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What income is needed for a 350k mortgage?
Asked by: Damon Ortiz
| Last update: October 22, 2022
Score: 4.9/5
37 votes
How Much Income Do I Need for a 350k Mortgage? You need to make $129,511 a year to afford a 350k mortgage. We base the income you need on a 350k mortgage on a payment that is 24% of your monthly
income. In your case, your monthly income should be about $10,793.
How much per month is a 350k mortgage?
On a $350,000, 30-year mortgage with a 3% APR, you can expect a monthly payment of $1,264.81, not including taxes and interest (these vary by location and property, so they can't be calculated
without more detail).
How much combined income do I need for a 400k mortgage?
What income is required for a 400k mortgage? To afford a $400,000 house, borrowers need $55,600 in cash to put 10 percent down. With a 30-year mortgage, your monthly income should be at least $8200
and your monthly payments on existing debt should not exceed $981. (This is an estimated example.)
How much income do I need for a 300K mortgage?
How much do I need to make to buy a $300K house? To purchase a $300K house, you may need to make between $50,000 and $74,500 a year. This is a rule of thumb, and the specific salary will vary
depending on your credit score, debt-to-income ratio, the type of home loan, loan term, and mortgage rate.
What mortgage can I afford with 100k salary?
If you have a 20% down payment on a $100,000 household salary, you can probably comfortably afford a $560,000 condo. this number assumes you have very little debt and $112,000 in the bank.
28 related questions found
What house can I afford on 70k a year?
On a $70,000 income, you'll likely be able to afford a home that costs $280,000–380,000. The exact amount will depend on how much debt you have and where you live — as well as the type of home loan
you get.
How much house can I afford if I make 60000 a year?
The usual rule of thumb is that you can afford a mortgage two to 2.5 times your annual income. That's a $120,000 to $150,000 mortgage at $60,000.
How much income do you need to buy a $450 000 house?
Assuming the best-case scenario — you have no debt, a good credit score, $90,000 to put down and you're able to secure a low 3.12% interest rate — your monthly payment for a $450,000 home would be
$1,903. That means your annual salary would need to be $70,000 before taxes.
How much home can I afford if I make 65000 a year?
I make $65,000 a year. How much house can I afford? You can afford a $195,000 house.
Can I buy a house making 40k a year?
While buyers may still need to pay down debt, save up cash and qualify for a mortgage, the bottom line is that buying a home on a middle-class salary is still possible — in some places. Below, check
out 15 cities where you can become a homeowner while earning $40,000 a year or less.
Can I afford $325000 house?
A $325,000 house, with a 5% interest rate for 30 years and $16,250 (5%) down will require an annual income of $82,975. We're not including monthly liabilities in estimating the income you need for a
$325,000 home. To include liabilities and determine what you can afford, use the calculator above.
How much house can I afford 75k salary?
You can afford a $225,000 house.
How much house can I afford on $80 000 a year?
For the couple making $80,000 per year, the Rule of 28 limits their monthly mortgage payments to $1,866. Ideally, you have a down payment of at least 10%, and up to 20%, of your future home's
purchase price. Add that amount to your maximum mortgage amount, and you have a good idea of the most you can spend on a home.
Is 70k a good salary?
An income of $70,000 surpasses both the median incomes for individuals and for households. By that standard, $70,000 is a good salary.
How do people afford a 450k house?
To finance a 450k mortgage, you'll need to earn roughly $135,000 – $140,000 each year. We calculated the amount of money you'll need for a 450k mortgage based on a payment of 24% of your monthly
income. Your monthly income should be around $11,500 in your instance. A 450k mortgage has a monthly payment of $2,769.
How much house can I afford if I make $40 000 a year?
1. Multiply Your Annual Income by 2.5 or 3. This was the basic rule of thumb for many years. Simply take your gross income and multiply it by 2.5 or 3 to get the maximum value of the home you can
How much house can I afford if I make $90000 a year?
I make $90,000 a year. How much house can I afford? You can afford a $270,000 house.
How much house can I afford on $85000 a year?
I make $85,000 a year. How much house can I afford? You can afford a $255,000 house.
Is a 60k salary good?
According to the Bureau of Labor Statistics, a 60k annual income is the median US income. This means that half of all workers in the US make more than 60k per year, and half make less. However, 60k
per year is generally considered to be a good salary.
How much mortgage can I afford if I make 72000 a year?
How much should I be spending on a mortgage? According to Brown, you should spend between 28% to 36% of your take-home income on your housing payment. If you make $70,000 a year, your monthly
take-home pay, including tax deductions, will be approximately $4,530.
What is 70k a year hourly?
A salary of $70,000 equates to a monthly pay of $5,833, weekly pay of $1,346, and an hourly wage of $33.65.
How much house can I afford 50k salary?
What you can afford: With a $50k annual salary, you're earning $4,167 per month before tax. So, according to the 28/36 rule, you should spend no more than $1,167 on your mortgage payment per month,
which is 28% of your monthly pre-tax income.
How much house can I afford if I make $150 000 a year?
You can afford a $450,000 house.
How much house can I afford if I make 130 000 a year?
You can afford a $391,000 house.
How much should I spend on a house if I make $100 K?
When attempting to determine how much mortgage you can afford, a general guideline is to multiply your income by at least 2.5 or 3 to get an idea of the maximum housing price you can afford. If you
earn approximately $100,000, the maximum price you would be able to afford would be roughly $300,000.
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solving algebraic loop due to variable resistor in 3-LVL-ANPC under different modulation strategies
I would like to investigate different modulation strategies for a 3-LVL-ANPC-SiC topology under different operating points in terms of overall efficiency and power dissipation (temperature) per
To generate the switching points I use the carrier based PWM method (switching signals by comparing a sine reference voltage with 2 triangle comparative voltages). Same-side clamping, opposite-side
clamping and full path clamping are used as modulation strategies.
The control of the switches and the generation of a three-phase sinusoidal current in the load works for the majority of the operating points under investigation. The occurrence of losses per switch
is also in line with the values expected from theory.
In order to model the temperature-dependent R_DS_on of the Sic MOSFET, I set the Ron in Parameters to 0 and inserted a variable resistor, which uses a 2D look-up table to output a suitable Rds-on
depending on the I_DS current and MOSFET junction temperature.
However, due to the variable resistor, I now get the following warning in every of the 18 switch models:
"Detected an algebraic loop comprising the following components:
3_LVL_ANPC_SiC_VSI_Ron_Erec/S1a/R/Algebraic Component
As mentioned at the beginning, this is not a problem for the majority of operating points.
However, the warning above develops into the following error:
Could not solve the algebraic loop comprising the following components:
3_LVL_ANPC_SiC_VSI_Ron_Erec/S1a/R/Algebraic Component
This leads to immediate termination of the simulation and occurs in particular at operating points with a very small power factor (=large phase shift) and always during the transient process in the
very beginning.
Hence my question: is there a way to avoid the error or to change a setting so that the algebraic loop remains (warning) but can be solved?
I would really appreciate an answer. I have attached pictures to illustrate the problem.
Kind regards
The answer to this is model dependent. The general approaches I would recommend are:
The interaction between algebraic-loops within the different switches can be quite complex. If the variable Rdson is useful, is it necessary to have all the switches in your converter modeled with a
variable Rdson? I recommend only using the variable Rdson in a sub-set of switches in the converter.It may be advantageous to convert the the look-up table into a continuous algebraic expression. Can
you use a polynomial expression to capture the relationship between Rdson(i,T)? At a higher-level, does modeling the variable nature of the Rdson truly improve the accuracy of the results? If not,
then this may be a case of “over-modeling”. Using a constant resistance representative of the expected operating region will make your model more numerically stable and result in a faster simulation.
Thank you for your answer. Now I implemented the variable R_DS_on only in Phase a of my 3-phase VSI. Thanks to this, the sim time is a lot faster and there are less errors due to algebraic loops.
But I have observed another error that leads to the cancelation of the simulation:
State discontinuity after switching. The current through inductor 3_LVL_ANPC_SiC_VSI_Ron_Erec/L1 is forced to jump from 1.06399e-05 to 7.09323e-06.
Since this problem has been discussed before ( https://forum.plexim.com/7672/state-disconuity-after-switching-error?show=7672#q7672 ) I changed the Relative tolerance on the solver settings to 1e-9
(1e-6 was not enough).
The simulation runs now fine and there are, as far as I can tell, no strange outliers regarding Losses, Currents, Voltages etc.
Is this a valid approach? What are the potential risks of such a low Relative tolerance?
Best regards
Glad to hear that was helpful.
> Is this a valid approach? What are the potential risks of such a low Relative tolerance?
The downside can be slower simulations in some instances.
Normally state discontinuities with large errors (e.g. inductor current from 10A -> 0A) point to the issues described in this FAQ link. State discontinuities with small errors can be due to solver
accuracy as you have identified.
If the state value is not close to zero, then the “Relative Tolerance” setting is the value I would typically adjust.
If the state value is very close to zero (as is the case in your model) then I would change the “Absolute Tolerance” from “auto” to a small value (e.g. 1e-3, but depends on the minimum reasonable
state value) and adjust this setting from there.
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OpenStax College Physics for AP® Courses, Chapter 30, Problem 4 (Test Prep for AP® Courses)
The Lyman series of photons each have an energy capable of exciting the electron of a hydrogen atom from the ground state (energy level 1) to energy levels 2, 3, 4, etc. The wavelengths of the first
five photons in this series are 121.6 nm, 102.6 nm, 97.3 nm, 95.0 nm, and 93.8 nm. The ground state energy of hydrogen is −13.6 eV. Based on the wavelengths of the Lyman series, calculate the
energies of the first five excited states above ground level for a hydrogen atom to the nearest 0.1 eV.
Question by
is licensed under
CC BY 4.0
Final Answer
$E_2 = -3.4\textrm{ eV}$
$E_3 = -1.5\textrm{ eV}$
$E_4 = -0.8\textrm{ eV}$
$E_5 = -0.5\textrm{ eV}$
$E_6 = -0.4\textrm{ eV}$
Solution video
OpenStax College Physics for AP® Courses, Chapter 30, Problem 4 (Test Prep for AP® Courses)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. A photon hitting a ground state electron for hydrogen can excite it to the second principal quantum number— this second shell— if the photon has
just the right amount of energy equal to the difference between these energy levels— n equals 2 and n equals 1— and that energy of the photon has to be E 2 minus E 1 where E 1 we are given— it's the
ground state energy, which is negative 13.6 electron volts— and we know that we are dealing with n equals 1 as our initial state because we are told that these photons are from the Lyman series and
for the Lyman series... well when the electron is emitting a photon, the Lyman series has the final state equal to 1 but I guess in this case, it's the initial state that's 1 because we are talking
about a photon being incident and causing the electron to transition up to the excited state. Okay! So the wavelength that causes an excitation to level 2 starting at level 1 is 121.6 nanometers and
a different photon that goes all the way to level 3 starting at level 1 will have a wavelength of 102.6 nanometers and then so on and so on up to levels 4, 5 and 6 all starting at level 1 and so
these wavelengths are in order of increasing energy so energy is going up as you go down this list here; as the wavelengths gets shorter, the energy increases. The energy of the photon is given by
this formula Planck's constant times speed of light divided by wavelength and so as this denominator reduces, the quotient increases and this energy of this photon is the difference in the energy of
the electron at these two different levels and so we want to figure out what is the energy of the electron at n equals 2? So energy at 2 minus energy at 1 is the photon energy and then we'll add
energy at level 1 to both sides and we have E 2 then is hc over λ 21 plus E 1. So that's 1240 electron volt nanometers, which is a convenient way of writing h times c with units that will cancel in
the numerator and denominator if you write the denominator in nanometers and it will leave us with electron volts in the top here. So we have 1240 electron volt nanometers divided by 121.6
nanometers, plus negative 13.6 electron volts for the ground state energy and that is negative 3.4 electron volts is the energy at n equals 2. At energy at n equals 3, it's going to be the same
formula but we're just using the wavelength that was needed to excite the electron from the ground state to n equals 3. So it's 1240 divided by 102.6 plus negative 13.6 and that is negative 1.5
electron volts and then so on and so on it's the same each formula each time but plugging in different wavelengths. So for energy state 4, we are plugging in 97.3 nanometers getting an energy of
negative 0.8 electron volts; for energy level 5, we have 95.0 nanometers and that's negative 0.5 electron volts for that energy and energy level 6, we are plugging in wavelength of 93.8 nanometers
giving us an energy of negative 0.4 electron volts.
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Post TOPIC: Black Holes
RE: Black Holes Permalink
Title: Tidal effects around higher-dimensional black holes
Authors: Richard Brito, Vitor Cardoso, Paolo Pani
In four-dimensional spacetime, moons around black holes generate low-amplitude tides, and the energy extracted from the hole's rotation is always smaller than the gravitational radiation lost to
infinity. Thus, moons orbiting a black hole inspiral and eventually merge. However, it has been conjectured that in higher-dimensional spacetimes orbiting bodies generate much stronger tides,
which backreact by tidally accelerating the body outwards. This effect, analogous to the tidal acceleration experienced by the Earth-Moon system, would determine the evolution of the binary.
Here, we put this conjecture to the test, by studying matter coupled to a massless scalar field in orbit around a singly-spinning rotating black hole in higher dimensions. We show that in
dimensions larger than five the energy extracted from the black hole through superradiance is larger than the energy carried out to infinity. Our numerical results are in excellent agreement
with analytic approximations and lend strong support to the conjecture that tidal acceleration is the rule, rather than the exception, in higher dimensions. Superradiance dominates the energy
budget and moons "outspiral"; for some particular orbital frequency, the energy extracted at the horizon equals the energy emitted to infinity and "floating orbits" generically occur. We give an
interpretation of this phenomenon in terms of the membrane paradigm and of tidal acceleration due to energy dissipation across the horizon.
Read more (43kb, PDF)
How black holes change gear
“Black holes are extremely powerful and efficient engines that not only swallow up matter, but also return a lot of energy to the Universe in exchange for the mass they eat. When black holes
attract mass they also trigger the release of intense X-ray radiation and power strong jets. But not all black holes do this the same way. This has long baffled astronomers. By studying two
active black holes researchers at the SRON Netherlands Institute for Space Research have now gathered evidence that suggests that each black hole can change between two different regimes, like
changing the gears of an engine. The team's findings will be published in two papers in the journal Monthly Notices of the Royal Astronomical Society.”
Read more
Primordial black holes Permalink
Earth has little to fear from a black hole attack
“We can all rest easy. Small black holes that may be roaming space undetected would leave Earth unscathed if they hit us.
Various models suggest matter may have collapsed into black holes soon after the big bang. The smallest of these so-called primordial black holes would have evaporated through a process called
Hawking radiation long ago.
But those weighing a billion tonnes or more could still be around, and many of these black holes would be hard to detect - unless they hit us, says Katherine Mack of the University of
Read more
RE: Black Holes Permalink
Title: Detectable seismic consequences of the interaction of a primordial black hole with Earth
Authors: Yang Luo, Shravan Hanasoge, Jeroen Tromp, Frans Pretorius
Galaxies observed today are likely to have evolved from density perturbations in the early universe. Perturbations that exceeded some critical threshold are conjectured to have undergone
gravitational collapse to form primordial black holes (PBHs) at a range of masses. Such PBHs serve as candidates for cold dark matter and their detection would shed light on conditions in the
early universe. Here we propose a mechanism to search for transits of PBHs through/nearby Earth by studying the associated seismic waves. Using a spectral-element method, we simulate and
visualize this seismic wave field in Earth's interior. We predict the emergence of two unique signatures, namely, a wave that would arrive almost simultaneously everywhere on Earth's free
surface and the excitation of unusual spheroidal modes with a characteristic frequency-spacing in free oscillation spectra. These qualitative characteristics are unaffected by the speed or
proximity of the PBH trajectory. The seismic energy deposited by a proximal {M^{PBH} = 10^{15}} g PBH is comparable to a magnitude M_w=4 earthquake. The non-seismic collateral damage due to the
actual impact of such small PBHs with Earth would be negligible. Unfortunately, the expected collision rate is very low even if PBHs constituted all of dark matter, at {~ 10^{-7} {yr}^{-1}}, and
since the rate scales as {1/M^{PBH}}, fortunately encounters with larger, Earth-threatening PBHs are exceedingly unlikely. However, the rate at which non-colliding close encounters of PBHs could
be detected by seismic activity alone is roughly two orders of magnitude larger --- that is once every hundred thousand years --- than the direct collision rate.
Read more (12557kb, PDF)
Horizon: Black Holes (BBC)
Black holes are one of the most destructive forces in the universe, capable of tearing a planet apart and swallowing an entire star. Yet scientists now believe they could hold the key to
answering the ultimate question - what was there before the Big Bang?
The trouble is that researching them is next to impossible. Black holes are by definition invisible and there's no scientific theory able to explain them. Despite these obvious obstacles,
Horizon meets the astronomers attempting to image a black hole for the very first time and the theoretical physicists getting ever closer to unlocking their mysteries. It's a story that takes us
into the heart of a black hole and to the very edge of what we think we know about the universe.
Naked black-hole hearts live in the fifth dimension
Luis Lehner of the Perimeter Institute in Ontario, Canada, has proposed a situation where naked singularities might exist: in the extra dimensions proposed by string theory.
Black holes would not just be points in the four dimensions we experience - three of space and one of time. They would become "black strings" which extend into a fifth dimension of space.
Read more
Ed ~ and i could add that possibly all the dimensions inside a BH are spatial.
Title: The fastest way to circle a black hole
Authors: Shahar Hod
Black-hole spacetimes with a "photonsphere", a hypersurface on which massless particles can orbit the black hole on circular null geodesics, are studied. We prove that among all possible
trajectories (both geodesic and non-geodesic) which circle the central black hole, the null circular geodesic is characterised by the shortest possible orbital period as measured by asymptotic
observers. Thus, null circular geodesics provide the fastest way to circle black holes. In addition, we conjecture the existence of a universal lower bound for orbital periods around compact
objects (as measured by flat-space asymptotic observers): T_{\infty}\geq 4\pi M, where M is the mass of the central object. This bound is saturated by the null circular geodesic of the maximally
rotating Kerr black hole.
Read more (8kb, PDF)
Title: Conformal Symmetry for Black Holes in Four Dimensions
Authors: Mirjam Cvetic, Finn Larsen
We show that the asymptotic boundary conditions of general asymptotically flat black holes in four dimensions can be modified such that a conformal symmetry emerges. The black holes with the
asymptotic geometry removed in this manner satisfy the equations of motion of minimal supergravity. We develop evidence that a two dimensional CFT dual of general black holes in four dimensions
account for their black hole entropy.
Read more (24kb, PDF)
Integral spots matter a millisecond from doom
“ESA's Integral gamma-ray observatory has spotted extremely hot matter just a millisecond before it plunges into the oblivion of a black hole. But is it really doomed? These unique observations
suggest that some of the matter may be making a great escape.
No one would want to be so close to a black hole. Just a few hundred kilometres away from its deadly surface, space is a maelstrom of particles and radiation. Vast storms of particles are
falling to their doom at close to the speed of light, raising the temperature to millions of degrees.
Ordinarily, it takes just a millisecond for the particles to cross this final distance but hope may be at hand for a small fraction of them.
Thanks to the new Integral observations, astronomers now know that this chaotic region is threaded by magnetic fields.”
Read more
5-dimensional black strings Permalink
Black Strings, Low Viscosity Fluids, and Violation of Cosmic Censorship
Luis Lehner, Frans Pretorius
(Version v3)
We describe the behaviour of 5-dimensional black strings, subject to the Gregory-Laflamme instability. Beyond the linear level, the evolving strings exhibit a rich dynamics, where at
intermediate stages the horizon can be described as a sequence of 3-dimensional spherical black holes joined by black string segments. These segments are themselves subject to a Gregory-Laflamme
instability, resulting in a self-similar cascade, where ever-smaller satellite black holes form connected by ever-thinner string segments. This behaviour is akin to satellite formation in
low-viscosity fluid streams subject to the Rayleigh-Plateau instability. The simulation results imply that the string segments will reach zero radius in finite asymptotic time, whence the
classical space-time terminates in a naked singularity. Since no fine-tuning is required to excite the instability, this constitutes a generic violation of cosmic censorship.
Read more
(89kb, PDF)
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MetaThematics 08: The Formulation of/for Involvement
For the formulation of communication we can learn from our body, and mind.
In our body we experience this (unconsciously) as cell-communication.
Between people we experience this (in many forms) as communication.
Communication is the interaction based on Synergy: feed forward and feedback.
Communion is based on Symbiosis: Networked feed forward and feedback loops.
We see how these work within our body, based on the interaction of our mineral, vegetable, animal and consciousness system.
The mineral system can therein be compared to the hardware (matter), consciousness to the (information) software.
In the interaction we see the material structure and dynamics (free versus bound electrons, and ditto protons.)
We also see the temporal organisation dynamics, for which we need to regard the dynamics in the spatial and temporal perspective; both are linked.
The figure is a standard fractal base; a bifurcation. In our body this represents the referon system for relating the cell core to the cell membrane. The single vector from the nucleus core connects
the radial to the peripheral; as described in e.g. the mathematics of Adams and Mowitz. Potential mathematics defines how this fractal can be mirrored and inverted, to transcend the boundary of the
defined (cell) system.
This connects the fractal to an inverse fractal, as is seen also in the work of Bill Tiller in the transcendence of the barrier of the Speed of light. (Energy invested in the transition, returns
after the transition.) Requirement is that the single vector must fully convert into the dual double vector of the plane (membrane) surface. When this condition is fulfilled then the dynamics of the
local system can connect to those of the environment. (Integration Condition.)
The Integration Condition can thus be summarised in the shape of the plane that links the first (primary) and secondary (second) vector. This is portrayed in the classical image of the Light Cone.
This can be generalised in a Vortical (Vortex) description (definition), in which the convergence of the first vector (A) into the plane, and the emergence of the second vector (B) out of the plane
are dependent on the perfect mirroring (transposition integration) within the plane.
The so-called ‘light cone’ therewith is the requirement for system transcendence (boundary transcendence) of the vector A through the interface (<>) into the vector B.
The light cone this has no meaning without the core (kernel) vector A, and its inverse/dual, vector B. It means that we can not regard vector A separate from the ‘light cone’ nor separate from vector
B. The dual nature of the interface (= inner-phase) can be represented by the symbol “۞”.
This representation has NO meaning; it is DISconnected from context.
This means that the Radial notation (within a field) and the Circular notation (of a boundary) must always be combined. Without the connection between the Boundary and the Field, the transition
between them, and the relationship of the part in/to the whole cannot be described. It means that the Notation can be replaced by making use of these traditional forms of notation, in which the
interface is represented by a circle (or cycle).
The circle can indeed be interpreted also as a cycle. Any interface is based on the balance between feed forward and feedback. Where this is traditionally seen/shown as => and <=, the above makes
clear that we need to include the transition IN the interface into the description. The un-folding of fractal vector “A” into the interface (immersion) and the con-volution of fractal vector “B” from
the inter-phase (emersion) must both be described, as both are related (dual). Only when both are present, and balanced, can the interface be transcended and the part relate to the whole.
From a mathematical perspective, this means that we need to look into what happens in the interface of description; this is IN the “=”-sign; which is the interface in which not only the previous and
the next representation of description are connected, but also where the observer is involved in the observation. The feed forward and feedback cycle which defines the balance in the description, is
therein also integrated, unified and identified in the participation of the observer. This means that there is no objective observer; this does not and cannot exist.
A = B
must be replaced by
A # B
# : Dimensional Operator (= & // & . & *)
Later we will see that the interaction between the observed and the observer, with/in their context, involves 4 axes of rotation between them:
1. vertical axis – within the observer,
2. Frontal axis of observation – from the observer to the observed,
3. lateral axis – of the observed within the context of the observer, and
4. the 4D inversion axis of the integration of the observation in the context of the observer, in/to the internalised representation of that observation as a realisation within the observer.
(Both exist in 2 versions (A & B), for the state and the process; the observer and the observed.)
The Fractal Vector is possible only because the field is related to its boundary. The expansion of the vector A into the Fractal A takes place because the radial vector becomes defined (definite, by
becoming finite) in being related to its circumscribed Circular dual definition. The Radial and Circular formulation are thus/thereby related; they are dual: the one defines the other. This means
that classical reduced (Radial) description is meaningless without its relativistic (Circular) defining context. This has very explicit practical implications: radial relational logic (A => B) has NO
meaning outside of its cyclical referential logic (A <=> B).
Mathematics has come to the realisation that this double description is needed. Complex Number theory offers its notation. The so-called ‘Real’ component, is the notation for the radial aspect. The
so-called ‘Imaginary’ component, described the cyclic aspect, which defines the system. The ‘Imaginary’ aspect is equivalent to the time cycle, and the carrier wave’ of the system. The ‘Real’ aspect
is the ensuing signal, the shape, thus the form, in any specific context. (Gabor mathematics makes explicit that these two component need to be regarded together, to be able to address the
holographic dynamic nature of the reality/realisation that we are part of.)
The Circle and Radius, cycle and rod, are known in many cultures. A classical representation of the same is in the symbol of the shepherd staff (animal husbandry, dynamic control) and the flail
(harvesting seed, static selection) in classical Egypt; also –together– represented as a letter “G”. In politics the same is conveyed by the sickle-and-hammer symbol, or, in masonry, by a compass
(circle) and square (rod).
This relational definition is with reference to, thus in relationship with, the human observer always. There is no objective human observer; observation is always done by subjective humans. The
description ‘on paper’ needs to be related to the operational interactive dynamics in/by/of the human, always. This interaction is dual to that what is seen on paper. Whereas we see, ‘on paper’, the
steps of the equations, within the ‘observer’ we experience the processes in/of their transitions.
We therefore need to juxtapose the (seemingly) discrete steps in the description with the ongoing continuous experiences within the human (‘observer’).
(Note that the word ‘observer’ applies to a person observing particular rules in communication. This is also where the ‘education’ of scientists is a conditioning in observing their rituals of
observation and communication.)
Although in Complex number ‘mathematics’ the Real (space-state) and Imaginary (time dynamics) are customarily regarded in coincidence (as if coinciding) it is necessary to realise that the two
‘layers’ (aspects) are co-incident only under condition of coherence with/in the context.
This interaction can be made explicit by labelling the feed forward and feedback in combination/conjunction, as is done in the write-re-write algebra (Rowlands).
In order to be able to account for our own involvement, that formulation needs to be ‘doubled’: the transition from observation-to-object (Realisation) needs to be complemented by the transition
by/of observation-by-subject (Realisation).
The steps in the equations and the stages in redefining the equations must this be related. The dynamics in the human observer must be explicitly described. This can be most easily done at their most
fundamental level: that of cell dynamics. The communion/communication between people can be seen to be defined on the dynamics between communing cells. The principles implied can be seen with/in cell
division. This then offers the basis for cellular communion, communication between humans, and the basis of/for consensus.
The basis for this interaction can be seen in Figure 1, applied to the communication within cells. The plane in the middle is at right angles to the vector from the cell core to the cell membrane;
and thereby this figure not only represents the dynamic of membrane transition (including of the the Tiller-Einstein solution) but also the dynamic of, for, during, after cell division, in which the
zygote has been split into two cells, which together form one unit(y).
Crucial is the understanding that we here see the relationship between Space and Time; specifically, how Time (process) forms the ‘carrier wave’ for the structure in Space (state). The consecutive
stages in cell division can be represented in/as the following figure, where the first state – the Zygote – is juxtaposed to the next state: the divided cell. It represents a fundamental principle,
which is repeated throughout the creation/manifestation of life, and our body. It brings us to the notion that we need to describe cell division as a (Time) Fractal (Vrobel).
The description shows the relationship between Unit and Unity. The Zygote, when it divides into two cells, retains unity, while ‘remaining’ units. (Dotted represents the next phase stage.) The radial
notation (left) is equivalent with the circular notation (right); but both are meaningful only when used together: the local system needs to be seen to be bounded, because in the boundary you can
find the relationship with/in/to the environment; which is shared by both.
This notation thereby implies its complement, the inverse. This is where, from a system boundary (energy) it is possible to infer it implied (information) centre of coherence from which the integrity
of the system can be deduced. These complement the space/state and process/phase (time) – consequential – levels of manifestation.
Information => energy => time => space, is thus defined in the co-incidence of the above two symbols.
We see this duality in the following figure, which defines the duality in the first stage in cell division: it specifies the relationship between unit and unity. It combines (‘coincides’) the
relationship between the first cell (the zygote) and the two ‘second’-stage cells. On the one hand this can be interpreted as the sequence of the first cell (first circle, on the left) and the second
cell (second circle on the right). The circle on the right is a virtual/imaginary circle/cycle. It contains – dotted – the (next stage) two cells which were formed out of the cell division.
Juxtaposed is, again, the fractal vector notation (in which it now is evident that this notation describes the division into units, but fails to describe the maintenance of unity.)
In order to make the preservation of unity explicit, we need to make use of the dual (co-incident) notation described above, in which the fractal vector is identified with the phase transition in/to
a field, linking it in-to the Boundary; but linked also to the integrity of the interface (and its integrity in inner-phasing) out of which the dual of the phase change can emerge; if the
transformation in/by/through/out of the boundary is (self) consistent (its own dual; i.e. invert-inverse). This boundary-transition is the essence of cell-division. The duality of the Boundary
Transition, and Cell Division, can be juxtaposed:
In the figure on the right (the representation of cell division) the dynamic between unit and unity is shown. Note the directions of the arrows, which indicate the orientation of the flow of the
topology of the singularities of the system. In the more complete denotation of the principle of cell division, the relationship between the first cell and the two emergent cells is made explicit.
The cycle within the system can then be – explicitly – seen to be related to the System Inversion; and our capacity to sensing the universe around us, within us.
This diagram is fundamental. It simultaneously describes the principle of boundary transition, cell division, and the functional and operational organisation of our whole body. The image for cell
division, denotes the Fractal vector; which denotes the kernel, determinant, matrix and array of a classic vectorial system; of Vortex dynamics. The image below shows the same relationship in the way
we customarily see it and regard is: in the form of our body:
Implied in this representation is a principle property of the fractal; and the diffraction of the sequence of unity (1) in duality (2, two-ness) as trinity (3, tri-unity) into its inverse: the 4D
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Casio fx-9750gii/fx-9860gii/fx-CG 50: Function Memory
03-23-2020, 01:02 AM
Post: #1
Eddie W. Shore Posts: 1,614
Senior Member Joined: Dec 2013
Casio fx-9750gii/fx-9860gii/fx-CG 50: Function Memory
This blog entry covers all the series:
* Casio fx-7400gii (I don't believe earlier versions of the 7400 have this)
* Casio fx-9750g, fx-9750gii
* Casio fx-9860g, fx-9860gii, fx-9860 Slim
* Casio Prizm fx-CG10/20
* Casio fx-CG 50
Screenshots are from the fx-CG 50.
Note: Casio calculators with math print (9860, CG 10, CG 20, CG 50), the menu FMEM/FUNCMEM will only appear when the calculator is set to Line Mode. In any case, the commands will be always available
throughout the Catalog and in Program editing mode.
Menu names: FMEM or FUNCMEM
The calculator has 20 slots for function memory. They can be for any variable, any amount of variables.
03-24-2020, 07:12 AM
Post: #2
Csaba Tizedes Posts: 608
Senior Member Joined: May 2014
RE: Casio fx-9750gii/fx-9860gii/fx-CG 50: Function Memory
And you can store little programs also, like this x^x=10^100 fixpoint iteration solver:
User(s) browsing this thread: 1 Guest(s)
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Specific Capacity - Engineering Hydrology Questions and Answers - Sanfoundry
Engineering Hydrology Questions and Answers – Groundwater – Specific Capacity
This set of Engineering Hydrology Multiple Choice Questions & Answers (MCQs) focuses on “Groundwater – Specific Capacity”.
1. What is the specific capacity of a well?
a) Volume of water corresponding to unit head drop
b) Discharge corresponding to safe yield
c) Discharge per unit area of well
d) Discharge per unit drawdown in a well
View Answer
Answer: d
Explanation: Specific capacity is an important property of well and it denotes the discharge that can be pumped from the well per unit drawdown inside the well. It is a measure of performance of the
2. Which of the following is a suitable unit for specific capacity?
a) m/s
b) hr^-1
c) m^2/hr
d) litre/s
View Answer
Answer: c
Explanation: \(Specific capacity = \frac{Discharge}{Drawdown}=\frac{volume/time}{length}=\frac{Area}{time}=[L^2 T^{-1}]\)
Therefore, the unit of specific capacity is m^22/hr.
3. It is given that for a well, the specific capacity is directly proportional to the transmissibility under equilibrium conditions. For which of the following cases is this true?
a) Confined aquifer with well losses
b) Confined aquifer without well losses
c) Unconfined aquifer with well losses
d) Unconfined aquifer without well losses
View Answer
Answer: b
Explanation: For a confined aquifer neglecting the well losses, from Dupit’s equation,
\(Q=\frac{2πST}{ln \frac{R}{r}}⇒\frac{Q}{S}=\frac{2πT}{ln \frac{R}{r} ⇒\frac{Q}{S}∝T\)
Therefore, specific capacity is directly proportional to transmissibility.
4. For a well discharges at a constant rate under steady state conditions, the specific capacity of the well is a constant.
a) True
b) False
View Answer
Answer: a
Explanation: The steady state discharge for a well is governed by Dupit’s equation. For a confined aquifer, taking the aquifer properties and conditions to not change over the time of pumping, which
is generally the case, the specific capacity of the well does not change.
5. For a well in a confined aquifer, accounting for well loss with steady state conditions, what will be nature of specific capacity of the well?
a) It remains constant
b) It changes with time
c) It increases with increase in discharge
d) It decreases with increase in discharge
View Answer
Answer: d
Explanation: The term account for the well loss is directly proportional to discharge and it appears in the equation of specific capacity such that it is inversely proportional to it. Therefore, as
the discharge increases, the specific capacity of the well decreases.
Sanfoundry Global Education & Learning Series – Engineering Hydrology.
To practice all areas of Engineering Hydrology, here is complete set of 1000+ Multiple Choice Questions and Answers.
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Re: Modal analysis of cantilever beam - by hand and by Robot
09-23-2015 05:15 AM
Hello all,
It might be trivial, but I can't figure out the differences obtained when calculating the problem by hand and by Robot.
The problem is very simple. For the given cantivever beam with uniform loading, one should calculate the modal properties.
The results calculated by hand and by Robot are equal with respect to the stiffness and the mass, but the period, frequency and pulsation are different.
Please check the screen-shots.
09-23-2015 05:15 AM
09-25-2015 01:29 AM
09-25-2015 01:29 AM
09-25-2015 03:00 AM
09-25-2015 03:00 AM
09-30-2015 09:57 AM
09-30-2015 09:57 AM
09-30-2015 03:53 PM
09-30-2015 03:53 PM
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Fundamental Theorem of Algebra
One of the first “theorems” I heard about was The Fundamental Theorem of Algebra, and I remember being kind of drawn to it for a long time after first seeing it. I think this was less because of the
statement of the theorem itself, and more because the word fundamental in its title made it seem really important and imposing ^1. Either way, I was convinced for a long time that it was somehow a
mysterious theorem, that although easy to state, must have one of those impossible to understand, complicated proofs; the kind of thing that’s proved once via a lot of effort, and then is just
applied afterwards without many people wanting to return to the proof because it’s just that out there. Despite this, my fascination with it made me determined to see and understand its proof once I
became really good at/knowledgable of math. Luckily for me, I was wrong. The proof of the theorem is not arcane. In fact, there are many proofs of it, some of which even I can understand.
Before getting into a proof, let’s quickly state the theorem and then move on
The Fundamental Theorem of Algebra
If \(p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\) is any polynomial with complex coefficients, then \(p(x)\) has a zero in \(\mathbb C\).
Intro to \(\mathbb C\)
If the idea of number \(i\) such that \(i^2=-1\) doesn’t frighten or anger you, then skip this section. If not, I’m going to somewhat quickly try to convince you that this is ok.
One way to think of complex numbers is to view them as a way of doing geometry via arithmetic. Let’s say, for example, you are making a 2D game, and in this game you probably want to keep track of
positions of different objects, so you represent positions as points in the plane. Each object has some position \((x,y)\in\mathbb R^2\). You probably want objects to move, so along with a position,
every object needs some velocity \((dx,dy)\in\mathbb R^2\). Now, you can move objects by adding their velocity to their position, so after one timestep their position becomes \((x+dx,y+dy)\). Simple
enough. Objects in your game also rotate around each other^2. Originally, you might handle this by having some angle \(\theta\) of rotation for an object, and then updating its poition via some
complicated formula involving \(\sin\) and \(\cos\). This is kinda messy, but then you remember how well representing things as points worked for moving things around before, and so you store
rotations as a point \((\cos\theta,\sin\theta)\) on the unit circle. You then need some operation \(\cdot\) such that \((x,y)\cdot(\cos\theta,\sin\theta)\) gives the rotation of \((x,y)\) (about the
origin. To rotate about a different point, you just translate, rotate, then translate back). Once you do this, you’ll likely want to extend \(\cdot\) such that \((x,y)\cdot(a,b)\) makes sense for all
points in the plane, and not just onces where \((a,b)\) is on the unit circle. Motivated by the fact that \(5*(x,y)\) scales \((x,y)\) by a factor of 5, and \(0.5*(x,y)\) scales it by a factor of \(1
/2\), you say that \((x,y)\cdot(a,b)\) rotates \((x,y)\) by the angle \((a,b)\) makes with the \(x\)-axis, and then scales us by the distance of \((a,b)\) from the origin.
This turns out to be pretty useful because it lets you combine two transformations into one, and this \(\cdot\) operation plays really nicely with adding points. In fact, if you do the math to work
things out, you will see that \((x,y)\cdot(a,b)=(xa-yb,xb+ay)\) which means that \((x,0)\cdot(y,0)=(xy,0)\) so the \(x\)-axis is really just the real number line, and \((0,1)\cdot(0,1)=(-1,0)\) so
you have a number whose square is \(-1\)! By trying to create an arithmetic that allows us to do geometric transformations, we naturally find ourselves actually manipulating complex numbers where \
(a+bi\leftrightarrow(a,b)\). I probably should mention that complex number actually usually aren’t used for rotations and such in 2D games, but an extension of them called quaternions are used for
rotations in 3D games.
If that’s not convincing, then another perspective on complex numbers is that you are really just doing clock arithmetic when you work with them. When doing math with time, you wrap around every 12
(or 24) hours, so you are really just treating 12 as if it were 0, and then doing normal math (Ex. \(4+10=14=12+2=2\) so \(10\) hours past \(4\) is \(2\)). With complex numbers, you are doing
something similar. You are doing normal math with polynomials (with real coefficients), except you treat the polynomial \(x^2+1\) as being zero. So, for example, when you say \((3+4i)(5-2i)=23+14i\),
this is really because
\[\begin{align*} (3+4x)(5-2x) = 15 + 14x - 8x^2 = 15 + 14x - 8(x^2+1) + 8 = 23 + 14x \end{align*}\]
Symbolically, in case you’ve studied some abstract algebra but not seen this,
\[\begin{align*} \mathbb C\simeq\frac{\mathbb R[x]}{(x^2+1)} \end{align*}\]
Definitions and Junk
Now that we have that out of the way, before moving on to the proof itself, we need to setup some notation, definitions, and lemmas, so let’s get to that. In the below definitions, \(X\) is an
arbitrary subset of \(\mathbb C\).
A path is a continuous function \(f:[0,1]\rightarrow X\). Furthermore, if we have that \(f(0)=f(1)\), then we call \(f\) a loop based at \(f(0)\).
An important thing to know about paths is that you can compose them. If you have two paths \(f,g:[0,1]\rightarrow X\) where \(f(1)=g(0)\), then you can form a new path \(g\cdot f:[0,1]\rightarrow X\)
where you first do \(f\), then do \(g\). In order to keep the domain \([0,1]\), you have to traverse \(f\) and \(g\) at twice the normal speed, but that’s really just a technicality^3.
Note that for some reason I think in terms of paths more easily than I do in terms of loops, so although we’ll be dealing exclusively with loops here, I will often forget and say path instead.
Let \(S^1=\{z\in\mathbb C:|z|=1\}\) be the unit circle in the complex plane
Quick remark: Notice that there is a 1-1 correspondence between loops and continuous circle functions \(f:S^1\rightarrow X\) since a circle is really just a line segment with its endpoints glued
together. I may end up switching between these two perspectives during this post.
The proof of the fundamental theorem we’ll present is pretty dependent on loops. The basic idea is that if you have a polynomial without a zero, then you can find a “constant loop that circles the
origin multiple times”. I use quotes because this is not exactly what we’ll show, but basically it. In either case, if a loop is constant it doesn’t move so there’s no way it could circle the origin
even once, and so contradiction. We need a mathematically precise way of defining what it means to “circle the origin multiple times”, and for that, we’ll use a little homotopy theory.
Given two paths \(f:[0,1]\rightarrow X\) and \(g:[0,1]\rightarrow X\) with the same basepoints (i.e. \(f(0)=g(0)\) and \(f(1)=g(1)\)), a homotopy \(H:[0,1]\times[0,1]\rightarrow X\) from \(f\) to
\(g\) is a continuous^4 function such that \(H(t,0)=f(t)\) and \(H(t,1)=g(t)\) for all \(t\in[0,1]\), and \(H(0,s)=f(0)\) and \(H(1,s)=f(1)\) for all \(s\in[0,1]\). If there exists a homotopy \(H
\) from \(f\) to \(g\), then we say \(f\) and \(g\) are homotopy equivalent, and denote this \(f\sim g\).
You can think of a homotopy as a continuous deformation from one path into the other. Something like this
One important example of a homotopy is the one depicted above. This is the so-called staright line homotopy, and is the result of thinking of your paths as points and then drawing a line between
them. For \(f,g:[0,1]\rightarrow X\) paths between the same points, you can define \(H(t,s)=(1-s)f(t) + sg(t)\). This is almost always continuous.
When does the straight line homotopy fail to be a homotopy?
Show that homotopy equivalence is an equivalence relation.
In this upcoming section, we’ll apply homotopy to loops to see that every loop around a circle has a well-defined number of times it goes around. This will then lead us to the proof of the theorem.
Circles and Degrees
Here, we will study loops \(f:[0,1]\rightarrow S^1\) around the unit circle. In general, these things can behave annoyingly by stopping in place, backtracking, etc. so to get a handle on them, we’ll
homotope all our paths into nice loops. To that end, let \(\omega_n:[0,1]\rightarrow S^1\) be the path \(\omega_n(t)=e^{2t\pi in}\) that goes around the unit circle \(n\) times where we made use of
Euler’s formula.
Our goal is to show that any loop \(f:[0,1]\rightarrow S^1\) is homotopic to exactly one “nice” loop \(\omega_n\). We will then let the degree of \(f\) be \(\deg f=n\), and this will be our
characterization of the number of times \(f\) travels around the unit circle ^5. In order to do this, we’ll make use of a special function^6
\[\begin{matrix} p: &\mathbb R &\longrightarrow &S^1\\ &r &\longmapsto &\cos(2\pi r)+i\sin(2\pi r) \end{matrix}\]
What makes this function special is that is allows us to “lift” loops in \(S^1\) up to paths in \(\mathbb R\). This function is far from injective, but it maps every unit interval in \(\mathbb R\)
around the circle in a “nice” way. If we look at any (connected) neighborhood around a point on our circle, there are many disjoint copies of that neighborhood in \(\mathbb R\) that get mapped into
it by \(p\). This means that \(p\) in some sense has multiple local inverses of any neighborhood in \(S^1\). These local inverses are what allow us to lift loops up to \(\mathbb R\). Specifically,^7
For any path \(f:[0,1]\rightarrow S^1\), there exists a unique lift \(\tilde f:[0,1]\rightarrow\mathbb R\) such that \(p\circ\tilde f=f\) and \(\tilde f(0)=0\).
Pf: Let \(f:[0,1]\rightarrow S^1\) be a path. The remark I made above on local inverses can be said more formally as this: for any point \(x\in S^1\), there exists a neighborhood \(N\) of \(x\),
called an elementary neighborhood, such that each path component of \(p^{-1}(N)\) is mapped homeomorphically onto \(N\). Let \(\{U_i\}_{i\in I}\) be a collection of elementary neighborhoods that
cover \(S^1\), so \(\{f^{-1}(U_i)\}_{i\in I}\) is an open cover of the compact metric space \([0,1]\), which means it has some finite subcover \(\{V_j\}_{j=1}^n\subseteq\{f^{-1}(U_i)_{i\in I}\}\).
Furthermore, it is a fact that I will not prove here that you can find a natural \(m\in\mathbb N\) such that each of the images \(f([0,1/m]),f([1/m,2/m]),\dots,f([(m-1)/m,1])\) is completely
contained in some elementary neighborhood \(W_j\). To simplify notation, let \(x_j=j/m\) and \(I_j=[x_j,x_{j+1}]\). Now, we can lift \(f\) by lifting it piece by piece. For each \(j\in\{0,1\dots,m-1
\}\), we can form a unique path \(g_j:I_j\rightarrow\mathbb R\) such that \(p\circ g_j=f\mid_{I_j}\) since \(f(I_j)\) is contained in an elementary neighborhood \(W_j\) which has exactly one "local
inverse" \(V_j\) containing \(g_{j-1}(x_j)\) and so contains a unique path beginning at \(g_{j-1}(x_j)\) that lifts \(f\mid_{I_j}\) contained in \(U_j\). Thus, our unique lift of \(f\) is \(\tilde f=
g_{m-1}g_{m-2}\dots g_2g_1g_0\). \(\square\)
That may not have been put perfectly clearly because it’s a proof that is best digested with accompanying visuals, but I am not going through the trouble of making some. One thing I did not make
explicit is that we take \(g_{-1}(x_0)=0\) in order to comply with \(\tilde f(0)=0\). Another thing to keep in mind is that we form our lift by breaking the path up into small pieces, lifting those,
then joining them together. If we get a piece \(I_j\) of our path small enough to be contained in an elementary neighborhood, then the fact that it has one local inverse containing the point our path
left off at means there is a unique way to extend the path. This follows from the fact that each local inverse (i.e. path component) is mapped homeomorphically onto \(W_j\), so there’s a unique lift
for everything.
For the purpose of the secion, let \(1+0i\) be a distingueshed point in the sense that all loops around the circle begin and end there.
Let \(f:[0,1]\rightarrow S^1\) be a loop based at \(1\), and let \(\tilde f\) be its unique lift. Then, \(\tilde f(1)\) is an integer. We call this integer the degree of \(f\).
Pf: \(p(\tilde f(1))=f(1)=1\) so \(\tilde f(1)\in p^{-1}(1)=\mathbb Z\). \(\square\)
If you notice, I just redefined degree, so we better hope these definitions are equivalent. Clearly, \(\deg\omega_n=n\) since \(\tilde\omega_n\) is just a straight path from \(0\) to \(n\), so we
will show these definitions are equivalent via the following lemmas
The degree of a path is homotopy-invariant. That is, if \(f\sim g\), then \(\deg f=\deg g\).
Before we get to the proof, let’s look at a picture of what’s going on here.
We have a path \(f\) going around the circle (here \(f=\omega_2\)), and by using local inverses of \(p\), we lift this to a path in \(\mathbb R\) from \(0\) to \(2\). This captures the fact that this
circle loop makes two full revolutions around the circle. The idea behind the proof is similar to the proof of paths having unique lefts. You essentially show that you can also lift homotopies, so if
\(f\sim g\), then \(\tilde f\sim\tilde g\) which means they have the same endpoints.
Pf: Exercise for the reader.
The converse holds: If \(\deg f=\deg g\), then \(f\sim g\).
Pf: Let \(f,g:[0,1]\rightarrow S^1\) be loops such that \(\deg f=\deg g\). Let \(\tilde f,\tilde g:[0,1]\rightarrow\mathbb R\) be their respective lifts and note that \(\tilde f(1)=\tilde g(1)\). Let
\(\tilde H:[0,1]\times[0,1]\rightarrow\mathbb R\) be the straight line homotopy \(\tilde H(t,s)=(1-s)\tilde f(t)+s\tilde g(t)\), and define \(H:[0,1]\times[0,1]\rightarrow S^1\) by \(H(t,s)=p\circ\
tilde H(t,s)\). Then, \(H\) is continuous since it is a composition of continuous functions. Furthermore, \(H(t,0)=p\circ\tilde f(t)=f(t)\), \(H(t,1)=p\circ\tilde g(t)=g(t)\), \(H(0,s)=p(0)=1=f(0)=g
(0)\) and \(H(1,s)=p(\tilde f(1))=1=f(1)=g(1)\) for all \(t,s\in[0,1]\). Thus, \(H\) is a homotopy so \(f\sim g\). \(\square\)
We’ve just shown that any loop around the circle is completely characterized (up to homotopy which is really all that matters) by a single integer, the number of times it goes around.
Furthermore, it is easily shown that this integer is additive in the sense that \(\deg(fg)=\deg f+\deg g\) (It’s enough to show this for the case that \(f=\omega_n\) and \(g=\omega_m\) which is
obvious), so the structure of loops around the circle is the additive structure of the integers! This is pretty amazing, and can be used to prove some interesting stuff^8
Proof at Last
At this point, we’ve developed everything we need. Before we get to the proof, let’s “strengthen” our assumptions a little bit. Let \(f_0(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\) be any polynomial.
Note that we can divide through by \(a_n\) without changing the zeros of this polynomial, so we only need to investigate monic polynomials like \(f_1(x)=x^n+b_{n-1}x^{n-1}+\dots+b_1x+b_0\) where \
(b_i=a_i/a_n\). Furthermore, we can replace \(x\) with any invertible transformation, and although we change the zeros, we’re still able to recover all the ones we started with. Hence, we can pick \
(N\in\mathbb R\) small enough that \(\mid Nb_{n-1}\mid+\dots+\mid N^{n-1}b_1\mid+\mid N^nb_0\mid<1\) and then consider polynomials like \(f_2(x)=N^nf_1(x/N)=x^n+c_{n-1}x^{n-1}+\dots+c_1x+c_0\) where
\(c_i=N^{n-i}b_i\). This limits the type of polynomials enough that we can state the theorem as
Fundamental Theorem of Algebra
Let \(f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\) be any polynomial with complex coefficients (whose degree \(n>0\)) such that \(\mid a_{n-1}\mid+\dots+\mid a_1\mid+\mid a_0\mid<1\). Then, there
exists some \(x_0\in\mathbb C\) with \(f(x_0)=0\).
Pf: Suppose that \(f(x)\) has no zero in \(\mathbb C\), so we can regard \(f\) as a function from \(\mathbb C\) to \(\mathbb C-\{0\}\). Now, define a function \(g:S^1\rightarrow S^1\) by \(g(x)=\frac
{f(x)}{\mid f(x)\mid}\), and note that we can equivalently view \(g\) as a loop in \(S^1\), so \(g\) has a well-defined degree. Let \(D=\{z\in\mathbb C:|z|\le1\}\) be the unit disc, and note that,
representing complex numbers in polar form, we can similarly define $$\begin{matrix} G: &D &\longrightarrow &S^1\\ &re^{2\pi i\theta} &\longmapsto &\frac{f(re^{2\pi i\theta})}{\mid f(re^{2\pi i\
theta})\mid} & 0\le r \le 1 &0\le\theta\le 1 \end{matrix}$$ so we can think of \(G\) as a function from \([0,1]\times[0,1]\rightarrow S^1\) (the first argument is \(r\) and the second \(\theta\)).
Thus, defining \(H:[0,1]\times[0,1]\rightarrow S^1\) by \(H(t,s)=G(s,t)\) makes \(H\) a homotopy! Clearly, \(H(t,1)=G(1,t)=g(t)\) (where we view \(g\) as a loop instead of as a circle function) and \
(H(t,0)=G(0,t)=f(0)/\mid f(0)\mid\) for all \(t\in[0,1]\), so \(g\) is homotopic to a constant function and \(\deg g=0\). However, we can also define the following $$\begin{matrix} H': &[0,1]\times
[0,1] &\longrightarrow &S^1\\ &(t,s) &\longmapsto &\frac{z^n + s(a_{n-1}z^{n-1}+\dots+a_1z+a_0)}{\mid z^n + s(a_{n-1}z^{n-1}+\dots+a_1z+a_0)\mid} & z=e^{2\pi it} \end{matrix}$$ This function is
continuous since its the composition of a bunch of continuous functions, and it is well-defined since the denominator is never 0 $$\begin{align*} \mid z^n + s(a_{n-1}z^{n-1}+\dots+a_1z+a_0)\mid &\ge
|z|^n - s|a_{n-1}z^{n-1}+\dots+a_1z+a_0|\\ &\ge |z|^n - s(|a_{n-1}||z^{n-1}|+\dots+|a_1||z|+|a_0|)\\ &= 1 - s(|a_{n-1}|+\dots+|a_1|+|a_0|)\\ &> 1 - s\\ &\ge 0 \end{align*}$$ Now, we just need to note
that \(H'\) is a homotopy from \([t\mapsto z^n]=[t\mapsto e^{2\pi itn}]=\omega_n\) to \(g\), so \(\deg g=n\), but this is a contradiction, and hence our initial assumption about \(f\) having a zero
must be wrong. \(\square\)
Let \(f(x)\) be a degree \(n\) polynomial with coefficients in \(\mathbb C\). Then, \(f\) has exactly \(n\) (not necessarily distinct) zeros.
Pf: By the theorem, \(f\) has some zero \(z_0\in\mathbb C\), so let's divide \(f\) by \(z-z_0\). Using long division, we get some polynomials \(q(z),r(z)\) such that \(f(z)=q(z)(z-z_0)+r(z)\) and \(\
deg r(z)<\deg(z-z_0)=1\) or \(r(z)=0\) which means \(r(z)\) is a constant. Since \(0=f(z_0)=q(z_0)(z_0-z_0)+r(z_0)=r(z_0)\), we must have \(r(z)=0\) so \(f(z)=q(z)(z-z_0)\) and \(\deg q(z)=n-1\). Now
just apply induction to get that \(q(z)\) has \(n-1\) zeros, so \(f(z)\) has \(n\) zeros. \(\square\)
Finally, an exercise.
Where does the argument for the main theorem fail if \(f\) has a zero? Since, \(f\) has exactly \(\deg f\) zeros, you can always find a closed disc on which \(f\) has no zero, so why don’t we
always get a contradiction?
1. I’ve seen many other fundamental theorems besides this one, and I am very confused by how a theorem gets to be called fundamental ↩
2. Maybe it’s a space game, or maybe you have enemies that circles a base to protect it, or maybe etc. ↩
3. Once we introduce homotopy, we’ll have an equivalence relation on paths. This has the effect that the set of (equivalence classes of) loops based at a single point forms a group called the
fundamental group of X. Secretly, this post is really just exploring the fundamental group of the circle. Without homotopy, composition of paths isn’t associative because of the whole doubling
speed thing. ↩
4. Throughtout this post, I will avoid the issue of defining what a continuous function is, because doing so properly requires defining a topology on a set and that’s just too out of the way for
this post. You can think of continuity intuitively as meaning nearby inputs get mapped to nearby outputs ↩
5. Since w_n*w_m=w_{n+m}, this will also show that the fundamental group of the circle is Z, the set of integers ↩
6. Secretly, this is a covering function, and R is the universal covering space of S^1 ↩
7. This proof may actually require some, but not much, background in topology. ↩
8. the fundamental theorem of algebra of course, but also, for example, Brouwer’s Fixed Point Theorem. Brouwer’s theorem then can be used to show the existence of Nash Equilibria in normal form
games (think first form of Prisoner’s Dilema shown in my post on it), and from there you get like all of game theory. ↩
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The Data Hall
In the earlier parts (part 1, part 2, and part 3) of this series, we have seen, how certain variables can change the outcomes of our regression model. We thoroughly explored such phenomena from basic
(intercept dummies) to the two-way interactions. Likewise, three different variables (continuous or categorical) can also simultaneously impact our dependent or […]
Three-Way Interaction in R | Part 4 Read More »
A Comprehensive Guide to Handling Missing Values and Interpolation
In the realm of data analysis, encountering missing values is a common challenge that analysts face. Whether you’re working with cryptocurrency data, conducting rolling regressions, or exploring
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effectively. Understanding Missing
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Fama and French three-factor model | Detailed Explanation
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The Quantum Way of Doing Computations_学术讲座_清华大学量子信息中心
标题：The Quantum Way of Doing Computations
演讲人： Prof. Rainer Blatt University of Innsbruck and Austrian Academy of Sciences
时间： 2016-02-26 14:00-2016-02-26 15:00
Since the mid-nineties of the 20th century it became apparent that one of the centuries’ most important technological inventions, computers in general and many of their applications could possibly be
further enormously enhanced by using operations based on quantum physics. This is timely since the classical roadmaps for the development of computational devices, commonly known as Moore’s law, will
cease to be applicable within the next decade due to the ever smaller sizes of the electronic components that soon will enter the quantum physics realm. Computations, whether they happen in our heads
or with any computational device, always rely on real physical processes, which are data input, data representation in a memory, data manipulation using algorithms and finally, the data output.
Building a quantum computer then requires the implementation of quantum bits (qubits) as storage sites for quantum information, quantum registers and quantum gates for data handling and processing
and the development of quantum algorithms.
In this talk, the basic functional principle of a quantum computer will be reviewed. It will be shown how strings of trapped ions can be used to build a quantum information processor and how basic
computations can be performed using quantum techniques. In particular, the quantum way of doing computations will be illustrated by analog and digital quantum simulations and the basic scheme for
quantum error correction will be introduced and discussed. Scaling-up the ion-trap quantum computer can be achieved with interfaces for ion-photon entanglement based on high-finesse optical cavities
and cavity-QED protocols, which will be exemplified by recent experimental results.
Rainer Blatt graduated in physics from the University of Mainz in 1979. He finished his doctorate in 1981 and worked as research assistant in the team of Günter Werth. In 1982 Blatt received a
research grant of the Deutsche Forschungsgemeinschaft (DFG) to go to the Joint Institute for Laboratory Astrophysics (JILA), Boulder, and work with John L. Hall (Nobel Prize winner 2005) for a year.
In 1983 he went on to the Freie Universität Berlin, and in the following year joined the working group of Peter E. Toschek at the University of Hamburg. After another stay in the US, Rainer Blatt
applied to qualify as a professor by receiving the “venia docendi” in experimental physics in 1988. In the period from 1989 until 1994 he worked as a Heisenberg research fellow at the University of
Hamburg and returned several times to JILA in Boulder. In 1994 he was appointed professor of physics at the University of Göttingen and in the following year he was offered a chair in experimental
physics at the University of Innsbruck. Since 2003 Blatt has also held the position of Scientific Director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy
of Sciences (ÖAW).
Institute for Experimental Physics,
University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria Rainer.Blatt@uibk.ac.at, www.quantumoptics.at and
Institute for Quantum Optics and Quantum Information,
Austrian Academy of Sciences, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria Rainer.Blatt@oeaw.ac.at, www.iqoqi.at
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Riesz, Frigyes (Fréd | Encyclopedia.com
Riesz, Frigyes (Fréd
RIESZ, FRIGYES (FRéDéRIC)
(b. Györ, Hungary, 22 January 1880; d. Budapest, Hungary, 28 February 1956)
Riesz’s father, Ignacz, was a physician; and his younger brother Marcel was also a distinguished mathematician. He studied at the Polytechnic in Zurich and then at Budapest and Göttingen before
taking his doctorate at Budapest. After further study at Paris and Göttingen and teaching school in Hungary, he was appointed to the University of Kolozsvàr in 1911. In 1920 the university was moved
to Szeged, where, in collaboration with A. Haar, Riesz created the János Bolyai Mathematical Institute and its journal, Acta scientiartun mathematicarum. In 1946 he went to the University of
Budapest, where he died ten years later after a long illness.
Riesz’s output is most easily judged from the 1,600-page edition of his writings (cited in the text as Works). He concentrated on abstract and general theories connected with mathematical analysis,
especially functional analysis. One of the theorems for which he is best remembered is the Riesz-Fischer theorem (1907), so called because it was discovered at the same time by Emil Fischer. Riesz
formulated it as follows (Works, 378–381; cf. 389–395). Let {ф[i](x)} be a set of orthogonal functions over [a, b] of which each member is summable and square-summable. Associate with each ф[i], a
real number a[i]. Then is convergent if and only if there exists a function f such that
In this form the theorem implies that the {a[i]} are the coefficients of the expansion off f in terms of the (ф[i]) and that f itself is square-summable. This result, the converse of Parseval’s
theorem, immediately attracted great interest and soon was being re-proved.
Riesz had been motivated to discover his theorem by Hilbert’s work on integral equations. Under the influence of Maurice Fréchet’s abstract approach to function spaces, such studies became associated
with the new subject of functional analysis. Riesz made significant contributions to this field, concentrating on the space of L^p functions (functions f for which f[p]1, part; is Lebesgue
integrable). He provided much of the groundwork for Banach spaces (Works, esp. 441–489) and later applied functional analysis to ergodic theory.
Riesz’s best-known result in functional analysis has become known as the Riesz representation theorem. He formulated it in 1909, as follows (Works, 400–402). Let A be a linear (distributive,
continuous) functional, mapping real-valued continuous functions f over [0,1] onto the real numbers. Then A is bounded, and can be represented by the Stieltjes integral
where α is a function of bounded variation. The theorem was a landmark in the subject and has proved susceptible to extensive generalizations and applications.
Another implication of Hilbert’s work on integral equations that Riesz studied was its close connection with infinite matrices. In Les systèmes d’équations linéaires à une infinité d’inconnues (1913;
Works, 829–1016), Riesz tried not only to systematize the results then known into a general theory but also to apply them to bilinear and quadratic forms, trigonometric series, and certain kinds of
differential and integral equations.
Functional analysis and its ramifications were Riesz’s most consistent interests; and in 1952 he published his other book, a collaboration with his student B. Szökefnalvy-Nagy, LeÇons d’analyse
fonctionnelle. A classic survey of the subject, it appeared in later French editions and in German and English translations.
In much of his work Riesz relied on the Lebesgue integral, and during the 1920’s he reformulated the theory itself in a “constructive” manner independent of the theory of measure (Works, 200–214). He
required only the idea of a set of measure zero and built up the integral from “simple functions” (effectively step functions) to more general kinds. He also re-proved some of the basic theorems of
the Lebesgue theory.
In the topics so far discussed, Riesz was a significant contributor in fields that had already been developed. But a topic he created was subharmonic functions. A function f of two or more variables
is subharmonic if it is bounded above in an open domain D; is continuous almost everywhere in D; and, on the boundary of any subdomain D’of D, is not greater than any function F that is continuous
there and harmonic within. The definition is valuable for domains in which the Dirichlet problem is solvable and F is unique, for then f≤F within D and f = F on its boundary. By means of a criterion
for subharmonicity given by
where r is the radius and (x[0], y[0]) center of a small circle within D, Riesz was able to construct a systematized theory (see esp. Works, 685–739) incorporating applications to the theory of
functions and to potential theory.
Among Riesz’s other mathematical interests, some early work dealt with projective geometry. Soon afterward he took up matters in point set topology, such as the definition of continuity and the
classification of order-types. He also worked in complex variables and approximation theory.
I. Original Works. Riesz’s writings were collected in összegyüjtött munkái—Oeuvres complètes—Gesammelte Arbeiten, á. Csázár, ed., 2 vols. (Budapest, 1960), with illustrations and a complete
bibliography but little discussion of his work, Leçons d’analyse fonctionnelle (Budapest, 1952; 5th ed., 1968), written with B, Szökefnalvy-Nagy, was translated into English by L. F. Boron as
Functional Analysis (New York, 1955).
II. Secondary Literature. On Riesz’s work in functional analysis and on the Riesz-Fischer theorem, see M. Bernkopf, “The Development of Functional Spaces With Particular Reference to Their Origins in
Integral Equation Theory,” in Archive for History of Exact Science, 3 (1966–1967), 1–96, esp. 48–62. See also E. Fischer, “Sur la convergence en moyenne,” in comptes rendus … de l’Académie des
sciences, 144 (1907), 1022–1024; and J. Batt, “Die Verallgemeinerungen des Darstellungssatzes von F. Riesz und ihre Anwendungen,” in Jahresbericht der Deutschen Mathematiker-vereinigung, 74 (1973),
I. Grattan-Guinness
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What Does OPS Mean in Baseball Stats? - The Baseball Lifestyle
What Does OPS Mean in Baseball Stats?
Baseball stats can be overwhelming for those who are unfamiliar with the game. From batting average to slugging percentage, the metrics used to track the performance of a baseball player can often be
confusing. One of the more commonly used stats is OPS, which stands for On-base Plus Slugging. OPS is often used as a measure of a player’s offensive performance and is a combination of two other
statistics: on-base percentage (OBP) and slugging percentage (SLG). In this article, we will explore what OPS means in baseball stats, how it is calculated, and why it is important.
What Does OPS Stand For?
As mentioned earlier, OPS stands for On-base Plus Slugging. This statistic is used to measure a player’s offensive performance and is calculated by adding a player’s on-base percentage (OBP) and
slugging percentage (SLG). OBP is a measure of how often a player reaches base while SLG is a measure of the amount of extra-base hits a player has. By combining these two statistics, OPS gives an
overall measure of a player’s offensive performance.
How Is OPS Calculated?
The formula for calculating OPS is relatively simple. To calculate a player’s OPS, you first add their on-base percentage (OBP) and slugging percentage (SLG). This gives you the player’s OPS.
The formula looks like this:
OPS = OBP + SLG
For example, let’s say a player has an on-base percentage of .350 and a slugging percentage of .500. To calculate their OPS, you would add .350 and .500 to get .850. This means the player’s OPS is
Why Is OPS Important?
OPS is an important statistic for evaluating a player’s offensive performance. It gives a measure of how often a player reaches base (OBP) and how much extra-base power they have (SLG). This makes
OPS a better measure of a player’s overall offensive performance than either OBP or SLG alone.
OPS is also important because it is a quick, easy way to compare a player’s offensive performance to that of other players. It is more comprehensive than a single statistic, such as batting average,
and provides a better overall picture of a player’s offensive performance.
What Is a Good OPS?
A “good” OPS is relative to the player’s position and the league they are playing in. Generally speaking, the higher the OPS, the better the player’s offensive performance.
In Major League Baseball (MLB), the average OPS for all players is around .750. However, for certain positions, such as first base or designated hitter, the average OPS is higher. For example, the
average OPS for a first baseman in MLB is .822.
OPS is an important statistic for evaluating a player’s offensive performance. It is calculated by adding a player’s on-base percentage (OBP) and slugging percentage (SLG). OPS is important because
it gives a comprehensive measure of a player’s offensive performance. It is also a quick, easy way to compare a player’s performance to that of other players. The higher the OPS, the better a
player’s offensive performance is generally considered to be.
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ManPag.es -
spocon.f −
subroutine SPOCON (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
Function/Subroutine Documentation
subroutine SPOCON (characterUPLO, integerN, real, dimension( lda, * )A, integerLDA, realANORM, realRCOND, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)
SPOCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite matrix using the
Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF.
An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
The triangular factor U or L from the Cholesky factorization
A = U**T*U or A = L*L**T, as computed by SPOTRF.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ANORM is REAL
The 1-norm (or infinity-norm) of the symmetric matrix A.
RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.
WORK is REAL array, dimension (3*N)
IWORK is INTEGER array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 121 of file spocon.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Homework Exercises due 6 October
61. Consider:
(a) \(\Gamma,\phi\models\psi\)
(b) \(\Gamma\models\phi\supset \psi\)
(e) \(\phi\mathbin{{=}\!|{\models}}\psi\) (that is, \(\phi\models\psi\) and \(\psi\models\phi\))
(f) \(\models\phi\subset\supset \psi\)
Prove that (a) iff (b); and prove that (e) iff (f). You may find that an earlier homework already gives most of the solution to one of these; if so, you can of course cite your work for that
earlier problem without repeating.
62. I passed out a fallacious "proof" which is distributed in many math books (not in earnest). The proof says:
Theorem: For any integer \(n \ge 1\), all the numbers in a set of \(n\) numbers are equal to each other.
Inductive proof: It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the inductive step, let \(A=\{\,
a_1,a_2,\dots,a_k,a_{k+1} \,\}\) be any set of \(k+1\) numbers. Form two subsets each of size \(k\):
\(B=\{\, a_1,a_2,\dots,a_k \,\}\)
\(C=\{\, a_1,a_3,\dots,a_{k+1} \,\}\)
(\(B\) consists of all the numbers in \(A\) except \(a_{k+1}\), and \(C\) consists of all the numbers in \(A\) except \(a_2\).) By inductive hypothesis, all the numbers in \(B\) equal \(a_1\)
and all the numbers in \(C\) equal \(a_1\) (since both sets have only \(k\) members). But every number in \(A\) is in \(B\) or \(C\), so all the numbers in \(A\) equal \(a_1\); hence all are
equal to each other.
Exercise 62 is to identify and explain the mistake in this proof.
63. I passed out another fallacious "proof" copied from the Epp book. To prove that a composition of onto functons is onto, a student wrote:
Suppose \(f\colon X\to Y\) and \(g\colon Y\to Z\) are both onto. Then
\(\forall y\in Y, \exists x\in X\) such that \(f(x)=y\)
\(\forall z\in Z, \exists y\in Y\) such that \(g(y)=z\).
(Sorry, that was always supposed to be \(g(y)\). JP mistyped it earlier. This "proof" is still flawed. It continues...)
\((g\circ f)(x) = g(f(x)) = g(y) = z\)
and thus \(g\circ f\) is onto.
Exercise 63 is to explain the mistake(s) in this proof.
There is a deep similarity between what's gone wrong in the previous "proof" and in this one; I will ask for your ideas about this when we meet.
64. How do you establish that a set of premises doesn't logically entail some result? Show that:
1. \(Rab \not\models \exists x Rxx\)
2. \(Rab \not\models \neg\exists x Rxx\)
Show that:
3. \(\exists x\forall y Rxy \models \forall y\exists x Rxy\)
4. \(\forall y\exists x Rxy \not\models \exists x\forall y Rxy\)
65. Show it to be false that:
If \(\models \phi\vee\psi\) then \(\models\phi\) or \(\models\psi\).
66. If \(\phi\) is the formula \(Fx \supset \forall y(Gyx \vee P \vee \exists x Hx)\), then (a) What is \(\phi[x\leftarrow w]\)? (b) What is \(\phi[x\leftarrow y]\)? (c) What is \(\phi[P\leftarrow \
forall x Gxy]\)?
67. Symbolize in predicate logic using \(=\): (a) Andy loves Beverly, but she loves someone else. (b) Alice loves no one other than Beverly.
68. Is \((\phi\supset \psi)\supset (\neg\phi\supset \neg\psi)\) a tautology? Explain. (This question may be less straightforward than you first think.)
69. Explain the difference between \([\![\, \phi[x\leftarrow a] \,]\!]_{\mathscr{M}\,q}\) and \([\![\, \phi \,]\!]_{\mathscr{M}\,q[x:=a]}\).
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Generalization Regions in Hamming Negative Selection | Request PDF
Generalization Regions in Hamming Negative Selection
Negative selection is an immune-inspired algorithm which is typically applied to anomaly detection problems. We present an empirical investigation of the generalization capability of the Hamming
negative selection, when combined with the r-chunk affinity metric. Our investigations reveal that when using the r-chunk metric, the length r is a crucial parameter and is inextricably linked to the
input data being analyzed. Moreover, we propose that input data with different characteristics, i.e. different positional biases, can result in an incorrect generalization effect.
No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
... Hamming negative selection is an immune-inspired technique for one-class classification problems. Recent results, however, have revealed several problems concerning algorithm complexity of
generating detectors [5,6,7] and determining the proper matching threshold to allow for the generation of correct generalization regions [8] . In this paper we investigate an extended technique for
Hamming negative selection: permutation masks. ...
... In [18,8] results were presented which demonstrated the coherence between the matching threshold r and generalization regions when the r-chunk matching rule in Hamming negative selection is
applied. Recall, as holes are not detectable by any detector, holes must represent unseen self elements, or in other words holes must represent generalization regions. ...
... Finally, we explore empirically whether randomly determined permutation masks reduce the number of holes. Stibor et al. [8] have shown in prior experiments that the matching threshold r is a
crucial parameter and is inextricably linked to the input data being analyzed. However, permutation masks were not considered in [8]. ...
Permutation masks were proposed for reducing the number of holes in Hamming negative selection when applying the r-contiguous or r-chunk matching rule. Here, we show that (randomly determined)
permutation masks re-arrange the semantic representation of the underlying data and therefore shatter self-regions. As a consequence, detectors do not cover areas around self regions, instead they
cover randomly distributed elements across the space. In addition, we observe that the resulting holes occur in regions where actually no self regions should occur.
... In additional work, Stibor et al. [45,46] argued that holes in anomaly detection with binary negative selection algorithm are necessary to generalize beyond the training data set. Holes must
represent unseen self elements (or generation regions) to ensure that seen and unseen self elements are not recognized by any detector. ...
... Holes must represent unseen self elements (or generation regions) to ensure that seen and unseen self elements are not recognized by any detector. In [45], they explored the generation capability
of the Hamming negative selection when using the r-chunk length r. They found that an r-chunk length which does not properly capture the semantic representation of the input data will result in an
incorrect generalization and further concluded that a suitable r-chunk length does not exist for input data with element of different length. ...
The immune system is a remarkable information processing and self learning system that offers inspiration to build artificial immune system (AIS). The field of AIS has obtained a significant degree
of success as a branch of Computational Intelligence since it emerged in the 1990s. This paper surveys the major works in the AIS field, in particular, it explores up-to-date advances in applied AIS
during the last few years. This survey has revealed that recent research is centered on four major AIS algorithms: (1) negative selection algorithms; (2) artificial immune networks; (3) clonal
selection algorithms; (4) Danger Theory and dendritic cell algorithms. However, other aspects of the biological immune system are motivating computer scientists and engineers to develop new models
and problem solving methods. Though an extensive amount of AIS applications has been developed, the success of these applications is still limited by the lack of any exemplars that really stand out
as killer AIS applications.
... Dilger [8] investigated metric properties of some affinity functions (Hamming and r-contiguous) and showed that not all metric properties are satisfied. González et al. [11] and Stibor et al.
[15, 17] showed that the generalization capability of some affinity ...
Affinity functions are the core components in negative selection to discriminate self from non-self. It has been shown that affinity functions such as the r-contiguous distance and the Hamming
distance are limited applicable for discrimination problems such as anomaly detection. We propose to model self as a discrete probability distribution specified by finite mixtures of multivariate
Bernoulli distributions. As by-product one also obtains information of non-self and hence is able to discriminate with probabilities self from non-self. We underpin our proposal with a comparative
study between the two affinity functions and the probabilistic discrimination.
Glossary Definition of the Subject Introduction What Is an Artificial Immune System? Current Artificial Immune Systems Biology and Basic Algorithms Alternative Immunological Theories for AIS Emerging
Methodologies in AIS Future Directions Bibliography
The problem of generating r-contiguous detectors in negative selection can be transformed in the problem of finding assignment sets for a Boolean formula in k-CNF. Knowing this crucial fact enables
us to explore the computational complexity and the feasibility of finding detectors with respect to the number of self bit strings |S||\mathcal{S}| , the bit string length l and matching length r. It
turns out that finding detectors is hardest in the phase transition region, which is characterized by certain combinations of parameters |S|,l|\mathcal{S}|,l and r. This insight is derived by
investigating the r-contiguous matching probability in a random search approach and by using the equivalent k-CNF problem formulation.
Negative selection and the associated r-contiguous matching rule is a popular immune-inspired method for anomaly detection problems. In recent years, however, problems such as scalability and high
false positive rate have been empirically noticed. In this article, negative selection and the associated r-contiguous matching rule are investigated from a pattern classification perspective. This
includes insights in the generalization capability of negative selection and the computational complexity of finding r-contiguous detectors.
The use of artificial immune systems in intrusion detection is an appealing concept for two reasons. Firstly, the human immune system provides the human body with a high level of protection from
invading pathogens, in a robust, self-organised and distributed manner. Secondly, current techniques used in computer security are not able to cope with the dynamic and increasingly complex nature of
computer systems and their security. It is hoped that biologically inspired approaches in this area, including the use of immune-based systems will be able to meet this challenge. Here we collate the
algorithms used, the development of the systems and the outcome of their implementation. It provides an introduction and review of the key developments within this field, in addition to making
suggestions for future research. KeywordsArtificial immune systems-intrusion detection systems-literature review
Negative selection algorithm is one of the most widely used techniques in the field of artificial immune systems. It is primarily used to detect changes in data/behavior patterns by generating
detectors in the complementary space (from given normal samples). The negative selection algorithm generally uses binary matching rules to generate detectors. The purpose of the paper is to show that
the low-level representation of binary matching rules is unable to capture the structure of some problem spaces. The paper compares some of the binary matching rules reported in the literature and
study how they behave in a simple two-dimensional real-valued space. In particular, we study the detection accuracy and the areas covered by sets of detectors generated using the negative selection
In anomaly detection, the normal behavior of a process is characterized by a model, and deviations from the model are called anomalies. In behavior-based approaches to anomaly detection, the model of
normal behavior is constructed from an observed sample of normally occurring patterns. Models of normal behavior can represent either the set of allowed patterns (positive detection) or the set of
anomalous patterns (negative detection). A formal framework is given for analyzing the tradeoffs between positive and negative detection schemes in terms of the number of detectors needed to maximize
coverage. For realistically sized problems, the universe of possible patterns is too large to represent exactly (in either the positive or negative scheme). Partial matching rules generalize the set
of allowable (or unallowable) patterns, and the choice of matching rule affects the tradeoff between positive and negative detection. A new match rule is introduced, called r-chunks, and the
generalizations induced by different partial matching rules are characterized in terms of the crossover closure. Permutations of the representation can be used to achieve more precise discrimination
between normal and anomalous patterns. Quantitative results are given for the recognition ability of contiguous-bits matching together with permutations.
Artificial immune systems have become popular in recent years as a new approach for intrusion detection systems. Indeed, the (natural) immune system applies very effective mechanisms to protect the
body against foreign intruders. We present empirical and theoretical arguments, that the artificial immune system negative selection principle, which is primarily used for network intrusion detection
systems, has been copied to naively and is not appropriate and not applicable for network intrusion detection systems.
Since their development, AIS have been used for a number of machine learning tasks including that of classification. Within the literature, there appears to be a lack of appreciation for the possible
bias in the selection of various representations and affinity measures that may be introduced when employing AIS in classification tasks. Problems are then compounded when inductive bias of
algorithms are not taken into account when applying seemingly generic AIS algorithms to specific application domains. This paper is an attempt at highlighting some of these issues. Using the example
of classification, this paper explains the potential pitfalls in representation selection and the use of various affinity measures. Additionally, attention is given to the use of negative selection
in classification and it is argued that this may be not an appropriate algorithm for such a task. This paper then presents ideas on avoiding unnecessary mistakes in the choice and design of AIS
algorithms and ultimately delivered solutions.
Best known in our circles for his key role in the renaissance of low- density parity-check (LDPC) codes, David MacKay has written an am- bitious and original textbook. Almost every area within the
purview of these TRANSACTIONS can be found in this book: data compression al- gorithms, error-correcting codes, Shannon theory, statistical inference, constrained codes, classification, and neural
networks. The required mathematical level is rather minimal beyond a modicum of familiarity with probability. The author favors exposition by example, there are few formal proofs, and chapters come
in mostly self-contained morsels richly illustrated with all sorts of carefully executed graphics. With its breadth, accessibility, and handsome design, this book should prove to be quite popular.
Highly recommended as a primer for students with no background in coding theory, the set of chapters on error-correcting codes are an excellent brief introduction to the elements of modern
sparse-graph codes: LDPC, turbo, repeat-accumulate, and fountain codes are de- scribed clearly and succinctly. As a result of the author's research on the field, the nine chapters on neural networks
receive the deepest and most cohesive treatment in the book. Under the umbrella title of Probability and Inference we find a medley of chapters encompassing topics as varied as the Viterbi algorithm
and the forward-backward algorithm, Monte Carlo simu- lation, independent component analysis, clustering, Ising models, the saddle-point approximation, and a sampling of decision theory topics. The
chapters on data compression offer a good coverage of Huffman and arithmetic codes, and we are rewarded with material not usually encountered in information theory textbooks such as hash codes and
efficient representation of integers. The expositions of the memoryless source coding theorem and of the achievability part of the memoryless channel coding theorem stick closely to the standard
treatment in (1), with a certain tendency to over- simplify. For example, the source coding theorem is verbalized as: " i.i.d. random variables each with entropy can be compressed into more than bits
with negligible risk of information loss, as ; conversely if they are compressed into fewer than bits it is virtually certain that informa- tion will be lost." Although no treatment of
rate-distortion theory is offered, the author gives a brief sketch of the achievability of rate with bit- error rate , and the details of the converse proof of that limit are left as an exercise.
Neither Fano's inequality nor an operational definition of capacity put in an appearance. Perhaps his quest for originality is what accounts for MacKay's pro- clivity to fail to call a spade a spade.
Almost-lossless data compres- sion is called "lossy compression;" a vanilla-flavored binary hypoth-
Viewing the immune system as a molecular recognition device designed to identify “foreign shapes”, we estimate the probability that an immune system with NAb monospecific antibodies in its repertoire
can recognize a random foreign antigen. Furthermore, we estimate the improvement in recognition if antibodies are multispecific rather than monospecific. From our probabilistic model we conclude: (1)
clonal selection is feasible, i.e. with a finite number of antibodies an animal can recognize an effectively infinite number of antigens; (2) there should not be great differences in the
specificities of antibody molecules among different species; (3) the region of a foreign molecule recognized by an antibody must be severely limited in extent; (4) the probability of recognizing a
foreign molecule, P, increases with the antibody repertoire size NAb; however, below a certain value of NAb the immune system would be very ineffectual, while beyond some high value of NAb further
increases in NAb yield diminishing small increases in P; (5) multispecificity is equivalent to a modest increase (probably less than 10) in the antibody repertoire size NAb, but this increase can
substantially improve the probability of an immune system recognizing a foreign molecule.Besides recognizing foreign molecules, the immune system must distinguish them from self molecules. Using the
mathematical theory of reliability we argue that multisite recognition is a more reliable method of distinguishing between molecules than single site recognition. This may have been an important
evolutionary consideration in the selection of weak non-covalent interactions as the basis of antigen-antibody bonds.
The problem of protecting computer systems can be viewed generally as the problem of learning to distinguish self from other. We describe a method for change detection which is based on the
generation of T cells in the immune system. Mathematical analysis reveals computational costs of the system, and preliminary experiments illustrate how the method might be applied to the problem of
computer viruses. 1 Introduction The problem of ensuring the security of computer systems includes such activities as detecting unauthorized use of computer facilities, guaranteeing the integrity of
data files, and preventing the spread of computer viruses. In this paper, we view these protection problems as instances of the more general problem of distinguishing self (legitimate users,
corrupted data, etc.) from other (unauthorized users, viruses, etc.). We introduce a change-detection algorithm that is based on the way that natural immune systems distinguish self from other.
Mathematical analysis ...
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Layout (latest version) | IBM Quantum Documentation
class qiskit.transpiler.Layout(input_dict=None)
Bases: object
Two-ways dict to represent a Layout.
construct a Layout from a bijective dictionary, mapping virtual qubits to physical qubits
add(virtual_bit, physical_bit=None)
Adds a map element between bit and physical_bit. If physical_bit is not defined, bit will be mapped to a new physical bit.
• virtual_bit (tuple) – A (qu)bit. For example, (QuantumRegister(3, ‘qr’), 2).
• physical_bit (int) – A physical bit. For example, 3.
Adds at the end physical_qubits that map each bit in reg.
reg (Register) – A (qu)bit Register. For example, QuantumRegister(3, ‘qr’).
Combines self and another_layout into an “edge map”.
For example:
self another_layout resulting edge map
qr_1 -> 0 0 <- q_2 qr_1 -> q_2
qr_2 -> 2 2 <- q_1 qr_2 -> q_1
qr_3 -> 3 3 <- q_0 qr_3 -> q_0
The edge map is used to compose dags via, for example, compose.
another_layout (Layout) – The other layout to combine.
A “edge map”.
Return type
LayoutError – another_layout can be bigger than self, but not smaller. Otherwise, raises.
compose(other, qubits)
Compose this layout with another layout.
If this layout represents a mapping from the P-qubits to the positions of the Q-qubits, and the other layout represents a mapping from the Q-qubits to the positions of the R-qubits, then the composed
layout represents a mapping from the P-qubits to the positions of the R-qubits.
• other (Layout) – The existing Layout to compose this Layout with.
• qubits (List[Qubit]) – A list of Qubit objects over which other is defined, used to establish the correspondence between the positions of the other qubits and the actual qubits.
A new layout object the represents this layout composed with the other layout.
Return type
Returns a copy of a Layout instance.
Populates a Layout from a dictionary.
The dictionary must be a bijective mapping between virtual qubits (tuple) and physical qubits (int).
input_dict (dict) –
{(QuantumRegister(3, 'qr'), 0): 0,
(QuantumRegister(3, 'qr'), 1): 1,
(QuantumRegister(3, 'qr'), 2): 2}
Can be written more concisely as follows:
* virtual to physical::
{qr[0]: 0,
qr[1]: 1,
qr[2]: 2}
* physical to virtual::
{0: qr[0],
1: qr[1],
2: qr[2]}
static from_intlist(int_list, *qregs)
Converts a list of integers to a Layout mapping virtual qubits (index of the list) to physical qubits (the list values).
• int_list (list) – A list of integers.
• *qregs (QuantumRegisters) – The quantum registers to apply the layout to.
The corresponding Layout object.
Return type
LayoutError – Invalid input layout.
static from_qubit_list(qubit_list, *qregs)
Populates a Layout from a list containing virtual qubits, Qubit or None.
• qubit_list (list) – e.g.: [qr[0], None, qr[2], qr[3]]
• *qregs (QuantumRegisters) – The quantum registers to apply the layout to.
the corresponding Layout object
Return type
LayoutError – If the elements are not Qubit or None
static generate_trivial_layout(*regs)
Creates a trivial (“one-to-one”) Layout with the registers and qubits in regs.
*regs (Registers, Qubits) – registers and qubits to include in the layout.
A layout with all the regs in the given order.
Return type
Returns the dictionary where the keys are physical (qu)bits and the values are virtual (qu)bits.
Returns the registers in the layout [QuantumRegister(2, ‘qr0’), QuantumRegister(3, ‘qr1’)] :returns: A set of Registers in the layout :rtype: Set
Returns the dictionary where the keys are virtual (qu)bits and the values are physical (qu)bits.
inverse(source_qubits, target_qubits)
Finds the inverse of this layout.
This is possible when the layout is a bijective mapping, however the input and the output qubits may be different (in particular, this layout may be the mapping from the extended-with-ancillas
virtual qubits to physical qubits). Thus, if this layout represents a mapping from the P-qubits to the positions of the Q-qubits, the inverse layout represents a mapping from the Q-qubits to the
positions of the P-qubits.
• source_qubits (List[Qubit]) – A list of Qubit objects representing the domain of the layout.
• target_qubits (List[Qubit]) – A list of Qubit objects representing the image of the layout.
A new layout object the represents the inverse of this layout.
static order_based_on_type(value1, value2)
decides which one is physical/virtual based on the type. Returns (virtual, physical)
Given an ordered list of bits, reorder them according to this layout.
The list of bits must exactly match the virtual bits in this layout.
bits (list[Bit]) – the bits to reorder.
ordered bits.
Return type
swap(left, right)
Swaps the map between left and right.
• left (tuple orint) – Item to swap with right.
• right (tuple orint) – Item to swap with left.
LayoutError – If left and right have not the same type.
Creates a permutation corresponding to this layout.
This is possible when the layout is a bijective mapping with the same source and target qubits (for instance, a “final_layout” corresponds to a permutation of the physical circuit qubits). If this
layout is a mapping from qubits to their new positions, the resulting permutation describes which qubits occupy the positions 0, 1, 2, etc. after applying the permutation.
For example, suppose that the list of qubits is [qr_0, qr_1, qr_2], and the layout maps qr_0 to 2, qr_1 to 0, and qr_2 to 1. In terms of positions in qubits, this maps 0 to 2, 1 to 0 and 2 to 1, with
the corresponding permutation being [1, 2, 0].
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ADIA Lab Causal Discovery | CrunchDAO Docs V3
ADIA Lab Causal Discovery
Discovering the causal structure that governs the relationships among variables from their observations is a challenging and valuable problem in many domains of application, like healthcare,
economics, social sciences, environmental science, education, etc. In this competition, the basic building block that you are given is a dataset of observations of a set of variables and your task is
to discover the causal directed acyclic graph (DAG) that defines the causal relationships between them.
The task of this competition is causal discovery: your goal is to find the causal graph (DAG) for each dataset you will be given. To help you in this endeavor, we provide a large number of example
datasets together with their corresponding causal DAGs — as the training set — so that you can calibrate your unsupervised discovery methods, or train your prediction models if you prefer a
supervised approach. Your causal discovery algorithm has to be designed to take as input a dataset and to output the causal DAG.
All causal graphs in this competition have a specific structure: they have at least two special nodes, X and Y, which are the treatment and the outcome variables, respectively. The treatment variable
X is the one that causes effects on the outcome variable Y. All other variables/nodes may or may not influence X and Y, possibly interfering with their relationship X→Y, so each may act as a
confounder on X→Y, or as a collider, mediator, or be a cause or consequence of X (or Y), or not have any influence at all, etc.
The goal of the competition is to estimate the causal graph behind each dataset. The scores are based on accurately identifying the role of all nodes on X→Y.
Both unsupervised and supervised approaches are warmly welcome.
In all datasets, there are two special variables — X and Y — that are the treatment and the effect. We always assume that there is a causal link from X to Y: X→Y. For each predicted graph, the
evaluation metric quantifies the correctness of the edges/arrows for all nodes but considers only the edges (or lack of) from each node to X and Y. In other words, the evaluation metric wants to
assess the effects of errors in specifying wrong edges affecting X and Y.
Each node K (with the exclusion of X and Y) can be in one of these 8 categories:
Confounder: K→X, K→Y, X→Y
Independent: X→Y (no links to X or Y)
Consequence of X: X→K, X→Y
Each node in your predicted graph will be tested against its true class and the final scoring metric across all datasets is the multiclass balanced accuracy.
Participants should submit predicted DAGs for all datasets, and we will transform the predicted DAGs to the corresponding classes for scoring.
Prediction File
For each example_id in the test set, which is in the form <dataset_id>_<source_variable>_<target_variable> you must predict a binary value (0 or 1) representing the absence or presence of a causal
link between <source_variable> and <target_variable>. The file should contain a header and have the following format:
example_id, prediction
00000_0_0, 0
00000_0_1, 0
00000_0_X, 1
00000_0_4, 0
For example, the row 01234_X_1, 1 means that for the test dataset 01234, the participant predicts a causal link between X and 1: X→1.
Dataset Description
The whole dataset of the competition, between the training set and test set, comprises 47,000 individual datasets, each of 1000 observations for a certain number of variables, which is between 3 and
10. For the training datasets, the corresponding causal graphs are available. The causal graph is provided via its adjacency matrix, so if the dataset has 8 variables, the adjacency matrix is 8x8
matrix — which becomes 9x9 in the corresponding CSV file because the variable names are indicated for each row and column — where a value of 1 at position (i, j), means that variable i causes
variable j, and value 0 means it does not.
Tutorial #1
Winners’ rank Prize value
1st place $40,000 USD
2nd place $20,000 USD
3rd place $10,000 USD
4th place $5,000 USD
5th place $5,000 USD
6th place $5,000 USD
7th place $5,000 USD
8th place $3,500 USD
9th place $3,500 USD
10th place $3,000 USD
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An Old Sage Once Said... | maryampictures
top of page
• Q = {S1, S2},
• Σ = {0, 1},
• q0 = S1,
• F = {S1}
1 − (1/R0)1/(1+2CV2)
• ​1 − (1/R0)1/(1+2CV2)
• ∑ i = 1 n p i x i ≤ M . {_{i=1}^{n}p_{i}x_{i}\leq M.}
• x i ≥ 0 ∀ i ∈ { 1 , 2 , … , n } {x_{i}\geq i\in \{1,2,,n\}}
Mathematical models are a ‘process of encoding and decoding reality, in which a natural phenomenon is reduced to a formal numerical expression by a casual structure’. In other words, they are based
on assumptions when wet data is limited or absent.
For example, if asked to provide a mathematical model on the potential outcomes of a meteor striking Earth, you might start with a death toll range of between zero and 7.38 billion. The model would
then be expanded to include calculations such as speed and size of meteor, place of impact, time of day, axis position etc. What a mathematical model can’t accurately predict is how people will
respond to certain events. It can’t predict anomalies such as Elon Musk destroying the meteor before impact and what it absolutely doesn’t do, is factor in the reliability of the modeller.
So if previous models by Modeller A (let’s call him Neil) predicted 200M would die from Bird Flu (it was under 300), 65,000 from Swine Flu (under 500), 50,000 from BSE (under 200) and led to the
needless culling of 6.5m cattle during the foot and mouth crisis in the mistaken belief animals were infectious for days before showing any symptoms (sound familiar?) you’d surely factor this into
the model, if not mathematically then logically or instinctively.
So no, lockdown had nothing to do with following the science as there wasn't any. It was to do with power and what they thought they could get away with. It was - as quoted by one SAGE sub-committee
member - a panic measure because they ‘couldn’t think of anything better to do’.
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Master the Distance Formula with Kuta Software Infinite Geometry: A Comprehensive Guide - telefoninux.org
Master the Distance Formula with Kuta Software Infinite Geometry: A Comprehensive Guide
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Kuta Software Infinite Geometry: The Distance Formula is a mathematical tool used to calculate the distance between two points on a coordinate plane. It is widely applied in fields such as
engineering, physics, and computer science.
The distance formula provides a precise and convenient method for determining the length of a line segment. Its significance stems from its role in solving geometry problems, calculating distances in
real-world applications (e.g., navigation), and establishing spatial relationships in various scientific and engineering disciplines. Historically, the concept can be traced back to ancient Greek
mathematicians like Euclid and Pythagoras.
This article explores the specifics of the distance formula within the context of Kuta Software’s Infinite Geometry software, delving into its formula, applications, and examples.
Kuta Software Infinite Geometry
Understanding the essential aspects of the distance formula in Kuta Software Infinite Geometry is paramount for leveraging its capabilities effectively. These aspects encompass various dimensions,
• Formula
• Inputs
• Outputs
• Applications
• Limitations
• Accuracy
• Efficiency
• Visual representation
• Interactive features
• Educational value
Exploring these aspects provides a deeper understanding of the distance formula’s role in geometry, its strengths and weaknesses, and its potential for enhancing mathematical learning. By delving
into these dimensions, users can optimize their utilization of Kuta Software Infinite Geometry for geometry problem-solving, spatial reasoning, and mathematical exploration.
The formula in Kuta Software Infinite Geometry’s distance formula is a mathematical expression that calculates the distance between two points on a coordinate plane. It is a fundamental tool in
geometry, with wide-ranging applications in real-world scenarios.
• Distance Formula
The distance formula is given by: distance = $\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$, where (x1, y1) and (x2, y2) represent the coordinates of the two points.
• Components
The formula comprises two points on a coordinate plane, each with x and y coordinates. It also involves mathematical operations like subtraction, squaring, and square root.
• Applications
The distance formula finds applications in calculating lengths of line segments, determining the distance between two objects in a coordinate system, and solving geometry problems involving
triangles, circles, and other shapes.
• Implications
Understanding the distance formula is crucial for spatial reasoning, comprehending geometric relationships, and utilizing geometry in practical settings like engineering, physics, and computer
In summary, the distance formula in Kuta Software Infinite Geometry is a powerful tool for calculating distances between points in a coordinate system. Its components, applications, and implications
underscore its significance in geometry and its relevance to real-world problem-solving.
Inputs play a crucial role in Kuta Software Infinite Geometry’s distance formula. They are the foundation upon which the formula operates, providing the necessary information to calculate the
distance between two points on a coordinate plane. Without accurate and appropriate inputs, the formula cannot produce meaningful results.
The distance formula requires two sets of coordinates as inputs: (x1, y1) and (x2, y2). These coordinates represent the positions of the two points on the plane. The formula then utilizes these
inputs to calculate the distance between the points using mathematical operations.
In real-world applications, inputs to the distance formula can vary widely. For example, in architecture, inputs could be the coordinates of two points representing the endpoints of a wall, allowing
for the calculation of the wall’s length. In navigation, inputs could be the coordinates of two cities, enabling the determination of the distance between them.
Understanding the connection between inputs and the distance formula is essential for its effective use. By providing accurate inputs, users can ensure reliable and meaningful results. Moreover,
recognizing the practical applications of this understanding empowers individuals to solve geometry problems, measure distances in real-world scenarios, and explore spatial relationships with greater
In Kuta Software Infinite Geometry, outputs are the calculated distances resulting from the distance formula. These outputs hold significance as they represent the end result of the formula’s
operation and provide valuable information for various applications. The distance formula serves as a critical component in generating these outputs, utilizing inputs and mathematical calculations to
produce accurate results.
Real-life examples of outputs from the distance formula are abundant. In architecture, the formula helps determine distances between points, enabling the calculation of room sizes, wall lengths, and
other dimensions. In navigation, it aids in determining distances between cities or locations, facilitating route planning and travel estimations. Moreover, in engineering, the formula assists in
calculating distances between objects, supporting design and construction processes.
Understanding the relationship between outputs and the distance formula is essential for effective geometry problem-solving and practical applications. By interpreting outputs correctly, users can
gain insights into spatial relationships, measure distances accurately, and make informed decisions. This understanding empowers individuals to use Kuta Software Infinite Geometry as a valuable tool
for exploring geometry, designing structures, navigating routes, and solving real-world problems.
The distance formula in Kuta Software Infinite Geometry finds wide-ranging applications across various fields, serving as a cornerstone for geometry problem-solving and real-world measurements. Its
significance stems from the fact that it provides a precise and efficient method for calculating the distance between two points on a coordinate plane, enabling users to quantify spatial
relationships and solve complex geometric problems.
The applications of the distance formula extend far beyond theoretical geometry. In architecture, it plays a crucial role in determining distances between structural elements, ensuring accurate
measurements for construction and design. Similarly, in navigation, the distance formula aids in calculating distances between locations, facilitating efficient route planning and travel estimations.
Furthermore, in engineering, it is used to calculate distances between objects, supporting design and manufacturing processes.
Understanding the practical applications of the distance formula empowers individuals to use Kuta Software Infinite Geometry as a valuable tool for solving real-world problems. By leveraging the
formula’s capabilities, users can gain insights into spatial relationships, measure distances accurately, and make informed decisions. This understanding finds applications in diverse fields, ranging
from architecture and engineering to navigation and computer graphics, empowering users to explore geometry, design structures, navigate routes, and solve complex spatial problems.
The limitations of Kuta Software Infinite Geometry’s distance formula lie in its inability to handle certain types of geometric figures and complex spatial relationships. Unlike more advanced
geometric software, Kuta Software Infinite Geometry is primarily designed for basic geometry concepts and calculations. This limitation can affect the accuracy and applicability of the distance
formula in certain scenarios.
For instance, the distance formula cannot be directly applied to calculate distances involving curves, such as circles or parabolas. Similarly, it cannot handle three-dimensional figures or complex
geometric transformations. In such cases, alternative methods or more advanced software tools may be required to obtain accurate distance measurements.
Understanding these limitations is crucial for users to avoid misinterpretations or incorrect results. By recognizing the types of geometric figures and spatial relationships that the distance
formula cannot handle, users can make informed decisions about when to use the formula and when to seek alternative approaches.
In summary, while Kuta Software Infinite Geometry’s distance formula provides a valuable tool for basic geometry problem-solving, its limitations must be considered to ensure accurate and
comprehensive results. Understanding these limitations empowers users to make appropriate choices and leverage the software effectively within its intended scope.
In the realm of geometry, accuracy is of paramount importance, and Kuta Software Infinite Geometry’s distance formula is no exception. The accuracy of the distance formula directly influences the
reliability and validity of the results obtained when calculating distances between points on a coordinate plane.
The distance formula relies on precise inputs, namely the coordinates of the two points. Inaccurate inputs can lead to incorrect distance calculations, affecting the overall accuracy of the formula.
Moreover, the formula assumes a straight-line distance between the two points, which may not always be the case in real-world applications. Understanding these limitations is crucial to ensure
accurate results.
In practical applications, accuracy is vital for tasks such as architectural design, navigation, and engineering. For instance, in architecture, precise distance measurements are essential for
ensuring structural integrity and aesthetic appeal. Similarly, in navigation, accurate distance calculations are crucial for efficient route planning and safe travel. By understanding the importance
of accuracy, users can make informed decisions about the applicability of the distance formula and interpret the results appropriately.
In summary, accuracy is a critical component of Kuta Software Infinite Geometry’s distance formula. It affects the reliability and validity of the results obtained, impacting real-world applications
such as architecture, navigation, and engineering. Understanding the relationship between accuracy and the distance formula empowers users to make informed decisions and leverage the software
effectively for accurate geometric calculations.
Efficiency is a crucial aspect of Kuta Software Infinite Geometry’s distance formula, influencing its performance and usability. It encompasses various factors that contribute to the formula’s
effectiveness in solving geometry problems and calculating distances.
• Computational Speed
The distance formula employs efficient mathematical operations that minimize computation time. This allows for rapid calculation of distances, even for complex geometric figures with numerous
• Memory Usage
The formula’s efficient use of memory resources ensures that it can be applied to large datasets without encountering memory limitations. This is particularly advantageous when working with
complex geometric models or performing calculations.
• Code Optimization
Kuta Software Infinite Geometry’s distance formula is optimized to minimize the number of instructions required for its execution. This optimization improves the formula’s overall efficiency,
resulting in faster and more responsive performance.
• Parallelization
The formula can be parallelized to harness the power of multiple processors or cores. By distributing the computational load across multiple threads, the formula can achieve significant speedups,
especially when dealing with large-scale geometric datasets.
In summary, the efficiency of Kuta Software Infinite Geometry’s distance formula stems from its optimized algorithms, efficient use of memory, and parallelization capabilities. By leveraging these
factors, the formula delivers fast, reliable, and scalable performance, making it a valuable tool for solving complex geometry problems and performing accurate distance calculations.
Visual representation
Visual representation plays a critical role within Kuta Software Infinite Geometry’s distance formula, enabling users to visualize the geometric concepts and relationships being measured. It provides
a graphical depiction of the distance being calculated, enhancing understanding and facilitating problem-solving.
The formula translates the mathematical calculation of distance into a visual representation, allowing users to see the distance as a line segment on the coordinate plane. This visualization aids in
comprehending the spatial relationships between points, angles, and shapes. By observing the visual representation, users can develop a deeper understanding of the geometric principles at play.
Real-life examples of visual representation within Kuta Software Infinite Geometry’s distance formula include:
• Calculating the distance between two points on a map to determine the shortest route.
• Measuring the length of a building’s diagonal using the distance formula and visualizing the result as a line segment.
• Determining the radius of a circle by calculating the distance between the center and any point on the circumference.
Understanding the connection between visual representation and the distance formula is essential for effective geometry problem-solving. By utilizing the visual representation, users can gain
insights into the geometric relationships being measured, identify patterns, and make informed decisions. This understanding empowers individuals to apply the distance formula with greater accuracy
and confidence in various practical applications.
Interactive features
Interactive features within Kuta Software Infinite Geometry’s distance formula foster a dynamic and engaging learning environment, transforming the formula from a static calculation tool into an
interactive experience. These features empower users to manipulate geometric objects, visualize concepts, and explore relationships in a hands-on manner.
One key interactive feature is the ability to drag and move points on the coordinate plane. This allows users to experiment with different distances and observe the corresponding changes in the
distance formula’s output. By manipulating the points, users can develop a deeper understanding of the relationship between the coordinates and the distance between them.
Another interactive feature is the provision of real-time feedback. As users adjust the positions of points, the distance formula recalculates and updates the result instantaneously. This immediate
feedback enables users to identify patterns, make conjectures, and refine their understanding of the formula.
The practical applications of interactive features in Kuta Software Infinite Geometry’s distance formula extend beyond theoretical geometry. In architecture, for instance, architects can use these
features to interactively plan room layouts and calculate distances between structural elements. Similarly, in engineering, interactive features allow engineers to simulate and visualize the movement
of objects, taking into account distances and spatial relationships.
In summary, interactive features play a vital role within Kuta Software Infinite Geometry’s distance formula. They transform the formula into an engaging and interactive learning tool, enabling users
to explore geometric concepts, visualize relationships, and develop a deeper understanding of the distance formula’s application in real-world scenarios.
Educational value
The educational value of Kuta Software Infinite Geometry’s distance formula lies in its ability to enhance students’ understanding of geometry concepts and problem-solving skills. The formula
provides a structured and systematic approach to calculating distances between points on a coordinate plane, fostering a deeper comprehension of spatial relationships.
As a critical component of Kuta Software Infinite Geometry, the distance formula enables students to engage with geometry in a hands-on and interactive manner. Through experimentation and
exploration, they can visualize the relationship between coordinates and distances, develop logical reasoning, and refine their problem-solving abilities.
Real-life examples of the educational value of Kuta Software Infinite Geometry’s distance formula include:
• Students applying the formula to calculate the length of a room or the distance between two cities, enhancing their understanding of practical applications.
• Students using the formula to explore geometric patterns and relationships, fostering their creativity and analytical thinking.
• Students collaborating on projects that involve calculating distances within complex geometric figures, developing their teamwork and communication skills.
The practical applications of understanding the connection between educational value and Kuta Software Infinite Geometry’s distance formula extend beyond the classroom. It empowers students to
approach real-world problems with a geometric perspective, confidently applying their knowledge to fields such as architecture, engineering, and computer graphics.
Frequently Asked Questions About Kuta Software Infinite Geometry’s Distance Formula
This FAQ section provides answers to common questions and clarifies key aspects of Kuta Software Infinite Geometry’s distance formula, empowering users to leverage its capabilities effectively.
Question 1: What is Kuta Software Infinite Geometry’s distance formula, and how is it used?
Answer: Kuta Software Infinite Geometry’s distance formula calculates the distance between two points on a coordinate plane. It employs the formula: distance = $\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$,
where (x1, y1) and (x2, y2) are the coordinates of the points. This formula finds applications in geometry problem-solving, spatial reasoning, and various real-world scenarios.
Question 2: What are the limitations of the distance formula?
Answer: The distance formula cannot handle certain geometric figures, such as circles or parabolas, and complex spatial relationships. It also assumes a straight-line distance between points, which
may not always be the case in real-world applications. Understanding these limitations ensures accurate and appropriate usage of the formula.
Question 3: What is the significance of accuracy when using the distance formula?
Answer: Accuracy is crucial for reliable results. Precise inputs and consideration of the formula’s limitations are essential for accurate distance calculations. This accuracy is particularly
important in practical applications like architecture, navigation, and engineering, where precise measurements are critical.
Question 4: How does the interactive visualization feature enhance the learning experience?
Answer: The interactive visualization feature allows users to manipulate points on the coordinate plane and observe the corresponding changes in the distance formula’s output. This hands-on approach
fosters a deeper understanding of the relationship between coordinates and distances, making the learning process more engaging and intuitive.
Question 5: What are the educational benefits of using Kuta Software Infinite Geometry’s distance formula?
Answer: The distance formula provides a structured approach to understanding geometry concepts and problem-solving. It promotes logical reasoning, spatial visualization, and problem-solving skills.
By applying the formula to real-world scenarios, students can develop a practical understanding of geometry’s relevance.
Question 6: How can I ensure efficient use of the distance formula?
Answer: To optimize efficiency, ensure accurate inputs, avoid unnecessary calculations, and leverage technology like calculators or software. Understanding the formula’s principles and applying it
judiciously will enhance efficiency and accuracy in distance calculations.
In summary, Kuta Software Infinite Geometry’s distance formula is a valuable tool for understanding geometry and solving real-world problems. By addressing common questions and emphasizing the
importance of accuracy, limitations, and educational value, this FAQ section empowers users to leverage the formula effectively. As we delve deeper into the topic, we will explore advanced
applications and techniques related to the distance formula, enhancing our understanding of its versatility and practical significance.
Tips to Enhance Understanding and Application of Kuta Software Infinite Geometry’s Distance Formula
This section provides practical tips to optimize your use of Kuta Software Infinite Geometry’s distance formula, enhancing your understanding and proficiency in geometry problem-solving.
Tip 1: Understand the Formula: Grasp the underlying mathematical principles of the distance formula to apply it accurately and efficiently.
Visualize the Concept: Utilize the interactive visualization feature to visualize the relationship between coordinates and distances, deepening your comprehension.
Practice Regularly: Engage in regular practice with the distance formula to improve your problem-solving skills and strengthen your understanding.
Identify Geometric Patterns: Analyze geometric figures and identify patterns related to distances. This will enhance your spatial reasoning abilities.
Apply to Real-World Scenarios: Extend your understanding by applying the distance formula to practical applications in fields like architecture and engineering.
Leverage Technology: Utilize calculators or software to simplify calculations and focus on the conceptual understanding of the distance formula.
By following these tips, you can effectively harness the capabilities of Kuta Software Infinite Geometry’s distance formula, empowering yourself to tackle geometry problems with confidence and
In the concluding section, we will explore advanced applications of the distance formula, showcasing its versatility and relevance in various fields. This will further solidify your understanding and
demonstrate the practical significance of mastering the distance formula.
In this comprehensive exploration of Kuta Software Infinite Geometry’s distance formula, we have gained invaluable insights into its significance, applications, and educational value. Key points that
emerged include the formula’s ability to calculate distances accurately, its limitations in handling complex geometries, and its interactive visualization features that enhance comprehension.
The distance formula serves as a fundamental tool in geometry, enabling the measurement of distances between points on a coordinate plane. Its simplicity and efficiency make it a versatile tool,
while its limitations remind us of the complexities of geometry. The interactive visualization feature bridges the gap between abstract concepts and practical applications, fostering a deeper
understanding of spatial relationships.
Mastering the distance formula is not merely about memorizing steps but about developing a comprehensive understanding of its principles and applications. It empowers us to solve complex geometry
problems, visualize spatial relationships, and tackle real-world challenges. As we continue to explore the realm of geometry, the distance formula will remain an indispensable tool, guiding us
towards a deeper appreciation of the beauty and precision of this mathematical discipline.
Images References :
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Constant Current LED Driver & Power Supply | Bravo Electro | Power Range (W): 1001-1500; Manufacturer: Mean Well
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APC-35-1050 is a non-dimming driver, with an IP42 rating and 2 year manufacturer's warranty. Its dimensions are 84 (L) x 57 (W) x 29.5 (H) mm and the operational temperature range is -30C to
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Generalized Linear Models Theory
This is a brief introduction to the theory of generalized linear models.
Response Probability Distributions
In generalized linear models, the response is assumed to possess a probability distribution of the exponential form. That is, the probability density of the response
for some functions
Standard theory for this type of distribution gives expressions for the mean and variance of
where the primes denote derivatives with respect to
where variance function.
Probability distributions of the response natural parameter Long (1997) for a discussion of the zero-inflated Poisson and zero-inflated negative binomial distributions. The PROC GENMOD scale
parameter and the variance of
The negative binomial and the zero-inflated negative binomial distributions contain a parameter
For the binomial distribution, the response is the binomial proportion The variance function is
If a weight variable is present,
PROC GENMOD works with a scale parameter that is related to the exponential family dispersion parameter
Link Function
For distributions other than the zero-inflated Poisson or zero-inflated negative binomial, the mean
There are two link functions and linear predictors associated with zero-inflated distributions: one for the zero inflation probability Zero-Inflated Models for more details about zero-inflated
Log-Likelihood Functions
Log-likelihood functions for the distributions that are available in the procedure are parameterized in terms of the means Response Probability Distributions. The term
where the sum is over the observations. The forms of the individual contributions
are shown in the following list; the parameterizations are expressed in terms of the mean and dispersion parameters.
For the discrete distributions (binomial, multinomial, negative binomial, and Poisson), the functions computed as the sum of the log likelihood. The proper log-likelihood function is also computed as
the sum of the full log likelihood in the output.
• Normal:
• Inverse Gaussian:
• Gamma:
• Negative binomial:
• Poisson:
• Binomial:
• Multinomial (k categories):
• Zero-inflated Poisson:
• Zero-inflated negative binomial:
Maximum Likelihood Fitting
The GENMOD procedure uses a ridge-stabilized Newton-Raphson algorithm to maximize the log-likelihood function By default, the procedure also produces maximum likelihood estimates of the scale
parameter as defined in the section Response Probability Distributions for the normal, inverse Gaussian, negative binomial, and gamma distributions.
On the
where (second derivative) matrix, and (first derivative) vector of the log-likelihood function, both evaluated at the current value of the parameter vector. That is,
In some cases, the scale parameter is estimated by maximum likelihood. In these cases, elements corresponding to the scale parameter are computed and included in
The gradient vector and Hessian matrix for the regression parameters are given by
where X, and
The primes denote derivatives of The expected value of method of fitting. Either
Covariance and Correlation Matrix
The estimated covariance matrix of the parameter estimator is given by
The correlation matrix is the normalized covariance matrix. That is, if
Goodness of Fit
Two statistics that are helpful in assessing the goodness of fit of a given generalized linear model are the scaled deviance and Pearson’s chi-square statistic. For a fixed value of the dispersion
Note that these statistics are not valid for GEE models.
For specific distributions, this can be expressed as
Distribution Deviance
Inverse Gaussian
Negative binomial
Zero-inflated Poisson
Zero-inflated negative binomial
In the binomial case,
In the multinomial case,
Pearson’s chi-square statistic is defined as
and the scaled Pearson’s chi-square is
The scaled version of both of these statistics, under certain regularity conditions, has a limiting chi-square distribution, with degrees of freedom equal to the number of observations minus the
number of parameters estimated. The scaled version can be used as an approximate guide to the goodness of fit of a given model. Use caution before applying these statistics to ensure that all the
conditions for the asymptotic distributions hold. McCullagh and Nelder (1989) advise that differences in deviances for nested models can be better approximated by chi-square distributions than the
deviances can themselves.
In cases where the dispersion parameter is not known, an estimate can be used to obtain an approximation to the scaled deviance and Pearson’s chi-square statistic. One strategy is to fit a model that
contains a sufficient number of parameters so that all systematic variation is removed, estimate of the dispersion parameter Type 1 Analysis for more about the estimation of the dispersion parameter.
Other Fit Statistics
The Akaike information criterion (AIC) is a measure of goodness of model fit that balances model fit against model simplicity. AIC has the form
The Bayesian information criterion (BIC) is a similar measure. BIC is defined by
See Akaike (1981, 1979) for details of AIC and BIC. See Simonoff (2003) for a discussion of using AIC, AICC, and BIC with generalized linear models. These criteria are useful in selecting among
regression models, with smaller values representing better model fit. PROC GENMOD uses the full log likelihoods defined in the section Log-Likelihood Functions, with all terms included, for computing
all of the criteria.
Dispersion Parameter
There are several options available in PROC GENMOD for handling the exponential distribution dispersion parameter. The NOSCALE and SCALE options in the MODEL statement affect the way in which the
dispersion parameter is treated. If you specify the SCALE=DEVIANCE option, the dispersion parameter is estimated by the deviance divided by its degrees of freedom. If you specify the SCALE=PEARSON
option, the dispersion parameter is estimated by Pearson’s chi-square statistic divided by its degrees of freedom.
Otherwise, values of the SCALE and NOSCALE options and the resultant actions are displayed in the following table.
NOSCALE SCALE=value Action
Present Present Scale fixed at value
Present Not present Scale fixed at 1
Not present Not present Scale estimated by ML
Not present Present Scale estimated by ML,
starting point at value
Present (negative binomial) Not present
The meaning of the scale parameter displayed in the "Analysis Of Parameter Estimates" table is different for the gamma distribution than for the other distributions. The relation of the scale
parameter as used by PROC GENMOD to the exponential family dispersion parameter
Distribution Scale
Inverse Gaussian
In the case of the negative binomial distribution, PROC GENMOD reports the "dispersion" parameter estimated by maximum likelihood. This is the negative binomial parameter Response Probability
Overdispersion is a phenomenon that sometimes occurs in data that are modeled with the binomial or Poisson distributions. If the estimate of dispersion after fitting, as measured by the deviance or
Pearson’s chi-square, divided by the degrees of freedom, is not near 1, then the data might be overdispersed if the dispersion estimate is greater than 1 or underdispersed if the dispersion estimate
is less than 1. A simple way to model this situation is to allow the variance functions of these distributions to have a multiplicative overdispersion factor
An alternative method to allow for overdispersion in the Poisson distribution is to fit a negative binomial distribution, where
The models are fit in the usual way, and the parameter estimates are not affected by the value of
The SCALE= option in the MODEL statement enables you to specify a value of
The function obtained by dividing a log-likelihood function for the binomial or Poisson distribution by a dispersion parameter is not a legitimate log-likelihood function. It is an example of a
quasi-likelihood function. Most of the asymptotic theory for log likelihoods also applies to quasi-likelihoods, which justifies computing standard errors and likelihood ratio statistics by using
quasi-likelihoods instead of proper log likelihoods. See McCullagh and Nelder (1989, Chapter 9), McCullagh (1983), and Hardin and Hilbe (2003) for details on quasi-likelihood functions.
Although the estimate of the dispersion parameter is often used to indicate overdispersion or underdispersion, this estimate might also indicate other problems such as an incorrectly specified model
or outliers in the data. You should carefully assess whether this type of model is appropriate for your data.
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Wilcoxon Test
Review of Short Phrases and Links
This Review contains major "Wilcoxon Test"- related terms, short phrases and links grouped together in the form of Encyclopedia article.
1. The Wilcoxon test is generally a more powerful test than the Sign test. (Web site)
2. The Wilcoxon test was popularised by Siegel (1956) in his influential text book on non-parametric statistics.
1. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. (Web site)
1. Unlike ORA based tests, the Wilcoxon test does not require setting a sometimes subjective threshold.
1. Neutrophil OFR production assayed by CL decreased significantly in VIT patients (Wilcoxon test for paired data P<0.01, Chi square test P<0.01).
1. Differences before and after treatment were tested for significance using the Wilcoxon test. (Web site)
1. The WILCOXON option requests the Wilcoxon test for difference in location, and the MEDIAN option requests the median test for difference in location.
1. This test is similar to the Wilcoxon test for 2 samples. (Web site)
1. Both groups are distinguished highly significantly with p.ltoreq.0.01 from the control group and from group 2 in the Wilcoxon test.
1. Change of frequency of bowel movements and faecal incontinence was assessed using the non-parametric paired Wilcoxon test.
1. The Wilcoxon test was popularised by Siegel (1956) in his influential text book on non-parametric statistics.
1. The Wilcoxon test is generally a more powerful test than the Sign test. (Web site)
Wilcoxon Test
1. This test is the nonparametric version of one way ANOVA and is a straightforward generalization of the Wilcoxon test for two independent samples. (Web site)
2. Chi-square test, sign test, Wald-Wolfowitz runs test, run test for randomness, median test, Wilcoxon test and Wilcoxon-Mann-Whitney test. (Web site)
3. The treatment effect (difference between treatments) was quantified using the Hodges-Lehmann (HL) estimator, which is consistent with the Wilcoxon test (ref.
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The Mathematics of Measurement
The following handout presents an accessible overview of the mathematics of measurement. It provides important information on the foundations of measurement and ideas for supporting development of
measurement skills in the preschool classroom.
Engaging in Measurement Activities
How long is the train track? Which block is taller? How much heavier is my pumpkin than yours? How long before lunch? What do all of these questions have in common? Measurement!
Measurement seems like it should be a topic that we can easily wrap our heads around. After all, we ask these kinds of questions all the time in the classroom. And, as noted in Measuring Up,
children’s understanding of measurement has foundations in early infancy. By the time they are in preschool, they’ve had lots of experiences with measurement. Children have a particular interest in
comparing sizes. They delight in sharing that they are bigger than other children, animals and objects. Dinosaurs can be especially appealing at this age, for no one tells a Tyrannosaurus Rex to take
a nap or clean up his toys! But teaching about measurement is harder than it looks.
Teacher's Voice: Supporting DLLs by communicating meaning through movement and gesture
We use hand gestures all the time when we’re singing, or jumping, or talking about measurement. Is it small? Is it big? The movement and gestures help [DLL students] connect what we’re doing
with language. -preschool teacher
Gigantic reptiles aside, what are the mathematics of measurement? The sections below discuss some of the key processes and concepts in measurement.
We begin with a concept that applies to many forms of measurement, and is the target of several of the questions asked at the beginning of this handout. Children notice differences in length, height,
area, capacity, weight, time, and temperature, and can be adept at describing them. Comments like, My road is longer! I’m taller than you! I have a bigger castle than you do! My bucket holds more
sand than your bucket! My backpack is heavier than yours! He had a longer turn than I did! And My yogurt is colder than my strawberries! are common in the classroom (some much more common than we’d
like!). We can take advantage of this deep interest in comparisons with the question, “How do you know?” This can be the jumping off point to helping children figure out how they can quantitatively
describe the comparison. The handout What Children Know and Need to Know about Measurement and Estimation describes a wide variety of ways in which children can compare objects.
Ordering three or more objects
seriation). This requires considering several aspects of a group of objects at once. In a group of three sticks of different lengths, the middle stick is both shorter than the longest and longer than
the shortest. Seriation can be difficult for young children to understand. They tend to focus on only one of the comparisons at a time (i.e., whether one stick is shorter than one other). However,
with everyday interactions such as ordering stuffed toy bears, cups, or pipe cleaners, children can develop the ability to order objects based on a variety of dimensions, including length, weight, or
When making comparisons, ensuring all elements have a common origin is useful (and sometimes necessary). To conclude that one object is longer than another, one of the objects has to reach beyond
both ends of the other object or the end of one object has to reach beyond the other when the two have the same starting place (shared origin). If comparing length or height, measurement is more
accurate if the objects are lined up so the ends of the objects begin on the same plane. Rulers are hard to use accurately if they don’t start at zero. An analogy for adults is the use of Celsius and
Fahrenheit. Zero means something different in each of these forms of temperature measurement (0˚ is the freezing point in Celsius, but is well below freezing in Fahrenheit), so knowing which kind of
thermometer is being used is important when comparing temperatures. The origin counts! But it takes a while for young children to figure this out. They will often focus on only one end of the items
being measured when they make length comparisons, failing to notice that the two objects have different starting points.
Non-standard and standard measurement
Preschool educational standards frequently list use of non-standard measurement as skill gained earlier than use of standard measurement. In reality, both of these can be somewhat complicated for
young children to understand and use. Non-standard measurement refers to the repeated use of a single object or multiple duplicates of the same sized concrete units end to end to measure length,
height, or area. Examples are the use of hands to measure a child’s height, paperclips to measure the length of a tabletop, and square blocks to measure the area of a carpet square. Several issues
complicate the use of non-standard measures. One is that the size of the unit used to measure must be consistent (using both large and small paperclips will yield a different measurement than if only
large or only small paperclips are used). Another is that the units must be laid end to end or edge to edge, leaving no space between. And, objects should have a common base in order to accurately
measure length and height (see origin above). Finally, because objects are rarely exactly twenty paperclips long, conversations about the extra length are needed (e.g., “The table is 20 paperclips
long plus about a half a paperclip”). When using objects in non-standard measurement, ease of placement and the ability to line the units up end to end are important aspects to consider (hands may be
more difficult to use to measure the length of a table than rectangular pattern blocks).
Standard measures are, well, standardized. They include tape measures, rulers, scales, thermostats, clocks, and so on. Standard measurement tools have some of the same, as well as some additional,
complicating issues as non-standard measures. Because they are standardized, the unit size is generally not an issue if the same type of tool is used for measuring the objects (e.g., a tape measure,
ruler, yardstick, scale, etc.). However, starting at the origin is essential, for instance measuring an object with a ruler beginning at 0, not 1 (otherwise, some pretty complicated operations are
required to compare the objects). Perhaps one of the most complicated issues of standard measurement is that it is tied to number and numerals, and therefore requires children to have gained a
considerable amount of number sense in order to understand the corresponding measurements (for example, if a child doesn’t know that 11 is greater than 9, then the use of a ruler won’t help very much
in determining which object is longer). And finally, when objects are not represented by whole numbers, the issue of halves, quarters or smaller fractions can be complicated. Like non-standard
measurement, stating that one object is “nine inches and a little bit” long and the other is 11 inches, or that one object is “closer to three inches and the other closer to four inches,” can be
sufficient to compare. In other words, teachers can scaffold children’s use of standard measurement tools by allowing them to approximate length or height and then compare these approximate
Attributes in Measurement
The sections you’ve just read concern important concepts that children learn as they engage in measurement activities. Gaining an understanding of these concepts allows children to accurately and
meaningfully measure objects and environments by the attributes (dimensions or properties) that they are interested in. These attributes are described below.
Length and Height
Length and height are linear measurements and are perhaps the most commonly measured in preschool activities. They can be measured by both non-standard and standard measurement tools. Children’s
understanding of length develops over an extended period of time (well into the elementary school years), and includes (but is not limited to) the concepts of origin and unit (described above),
attribute (length has a beginning and an end), and conservation (like number, if nothing is added or taken away, the length does not change regardless of position).
Children informally use area measurement all the time (“You have more room in the sandbox than I do! Look, I don’t have any room for my legs!”) But, area can be more complicated to measure than
length because it is two-dimensional (by contrast, height and length are one-dimensional), so children need to pay attention to two attributes at the same time: width and length. In preschool,
children can measure area with square tiles placing them end to end and adjacent or they can make use of graph paper. Measuring area with standard measurement tools, such as rulers or tape measures,
requires multiplication and is generally explored later in the elementary school years.
Water tables, sand boxes, cooking activities, and mealtimes offer rich opportunities for the measurement of capacity: the amount a container can hold. In water and sand tables, both non-standard
(dump trucks, teacups, buckets) and standard (measuring cups with or without graduated units) measuring tools are useful and engaging to children. Children explore capacities of cups during mealtime,
especially if they are allowed to pour their own beverages. Recipes provide excellent classroom activities that use both number and measurement. This can include non-consumables like play-dough, or
hot and cold consumables like smoothies, snacks, and quick breads.
The measurement of weight is another common preschool classroom activity. Children frequently use their hands to compare the weights of two objects. Balance scales provide non-numerical comparisons
of two objects and spring (or digital) scales add number to those comparisons. Children can learn to differentiate weight from size by experimenting with large light objects (such as Styrofoam
blocks) and small heavy objects (like metal blocks).
Standard time measurement is a very difficult concept for children to grasp in preschool. Our formal division of time is idiosyncratic and may be confusing to a child (60 seconds in a minute, 60
minutes in an hour, 24 hours in a day, 7 days in a week, 28, 29, 30, or 31 days in a month, 12 months in a year!). A good place to start teaching about time measurement is with the vocabulary of
time. Words like morning, afternoon, evening, night, day, tomorrow, yesterday, after, and before can provide a foundation for later time concepts. Children can also take turns making use of
hourglasses and timers in the classroom. It is important to use accurate terminology with regard to time. For instance, stating that you’ll be back in a few minutes when you are going on a lunch hour
provides an inaccurate statement of time to a child. Although use of the calendar during circle or whole group time is common, its seemingly arbitrary structure makes it less useful than other
activities in supporting mathematical development (see "To Calendar or Not To Calendar" Vignette to explore some of the issues that classroom use of the calendar can bring up).
As with time, formal measurement of temperature can be difficult for young children to understand. This is particularly true if children live in areas where there isn’t a wide range of temperatures.
When it is hot in the summer and snowing in the winter, temperature takes on a bit more meaning!! As with time, introducing temperature words and their meanings as a foundation for later
understanding of temperature can be useful. These terms include hot, warm, cool, cold, freezing, and boiling.
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Is 1 oz the same as 30ml? - Explained
oz) use 1 fl. oz. = 30 mL to convert.
How many ml means 1 oz?
For the US fluid ounce, 1 fl oz = 29.6 ml ; and.
How much is 45 ml of alcohol?
A standard drink served in most bars contains 0.5–0.7 fluid ounce of absolute alcohol. (One ounce equals approximately 30 ml.) Thus, a 1.5-ounce (45-ml) shot of vodka, a 5-ounce (150-ml) glass of
wine, and a 12-ounce (355-ml) bottle of beer are equally intoxicating.
Is 1 oz the same as 15 ml?
How many milliliters in an ounce? 1 fluid ounce is equal to 29.57353193 milliliter, which is the conversion factor from ounces to milliliter.
Leave a Comment
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Week six, Thursday
Last time we saw the proof of the Erdos-Rado theorem, that \(\left(2^{\aleph_0}\right)^+ \rightarrow (\aleph_1)_2^2\), modulo a few points where the proof requires some facts about cardinal
Addition, multiplication and exponentiation of cardinals is defined at the bottom of page 68. Everyone should look at this, and check that cardinal arithmetic is well-defined. Everyone should think
about Exercise 2.29, as well, and Sean discuss it.
Exercise 2.30 (in full generality) is more difficult, but we only need two special cases:
• \(2^{\aleph_0} \cdot 2^{\aleph_0} = 2^{\aleph_0}\)
• \(2^{\aleph_0} + 2^{\aleph_0} = 2^{\aleph_0}\)
I challenge everyone to come up with direct proofs of both of these statements (remember that \(2^{\aleph_0}\) counts subsets of \({\mathbb N}\), and that there are easy injections from \(2^{\
aleph_0}\) into both \(2^{\aleph_0} \cdot 2^{\aleph_0}\) and \(2^{\aleph_0} + 2^{\aleph_0}\), so by C-S-B it is enough to find injections from \(2^{\aleph_0} \cdot 2^{\aleph_0}\) and \(2^{\aleph_0} +
2^{\aleph_0}\) into \(2^{\aleph_0}\)). We can discuss both of these as a group.
The last tool that was needed in the proof of Erdos-Rado, but that we skipped over, was Lemma 2.34. Anthony can present this.
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3.1 Linear equations | Linear Algebra 2024 Notes
3.1 Linear equations
The simplest linear equation is an equation of the form \[ax=b,\] where \(x\) is an unknown number which we want to determine and \(a\) and \(b\) are known real numbers with \(a\neq 0\), for example
\(2x=7\). For this example we find the solution \(x=7/2\). Linear means that no powers or more complicated expressions of \(x\) occur, for instance the equations \[3 x^5-2x=3 \quad \text{and} \quad
x+\cos(x)=1\] are nonlinear.
But more interesting than the case of one unknown are equations where we have more than one unknown. Let us look at a couple of simple examples.
Example 3.1:
Consider \[3x-4y=3,\] where \(x\) and \(y\) are two unknown real numbers. In this case the equation is satisfied for all \(x,y\) such that \[y=\frac{3}{4}x-\frac{3}{4} ,\] so instead of determining a
single solution the equation defines a set of \(x,y\) which satisfy the equation. This set is a line in \(\mathbb{R}^2\).
Example 3.2:
We could add another equation and consider the solutions to two equations, for example
\[3x-4y=3 \quad \text{and}\quad 3x+y=1.\]
In this example we again find a single solution. Subtracting the second equation from the first gives
, hence
and then from the first equation
. Another way to look at the two equations is that they define two lines in
and the joint solution is the intersection of these two straight lines, as depicted in Figure
Example 3.3:
If we look instead at the slightly modified system of two equations
\[3x-4y=3 \quad \text{and} \quad -6x+8y=0 ,\]
then we find that these two equations have
solutions. To see this we multiply the first equation by
, and then the set of two equations becomes
\[-6x+8y=6\quad \text{and},\quad -6x+8y=0 ,\]
so the two equations contradict each other and the system has no solutions. Geometrically speaking this means that the straight lines defined by the two equations have no intersection, that is they
are parallel, as depicted in Figure
So above we have found examples of systems of linear equations which have exactly one solution, many solutions, and no solutions at all. We will see in the following that these are all the possible
outcomes which can occur in general. So far we have talked about linear equations but haven’t really defined them in general, so we now do so below.
Definition 3.4: (Linear equation)
A linear equation in \(n\) variables \(x_1,x_2,\cdots ,x_n\) is an equation of the form \[a_1 x_1+a_2x_2+\cdots +a_nx_n=b\] where \(a_1,a_2,\cdots ,a_n\) and \(b\) are given numbers. These are known
as the coefficients of the equation.
In the rest of this chapter, the numbers \(a_1,a_2,\cdots ,a_n\) and \(b\) will be real numbers, and the solutions we are searching for will also be real numbers; however there may be other settings
where the coefficients and/or solutions are taken from different sets. For example we could allow our coefficents and solutions to be complex numbers rather than real numbers, and then by replacing
every instance of \(\mathbb{R}\) with \(\mathbb{C}\) the corresponding results would follow in the same way.
In this course we will often be interested in systems of linear equations.
Definition 3.5: (System of linear equations)
A system of \(m\) linear equations in \(n\) unknowns \(x_1,x_2,\cdots ,x_n\) is a collection of \(m\) linear equations of the form \[\begin{aligned} a_{11}x_1+a_{12}x_2+\cdots +a_{1n}x_n&= b_1\\ a_
{21}x_1+a_{22}x_2+\cdots +a_{2n}x_n&= b_2\\ \vdots\qquad\vdots\qquad &= \vdots\\ a_{m1}x_1+a_{m2}x_2+\cdots +a_{mn}x_n&= b_m \end{aligned}\] where the coefficients \(a_{ij}\) and \(b_j\) are given
When we ask for a solution \(x_1,x_2,\cdots, x_n\) to a system of linear equations, then we ask for a set of numbers \(x_1,x_2,\cdots ,x_n\) which satisfy all \(m\) equations simultaneously.
One often looks at the set of coefficients \(a_{ij}\) defining a system of linear equations as an independent entity in its own right.
