo J. Colombo
Ashraf Owis Feedback optimal control using Hamiltonian dynamics
The aim of this work is to combine multiobjective optimization with the feedback optimal control problem via solving the Hmailton Jacobi Bellman (HJB) PDE. A method used to solve the feedback optimal
control problem has been introduced by Park, Guibout and Scheeres, 2006. This method is called the generating function technique which allows to solve the associated Hamilton-Jacobi-Bellman equation
directly. Our aim is to intertwine this technique with numerical methods for feedback optimal control problem. In order to overcome the nonlinearity of the most of the physical problems we transform
the nonlinear feedback optimal control problem to iterative linear problems by a sequence approximating series method. The solutions derived by the proposed method have the property to be robust to
perturbations and errors in the initial conditions: in this way we do not solve a single optimal control problem but rather we find the whole family of optimal control laws. We can apply this
technique with both soft and hard constraints according to the physical model of the problem in hand.
Poster session
Manuel Lainz Valcázar Contact Hamiltonian systems with nonholonomic constraints
We study the dynamics and geometry of contact Hamiltonian systems with nonholonomic constraints. The equations of motion can be obtained from Herglotz's variational principle when we constrain the
variations to lie on a given distribution. We prove that this dynamics can also be understood as a projection of the non-constrained dynamics. Finally, we construct a bracket that provides the
dynamics on the constrained system that generalizes the Jacobi bracket of the contact manifold on the unconstrained case. This new bracket does not fulfill the Jacobi identity.
Miguel Berbel A category towards Lagrangian reduction by stages in Field Theory
The symmetry of a Lagrangian mechanical system, given by the action of a Lie group G, may sometimes come from different sources. In this context, it is natural to reduce by stages, that is, reducing
first by a normal subgroup N of G and afterwards by the quotient. In order to perform successive steps of reduction it was introduced [2] the category of Lagrange- Poincaré bundles. The aim of this
poster is to explore if there exists a category of bundles playing an analogous role in the Field Theory setting developed in [1,3].
[1] M. Castrillón López, P. L. García Pérez, T.S. Ratiu: Euler-Poincaré Reduction on Principal Bundles. Lett. Math. Phys 58, 167-180, 2001.
[2] H. Cendra, J.E. Marsden, T.S. Ratiu: Lagrangian reduction by stages. Mem. Amer. Math. Soc. 152, no. 722 (2001).
[3] F.Gay-Balmaz, D. Holm, and T. S. Ratiu: Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions J. Braz. Math. Soc. 42(4), (2011), 579–606.
Omayra Yago Nieto Path planning and formation control for a team of quadrotors
In this poster, we study how the equations of motions for a quadrotor arise naturally from variational principles on Lie groups for systems with external forces. We extend the analysis to the motion
of four quadrotors in formation, where a constraint is introduced to keep the formation between the quadrotors. We formulate the problem of inexact interpolation for the centroid dynamics of the
team, that is, a trajectory planning problem, for the quadrotors moving in the space while they keep the formation and the planned (smooth) trajectory passes close enough to the centroid of the
formation. This problem can bee seen as an optimization problem with constraints and moreover, as a constrained higher-order variational problem. We derive necessary conditions for optimal solutions.
This is joint work together with Leonardo Colombo.
Antonio Bueno Rotational symmetry of hypersurfaces with prescribed linear mean curvature
A hypersurface M immersed in the Euclidean space has prescribed linear mean curvature if its mean curvature function is given as a linear function defined in the n-sphere, depending on its Gauss map.
These hypersurfaces are related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. They also can be characterized as critical points of
a variational problem involving the weighted area and volume functionals. In this work, we give a classification of such hypersurfaces that are rotational. This is a joint work with Irene Ortiz.
Irene Ortiz Cylindrical flat hypersurfaces in R^{n+1} with linear prescribed mean curvature
In this work we study hypersurfaces in R^{n+1} with linear prescribed mean curvature which have constant curvature. By classical theorems of Liebmann, Hilbert and Hartman-Nirenberg, any such
hypersurface must be flat, hence invariant by an (n − 1)-group of translations and described as the riemannian product α × R^{n+1}, where α is a plane curve called the base curve. We classify such
hypersurfaces by giving explicit parametrizations of the base curve. This is a joint work with Antonio Bueno.