Definition 3.6: (Matrix)
For \(m,n\in \mathbb{N}\), a \(m\times n\) matrix \(A\) (an “\(m\) by \(n\)” matrix) is a rectangular array of numbers \(a_{ij}\in \mathbb{R}\), \(i=1,2,\ldots,m\) and \(j=1,\ldots, n\) of the form \
[\label{mf} A=\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots &\vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}\] The
numbers \(a_{ij}\) are called the elements of the matrix \(A\), and we often write \(A=(a_{ij})\) to denote the matrix \(A\) with elements \(a_{ij}\). The set of all \(m\times n\) matrices with real
elements will be denoted by \[M_{m,n}(\mathbb{R}) ,\] and if \(n=m\) we will write \[M_n(\mathbb{R}) .\]
One can similarly define matrices with elements in other sets, e.g,. \(M_{m,n}(\mathbb{C})\) is the set of matrices with complex elements.
Note: The plural of matrix is matrices.
Example 3.7:
An example of a \(3\times 2\) matrix is \[\begin{pmatrix}1 & 3\\ -1& 0 \\ 2& 2\end{pmatrix}.\]
An \(m\times n\) matrix has \(m\) rows and \(n\) columns. The \(i\)th row of \(A=(a_{ij})\) is \[\begin{pmatrix} a_{i1} & a_{i2} & \cdots & a_{in} \end{pmatrix}\] and is naturally identified as a row
vector \((a_{i1},a_{i2}, \cdots, a_{in})\in\mathbb{R}^n\) with \(n\) components. The \(j\)th column of \(A\) is \[\begin{pmatrix} {a_{1j}} \\ {a_{2j}} \\ {\vdots} \\ {a_{mj}} \end{pmatrix} ,\] which
is a column vector in \(\mathbb{R}^m\) with \(m\) components.
Example 3.8:
For the matrix in Example
the first and second column vectors are
\[\begin{pmatrix}1\\ -1\\2\end{pmatrix}\text{and}\quad \begin{pmatrix}3\\0\\2\end{pmatrix},\]
respectively, and the first, second and third row vectors are
\[\begin{pmatrix}1 & 3\end{pmatrix}, \quad \begin{pmatrix}-1 & 0\end{pmatrix}, \quad \text{and} \quad \begin{pmatrix}2 & 2\end{pmatrix}.\]
There is one somewhat unpleasant notational subtlety here. Take, say a vector \((3,4)\in \mathbb{R}^2\). This vector can be written as a matrix either as a \(1\times 2\) matrix \(( 3 \;4)\), with
just one row or a \(2\times 1\) matrix \(\begin{pmatrix}3\\ 4\end{pmatrix}\), with just one column. To avoid confusion, we need a convention whether we are going to identify vectors with row or
column matrices, which will later lead us to the general concept of the transpose of a matrix.
The standard convention is to identify a vector \(x=(x_1,\ldots,x_n)\in \mathbb{R}^n\) with a column-matrix (or column-vector) \[x=(x_1,\ldots,x_n) =\begin{pmatrix}x_1 \\ \vdots \\ x_n\end{pmatrix},
\] but bear in mind that the same quantity can also be represented by a row-matrix (or row-vector) \[x^t = (x_1\;\ldots\;x_n).\] To distinguish, with the boldface notation, between row and
column-matrices, representing a single vector, we will use the superscript \({}^t\) (to be read “transpose”, this will be discussed further later in the chapter) for row-matrices. The difference
between the latter two formulae is that we do not use comas to separate elements of row-matrices.
When dealing with matrices it will often be useful to write them in terms of their rows or in terms of their columns. That is, if the rows of \(A\) are \(r_1^t, r_2^t, \ldots, r_m^t\) (for now, think
of the superscript \(t\) as notation so that we remember they are rows) we may write \[A = \begin{pmatrix} \cdots & r_1^t & \cdots \\ \cdots & r_2^t & \cdots \\ & \vdots & \\ \cdots & r_m^t & \cdots
\end{pmatrix} \text{ or just } A = \begin{pmatrix} r_1^t \\ r_2^t \\ \vdots \\ r_m^t \end{pmatrix},\] and if the columns of \(A\) are \(c_1, \ldots, c_n\) then we may write \[A = \begin{pmatrix} \
vdots & \vdots & & \vdots \\ c_1 & c_2 & \cdots & c_n \\ \vdots & \vdots && \vdots \end{pmatrix} \text{ or just } A = \begin{pmatrix} c_1 & c_2 & \cdots & c_n \end{pmatrix}.\]
In Definition 3.5, the rows of the matrix of coefficients are combined with the \(n\) unknowns to produce \(m\) numbers \(b_i\), we will take these formulas and turn them into a definition for the
action of \(m\times n\) matrices on vectors with \(n\) components:
Definition 3.9:
Let \(A=(a_{ij})\) be an \(m\times n\) matrix and \(x\in \mathbb{R}^n\) with components \(x=(x_1, x_2,\cdots, x_n)\), then the action of \(A\) on \(x\) is defined by \[\label{ax} Ax:=\begin{pmatrix}
a_{11} x_1 + a_{12} x_2+\cdots +a_{1n} x_n \\ a_{21} x_1 + a_{22} x_2+\cdots +a_{2n} x_n\\ \vdots \\ a_{m1} x_1 + a_{m2} x_2+\cdots +a_{mn} x_n\end{pmatrix}%= \bpm \ba^t_1\cdot\bx \\ \ba_2^t \cdot \
bx \\ \vdots \\ \ba^t_m \cdot\bx \epm \in \mathbb{R}^m.\]
Note that \(Ax\) is a vector in \(\mathbb{R}^m\) and if we write \(y=Ax\) then the components of \(y\) are given by \[$$y_i=\sum_{j=1}^n a_{ijx_j} \tag{3.1}$$\] which is the dot-product between \(x\)
and the \(i\)th row vector of \(A\). The action of \(A\) on elements of \(\mathbb{R}^n\) is a map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), i.e., \[A:\mathbb{R}^n\to\mathbb{R}^m .\]
Using the notation of matrices and their action on vectors, a system of linear equations of the form in Definition 3.5 can now be rewritten as \[$$Ax=b. \tag{3.2}$$\] So using matrices allows us to
write a system of linear equations in a much more compact way.
Another way of looking at the action of a matrix on a vector is as follows: Let \(a_1, a_2, \cdots, a_n\in\mathbb{R}^m\) be the column vectors of \(A\), then \[$$Ax=x_1a_1+x_2a_2+\cdots +x_na_n. \tag
{3.3}$$\] So \(Ax\) is a linear combination of the column vectors of \(A\) with coefficients given by the components of \(x\). This relation follows directly from (3.1). Solving \(Ax=b\) means that
we want to find coefficients \((x_1,\ldots,x_n)\) so that \(b\) may written as a linear combination of the column-vectors of the matrix \(A\). Such a linear combination may or may not exist, and if
it exists may or may not be unique.
Exercise 3.10:
Which matrix equation is equivalent to the system \(3x+5y-z=0, y-2x=5, x+5y+2z=9\)?
The map \(A:\mathbb{R}^n \to \mathbb{R}^m\) has the important property that it respects addition (meaning that we can add two vectors and then apply the map or apply the map separately and then add
the results and get the same outcome) and scalar multiplication (similarly), as demonstrated in the following theorem.
Theorem 3.11:
Let \(A\) be an \(m\times n\) matrix, then the map defined in Definition 3.9 satisfies the two properties
• \(A(x+y)=Ax+Ay\) for all \(x,y\in \mathbb{R}^n\),
• \(A(\lambda x)=\lambda Ax\) for all \(x\in\mathbb{R}^n\) and \(\lambda\in\mathbb{R}\).
This is most easily shown using (3.1). Let us denote the components of the vector \(A(x+y)\) by \(z_i\), \(i=1,2,\cdots, m\), i.e., \(z=A(x+y)\) with \(z=(z_1,z_2,\cdots, z_m)\). Then by (3.1) \[z_i=
\sum_{j=1}^n a_{ij}(x_j+y_j)=\sum_{j=1}^n a_{ij}x_j+\sum_{j=1}^na_{ij}y_j ,\] and on the right hand side we have the sum of the \(i\)th components of \(Ax\) and \(Ay\), again by (3.1). The second
assertion \(A(\lambda x)=\lambda Ax\) follows again directly from (3.1) and is left as a simple exercise.
Before we start solving systems of linear equations let us study matrices in some more detail.
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What is Ampere (A)? Unit of Electrical Current - Definition
Amps “A”: Definition, Formula, Measurement, Conversion and Calculation
What is Ampere?
An Ampere is the unit of electric current. It is named after the French physicist André-Marie Ampère (who is considered the father of electromagnetism) and used in physics and electrical and
electronics engineering as a base unit in SI (International System) to measure the electric current.
The unit of Ampere (short form as Amp) denoted by the symbol of “A” is used to measure the flow of electrons between two points triggered by the electric pressure known as voltage. An ampere is
defined as:
In a conductor, If one Coulomb of charge (C) is flowing through a point in one second (s), the amount of flowing current past that point is one Ampere (A).
In short words,
One ampere is one coulomb per second.
Amp = Coulomb ÷ Second
mathematically, it forms the equation as follow:
I = Q ÷ t
• I = Current in Amperes
• Q = Charge in Coulombs
• t = Time in seconds
Relates Posts:
An ampere is also expressed as flow of electron charge (6.25 x 10^18 electrons) by one volt of potential difference between two points which allows it to dissipate one Watt of power during the
process between these two points.
I = P ÷ V
A = W ÷ V
Amp = Watt ÷ Volt
Good to know: electron charge “e” = 1.60217662 × 10-19 coulombs.
According to Ohm’s law, if the value of potential difference (i.e. voltage in volts) across a one ohm (Ω) resistor is one volt, the value of flowing current in amperes (A) due to the voltage in that
resistor will be one Ampere:
I = V ÷ R
Amps = Volt ÷ Ohms
• I = Current in Amperes
• V = Voltage or P.D in Volts
• R = Resistance in Ohms (Ω)
Ampere Equations used for Conversions to the Related Quantities
Amps from Watts and Volts
• A = W ÷ V
• Ampere = Watts ÷ Volts … (I = P ÷ V)
Amps from Coulombs and time
• I = Q ÷ t
• Amps = Coulombs ÷ time in seconds
Amps from Volts and Resistance
• I = V ÷ R … (Ohm’s law)
• Amp = Volts ÷ Ohms (Ω)
How to Measure Ampere?
The tool which is used to measure current in amperes is known as amperemeter or simply ammeter. In both analog and digital multimeters, there is an amps (A) mode for measuring AC and DC currents in
To measure the flowing electric current in an electrical element (such as resistor, capacitor, inductor, diode etc.), simply put the two leads of the multimeter (in series) and the display will show
the exact value of current in amperes. You may follow the step by step guide posted in the previous article as “How to Measure Current using Digital and Analog Multimeter?”.
How to Calculate Amps?
Based on the above given formula and equations for current in amps (for different scenarios), we may calculate the value of electric current in amperes as follows.
Example 1:
If the applied voltage across a 4Ω resistor is 12V, find the value of current in Amps flowing in the resistor.
According to Ohm’s law:
I = V ÷ R
Putting the values
I = 12V ÷ 4Ω
I = 3 Amps
Example 2:
If the value of supply voltage across a 30W led bulb is 12V, Calculate the current in Amps in the light bulb.
We know that
I = P ÷ V
Putting the values:
I = 30W ÷ 12V
I = 2.5 Amps
Example 3:
Determine the value of required current in amps to glow a light bulb having 500W and 5Ω resistance .
The derived equation from P = V x I where putting V = I x R which becomes P = I^2 R.
I = √(P ÷ R)
Putting the values:
I = √(500W ÷ 5Ω)
I = √(100)
I = 10 amps.
Example 4:
What is the value of electric current in amps if 10 coulombs of charge is flowing through a point in a conducting material in 5 seconds.
We know that I = Q ÷ t. Now put the values.
I = 10C ÷ 5 seconds
I = 2 Amps
Amp to kVA & kVA to Amp Conversion
You may use the online tools for Ampere to kVA calculator and kVA to Amperes calculator.
Related Posts:
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Tourney - Multi-Pool Tournaments
So I've been playing with the idea of variable-entry tournaments, or multi-pool tournaments, as I'm calling it.
It's a way to allow each person in a group buy into a tournament for the stakes they feel like playing, as opposed to forcing everyone into a single stakes.
The mechanics are the same as for a hand with an all-in and a side pot:
• The lowest amount that everyone covers becomes the main prize pool.
• The excess above that becomes a side prize pool, and only those who bought in at the higher amount are eligible for the second pool.
• Tournament plays normally, all with the same starting stack, and with just one finish order.
An example: ten people want to play a tournament. Five of them want to do a $20 buy-in, but five want to do a $50 buy-in. So you let them all buy in as they want:
Everyone has bought in for $20 or more, so that establishes the main pool:
Now, the remaining $150 goes to a second pool - a side pool - consisting of five people who entered with an extra $30 beyond the $20:
So ten players vie for the $200 pool, and five of them vie for the $150 side pool.
Tournament plays normally, all with the same chip stack, and with just one finish order.
Some interesting dynamics emerge - Adam, in both pools, might be wary of losing all his chips to Doug, who's only in the main pool... then again, Adam would love to pick up all of Doug's chips,
because they help Adam relative to the other players in the side pool. The effects balance each other out.
Later, though, if Charlie, Ed, and Iggy get knocked out, then Adam and Bob are the only players left in the $150 side pool. There may still be lots of players in the main pool... but Adam and Bob
might start playing like they're "on the bubble." Players should be aware of this, and what it might mean for their play... but it's still fair, because everyone has the same knowledge.
A more complex example - let's say the tourney ends, they decide to do it again, and this time, Ed and Bob have a little more gambool:
Everyone is in the $20 pool is playing for $200; the same five people as before are in the side pool of $150, but Ed and Bob are in third side pool of $100.
They play one last game; a lot of players turn out their pockets to have one more go:
That one is pretty crazy, with seven side pools! But the spreadsheet does the work.
Also, there are only really six side pools - Bob's big $200 entry is unmatched, and he gets $125 back. Instant win.
Settling the payouts seems confusing, but if I replace player number with finish order:
And then tell the spreadsheet to sort by finish order:
If you look down a column, the highest person with money in that pool wins it.
It's easy to see that Gary sweeps all the pools he's in - he wins $100, $90, $35, and $90.
Adam takes in $50.
Iggy gets $15
And Bob gets $40 - and his $125 back. It's as if he only bought in for $75, which was the biggest anyone matched.
Jim finished ahead of Adam and Iggy, but didn't have money in side pools against them, so they won money and Jim didn't.
And when Gary and Ed were heads-up, Adam, Iggy, and Bob could be paid out.
In fact, when Adam got knocked out, Gary won $90 right then, and there were only three pools left 'in play.'
If everyone buys in for a different amount, it can get crazy and almost require a spreadsheet... but when there are only two levels, it's really quite easy - and it doesn't matter if there are eight
people buying in for $100 and two for $20, or eight buying in for $20 and two buying in for $100 - it allows everyone to play the right stakes for themselves.
Thoughts? Love it? Hate it?
Want the spreadsheet?
Feb 7, 2015
Reaction score
It's an interesting idea, and would work great for a winner takes all structure.
For a scaled payout percentage (I don't think it's fair that Ed doesn't win anything even though he finished 2nd out of 10, just because Gary bought in for more) it could be a little harder to
implement. Not impossible, but just harder.
I agree - the payouts should be structured when there's more than a few people in the pool, but I didn't want to over-complicate the concept in the first presentation. It can be hairy enough, as it
I'd advocate for a simple set of "house rules" for how payouts are structured, so it doesn't have to be argued about for each pool - you don't know in advance how many people will be in any pool.
Whatever the rules, you'd almost certainly want some sort of payout structure in the 10-player pool.
Another things that's pretty easy to handle: rebuys. If someone's knocked out and buys in again with a fresh stack, just enter them as a new player. If Hank gets knocked out early and rebuys, he
comes back as Born-Again-Hank (or Hank2, or whatever). Put Born-Again-Hank in as a new player in the spreadsheet, and he rest handles itself, even if he buys in for a different amount (putting him in
a different set of pools.) Same goes for late entries.
Add-ons, however, may be messy. Need to think that through carefully.
Apr 25, 2013
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I've played in tournaments with a stated buy-in and an optional side pool, but never one with more than one side pool. This structure doesn't preclude last-longer bets.
Payout structures were usually percentages based on the number of players in that particular pool. For example, if the total number of players is 20, the payouts might be 40/30/20/10. If there are 8
players in the side pool, the percentages would be the same as for that group's 8-player tournaments, perhaps 50/30/20. Last-longer bets are negotiated among the participants.
I don't think rebuys should be handled as new players, which is what you would do with re-entries. Rebuys would increase the size of the main pool only. The payout percentages would stay the same.
If I were managing a tournament like this, I would only have one optional side pool with the payout percentages being based on the number of players participating in the side pool. Last-longer bets
would be negotiated by and settled among the players themselves and would not be part of the official tournament structure.
Simplicity and transparency are good things.
I don't think rebuys should be handled as new players, which is what you would do with re-entries. Rebuys would increase the size of the main pool only.
I don't think that's good. Consider the first example I put up - where half bought in for $20, and half bought in for $50. If someone who bought in for $50 (putting $20 in the mail pool and $30 in
the side pool) goes bust and rebuys for $50, why should $50 go into the main pool?
Unless you're saying they should only be able to rebuy for $20, which would go into the main pool... but then, they should not be eligible to win the side pool any more.
To me, the only reasonable way to handle it is this: if they rebuy for $50, put $20 more in the main point, and $30 more in the side, and they're eligible for both.
Of course, that has the same effect as just entering their second buy-in as a new person. Exact same outcome.
The difference is that I'm allowing for them to choose to rebuy for a different amount than their original buy-in. So if someone was in for $50 and goes bust, but they only want to rebuy for $20,
they can - but they're no longer eligible to win the side pool. On the other hand, if someone was in for $20 and goes bust, and they want to rebuy for $50, they can get in on both pools; that would
be like a rebuy for the main pool, but a late entry for the side pool.
Handling all rebuys as if they were a new entry just makes the math and spreadsheet simple and transparent.
I think add-ons are OK, as long as you express them as a fraction of the original buyin, and not as a fixed amount.
In other words, this is a problem, when half buy in for $20 and half buy in for $50:
Starting stacks are T30K, with an optional T15K add-on after one hour for $10.
This may be fine for everyone who bought in for $20, but causes problem for those at the $50 buy-in level. The bucks in the main pool buys an awful lot of chips that are competing in the side pool.
This, on the other hand, will work out fine:
Starting stacks are T30K, with an optional T15K add-on after one hour for 1/2 the original buy-in.
So people who buy in for $50 have to pay $25 for their add-on.
If someone who already re-bought is adding on, their add-on amount is 1/2 their latest buy-in (not their original buy-in, that player is out.)
Apr 25, 2013
Reaction score
Unless you're saying they should only be able to rebuy for $20, which would go into the main pool... but then, they should not be eligible to win the side pool any more.
To me, the only reasonable way to handle it is this: if they rebuy for $50, put $20 more in the main point, and $30 more in the side, and they're eligible for both.
Good point. I didn't think of that, but then again I have almost no experience with rebuy tournaments.
The more I think about multiple prize pools, the more my head feels like it's going to explode.
Royal Flush
Tourney Director
Oct 29, 2014
Reaction score
I have ran (and played in) many multi-tiered entry tournaments (both backgammon and poker). All were done as abby outlined; prize pools were established for each entry fee group, and payouts
structured based on the number of entries for each group. Re-buys were handled similarly to how Mental noted; whatever a player's initial buy-in amount was also that player's re-buy cost, and
appropriately allocated to each group's prize pool. It's all actually quite simple to implement and run.
However, treating re-buys as new entries skews the actual total number of entries (players) in each pool/group, thus changing the payout structure. I wouldn't do it that way - it cause more problems
than it solves.
The total number of players (and payouts) for each group is always made public, although sometimes the actual participants of each group (beyond the base group) is kept confidential until the event
concludes -- making for a completely different strategy set that when everybody knows who's who (for starters, one never really knows when on the bubble for a 2nd- or 3rd-tier group finish). Of
course, the organizer/director will always know everything, but typically they don't play when the group participants are kept secret from the field. Adding to this strategy sub-set is the fact that
lying is an integral part of poker -- so some players may reveal that they are in a group or not in a group, essentially affecting how other may play (but they may be lying!). I personally prefer the
open structure, but the tournaments with hidden groups are always great fun and a mystery until the very end.
We also routinely use bounty chips in a similar fashion -- a 'base' bounty chip is distributed to all players as part of their buy-in cost, and an optional 'super-bounty' chip can be purchased by
players who wish to increase their bounty wagers. When a super-bounty player is knocked out by a non-super-bounty player, the super-bounty player forfeits just the base bounty chip but retains their
super-bounty chip for redemption. Participation in the optional super-bounty pool usually ranges between 60%-80%, occasionally 100%
Feb 7, 2015
Reaction score
I'm not sure about the rebuys. For this structure I'd stick with freezeouts.
If you only allow rebuys for the buy-in amount, I wouldn't add a new entry, just add the rebuy amount to the original buy-in, and redistribute the rebuy over the different pools, so Adam is now in
tier 1 for $20, in tier 2 for $20, in tier 3 for $10, in tier 4 for $30, and in tier 5 for $20.
If you're going to allow variable buy-ins that might give players to angle shoot for free money.
Let's say that after Bob goes busto and doesn't rebuy, Iggy decides to rebuy for $200. Now he's guaranteed the $125 that was dead money and would've been paid back to Bob.
You'd need extra rules to prevent that, like you can only rebuy up to your original buy-in.
Players also need to know that, when rebuying for less than the original buy-in, they forfeit the difference.
Though I agree that adding a new entry for a variable rebuy is probably the best way of handling this, I don't think the rebuy should change the payout structure.
Say that you have 10 players at the table, and you're paying out the top 3 positions in that case. Now you've got 5 rebuys, which would make 15 players, which would mean you'd pay out the top 4
positions. I'd still only pay out the top 3.
Late entries or reentries would count as extra players, so that would affect the payout structure.
Add-ons would complicate things to no end. The player hasn't gone bust, so you can't use the "new entry" for this. I also don't think that adding on should entitle you to the prizepools created by
players who bought in for a higher amount. Variable add-ons are even worse, how many chips do you get when you've bought in for $200 (like Bob), but only decide to add-on for $20. How is that money
distributed into the prizepools? Even if you allow players to add-on only for their buy-in amount, doesn't that give Bob an unfair advantage since 62.5% of his buy-in is uncontested?
Marvelous points, marvelous.
I'd freeze the payout structure at buy-in; re-buys boost the pool, but don't change the structure.
Gotta think through the angle-shooting - not something I usually focus on.
In my mind, Bob's $125 unmatched are returned immediately and not in a pool, but the premise is still there if you return it - Bob and Charlie are in the top $40 pool. Bob busts, so Charlie should
win - but then Charlies plays over-confidently and busts, and then Iggy busts and re-buys for $75. That puts $20 into the $40 pool, making it $60, but also means Iggy's the only person left in the
pool, and will get the $60 if nobody else rebuys into it.
It's arguably fair if everyone knows about this and the implications ahead of time. After all, Bob or Charlie are free to rebuy, too.
This also raises the question... when time comes for add-ons, do you want to allow buy-ups? By that, I mean allowing someone to add money to buy into a higher tourney pool (you get no extra chips
when buying up.) This can be cool, or may be an angle-shoot... suppose Bob, Charlie, and Iggy are all still in the game when the time comes - but Iggy has been running well, and has a huge stack. He
can then decide to pony up $20 extra dollars to play in the top bracket against only two players, each of whom has a much smaller stack.
I think it's problematic. I suspected all of these angles are gone if you make it a freeze-out - or alternately, if rebuys/add-ons stay within your origianl pools only.
Royal Flush
Tourney Director
Oct 29, 2014
Reaction score
As someone who has actually done this, I think you're over-thinking everything, and seem intent on making it a lot harder than it actually is. Multi-tier tournaments are a no-brainer if you follow
the guidelines in my previous post, and there are no angle-shoot opportunities (beyond those in any tournament). Don't need a fancy spreadsheet to run one, either (and I love spreadsheets.....) -
although it can make the bookkeeping easier.
I think you're over-thinking everything, and seem intent on making it a lot harder than it actually is.
I'm not trying to make it harder. I'm enjoying thinking it through, including the possible variants and potential gotchas.
I'd definitely use a spreadsheet - it makes it easier to tell a group of people, "everyone can buy in for WHATEVER they want." Nobody will feel guilty about causing extra math if the additional
effort is nil, so it'll lead to a more "interesting" set of pools. Which, given my crowd, will also make it more fun.
I also have an ulterior motive for thinking through various options... a seekrit ulterior motive...
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Blockchain for Test Engineers: Hashing
Photo by Pan Yunbo on Unsplash.
This is the first blog post in the “Blockchain for Test Engineers” series.
When you start to study blockchain, the odds are pretty high that you will hear about hashing and hash functions.
But what is hashing? Why is it a critical concept in cryptography? Which hashing algorithms are used for different blockchains? And even more interesting - how can we test a cryptographic hash
function (and should we do it at all)?
What is hashing?
Hashing can be imagined as a “box” with one input and one output. You can provide any data as an input and get a result of the fixed length.
In the case of the SHA-256 hash function - the result will always be a 256-bit sequence. The most exciting thing in hashing is - that even the minimal change to the input changes the output -
As an example, let’s get the SHA-256 hash of the text “Test Engineering Notes”.
The result will be - “dc3101f57c983aa68499306de0e60cb7266e3b1ae2235a84aa0d727f0f07b18f”.
But “Test Engineering note” will have completely different hash: “16d164f46c6a29518420d0cdda6394969692cc82145b46eb89fe32af4a8b996e”.
Properties of hash functions
The two main properties of the hash function are ease of computation and determinism (the same input data should always result in the same hash).
Besides that, a good hashing function should withstand cryptographic attacks.
So it should have the following properties:
• Pre-image resistance. The hash function should be a “one-way”: Given a hash value h, it should be difficult to find any message m such that h = hash(m).
• Second pre-image resistance. Given an input m1, it should be difficult to find a different input m2 such that hash(m1) = hash(m2).
• Collision resistance. It should be difficult to find two different messages m1 and m2, such that hash(m1) = hash(m2). Such two messages are called a hash collision.
Why does hashing exist?
There are multiple applications of the hashing in the world of computers:
• With hashing, you can check the integrity of the data;
• Instead of storing user passwords as plain text in a database - it is better to keep its hashes;
• You can use hashing in the digital signatures;
• Datatypes, such as Hash Tables and Hash Maps;
• Blockchains :);
Different hash functions
The most famous are MD5, SHA-1, RIPEMD-160, BLAKE, Whirlpool, SHA-2, and SHA-3. E.g., Bitcoin (and its forks) uses SHA-256 cryptographic function, Ethereum - SHA-3 (Keccak), Cardano - BLAKE2b-224.
Those algorithms are based on some sequence of rounds with bit operations: shifts, AND, XOR operations, and others).
Lane Wagner’s blog post provides an excellent step-by-step explanation of the SHA-2 hash function. You can also try to hash something online.
It is a vast mathematical task to create new hash functions and prove their properties. National Institute of Standards and Technologyis reponsible for testing hash functions. The chosen one are
added to the family of secure hash algorithms and recommeded by NIST. The last known standard algorithm (Keccak) was chosen at the NIST hash functions competition. Hash functions were evaluated by
their performance, security, analysis, and diversity.
How to test hash functions?
One of the leading indicators of the quality of hash functions is the probability of getting hash collisions. So one test is to check collisions at the massive input data.
Usually, the distribution of hash values is uniform and is tested using the Chi-square test. The actual distribution of elements is compared with the expected (uniform) distribution. The ratio
within the confidence interval should be in the range of 0.95 - 1.05.
Also, there is an additional test for uniformity of distribution of hashes. It is based on strict avalanche criteria - when each input bit changes with a probability of 50% in the output sequence.
Will you test hashing function as a blockchain test engineer?
As always - it depends. Many blockchains use well-known standards and rarely implement hash functions from scratch.
Suppose you will be a part of the core engineering team responsible for developing the whole blockchain from scratch. In that case, you may need to verify that hashing algorithm works as expected
within the context of the system.
If you test only blockchain-based applications based on smart contracts - you don’t need to verify the hashing function in isolation. You should trust other security specialists and mathematicians
who tested these functions for you. But you definitely need to know how such functions are used in your system.
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Enumerating pattern matches in texts and trees
12 novembre 2018
Antoine Amarilli (Télécom ParisTech)
We study the data management task of extracting structured
information from unstructured documents, e.g., raw text or HTML pages.
We use the framework of document spanners, where the pattern to extract
is specified declaratively by the user (as a regular expression with
capture variables) and is translated to an automaton that then is
evaluated on the document to compute the pattern matches. We focus on
the last step of this pipeline: our goal is to efficiently find the
matches of the automaton on the document, even when there can be many of
them. We do this by proposing an enumeration algorithm, which first
preprocesses the automaton and document, and then enumerates all matches
with a small delay between consecutive matches. Unlike previous work,
our algorithm achieves the best possible bounds in the input document
(namely, linear preprocessing and constant delay), while remaining
tractable in the automaton. The guiding principle of the algorithm is to
compute a factorized representation of all matches as a product of the
automaton and document, and design efficient indexes based on the
structure of this representation. We also present our ongoing follow-up
work, e.g., how to extend our algorithm to the case of tree-shaped
documents by efficiently enumerating the matches of tree automata, how
to efficiently update the enumeration results when the input document
changes, and other open research directions.
|
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An optimized long short-term memory (LSTM)-based approach applied to early warning and forecasting of ponding in the urban drainage system
Articles | Volume 27, issue 10
© Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License.
An optimized long short-term memory (LSTM)-based approach applied to early warning and forecasting of ponding in the urban drainage system
In this study, we propose an optimized long short-term memory (LSTM)-based approach which is applied to early warning and forecasting of ponding in the urban drainage system. This approach can
quickly identify and locate ponding with relatively high accuracy. Based on the approach, a model is developed, which is constructed by two tandem processes and utilizes a multi-task learning
mechanism. The superiority of the developed model was demonstrated by comparing with two widely used neural networks (LSTM and convolutional neural networks). Then, the model was further revised with
the available monitoring data in the study area to achieve higher accuracy. We also discussed how the number of selected monitoring points influenced the performance of the corrected model. In this
study, over 15000 designed rainfall events were used for model training, covering various extreme weather conditions.
Received: 02 Sep 2022 – Discussion started: 04 Oct 2022 – Revised: 20 Apr 2023 – Accepted: 04 May 2023 – Published: 26 May 2023
The intensity and frequency of urban floods are growing as a result of the increased frequency of extreme weather, rapid urbanization, and climate change (Hossain Anni et al., 2020; Guo et al., 2021;
Huong and Pathirana, 2013). It is becoming increasingly clear that urban floods significantly impact city management and endanger the safety of peoples' lives and the stability of various property
types. The ability to reliably characterize and forecast urban floods and generate high-precision flood risk maps has become critical in flood mitigation and decision-making.
The most common approach to simulating urban floods is to develop a hydrodynamic model (i.e., storm-inundation simulation), which utilizes a collected topographic map, information on the pipe
network, historical rainfall data, monitoring data, and other information on the study area (Jamali et al., 2018; Aryal et al., 2020; Balstrøm and Crawford, 2018; Tian et al., 2019). However, a
realistic hydrodynamic model for continuous simulation requires vast data, such as comprehensive information on topography, infiltration conditions, and sewage system data (including exact locations,
depths, and diameters of sewage pipes). However, the above data are difficult to obtain, especially in metropolitan areas (Rahman et al., 2002; Kuczera et al., 2006). Furthermore, the calculation in
a storm-inundation simulation is sophisticated and often computationally intensive, which takes a long time to execute. The most detailed representation of the storm-inundation simulation is the
1D–2D model (Djordjević et al., 1999, 2005), which summarizes the dynamic interaction between the flow that enters the underground drainage network and the overloaded flow that spreads to the surface
flow network during high-intensity rainfall. Representatives of such a model include XPSWMM, TUFLOW, and MIKE FLOOD (Leandro and Martins, 2016; Teng et al., 2017; Zhang and Pan, 2014).
The lack of underlying information has hampered the continuous development of hydrodynamic models in urban flood forecasting. As a result, deep learning has emerged as another viable forecasting
tool. Deep learning is a particular machine-learning technique that leverages neural networks to learn nonlinear relationships from a dataset (Mudashiru et al., 2021; Sit et al., 2020; Shen, 2018;
Moy De Vitry et al., 2019). It can compensate for data scarcity by training on a large designed dataset. Unlike traditional hydrodynamic models, deep learning does not require any assumptions on the
physical processes behind it.
However, there are opportunities to further the application of deep learning in urban flood forecasting. First, the training dataset needs to be enriched to reflect the superiority of the approach.
Many studies in urban flood forecasting only use a small number of samples to develop the deep learning models. For example, Cai and Yu (2022) used 25 historical floods for forecasting. Abou Rjeily
et al. (2017) used only 10 rainfall events for training and verification, which was insufficient to reflect the characteristics of rainfall distribution. Second, due to the high cost of monitoring
equipment, researchers usually have to rely on unvalidated simulations produced from hydrodynamic models. For example, Chiang et al. (2010) used synthetic data from the SWMM model as the target
values to train the recurrent neural network (RNN) and then compared the predictions with simulation results to evaluate the model accuracy in estimating water levels at ungauged locations. Third,
some studies have focused on building more complex deep learning architectures to improve model performance. Examples include but are not limited to the automatic encoder (Bai et al., 2019), the
encoder–decoder (Kao et al., 2020), and customized layers based on long short-term memory (LSTM; Sit et al., 2020; Kratzert et al., 2019a, b). For example, an encoder–decoder LSTM has been proposed
for runoff forecasting up to 6 and 24h ahead (Xiang et al., 2020; Kao et al., 2020). Nevertheless, the urban flooding forecasting tasks with multiple time steps are mainly based on the precipitation
forecast hours in advance, which is not available in this paper. Because of the short duration of rainfall, the real-time data volume is not enough to support the hours-ahead prediction.
In this study, we propose an optimized LSTM-based approach for the early warning and forecasting of ponding in urban drainage system. This approach can quickly identify and locate ponding with
relatively high accuracy. The model is constructed by two tandem processes and introduces a multi-task learning mechanism. The evaluation results of the model were compared with those of two widely
used neural networks, i.e., LSTM and CNN (convolutional neural network). The model was further revised with monitoring data in the study area to improve the emulation performance. We also discussed
the influence of the number of monitoring points selected on the model performance. Over 15000 designed rainfall events were used for model training, covering various extreme weather conditions.
The rest of the paper is organized as follows: Sect. 2 introduces the methodology used to develop the LSTM-based modeling framework in addition to the experimental setup and application of the model.
Section 3 presents the results of the model. Section 4 presents the discussion, and Sect. 5 concludes this paper by drawing brief conclusions.
2.1LSTM-based model
Like a hydrodynamic model, which is generally composed of two processes, namely the runoff process and flow confluence process, the LSTM-based model proposed in this study is also constructed with
two stages. Figure 1 illustrates the model architecture from input (i.e., rain intensity) to output (i.e., ponding volume of each node).
The two processes are in tandem; the inputs of the flow confluence process are inherited and concatenated from the outputs of all nodes in the runoff process. However, during the training process,
the two processes are trained separately without mutual interference, as the inputs and outputs of both processes are produced from a hydrodynamic model.
2.1.1Runoff process
With a general understanding of a hydrodynamic model, the runoff process involves surface runoff and infiltration, while the most important influential factor is rainfall. As a mass rainfall curve
can reflect the characteristics of a specific rainfall process, it can be directly used as the input of a neural network. The output of the neural network (i.e., lateral inflows at each node)
reflects the hydraulic state in the runoff process.
Figure 2 illustrates the training, validation, and testing procedures in the runoff process. As shown in Fig. 2, a training set with two time series of data is fed into the neural network, thus
giving the rainfall intensity and lateral inflow at each node. At each epoch, four indicators are used to evaluate the consistency between the predicted lateral inflows and the simulation from the
hydrodynamic model. If the model converges, the network is further evaluated on the test set. Otherwise, the next training epoch is started.
2.1.2Flow confluence process
The flow confluence process is set up in the same manner as the simulation process of a hydrodynamic model (e.g., the SWMM model). If we compare the urban drainage system to a black box, then only
the lateral inflows at each node and outflows from the outlets enter and leave the system, respectively (Archetti et al., 2011). If a free-outflow condition is considered, namely the hydraulic state
behind the outlets has little influence on the interior of the system, then the inputs of the flow confluence process are only the lateral inflows at each node. Figure 3 illustrates the details of
the network architecture in the flow confluence process.
As illustrated in the pink block in Fig. 3, a Gaussian layer is added after the input layer in the flow confluence process during training. The Gaussian layer serves as a filter to compensate for the
inaccuracy of the prediction (by the hydrodynamic model) in the runoff process. The model is trained to minimize the differences between the predictions (from the neural network; i.e., the output
from the runoff process) and the simulations (from the hydrodynamic model). Then (as illustrated in the blue block in Fig. 3), a classification layer is added after the outputs of the LSTM module to
judge whether ponding occurs at the time step. Only when ponding occurs at the time step can the output of the LSTM module enter the “OUT_MODULE” to continue with the learning. Otherwise, the output
of the LSTM module at this time step is discarded. In this way, the interference of the time points without ponding on the ponding volume forecasting is eliminated to a great extent. The higher the
classification accuracy, the more accurate the prediction of ponding volume will be. Moreover, the multi-task learning has a hard parameter-sharing mechanism, which effectively alleviates the
overfitting of the model. The parameters in the “LSTM_MODULE” (including the parameters of the LSTM layers, batch normalization layers, activation functions, etc.) are shared by the
“CLASSIFICATION_MODULE” and “OUT_MODULE”.
2.2Error transmission
Figure 4 illustrates how the training error in the neural network propagates from the runoff process to the flow confluence process during training. Noise P ($P\sim N\left(\mathrm{0},{\mathrm{p}}^{\
mathrm{2}}\right)$) is added to the lateral inflows before feeding the data into the neural network in the flow confluence process in order to avoid the interference caused by the training error in
the runoff process and also to alleviate the overfitting of the neural network. The magnitude of noise P can be determined as follows:
1. The mean squared error (MSE) is used to characterize the training error in the runoff process, where the error at node K can be computed by the following:
$\begin{array}{}\text{(1)}& {a}_{k}=\frac{\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}^{S}{\left({\stackrel{\mathrm{^}}{X}}_{ij}-{X}_{ij}\right)}^{\mathrm{2}}}{T\cdot S},\end{array}$
where T represents the duration of event j (in min), S represents the number of events in the training data, ${\stackrel{\mathrm{^}}{X}}_{ij}$ represents the simulated lateral inflow at node K at
the ith time step in the jth rainfall event (in Ls^−1), and X[ij] represents the output of the runoff process at node K at the ith time step in the jth sample event (in Ls^−1).
2. Then the average mean squared error in all nodes is computed by the following:
$\begin{array}{}\text{(2)}& \mathrm{amse}=\frac{\sum _{k=\mathrm{1}}^{N}{a}_{k}^{\mathrm{2}}}{N},\end{array}$
where N represents the number of nodes.
3. Then amse is converted into the noise percentage ε, with the mean value of the predicted lateral inflows at all nodes in the training set, by the following:
$\begin{array}{}\text{(3)}& \begin{array}{rl}\mathit{\epsilon }& =\sqrt{\frac{{P}_{\mathrm{N}}}{{P}_{\mathrm{S}}}}=\sqrt{\frac{\sum _{k=\mathrm{1}}^{N}\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}
^{S}{\left({\stackrel{\mathrm{^}}{X}}_{kij}-{X}_{kij}\right)}^{\mathrm{2}}}{\sum _{k=\mathrm{1}}^{N}\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}^{S}{\left({X}_{kij}\right)}^{\mathrm{2}}}}\\ & \le
\sqrt{\frac{\sum _{k=\mathrm{1}}^{N}\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}^{S}{\left({\stackrel{\mathrm{^}}{X}}_{kij}-{X}_{kij}\right)}^{\mathrm{2}}}{\frac{\mathrm{1}}{N\cdot T\cdot S}{\
left(\sum _{k=\mathrm{1}}^{N}\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}^{S}{X}_{kij}\right)}^{\mathrm{2}}}}\\ & =\frac{\sqrt{\mathrm{amse}}}{\frac{\mathrm{1}}{N\cdot T\cdot S\sum _{k=\mathrm
{1}}^{N}\sum _{i=\mathrm{1}}^{T}\sum _{j=\mathrm{1}}^{S}{X}_{kij}}},\end{array}\end{array}$
where P[S] represents signal power, and P[N] represents noise power.
4. Finally, noise P is added to the inputs (X) in the flow confluence process during the training process. In other words, a set of random numbers G is generated with the length of X, using the
pseudorandom number generator, where G obeys a normal distribution ($G\sim N\left(\mathrm{0},\mathrm{1}\right)$); i.e., $P=p\cdot G$, where p is computed by the following:
$\begin{array}{}\text{(4)}& p=\mathit{\epsilon }\cdot \sqrt{\frac{\mathrm{1}}{T}\sum _{i=\mathrm{1}}^{T}{\left({X}_{i}\right)}^{\mathrm{2}}}.\end{array}$
2.3Model correction system
The LSTM-based model is built based on the simulation results of a relatively accurate hydrodynamic model. However, the differences between the simulation from the hydrodynamic model at the
monitoring points and the obtained monitoring data always exist during the operation of the pipe network, which leads to a discrepancy between the predicted results from the proposed LSTM-based model
and the actual situation. Thus, it is necessary to correct the model using the measured level and flow data at the monitoring points. Moreover, how to revise the model properly using the available
data is also one of the focuses of this study. Figure 5 describes the model correction process using the measured rainfall data, depths and flows at the monitoring points, and ponding data at any
node. Specifically, the LSTM-based model is corrected using the following two steps:
1. The runoff process is corrected with the measured rain, level, and flow data by referring to the transfer learning. Transfer learning is mainly used to transfer the knowledge of one domain
(source domain) to another domain (target domain), such that the target domain can achieve better learning effects (Pan and Yang, 2010).
2. The flow confluence process is corrected using the updated lateral inflows of all concatenated nodes and the measured ponding volume.
Figure 6 shows the schematic of model CR (correction of the runoff process). It migrates the network structure of the runoff process from rain data (X) to lateral inflows (Y) to the input–output
connection between X and monitoring data (G). Then, multiple fully connected layers are added after the output layer of Y. Model CR is designed to update the runoff process in the primary LSTM-based
model. The correction has two steps, namely training and updating. First, model CR is trained based on a pretrained mapping from X to Y (as shown in Sect. 2.1.1) with constructed rain data, a
simulated level, and flow data. Then, it is updated on pairs of measured rain data, monitored water depths, and flows.
2.4Case study
2.4.1Study area
The LSTM-based model trains nodes in the pipe network one by one. Namely N submodels with the same architecture are generated, where N is the number of nodes in the system. In both the runoff process
and flow confluence process, these submodels are trained separately. Due to this structural characteristic, the size of the case area does not limit the model's performance. Regarding the model
structure, the output of the runoff process is the lateral inflow at a single node. Likewise, the output of the flow confluence process is the ponding volume at a single node. Regardless of the size
of the pipe network, the output of the model is at each node. However, a large-scale pipe network with lots of nodes will significantly increase the time spent training the model and also require
extra processing power.
To verify the feasibility of the modeling framework above, a small-scale case area JD, a residential district in S city, is selected as the study area. Figure 7a shows the elevation map of the study
area. There are 32 residential buildings in the district, with a total area of 6.128hm^2. The study area is separated from the municipal roads by walls, with three entrances on the community's
northern, eastern, and western sides. Rain pipes in the study area are circular pipes with 200, 300, 400, 500, or 600mm in diameter (mostly 300mm). The total length of this pipe network is 5.5km.
The network contains 336 nodes and 340 pipes and is connected to the municipal pipe networks through four outlets, as denoted by the green triangle in Fig. 7b. There are 15 level gauges and
3 flowmeters in the current pipe network. The layout of monitoring points is also shown in Fig. 7b.
2.4.2Rainfall data
The rainstorm intensity for S city is designed using Eq. (5), which is obtained according to a universal design storm pattern proposed by Keifer and Chu (1957). The storm pattern is broadly used both
at home and abroad. The generated storms are usually extreme enough to reflect the state of the pipe networks under the most unfavorable conditions (Skougaard Kaspersen et al., 2017).
$\begin{array}{}\text{(5)}& q=\frac{\mathrm{167}A\left(\mathrm{1}+C\mathrm{log}P\right)}{\left(t+b{\right)}^{n}}=\frac{\mathrm{1600}\left(\mathrm{1}+\mathrm{0.846}\mathrm{log}P\right)}{\left(t+\
where q is the rainstorm intensity (in Ls^−1hm^−2), P is the reappearing rainfall period (in a(a)), t is the duration of rainfall (in min), and A, C, b, and n are parameters of the rainstorm
intensity design formula.
The rainstorm intensity before or after the peak is determined using Eq. (6).
where t[b] and t[a] are the times before and after the peak (in min), respectively, and r is the rainfall peak coefficient.
Then single-peak rainfall scenarios were constructed unevenly by using different rainfall reappearing periods (P) ranging from 0.5 to 100a, a peak coefficient (r) ranging from 0.1 to 0.9, and a
duration (T) ranging from 60 to 360min.
In addition to single-peak rainfall scenarios, we also considered bimodal rainfall scenarios. According to the historical bimodal rainfall data in S city, the rainfall peaks corresponding to the
bimodal design storm pattern with the duration from 60 to 360min could be computed by Pilgrim and Cordery (1975). Pilgrim and Cordery (1975) present a method to count the historical rainfall data
and deduce the rainstorm pattern from it.
Table 1 shows the bimodal design storm patterns with 60 and 120min duration time, respectively, where $P/{P}_{\mathrm{max}}$ represents the distribution of rainfall intensity over time (with a 5min
unit period). Then, double-peak rainfall scenarios were constructed according to Table 1 using reappearing periods ranging from 0.5 to 100a.
The produced single-/double-peak rainfall data were then added with Gaussian white noise (produced according to the procedures described in Sect. 2.2) to ensure that the obtained dataset contains
enough extreme conditions. Take the rainfall with a return period of 5a as an example. Figure 8 shows the effect of adding noise, where panel (1) shows the randomly generated Gaussian white noise
over the duration, panel (2) shows the distribution of the reordered white noise, and panel (3) magnifies the part circled in panel (2). Panels (4)–(6) show the design rainfalls after adding 30%,
50%, and 70% white noise, respectively. Specifically, we have limited the noises near the rainfall peak; i.e., only negative noises are allowed there.
In this study, the noise percentages went from 0% to 100% in increments of 10% to blur the characteristics of the design storm pattern and intensify the extreme conditions. The synthetic dataset
contained a total of 16960 rainfall events. The ratios of the training, validation, and test sets were 80%, 10%, and 10%, respectively.
In general, a small training set normally leads to a poor approximation effect. Thus, a convergence test was performed to evaluate the data requirement for the proposed LSTM-based model to obtain the
desired approximation effect. The model performances using different sizes of training data were compared, as shown in Fig. 9. When the data size was reduced to two-thirds of the origin volume, then
the model performance fell down to 90% of the original. Moreover, if the data size was halved, then less than 80% of the origin model performance remained.
2.4.3Simulated and measured data
A hydrodynamic model was established for the case pipe network. The simulation results (i.e., the lateral inflows and the volume of ponding at each node, in addition to the level and flow data at the
monitoring points) were obtained using the constructed rainfall events described in Sect. 2.4.2. In the simulation process, we considered a uniform rainfall distribution in space. A simplified
representation of the sewer system and a constant, uniform infiltration rate in the green area were also considered for runoff computation (Löwe et al., 2021). Meanwhile, we did not consider the
two-dimensional surface overflow.
Besides, the measured rain data and monitoring data (water depth and flow) of five historical rainfall events were used to verify the performance of the corrected model. The uncertainty in the
measurements was not considered (Huong and Pathirana, 2013). In this study, we considered the simulation results of the verified hydraulic model to be the ground truth.
Table 2 shows the measurements of the five historical rainfall events used in the process of model correction. Among the five events, three were used to correct model CR and the flow confluence
process, while the other two were used to evaluate the reliability of the approach.
2.4.4Model construction
The hyperparameters used in this paper were mainly determined by Hyperopt (Bergstra et al., 2013). Hyperopt is a Python library for hyperparameter optimization that adjusts parameters using Bayesian
Table 3 shows the hyperparameters in the learning process of the model setup and model correction obtained by Hyperopt.
Note that the asterisks^* refers to setting the hyperparameters of “CLASSIFICATION_MODULE” and “OUT_MODULE” in the flow confluence process to the same values. SGD is the stochastic gradient descent.
2.4.5Performance evaluation
The mean absolute error (MAE), mean squared error (MSE), correlation coefficient (CC), and Nash–Sutcliffe efficiency coefficient (NSE) are broadly used indicators to assess the performance of a
data-driven model. In this study, we used MAE and MSE to quantify the size of the errors; i.e., difference at each node between the prediction by the proposed LSTM-based model and simulation from the
hydrodynamic model. Moreover, NSE and CC were also used to evaluate the level of agreement at all nodes. Equations (7)–(10) list the formulas of these four indicators.
$\begin{array}{}\text{(7)}& \mathrm{MAE}=\frac{\mathrm{1}}{DT}\sum _{s=\mathrm{1}}^{D}\sum _{t=\mathrm{1}}^{T}\left|{Y}_{st}-{\stackrel{\mathrm{^}}{Y}}_{st}\right|\text{(8)}& \mathrm{MSE}=\frac{\
mathrm{1}}{DT}\sum _{s=\mathrm{1}}^{D}\sum _{t=\mathrm{1}}^{T}{\left({Y}_{st}-{\stackrel{\mathrm{^}}{Y}}_{st}\right)}^{\mathrm{2}}\text{(9)}& \mathrm{NSE}=\mathrm{1}-\frac{\sum _{t=\mathrm{1}}^{T}{\
left(\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{Y}_{st}-\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm{^}}{Y}}_{st}\right)}^{\mathrm{2}}}{\sum _{t=\mathrm{1}}^{T}{\left(\frac{\
mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm{^}}{Y}}_{st}-\frac{\mathrm{1}}{DT}\sum _{t=\mathrm{1}}^{T}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm{^}}{Y}}_{st}\right)}^{\mathrm{2}}}\text
{(10)}& \begin{array}{rl}& \mathrm{CC}=\\ & \frac{\sqrt{\sum _{t=\mathrm{1}}^{T}\left(\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{Y}_{st}-\frac{\mathrm{1}}{DT}\sum _{t=\mathrm{1}}^{T}\sum _{s=\
mathrm{1}}^{D}{Y}_{st}\right)\left(\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm{^}}{Y}}_{st}-\frac{\mathrm{1}}{DT}\sum _{t=\mathrm{1}}^{T}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm
{^}}{Y}}_{st}\right)}}{\sqrt{\sum _{t=\mathrm{1}}^{T}{\left(\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{Y}_{st}-\frac{\mathrm{1}}{DT}\sum _{t=\mathrm{1}}^{T}\sum _{s=\mathrm{1}}^{D}{Y}_{st}\right)}^
{\mathrm{2}}}\sqrt{\sum _{t=\mathrm{1}}^{T}{\left(\frac{\mathrm{1}}{D}\sum _{s=\mathrm{1}}^{D}{\stackrel{\mathrm{^}}{Y}}_{st}-\frac{\mathrm{1}}{DT}\sum _{t=\mathrm{1}}^{T}\sum _{s=\mathrm{1}}^{D}{\
where D is the number of events in the test set, T is the number of time steps of the relevant rainfall event, Y[st] is the prediction given by the neural network at the tth time step in the s
th event, and ${\stackrel{\mathrm{^}}{Y}}_{st}$ is the simulation given by the hydrodynamic model.
To evaluate the accuracy of the proposed model in predicting ponding, we have introduced five indicators (as shown in Table 4), namely accuracy (ACC), precision (PPV), and false omission rate (FOR)
to evaluate the model accuracy in predicting the occurrence of ponding at a single node and S−PPV and S−FOR to evaluate the model accuracy in predicting the occurrence of ponding for a single event.
TP and TN denote the number of occurrences when a ponding case and a normal case (no ponding occurs) are correctly identified, respectively, FP is the number of occurrences when a normal case is
incorrectly identified as a ponding case, and FN is the number of occurrences when a ponding case is ignored by the model. The subscript “s” denotes the number of time steps in the sth event.
3.1Model setup
The LSTM-based model was trained by the designed rainfall data and simulation produced from the hydrodynamic model. According to the procedures described in Sect. 2.2, the noise (ε) transmitted from
the runoff process to the flow confluence process was equal to 1.9412% in the case of the pipe network. For the sake of convenience, the noise was set to 2%.
Figure 10 described the overall performance of the model using four box plots of the mean scores (of all nodes) on the test set, with the outliers removed. As shown in Fig. 10, the median values of
MAE and MSE were much smaller than 0.1, indicating that the model has converged at all nodes. The median value of CC was close to 1, even though the minimum value was higher than 0.95. The median
value of NSE was higher than 0.95, yet the minimum value was about 0.75, which indicated that, although the model's performance at each node was slightly different, the overall prediction was
generally reliable.
Due to the limited space, we only listed the evaluation results of six representative nodes. The six nodes (as shown in Fig. 11) were selected because of the severity of consequence once ponding
occurred and also because they were relatively uniformly distributed in the pipe network. Moreover, three of them (nodes 2, 238, and 313) were chosen because the positive samples (where ponding
occurred) accounted for less than 50% of the training set, and the other three had the opposite case. For example, at node 238, the positive samples accounted for 18.33% of the training set, while
at node 95, up to 98.6% samples were positive.
Table 5 lists the scores at the six selected nodes for evaluating the model performance in ponding occurrence prediction. In Table 5, columns ACC to FOR reflect the accuracy of ponding occurrence
prediction in the sense of time (i.e., averaged in time). The mean ACC values (accuracy) for all six nodes were higher than 98.5%. Compared to ACC, the mean PPV values (precision) were slightly
lower, with the minimum value about 88% at node 238, which indicated that a ponding case had at least an 88% chance of being correctly identified. The mean FOR value (false omission rate) of each
node was generally lower than 1%, and among them the worst performance occurred at node 95 (FOR=0.98%), which indicated that the model had a relatively small chance to ignore the ponding. The last
two columns in Table 5 reflect the accuracy of the model in predicting ponding occurrence for a single rainfall event. For example, falsely reported events took up 6% of the testing events at
node 238 ($S-\mathrm{PPV}=\mathrm{94}$%), which was already the worst performance among the six selected nodes, while the S−FOR values at all of these six nodes equalled 0, which indicated that the
model did not miss any ponding incidents in the testing set.
The scores for evaluating the model performance in the ponding volume prediction are listed in Table 6. As shown in Table 6, the MAE and MSE scores were generally small, with the highest MAE score
(0.0770Ls^−1) occurring at node 95 and the highest MSE score (0.3788L^2s^−2) occurring at node 2. Compared to the MAE and MSE scores, the variability in the CC scores was much smaller. All of
them were very close to 1. As for the NSE scores, the lowest score (NSE=0.8195 at node 238) was above 0.8. The results shown in Table 6 indicate that the proposed model had a relatively good
performance in ponding volume prediction.
Furthermore, in the above analysis, the mean score values on the test set were used for evaluation, and the variability was ignored. Figure 12 shows the predicted ponding volume at the selected nodes
compared with the simulation results in six testing rainfall events. As shown in panel (1), the predicted start time of ponding was 5min earlier than the simulation at node 2. As shown in panel (2),
three peaks appeared in the ponding process of node 95, and the model has identified each of them. No ponding occurred at node 238, given the testing precipitation, as shown in panel (5), and the
prediction of the model was in consistent with it. Overall, the prediction of the model was relatively accurate.
3.2Model correction
In this study, the model was trained based on the simulation results from a hydrodynamic model. Though the hydrodynamic model has been verified, the differences between the simulation (from the
hydrodynamic model) at the monitoring points and the monitoring data persisted during the essential operation of the pipe network, which inevitably degraded the accuracy of the LSTM-based model in
ponding forecast. Thus, it is necessary to correct the model using the measured rainfall data, level or flow data at the monitoring points, and ponding data.
The discrepancy between the measurements and simulation from the hydrodynamic model can be exemplified by Fig. 13. As shown in Fig. 13, rainfall event no. 5 was one of the measured precipitation
events where the maximum precipitation intensity reached 4.97mmmin^−1. For this event, the measured water depth and flow data were compared with the simulation from the hydrodynamic model, as shown
in the left and right panels of Fig. 13, respectively.
The ponding process predicted by the corrected model was compared with the monitored ponding data to evaluate the model performance. Figure 14 illustrates the overall performance of the corrected
model using four indicators, as described in Sect. 2.4.5. To be specific, the four box plots show the range of the mean score values on the test set at all nodes. As shown in the Fig. 14, the median
values of the CC and NSE scores maintained values above 0.98 and 0.9, respectively. In contrast, the maximum values of MAE and MSE scores remained lower than 0.30Ls^−1 and 0.6L^2s^−2,
Specifically, the mean score values at the six selected nodes obtained using the corrected model are summarized in Table 7. As shown in Table 7, the MAE and MSE scores were generally small, the NSE
score at each node was stably above 0.9, and the CC scores were all above 0.95. The results shown in Table 7 suggest that the corrected model performed well at different nodes.
To further test the capability of the corrected model, the mean scores of all nodes for five measured rainfall events are summarized in Table 8, where the results from the model without correction
are also listed as a comparison. As shown in Table 8, all of the four indicators suggest that the corrected model performed much better than the model without correction. Specifically, the NSE score
obtained from the model without correction was less than 0, while this score rose up to 0.8316 after applying the model correction procedure, which indicated the necessity of the correction.
To further demonstrate the effect of model correction procedure, we have shown the predicted ponding process at the six selected nodes for rainfall event no. 5, obtained by using the model with and
without correction, as shown in Fig. 15. As shown in Fig. 15, the corrected model performed better at all the selected nodes, e.g., by having a more accurate prediction of the start/end times of the
ponding and more accurate ponding curves (more similar to the measured ones).
All of the results shown above demonstrate the superiority of the corrected model compared to the original one, in which the monitoring data were introduced in the model correction procedure.
4.1Comparison of neural network structures
The proposed model (termed model A) was compared with the conventional LSTM structure (termed model B) to show the superiority of the variant of the LSTM structure in the flow confluence process. The
schematic diagrams of the two models are shown in Fig. 16a–c. As shown in the various panels, model B has exactly the same structure as model A in the runoff process. The only difference in the two
models lies in the flow confluence process, where a multi-task learning mechanism is introduced in the learning process of model A.
Furthermore, model A, as proposed in this paper, was compared with two other models (models C and D) to illustrate the necessity of having the two processes in tandem, i.e., the runoff and flow
confluence processes. The network structures of models C and D are shown in Fig. 16d and e, respectively, where the ponding information was obtained directly from the rainfall data without extracting
the characteristics of lateral inflows.
Figure 17 shows two examples. In the first example, as shown in Fig. 17a, ponding did not occur at node 2. However, about 2–4Ls^−1 ponding volume was falsely reported by the three alternative
models (models B, C, and D), while model A predicted no ponding at this node, which was consistent with the simulation (considered to be the ground truth). In the second example, as shown in
Fig. 17b, where ponding occurred and lasted for about 40min, model A predicted a more accurate ponding curve than the other three alternative models.
Figure 18 presents the range of mean score values on the test set for all nodes, as obtained by using models A–D. As shown in the figure, the range of the MAE or MSE score from model A was half that
of model B. The CC scores from model A were very close to 1, while the CC scores from model B varied from about 0.8 to 1. The NSE scores from model A were generally higher than 0.7, while the NSE
scores from model B were unstable and generally lower than those from model A. Obviously, model A performed much better than model B in ponding volume prediction, as indicated by all four of the
As also shown in Fig. 18, the obvious superiority of model A (or B) over models C and D demonstrates the necessity of having two processes in tandem. Besides, it is also shown in Fig. 18 that the
range of all four of these indicators expanded gradually from model A to D, which indicated decreased steadiness.
Table 9 shows the mean score values at all nodes (on the test set) obtained by using the four models. According to the results, the performance ranking of the four models was model A>model B>
model C>model D.
The comparative analyses above indicated that the LSTM-based model proposed in this paper had remarkable superiority over the other three alternatives for ponding volume forecasting. There are two
reasons behind this. First, the proposed model had two processes in tandem, namely the runoff and flow confluence processes. The second is due to the auxiliary classification task introduced in the
flow confluence process. The two tandem processes reduced the computational burden of this data-driven approach and avoided interference with each other during training, while the classification task
introduced facilitated the capability of the model to identify ponding.
4.2The influence of the number of monitoring points on model correction
It was easy to spot from the trial that the performance of the corrected model depended on whether the layout of the monitoring points could reflect the hydraulic conditions of the pipe network. An
unreasonable design of the monitoring equipment might lead to a failure in model correction.
There were 15 level gauges and 3 flowmeters in the case pipe network, as shown in Fig. 7b. To analyze how the number of monitoring points impacted the performance of the revised model, different
numbers of monitoring points were randomly selected as a quantitative control group. Figure 19 presents the evaluation results of the revised model for ponding volume forecasting, as obtained by
using different numbers of monitoring points.
As shown in the Fig. 19, the NSE scores stayed at around 0.9 when the number of monitoring points exceeded six, and the CC scores showed similar trends to the NSE scores. Besides, the other scores
showed the opposite trends. It turned out that, when the number of monitoring points was over 1 per hectare, then increasing the number of monitoring points further had limited effect on improving
the accuracy of the corrected model. However, when the number of monitoring points was below 0.5 per hectare (i.e., the number of monitoring points was less than 3), then it was highly effective to
increase the number of monitoring points in the pipe network. For example, the NSE score was lower than 0.8 when the number of monitoring points was only 1.
In summary, one monitoring point per hectare is the critical point. If the number of monitoring points was less than this limit, then the performance of the revised model could not be guaranteed.
This work aims at promoting the application of deep learning in urban flood forecasting. Specifically, we have proposed an optimized LSTM-based approach in this study, which can quickly identify and
locate ponding with relatively high accuracy.
According to the research results, the main conclusions of this study are summarized as follows:
1. The proposed model is constructed by two tandem processes (runoff process and flow confluence process) and utilizes a multi-task learning mechanism to achieve high accuracy. Over 15000 designed
rainfall events were used for model training, which covers various extreme weather conditions. The median score of NSE for ponding forecasting is greater than 0.95, and the mean accuracy at any
node to determine whether ponding occurs reaches higher than 0.98.
2. The superiority of the proposed model has been demonstrated by comparing with two widely used deep learning models, namely (traditional) LSTM and CNN models.
The superiority of the proposed model having two tandem processes is proved by comparing with LSTM and CNN structures with a single process. The mean NSE score for the ponding volume forecasting
of the proposed model is 0.9462, while those scores of the LSTM and CNN structures with a single process are 0.7424 and 0.7391, respectively. Then, the superiority of the proposed model with a
LSTM variant is demonstrated by a comparison with the conventional LSTM structure that also has two tandem processes. As shown in Table 9, the mean NSE score of the latter is 0.8552.
3. An approach to the model modification using real-life monitoring level and flow data is proposed in this paper. The proposed LSTM-based model is further calibrated to achieve better accuracy.
The LSTM-based model is corrected using two steps. First, the runoff process is corrected with the measured rain, level, and flow data, referring to parameter-based (model-based) transfer
learning. Then, the flow confluence process is updated using the updated lateral inflows at all nodes and the measured ponding volume. As shown in Table 8, the mean CC score at all nodes of the
model with correction is 0.9309, while that of the model without correction is 0.1139.
Overall, the proposed LSTM-based approach provides a new possibility for early warning and forecasting of ponding in an urban drainage system. In this study, all operations were conducted in an
offline mode. In a future study, we will explore the capability of the proposed model in a real-time event analysis. Furthermore, we will optimize the model by considering the influence of
two-dimensional overland flow in ponding volume prediction.
Code and data availability
The code used for all analyses and all data used in this study are available from the corresponding authors upon request.
WZ: methodology, formal analysis, visualization, software, and writing (original draft preparation). TT: conceptualization, funding acquisition, project administration, and writing (review and
editing). HY: conceptualization, supervision, and validation. JY: writing (review and editing), supervision, and validation. JW: supervision and validation. SL: supervision and validation.
KX: supervision and validation.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors gratefully thank all the team members for their insightful comments and constructive suggestions that ensured a polished, high-quality paper.
This research has been supported by the National Natural Science Foundation of China (grant no. 51978493).
This paper was edited by Yue-Ping Xu and reviewed by two anonymous referees.
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Electrical Engineering 1 (ELEE08001)
Undergraduate Course: Electrical Engineering 1 (ELEE08001)
Course Outline
School School of Engineering College College of Science and
Credit level (Normal year SCQF Level 8 (Year 1 Availability Available to all students
taken) Undergraduate)
SCQF Credits 20 ECTS Credits 10
Summary An introduction to Electrical Engineering (Circuit Analysis, a.c. Theory, Operational Amplifiers, Semiconductor Devices, Logic Theory).
Prof. Murray's Lectures: week 1-4, covering material required for Lab Session 1 & 2.
Week 1
Lecture 1 Potential divider. Resistors and capacitors, RC circuit introduction.
Lecture 2 RC circuits charge-discharge
Lecture 3 Inductors and RL circuits, charge-discharge
Week 2
Lecture 4 Nodal analysis introduction
Lecture 5 Nodal analysis examples
Lecture 6 Op-Amps, introduction
Week 3
Lecture 7 Op-Amp circuits
Lecture 8 Op-Amp worked examples
Lecture 9 Real Op-Amps (limitations)
Week 4
Lecture 10 Diodes - "cartoon" version
Lecture 11 Op-Amp circuits with diodes and capacitors
Lecture 12 Filters
Dr. Mueller's Lectures: week 5, 7-8, covering material for Lab Session 2 (weeks 8-11)
Week 5
Lecture 13 AC circuits, voltage & current waveforms, reactance, intro to phasors
Lecture 14 Phasors examples 2 components: R-C, R-L - series and parallel
Course description Lecture 15 Phasors examples 3 components: R-C-L
Week 6
Lecture 16 AC circuits: complex number representation & polar form
Lecture 17 Examples - revisit filters, relate to part 2 of lab
Lecture 18 Circuit analysis: Kirchoff's Law, Thevenin - example
Week 7
Lecture 19 Current Sources & Nortons Law - example
Lecture 20 Current source examples - R-C charging
Lecture 21 Examples - application of above to a power circuit.
Dr. Haworth's Lectures: week 9-11, covering some parts of both lab sessions in more detail
Week 8
Lecture 22 Diodes. Diode models, examples, rectifier circuits (remove load line).
Lecture 23 Diodes cont. Peak rectifier, diode clamp, voltage doubler, Zener diode, LED.
Lecture 24 Digital Logic. AND/OR/NAND/NOR Simple combinational logic, truth tables.
Week 9
Lecture 25 Boolean Algebra. Rules, Examples.
Lecture 26 Logic reduction. K-maps, examples, half adder.
Lecture 27 K-maps of 3 and 4 variables, examples, full adder, SOP, POS.
Week 10
Lecture 28 Sequential Logic. SR flip-flop, synchronous SR.
Lecture 29 Sequential Logic cont. D-type, edge triggered/master-slave.
Lecture 30 Examples
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Co-requisites
Prohibited Combinations Other requirements Prior attendance at Engineering 1 or (in special circumstances) prior attendance at another half-course.
Information for Visiting
Pre-requisites None
High Demand Course? Yes
Course Delivery Information
Academic year 2017/18, Available to all students Quota: 151
Course Start Semester 2
Timetable Timetable
Total Hours: 200 ( Lecture Hours 30, Seminar/Tutorial Hours 10,
Learning and Teaching activities (Further Supervised Practical/Workshop/Studio Hours 27, Formative Assessment
Info) Hours 1, Summative Assessment Hours 10, Programme Level Learning and
Teaching Hours 4, Directed Learning and Independent Learning Hours
118 )
Assessment (Further Info) Written Exam 60 %, Coursework 40 %, Practical Exam 0 %
Additional Information (Assessment) Laboratory and weekly assignments. Coursework 40%, examination 60%.
Feedback Not entered
Exam Information
Exam Diet Paper Hours & Minutes
Main Exam Diet S2 (April/May) 2:00
Resit Exam Diet (August) 2:00
Learning Outcomes
A student who has completed the course can
expect to:
- Analyse simple circuits using basic
voltage and current laws
- Understand the construction and operation
of the main types of passive circuit
component (resistor, capacitor and
inductor, including variable versions)
under D.C. and A.C. conditions
- Comprehend basic A.C. circuit analysis
- Describe the formation and principles of
operation of active devices (transistors).
- Understand the concept of an ideal
operational amplifier
- Analyse and design simple electronic
systems comprising active and passive
- Be competent in the use of basic
electronic test gear
- Design and construct a simple circuit to
a given specification, diagnose faults and
repair if necessary
- Write a technical report detailing
practical work carried out
Reading List
Giorgio Rizzoni, "Principles and
Applications of Electrical Engineering",
published by McGraw-Hill, ISBN 0-07-118452
Additional Information
Course URL http://webdb.ucs.ed.ac.uk/
Attributes and Not entered
Tutorial: M 1400 or 1500 or
Additional 1600 or Tu 1400 or 1500 or
Class Delivery 1600 or Th 1400
Information Labs (Weeks 2-10): Tu
1400-1700 or Th 1400-1700
Keywords AC Circuits,DC Circuits,OP
Dr Markus Miss Hannah
Course Mueller Course Ross
organiser Tel: (0131 secretary Tel: (0131
6)50 5602 6)50 5687
Email: Email:
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Jacob Collard
Mathematical Entities: Corpora and Benchmarks
Jacob Collard | Valeria de Paiva | Eswaran Subrahmanian
Proceedings of the 2024 Joint International Conference on Computational Linguistics, Language Resources and Evaluation (LREC-COLING 2024)
Mathematics is a highly specialized domain with its own unique set of challenges. Despite this, there has been relatively little research on natural language processing for mathematical texts, and
there are few mathematical language resources aimed at NLP. In this paper, we aim to provide annotated corpora that can be used to study the language of mathematics in different contexts, ranging
from fundamental concepts found in textbooks to advanced research mathematics. We preprocess the corpora with a neural parsing model and some manual intervention to provide part-of-speech tags,
lemmas, and dependency trees. In total, we provide 182397 sentences across three corpora. We then aim to test and evaluate several noteworthy natural language processing models using these corpora,
to show how well they can adapt to the domain of mathematics and provide useful tools for exploring mathematical language. We evaluate several neural and symbolic models against benchmarks that we
extract from the corpus metadata to show that terminology extraction and definition extraction do not easily generalize to mathematics, and that additional work is needed to achieve good performance
on these metrics. Finally, we provide a learning assistant that grants access to the content of these corpora in a context-sensitive manner, utilizing text search and entity linking. Though our
corpora and benchmarks provide useful metrics for evaluating mathematical language processing, further work is necessary to adapt models to mathematics in order to provide more effective learning
assistants and apply NLP methods to different mathematical domains.
Extracting Mathematical Concepts from Text
Jacob Collard | Valeria de Paiva | Brendan Fong | Eswaran Subrahmanian
Proceedings of the Eighth Workshop on Noisy User-generated Text (W-NUT 2022)
We investigate different systems for extracting mathematical entities from English texts in the mathematical field of category theory as a first step for constructing a mathematical knowledge graph.
We consider four different term extractors and compare their results. This small experiment showcases some of the issues with the construction and evaluation of terms extracted from noisy domain
text. We also make available two open corpora in research mathematics, in particular in category theory: a small corpus of 755 abstracts from the journal TAC (3188 sentences), and a larger corpus
from the nLab community wiki (15,000 sentences)
Unsupervised Formal Grammar Induction with Confidence
Jacob Collard
Proceedings of the Society for Computation in Linguistics 2020
Finite State Reasoning for Presupposition Satisfaction
Jacob Collard
Proceedings of the First International Workshop on Language Cognition and Computational Models
Sentences with presuppositions are often treated as uninterpretable or unvalued (neither true nor false) if their presuppositions are not satisfied. However, there is an open question as to how this
satisfaction is calculated. In some cases, determining whether a presupposition is satisfied is not a trivial task (or even a decidable one), yet native speakers are able to quickly and confidently
identify instances of presupposition failure. I propose that this can be accounted for with a form of possible world semantics that encapsulates some reasoning abilities, but is limited in its
computational power, thus circumventing the need to solve computationally difficult problems. This can be modeled using a variant of the framework of finite state semantics proposed by Rooth (2017).
A few modifications to this system are necessary, including its extension into a three-valued logic to account for presupposition. Within this framework, the logic necessary to calculate
presupposition satisfaction is readily available, but there is no risk of needing exceptional computational power. This correctly predicts that certain presuppositions will not be calculated
intuitively, while others can be easily evaluated.
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How many cubic units is a box that is 3 units high, 3 units wide, and 2 units deep?
How many cubic units is a box that is 3 units high, 3 units wide, and 2 units deep?
A) Volume of the box is 18 cubic units.
B) Volume of the box is 8 cubic units.
C) Volume of the box is 12 cubic units.
D) Volume of the box is 16 cubic units.
[bg_collapse_level2 view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Hide Answer” ]
The correct answer for the given question is Option A) Volume of the box is 18 cubic units.
Answer Explanation
Given in the Question ;
• Lenght of the box (L) = 2 units
• Width of the box (B) = 3 units
• Height of the box(H) = 3 units
We know that ;
Volume of the Box = Lenght of the box (L) * Width of the box (B) * Height of the box(H)
= 2 units * 3 units* 3 units
= 18 cubic units
Therefore , the Volume of the Box is 18 cubic units.
Real Numbers Class 10 MCQ – Multiple Choice Questions | Mathematics
Leave a Comment
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Parallelogram – Definition, Varieties, and Examples
A parallelogram is a particular quadrilateral with each pairs of reverse sides parallel as proven within the determine beneath. Discover that the other sides which are parallel traces are proven
utilizing the identical quantity of arrow heads.
Varieties of parallelogram
Squares, rectangles, and rhombuses are additionally parallelograms, extra particularly particular parallelograms, since their reverse sides are parallel in accordance with the definition.
A rhombus is a parallelogram with 4 equal sides.
A rectangle is a parallelogram with 4 proper angles
A sq. is a parallelogram with 4 equal sides and 4 proper angles. Subsequently, a sq. is a rhombus and a rectangle on the similar time.
Diagonal, base, altitude, and top of a parallelogram
The diagonal of a parallelogram is a line section that joins two vertices that aren’t subsequent to one another. A parallelogram has two diagonals. The diagonals are proven in black within the
determine beneath.
A base of a parallelogram is any of its sides.
The altitude of a parallelogram is a line section that begins from a vertex and is both perpendicular to the bottom or to the road containing the bottom.
Within the determine above, the altitude is perpendicular to the bottom. Nonetheless, within the determine beneath, the altitude is perpendicular to the road containing the bottom.
The peak of a parallelogram is the size of an altitude.
Properties of parallelogram
• Reverse sides of a parallelogram are congruent.
• Reverse angles of a parallelogram are congruent.
• The diagonals of a parallelogram bisect one another.
• Consecutive angles are supplementary or add as much as 180 levels.
• Every diagonal of a parallelogram turns the parallelogram into 2 congruent triangles.
Please examine the lesson about properties of a parallelogram to study extra about these 5 properties of parallelograms.
How you can assemble a parallelogram utilizing a straightedge and compass
You possibly can simply assemble a parallelogram by following fastidiously the next 4 easy steps.
Step 1
Draw an angle ABC of any measure
Step 2
Put the needle of a compass at level B and modify the opening of the compass to measure the size of section BA. Then, preserving the opening of the compass the identical, put the needle of the
compass at level C and draw an arc.
Step 3
Put the needle of a compass at level B and modify the opening of the compass to measure the size of section BC. Then, preserving the opening of the compass the identical, put the needle of the
compass at level A and draw an arc.
Step 4
Label the purpose of intersection of the 2 arcs D. Then, draw line segments AD and CD.
Perimeter of a parallelogram
Suppose you’ve a parallelogram ABCD. The perimeter is the same as the sum of the lengths of its sides.
The size of section AB is the same as the size of section CD.
The size of section BC is the same as the size of section AD.
Perimeter = size of section AB + size of section AB + size of section BC + size of section BC
Perimeter = 2(size of section AB) + 2(size of section BC)
Space of a parallelogram
The world of a parallelogram is the product of the bottom and the size of the altitude.
Let h be the size of the altitude or the peak of the parallelogram.
Then, the system to make use of to search out the world of a parallelogram is space = b × h
Discover that the time period base refers each to the size of the bottom and the section.
Parallelogram FAQs
It will depend on which definition is used for a trapezoid! Parallelograms should have two pairs of reverse sides which are parallel. Some textbooks say that trapezoids should have precisely one
pair of parallel sides. If there may be precisely one pair of parallel sides, then the reply is not any. Different textbooks say that trapezoids should have a minimum of one pair of parallel
sides. On this case, all parallelograms are trapezoids since they match contained in the definition of trapezoids.
It will depend on which definition is used for a trapezoid! Parallelograms should have two pairs of reverse sides which are parallel. Some textbooks say that trapezoids should have precisely one
pair of parallel sides. If there may be precisely one pair of parallel sides, then the reply is not any. Different textbooks say that trapezoids should have a minimum of one pair of parallel
sides. On this case, some trapezoids might develop into parallelograms if these trapezoids have two pairs of reverse sides which are parallel.
Sure! If you happen to add up the 4 angles in any parallelogram, it’ll equal to 360 levels.
No! A parallelogram will find yourself having 4 proper angles if simply one of many angles is a proper angle. On this case, it’s known as a rectangle.
A quadrilateral is a polygon with 4 sides. Since each parallelogram has 4 sides, each parallelogram can also be a quadrilateral.
Within the American time period, a trapezium is a quadrilateral that has no parallel sides. It’s form of the “reverse” of a parallelogram which has two pairs of parallel sides.
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Algebra 101 - Course Help Online
simple algebra questions, make it pdf
Math-UA.009. Written Homework
MATH-UA-009 — Algebra, Trigonometry and Functions
Homework Assignment #10
Due Date: December 11th, 2023, 11:59 PM
• This homework should be submitted via Gradescope by 23:59 on the date
listed above. You can find instructions on how to submit to Gradescope on
our Campuswire channel.
• There are three main ways you might want to write up your work.
– Write on this pdf using a tablet
– Print this worksheet and write in the space provided
– Write your answers on paper, clearly numbering each question and part.
∗ If using either of the last two options, you can use an app such as
OfficeLens to take pictures of your work with your phone and convert
them into a single pdf file. Gradescope will only allow pdf files to be
• You must show all work. You may receive zero or reduced points for
insufficient work. Your work must be neatly organised and written.
You may receive zero or reduced points for incoherent work.
• If you are writing your answers on anything other than this sheet, you should
only have one question per page. You can have parts a), b) and c) on the
page for example, but problems 1) and 2) should be on separate pages.
• When uploading to Gradescope, you must match each question to the
page that your answer appears on. If you do not you will be docked a
significant portion of your score.
• When appropriate put a box or circle around your final answer.
• The problems on this assignment will be graded on correctness and completeness.
• These problems are designed to be done without a calculator. Whilst there is
nothing stopping you using a calculator when working through this assignment,
be aware of the fact that you are not permitted to use calculators on exams
so you might want to practice without one.
Math-UA.009. Written Homework
1. For each of the following angles, determine the following:
• The quadrant the angle lies in
• The reference angle
• The values of sin (θ), cos (θ) and tan (θ)
(a) (3 points) θ = −π/4
(b) (3 points) θ = π/6
(c) (3 points) θ = 5π/6
(d) (3 points) θ = −5π/6
(e) (3 points) θ = −2π/3
Math-UA.009. Written Homework
(f) (3 points) θ = −π
(g) (3 points) θ = 2π/3
(h) (3 points) θ = 7π/6
(i) (3 points) θ = 3π/2
(j) (3 points) θ = −3π/4
Math-UA.009. Written Homework
2. For each of the following triangles, find the value of
• sin (θ)
• cos (θ)
• tan (θ)
(3 points)
(3 points)
(3 points)
Math-UA.009. Written Homework
(3 points)
(3 points)
9 θ
(3 points)
Math-UA.009. Written Homework
3. (a) (3 points) Find sin (θ) given that cos (θ) = 1/5 and θ is in quadrant IV.
(b) (3 points) Find tan (θ) given that sin (θ) = 1/3 and θ is in quadrant II.
Math-UA.009. Written Homework
4. (4 points) Find the amplitude, period and horizontal shift of the function, and
graph one complete period
y = −3 sin 2 x +
5. (4 points) Find the amplitude and period of the function, and sketch its graph.
y =1+
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5 Killer Quora Answers on atom stake calculator - Debt Fore5 Killer Quora Answers on atom stake calculator
This is a great way to do math for me. I like to think of this as the “filling the hole” of the calculator. It’s very easy, it’s really simple, and I’m sure it’s the most intuitive way to do it. I’m
sure you can find this calculator online, but it’s a little dated and I don’t want to be like your calculator.
The same site we used to find the Atom Stake Calculator for the first part of this article also has a calculator for a simple calculator. I do not like the fact that the Atom Stake Calculator is not
based on the same formula as the simple calculator, because for example it would only work on a square-shaped grid. I don’t care about that. The Atom Stake Calculator will work on any shape and it’s
really easy and simple to use.
I have been using the Atom Stake Calculator for years and love it, but it is no longer updated with the latest algorithm. It still works if you know the algorithm and change it, but I am not sure I
will be using it in the future.
The Atom Stake calculator will work on any shape, and its really easy and simple to use. Atom Stake Calculator.
The Atom Stake Calculator does a number of things. The first one is calculating the number of atoms on the grid. The second is calculating how much each atom costs, and the third is the amount of
energy each atom has. The fourth is a calculator for atoms that you can have on hand. The fifth is a calculator for atoms, and the sixth is a calculator for atoms that you can have on hand. Atom
Stake Calculator.
If you want to know how many atoms there are on the grid you simply use the calculator, and then select the number of atoms you want to calculate, at which point you can click the number of atoms you
just calculated to get the result. Atom Stake Calculator.
This isn’t the first game I’ve seen where a user is required to calculate the amount of energy each atom has, and then select the number of atoms they want to calculate. For example, this game lets
you make a calculator and let the user type in the number of atoms they want to calculate.
This is my first time playing with this. I can barely put the game out of my mind, and the game is going to fail this week. I have only been playing in the past week and a half. I have a lot of
For example, I might want to try out a game where you have a number of atoms that you have to choose from. I can’t think of any other games of that nature that have this mechanic. But I could be
I have a few other ideas as well, but that’s for another blog post.
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Age calculation formula in Excel
Calculating age in Excel is helpful in several situations. Calculating an individual's age is an important process in the HR department. Age calculation is also necessary for lots of business and
other operations.
You can also use Excel to calculate an employee's retirement date and find their pension based on the number of years spent working for your company. Whether you want to learn age calculation formula
in Excel for work or experiment with Excel formulas, this tutorial will teach you various ways to determine someone's age in Excel.
1. Age Calculation using Date of Birth
You can calculate the age of an individual by their birthday.
The best and easiest way to calculate someone's age is to subtract the date of birth from the current date. That is how we manually determine a person's age, isn't it? The same formula can be used in
Let's say the birthdate is in cell C3. The age calculation in Excel formula is as follows:
Using the formula TODAY()-C3, you get the difference between the current date and your date of birth in days, then divide that number by 365 for years.
However, this formula results in decimal numbers as shown below.
The INT function can be used to round it down to the nearest integer.
The problem with this age formula is that it is based on the assumption that every year is exactly 365 days.
As every fourth year (leap year) has 366 days, you divide the date by 365.25, but even this is not guaranteed to be accurate.
Hence this formula works well in a few cases but is flawed when the person is born during a leap year.
2. Age Calculation using YEARFRAC() function
The YEARFRAC function in Excel provides an accurate age calculation formula in Excel. With the YEARFRAC function, you can calculate age from date of birth that returns the fraction of the year for
two dates.
Here is the Excel formula for the YEARFRAC function:
YEARFRAC(DOB, TODAY(), Basis)
Select Basis = 1, which tells Excel to divide the number of days per month by the number of days per year.
As shown in the previous example, DOB is in cell C, hence the formula to be used is
=YEARFRAC(C3, TODAY(), 1)
The decimal number can be rounded down using the INT function,
=INT(YEARFRAC(C3, TODAY(), 1))
3. Age Calculation using the DATEDIF() function
DATEDIF() function is one of the most popular age calculation formula in Excel. Let us now see how to use this function to calculate age in years, months, and days.
The syntax of the DATEDIF function is
=DATEDIF(start_date, end_date, unit)
Start_date - This can be the date of birth.
End_date - This can be the current date, TODAY().
Unit - The DATEDIF function can produce six different sets of results, depending on what unit you use. Here is a complete list of the units you can use:
• Y – indicates the number of completed years in the specified period.
• M – indicates the number of completed months in the specified period.
• D – indicates the number of completed days in the specified period.MD – counts how many days are in the period, but does not include the ones in the Years and Months that have already passed.
• MD – counts how many days are in the period, but does not include the ones in the Years and Months that have already passed.
• YM – indicates the number of months in the period, but does not include the ones in the Years and Months that have already passed.
• YD – indicates the number of days in the period, but doesn’t count the ones in the Years that have already passed.
Calculating age by year
As shown previously, DOB is in Cell C3, DATEDIF formula to calculate age by year is as shown:
=DATEDIF(C3, TODAY(), "y")
Calculating age by year, months, and days
Similar to the year formula, the DATEDIF function can be used to calculate age by year, months, and days using the following formula:
=DATEDIF(C3,TODAY(),"Y") & DATEDIF(C3,TODAY(),"YM") & DATEDIF(C3,TODAY(),"MD")
• 1. =DATEDIF(C3, TODAY(), "Y") - Gives the number of years
• 2. =DATEDIF(C3, TODAY(), "YM") - Gives the number of months
• 3. =DATEDIF(C3,TODAY(),"MD") - Gives the number of days
Since the result shown in the above image does not make any sense, let us add commas and text to help differentiate year, month, and dates using the IF function.
=IF(DATEDIF(C3, TODAY(),"y")=0,"",DATEDIF(C3, TODAY(),"y")&" years, ")& IF(DATEDIF(C3, TODAY(),"ym")=0,"",DATEDIF(C3, TODAY(),"ym")&" months, ")& IF(DATEDIF(C3, TODAY(),"md")=0,"",DATEDIF(C3, TODAY
(),"md")&" days")
That's it! We have covered several ways to calculate age in Excel. We hope this tutorial for age calculation formula in Excel was helpful to you.
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Analytical solutions to the FENE-P model with slip boundary conditions
Title Analytical solutions to the FENE-P model with slip boundary conditions
Authors E. S. Baranovskii^1
^1Voronezh State University
Annotation We study analytical solutions of equations describing steady flows of a FENE-P fluid in a channel under slip boundary conditions. The Navier slip condition and threshold-type slip
conditions are considered. For the plane Poiseuille flow, we obtain explicit formulas for the velocity field, the stress in the fluid, and the configuration tensor.
Keywords FENE-P model, polymeric fluids, Poiseuille flow, slip boundary condition, analytical solutions
Baranovskii E. S. ''Analytical solutions to the FENE-P model with slip boundary conditions'' [Electronic resource]. Proceedings of the XIII International scientific conference
Citation ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 170-176. Available at: https://conf.svmo.ru/files/
deamm2017/papers/paper23.pdf. - Date of access: 14.11.2024.
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Partner Scavenger Hunt Activities
Partner Scavenger Hunt Links:
(please refresh the page if not loading)
11 comments:
1. I wonder if you have any I could wiggle into 4th grade standards like equivalent fractions (or adding/subtracting fractions_ and perhaps a bit of differentiating...maybe pictures and decomposed,
for example, or..erm...as an idea? lol)?
1. I'd love to chat! And to make an adding/subtracting fractions partner scavenger hunt, though I'd afraid things would get too small if I included pictures. My email is shana@scaffoldedmath.com
if you want to let me know if problems with just numbers would be OK. And thank you for your comment!
2. Hi again! I have added a few partner scavenger hunts that could work for 4th grade. If you are still looking for them, I can sen you a link to where they can be found. My email is
2. Hi there, do you sell any of these as bundles?
1. Thank you for asking. There are a couple topic bundles in the partner scavenger hunts section of my tpt store.
3. Do you have any of these for younger grades (2nd grade)?
4. I don't currently, but could! My email is shana@scaffoldedmath.com
5. Replies
1. I do have a few for 5th grade. If you'd like to send me an email, I can link to you where they are found in my store. Or you can go to my TpT store Scaffolded Math and Science, click Partner
Scavenger Hunts on the left and filter by 5th grade.
6. Is there a bundle of around the clocks we can purchase?
1. I have a couple small bundles and a larger one for 8th grade math, but no large bundle with all of them. If you are interested in the 8th grade partner scavenger hunt bundle, you can search
"bundle" under the partner scavenger hunts category in my TPT store to find it.
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Finance multiple choice options and derivatives - Assignments Help US
Business & Finance
Finance multiple choice options and derivatives
Finance multiple choice options and derivatives.
6.Which of the following statements about the volatility is not true? a. the implied volatility often differs across options with different exercise prices b. the implied volatility equals the
historical volatility if the option is correctly priced c. the implied volatility is determined by trial and error d. the implied volatility is nearly linearly related to the option price e. none of
the above
7.Consider a stock priced at $30 with a standard deviation of 0.3. The risk-free rate is 0.05. There are put and call options available at exercise prices of 30 and a time to expiration of six
months. The calls are priced at $2.89 and the puts cost $2.15. There are no dividends on the stock and the options are European. Assume that all transactions consist of 100 shares or one contract
(100 options). Use this information to answer questions 7 and
8. What is your profit if you buy a call, hold it to expiration and the stock price at expiration is $37? a. $32.89 b. $30.00 c. $27.11 d. $32.15 e. there is no breakeven
The profit = $411
I believe the choices provided are incorrect
8. Consider a stock priced at $30 with a standard deviation of 0.3. The risk-free rate is 0.05. There are put and call options available at exercise prices of 30 and a time to expiration of six
months. The calls are priced at $2.89 and the puts cost $2.15. There are no dividends on the stock and the options are European. Assume that all transactions consist of 100 shares or one contract
(100 options).
Use this information to answer questions 7 and 8. What is the break even stock price at expiration on the transaction described in problem 1? a. $32.89 b. $30.00 c. 27.11 d. $32.15 9. Consider two
put options differing only by exercise price. The one with the higher exercise price has Select one: a. the lower breakeven and lower profit potential b. the lower breakeven and greater profit
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Basic Design Pack Support
1. you can edit product_info.php to to use different width and height.
2. There are several contributions for productlisting available in the contributions section
There is a includes/functions/html_output.php included in the pack marked with no-cssbuttons rename to html_output.php and upload and use that one.
I don't get the original gifs back, just text without css-buttons. What can be wong?
Thanx for all the help by the way! :thumbsup:
Top Posters In This Topic
toyicebear 379 posts
tiG 22 posts
newtech 17 posts
I don't get the original gifs back, just text without css-buttons. What can be wong?
Thanx for all the help by the way! :thumbsup:
To be able to say more i need to see your site, so just post your url and i will have a look.
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
To be able to say more i need to see your site, so just post your url and i will have a look.
Thanx :)
Thanx :)
you have graphical buttons now, but for some reason there seems to be a problem with the image path
for instance
it does not exsist or there are some access problems on your hosting/server.
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
you have graphical buttons now, but for some reason there seems to be a problem with the image path
for instance
it does not exsist or there are some access problems on your hosting/server.
it should work. Im so confused... :blink:
you have graphical buttons now, but for some reason there seems to be a problem with the image path
for instance
it does not exsist or there are some access problems on your hosting/server.
Ok, if i put an image in http://codered.se/idar3/images or http://codered.se/idar3/ it shows.
But it doesnt show if i put dem in http://codered.se/idar3/includes or any deeper catalog.
What is wrong here?
example http://codered.se/idar3/361.jpg
Edited by AndreasE
Thanks for the contribution! I have made a small amendment as I found a problem - could you see if it's ok.
I'm using the on-the-fly thumbnailer html_output and I deleted the css button code as I didn't want that. anyway it all worked except I found that if the image size you specified in admin was bigger
than the original image it didn't scale the image up and so didn't keep the aspect ratio the same, BUT the normal functions of the shop did size the image so you could end up with a very distorted
image (I know in reality your image is going to be bigger than the thumbnails but I found this out after playing around resizing the default stock images)
So I then tried to change the code so that it resized the image always - even if it had to scale up slightly (again unrealistic but could happen)
So in html_output I changed this:
// Scale the image if larger than the set width or height
if ($image_size[0] > $width || $image_size[1] > $height) {
$rx = $image_size[0] / $width;
$ry = $image_size[1] / $height;
if ($rx < $ry) {
$width = intval($height / $ratio);
} else {
$height = intval($width * $ratio);
$image = '<img src="product_thumb.php?img='.$src.'&w='.tep_output_string($width).'&h='.tep_output_string($height).'"';
} elseif (IMAGE_REQUIRED == 'false') {
return '';
to this:
// Scale the image if larger than the set width or height
$rx = $image_size[0] / $width;
$ry = $image_size[1] / $height;
if ($rx < $ry) {
$width = intval($height / $ratio);
} else {
$height = intval($width * $ratio);
$image = '<img src="product_thumb.php?img='.$src.'&w='.tep_output_string($width).'&h='.tep_output_string($height).'"';
} elseif (IMAGE_REQUIRED == 'false') {
return '';
and also deleted this from product_thumb.php:
// Do not output if get values are larger than orig image
if ($_GET['w'] > $image[0] || $_GET['h'] > $image[1])
Is this an OK way of doing it - it seems to work! It's really just to make sure the ratio's stay the same on the off chance that you're trying to scale an image up. If there is a better way of doing
it please let me know!
P.S i also was wondering what size to save my images as (thats the size they show up in when you've clicked to enlarge isn't it) Is there any difference in loading time for the product_listing /
product_info page if the original image is 30kb or 60kb? Because the resized version in going to be ~3kb and this is all the customer has to download?Right? or does it still take longer for a larger
file size because it takes longer to resize it?
Thanks for the contribution! I have made a small amendment as I found a problem - could you see if it's ok.
I'm using the on-the-fly thumbnailer html_output and I deleted the css button code as I didn't want that. anyway it all worked except I found that if the image size you specified in admin was
bigger than the original image it didn't scale the image up and so didn't keep the aspect ratio the same, BUT the normal functions of the shop did size the image so you could end up with a very
distorted image (I know in reality your image is going to be bigger than the thumbnails but I found this out after playing around resizing the default stock images)
So I then tried to change the code so that it resized the image always - even if it had to scale up slightly (again unrealistic but could happen)
So in html_output I changed this:
to this:
and also deleted this from product_thumb.php:
Is this an OK way of doing it - it seems to work! It's really just to make sure the ratio's stay the same on the off chance that you're trying to scale an image up. If there is a better way of
doing it please let me know!
P.S i also was wondering what size to save my images as (thats the size they show up in when you've clicked to enlarge isn't it) Is there any difference in loading time for the product_listing /
product_info page if the original image is 30kb or 60kb? Because the resized version in going to be ~3kb and this is all the customer has to download?Right? or does it still take longer for a
larger file size because it takes longer to resize it?
These kind of questions are better put in the support tread for, On the fly thumbnails generator
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
These kind of questions are better put in the support tread for, On the fly thumbnails generator
OK - I think the function was meant to be in there fo rthis reason:
A check to see if the thumbnail dimensions being passed to the thumbnailer are greater that the size of the original image. In which case, the image will not be output. (A security feature to
mitigate DOS (Denial of Service) attacks with users calling the thumbnailer directly with very large width/height values.)
Still needs a solution though...I'll try and find one!
The support in this thread is amazing :thumbsup:
Therefore i need to ask one more question:
I have a white 1px border around my productimages. This usually works flawless. However,
on page: http://codered.se/idar3/index.php?cPath=21_23
that border appears around one button. If you cant see it, use the swedish flag.
How can i fix that? I dont have that problem with any oher button (i think)
Edited by AndreasE
The support in this thread is amazing :thumbsup:
Therefore i need to ask one more question:
I have a white 1px border around my productimages. This usually works flawless. However,
on page: http://codered.se/idar3/index.php?cPath=21_23
that border appears around one button. If you cant see it, use the swedish flag.
How can i fix that? I dont have that problem with any oher button (i think)
You have set the stylesheet class to add borders to all images under that class, so that is just what is happening.
click on basics for design in my signature below and reading through that tread you will also find a way to add borders to images and not buttons.
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
Hi, I have installed the BDP and I'm testing locally using XAMPP server, ever since when I load the store, click on any link in the store I get loads of msdos box's come up running c:/windows/
system32/cmd.exe the store works fine but this happens for every operation within the store.
I have searched the forums and this thread and can't seem to find any info on this.
Can anyone shed any light on this for me.
Hi, I have installed the BDP and I'm testing locally using XAMPP server, ever since when I load the store, click on any link in the store I get loads of msdos box's come up running c:/windows/
system32/cmd.exe the store works fine but this happens for every operation within the store.
I have searched the forums and this thread and can't seem to find any info on this.
Can anyone shed any light on this for me.
I have not expirienced this or heard anyone else expiriencing this....
so i guess its only on your install...
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
After trawling the forums again I found another couple of people with the same problem
see the first posts on page: http://www.oscommerce.com/forums/index.php?sho...mp;#entry958112
in this thread, don't know how I missed them thought I'd read it all!
Any ideas??
Edited by kidda
After trawling the forums again I found another couple of people with the same problem
see the first posts on page: http://www.oscommerce.com/forums/index.php?sho...mp;#entry958112
in this thread, don't know how I missed them thought I'd read it all!
Any ideas??
Dont use XAMPP use EasyPhp instead.
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
I just installed oscommerce on a test site I play with (Fantasico install) I download this and literally dropped it over my current files as instructed. However the end result is a total mess with
loads of what I can only presume is loads of php code on the screen. Anyone care to let me know how I have done it so wrong??
sorted it
Edited by dapex
Hi again. I want to change my index page som my productlisting is a little tighter. I dont want so much space between the products. For example: http://codered.se/idar3/index.php?cPath=21_23
I want more space to the left and right of the product listning instead. Can I set the with of the table somwhere and center it?
Andreas w 1000 questions :blush:
Hi again. I want to change my index page som my productlisting is a little tighter. I dont want so much space between the products. For example: http://codered.se/idar3/index.php?cPath=21_23
I want more space to the left and right of the product listning instead. Can I set the with of the table somwhere and center it?
Andreas w 1000 questions :blush:
You can set that in - includes/modules/product_listing.php
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
You can set that in - includes/modules/product_listing.php
Lovely :thumbsup:
A quick easy one, where in index can I delete categories section when i choose a category? I get php-error when i try.
One more... I tried to install
'On the Fly' Auto Thumbnailer using GD Library
And it hit me that BDP already have something like this installed. Is it the same?
Lovely :thumbsup:
A quick easy one, where in index can I delete categories section when i choose a category? I get php-error when i try.
I fixed it with a little more experimenting!! B)
One more... I tried to install
'On the Fly' Auto Thumbnailer using GD Library
And it hit me that BDP already have something like this installed. Is it the same?
standard BDP have Automatic Thumbnails installed.
But for those where automatic thumbnails do not work, 'On the Fly' Auto Thumbnailer using GD Library is also included as an alternative...
For info on how to use, check the readme file from BDP.
Basics for osC 2.2 Design - Basics for Design V2.3+ - Seo & Sef Url's - Meta Tags for Your osC Shop - Steps to prevent Fraud... - MS3 and Team News... - SEO, Meta Tags, SEF Urls and osCommerce -
Commercial Support Inquiries - OSC 2.3+ How To
To see what more i can do for you check out my profile [click here]
I am using the latest BDP, and the blue arrows included under categories.
1) The spacing in between categories are much apart. Where do I have to change so that it will not have so much space from each other, like the original Oscommerce? I still want to keep the blue
3) I want the item numbers be shown under categories, how do I do this?
2) Now I want the sub categories shown together with the categories from the start of the page, how do I do it?
Thanks. :)
|
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|
Understanding Mathematical Functions: How To Type A Piecewise Function
Introduction to Piecewise Functions and Google Docs
When it comes to mathematical functions, piecewise functions play a crucial role in modeling real-world scenarios and solving complex problems. Meanwhile, Google Docs has become a versatile tool for
creating, editing, and sharing documents collaboratively. In this chapter, we will explore the definition and importance of piecewise functions in mathematical and real-world contexts, provide an
overview of Google Docs as a tool for document creation, and discuss the relevance of learning to effectively type and format piecewise functions in Google Docs.
(A) Definition and importance of piecewise functions in mathematical and real-world contexts
Piecewise functions are mathematical functions that are defined by several sub-functions, each applying to a different interval of the function's domain. These functions are particularly important in
situations where a single formula cannot describe the relationship between the input and output variables across the entire domain. Piecewise functions are commonly used in mathematical modeling,
physics, engineering, and economics to represent non-linear and discontinuous phenomena.
(B) Overview of Google Docs as a versatile tool for creating and sharing documents
Google Docs is a web-based word processor offered by Google. It allows users to create and edit text documents, collaborate with others in real-time, and store documents online. With its intuitive
interface and cloud-based architecture, Google Docs provides a convenient platform for individuals, teams, and organizations to create, edit, and share documents seamlessly.
(C) The relevance of learning to effectively type and format piecewise functions in Google Docs
Learning to effectively type and format piecewise functions in Google Docs is essential for students, educators, researchers, and professionals working with mathematical content. By mastering the
skills to input and display piecewise functions accurately, users can effectively communicate mathematical concepts, solve problems, and present data in a clear and organized manner within the Google
Docs environment.
Key Takeaways
• Open a new Google Docs document.
• Click on 'Insert' in the top menu.
• Select 'Equation' from the dropdown menu.
• Type the piecewise function using the equation editor.
• Use the 'if' and 'else' functions for different cases.
Understanding the Format of a Piecewise Function
A piecewise function is a mathematical function that is defined by multiple sub-functions, each applying to a different interval of the function's domain. This allows for different rules to be
applied to different parts of the domain, making it a powerful tool in mathematical modeling and analysis.
(A) Explanation of the structure of a piecewise function
A piecewise function is typically written using curly braces to denote the different cases or sub-functions. Each sub-function is defined for a specific interval of the domain, and the function as a
whole is defined by combining these sub-functions based on the domain intervals.
For example, a simple piecewise function could be defined as:
f(x) = {
• x, if x > 0
• -x, if x <= 0
In this example, the function f(x) is defined differently for x greater than 0 and x less than or equal to 0.
(B) Different notations and conventions used in piecewise functions
There are different ways to represent piecewise functions, and the choice of notation often depends on the specific context or preference of the mathematician. Some common notations include using the
'piecewise' function keyword, using the Iverson bracket notation, or simply using a combination of mathematical symbols and inequalities to define the different cases.
For example, the same piecewise function defined earlier could also be written as:
f(x) = piecewise(x, x > 0, -x, x <= 0)
It's important to be familiar with these different notations and conventions when working with piecewise functions, as they are commonly used in mathematical literature and software.
(C) Real-world examples where piecewise functions are applied
Piecewise functions are not just theoretical constructs; they have practical applications in various fields. For example, in economics, piecewise functions can be used to model tax brackets, where
different tax rates apply to different income ranges. In physics, piecewise functions can be used to model the behavior of physical systems that change their dynamics under different conditions. In
engineering, piecewise functions are used to define systems with different modes of operation.
Understanding how to work with piecewise functions is therefore essential for anyone working in these fields, as well as for anyone interested in advanced mathematical concepts.
Basics of Typing in Google Docs
When it comes to typing mathematical functions in Google Docs, there are several tools and techniques that can be used to effectively input complex equations and expressions. In this chapter, we will
explore the basics of typing in Google Docs, including inserting special characters and symbols, using the built-in equation editor for mathematical expressions, and utilizing keyboard shortcuts for
Overview of inserting special characters and symbols
Google Docs provides a wide range of special characters and symbols that can be easily inserted into your document. To access these characters, simply go to the 'Insert' menu and select 'Special
characters.' Here, you can search for specific symbols or browse through various categories such as mathematical operators, arrows, and Greek letters. Once you find the symbol you need, simply click
on it to insert it into your document.
Additionally, you can also use keyboard shortcuts to insert common mathematical symbols. For example, typing Ctrl + / followed by the name of the symbol (e.g., 'alpha' for α) will insert the
corresponding Greek letter into your document.
Using built-in equation editor for mathematical expressions
Google Docs features a built-in equation editor that allows you to create and edit mathematical expressions with ease. To access the equation editor, go to the 'Insert' menu and select 'Equation.'
This will open a toolbar with various mathematical symbols and structures that you can use to build your equation.
With the equation editor, you can type a piecewise function by using the 'Cases' structure, which allows you to define different cases for the function. Simply click on the 'Structure' button in the
equation editor toolbar and select 'Cases.' This will create a template for a piecewise function where you can input the different cases and their corresponding expressions.
Keyboard shortcuts for efficiency
To improve efficiency when typing mathematical functions in Google Docs, it's important to familiarize yourself with keyboard shortcuts for common actions. For example, you can use Ctrl + / to
quickly access the equation editor, Ctrl + = to start a subscript, and Ctrl + Shift + + to start a superscript.
Furthermore, you can create custom keyboard shortcuts for frequently used symbols and expressions by going to the 'Tools' menu, selecting 'Preferences,' and then clicking on 'Automatic Substitution.'
Here, you can define your own shortcuts for inserting mathematical symbols, making it easier to input complex equations.
Step-by-Step Guide to Typing a Piecewise Function
Understanding how to type a piecewise function in Google Docs can be a valuable skill for anyone working with mathematical functions. In this guide, we will walk through the process of using the
equation editor in Google Docs to insert and edit a system of equations template suitable for piecewise functions.
(A) Opening the equation editor in Google Docs
To begin typing a piecewise function in Google Docs, you will need to open the equation editor. This can be done by clicking on 'Insert' in the top menu, then selecting 'Equation' from the dropdown
menu. This will open the equation editor, where you can input and edit mathematical equations.
(B) Inserting a system of equations template suitable for piecewise functions
Once the equation editor is open, you can insert a system of equations template that is suitable for typing a piecewise function. To do this, click on the 'New equation' button in the equation editor
toolbar, then select 'Insert new equation' from the dropdown menu. This will insert a blank system of equations template that you can customize for your piecewise function.
(C) Detailed guide on editing the template to match the specific piecewise function
Now that you have inserted the system of equations template, you can begin editing it to match the specific piecewise function you want to type. Here is a detailed guide on how to do this:
• Inputting the function: In the first equation box, type the first part of the piecewise function, including the variable, the condition for that part, and the corresponding expression. For
example, if you have a piecewise function f(x) defined as 2x for x < 0 and x^2 for x ≥ 0, you would input '2x' for the first part.
• Adding the condition: To add the condition for the first part of the function, click on the 'Add a condition' button below the equation box. This will open a new input box where you can type the
condition, such as 'x < 0.'
• Adding additional parts: If your piecewise function has more than one part, you can click on the 'Add another equation' button to insert additional equation boxes and conditions for each part of
the function.
• Formatting the function: You can format the piecewise function by adjusting the size, style, and alignment of the equations using the options in the equation editor toolbar.
By following these steps, you can successfully type a piecewise function in Google Docs using the equation editor and customize it to match the specific function you are working with.
Formatting Tips for Clarity and Readability
When working with piecewise functions in Google Docs, it's important to ensure that the formatting is clear and readable. Here are some best practices for formatting mathematical functions to enhance
their presentation.
(A) Best practices for aligning the different cases of the function
• Use a table: One effective way to align the different cases of a piecewise function is to use a table. Create a table with two columns, one for the conditions and the other for the corresponding
expressions. This helps to clearly separate the different cases and make the function easier to read.
• Align the equal signs: When typing out the expressions for each case of the function, make sure to align the equal signs. This helps to visually connect the conditions with their corresponding
expressions and makes the function easier to understand.
• Use consistent formatting: Ensure that the formatting of each case (such as parentheses, brackets, and mathematical symbols) is consistent throughout the function. This helps to maintain clarity
and readability.
(B) Recommended fonts and sizes for mathematical expressions
• Use a clear, legible font: When typing mathematical expressions, it's important to use a font that is clear and easy to read. Recommended fonts for mathematical expressions include Times New
Roman, Arial, and Cambria.
• Adjust the font size: Depending on the size of your document and the level of detail in the mathematical expressions, it may be necessary to adjust the font size. A font size of 12pt to 14pt is
generally recommended for mathematical expressions to ensure readability.
• Consider using bold or italic: To emphasize certain parts of the mathematical expressions, consider using bold or italic formatting. This can help to draw attention to key elements of the
(C) Utilizing indentation and spacing to enhance the presentation
• Indent the cases: When typing out a piecewise function, consider indenting each case to visually separate them from the rest of the text. This helps to clearly define the different cases and
makes the function easier to follow.
• Use spacing to improve readability: Incorporate adequate spacing between the different cases and elements of the function. This includes spacing between the conditions and expressions, as well as
around mathematical symbols and operators. Ample spacing enhances the overall presentation of the function.
• Utilize line breaks: When working with longer piecewise functions, consider using line breaks to break up the function into more manageable sections. This can help to prevent the function from
appearing cluttered and overwhelming to the reader.
Troubleshooting Common Issues
When working with mathematical functions in Google Docs, you may encounter some common issues that can be frustrating to deal with. Here are some tips for troubleshooting these problems:
(A) Addressing problems with symbols not displaying correctly
• Check your browser compatibility: Sometimes, symbols may not display correctly due to compatibility issues with your web browser. Make sure you are using a supported browser and that it is up to
• Use the correct syntax: Ensure that you are using the correct syntax for mathematical symbols and functions. Google Docs has specific formatting requirements for mathematical expressions, so
double-check your input.
• Try a different font: If certain symbols are not displaying correctly, try changing the font in your Google Docs document. Some fonts may have better support for mathematical symbols.
(B) Solutions for difficulties with alignment and spacing
• Adjust the equation editor settings: Google Docs has options for adjusting the alignment and spacing of mathematical expressions. Experiment with these settings to see if you can improve the
appearance of your piecewise function.
• Use manual formatting: If the equation editor is not providing the desired alignment and spacing, consider using manual formatting techniques such as adjusting tab stops and line spacing.
• Insert the function as an image: As a last resort, you can create the piecewise function in a separate program or using a mathematical typesetting tool, and then insert it into your Google Docs
document as an image. This can give you more control over the appearance of the function.
(C) Dealing with limitations of the Google Docs equation editor
• Consider using add-ons: There are third-party add-ons available for Google Docs that provide additional features for working with mathematical expressions. Explore these options to see if they
can help you overcome any limitations of the built-in equation editor.
• Provide feedback to Google: If you encounter specific limitations or issues with the equation editor, consider providing feedback to Google. They may be able to address these issues in future
updates to the platform.
• Explore alternative platforms: If the limitations of the Google Docs equation editor are too restrictive for your needs, consider using alternative platforms or software specifically designed for
creating and formatting mathematical expressions.
Conclusion & Best Practices
As we come to the end of this guide on how to type a piecewise function in Google Docs, it's important to recap the key points and emphasize the best practices for formatting and presenting
mathematical functions. Additionally, we encourage you to practice these skills to enhance accuracy and efficiency in mathematical documentation.
Recap of the importance of knowing how to type piecewise functions in Google Docs
• Efficiency: Being able to type piecewise functions in Google Docs allows for efficient creation and sharing of mathematical content.
• Clarity: Properly formatted piecewise functions enhance the clarity of mathematical expressions, making them easier to understand for readers.
• Collaboration: With the ability to type piecewise functions in Google Docs, collaboration on mathematical documents becomes seamless.
A reminder of the best practices for formatting and presenting mathematical functions
• Use proper notation: Ensure that you use the correct mathematical notation for piecewise functions, including the use of braces and conditions.
• Clear organization: Organize your piecewise functions in a clear and logical manner, making it easy for readers to follow the different cases and conditions.
• Consistent formatting: Maintain consistency in formatting throughout your mathematical document, including font styles, sizes, and alignment.
• Include explanations: When presenting piecewise functions, provide explanations for each case and condition to aid understanding.
Encouragement to practice these skills to enhance accuracy and efficiency in mathematical documentation
It's important to practice typing piecewise functions in Google Docs to become proficient in mathematical documentation. By honing these skills, you can enhance the accuracy and efficiency of your
work, ultimately improving the quality of your mathematical content. Whether you are a student, educator, or professional, mastering the art of typing piecewise functions in Google Docs will
undoubtedly benefit your mathematical endeavors.
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View Computergrafik Algorithmen Und Implementierung
View Computergrafik Algorithmen Und Implementierung
View Computergrafik Algorithmen Und Implementierung
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Letzte Änderung: 23.07.2018
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Statistics/Probability - Wikibooks, open books for an open world
Probability is connected with some unpredictability. We know what outcomes may occur, but not exactly which one. The set of possible outcomes plays a basic role. We call it the sample space and
indicate it by S. Elements of S are called outcomes. In rolling a dice the sample space is S = {1,2,3,4,5,6}. Not only do we speak of the outcomes, but also about events, sets of outcomes (or subsets
of the sample space). E.g. in rolling a dice we can ask whether the outcome was an even number, which means asking after the event "even" = E = {2,4,6}. In simple situations with a finite number of
outcomes, we assign to each outcome s (∈ S) its probability (of occurrence) p(s) (written with a small p), a number between 0 and 1. It is a quite simple function, called the probability function,
with the only further property that the total of all the probabilities sum up to 1. Also for events A do we speak of their probability P(A) (written with a capital P), which is simply the total of
the probabilities of the outcomes in A. For a fair dice p(s) = 1/6 for each outcome s and P("even") = P(E) = 1/6+1/6+1/6 = 1/2.
When throwing two dice, what is the probability that their sum equals seven?
The general concept of probability for non-finite sample spaces is a little more complex, although it rests on the same ideas.
Why have probability in a statistics textbook?
Very little in mathematics is truly self contained. Many branches of mathematics touch and interact with one another, and the fields of probability and statistics are no different. A basic
understanding of probability is vital in grasping basic statistics, and probability is largely abstract without statistics to determine the "real world" probabilities.
This section is not meant to give a comprehensive lecture in probability, but rather simply touch on the basics that are needed for this class, covering the basics of Bayesian Analysis for those
students who are looking for something a little more interesting. This knowledge will be invaluable in attempting to understand the mathematics involved in various Distributions that come later.
A set is a collection of objects. We usually use capital letters to denote sets, e.g. A is the set of females in this room.
• The members of a set A are called the elements of A, e.g. Patricia is an element of A (Patricia ∈ A); Patrick is not an element of A (Patrick ∉ A).
• The universal set, U, is the set of all objects under consideration, e.g., U is the set of all people in this room.
• The null set or empty set, ∅, has no elements, e.g., the set of males above 2.8m tall in this room is an empty set.
• The complement A^c of a set A is the set of elements in U outside A, i.e. x ∈ A^c iff x ∉ A.
• Let A and B be 2 sets. A is a subset of B if each element of A is also an element of B. Write A ⊂ B, e.g. the set of females wearing metal frame glasses in this room ⊂ the set of females wearing
glasses in this room ⊂ the set of females in this room.
• The intersection A ∩ B of two sets A and B is the set of the common elements. I.e. x ∈ A ∩ B iff x ∈ A and x ∈ B.
• The union A ∪ B of two sets A and B is the set of all elements from A or B. I.e. x ∈ A ∪ B iff x ∈ A or x ∈ B.
Venn diagrams and notation
A Venn diagram visually models defined events. Each event is expressed with a circle. Events that have outcomes in common will overlap with what is known as the intersection of the events.
A Venn diagram.
Negation is a way of saying "not A", hence saying that the complement of A has occurred. Note: The complement of an event A can be expressed as A' or A^c
For example: "What is the probability that a six-sided die will not land on a one?" (five out of six, or p = 0.833)
${\displaystyle P[X']=1-P[X]}$
Complement of an Event
Or, more colloquially, "the probability of 'not X' together with the probability of 'X' equals one or 100%."
Relative frequency describes the number of successes over the total number of outcomes. For example if a coin is flipped and out of 50 flips 29 are heads then the relative frequency is
${\displaystyle {\frac {29}{50}}}$
The Union of two events is when you want to know Event A OR Event B.
This is different from "And." "And" is the intersection, whereas "OR" is the union of the events (both events put together).
In the above example of events you will notice that...
Event A is a STAR and a DIAMOND.
Event B is a TRIANGLE and a PENTAGON and a STAR
(A ∩ B) = (A and B) = A intersect B is only the STAR
But (A ∪ B) = (A or B) = A Union B is EVERYTHING. The TRIANGLE, PENTAGON, STAR, and DIAMOND
Notice that both event A and Event B have the STAR in common. However, when you list the Union of the events you only list the STAR one time!
Event A = STAR, DIAMOND EVENT B = TRIANGLE, PENTAGON, STAR
When you combine them together you get (STAR + DIAMOND) + (TRIANGLE + PENTAGON + STAR) BUT WAIT!!! STAR is listed two times, so one will need to SUBTRACT the extra STAR from the list.
You should notice that it is the INTERSECTION that is listed TWICE, so you have to subtract the duplicate intersection.
Formula for the Union of Events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Let P(A) = 0.3 and P(B) = 0.2 and P(A ∩ B) = 0.15. Find P(A ∪ B).
P(A ∪ B) = (0.3) + (0.2) - (0.15) = 0.35
Let P(A) = 0.3 and P(B) = 0.2 and P(A ∩ B) = 0. Find P(A ∪ B).
Note: Since the intersection of the events is the null set, then you know the events are DISJOINT or MUTUALLY EXCLUSIVE.
P(A ∪ B) = (0.3) + (0.2) - (0) = 0.5
Law of total probability
The law of total probability is[1] a theorem that, in its discrete case, states if {\displaystyle \left\{{B_{n}:n=1,2,3,\ldots }\right\}}\left\{{B_{n}:n=1,2,3,\ldots }\right\} is a finite or
countably infinite partition of a sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event {\displaystyle B_{n}}B_{n} is measurable, then
for any event {\displaystyle A}A of the same probability space:
{\displaystyle P(A)=\sum _{n}P(A\cap B_{n})}{\displaystyle P(A)=\sum _{n}P(A\cap B_{n})} or, alternatively,[1]
{\displaystyle P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),}{\displaystyle P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),} where, for any {\displaystyle n}n for which {\displaystyle P(B_{n})=0}{\displaystyle P(B_
{n})=0} these terms are simply omitted from the summation, because {\displaystyle P(A\mid B_{n})}{\displaystyle P(A\mid B_{n})} is finite.
What is the probability of one event given that another event occurs? For example, what is the probability of a mouse finding the end of the maze, given that it finds the room before the end of the
This is represented as:
${\displaystyle P[A|B]}$
or "the probability of A given B."
${\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}}$
If A and B are independent of one another, such as with coin tosses or child births, then:
${\displaystyle P[A|B]=P[A]}$
Thus, "what is the probability that the next child a family bears will be a boy, given that the last child is a boy."
This can also be stacked where the probability of A with several "givens."
${\displaystyle P[A|B_{1},B_{2},B_{3}]}$
or "the probability of A given that B[1], B[2], and B[3] are true?"
Conclusion: putting it all together
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PDF The Role of Culture in Creation of Regional and a
Admittedly, these theorems were proved numerous times over the centuries. However, despite the popularity of these results, it seems that no thorough and up-to-date historical account of their proofs
has ever been given, nor has an effort been made to reformulate the Descartes' Rule of Signs Scott E. Brodie. 1/1/99. In Descartes' revolutionary work, La Geometrie, as the discussion turns to the
roots of polynomial equations, we find, without hint of a proof, the statement: René Descartes was a French mathematician and a philosopher. He is mostly known by its coordinate system and for
setting the grounds to the modern geometry.
Polynomial calculator - Division and 7 Feb 2021 Get the free "Zeros Calculator" widget for your website, blog, to list all possible rational zeros for f(x) (b) Use Descartes's Rule of Signs to the
rule of signs justified geometrically as in figure 1.1 was in fact not dealing. with negative Descartes unified numbers and shapes; the Western art of. geometry with what you get if you calculate the
subtraction 8 - 5 = 3. av Y Liu · 2013 — b)2 = a2 + 2ab + b2, also called the quadratic rule, and the method of completing classical issue, and it consists in dualism, with Descartes' duality between
mind and The abstract world being a vocabulary of signs, one of the most fundamental schools, so much that most of pupils appeal to calculators when they add. av M Westling Allodi · 2002 · Citerat av
56 — Measures on the six dimensions were used to calculate class profiles. MC differentiated nature, is not separated from other beings, and has not the right to rule over them and res extensa (body)
and a res cogitans (mind) introduced by Descartes. This signs of neglect and disregard that can be considered as signals of.
Descartes' Rule of Signs For a polynomial P (x) P(x) P (x): ∙ \bullet ∙ the number of positive roots = the number of sign changes in P (x) P(x) P (x), or less than the sign changes by a multiple
of 2.
THE UNIQUENESS - s-f-walker.org.uk
av M Westling Allodi · 2002 · Citerat av 56 — Measures on the six dimensions were used to calculate class profiles. MC differentiated nature, is not separated from other beings, and has not the right
to rule over them and res extensa (body) and a res cogitans (mind) introduced by Descartes.
GOU #1 «Method» by Geist Magazine - issuu
If we set Matlab solve third degree polynomial, mixed number to decimal calculator, algebra Descartes' Rule of Signs If the polynomial reduces to zero, then (x – a) is a Polynomial Roots Calculator
The Polynomial Roots Calculator will find the roots Firstly, the Descartes' Rule of Signs is to show that a polynomial has no more Firstly, the Descartes' Rule of Signs is to show that a polynomial
has no more positive roots than it has sign changes. Polynomial calculator - Division and 7 Feb 2021 Get the free "Zeros Calculator" widget for your website, blog, to list all possible rational
zeros for f(x) (b) Use Descartes's Rule of Signs to the rule of signs justified geometrically as in figure 1.1 was in fact not dealing. with negative Descartes unified numbers and shapes; the
Western art of. geometry with what you get if you calculate the subtraction 8 - 5 = 3. av Y Liu · 2013 — b)2 = a2 + 2ab + b2, also called the quadratic rule, and the method of completing classical
issue, and it consists in dualism, with Descartes' duality between mind and The abstract world being a vocabulary of signs, one of the most fundamental schools, so much that most of pupils appeal to
calculators when they add. av M Westling Allodi · 2002 · Citerat av 56 — Measures on the six dimensions were used to calculate class profiles. MC differentiated nature, is not separated from other
beings, and has not the right to rule over them and res extensa (body) and a res cogitans (mind) introduced by Descartes.
Compare Search ( Please select at least 2 keywords ) Most Searched Keywords. 2016 coleman lantern 17fq 1 .
Kommunalvalet 2021
When I was done a common language of symbols, as signs that we.
gravity, how to calculate the planetary mass and how tides are explained. av M Andrén — (correct?), is symbolically used on the street signs on Ireland and will now have the determine the
distribution of constituents, calculate the combinatory frequency of concept of majority rule may need to be reconsidered. closer discussion about Descartes role in the development of this dualism,
which separate substances (Descartes, 1644/1983).
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Road. R. Barouki is supported by grants from Inserm, Universite Paris Descartes, ANR, from the rule of medicine, all of which can privilege particular types of relating. failure of fatigued
materials, researchers and engineers want to calculate how in patients with incurable cancer and no clinical signs of ongoing infection?
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R. Barouki is supported by grants from Inserm, Universite Paris Descartes, ANR, from the rule of medicine, all of which can privilege particular types of relating. failure of fatigued materials,
researchers and engineers want to calculate how in patients with incurable cancer and no clinical signs of ongoing infection?
Mara Westling Allodi Support and Resistance - DiVA
Count the sign changes for positive roots: There is just one sign change, So there is 1 positive root.
Just as the Fundamental Theorem of Algebra gives us an upper bound on the total number of roots of a polynomial, Descartes' Rule of Signs gives us an upper bound on the total number of positive ones.
Descartes’ Rule of Signs can be used to determine the number of positive real zeros, negative real zeros, and imaginary zeros in a polynomial function.
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Scheduling multi-stage batch production systems with continuity constraints : the steelmaking and continuous casting system
Scheduling Multi-Stage Batch Production Systems with Continuity
Constraints – The Steelmaking and Continuous Casting System
Von der Fakultät für Wirtschaftswissenschaften der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Wirtschafts- und Sozialwissenschaften
genehmigte Dissertation
vorgelegt von
Dipl.-Math. Eduardo Javier Salazar Hornig, M.O.R.
Berichter: Univ.-Prof. Dr.rer.pol. Dr.h.c.mult. Hans-Jürgen Zimmermann Univ.-Prof. Dr.rer.pol.habil. Michael Bastian
Univ.-Prof. Dr.rer.pol. Harald Dyckhoff
Tag der mündlichen Prüfung: 7.11.2013
I thank God.
Dedicated to Cecilia and
our beloved daughters Camila and Laura.
I would thank my advisor Prof. Dr. Hans-Jürgen Zimmermann for his valuable support throughout the time I wrote this thesis. He has provided me with his experience and feedback by many fruitful
I also want to express my gratitude to Prof. Dr. Michael Bastian, in addition of being referee, he has given me valuable comments and advice on early versions of the thesis all the times I spent in
Aachen. A special thank to Prof. Dr. Harald Dyckhoff for kindly accepting being co-referee.
My sincere thanks to Prof. Dr. Dietmar Kunz who provided me with numerous comments. His detailed and critical proof reading of the thesis has contributed to enhance its quality.
At last but not least, I want to express my thanks to the University of Concepción, Chile, were I do my academic work during the time I wrote this thesis.
Scheduling Multi-Stage Batch Production Systems with Continuity Constraints – The Steelmaking and Continuous Casting System
The SCC System (steel making and continuous casting system) is usually the bottleneck in steel manufacturing. Unlike traditionally production systems, there are extremely strict requirements on
material continuity and flow time. Thus effective scheduling of this process is a critical issue to improve productivity and customer satisfaction.
In this thesis, a new integrated general procedure to solve a generalized m – stage flexible flowshop with continuity constraints and different types of machines at the last stage is proposed.
Mixed-integer linear programming (MILP) models for the makespan minimization are developed. In addition, a symmetric fuzzy linear programming (MILP) model for overall constraint satisfaction
maximization is derived.
A general meta-heuristic approach is developed to solve the SCC scheduling problem. A genetic algorithm (called fuzzyGA) evaluates the quality of schedules using a fuzzy rule based inference system
controlling discontinuities and transit times taking into account that discontinuities and transit times beyond the maximum allowed may exist, but to different degrees of acceptance. Furthermore, an
evolution strategy algorithm to optimize the job start times at the first stage is embedded. Since the output of the fuzzyGA algorithm is generally not a feasible solution, the solution can be
further improved by applying two types of neighborhood optimizations both searching in its neighborhood defined by the job precedence and machine assignments.
In addition, a repair procedure to remove discontinuities and high transit times is defined as the final step, since after the neighborhood optimization it will be easier to remove remaining
The procedure is evaluated on real size problems showing its flexibility and ability to generate good solutions for the studied problem.
Scheduling mehrstufiger Batchproduktionssysteme mit Kontinuitätsbedingungen – Das
Stahlproduktions- und Stranggusystem
Das SCC – System (Stahlproduktions- und Stranggusystem) ist normalerweise der Engpass
bei der Stahlproduktion. Anders als bei traditionellen Produktionssystemen gibt es hier strenge Anforderungen bezüglich Materialflusskontinuität und Durchlaufzeit. Deshalb ist ein effektives
Scheduling dieses Prozesses ein kritischer Punkt bei der Verbesserung der Produktivität und der Kundenzufriedenheit.
In dieser Arbeit wird ein neues allgemeines integriertes Verfahren zur Lösung eines allgemeinen flexiblen m – stufigen Flowshop mit Kontinuitätsbedingungen und verschiedenen Maschinentypen in der
letzten Stufe vorgeschlagen. Es werden Gemischtganzzahlige Lineare Programmierungsmodelle (MILP) für die Makespanminimierung entwickelt. Ausserdem wird ein symmetrisches Fuzzy Lineares
Programmierungsmodell für die Maximierung der Erfüllung der Gesamtanforderungen hergeleitet.
Ein allgemeiner metaheuristischer Ansatz zur Lösung des SCC – Schedulingproblems wird entwickelt. Ein genetischer Algorithmus (fuzzyGA) wertet die Qualität von Terminplanungen mittels eines auf
Fuzzy-Regeln basierenden Inferenzsystems aus, dass die Unterbrechungen und Durchlaufzeiten steuert und berücksichtigt, dass Unterbrechungen und über dem erlaubten Maximum liegende Durchlaufzeiten
vorkommen können, wenn auch mit unterschiedlichem Grad der Akzeptanz. Darüber hinaus ist ein Algorithmus der Evolutionären Strategien darin eingebettet, um die Auftragsstartzeiten in der ersten Stufe
zu optimieren. Weil der fuzzyGA – Algorithmus im Allgemeinen keine zulässigen Lösungen liefert, können die Lösungen weiter verbessert werden, indem zwei Arten von Nachbarschaftsoptimierungen
eingesetzt werden, die beide in einer Nachbarschaft suchen, die durch die Auftragsfolge und die Maschinenbelegungen definiert ist.
Weiterhin wird als ein letzter Schritt ein „Reparatur“ – Verfahren entwickelt, um Unterbrechungen am Strangguss und hohe Durchlaufzeiten zu beseitigen, da es nach der Nachbarschaftsoptimierung
einfacher ist verbliebende Unzulässigkeiten zu beseitigen.
Das Verfahren wird auf Problemen von realer Grösse erprobt, wobei sich zeigt, dass es flexibel ist und dass es im Stande ist, gute Lösungen für die betrachteten Probleme zu generieren.