Fahim Kistosil Rough Paths Perturbation Of The Equation of Semilinear Volterra Equations
In the classical theory of thermodynamics, thermal signals propagate with infinite speed, local actions and cumulative behaviour are neglected; the history, even the very recent history is not taken
into account. A way to introduce a memory is first to introduce a so called memory function. By this memory function β the history will be taken into account by taking a averaging the past with β and
leads to a so-called Volterra equation. The theory of rough path introduced by Terry Lyons in his seminal work as an extension of the classical theory of controlled differential equations. In this
work we show existence and uniqueness of mild solution for an infinitesimal semilinear Volterra equations driven by a rough path perturbation. The first step we give some maximal regularity results
of the Ornstein Uhlenbeck process with memory term driven by a rough path using the Nagy Dilation Theorem.
Samreen Kahn Inversion of Covering Maps and its Polar Decomposition
My talk will focus on inverting the covering map from SL(2,R) to So+(2,1) and simultaneously provide the polar decomposition for matrices in So+(2,1). Extensions to higher dimensions will also be
Abraham Bobadilla Rolling and holonomy of pseudo-Riemannian manifolds
In Riemannian geometry, the rolling system is a well-known framework for comparing certain geometric information of two Riemannian manifolds. Typically, one is well understood and something needs to
be said about the other one. In a series of papers, Chitour, Kokkonen, Godoy (2015, 2016, 2017) and others, found a deep link between the holonomy of an appropriate vector bundle connection and the
controllability of the rolling system. In this poster, I will present some results that extend this, but for rolling pseudo-Riemannian manifolds, as introduced by Markina and Silva-Leite (2016).
These results are part of my Ph.D. thesis in Mathematics at Universidad de La Frontera (Temuco, Chile).
Mauricio Godoy Molina Submersions and curves of constant geodesic curvature
A classic point of view to study sub-Riemannian manifolds is to find a Riemannian metric taming the sub-Riemannian one. Controlling this taming is necessary if one hopes to be able to prove
something. One possibility is to consider sub-Riemannian metrics obtained via Riemannian submersions. The aim of this talk is to present necessary and sufficient conditions for when sub-Riemannian
normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. Additionally, it is possible to describe a canonical extension of
the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This is joint work with Erlend Grong (Orsay) and Irina Markina (Bergen).
Rodrigo T. Sato Martín de Almagro A variational derivation of the forced Euler-Poincaré equations and applications to error analysis
Abstract. In this poster, we describe a variational derivation of the forced Euler-Poincaré equations [1] using a duplication of variables technique originated in [2, 3]. We show that the underlying
geometry is related with the notion of a Poisson groupoid. This is applied to the variational construction of geometric integrators for forced systems and allows us to apply variational error
analysis, extending the results of [4] to the reduced case. This is a joint work with David Martín de Diego.
[1] D. Martín de Diego and R. T. Sato Martín de Almagro. Variational order for forced Lagrangian systems II: Euler-poincaré equations with forcing. (preprint, arXiv:1906.09819), 2019.
[2] C. R. Galley. Classical mechanics of nonconservative systems. Phys. Rev. Lett., 110:174301, 4 2013.
[3] C. R. Galley, D. Tsang, and L. C. Stein. The principle of stationary nonconservative action for classical mechanics and field theories. (preprint, arXiv:14 12.3082), 12 2014.
[4] D. Martín de Diego and R. T. Sato Martín de Almagro. Variational order for forced Lagrangian systems. Non-linearity, 31(8):3814-3846, 2018.
The workshop is supported by:
Klaus-Inhülsen Stiftung
Fakultät für Mathematik und Informatik
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Time Clock Rounding Calculator | Types of Time Clock Rounding & their Rules
Example 1: What is 15:03 rounded to 1/10th of an hour?
Given Clock Time is 15:03
As we want to round to it 1/10th of an hour we will see whether it falls in the first or second class of 3-minute split interval
Since it is 03 minutes we will round down
Therefore, 15:03 rounded to the 1/10th of an hour is 15.
Example 2: An Employee punches in the office at 8:56 AM? Find the office time rounded to 5-minutes?