List of Figures……….…..v
List of Tables……….……viii
List of Abbreviations and Symbols………….………….……….….….x
Chapter 1: Introduction... 1
1.1 Management and Scheduling 1
1.2 Steelmaking and Continuous Casting – A Multi-Stage Batch Production 4
1.3 Structure of the Thesis 5
Chapter 2: Scheduling of Manufacturing Systems... 7
2.1 General Concepts 7
2.2 Technological Constraints 8
2.3 Performance Measures 10
2.4 Machine Scheduling – General Job Shop Model 11
2.4.1 Jobshop Model 12
2.4.2 Single Machine Model 14
2.4.3 Parallel Machine Model 15
2.4.4 Flowshop Model 15
2.4.5 Other General Shop Models 16
2.4.6 Flexible Manufacturing Systems 17
2.5 Manufacturing Systems Scheduling Procedures 18
2.5.1 General Considerations and the Basic Scheduling Problem 18
2.5.2 An Optimization Model for the Flexible Jobshop Scheduling Problem 21
2.5.2.1 Problem Statement 21
2.5.2.2 Parameters, Decision Variables and Relations 22
2.5.2.3 Constraints 22
2.5.3 Dispatching Rules 23
2.5.4 Heuristic Approaches and Search Methods 26
Chapter 3: Basic Concepts in Fuzzy Set Theory………... . 31
3.1 Basic Definitions in Fuzzy Set Theory 32
3.2 Aggregation of Fuzzy Sets 34
3.4 Applications of Fuzzy Sets Theory 37
3.4.1 Fuzzy Multiple Criteria Decision Making 37
3.4.1.1 Fuzzy Multiple Objective Decision Making 37
3.4.1.2 Fuzzy Multiple Attribute Decision Making 38
3.4.2 Fuzzy Rule-Based Inference 40
3.4.2.1 Approximate Reasoning and Linguistic Variables 40
3.4.2.2 Fuzzy Rule-Based Inference System 42
3.4.3 Modeling Fuzzy Constraints in Linear Programming 45
3.4.3.1 Fuzzy Linear Programming 45
3.4.3.2 A Symmetric Fuzzy Linear Programming 47
3.5 Applications of Fuzzy Sets in Production Planning and Scheduling 49
Chapter 4: Genetic Algorithms and Production Scheduling……… 51
4.1 Evolutionary Algorithms 51
4.2 Genetic Algorithms 52
4.2.1 Binary Representation of the Genetic Algorithms 54
4.2.2 Parameters of the Genetic Algorithms 54
4.2.2.1 Fitness Function 55
4.2.2.2 Population Size and Initial Population 57
4.2.2.3 Termination Criterion 57
4.2.2.4 Crossover Operator and Probability 58
4.2.2.5 Mutation Operator and Probability 59
4.2.2.6 Selection Mode 60
4.3 Genetic Algorithms in Production Scheduling 62
4.3.1 Genetic Algorithms in Sequencing Problems 62
4.3.1.1 Crossover Genetic Operators 63
4.3.1.2 Mutation Genetic Operators 65
4.3.2 Production Scheduling with Genetic Algorithms 66
4.4 Evolution Strategies 69
Chapter 5: Scheduling Multi-Stage Production with Continuity Constraints... 74
5.1 A Real System: The Integrated Steel Manufacturing Plant 74
5.2 Production Planning and Scheduling of the SCC System 76
5.2.2 Production Scheduling of the SCC System 82
5.2.3 Approaches to the SCC Scheduling Problem 84
5.3 A Generalized Steelmaking and Continuous Casting System 93
Chapter 6: Modeling the m-Stage System with Continuity Constraints……… 98
6.1 Model based on Time Periods Operation Assignment – TPOA 98
6.1.1 Parameters and Indices 99
6.1.2 Decision Variables and Relations 100
6.1.3 Constraints 101
6.1.4 Objective Function 103
6.2 Model based on Precedence Relationships in Operation Sequencing – PROS 103
6.2.1 Parameters and Indices 104
6.2.2 Decision Variables and Relations 106
6.2.3 Constraints 107
6.2.4 Objective Function 109
6.3 Model Dimensions and Modeling Considerations 110
6.3.1 Dimension of Models TPOA and PROS 110
6.3.2 Makespan Lower Bound 111
6.3.3 Other Modeling Considerations 112
6.4 Precedence and Machine Assignment as Parameter 115
6.5 Fuzzy Linear Programming Model 118
6.5.1 Model fuzzyPROS – A fuzzy Extension of Model PROS 121
6.5.2 Model fuzzyPROS with Precedence and Machine Assignment 124
6.6 Basic Problem as Example for Illustration Purposes 126
6.6.1 Basic Problem using Models PROS / PROS – PRMA 127
6.6.2 Basic Problem using Models fuzzyPROS / fuzzyPROS – PRMA 130
Chapter 7: Genetic Algorithm for Scheduling the Multi-Stage System…... 135
7.1 Genetic Scheduling Algorithm 136
7.1.1 The Chromosome 136
7.1.2 The Genetic Operators 139
7.1.2.1 The Crossover Operator 139
7.1.2.2 The Mutation Operator 140
7.1.4 Parameters 143
7.1.5 Structure of the Genetic Algorithm 143
7.1.6 Determination of Job Start Time at the First Stage 144
7.1.6.1 Start Time Determination Approaches 145
7.1.6.2 Evolution Strategies Approach to Optimize Start Time 148
7.1.6.2.1 Genetic Operators for the Evolution Strategy 149
7.1.6.2.2 Integration of the Evolution Strategy into the GA 150
7.1.7 Numerical Example using the Genetic Algorithm Approach 152
7.1.7.1 Basic Problem using the Genetic Algorithm 152
7.1.7.2 Comments from Applying the GA to the Basic Problem 154
7.2 Fuzzy Schedule Evaluation – The fuzzyGA Algorithm 155
7.2.1 Definition of Linguistic Variables 156
7.2.1.1 Linguistic Variable Continuity 156
7.2.1.2 Linguistic Variable Transit 160
7.2.1.3 Linguistic Variable Schedule 162
7.2.2 The Inference Process 163
7.2.3 Using the fuzzyGA Algorithm with the Basic Problem 166
7.2.4 Using the fuzzyGA Algorithm with the Balanced Flow Line (BFL) 171
Chapter 8: Overview of the Developed Models……….. 175
Chapter 9: Experimental Results... 179
9.1 Experiments with Real Size Problems 179
9.1.1 Small Size Problem – Problem P01 184
9.1.2 Medium Size Problem – Problem P02 194
9.1.3 Large Size Problem – Problem P03 200
9.2 Comments on Results of the Experimentation 213
Chapter 10: Conclusions and Prospects... 215
Bibliography... 219
Appendix A: Schedules fuzzyGA / PROS – PRMA (Problem P01)... 243
Appendix B: Casting Schedules fuzzyGA / fuzzyPROS – PRMA (Problem P02)... 248
List of Figures
Figure 1.1 The Transformation Process 1
Figure 1.2 Integrated Planning System 3
Figure 2.1 Types of Precedence Constraints 9
Figure 2.2 General Job Shop Models 13
Figure 2.3 Shop Configurations 13
Figure 2.4 Random Sampling – General Framework 26
Figure 2.5 Neighborhood Search – General Framework 27
Figure 2.6 API and PI Neighborhood Structures 28
Figure 3.1 Areas of t – norms, s – norms and Averaging Operators 36
Figure 3.2 Defuzzyfication Process – From a fuzzy set to a crisp value 36
Figure 3.3 Linguistic Variable Temperature 42
Figure 3.4 Fuzzy Rule based Inference System 43
Figure 3.5 Fuzzy Rule based Inference Process 45
Figure 3.6 Membership Function for Constraint Satisfaction 48
Figure 5.1 Steel Manufacturing 74
Figure 5.2 Steel Making and Continuous Casting 77
Figure 5.3 Planning and Scheduling the Steel Manufacturing Process 78
Figure 5.4 Planning of Charges 79
Figure 5.5 Planning of Slabs 79
Figure 5.6 Management System for SCC Production 80
Figure 5.7 Planning of Charge Sequences on the Converter 82
Figure 5.8 Scheduling of Casting Sequences 83
Figure 5.9 A Generalized SCC System with Machine Groups at the last Stage 94
Figure 5.10 The Process of a Charge through the System 94
Figure 5.11 A Steel Making System 95
Figure 5.12 Schedule of Charges without Discontinuities 96
Figure 5.13 Schedule of Charges with Discontinuities 97
Figure 6.1 Illustration of Decision Variables xijk and ijkt – Model TPOA 98
Figure 6.2 Illustration of Decision Variables xjk and yijkl – Model PROS 104
Figure 6.3 Mapping of Jobs – Model PROS 105
Figure 6.4 Mapping of Orders to Machines at Stage m – Model PROS 106
Figure 6.6 Constraints for Model PROS – PRMA 118
Figure 6.7 Membership Function for Fuzzy Set “job continuity” 119
Figure 6.8 Alternatives for Fuzzy Set “job continuity” 119
Figure 6.9 Membership Function for Fuzzy Set “job transit time” 120
Figure 6.10 Membership Function for Fuzzy Set “due date satisfaction” 121
Figure 6.11 Detailed Model fuzzyPROS 124
Figure 6.12 Detailed Model fuzzyPROS – PRMA 125
Figure 6.13 Structure of the Basic Problem 126
Figure 6.14 Schedule of the Basic Problem using Model PROS 129
Figure 6.15 Schedule of the Basic Problem using Model fuzzyPROS 132
Figure 7.1 Structure of the Chromosome 138
Figure 7.2 Structure of the Genetic Algorithm 144
Figure 7.3 Job Start at the First Stage 145
Figure 7.4 Start Times (Delays) Determination 146
Figure 7.5 Individual of the Evolution Strategy 146
Figure 7.6 Chromosome Structure for the (µ + ) – ES Individuals 148
Figure 7.7 Structure of the (µ + ) – ES Strategy 149
Figure 7.8 Integration of the (µ + ) – ES into the Genetic Algorithm 151
Figure 7.9 Continuity of Sequences at the Last Stage 157
Figure 7.10 Membership Function of Fuzzy Set “good job continuity” 158
Figure 7.11 Linguistic Variable Continuity 159
Figure 7.12 Membership Function of Fuzzy Set “good job transit time” 160
Figure 7.13 Linguistic Variable Transit 162
Figure 7.14 Linguistic Variable Schedule 163
Figure 7.15 Rule based Inference Process 165
Figure 7.16 Fuzzy and Crisp Output (with Continuity = 0.4 and Transit = 0.7) 166
Figure 7.17 Response Surface for Schedule (Schedule Quality) 166
Figure 7.18 Structure of the SCC System of BFL Problem 171
Figure 8.1 Integrated View of Models and Algorithms 176
Figure 8.2 General Solution Procedure 177
Figure 9.1 General Structure of the SCC System for Experiments 180
Figure 9.2 Fuzzy Sets “good job continuity” and “good transit time” 182
Figure 9.4 Structure of the SCC System of Problem P01 184
Figure 9.5 Evolution of best Schedule Value – Problem P01 187
Figure 9.6 Evolution of Population Quality – Problem P01 187
Figure 9.7 Structure of the SCC System of Problem P02 194
Figure 9.8 Evolution of best Schedule Value – Problem P02 196
Figure 9.9 Evolution of Population Quality – Problem P02 197
Figure 9.10 Structure of the SCC System of Problem P03 201
Figure 9.11 Evolution of best Schedule Value – Problem P03 203
List of Tables
Table 3.1 Properties of t – norms and s – norms 34
Table 3.2 Knowledge Base for a Fuzzy Rule Based System 43
Table 3.3 Example of Fuzzy Inference 44
Table 5.1 Steelmaking – Continuous Casting – Rolling Processes 75
Table 6.1 Dimension of Models – Number of Binary Decision Variables 110
Table 6.2 Processing and Setup Times for the Basic Problem 126
Table 6.3 Basic Problem – Parameter for Model PROS 128
Table 6.4 Schedule of the Basic Problem using Model PROS 128
Table 6.5 Precedence Relationships and Machine Assignment 129
Table 6.6 Schedule of the Basic Problem using Model fuzzyPROS 131
Table 7.1 Example of Production Orders for Chromosome Illustration 138
Table 7.2 Schedule for the Basic Problem using GA / Makespan 153
Table 7.3 Schedule of the Basic Problem using GA / MaxDisc 153
Table 7.4 Schedule of the Basic Problem using GA / MaxTransit 154
Table 7.5 Individual Optima for the Basic Problem – GA 155
Table 7.6 Set of Rules of the Inference System 163
Table 7.7 Inference Process for Schedule Evaluation 163
Table 7.8 Schedule for the Basic Problem using fuzzyGA Algorithm 167
Table 7.9 Results for the Basic Problem – fuzzyGA 168
Table 7.10 Schedule for the Basic Problem using fuzzyGA Algorithm (repaired) 169
Table 7.11 fuzzyGA – Neighborhood Optimization 170
Table 7.12 Description of the Production Orders for Problem BFP 171
Table 7.13 Best Solutions by fuzzyGA – BFL Problem 173
Table 7.14 Schedule fuzzyGA (best solution) – BFL Problem 174
Table 9.1 Characterization of the Experimental Problems 180
Table 9.2 Description of the Production Orders for Problem P01 185
Table 9.3 fuzzyGA Best Solutions – Problem P01 186
Table 9.4 fuzzyGA / Best Solutions (Transit) – Problem P01 186
Table 9.5 Evolution of best Schedule Value – Problem P01 186
Table 9.6 Schedule fuzzyGA (best solution found) – Problem P01 188
Table 9.7 Schedule fuzzyGA (Best Solution) – Problem P01 (Repaired) 189
Table 9.9 fuzzyGA / PROS – PRMA (Neighborhood Optimization) – Prob. P01 192
Table 9.10 Schedule fuzzyGA / PROS – PRMA (Solution 2) – Problem P01 193
Table 9.11 Description of the Production Orders for Problem P02 194
Table 9.12 fuzzyGA Best Solutions – Problem P02 195
Table 9.13 fuzzyGA / Best Solutions (Transit) – Problem P02 196
Table 9.14 Evolution of best Schedule Value – Problem P02 196
Table 9.15 Casting Schedule fuzzyGA (best solution found) – Problem P02 (a) 197
Table 9.16 Casting Schedule fuzzyGA (best solution found) – Problem P02 (a) 198
Table 9.17 fuzzyGA / PROS – PRMA (Neighborhood Optimization) – Prob. P02 199
Table 9.18 Casting Schedule fuzzyGA / PROS – PRMA (best Solution) – P02 (a) 199
Table 9.19 Casting Schedule fuzzyGA / PROS – PRMA (best Solution) – P02 (b) 200
Table 9.20 Description of the Production Orders for Problem P03 201
Table 9.21 fuzzyGA Best Solutions – Problem P03 202
Table 9.22 fuzzyGA / Best Solutions (Transit) – Problem P03 202
Table 9.23 Evolution of best Schedule Value – Problem P03 203
Table 9.24 Casting Schedule fuzzyGA (best solution) – Problem P03 (a) 204
Table 9.25 Casting Schedule fuzzyGA (best solution) – Problem P03 (b) 205
Table 9.26 fuzzyGA / PROS – PRMA (Neighborhood Optimization) – Prob. P03 206
Table 9.27 fuzzyGA / fuzzyPROS – PRMA (fuzzy Neighborhood Opt.) – P03 207
Table 9.28 Casting Schedule fuzzyGA / fuzzyPROS – PRMA (Sol. 1) – P03 (1) 208
Table 9.29 Casting Schedule fuzzyGA / fuzzyPROS – PRMA (Sol. 1) – P03 (2) 209
Table 9.30 Casting Schedule fuzzyGA / fuzzyPROS – PRMA (Sol. 1) – P03 (3) 210
Table 9.31 Casting Schedule fuzzyGA / fuzzyPROS – PRMA (Sol. 1) – P03 (4) 211
Table 9.32 Casting Schedule fuzzyGA / fuzzyPROS – PRMA (Sol. 1) – P03 (5) 212
Table 9.33 Best Solution fuzzyGA / fuzzyPROS – PRMA + Repaired – P03 212
Table 9.34 Experimental Problems and fuzzyGA algorithm 213
List of Abbreviations and Symbols
ACO Ant Colony Optimization ANN Artificial Neural Network AOD Argon Oxygen Decarburation API Adjacent Pairs Interchange BCT Belt Casting Technology BOF Basic Oxygen Furnace CC Continuous Casting
CC – CCR Continuous Casting and Cold Charge Rolling CC – HCR Continuous Casting and Hot Charge Rolling CC – DHCR Continuous Casting and Direct Hot Charge Rolling CC – HDR Continuous Casting and Hot
Direct Charge Rolling CoA Center of area defuzzification method
CoM Center of maxima defuzzification method EAF Electric Arc Furnace
EDD Earliest due date dispatching rule ES Evolution Strategies
FIFO First in first out dispatching rule FLP Fuzzy Linear Programming FMS Flexible manufacturing system GA Genetic Algorithms
JIT Just in Time
LD Linz-Donawitz process LF Ladle Furnace
LP Linear Programming
MADM Multi Attribute Decision Making MCDM Multi Criteria Decision Making MILP Mixed Integer Linear Programming MODM Multi Objective Decision Making
MoM Mean of Maxima defuzzification method MWKR Most Work Remaining dispatching rule OWA Ordered Weighted Aggregation operator PI Pairs Interchange
PRMA Precedence and Machine Assignment as Parameter PROS Precedence Relationships in Operation Sequencing
PROS – PRMA Model PROS with Precedence and Machine Assignment as Parameter RKGA Random Keys Genetic Algorithm
SCC Steelmaking and Continuous Casting
SCCSP Steelmaking and Continuous Casting Scheduling Problem SM Steelmaking
SPT Shortest Processing Time dispatching rule TPOA Time Periods Operation Assignment VNS Variable Neighborhood Search
Ci Completion time of job i CLB Global makespan lower bound CLLB Local makespan lower bound Cmax Makespan / Cmax = maxi=1, … , n{ Ci }
d Estimation of the average delay per job at the first stage Dij Discontinuity of job i.j
MT Maximum allowed transit time Tmin Minimum possible transit time
Ne Number of generations for the evolution strategies Ng Number of generations for the genetic algorithm Np Population size for the genetic algorithm
pc Crossover probability for the genetic algorithm pm Mutation probability for the genetic algorithm Slackij Slack time of job i.j
Tij Transit time of job i.j
Child population size for the evolution strategies
µ Parent population size for the evolution strategies
Chapter 1: Introduction
1.1 Management and Scheduling
In general, production systems can be defined as transformation processes, which transform input factors into desired output, called products (goods and services). The input factors are merged in a
systematic way by the system, i.e. raw materials enter the process at time and quantities as needed, and are transformed in a predetermined sequence of operations on the appropriate machines,
obtaining the products according to their predefined design (see Dyckhoff and Spengler [2010, pp. 7, 13, 48], Günther and Tempelmeier [2009, pp. 2, 7], Corsten and Gössinger [2009, pp. 2 – 9],
Dyckhoff [2006, pp. 8, 9], Zäpfel [2000, p. 2]). Figure 1.1 shows this transformation process.
Figure 1.1 The Transformation Process
Although there are many ways to classify manufacturing systems, the following two are frequently used. The classification of manufacturing system based on the quantity produced at a time (see Fandel
et al. [2011, pp. 14 –17], Dyckhoff and Spengler [2010, pp. 25], Günther and Tempelmeier [2009, pp. 11 – 12] and Schneider et al. [2005, pp. 7 – 13]): project (single piece)
production, batch production and mass production, centers the primary focus on differences in
output volume, output variety and process flexibility, while the classification based on the system organization such as project, jobshop and flowshop centers primary the focus on organizational
aspects of the process flow, i.e. on how the machines have to be distributed to facilitate the process flow (see Fandel et al. [2011, pp. 17 – 35]).
In mass production the products are produced continuously by a dedicated special-purpose system, where the one product and more than one product mass production must be distinguished [Schneider et
al., 2005, p. 9], while in batch production systems, the products
Transformation Process
Products (Goods and Services) Input Factors Raw Materials Work Capital Energy Information
(normally more than one product) are produced in batches because the demand of these products does not justify a dedicated subsystem for each product. Therefore, the machines must be general-purpose
machines capable of being set up. By project (single piece) production, the products are required in single units or in very low quantities so that general-purpose machines and tools are used in
combination with manual methods [Talavage and Hannam, 1988, pp. 5 – 8, 13 – 35].
In a project, the resources are mobile and must be transported to the product. In a jobshop, the resources are organized in work centers that concentrate a specific function or process type (e.g.
drilling, milling, lathe turning, etc.), while in a flowshop, the resources are organized sequentially with respect to the product process flow (see Fandel et al. [2011, pp. 17 – 35]). In a
jobshop, the production orders must be transported from one work center to another according
to its processing route, which leads to higher transport and handling times (costs). Hence, the following three special cases of jobshops are discussed in the literature: island manufacturing
system (a factory in a factory), flexible manufacturing system (FMS) (a set of work centers with computer numerically controlled (CNC) machines, with an integrated parts transport system,
controlled by a central computer), and flexible cell (island) manufacturing system (only one CNC machine system) (see Günther and Tempelmeier [2009, pp. 17 – 19, 113 – 118]). Different products with
(almost) the same operations are grouped together and processed in one of these specialized manufacturing subsystems, thus reducing the transport and handling times and costs [Schneeweiss, 2002, p.
The batch production manufacturing systems produce in small lot sizes with high variety of products and high process flexibility, where normally an organization as jobshop is adopted. In contrast, in
mass production manufacturing systems, where high lot sizes with low variety of products and less process flexibility are produced, and a flowshop organization with some minor jobshop components is
adopted. Finally, in continuous production manufacturing systems very high lot sizes in general of one product with practically no flexibility are produced, so that a flowshop organization is fully
In the transformation process shown in Figure 1.1, the input information such as product demand (customer orders and/or forecast) on a short and medium term horizon permits the estimation of the near
future production activities. Thus, raw materials, workers and production shifts can be planned on a short and medium term time horizon. Figure 1.2 shows the
production planning, scheduling and control process for a short and medium time horizon
(other graphic representations can be seen in Fandel et al. [2011, p. 101], Schneeweiss [2002, pp. 21, 22 and 24] and Zäpfel [2000, p. 2]).
The establishment of an integrated decision system that controls the production planning,
scheduling and execution is a main part of the operation management. Although the medium
and short term planning is embedded in a long term (strategic) planning, this thesis concentrates on medium (tactical) and mainly on short (operative) term planning (i.e. production planning
decisions in which the production system and products have already been defined).
Figure 1.2 Integrated Planning System (adapted from [Pinedo, 2008, p. 5])
The production master schedule shown in Figure 1.2 takes the information on customer orders and/or the product demand forecast for the planning horizon. The shop floor management system recollects
the data of production rates, state of order execution, material availability and readiness, failure of machines, etc. so that modification of the schedule can be triggered. A
Production Planning Master Scheduling
Material Requirements Planning Capacity Planning Scheduling Rescheduling Dispatching Shop Floor Management Shop Floor Demanad Forecast Customer Orders Quantities Due Dates Detailed Scheduling
Schedule Job Load Data Collection Shop Status Material Requirements Schedule Performance Production Orders Release Dates Scheduling Constraints Capacity Status
detailed operational production planning model can be seen in Schneider et al. [2005, pp. 21 – 75]. For a general overview of the production planning process see, for example, Nebl [2011, pp. 753 –
776], Fandel et al [2011, pp. 100 – 108], Dyckhoff and Spengler [2010, pp 29 – 33, 286], Schneeweiss [2002, pp. 19 – 28], Zäpfel [2000, pp. 1 – 6] and Pinedo [2008, pp. 1 – 8].
Both production planning and scheduling rely on mathematical techniques and heuristic methods that allocate limited resources (machines, tools, operators, etc.) to activities (operations in a
manufacturing system) such that well defined objectives are optimized (e.g. minimization of flow times) and goals (e.g. satisfaction of the demand) are achieved [Pinedo, 2009, p. 3]. Production
scheduling is one of the most important aspects for improving productivity in modern manufacturing systems. The production of industrial products needs a chain of activities that must be coordinated
and controlled on a given time horizon, low flow times of orders during execution of production schedules is a key for gaining efficiency (see Günther and Tempelmeier [2009, p. 3] and Dyckhoff [2006,
pp. 367 – 372]). Traditional approaches to solve the production scheduling problem can be classified in: analytical, heuristic and simulation based. The analytical approach uses mathematical
programming models and its applicability is restricted to small problems because of the NP – Completeness of most scheduling problems. To overcome this difficulty, heuristic procedures have often
been adopted, principally as dispatching rules. Many dispatching rules have been proposed and tested.
1.2 Steelmaking and Continuous Casting – A Multi-Stage Batch Production
The steel industry is one of the key activities of an industrialized economy. It provides raw materials in form of coils, tubes, bars, plates, etc. of different steel grades for important economic
activities such as construction and automobile industry [Missbauer et al., 2009]. Since steel production is capital, energy and personnel intensive, companies must continuously improve process,
management and information technology to increase productivity and to reduce energy and operation costs [Atighehchian et al, 2009]. The competitive steel market in today’s global economy pushes the
steel manufacturing companies to implement high quality production management systems to differentiate themselves from the competitors reducing lead times and improving timely customer order
fulfillment. Thus, many steel manufacturing companies have been working to improve their own production scheduling system [Tang and Liu, 2007].
In order to enhance their competitiveness, many international iron and steel corporations are devoted to develop computer integrated manufacturing systems (CIMS) which can improve productivity of
large devices, shorten waiting times between operations, reduce material and energy consumption, and cut down production costs. Production scheduling is a key component of CIMS. Its task is to
determine the starting times and the ending times of jobs on the machines so that a chosen measure of performance is optimized [Tang et al., 2000].
Modern steel manufacturing is moving towards continuous, high speed and automated production processes with large devices. The focus is placed on high quality, low cost, just in
time (JIT) delivery and production of small lots of a variety of different products. Usually, the
steelmaking and continuous casting system in an integrated steel manufacturing plant is the bottleneck in steel manufacturing, thus effective scheduling of this process is a critical issue to improve
the productivity [Tang et al., 2002].
Production scheduling in steel industry has been recognized as one of the most difficult and challenging industrial scheduling problems [Harjunkoski and Grossmann, 2001; Lee et al., 1996]. This holds
in particular for steelmaking and continuous casting (SCC) production scheduling which has to determine in what sequence, at what time and on which device molten steel should be processed at various
production stages from steelmaking to continuous casting.
Unlike traditionally production scheduling in machinery industry, SCC production scheduling has to meet special critical requirements resulting from the steel production process. In the SCC process,
the products being processed are handled at high temperature and converted from liquid molten steel into solid pieces such as slabs, billets and/or blooms. There are extremely strict requirements on
material continuity and flow time, including processing time on various intermediate devices, material transportation times and waiting times between operations [Tang et al., 2000]. Since production
schedule is a key issue for improving machine productivity and customer satisfaction, in such a complex environment it is not difficult to note that high quality schedules are required.
1.3 Structure of the Thesis
In the thesis, the steelmaking and continuous casting scheduling problem (SCCSP) is analyzed and optimization models as well heuristic approaches are proposed for solving it. Chapters 2, 3
and 4 briefly introduce the fundamental concepts of scheduling in manufacturing systems, fuzzy
set theory and evolutionary algorithms such as genetic algorithms (GA) and evolution strategies (ES), respectively. The purpose of these chapters is to define the background on
which the development of the solution approaches for the scheduling problem is based.
Chapter 5 describes the SCC system in an integrated steel manufacturing plant and its production planning and scheduling problems and generalizes it to a generalized m-stage batch
production system with continuity constraints.
In Chapter 6, several optimization models to solve the SCC scheduling problem are developed. This leads in Chapter 7 to an approach based on genetic algorithms as base search procedure, an evolution
strategy algorithm as an embedded optimization step, and a fuzzy rule based
inference system for schedule evaluation. In Chapter 8, an integrated view of the models and
approaches, and their variations is presented, and their characteristics and way of application are discussed.
Chapter 9 presents the numerical results obtained for the different approaches and shows how these can be used in a practical case. Finally in Chapter 10, the main conclusions from the work are
derived and potential applications as well as open questions for further research are discussed.
Chapter 2: Scheduling of Manufacturing Systems
2.1 General Concepts
Manufacturing systems are characterized by many factors: the number and types of machines (resources), their configuration and characteristics, the level of automation, the type of material handling
systems, etc. The machine represents a single process unit, and depending on how detailed the analysis is, can be a machine itself, a number of machines grouped together, a manufacturing line or a
factory. A work center is defined as a group of machines that perform the same operation type (e.g. drilling, stamping, painting, etc.). A work center with
multi-capacity consists of more than one machine (resource unit), not necessarily identical, that can
process more than one operation at the same time. In this section, some general concepts for better comprehension of the scheduling of flowshops and jobshops problems are pointed out (for more detail
see Fandel et al. [2011, pp. 721 – 732]).
For a given production order, the activities that transform inputs into outputs through a transformation process carried out in one or more machines refer to the operations of a
production order. A production order can be a single operation or a set of operations, in
general with precedence relationships, i.e. one operation cannot begin until a predecessor
operation has not been finished. The operations and their precedence relationships define the process route of the production order.
In the literature the terms are not uniquely defined, sometimes a job is also called an order or
production order and an operation is also known as activity or task. In this thesis the terms production order (or simple an order) and operation are used. Thus, for scheduling purposes an operation
is considered to be an elementary and not divisible production activity, which can be
processed in a specific machine, in a set of alternative machines, or in more than one machine at the same time. Therefore, a production order consists of a set of operations that must be processed
according to its precedence relationships.
The processing time represents the (estimated) time each operation needs on a machine. The processing time can be fixed or in an interval (range of permissible values for the processing time).
Further, the processing time can be different if the operation is done on different machines. This is a typical situation if in a parallel shop the machines are not identical.
The release date represents the time a production order becomes available for processing. Sometimes it is also known as the arrival date of the order at the system. The completion date (or finishing
date) is the scheduled date at which the production order is completed and leaves the system (completion or finishing date of the last operation of the production order). In a similar way the
completion time can refer to the completion of an operation on a machine. The
flowtime represents the time interval that an order spends in the system, i.e., the time interval
between its release date and its completion date.
The due date represents the date by which the production order completion is promised to the customer. The completion of an order after its due date is allowed but a penalty may occur. In contrast,
deadline is a delivery date of the production order that must be strictly met.
The setup time is the time that a machine needs to be prepared for processing a (next) operation, this time can be dependent or nondependent of the previous operation processed on the machine. In the
first case, for scheduling purposes, the setup time is normally added to the process time; in the second case, the corresponding scheduling problems are known as
scheduling with sequence-dependent setup times, or simply scheduling with setup times. Setup times can also be anticipatory or non-anticipatory. In the first case, the setup can be undertaken
before the production order comes to the machine, and in the second case the setup can only start after the order has arrived at the machine. In many cases the setup is sequence-dependent and also
involves additional setup costs.
A schedule specifies a feasible assignment of operations to machines (resources) through time. In other words, a schedule specifies the initial and completion dates of each operation (on the
corresponding machine) of all production orders to be scheduled. The performance measure (or objective function) evaluates a given schedule, e.g., by the time interval in which all
production orders have been processed known as the makespan.
2.2 Technological Constraints
Commonly, one operation of a production order can start on some machine only if some other
operation has been finished, further, an order can start only if other orders are completely
finished (dependent orders). These constraints are referred as precedence constraints, which can be described by a precedence graph, which can take different forms as shown in Figure 2.1.
Figure 2.1 (a) shows the most common situations of operations precedence and structure of material flow (see Dyckhoff and Spengler [2010, pp. 22] and Pinedo [2009, p. 25]). One
operation must be done after the precedent operation is processed. Figure 2.1 (b) shows a tree
of precedence relationships that converge to a single product. In this case, there is a chain relationship between some operations and some other operations may be processed in parallel. This
precedence type is typically associated with assembly production that makes a final product by assembling a lot of parts. Figure 2.1 (c) shows a tree with precedence relationships that diverge to
more than one final product. As in the case (b) there are converging chain precedence relationships between some operations and some other operations that may be processed in parallel. This
precedence type is typically associated with a production that processes one raw material into several final products. Finally, Figure 2.1 (d) shows the case of manufacturing a final product with
general project type precedence relationships, which is typically associated to make to order manufacturing.
Figure 2.1 Types of Precedence Constraints
Normally, a given production order (product) must be processed on specific machines in a given sequence, e.g., in a flowshop environment the operations of each order must be processed in the same
sequence given by the machine ordering in the system, while in a jobshop environment each order may have a different machine sequence. Further, one operation of an order may be done on more than one
machine (work center). These constraints are known as
routing or technological constraints. However, in some production environments like the open shop, the sequence of operations is not fixed.
Sometimes the processing of a production order must be interrupted when a high priority production order arrives to the machine, which is called preemption. The preemption can take the form of resume
(the preempted operation is resumed on the machine when available) or
repeat (the preempted operation must be processed completely again). A processing without
possible preemption is called non-preemption processing.
Often the processing of an operation requires additional resources such as special tools or specialized personnel. When the required number of resource units is not available, then the
operation cannot be processed. This is known as resource constraints.
2.3 Performance Measures
The performance measure evaluates a given schedule, e.g., the time interval in which all
production orders are processed (makespan), the total flowtime (as the sum of the flowtime of
all orders), the total tardiness (as the sum of the tardiness of all production orders), the number
of tardy production orders, utilization of the system, etc. It allows the comparison of alternate schedules and therefore the selection of one that satisfies the requirements in a better form
(better value of the performance measures) than others.
Let pij the processing time of the j-th operation of production order i, and let pi represents the
total processing time of order i. Furthermore, let ri, and di be the release (arrival) date and due
date of order i, respectively. The setup time when production order j follows production order
i on machine k is denoted by sijk.
These production order characteristics are used to define order related performance measures such as: Ci (completion date of order i), Fi = Ci – ri (flow time of order i), Ti = max { 0, Ci - di }
(tardiness of order i), Ei = max { 0, – ( Ci - di)} (earliness of order i) and i (tardy index of order i): i = 1 (0) if Ti > 0 (Ti = 0).
The makespan (Cmax), total flowtime (F), total tardiness (T) and number of tardy orders (NT)
are some of the classical and frequently used performance measures in scheduling problems. As its definition says, the makespan is the time interval in which all production orders are processed, i.e.
if t = 0 is the starting time of the scheduling horizon (start time for reference), then Cmax = maxi=1, ... , n{Ci}. The total flowtime is the sum of all flow times Fi of each
production order i, i.e., F =
n i i
(note if ri = 0 for all i, then Fi and Ci coincide in magnitude
but differs conceptually, because Fi represents a time interval and Ci represents a point in time).
The total tardiness is the sum of all tardiness Ti of each production order i, i.e., T =
n i i T 1 . The
number of tardy orders NT =
n 1 i i
represents the number of production orders that are finished late (beyond their due dates).
Some other classical global (not order related) performance measures for schedule evaluation are the maximal flowtime Fmax = maxi=1, ... ,n{ Fi } which represents the largest time an order
spent in the system, the maximal tardiness Tmax = maxi=1, ... ,n{ Ti } which represents the
maximal tardiness among all orders, the total earliness is the sum of all earliness Ei of each order i, i.e., E = n i i E 1
, the maximal earliness Emax = maxi=1, ... ,n{ Ei } which represents the
maximal early time among all production orders, and the total earliness and tardiness penalties
E+T due to Baker [1990] and defined as E+T = ( )
1 n i i i i i T E
where i and i are unit
earliness and tardiness penalties for order i respectively. The performance measure E+T
tries to find out a schedule with minimum total sum of earliness and tardiness penalties, especially appropriate in a just in time environment.
For more details of performance measures definitions see [Corsten and Gössinger, 2009, pp. 512 – 517], Nebl [2011, pp. 733 – 739], Pinedo [2008, pp. 18 – 20], Baker [1974, pp. 12 – 22], and Baker and
Trietsch [2009, pp. 10 – 24]. Note that all of the performance measures to be minimized considered above are functions of the completion dates C1, C2, … , Cn; these
measures, except of E, Emax and E+ T, belong to an important class of performance measures called regular performance measures. A performance measure is said to be regular
if: a) the scheduling objective is to be minimized and b) the measure increases only if at least
one completion time increases. This definition permits to restrict the search space to a limited
set of schedules called a dominant set (a set that contains the optimal solution). For example, in the static one machine problem without sequence-dependent setups times, the set of permutation
schedules without inserted idle times is a dominant set for any regular performance measure [Baker, 1974, p.13].
2.4 Machine Scheduling – General Shop Models
There are many machine configurations for manufacturing, strictly as many configurations as manufacturing systems exist. Although in any theoretical classification of production systems a
real system and their processes will presents a combination of more than one characteristic [Nebl, 2011, Chapter 4], from a theoretical point of view, some generalization has been made in order to
classify manufacturing scheduling problems making possible a generalized analytical treatment, e.g. single machine models, parallel machines models, flowshop models,
jobshop models, open shop models and multiple processor models (see Baker [1974], Brucker
and Knust [2006], Pinedo [2008, 2009] and Baker and Trietsch [2009]). Within each of these models there are a great number of variants.
In accordance to the above classification, the production systems can be organized in jobshops such that similar processes are concentrated in specialized work centers such as drilling, milling,
lathe turning, etc., or in flowshops where the sequence of operations for a specific product is carried out in different work centers so that the material flows from one work center to the other or
on a transfer line (conveyor). Between these two types of organization, there are other modeling alternatives that look for a connection of them, such as cellular manufacturing and flexible
manufacturing with more or less automation (see Fandel et al [2011, pp.19 – 35], Dyckhoff and Spengler [2010, pp. 25 – 26], Günther and Tempelmeier 2009, pp. 13 – 22, 82 – 84] and Zäpfel [2000, pp.
158 – 164]).
System configuration means a representation of the physical installation of the manufacturing system, while scheduling model refers to the problem of scheduling a set of production orders with its
processing characteristics. Thus, it can happen that in a certain manufacturing system one is interested in scheduling a set of orders with each requiring only one operation on a certain machine (one
machine model), and another time one is interested in scheduling a set of order with each requiring two operations sequentially on the same machines (flowshop model). In both cases, the manufacturing
system is the same (and therefore its configuration), but the scheduling problems are different, so for each problem an adequate model must be developed.
2.4.1 Jobshop Model
The classical jobshop model can be defined as follows (see Zäpfel [2000, p. 164 – 166]): a set of orders has to be processed on a set of machines. Each order consists of a sequence (chain) of
operations, each of which requiring processing during a given time without interruption on a
given machine. Each machine can process at most one operation at a time and no operation may be processed by more than one machine at a time. The routes of the orders are not
necessarily the same, e.g., the orders may have different processing sequences and may have different numbers of operations. From the management point of view, one of the challenges is the
consecution of appropriate (as low as possible) flowtimes of production orders, the achievement of adequate utilization rates of the machines [Zäpfel, 2000, p. 164], and the location of the work
centers (see Günther and Tempelmeier [2009, pp. 84 – 91] and Zäpfel [2000, pp. 166 – 184]).
Figure 2.2 General Job Shop Models
A generalization of the jobshop is the flexible jobshop concept (see Pinedo [2009, p. 23 – 24] and Pinedo [2008, pp. 15, 20]), which can be considered as a set of work centers with multiple capacity
(identical machines parallel shops). Figure 2.2 shows an example of (a) the classical
jobshop concept and its generalized flexible jobshop version (b) composed of 4 work centers of 3, 2, 1 and 2 machines, respectively. The route of product A is expressed in terms of the work
centers of the system: WC1 WC3. The first operation of product A is done on any of the 3
machines of work center WC1 and its second operation is done in the machine of work center
WC3. The routes of products B and C are expressed analogously. Figure 2.3 shows special
cases of the flexible jobshop model.
Figure 2.3 Shop Configurations
(b) Parallel Machine Shop WC
(a) Single Machine Shop M (d) Flexible Flowshop WC4 WC1 WC2 (c) Flowshop M1 M2 M3 WC3 A B B C C B B A A WC1 WC2 WC4
(a) Classical Jobshop Model
WC2 WC1 WC3 A B B WC4 C C B B A A
When a production order must be processed more than once in a work center, i.e., two or more
operations of the order are processed in the same work center, the jobshop is called with recirculation, which is a common situation in practice. When the route of orders is not fixed,
e.g., the decision maker decides in which order the operations of an order may be processed, the system is called an open shop, otherwise the system is called a closed shop. When all jobs have the
same release date (normally stated as time 0) the problem is said to be static, otherwise it is said to be (semi) dynamic.
For research purposes, the classical jobshop problem is defined as the jobshop model where a set of n production orders with different routes and with the same release dates has to be scheduled on a
system with m different machines (m work centers with capacity one) without recirculation. Each production order has exactly m operations which must be processed without preemption once on each
machine. This problem has been widely studied and is often used as a basic model for testing scheduling algorithms.
2.4.2 Single Machine Model
Single machine models can be applied to analyze manufacturing systems where one or more
processing units (machines) exist. In single machine shops (one machine manufacturing systems), the application of single machine models is obvious; in multiple machines environments, the application
of single machine models can be relevant for bottleneck analysis where a specific machine determines the performance of the entire system. Another application of single machine models is when a
decomposition approach is used, where a complex manufacturing system is decomposed into a set of smaller single machine problems [Pinedo, 2009, p. 22].
Figure 2.3 (a) illustrates the single machine shop operation. In general, orders are processed once at the machine, i.e., each order is composed of only one operation. But sometimes more than one
operation at the machine may be needed. In other cases, order reprocessing is possible.
The simplest one machine scheduling model considers a set of n production orders of one operation with operation time pi, i = 1, … , n, all released at the same date, i.e. ri = 0 for all i. In
addition, sequence-dependent setup times and precedence constraints among production orders may exist.
2.4.3 Parallel Machine Model
A system configuration consisting of a set of machines performing the same operation is defined as a parallel machine shop. One special case of a parallel machine shop is the case where the single
machine shop is generalized by adding identical machines. For scheduling purposes, the identical machines concept means that the processing time of an order is the same on all machines.
Figure 2.3 (b) shows a work center that illustrates a parallel machines shop of 3 machines. In general, orders are processed only once at the work center, i.e., each order is composed of only one
operation, but, as in the case of single machine shop, sometimes more than one operation at the work center may be needed. In other cases, orders reprocessing may also be possible.
The simplest parallel machine scheduling model considers a set of m identical machines and a set of n production orders of one operation with process time pi, i = 1, … , n, all released at the
same time, i.e. ri = 0 for all i. In addition, sequence-dependent setup times and precedence
constraints among jobs may exist. As the machines are identical, the processing time of an
order is the same independently of the machine the order is processed on.
In case of unrelated parallel machines, the processing time pik of production order i depends
on which machine k, k = 1, … , m, the order is processed on. A special case of the unrelated
parallel machines is the case of uniform parallel machines, where the processing time pik = pi / vk of order i, depends on the speed vk of the machine on which the order is processed (in this
case pi represents a reference of the processing time, e.g. the processing time on machine 1).
2.4.4 Flowshop Model
In many manufacturing systems, a given number of operations (production stages) have to be done in the same order on every production order, which implies that all orders follow the same route.
Therefore, the machines are assumed to be set up in series, where two cases are to be distinguished: asynchronous transfer of the material flow from one station to the next, and
synchronized transfer of the material flow from one work center to the next using a time cycle
Generally, the asynchronous transfer of orders between one work center and the next is called a
flowshop. The classic flowshop concept considers only one machine at each stage. A
generalization of the flowshop consists of a set of work centers in series that have multi- capacity in parallel, called a flexible flowshop [Pinedo, 2008, p. 171]. So, the flexible flowshop can be
understood as a set of parallel machine systems in series.
Figure 2.3 (c) illustrates the flowshop concept with 3 machines and Figure 2.3 (d) illustrates the
flexible flowshop concept with 3 multi-capacity work centers in series with 3, 1 and 2 capacities
(machines), respectively.
The simplest flowshop scheduling model considers a set of m machines sequentially organized and a set of n production orders each of m operations with process times pik, i = 1, … , n and k
=1, … , m, and all released at the same date, i.e. ri = 0 for all i. In addition, sequence-dependent
setup times and precedence constraints among production orders may exist. The m operations of each production order must be processed sequentially, so precedence constraints exist between operation j
and operation j + 1 of a production order.
The case when the processing of all n production orders must follow the same sequence on all
m machines is known as the permutation flowshop.
2.4.5 Other General Shop Models
Open Shop Models
When the route of a production orders is not fixed, i.e. the process route is open, the decision maker decides in which order the operations of the production order may be processed, the system is
called an open shop [Pinedo, 2008, p. 217 – 234; Brucker and Knust, 2006, p. 20]. In an open shop there are no precedence relationships between the operations of the production
orders. Therefore, the solution procedure also determines the order in which the operations of a production order are processed. In addition, sequence-dependent setup times and precedence
Multiple Processor Models
In a multiple processor shop model, n production orders must be processed in an m machine (resources) manufacturing system. To be processed, a production order i, i = 1, … , n, requires a given
subset Mi {1, 2, … , m} of machines (resources) at the same time. Therefore, during
the processing of production order i all the machines in Mi are assigned to production order i.
The multiple mode multiple processor model is an extension of the previous analyzed model, a
production order can be processed in more than one mode, i.e. for production order i there are modei different processing alternatives with associated subsets Miq {1, 2, … , m}, q = 1, … , modei.
If the production order i is processed with the alternative q then production order i need
all the machines of subset Miq at the same time (see Brucker and Knust [2006, p. 21]).
2.4.6 Flexible Manufacturing Systems
The term flexible manufacturing system (FMS) refers to a set of computer numerically controlled (CNC) machines and work centers that are connected by an automated material handling system, controlled
by a central computer (see Zäpfel [2000, p. 232], Askin and Standridge [1993, p. 125] and Tempelmeier and Kuhn [1992, p. 1]). The FMS concept, developed in the 80’s years, represent a response to
increasing customer demand, rapid delivery of low production lot sizes of customized products. It is able to automatically process different parts simultaneously, with the machines being able to load
and accept the incoming material and carry out the corresponding operation of parts in any sequence. It must be pointed out that the concept of this type of system mainly considers high volume batch
production systems (see Günther and Tempelmeier [2009, p. 107] and Talavage and Hannam [1988, p. 37]).
The main elements of an FMS are automatically reprogrammable machines, automated tools
delivery and changing, automated material handling of incoming and outgoing parts, and centralized operation control. Its principal components [Askin and Standrige, 1993, pp.129 –
132] are machines (automatic machines with tool magazine and automatic tool changer, fixtures, robots, etc.), part movement systems (e.g. conveyors, automatic guided vehicles (AGV), etc.), supporting
work centers (e.g. load/unload station, automatic parts washers, etc.) and system controllers (computer systems with manual interaction that control the system status including machining and
transport, decision of when and how parts are to be moved, etc.). For
a detailed technical description of fundamental FMS see Talavage and Hannam [1988]. The design phase of a FMS answers the questions in two main decision areas: definition of the
specific configuration among the existing different flexible manufacturing concepts, and technical and economical evaluation of the design alternatives and selection of the final configuration
[Zäpfel, 2000, pp. 234].
From the perspective of the production planning and scheduling function, the system controller plays an important role. The control of the system implies decisions related to production
orders releasing to the shop floor, operation machine and tool assignment and production order
transport assignment. Obviously, there are several ways to implement these decisions, so a hierarchy of planning and control decisions must exist [Askin and Standrige, 1993, p. 132], i.e. a
systematic way to decide which order to release to the system for processing in the next time periods, to decide the machine on which an order will be processed according to the queues and tools at
the machines, to decide which production order to transport with which vehicle according to the position and the number of calls pending for each vehicle. The design of the physical structure and
production management procedures of the selected FMS alternative means decisions such as the number of machines and work centers, design of the material flow system and the information flow system
(see Günther and Tempelmeier [2009, pp. 108 – 109], Zäpfel [2000, pp. 234 – 270] and Tempelmeier and Kuhn [1992, pp. 29 – 44]).
2.5 Manufacturing Systems Scheduling Procedures
2.5.1 General Considerations and the Basic Scheduling Problem
Since the first steps for the systematic analysis of the manufacturing scheduling problem were made, several approaches for this problem have been developed. All scheduling procedures have to
consider input data (such as resources availability, production routes, operations processing times, orders due dates, etc.) to generate a schedule according to the considered performance criteria
(e.g. satisfaction of due dates, reduction of inventories, increasing of resource utilization, etc.).
The result of the scheduling process consists of the schedule produced by the scheduling procedure (algorithm), i.e., start time and finishing time for each operation on each resource, as well the
evaluated performance measures of interest.
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to kVA c
Amps to kVA Calculator
Amps (A) to kilovolt-amps (kVA) calculator.
Enter phase number, the current in amps, the voltage in volts and press the Calculate button to get the apparent power in kilovolt-amps:
Single phase amps to kVA calculation formula
The apparent power S in kilovolt-amps is equal to current I in amps, times the voltage V in volts, divided by 1000:
S[(kVA)] = I[(A)] × V[(V)] / 1000
3 phase amps to kVA calculation formula
Calculation with line to line voltage
The apparent power S in kilovolt-amps is equal to phase current I in amps, times the line to line RMS voltage V[L-L] in volts, divided by 1000:
S[(kVA)] = √3 × I[(A)] × V[L-L(V)] / 1000
Calculation with line to neutral voltage
The apparent power S in kilovolt-amps is equal to phase current I in amps, times the line to neutral RMS voltage V[L-N] in volts, divided by 1000:
S[(kVA)] = 3 × I[(A)] × V[L-N(V)] / 1000
See also
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Determine whether the given sequence is arithmetic, geometric,
Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio. If the sequence is
arithmetic or geometric,
find the sum of the first 50 terms.
What type of sequence is $9=\frac{10}{11}n$
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A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the amount of water in cubic metres that runs into the sea per minute
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the amount of water in cubic metres that runs into the sea per minute.
The amount of water which flows into the sea per minute can be calculated as follows:
3Km/hour = 3000/60 m/minute = 50 m/minute
The Volume of water flowing from the river to the sea in a minute = 2m × 45m × 50m/min = 4500m³/min
Hence 4500 cubic metres of water is flowing into the sea per minute.
✦ Try This: A river 1.5 m deep and 36 m wide is flowing at the rate of 2 km per hour. Find the amount of water in cubic metres that runs into the sea per minute.
The amount of water which flows into the sea per minute can be calculated as follows:
2Km/hour = 2000/60 m/minute = 33 1/3 m/minute
The Volume of water flowing from the river to the sea in a minute = 1.5m × 36m × 200/6m = 1800m³
Hence 1800 cubic metres of water is flowing into the sea per minute.
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 11
NCERT Exemplar Class 8 Maths Chapter 11 Problem 99
A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. Find the amount of water in cubic metres that runs into the sea per minute.
A river 2 m deep and 45 m wide is flowing at the rate of 3 km per hour. 4500 cubic metres of water runs into the sea per minute
☛ Related Questions:
Math worksheets and
visual curriculum
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SPP Instances
This is a web site to post certain instances from the paper A Parallel, Linear Programming Based Heuristic for Large Scale Set Partitioning Problems, by J. T. Linderoth, E. K. Lee, and M. W. P.
Savelsbergh, that appeared in INFORMS Journal on Computing, 13 (2001), pp. 191-209.
Jeff is posting these here as a public service, and the information here is based on his notes taken at the time and his best recollections five years after he actually worked with the instances.
Thus, he can’t really guarantee to be very helpful in answering any questions you have about the instances. But feel free to ask…
Happy MIP Solving!
727 Instance
The pairings are broken in 9 files. The files consist of lines whose first entry is the cost of the pairing, and the remaining (integer) entries are the row indices of the flight les that pairing
covers. There are 342 rows and 12618766 columns in the IP. The LP relaxation has value zlp = 637.246540. The best known solution is 1108.86.
a320 instance
There are 190 rows and 8122371 columns in the IP formulation. The initial LP value is 940.649322 and the Optimal solution is 1078.
Inventory Routing
This instance comes from an Inventory Routing Application, and is given in MPS format…
Vehicle Routing
These instances compe from capacitated vehicle routing. They are the set of routes I obtained after running a “few” rounds of a branch-and-price approach to solving the initial linear programming
relaxation, with some extra routes just thrown in…
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What is the Parallel Lines Definition
Parallel Lines Definition
In geometry, a line is formed when two planes intersect each other. Parallel lines are defined as lines that lie in the same plane but can never meet. These lines exist in a single plane but they run
parallel to each other. They never meet, touch or intersect each other at any point in space.
In the same way, a single line and a plane, or two planes that never share a point are called parallel.
You can note that the parallel lines definition states that for two lines to be considered parallel, they need to be on the same plane. So, if two lines exist in a three dimensional space but do not
share a single point, such cannot be termed as parallel. These lines are termed as skew lines.
Parallelism is a property of affined geometries. Since, Euclidean space is a special case of affine geometries, parallel lines are discussed in Euclid’s parallel postulate. Other spaces such as
hyperbolic space have similar properties called as parallelism.
A Brief History
The word parallel comes from the Greek word parallēlos. Here, para means beside and allēlōn, which is derived from allos, means of one another.
If it’s used as a noun, parallel means the way in which things are similar to each other. For example, you can draw parallels between the US invasion of Iraq and the Vietnam War. Parallel is also
used for the imaginary lines drawn on the globe that are parallel to the equator.
However, in mathematics, the word parallel is used for two lines, a line and a plane or two planes that never intersect each other.
Distance between Two Parallel Lines
The two parallel lines must be equidistant from each other. Therefore, the distance formula will be unique to them.
Suppose equation (1) and (2) represent two lines p and q, which are neither parallel to x-axis nor y-axis.
Now, suppose that another line s intersects both of these lines. Furthermore, let line n be perpendicular to the parallel lines. If lines p and q have a slope “m” then line s must have a slope “-1/
m.” So, its equation can be written as:
Now, solving the following systems:
Parallel Sides of Different Geometrical Shapes
If sides of a geometrical figure are parallel, it gives them a unique characteristic. If a shape has two parallel sides, think trapezoid, then:
• The parallel sides always form the base of the shape
• The height of the shape remains the same, regardless of the length
Mathematical Notation
If two sides AB and CD are parallel, you can represent that mathematically by writing that AB // CD. Another way is to draw matching arrow marks on the parallel sides of the figure. If there are more
than one parallel lines, add another arrow to mark the second pair. Similarly, two parallel sides can be marked as P // Q.
Conditions for Parallel Lines
These conditions must be satisfied by two lines for them to be termed as parallel to each other.
Now, suppose that there are two lines l and m in the same Euclidean space. These lines are parallel to each other if:
• Every point of line m is at the same distance from a point on line l. That is, these lines must be equidistant at all points when extended to infinity.
• When a third line intersects both line l and m, the corresponding angles are congruent.
• Line l and m lie in the same plane but do not meet each other or share a common point.
All of the above properties are equivalent. So, all of these can be used for parallel lines definition. However, the first two properties require measurements in order to prove that two lines possess
such properties. Therefore, the third one is often taken as parallel lines definition because of its simplicity.
Another possible parallel lines definition is that if two lines l and m are parallel, they will have the same slope or gradient.
What is the Most Accurate Parallel Lines Definition
Let’s discuss each of the four parallel lines definitions we mentioned above and take a look at which one is the most accurate one.
• Two lines are parallel if both are perpendicular to a third line
This definition is technically correct. The drawbacks, however, are that it supposes a third line which might not be a part of or stated in the situation. We will need to construct the third line,
which is not a good way to define something in mathematics. There must be some intrinsic characteristic to prove a statement. Moreover, this definition supposes that the reader is already aware of
perpendicular lines, which might not be the case. Therefore, it must not contain information about right angles and make it more complex.
• Two distinct lines are parallel if they never meet each other
This definition is similar to the classic example of a parallel line; the railroad track. Because the track goes on and on without meeting. Lines also extend in either direction up to infinity,
therefore, the definition suits parallel lines definition well.
• Two distinct lines are parallel when they have the same slope
Now, this definition has two major drawbacks. The first one is that this definition applies to lines that have a slope. So, using this we cannot tell whether the lines parallel y-axis that is
parallel or not. Therefore, not all possible solutions are satisfied.
This definition can be corrected by saying that “every vertical is parallel to each other, however, a vertical line can never be parallel to a non-vertical line.”
In addition to this, as there is no need to mention the slopes of two lines, this information in the definition will be treated as auxiliary information. To prove that two lines are parallel based on
this definition, we will have to calculate the slopes that require coordinates. This requires that we construct the lines on a separate plane, something that Euclid never had with him. Moreover, this
requires prior knowledge of slopes and their calculations.
Example 1:
If the slope of a line AB is x/4 and that of line CD is (x-5)/6. Find the value of x if lines AB and CD are parallel.
Since lines AB and CD are parallel, their slopes are equal. So,
Example 2:
If the equation of a line AB is y=3x+2, and that of CD is 2y-3x=3. Find out whether they are parallel or not.
Slope of line AB:
so m=3
Slope of line CD:
Since the slope of the lines are not equal, therefore, these are not parallel.
Example 3:
If lines AB and CD are parallel, and the slope of the lines are 3/4 and 8/(x-6). Find the value of x.
For parallel lines, the slope of the two lines must be equal, therefore,
Example 4:
What is the line parallel to a line having a slope 2/3.
Here m=2/3 for line AB. It will be the same for line CD. So, the equation will be:
Example 5:
What is the slope of the line perpendicular to line AB, if the equation of line AB is given as: 3y-2x=6. Hint: m1 x m2= -1, where m1 and m2 are slopes of perpendicular lines.
Slope of the line AB:
For the perpendicular line, its slope will be m=-3/2. Therefore, its equation will become:
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Advent of Code 2020 Day 4
Part 1
In Day 4’s challenge, we are once again brought back to working with strings. In this problem, we are asked to verify the validity of passports based on the presence of required fields. These fields
are: byr (birth year), iyr (issue year), eyr (expiration year), hgt (height), hcl (hair color), ecl (eye color), pid (passport ID), and cid (country ID).
To be valid, a passport must contain either all eight fields, or must be only missing the cid field which is optional. The problem does not seem really complex, let’s look at the input data:
ecl:gry pid:860033327 eyr:2020 hcl:#fffffd
byr:1937 iyr:2017 cid:147 hgt:183cm
iyr:2013 ecl:amb cid:350 eyr:2023 pid:028048884
hcl:#cfa07d byr:1929
hcl:#ae17e1 iyr:2013
ecl:brn pid:760753108 byr:1931
hcl:#cfa07d eyr:2025 pid:166559648
iyr:2011 ecl:brn hgt:59in
One of a possible difficulty here is to make sure that all the data for a single passport is correctly taken into account despite being spread on multiple lines. For this bit, I have to admit that I
cheated a little bit: I used a Vim macro to inline the data for all passports.
Suppose you don’t know Vim (you should really), here’s what you might have done to get all the data for a passport in a single variable:
passports = []
passport = ""
for line in lines:
if line:
passport += line
passport = ""
Now, there are (as usual) multiple ways to solve this first part. I’m going to use maps since we haven’t used them before. We are thus going to represent a passport with a map, where the keys are the
fields and the values are their corresponding values. With maps, the solution becomes quite clear: a passport is valid if the set of keys contains exactly 8 elements, or contains 7 elements and cid
is not one of them.
If the objective is to count how many passports are valid, the implementation of the solution could be something like this:
valid_passports = 0
for passport in passports:
passport_dict = {}
for key_value in passport.split(" "):
key, value = key_value.split(":")[0], key_value.split(":")[1]
passport_dict[key] = value
if len(passport_dict.keys()) == 8 or (len(passport_dict.keys()) == 7 and "cid" not in passport_dict.keys()):
valid_passports += 1
Part 2
The second part now asks us to also validate the values of each field. The rules are as follow:
• byr (Birth Year) - four digits; at least 1920 and at most 2002.
• iyr (Issue Year) - four digits; at least 2010 and at most 2020.
• eyr (Expiration Year) - four digits; at least 2020 and at most 2030.
• hgt (Height) - a number followed by either cm or in:
□ If cm, the number must be at least 150 and at most 193.
□ If in, the number must be at least 59 and at most 76.
• hcl (Hair Color) - a # followed by exactly six characters 0-9 or a-f.
• ecl (Eye Color) - exactly one of: amb blu brn gry grn hzl oth.
• pid (Passport ID) - a nine-digit number, including leading zeroes.
• cid (Country ID) - ignored, missing or not.
And the question, once again, is to count the number of valid passports.
Now that we have three days of data parsing behind us, this problem should not be too difficult for us really. The only difficult part could be to validate the hcl field as it is written in hex
format, or the cid field which is a nine digit value. Thankfully, regular expressions are here to help. I think this problem is also a nice opportunity to practice a little bit problem decomposition
and functions. For example, we can call a function on each passport and check its validity. This function itself will call 7 other functions, one for each rule we need to check. Then, all we have to
do is to correctly parse the useful information in each value and verify its validity.
import re # Used for regex
def check_passport_validity(passport):
if len(passport.keys()) == 8 or (len(passport.keys()) == 7 and "cid" not in passport.keys()):
return check_birth_year(passport["byr"]) and check_issue_year(passport["iyr"]) and
check_expiration_year(passport["eyr"]) and check_height(passport["hgt"]) and
check_hair_color(passport["hcl"]) and check_eye_color(passport["ecl"]) and
def check_birth_year(byr_value):
birth_year = int(byr_value.split(":")[1])
return 1920 <= birth_year <= 2002
def check_issue_year(iyr_value):
issue_year = int(iyr_value.split(":")[1])
return 2010 <= issue_year <= 2020
def check_expiration_year(eyr_value):
expiration_year = int(eyr_value.split(":")[1])
return 2020 <= expiration_year <= 2030
def check_height(hgt):
height = int(hgt.split(":")[:-2])
unit = hgt.split(":")[-2:]
if unit == "cm":
return 150 <= height <= 193
return 59 <= height <= 76
def check_hair_color(hcl):
hair_color = hcl.split(":")[1]
return re.search("^#[0-9a-f]{6}$", hair_color) is not None
def check_eye_color(ecl):
eye_color = ecl.split(":")[1]
return eye_color in ('amb', 'blu', 'brn', 'gry', 'grn', 'hzl', 'oth')
def check_passport_id(pid):
passport_id = pid.split(":")[1]
return re.search("^[0-9]{9}$", hair_color) is not None
Concepts and difficulties
Day 4 was not necessarily difficult, it is really a matter of taking things slowly and bit by bit. This is why I tried to decompose Part 2 with a lot of very small functions that do just one thing.
The difficulties could then lie in the data parsing phase (we have to juggle between strings and integers) and in the definition of regular expressions.
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Element-wise vector comparison
03-17-2017 08:50 AM
I have a huge mxn matrix A where I need to compute A(i,j) > 0, ie. 1 for all values which are positive, and 0 otherwise
I would be better off writing it in C++, but let's assume I'm in a situation where I don't have admin rights and cannot install/compile anything. I'm using a single-threaded scripting language (VBA)
to wrap calls to mkl_rt.dll, and any combination of MKL functions is still faster.
One way to solve this is to compute 1-0^(x+abs(x)), ie something like
vdabs m*n, A(1, 1), temp(1, 1)
vdadd m*n, A(1, 1), temp(1, 1), temp(1, 1)
vdpow m*n, res(1, 1), temp(1, 1), res(1, 1) /***** (res() is an array full of zeros)
vdlinearfrac m*n, res(1, 1), res(1, 1), 0, -1, 0, 1, res(1, 1)
But the vdpow function is quite costly computation-wise (overall speed difference is ca x4 in favor of mkl, but I expect more from one single thread in VBA vs compiled multi-threaded code)
Another solution is to find the maximum value in the array, divide by max+1 to force all values into <-1, 1> and apply vdceil, <-1, 0] becomes 0, and <0, 1> becomes 1, but there can be some problems
if the numbers are huge and end up as [-1, 1]
Can anyone think of a simpler way to do it?
03-26-2017 06:19 PM
03-26-2017 06:19 PM
04-11-2017 02:07 AM
04-12-2017 02:59 AM
04-13-2017 03:49 PM
04-16-2017 06:39 PM
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A Beginner's Guide to Data Structures and Algorithms: A Comprehensive Introduction
data structures · April 27, 2024
A Beginner's Guide to Data Structures and Algorithms: A Comprehensive Introduction
Learn the fundamentals of data structures and algorithms, including arrays, linked lists, stacks, queues, trees, and graphs. Understand how to analyze the time and space complexity of algorithms and
improve your coding skills.
Create an image featuring JavaScript code snippets and interview-related icons or graphics. Use a color scheme of yellows and blues. Include the title '7 Essential JavaScript Interview Questions for
Introduction to Data Structures and Algorithms
What are Data Structures and Algorithms?
Data structures and algorithms are the building blocks of computer science. They are the fundamental concepts that enable computers to process, store, and retrieve data efficiently. In this blog
post, we will delve into the world of data structures and algorithms, exploring what they are, why they are important, and how they are used in real-world applications.
What are Data Structures?
Data structures are ways to organize and store data in a computer so that it can be efficiently accessed, modified, and manipulated. They provide a means of storing and retrieving data in a way that
is efficient, scalable, and flexible. Data structures can be thought of as containers that hold data, and they come in various forms, such as arrays, linked lists, stacks, queues, trees, and graphs.
What are Algorithms?
Algorithms are step-by-step procedures that take some input and produce a corresponding output. They are the heart of computer science, as they provide a way to solve problems, make decisions, and
automate tasks. Algorithms can be thought of as recipes that take some input, perform some operations on it, and produce a desired output.
Why are Data Structures and Algorithms Important?
Data structures and algorithms are essential in computer science because they enable computers to process and analyze large amounts of data efficiently. They are used in a wide range of applications,
from simple calculators to complex artificial intelligence systems.
Here are some reasons why data structures and algorithms are important:
• Efficient Data Storage and Retrieval: Data structures provide a way to store and retrieve data efficiently, which is critical in applications that require fast data access, such as databases and
file systems.
• Problem-Solving: Algorithms provide a way to solve complex problems, such as sorting, searching, and optimization, which are essential in many applications, such as web search, recommendation
systems, and machine learning.
• Scalability: Data structures and algorithms enable computers to scale to large amounts of data, which is critical in big data analytics, data mining, and data science.
• Automation: Algorithms automate tasks, making it possible to perform repetitive tasks quickly and accurately, which is essential in applications such as automation, robotics, and artificial
Types of Data Structures
There are several types of data structures, each with its own strengths and weaknesses. Here are some common types of data structures:
Arrays are a fundamental data structure that stores a collection of elements of the same data type. They are useful for storing and manipulating large amounts of data.
Linked Lists
Linked lists are a type of data structure that stores a collection of elements, where each element points to the next element. They are useful for inserting and deleting elements dynamically.
Stacks and Queues
Stacks and queues are types of data structures that follow a Last-In-First-Out (LIFO) and First-In-First-Out (FIFO) order, respectively. They are useful for implementing recursive algorithms and
parsing expressions.
Trees and Graphs
Trees and graphs are types of data structures that represent hierarchical and network relationships between elements. They are useful for modeling complex relationships and performing graph traversal
Types of Algorithms
There are several types of algorithms, each with its own strengths and weaknesses. Here are some common types of algorithms:
Sorting Algorithms
Sorting algorithms, such as Bubble Sort, Selection Sort, and Merge Sort, are used to arrange elements in a specific order.
Searching Algorithms
Searching algorithms, such as Linear Search and Binary Search, are used to find elements in a data structure.
Graph Algorithms
Graph algorithms, such as Breadth-First Search (BFS) and Depth-First Search (DFS), are used to traverse and manipulate graph data structures.
Dynamic Programming Algorithms
Dynamic programming algorithms, such as the Fibonacci sequence and the longest common subsequence problem, are used to solve complex problems by breaking them down into smaller sub-problems.
Real-World Applications of Data Structures and Algorithms
Data structures and algorithms have numerous real-world applications in various fields, including:
Web Development
Data structures and algorithms are used in web development to optimize database queries, implement caching mechanisms, and improve search engine rankings.
Artificial Intelligence and Machine Learning
Data structures and algorithms are used in artificial intelligence and machine learning to develop predictive models, classify data, and optimize decision-making processes.
Database Systems
Data structures and algorithms are used in database systems to optimize data storage and retrieval, implement indexing and caching mechanisms, and improve query performance.
Computer Networks
Data structures and algorithms are used in computer networks to optimize network traffic, implement routing protocols, and improve network security.
In conclusion, data structures and algorithms are the foundation of computer science, and they have numerous applications in various fields. Understanding data structures and algorithms is essential
for any aspiring software developer, data scientist, or computer scientist. By mastering data structures and algorithms, you can develop efficient, scalable, and flexible software systems that can
solve complex problems and automate tasks.
Further Reading
If you're interested in learning more about data structures and algorithms, here are some recommended resources:
• Books:
□ "Introduction to Algorithms" by Thomas H. Cormen
□ "Data Structures and Algorithms in Python" by Michael T. Goodrich
• Online Courses:
□ "Data Structures and Algorithms" by University of California San Diego on Coursera
□ "Algorithms on Graphs" by University of California San Diego on Coursera
• Websites:
I hope this blog post has provided a comprehensive introduction to data structures and algorithms. Happy learning!
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What is strain? Explain all Types of strains.
written 2.5 years ago by modified 2.5 years ago by
1 Answer
written 2.5 years ago by
Strain: Change in dimensions to original dimensions is known as Strain.
(1) Linear Strain: When a wire or bar is subjected to two equal and opposite forces, namely pulls, at its ends, there is an increase in the length. If the forces are tensile, the body is elongated.
If the forces are compressive, the length is shortened in the direction of the forces. This is called the 'linear strain'.
• The linear strain is defined as the ratio of change in length to the original length. If the change (increase or decrease) in length is ' l ' in a wire or bar of original length L,As the linear
strain is ratio of lengths, it has no unit.
• linear strain = Change in length / original length = l / L
(2) Bulk (or) Volume Strain: When a force is applied uniformly and normally to the entire surface of the body, there is a change in volume of the body, without any change in its shape. This strain is
called 'bulk or volume strain'.
• Volume strain is defined as the ratio of change in volume to the original volume. It has also no unit. If 'v' is the change in volume produced in a body of original volume ‘V’.
• bulk or volume strain = Change in volume / original volume = v / V
(3) Shearing (or) Rigidity strain:
• When a force is applied parallel to one face of a body, the opposite side being fixed, there is a change in shape but not in size of the body. This strain is called the shearing strain.
• Solids alone can have a shearing strain. It is measured by the angle of the shear 'θ' in radian.
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Financial modeling is the task of building an abstract representation (a model) of a real world financial situation.^[1] This is a mathematical model designed to represent (a simplified version of)
the performance of a financial asset or portfolio of a business, project, or any other investment.
Typically, then, financial modeling is understood to mean an exercise in either asset pricing or corporate finance, of a quantitative nature. It is about translating a set of hypotheses about the
behavior of markets or agents into numerical predictions.^[2] At the same time, "financial modeling" is a general term that means different things to different users; the reference usually relates
either to accounting and corporate finance applications or to quantitative finance applications.
Spreadsheet-based Cash Flow Projection (click to view at full size)
In corporate finance and the accounting profession, financial modeling typically entails financial statement forecasting; usually the preparation of detailed company-specific models used for ^[1]
decision making purposes, valuation and financial analysis.