Given Clock Time is 8:56 AM
Since we want to round the clock time using 5-minute rounding we will first see whether the time falls under first half of the 2.5 mins interval or second half of 2.5 minutes interval
8:52 AM rounded to 5-minutes is 8:50 AM
Example 3: Round 7:05 to 1/4th of an hour?
Given Clock Time = 7:05
Since we want to round to 1/4th of an hour we will see whether the given clock time falls in the first half or second half of 7.5 minute split interval and thereby round up or round down accordingly.
Since 5 minutes falls in the first 7.5 minute interval we will round down i.e. 7:05 rounded to 15 minutes is 7:00
Seek help regarding several mathematical concepts that you feel daunting from roundingcalculator.guru and understand them clearly in no time.
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10 Examples of Dependent Events
| 10 Examples of Dependent Events
10 Examples of Dependent Events
October 22, 2023
In mathematics, Dependent events in probability are events where the outcome of one event affects the outcome of the other event.
In this article, we will discuss ten examples of dependent events in mathematics.
Examples of Dependent Events
These are 10 examples of dependent events.
1: Drawing Cards from a Deck
Drawing a card and not replacing it affects the probabilities of drawing specific cards in subsequent draws.
2: Marbles from a Bag
When drawing marbles from a bag without replacement the probability of drawing a certain color or type of marble, making the events dependent.
3: Rolling Dice
When you roll dice, the chance of getting certain results in a row.
For example, getting two sixes on two dice, is affected by what you rolled in the first throw.
4: Test Questions
In a multiple-choice test, the probability of answering the second question correctly depends on whether the first question was answered correctly.
5: Sequential Coin Flips
Flipping a coin multiple times and recording the outcomes sequentially results in dependent events.
6: Playing Cards in a Game
In card games like Poker, the probability of drawing specific hands depends on the cards that have been revealed. It is dependent events.
7: Choosing Socks from a Drawer
Picking two socks from a drawer, the chance of the second sock matching the first is influenced by your first pick.
8: Drawing Balls from Urns
When you draw balls from an urn without putting them back, the likelihood of picking particular colors or numbers in the next draws changes.
9: Lottery Draws
In some lottery formats, if you win a smaller prize, the probability of winning a larger prize in the same draw becomes zero. It is dependent events.
10:Bike having a flat tire
Getting a flat makes you late to work, so the events are dependent.
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Binary Operations Explained in 5 Minutes
Binary Operations are all over mathematics. It's hard to avoid them. They're a fundamental concept.
Suppose we have a set of n objects S={x1,x2,x3,..,xn}. Then a binary operation takes two objects from the set and produces a new object, like this B(xi,xj)=z
The new object z may be in the set S or not. It could be a totally different object. When z is in the set we say that S is "closed" under the B operation. The point is that B operates on just two
objects and this is why it's called a binary operation. Addition of numbers is an example of a binary operation. In this case order does not matter, so x+y=y+x, but this is not true in general and B
(x,y) can be different than B(y,x). Binary operations for which B(x,y)=B(y,x) are known as Abelian operations after the Norwegian mathematician Niels Henrik Abel.
Can we generalize binary operations? Perhaps..
If can regard (xi,xj) as a sequence of two items from S then B is an operation that acts on sequences of length 2. So one way to generalize B would be to define an operation which acts on sequences
of length, like this..
A is not a binary operation, it's an n-ary operation. Binary operations only act of sequences of length 2, but A acts on sequences of length n. This interpretation gets us to thinking about
sequences. What exactly is a sequence?
A sequence is simply an ordered list of mathematical objects. The order is important. For example, (a,b) and (b,a) are different sequences. The other thing about a sequence is that, unlike a set, a
sequence can contain duplicate objects. For example (a,a), (a,a,a), (a,a,b,b,b) and (a,a,a,...) are valid sequences. Also, it's easy to see that a sequence can be finite or infinite.
So now we can think of A as acting on a single object, a sequence of length n, and binary operations are the special case of n=2.
Let's summarize. We have a set S containing m objects. Then we define the "sequence set of S of order n" as the set of all sequences of length n made using the objects in S. Let's call this set Sn.