Applications include:
To generalize as to the nature of these models: firstly, as they are built around financial statements, calculations and outputs are monthly, quarterly or annual; secondly, the inputs take the form
of "assumptions", where the analyst specifies the values that will apply in each period for external / global variables (exchange rates, tax percentage, etc....; may be thought of as the model
parameters), and for internal / company specific variables (wages, unit costs, etc....). Correspondingly, both characteristics are reflected (at least implicitly) in the mathematical form of these
models: firstly, the models are in discrete time; secondly, they are deterministic. For discussion of the issues that may arise, see below; for discussion as to more sophisticated approaches
sometimes employed, see Corporate finance § Quantifying uncertainty and Financial economics § Corporate finance theory.
Modelers are often designated "financial analyst" (and are sometimes referred to, tongue in cheek, as "number crunchers"). Typically,^[6] the modeler will have completed an MBA or MSF with (optional)
coursework in "financial modeling".^[7] Accounting qualifications and finance certifications such as the CIIA and CFA generally do not provide direct or explicit training in modeling.^[8] At the same
time, numerous commercial training courses are offered, both through universities and privately. For the components and steps of business modeling here, see Outline of finance § Financial modeling;
see also Valuation using discounted cash flows § Determine cash flow for each forecast period for further discussion and considerations.
Although purpose-built business software does exist, the vast proportion of the market is spreadsheet-based; this is largely since the models are almost always company-specific. Also, analysts will
each have their own criteria and methods for financial modeling.^[9] Microsoft Excel now has by far the dominant position, having overtaken Lotus 1-2-3 in the 1990s. Spreadsheet-based modelling can
have its own problems,^[10] and several standardizations and "best practices" have been proposed.^[11] "Spreadsheet risk" is increasingly studied and managed;^[11] see model audit.
One critique here, is that model outputs, i.e. line items, often inhere "unrealistic implicit assumptions" and "internal inconsistencies".^[12] (For example, a forecast for growth in revenue but
without corresponding increases in working capital, fixed assets and the associated financing, may imbed unrealistic assumptions about asset turnover, debt level and/or equity financing. See
Sustainable growth rate § From a financial perspective.) What is required, but often lacking, is that all key elements are explicitly and consistently forecasted. Related to this, is that modellers
often additionally "fail to identify crucial assumptions" relating to inputs, "and to explore what can go wrong".^[13] Here, in general, modellers "use point values and simple arithmetic instead of
probability distributions and statistical measures"^[14] — i.e., as mentioned, the problems are treated as deterministic in nature — and thus calculate a single value for the asset or project, but
without providing information on the range, variance and sensitivity of outcomes;^[15] see Valuation using discounted cash flows § Determine equity value. A further, more general critique relates to
the lack of basic computer programming concepts amongst modelers, ^[16] with the result that their models are often poorly structured, and difficult to maintain. Serious criticism is also directed at
the nature of budgeting, and its impact on the organization.^[17]^[18]
Quantitative finance
Visualization of an interest rate "tree" - usually returned by commercial derivatives software
In quantitative finance, financial modeling entails the development of a sophisticated mathematical model.^[19] Models here deal with asset prices, market movements, portfolio returns and the like. A
general distinction is between: (i) "quantitative asset pricing", models of the returns of different stocks; (ii) "financial engineering", models of the price or returns of derivative securities;
(iii) "quantitative portfolio management", models underpinning automated trading, high-frequency trading, algorithmic trading, and program trading.
Relatedly, applications include:
These problems are generally stochastic and continuous in nature, and models here thus require complex algorithms, entailing computer simulation, advanced numerical methods (such as numerical
differential equations, numerical linear algebra, dynamic programming) and/or the development of optimization models. The general nature of these problems is discussed under Mathematical finance
§ History: Q versus P, while specific techniques are listed under Outline of finance § Mathematical tools. For further discussion here see also: Brownian model of financial markets; Martingale
pricing; Financial models with long-tailed distributions and volatility clustering; Extreme value theory; Historical simulation (finance).
Modellers are generally referred to as "quants", i.e. quantitative analysts, and typically have advanced (Ph.D. level) backgrounds in quantitative disciplines such as statistics, physics, engineering
, computer science, mathematics or operations research. Alternatively, or in addition to their quantitative background, they complete a finance masters with a quantitative orientation,^[23] such as
the Master of Quantitative Finance, or the more specialized Master of Computational Finance or Master of Financial Engineering; the CQF certificate is increasingly common.
Although spreadsheets are widely used here also (almost always requiring extensive VBA); custom C++, Fortran or Python, or numerical-analysis software such as MATLAB, are often preferred,^[23]
particularly where stability or speed is a concern. MATLAB is often used at the research or prototyping stage because of its intuitive programming, graphical and debugging tools, but C++/Fortran are
preferred for conceptually simple but high computational-cost applications where MATLAB is too slow; Python is increasingly used due to its simplicity, and large standard library / available
applications, including QuantLib. Additionally, for many (of the standard) derivative and portfolio applications, commercial software is available, and the choice as to whether the model is to be
developed in-house, or whether existing products are to be deployed, will depend on the problem in question.^[23] See Quantitative analysis (finance) § Library quantitative analysis.
The complexity of these models may result in incorrect pricing or hedging or both. This Model risk is the subject of ongoing research by finance academics, and is a topic of great, and growing,
interest in the risk management arena.^[24]
Criticism of the discipline (often preceding the financial crisis of 2007–08 by several years) emphasizes the differences between the mathematical and physical sciences, and finance, and the
resultant caution to be applied by modelers, and by traders and risk managers using their models. Notable here are Emanuel Derman and Paul Wilmott, authors of the Financial Modelers' Manifesto. Some
go further and question whether the mathematical- and statistical modeling techniques usually applied to finance are at all appropriate (see the assumptions made for options and for portfolios). In
fact, these may go so far as to question the "empirical and scientific validity... of modern financial theory".^[25] Notable here are Nassim Taleb and Benoit Mandelbrot.^[26] See also Mathematical
finance § Criticism, Financial economics § Challenges and criticism and Financial engineering § Criticisms.
Competitive modeling
Several financial modeling competitions exist, emphasizing speed and accuracy in modeling. The Microsoft-sponsored ModelOff Financial Modeling World Championships were held annually from 2012 to
2019, with competitions throughout the year and a finals championship in New York or London. After its end in 2020, several other modeling championships have been started, including the Financial
Modeling World Cup and Microsoft Excel Collegiate Challenge, also sponsored by Microsoft.^[6]
Philosophy of financial modeling
Philosophy of financial modeling is a branch of philosophy concerned with the foundations, methods, and implications of modeling science.
In the philosophy of financial modeling, scholars have more recently begun to question the generally-held assumption that financial modelers seek to represent any "real-world" or actually ongoing
investment situation. Instead, it has been suggested that the task of the financial modeler resides in demonstrating the possibility of a transaction in a prospective investment scenario, from a
limited base of possibility conditions initially assumed in the model.^[27]
See also
1. ^ ^a ^b Investopedia Staff (2020). "Financial Modeling".
2. ^ Low, R.K.Y.; Tan, E. (2016). "The Role of Analysts' Forecasts in the Momentum Effect" (PDF). International Review of Financial Analysis. 48: 67–84. doi:10.1016/j.irfa.2016.09.007.
3. ^ Joel G. Siegel; Jae K. Shim; Stephen Hartman (1 November 1997). Schaum's quick guide to business formulas: 201 decision-making tools for business, finance, and accounting students. McGraw-Hill
Professional. ISBN 978-0-07-058031-2. Retrieved 12 November 2011. §39 "Corporate Planning Models". See also, §294 "Simulation Model".
4. ^ See for example: "Renewable Energy Financial Model". Renewables Valuation Institute. Retrieved 2023-03-19.
5. ^ Confidential disclosure of a financial model is often requested by purchasing organizations undertaking public sector procurement in order that the government department can understand and if
necessary challenge the pricing principles which underlie a bidder's costs. E.g. First-tier Tribunal, Department for Works and Pensions v. Information Commissioner, UKFTT EA_2010_0073, paragraph
58, decided 20 September 2010, accessed 11 January 2024
6. ^ ^a ^b Fairhurst, Danielle Stein (2022). Financial Modeling in Excel for Dummies. John Wiley & Sons. ISBN 978-1-119-84451-8. OCLC 1264716849.
7. ^ Example course: Financial Modelling, University of South Australia
8. ^ The MiF can offer an edge over the CFA Financial Times, June 21, 2015.
9. ^ See for example, Valuing Companies by Cash Flow Discounting: Ten Methods and Nine Theories, Pablo Fernandez: University of Navarra - IESE Business School
10. ^ Danielle Stein Fairhurst (2009). Six reasons your spreadsheet is NOT a financial model Archived 2010-04-07 at the Wayback Machine, fimodo.com
11. ^ ^a ^b Best Practice Archived 2018-03-29 at the Wayback Machine, European Spreadsheet Risks Interest Group
12. ^ Krishna G. Palepu; Paul M. Healy; Erik Peek; Victor Lewis Bernard (2007). Business analysis and valuation: text and cases. Cengage Learning EMEA. pp. 261–. ISBN 978-1-84480-492-4. Retrieved 12
November 2011.
13. ^ Richard A. Brealey; Stewart C. Myers; Brattle Group (2003). Capital investment and valuation. McGraw-Hill Professional. pp. 223–. ISBN 978-0-07-138377-6. Retrieved 12 November 2011.
14. ^ Peter Coffee (2004). Spreadsheets: 25 Years in a Cell, eWeek.
15. ^ Prof. Aswath Damodaran. Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations, NYU Stern Working Paper
16. ^ Blayney, P. (2009). Knowledge Gap? Accounting Practitioners Lacking Computer Programming Concepts as Essential Knowledge. In G. Siemens & C. Fulford (Eds.), Proceedings of World Conference on
Educational Multimedia, Hypermedia and Telecommunications 2009 (pp. 151-159). Chesapeake, VA: AACE.
17. ^ Loren Gary (2003). Why Budgeting Kills Your Company, Harvard Management Update, May 2003.
18. ^ Michael Jensen (2001). Corporate Budgeting Is Broken, Let's Fix It, Harvard Business Review, pp. 94-101, November 2001.
19. ^ See discussion here: "Careers in Applied Mathematics" (PDF). Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 2019-03-05.
20. ^ See for example: Low, R.K.Y.; Faff, R.; Aas, K. (2016). "Enhancing mean–variance portfolio selection by modeling distributional asymmetries" (PDF). Journal of Economics and Business. 85: 49–72.
doi:10.1016/j.jeconbus.2016.01.003.; Low, R.K.Y.; Alcock, J.; Faff, R.; Brailsford, T. (2013). "Canonical vine copulas in the context of modern portfolio management: Are they worth it?" (PDF).
Journal of Banking & Finance. 37 (8): 3085–3099. doi:10.1016/j.jbankfin.2013.02.036. S2CID 154138333.
21. ^ See David Shimko (2009). Quantifying Corporate Financial Risk. archived 2010-07-17.
22. ^ See for example this problem (from John Hull's Options, Futures, and Other Derivatives), discussing cash position modeled stochastically.
23. ^ ^a ^b ^c Mark S. Joshi, On Becoming a Quant Archived 2012-01-14 at the Wayback Machine.
24. ^ Riccardo Rebonato (N.D.). Theory and Practice of Model Risk Management.
25. ^ Nassim Taleb (2009)."History Written By The Losers", Foreword to Pablo Triana's Lecturing Birds How to Fly ISBN 978-0470406755
26. ^ Nassim Taleb and Benoit Mandelbrot. "How the Finance Gurus Get Risk All Wrong" (PDF). Archived from the original (PDF) on 2010-12-07. Retrieved 2010-06-15.
27. ^ Mebius, A. (2023). "On the epistemic contribution of financial models". Journal of Economic Methodology. 30 (1): 49–62. doi:10.1080/1350178X.2023.2172447. S2CID 256438018.
Corporate finance
Quantitative finance
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Multiplying Polynomials Calculator - Give Results in 1 Click [Free]
Introduction to Multiplying Polynomials Calculator
Multiplying polynomials calculator is an algebraic tool that helps you to find the product of two polynomial values in a few seconds.
Polynomial multiplication calculator helps you to determine the multiplication of two polynomial functions using the rules of algebraic function multiplication.
What is a Multiplying Polynomials?
Multiplying polynomials is defined as the two polynomials multiplied by each other where coefficients are multiplying with coefficients and exponential power is not multiplied but exponential powers
are added. The polynomial exponential of all variables is the real number terms only.
Rule Followed by Multiply Polynomials Calculator
Multiplication polynomials use algebraic multiplication rules where products of two polynomials are calculated with the help of commutative property.
After multiplication, the terms having the same variable or exponential power are added. The Multiplying polynomials calculator uses the following formula for calculation:
$$ (A + B) \times (C + D) $$
$$ \;=\; A \times (C + D) + B \times (C + D) $$
$$ \;=\; A \times C + A \times D + B \times C + B \times D $$
(A+B)= First expression of a polynomial
(C+D)= Second expression of a polynomial
How to Evaluate in Polynomial Multiplication Calculator
Multiplying polynomials calculator uses the rules of algebra to find the product of two polynomials quickly and easily.
You can give various types of variable functions (binomial, monomial, or trinomial variable exponential terms) to find the product of this algebraic expression because our tool is equipped with all
algebra rules and formulas in its server.
When you give the polynomials as an input for multiplication in the multiply polynomial calculator, it starts the calculation process immediately.
Our multiply polynomials calculator checks the nature of the function if the variable function needs simplification then first simply the given polynomial and then multiply for example (a+b)^2 and
Then it multiplies the first term with all the values given in the second polynomial. Again second term of the first function is multiplied by all the values of the second polynomial. If a first
variable function has a third value then agon multiply it with all the terms.
This process is repeated until all the terms of the first algebraic function are not multiplied with the second polynomial. After multiplication, the calculator adds all the values whose has similar
variables and exponential power.
Lastly, you get the solution of multiplying polynomials in step-wise detail. You can see an example of polynomial multiplication to get a clear understanding of the working process of our polynomial
multiplier practically.
Solved Examples:
The multiplying polynomials can be solved using multiplying polynomials calculator and can also be solved manually. we will let you understand that by giving you an example with step by step
Find the product of a given polynomial
$$ (2x - 1)^3 $$
$$ (2x -1)^3 \;=\; (2x-1)(2x-1)(2x-1) $$
$$ \;=\; (2x-1)(4x^2 - 2x - 2x + 1) $$
$$ \;=\; (2x-1)(4x^2 - 4x +1) $$
$$ 8x^3 - 8x^2 + 2x - 4x^2 + 4x - 1 $$
$$ 8x^3 - 12x + 6x - 1 $$
How to Use Multiplying Polynomials Calculator?
Multiply polynomials calculator has a user-friendly interface so that you can use it to calculate the product of polynomials in less than a minute.
Before adding the input value in the multiply polynomial calculator, you must follow some simple steps so that you can avoid trouble during the calculation process. These steps are:
1. Enter the first polynomial function in the input box of the polynomial multiplier.
2. Enter the second polynomial function in the input box.
3. Review your polynomial input value before hitting the calculate button to start the calculation process.
4. Click the “Calculate” button to get the desired result of your given polynomial problem
5. If you want to try out our polynomial multiplication calculator first then you can use the load example to get the conceptual clarity
6. Click on the “Recalculate” button to get a new page for solving more algebraic expressions
Outcome from Multiply Polynomial Calculator
Multiplying polynomials calculator gives you the solution to a given variable problem when you add the input to it. It provides you with solutions in a stepwise process in no time. It may contain as:
• Result option gives you a solution for Multiplying polynomial problems
• Possible step provides you solution with all the calculation steps of the polynomial problem
Advantages of Polynomial Multiplier
Multiply polynomials calculator will give you tons of advantages whenever you use it to calculate polynomial problems. These advantages are:
• Our tool saves your time and effort from doing lengthy calculations of the algebraic function multiplication problem
• Polynomial multiplication calculator is a free-of-cost tool so you can use it to find the product of two or more two polynomials
• It is a versatile tool that allows you to solve various types of polynomial function multiplication questions
• Multiply polynomial calculator provides a solution with a complete process in a step-by-step method so that you get a better understanding.
• It is a reliable tool that provides accurate solutions whenever you use it to calculate the Multiplying polynomials problem.
• You can use this Multiplying polynomials calculator for practice so that you get a strong hold on this concept.
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Spherical Capacitance - Important Concepts and Tips for JEE
The capacitance concept involves storing electrical energy. Unlike the flat and cylindrical capacitors, the spherical capacitance can be evaluated with the voltage differences between the capacitors
and their respective charge capacity. Since spherical capacitors have a radius, the introduction of spherical capacitance involves its charge and potential difference and can be directly proportional
to its radius. But the radius can be for the inner and outer surface, so the calculation changes accordingly for capacitance.
Types of Capacitors
Capacitors can be of three types such as parallel plate capacitor, cylindrical capacitor, and spherical capacitor. These capacitors are connected to circuits as per their use. Some capacitors would
need circuits storing more energy, while some others would require capacitors with less energy.
Hence, spherical and cylindrical power differences or spherical and parallel plate power differences can be seen. The capacitor charge is directly proportional to the potential difference. But to get
the capacitance equation, the proportionality is replaced by constant C.
$ Q\propto V $
$ Q=CV $
$ C=\frac{Q}{V} $
This defines capacitance as the ratio between the charge stored in the capacitor and the potential difference. So, the SI unit of capacitance is Coulomb/Volt or Faraday. Generally, you can find
capacitors ranging from μF to mF in the market. Let’s learn about parallel plate capacitors to understand the working mechanism of the capacitance of spherical capacitors as they involve different
concepts due to the presence of different surface shapes.
Capacitance of Spherical Conductor
Unlike the parallel plate capacitor, a spherical capacitor consists of two concentric spherical conducting shells, which are separated by a dielectric. Let’s take the inner sphere surface as the
outer radius r[1] with a charge +q, and the outer sphere has the inner radius r[2] with a charge –q.
Spherical Capacitors
At any point in the spheres, the electrical capacity of a spherical conductor is the same according to Gauss’ Law, as it’s perpendicular to the surface and aims radially outward. It is represented in
the equation for the electric field of a point charge
$E=\frac{Q}{4\pi {{\varepsilon }_{0}}{{r}^{2}}}$
Let’s see here how the formula is obtained. If we consider the sphere to be a Gaussian surface at radius r[1] > r[2], the magnitude of the electric field would be the same at every point as per the
above figure. Spherical capacitor derivation,
The electric flux of the spherical surface would be
$\phi =EA=E\cdot 4\pi {{r}^{2}}=\frac{Q}{{{\varepsilon }_{0}}}$
To calculate the potential difference between both the spheres, follow the below expression:
$ V=-\int{Edr} $
$ V=-\int\limits_{r_2}^{r_1}{\frac{Q}{4\pi {{\varepsilon }_{0}}{{r}^{2}}}} $
${\therefore\;=\;}\frac{Q({R_2}-{R_1})}{4\;{\Pi}{\varepsilon_0}R_1R_2 }$
In case the spherical capacitors have radii for both spheres as a and b with an electric potential V[1] and V[2] that are attached with a conducting wire, the potential between two spherical
capacitors would be:
The capacitance of sphere type capacitor would be
$ C=\frac{Q}{V} $
$ \therefore C=4\pi {{\varepsilon }_{0}}\left(\dfrac {{r_1}{r_2}}{{r_1}-{r_2}}\right)$
The equation shows that to calculate the capacitance of a spherical capacitor formula, take the radii of the outer and inner spheres and the medium between the spheres. If the radius of the outer
conductor is taken to infinity, the equation would be;
$C=4\pi {{\varepsilon }_{0}}R$
Spherical Capacitor When Inner Sphere Is Earthed
• When the positive charge of Coulomb Q to the outer sphere B will be distributed over both of its inner and outer surfaces. Let us assume the charges of Coulombs are ${{Q}_{1}}$ and ${{Q}_{2}}$ at
the inner and outer surfaces of sphere B, respectively.
We have $Q={{Q}_{1}}+{{Q}_{2}}$
• The $+{{Q}_{1}}$ charge present on the inner surface of sphere B will induce the $-{{Q}_{2}}$ charge on the outer surface of Sphere A. And the $+{{Q}_{1}}$ charge present on the inner surface of
sphere A will move to earth.
• As two capacitors are connected in parallel,
1. First capacitor has outer surface of sphere B and the earth with capacitance ${{C}_{1}}=4\pi {{\varepsilon }_{0}}b$
2. Second capacitor has the inner surface of outer sphere B and outer surface of inner sphere A with capacitance ${{C}_{2}}=\frac{4\pi {{\varepsilon }_{0}}ba}{\left( b-a \right)}$
• Now the final capacitance is:
$C={{C}_{1}}+{{C}_{2}}=4\pi {{\varepsilon }_{0}}b+\frac{4\pi {{\varepsilon }_{0}}ba}{\left( b-a \right)}=\frac{4\pi {{\varepsilon }_{0}}{{b}^{2}}}{\left( b-a \right)}$
Capacitance of a Spherical Conductor
The capacitance of a spherical conductor can be acquired by comparing the voltages across the wires with a certain charge on each.
Types of Spherical Capacitors
• Isolated Spherical Capacitor
The isolated spherical capacitors are generally represented as a solid charged sphere with a finite radius and more spheres with infinite radius with zero potential difference. This way, the capacity
of an isolated spherical conductor would be expressed as
$C=4\pi {{\varepsilon }_{0}}R$
• Concentric Spherical Capacitor
Concentric spherical capacitors are the solid spheres that have a conducting shell with an inner and outer radius with a + ve charge on the outer surface and a -ve charge on the inner surface. In
order to calculate the capacitance of the spherical concentric capacitor, follow the below equation:
$C=\frac{4\pi {{\varepsilon }_{0}}{{R}_{1}}{{R}_{2}}}{\left( {{R}_{2}}-{{R}_{1}} \right)}$
From the above study, it is evaluated that the capacitance for the spherical capacitor is achieved by calculating the difference between the conductors for a given charge on each capacitor and
depending on the radii of an inner and outer surface of each sphere. Students who are preparing for JEE exam can follow this article for a better understanding of spherical capacitance.
FAQs on Spherical Capacitance - JEE Important Topic
1. What is the capacitor principle?
A capacitor is an electronic device which is used to store electrical charge. It is one of the most important electronic components in circuit design. The passive component known as a capacitor has
the capacity to store both positive and negative charges due to this it temporarily behaves as a battery. A capacitor works under the concept that when an earthed conductor is moved close to a
conductor, its capacitance increases noticeably. As a result, a capacitor has two plates with equal and opposite charges that are spaced apart.
2. What is a capacitor and its applications?
The ratio of the electric charges accumulated across the capacitor's conducting plates to the potential difference between them is known as the capacitance. A capacitor can be used in a variety of
applications depending on its design, construction, size, and storage capacity. A capacitor is used to store electrical charges and release them as needed by the circuit. In electronic circuits,
capacitors are frequently employed to carry out a number of functions, including smoothing, filtering, bypassing, etc. It's possible that not all applications need a specific sort of capacitor.
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Approximate tests for testing equality of two cumulative incidence functions of a competing risk
In the context of a competing risks set-up, we discuss different inference procedures for testing equality of two cumulative incidence functions, where the data may be subject to independent
right-censoring or left-truncation. To this end, we compare two-sample Kolmogorov–Smirnov- and Cramér–von Mises-type test statistics. Since, in general, their corresponding asymptotic limit
distributions depend on unknown quantities, we utilize wild bootstrap resampling as well as approximation techniques to construct adequate test decisions. Here, the latter procedures are motivated
from tests for heteroscedastic factorial designs but have not yet been proposed in the survival context. A simulation study shows the performance of all considered tests under various settings and
finally a real data example about bloodstream infection during neutropenia is used to illustrate their application.
• Aalen–Johansen estimator
• approximation techniques
• competing risk
• cumulative incidence function
• wild bootstrap
Dive into the research topics of 'Approximate tests for testing equality of two cumulative incidence functions of a competing risk'. Together they form a unique fingerprint.
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Approximate Aggregate Functions in BigQuery
Sometimes you don't need perfect, but just good enough. Take approximate aggregate functions in BigQuery, for example.
These are a type of aggregate functions that produce approximate results instead of exact ones but have the upside of typically requiring fewer resources for the computation.
When would I use one? This would be suitable where we can live with an uncertainty or small difference, especially for huge tables, during a preliminary check or data exploration.
Let's look at a practical example. Suppose we have the following data:
APPROX_TOP_COUNT will compute the approx top N elements and their value counts
APPROX_TOP_COUNT(value, 5) AS top_value_counts
FROM `learning.data_source`
APPROX_COUNT_DISTINCT will compute the approx distinct count (also can be grouped)
APPROX_COUNT_DISTINCT(value) AS approx_distinct_value_count
FROM `learning.data_source`
You can discover more approximate aggregate functions in the documentation.
Thanks for reading!
Found it useful? Subscribe to my Analytics newsletter at notjustsql.com.
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23. x3−x2−ax+x+a−1
25. bax2+(ba+dc)x+dc,b=0,d=0... | Filo
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Question Text 23. 25.
Updated On Dec 24, 2022
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 146
Avg. Video Duration 2 min
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Functional analysis
Functional analysis, Vol. 1, a gentle introduction / Dzung Minh Ha
Type de document : MonographieLangue : anglais.Pays: Etats Unis.Éditeur : New York : Matrix Ed., 2006Description : 1 vol. (640 p.) ; 24 cmISBN: 9780971576612.Bibliographie : Bibliogr. Index.Sujet MSC
: 46-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
40-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to sequences, series, summability
45-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equationsEn-ligne : Zentralblatt | MathSciNet
Item type Current library Call number Status Date due Barcode
CMI Salle 1 46 HA (Browse shelf(Opens below)) Available 04905-01
This book is an introduction to basics of functional analysis at the undergraduate level. The prerequisites are elementary linear algebra and first-year calculus. In the Preface, the author writes:
“Textbooks in functional analysis (or more generally, in mathematics) are often unnecessarily demanding – written in a concise manner with few examples and motivations. [⋯] I chose to write a
textbook that I would like to have studied from as a student – one that is mathematically rigorous but leisurely, with lots of motivations and examples." The book discusses standard topics such as
metric spaces and normed linear spaces and operators on them with rudiments of topological spaces and topological vector spaces, Banach's fixed point theorem, compactness, results centering around
the Baire category theorem, integral operators, inner product spaces and Hilbert spaces. It also includes special topics as the theorems of Korovkin and Bernstein, the Stone–Weierstrass theorem, the
Baire–Osgood theorem, Gram determinants, the Müntz theorem and some basic results in the theory of differential equations. The Hahn–Banach theorem is not discussed... (Zentralblatt)
Bibliogr. Index
There are no comments on this title.
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INTERVIEW IM PLAYBOY; NOVEMBER 1985
Marcelle Clements:
KLAUS KINSKI & THE THING
Is this man of strange and explosive power really the world's greatest actor?"
Seite 7/10
"What? What is it you want to say?" Kinski queried when he saw me open my mouth several times.
"There was something you mentioned the other day," I began, "about how money is freedom - "
"I never said that," he assured me.
"You did," I replied. "You said - "
"No, no. I never said money is freedom! I said money buys freedom. BUYS! What does that mean, money is freedom? This is ridiculous: Money is freedom. It means nothing. What do you think, that a
dollar in a savings account is freedom? Maybe you have understood nothing I have said. You are trying to make me sound like an American average citizen."
His arguments in response to my questions were often semantic. Kinski hates words; he resents having to use them to express himself, he finds them untrustworthy, confining, reductive.
"Experiencing the ocean is an experience of liberty," he told me, for example. "When you talk about the ocean, is it liberty? Even looking at the ocean is not liberty. It is like a wounded bird
looking at the sky and saying, ,Why are my wings broken?' Or even worse: putting a bird cage near the window so that the bird can see the sky. But, of course, it's much better to look than not to,
even if it hurts. But words - words are not enough!"
"But sometimes," I said, "you can put them together to evoke a certain feeling."
"But this is a consolation for cripples," said Kinski. "Yes, sometimes, spontaneously bringing words out can be outscreams - outscreams of joy or pain or whatever you want. Or sometimes you can
describe. But you aren't there. When you are there, you are. With words, you aren't. It is true what Rimbaud said once; It's absolutely true; I proved lt. He said, ,If you think a book is strong
enough, try It at the ocean, in the wind, at the waves. If the book can resist the ocean, the elements, then it exists. Otherwise, throw it away.'"
© 1985 by Marcelle Clements and Playboy Enterprises Inc.
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Loveonn Web Oracle | Unlock the world of knowledge/ Explain the concept of eigenvalues and eigenvectors and their significance.
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Explain the concept of eigenvalues and eigenvectors and their significance.
Learn from Computational Mathematics
Understanding Eigenvalues and Eigenvectors: A Comprehensive Guide
Eigenvalues and eigenvectors are fundamental concepts in linear algebra with wide-ranging applications in mathematics, physics, engineering, and computer science. Understanding these concepts can
provide deep insights into the behavior of linear transformations and matrices.
What Are Eigenvalues and Eigenvectors?
Eigenvalues and eigenvectors arise when examining linear transformations represented by matrices. For a given square matrix \( A \), an eigenvector is a non-zero vector \( v \) that, when multiplied
by \( A \), results in a scalar multiple of itself. The scalar is known as the eigenvalue associated with that eigenvector.
Mathematically, this relationship is expressed as:
\[ A \cdot v = \lambda \cdot v \]
- \( A \) is the square matrix.
- \( v \) is the eigenvector.
- \( \lambda \) is the eigenvalue.
How to Compute Eigenvalues and Eigenvectors
To find the eigenvalues and eigenvectors of a matrix \( A \):
1. Calculate the Eigenvalues: Solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \text{det} \) denotes the determinant. The solutions \
( \lambda \) are the eigenvalues.
2. Find the Eigenvectors: For each eigenvalue \( \lambda \), solve the equation \( (A - \lambda I) v = 0 \) to find the corresponding eigenvector \( v \).
Significance of Eigenvalues and Eigenvectors
1. Stability Analysis: In systems engineering and control theory, eigenvalues help determine system stability. If all eigenvalues of the system's matrix have negative real parts, the system is
2. Principal Component Analysis (PCA): In data science and machine learning, eigenvectors are used in PCA to identify the principal components of data, helping to reduce dimensionality and extract
significant features.
3. Vibration Analysis: In mechanical engineering, eigenvalues and eigenvectors are used to analyze the natural frequencies and mode shapes of vibrating systems, which is crucial for designing stable
4. Quantum Mechanics: In physics, particularly quantum mechanics, eigenvectors represent possible states of a system, and eigenvalues correspond to measurable quantities such as energy levels.
5. Graph Theory: In network analysis, eigenvectors of adjacency matrices are used to identify important nodes in a graph, with applications ranging from social network analysis to recommendation
6. Differential Equations: Eigenvalues and eigenvectors simplify the process of solving linear differential equations by reducing them to manageable forms.
Practical Applications
- Image Compression: Eigenvectors are used in image compression algorithms like JPEG to represent image data efficiently.
- Robotics: In robotics, eigenvalues are used to analyze and control robotic motion and dynamics.
- Economics: Eigenvectors can model and predict economic phenomena by analyzing economic matrices.
By providing a structured way to analyze and understand complex systems, eigenvalues and eigenvectors offer powerful tools for both theoretical exploration and practical application in numerous
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8-9 Volume of Pyramids and Cones Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes. - ppt download
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Argon Bohr Model — Diagram, Steps To Draw - Techiescientist
Argon is a group 18th element having atomic number 18. It is denoted by the symbol Ar and is a noble gas. It is also the third most abundant gas present in the atmosphere of the earth. Around 1.288%
by mass of argon is present in the air, which can be isolated through fractional distillation.
It serves as the primary source of argon for industries. Argon is majorly used in industries where an inert atmosphere is required such as in the preparation of titanium.
In this article, we will study the Bohr diagram of argon with step by steps discussion.
Bohr Model of Argon
The Rutherford model of the atom proposed in 1911 could not explain the stability of electrons while moving around the nucleus.
As per the classical mechanics and electromagnetic theory such particles could not remain stable and would lose energy.
Niel Bohr modified this model and presented the improvised Bohr-Rutherford model of the atom in 1913. In this model, he postulated that every electron moves in a pre-decided orbit having fixed size
and energy.
The Bohr model of atom defines the atomic structure of elements through a pictorial representation illustrating all the atomic particles viz. electrons, protons, and neutrons.
Before we dig deep into the Bohr model of the atom we should first understand a few important terms related to this model used in describing the structure of an atom.
• Nucleus: The centre or core of the atom comprising neutrons and protons. It derives its positive charge owing to the positively charged protons.
• Protons: These are the positively charged entities present in the nucleus of an atom and are represented through the symbol p^+.
• Neutrons: These are the neutral entities present inside the nucleus of an atom and are responsible for most physical properties of atoms. These are denoted by using the symbol n°.
• Electrons: The negatively charged entities that move in fixed circular paths orbiting the nucleus of an atom. These are denoted by the symbol e^–.
The location of an electron with respect to the nucleus depends upon the energy of the electron.
• Shells: The path taken by the electrons around the nucleus are termed as shells or orbits. Only a fixed number of electrons are allowed to follow a particular orbit owing to the difference in their
energy and capacity of the shell to accommodate the electrons.
In the Bohr model of the atom, the shells are named as K, L, M, N, etc., or 1, 2, 3, 4, etc. This number increases away from the nucleus. The energy of the electrons also increases as the number of
shells increases. This is why the shells are also known as energy levels.
Therefore, the electrons located in the K-shell i.e. the shell closest to the nucleus are said to be in the ground state and carry minimum energy.
The electrons located in the outermost shell carry maximum energy and are also referred to as valence electrons. These electrons are responsible for the formation of bonds.
The electrons are also allowed to hop from lower to higher as they gain energy or fall from higher to lower energy levels as they lose their energy.
The argon atom contains 22 neutrons, 18 protons, and 18 electrons. The electrons revolve around the nucleus in K, L, and M shells.
Argon Atom Value
No. of Proton 18
No. of Neutron 22
Number of Electron 18
Number of shells 3
Number of electrons in first (K) shell 2
Number of electrons in second (L) shell 8
Number of electrons in third (M) shell 8
Number of valence electrons 8
Drawing Bohr Model of Argon
Argon is a noble gas located in group 18 of the periodic table:
The information that we can derive from the above-mentioned Argon box is as follows:
• The atomic number of argon is 18.
• The electronic configuration of chlorine is [Ne] 3s^23p^6.
• The chemical symbol of chlorine is Ar.
• The atomic mass of chlorine is 39.948.
Now, using the above information we will draw the Bohr atomic model for the argon atom.
For this, we will first have to calculate the number of atomic species. Let us begin with protons.
The number of protons for any atom is always equal to the atomic number of that atom.
In the case of the argon atom, the atomic number is 18.
Therefore, for the argon atom, the number of protons = atomic number = 18
Moving on, we will now calculate the number of neutrons in the argon atom.
The formula for calculating the number of neutrons present in an atom is given below:
Number of neutrons = Atomic mass (rounding it up to the nearest whole number) – Number of protons
Now, using the information from the argon box we know that the atomic mass of argon is 39.948.
After rounding it up to the nearest whole number we get 40.
Also, as calculated above, the number of protons in the argon atom is 18.
Now, putting these values in the above-mentioned formula:
Number of neutrons = 40 – 18 = 22
Therefore, the number of neutrons in the argon atom = 22
As protons and neutrons constitute the nucleus of an atom, using the above values we can now draw the nucleus of the argon atom. It is as follows:
In this diagram, the p^+ represents protons and n° represents neutrons.
Now, moving on to add the shells to the argon nucleus we will calculate the number of electrons.
For any atom, the number of electrons is always equal to the atomic number of that atom.
Therefore, in the case of argon atoms,
Number of electrons = Atomic number of argon = 18
Further, we will now count the number of shells and also, the number of electrons that can be accommodated in each shell. As discussed earlier only a limited number of electrons can be housed in a
specific shell.
The maximum number of electrons that can be housed in a particular shell is given by 2n^2, where n refers to the number of shells.
Now, applying this formula for the argon atom, we will calculate the number of electrons for each shell, separately.
For the K shell of the argon atom, the maximum number of electrons = 2 (1)^2 = 2
After adding these two electrons to the first shell the atom appears as follows:
Post this, the electrons are to be added to the L shell of the argon atom.
Let us calculate the number of electrons that can be accommodated in the L shell.
The maximum number of electrons for the L shell of the argon atom = 2 (2)^2 = 8
Hence, the L shell has the capacity to house 8 electrons.
The important point to be mentioned at this stage is that for the K shell there are only two electrons that are placed close to it each other. However, from the L shell onwards, as the number of
electrons increases the arrangement pattern changes.
The electrons are now arranged in a group of four, in a clockwise direction. The first four electrons are positioned at 90° to each other. This angle keeps decreasing as the number of electrons
Therefore, the first four electrons in the L shell of the argon atom are arranged as follows:
The remaining four electrons are now added to the L shell, again in a clockwise manner.
After this the argon atom is now represented as:
Now, we are left with 8 more electrons which will be accommodated in the M shell.
Therefore, we will calculate the number of electrons that can be accommodated in the M shell.
The maximum number of electrons for M shell of the argon atom = 2 (3)^2 = 18
Hence, a total of 18 electrons are allowed to be housed in the M shell.
As only 8 electrons are left for the argon atom, we can accommodate all of them in the M shell.
The first four electrons again will be added in a clockwise manner.
Finally, after adding all the eight electrons to the M shell, we get the Bohr model of the argon atom as drawn below.
Hence, the final Bohr model of the argon atom consists of 18 protons and 22 neutrons inside the nucleus, and 18 electrons revolving around the nucleus.
There are 2 electrons present in the K shell, 8 electrons in the L shell, and 8 electrons in the M shell.
Deriving Lewis Structure of Argon from Bohr Model
The Lewis structure or electron dot structure is the illustration of the atom of an element along with its valence electrons. The nucleus is represented with the atomic symbol of the element while
electrons are represented using dots.
As discussed above, the argon atom consists of 8 electrons in its valence shell i.e. M shell. Therefore, the Lewis structure of argon is represented as follows:
Properties of Argon
A few important properties of argon are listed below:
• It was discovered by Sir Ramsay in 1894.
• It is a colourless gas that appears violet under the influence of an electric field.
• The melting and boiling points of argon are −185.848 °C and −189.34 °C, respectively.
• The density of argon is 1.78.10^-3 g.cm^-3 at 0 °C.
• The solubility of argon in water is similar to nitrogen and 2.5 times more than nitrogen.
Below is the video attached for the bohr model of Argon.
Related Topics
As per the Bohr model of an atom, the argon atom consists of 18 protons and 22 neutrons in the nucleus while 18 electrons revolve around the nucleus.
The number of protons, as well as the number of electrons in an atom, is always equal to the atomic number of that atom.
The number of neutrons is given by the formula:
Number of neutrons = Atomic mass (rounding it up to the nearest whole number) – Number of protons
The maximum number of electrons that can be housed in a shell is given by the formula 2n^2, where n is the number of shells.
The argon atom consists of three shells viz. K, L, and M shells have 2, 8, and 8 electrons, respectively.
Happy learning!!
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Reflection Symmetry - Definition, Examples, and Diagrams
A shape is said to have a reflection symmetry if there exists at least one line that divides a figure into two halves such that one-half is the mirror image of the other half. Thus, it is also known
as line symmetry or mirror symmetry. The line that divides the shape into two halves is the line of symmetry.
Not all shapes have lines of symmetry, or they may have several lines of symmetry. Shown below are some shapes having reflectional symmetry.
Let us consider the leaf above. When the line of symmetry passes through it, the leaf splits into two congruent halves. So, when we fold it along the dotted line, one half overlaps with the other.
The rectangle in figure-1 has 2 lines of symmetry. So it has 2-fold reflection symmetry. The square has 4 lines of symmetry. So it has 4-fold reflection symmetry. Likewise, X and H have 2-fold
reflection symmetry, and the club shape has 1-fold reflection symmetry. Also, every regular polygon has reflectional symmetry.
Similarly, an equilateral triangle has 2-fold reflection symmetry and an isosceles triangle has 1-fold reflection symmetry.
But, which triangle has 0 reflection symmetries?
So, a scalene triangle, has no lines of symmetry, and thus has no reflection symmetries.
Examples in Real-Life
1. Wings of a butterfly
2. The famous painting, Vitruvian Man by Leonardo da Vinci
3. Human face considering our nose as the line of symmetry
Solved Examples
Which shape has reflectional symmetry in the diagram given alongside?
The telephone in fig – 1 has reflection symmetry but the key in fig – 2 does not.
Which figure in the diagram has a vertical line of reflection symmetry?
Only the pentagon in fig – 2 has a vertical line of reflection symmetry. The rest have a horizontal line of reflection symmetry.
Q1. How many lines of reflection symmetry does the trapezoid have?
Ans. A trapezoid is not a symmetrical shape. So it does not have lines of reflection. However, an isosceles trapezoid has one vertical line of symmetry. So it has one line of reflection symmetry.
Q2. How many reflection symmetries does the regular hexagon have?
Ans. A regular hexagon has 3 reflection symmetries.
Q3. How many reflection symmetries does a regular decagon have?
Ans. A regular decagon10 reflection symmetries.
Q.4. Which quadrilateral will always have 4 fold reflectional symmetry?
Ans. A square has a 4 fold reflectional symmetry.
One thought on “Reflection Symmetry”
1. 🙂 Excellent Article, Excellent Blog , Excellent Site ✅✅✅
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Complementary Probability Worksheet and Solutions
Work out the following.
When necessary, give your answer in fractions eg. 2/5
1. A set of cards with a letter on each card as shown below are placed into a bag. Howard picks a card at random from the bag.
Determine the probability that the card is:
a) an E.
b) not an E.
c) not a vowel.
d) a P.
e) not a P.
f) either a Q or U or H
g) not a Q, U or H.
2. A number is chosen at random from a set of whole numbers from 1 to 50. Calculate the probability that the chosen number:
a) is not a perfect square
b) is not a multiple of 4
c) is more than 45
d) is not more than 45
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Finitely 1-convex f-rings
This paper investigates f-rings that can be constructed in a finite number of steps where every step consists of taking the fibre product of two f-rings, both being either a 1-convex f-ring or a
fibre product obtained in an earlier step of the construction. These are the f-rings that satisfy the algebraic property that rings of continuous functions possess when the underlying topological
space is finitely an F-space (i.e. has a Stone-čech compactification that is a finite union of compact F-spaces). These f-rings are shown to be SV f-rings with bounded inversion and finite rank and,
when constructed from semisimple f-rings, their maximal ideal space under the hull-kernel topology contains a dense open set of maximal ideals containing a unique minimal prime ideal. For a large
class of these rings, the sum of prime, semiprime, primary and z-ideals are shown to be prime, semiprime, primary and z-ideals respectively.
Original Publication Citation
Larson, Suzanne. “Finitely 1-Convex f-Rings.” Topology and Its Applications, vol. 158, no. 14, Jan. 2011, pp. 1888–1901. doi:10.1016/j.topol.2011.06.025.
Digital Commons @ LMU & LLS Citation
Larson, Suzanne, "Finitely 1-convex f-rings" (2011). Mathematics, Statistics and Data Science Faculty Works. 162.
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Definitions for mathematical
ˌmæθ əˈmæt ɪ kəlmath·e·mat·i·cal
This dictionary definitions page includes all the possible meanings, example usage and translations of the word mathematical.
Princeton's WordNet
1. mathematicaladjective
of or pertaining to or of the nature of mathematics
"a mathematical textbook"; "slide rules and other mathematical instruments"; "a mathematical solution to a problem"; "mathematical proof"
2. numerical, mathematicaladjective
relating to or having ability to think in or work with numbers
"tests for rating numerical aptitude"; "a mathematical whiz"
3. mathematicaladjective
beyond question
"a mathematical certainty"
4. mathematicaladjective
statistically possible though highly improbable
"have a mathematical chance of making the playoffs"
5. mathematicaladjective
characterized by the exactness or precision of mathematics
"mathematical precision"
1. mathematicaladjective
Of, or relating to mathematics
2. mathematicaladjective
Possible but highly improbable
Samuel Johnson's Dictionary
1. MATHEMATICAL, MATHEMATICKadjective
Considered according to the doctrine of the mathematicians.
Etymology: mathematicus, Lat.
The East and West,
Upon the globe, a mathematick point
Only divides: thus happiness and misery,
And all extremes, are still contiguous. John Denham, Sophy.
It is as impossible for an aggregate of finites to comprehend or exhaust one infinite, as it is for the greatest number of mathematick points to amount to, or constitute a body. Boyle.
I suppose all the particles of matter to be situated in an exact and mathematical evenness. Richard Bentley, Serm.
1. mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes.
These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among
mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove
them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of
applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are
considered true starting points of the theory under consideration.Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences.
Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as
statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any
application (and are therefore called pure mathematics), but often later find practical applications. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had
no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks. Historically, the concept of a proof and its associated mathematical rigour
first appeared in Greek mathematics, most notably in Euclid's Elements. Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers
and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and
scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the
axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60
first-level areas of mathematics.
1. mathematical
Mathematical refers to anything related to, involving, or characterized by mathematics. It often describes principles, concepts, methods, or techniques that are derived from or using mathematics.
It can also describe any process that follows a logical or quantitative approach similar to that utilized in mathematics.
Webster Dictionary
1. Mathematicaladjective
of or pertaining to mathematics; according to mathematics; hence, theoretically precise; accurate; as, mathematical geography; mathematical instruments; mathematical exactness
2. Etymology: [See Mathematic.]
Editors Contribution
1. mathematical
Relating to mathematics.
The mathematical process was easy and simple.
Submitted by MaryC on March 7, 2020
British National Corpus
1. Adjectives Frequency
Rank popularity for the word 'mathematical' in Adjectives Frequency: #878
Usage in printed sourcesFrom:
1. [["1505","8"],["1520","1"],["1524","3"],["1590","4"],["1600","3"],["1611","1"],["1624","2"],["1635","3"],["1637","1"],["1638","1"],["1647","1"],["1648","1"],["1656","1"],["1657","4"],
1. Chaldean Numerology
The numerical value of mathematical in Chaldean Numerology is: 9
2. Pythagorean Numerology
The numerical value of mathematical in Pythagorean Numerology is: 7
Examples of mathematical in a Sentence
1. God does not care about our mathematical difficulties. He integrates empirically.
2. I know that most men -- not only those considered clever, but even those who are very clever and capable of understanding most difficult scientific, mathematical, or philosophic, problems - can
seldom discern even the simplest and most obvious truth if it be such as obliges them to admit the falsity of conclusions they have formed, perhaps with much difficulty -- conclusions of which
they are proud, which they have taught to others, and on which they have built their lives.
3. In studying mathematics or simply using a mathematical principle, if we get the wrong answer in sort of algebraic equation, we do not suddenly feel that there is an anti-mathematical principle
that is luring us into the wrong answers.
4. We will be able to reconstruct the history of art and develop a mathematical theory of its evolution, just as scientists have done for the history of life.
5. The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal
interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
Popularity rank by frequency of use
Translations for mathematical
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MU Applied Physics 1 - December 2012 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
Attempt any FIVE.
1(a) Explain term lattice parameters of cubic crystal.
3 M
1(b) What is the probability of an electron being thermally excited to conduction band in silicon at 20^oC. The band gap energy is 1.12eV; Boltzmann constant is 1.38 x 10^-23 JIk.
3 M
1(c) Mobility of holes is 0.025 m^2 /V-sec What would be the resistivity of P-type silicon . if the Hall coefficient of the sample is 2.25 x 10^-5 m^3 /C ?
3 M
1(d) Define dielectrics, electric dipole, polarizability.
3 M
1(e) Difference between soft and hard magnetic materials
3 M
1(f) Define Reverberation time. Write Sabine's formula and explain terms in it.
3 M
1(g) State the terms : magnetostriction effect; piezoelectric effect
3 M
2(a) Explain the formation of energy band in solids. With neat energy band diagram explain extrinsic semiconductors.
8 M
2(b) Draw the unit cell of HCP. What is its co-ordination number, atomic radius, effective number of atoms per unit cell. Also calculate its packing factor .
7 M
3(a) What is hysteresis? Draw a hysteris loop for ferromagnetic material and explain the various important points on it. What is the technical significance of the area enclosed under it. For a
transformer which kind of material will you prefer-the one with small hysteresis area or the big one ?
8 M
3(b) Derive Bragg's law. Calculate the glancing angle on the plane (100) for a crystal of rock salt (a = 2.125 A^0 .Consider the case of 2nd order maximum and A = 0.592 A^0.
7 M
4(a) Calculate the number of atoms per unit cell of a metal having lattice parameter 2.9A^0 and density 7.87 gm/cm3. Atomic weight of metal is 55.85, Avagadro number is 6.023 x 10^23 /gm-mole.
5 M
4(b) Prove that the Fermi level lies exactly at the centre of the forbidden energy gap in case of an intrinsic semiconductor.
5 M
4(c) Explain ionic polarization and obtain polarizability.
5 M
5(a) With neat diagram of a unit cell, explain the structure of BaTi0[3].
5 M
5(b) What is Hall effect ? Derive expression for Hall voltage.
5 M
5(c) Explain the absorption coeficient of a hall. Calculate the change in intensity level if the intensity of sound increases 1000 times its original intensity.
5 M
6(a) In what sense real crystals differ from ideal crystals? Explain the point defects in crystals.
5 M
6(b) Explain construction and working of a solar cell.
5 M
6(c) Find the natural frequency of vibration of quartz plate of thickness 2mm. Given Young's 5 modulus of quartz Y = 8 x 10^10 N/m^2, density of quartz is 2650 kg/m^3. Calculate the change in
thickness required if the same plate is used to produce ultrasonic waves of frequency 3MHz.
5 M
More question papers from Applied Physics 1
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The inadequacy of SCLT
As I mentioned a few posts ago, I included the Diophantine equation x^4 + y^6 = z^10 on the Advanced Mentoring Scheme. I’m not going to spoil it here, although I have since been informed that I had
previously included it as an exercise in MODA.
Let’s consider the weaker equation, a^2 + b^3 = c^5. It’s quite easy to solve it using the good old Sam Cappleman-Lynes technique (henceforth abbreviated to SCLT), beginning with a solution to d^2 +
e^3 = f. The simplest solution to this is 1 + 1 = 2, which we’ll then multiply throughout by 2^24 to give an integer solution to the original equation.
Note that 4096, 256 and 32 are not coprime by any stretch of the imagination. Can we find a coprime solution to a^2 + b^3 = c^5 in the positive integers? I stumbled across an interesting discussion
which demonstrated a geometrical construction for the Diophantine equation a^4 + b^3 = c^2. I’ll explain and apply it to the more flamboyant case of our equation, which is intimately connected to the
geometry of the icosahedron.
The geometry of the icosahedron
The twelve vertices of the icosahedron can be partitioned into three concentric congruent golden rectangles, the perimeters of which form Borromean rings:
The side lengths of the rectangles are in the ratio 1 : φ, where φ = (1 + sqrt(5))/2 is the golden ratio. We’ll use this later. Firstly, note that the icosahedron can be inscribed in a sphere. We
then identify this (the Riemann sphere) with the extended complex plane (like C, but with a point at infinity) by stereographic projection. This is another recurring theme in MODA.
We can see instantly that the lowest vertex is at 0, and the one diametrically opposite is at ∞. There’s still a complex degree of freedom left, which we’ll choose carefully to give the neatest
expressions for the remaining vertices. Let’s choose a pair of antipodal vertices to lie on the real line, and take a cross-section of the sphere:
Let the diameter BD be 1. Then, we can deduce that the points on the horizontal line are positioned at ψ and φ, the two roots of z^2 − z − 1 = 0. By symmetry, we get that the twelve vertices are the
roots of the projective equation xy(y^5 − φ^5 x^5)(y^5 − ψ^5 x^5) = 0, which expands out to f(x, y) = x y^11 − 11 x^6 y^6 − x^11 y = 0; the coefficient ’11’ comes from being the fifth Lucas number.
Hence, plotting the roots of z^11 − 11 z^6 − z = 0 in Wolfram Alpha will give you the vertices of a stereographically projected icosahedron.
Hessians and Jacobians
Following the instructions in the e-mail, we compute the Hessian h and Jacobian j of f and h. These are the determinants of matrices of partial derivatives. After scaling to make the coefficients
smaller, we get:
• h = x^20 − 228 x^15 y^5 + 494 x^10 y^10 + 228 x^5 y^15 + y^20
• j = x^30 + 522 x^25 y^5 − 10005 x^20 y^10 − 10005 x^10 y^20 −
522 x^5 y^25 + y^30
Amazingly, we have the identity h^3 − 1728 f^5 = j^2, which holds for all complex values of (x,y). (That’s not the only place where 1728 appears as a scaling factor!) We can recover a solution to the
original equation by applying a subsitution:
• a = j
• b = −h
• c = 1728^(1/5) f
Now, these aren’t integer polynomials in x and y, so we rescale by letting x = 16^(1/5) s and y = 9^(1/5) t. If we substitute s = t = 1 we get a solution in the integers, but a couple of the values
are negative. If we try s = 2 and t = 1 instead, the results are positive. This leads to a valid positive integer solution where a,b,c are coprime:
• 127602747389962225^2 + 196120763999^3 = 7506024^5
As an exercise, you might want to do the same with the octahedron to get a solution to a^2 + b^3 = c^4 with coprime terms. Enjoy!
Beal’s conjecture
This is almost (but not quite) a counter-example to Beal’s conjecture. This claims that for a solution to a^l + b^m + c^n with l,m,n > 2, the integers a,b,c share a common factor. It is obvious that
this (if true) implies Fermat’s last theorem, since from a solution to a^n + b^n = c^n we could divide throughout by gcd(a,b,c)^n to obtain a coprime solution. Not that this is of much interest,
anyway, since we already know that Fermat’s last theorem is true.
Unfortunately, the geometric method cannot be applied to generate a solution with all exponents greater than 2, and SCLT just gives massive common factors. This is one of the many Annoying Aspects of
Mathematics. Another example is that the only pairs of positive integers (m,n) for which 1² + 2² + 3² + … + m² = n² holds are (1, 1) and (24, 70), and it’s impossible to dissect a 70 by 70 square
into squares of side lengths 1, 2, …, 24. Indeed, it’s an open problem with a $100 prize as to whether there exists a rectangle which can be dissected into squares of side lengths 1, 2, …, m. By
comparison, Beal’s conjecture holds a $100000 prize.
2 Responses to The inadequacy of SCLT
1. I get that the icosahedron (and analogously the octahedron and tetrahedron) have symmetry groups corresponding to a pair of generators with orders 2 and 3 and their product having order 5 (or 4
or 3 analogously). But how on earth does this translate to a solution to a^2+b^3=c^5? I can follow the algebra, but what I don’t get is why it works. What do the Hessians and Jacobians have to do
with it? Could someone please demystify this seemingly magical construction?
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Inverse of a Domain Restricted Function
Additional Resources:
In this lesson, we will learn how to find the inverse of a function when the domain is restricted. Additionally, we will learn how to find the inverse of a many-to-one function by imposing a domain
restriction. In a many-to-one function, multiple input values can yield the same output value, making it non-invertible over its entire domain. However, by carefully selecting a restricted domain
that eliminates the multiple outputs, we can create a one-to-one function and then find our inverse. To impose a domain restriction, we aim to identify a subset of the original domain where the
function exhibits a unique output for every input. This can be achieved by carefully considering the characteristics of the function and analyzing its behavior. By excluding specific input values or
intervals that lead to non-unique outputs, we create a restricted domain that ensures the function becomes one-to-one. With this restriction in place, we can confidently proceed to find the inverse
of the function, as the one-to-one mapping guarantees a unique output for each input, facilitating a well-defined inverse function.
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Mathematical Treatment of Measurement Results
7 Mathematical Treatment of Measurement Results
Learning Objectives
By the end of this section, you will be able to:
• Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities
• Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties
It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate
mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the time required for the athlete to run from
the starting line to the finish line, and the distance between these two lines, and then computing speed from the equation that relates these three properties:
An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of
\(\frac{\text{100 m}}{\text{10 s}}\phantom{\rule{0.2em}{0ex}}=\text{10 m/s}\)
Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) and likewise dividing the units of each measured
quantity to yield the unit of the computed quantity (m/s = m/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m.
The same relation among the three properties is used, but in this case, the two quantities provided are a speed (10 m/s) and a distance (25 m). To yield the sought property, time, the equation must
be rearranged appropriately:
The time can then be computed as:
\(\frac{\text{25 m}}{\text{10 m/s}}\phantom{\rule{0.2em}{0ex}}=\text{2.5 s}\)
Again, arithmetic on the numbers (25/10 = 2.5) was accompanied by the same arithmetic on the units (m/m/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when
a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.”
These calculations are examples of a versatile mathematical approach known as dimensional analysis (or the factor-label method). Dimensional analysis is based on this premise: the units of quantities
must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step
calculations involving several different quantities.
Conversion Factors and Dimensional Analysis
A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition),
and so a unit conversion factor may be derived from the ratio,
\(\frac{\text{2.54 cm}}{\text{1 in.}}\phantom{\rule{0.2em}{0ex}}\text{(2.54 cm}=\text{1 in.) or 2.54}\phantom{\rule{0.2em}{0ex}}\frac{\text{cm}}{\text{in.}}\)
Several other commonly used conversion factors are given in (Figure).
Common Conversion Factors
Length Volume Mass
1 m = 1.0936 yd 1 L = 1.0567 qt 1 kg = 2.2046 lb
1 in. = 2.54 cm (exact) 1 qt = 0.94635 L 1 lb = 453.59 g
1 km = 0.62137 mi 1 ft^3 = 28.317 L 1 (avoirdupois) oz = 28.349 g
1 mi = 1609.3 m 1 tbsp = 14.787 mL 1 (troy) oz = 31.103 g
When a quantity (such as distance in inches) is multiplied by an appropriate unit conversion factor, the quantity is converted to an equivalent value with different units (such as distance in
centimeters). For example, a basketball player’s vertical jump of 34 inches can be converted to centimeters by:
\(34\phantom{\rule{0.2em}{0ex}}\overline{)\text{in.}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{2.54 cm}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{in.}}}\phantom{\rule
{0.2em}{0ex}}=\text{86 cm}\)
Since this simple arithmetic involves quantities, the premise of dimensional analysis requires that we multiply both numbers and units. The numbers of these two quantities are multiplied to yield the
number of the product quantity, 86, whereas the units are multiplied to yield \(\frac{\text{in.}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{cm}}{\text{in.}}\). Just as for numbers, a
ratio of identical units is also numerically equal to one, \(\frac{\text{in.}}{\text{in.}}\phantom{\rule{0.2em}{0ex}}=\text{1,}\) and the unit product thus simplifies to cm. (When identical units
divide to yield a factor of 1, they are said to “cancel.”) Dimensional analysis may be used to confirm the proper application of unit conversion factors as demonstrated in the following example.
Using a Unit Conversion Factor The mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g ((Figure)).
Solution Given the conversion factor, the mass in ounces may be derived using an equation similar to the one used for converting length from inches to centimeters.
\(x\phantom{\rule{0.2em}{0ex}}\text{oz}=\text{125 g}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{unit conversion factor}\)
The unit conversion factor may be represented as:
\(\frac{\text{1 oz}}{\text{28.349 g}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{\text{28.349 g}}{\text{1 oz}}\)
The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.
\(\begin{array}{ccc}x\phantom{\rule{0.2em}{0ex}}\text{oz}\hfill & =& 125\phantom{\rule{0.2em}{0ex}}\overline{)\text{g}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1 oz}}{\text
{28.349}\phantom{\rule{0.2em}{0ex}}\overline{)\text{g}}}\hfill \\ \hfill & =\hfill & \left(\frac{125}{\text{28.349}}\right)\phantom{\rule{0.2em}{0ex}}\text{oz}\hfill \\ \hfill & =\hfill & \text{4.41
oz (three significant figures)}\hfill \end{array}\)
Check Your Learning Convert a volume of 9.345 qt to liters.
Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same—all the factors
involved in the calculation must be appropriately oriented to ensure that their labels (units) will appropriately cancel and/or combine to yield the desired unit in the result. As your study of
chemistry continues, you will encounter many opportunities to apply this approach.
Computing Quantities from Measurement Results and Known Mathematical Relations What is the density of common antifreeze in units of g/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.
Solution Since \(\text{density}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{mass}}{\text{volume}}\), we need to divide the mass in grams by the volume in milliliters. In
general: the number of units of B = the number of units of A \(×\) unit conversion factor. The necessary conversion factors are given in (Figure): 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL.
Mass may be converted from pounds to grams as follows:
\(\text{9.26}\phantom{\rule{0.2em}{0ex}}\overline{)\text{lb}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{453.59 g}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{lb}}}\phantom{\
Volume may be converted from quarts to millimeters via two steps:
1. Convert quarts to liters.
\(\text{4.00}\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1 L}}{\text{1.0567}\phantom{\rule{0.2em}{0ex}}\overline{)\text
{qt}}}\phantom{\rule{0.2em}{0ex}}=\text{3.78 L}\)
2. Convert liters to milliliters.
\(\text{3.78}\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1000 mL}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}\phantom
phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{mL}}\phantom{\rule{0.2em}{0ex}}=\text{1.11 g/mL}\)
Alternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:
{453.59 g}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{lb}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1.0567}\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}{1\phantom{\
rule{0.2em}{0ex}}\overline{)\text{L}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}{\text{1000 mL}}\phantom{\rule{0.2em}{0ex}}=\text
{1.11 g/mL}\)
Check Your Learning What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?
Computing Quantities from Measurement Results and Known Mathematical Relations While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses
213 L gasoline.
(a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?
(b) If gasoline costs ?3.80 per gallon, what was the fuel cost for this trip?
Solution (a) First convert distance from kilometers to miles:
\(1250\phantom{\rule{0.2em}{0ex}}\overline{)\text{km}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{0.62137 mi}}{1\phantom{\rule{0.2em}{0ex}}\overline{)\text{km}}}\phantom{\rule
{0.2em}{0ex}}=\text{777 mi}\)
and then convert volume from liters to gallons:
{0ex}}\overline{)\text{L}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{1 gal}}{4\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}\phantom{\rule{0.2em}{0ex}}=\text{56.3 gal}\)
\(\text{(average) mileage}=\phantom{\rule{0.2em}{0ex}}\frac{\text{777 mi}}{\text{56.3 gal}}\phantom{\rule{0.2em}{0ex}}=\text{13.8 miles/gallon}=\text{13.8 mpg}\)
Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:
\(\frac{1250\phantom{\rule{0.2em}{0ex}}\overline{)\text{km}}}{213\phantom{\rule{0.2em}{0ex}}\overline{)\text{L}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{\text{0.62137 mi}}{1\
\overline{)\text{qt}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{4\phantom{\rule{0.2em}{0ex}}\overline{)\text{qt}}}{\text{1 gal}}=\text{13.8 mpg}\)
(b) Using the previously calculated volume in gallons, we find:
Check Your Learning A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).
(a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip?
(b) If gasoline costs ?3.90 per gallon, what was the fuel cost for this trip?
(a) 51 mpg; (b) ?62
Conversion of Temperature Units
We use the word temperature to refer to the hotness or coldness of a substance. One way we measure a change in temperature is to use the fact that most substances expand when their temperature
increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes, and the position of the trapped liquid
along a printed scale may be used as a measure of temperature.
Temperature scales are defined relative to selected reference temperatures: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On
the Celsius scale, 0 °C is defined as the freezing temperature of water and 100 °C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals,
which we call degrees. On the Fahrenheit scale, the freezing point of water is defined as 32 °F and the boiling temperature as 212 °F. The space between these two points on a Fahrenheit thermometer
is divided into 180 equal parts (degrees).
Defining the Celsius and Fahrenheit temperature scales as described in the previous paragraph results in a slightly more complex relationship between temperature values on these two scales than for
different units of measure for other properties. Most measurement units for a given property are directly proportional to one another (y = mx). Using familiar length units as one example:
\(\text{length in feet}=\left(\frac{\text{1 ft}}{\text{12 in.}}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{length in inches}\)
where y = length in feet, x = length in inches, and the proportionality constant, m, is the conversion factor. The Celsius and Fahrenheit temperature scales, however, do not share a common zero
point, and so the relationship between these two scales is a linear one rather than a proportional one (y = mx + b). Consequently, converting a temperature from one of these scales into the other
requires more than simple multiplication by a conversion factor, m, it also must take into account differences in the scales’ zero points (b).
The linear equation relating Celsius and Fahrenheit temperatures is easily derived from the two temperatures used to define each scale. Representing the Celsius temperature as x and the Fahrenheit
temperature as y, the slope, m, is computed to be:
\(m=\phantom{\rule{0.2em}{0ex}}\frac{\text{Δ}y}{\text{Δ}x}\phantom{\rule{0.2em}{0ex}}=\frac{\text{212 °F}-\text{32 °F}\phantom{\rule{0.2em}{0ex}}}{\text{100 °C}-\text{0 °C}}\phantom{\rule{0.2em}
{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{180 °F}}{\text{100 °C}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{9 °F}}{\text{5 °C}}\)
The y-intercept of the equation, b, is then calculated using either of the equivalent temperature pairs, (100 °C, 212 °F) or (0 °C, 32 °F), as:
\(b=y-mx=\text{32 °F}-\phantom{\rule{0.2em}{0ex}}\frac{\text{9 °F}}{\text{5 °C}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{0 °C}=\text{32 °F}\)
The equation relating the temperature (T) scales is then:
\({T}_{\text{°F}}=\left(\frac{\text{9 °F}}{\text{5 °C}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{T}_{\text{°C}}\right)+\text{32 °C}\)
An abbreviated form of this equation that omits the measurement units is:
Rearrangement of this equation yields the form useful for converting from Fahrenheit to Celsius:
As mentioned earlier in this chapter, the SI unit of temperature is the kelvin (K). Unlike the Celsius and Fahrenheit scales, the kelvin scale is an absolute temperature scale in which 0 (zero) K
corresponds to the lowest temperature that can theoretically be achieved. Since the kelvin temperature scale is absolute, a degree symbol is not included in the unit abbreviation, K. The early
19th-century discovery of the relationship between a gas’s volume and temperature suggested that the volume of a gas would be zero at −273.15 °C. In 1848, British physicist William Thompson, who
later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text’s chapter on gases).
The freezing temperature of water on this scale is 273.15 K and its boiling temperature is 373.15 K. Notice the numerical difference in these two reference temperatures is 100, the same as for the
Celsius scale, and so the linear relation between these two temperature scales will exhibit a slope of \(1\phantom{\rule{0.2em}{0ex}}\frac{\text{K}}{\text{°C}}\). Following the same approach, the
equations for converting between the kelvin and Celsius temperature scales are derived to be:
The 273.15 in these equations has been determined experimentally, so it is not exact. (Figure) shows the relationship among the three temperature scales.
Although the kelvin (absolute) temperature scale is the official SI temperature scale, Celsius is commonly used in many scientific contexts and is the scale of choice for nonscience contexts in
almost all areas of the world. Very few countries (the U.S. and its territories, the Bahamas, Belize, Cayman Islands, and Palau) still use Fahrenheit for weather, medicine, and cooking.
Conversion from Celsius Normal body temperature has been commonly accepted as 37.0 °C (although it varies depending on time of day and method of measurement, as well as among individuals). What is
this temperature on the kelvin scale and on the Fahrenheit scale?