How big is it? That's easy, it has m^n members. Sn is interesting because no matter what the size of S, Sn can be very big. For example, even if S has just one member, S={a}, the size of Sn is
infinite. Our operator A operates on Sn.
Like this post? Please click G+1 below to share it.
Content written and posted by Ken Abbott
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Yale Alumni Magazine - Dan Spielman
Infinite complexity
Sometimes, even mathematicians rely on intuition. Professor Daniel Spielman’s helped him solve an unsolvable problem.
Richard Panek teaches writing at Goddard College. He is the author of several books on science, most recently The Trouble with Gravity: Solving the Mystery beneath Our Feet.
One day in 2008, a visiting professor dropped by the office of mathematician Daniel Spielman ’92 and asked, as he usually did during the alternating semesters he was teaching at Yale, what Spielman
was working on.
Spielman told him. Gil Kalai, a mathematician himself then on leave from Hebrew University, said Spielman’s work reminded him of an old, enduring problem in theoretical physics. Kalai described it.
Spielman said he might be able to solve it.
You don’t understand, Kalai said. The question isn’t whether Daniel Spielman can solve the problem. The question isn’t even if anyone can solve the problem—because the consensus among theoretical
physicists had long ago coalesced around an answer of No. The problem has no solution.
Spielman listened politely. He was no theoretical physicist; he freely acknowledged his ignorance of the subject. What he was, instead, was a mathematician, and he had a mathematician’s hunch.
This past spring the National Academy of Sciences rewarded Spielman and two collaborators with the 2021 Michael and Sheila Held Prize, one of the highest honors in computer science. Spielman’s hunch
had paid off: one area of math can apply to disparate fields in ways you’d never anticipate.
“This happens a lot,” says Spielman, Sterling Professor of Computer Science and professor of statistics and data science. “There are all sorts of places where people have been burrowing really deeply
into tunnels in one area of mathematics, and at some point they come along and they hit another area of mathematics, and you discover connections.”
The tunnel that Spielman was digging when he had his fateful discussion with Kalai involved “networks”—social networks, computer networks, communication networks. In graphing a social network, for
instance, mathematicians will represent a single person as a dot, or a “node.” If they connect that node to another node, they draw a straight line, or an “edge.” Count the edges extending from any
single node and you know its number of connections. Mathematicians, Spielman says, “call this ‘degree.’ Physicists call it ‘valence.’ Other people call it ‘How many friends do you have?’”
If the network you’re studying is a small town, the graph might be relatively simple. But what if you’re graphing a shipping service’s possible routes? Plot the collection points (front porches,
outlet stores, drop boxes), then the airport hubs through which the packages pass, then the possible destinations (every street address and PO box in the United States and Canada). Then add up every
conceivable path a single package might take. The answer might not be an infinity, and the time an algorithm needs to plot the most efficient path might not be an eternity. But if you want to send a
birthday gift to Grandma that absolutely, positively has to be there overnight?
The possible solution that Spielman explored was “sparsificaton”—speeding up the algorithm by simplifying networks. “Which meant,” Spielman says, “dropping most connections”—or, in this metaphor,
rerouting traffic onto only a few major roads.
As a practical matter, shipping companies wouldn’t survive without such shortcuts. Rather than figure out the most efficient path of each package in advance, they can leave the local logistics to
regional dispatch centers, which might be operating according to algorithms of their own.
For Spielman, however, the challenge wasn’t to deliver the goods. It was more abstract: take a virtually infinitely complex system and see how far he could sparsify it without sacrificing its
integrity. In 2008—around the time he was meeting with Kalai—he and his student Nikhil Srivastava ’10PhD (now at the University of California–Berkeley) were finishing “Graph Sparsification by
Effective Resistances,” which appeared in Society for Industrial and Applied Mathematics Journal on Computing.
“We were able to show you can actually approximate any network,” Spielman says. “This was a sort of crazy result. It wasn’t necessarily practical”—it wouldn’t guarantee the most efficient route door
to door, so to speak—“but it was mathematically very intriguing.”
In using this word, Spielman is evoking the fundamental distinction in his field between math that is promising and math that is practical—between an algorithm that, after further burrowing, might
eventually solve real-life problems or intersect with another tunnel, and an algorithm that’s ready to use today.
Another tunnel is what his intuition whispered to him while Kalai was describing the long-standing problem that theoretical physicists believed had no solution. The Kadison-Singer problem—formulated
by mathematicians Richard Kadison and Isadore Singer in 1959—asks whether a mathematically complete description of a quantum subsystem might allow a mathematically complete description of the quantum
system as a whole.
Spielman himself struggles to find a layperson’s explanation. The best he can do, he says, is: the Kadison-Singer problem “asks whether a large number of measurements made on a part of a quantum
system uniquely determine the system as a whole.” (He also says, “If given a choice between attempting to explain the original KS problem and defending myself from a pride of hungry tigers, I’d give
myself better odds with the tigers.”)
One reason that theoretical physicists thought the answer must be No is that the core principle of quantum mechanics is uncertainty—for instance, the impossibility of simultaneously measuring a
particle’s position and velocity. You can’t use a subsystem of uncertainty to capture the system as a whole … can you?
Why not? thought Spielman. Wasn’t he doing the same thing, sort of? True, he wasn’t working in the quantum realm. But pruning a virtual infinity of choices in a system to describe a subsystem
couldn’t be all that different from using a subsystem to describe a system.
His confidence only grew when Kalai referred Spielman to a paper he thought might help. It included a statement about vectors (or, in their terminology, degrees) and matrices (systems) that Spielman
and his collaborators had already assumed was true. Tunnels were, indeed, converging.
“It looked like something we understood incredibly well,” Spielman says. Maybe he would never comprehend the implications for the quantum realm. “But at least this one version was something I could
understand. So it was at least clear that the problem was fundamental in many different areas.”
The decision was easy. We’ll work on this, he thought. It’ll go pretty fast.
Five years passed.
Five not altogether unpleasant years: in 2008, Spielman and his collaborator Shang-Hua Teng won the Gödel Prize; in 2010 he received the Nevanlinna Prize; in 2012 he joined the inaugural class of
Simons Investigators, a fellowship providing $660,000 in research funding over five years; in 2012, he received a MacArthur Fellowship. All these honors, though, recognized work that was receding
farther and farther into the past, while the Kadison-Singer problem threatened to swallow his (and his colleagues’) future.
Vacations became distractions. Holidays became hurdles. If his wife told him she was going out for drinks with some nodes of hers, Spielman would think, Good. I’ll stay home and work on the
Kadison-Singer problem, and I don’t have to worry about my wife having fun.
“It always felt like we were making progress,” he says. “We kept coming up with conjectures that were pretty and looked true.” Still, he couldn’t ignore the possibility that their tunnel was going in
circles, at least regarding the Kadison-Singer problem.
“Maybe we should find a way out,” Spielman finally decided. He and his collaborators, Adam Marcus (now at the École polytechnique fédérale de Lausanne in Switzerland) and Srivastava, collated the
hundreds of pages of emails they’d sent one another over the years and asked: Can we use them to do anything else that’s important?
Maybe, Spielman thought, they could use their math to find new applications for Ramanujan graphs—a sometime inspiration for Spielman’s past work, including his doctoral thesis at MIT. (He majored in
mathematics and computer science at Yale.) He knew that Ramanujan graphs were extremely efficient. “If you want to be able to transmit messages through a network, these are the networks you would
want,” Spielman says. “There’s no interference”—no bottlenecks where two messages block each other—“and there’s short paths between everything.” Srivastava soon located a paper that suggested a way
to generate Ramanujan graphs. It was still missing a crucial step—but Spielman’s team realized they could derive that step from the techniques they’d been developing on their own for the
Kadison-Singer problem.
“About a week later,” Spielman says, “we knew how to make Ramanujan graphs” using their own math. “It was a shockingly fast development. At which point I thought, ‘Okay, this is great. Even if we
haven’t solved the Kadison-Singer problem, we’ve developed all this new mathematics, and we’ve used it do something, and maybe someone else will use it to solve the Kadison-Singer problem.”
Fortunately for Spielman and his two collaborators, nobody else did.
Instead, they did. “It was only by taking what we thought was a diversion,” Spielman says, “that a few months later we realized we actually had enough to solve the Kadison-Singer problem.”