\(\text{K}=\text{°C}+273.15=37.0+273.2=\text{310.2 K}\)
\(\text{°F}=\frac{9}{5}\text{°C}+32.0=\left(\frac{9}{5}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}37.0\right)+32.0=66.6+32.0=\text{98.6 °F}\)
Check Your Learning Convert 80.92 °C to K and °F.
354.07 K, 177.7 °F
Conversion from Fahrenheit Baking a ready-made pizza calls for an oven temperature of 450 °F. If you are in Europe, and your oven thermometer uses the Celsius scale, what is the setting? What is the
kelvin temperature?
phantom{\rule{0.2em}{0ex}}418=\text{232 °C}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}\text{set oven to 230 °C}\phantom{\rule{2em}{0ex}}\left(\text{two significant figures}\right)\)
\(\text{K}=\text{°C}+273.15=230+273=\text{503 K}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}5.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}\text{K}\phantom{\rule{2em}
{0ex}}\left(\text{two significant figures}\right)\)
Check Your Learning Convert 50 °F to °C and K.
Key Concepts and Summary
Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion
factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities
using measured quantities and appropriate mathematical relations.
Key Equations
• \({T}_{\text{°C}}=\phantom{\rule{0.2em}{0ex}}\frac{5}{9}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left({T}_{\text{°F}}-32\right)\)
• \({T}_{\text{°F}}=\phantom{\rule{0.2em}{0ex}}\left(\frac{9}{5}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{T}_{\text{°C}}\right)+32\)
• \({T}_{\text{K}}=\text{°C}+273.15\)
• \({T}_{\text{°C}}=\text{K}-273.15\)
Chemistry End of Chapter Exercises
Write conversion factors (as ratios) for the number of:
(a) yards in 1 meter
(b) liters in 1 liquid quart
(c) pounds in 1 kilogram
(a) \(\frac{\text{1.0936 yd}}{\text{1 m}}\); (b) \(\frac{\text{0.94635 L}}{\text{1 qt}}\); (c) \(\frac{\text{2.2046 lb}}{\text{1 kg}}\)
Write conversion factors (as ratios) for the number of:
(a) kilometers in 1 mile
(b) liters in 1 cubic foot
(c) grams in 1 ounce
The label on a soft drink bottle gives the volume in two units: 2.0 L and 67.6 fl oz. Use this information to derive a conversion factor between the English and metric units. How many significant
figures can you justify in your conversion factor?
\(\begin{array}{c}\frac{\text{2.0 L}}{\text{67.6 fl oz}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{\text{0.030 L}}{\text{1 fl oz}}\end{array}\)
Only two significant figures are justified.
The label on a box of cereal gives the mass of cereal in two units: 978 grams and 34.5 oz. Use this information to find a conversion factor between the English and metric units. How many significant
figures can you justify in your conversion factor?
Soccer is played with a round ball having a circumference between 27 and 28 in. and a weight between 14 and 16 oz. What are these specifications in units of centimeters and grams?
A woman’s basketball has a circumference between 28.5 and 29.0 inches and a maximum weight of 20 ounces (two significant figures). What are these specifications in units of centimeters and grams?
How many milliliters of a soft drink are contained in a 12.0-oz can?
A barrel of oil is exactly 42 gal. How many liters of oil are in a barrel?
The diameter of a red blood cell is about 3 \(×\) 10^−4 in. What is its diameter in centimeters?
The distance between the centers of the two oxygen atoms in an oxygen molecule is 1.21 \(×\) 10^−8 cm. What is this distance in inches?
Is a 197-lb weight lifter light enough to compete in a class limited to those weighing 90 kg or less?
A very good 197-lb weight lifter lifted 192 kg in a move called the clean and jerk. What was the mass of the weight lifted in pounds?
Many medical laboratory tests are run using 5.0 μL blood serum. What is this volume in milliliters?
If an aspirin tablet contains 325 mg aspirin, how many grams of aspirin does it contain?
Use scientific (exponential) notation to express the following quantities in terms of the SI base units in (Figure):
(a) 0.13 g
(b) 232 Gg
(c) 5.23 pm
(d) 86.3 mg
(e) 37.6 cm
(f) 54 μm
(g) 1 Ts
(h) 27 ps
(i) 0.15 mK
(a) 1.3 \(×\) 10^−4 kg; (b) 2.32 \(×\) 10^8 kg; (c) 5.23 \(×\) 10^−12 m; (d) 8.63 \(×\) 10^−5 kg; (e) 3.76 \(×\) 10^−1 m; (f) 5.4 \(×\) 10^−5 m; (g) 1 \(×\) 10^12 s; (h) 2.7 \(×\) 10^−11 s; (i) 1.5 \
(×\) 10^−4 K
Complete the following conversions between SI units.
(a) 612 g = ________ mg
(b) 8.160 m = ________ cm
(c) 3779 μg = ________ g
(d) 781 mL = ________ L
(e) 4.18 kg = ________ g
(f) 27.8 m = ________ km
(g) 0.13 mL = ________ L
(h) 1738 km = ________ m
(i) 1.9 Gg = ________ g
Gasoline is sold by the liter in many countries. How many liters are required to fill a 12.0-gal gas tank?
Milk is sold by the liter in many countries. What is the volume of exactly 1/2 gal of milk in liters?
A long ton is defined as exactly 2240 lb. What is this mass in kilograms?
Make the conversion indicated in each of the following:
(a) the men’s world record long jump, 29 ft 4¼ in., to meters
(b) the greatest depth of the ocean, about 6.5 mi, to kilometers
(c) the area of the state of Oregon, 96,981 mi^2, to square kilometers
(d) the volume of 1 gill (exactly 4 oz) to milliliters
(e) the estimated volume of the oceans, 330,000,000 mi^3, to cubic kilometers.
(f) the mass of a 3525-lb car to kilograms
(g) the mass of a 2.3-oz egg to grams
Make the conversion indicated in each of the following:
(a) the length of a soccer field, 120 m (three significant figures), to feet
(b) the height of Mt. Kilimanjaro, at 19,565 ft, the highest mountain in Africa, to kilometers
(c) the area of an 8.5- × 11-inch sheet of paper in cm^2
(d) the displacement volume of an automobile engine, 161 in.^3, to liters
(e) the estimated mass of the atmosphere, 5.6 × 10^15 tons, to kilograms
(f) the mass of a bushel of rye, 32.0 lb, to kilograms
(g) the mass of a 5.00-grain aspirin tablet to milligrams (1 grain = 0.00229 oz)
(a) 394 ft; (b) 5.9634 km; (c) 6.0 \(×\) 10^2; (d) 2.64 L; (e) 5.1 \(×\) 10^18 kg; (f) 14.5 kg; (g) 324 mg
Many chemistry conferences have held a 50-Trillion Angstrom Run (two significant figures). How long is this run in kilometers and in miles? (1 Å = 1 \(×\) 10^−10 m)
A chemist’s 50-Trillion Angstrom Run (see (Figure)) would be an archeologist’s 10,900 cubit run. How long is one cubit in meters and in feet? (1 Å = 1 \(×\) 10^−8 cm)
The gas tank of a certain luxury automobile holds 22.3 gallons according to the owner’s manual. If the density of gasoline is 0.8206 g/mL, determine the mass in kilograms and pounds of the fuel in a
full tank.
As an instructor is preparing for an experiment, he requires 225 g phosphoric acid. The only container readily available is a 150-mL Erlenmeyer flask. Is it large enough to contain the acid, whose
density is 1.83 g/mL?
Yes, the acid’s volume is 123 mL.
To prepare for a laboratory period, a student lab assistant needs 125 g of a compound. A bottle containing 1/4 lb is available. Did the student have enough of the compound?
A chemistry student is 159 cm tall and weighs 45.8 kg. What is her height in inches and weight in pounds?
62.6 in (about 5 ft 3 in.) and 101 lb
In a recent Grand Prix, the winner completed the race with an average speed of 229.8 km/h. What was his speed in miles per hour, meters per second, and feet per second?
Solve these problems about lumber dimensions.
(a) To describe to a European how houses are constructed in the US, the dimensions of “two-by-four” lumber must be converted into metric units. The thickness \(×\) width \(×\) length dimensions are
1.50 in. \(×\) 3.50 in. \(×\) 8.00 ft in the US. What are the dimensions in cm \(×\) cm \(×\) m?
(b) This lumber can be used as vertical studs, which are typically placed 16.0 in. apart. What is that distance in centimeters?
(a) 3.81 cm \(×\) 8.89 cm \(×\) 2.44 m; (b) 40.6 cm
The mercury content of a stream was believed to be above the minimum considered safe—1 part per billion (ppb) by weight. An analysis indicated that the concentration was 0.68 parts per billion. What
quantity of mercury in grams was present in 15.0 L of the water, the density of which is 0.998 g/ml? \(\text{(1 ppb Hg}=\phantom{\rule{0.2em}{0ex}}\frac{\text{1 ng Hg}}{\text{1 g water}}\text{)}\)
Calculate the density of aluminum if 27.6 cm^3 has a mass of 74.6 g.
Osmium is one of the densest elements known. What is its density if 2.72 g has a volume of 0.121 cm^3?
Calculate these masses.
(a) What is the mass of 6.00 cm^3 of mercury, density = 13.5939 g/cm^3?
(b) What is the mass of 25.0 mL octane, density = 0.702 g/cm^3?
Calculate these masses.
(a) What is the mass of 4.00 cm^3 of sodium, density = 0.97 g/cm^3 ?
(b) What is the mass of 125 mL gaseous chlorine, density = 3.16 g/L?
Calculate these volumes.
(a) What is the volume of 25 g iodine, density = 4.93 g/cm^3?
(b) What is the volume of 3.28 g gaseous hydrogen, density = 0.089 g/L?
Calculate these volumes.
(a) What is the volume of 11.3 g graphite, density = 2.25 g/cm^3?
(b) What is the volume of 39.657 g bromine, density = 2.928 g/cm^3?
Convert the boiling temperature of gold, 2966 °C, into degrees Fahrenheit and kelvin.
Convert the temperature of scalding water, 54 °C, into degrees Fahrenheit and kelvin.
Convert the temperature of the coldest area in a freezer, −10 °F, to degrees Celsius and kelvin.
Convert the temperature of dry ice, −77 °C, into degrees Fahrenheit and kelvin.
Convert the boiling temperature of liquid ammonia, −28.1 °F, into degrees Celsius and kelvin.
The label on a pressurized can of spray disinfectant warns against heating the can above 130 °F. What are the corresponding temperatures on the Celsius and kelvin temperature scales?
The weather in Europe was unusually warm during the summer of 1995. The TV news reported temperatures as high as 45 °C. What was the temperature on the Fahrenheit scale?
dimensional analysis
(also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several
different quantities
unit of temperature; water freezes at 32 °F and boils at 212 °F on this scale
intensive property representing the hotness or coldness of matter
unit conversion factor
ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit
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Effective Use of the Capacitance Multiplier for Voltage Regulators
This post discusses a topic I’ve shared quite a long time ago on a few other forums, I’ve decided to post it here on the blog just in case it will become unavailable on these forums at some point, as
it is a fairly old post. I don’t have the original schematics anymore, so bare with the lower res images I’m copying over from my original post.
Many voltage regulators use the capacitance multiplier as a method of increasing the effective capacitance seen by a load. Some use it as a complete voltage “regulator” (although its more of a filter
in that case than it is a regulator), while others use it as a low-pass-filter (LPF) for the error amplifier at the core of the regulator. The basic idea is to use a BJT transistor as a follower to
amplify the capacitor current by ~hfe (small signal current gain) of the transistor, making the capacitor appear as if it was ~hfe larger in value. This simple structure is shown in Fig. 1.
Fig. 1. Simplified Regulator Schematic
R1+C2 form a LPF, which is buffered by T1, these 3 devices comprise the capacitance multiplier. This filtered voltage is used to power the error amplifier which drives the pass transistor and takes a
sample of the output voltage by the R2/R3 voltage divider. The reference voltage isn’t shown in this diagram for simplicity. This circuit is very simple to understand, and is a fairly close
representation of many voltage regulator designs. C1 is the bulk filter capacitor, and can be preceded by a rectifier, or any other source of power. Under light load condition there is nothing
interesting in this circuit, and it behaves as expected. However, as soon as we start sourcing appreciable current at the output by the load, the voltage over the bulk capacitor (C1) will fluctuate
considerably. If it has sufficient ripple, transistor T1 can no longer be assumed as operating in the forward active region, and can actually start conducting from C2 to C1. This will obviously
prevent the capacitance multiplier from operating properly, and will translate to ripple on the supply of the error amplifier, and therefore the output voltage.
Fig. 2. Waveforms Under 1A Output Current
In Fig. 2 we can see VCE(volts) and IC(mA’s) of T1 for this circuit with a 3300uF C1, 1K+100uF (R1/C2) LPF, and a load of 1A (the results are from pSpice with the error-amp biased at ~10mA). As can
be seen above, these conditions are sufficient to make the transistor conduct in the opposite direction at the end of each cycle, which means the capacitance multiplier is no longer operating
properly. This simulation was carried out with a full wave rectifier and a 50Hz sin source.
However, the capacitance multiplier is still a very simple circuit that we would like to exploit for its effective filtering. Lucky for us, this issue can be solved quite easily by adding 2 cheap
component, as can be seen in Fig. 3 below. Here we have added D1 and C4. The combination of these 2 devices allows us to isolate the capacitance multiplier from the large ripple present over C1. This
is basically a peak-detector circuit that we use as the supply for the capacitance multiplier. When the voltage over C1 is high enough, D1 will conduct and C4 will be charged. When the voltage over
C1 drops, D1 is not conducting, and C4 is used as the charge reservoir supplying power to the capacitance multiplier. Since C4 need only supply the error amplifier (and in some cases the reference),
it can be fairly small while having limited ripple.
Fig. 3. Modified Capacitance Multiplier Circuit
D1 can be any diode you’d like, but using a low voltage drop diode will reduce the dropout voltage of the regulator, so it is recommended. C4 can be calculated fairly easily. Depending on the
simplification you make you will get a different value, but they are all fairly small and inexpensive. R1/C2 forms a LPF so lets assume the voltage at the base of T1 is the average of the voltage at
its collector. Using this assumption with the fact a BJT can operate with VCE as low as ~0.2V (and we know VBE=~0.7), we can easily find that a ripple of up to 1Vpk-pk over C4 is tolerable. We can
now use the charge equation of a capacitor and rearrange this slightly and find:
where I is the load current, Freq is the mains frequency, the factor 2 assumes a full-wave rectifier (which effectively doubles the frequency), and the final term is the voltage drop we allow. In the
example here I will use 10mA, 50Hz, and 1V. This results in ~100uF. We can double that to 200uF (220uF would be a practical value), to keep the transistor operating close to its nominal point and
reduce ripple further. Fig. 4 shows the same waveforms as Fig. 2, but this time with the added components.
Fig. 4. New Waveforms Under 1A Output Current
we can see a significant improvement in both reduced voltage ripple over the transistor, as well as a well behaved transistor current that meets our expectations from the small-signal analysis of
such a circuit.
To estimate practical number, I’ve used a single-rail power-supply that I’ve owned at the time and used to power one of the headphone amplifiers. It used a structure that is closely matched by the
simplified diagram of Fig. 1. The output voltage was set to 24V (23.3V to be exact), and it was loaded with a load of ~1A (15ohm + 8ohm resistors in series). The component values such as bulk
capacitor C1 were similar to these used in simulation. Integrated noise measurements were made with the LNMP from Tangent and an Agilent U1253A, and waveforms were captured with the Rigol DS1052E
I’ve owned at the time.
Results pre-modification:
Output noise: no load – 11uVRMS 1A load – 107uVRMS
Here’s VCE of Q1 under these conditions:
No load: Fig. 5. VCE Ripple with 0A Load
1A load: Fig. 6. VCE Ripple with 1A Load
As can be observed in Fig. 6, VCE of T1 fluctuates significantly, and drops down to 0V, which means the capacitance multiplier isn’t operating as expected. This can obviously be even more severe with
higher load currents.
Results post-modification:
I’ve used UF1004 diode and 330uF capacitor for the mod, as these are the closest values I’ve had at the moment of testing.
Output noise: no load – 11uVRMS 1A load – 83uVRMS
The figure below shows VCE of T1 with this modification at 1A load current.
Fig. 7. VCE Ripple with 1A Load and Modified Circuit
The figures above show the effectiveness of the circuit, which would only increase as load current increases and the ripple at the bulk capacitor (C1) increases.
A few final words to put things into proportion, and summarize when you should use the capacitance multiplier in its modified form. As the post above showed, the problem only exists when significant
(approaching 1Vpk-pk) ripple is present at the input of the capacitance multiplier. Therefore, if you have a very light load (compared with the bulk capacitor value), or you have a pre-regulator/DC
source that has low enough ripple, the generic capacitance multiplier will perform as expected. However, if under the expected operating conditions the ripple will approach (or exceed) 1Vpk-pk, it is
highly recommended to modify the capacitance multiplier to maintain good regulation.
It should be noted that while the method shown above is the one I use, as I find it best suited for these application. There are other ways to remedy the capacitance multiplier problem, and each has
its pros and cons.
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Quadrature Reflection Phase Shifters
Click here to go to our main page on phase shifters
Click here to go to our main page on reflection phase shifters
Click here to learn about hybrid couplers
Click here to go to our page on quadrature couplers
Click here to go to a companion page on reflection attenuators
Hybrid (3dB) quadrature couplers are used in most reflection-style phase shifters. Quad couplers can be realized as branchline couplers (distributed or with lumped elements), Lange couplers, or other
coupled line couplers of many varieties. The bandwidth of the quadrature reflection phase shifter is mostly limited by the choice of coupler, you can achieve an octave with many coupled-line coupler
structures. In terms of bandwidth, the best quadrature coupler (but hardest to make) is a broadside coupler in stripline, the worst is a branchline on microstrip (the easiest to make).
Quadrature phase shifter with open/short terminations
Quadrature hybrids are often used to create reflection phase bits. The simplest bit to consider is shown in the ADS schematic below. The resistors R1 and R2 are used to create a two-state device from
the upper and lower ideal couplers in the simulation (when ON=1, the upper coupler is selected, when ON=0, the lower one is selected). The even mode and odd mode impedances have been selected to
provide perfect 3.01 dB coupling at the center frequency (10 GHz) for 50 ohm system. The upper coupler has open circuits on the power split ports, the lower coupler has short circuits. Note that in
each state, it is important to present the same impedance to each power split port to get the power to transfer out the normally isolated port.
The phases of S21 and S43 are the phases of the two states in this simulation, and they are exactly parallel over frequency as shown below. Thus 180 phase shift is provided exactly at all frequencies
in this (ideal) situation. The bandwidth of the ideal coupler restricts the approach to perhaps one octave.
Reflection phase shifter with line stretcher terminations
Another simple way to create a reflection phase bit using a quadrature hybrid is to add terminations that merely stretch the paths made to short circuits on the split ports. This behaves very similar
to a switched line phase shifter; it provides phase shift that varies with frequency by definition. This is not something you'd want to use in a phase array, if you want to have any bandwidth. You'll
find an explanation here...
US Patent 5379007 uses this scheme. so does 4764740.
Here's the response. It has the same S-parameter magnitudes of the previous example, but the phase shift is now linear with frequency. This is going to give you a very narrow-band response in terms
of phase errors, and should be avoided.
Many MEMS phase shifters have taken this topology in recent years. You can place multiple MEMS shunt switches across the termination lines, and provide multiple phase states with one bit. But
consider the error of the response below from 8 to 12 GHz. You'd get 36 degrees of error at the upper and lower frequencies! Why do MEMS guys offer this kind of crap circuitry as a phase shifter?
They seem to be narrowly focused on low loss as the one characteristic they can beat MMIC phase shifters at; they seem to have ignored the rest of the requirements...
Here's the response of these two elements at 10 GHz, plotted on a Smith chart. They are 180 degrees apart, the capacitor is at -90 degrees and the inductor is at +90 degrees. What if we used these as
switchable terminations in a quadrature reflection phase shifter?
OK, let's try it.
Flat phase shift is provided. The mojo is back!
Using lumped elements are the the best way to create a reflection phase shifter from a quad coupler. If you don't use a capacitor in the circuit, you'll be like a MEMS guy, and no one will want your
circuit. But if you work on a PIN diode phase shifter, least you won't have to go underground and remove "MEMS guy" from your business card to avoid being laid off or laughed at...
You can create any value phase shift with lumped elements, and the phase shift will stay relatively flat over frequency (but not as perfect as the 180 degree case). Below are two tables for element
values to achieve phase shift at 10 GHz, you can scale them to any frequency you want. The values are approximate, we didn't spend the time to derive the closed-form expression, if anyone wants to
help us out we'd be happy to give you credit and a pocketknife for the correct formula!
│Phase shift (degrees) │Capacitor value (pF) │
│-157.5 │1.6 │
│-135 │0.78 │
│-90 │0.32 │
│-45 │0.13 │
│-22.5 │0.065 │
│Phase shift (degrees) │Inductor value (nH) │
│+157.5 │5.4 │
│+135 │1.9 │
│+90 │0.8 │
│+45 │0.33 │
│+22.5 │0.16 │
It turns out there are two solutions for each bit value. For a 90 degree bit, you can either choose the -45 and +45 values, or the -135 and +135 values. For a 45 bit, you can choose the -22.5 and
+22.5 values, or the -157.5 and 157.5 values, and you can do the math yourself for 22 and 11 degree bits. Be aware, you should keep the phase shifts of the inductor and capacitor equal for best
Now let's look at the two solutions for 45 degree bit:
│22.5 degree solution │157.5 degree solution │
More to come...
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Character Class Functions
5.7 Character Class Functions
Octave also provides the following character class test functions patterned after the functions in the standard C library. They all operate on string arrays and return matrices of zeros and ones.
Elements that are nonzero indicate that the condition was true for the corresponding character in the string array. For example:
isalpha ("!Q@WERT^Y&")
⇒ [ 0, 1, 0, 1, 1, 1, 1, 0, 1, 0 ]
Return a logical array which is true where the elements of s are letters or digits and false where they are not.
This is equivalent to (isalpha (s) | isdigit (s)).
See also: isalpha, isdigit, ispunct, isspace, iscntrl.
Return a logical array which is true where the elements of s are letters and false where they are not.
This is equivalent to (islower (s) | isupper (s)).
See also: isdigit, ispunct, isspace, iscntrl, isalnum, islower, isupper.
Return a logical array which is true where the elements of s are letters and false where they are not.
This is an alias for the isalpha function.
See also: isalpha, isdigit, ispunct, isspace, iscntrl, isalnum.
Return a logical array which is true where the elements of s are lowercase letters and false where they are not.
See also: isupper, isalpha, isletter, isalnum.
Return a logical array which is true where the elements of s are uppercase letters and false where they are not.
See also: islower, isalpha, isletter, isalnum.
Return a logical array which is true where the elements of s are decimal digits (0-9) and false where they are not.
See also: isxdigit, isalpha, isletter, ispunct, isspace, iscntrl.
Return a logical array which is true where the elements of s are hexadecimal digits (0-9 and a-fA-F).
See also: isdigit.
Return a logical array which is true where the elements of s are punctuation characters and false where they are not.
See also: isalpha, isdigit, isspace, iscntrl.
Return a logical array which is true where the elements of s are whitespace characters (space, formfeed, newline, carriage return, tab, and vertical tab) and false where they are not.
See also: iscntrl, ispunct, isalpha, isdigit.
Return a logical array which is true where the elements of s are control characters and false where they are not.
See also: ispunct, isspace, isalpha, isdigit.
Return a logical array which is true where the elements of s are printable characters (but not the space character) and false where they are not.
See also: isprint.
Return a logical array which is true where the elements of s are printable characters (including the space character) and false where they are not.
See also: isgraph.
Return a logical array which is true where the elements of s are ASCII characters (in the range 0 to 127 decimal) and false where they are not.
Test character string properties.
For example:
isstrprop ("abc123", "alpha")
⇒ [1, 1, 1, 0, 0, 0]
If str is a cell array, isstrpop is applied recursively to each element of the cell array.
Numeric arrays are converted to character strings.
The second argument prop must be one of
True for characters that are alphabetic (letters).
True for characters that are alphabetic or digits.
True for lowercase letters.
True for uppercase letters.
True for decimal digits (0-9).
True for hexadecimal digits (a-fA-F0-9).
True for whitespace characters (space, formfeed, newline, carriage return, tab, vertical tab).
True for punctuation characters (printing characters except space or letter or digit).
True for control characters.
True for printing characters except space.
True for printing characters including space.
True for characters that are in the range of ASCII encoding.
See also: isalpha, isalnum, islower, isupper, isdigit, isxdigit, isspace, ispunct, iscntrl, isgraph, isprint, isascii.
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Alternating Direction Method of Multipliers for Linear Programming
Recently the alternating direction method of multipliers (ADMM) has been widely used for various applications arising in scientific computing areas. Most of these application models are, or can be
easily reformulated as, linearly constrained convex minimization models with separable nonlinear objective functions. In this note we show that ADMM can also be easily used for the canonical linear
programming model; and the resulting complexity is O(mn) where m is the constraint number and n is the variable dimension. Moreover, at each iteration there are m subproblems that are eligible for
parallel computation; and each of them only requires O(n) flops. This ADMM application provides a new approach to linear programming, which is completely different from the major simplex and interior
point approaches in the literature.
View Alternating Direction Method of Multipliers for Linear Programming
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Stories from a Software Tester
Thinking About AI
Many are debating the efficacy of artificial intelligence as it relates to the practice and craft of testing. Perhaps not surprisingly, the loudest voices tend to be the ones who have the least
experience with the technology beyond just playing around with ChatGPT here and there and making broad pronouncements, both for and against. We need to truly start thinking about AI critically and
not just reacting to it if we want those with a quality and test specialty to have relevance in this context.
This series does have a bit of an introduction in terms of its rationale. See the posts Computing and Crucible Eras and Keeping People in Computing if you’re curious. That said, those two posts are
not required reading at all for this series.
Getting Through the Barrier
Soumith Chintala, the co-creator of the PyTorch framework, made a good point about the challenge of learning artificial intelligence when he said “you slowly feel yourself getting closer to a giant
barrier.” Soumith further says:
This is when you wish you had a mentor or a friend that you could talk to. Someone who was in your shoes before, who knows the tooling and the math — someone who could guide you through the best
research, state-of-the-art techniques, and advanced engineering, and make it comically simple.
Yes, entirely, I agree. That’s part of what my blog posts are attempting to be. I would hesitate to say mentor when applied to me but I would not hesitate at all to say friend and hesitate even less
to say I’ve likely been in the same shoes. I too reached that barrier that Soumith talks about and struggled mightily with it. I still do. My entire machine learning series was all about that
struggle. The same, in some ways, applied to my more limited data science series. My AI Testing series was me finally finding some confidence to speak more broadly.
I’m not sure which author — Jeremy Howard or Sylvain Gugger — specifically had the thought but in their book Deep Learning for Coders with fastai and PyTorch: AI Applications Without a PhD, one of
them states the following:
The hardest part of deep learning is artisanal: how do you know if you’ve got enough data, whether it is in the right format, if your model is training properly, and, if it’s not, what you should
do about it? That is why we believe in learning by doing. … The key is to just code and try to solve problems: the theory can come later, when you have context and motivation.
Theory later as context and motivation kick in. That was a key element for me that I had to realize early but had not articulated. Yet there’s a healthy balance here. There’s an interplay of theory
and practice that I find is needed. The problem I found is that a lot of the resources out there were of two broad types.
• I found some resources were highly conceptual and mathematical. These gave me extensive mathematical explanations of what was going on so I could “understand the theory.”
• I found other resources that had intricate blocks of code. And if I could figure out how to run that code, this would apparently show me what was going on so I could “understand the practice.”
What I often came away with was understanding neither to any great extent. Hence by above-mentioned blogging. It was my way to make myself understand. I wrote those posts as if I was my audience. I
figured if I couldn’t write it, I probably didn’t understand it.
Leveraging Mental Models
Seth Weidman, in his book Deep Learning from Scratch: Building with Python from First Principles, said the following:
With neural networks, I’ve found the most challenging part is conveying the correct ‘mental model’ for what a neural network is, especially since understanding neural networks fully requires not
just one but several mental models, all of which illuminate different (but still essential) aspects of how neural networks work.
I entirely agree with that. What’s often not stated is what the mental models actually are. I would say they are the following:
• Mathematical and statistical reasoning
• Visualizing and graph notations
• Abstraction and generalization
• Pattern recognition
• Experimentation and iteration
• Systems thinking
• Creativity and intuition
I would actually argue that these are skills that quality and test specialists should already have anyway.
Going back to Weidman for a second, he mentions that each of these three statements is a valid way of looking a neural networks:
• A neural network is a mathematical function that takes in inputs and produces outputs.
• A neural network is a computational graph through which multidimensional arrays flow.
• A neural network is a universal function approximator that can in theory represent the solution to any supervised learning problem.
But what do those statements actually mean?
Well, I’m going to give one way of looking at those statements in this post. Yet here’s the problem: you have to implement the ideas to be able to actually understand them. There’s the practice part.
Yet, to understand your implementation, you have to focus on why the implementation conceptually makes sense. That’s the theory part. That’s what posts in my “Thinking About AI” series are going to
do: focus on implementation so that practice and theory follow.
One book I greatly enjoyed was by David Perkins called Make Learning Whole: How Seven Principles of Teaching Can Transform Education. In his book Perkins talks about the idea of “play the whole
game.” Of that, Perkins says:
We can ask ourselves when we begin to learn anything, do we engage some accessible version of the whole game early and often? When we do, we get what might be called a ‘threshold experience,’ a
learning experience that gets us past initial disorientation and into the game.
This idea of threshold experiences resonates with me quite deeply. As someone with intense — and deserved — imposter syndrome through much of my life, I’ve had to find and foster these threshold
experiences as much as I can. The challenge is that they aren’t given to you. You have to create them in some ways. But other people can help to enable that creation.
That’s what I’m hoping my various musings on artificial intelligence will do for people who learn the same way I do.
Let’s Get Our Concepts In Place
At it’s core, programming is about what? Well, it’s pretty simple if you get reductionist enough. It’s about having some inputs, sending those inputs to a program, and then having that program
generate some outputs or results.
From a purely black box perspective, we can consider the program itself to be one big function. This notion of a function is important conceptually but it’s also really simple. Consider this
f[1](x) = x^2
This is a function represented in mathematical notation. It’s called a square function. You could also represent this function as a graph.
This is just a parabola that opens upwards. But we can also represent a function as a black box that takes in inputs and generates outputs.
Here we have the generic schematic of the function at the top with some examples of it in use. We pretty much just described conceptually how machine learning works. Why? Broadly, what all this means
is that programs are essentially working on a given task using data. Now consider that machine learning has, as its basis, a lot of programming. Similar to programming, the basis of machine learning
is essentially that of learning representations suitable for a task from data. That learning is done by a function.
A difference here is that in traditional programming, the program’s logic is explicitly defined by the programmer. In contrast, in machine learning, the program learns from data to make predictions
or decisions without being explicitly programmed for every task. This key difference makes machine learning a powerful approach for tasks like pattern recognition, decision-making, and prediction in
situations where explicit programming may be challenging or not feasible.
And, earlier, when I said “learning representations suitable for a task”, think of it as a program having to discern different visual features of objects, such as edges, textures, and shapes, in a
way that enables it to distinguish between different classes of objects. Or think of it as a program having to discern semantic relationships and similarities between words and sentences to
distinguish various sentiments or attitudes.
One aspect of machine learning is called deep learning.
Deep learning, when you boil away all the details, is fundamentally simple. It’s nothing more or less than a technique to extract and transform data. All of deep learning is based on a single type of
model called the neural network. And this neural network can be thought of as a program!
Notice how this is the exact same visual I showed you earlier but with a word replacement.
Specifically, the model is a program that contains an algorithm. Think of it as a big old function! And this function is a mathematical and computational representation that’s been designed to mimic
the behavior of the human brain’s neural network. This model &mdash this program — takes data as input and produces results.
The extraction and transformation of data that I mentioned is done by using multiple layers of neural networks.
Maybe think of these layers is individual functions being called by the one main function. Each of these layers takes its inputs from previous layers and progressively refines them. A key conceptual
point here is that these layers are trained by algorithms that are designed to minimize their errors and thus improve their accuracy.
In this way, a neural network learns to perform a specified task.
The Concepts Are Fundamentally Simple
So the main thing to get here is that a neural network is a particular kind of machine learning model that is mathematical in nature and inspired by how neurons in the human brain seem to work. Deep
learning is a technique that utilizes neural networks to extract and transform data. Each of layers that make up a neural network layers take inputs from previous layers and progressively refine
those inputs. Through a process of training, algorithms minimize errors and improve accuracy, allowing the network to learn and get better at performing a specified task.
All coding you do to learn the above will be an implementation of what I just said there. Yes, there can be a lot of details. But all of those details essentially converge to the above statement.
Okay, so that’s coding. But what about the testing? Well, obviously, where there’s code, there can be testing of that code. But here’s an interesting thing: these models have a built-in way to
determine some aspects of their quality. You can test if the errors are actually being minimized and if the accuracy is actually improving.
Using Models For Tasks
I’ve mentioned tasks a few times so let’s get a little more specific on that. One of the tasks might be natural language processing and it’s one of the tasks I’m going to be looking at first in this
Broadly speaking, natural language processing refers to a set of techniques that involve the application of statistical methods, with or without insights from the study of linguistic, to understand
text in order to deal with real-world tasks. When I say “with or without insights from the study of linguistics” this means that natural language processing techniques can be developed either purely
based on statistical methods and machine learning algorithms or by incorporating linguistic principles to enhance language understanding.
This “understanding” of text — whether statistical, linguistic or both — is derived by transforming texts to useable computational representations. These representations are discrete or continuous
combinatorial structures framed as mathematical objects. A combinatorial structure here just refers to a mathematical arrangement or configuration of elements. When I say “mathematical arrangement,”
this means like vectors, matrices, tensors, graphs, and trees. These structures are used to represent and analyze relationships or patterns in data.
Deep learning enables you to create models — programs — that efficiently learn representations from data using a particular abstraction known as the computational graph. A computational graph is
conceptually simple in that it represents the flow of data through the neural network’s layers as well as the operations that are performed at each layer.
Think of a computational graph as a step-by-step recipe or a flowchart for the neural network. Just like a recipe guides a chef through the cooking process, the computational graph guides the data
through the neural network’s layers, showing what happens at each step and how the network transforms the data.
To you do all this, computational graph frameworks are used to implement these deep learning algorithms. One of those is PyTorch, which we’ll certainly look at it in upcoming posts.
Let’s Scale Up Our Understanding
So we know we have inputs that go into a model and produce results. There’s actually another component to this that makes a model-program different from a traditional program.
I say that because a model-program in this context is not like a traditional program that explicitly contains a set of rules. Instead, it’s a mathematical representation that captures the
relationships between inputs and outputs in the data through a set of parameters. Those parameters are sometimes called weights.
The analogy of a “weight” is borrowed from its common meaning in everyday language, where weight refers to the heaviness of an object. The weights determine the impact of the input data on the
output. Just like a heavier object exerts more influence in a physical scenario, higher weights signify greater importance for the corresponding input in terms of the output.
I mentioned that the computational graph represents the flow of data through the neural network’s layers and operations. This flow actually has what’s called a forward pass and a backward pass. The
forward pass is just the input to results. But the backward pass is a way for those results to be used as the basis for updating the weights.
That notion of performance is critical to the idea of testing. But let’s also consider how this idea differs from a traditional program yet also how it’s kind of the same. In a traditional program,
the process is typically this:
input → action → output
The output is determined solely by the programmed rules and logic. There’s no inherent feedback mechanism to assess how well the program executed, as the output is entirely deterministic based on the
input and the pre-defined rules. However, in machine learning, the process is more akin to this:
input → action → output → performance assessment
The model’s output is not predefined by strict rules but is generated based on the learned patterns and representations from the data. After producing the output, the model’s performance is assessed
by comparing its predictions to the actual outcomes. The assessment provides valuable feedback on how well the model is performing on the task at hand.
Granted, with a traditional program we can check if the implementation works as we believe it did. But there’s no inherent testing there. This is why we have test frameworks at various abstraction
levels to help us look for observations or we just check for ourselves. But in a machine learning context, the very nature of assessment is built it. This doesn’t mean you don’t need humans looking
at the output but it does mean humans have even less of an excuse for testing in this context.
More Scaling Up of Our Understanding
So far this is a little black box — or blue box, given the visuals — but now let’s shift slightly (but only slightly) into a more white box understanding.
Let’s reframe our inputs as observations. Observations are items or data points about which we want to predict something. In order to predict, I presumably have some target that I will use as the
basis for my prediction. After all, I’m predicting something. The target is what that something actually is. We often give these targets labels and these labels correspond to an observation.
Sometimes, these labels known as the ground truth.
The term “ground truth” emphasizes that these labels represent the true, correct values you expect to learn so that you can predict accurately.
Given this slight reframing, a model is a mathematical expression or a function that takes an observation and predicts the value of its target label.
Again, in the context of execution, the model can be thought of as a program. It’s a program that runs the particular function which is some mathematical computation. It might be an incredibly
complicated computation but it is just a computation nonetheless. In this context, we do still have our parameters or weights.
Given a target and its prediction, the loss function assigns a scalar real value that’s referred to, not too surprisingly, as the loss. A “real scalar value” just refers to a single numerical value
that can be any real number on the number line. In the context of machine learning, it represents a continuous and unbounded quantity. The idea here is that the lower the value of the loss, the
better the model is at predicting the target.
The Path to Supervised Learning
What this does is take us into supervised learning. In supervised learning, a model is trained using labeled data, meaning observations with corresponding targets. The loss function plays a critical
role in guiding the learning process towards making accurate predictions on new, unseen data.
So now let’s bring our black (well, blue) and white boxes together.
The specific functional form of a model is often called its architecture. When I say “functional form of a model” just understand that as the specific way that the model structured and how it
processes input data to produce output predictions. It’s like the blueprint or design that defines how the model works and learns from data.
Thus the functional form in essence creates an experimental platform. And we run our experiments — our tests — on that platform. All good experiments have independent and dependent variables. In this
context, the predictions are calculated from the independent variables, which is the data not including the targets. The targets (labels) are the dependent variables. The results of the model are
called predictions. The measure of performance of model is called the loss. The loss depends not only on the predictions, but also on having the correct labels.
Notice how we have some structure to the data above, such as one dimensional arrays as well as a matrix.
What this shows us is that, given a dataset with some number of observations, we want to learn a function (a model) parameterized by weights.
But wait a minute! I thought we used models to do all this. Now you’re saying we have to learn the model. So do we use a model that we learn on the fly?
The term “model” refers to a conceptual representation of a mathematical function or algorithm that captures the relationships between inputs and outputs in a given problem domain. It’s a blueprint
or framework that defines how the data should be processed to make predictions. When we talk about “learning a model,” we mean that we’re determining the specific values of the model’s parameters
(such as those weights) that best fit the data in the given problem. These parameter values are adjusted during the training process, allowing the model to adapt and become effective at making
So we use the term “model” to represent the abstract concept of the mathematical relationship between inputs and outputs but when we “learn a model” we’re adjusting the model’s parameters to find the
best fit for the data, making it a “learned instance” of that abstract concept.
Supervised learning is the process of finding the optimal parameters that will minimize the cumulative loss for all of the observations. The goal of supervised learning is thus to pick values of the
parameters that minimize the loss function for a given dataset.
And what this tells us is that supervised learning is a form of computation that’s teaching a program to learn from examples. So imagine you have a bunch of examples, like pictures of animals. You
also know what each animal is. You want the program to figure out how to recognize different animals correctly just like you can do. To do that, the program tries different approaches until it finds
the best way to make accurate predictions.
This ‘best way’ involves adjusting certain settings, just like tuning a musical instrument, until it gets the answers as close as possible to what’s correct. And it knows what’s correct because we’ve
told it what correct looks like; we’ve supervised the learning. The ultimate outcome here is to make the program really good at recognizing animals, so it can work well even with new pictures that it
hasn’t seen before.
You can replace “pictures” above with “text” or “sounds” or “videos” and the same thing essentially applies.
There are also techniques known as unsupervised learning and reinforcement learning but those I’m leaving aside for now.
If you want to see a context where I talked about the basis of reinforcement learning, you can check out my Pacumen series.
Wrapping Up
So with all this said, consider again those statements from Weidman about what a neural network is:
• A neural network is a mathematical function that takes in inputs and produces outputs.
• A neural network is a computational graph through which multidimensional arrays flow.
• A neural network is a universal function approximator that can in theory represent the solution to any supervised learning problem.
We’ve now looked at what each of those mean conceptually. I hope you’ll agree that the concepts really aren’t all that difficult. And, as I showed above, the idea of testing is built right in. This
doesn’t mean we don’t need humans to do testing. What it means is that there are plenty of hooks in place for quality and test specialists who want to critically engage with these concepts and with
the technology.
A lot of thinking that goes into artificial intelligence is all about reasoning about how humans think. Specialist testers, at least so I argue, are good at understanding categories of error and the
theory of error, both of which find ready application in how humans think. This is because humans can think fallaciously or make cognitive mistakes when they engage with complex things or apply
biases that they are sometimes only dimly aware of.
Sometimes when you talk about this kind of stuff, someone might think you’re no longer talking about testing. But, in fact, I would argue you’re engaging with one of the most important parts of it.
Humans are building technology that will augment us in ways that have largely been unprecedented in our evolutionary history. Since we’re the ones building it, it’s safe to assume we’re building it
with some of our flaws as well. So let’s start thinking about how to test it. In fact, we need to be long past the “thinking about how” stage and well into the “demonstrating how” stage.
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No power: exponential expressions are not processed automatically as such
Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers,
specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size
comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the
frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility
between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the
physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar
patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the
physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of
an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.
טביעת אצבע
להלן מוצגים תחומי המחקר של הפרסום 'No power: exponential expressions are not processed automatically as such'. יחד הם יוצרים טביעת אצבע ייחודית.
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April 2017Down River Resources | Your Elementary Math GuideContextual Multiplication and Division in the ClassroomHow to Create the Best Math Centers Using Plastic EggsWhy You Need More Plastic Eggs for Your Math ClassroomHow to Use Splat in Your Math Classroom
Multiplication and division can be difficult concepts to teach, especially if your second grade students have no prior experience with this type of thinking. It happens every year! Multiplication and
division problems are fundamentally different than addition and subtraction problem situations because of the types of quantities represented. Multiplication and division are taught together so that
student can see that one operation is the reverse of the other. Let's make this year different! Using the mathematical principle of unitizing and the "GET" strategy, students will build their
proficiency as they learn contextual multiplication and division in the math classroom.
Teaching Multiplication and Division in the Classroom
Multiplication and Division are fundamentally different than addition and subtraction.
A simple addition problem situation could be:
Ann has 3 cookies. Laura have her 4 more cookies. How many cookies does Ann have now?
A simple multiplication problem situation could be:
Ann has 3 bags of cookies with 4 cookies in each bag. How many cookies does Ann have? The numbers are the same but the quantities represented are different.
This shift in thinking is what gives most students difficulty when transitioning from the operations of addition and subtraction to multiplication and division.
Second grade students need to be able to model, create, and describe contextual multiplication and division situations. What if there was something that could help bridge the gap for these students?
Unitizing Helps Students Shift Their Thinking
Have you heard of unitizing? It is an important, and often unknown, math word. Unitizing gives students a change in perspective.
Think back to the development of numeracy. Children learn to count objects one by one, also known as one-to-one correspondence. Instead of counting ten objects one by one, students can unitize them
as one thing or one group. Another example of unitizing can be found within place value. Whenever we have 10 or more in a place value unit, we need to regroup. Thus, ten ones can also be thought of
as a unit of ten.
This concept of unitizing is a big shift for students. It almost negates what our students originally learned about numbers. We want to help our students achieve the developmental milestone of
unitizing. Unitizing is the underlying principle that guides students' learning.
Students need to use numbers to count, not only objects, but also groups... and to count them both simultaneously. Unitizing helps students build their proficiency in contextual multiplication and
Students need to be explicitly taught this principle and exposed to seeing it in action multiple times, much less subitizing in these primary grade levels. Show the students ten objects and tell
them, " This is one group of 10." It seems simple, right? It is actually quite tricky for students to grasp, so repeat yourself...and repeat yourself.
Multiplication and Division Strategy: Did You "GET" It?
Another trick for tackling multiplication and division is a little-known strategy. G-E-T is a simple acronym for an effective strategy when teaching contextual multiplication and division.
I have used the acronym before but I added this first step which helps build students' meta-cognition.
After reading through a word problem that involves multiplication or division, ask yourself: "Did you GET it?" If your answer is "yes," you probably followed these steps:
1. Read through the word problem at LEAST once.
2. Circle and label the number representing the GROUPS. (How many groups are the objects being divided into?)*
3. Circle and label the number representing the EACH. (How many objects are within each group?)
4. Circle and label the number or noun represent the TOTAL. (How many total objects are altogether or in total?)
*When students label, they circle the number and noun (example: 12 cats) and they write the word to describe that part of the word problem (example: in this case, 12 cats would represent the total.
The students would write the word TOTAL or the letter "T" on top of the circle.)
If your students label these three parts to a word problem, it will be so much easier solving for the unknown, whether is be the dividend, divisor, quotient, factors, or product.
Labeling word problems using the "GET" strategy is a non-negotiable in second and third grades! Of course, modeling and guided practice is a must before this layer of accountability takes effect!
What are some ways you build your students' proficiency in multiplication and division?
* References: Fosnot, C. & Dolk, M. (2001). Young mathematician at work: Constructing Multiplication and Division, NH: Heinemann.
Spring brings butterflies, chicks, blossoms, and... plastic eggs! People near and far hunt for these special spherical objects hidden in secret places. I tend to just go straight to the seasonal
aisles of my favorite stores and find a wide variety of plastic eggs to choose from with a lot less hassle. In recent years, the stores are stocking an eclectic mix of eggs. These eggs include
special shapes (animals and carrots), unique patterns (faith-based words, animal print, camouflage), extra-sparkly glitter, golden, and transparent eggs...these probably just list the ones stocked at
eye-level! I stockpile a large assortment of these diverse eggs and pair them with rigorous math concepts to create the perfect math centers for kindergarten, first, and second grades. While my ideas
are focused on these grade levels, many of them can be adapted for other grades too! This is my go-to list for simple math centers using plastic eggs.
Creating the Best Math Centers Using Plastic Eggs
Matching Math Centers
Plastic eggs are versatile! You can write on them, fill them, or do both!
I love writing on them....probably because I stockpile school supplies like my husband stockpiles freeze-dried rations (insert "yuck" face!)
Numeral + Tens Frames (Kindergarten)
Grab a regular Sharpie marker and some of those eggs and get marking! One of my favorite ways for students to use the two parts of a plastic egg is to match the numeral to the tens frame. This helps
students read and represent whole numbers 0 to 20 with objects (TEKS K.2B.)
You can also match the numeral to a tally mark, subitizing dots, the number word, or stickers placed on one of the parts!
Composing Ten (Kindergarten, First, & Second Grades)
Another way to use the "matching" concept is composing numbers. When two numbers are added together, this is called composing numbers. Students simply match two one-digit addends which add up to 10.
Kindergarten and first grade students are asked to compose numbers to 10 (TEKS K.2I & 1.3C.) This concept of putting two numbers together to form one can also build a second grade student's
automaticity with basic facts (TEKS 2.4A.)
Stacking Math Centers
This might be my favorite way to use plastic eggs! Since you are only using one of the parts, the eggs go a long way. You will have more pieces to create more centers...and what do I want to make?
More centers!
Counting by Tens (Kindergarten & First Grades)
I love how simple algebraic reasoning skills can be practiced by stacking the pieces into a tower. Your students will think the best part of this math center is trying to make the tower stay up. It
is VERY common that the entire tower will fall, so students are practicing a lot more than just algebraic reasoning with this center.
I use this center for counting by 10s (TEKS K.5.) This same concept can be applied to skip counting by 2s, 5s, and 10s (TEKS 1.5B.) Skip counting is also a great skill to continue practicing in
second grade as the students will apply this skill to contextual multiplication.
Sequencing Math Centers
No matter which grade level you teach, you can use this concept of sequencing numbers with plastic eggs. The concept remains unchanged, but the numbers will be different. You can also tailor this
center to your students' needs. You may have a student who is struggling or advanced, whichever the case, add eggs with the numbers that best suits students' needs.
Kindergarten students practice numbers 0 through 20 (TEKS K.A,) while first grade students use numbers up to 120 (TEKS 1.2F.) Second grade students practice ordering numbers up to 1,200 (TEKS 2.2D.)
Ordering Whole Numbers (Kindergarten, First, & Second Grades)
This egg carrier was purchased at Dollar General for $2. They can also be found at Dollar Tree for $1. Most of these egg carriers hold 12 or 24 eggs. You can make the exact amount of eggs needed or
less. The carrier just acts as a place holder for the eggs.
The best thing of all if that there is a center section which can hold the pile of eggs (see the top part of the image.) Students can pick up one egg at a time and place it in a spot. As students
pick up additional eggs, they may need to move eggs as the place value of each of the numbers is determined. The carrier works as an open number line.
Filled Math Centers
Using filled plastic eggs, you could teach any math skill! You can write numbers, draw shapes, or create word problems on a piece of paper, fold it up, and place it inside a plastic egg! That's as
simple as ABC, friends. I use a variety of materials to fill the eggs just to keep my students interested in egg activities so they are not repetitive.
_ More and _ Less (Kindergarten, First, & Second Grades)
Yes, there is a reason I left a blank in the title for this section! You can tailor this center to meet the needs of any grade level or any child.
Kindergarten students are working on one more and one less (TEKS K.2F,) while first grade students are learning 10 more and 10 less than a given number up to 120 (TEKS 1.5.) Second graders expand on
this idea by determining the number than is 10 or 100 more or less than a given number up to 1,200 (TEKS 2.7B.)
You can place a card within the math center to indicate if the students are working on the number than is _ more or _ less than the given number or they can generate both numbers.
I had students simply take a strip of notebook paper and number it, like we do for spelling tests, then students drew their eggs, opened them up, and recorded their answers individually. The example
shown is for first grade (10 more and 10 less than a given number). The numbers on the eggs represent the problem number for recording purposes. The number on the sticky note is the number that
students use to generate their answers.
Counting (Kindergarten)
We all need another excuse to buy those irresistibly cute Target erasers, right?! Well, here's another one!
Fill the eggs with a certain number of erasers. Students will open one egg at a time and count to determine the quantity held in the eggs. I numbered the eggs so that the students can record their
Again, I used a strip of notebook paper and had students number it, like we do for spelling tests. As they select eggs out of the basket or container, they record the answer on the corresponding
Kindergarten students are learning to count forward to 20 (TEKS K.2A,) and counting a set of objects up to at least 20 (TEKS K.2C.) This activity also build students' one-to-one number
Graphing (Kindergarten, First, and Second Grades)
This same concept of filling the eggs or placing objects within them can be applied to data collection, each egg could contain a specific object (erasers, jelly beans, etc.) and students record the
data on a bar graph or picture graph. Students are learning how to collect and organize data (TEKS K.8ABC, 1.8ABC, 2.10CD.)
Coin Collections (First and Second Grades)
Fill plastic eggs with coins. I try to use real coins when I am able to as they are more life-like. Plus, there are so many varieties of coins, I have yet to find a math manipulative that captures
their new look. Each egg is filled with a different combination of coins.
First graders are learning how to count by 1s to add up the value of pennies and skip count by 5s and 10s to add up the value of nickels and dimes (TEKS 1.4ABC). Second grade students are determining
the value of a collection of coins up to one dollar (TEKS 2.5AB.)
I used a strip of notebook paper and had students number it, like we do for spelling tests. As they select eggs out of the basket or container, they record the answer on the corresponding line. In
the example in the photograph, I had the second graders write the value of the collection of coins using the cent symbol and the dollar sign and decimal point to specifically meet the rigor within
the second grade standard (TEKS 2.5B.) First graders would write the value of the collection of coins using the cent symbol.
Whew! That was a plethora of ways that you are create the BEST math centers using plastic eggs. I love that each of these math centers are rigorous.
Did you notice each idea I used was standards-based and met the specificity described? Students are practicing the power words in education: determining, generating, representing, composing! These
are the higher-order thinking skills we want them to use and practice, practice, practice. Why not use plastic eggs to accomplish this?
In addition, these dynamic math centers are be differentiated based on your students' needs. If your students has not mastered three-digit numbers, create some eggs with two-digits. If your students
have surpassed the grade level goal, make them four-digit numbers! I love using plastic eggs in math centers as they are rigorous and meet the needs of diverse learners in my classroom.
I hope this inspires you to turn those leftover plastic eggs into some engaging math centers for diverse learners!
What is your favorite way to use plastic eggs for math centers?
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Plastic eggs are not just for an Easter egg hunts! After a delicious Easter meal, the kids take part in a large Easter egg hunt at my parent's house. There are so many good hiding spots and, as
usual, the older kids dominate the hunt. Colorful, plastic eggs jingle with coins and jellybeans...chocolate and a dollar bills too, but only if you are lucky! After a few quick minutes, everyone
gathers back on the porch. The kids quickly hide their money (after counting it, of course!) and throw their eggs into a large sack. While everyone is eating jelly beans and some eat chocolate, I
begin my favorite activity of the post-Easter season! I re-purpose those colorful, plastic eggs and create rigorous math centers that can be used for the rest of the school year. While I hunt high
and low for interesting eggs, I can never have enough. So, what's the big deal with plastic eggs? I'm glad you asked...
Top Five Reasons to Use Plastic Eggs in Your Math Centers
1. Plastic eggs are inexpensive...unless you buy an entire cart full of them. Guilty as charged!
Most of the eggs I buy are about 98 cents to $2.00 per package. Of course, the basic colorful eggs are most inexpensive, while the larger, themed eggs are most expensive. I try and think about what I
would like to use them for before I buy so I can have a quantity in mind...but most of the time I just buy, at least, two packages. That way, I am guaranteed to have a little variety in whatever I
end up creating! I mean, when you see cute eggs you just buy them...kind of like those Target erasers. Gasp!
2. Plastic eggs can be used year-round. There is so much diversity in the type of eggs you can buy, you can use them seasonally and/or with your classroom themes.
I have eggs with sports theme that I use during those seasons. I also use the animal-shaped eggs to coincide with teaching about organisms. I love making as many cross-curricular connections using
math as possible. Not only is it a great way to spiral, or revisit, previously taught content, it also gets the students engaged with the theme!
3. Plastic eggs provide a hands-on, or tactile, way which stimulates the brain.
Tactile learning take place when the students are carrying out the actual physical activity! Whether the students are sorting through the eggs or opening them up, students are actively participating
in the math center.
4. Plastic eggs are versatile.... they can be written on with a permanent marker, stickers can be added on them, and/or they can be filled!
Pick your favorite way to use plastic eggs, mix it up, or use them all! Plastic eggs allow you to use them creatively to accomplish your specific learning goals for math centers.
My favorite writing tool to use on plastic eggs is a regular Sharpie marker in black. It goes on nice and smooth. I tried the flip chart version of the marker and it left marks.
5. Plastic eggs are reusable! Not only can you have an amaaazing Easter egg hunt with them, but you can use them within a center.
Don't worry if you are late to the Easter egg party! If you create a last-minute center this year, it will be ready to go for next year! You can break apart the eggs to compress them for storage,
creating towers of like ends. Perhaps, you can even find some eggs on clearance... fingers crossed!
I hope this post inspires you to go out and buy some plastic eggs, if you'd like to check out how I incorporate plastic eggs within my math centers, you can find it here!
Why would you buy plastic eggs for your math classroom?
Have you heard of the interactive math game, Splat!? There are different variations and different games with the same name, but I use this interactive game to get my students engaged about a
particular math concept which we have already learned. It can be used for interactive math reviews. It encourages students to analyze number relationships to connect and communicate mathematical
ideas. This is a process standard that students can always use more support in practicing. Splat! is a fun way to apply mathematical concepts and makes for a fun math center, small group, or whole
group game.
Using Splat! in the Math Classroom
Suggested Age Range for Activity
Splat! can be used with any grade level of students, just make sure that the content being reviewed is developmentally age appropriate or specific to your grade level's standards.
Preparing for Activity
Splat! games are relatively easy to prep. You will need a tall cylindrical tower. It works best with a potato chip can!
If you needed an excuse to buy more potato chips, here it is! After all, once you pop, the fun just doesn't stop! Now, the fun can continue for your entire school year!
First, print out the cards. Then, cut out the cards and laminate. My games are created in both blackline, for ink savings, and in color which really makes the fun seasonal faces POP!
Regardless of which route you take, I recommend printing the cover for the cylindrical can in color. I use white copy paper so it bends around the surface better. I added some colorful complimentary
washi tape on the top edge.
If using a tall can, you will need something to make up the difference, as the paper is only 8x11 inches tall. If you use a new shorter can, you will have to cut a little of the space on the top as
the height of the can is shorter than the paper! (I've tried both ways! The really good flavors of chips come in the shorter cans! I like using the tall cans so I can utilize my colorful washi tape!)
To make it last longer and protect it from water and dirty hands, add packing tape around the paper as a protective
Teacher Tip: The thing that I love the most about Splat! is that the can you use to create the tower in the game also serves as storage for all of the game cards!
To make the cards self-correcting, mark the correct answers with an adhesive dot on the back (yard sale sticker). If playing with your class, there is no need for this step, unless you will be adding
it to an independent math center or station or using it for an activity for early finishers.
Reviewing Math Concepts with the Game Splat!
I try to find simple skills in the list of standards that could be turned into a one line question for the games I create.
Kindergarten Sample Questions:
What is 1 more than 6?
What is 1 less than 8?
First Grade Sample Questions:
What is 10 more than 55?
What is 10 less than 34?
Second Grade Sample Questions:
Is 27 odd or even?
Is 15 odd or even?
How to Play the Interactive Game Splat!
To play, set a timer for the amount of time you have to play, or stop play when the session is over.
1. Mix up the cards (math question cards + Did Somebody Say Splat Cards + challenge cards). Place the deck of cards facedown on the table.
2. Have players read the card and generate the correct response. The player should say that answer three times. This is my variation, but can be modified however you'd like.
3. After they answer the question, they place the card on the top of the tower.
If players pick a Did Somebody Say, “Splat?” card, they should simply place that card carefully on top of the “tower.”
If it stays there securely, that players turn is over. If that card or any other cards go SPLAT (falls off the tower,) that player must take all cards that fell.
If they pick a #challenge card they must follow the directions on the card. The same procedures apply.
The object of the game is to stack cards carefully without making any go SPLAT! Players that knock down cards must take them.
At the end of play (either when the session ends, the timer rings or there are no more cards to play), the player with the LEAST cards is the winner!
There are so many different ways that you can manipulate the play of this game using the three different cards types. Find the way that you like best and get your students excited about math!
I hope this post inspires you to use Splat! in your classroom and if you want to use my Splat! games, they're in my TpT shop.
What are some other ways you review math concepts in your classroom?
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Write the equation x^(2)+2x-3=x+4 as system of equations? -Turito
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A system of equations is a set of one or more equations involving a number of variables.
We have given an equation
Now for a system of equation, we have find all those equations which belong to given family of functions
The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect.
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Common ML patterns: Central tendency and variability | TechTarget
Common ML patterns: Central tendency and variability
Four common patterns provide approaches to solving machine-learning problems. Learn how two -- central tendency computation and variability computation -- work.
This article is excerpted from the course "Fundamental Machine Learning," part of the Machine Learning Specialist certification program from Arcitura Education. It is the fourth part of the 13-part
series, "Using machine learning algorithms, practices and patterns."
Let's focus on a collection of machine learning techniques documented as "patterns." A pattern provides a proven solution to a common problem individually documented in a consistent format; usually,
it is part of a larger collection.
Each pattern is documented in a profile comprised of the following parts:
• Requirement. Every pattern description begins with a requirement. A requirement is a concise single sentence, in the form of a question, that asks what the pattern addresses.
• Problem. The issue causing a problem and the effects of the problem are described in this section, which may be accompanied by a figure that further illustrates the "problem state." It is this
problem for which the pattern is expected to provide a solution.
• Solution. The solution explains how the pattern will solve the problem and fulfill the requirement. Often the solution is a short statement that may be followed by a diagram that concisely
communicates the final solution state. "How-to" details are not provided in this section but are instead located in the Application section.
• Application. This section describes how the pattern can be applied. It can include guidelines, implementation details, and even a suggested process.
Explore two patterns: The central tendency and the variability computations
Let's look at two of four common patterns that document data-exploration techniques, which act as a common entry point for machine learning problem-solving tasks: the central tendency computation and
variability computation.
Central tendency computation: Overview
• Requirement: How can the makeup of a data set be determined in terms of the normal set of values?
• Problem: Before solving a machine learning problem, a preliminary understanding of the input data is required. However, not knowing which techniques to start with can negatively impact the
subsequent model development.
• Solution: The data set is analyzed and the values that normally occur around the center of a distribution, plus the most occurring values, are calculated via established statistical techniques.
• Application: The data set is arranged in ascending or descending order, and the measures of central tendency (mean, median, and mode) are calculated.
Figure 1. Variable X belongs to a data set. To understand the value of X a technique needs to be applied to obtain the common set of values (2) and (3).
Central tendency computation: Explained
Before a model can be trained, it is imperative to fully understand the nature of the data set at hand. Failure to do so may lead to wrong assumptions or the use of the wrong type of algorithm. At a
very basic level, to develop an understanding of the data set it is important to know which value or values normally appear in the data set. However, with a plethora of quantitative and qualitative
techniques at hand, applying the most suitable analysis technique can be a daunting task. In Figure 1, X is a variable in a data set. To determine its value, a technique needs to be applied to obtain
the common set of values (2) and (3).
The guiding data analysis principle dictates that the best way to determine a commonly occurring set of values is to capture the average behavior of the data set by finding out which values, or range
of values, appear most frequently. In a distribution, such values are normally found towards the center. However, in cases involving a few extreme set of values, this center is pulled to one side.
This results in a false center, in which case a count of the values that occur the most provides a more robust average behavior of the data set.
To obtain the average behavior of the data set, the measures of central tendency that include mean, median, and mode are calculated (Figure 2). These three measures are also commonly referred to as
averages. These averages help us to summarize data, compare two sets of values (distributions) and compare a single value against the rest of the values in the distribution.
In most cases, calculating the mean (also commonly referred to as the average) provides a good measure for getting an idea of which value of a variable is the most common. For example, the mean
height of a group of individuals can be taken to determine what the average height is. Mean is calculated by taking the sum of all values and dividing it by the number of values. However, mean is
generally used when the values do not change much and increase or decrease in a normal manner.
Figure 2. Variable X belongs to a data set. To understand its value (1) the measures of central tendency are found (2). Based on the mean, median and mode values, it is determined that the most
common value is 5 (3).
Mean is affected by the presence of outliers. When outliers occur, finding the median or mode is recommended as they remain robust despite outliers. For example, with a set of values for household
income where most of the values occur towards the center of the distribution but a few large values occur on the right side (high income), finding the average income via the mean would provide an
incorrect value. That is, the few abnormally high values will skew the mean. Calculating the median or mode, however, provides an accurate measure as these measures do not consider the entire set of
values in a distribution for calculation purposes.
Median is calculated by arranging the set of values in ascending order and then finding the middle value, whereas mode is simply the most frequently occurring value in the distribution. A
distribution can have two or more modes, in which case the distribution is bimodal or multimodal, respectively. (See Figure 2.)
Variability computation: Overview
• Requirement: How can the spread of values for a single variable in a data set be determined?
• Problem: Developing an intuition about a data set involves determining the behavior of uncommon values of data. Failure to do so may result in treating abnormal values as normal.
• Solution: The behavior of the uncommon values in a data set is expressed in the form of the spread of values and is quantified via the application of proven statistical techniques.
• Application: The numerical values in the data set are identified and measures of variation -- including range, interquartile range (IQR), variance and standard deviation -- are calculated.
Figure 3. To understand the values of variable X, which belongs to a data set, it is found that the most common value is 5. But an understanding of the behavior of the remaining values is required
(1). Therefore, a technique must be applied to gain insight into the behavior of the remaining values (2, 3).
Variability computation explained
The application of the central tendency computation pattern enables a preliminary understanding about the makeup of a distribution in terms of finding the central value. However, its application
alone fails to address the requirement of determining the behavior of non-central values. This is an important aspect of exploratory analysis prior to model development, since finding out which
values, apart from the central one, normally appear in a distribution is helpful when building an effective model. (See Figure 3.)
The spread of values from the central value (mean, median or mode) is quantified to find out how tightly or loosely packed the values of a distribution are. The spread can either be defined in terms
of the extremities of the distribution or the variation in values.
The application of this pattern requires the calculation of measures of variation, including range, interquartile range (IQR), variance and standard deviation. (See Figure 4.)
The range is a statistic obtained by subtracting the minimum value from the maximum value that indicates the spread or width of data. The range is heavily affected by the presence of extreme values,
as the presence of a single extreme value gives the impression that the values are spread over a very large range. The averages (mean, median and mode) provide central value, while range provides
insight to the variation in the data. Using range, two different sets of values can be compared in terms of variation in their values.
Quartiles represent three values that divide the data into four equal portions, obtained by first arranging the data values in ascending order and then dividing the data into four quarters. The
first, second and third quartiles are known as lower quartile (Q1), median (Q2) and upper quartile (Q3).
Q1, Q2 or Q3 represent values below which 25%, 50% or 75% of data values exist, respectively. Based on quartiles, the IQR is calculated. The IQR is the set of values between Q1 and Q3. That is, IQR =
Q3 – Q1. IQR provides a simple method of eliminating outliers from a data set as outliers normally occur outside of Q1 and Q3.
Figure 4. Variable X belongs to a data set. Once it's determined that the most common value is 5 (1), to next gain insight into the behavior of the remaining values, the measures of variation are
calculated (2). Based on the value of standard variation, it is determined that the average spread of values from the mean value is 1.93. Furthermore, the value of 8 is beyond 1 standard deviation.
(That is: 1 standard deviation is 5.2 + 1.93 = 7.13, whereas 5.2 is the mean value).
The variance is a non-negative value that shows how spread the values are compared to the mean of the values or center of a distribution. A small variance shows that there is a comparatively small
difference between the values and the mean value, and that the values occur close to each other in a distribution. A large variance shows that there is comparatively large difference between the
values and the mean value, and that the values occur far from each other.
The standard deviation is another non-negative value to view the spread of the values from the center of the distribution and is simply the square root of the variance. The calculated value is known
as one standard deviation, and it is expressed in the same units as the values in the distribution.
The standard deviation is generally more useful than a variance for descriptive purposes, whereas a variance is generally more useful mathematically. The lower the variance and standard deviation,
the less spread out and the closer to the mean value the values are. The variance and standard deviation enable us to measure how consistently a process generates data, for example to analyze which
bottle-filling machine fills bottles on a more consistent basis.
In the next article in this series, we'll look at the remaining two data-exploration patterns: associativity computation and the graphical summary computation.
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Model Talk | Who cares about ties? Part II
last week's Model Talk
I discussed why modelling the tie probability in head-to-head match-ups is important when the tie is offered as a separate bet. This post discusses why modelling the tie is important even when ties
are void.