To a layperson the word solve can be misleading. Spielman didn’t find the answer to the Kadison-Singer problem itself. Rather, he discovered that the answer to the question he had addressed in that
long-ago office meeting with Kalai—the answer to the question of whether a solution to the Kadison-Singer problem was even possible—the answer that physicists had come to believe was No, was Yes.
Which is to say, mathematically, Spielman’s result is promising.
The citation for the Held Prize recognizes both achievements: generating new constructions of Ramanujan graphs and showing that the Kadison-Singer problem is potentially solvable. But the Held Prize
citation also places their work within a specific context—one that a scientist harboring a mathematician’s hunch might especially appreciate: the collaboration “uncovered a deep new connection
between linear algebra, geometry of polynomials, and graph theory that has inspired the next generation of theoretical computer scientists.”
Deeper tunnels. More degrees. New nodes.
Since the paper went online in 2013, mathematicians in multiple fields have been trying to wrest something practical from the promise inherent in the Kadison-Singer problem. Spielman and his
collaborators have themselves written a paper, still undergoing peer review, that they think might advance that discussion.
In the meantime, the burrowing continues.
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Special Relativity
the laws of physics are the same for all (inertial) observers
the speed of light in a vacuum has the same value for all (inertial) observers.
These two postulates taken together tell us that space and time are not absolute but must be relative, i.e. change depending on the observer. Otherwise, it isn't possible that different moving
observers always measure exactly the same value for the speed of light. This was discovered by the famous Michelson Morley experiment.
It's About Time: Understanding Einstein's Relativity by Mermin is a great book for laymen on special relativity.
Special relativity is Einstein’s description of how some of the basic measurable quantities of physics – time, distance, mass, energy – depend on the speed of the measuring apparatus relative to the
object being studied. It shows how they must change in order to guarantee that Galileo’s principle of relativity (that the laws of physic should be the same for every experimenter, regardless of
speed) should hold even at speeds near that of light. Gravity from the Ground Up by Bernard Schutz
Special relativity is Einstein’s description of how some of the basic measurable quantities of physics – time, distance, mass, energy – depend on the speed of the measuring apparatus relative to the
object being studied. It shows how they must change in order to guarantee that Galileo’s principle of relativity (that the laws of physic should be the same for every experimenter, regardless of
speed) should hold even at speeds near that of light.
The diagram below illustrates the difference between distance in Euclidean space (left-hand side) and "distance" (actually proper time) in space-time according to Einstein's special theory of
relativity (right-hand side). Time dilation is shown in red. For a more detailed explanation of this diagram see Fun with Symmetry.
Mathematically, special relativity is the statement that all laws of physics are invariant under the Poincare group.
In special relativity spacetime denotes the configuration space of a single point particle. We denote the Minkowski spacetime by $Q$, i.e., $\mathbb{R}^{3+1}\ni(x^0,\ldots,x^n)$ with the Minkowksi
\[ \eta_{\mu\nu} = \left( \begin{array}{ccccc} 1 & 0 & 0 & \ldots & 0 \\ 0 & -1 & 0 & \ldots & 0 \\ 0 & 0 & -1 & & 0 \\ \vdots & \vdots & & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & -1 \end{array}\
right). \]
What interests us most of the time is path of a particle \[ q:\mathbb{R}(\ni t) \longrightarrow Q \] where $t$ is a parameter for the path, not necessarily $x^0$, and not necessarily the proper time
To derive the equations of motion for such a free particle, we need the corresponding action \[ S(q) = \int_{t_0}^{t_1} L\Bigl(q^i(t),\dot{q}^i(t)\Bigr)\,dt \] for $q\colon [t_0,t_1]\rightarrow Q$. A
useful hint towards the correct Lagrangian is that it should be independent of the arbitrary parametrization parameter $t$. The obvious candidate for the action $S$ is mass times the "proper time" \[
S = m\int_{t_0}^{t_1} \sqrt{\eta_{ij}\dot{q}^i(t)\dot{q}^j(t)}\,dt . \]
Using $$\label{eq:relativistic-Lagrangian} L = m \sqrt{\eta_{ij}\dot{q}^i\dot{q}^j}$$ and the Euler-Lagrange equations, we get \begin{align*} p_i = \frac{\partial L}{\partial\dot{q}^i} &= m\frac{\
partial}{\partial\dot{q}^i}\sqrt{\eta_{ij}\dot{q}^i\dot{q}^j} \\ &= m\frac{2\eta_{ij}\dot{q}^j}{2\sqrt{\eta_{ij}\dot{q}^i\dot{q}^j}} \\ &= m\frac{\eta_{ij}\dot{q}^j}{\sqrt{\eta_{ij}\dot{q}^i\dot{q}^
j}} \, = \frac{m\dot{q}_i}{|{\dot{q}}|}. \end{align*} The Euler–Lagrange equations say \[ \dot{p}_i = F_i = \frac{\partial L}{\partial q^i} = 0. \]
We can understand this better when we use the proper time as our parameter $t$ so that \[ \int_{t_0}^{t_1}|{\dot{q}}|dt = t_1-t_0, \quad \forall\,t_0,t_1 . \] This fixes the parametrization up to an
additive constant. Thus $|{\dot{q}}|=1$, so that \[ p_i = m\frac{\dot{q}_i}{|{\dot{q}}|} = m\dot{q}_i \] and the Euler–Lagrange equations say \[ \dot{p}_i =0 \Rightarrow m\ddot{q}_i = 0 . \] We can
therefore conclude that our free particle moves unaccelerated along a straight line.
We believe that special relativity at the present time stands as a universal theory describing the structure of a common space-time arena in which all fundamental processes take place. All the laws
of physics are constrained by special relativity acting as a sort of ”super law”. One more derivation of the Lorentz transformation by Jean-Marc Levy-Leblond
We believe that special relativity at the present time stands as a universal theory describing the structure of a common space-time arena in which all fundamental processes take place. All the laws
of physics are constrained by special relativity acting as a sort of ”super law”.
One more derivation of the Lorentz transformation by Jean-Marc Levy-Leblond
The Michelson-Morley experiment demonstrated that the speed of light is the same in all frames of reference. Special relativity is the correct theory that incorporates this curious fact of nature.
Special relativity is an extension of classical Newtonian mechanics. In particular for objects that move very fast (close to the speed of light) predictions that we get using Newtonian mechanics turn
out to be wrong and only special relativity yields the correct results. For slowly moving objects both, Newtonian mechanics and special relativity yield approximately the same results.
Special relativity predicts many effects like the relativity of time or of the dilation of fast moving objects which are all in agreement with experiments. An important application of special
relativity is the GPS system. Without the correct equations of special relativity, the clocks on GPS satellites wouldn't show the correct time and thus precise navigation would be impossible.
Consider the continuous motion of an object in discrete space and time, in which there is a minimum length, denoted by LU , and a minimum time interval, denoted by TU . If the object moves with a
speed larger than LU TU / , then it will move more than a minimum length LU during a minimum time interval TU , and thus moving LU will correspond to a time interval shorter than TU during the
motion. Since TU is the minimum time interval in discrete space and time, which means that the duration of any change cannot be shorter than TU , the motion with a speed larger than LU TU / will be
prohibited. As thus, there is a maximum speed in discrete space and time, which equals to the ratio of minimum length and minimum time interval." " Now I will further argue that the maximum speed c
is invariant in all inertial frames in discrete space and time. According to the principle of relativity, the discrete character of space and time, in particular the minimum time interval TU and the
minimum length LU , should be the same in all inertial frames. If the minimum sizes of space and time are different in different inertial frames, then there will exist a preferred Lorentz frame. This
contradicts the principle of relativity. Thus, LU TU c ≡ / will be the maximum speed in any inertial frame (see also Rindler 1977; 1991)" "In this meaning, Galileo’s relativity is a theory of
relativity in continuous space and time, while Einstein’s relativity is a theory of relativity in discrete space and time.http://image.sciencenet.cn/olddata/kexue.com.cn/upload/blog/file/2010/8/
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Steps to Miles Calculator
About Steps to Miles Calculator
How many steps is 1 miles?