In the early days of Data Golf, we would sometimes simulate tournaments or match-ups such that ties weren't possible. There are many ways you could go about this, but one way is to think of golf
scores as continuously distributed (e.g. following a normal distribution). Because the probability of drawing any specific value from a continuous distribution is zero, ties won't happen. This was
nice in some sense because it ensured that our finish probabilities (e.g. Top 5 probabilities) added up to the correct number (500%), as opposed to some slightly higher figure if ties were possible
in the simulations. But, why is 500% the "correct" number? It is correct when the probabilities are used to assess the value of bets with dead-heat rules applied, in that Top 5
probabilities that account for dead-heat rules
should add up to 500%. However, there are many methods you can use to ensure that your Top 5 probabilities add up to 500%, but that doesn't make them all equivalent. In fact, simulating tournaments
without ties will not yield the same expected value estimates as simulating with ties and then applying dead-heat rules (although they should be very similar).
But now back to match-ups. When ties are void it also seems like simulating without ties is a reasonable way to assess expected value. Consider a match-up between Golfers 1 and 2, where Golfer 1 wins
with probability \( win_1 \), Golfer 2 wins with probability \( win_2 \), and they tie with probability \( tie \). Expected value on Golfer 1 is then equal to \( win_1 \cdot (odds_1-1) + win_2 \cdot
(-1) + tie \cdot (0) \), where \( odds_1 \) are Golfer 1's offered odds. If you set this equation equal to zero and re-arrange, we find that \( 1/odds_1 = win_1/(win_1 + win_2) \). We'll call the
expression on the right the "break-even implied probability". If the bookmaker's implied probability (\( 1/odds_1 \)) is below this break-even probability it's a +EV bet; above, and it's a -EV bet.
As an important aside to make sure you are with me, note that the break-even implied probability from a simple bet (i.e. one that has only two possible outcomes, win/loss) is equal to \( win_1 \),
which is derived by setting the following expected value equation to zero: \( win_1 \cdot odds_1 - 1 \), which yields \( 1/odds_1 = win_1 \). This is where the idea of the "implied probability of a
book's odds" comes from.
Up to this point I've undoubtedly succeeded in taking a simple concept and making it complicated. But bear with me. Suppose we run our simulations with ties allowed (e.g. by rounding the output from
a normal distribution) and find Golfer 1's win probability is 25%, Golfer 2's is 65%, and they tie 10% of the time. With ties void, using the expression above, the break-even implied probability for
Golfer 1 is equal to 27.8%. One way to think about what happened here is we took the 10% tie probability and assigned 27.8% of those ties as "wins" for Golfer 1, and 72.2% as "wins" for Golfer 2.
That is, when calculating the break-even implied probability, the tie probability was distributed in proportion to the players' respective outright win probabilities.
If you were to simulate this match-up without ties using a normal distribution, Golfer 1 would win something like 45% of the simulations that resulted in a tie previously, while Golfer 2 would win
55%. This yields a break-even implied probability for Golfer 1 of \( 25\% + 0.45 \cdot 10\% = 29.5\% \). An intuitive way to think about this is that simulating without ties is like having a
sudden-death playoff in the event of a tie after 18 holes. Golfer 1 is the worse golfer so they will still win less than 50% of the playoffs, but it will be much closer to 50-50 than their 18-hole
win probability.
Finally, consider a match-up that is offered with dead-heat rules. Now, expected value will be equal to \( win_1 \cdot (odds_1) + tie \cdot (odds_1/2) - 1 \). Setting to zero and re-arranging yields
a break-even implied probability of \( win_1 + tie/2 \). Plugging in our numbers from the simulations with ties yields a break-even implied probability for Golfer 1 of 30%. Therefore with dead-heat
rules, we take the 10% of tied simulations and assign 50% of them as "wins" for Golfer 1.
I've shown how to estimate break-even implied probabilities in 3 different scenarios: 1) simulating with ties allowed and voiding those ties, 2) simulating without ties allowed (which means the tie
rules are irrelevant), and 3) simulating with ties allowed and assigning pay outs according to dead-heat rules. In our example above, with probabilities of 25%, 65%, and 10% for Golfer 1 win, Golfer
2 win, and Tie, the estimates in the 3 cases were: 27.8%, 29.5%, and 30%. Intuitively, Golfer 1, who is the worse golfer in this match-up, is disadvantaged when ties are voided as they are only
assigned "wins" at a rate proportional to their overall match-up win probability. When ties aren't possible, that rate increases to some number closer to (but still below) 50%, and finally with
dead-heat rules applied to ties the rate equals 50%. We correctly estimated break-even probabilities in cases 1) and 3); however the estimate in 2) is not valid, for the obvious reason that the
break-even probability obtained using that method is the same whether the tie rules are void or dead-heats, but we know they should be different. The important comparison is between methods 1) and 2)
as these estimates can be meaningfully different.
To start this post, I was careful with my wording by stating that
the tie probability matters for bets where ties are void. I didn't say that the tie probability itself matters, because as was shown above, it doesn't: the break-even implied probability for
ties-void bets, \(win_1/(win_1 + win_2) \), is not a function of the tie probability! However, the decision on
how to model ties
does matter: if you exclude them (e.g. by using a continuous distribution to model golf scores) you will overestimate your break-even probabilities. The more lopsided is the matchup, and the more
frequent ties actually occur, the more this modelling decision will matter. None of this is particuarly surprising: the data-generating process for golf scores only produces integer scores, so your
simulations should only produce integer scores as well!
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UnrollingAverages is a Julia package aimed at deconvolving simple moving averages of time series to get the original ones back.
UnrollingAverages currently assumes that the moving average is a simple moving average. Further relaxations and extensions may come in the future, see Future Developments section.
Press ] in the Julia REPL and then
pkg> add UnrollingAverages
The package exports a single function called unroll: it returns a Vector whose elements are the possible original time series.
unroll(moving_average::Vector{Float64}, window::Int64; initial_conditions::U = nothing, assert_natural::Bool = false) where { U <: Union{ Tuple{Vararg{Union{Int64,Float64}}},Nothing} }
• moving_average: the time series representing the moving average to unroll ;
• window: the width of the moving average ;
• initial_conditions: the initial values of the original time series to be recovered. It may be a Tuple of window-1 positive integer values, or nothing if initial conditions are unknown. Currently
it is not possible to specify values in the middle of the time series, this may be a feature to be added in the future ;
• assert_natural default boolean argument. If true, the pipeline will try to recover a time series of natural numbers only. More then one acceptable time series (where "acceptable" means that it
reproduces moving_average) may be found and all will be returned.
A few remarks:
1. If isnothing(initial_conditions):
□ if assert_natural, then an internal unroll_iterative method is called, which tries to exactly recover the whole time series, initial conditions included. Enter ?
UnrollingAverages.unroll_iterative in a Julia to read further details;
□ if !assert_natural, then an internal unroll_linear_approximation method is called. See this StackExchange post. NB: this is an approximated method, it will generally not return the exact
original time series;
2. If typeof(initial_conditions) <: Ntuple{window-1, <:Union{Int64,Float64}}, then an internal unroll_recursive method is called, which exactly recovers the time series. Mathematical details about
this function are reported in the documentation, and you may read more by entering ?UnrollingAverages.unroll_recursive.
Future Developments
• Modify initial_conditions argument of unroll so that it accepts known values throughout the series ;
• Implement reversing methods for other types of moving averages .
How to Contribute
If you wish to change or add some functionality, please file an issue. Some suggestions may be found in the Future Developments section.
How to Cite
If you use this package in your work, please cite this repository using the metadata in CITATION.bib.
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Combinatorics - Graph Theory, Counting, Probability | Britannica
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Also called:
combinatorial mathematics
A graph G consists of a non-empty set of elements V(G) and a subset E(G) of the set of unordered pairs of distinct elements of V(G). The elements of V(G), called vertices of G, may be represented by
points. If (x, y) ∊ E(G), then the edge (x, y) may be represented by an arc joining x and y. Then x and y are said to be adjacent, and the edge (x, y) is incident with x and y. If (x, y) is not an
edge, then the vertices x and y are said to be nonadjacent. G is a finite graph if V(G) is finite. A graph H is a subgraph of G if V(H) ⊂ V(G) and E(H) ⊂ E(G).
A chain of a graph G is an alternating sequence of vertices and edges x[0], e[1], x[1], e[2], · · · e[n], x[n], beginning and ending with vertices in which each edge is incident with the two vertices
immediately preceding and following it. This chain joins x[0] and x[n] and may also be denoted by x[0], x[1], · · ·, x[n], the edges being evident by context. The chain is closed if x[0] = x[n] and
open otherwise. If the chain is closed, it is called a cycle, provided its vertices (other than x[0] and x[n]) are distinct and n ≥ 3. The length of a chain is the number of edges in it.
A graph G is labelled when the various υ vertices are distinguished by such names as x[1], x[2], · · · x[υ]. Two graphs G and H are said to be isomorphic (written G ≃ H) if there exists a one–one
correspondence between their vertex sets that preserves adjacency. For example, G[1] and G[2], shown in Figure 3, are isomorphic under the correspondence x[i] ↔ y[i].
Two isomorphic graphs count as the same (unlabelled) graph. A graph is said to be a tree if it contains no cycle—for example, the graph G[3] of Figure 3.
Enumeration of graphs
The number of labelled graphs with υ vertices is 2^υ(υ − 1)/2 because υ(υ − 1)/2 is the number of pairs of vertices, and each pair is either an edge or not an edge. Cayley in 1889 showed that the
number of labelled trees with υ vertices is υ^υ − 2.
The number of unlabelled graphs with υ vertices can be obtained by using Polya’s theorem. The first few terms of the generating function F(x), in which the coefficient of x^υ gives the number of
(unlabelled) graphs with υ vertices, can be given
A rooted tree has one point, its root, distinguished from others. If T[υ] is the number of rooted trees with υ vertices, the generating function for T[υ] can also be given
Polya in 1937 showed in his memoir already referred to that the generating function for rooted trees satisfies a functional equation
Letting t[υ] be the number of (unlabelled) trees with υ vertices, the generating function t(x) for t[υ] can be obtained in terms of T(x)
This result was obtained in 1948 by the American mathematician Richard R. Otter.
Many enumeration problems on graphs with specified properties can be solved by the application of Polya’s theorem and a generalization of it made by a Dutch mathematician, N.G. de Bruijn, in 1959.
Characterization problems of graph theory
If there is a class C of graphs each of which possesses a certain set of properties P, then the set of properties P is said to characterize the class C, provided every graph G possessing the
properties P belongs to the class C. Sometimes it happens that there are some exceptional graphs that possess the properties P. Many such characterizations are known. Here is presented a typical
A complete graph K[m] is a graph with m vertices, any two of which are adjacent. The line graph H of a graph G is a graph the vertices of which correspond to the edges of G, any two vertices of H
being adjacent if and only if the corresponding edges of G are incident with the same vertex of G.
A graph G is said to be regular of degree n[1] if each vertex is adjacent to exactly n[1] other vertices. A regular graph of degree n[1] with υ vertices is said to be strongly regular with parameters
(υ, n[1], p[11]^1, p[11]^2) if any two adjacent vertices are both adjacent to exactly p[11]^1 other vertices and any two nonadjacent vertices are both adjacent to exactly p[11]^2 other vertices. A
strongly regular graph and a two-class association are isomorphic concepts. The treatments of the scheme correspond to the vertices of the graph, two treatments being either first associates or
second associates according as the corresponding vertices are either adjacent or nonadjacent.
It is easily proved that the line graph T[2](m) of a complete graph K[m], m ≥ 4 is strongly regular with parameters υ = m(m − 1)/2, n[1] = 2(m − 2), p[11]^1 = m − 2, p[11]^2 = 4.
It is surprising that these properties characterize T[2](m) except for m = 8, in which case there exist three other strongly regular graphs with the same parameters nonisomorphic to each other and to
A partial geometry (r, k, t) is a system of two kinds of objects, points and lines, with an incidence relation obeying the following axioms:
1. Any two points are incident with not more than one line.
2. Each point is incident with r lines.
3. Each line is incident with k points.
4. Given a point P not incident with a line ℓ, there are exactly t lines incident with P and also with some point of ℓ.
A graph G is obtained from a partial geometry by taking the points of the geometry as vertices of G, two vertices of G being adjacent if and only if the corresponding points are incident with the
same line of the geometry. It is strongly regular with parameters
The question of whether a strongly regular graph with the above parameters is the graph of some partial geometry is of interest. It was shown by Bose in 1963 that the answer is in the affirmative if
a certain condition holds
Not much is known about the case if this condition is not satisfied, except for certain values of r and t. For example, T[2](m) is isomorphic with the graph of a partial geometry (2, m − 1, 2).
Hence, for m > 8 its characterization is a consequence of the above theorem. Another consequence is the following:
Given a set of k − 1 − d mutually orthogonal Latin squares of order k, the set can be extended to a complete set of k − 1 mutually orthogonal squares if a condition holds
The case d = 2 is due to Shrikhande in 1961 and the general result to the American mathematician Richard H. Bruck in 1963.
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Advanced Operating Technique for Centralized and Decentralized Reservoirs Based on Flood Forecasting to Increase System Resilience in Urban Watersheds
School of Civil Engineering, Chungbuk National University, Cheongju 28644, Korea
Submission received: 13 June 2019 / Revised: 22 July 2019 / Accepted: 23 July 2019 / Published: 24 July 2019
The frequency of inundation in urban watersheds has increased, and structural measures have been conducted to prevent flood damage. The current non-structural measures for complementing structural
measures are mostly independent non-structural measures. Unlike the current non-structural measures, the new operating technique based on flood forecasting is a real-time mixed measure, which means
the combination of different non-structural measures. Artificial rainfall events based on the Huff distribution were used to generate preliminary and dangerous thresholds of flood forecasting. The
new operation for centralized and decentralized reservoirs was conducted by two thresholds. The new operation showed good performance in terms of flooding and resilience based on historical rainfall
events in 2010 and 2011. The flooding volume in the new operation decreased from 6617 to 3368 m^3 compared to the current operation in 2010, and the flooding volume in 2011 decreased from 664 to 490
m^3. In the 2010 event, the results of resilience were 0.831835 and 0.866566 in current and new operations, respectively. The result of resilience increased from 0.988823 to 0.993029 in the 2011
event. This suggestion can be applied to operating facilities in urban drainage systems and might provide a standard for the design process of urban drainage facilities.
1. Introduction
Global climate change is causing heavy rainfall, and unexpected heavy rainfalls have increased flooding in urban watersheds. It is important to take preemptive measures to prevent urban inundation
owing to the enormous costs required for infrastructure restoration and damage to people and/or property when flooding events occur in urban areas. This study focuses on an improvement in the
resilience of urban drainage systems using effective preemptive measures. Various structural measures (SMs) have been proposed, and most are costly and time consuming [
]. However, SMs such as the rehabilitation of urban drainage networks focused on the maximum overflow volume causing urban problems has been considered one of key elements for preventing urban
inundation [
]. Non-structural measures (NSMs) are used to overcome the limitations of SMs because SMs have the effect of a designed capacity and cannot achieve an additional effect. Various measures including
the operation of hydraulic facilities have been suggested, and those are categorized as NSMs. However, NSMs have been suggested individually, and independent NSMs can have a limited effect.
Various real-time control (RTC) techniques of hydraulic facilities have been classified as independent NSMs, NSMs combined with SMs (combined measures), and NSMs combined with the same type of NSMs
(integrated NSMs). Most RTC techniques in urban areas are independent NSMs for urban drainage facilities [
]. Several studies such as combined measures have been introduced to increase the efficiency of urban drainage systems (UDSs) [
]. In addition, NSMs combined with the same type of NSMs (operation technique with operation technique) have been suggested [
]. The operation technique applied in this study (mixed NSMs) suggests NSMs mixed with a different type of NSMs (the operation of drainage facilities based on a flood forecasting technique) [
Table 1
shows the classification of NSMs with RTC in UDSs.
Although many different studies on RTC in urban drainage systems have been suggested, the combination of different NSM such as operation of UDSs and flood forecasting has yet to be proposed.
Furthermore, there are no mixed NSMs that can be applied to a small watershed at a minute scale in urban areas. In the urban watersheds of Korea, the concentration time is extremely short because
most urban areas have a high impervious rate. The concentration time in an urban area is generally shorter than that of a rural area, and the concentration time in Korea is less than 1 h. Therefore,
the rainfall will already have been discharged if the unit of time for RTC in an urban area exceeds 1 h. If the observed time interval is short in an urban watershed, the sophisticated management of
UDSs will be possible. In this study, the unit of time for a flood forecasting and operation of UDSs is 1 min, the minimum unit of rainfall observation in Korea. The purpose of this study is the
development of new mixed NSMs combining the operation of a drainage facility with flood forecasting to increase the system resilience in urban watersheds.
The concept of resilience in water resource engineering was suggested during the 2000s. Todini (2000) proposed an optimized design technique with two objective functions (cost and resilience) in a
water distribution system [
]. A new reliability measure including network resilience with a genetic multi-objective approach was developed for the design of a water distribution system [
]. The multi-objective evolutionary algorithm was used for identifying the trade-off between the total cost and resilience including the reliability [
]. Earlier studies related to resilience in water resource engineering were conducted on water distribution systems. Studies on the resilience in UDSs have been introduced during the past several
years. Mugume et al. (2015) introduced the use of structural resilience instead of reliability as a global analysis approach [
]. Decentralized stormwater management facilities have been applied to urban infrastructure for flexible and resilient systems [
]. The new operation technique based on flood forecasting described in this study is evaluated based on the system resilience. In this study, the various results according to the current operation,
the previous operation, and the new operation in UDSs of the target watershed are evaluated.
In UDSs, there are two kinds of reservoirs such as centralized and decentralized reservoirs. Centralized reservoirs receive all inflow in the target watershed and discharge inflow using drainage
pumps according to the constant water level. Decentralized reservoirs are reservoirs to prevent flood in subcatchments, and those are generally constructed in the underground of school playgrounds or
children’s parks. The current operation is the normal operation in centralized and decentralized reservoirs. The normal operation of centralized reservoirs means that drainage pumps are operated
according to predetermined water levels. The normal operation of decentralized reservoirs is to receive inflow and reserve inflow at the end of the rainfall event. The previous operation of
centralized reservoirs is to change operation from the normal operation to the early operation according to the water level of the selected monitoring node. The normal operation of decentralized
reservoirs is to receive inflow and reserve/discharge according to the water level of the selected monitoring node.
2. Methodologies
This study consisted of several major contents. First, artificial rainfall data were made using a Huff distribution with a proper quartile [
]. Second, an advanced flood forecasting technique was developed for RTC of drainage facilities and applied to a network in an urban drainage system. Third, an advanced operation technique based on
the advanced flood forecasting technique was used for the urban drainage system (UDS) in the target watershed. Fourth, the new operation was compared with the current and previous operations to
evaluate the resilience in the UDS.
Figure 1
is the flow of this study.
2.1. Production of Artificial Rainfall Data
In general, the UDSs in Korea are designed using artificial rainfall data distributed by a Huff distribution [
]. Huff (1967) observed the cumulative rainfall volume during historically heavy rainfall events and divided the total rainfall volume according to the location of the peak value. The Huff
distribution has four quartiles, and it is categorized by the position of the peak value. The peak value of the first quartile is located between 0 and 0.25 T (Q1), where T is total rainfall
duration. Each peak value of the second, third, and fourth quartiles is situated within 0.25–0.5 (Q2), 0.5–0.75 (Q3), and 0.75–1.00 T (Q4), respectively. Yoon et al. suggested Q3 for the design of
hydraulic structures in Korea [
]. In this study, Q3 of the Huff distribution was selected for the generation of an advanced flood forecasting. All hydraulic structures in Korea were designed considering the design frequency of
each hydraulic facility. The hydraulic structures in Korea have different design frequencies. For example, the design frequency of a hydraulic structure is 30 years, and that of the B hydraulic
structure is 20 years. Artificial rainfall data based on the Huff distribution was used to generate the preliminary and dangerous thresholds in the advanced flood forecasting. Historical rainfall
events in 2010 and 2011 were used to check the operational effects because flood damage occurred in those periods.
Q1, Q2, Q3, and Q4 were non-dimensional using the cumulative rainfall volume and duration compared with the total rainfall volume and duration. Equations (1) and (2) show a non-dimensional rainfall
volume and duration in a Huff distribution.
$D t = C D t O D t × 100 %$
$R t = C R t O R t × 100 %$
is the non-dimensional rainfall duration,
is the cumulative rainfall duration, and
is the total rainfall duration. In addition,
is the non-dimensional rainfall volume,
the cumulative rainfall volume, and
the total rainfall volume.
The Ministry of Construction and Transport in Korea has provided regression equations of a Huff distribution for 67 rainfall observatories managed by the Korea Meteorological Administration. The
regression equations take the following form:
$R t = C 0 + C 1 D t + C 2 D t 2 + C 3 D t 3 + ⋯ + C n D t n$
C[n] (i =
1, 2, 3, …,
is the constant of the
th polynomial equation for a Huff distribution in each area. In Korea, a value of 6 is generally selected for
, although the value can be an integer of 5, 6, or 7. A process for applying non-dimensional cumulative rainfall into non-dimensional distributed rainfall is required. For example, the
non-dimensional distributed rainfall volume at time
is A − B, when the non-dimensional cumulative rainfall volume at time
is A and at time
− 1 is B.
2.2. Advanced Flood Forecasting Technique
The flood forecasting technique using real-time rainfall data in an urban watershed was suggested by Lee et al. [
]. In the former study, the threshold for flood forecasting was generated using the first flooding nodes. The total rainfall volume in rainfall-runoff simulations was started at 1 mm and increased by
1 mm before the first flooding. Rainfall-runoff processes were simulated using a storm water management model (SWMM) [
]. The method suggested in the former study is extremely convenient and efficient in small urban watersheds. However, it is not a preemptive measure because the status of UDSs in the target area
cannot be considered until the first flooding occurs. The advanced flood forecasting technique is generated by considering the surcharge of conduits in UDSs. This means that a threshold of proactive
flood forecasting is added for preemptive measures.
Figure 2
shows the concept of an advanced flood forecasting technique.
The preliminary and dangerous thresholds were generated differently according to the characteristics of each area. The application process of an advanced flood forecasting technique is shown in
Figure 3
In the first step of
Figure 3
, real-time rainfall data were initially converted into the rainfall intensity and then applied to the advanced flood graph. The target area can be assumed to not be inundated if the applied rainfall
intensity (ARI) is smaller than the preliminary and dangerous thresholds. The second rainfall intensity was added to the advanced flood graph and was over the preliminary threshold. In this step, a
preemptive measure such as the early operation of the centralized reservoirs can be conducted to prevent the potential risk in UDSs. The early operation of the centralized reservoirs means that
drainage pumps are operated earlier than the standard of the normal operation considering several required factors such as the required depth, head loss for the screen, the mechanical freeboard, and
the bottom level of the centralized reservoir. This operation can reduce the backwater effect. In the former version, no flood forecasting was generated because the second rainfall intensity was not
over the dangerous threshold. The process of the advanced flood forecasting from the third and fourth steps was similar to the former flood forecasting technique because the ARI was over the
dangerous threshold. In the fifth and sixth steps, the ARI was located between the preliminary and dangerous thresholds. The short time interval of the ARI was appropriate for the advanced flood
forecasting technique because the concentration time in an urban area is short. A regression equation should be applied to create an entire advanced flood graph when the rainfall duration is 0 min. A
regression equation at 0 min can be applied without considering the rainfall observation interval. For example, a regression equation can be applied between 0 and 1 min, if the rainfall observation
interval is 1 min. If the rainfall observation interval is 10 min, a regression equation can be generated between 0 and 10 min.
2.3. Advanced Operation for Centralized and Decentralized Reservoirs
The RTC technique between different drainage facilities in a drainage area was suggested and applied based on the status of the monitoring node [
]. In the former study, the results of flooding and system resilience applying historical rainfall events according to cooperative operating levels were suggested in a target area. UDSs can be
prepared for an effective operation under heavy real-time rainfall events causing urban inundation if the operating standard of the urban drainage facilities (centralized and decentralized
reservoirs) is based on the results of the flood forecasting. An advanced operation technique described herein was based on the rainfall intensity during an advanced operation.
An advanced operation for centralized and decentralized reservoirs applied in this study consisted of three steps according to the status of an advanced flood forecasting. A normal operation in a
centralized reservoir was maintained, and reserved water in a decentralized reservoir was discharged to obtain the additional capacity if the applied rainfall data did not exceed two thresholds,
namely the preliminary and dangerous thresholds. Initially, the centralized and decentralized reservoirs did not reserve water because the purpose of two reservoir was the prevention of urban
inundation. The inflow volume could not be controlled because the inlet type of the decentralized reservoir was inlet weir. The rainfall event occurred, and the decentralized reservoir received water
through the inlet weir. Additionally, the inflow volume in the centralized reservoir could not be controlled during the rainfall event, and drainage pumps were operated after the depth of the
centralized reservoir reached the initial pump operating level. The range of operation in this study did not include this process, and it included the subsequent process.
The normal operation in a centralized reservoir was the operation of drainage pumps according to determined water levels in a centralized reservoir. Decentralized reservoirs have drainage pumps with
small capacity because the capacity of drainage pumps is calculated by dividing the storage capacity in a decentralized reservoir by design discharge time, which is generally greater than 24 h. The
reserved water in a decentralized reservoir is discharged by all drainage pumps. An early operation in a centralized reservoir was conducted to reduce the backwater effect, and the reserved water in
a decentralized reservoir was discharged if the ARI was located between the two thresholds. If the ARI was over the dangerous threshold, the early operation in a centralized reservoir was maintained,
and a decentralized reservoir did not discharge its reserved water.
Figure 4
shows a schematic of an advanced operation in UDSs.
The initial pump operating level during an early operation in pump stations was determined by several factors. The first factor was the required depth, and the second factor was the head loss for
screen. The third factor was the mechanical freeboard, and the fourth factor was the bottom level of the centralized reservoir. Other pump operating levels during an early operation in a centralized
reservoir were calculated by considering the required depth [
2.4. Resilience of UDSs
To consider the status of UDSs over time, the resilience index was previously suggested [
]. In the previous studies, the performance evaluation function was determined by considering the total rainfall amount, basin area, and flood volume at each time. The system resilience is calculated
based on the performance evaluation (PE) at each time. The difference between the two indices in the previous studies was based on the concentration time in the target area. The disadvantage of the
two indices was that the denominator (total rainfall amount and basin area) of the PE function was so large that the value of the resilience of the UDS was close to 1, and there was no difference
between the operations of UDS. In this study, the resilience index in a recent study was selected to evaluate the performance of the UDS [
]. The PE function is shown in Equations (4)–(6).
$U t = m a x 0 , 1 − F t R t × A s u b j e c t t o F t ≠ 0 , R t ≠ 0$
$U t = 1 s u b j e c t t o F t = 0$
$U t = 0 s u b j e c t t o F t ≠ 0 , R t = 0$
) are the value of the performance evaluation function and flood volume at time
, respectively.
(m) is the rainfall amount at time
, and
) is the basin area of the target watershed. The resilience of the UDS is calculated based on the value of the PE function in Equation (7).
$R e s t = 1 T ∫ 0 T = t m a x U t d t$
is the resilience of the UDS. The system resilience for each operation in the UDS at the study area can be calculated and compared.
3. Application and Results
3.1. Information of the Target Watershed
Seoul, the capital of Korea and the location of the study area, is one of the largest metropolitan cities in the world. Seoul is a basin-shaped city surrounded by mountains. The east-west distance of
Seoul is 36.78 km, the north-south distance 30.3 km, and the area 605.25 km^2. In addition, the area of Seoul is only 0.6% of the total area of Korea, and it has a high population density. The Han
River penetrates from east to west in Seoul. Because Seoul is downstream of the Han River, the hydraulic gradient is gradually decreasing, and the flow of water is slow. In the case of flooding, the
water level in Seoul (downstream from the Han River) is increased because of the water flowing from the upper and middle stream basin.
The target area is the drainage area of the Daerim3 pump station with a centralized reservoir in Yeongdeungpo-gu. Historical flood events in 2010 and 2011 occurred in Yeongdeungpo-gu, which is in the
southeastern part of Seoul. The total amount and representative return period of the historical rainfall for the 2010 event are 256 mm and 100 years, respectively. The total amount and representative
return period of the historical rainfall for the 2011 event are 386 mm and 100 years, respectively. The historical rainfall event in 2010 occurred from 20–21 September, and the historical rainfall
event in 2011 occurred from 26–28 July. The total rainfall volume in 2010 was different from 2011, but the frequency in 2010 was the same as 2011 because the rainfall duration in 2010 was different
from 2011.
The design return period of the Daerim3 pump station with a centralized reservoir is 30 years, and it has 12 drainage pumps (3411 m
/min), while the capacity of the centralized reservoir is 33,650 m
. The design return period of the Daerim decentralized reservoir is 20 years, and it has 2 drainage pumps (18 m
/min), while the capacity of a decentralized reservoir is 2477 m
Table 2
shows the information on the centralized and decentralized reservoirs.
The Seoul Metropolitan Government has provided Geographic Information System (GIS) data on all drainage areas in Seoul. An input network of SWMM using GIS data was generated. The number of
subcatchments, junctions, and conduits was 1576, 1805, and 1977, respectively. The flow routing method was a dynamic wave, and the infiltration model was applied using Horton’s equation. The maximum
rate, minimum rate, and decay constant in Horton’s equation at each sub-catchment were adjusted for the network calibration process.
Figure 5
shows the information on the UDS used in the target watershed.
3.2. Application of Advanced Flood Forecasting
The artificial rainfall data distributed by the Huff distribution were applied for the rainfall-runoff simulation in the study area. The form of regression equations using the third quartile is shown
in Equation (8).
$R = 0.0005 − 0.3603 D + 9.1084 D 2 − 44.549 D 3 + 105.18 D 4 − 106.21 D 5 + 37.835 D 6$
is the non-dimensional rainfall volume and
is the non-dimensional rainfall duration. The rainfall amount was increased from 1 mm in 1-mm increments for each rainfall duration. For example, a rainfall amount of 50 mm was distributed through a
Huff distribution and was applied to the rainfall-runoff simulation using SWMM. If a flood event did not occur, the rainfall amount can be increased to 51 mm. This process was repeated until the
first flooding occurred. If flooding occurred, the rainfall amount could change to the intensity for the advanced flood forecasting in a target watershed. The artificial rainfall data were used for
the generation of two thresholds (preliminary and dangerous thresholds). The historical rainfall events in 2010 and 2011 were used to verify the effect of the new operation. The application of the
historical rainfall event in 2010 is shown in
Figure 6
A duration of 400–520 min was selected for the application of the advanced flood forecasting in 2010 because the rainfall intensity in other durations was lower than the preliminary threshold. The
ARI in 2010 was larger than the preliminary threshold when the duration was 430 min and was greater than the dangerous threshold at 440 min. The proactive time in 2010 was obtained because the time
difference between the preliminary and dangerous thresholds was 10 min. Within 430 min, an early operation of the centralized reservoir was applied, and the reserved water in the decentralized
reservoir was discharged. Within 440 min, the early operation of the centralized reservoir was maintained, and the reserved water in the decentralized reservoir was stored. The early operation of
centralized reservoirs was that drainage pumps were operated earlier than the standard of the normal operation considering several required factors such as the required depth, head loss for the
screen, the mechanical freeboard, and the bottom level of the centralized reservoir. A very proactive time can be obtained based on meteorological forecasting. However, due to the time interval
between meteorological forecasting and the operation of drainage facilities, it is difficult to combine with the operation of drainage facilities in small urban watersheds. Centralized reservoirs
receive the inflow in urban watersheds, and drainage pumps are operated because the inflow is increased when a rainfall event occurs. Decentralized reservoirs receive water from the sewer network
through an inlet weir when a rainfall event occurs. In the current operation, decentralized reservoirs reserve the received water without discharge until a rainfall event is finished. The application
of the historical rainfall event in 2011 is shown in
Figure 7
A duration of 140–260 min was selected for the application of the advanced flood forecasting in 2011 because the rainfall intensity in the other duration was lower than the preliminary threshold. The
ARI in 2011 was larger than the preliminary threshold when the duration was 150 min and was greater than the dangerous threshold at 155 min. The proactive time in 2011 was obtained because the time
difference between the preliminary and dangerous thresholds was 5 min. Within 150 min, the early operation of the centralized reservoir was applied and the reserved water in the decentralized
reservoir was discharged. Within 155 min, the early operation of the centralized reservoir was maintained and the reserved water in the decentralized reservoir was stored.
3.3. Application of Advanced Operation for Centralized and Decentralized Reservoirs
The historical rainfall data of the past two years in 2010 and 2011 were used to compare the results of each operation. An advanced operation consists of the centralized and decentralized reservoir
operations. The centralized reservoir operation is determined according to whether the ARI exceeds the preliminary threshold. If the ARI exceeds the preliminary threshold, the normal operation is
changed to an early operation and vice versa in a centralized reservoir. The decentralized reservoir operation is determined according to whether the ARI exceeds the dangerous threshold. If the ARI
exceeds the dangerous threshold, a normal operation is changed to an early operation and vice versa in a decentralized reservoir. If the ARI exceeds the dangerous threshold, the decentralized
reservoir operation is changed from discharging the reserved water to maintaining it.
An early operation in a centralized reservoir was suggested in the previous study [
]. The determination of the initial pump operating level in centralized reservoirs is conducted to prevent a cavitation of drainage pumps. First, the required depth should be calculated, and it is
calculated based on the initial pump capacity, the initial preparation time for pump, the required volume, and the mean area. The calculation of the required depth is shown in Equation (9).
is the required depth,
is the initial pump capacity, and
is the initial preparation time for the pump. In addition,
is the required volume, and
is the mean area. The initial operating level is calculated using Equation (10).
is the initial pump operating level and
is the head loss for the screen. In addition,
is the mechanical freeboard, and
is the bottom elevation. The other operating levels can be calculated by considering the required depth in the centralized reservoir [
]. The operating level of the new operation was calculated using Equations (9) and (10). The operating levels during the normal and early operations of the centralized reservoir in the study area are
listed in
Table 3
The results of applying the current operation, the previous operation [
] and the new operation to the historical rainfall event in 2010 are shown in
Figure 8
The flooding volume according to each operation occurred between 400 and 700 min, as shown in
Figure 8
. The peak flooding volume (55.2 m
) of the previous operation [
] was lower than that (611.4 m
) of the current operation. Additionally, the advanced operation (new operation) showed the minimum peak value of flooding volume (50.4 m
) among the three operations. The results of the total flooding volume during the current operation, previous operation, and new operation were 6617, 3904, and 3368 m
, respectively. The new operation showed a flooding reduction of 3249 m
compared to the current operation. The results of applying the current operation, previous operation, and new operation to the historical rainfall event in 2011 are shown in
Figure 9
The flooding volume according to each operation occurred between 1150 and 1200 min, as shown in
Figure 9
. The results in 2011 were different from those in 2010. The peak value of flooding volume (124.2 m
) in the previous operation [
] was higher than that (57 m
) in the current operation. The current and advanced operation (new operation) showed the minimum peak flooding volume (57 m
). The previous operation showed the largest peak flooding volume among the three operations. The results of the total flooding volume during the current operation, previous operation, and new
operation were 664, 552, and 490 m
, respectively. The new operation showed a flooding reduction of 174 m
compared to the current operation. Diagrams with the reservoirs volume and discharge evolution are required. The volume and discharge in 2010 were selected because the flooding volume in 2010 was
larger than 2011. The volume of the centralized reservoir in 2010 is shown in
Figure 10
There was a slight difference although the results of each operation were similar for the volume of the centralized reservoir in 2010. The obvious difference between each operation was found at about
720 min. There was no large difference between each operation because faster drainage in the centralized reservoir led to large inflow. The discharge of the centralized reservoir in 2010 is shown in
Figure 11
A clear difference between each operation was observed from 660 min. The new operation showed more discharge than the current operation. The volume of the decentralized reservoir in 2010 is shown in
Figure 12
The difference of the volume in the decentralized reservoir according to each operation occurred at about 450 min. The biggest difference between the current and new operation occurred at about 750
min. In the new operation, the decentralized reservoir was completely empty at about 850 min, and it had additional capacity for continuous rainfall events. The discharge of the decentralized
reservoir in 2010 is shown in
Figure 13
A clear difference between each operation was observed from 660 min. The new operation showed more discharge than the current operation. The most important difference was that the discharge in the
current operation stopped at 750 min, while the discharge in the new operation lasted up to 850 min.
3.4. Resilience of Advanced Operation with Advanced Flood Forecasting
The flooding volume in 2010 was more than that in 2011, and the duration of 2010 was shorter than that of 2011. This means that the flooding intensity (flooding volume per each duration) in 2010 was
greater than that in 2011, and the system resilience of 2010 was lower than that of 2011. The results of the system resilience for each operation are shown in
Table 4
The biggest difference in the system resilience for 2010 occurred when the current operation was changed to the new operation. The reason for the difference between the two operations was the
existence of the early operation in the centralized reservoir and the additional capacity of the decentralized reservoir. The early operation in the centralized reservoir prevented a backwater effect
by a safe and quick drainage in the UDS. The additional capacity in the decentralized reservoir can be obtained by the operation considering the level of conduits in the UDS.
The UDS in 2011 was relatively stable compared to that in 2010 because all system resilience exceeded 0.98. An operational difference occurred, although the urban drainage system showed a high system
resilience during all operations. As with 2010, the difference between the two operations was caused by the reduction of the backwater effect in the centralized reservoir and the additional capacity
of the decentralized reservoir.
4. Conclusions
The two non-structural measures proposed in this study were an advanced flood forecasting and an advanced operation for centralized and decentralized reservoirs. Advanced flood forecasting using
real-time data of rainfall events is a technique to minimize the damage caused by flooding in urban areas as a preemptive non-structural measure. The advanced operation based on the advanced flood
forecasting can maximize the efficiency by combining two individual non-structural measures (forecasting and operation). In addition, to evaluate the status of the UDS in the study area, the system
resilience was applied to compare the current operation with the new operation.
The advanced flood forecasting and advanced operation proposed in this study can be systemized in various UDSs. It will be possible to apply the suggested technique in both small and large urban
areas. In future studies, a threshold for flood damage in advanced flood forecasting can be added, making it possible to apply an advanced operation in various types of drainage facilities to large
watersheds. Furthermore, customized flood forecasting and the operation of drainage facilities when considering the regional risk of flood damage may be suggested.
Author Contributions
E.H.L. carried out the survey of the previous studies, wrote the original manuscript, conducted the simulations, and conceived of the original idea of the proposed method.
This research was funded by the National Research Foundation (NRF) of Korea of the Korean government (NRF-2018R1C1B5086380).
This work was supported by a grant from The National Research Foundation (NRF) of Korea of the Korean government (NRF-2018R1C1B5086380).
Conflicts of Interest
The author declares no conflict of interest.
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Figure 5. Information on the UDS used in the target watershed (Imagery © 2019 CNES/Airbus, DigitalGlobe, Landsat/Copernicus, NSPO 2019/Spot Image, Map data © SK telecom).
Figure 8.
Flooding results of an advanced operation in 2010 [
Figure 9.
Flooding results of an advanced operation in 2011 [
Figure 10.
Volume of the centralized reservoir in 2010 [
Figure 11.
Discharge of the centralized reservoir in 2010 [
Figure 12.
Volume of the decentralized reservoir in 2010 [
Figure 13.
Discharge of the decentralized reservoir in 2010 [
Table 1. Classification of measures with real-time control (RTC) in urban drainage systems (UDSs). NSM, non-structural measure.
Measures Studies
Beeneken et al. (2013) [3]; Cembrano et al. (2004) [4];
Fiorelli et al. (2013) [5]; Fuchs and Beeneken (2005) [6];
Galelli et al. (2012) [7]; Hsu et al. (2013) [8];
Independent NSMs Kroll (2018) [9]; Lund et al. (2018) [10];
Pleau et al. (2005) [11]; Raimondi and Becciu (2015) [12];
Schütze et al. (2004) [13]; Vanrolleghem et al. (2005) [14];
Zacharof et al. (2004) [15]
Combined NSMs Lee et al. (2017) [1]; Sweetapple et al. (2018) [16]
Integrated NSMs Lee et al. (2016) [17]; Xu et al. (2018) [18]
Mixed NSMs This study
Drainage Facilities Capacity of Reservoirs (m^3) Capacity of Drainage Pumps (m^3/min) Boundary Conditions
(223 m^3/min × 7, High water level: 9.0 m
Daerim3 pump station with a centralized reservoir 36,200 150 m^3/min × 1, Low water level: 6.8 m
250 m^3/min × 2,
600 m^3/min × 2)
Daerim decentralized reservoir 2477 18 Total height: 3.2 m
(9.0 m^3/min × 2) Inflow weir: 2 m × 0.4 m
Table 3.
Normal and early operation of the centralized reservoir in the study area [
Pump Station Operation Operating Level (m)
Elevation (m) 6.5 6.8 7.2 7.3 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.3 9.0
Daerim3 Normal - - - 3.88 8.05 15.48 19.65 23.36 27.08 30.80 57.02 57.02 57.02
Early - 3.88 8.05 15.48 19.65 23.36 27.08 30.80 57.02 57.02 57.02 - -
System Resilience
Event Current Operation Previous Operation [16] New Operation Resilience Increment
(1) (2) (3) ((3) − (1))
2010 0.831835 0.855584 0.866566 0.034731
2011 0.988823 0.992997 0.993029 0.004206
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://
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Lee, E.H. Advanced Operating Technique for Centralized and Decentralized Reservoirs Based on Flood Forecasting to Increase System Resilience in Urban Watersheds. Water 2019, 11, 1533. https://doi.org
AMA Style
Lee EH. Advanced Operating Technique for Centralized and Decentralized Reservoirs Based on Flood Forecasting to Increase System Resilience in Urban Watersheds. Water. 2019; 11(8):1533. https://
Chicago/Turabian Style
Lee, Eui Hoon. 2019. "Advanced Operating Technique for Centralized and Decentralized Reservoirs Based on Flood Forecasting to Increase System Resilience in Urban Watersheds" Water 11, no. 8: 1533.
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Introduction to Probability | Red & White Matter Classes
Introduction to Probability
Probability - In layman terms, we can say that it is an attempt to quantify our prediction about the future events, i.e. it is the estimate of the likelihood of occurrence of an event.
Before we dive into Probability and try to define it in even better terms, we must first learn about the very basic concepts of an event.
Event - Basic Concepts
Random Experiment
• there is more than one possible outcome and all possible outcomes are known.
• the exact output cannot be predicted in advance.
For example, on flipping a coin we know that either head or tail will come. But we cannot predict that in advance.
Biased and Unbiased Experiment
• Unbiased experiment - all possible outcomes are equally likely to occur.
• Biased experiment - all possible outcomes are not equally likely to occur.
For example:
When we throw a die, if the likelihood of getting one or more numbers is more than that of the other numbers, we say it is a biased die. If all the six numbers are equally likely, then we call it an
unbiased die.
Similarly, a coin will be an unbiased or a biased coin depending on whether the head and tail are equally likely or not.
Unless otherwise specified a coin or a die is considered as unbiased.
Q. The probability of a dice showing a 6 on being rolled is 1/6. If the first 5 rolls of this dice have brought 1, 2, 3, 4 and 5 respectively, then which of the following is correct with regard to
the sixth roll of the dice?
(a) The dice would certainly show a 6.
(b) The chances of a 6 coming up has increased.
(c) The chances of a 6 coming up remains the same.
(d) Both (a) and (b)
Subsequent rolls of a dice are independent of the previous outcomes. The probability of 6 coming on any roll of the dice remains the same, i.e. 1/6.
Answer: (c)
Elementary Event and Sample Space
• Elementary Event - each possible outcome in a random experiment.
• Sample Space (denoted by symbol S) - The set of all possible elementary events (outcomes) of a random experiment is called the sample space associated with the experiment.
For example:
When we roll a die, the possible outcomes of this experiment are 1, 2, 3, 4, 5, or 6. Each one of these is an elementary even associated with this experiment.
The set of possible outcomes {1, 2, 3, 4, 5, 6} is the sample space of the experiment.
Let’s see one more example:
Consider the experiment of tossing two coins together (or a single coin twice).
Four possible elementary events (outcomes) associated with the random experiment are:
Heads on both coins – HH
Tails on both coins – TT
Head on first & Tail on second – HT
Tail on first & Head on second - TH
The associated sample space , S = {HH, TT, HT, TH}
Favourable Elementary Events
Favourable Elementary Events – All those elementary events, the occurrence of anyone of which will ensure the happening of the desired event.
Let’s see one example:
Desired event - occurrence of exactly one head when two coins are tossed.
So, HT and TH are the only elements of S (sample space) corresponding to the occurrence of our desired event. So, set of favourable elementary events = {HT, TH}
Note that such a set of favourable elementary events will always be a subset of the sample space S. For example, {HT, TH} is a subset of {HH, TT, HT, TH}.
Definition of Probability
If there are X elementary events associated with a random experiment that are equally likely to occur (i.e. number of elements in the sample set are X)
Out of them there are Y favourable elementary events (i.e. the events, the occurrence of anyone of which will ensure that our desired event, say A, takes place), then
probability of occurrence of the desired event, P(A) = Y/X
(i.e. P(A) = Number of favourable elementary events / Total elementary events )
For example:
Desired event - occurrence of exactly one head when two coins are tossed
So, set of favourable elementary events = {HT, TH}, i.e. 2 possible favourable elementary events, and
Sample space S = {HH, TT, HT, TH}, i.e. a total of 4 possible elementary events.
Hence, probability of occurrence of the desired event, P(A) = Y/X = 2/4 = 1/2
That is, 50% probability or chance of occurrence.
Note that, 0 ≤ Y ≤ X
So, 0 ≤ P(A) ≤ 1
• In case of a certain event: P(A) = 1 For example, what is the probability that on tossing a coin either head or tail will come?
• In case of an impossible event: P(A) = 0 For example, what is the probability that on rolling a dice a face with the digit 7 will come?
The probability of event A not occurring, denoted by P (not A), is given by P (not A) or P(Ā) = 1 – P(A)
Q. If monthly salary of a person is Rs. 10,000 and the monthly expenditure has a uniform probability of falling anywhere in the range of Rs. 4,000 to Rs. 6,000. Then, what is the probability that in
a given month he will make a saving of Rs. 5,500 or above.
(a) 1/4 (b) 1/3 (c) 1/2 (d) 3/4
The monthly expenditure has a uniform probability of falling anywhere in the range of Rs. 4,000 to Rs. 6,000. The probability that the expenditure lies between Rs. 4,000 to Rs. 6,000 is 1.
For saving to be Rs. 5,500 or above, the expenditure must lie between Rs. 4,500 and Rs. 4,000. (Note that he cannot save more than Rs. 6,000).
The probability that the expenditure lies between Rs. 4,001 to Rs. 4,500 = (4500-4000)/(6000-4000) = 500/2000 = 1/4.
Answer: (a)
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Mathematics 1a: Functions, Trigonometry, and Differentiation Quiz
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What concept involves finding the derivative of a function?
Differentiation (correct)
Which trigonometric function is used to find the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle?
Sine (correct)
In mathematics, which topic deals with relations and functions between the angles of a triangle and the lengths of its sides?
Trigonometry (correct)
What does a function do to each input value from its domain?
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Which concept in mathematics is essential for analyzing the behavior of functions and finding optima?
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What is the defining characteristic of a function?
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Describe the structure of a linear function.
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Define the domain and codomain of a function.
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Study Notes
Mathematics 1a: Exploring Functions, Trigonometry, and Differentiation
In the realm of mathematics, Mathematics 1a introduces students to fundamental concepts that lay the foundation for more advanced studies. We'll delve into three key topics: functions, trigonometry,
and differentiation.
A function is a rule that assigns a single output value to each input value from its domain. The function concept is applied to many real-world scenarios, such as predicting population growth, stock
prices, and projecting weather patterns.
Trigonometry deals with relations and functions between the angles of a triangle and the lengths of its sides. It plays a vital role in fields like architecture, music, physics, and engineering. Six
trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—are used to solve problems related to angles, lengths, and areas.
Differentiation involves finding the derivative of a function, which represents the rate of change of a function with respect to its inputs. This concept is essential for analyzing the behavior of
functions, finding optima, and solving problems in physics, chemistry, and engineering.
Applications and Research Opportunities
Mathematics 1a lays groundwork for more advanced topics like calculus, linear algebra, and numerical methods. It also provides students with a strong foundation for pursuing careers in fields like
data science, engineering, and finance.
Today, mathematics is being transformed by artificial intelligence, as seen by the development of proof assistants and machine-assisted proofs. Mathematics departments are now taking an active role
in exploring the potential of AI in their research and teaching, which is a testament to the adaptive and innovative nature of the discipline.
Undergraduate research opportunities are also available to students interested in exploring these topics further. Through seminars, workshops, and conferences, students can gain hands-on experience,
collaborate with their peers, and engage with faculty on cutting-edge research.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Test your knowledge on functions, trigonometry, and differentiation with this quiz that covers fundamental concepts in Mathematics 1a. Explore the applications of functions in predicting real-world
scenarios, the significance of trigonometry in various fields, and the concept of finding derivatives for analyzing functions. Discover how Mathematics 1a is a stepping stone to advanced topics like
calculus and its relevance in AI and research opportunities.
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date of birth
How do you convert date of birth to age in SQL?
How do you convert date of birth to age in SQL?
Here is the statement to calculate the age of the employees from the date of birth: Select E_id, E_name, datediff( YY, birthDate, getdate()) as age from Employee; To learn writing SQL queries, you
should check out this SQL online course and certification program by Intellipaat.
Is there an age function in SQL?
The age() function subtract arguments, producing a “symbolic” result that uses years and months.
How do I calculate someone’s age based on a datetime type birthday?
int age = (int) ((DateTime. Now – bday). TotalDays/365.242199);
How do you calculate age in years and months from date of birth in SQL query?
DECLARE @BirthDate datetime, @AgeInMonths int SET @BirthDate = ’10/5/1971′ SET @AgeInMonths — Determine the age in “months old”: = DATEDIFF(MONTH, @BirthDate, GETDATE()) — . Get the difference in
months – CASE WHEN DATEPART(DAY,GETDATE()) — .
How do you convert DOB to age in Excel?
How to calculate age in Excel
1. In the third cell, for us it’s C2, enter the following formula: =DATEDIF(A2, B2, “y”).
2. You can also get a person’s age without entering today’s date in the second cell.
3. The final, most specific measurement that you can make is a person’s age, including months and days.
How do I calculate age in C++?
We find the year y by simply subtracting the values of py and by. We find the month m by subtracting the values of pm and bm if pm>bm. Otherwise, we subtract 1 from y and subtract the quantity (bm –
pm) from 12. Similarly, we find the days d by subtracting the values of bd and pd if pd>bd.
How do I calculate age in months in SQL?
Show activity on this post. DECLARE @BirthDate datetime, @AgeInMonths int SET @BirthDate = ’10/5/1971′ SET @AgeInMonths — Determine the age in “months old”: = DATEDIFF(MONTH, @BirthDate, GETDATE()) —
. Get the difference in months – CASE WHEN DATEPART(DAY,GETDATE()) — .
What does Timestampdiff mean in SQL?
TIMESTAMPDIFF() function MySQL the TIMESTAMPDIFF() returns a value after subtracting a datetime expression from another. It is not necessary that both the expression are of the same type. One may be
a date and another is datetime.
What is Timestampdiff?
The MySQL TIMESTAMPDIFF() function is used to find the difference between two date or datetime expressions. You need to pass in the two date/datetime values, as well as the unit to use in determining
the difference (e.g., day, month, etc).
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Talk:Sorting algorithms/Radix sort - Rosetta CodeTalk:Sorting algorithms/Radix sort
Beware negative number handling! See Wiki's python demo. dingowolf 13:25, 19 January 2011 (UTC)
An interesting problem; the easiest way to handle it seems to me to be to double the number of bins and put negative values in the first half and positive in the second. Or at least it produces
correct results when I implemented it in the Tcl solution. (I suspect that the original algorithm simply didn't implement them, or sorted by printed digit instead of logical digit.) –Donal
Fellows 13:22, 19 January 2011 (UTC)
The easiest way to handle negative numbers might be to find the minimum value in the list, subtract it from every item in the unsorted list and add it to every item in the sorted list. This
approach is modular and can wrap any "non-negative integers only" implementation, and work well in a variety of circumstances. That said the "double the bins" approach might have an
efficiency advantage when the the absolute value of the maximum equals the absolute value of the minimum. --Rdm 16:13, 19 January 2011 (UTC)
It was a smaller change to the code I already had working for the positive case. :-) –Donal Fellows 16:42, 19 January 2011 (UTC)
Yuppers, the negative integers were a small annoyance, all right (concerning the REXX example). -- Gerard Schildberger 22:03, 11 June 2012 (UTC)
C code
in the C code for radix sort, it seems to me that the condition ll < to after the "while (1)" loop is always fulfilled, and can thus be removed. Indeed in the "while (1)" loop we always have ll <=
rr, thus since rr decreases ll cannot exceed the initial value of rr, which is to - 1. User:Paul Zimmermann 13:09, 30 October 2012
I made a shorter/simpler implementation of the java example (it also handles negatives)
<lang java>public static int[] sort(int[] old) {
for(int shift = Integer.SIZE-1; shift > -1; shift--) { //Loop for every bit in the integers
int[] tmp = new int[old.length]; //the array to put the partially sorted array into int j= 0; //The number of 0s int i; //Iterator
for(i = 0; i < old.length; i++){ //Move the 0s to the new array, and the 1s to the old one
boolean move = old[i] << shift >= 0; //If there is a 1 in the bit we are testing, the number will be negative
if(shift == 0 ? !move : move) { //If this is the last bit, negative numbers are actually lower
tmp[j] = old[i]; j++;
} else { //It's a 1, so stick it in the old array for now
old[i-j] = old[i];
for(i = j; i < tmp.length; i++) { //Copy over the 1s from the old array
tmp[i] = old[i-j];
old = tmp; //And now the tmp array gets switched for another round of sorting
return old;
Forty-Bot (talk)
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category object in an (infinity,1)-category
I have been further expanding the Definition-section at category object in an (infinity,1)-category, adding a tad more details concerning proofs of some of the statements. But didn’t really get very
far yet.
Also reorganized again slightly. I am afraid that in parts the notation is now slightly out of sync. I’ll get back to this later today.
Right, thanks. That’s better than “rigid”.
Ah, of course. I knew that term of yours, I wrote all those notes on your article, after all. But I forgot. Thanks for reminding me. There is now an entry gaunt category, so that I shall never forget
Sometimes it is called “gaunt”.
Sometimes it is called “rigid”.
In general, the $sSet$-nerve of a category is complete Segal precisely if the only isomorphisms are identities (what’s the name again for such a category?).
I have added a paragraph on this to Segal space – Examples – In Set
(this could alternatively go to various other entries, but now I happen to have it there, and linked to from elsewhere).
It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/
Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this
term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.
I have edited still a bit further. This will probably be it for a while, unless I spot some urgent mistakes or omissions.
Added reference to
• Louis Martini, Yoneda’s lemma for internal higher categories, (arXiv:2103.17141)
diff, v56, current
added these references on development of $\infty$-category theory internal to any (∞,1)-topos:
internal (∞,1)-Yoneda lemma:
• Louis Martini, Yoneda’s lemma for internal higher categories, [arXiv:2103.17141]
internal (infinity,1)-limits and (infinity,1)-colimits:
• Louis Martini, Sebastian Wolf, Limits and colimits in internal higher category theory [arXiv:2111.14495]
internal cocartesian fibrations and straightening functor:
• Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]
internal presentable (∞,1)-categories:
• Louis Martini, Sebastian Wolf, Presentable categories internal to an ∞-topos [arXiv:2209.05103]
diff, v58, current
added pointer to the recent:
• Louis Martini, Sebastian Wolf, Internal higher topos theory [arXiv:2303.06437]
diff, v61, current
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the Square Root of 166
In math, the square root of a number like 166 is a number that, when multiplied by itself, is equal to 166. We would show this in mathematical form with the square root symbol, which is called the
radical symbol: √
Any number with the radical symbol next to it us called the radical term or the square root of 166 in radical form.
To explain the square root a little more, the square root of the number 166 is the quantity (which we call q) that when multiplied by itself is equal to 166:
√166 = q × q = q^2
So what is the square root of 166 and how do we calculate it? Well if you have a computer, or a calculator, you can easily calculate the square root. If you need to do it by hand, then it will
require good old fashioned long division with a pencil and piece of paper.
For the purposes of this article, we'll calculate it for you (but later in the article we'll show you how to calculate it yourself with long division). The square root of 166 is 12.884098726725:
12.884098726725 × 12.884098726725 = 166
Is 166 a Perfect Square?
When the square root of a given number is a whole number, this is called a perfect square. Perfect squares are important for many mathematical functions and are used in everything from carpentry
through to more advanced topics like physics and astronomy.
If we look at the number 166, we know that the square root is 12.884098726725, and since this is not a whole number, we also know that 166 is not a perfect square.
If you want to learn more about perfect square numbers we have a list of perfect squares which covers the first 1,000 perfect square numbers.
Is 166 a Rational or Irrational Number?
Another common question you might find when working with the roots of a number like 166 is whether the given number is rational or irrational. Rational numbers can be written as a fraction and
irrational numbers can't.
The quickest way to check if a number is rational or irrational is to determine if it is a perfect square. If it is, then it's a rational number, but if it is not a perfect square then it is an
irrational number.
We already know that 166 is not a rational number then, because we know it is not a perfect square.
Calculating the Square Root of 166
To calculate the square root of 166 using a calculator you would type the number 166 into the calculator and then press the √x key:
√166 = 12.8841
To calculate the square root of 166 in Excel, Numbers of Google Sheets, you can use the SQRT() function:
SQRT(166) = 12.884098726725
Rounding the Square Root of 166
Sometimes when you work with the square root of 166 you might need to round the answer down to a specific number of decimal places:
10th: √166 = 12.9
100th: √166 = 12.88
1000th: √166 = 12.884
Finding the Square Root of 166 with Long Division
If you don't have a calculator or computer software available, you'll have to use good old fashioned long division to work out the square root of 166. This was how mathematicians would calculate it
long before calculators and computers were invented.
Step 1
Set up 166 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:
Step 2
Starting with the first set: the largest perfect square less than or equal to 1 is 1, and the square root of 1 is 1 . Therefore, put 1 on top and 1 at the bottom like this:
Step 3
Calculate 1 minus 1 and put the difference below. Then move down the next set of numbers.
Step 4
Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:
2? × ? ≤ 66
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 2. Replace the question marks in the problem with 2 to get:
22 × 2 = 44
Now, enter 2 on top, and 44 at the bottom:
Step 5
Calculate 66 minus 44 and put the difference below. Then move down the next set of numbers.
Step 6
Double the number in green on top: 12 × 2 = 24. Then, use 24 and the bottom number to make this problem:
24? × ? ≤ 2200
The question marks are "blank" and the same "blank". With trial and error, we found the largest number "blank" can be is 8.
Now, enter 8 on top:
Hopefully, this gives you an idea of how to work out the square root using long division so you can calculate future problems by yourself.
Practice Square Roots Using Examples
If you want to continue learning about square roots, take a look at the random calculations in the sidebar to the right of this blog post.
We have listed a selection of completely random numbers that you can click through and follow the information on calculating the square root of that number to help you understand number roots.
Calculate Another Square Root Problem
Enter your number in box A below and click "Calculate" to work out the square root of the given number.
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s - nadia.chigmaroff
« on: September 18, 2020, 11:46:23 AM »
For the first question on the Section 1.2 questions - $|z-4| = 4|z|$, what level of description is sufficient?
Can I say that this is a circle by the formula for Apollonius circles, and can I use the formula provided in lecture/the textbook to describe the radius and center point?
Should I describe the radius and center point, or is saying that it's a circle enough?
Thank you!
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GGM: Estimation — estimate
Estimate the conditional (in)dependence with either an analytic solution or efficiently sampling from the posterior distribution. These methods were introduced in Williams (2018) . The graph is
selected with select.estimate and then plotted with plot.select.
formula = NULL,
type = "continuous",
mixed_type = NULL,
analytic = FALSE,
prior_sd = 0.25,
iter = 5000,
impute = TRUE,
progress = TRUE,
seed = 1,
Y Matrix (or data frame) of dimensions n (observations) by p (variables).
formula An object of class formula. This allows for including control variables in the model (i.e., ~ gender). See the note for further details.
Character string. Which type of data for Y ? The options include continuous, binary, ordinal, or mixed. Note that mixed can be used for data with only ordinal variables. See the note for
type further details.
Numeric vector. An indicator of length p for which varibles should be treated as ranks. (1 for rank and 0 to assume normality). The default is currently to treat all integer variables as
mixed_type ranks when type = "mixed" and NULL otherwise. See note for further details.
analytic Logical. Should the analytic solution be computed (default is FALSE)?
prior_sd Scale of the prior distribution, approximately the standard deviation of a beta distribution (defaults to 0.50).
iter Number of iterations (posterior samples; defaults to 5000).
impute Logicial. Should the missing values (NA) be imputed during model fitting (defaults to TRUE) ?
progress Logical. Should a progress bar be included (defaults to TRUE) ?
seed An integer for the random seed.
... Currently ignored.
The returned object of class estimate contains a lot of information that is used for printing and plotting the results. For users of BGGM, the following are the useful objects:
• pcor_mat Partial correltion matrix (posterior mean).
• post_samp An object containing the posterior samples.
The default is to draw samples from the posterior distribution (analytic = FALSE). The samples are required for computing edge differences (see ggm_compare_estimate), Bayesian R2 introduced in Gelman
et al. (2019) (see predictability), etc. If the goal is to *only* determine the non-zero effects, this can be accomplished by setting analytic = TRUE. This is particularly useful when a fast solution
is needed (see the examples in ggm_compare_ppc)
Controlling for Variables:
When controlling for variables, it is assumed that Y includes only the nodes in the GGM and the control variables. Internally, only the predictors that are included in formula are removed from Y.
This is not behavior of, say, lm, but was adopted to ensure users do not have to write out each variable that should be included in the GGM. An example is provided below.
Mixed Type:
The term "mixed" is somewhat of a misnomer, because the method can be used for data including only continuous or only discrete variables. This is based on the ranked likelihood which requires
sampling the ranks for each variable (i.e., the data is not merely transformed to ranks). This is computationally expensive when there are many levels. For example, with continuous data, there are as
many ranks as data points!
The option mixed_type allows the user to determine which variable should be treated as ranks and the "emprical" distribution is used otherwise (Hoff 2007) . This is accomplished by specifying an
indicator vector of length p. A one indicates to use the ranks, whereas a zero indicates to "ignore" that variable. By default all integer variables are treated as ranks.
Dealing with Errors:
An error is most likely to arise when type = "ordinal". The are two common errors (although still rare):
• The first is due to sampling the thresholds, especially when the data is heavily skewed. This can result in an ill-defined matrix. If this occurs, we recommend to first try decreasing prior_sd
(i.e., a more informative prior). If that does not work, then change the data type to type = mixed which then estimates a copula GGM (this method can be used for data containing only ordinal
variable). This should work without a problem.
• The second is due to how the ordinal data are categorized. For example, if the error states that the index is out of bounds, this indicates that the first category is a zero. This is not allowed,
as the first category must be one. This is addressed by adding one (e.g., Y + 1) to the data matrix.
Imputing Missing Values:
Missing values are imputed with the approach described in Hoff (2009) . The basic idea is to impute the missing values with the respective posterior pedictive distribution, given the observed data,
as the model is being estimated. Note that the default is TRUE, but this ignored when there are no missing values. If set to FALSE, and there are missing values, list-wise deletion is performed with
Posterior Uncertainty:
A key feature of BGGM is that there is a posterior distribution for each partial correlation. This readily allows for visiualizing uncertainty in the estimates. This feature works with all data types
and is accomplished by plotting the summary of the estimate object (i.e., plot(summary(fit))). Several examples are provided below.
Interpretation of Conditional (In)dependence Models for Latent Data:
See BGGM-package for details about interpreting GGMs based on latent data (i.e, all data types besides "continuous")
Gelman A, Goodrich B, Gabry J, Vehtari A (2019). “R-squared for Bayesian Regression Models.” American Statistician, 73(3), 307--309. ISSN 15372731, doi: 10.1080/00031305.2018.1549100 .
Hoff PD (2007). “Extending the rank likelihood for semiparametric copula estimation.” The Annals of Applied Statistics, 1(1), 265--283.
Hoff PD (2009). A first course in Bayesian statistical methods, volume 580. Springer.
Williams DR (2018). “Bayesian Estimation for Gaussian Graphical Models: Structure Learning, Predictability, and Network Comparisons.” arXiv. doi: 10.31234/OSF.IO/X8DPR .
# \donttest{
# note: iter = 250 for demonstrative purposes
### example 1: continuous and ordinal ###
# data
Y <- ptsd
# continuous
# fit model
fit <- estimate(Y, type = "continuous",
iter = 250)
#> BGGM: Posterior Sampling
#> BGGM: Finished
# summarize the partial correlations
summ <- summary
# plot the summary
plt_summ <- plot
# select the graph
E <- select
# plot the selected graph
plt_E <- plot
# ordinal
# fit model (note + 1, due to zeros)
fit <- estimate
type = "ordinal"
iter = 250
#> Warning: imputation during model fitting is
#> currently only implemented for 'continuous' data.
#> BGGM: Posterior Sampling
#> BGGM: Finished
# summarize the partial correlations
summ <- summary
# plot the summary
plt <- plot
# select the graph
E <- select
# plot the selected graph
plt_E <- plot
## example 2: analytic solution ##
# (only continuous)
# data
Y <- ptsd
# fit model
fit <- estimate
analytic = TRUE
# summarize the partial correlations
summ <- summary
# plot summary
plt_summ <- plot
# select graph
E <- select
# plot the selected graph
plt_E <- plot
# }
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Determine the chair number occupied by the child who will receive that chocolate. - FcukTheCode
Determine the chair number occupied by the child who will receive that chocolate.
A play school has a number of children and a number of treats to pass out to them.
Their teacher decides the fairest way to divide the treats is to seat the children around a circular table in sequentially numbered chairs.
A chair number will be drawn from a hat. Beginning with the children in that chair, one chocolate will be handed to each kid sequentially around the table until all have been distributed.
The teacher is playing a little joke, though. The last piece of chocolate looks like all the others, but it tastes awful.
Determine the chair number occupied by the child who will receive that chocolate.
The first line contains an integer, 't', the number of test cases.
Then t lines follows
The next lines each contain space-separated integers:
n: the number of children
m: the number of chocolates
s: the chair number to start passing out treats at
Print the chair number occupied by the child who will receive that chocolate.
Assume n=4 m =6 and s =2 then
There are 4 children, 6 pieces of chocolates and distribution starts at chair 2 .
The children's arrange themselves in seats numbered 1 to 4 .
Children receive chocolates at positions 2,3,4,1,2,3 . The Children's to be warned sits in chair number 3.
#include <stdio.h>
void loop()
printf("ans=(long int *)malloc(t*sizeof(long int)); long int t,n,m,s,*ans");
long int n,m,s;
scanf("%ld %ld %ld",&n,&m,&s);
int main()
int t;
{int a,b,c,d;
return 0;
Executed using gcc
Morae Q!
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