You may have heard advice to count your steps at some time if you are concerned about your fitness. It's never been simpler to keep track of how many steps you take each day with the popularity of
health gadgets like smartwatches. An average person's stride is between 2.1 and 2.5 feet long. Accordingly, it takes more than 2,000 steps to cover a mile, and 10,000 steps equal roughly 5 miles. A
sedentary person might only take 1,000–3,000 steps each day on average. Increasing their step count has numerous health advantages for them.
How many km is 10,000 steps?
The average person walks 2,000 to 2,500 steps each mile, according to fitness trackers or phone motion sensors. Due to the longer stride length of running steps, you can take 1,000–2,000 steps per
mile. 10,000 steps is equivalent to 4 to 5 miles. The average number of steps each mile depends on the length of your stride. You can start to imagine how far you need to walk to reach 10,000 steps
per day by knowing how many steps are typically required to cover a mile. The opposite is also true. If you consider how many you manage to log during your daily activities, miles might not seem so
long. You'll reach your daily target if you keep moving.
How to track your steps?
You can easily keep track of your daily steps by using a pedometer or fitness tracker. To begin, put the pedometer on every day for a week. Put it on the moment you wake up and wear it till you go to
sleep. In a journal or notepad, track your daily steps. Your average daily step count will be known towards the end of the week. Your daily step count may surprise you, depending on how active you
are. It is appropriate to aim to average 10,000 steps per day by increasing your average daily steps by 500 per week.
How to convert steps to miles?
A pedometer's primary purpose is to track your steps when you walk, run, or jog. You can achieve specific fitness targets, such as a daily walking goal, with the use of a pedometer. 10,000 daily
steps count as "active." Calculating how many kilometres you walk each day is simple.
Walk one loop around a 400-meter track while keeping track of your steps to do a manual conversion of walking steps to miles. To calculate steps per mile, multiply your total number of steps by 4.
Use the odometer on your automobile to measure 1 mile while driving, 1 mile when walking, and then add up all of your steps. You may determine how many miles you walk each day by dividing your total
daily steps by your average steps per mile once you know your average steps per mile. For instance, if you take 500 steps on a track that is 400 metres long, you are walking at a pace of 2,000 steps
per mile. If your pedometer registers 10,500 steps during the day, that means you walked 5.2 miles.
Utilizing a step counting app on your smartphone is an additional choice. While these aren't as precise as a pedometer, they work well enough to give you a reasonable estimate of the number of steps
you take per mile.
What factors affect steps count in a mile?
Your height, weight, how quickly your legs are moving, and hundreds of other characteristics, including the length of your legs and your walking style, all influence how many steps you take in a
mile. Due to the fact that each person's step count will produce a different distance measurement, the basic step is an imperfect way to estimate distance.
Longer legs are more common in taller persons, which allows them to go farther with each step. Studies have shown that even at equal heights, men may take a few more steps per mile than women since
they are generally taller than women. Male and female steps would be extremely close at a similar height, therefore for our purposes, we'll just focus on height.
Steps per mile are also influenced by height and step size. To calculate steps per mile depending on walking pace, we'll make the following assumptions: average heights for men and women; average
step lengths.
Stride length and step length are frequently conflated. Step length is the distance between the heels of each foot when walking (the distance of one step). The distance between the heels of one foot
and the same foot when taking two full steps is known as the stride length. Adults typically take steps that are between 2.2 and 2.5 feet long (26-30 inches). That equates to 66–76 cm. Women's
average height is approximately 26 inches (66 cm), whereas men's average height is approximately 31 inches (79 cm).
The majority of people who set a 10,000 step target don't try to do it all at once. It is undoubtedly possible to walk five miles (8 km) and accumulate ten thousand steps, but at a quick pace of 3.5
mph (5.6 kph), that will only take around one to two miles. It's important to keep in mind that the typical American day includes up to 5,000 steps or more. To reach 10,000 steps, you may only need
to walk 2.5 miles, which you may do in less than an hour. A lot of calories can be burned during that hour which can also help you in improving your health.
Individuals should evaluate their personal baseline step count and then develop a plan to gradually increase how frequently and how quickly they walk in a way that is safe for them.
It can be useful to know how many steps there are in a mile. There is growing evidence that tracking your step count, whether it be during a lengthy run or a few laps around the park, can assist keep
your general health in check in addition to helping you manage your fitness regimen.
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